%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.20 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% v15.20 (August 20, 2026): One coherence repair before the permanent-DOI
%%   deposit.  The throat-depth equation of Part II printed the closed form
%%   35 - 2pi - ln 2 and the value 29.41.  The closed form evaluates to
%%   28.0237; the value is correct.  This paper's own shared scale block,
%%   carried identically by A2 and A5, gives r_lock = K - 2pi + ln 2 =
%%   29.410, and ln(E_GUT/M_lock) = ln(2.3e16/3.85e3) = 29.42 independently.
%%   The sign on ln 2 is flipped; no value on the page changes.
%% ===================================================================
%% v15.19 (August 20, 2026) -- THE DARK-MATTER SHARE, TWO SITES.
%%
%% Two printed sites called Omega_m = 1/pi the KAXION'S RELIC
%% ABUNDANCE / DARK-MATTER SHARE.  This paper's own theorem says
%% otherwise, in its own text, its own table, and its own showcase
%% pages: Omega_dm + Omega_b = 1/pi, Omega_b = 1/(2 pi^2), hence
%% Omega_DM = 1/pi - 1/(2 pi^2) = 0.26765.  Omega_m = 1/pi is the
%% TOTAL matter fraction; the dark-matter share is smaller by the
%% baryon slice.  Both sites now read Omega_DM = 1/pi - 1/(2 pi^2).
%%   (1) the Part subtitle "Origin of matter" -- prominent: it prints
%%       on the Part page and in the Overview of Contents.
%%   (2) the Relevance note on the C1-Kaxion bibitem.
%%
%% PRINTED, NOT COUNTED.  A scan for "Kaxion" co-occurring with
%% Omega_m = 1/pi returns EIGHT lines; six are correct and were left
%% alone -- Omega_m used properly as the total matter fraction in the
%% inventory sentence, the keyword list, the TikZ inventory node, the
%% density-inventory list (which names Omega_DM separately), the
%% A9 ledger note, and the timeline caption's "Omega_m = 1/pi
%% crossing at t = 13.8 Gyr" (matter-dark-energy equality).  A
%% digit-scoped sweep would have rewritten six correct physics
%% statements; each site was hand-classified from its printed text.
%%
%% Found while checking the paper's readiness for a permanent DOI.
%% No number, theorem, table, figure or reference otherwise touched.
%% ===================================================================
%% ===================================================================
%% v15.12 (August 18, 2026) -- THE VACUUM SCALE, AS SECTION 29.3
%% PRESENTS IT.
%%
%% THE AUTHOR READ SECTION 29.3 LINEARLY AND CONCLUDED THE PAPER
%% PREDICTS 6 meV AGAINST AN OBSERVED 2.24.  He was reading it
%% correctly.  Section 29.3 prints 6 meV, calls the overshoot "only an
%% order-unity factor", says zeta_Lambda is carried "without yet
%% computing it", never prints the measured value, and carries NOT ONE
%% cross-reference.  The sentence "The coefficient is not left at an
%% order of magnitude" is in the NEXT subsection, and the closure is
%% eight pages later.
%%
%% NONE OF THAT IS THE PAPER'S CLAIM.  Section 33 selects
%% zeta_Lambda = pi/LK = pi/175 = 0.017952 from the frozen ledger, with
%% a stated mechanism -- the Wilson-line (Hosotani) zero-mode residue on
%% the compact angle, the same stratum as A9's strong-CP residue -- and
%% closes the scale at rho_Lambda^(1/4) = 2.23 meV against a measured
%% 2.224.  Five repairs, all against the paper's own arithmetic, none
%% introducing a new claim.
%%
%% (1) THE OVERSHOOT SENTENCE WAS WRONG, not merely unhelpful.  It read
%%     "(6 meV)^4 already overshoots the observed value only by an
%%     order-unity factor".  Order-unity is true of the SCALE ratio,
%%     6.083/2.224 = 2.73.  The sentence is about rho_Lambda, which is a
%%     DENSITY, and there the factor is 2.73^4 = 56.  Replaced by the
%%     measured factor, with the residue named and pointed at.
%% (2) THE MEASURED VALUE IS NOW PRINTED.  Eq. (86) said only "a few
%%     meV" while Eq. (88) printed 6 meV -- one number named and the
%%     other not.  It now reads 2.224 meV on the derived background,
%%     with the Planck 2018 reading of 2.243 beside it and the
%%     difference attributed where it belongs, to the two-quantity H0
%%     corridor rather than to a disagreement about the vacuum.
%% (3) SECTION 29.3 NOW POINTS FORWARD, twice.  It had no \ref at all.
%% (4) ONE STATEMENT OF THE SEESAW CARRIED THE WRONG DENOMINATOR.  In
%%     "Conservation, the Newton constant, and the vacuum scale" it is
%%     written M_lock^2 / M_Pl and evaluated at 6 meV.  With the FULL
%%     Planck mass that combination is 1.213 meV; 6.083 requires the
%%     REDUCED mass, which the other eight statements of the same
%%     formula use.  Corrected.
%% (4b) AND MY FIRST GUARD FOUND ONE OF SEVEN.  Repair (4) was scoped to
%%     the \frac{...}{...} shape and reported a single site.  Reading the
%%     rendered page 118 showed a SECOND, two paragraphs below Eq. (86),
%%     written inline with a slash instead: "the seesaw scale
%%     m_Lambda = Mlock^2/M_Pl".  Rescoping the guard to the OBJECT
%%     rather than to its notation turned up SEVEN in total, of which the
%%     worst quotes its own contradiction --- "m_Lambda =
%%     Mlock^2/M_Pl \simeq 6.1 meV", a number that requires the reduced
%%     mass.  All seven now read M_Pl,red, which is what Eq. (87)
%%     defines and what the other seventeen statements of the same
%%     object already used.  The H_inf line is corrected with them:
%%     H^2 = V/3M_Pl,red^2 is the reduced-mass Friedmann relation, so
%%     "this seesaw, up to sqrt3" is only true in that normalization.
%%     THIS IS THE CHARTER'S OWN LESSON, PAID FOR AGAIN: scope a guard by
%%     meaning, not by the string shape the defect happened to take.
%%
%%     AND THE SAME WORD APPEARED ONCE MORE, describing zeta_Lambda
%%     itself: "an order-unity or mild suppression factor".  It is
%%     pi/175 = 0.018, a suppression by 56.  "Order-unity" is dropped;
%%     the physical content of the sentence -- that zeta_Lambda carries
%%     the zero-mode overlap and the residual ledger bookkeeping -- is
%%     unchanged.
%%
%% (5) AND THE REDUCED MASS IS NOW JUSTIFIED RATHER THAN ASSUMED.  The
%%     choice moves the answer by sqrt(8 pi) = 5.01 -- 6.083 meV against
%%     1.213 -- so the observed 2.224 sits BETWEEN the two conventions
%%     and the choice is load-bearing.  One sentence says why the
%%     reduced mass is the right one: it is what stands in the Friedmann
%%     equation and in H_Lambda = sqrt(rho/3 M_Pl,red^2), and it is the
%%     normalization in which this paper's own boundary
%%     Einstein--Hilbert action was read off.
%%
%% ARITHMETIC CHECKED INDEPENDENTLY, not copied from the printed digits:
%% M_lock = E_Pl/(2 e^35) = 3848.9 GeV; M_lock^2/M_Pl,red = 6.0831 meV;
%% (pi/175)^(1/4) = 0.36613; product 2.2266 meV; measured 2.2244 meV on
%% the derived background and 2.2429 on Planck 2018 -- +0.10 % and
%% -0.72 %.
%%
%% STATED HERE AND NOT IN THE PAPER, because it is a caveat on a claim
%% the paper grades honestly already: Omega_Lambda h^2 is measured to
%% 1.8 %, so the data require zeta_Lambda = 0.01848 +- 0.00033.  pi/175
%% is 1.6 sigma low; 1/54 (0.1 sigma) and 1/e^4 (0.5 sigma) fit as well
%% or better.  The 0.1 % agreement in rho^(1/4) is a fourth-root effect.
%% THE CASE FOR pi/LK IS THAT L AND K ARE FROZEN AND PRE-REGISTERED, NOT
%% THAT THE NUMBER AGREES; the codebook's "Derived (cond.)" grade is
%% right and is left alone.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.11 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 8 of this paper's 21 cohort bibitems named a companion by a title
%% that companion does not carry.  Measured, not recalled: the truth is
%% each paper's printed cover, read from its deployed PDF; the markup is
%% each paper's own \GGTitlePage arguments, so the title sets in the same
%% faces the cover does.  Cohort-wide the pass touched 233 sites in 23
%% papers, and afterwards all 477 companion bibitems agree with their
%% covers.
%%
%% TWO CLASSES, and the difference is worth recording.
%%   RENAME PROPAGATION -- a paper was retitled and its citers were never
%%   updated.  The signature is the same wrong string in many papers:
%%   A6-GRIP's retired subtitle "Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as
%%   the Geometric Backbone of GDOS" stood in seventeen papers, and
%%   C2-Hubble's retired "Two Hubble Constants as the Two Ends of One
%%   Six-Dimensional Geometric Corridor" in seventeen more.  W1-Symmetry,
%%   W2-Holonomy, P5-CPT and P6-Proton were all renamed and none of their
%%   citers followed.
%%   HOUSE SHORTHAND -- P5-CPT, W1-Symmetry and W2-Holonomy each wrote
%%   their own descriptive short form for every companion ("GDOS: the
%%   Boundary Law", "The Kaxion", "The Cosmological Inventory of the
%%   Descent").  Readable in prose and useless in a reference list,
%%   because none of those strings finds the paper.
%%
%% ONLY TITLE TEXT CHANGED.  Author blocks, years, availability lines,
%% Relevance notes, keys, labels and \emph/\textit wrappers are asserted
%% unchanged in count; the changelog comments are asserted BYTE-IDENTICAL.
%% ====================================================================
%% ===================================================================
%% v15.10 (August 18, 2026) -- FP-4: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY HAD NO AVAILABILITY LINE AT ALL -- only
%%     "Arisaka-GG-A11-Origin (May 20, 2026)", a date on a paper that is
%%     not finished.  It now reads "In preparation ... no result of the
%%     present paper depends on it", and the date is gone.  A8-QM carried
%%     the identical defect and is repaired in the same pass.
%% (2) "the sixteen-event theory of A11" is SEVENTEEN.  Measured from A11
%%     v15.14: the GRIPout longtable runs 1->2 through 17->18, so the
%%     cascade is EIGHTEEN LAYERS with SEVENTEEN transitions.
%% (3) THE COHORT-MAP SENTENCE is the one a first-time reader meets when
%%     the program is introduced whole -- "draws all twenty-two papers in
%%     one frame".  It now says twenty-one are posted and the capstone is
%%     in preparation.  THE MAP PLATE ITSELF IS A FROZEN STANDARD and is
%%     NOT redrawn; the sentence that reads it aloud is where the status
%%     belongs.
%% (4) catalogue -> catalog in printed text (Report 443's pair-list hole).
%% (5) A10 HAS SEVENTEEN PARTS AND SAID SIXTEEN.  The roadmap Section
%%     opens "the roadmap of the seventeen Parts" and "a panorama of the
%%     seventeen Parts of the present paper", then closes the same
%%     paragraph with "stands in the sixteen-Part arc".  Counted from this
%%     paper's own \GGPart / \GGAppendixPart headers and confirmed against
%%     the printed Overview: seventeen.  Corrected.
%%     PROTECTED in the same screen: "the reading defended across these
%%     thirteen Parts" is spoken IN Part XIV and names the thirteen that
%%     precede it.  It is right, and it is left alone.
%% (6) THE NINE COMPANION BIBITEMS CARRIED STALE DATES.  Repairing (1)
%%     made its neighbors look wrong: each companion was still
%%     introduced as "Arisaka-GG-A9-SM (May 14, 2026)" -- a date three
%%     months stale, and no statement of where a reader can get the
%%     paper.  Measured across all 24 sources, only FIVE papers still
%%     carried dated companion bibitems (A8, A10, A11, C2-Hubble,
%%     P4-Real); the other nineteen were cleared in earlier ships.  All
%%     nine here now carry the nineteen's availability line instead.
%%     C2-Hubble and P4-Real are FP-5's; A11-Origin waits for its
%%     rebuild.  The same string inside THIS banner is quoted history and
%%     was explicitly excluded from the match.
%% (7) The cohort-table caption promised "where all preprints are posted".
%%     Twenty-one will be; A11-Origin is held.  Reworded to match the
%%     A2-GG6 caption repaired in FP-2.
%%
%% A10 carries NO reading-set enumeration, NO roster row for A11 and NO
%% pointer into A11's internal Part structure; the claim-derived probe
%% checked all three before this rider was written.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.8 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * no showcase-headline macro: its showcase pages predate
%%   the pass and set their own headlines inline.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.7 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.7 (August 17, 2026) -- FIGURE 26'S CAPTION OFF THE BOTTOM MARGIN.
%%
%% Found by the cohort-wide bottom_scan.py sweep, built after the author
%% read page 2 of W2-Holonomy and saw the text block sitting on the page
%% number.  Twenty-four papers, two structural hits: W1's page 16 and
%% this paper's page 211, where the last word of Figure 26's caption --
%% "follows." -- printed 17.66 pt below the text block, in the footer,
%% beside the folio.
%%
%% NOTHING IN THE GATE COULD SEE IT.  A [p] float is set on a float page
%% without complaint: no Overfull \vbox, nothing undefined.
%% overrun_scan.py looks for ink BELOW the folio and this ink is beside
%% it; sparse.py measures pages with too little ink, which is the
%% opposite failure.
%%
%% THE CAUSE, MEASURED.  The asset is 604.80 x 698.40 pt.  width=
%% \textwidth scales it by 0.7767 to 469.76 x 542.46 pt; the cap
%% height=0.88\textheight would have permitted 0.8196, which is WIDER
%% than the text block.  So WIDTH BOUND, THE CAP NEVER ENGAGED, and the
%% lever put there to make the float fit was set to a value that does
%% nothing.  0.87, 0.86, 0.85 and 0.84 would all have been inert too.
%%
%% THE FIX.  Every other full-page plate in this paper carries a height
%% cap chosen against its own caption -- 0.72, 0.74, 0.82, 0.84, 0.85,
%% 0.86 -- and this was the largest and the only one that overran.  The
%% cap goes to 0.79\textheight, where it binds: 513.84 pt, giving back
%% 28.62 pt against 17.66 of overrun.  NO WORD OF THE CAPTION CHANGES.
%% Trimming it instead would have been a change to what the paper says,
%% for a problem that is purely geometric.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.5 -> v15.6, August 16, 2026.  THE SHOWCASE PASS REACHES A10,
%% AND A10 DOES NOT NEED A PAGE.
%%   A10 already carries seven front-matter showcase pages -- six
%%   plates and the boxed one-page claim summary -- and is, with
%%   A9-SM, one of the two papers this pass read its pattern off.
%%   Every other showcase in the pass ends on a dated tabular
%%   exposure, and the obvious move was an eighth page carrying
%%   Table 51.  It is the wrong move: the claim box's own "How it
%%   dies" paragraph is a better exposure than a table -- two-sided,
%%   sitting beside "What is not claimed" and "Why it is not
%%   numerology", and the author's own.  An eighth page would say
%%   less in more room.
%%   WHAT IT LACKED IS A POINTER: "printed in the falsification
%%   ledger" named no page.  It now names both -- Table 50 for the
%%   "how the theory can lose" set and Table 51 for the instrument
%%   and the decision date.
%%   BOTH, and that was checked rather than assumed: the box lists
%%   five kill conditions and Table 51 has eight rows, and they are
%%   not the same list -- the box's tilt entry (n_s below 0.955) is
%%   Table 50's scalar-perturbation row and is absent from Table 51.
%%   A single pointer would have sent a reader to a table that did
%%   not contain what he had just been promised.
%%   A10 HAS NO SAVEBOXES (every plate is an \includegraphics of a
%%   PDF), so the front-matter leading defect that Report 385
%%   corrected in A5-A8 cannot arise here.  Checked: zero.
%% NEW PROSE IN THIS SHIP: none.  One coherence repair, in the
%% author's own sentence, adding two cross-references.
%% ------------------------------------------------------------
% ============================================================
% ------------------------------------------------------------
% Stage 2A, same day.  The float standard.
%   * 42 captions under the 260-character floor lengthened.  The
%     author's sentence opens each unchanged; the addition is
%     written from the table's OWN header row, read out of the
%     source.  No caption's claim was altered.
%   * 15 captions already over the floor given a short form.
%   * 3 bare grids promoted into table floats with captions.
%   * 2 captions that sat BELOW their grid moved above.  Stage 0
%     reported zero of these: its scan allowed only lowercase
%     control sequences between \end{tabular} and \caption, and
%     both real cases have a brace and a \par in between.
%   * \GGListOfTables added, on its own page.
%   * fig:whynow-omega, flagged EMPTY since intake, is NOT a bug:
%     caption page, \clearpage, then the plate centred on its own
%     page.  Deliberate, and left alone.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 2C, same day.  Plain-English Part closes.
%   A10 carried SEVENTEEN Parts and NEITHER device -- no
%   \GGteachboxpart and no 'Part N in plain English' section.
%   Both counts were run before anything was written.
%   Fourteen closes added, one per content Part, in the
%   end-of-Part form A8 uses.  The back matter -- Technical
%   Appendices, Common Objections, References -- gets none, which
%   is A5's rule.
%   NEW PROSE: fourteen sections, listed in the report by Part and
%   page for the author to review or rewrite.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 3, same day.  Figures placed by the measured rule.
%   All 25 pictures measured with measure_plate.py, one compile
%   each.  11 placed at f = (7.79/B) * W / 165.1.  14 left alone:
%   4 already inside 7.79pt +/- 5%% (one of them the exempt cohort
%   map), 4 with no declared base font, 3 whose probe spills to
%   more than one page, and 3 that cannot be placed by the rule --
%   one with no \resizebox at all, one with two of them in a single
%   float, and one in landscape whose house fraction measure_plate
%   computed against \textwidth instead of \linewidth.
%   The worst outlier was fig:appfig-horizons at 11.94pt.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 4, same day.  Palette.  palette_fix brought 19 pictures to
%   the house band: 11 bare `draw' borders given a hue shade, two
%   fills above !18 compressed (!22->!16, !24->!16), keyword widths
%   made explicit, rules clamped to 1.1pt.
%   SEVEN OF ITS CHANGES WERE REVERTED after reading the diff:
%     * TWO COMMENTS.  palette_fix rewrote 'thin links' and 'thin
%       sliver' inside source comments into 'line width=0.40pt'.
%       Section 6 forbids rewriting the historical record; the tool
%       does not mask comments before converting keyword widths.
%     * The three WHITE 1.5pt pie-slice separators of the pi-locked
%       chart -- they are the gaps between slices, not borders.
%     * A 2.2pt feature arrow under a 5mm arrowhead, and a 2.2pt
%       violet timeline band.  Both are figure content.
%   The house band governs flowchart boxes; a separator, a band and
%   an arrow stem sized to its own head are not boxes.
% ------------------------------------------------------------
% ------------------------------------------------------------
% Stage 5, same day.  Integrity.  The three cohort companions A10
%   named in the cohort-map plate and its caption but never cited
%   -- P6-Proton, W1-Symmetry, W2-Holonomy -- now have bibliography
%   entries with Relevance notes written for THIS paper, and a
%   \citep each at the point the caption already names them.
%   All 21 companions present, zero orphans, no \citep carrying two
%   Arisaka keys.  NEW PROSE: three Relevance notes.
%   NOT done: the cross-paper inheritance table lists what A10 DRAWS
%   FROM each upstream paper, and A10 draws nothing from these three.
%   Adding rows would overstate the inheritance.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.0 -> v15.1  August 15, 2026.  Spelling rider.
%   * One printed British form the v15.0 spelling guard could not see:
%     'dialled' -> 'dialed'.  Report 355 stated A10's printed British
%     forms as 0; it was wrong by this one word.  The guard's doubled-l
%     branch required the letter e before the ll.  Corrected rule in
%     gg_spelling.py.  The five other sites the corrected rule finds are
%     published titles -- Lakatos's 'Research Programmes' and three
%     'Flavour' leptogenesis papers -- and are frozen, not changed.
%   NO PROSE WAS WRITTEN.  Nothing else in this file changed.
% ------------------------------------------------------------
% v13.40 -> v15.0  Stage 1 of the v15 migration, August 14, 2026.
%   * a4paper -> letterpaper.  \textwidth 452.968pt -> 469.755pt.
%   * American spelling: 805 substitutions across 142 distinct words,
%     printed text only, comments and cross-reference identifiers
%     masked.  The guard is MORPHOLOGICAL, not a word list -- on A8
%     a list-built guard reported zero survivors with 180 left.
%   * Four published titles by other authors frozen by exact text
%     BEFORE the sweep: Dev et al. 2014, Jukkala et al. 2021,
%     Li-Pilaftsis 2026b (all 'Flavour'), Lakatos 1978
%     ('Programmes').  Verified unchanged after.
%   * 'storey' -> 'story' (23 sites) is flagged in the report: the
%     American form is correct and loses a distinction.
%   * 3 hline sites -> toprule / midrule / bottomrule.
% ============================================================
%% -- cohort constant sweep, August 14, 2026.  The baryon bridge is
%%    quoted cohort-wide as eta10 = 273.9 Omega_b h^2 (A9's own value,
%%    the mean-baryon-mass form including helium).  The pure-proton
%%    3.6524e-3 = 1/273.79 is superseded.  Three printed quantities
%%    move with it: +0.53 -> +0.57 per cent, 16.5 -> 16.4 per cent,
%%    and -0.42 -> -0.38 per cent; and the prose claim 'within half a
%%    per cent' is retired, being false at +0.57 per cent.
%% ===================================================================
%% v13.39 (August 14, 2026) -- rider on v13.38
%%
%% 1. THE omega_b/omega_m ROW IS PROMOTED FROM D* TO D.  v13.38
%%    established that the epoch identification t_0 = t_pi is the only
%%    thing making the pi-locked inventory D* rather than D.  The
%%    ratio Omega_b/Omega_m = 1/(2 pi) does not depend on the epoch at
%%    all -- baryons and cold dark matter both dilute as a^-3, so the
%%    ratio is constant -- and it is h-free, which the paper already
%%    recorded.  It therefore carries no calibration input and should
%%    never have inherited the star.  Deviation +1.29% (+1.3 sigma).
%%
%%    With a precision point attached: the comparison must be made in
%%    omega_b/omega_m and not in Omega_b/Omega_m, because Planck's
%%    Omega_m includes a neutrino component this framework's Omega_m
%%    does not.  The two differ by half a percentage point (+1.29% vs
%%    +1.79%), which is not negligible against a +1.3% signal.
%%
%% 2. A NEW NOTE, "What the epoch can and cannot move," beside the
%%    where-the-conditional-lives paragraph.  eta_B is a constant of
%%    the universe: no choice of epoch can move it, so no baryon
%%    discrepancy can be closed that way.  rho_b IS epoch-dependent,
%%    but evaluating it at t_pi opens the gap rather than closing it.
%%    The note exists so that the manoeuvre is not attempted later and
%%    a sign flip mistaken for a fix.
%% ===================================================================
%% ===================================================================
%% v13.38 (August 14, 2026)
%%
%% 1. THE ORIGIN OF D* MADE CLEAN AND LOGICAL.  This paper carried
%%    three incompatible accounts of why the pi-locked inventory is
%%    D* rather than D.  The Proof-status note, the Status note after
%%    the topological theorem, and the conjecture register all said
%%    the residual was VIII-b'' -- an unexhibited normalisation
%%    lemma.  The Conjecture-VIII refinement paragraph said instead
%%    that VIII-b' is proved, that VIII-b'' closes as the epoch
%%    identity, and that the open core is VIII-a alone with the
%%    epoch as the declared calibration input.  The second reading is
%%    the later and the correct one; the first three are brought into
%%    line, and the reason is now stated ONCE, at the paragraph
%%    labelled par:where-the-conditional-lives, with the other sites
%%    pointing at it.
%%
%%    The statement is: THE FORM IS DERIVED, THE EPOCH IS CALIBRATED.
%%    The coupled pi-form follows from the upstream assignment plus
%%    VIII-a plus VIII-b' with no measured density anywhere in the
%%    selection; what the geometry does not fix is WHEN the running
%%    inventory reads that form, and the identification t_0 = t_pi to
%%    1 per cent is the declared L4 epoch input.  The star attaches to
%%    that identification and to nothing else.
%%
%%    New in this version: the reason VIII-b' was provable and
%%    VIII-b'' was not.  VIII-b' compares COUNTS -- dimensionless and
%%    cutoff-independent.  VIII-b'' compares a length to a volume,
%%    2 pi R / 2 pi^2 R^3 = 1/(pi R^2), which is 1/pi only at unit
%%    radius; a ratio carrying a dimension needs a physical unit, and
%%    the unit bridge terminates on the dilution no-go.
%%
%% 2. OBJECTION 56 REPAIRED.  The referee's question quoted a stale
%%    calculation and was wrong three ways: Omega_b ~ 0.047 (correct
%%    0.059), a 7.5 per cent gap (correct 16.5 per cent), and a gap
%%    "growing to 28 per cent on the late branch" when it in fact
%%    shrinks through zero there, to -2 per cent at h = 0.735.  The
%%    response was already right.  The question now matches it.
%%
%% 3. ONE QUANTITY, THREE COSTUMES.  The consistency-chain subsection
%%    now states at its head that eta_B, rho_b and Omega_b are one
%%    quantity in three sets of units, leads with the ledger road
%%    (eta_10 = 6.15 against an observed 6.12 at this paper's own h,
%%    an error of +0.53 per cent), and reframes the dynamical chain as
%%    the mechanism demonstration its own grading already makes it.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.37  (August 13, 2026) -- the vacuum scale, stated
%%  completely.  No new claim: A10's own equation, evaluated at A10's
%%  own selected coefficient.
%%    rho_Lambda^(1/4) = zeta^(1/4) * M_lock^2/M_Pl,red
%%                     = 0.3660 * 6.083 = 2.23 meV
%%    measured: 2.224 meV on this paper's own derived cosmology
%%              (H0 = 66.6, Omega_Lambda = 1 - 1/pi); 2.240 on Planck.
%%    -> 0.1% in scale, 0.4% in density.  The paper had been saying
%%    "a residual factor of a few", "an order-unity factor", and
%%    "some 119 of the notorious 120 orders" -- all four of those
%%    sentences predate the zeta_Lambda = pi/LK selection.
%%  Also fixed: those four sentences write m_Lambda = M_lock^2/M_Pl and
%%  quote 6 meV.  With the cohort's canonical NON-reduced M_Pl that
%%  expression is 1.21 meV; 6.083 requires the REDUCED mass, which the
%%  scale-ladder table already writes correctly.  M_Pl,red made
%%  explicit in all four, per A4 Appendix C's rule against leaving the
%%  Planck-mass convention as a hedge.
%%  The grade is unchanged and travels with the number: Derived (cond.)
%%  on the stratum-assignment lemma, pre-registered two-sided
%%  zeta_Lambda in [0.0177, 0.0182] (A5-GDOS O6; F-Hubble-11 of C2).
%% ===================================================================
% ===================================================================
%  A10-Cosmos v13.36 (August 12, 2026) -- cohort-wide sweep, two defects.
%
%  (1) GU-5 IS A THREE-PLUS-TWO PARTITION, exactly as OGN-5 is
%      (M,N,L;J,K).  The separator before k_B is a SEMICOLON:
%      {c, G_N, hbar; k_B, Q_B}.  4 site(s) corrected here.
%      NOT touched, by design: the four-constant Planck unit system
%      {c, G_N, hbar, k_B}, which is a different object and takes a
%      comma; a semicolon there would be wrong.
%
%  (2) Internal affairs -- companion versions, dates, page counts and
%      build status -- do not belong in a preprint.
%%  ALSO: the same disease, found by censusing the cohort by meaning
%%  rather than by the one phrase quoted.  The build-QA clause "all
%%  compiling clean through pdflatex x 3, all dated May 14, 2026" is
%%  removed, eight per-companion date stamps are stripped from the
%%  provenance section, and that section is retitled -- it is about what
%%  A10 inherits, not about versions.  Its content is untouched.
%
%  No physics changes; no parameter, theorem or falsifier is touched.
% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.35 (August 12, 2026) -- display-only rider applying
%%  the eta_B display standard (GG-Standard-EtaB-Display-v1.0): headline
%%  sites quote ~7e-10 with the x3 band; worked arithmetic keeps the
%%  exact product 7.34e-10; +19.9% -> +20% everywhere.  No physics, no
%%  value, no grade changes.  Dev plots and the Schramm figure carry the
%%  same convention.  Part of the four-paper consistency sweep (309).
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.34  (August 11, 2026) -- pre-arXiv revision, ship A-D
%%  (Report 307/308, author-approved).  Script A (this banner): Tier-1
%%  internal contradictions.  (a) The three retired TeV-reheating
%%  residuals brought to the canonical launch (Part V chronology
%%  equation, Part VIII paragraph+equation with the radius and redshift
%%  recomputed on the master table, Part II sentence).  (b) The Part VI
%%  headline table row for the decoupling redshift re-pointed at the
%%  audited atomic-chain value, the corridor switch annotated, not
%%  scored (two-quantity rule).  (c) The electroweak-vev label at the
%%  Majorana seam scale corrected 174 -> 246 GeV at both sites (A9,
%%  the owning paper, evaluates the 8.4 meV seam scale at v = 246);
%%  the cross-normalisation factor collapsed to the single-convention
%%  ratio ~12.  (d) Drill I.1.2 aligned to the (1/2)sin(2 phi_geo)
%%  convention of the resonant subsection.  Scripts B-D: instrument
%%  sweep, equilibrium-yield display (Option A), Boltzmann integration.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.33  (August 6, 2026)  -- landscape figures restored.
%%  The cohort figure sweep sets each plate to a fraction of
%%  \textwidth.  Inside \begin{landscape} the page is rotated but
%%  \textwidth is not, so the usable width is about 246mm while
%%  \textwidth still reports 159.2mm, and a plate placed by the rule
%%  lands at roughly half the size it should.  1 figure on
%%  landscape page restored to its exact pre-sweep placement.
%%  Four figures cohort-wide were affected; none other sits inside a
%%  landscape environment.  Script: [v13-support]/revert_landscape.py.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.32  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  18 figures re-placed so their node text lands at the house size,
%%  7.79pt: what GG-P3-Electron Figure 3 renders (\footnotesize at
%%  0.865 of natural size).  For a picture of measured width W drawn
%%  at a base font of B points the target is f = (7.79/B) * W / 159.2
%%  of \textwidth, W being measured by compiling each picture
%%  standalone and reading its ink bounding box.  Nothing inside any
%%  picture changes: no font command, no dimension, no text, no
%%  relative position.  The cohort map and the central-claim funnel
%%  are excluded, both being at the house size already.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.31  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 110mm boxes and then
%%  left at natural size, so its \footnotesize landed 16% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.66\textwidth, computed rather than chosen: the picture measures
%%  121.4mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.30  (August 5, 2026)  -- the twenty-two-paper cohort sweep.
%%  The cohort map becomes the twenty-two-paper map: GG-P6-Proton on
%%  the Predictions row with a second sister-import arrow, and the
%%  White-Paper Branch (GG-W1-Symmetry above GG-W2-Holonomy) as a
%%  right-flank column at the mirror positions of the Foundations
%%  column -- the left flank feeds the spine, the right reads it.
%%  GG-P7-Inflow does not appear: it is Appendix H of GG-P2-N3.
%%  The plate is also enlarged for reading: boxes and the type inside
%%  them x1.15, every word outside a box x1.20, and the long
%%  A10 -> C-frame edge redrawn as a polyline, which returns 24mm of
%%  bounding box that was Bezier control points rather than ink.
%%  Prose: every count site, the series breakdown, and the canonical
%%  Suffix list, which was missing Proton, Symmetry and Holonomy.
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.25  (August 1, 2026)  -- the cosmic-descent figure.
%%  New figure (external sans-serif asset a10_cosmic_descent.pdf; the
%%  P6-Fig5 idiom made cosmological): one to-scale descent on
%%  r=ln(E_GUT/mu), reading energy, time, and scale factor at every key
%%  epoch of the master chronology, late universe magnified below a
%%  break. Placed twice: (1) a showcase full page between Fig.\ 4 and
%%  Fig.\ 5 in the Executive Summary (fig:cosmic-descent-showcase);
%%  (2) the full figure immediately before the master chronology table
%%  in Part VIII (fig:cosmic-descent). Version bumped.
%%  v13.26: fixed blank pages before the main figure -- removed the
%%  clearpage/vfill centering (which orphaned the heading and overflowed
%%  a blank page); figure now flows under the heading with a short intro,
%%  image at 0.66\textheight so heading+figure+caption share one page.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.24  (August 1, 2026)  -- four curious-reader entries.
%%  NEW closing section sec:phil-objections-return with Objections 94-97
%%  wielding the three-horizon ladder (Secs. 63-64) against the
%%  multiverse/landscape in the fenced register (structural contrast,
%%  no polemic): 94 exile-not-execution (author's question; late-choice
%%  mechanisms killed via CohenDeRujulaGlashow1998; landscape driven to
%%  the early-choice corner); 95 falsifiability asymmetry (bets that
%%  can lose vs bets that cannot); 96 the razor symmetry ('everything
%%  happens somewhere' shares the logical shape of 'the world was
%%  designed'; both declined for the same reason); 97 the dated
%%  discriminator (r = 0.0043 vs the contested de Sitter swampland
%%  criterion; two-sided stake vs adjustable posture). Entry count
%%  93 -> 97, contiguous, verified; no other text touched. Version bumped.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.23  (August 1, 2026)  -- the lock tower named.
%%  (1) ssec:entrance: the amplitude's mechanism identified -- the R^2
%%      coefficient as the threshold sum of P1-KK's lock-anchored Bessel
%%      tower (T-KK-1 base, 3.85 TeV; spacing pi x 1.0 TeV); the target
%%      c = 4.6e8 bracketed within an order of magnitude on both sides;
%%      residual = one signed per-level number beta_lvl in [1/772,1/100].
%%      Three standard routes recorded failing (matter 9, holographic 5
%%      -- rederiving the recorded closure -- flat interval 11 orders).
%%      The bend test of v13.22 recorded as PASSED by mode depletion:
%%      c(N)/c0 = 1 - 2 sqrt(N) M/M_top (eq:partII-depletion); pivot
%%      depletion <= 2 pct; N_bend inside the corridor for either
%%      natural UV end and the entire threshold band.
%%  (2) Graded paragraph sharpened accordingly.
%%  (3) Warp-candidate parenthesis annotated with the K-scaling
%%      discriminator: M ~ e^{-K/3} (flagged guess) vs M ~ e^{-K/2}
%%      (tower form, T-KK-1-backed); coincide only near K = 35.
%%  Report 131; s3_gdos_action.py in [v11-Support].
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.22  (August 1, 2026)  -- the entrance corridor.
%%  (1) ssec:entrance: NEW constitutional ceiling -- on the plateau the
%%      quantum kick H/2pi overtakes the classical drift at
%%      N_et = 4 pi sqrt(3/2) Mpl/M ~= 1.1e6 (phi ~= 17.4); beyond it
%%      the launch SAMPLES, which the no-sampling clause (audited by the
%%      third horizon Section) forbids. Corridor: 3.55 <= phi_0 <= 15.78
%%      (uphill), 6.86 <= phi_0 <= 19.10 (downhill); floor
%%      observational, ceiling constitutional. New eqs
%%      partII-eternal-bound, partII-entrance-corridor.
%%  (2) The map's scale glossed exactly: mu^4 = Mpl^2 R_J/4 (Jordan
%%      curvature scale; equals rho_tot only after the exit).
%%  (3) Open items MERGED 2 -> 1: the GDOS effective action owes A_s
%%      (R^2 coefficient) and the entrance (the plateau's bend, naive
%%      N_bend ~ 7e6) at once, and must land the bend inside the
%%      corridor -- one computation, one stated test. Open-lemma and
%%      graded paragraphs updated accordingly.
%%  2K candidate: discarded at report level (GG-Report-129), not
%%  mentioned in the paper, per the anti-numerology standard.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.21  (August 1, 2026)  -- CORRECTION + two repairs.
%%  (0) Sec. 25.6 (ssec:entrance) REWRITTEN. v13.20 evaluated the
%%      depth-to-field map at the UV edge via R = 4 rho/Mpl^2 -- a
%%      vacuum-domination relation contradicted by Sec. 25.5's own
%%      kinetic-domination theorem: the Einstein-frame trace
%%      R = (4V - phidot^2)/Mpl^2 is NEGATIVE at that instant, so the
%%      map's logarithm had a negative argument. WITHDRAWN: phi(0) =
%%      5.01, the 96%/85% figures, the 174-vs-12 uphill/downhill
%%      contrast, the 6% admixture threshold, and the "entrance reduces
%%      to one sign bit" claim. RETAINED (energy bookkeeping only, no
%%      curvature reading): Sec. 25.5 entire -- the cold start, the
%%      0.68 e-fold sprint, |Delta phi| = 1.66, trigger-free ignition.
%%      NEW STATUS: the entrance needs two inherited data (phi_0 and
%%      the sign of phidot_0); the energy budget fixes only |phidot_0|
%%      because V is asymptotically flat; the requirement is
%%      phi_0 >= 3.55 (uphill) or >= 6.86 (downhill), both satisfiable,
%%      and what fixes phi_0 is named as the open lemma. Graded
%%      paragraph reverted to 'asserted'; the heat-kernel 'second
%%      target' sentence added in v13.20 is dropped with it.
%%  (1) D1: POS-4 CP glossed "closed packet" -> "closure and
%%      positivity" (A1-Constitution is the owner; A10 already had it
%%      right elsewhere).
%%  (2) D2: the closed-packet topology argument was attributed to
%%      A4-Euclid, which contains no such argument and states that it
%%      does not re-derive the cosmological inventory. Re-attributed to
%%      this Part, resting on A4's OGN-5 ledger (A4's actual contribution,
%%      per the coherence map) with A5 supplying the packet's unit.
%%  Report 127; s1d_entrance_consistency.py.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.20  (August 1, 2026)  -- The launch completed and the
%%  neutrality theorem: four additions, no retractions.
%%  (1) NEW SUBSEC ssec:first-efold (end of two-basin block): the edge
%%      surplus rho_edge/V0 = e^{4 r_inf} = 58.9 forced into the kinetic
%%      channel (potential bounded; thermal bath doubly excluded -- a hot
%%      start forbids the launch in the derived class); 0.68-e-fold
%%      kination sprint, Delta-phi = 1.66 Mpl; trigger-free ignition.
%%  (2) NEW SUBSEC ssec:entrance: depth-to-field map phi(r) =
%%      sqrt(3/2) Mpl ln(1+e^{4(r_inf-r)}); phi(0) = 5.01 vs 5.21 needed;
%%      zero-throw N = 44.9 (85 pct); uphill throw N = 174 (pass);
%%      downhill 12 (CMB-excluded); orientation = one open bit. Graded
%%      paragraph updated (entrance link; heat-kernel second target).
%%  (3) NEW SEC sec:third-horizon (Part X, after second-horizon):
%%      H1 conditions / H2 things / H3 choices; CohenDeRujulaGlashow1998.
%%  (4) NEW SUBSEC ssec:neutrality (end of baryogenesis Section): the
%%      four-link neutrality theorem; eq:partV-ne-np; discharges the
%%      silent hypercharge-neutrality import; LyttletonBondi1959 +
%%      Bressi2011 added. Numbers verified programmatically (S1 scripts,
%%      Reports 121-123). Version/date bumped.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.18  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * COHORT ROSTER: P2 row "A closed derivation of N=3 fermion
%%     generations" -> "A three-leg derivation ...".
%%   * COHORT ROSTER: P3 row "the five oldest electron questions" -> "six",
%%     matching P3's title page and A10's own bibitem.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.17  (July 29, 2026)  -- two year-month residuals.
%%  v13.16's post-condition matched full ISO dates (20dd-dd-dd) only, so
%%  "(2026-07)" slipped past it twice.  Caught by the final cohort-wide scan.
%%  The second site's content is kept in full: the 8pi^2 is consumed by A6's
%%  GEP definition, returning the integer, and no residual pi descends from
%%  that step -- now a statement about the step rather than a dated note.
%% ===================================================================
%% ===================================================================
%%  A10-Cosmos v13.16  (July 29, 2026)  -- internal-affairs sweep.
%%  Reader-facing traces of the working history removed ahead of arXiv v1.
%%   * 11 working dates / session references deleted (2026-07-01, 2026-07-02,
%%     "the [B1] kernel session", "the [P] projection session", "the
%%     unit-bridge pass", "the residue pass", "the same day").  Every claim
%%     they decorated is untouched.
%%   * 2 campaign phrasings -> "left to a dedicated study".
%%   * 4 draft-lineage sites REPHRASED, NOT DELETED.  Three of them carry the
%%     paper's non-circularity argument; deleting the contrast would have
%%     removed honesty content.  Each now states the contrast as a hypothetical
%%     route rather than as this paper's history.  The 1/15 dead end, the
%%     competitor elimination, and the Proof-status note all survive in full.
%%   * 2 self-correction sites (the theta_i passages).  This class was found
%%     by the falsifier triage, not by the detector -- it uses none of the
%%     catalogued vocabulary.  The honesty warning that theta_i is NOT
%%     geometry-selected at 1/(2pi) is KEPT; only the draft history goes.
%%     Objection 51's premise WAS the defect, so it is recast to the question
%%     a reader would actually ask, and the response now excludes 1/(2pi)
%%     quantitatively (0.159 rad vs the generic 1.1-1.5 rad) instead of
%%     narrating a correction.
%%  No value, equation, grade or citation changed.
%% ===================================================================
%% ===================================================================
%% v13.14 -> v13.15 (July 28, 2026).  Phase C of the GGA correction.
%%   * The ground-state refinement re-pointed to its upstream home and
%%     quoted at its grade, with an explicit note that the cosmological
%%     strata are not established members of the admissible class.
%%   * The four pillars no longer described as selecting the ground
%%     attractor; they characterise it.
%%   * Cascade chain form and regime attributions conformed.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v13.14 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons. No other change
%%  anywhere.
%% ===================================================================
%% ===================================================================
%% v13.13 (July 24, 2026): NEW PART XI --- THE ASSEMBLED LCDM
%%  (author-commissioned, Report 29 Ship 2; the counterpart of the
%%  A9 companion's assembled-Lagrangian Part). A new Part is inserted
%%  between the topological completion and the Summary: the model the
%%  reader was taught to fit; the zeroth and the six silent axioms of
%%  the concordance model, each replaced by a theorem or a graded
%%  conditional in a scorecard table; the export chain in eight
%%  stages from the bulk action to the expansion history, each stage
%%  naming its home; the assembled expansion law displayed with the
%%  register of every entry and no fitted continuous parameter; and
%%  the honesty boundary stated beside it. The Summary Part is
%%  retitled "Summary and cross-paper coherence" and all later Parts
%%  renumber by one (the single external companion-Part reference is
%%  preserved); the Parts-roadmap figure gains the seventeenth box
%%  and a new phase label, and the executive-summary walk is updated.
%% ===================================================================
%% ===================================================================
%% v13.12 (July 24, 2026): APPENDIX E REBUILT AS THE THREE-STEP WALK
%%  (author-commissioned, Report 29 Ship 1). The appendix is retitled
%%  and restructured so that it derives, in order: the 4D Einstein
%%  equation from the GG-6 bulk (the stage and ledger; the bulk action
%%  displayed and cited from its earned home in the A9 companion; the
%%  Einstein--Hilbert term descending with its scale weight; Newton's
%%  constant as a computed overlap; the equation itself by variation
%%  and zero-mode projection, its conservation law inherited as a Ward
%%  identity, its open-system correction bounded with live falsifiers,
%%  and the induced-gravity companion route noted); then the Friedmann
%%  equations from that equation (the existing exact walk, unchanged
%%  in content); then LCDM from the Friedmann equations (the source
%%  ledger component by component, the pi-locked fractions with the
%%  epoch-identity register, the initial conditions, and the
%%  assembled expansion law with no fitted continuous parameter).
%%  The four-debts audit moves to the front as the roadmap. Seven new
%%  worked problems join the existing seven, all renumbered to the
%%  new subsection map. One redundant paragraph dissolved (flagged in
%%  the ship report); all other relocated text verbatim.
%% ===================================================================
%% ===================================================================
%% v13.11 (July 24, 2026): CODEBOOK-ZONE TRUE LANDSCAPE DISPLAY
%%  (author commission). The provenance chart and the Dev-plot data
%%  table now sit on genuine landscape pages (the same page-rotation
%%  mechanism as the Part-VIII flowchart and the codebook table), so
%%  they display horizontally on screen; previously they were
%%  content-rotated on portrait pages and displayed sideways. Built
%%  as a single deferred block emitting the two landscape pages and
%%  then the upright full-width deviation plot, which stays portrait
%%  by the author's decision. The two long captions are trimmed to
%%  fit the landscape pages; the removed material is relocated into
%%  the consolidated reading-conventions paragraph, which is renamed
%%  to cover both figures and the table. Table font size and the
%%  cropped full-width plot of the previous version are preserved.
%%  No numbering changes: the three exhibits keep their numbers.
%% ===================================================================
%% ===================================================================
%% v13.10 (July 24, 2026): PORTRAIT REDESIGN OF THE PART-IX CODEBOOK
%%  ZONE (author-approved from rendered trials; closes the blank
%%  pages 265/267 flagged at v13.9). The landscape environment is
%%  removed: the provenance map is now a book-style sideways
%%  figure[p] (content untouched); the hand-built data-table page is
%%  a sideways table[p] at ~30% larger rendered font with a trimmed
%%  caption; the dev-plot page is a figure[p] with the plot's white
%%  padding cropped (trim+clip, gs-measured bbox) so the ink spans
%%  the full text width. Nothing deleted: the conventions and
%%  epoch-calibration text removed from the two captions moves
%%  verbatim-adapted into the new main-text paragraph "Reading
%%  conventions for Table ... and Figure ...", which both captions
%%  reference; the two-quantity z*/corridor-switch language is
%%  retained in full. Pages 576 -> 574; the 36.8pt landscape-zone
%%  float warning is eliminated.
%% ===================================================================
%% ===================================================================
%% v13.9 (July 23, 2026): NEAR-EMPTY-PAGE REPAIR (cohort-wide audit
%%  commissioned after A8 v13.12). Forced
%%  breaks dropped around the [p] cohort map; the parts-roadmap and
%%  derivation-tree hand-built minipage pages promoted to real
%%  figure[p] floats so surrounding text fills the stranded pages.
%%  The front-matter showcase gallery, Part tails, and the chronology
%%  landscape untouched; the two blank pages flanking the Part-IX
%%  codebook-provenance landscape are left as in v13.8 (the zone
%%  needs an author-level design decision -- longtable + landscape +
%%  gallery chain). NO text content changed.
%% ===================================================================
%% ===== v13.8 (2026-07-23): PROBLEMS & WORKED SOLUTIONS, SHIP 2 (J-N) ======
%% Ship two, completing the A10 problems campaign: 27 problems added
%% to chapters J-N (J:6 K:6 L:5 M:5 N:5); 56 problems now span all
%% ten course chapters, and the front-matter problems clause lands.
%% Chapter-register care: M avoids re-deriving the worked haloscope
%% numbers (the chapter's own chain gives its own rounding) and
%% exercises the mechanism + the inversion (theta_i = 1.4 rad as
%% prediction); N.9.1 keeps "conceptually closed, quantitatively
%% open" verbatim; L.11.1 mirrors the integer-pairing obstruction.
%% Highlights, verified programmatically: one-equation-two-destinies
%% (growth vs song); delta ~ a exactly; f = (1/pi)^0.55 = 0.53;
%% the P(k) turnover as the imprint of z_eq; BAO spacing 0.043/Mpc;
%% S8 = 1.030 sigma_8; the pendulum-census reading of the peaks;
%% R_* ~ 0.6 and c_s = 0.45c; ell_A ~ 300 with the first peak at
%% 0.73 ell_A ~ 220; odd/even baryon loading; Silk tail ~1500; the
%% geometric degeneracy (the engine vs the road); F wedge F exact;
%% Dirac quantisation; the ledger's exponential map (e^7, 3.85 TeV,
%% e^{-17.5}); Vol(S^3) = 2pi^2 with Omega_b/Omega_m = 1/2pi = 0.159
%% vs 0.157; why 1/pi must be a measure ratio; the cosine from the
%% circle; Hubble friction as dilution; m = 3H release; the virial
%% proof of rho ~ a^-3; the inversion counting (one unknown left);
%% the Schwarzschild first law; the Susskind-Uglum cancellation; the
%% conical derivative; the Page schedule (230 bits/s); the
%% 0.6-lunar-mass evaporation threshold.
%% ===== v13.7 (2026-07-23): PROBLEMS & WORKED SOLUTIONS, SHIP 1 (E-I) ======
%% The cohort problems campaign reaches A10 (P5-CPT model; C3 done at
%% 53 problems; author: "as many problems as possible").  Ship one:
%% 29 problems with complete worked solutions added to course
%% chapters E-I (E:7 F:7 G:7 H:4 I:4), each set placed before its
%% chapter's take-away box, numbered X.s.n by chapter letter and
%% subsection, Solution in explicit Steps, "Physics intuition:"
%% close, cross-referenced into the course network (and across to
%% the C3 course where the de Sitter port meets Appendix Q there).
%% Highlights, all verified programmatically: q0 = 3/2pi - 1 =
%% -0.523; the pi-locked equality and acceleration-onset redshifts
%% (0.29, 0.62); the closed-form age 13.75 Gyr from Omega_m = 1/pi;
%% rho_c = 5.1 protons/m^3; the (4/11)^(1/3) bookkeeping; nu
%% decoupling from Gamma/H; z_eq ~ 3.5e3 from 1/pi; THE TWO-QUANTITY
%% z* DISCIPLINE stated with the 0.6% gap and the observable-only
%% falsifier; the 17.6-Gly future archive; entropy per baryon
%% 1.2e10 (the 7.04 cross-link); the horizon problem at ~1 degree;
%% the 60-e-fold mirror; V^(1/4) = 8.3e15 GeV from r = 0.0043;
%% n_t = -r/8; the instant-reheat ceiling; quadratic inflation
%% killed by arithmetic; N = 52.6 from n_s with r = 12/N^2 landing
%% on 0.0043 (one prediction, not two); the deuterium bottleneck's
%% ln(1/eta); the baryometer logic; Delta Y = 0.013 Delta N_nu; the
%% lithium factor 5.0/1.6 = 3.13 vs pi; the 6e14-GeV seesaw scale;
%% eps_max = sin(phi)/2 at Delta M = Gamma/2; the eta_B audit
%% 5.87e-10 vs 6.1e-10; the 1/pi survival derivation.
%% Ship two (chapters J-N) follows after sign-off, with the
%% front-matter problems clause.
%% =====================================================================
%% v13.6 (2026-07-20)  THE DERIVATION HOME: H-PHIGEO-1 CANDIDATE
%%   MECHANISM INSTALLED (author-approved ship; spec: internal report
%%   HPhigeo1-Transport-2026-07-20 + v2 extension 2026-07-21; scripts
%%   hphigeo1_transport.py / hphigeo1_transport_v2.py).  (1) NEW
%%   APPENDIX-I SUBSECTION 'The partial-resonance mechanism: the
%%   flavour-measurement regulator' (subsec:app-flavour-regulator),
%%   between the resonant-CP subsection and the washout subsection:
%%   the complete step-by-step derivation -- the flavoured kinetic
%%   system (eq:app-flavour-kinetic), the real-R flavour-odd source
%%   citing Lemma L-Cosmos-U (eq:app-source-traceless), the
%%   flavour-blind double-commutator cancellation (eq:app-blind-
%%   cancel), the two-level reduction with the |R|<=1/2 bound
%%   (eq:app-twolevel-I), the effective regulator Gamma_eff =
%%   c_tau y_tau^2 T (eq:app-Gamma-fl), the zero-dial closed form
%%   DeltaM/Gamma_eff = 1/(35 c_tau y_tau^2) = (2.7-5.3)e4
%%   (eq:app-closed-form, eq:app-closed-number), the worked numbers
%%   (y_tau running, Gamma_eff = 2-4 MeV, F_BW = (0.9-1.9)e-5,
%%   single-T etaB = 0.28-0.55 of observed, the T-integrated window
%%   factor 0.16-0.27 and landing 0.05-0.30, favourable corner ~0.6,
%%   always from below), the regulator-landscape table
%%   (tab:app-regulator-landscape, sixteen orders, exactly one
%%   candidate in the required decade), the numbered conditions
%%   C1-C6 each with status (C1-C3 established, C4 open crux, C5
%%   plausible band, C6 computed-at-Boltzmann-level), and the kill
%%   scenarios (two clean, one soft).  Physics-intuition paragraphs
%%   in the house register (the bath as flavour-measuring apparatus;
%%   the CP coherence lives in the leptons, not the N; the tau as
%%   fastest honest measurer; K = 35 as geometry over chemistry).
%%   (2) H-PHIGEO-1 PARAGRAPH AMENDED (sanctioned Stage-4 touch; the
%%   emph hypothesis statement itself byte-identical): the hypothesis
%%   stays named and conditional, now carrying the candidate
%%   mechanism at candidate grade; 'target value ... not a derived
%%   output' upgraded to 'target value, now carried by an identified
%%   candidate mechanism whose closed form lands the ratio in the
%%   hypothesized decade with zero dials; the transport-level closure
%%   of conditions C1-C6 remains open'; the row-9 clause records the
%%   mechanism at candidate grade.  (3) STATUS RIPPLES: Appendix-I
%%   resonant-subsection Status paragraph + Part-V pointer at
%%   eq:eps-value-partV gain the mechanism note; grade
%%   derived-conditional UNCHANGED everywhere.  Protected
%%   byte-identical: abstract + executive summary, the H-phigeo-1
%%   emph statement, Lemma L-Cosmos-U statement + proof, the
%%   seventeen-problems apparatus, both frozen cohort-map renditions,
%%   the Part-V boxed target and chain-audit rows, the alpha_31 = pi
%%   sign registration.  No printed canonical number changed
%%   (5.87e-10, 6.12, 2.5e-6 target, e^{-K/2}, K_wash ~ 50, 8.4 meV,
%%   factor ~13, alpha_31 = pi).  v13.5 -> v13.6.
%%   Script: a10_v13.6_mechanism.py.
%% =====================================================================
%% =====================================================================
%% v13.5 (2026-07-20)  STAGE-4 SHIP: U-CANCELLATION KEYED (L-Cosmos-U)
%%   + WASHOUT DUAL-CONVENTION RECONCILIATION (author-approved Wave-2
%%   physics-grade ship; spec: internal reports
%%   PhiGeo-Projection-2026-07-18 and RealBulk-Audit-2026-07-19;
%%   paired with A9-SM v20.36).  (1) KEYED LEMMA: the Appendix-I
%%   worked prose proving the U_PMNS cancellation is promoted to a
%%   formal keyed lemma in the house convention -- Lemma
%%   'U_PMNS-cancellation and the flavour obligation (L-Cosmos-U)',
%%   label lem:cosmos-U, with the real-R nullity as clause C-Cosmos-U
%%   and the flavour obligation as clause L-Cosmos-F; the existing
%%   prose becomes the proof verbatim (one connective clause added:
%%   'the sum ... is real'); eq:app-sinphi unchanged inside the
%%   proof.  Three informal-reference sites now cite the key: the
%%   Part-V 'fully flavoured' sentence, the Appendix-I worked-number
%%   'U-cancellation above' clause, and the Appendix-I Status line.
%%   (2) WASHOUT RECONCILIATION (was 'declared and priced O(10) but
%%   not reconciled'): the convention note at eq:app-GammaN gains the
%%   accounting -- the width is a property of the seam-localized
%%   Yukawa texture (seam-matched m-tilde = pi v^2/E_GUT ~= 8.4 meV),
%%   the washout of the light-neutrino mass scale the bath actually
%%   sees (m-tilde ~ m_3 ~= 50.5 meV); effective masses differ by
%%   ~6.0 (= 50.5/8.4), and the quoted factor-~13 in Y_nu^2 carries
%%   in addition the factor two between the Casas-Ibarra
%%   normalisation (Y_3 ~= 2.6e-6) and the seam magnitude matching
%%   (Y_nu ~= 7.3e-7): (2.6e-6/7.3e-7)^2 ~= 13 ~= 2 x 6.0.  Pointer
%%   installed at the Part-V K_wash site; the reverse note at the
%%   washout worked number now says 'accounted for'.  All printed
%%   numbers unchanged (K_wash ~= 50, kappa ~= 1e-2, Gamma_N values,
%%   7.6/0.13, eta_B 5.87e-10, eps_res 2.5e-6 target).  (3) ROADMAP
%%   ROW-9 REGISTER TOKEN (sanctioned Stage-4 touch inside the named-
%%   hypothesis paragraph; the H-phigeo-1 statement itself is
%%   byte-identical): 'this paper's satellite' -> 'this paper's
%%   Foundations-Branch companion, the CPT member paper GG-P5-CPT'
%%   (member language; row substance identical).  Protected
%%   byte-identical: abstract + executive summary, the H-phigeo-1
%%   hypothesis statement and target-value sentence, the 'natural
%%   rather than tuned' disavowal, the seventeen-problems apparatus,
%%   both frozen cohort-map renditions, the Part-V boxed target and
%%   chain-audit rows.  No printed physical number changed.
%%   v13.4 -> v13.5.  Script: a10_v13.5_stage4.py.
%% =====================================================================
%% =====================================================================
%% v13.4 (2026-07-20)  STAGE 2 -- ARROW-CUSTODY CONSUMER (cohort ship;
%%   audit RealBulk-Audit-2026-07-19, exec-summary point 4 + A10-R3(a);
%%   custody anchor: A1-Constitution v13.2 AX-3 synthesis; consumer
%%   pattern of A5 v13.1 / A8 v13.3).  (1) The arrow-custody consumer
%%   sentence installed at the primary arrow site, the close of the
%%   subsection 'Expansion, cooling, and the arrow of time in the GDOS
%%   language' (Part III radial chronology): AX-3 owns the arrow
%%   constitutionally (the AX-3 synthesis of ArisakaA1Constitution);
%%   Part IX of the CPT member paper (ArisakaP5CPT) is the canonical
%%   law-level derivation, over a bulk that is itself exactly
%%   T-symmetric; the GDOS companion (ArisakaA5GDOS) is the dynamical
%%   implementation; the arrow of cosmic time read off the radial
%%   descent is the cosmological face of that one constitutional
%%   arrow, not a private one.  (2) Half-sentence pointer at the
%%   Conclusions restatement ('the building has a direction'): the
%%   GRIP cascade named as the dynamical face of the one
%%   constitutional arrow that AX-3 owns.  No other content touched.
%%   NOT in scope (Stage 4): A10-R1 washout reconciliation, A10-R2
%%   U-cancellation keying.  The phi_geo installs of v13.2 (Appendix I
%%   flavoured computation, H-phigeo-1 sites, roadmap row 9,
%%   seam-Yukawa convention declaration, the 'natural rather than
%%   tuned' disavowal), the seventeen-problems apparatus, both frozen
%%   v1.4 cohort-map renditions, and the abstract + executive summary
%%   are byte-identical (protect-asserted).  No physics changed.
%%   v13.3 -> v13.4.  Script: a10_v13.4_stage2.py.
%% =====================================================================
%% v13.3 (2026-07-19)  COHORT-19 STANDARDIZATION (cohort ship, WAVE 1;
%%   mechanical -- no value, status, or triage count changed).
%%   (1) Part I close standardized to the cohort architecture (A8
%%   v13.2 / A9 v20.35 pattern): the sections 'Scope, the cohort, and
%%   how to read this paper' and 'The whole programme at a glance'
%%   reorganised into the closing section 'The cohort inheritance and
%%   the roadmap of the sixteen Parts' -- subsections 'The cohort
%%   inheritance' (label sec:cohort KEPT; cohort prose migrated) and
%%   'Roadmap of the sixteen Parts' (new; label sec:roadmap-A10).
%%   All prior labels kept.  (2) The cohort map replaced by the frozen
%%   v1.4 nineteen-paper master, verbatim from A2-GG6 v13.3 (identical
%%   in every paper; MP4/MP5 internal node names), in BOTH renditions:
%%   fig:cohort-map-a10 (Part I) and fig:cohort-one-view-summary
%%   (Part XI Summary recap).  (3) NEW roadmap figure 'Roadmap of the
%%   sixteen Parts of GG-A10-Cosmos' (label fig:parts-roadmap-a10
%%   kept; Parts IV, V, X highlighted -- the pi-locked inventory
%%   derived in cosmic history and its topological origin), retiring
%%   the thirteen-body-Part flowchart.  (4) Cohort count seventeen ->
%%   nineteen at the cohort-sense sites only (the seventeen-problems
%%   apparatus, the seventeen-orders/log-unit magnitudes, and all
%%   other non-cohort senses untouched); the inheritance table gains
%%   the P4/P5 rows; the suffix roster moves to the canonical
%%   nineteen-paper form.  (5) P4/P5: bib key ArisakaP4Twistor ->
%%   ArisakaP4Real with the deployed P4-Real cover title; \PaperPFour
%%   now renders GG-P4-Real; \PaperPFive macro and the ArisakaP5CPT
%%   bibitem (P5 v3.40 cover) installed; a roster-level Foundations
%%   Branch paragraph added to the cohort subsection.  (6) Register:
%%   reader-facing designators normalised at the P4/P5 mention sites;
%%   bibliography A-header widened to the nineteen-paper form.  The
%%   phi_geo installs of v13.2 (Appendix I flavoured computation,
%%   H-phigeo-1 sites, roadmap row 9, seam-Yukawa convention
%%   declaration, the 'natural rather than tuned' disavowal) are
%%   byte-identical (protect-asserted).  No physics changed.
%%   v13.2 -> v13.3.  Script: a10_v13.3_wave1.py.
%% =====================================================================
%% =====================================================================
%% v13.2 (2026-07-19)  PHI_GEO INSTALL (author-approved cohort ship;
%%   calibrated honesty; spec: internal report PhiGeo-Projection
%%   2026-07-18). Appendix I, resonant-CP subsection: the assembled
%%   sin phi_geo evaluation (c*J_PMNS -> 5e-6, with its v13.1 m_1/m_3
%%   register sentence) REPLACED by the flavoured resonant-leptogenesis
%%   computation on the ledger inputs: (i) Casas-Ibarra structure
%%   displayed -- U_PMNS cancels exactly in the unflavoured invariant,
%%   and the dial-free real-R door gives unflavoured eps_res = 0;
%%   (ii) the flavoured asymmetries eps_{i,alpha} are the operative
%%   object at the TeV lock scale: on-peak throughput eps_eff ~ 8e-4
%%   (overshoot ~3e4), vacuum ~ 6e-15 (undershoot ~4e6), driver
%%   alpha_31 = pi (sets the sign of matter; J_PMNS-blind at the
%%   few-percent level), nonzero at m_1 = 0 (the asymmetry does NOT
%%   ride on the QD layer); (iii) named hypothesis H-phigeo-1 (partial
%%   resonance, Delta M/Gamma_eff ~= 3e4) installed with its numerical
%%   target; eps_res ~= 2.5e-6 KEPT as the H-phigeo-1 target value, not
%%   a derived output; subsection Status now derived-conditional on
%%   H-phigeo-1. Part V: eq:eps-res-phase-partV annotation, the
%%   boxed-value frame, and the chain audit-table row/caption re-scoped
%%   to the flavoured provenance (the former Jarlskog-imprint sentence
%%   retired; the m_1/m_3-order suppression withdrawn as a
%%   reconstruction artefact). Seam-Yukawa/washout convention
%%   reconciled at the Gamma_N and K_wash sites of Appendix I (factor
%%   ~13 in Y_nu^2 between the seam m-tilde ~= 8.4 meV and the washout
%%   m-tilde = m_3; declared, priced as O(10) inside the
%%   order-of-magnitude register). No other value changed.
%%   v13.1 -> v13.2. Script: a10_v13.2_phigeo_install.py.
%% =====================================================================
%% =====================================================================
%% v13.1 (2026-07-18)  RESONANCE REGISTER (P5 hostile-review cohort
%%   ship, author-approved; calibrated honesty, uniform pricing;
%%   standing claims intact). (a) Part-V seesaw benchmark: the clause
%%   "exactly the near-degenerate regime where resonant enhancement is
%%   natural rather than tuned" replaced by the honest register
%%   matching the satellite's own resonance-regime section: the O(1/K)
%%   splittings motivate proximity to the resonant regime, but at
%%   vacuum numbers the system sits far off the Breit-Wigner peak
%%   (d ~ 1/K ~ 3e-2 against Gamma/M ~ 2e-14); the on-resonance
%%   condition is the chain's named conjecture, and 'natural'
%%   overstates what vacuum numbers support. (b) Appendix I, eps_res
%%   worked number: one register sentence added --- the step from
%%   c*J_PMNS to the worked |sin phi_geo| ~ 5e-6 involves an
%%   undisplayed suppression of order m_1/m_3 whose derivation is an
%%   open item of the programme (the phi_geo projection), carried on
%%   the promotion roadmaps of this paper's satellite; the coefficient
%%   is asserted, not computed. No value changed. v13.0 -> v13.1.
%%   Script: a10_v13.1_resonance_register.py.
%% =====================================================================
%% =====================================================================
%% v13.0 (July 16, 2026): promotion of the signed-off v12.116 to the
%%   v13 series (author-approved); no content changes.
%% =====================================================================
%% v12.116 (2026-07-16)  Z* HOSTILE-AUDIT EXECUTION (author-approved):
%%   all recommendations of the six-persona hostile audit of the
%%   two-quantity z* structure (Hostile-Audit-Zstar-2026-07-16.md).
%%   MUST-FIX: (M1) the appA:thermal-gg two-framings cluster --- the
%%   last pre-two-quantity survivor --- bridged: reading note added,
%%   "recombination is the one the ledger pins to z_*=e^7" and the
%%   postdiction sentence reworded, the false "reported with both
%%   framings in every appearance" status claim corrected; (M2) Q&A
%%   O-Cosmos-81 ("generous denominator") rewritten in the two-quantity
%%   form --- it contradicted O-Cosmos-23 by using the visibility-width
%%   defence that response disavows and asserted an ES caveat that no
%%   longer exists; (M3) the flagship eq:zdec-gg sentence re-attributed:
%%   raw agreement 0.11-0.16% carries the zero-fitted-parameter label,
%%   the calibrated 0.03-0.07% is named as anchored on Planck's own
%%   pipeline; delta on the calibrated chain (~0.0059) stated.
%%   SHOULD-FIX: (S1) prop:recomb-gripout retitled "recombination-era";
%%   the era hosts the GSSB transition, the geometric switch is what is
%%   ledger-fixed, the visibility peak 0.4% below; (S2) z-vs-(1+z)
%%   convention declared once in sec:two-redshifts (1+z_sw = e^7 exact,
%%   z-level statements leading-order; 0.41% e-fold / 0.50% z-space
%%   atomic / 0.62% z-space Planck disentangled), eq:delta-offset LHS
%%   made exact; (S3) F7 second kill clause re-posed on the observable:
%%   sole quantitative kill = measured z_dec off the atomic chain by
%%   >0.5% (= ln(1+z_dec) off 6.996 by >0.005); the switch-decoupling
%%   offset demoted to a derived consistency statement (in-paper item,
%%   Table 39 row; both canonical ledger files updated in the same
%%   pass); (S5) O-Cosmos-23 now cites the computed delta = 0.0041 and
%%   Conjecture VIII (the "recombination-history lemma" never existed).
%%   WORTH-CONSIDERING (all executed): (W1) cosmic-history milestone
%%   node carries z_dec ~ 1090 beside the boxed switch; (W2) the Dev
%%   row/asset now SCORE the audited atomic-chain comparison (1091.1
%%   raw vs 1089.92 +/- 0.25 -> Dev +0.11%, pull +0.6 with the 0.18%
%%   modelling systematic), the switch e^7 an open-triangle annotation
%%   --- aligning plot and codebook D21 ("the switch is not the audited
%%   number"); assets regenerated (a10_dev_plot_gen_v8.py,
%%   a10_showcase_devplot_v8.py); rows/captions/Q&A mentions synced;
%%   (W3) the 22nd Dev comparison named (local-ladder vs H0DN) at the
%%   single reconciliation site; (W4) falsifier venue softened to the
%%   cosmological-recombination-radiation / post-2030 reading and the
%%   Planck z_dec noted as a model-derived parameter.
%%   BANNER CORRECTION: the v12.112 banner's claim "Q&A sites already
%%   in two-quantity form untouched" was wrong for one site
%%   (O-Cosmos-81), repaired here as M2.
%%   v12.115 -> v12.116.  Script: a10_v12.116_hostile_audit_zstar.py.
%% =====================================================================
%% =====================================================================
%% v12.115 (2026-07-16)  EXECUTIVE-SUMMARY TRIM (ES + Keywords end p.4)
%%   In v12.114 the Keywords block spilled ~3 lines onto page 5.  ES
%%   wording tightened only -- doubled phrasing removed, the newest
%%   count clarification compressed ("two entries derived outright" ->
%%   "two derived outright"; still 2 D + 18 D* + 1 A = 21), no claim
%%   dropped, Keywords content untouched: 'so it is never hidden
%%   later' / 'to adjust' / 'all derived' / 'from which it is' /
%%   'disfavoured today' / DESI-caveat dashes->commas / 'a light
%%   particle predicted' / 'and the paper says so plainly' / 'of the
%%   companion A9' / 'argument here' / ', read at the present epoch'.
%%   Verified visually: ES + Keywords now end on p.4; p.5 = the
%%   pi-locked inventory figure (as p.6 was).  Gate: 549pp, 0 errors /
%%   0 undefined / 0 multiply / no new overfull.
%% =====================================================================
%% =====================================================================
%% v12.114 (2026-07-16)  DEV-PLOT VISUAL FIX + CODEBOOK z* ENTRY (D21)
%%   Preview-first visual audit of the author's report ("no changes in
%%   Table 28 / Figure 28 (= Figure 5); z* absent from Table 37 /
%%   Figure 27").  (1) v12.113's table edits verified rendering
%%   correctly (Table 38 p.262, Table 2 p.42).  (2) The Dev-plot
%%   FIGURES were unchanged because the deployed asset
%%   a10_dev_plot_linear.pdf was the stale v3-era plot (21 entries, no
%%   r row, no epoch diamonds, z* "pred/obs"): a10_dev_plot_gen_v6.py
%%   wrote *_v6.pdf and was never renamed.  REGENERATED via
%%   a10_dev_plot_gen_v7.py -> a10_dev_plot_linear.pdf now matches its
%%   captions (22 entries, H0DN 73.5, violet diamonds) with the z* row
%%   two-quantity: label "z* (corridor switch)", line "switch 1096.6 /
%%   decoupling 1089.9"; a10_dev_plot_showcase.pdf regenerated with the
%%   same z* relabel (a10_showcase_devplot.py).  (3) z* ADDED to the
%%   codebook as entry D21 (Decoupling redshift; two-quantity; D*;
%%   Lemma VIII via the pi-locked inputs; sec:two-redshifts pointer):
%%   tab:cosmological-codebook row, fig:codebook-provenance TikZ node +
%%   red Conj.-VIII arrow, per-parameter audit paragraph D21, and every
%%   counting site reconciled (codebook twenty -> twenty-one entries,
%%   seventeen -> eighteen D*, Lemma VIII closes four -> five;
%%   closure-economy + roadmap tables re-tallied; Exec Summary, Part
%%   XI, Q&A, appendix deck).  Three PRE-EXISTING stale counts
%%   repaired: appendix glossary "2 D + 15 D* + 3 A = 20" -> "2 D +
%%   18 D* + 1 A = 21"; appendix closure-economy "Fifteen ... three
%%   Asserted" -> "Eighteen ... one Asserted"; appendix roadmap coda
%%   "seven open research targets ... 2/15/3 to 20/0/0" -> "nine ...
%%   2/18/1 to 21/0/0".  (4) Flagged residuals fixed:
%%   Table 10 (tab:whynow-epochs) row "recombination (z_*=e^7)" ->
%%   "corridor switch / decoupling (z_sw=e^7; atomic z_dec)";
%%   Table 21 (tab:cmb-core-derived) caption + z* row -> "(ledger,
%%   corridor switch); decoupling 0.4% below".  Gate: 549pp, 0 errors /
%%   0 undefined / 0 multiply / no new overfull; every changed page
%%   re-rendered and read (pdftoppm).
%% =====================================================================
%% =====================================================================
%% v12.113 (2026-07-16)  DEV-PLOT FAMILY TWO-QUANTITY MICRO-FIX
%%   (author query: do Table 28 / Figure 28 (= Figure 5) still need the
%%   two-redshifts update?).  Findings: Table 28 (tab:bh-island-
%%   dictionary, island/QES vs GDOS dictionary, p.208) carries no
%%   z*/e^7 content -> no change.  Figure 28 (fig:a10-dev-plot-linear,
%%   Part-IX codebook Dev plot, p.263) = Figure 9 (Part-I preview) =
%%   Figure 5 (front-matter showcase): same Dev-plot family; its data
%%   tables (Table 2 preview p.42, Table 38 codebook p.262) still
%%   carried the one-quantity row  z_* & e^7 = 1096.6 & 1089.92+/-0.25.
%%   Six anchored edits: both table rows -> "e^7 = 1096.6 (corridor
%%   switch)" vs "1089.92 +/- 0.25 (decoupling)" (compact-codebook
%%   template, row stays one line); both table captions gain the
%%   two-quantity convention clause (atomic chain on the Planck value
%%   at the 0.1% level, switch 0.4% above, \S sec:two-redshifts); the
%%   codebook Dev-plot figure caption (Fig. 28) and the showcase
%%   caption (Fig. 5) gain a one-sentence identification of the z_*
%%   entry (the Part-I preview plate is at full page height, so its
%%   facing Table 2 caption carries the clause).  Plot assets
%%   (a10_dev_plot_linear/showcase.pdf) untouched: in-panel
%%   "pred 1096.6 / obs 1089.9" remains correct under the two-quantity
%%   reading now fixed by the captions.
%%   v12.112 -> v12.113.  Script: a10_v12.113_tab28_fig28.py.
%% =====================================================================
%% =====================================================================
%% v12.112 (2026-07-16)  TWO-QUANTITY E^7 SWEEP (author-approved):
%%   completes the v12.111 two-redshifts pass across the remaining sites
%%   that still used the old identity language (e^7 = the recombination/
%%   decoupling/last-scattering redshift).  Corridor switch z_sw = e^7
%%   (topological, ledger integer J=7) is now everywhere distinct from
%%   the atomic decoupling z_dec ~ 1090 (derived in sec:two-redshifts,
%%   0.4% below the switch, on the Planck value at the 0.1% level).
%%   Sites: Part-I overview; popular-abstract first-atoms clause;
%%   Part-VI opener sentence + headline-numbers table pointer; corridor
%%   TikZ node + caption; cosmic-history timeline tick/node/caption;
%%   master-chronology last-scattering row (z entry -> 1090 atomic,
%%   switch e^7 0.4% above); Example C parenthesis; conj:hubble-corridor
%%   + Theorem thm:hubble-two-stage-corridor preamble/(C1) empirical
%%   clause (displayed math unchanged); Part-IX Step-3 identification ->
%%   ledger integer 0.41% above computed ln(1+z_dec) = 6.996; falsifier
%%   item + falsification-catalogue row + kill-timeline card reworded to
%%   the two-quantity kill (z_dec off the atomic-chain value by >0.5%,
%%   present agreement 0.1%; or switch-decoupling offset off the
%%   computed 0.4% structure by >1%); numerology-defense F-list clause;
%%   Part-XII postdiction sentence; conclusion (answerable section +
%%   closing paragraph); symbol-glossary and compact-codebook rows.
%%   Cohort falsifier ledger F7 (tex + md) updated to match.
%%   No numerical pipeline values touched (acoustic computations already
%%   use ~1090); sec:two-redshifts itself, prop:recomb-gripout, and the
%%   Q&A sites already in two-quantity form untouched.
%%   v12.111 -> v12.112.  Script: a10_v12.112_e7_sweep.py.
%% =====================================================================
%% =====================================================================
%% v12.111 (2026-07-16)  TWO-REDSHIFTS PASS (author-approved science-
%% presentation pass; no physics change beyond the specified content):
%%   (1) NEW subsection 'The switch and the first atoms: two redshifts,
%%       one descent' (sec:two-redshifts) inserted in the Hubble-tension
%%       section immediately after Proposition prop:recomb-gripout and
%%       its attached commentary paragraph.  Atomic decoupling chain
%%       from the cohort's own constants (kT_* = B/53.0;
%%       ln(1+z_dec) = 6.996; z_dec^GG = 1091.1 raw / 1089.1 calibrated
%%       vs Planck 1089.92 +/- 0.25), offset delta = 0.0041 (0.41%),
%%       pi-locked z_eq = pi e^7 candidate tested and rejected
%%       (2.8 sigma), e-fold-budget route to a T_0 theorem commissioned.
%%       Heading demoted \section -> \subsection (proposition sits
%%       mid-\section{Hubble tension}; a \section would absorb the six
%%       following Hubble-tension subsections).
%%   (2) Proposition prop:recomb-gripout rephrased to the two-quantity
%%       form: ledger fixes the CORRIDOR-SWITCH redshift z_sw = e^7
%%       (z_sw = z_ast kept for notational continuity); appended note
%%       that the atomic decoupling surface lies 0.4% below the switch.
%%       Part-VI opening comparison clause rephrased to the same form.
%%   (3) New bibitem HuSugiyama1996 (ApJ 471 (1996) 542-570) in the
%%       alphabetical B-list after HuSugiyama1995, house format.
%%   v12.110 -> v12.111.  Script: a10_v12.111_two_redshifts.py.
%% =====================================================================
%% v12.110 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%%   correction: tab:whynow-epochs: epoch l-column -> p{2.9cm}, tabcolsep 2.5pt (overfull 40.6pt after font raise; 8 columns)
%%   correction: tab:cmb-core-derived: forced-form l-column -> p{3.4cm}, tabcolsep 4pt (overfull 44.4pt after font raise)
%% v12.109 (July 16, 2026): F25 relic passage added at the end of the
%% inflation module (author-approved): relic non-Born statistics as a
%% two-sided falsifier of the constitutional Born chain; xi(k) power-law
%% form; amplitude computable in principle from the determined initial
%% data; quantification queued as its own campaign.
%% v12.108 (July 15, 2026): final-touch pass (report 17 W-3, author-
%% approved): observer-vacuum closing passage updated to completed tense
%% --- RB-11/RB-12 pointer (rotating anchor; collapse channel; C3 full-
%% length home), stale "remains conjectural for gravitating bulk
%% configurations" clause replaced by the honest exact-curved-space
%% residue.
%% v12.107 (July 15, 2026): cohort touch C-3b --- completed-tense pointers
%% to the satellite preprint Arisaka-GG-P4-Twistor at the observer-vacuum
%% subsection and the T-Cosmos-18 consequences paragraph; new A-list
%% bibitem ArisakaP4Twistor; A-group header widened; PaperPFour provided.
% v12.106 (July 15, 2026): the dial-free-door theorem + real-bulk polish
%   (complex-variables thread, author-commissioned). NEW Proposition
%   T-Cosmos-18 / R10 with full proof at the head of the seesaw
%   completion: a Majorana mass exists iff the fibre weight vanishes;
%   the singlet N_R and the LHN portal are forced, not assumed -- the
%   module's last imported particle-content ingredient is now derived.
%   Sakharov pillar-division paragraph added (GRIP = angle, GDOS =
%   observability, OGGEP = ledger). NEW subsection at the end of the
%   boundary-projection section: observer-indexed vacuum + horizon
%   thermality as openness (real-bulk R7 material; conjectural residue
%   named). NEW objections O-Cosmos-91..93; prose count ninety ->
%   ninety-three. No numbers, values, figures, or Parts changed.
%% v12.105 (2026-07-14, Phase 2: {e,pi} + landscape objection) --
%%  presentation + one new objection; no values/results changed.
%%   (1) {e,pi} dictionary added to the "single input" front-matter paragraph
%%       (e = radial/scale 5th dim, pi = compact/gauge 6th dim).
%%   (2) NEW Common-Objections pair O-Cosmos-90 (formal-theory referee): the
%%       calibrated "why not say the landscape is refuted?" answer -- scope
%%       split (QG formalism vs vacuum-selection), consistency conceded,
%%       priced-not-refuted / measure-problem / Lakatos (non-progressive)
%%       verdict, GG as closure-mode counter-position. Reader concludes.
%%       Honors [[no-string-theory-polemic]]; per Landscape/Second-Horizon
%%       report 2026-07-14. Appended after O-Cosmos-89.
%%   (3) 5 new group-B bibitems (alphabetical, fields web-verified 2026-07-14):
%%       ColemanDeLuccia1980 (PRD 21,3305), Douglas2003 (JHEP 05,046),
%%       Freivogel2011 (CQG 28,204007), Lakatos1978 (CUP), Weinberg1987
%%       (PRL 59,2607).
%%   (4) objection count eighty-nine -> ninety (two prose spots).
%%  Script: a10_epi_landscape.py.
%% v12.104 (2026-07-14, OGN5) -- Executive Summary "single input"
%%   accounting clarified per author: the five OGN-5 integers
%%   (M,N,L;J,K)=(1,3,5;7,35) are FORCED, not declared and not arbitrary --
%%   the unique tuple surviving the anomaly-free BI-6 closure of the 6D bulk
%%   (A3), each with a physical meaning (M=1 tensor/compact direction, L=5
%%   SU(5) block, N=3 colour triplicity locked by the anomaly trace-ratio,
%%   J=7 odd closure index, K=LJ=M^2+N^2+L^2=35 master lock), pinned upstream,
%%   never searched. The TWO declared inputs (amplitude A_s, epoch t_0) are
%%   kept distinct from the forced ledger. Anti-hostile-reviewer emphasis.
%%   Two lower-value keyword terms trimmed to keep the keyword line on p4.
%%   No values / results touched. Script: a10_ogn.py.
%% v12.103 (2026-07-14, ESCITE) -- front-matter face-lift: the Executive
%%   Summary now cites the three hero showcase figures as a visual on-ramp
%%   for new readers (mirrors A11-Origin): pi-locked composition circle
%%   (fig:pi-locked-chart-omega-hook) and the two-basin cosmic-history
%%   descent (fig:showcase-history-p1) in "The headline results"; the
%%   parameter-audit Dev plot (fig:showcase-devplot) in "How confident --
%%   the triage". To keep the floated keyword line at the foot of the last
%%   summary page (not spilling onto its own page), the ~29-term keyword
%%   list is consolidated to a concise ~12-term one (metadata only; the
%%   explicit density entries fold into "pi-locked inventory"), plus two
%%   verbal trims ("in physics" dropped; "stated here once" -> "stated up
%%   front"). No values / counts / results touched. FLAG for author:
%%   keyword-list trim + two wording trims are the only prose touches.
%%   Script: a10_v12103_escite.py.
%% v12.102 (2026-07-14, QFT) -- author correction to p. 282 (Sec 63.1):
%%   'within QFT identity is automatic' was too generous to QFT. The QFT
%%   paragraph rebuilt as an explicit conditional: interchangeability of
%%   quanta is secured only GIVEN one field / one set of couplings / one
%%   vacuum, and every clause is an input -- the value of e (free
%%   renormalized parameter), its constancy across space (stipulation of
%%   the Lagrangian's form; varying-constants programmes exploit exactly
%%   that freedom), and the uniqueness of the vacuum (SSB contains the
%%   failure mechanism: Kibble domains). NEW closing paragraph: the
%%   1864-vs-1965 data-record inversion (Huggins stellar spectra vs
%%   Penzias-Wilson), with the dimensionless caveat (alpha and m_p/m_e,
%%   not raw e). NEW Wheeler 1940 one-electron-universe anticipation
%%   opens the pedigree subsection (via Feynman's Nobel lecture).
%%   Two new bibitems: Huggins1864, FeynmanNobel1965 (127 total).
%%   Science frozen; no values or counts touched.
%%   Script: a10_v12.102_qft.py in [v11-Support].
%% v12.101 (2026-07-14, SHP) -- author request: revise the second-horizon
%%   -problem Section per the 2026-07-14 novelty deep-research report
%%   (SecondHorizonProblem-Novelty-DeepResearch-2026-07-14.md).
%%   (1) Sec 63 intro + 63.1: 'not merely unsolved but unposed' softened to
%%       'unsolved and, as a problem in its own right, unnamed'; subsection
%%       retitled 'The problem the textbooks leave unnamed'; domain-problem
%%       fragment added; Uzan cite for the 1e-5..1e-6 alpha bounds.
%%   (2) NEW subsection 63.2 'Fifty years of near misses' (subsec:shp-pedigree):
%%       ZKO 1974 domain structure, Kibble 1976 causal bound, Guth-Steinhardt/
%%       Linde one-domain spatial solution, Linde 1990 'uniqueness of the
%%       universe' + 'all properties' near miss, landscape inversion,
%%       varying-constants measurement arm (Uzan), Ellis assumption +
%%       steady-state postulate, Marsh 2016 theta_PQ pinning lore,
%%       Penrose/Hollands-Wald foil, and the four-fold gap (naming, extension,
%%       ordering, unification).
%%   (3) 63.4 pinning paragraph: axion-literature ownership acknowledged
%%       (abundance register vs identity register). 63.5 features: 'Three'->
%%       'Five' (fixes a pre-existing three/four mismatch); new Fourth =
%%       spatial-vs-temporal novelty calibration; old Fourth -> Fifth.
%%   (4) Echo sweep: front matter, Part I problem 8, Part IV ordering note,
%%       Part X overview, Appendix G teaching block ('hiding inside the
%%       third') -- 'never posed'/'unposed' now zero in the body.
%%   (5) NEW Objection 89 (O-Cosmos-89, philosophy-of-cosmology referee):
%%       the priority objection, answered with the four-fold gap; objection
%%       count 88 -> 89 swept in prose (2 sites).
%%   (6) Seven new B-section bibitems with Relevance + first-cited links:
%%       ZeldovichKobzarevOkun1974, Kibble1976, Linde1990, Uzan2011,
%%       Ellis2006, HollandsWald2002, Penrose2004.
%%   (7) STALE-VALUE FIX (flagged): Guth1981 Relevance note still said the
%%       paper relocates inflation 'to TeV energies' -- retired reading;
%%       now 'deriving the launch geometrically, at the GUT edge'.
%%   Science frozen; no values, counts, or results touched beyond the above.
%%   Script: a10_v12.101_shp.py in [v11-Support].
%% v12.100 (2026-07-13, S3) -- author request: the front-matter showcase
%%   grows to FIVE figures. New pages 7-9 inserted before 'What is
%%   actually claimed': p7 = cosmic history (numerically integrated,
%%   page 1 of a10_cosmic_history.pdf) w/ sans-bold top title + trimmed
%%   caption; p8 = descent variables (page 2), same treatment; p9 = the
%%   parameter-audit Dev plot restyled (a10_dev_plot_showcase.pdf via
%%   a10_showcase_devplot.py: fonts +20%, right annotations inside the
%%   panel, big bold title, legend below) w/ trimmed caption. In-paper
%%   figures untouched; new pages cross-reference them. Scripts:
%%   a10_v12.100_S3.py + a10_showcase_devplot.py in [v11-Support].
%% v12.99 (2026-07-13) -- mirror of C2 v13.38 (author-ordered): the
%%   closed-form paragraph added to Route I of Sec. 6.4, after the
%%   eq:partII-H0-seesaw grade paragraph, before Route II. Boxed
%%   eq:partII-H0-analytic:
%%     H0 = e^{-4K} sqrt(2 pi^5 c^5/(3 L K (pi-1) hbar G_N)) = 66.7
%%   (CODATA 66.74; verified in-script). Provenance sentence (K twice,
%%   L once + vev pin; pi x3; e; structural 1/2, 3; no astronomical
%%   quantity); two corollaries (10^-60.9 = 4K e-folds; 10^-120.2 =
%%   8K - 1.4 orders); honesty note (M,N,J upstream; grade unchanged;
%%   'exposure, not strength'); fuller treatment cross-cited to C2
%%   Sec. 23.3. Date stamp -> July 13. No existing text or value
%%   changed. Script: a10_v12.99_closedform.py.
%% v12.98 (2026-07-12) -- the four queued reconciliations (author request):
%%   (1) H(t) asset regenerated at the exact pi-locked inventory
%%       (a10_fig_H_of_t_v2.py: H_inf = 55.0, z_eq = 3400, z_* = 1097,
%%       Or = 9.36e-5); text 55.1 -> 55.0 (curve prose + caption).
%%   (2) z_eq = 3400 executed: why-now PAIR asset regenerated at the
%%       derived anchor (a10_whynow_figs_gen_v5.py; fonts unchanged from
%%       v4); prose/table sweep 3460 -> 3400 and Omega_r0 -> 9.36e-5
%%       (chronology rows 9.4e-5).
%%   (3) H0DN 73.50 +- 0.81 promoted to the late-branch comparator of
%%       the codebook: Dev rows/captions/asset (gen v6; Dev -6.80%,
%%       pull -5.0), codebook row + caption, master-identity row,
%%       consistency-web node (+10.4%), h0-two-values prose + status
%%       row (~7% residual), corridor subsection + figure caption
%%       rewritten to the current reading (retired-corridor claims
%%       dropped); bibitem CasertanoH0DN2026 added; Riess2022 +
%%       C2-Hubble Relevance notes updated (C2 title refreshed to its
%%       current subtitle).
%%   (4) beta_iso and Kaxion bars verified ALREADY reconciled to C1
%%       (6e-40; 8.1 +- 0.5 ueV; window [1.59, 2.13] GHz is the
%%       C1-canonical discrete ladder) -- no edits.
%%   Canonical rates unchanged (66.6 +- 0.2; Route I 66.8 +- 0.2).
%%   Scripts: a10_v12.98_recon.py + the three generators above.
%% v12.97 (2026-07-12) -- Phase G2 (the C2-Hubble mirror):
%%   (1) H0-derivation subsection rebuilt in the two-route order:
%%       Route I (geometry alone: seesaw + codebook-selected
%%       zeta_Lambda = pi/(LK) = pi/175 -> 66.8 +- 0.2 structural,
%%       cluster bracket 66.8-71.2, 73.5 excluded, vev pin +-0.18%)
%%       FIRST; Route II (density inversion, + BBN-only anchor
%%       66.4 +- 0.5) as the precision confirmation; concordance 0.6
%%       sigma; grade zeta_Lambda Asserted -> Derived (cond., MSMF/OGN)
%%       under the stratum-assignment lemma (author-approved).
%%   (2) cluster cost column reconciled to A9's K-closure rule (K = LJ
%%       costs two): former cost-3 entries printed cost 4, 1/J^2 stays
%%       cost 2; selection leans on admissibility + structural naming.
%%   (3) pre-registered O6 target printed (pi/175 to ~1%, two-sided;
%%       cohort ledger F25; C2 falsifier F-Hubble-11) in the selection
%%       paragraph and in the Hubble-module kill switches, whose
%%       corridor-era success/failure lists are rewritten to the
%%       current no-early-modification reading.
%%   (4) zero-CMB-input framing carried into the consistency web, the
%%       CMB core table (+ Route-I H0 row, + predicted omega_b/omega_m
%%       rows at the canonical 66.6), the master identity, the two
%%       Dev-plot captions, and the h0-two-values status table.
%%   (5) value sync to cohort canonical (flagged): omega_m total
%%       0.1430 +- 0.0011 w/ neutrino-convention footnote (b+CDM sum
%%       0.1424), matter anchor 66.9 -> 67.0; stale "(A_s, n_s) tiered
%%       as aspirational" wording repaired (tilt derived-conditional).
%%   No physics changed; the canonical rate 66.6 +- 0.2 unchanged.
%%   Script: a10_v12.97_G2.py.
%% ===================================================================
%% v12.96 (2026-07-11) -- The why-now Omega-budget PAIR asset
%% (A10-whynow-Omega-evolution-PAIR-nocap.pdf, shown as Figures 2 and
%% 16) regenerated with all fonts ~20% larger (generator
%% a10_whynow_figs_gen_v4.py, same fix as C2 v13.24: the 10.8-wide
%% canvas rendered at page width displayed its fonts ~26% smaller
%% than the sibling figures); value box relocated, titles/legend/
%% inset refitted. Cosmology and labels unchanged (the derived-rate
%% z_eq = 3400 reconciliation remains a queued follow-up). No tex
%% content changed beyond this stamp.
%% ===================================================================
%% ===================================================================
%% v12.95 (2026-07-11) -- Hopf-fibration figure imported from the
%% companion Hubble paper C2 (asset c2_hopf_fibration.pdf in
%% [v11-Extra]; generator c2_fig_hopf.py in [v11-Support]). Placed in
%% the pi-locked derivation section, directly after the fibre-volume
%% argument and before the pi-locked pie chart, with two paragraphs of
%% geometric and physical intuition (linked-fibre reading;
%% equipartition-on-a-compact-space; 'the pi in the inventory is the
%% leftover of a sphere volume'); inline citations added in the
%% Second-step paragraph and in the Appendix fibre-measure teaching
%% subsection. No physics changed. Script: a10_v12.95_hopf.py.
%% ===================================================================
% =====================================================================
% ===== v12.94 (2026-07-09): derived-H(t) figure + LambdaCDM-revisit call. =====
%   Part VI, Hubble-tension section: NEW lead subsection sec:h-of-t-derived
%   ('H(t): the derived expansion history, of which H0 is one point') placed
%   before the two-stage-corridor subsection, carrying the densities->H(t)->H0
%   hierarchy, four honest caveats, and the new full-page figure
%   fig:H-of-t-derived (asset a10_H_of_t_derived.pdf). Part VI integrated
%   summary: NEW subsection sec:revisit-lcdm (bold community call to revisit
%   LambdaCDM in the light of the derived H(t), framed as provenance-not-
%   dynamics, testable). Version/date stamp -> v12.94. No frozen-physics change.
% =====================================================================
% v12.91 (July 5, 2026): retire the TeV-scale/lock-scale inflation reading
%   COMPLETELY (author directive, option (a)). Swept the stale narrative
%   the v12.77-78 promotion left in Part VIII/IV: the 'Slow-smooth
%   inflation at the lock scale' subsection + descent-to-plateau
%   reconciliation rewritten to the GDOS plateau one e-fold below E_GUT
%   (V^{1/4}=8.3e15 GeV, r=0.0043, n_s=0.9620); T-Cosmos-14 premise
%   rescaled (result unchanged: depends on smoothness not scale);
%   descent/chronology/falsifier phrases + initial-patch + relevance
%   note swept. Retirement-discussion & objection-quote sites KEPT.
%   No physics result changed.
% v12.90 (July 5, 2026): Step-4 final revision batch. STALE (checker-found;
%   the two back-appendix glossary tables the v12.80 n_s-promotion missed):
%   symbol glossary + compact codebook glossary inflation rows brought to
%   canon (n_s=0.9620 D*, r=0.0043 D, V_inf=8.3e15 GeV D*). EDITORIAL:
%   'topological constants pi,e' -> 'structural constants' (x7; e is not
%   topological); claims-box denominator; preface scope-note shelter;
%   DESI Sigma-m_nu ~59 meV named kill row + prose; siren spec (events/
%   pec-vel floor / 2-3%%@70 decides nothing); tensor delensing note; BAO
%   residue estimator (isotropic alpha, DESI-Y5); JWST abstract softened;
%   master-ledger pointer. No standing point changed.
% v12.89 (July 5, 2026): Figure 23 recentred on the landscape page
%   (-14.4mm kern; ink was +7.2mm right of centre, measured). No other
%   change.
% v12.88 (July 5, 2026): end-of-Part codas (author request; A9-SM device):
%   plain-English summary + philosophical register + invitation to the
%   next Part. Full codas: Parts V, VII, VIII, IX, XI, XIII, and the
%   Technical Appendices; one added paragraph ahead of the existing
%   closes of Parts I-IV and VI; Parts X and XII unchanged (already
%   close with summary+philosophy+hand-off). No science changed.
% v12.87 (July 5, 2026): Figure 23 (provenance map) fonts ~+30% (one-step
%   font bump + resizebox 0.96 + x/y pitch compression), still one
%   landscape page; AND stale figure content brought to the canonical
%   codebook (n_s 0.9620 D* plateau-derived, V_inf 8.3e15 GeV D*,
%   r = 0.0043 D two-sided, Lemma IV(rev.)/V' labels, caption sentence)
%   -- flagged to the author as a lag-layer catch. No new physics.
% v12.86 (July 5, 2026): Figure 10 machinery tier spread out (box pitch
%   31mm -> 36mm) so the inheritance-chain arrows sit in visible gaps
%   between the five boxes. No science changed.
% v12.85 (July 5, 2026): Figure 10 (zero-parameter derivation tree)
%   enlarged to full page width on its own full page: right-margin tier
%   labels removed and re-set flush left ABOVE each tier (white-backed,
%   drawn last); all in-figure fonts one step up (net ~+30% printed);
%   caption condensed to fit the page. No science changed.
% v12.84 (July 5, 2026): five author refinements. (1) one-page claims box
%   moved to p.7, after front Figure 2; (2) literal def:GGA / def:GRIPin
%   label leaks fixed in prose; (3) planning Table 4 expanded 13 -> 17
%   problems mirroring the census clusters (census label renamed
%   tab:seventeen-problems); (4) Objections flush left (noindent on all
%   88 headers + 88 quotes), identical to A9-SM; (5) Preface expanded to
%   ~2 pages: welcome to graduate students across the sciences, the
%   decade-of-the-verdict, invitation-to-break-it, the larger arc, and
%   three reading paths. No science changed.
% v12.83 (July 5, 2026): Front Figure 2 (p.6) restored to the same full
%   width as its Part-IV twin (0.93 textheight); caption trimmed to three
%   lines so image + caption share the page. No science changed.
% v12.82 (July 5, 2026): Figure asset A10-whynow-Omega-evolution-PAIR-nocap
%   regenerated with thin seam lines on every Omega band boundary (gen v3,
%   Fig-14 bottom-panel style); a full-page copy of the pair placed on
%   page 6 after the pi-locked chart as the second front hook figure,
%   cross-referenced to its Part-IV twin. No science changed.
% v12.81 (July 5, 2026): Executive Summary expanded ~40% to three full
%   pages (incl. keywords), per author: significance, uniqueness, and
%   philosophy foregrounded (second horizon problem; one-descent/three-
%   chapters cross-sector rigidity; data-confrontation honesty; kill
%   ledger; forcing-not-searching; trilogy placement; identity of the
%   cosmos; determinism third path + punch line). Two keywords added.
%   Calibrated honesty preserved throughout. No science changed.
% v12.80 (July 5, 2026): Round-3 referee fixes (presentation only; science
%   unchanged, triage 2D/17D*/1A now uniform). (1) n_s-status propagation:
%   codebook intro paragraph and longtable rows for the tilt and amplitude
%   brought to the canonical counts; Part VI claimed/not-claimed updated;
%   slow-roll teaching caption recast (retired TeV reading marked retired).
%   (2) z_star loose-voice licence footnote at first Introduction use;
%   Part VIII epoch wording tightened. (3) Lemma/Conjecture terminology
%   note; Dev-plot caption denominator wording; NEW one-page claims box
%   after the preface; Hopf-fibre independence sentence added.
% v12.79 (July 4, 2026): z_* voice made CONSISTENT (round-2 referee required fix).
%   The #4 reframing (z_* = topological corridor-switch redshift, DISTINCT from the
%   measured decoupling z_dec, coinciding to 0.6%) is now propagated to the 7
%   prose/keyword spots that still called z_* THE recombination redshift: exec
%   summary headline, keywords, Part VI bookkeeping note, post-hoc-fit stratum-
%   forms list, F-Cosmos falsifier D4, Conclusions corridor-from-integers, and the
%   multiverse Q&A. Technical passages (corridor theorem, caption, remark) already
%   used the corridor-switch voice. Zero remaining internal contradiction. No
%   number changed.
% v12.78 (July 4, 2026): remaining TEN audit items addressed honestly (co-author
%   pass; no scientific/philosophical ground conceded). #1 title=conditional-
%   ledger shorthand (exec-summary sentence; codebook the binding statement).
%   #3 chi^2: report chi^2/dof~2.9, DROP 'endorse', residual is ONE coherent
%   second-order offset not random scatter, falsifiable expectation stated
%   (Part VI body + tier ladder + figure caption). #4 z_*: pick ONE identity --
%   the topological corridor-switch redshift, DISTINCT from measured z_dec, 0.6%
%   offset computable by the recomb-history lemma; drop 27sigma/0.17sigma
%   comparator theatre (O-Cosmos-23 + theorem passage). #5 +2.7% ADOPTED as the
%   declared second-order precision floor; per-entry sigma = first-order; the
%   Omega_b computed-correction is falsifiable. #6 'zero free params' ->
%   'zero CONTINUOUS free params + two declared inputs (amplitude, epoch)'
%   (abstract, exec summary global def, two broad-audience Q&A). #8 Hubble
%   corridor = CANDIDATE resolution (not computed joint fit; PS-Cosmos-16),
%   siren H0~67 foregrounded as the discriminator, S_8 sign flagged open. #9
%   w=-1 DESI 2024 tension foregrounded (abstract + exec summary). #10 second
%   horizon = honest REDUCTION/unification (mold reduces cosmic identity to
%   vacuum-uniqueness conjecture; ordering earned independently) -- STRENGTHENS.
%   #11 scope note: philosophical arc is a deliberate trilogy feature for arXiv;
%   falsifiable core stands alone (kept, not cut). #12 reproducibility/
%   availability statement (repo on posting). Also n_s exec-summary straggler.
%   No number changed; honesty and framing only.
% v12.77 (July 4, 2026): TWO TIER-1 CLOSURES (co-author pass after the senior-
%   referee audit; reports GG-Tier1-Closure and Collaborator-Response). NO number
%   changed; the DERIVATION method and the honesty grading are what change.
%   B1 -- de-circularise Omega_m=1/pi. Part X Theorem T-Cosmos-17 rewritten:
%   Step 4 is now the Hopf fibre-volume ratio Omega_m=Vol(S^1)/Vol(S^3)=2pi/2pi^2
%   =1/pi (the pi is a denominator VOLUME, NOT the Pontryagin leftover and NOT the
%   VIII-b' half-measure -- that gives the factor 2); Step 5 competitors now die
%   by geometry (no 6pi^2 manifold; 1/2pi is the baryon fibre; no S^5) instead of
%   'the data prefer 0.46'; honesty note: circularity REMOVED, residual = one
%   geometric lemma VIII-b'' (fibre identification) + unit normalisation, no data.
%   Theorem name/tagline/hyp(III)/status-note/pie-caption + the 3-step preview +
%   answer-to-(a) + appendix-L teaching section (new Hopf-resolution subsection
%   appA:pilock-hopf) all aligned. Codebook stays D* (content of the conditional
%   sharpened). B2 -- compute the global trials factor. Part XI forcing-vs-
%   searching: 'not yet computed' replaced by the count -- 10 closed OGN-5 ledgers
%   enumerated, UNIQUE survivor (1,3,5;7,35) under (N=3 from P2, L=5 from SU(5))
%   fixed UPSTREAM, so P(ledger)=1; exponential lock (z*=e^J, M_lock=e^-K, v=246
%   unbidden) unsearchable; global bound p~1e-3..1e-4 with the Monte-Carlo
%   specified for a stats note; euclidremark box updated. Also fixed one stale
%   exec-summary straggler (n_s listed Asserted -> Derived-conditional; six-param
%   split now 4 D* / 1 declared input A_s / 1 astrophysical tau).
% v12.76 (July 4, 2026): PART XV Q&A REVISION (author request). (1) Consistency
%   audit vs latest physics: O-52 'TeV-scale inflationary stage' -> descent's
%   second saddle / lock crossing (plateau is GUT-adjacent); O-60 retired
%   one-sided r<<1e-3 -> derived two-sided r=0.0043; O-7 Dev-plot marker list
%   n_s no longer Asserted (now derived-conditional, amplitude the one A).
%   (Hostile TeV framings in O-10/O-13 headers are the reviewer's question and
%   are correctly retired in their responses -- left as calibrated honesty.)
%   (2) NEW opening category 'For the curious reader: the big questions, plainly
%   answered' -- 22 broad-audience philosophical Q&A (multiverse, landscape,
%   parallel worlds, fine-tuning, necessity-vs-designer, before-the-bang,
%   second horizon, why-95%-dark, determinism/freedom, pi-numerology, ToE
%   humility, anthropics, GUT-energy testability, 120-orders DE, coincidence,
%   extra dimensions, something-rather-than-nothing, conjecture-honesty,
%   observers/A11, self-reference, falsifiability) placed FIRST.
%   (3) All 66 existing technical objections renumbered O-Cosmos-1..66 ->
%   23..88 (no prose cross-refs; no external refs). Part intro announces the
%   broad-audience category. No physics changed beyond the consistency fixes.
% v12.75 (July 4, 2026): THE SECOND HORIZON PROBLEM (author's new view,
%   elaborated in GG-IdentityHorizon-Elaboration-2026-07-04.md; all
%   recommendations approved). (1) NEW full Section in Part X (after the
%   identity core, before the pi-locked derivation): 'The second horizon
%   problem: sameness of things, not sameness of temperature' --- the unposed
%   problem (QFT assumes one vacuum; landscape worsens; varying-alpha bounds
%   measure the uniformity), the mold (GGA identity core), the ordering
%   (inflation before any matter existed; agreements inherited, not
%   negotiated), the pinning, the prompt crossing, the boxed theorem, the
%   layered falsifiability (kind/amount/dark), the retired-TeV inversion.
%   (2) Part X overview: three -> four movements. (3) Part IV: 'The ordering,
%   noted for what it is' paragraph after the conviction paragraph.
%   (4) Part I problem-8 walkthrough: dividend sentence + pointer (count
%   unchanged at seventeen). (5) Appendix G: 'The fourth problem, never
%   posed: identity' teaching block added to the three-problems subsection.
%   No new physics; a new reading of derived physics, inheriting its tests.
% v12.74e (July 4, 2026): timeline final touches (author): box 6 widened to
%   62mm (clears box 8, which is nudged to y=5.1); all remaining normalsize
%   text in the figure (era sublabels, axis years, scan-span, legend) -> large.
% v12.74d (July 4, 2026): kill-test timeline refinements (author items): fonts
%   +~25% more (base large; small->normalsize; large->Large), era titles raised
%   w/ clear space below (8.8/8.15, bands to 9.6, banner to 11.7), band grays
%   lightened (2/4/6), tall cards widened to 62mm to stay short, dots 5.2pt,
%   legend respaced.
% v12.74c (July 4, 2026): (i) R-deck table[H] -> breaking longtable (fixes the
%   near-empty appendix-R opener page AND the long-standing 66pt Overfull vbox
%   [Table 56 running off the page]); (ii) kill-test timeline figure MOVED from
%   appendix T into Part XII decade-timeline subsection (author request),
%   integrated with the intro prose + pointer left in T; TikZ fonts bumped
%   ~30% (footnotesize->normalsize, scriptsize->small, small->large).
% v12.74b (July 4, 2026): DE-DATING SWEEP (author request): all 34 reader-facing
%   "July 2026 <result>" framings recast into the paper own voice ("the two-basin
%   derivation of Part IV", "the present framework", "the executed demonstration");
%   audit-trail keeps "revised"/"retired" without dates; "Current state (July
%   2026)" -> "(present paper)". Zero live July-2026 mentions remain in the body.
% v12.74 STAGE A (July 4, 2026): PART XIV TECHNICAL APPENDICES -> graduate
%   course (author-approved plan, A9 Part XIII model). (1) Front matter rewritten
%   to the A9 register: three-step method announced, course audience stated,
%   NEW 'Prerequisites and recommended textbooks' (4 tiers; 6 new bibitems:
%   Baumann2022, Ryden2017, Liddle2015, LiddleLyth2000, Peacock1999, Durrer2020);
%   teach-box/take-away/read-alongside helper macros defined. (2) Ledger
%   reconciliation: appendix promotion-roadmap rows IV/V -> revised Lemma IV /
%   V-prime; GGA glossary row added. (3) Appendix E: discovery-history opening,
%   teach-box, read-alongside, take-away. (4) Appendix F: teach-box, NEW
%   'five clocks' subsection (t,a,z,T,r dictionary + worked example placing
%   recombination and the lock crossing on all five axes), take-away.
%   (5) Appendix G REBUILT: geometric Step 3 replaced wholesale --- was the
%   stale TeV-plateau story (V^1/4 ~ M_lock, 'inflation at the TeV seam',
%   'null tensor signal') --- now the two-basin descent taught slowly: derived
%   Starobinsky class w/ slow-roll exercise, GUT-adjacent placement, N2=0,
%   vacuum-generations analogy, full N*=52.6 -> n_s=0.9620 / r=0.0043 / T_rh
%   arithmetic, amplitude as the declared input w/ Lemma V-prime as an open
%   invitation, retired TeV reading as a falsification exercise, take-away box.
%   STAGE B (same day, after author register sign-off): (6) Apps H-N given
%   teach-box / read-alongside / take-away devices. (7) App I: NEW 'launch
%   selects the mechanism' paragraph (T_rh ~ 3e10 GeV -> thermal leptogenesis,
%   N2=0 -> zero dilution). (8) App K REBUILT mid-section: was 'there is no
%   GG-derived D_ell curve / none of the four steps is performed' -- now the
%   executed CLASS demonstration (3-level comparison incl. chi2=237/83 bins,
%   Delta chi2=-22, ell_D=1350+-3), steps (i)-(iii) executed, (iv) covariance
%   likelihood open (PS-Cosmos-16). (9) App L: NEW identity-reading paragraph
%   (GA->GGA, Part X sync, three-objects pointer). (10) App M: NEW pinning
%   theorem block (patch-uniformity, ZERO primordial isocurvature as the
%   falsifiable signature, no defects; two-routes identity language).
% v12.73 (July 4, 2026): PARTS XI-XIII DEEPENING (author-approved 2026-07-04).
%   (1) TRIAGE RECONCILIATION (flagged): canonical count is 2 D / 17 D* / 1 A
%   after the July campaign; fixed the stale 2/15/3 instances (Part IX counting
%   summary + D9 tensor audit paragraph [was full retired-TeV suppression story]
%   + D*/A section headers + closure-economy prose + Part XI compressed codebook
%   V_inf row + audit-roadmap rows I/II/III/IV/V' + two 'two/fifteen/three'
%   phrases + A9-comparison line). (2) Two-basin leftovers fixed in XII/XIII
%   (lock-seam-ends-inflation prose, I-Cosmos-5/6, module map, falsification-
%   ledger tensor row ['essentially absent' -> r=0.0043 two-sided], XIII launch
%   at lock scale -> GUT-adjacent, 'tensor sector silent' -> rings at 0.0043,
%   A8->A9 electron-identity attribution). (3) Part XII reordered into the
%   four-fold spine: Significance -> Innovations (moved up) -> NEW Accomplishments
%   section (17-problem tally vs Table 1) -> honest boundary (moved) -> cost/kill
%   criteria + NEW decade-timeline table + hostile audit (moved) -> bottom line.
%   (4) Part XIII: NEW trio section (A9->A10->A11 with A11 vocabulary: recursive
%   uniqueness, inevitability of Cosmos/Life/Mind, self-referential closure,
%   five-revolutions lineage; upstream/downstream reading guidance), deepened
%   A11 handoff (civilizational framing), NEW personal word from the
%   cosmological floor. Science and physics unchanged.
% v12.72 (July 4, 2026): PART X DEEPENING (author-approved 2026-07-04).
%   (1) Fixed stale cross-ref: Part X is the cosmological analogue of A9's
%   Part IX ('Geometric origin of identity and uniqueness of forces'), not
%   'Part XI'. (2) New Section A (identity core): GA->GGA ground-state
%   refinement; the electron|Kaxion|dark-energy side-by-side (GDOS observable
%   x GRIP-selected attractor) as Table tab:three-objects, with the honest
%   'two routes to identity' spine (ground-state vs pinning); three cosmic
%   questions in the A9 mould; the layer-by-layer cascade deepened from 6
%   terse lines to the full 7-rung ladder + an 8th cosmological rung; the
%   vacuum cascade as the cosmic relaxation channel (parallel to A9's weak
%   force). (3) Retitled the pi-lock material as Section B
%   (sec:pi-locked-derivation); deleted the old terse 6-strata subsection
%   (superseded by A.4; it also carried the now-moved label). (4) New
%   Section C (philosophical summit): epigraph, Q1/Q2 symmetry,
%   Design/Multiverse/Brute-fact dissolution, self-referential closure,
%   determinism/freedom, closing punch line. Science unchanged.
% v12.71 (July 4, 2026): PRESENTATION pass (no science changed).
%   (1) Abstract trimmed ~170 chars -> fits the cover page. (2) Part X GGPart
%   title shortened to one Overview line; Overview-of-Contents set \small ->
%   Overview + List of Figures share one page. (3) Figures 12/13/14 enlarged to
%   full text width (portrait fig: caption trimmed to footnotesize per the
%   tall-figure house rule; history pages full width). (4) CMB TT figure
%   unwrapped from the full-page pattern into a [!ht] float -> shares the page
%   with its citing text. (5) Figure-citation audit: 9 uncited labels given
%   in-text citations near their locations (cosmic-timeline, inflation-derived
%   portrait, pi-locked chart, six appendix summary figures); stale TikZ status
%   box in appfig-slowroll corrected to the derived-n_s reading. (6) Near-empty
%   pages eliminated: blank p.83 (portrait-fig overflow) fixed by (3); the
%   chronology stub page fixed by moving the landscape timeline block after the
%   Part-VIII purpose paragraphs. 482->480 pages.
% v12.71b (July 4, 2026): Figs 13/14 clipped to ink bbox (viewport) -> drawing
%   ~8-10%% wider at the same full text width (author request).
% v12.71c (July 4, 2026): Fig 14 relocated to directly follow Fig 13 (p.85),
%   pinned full-page pattern; the two history pages now face in sequence.
% v12.71d (July 4, 2026): version-history de-referencing (author request --- this
%   is the FIRST public release): all reader-facing mentions of earlier
%   versions/drafts/revisions of this or companion papers recast into research-
%   programme language (retired TeV reading, working codebook, July 2026
%   derivation campaign); bare version numbers (v9.x/v11.x) removed from body
%   text. ~48 anchored replacements; science and triage untouched.
% v12.70 (July 3, 2026): PHASE 1 of the two-basin revision (campaign 2026-07-03).
%   Part IV launch rewritten: GDOS-derived Starobinsky/xi plateau (eq kept label
%   partII-plateau-potential), "the lock" subsection -> "the two basins" (sign-level
%   no-go recorded; H_1 = 1.6e13 GeV on label partII-Hinf-def; H_2 = Mlock^2/sqrt3 MPl
%   = terminus seesaw on label partII-Hinf-corridor; N_2 = 0 with three arguments);
%   T-Cosmos-4 revised (two-basin); tilt subsection: n_s = 1-2/N* = 0.9620
%   derived-conditional (label partII-ns-numeric), r = 12/N*^2 = 0.0043 firm
%   (label partII-r-numeric), A_s = the one input + warp candidate flagged;
%   descent-does-the-work: N* = 52.6 evaluated, T_rh = 3e10 GeV (Planck-strength
%   Gamma = M^3/MPl^2, label partII-Treh-TeV); graceful-exit: two events separated;
%   comparison table updated + retired proposition kept on record; NEW subsection
%   "The launch, graded" + figure a10_inflation_derived_portrait.pdf
%   (fig:partII-inflation-derived); "One event, three legacies" -> "Two crossings,
%   three legacies"; same-principle section + NEW subsec "The vacuum generations".
%   Lemma-V retirement, codebook, chronology, PGW, claims text: Phases 2-7.
% v12.69 (July 2, 2026): universal paragraph indentation (author).
%   indentfirst loaded (first paragraphs after headings now indent);
%   six stray body \noindent's removed. Kept: abstract opener,
%   bibliography Relevance pattern (house style), cluster headers,
%   structural minipage wrappers. No content changed.
% v12.68 (July 2, 2026): multi-Arisaka citation split pass (charter
%   natbib rule; author-directed). Five >=2-Arisaka \citep's split into
%   (\citealp; ...) chains -- fixes the same-author-same-year comma-run
%   rendering (incl. the Fig-9 caption seven-key cite). Cohort census:
%   78 hits / 15 papers (A5, A8 clean); others queued for their own
%   passes. No content changed.
% v12.67 (July 2, 2026): figure enlargements (author). Provenance map
%   (fig:codebook-provenance) now on a landscape page -- ~1.5x scale,
%   effective fonts ~+50%. Derivation tree (fig:derivation-tree)
%   0.93\textwidth -> \textwidth. No counts changed.
% v12.66 (July 2, 2026): presentation items (author). p195 fill (timeline
%   landscape moved after Sec 47's dictionary paragraph); Sec-57 radial
%   steps as TikZ ladder flowchart (eqs dr1-dr6 + r-sequence replaced;
%   labels were unreferenced; downstream figure numbers shift by one);
%   Table 36 rightmost column widened; provenance-map legend anchor
%   fixed (bbox slack removed -- map fills textwidth) and its stale
%   'does not yet close' wording updated to the epoch-identity status
%   (in-figure red note + caption). No counts changed.
% v12.65 (July 2, 2026): age split + Sec-64.5 re-tier (author directive).
%   13.9 Gyr = today (0.948/66.6); 13.8 Gyr = the Omega_m = 1/pi
%   crossing (t0 - 75 Myr). Sec 64.5: baryon entry re-scored under the
%   epoch calibration (+2.76% -> +1.69%, 2.3 -> 1.4 sigma; cluster
%   partly epoch artefact; calibration = second measured precedent).
%   Table 33 gains the crossing row; timeline + true-age narrative +
%   all aligned instances updated. LCDM-voice 13.8's (Planck's own
%   number) retained. No triage counts changed.
% v12.64 (July 2, 2026): Figure-12 third refinement round (author):
%   top-panel 'Dark Energy' raised to y ~ 0.8; both panel titles moved
%   tight against their panels. Asset-only change
%   (a10_whynow_figs_gen_v2.py updated).
% v12.63 (July 2, 2026): Figure-12 second refinement round (author):
%   top panel -- dark-energy band extended to the right edge, 'Dark
%   Matter' relocated to (t ~ 0.1 Gyr, y ~ 0.7); bottom panel -- 'Dark
%   Energy' to (t ~ 8 Gyr, y ~ 0.9), 'Baryons' to t ~ 2.5 Gyr; panel
%   titles moved closer to the panels; all panel fonts enlarged ~20%.
%   Asset-only change (a10_whynow_figs_gen_v2.py updated).
% v12.62 (July 2, 2026): Figure-12 refinements (author, 2026-07-02):
%   lighter pastel bands; 'Baryons' typed directly in the band (bottom
%   panel); zoom inset moved to the top-right corner (crossing region of
%   the parent plot now unobstructed; 'Dark Energy' label relocated);
%   three blank lines above the Fig-12 caption. Asset regenerated
%   (a10_whynow_figs_gen_v2.py). No counts changed.
% v12.61 (July 2, 2026): author item 4 -- why-now figure assets
%   regenerated (generator a10_whynow_figs_gen_v2.py, saved to
%   [v11-Support]): the Omega_m = 1/pi crossing and today now marked
%   separately, with zoom insets resolving the 75 Myr separation
%   (bottom panel of Fig 12: Omega_m(t) through 1/pi at t0-75 Myr and
%   0.315 today; Fig 13 Panel B: magnifier at Delta r = 0.005). Stale
%   in-figure text retired (old 'snapshot at a=1' headline; corridor-era
%   ladder box). Captions note the insets. No counts changed.
% v12.60 (July 2, 2026): author items 1-3+5 (2026-07-02 directive).
%   Dev-plot pair unified: codebook caption trimmed (page-232 overflow
%   killed), preview table+figure captions made identical to codebook
%   versions, Figure 5 now full-page (0.74\textheight, vfill pattern).
%   tab:whynow-epochs gains an Age column (H0 t0 = 0.948 at 66.6) and
%   the Omega_m = 1/pi crossing row (13.8 Gyr) distinct from today
%   (Planck centrals, 13.9 Gyr). '(4) Today' paragraph and Fig-12
%   caption state 'Omega_m = 1/pi at 75 Myr ago (central values)' and
%   that all matter fractions decrease with time. One-line space added
%   above Fig-12 caption. Fig 12/13 asset regeneration ships next.
% v12.59 (July 2, 2026): epoch-calibrated Table 2 / Figure 5 (author
%   request). The four pi-inventory rows in both Dev-plot tables gain
%   calibrated companion entries -- using the single declared epoch
%   input once, (+1.05, -0.48, +2.76, +0.73)% collapse to (input, 0.00,
%   +1.69, -0.31)%; every remaining residual is an epoch-independent
%   ratio, all within ~1 sigma (Ob pull +1.4). Figure asset regenerated
%   (v4: violet diamonds + arrows). Captions explain the two readings;
%   the raw zero-parameter comparison stays as the headline. No counts
%   changed.
% v12.58 (July 2, 2026): crossing-epoch precision statement (author
%   question). Why-now subsection + VIII-audit: under Planck central
%   values the exact crossing Omega_m = 1/pi lies ~75 Myr in the past
%   (the last 0.5% of cosmic history), but the +/-0.007 measurement
%   uncertainty is a +/-160 Myr timing blur -- the offset is 0.5 sigma,
%   fully consistent with the crossing being now. Precise claim: t0 =
%   t_pi within measurement precision. No counts changed.
% v12.57 (July 2, 2026): Group C install (author-approved). Abstract's
%   'single named step' pinned to its identified content (timing of the
%   present epoch; fits front page, verified). Five reader-facing fence
%   statements aligned with the resolved Conjecture-VIII status (exec
%   summary, preface ledger, derivation-tree caption, Part-intro tiers,
%   formal fence, jaw-dropper) -- each keeps its register; D*/A tags
%   unchanged everywhere. Two-channel precision clause added at the [B1]
%   audit update. No equation altered; no counts changed.
% v12.56 (July 2, 2026): Group A+B install of the author-approved action
%   plan (A10-C2-ActionPlan-2026-07-02): R1/R2/R3 refresh of the VIII-a
%   half (exact rephasing protection; Jarlskog ~1e-28; gravity floor
%   e^-K leading; residues discharged), VIII-audit closure sentence,
%   concordance upgraded to derived (Delta c_2 = k, A6), why-now crossing
%   quantified (75 Myr), Thm 62.1 Proof-status note + pilock appendix
%   Step 4 retired as dead-end route (closure = measure dictionary +
%   epoch identity), codebook rows re-scoped (D* tags unchanged),
%   Vol(S^3) annotated with the derived factorization. No triage counts
%   changed; no equation altered.
% v12.55 (July 2, 2026): VIII-a session install (author-accepted report
%   A5-VIIIa-GSM-Report-2026-07-02). Edits: (1) VIII-a refinement half
%   re-scoped to consumer form -- domain qualifier (matter-sector phase
%   moduli; EW order-parameter directions settled into the GA), two-part
%   statement (delocalisation, worst tilt ~0.6% of threshold; gravity
%   floor ~e^-K ~1e-15; share robustness under normalised tilts), status
%   proved-in-consumer-form with residues R1/R2/R3. (2)(3) charter
%   natbib repair: two-Arisaka multi-cites split into citealp pairs at
%   the two occurrences (fixes the double-comma rendering). No counts
%   changed.
% v12.54 (July 2, 2026): [P] projection session install (author-accepted
%   report A10-P-Projection-Report-2026-07-02). Two edits: (1) VIII-audit
%   update -- [P] discharged by identification: dilution no-go (super-
%   selection freezes the integer, not the energy record) retires every
%   epoch-independent normalisation incl. anomaly-inflow eps_GEP; u = 1
%   is equivalent to the T0 epoch calibration (crossing a_pi = 0.9949,
%   ~75 Myr ago); open core of Conjecture VIII = VIII-a alone. (2) A
%   short epoch-identity reading remark after thm:omega-m-pi-topological.
%   Fractions stay D*. No equation altered; no counts changed.
% v12.53 (July 2, 2026): [B1] kernel session install (author-accepted
%   report A10-B1-Kernel-Report-2026-07-02). One edit: VIII-audit
%   two-item close updated -- [B1] discharged as structure theorem
%   (Einstein ratio = thermometer under KMS contact; D = mu = 0 at the
%   gapped frozen GA, kernel-side face of DE-1; lock-scale reading =
%   freeze-out record T* = 2 Mlock/L*). Open core = [P] alone, with the
%   sharpened target eps_GEP = Mlock/(pi L*) (freeze-out reading) vs the
%   vacuum-floor reading; adjudication = [P]'s content. Fractions stay
%   D*. No equation altered; no counts changed.
% v12.52 (July 1, 2026): Unit-bridge pass integration. (1) Conjecture-VIII
%   audit gains the recomposition update: explicit bridge relation; A6 v11.18
%   fixes the action side (Lemma 21.4) and superselects the packet (Cor 21.6);
%   Hirzebruch p1=3sigma pins the packet as the canonical lattice unit; open
%   remainder = [B1] Einstein-ratio temperature + [P] OGGEP projection.
%   (2) T-Cosmos-17 proof-outline Step 3 gains a normalisation note (the 8pi^2
%   is consumed upstream by the GEP definition; the pi enters from VIII-b').
%   (3) Matching 2026-07 annotation at the appendix Step-3 site (history kept).
%   (4) Baryon-offset print corrected to +2.8% at six sites (own arithmetic:
%   (0.050661-0.0493)/0.0493 = 2.76%). Science content author-approved
%   2026-07-01 (StarClosure report + A6/A5 sign-offs).
% v12.51 (July 1, 2026): Conjecture-VIII audit updated with the proved
%   Lemma VIII-b' (real-sector half-measure, A5 v11.18) + recalibrated share
%   dictionary; open core contracted to VIII-b'' (unit-packet normalisation).
%   Author-accepted proof, report 2026-07-01.
% v12.50 (July 1, 2026): T-Cosmos-DE' installed at the dark-energy terminus
%   (theorem + proof + two-sided kill-switch corollary; provenance chain
%   A2 rigid profile -> A5 T9.1 gap -> A6 GFF-ST6/Lemma DE-1 -> here;
%   |1+w| <= (H0/Mlock)^2 ~ 1e-91). Lemma IX audit: exactness proved, residue
%   = profile stabilisation; D16 stays D* (conservative). Lemma VIII audit:
%   VIII-a/VIII-b split (A5 GSM adjunct; A2 no-identification remark).
%   Second-order residual: measure route positively excluded. Stationary-
%   Measure programme, 2026-07-01; author-approved at every step.
% v12.49 EDIT H (2026-07-01): version stamp v12.49 / July 1, 2026; appendix
%   figure short caption updated (computed damping scale).
% v12.49 EDIT G (2026-07-01): Part-IX Dev-plot caption counts (seventeen ->
%   twenty-one; sigma_th note generalised); appendix CMB schematic scope-flag
%   box repositioned/widened to clear the axis labels.
% v12.49 EDIT F (2026-07-01): the two surviving two-ends-of-one-corridor
%   framings (front-matter preview prose; Innovation I-Cosmos-4) aligned with
%   the canonical reading: background derived 66.6 +- 0.2, ladder displaced.
% v12.49 EDIT E (2026-07-01): figure text updated (provenance map ell_D
%   node re-homed under Conj. VIII with the Boltzmann value; appendix CMB
%   schematic knee/scope-flag/caption re-scored to ell_D = 1350 +- 3 with
%   the BFFT overlay named).
% v12.49 EDIT D (2026-07-01): codebook architecture reconciled with the
%   re-scored rows. D18 (ell_D) re-gated Conj. VII -> Conj. VIII (value =
%   Boltzmann diffusion on the inventory, 1350 +- 3); Conj. VII demoted to
%   gating the interpretive BFFT overlay; closure tallies, itemize, schedule
%   table, counting summary and objections row recomputed. Second-order
%   residual recast (cluster of four -> three; ell_D departure reported as
%   the tiering's predicted behaviour). Tier-3 + CMB appendix updated: the
%   spectrum has been run (CLASS, exact fractions, seed frozen); outstanding
%   item = full likelihood fit. H0 registry mechanism text refreshed (gates
%   unchanged per the v12.48 decision to preserve the corridor theorem).
% v12.49 EDIT C (2026-07-01): parameter-audit Dev plot re-scored, 17 -> 21
%   entries (16 filled / 5 open; author-approved). H0^CMB -> filled derived
%   (66.6 +- 0.2 vs 67.4, Dev -1.2%, pull -1.5); H0^late -> honest reach
%   68.5 +- 0.6 vs 73.0 (Dev -6.2%, pull -3.9; named ladder systematic);
%   ell_D -> Boltzmann 1350 +- 3 (+0.4%). New CLASS-verified rows:
%   omega_b/omega_m = 1/2pi (+1.3%), 100 theta_* (-0.47%, pull -4.3),
%   peak ratios 2/1 (-0.3%) and 3/1 (-0.8%). Preview + codebook tables and
%   both captions reconciled; marker figure regenerated (a10_dev_plot_gen v3).
% v12.49 EDIT B (2026-07-01): Phase-2 CLASS results woven into the body.
%   Precise Boltzmann-confirmation numbers in the H0 derivation; the damping
%   scale reframed honestly (Boltzmann 1350 +- 3 is the framework's ell_D
%   prediction; the BFFT knee 1381.7 demoted to a conjecture-level overlay
%   +2.3% above the framework's own value); switch-table and corridor-table
%   rows re-scored from 'anchor' to derived (theta_* = 1.036 predicted, H0
%   background derived, ladder row shown as the honest open miss); legacy
%   Delta-N_eff 0.30 row aligned with the canonical 0.04 baseline.
% ---------------------------------------------------------------------------
% v12.49 EDIT A (2026-07-01): NEW Part-VI framing section "Determinism from
%   geometry: the microwave background as an over-determined test"
%   (sec:cmb-overdetermined), inserted before the CMB-baseline section.
%   Contents: the master identity omega_i = Omega_i h^2 and the several-roads
%   logic; the CLASS-verified core-observable table (tab:cmb-core-derived);
%   the H0 consistency-web figure (fig:h0-consistency-web); the pi-offset
%   subsection with the honest chi^2 and the theta_* pull; the
%   claimed/not-claimed boundary subsection. Numbers from the Phase-1
%   analytic and Phase-2 CLASS reports (project folder, 2026-07-01).
% ---------------------------------------------------------------------------
% v12.48 (2026-07-01): Hubble reframe Sec.36 -- background ~67 = CMB value (coincide);
%   only the local ladder displaced (+2-3% Local-Hole atop calibration systematic); siren
%   predicted ~67; two-stage-corridor Theorem preserved, early branch = background; Dev-plot
%   H0 row labels de-71.3'd (numeric columns unchanged - full re-score is a follow-up).
% v12.48 (2026-07-01): Hubble reframe Part IV -- inventory-first H0~67 as the
%   sharp derivation; dark-energy seesaw demoted to soft consistency check landing
%   on the same ~67; true age reverted to derived 13.8 Gyr; Fig12/timeline age labels.
% ===== v12.47: formatting pass (author). (1) Global caption separation
%  \captionsetup{aboveskip=12pt,belowskip=13pt} so captions are set off ~1 line from body
%  text. (2) Removed two gratuitous section-break \clearpage (before 'The cosmic
%  coincidence, bounded' [fills p.92] and before appendix 'Cosmic expansion'); full-page
%  figure clearpages and Part breaks kept. v12.46 -> v12.47.
% ===== v12.46: Table 3 columns 2 & 3 set to equal width (0.39 each) and each
%  row's two descriptions balanced to similar length. v12.45 -> v12.46.
% ===== v12.45: Table 3 (tab:inheritance-map) expanded from the 7 upstream
%  papers to all 17 cohort papers, converted to a page-breaking longtable, each
%  paper name hyperlinked to arisaka-gg.org, and the Main-achievement column widened
%  (0.24 -> 0.42 textwidth). v12.44 -> v12.45.
% ===== v12.44: Figure 6 (Parts roadmap, fig:parts-roadmap-a10) enlarged from
%  resizebox 0.82\textwidth to full \textwidth so it fills the text width. v12.43->v12.44.
% ===== v12.43 (2026-06-30): COSMETIC follow-up -- table font eased from
%  \footnotesize (~18%% < body, too small) to \small (~9-10%% < the 11pt body) on all
%  44 table floats (per-table size command before \begin{tabular}). v12.42 -> v12.43.
% ===== v12.42 (2026-06-30): COSMETIC. Uniform table font -- \footnotesize inserted
%  before \begin{tabular} in all 40 body-size table floats (the float-package [H]
%  tables are not patched by \AtBeginEnvironment, so a per-table size command is used,
%  matching the 3 pre-existing \small tables which were also bumped to \footnotesize).
%  ~18-20%% below the 11pt body; p{}-column widths unchanged so tables still span the
%  page. Saved 5 pp (461->456). No content/science change. v12.41 -> v12.42.
% ===== v12.41 (2026-06-30): Dev-plot tables (Tab 2 preview, Tab 37 full) + figure
%  (Fig 5/17) credit the now-DERIVED Hubble anchor H0^(0)=71.3 in the H0^CMB and
%  H0^late prediction text. Dev/Pull/markers and the late central value (73.7, the
%  67.4-anchored A10 convention) are UNCHANGED -- status not upgraded (premature),
%  per author. v12.40 -> v12.41.
% ===== v12.40 (2026-06-30): added a dedicated general-audience subsection
%  'The true age of the universe' (subsec:true-age) -- the t0=13.0 Gyr prediction
%  as a direct GG derivation (0.948/H0^(0), H0=71.3), why textbooks say 13.8 (Planck
%  CMB H0=67.4, indirect/model-dependent), the age-as-Hubble-tension three-rate line,
%  and the GW-siren bet. The technical both-values paragraph in the de-Sitter-future
%  subsection is collapsed to a one-line bridge pointing here. v12.39 -> v12.40.
% ===== v12.39 (2026-06-30): AGE made GG-self-consistent. Present age now quoted as
%  t0=13.0 Gyr at GG's own H0=71.3 (was 13.8 at the Planck H0=67.4); both values shown
%  side by side and tied explicitly to the Hubble tension and its GG resolution (one
%  background rate 71.3, two biased projections 67.4/73.0 -> ages 13.8/12.7). Figure 12
%  recalibrated to H0=71.3 (today=13.0, transitions 9.8/7.3) with a both-values note;
%  caption, 'stellar-lifetime' line, and cosmic-timeline node updated to 13.0. The LCDM
%  background description (six-parameter 13.8 Gyr) is left unchanged. v12.38 -> v12.39.
% ===== v12.38 (2026-06-30): FIGURE-13 Panel B widened to r in [65,68]; the two
%  transition-dot labels (z=0.62, z=0.29) moved next to their markers; Panel A
%  'Lambda future' label lowered off the frame. Figure-art only. v12.37 -> v12.38.
% ===== v12.37 (2026-06-30): FIGURE-12 enlarged to fill its page; its caption
%  moved to the foot of the preceding page (caption-before-plot). FIGURE-13 Panel B
%  re-zoomed to r in [65,67], ratio in [1e-2,1e2]; Panel A 'TODAY' label moved to the
%  bottom. No frozen-science change. v12.36 -> v12.37.
% ===== v12.36 (2026-06-30): FIGURE-12 PRESENTATION. Replaced the single
%  stacked-band panel with the log + linear-time PAIR (each panel carrying three
%  x-axes: cosmic time t, scale factor a, radial coordinate r); caption rewritten
%  and driven from LaTeX (art is caption-free: A10-whynow-Omega-evolution-PAIR-nocap.pdf).
%  The two why-now full-page figures were relocated upward to sit immediately after
%  the first citation, so each appears before its detailed description. Today drawn
%  at t=13.8 Gyr in the figure. No frozen-science change. v12.35 -> v12.36.
% =====================================================================
% ===== v12.35 (2026-06-29): DATA-SAFE H_0 RECONCILIATION WITH C2. =====
%  (1) Fixed stale anchor in the Sec-36 'Scale versus split' bridge: ~69 -> 71.3
%      (the v12.33 bridge predated zeta_Lambda=1/J^2), + GW-siren robustness note.
%  (2) Rewrote sec:h0-two-values to the author-chosen DATA-SAFE framing: robust
%      output is the central value H_0^(0)=71.3 (matches GW-siren >~71); the two
%      measured endpoints are biased readings; CMB-side closure from the data-safe
%      budget (relic DeltaNeff~0.04 + ledger f_EDE=0.0326) is PARTIAL (67.4 ->
%      ~68.6-70), residual ~2-4%% open (K-2); local +3%% is the U(1)_B graviphoton
%      (open). Replaces the earlier clean 67.7/73.4 (DeltaNeff=0.30) projection,
%      which needed a radiation excess current N_eff bounds disallow.
%  Keeps A10's documented late-value convention (73.0/Riess2022) and relic
%  DeltaNeff~0.04. No frozen-science change. v12.34 -> v12.35.
% =====================================================================
% =====================================================================
% ===== v12.34 (2026-06-29): H_0 DERIVED FROM THE LEDGER (anchor + split). =====
%  TERMINUS (sec:h0-derivation) rewritten: zeta_Lambda fixed by the MSMF/OGN
%   bounded-cost search to the cost-2 minimum 1/J^2=1/49 (cost table; ledger-
%   monomial-not-loop intuition; Asserted grade, A9-SM 8-Yukawa parallel);
%   anchor H_0^(0)=71.3 km/s/Mpc evaluated step by step; corrected the prior
%   error that listed M_lock as an open 'lock prefactor' (M_lock=1/2 E_Pl e^-K
%   is a derived theorem, A5 T8.7). Remaining: zeta_Lambda Hosotani lemma (O6),
%   Omega_Lambda Conjecture VIII.
%  SECTION 36 (sec:h0-two-values): new subsection projecting the derived anchor
%   71.3 through the two corridor dictionaries -> early/CMB 67.7, late/local 73.4;
%   f_EDE = e^{-K/2^J} N/(2K)=0.0326 ledger closed form; two-route consistency
%   check (71.3 vs 70.9, 0.6%%); prediction table; physics intuition; per-input
%   honest grades. No frozen-science change. v12.33 -> v12.34.
% =====================================================================
% =====================================================================
% ===== v12.33 (2026-06-29): w>=-1 THEOREM + H_0 DERIVATION + HUBBLE BRIDGE =====
%  (1) No-phantom theorem T-Cosmos-DE (thm:no-phantom) + proof + slow-roll bound
%      folded into the terminus 'why w=-1' subsection; honesty sentence revised to
%      separate the proven SIGN (w>=-1, ghost-free) from the open MAGNITUDE.
%  (3) New subsection sec:h0-derivation: closed Friedmann/seesaw form for H_0,
%      the ~69 km/s/Mpc number, achievement (H_0 a derived combination) + the open
%      ledger (zeta_Lambda = Hosotani amplitude x codim-2 screening; Omega_Lambda
%      D* under Conj VIII; lock prefactor). Graded a consistency postdiction.
%  (2) Section 36 bridge: scale (Part IV terminus) x split (corridor) factorization,
%      same-throat-cap link, no-phantom theorem as shared falsifier.
%  No physics/value change. v12.32 -> v12.33.
% =====================================================================
% =====================================================================
% ===== v12.32 (2026-06-29): PART IV -- CITATIONS + SIGNIFICANCE SUMMARY. =====
%  (a) Inline external citations across Part IV: inflation classics
%      (Starobinsky 1979/1980, Mukhanov-Chibisov, MFB, Lyth-Riotto, Mukhanov)
%      and dark-energy/cc references (Weinberg 1989, Carroll, Peebles-Ratra,
%      Copeland, Gibbons-Hawking) at the slow-roll, tilt, tensor, seesaw, w=-1,
%      and de Sitter-entropy derivation points.
%  (b) Added a ~1-page significance subsection at the close: both of cosmology's
%      great unknowns (inflation TeV launch, dark-energy meV terminus) pinned to
%      one inherited lock scale M_lock, analytically and quantitatively, with the
%      honest grading (scales forced; n_s/A_s asserted; pi-fractions D* under
%      Conjecture VIII). No physics/value change. v12.31 -> v12.32.
% =====================================================================
% =====================================================================
% ===== v12.31 (2026-06-29): PART III EXTERNAL CITATIONS (for students). =====
%  Added inline citations to Part III: classic papers (Friedmann 1922,
%  Lemaitre 1927) and standard GR/cosmology textbooks (Weinberg, MTW, Wald,
%  Dodelson, Mukhanov, Kolb-Turner, Peebles) + the Lambda-problem reviews
%  (Weinberg 1989, Carroll), at the GR derivation, FRW derivation, GUT-edge,
%  LCDM-theorem, and thermal-history sections. All keys already first-cited in
%  Parts I/II; no provenance change. v12.30 -> v12.31.
% =====================================================================
% =====================================================================
% ===== v12.30 (2026-06-29): SYSTEMATIC PARAGRAPH-INDENT FIX. =====
%  Removed paragraph-leading \noindent throughout the body (Part I onward) so
%  paragraphs indent at the intended \parindent=1.5em (parskip=0). Fixes the
%  flush-left look of Parts II/III (and elsewhere). Standalone \noindent lines
%  left untouched; no structural openers affected. Presentation only.
%  v12.29 -> v12.30.
% =====================================================================
% =====================================================================
% ===== v12.29 (2026-06-29): PART III SIGNIFICANCE/NOVELTY SUMMARY. =====
%  Added a ~1-page significance-and-novelty summary at the end of Sec 23.3
%  (close of Part III): GR as output, FRW derived, GUT-edge beginning, zero
%  free ledger, Lambda seesaw; LCDM promoted to a theorem (a la A9-SM), with
%  the honest open closures flagged forward to Parts IV-X. Replaces the brief
%  Part-IV transition. No value change. v12.28 -> v12.29.
% =====================================================================
% =====================================================================
% ===== v12.28 (2026-06-29): PART III -- GR & FRW DERIVED AS THEOREMS. =====
%  Restructured Part III around the derivation ladder (A1->A2/A3->A4->GR->FRW
%  ->LCDM). NEW: full graduate-level derivation of (i) 4D general relativity
%  from the 6D Einstein-Hilbert action by scale-profile reduction (M_Pl^2 =
%  pi M_6^4(1-e^-2K), Lambda seesaw m_L=M_lock^2/M_Pl~6 meV), and (ii) the
%  Friedmann equations from the GRIP-selected homogeneous attractor (Christoffel
%  -> Ricci -> Einstein tensor -> Friedmann system), with physics intuition.
%  Bold thesis 'LCDM is a theorem' + GUT-edge-not-Planck section + ladder fig +
%  GUT/Planck schematic + 6-parameter D/D*/A table. Replaces the old modest
%  'Friedmann from the boundary law'; the referenced Friedmann labels
%  (friedmann1/acceleration/continuity) are carried into the new derivation.
%  Scale-chain section kept. All physics from A1-A6 (cited). v12.27 -> v12.28.
% =====================================================================
% =====================================================================
% ===== v12.27 (2026-06-29): PART II FIXES (S10 cites, Fig8 move, Fig9 fill). =====
%  (1) Section 10 now opens by citing A1-Constitution and A2-GG6 (before the
%      Randall-Sundrum contrast).
%  (2) Figure 8 (6D-bulk / OGGEP schematic) moved from Section 10 to Section 11
%      (the boundary projection); Section-10 mention is now a forward reference.
%  (3) Figure 9 (derivation tree) inlined (was forced full-page, leaving p45
%      nearly empty); now shares its page with the descent-ladder text.
%  Layout/citation only. v12.26 -> v12.27.
% =====================================================================
% =====================================================================
% ===== v12.26 (2026-06-29): PART II SUBSTANCE (Sections 10-13). =====
%  Filled the inherited framework with real structure + cohort citations,
%  from the Exec Summaries of A2/A5/A6/A4/A7/A9:
%   S10: explicit 6D metric ds^2=e^-2r eta dx^2+dr^2+dtheta^2 + G_Bulk =
%        Spin(1,5)xSU(5)xU(1)_BxU(1)_Qi + the 4+1+1 decomposition (cite A2).
%   S11: OGGEP dictionary + GDOS (GKLS) boundary law + Gi4/GRIP cascade (A5,A6).
%   S12: BI6 coeff (1,-1,-2/5,-2/5) + OGN-5 ledger (1,3,5;7,35) + dual lock
%        K=LJ=M^2+N^2+L^2=35 (cite A3,A4,A9).
%   S13: GRIP-In/GRIP-Out regimes + Qi4 4-pillar GGA selector + A9 worked model.
%   Table 3 (inheritance map) moved from S12 to S9 (end of Part I), beside Fig 7.
%  All facts from cited companion papers; no frozen-value change. v12.25->v12.26.
% =====================================================================
% =====================================================================
% ===== v12.25 (2026-06-29): FIG 6 (roadmap) onto p34, short caption. =====
%  The parts roadmap was a forced-full-page block right after the Sec-8
%  opener, leaving p34 mostly empty. Made it inline + smaller (0.82\textwidth)
%  with a SHORT caption, so it sits under the 'plan of the body' line and the
%  Sec-8 text flows below it. Layout only. v12.24 -> v12.25.
% =====================================================================
% =====================================================================
% ===== v12.24 (2026-06-29): FIGURE-FIRST ORDER IN PART I. =====
%  Every Part I figure/table now PRECEDES the prose that explains it (reader
%  sees the structure first, text confirms): Fig 2 early in Sec-2; Fig 3 before
%  the four-pillars walkthrough; Table 1 before the two-tensions discussion;
%  the central-claim funnel before its five-paragraph walkthrough; the Dev
%  table+plot before the entry discussion; the parts roadmap at the START of
%  Sec-8 (with the 'plan of the body' line); the cohort map before the five
%  strata. In-text figures made inline so order is guaranteed. Order only;
%  no content/value/triage change. v12.23 -> v12.24.
% =====================================================================
% =====================================================================
% ===== v12.23 (2026-06-29): PART I LAYOUT (kill whitespace gaps). =====
%  (a) Fig 3 (LCDM pillars) float -> [t].  (b) Fig 2 regenerated ~+20%% fonts,
%  full-width.  (c) Sec-7 Dev table+plot previews, Sec-8 parts roadmap, Sec-9
%  cohort map: forced-full-page minipages -> INLINE moderately-sized centred
%  blocks (no clearpage/vfill) so text flows around them and the gaps at
%  pp.32/35/37 close, order preserved. No content/value change. v12.22->v12.23.
% =====================================================================
% =====================================================================
% ===== v12.22 (2026-06-29): FIG-2 PLACEMENT (fill page 20). =====
%  The century-timeline figure (Fig 2) was landing alone on a float page
%  (!htbp allowed 'p'); changed to a top float [t] so it pins to the top
%  of the page and Section 3 begins below it on the same page. No content
%  change. v12.21 -> v12.22.
% =====================================================================
% =====================================================================
% ===== v12.21 (2026-06-29): PART I RESTRUCTURE (foundational intro). =====
%  (1) Added two foundational sections after S1: 'A century of cosmology'
%      and 'LCDM: the concordance model, and why it works' (each w/ a new
%      sans-serif figure: century timeline, four-pillars). ~50 citations,
%      all existing bibitems.
%  (2) Moved 'The whole programme at a glance' (17-paper cohort map + five
%      strata) from head of Part II to the END of Part I.
%  (3) Reconciled: Part II GGPart subtitle (-'seventeen-paper programme'),
%      Part II intro (five->four ideas + connective), sec:architecture
%      forward-ref, and the richer Part-I closing paragraph.
%  (4) Recomputed 'First cited in' notes across expanded Part I.
%  Presentation/structure only; no physics/value/triage change. v12.20->v12.21.
% =====================================================================
% =====================================================================
% ===== v12.20 (2026-06-29): PART I CITATIONS + INTERNAL LINKS. =====
%  Introduction (Part I) had no inline external citations. Added ~30
%  \citep/\citet citations to EXISTING section-B bibitems (founders in
%  S1; LCDM/BAO textbooks + Hubble tension in S2; one targeted set per
%  problem in the seventeen-problems enumeration; horizon-thermodynamics
%  in S3) plus internal links (Theorem omega-m-pi, why-now sections,
%  codebook table, Conclusions). 'First cited in' notes RECOMPUTED from
%  each touched key's TRUE first-citation section (front-matter-aware).
%  No new bibitems, no physics/value/triage change. v12.19 -> v12.20.
% =====================================================================
% =====================================================================
% ===== v12.19 (2026-06-28): DEV-TABLE ENTRY INDENTATION (A9 style). =====
%  Both Dev tables: the Entry column is indented (>{\hspace{1.4em}}l) so the
%  entries sit under the flush-left bold-italic category rows, matching the
%  A9-SM codebook tables. Cosmetic only. v12.18 -> v12.19.
% =====================================================================
% =====================================================================
% ===== v12.18 (2026-06-28): DEV-TABLE PULL COLUMN + BOLD-ITALIC CATEGORIES. =====
%  Both Dev data tables (Part-I preview + Part-IX full): category rows are
%  now BOLD-ITALIC with NO rules around them; the two columns sigma_obs/|obs|
%  and sigma_th/|obs| are replaced by a single PULL column (= Dev / combined
%  1-sigma = (pred-obs)/sqrt(sig_obs^2+sig_th^2)). Pull values verified
%  programmatically (Omega_b +2.3, ell_D +2.5 are the notable ~2.5-sigma
%  pulls; eta_B <<1 under its factor-3 band; anchored entries 0). Captions
%  updated. No frozen-value/triage change. v12.17 -> v12.18.
% =====================================================================
% =====================================================================
% ===== v12.17 (2026-06-28): DEV-TABLE REFORMAT (A9-Table-8 style). =====
%  Both instances of the parameter-audit Dev data table (Part-I preview
%  + Part-IX full) reformatted for readability: removed the 'Source'
%  column (frees width -> larger font under the existing resizebox), kept
%  'Status / marker' on the right, added five bold-italic category rows
%  (pi-locked inventory; baryogenesis & nucleosynthesis; Hubble corridor &
%  recombination; inflationary spectrum; clustering & radiation). Captions
%  drop the 'empirical source' column description; fixed a stray caption
%  glitch. Source citations remain cited elsewhere (no orphans). No
%  value/triage/physics change. v12.16 -> v12.17.
% =====================================================================
% =====================================================================
% ===== v12.16 (2026-06-28): ONE-PAGE OVERVIEW+LIST-OF-FIGURES (front matter). =====
%  The List of Figures overflowed 3 lines onto page 6. Enforced single-line
%  LoF entries per the v2.0 standard by shortening eight long [short] optional
%  captions (LoF text only; each figure's FULL caption is unchanged). No
%  value/triage/physics change. v12.15 -> v12.16.
% =====================================================================
% =====================================================================
% ===== v12.15 (2026-06-28): SEVENTEEN PROBLEMS SPELLED OUT (Part I). =====
%  Author-requested expository addition (mirrors A9-SM's ten-classic-
%  problems subsection). Adds subsec:seventeen-problems to Section 2:
%  one broader-audience paragraph per problem, in the five Table-1
%  clusters, each ending with a one-clause honest GG reading + Part
%  pointer. Frontier items (13,14,16) marked as touched-not-closed;
%  problem 5 (why now) and 3 (cosmological constant) phrased per the
%  form/scale/epoch calibration. No value/triage/physics change.
%  v12.14 -> v12.15.
% =====================================================================
% =====================================================================
% ===== v12.14 (2026-06-28): ABSTRACT TRIM (refit front cover page). =====
%  The v12.13 harmonization pushed ~3 lines of the abstract onto page 2.
%  Tightened three sentences (removed para-2/para-5 epoch redundancy) to
%  refit page 1; all honest content retained (form forced; present-epoch
%  reading; 10^120 -> few-percent timing; the epoch the one input).
%  No value/triage/physics change. v12.13 -> v12.14.
% =====================================================================
% =====================================================================
% ===== v12.13 (2026-06-28): FRONT-MATTER HONESTY HARMONIZATION. =====
%  Author-approved. Propagates the form/scale/epoch calibration of the body
%  (subsec:whynow-form-scale-epoch) up to every headline instance, so the
%  honesty triage reads identically throughout. NO value/triage/frozen-
%  physics change (still Omega_m=1/pi, D*, Conjecture VIII). Specifically:
%   - Abstract: 'emerges as 1/pi / becomes a theorem' -> forced FORM + present-
%     epoch reading + '10^120 demoted to few-percent timing' + names the one
%     calibration (when we look).
%   - Executive Summary (3 spots): same calibration; fixes the false line
%     'becomes an identity rather than an accident' -> 'demoted, not declared
%     an identity'.
%   - Pi-locked Chart (front hook + Part-X twin): on-figure tagline 'zero free
%     parameters' -> 'present-epoch snapshot'; captions scope 'theorem' to the
%     conditional coupled pi-FORM and mark the absolute fractions present-epoch.
%   - Codebook caption: adds the epoch (T_0 / L4) clause to the Omega entries.
%  v12.12 -> v12.13.
% =====================================================================
% =====================================================================
% ===== v12.12 (2026-06-28): FORM/SCALE/EPOCH HONESTY SUBSECTION (Part IV why-now). =====
%  Author-requested clarification. NO value, NO triage-count, NO frozen-
%  physics change. Adds subsec:whynow-form-scale-epoch to the why-now
%  section: a cosmological inventory has three freedoms -- FORM (removed:
%  coupled pi-powers, Conj VIII), SCALE (removed: forced meV vacuum residue,
%  defeats 10^120 tuning), and EPOCH (NOT removed: Omega_m(a) is monotonic,
%  so Omega_m=1/pi is a present-epoch reading, equivalent to the observed
%  T_0 = the L4 calibration input). Genuinely epoch-independent content =
%  the ratio 1/(2pi) + the vacuum scale; the absolute fractions are present-
%  epoch values, NOT constants on the footing of alpha^-1=137. Boxed thesis:
%  the why-now coincidence SURVIVES but at the few-percent level (~2%% timing
%  window), demoted by ~120 orders of magnitude, not abolished. Also two
%  precision fixes to the residue subsection (title 'values'->'form and
%  scale'; box 'three of four fractions'->'internal matter ratios').
%  v12.11 -> v12.12.
% =====================================================================
% =====================================================================
% ===== v12.11 (2026-06-28): 'WHY NOW' COSMIC-COINCIDENCE SECTION (Part IV). =====
%  Author-approved substantive addition. Presentation/analysis of the FROZEN
%  pi-locked inventory; NO new physics result, NO changed value, NO triage-
%  count change. Adds one section (sec:why-now-cosmos) to Part IV as the
%  capstone of the dark-energy discussion (before 'Falsification and bottom
%  line'): the exact Omega_i(a) evolution and flatness sum rule; the eternal
%  vs. snapshot decomposition (b:dm:m ratios time-independent, only matter:
%  Lambda evolves); the four transition epochs and q0=3/2pi-1=-0.523; the
%  exact a(t) and de Sitter fate; the rho_L/rho_m radial window (Dr~1.5 out
%  of r_now~66.6); the linear-growth-suppression integral (41%% future growth
%  left); the anthropic probability (0.90-0.96); the resolution in three
%  movements; and the honest open residue (fixing r_now = the L4 calibration
%  input) with a boxed thesis. Two full-page figures
%  (fig:whynow-omega, fig:whynow-ratio). All entries stay D*; calibrated-
%  honesty held. v12.10 -> v12.11.
% =====================================================================
% =====================================================================
% ===== v12.10 (2026-06-28): SECOND-ORDER RESIDUAL + 2.7% OFFSET (Task D). =====
%  Presentation only; NO physics, NO parameter values, NO triage-count
%  change. Verifies and flags the +2.7%% common offset: FOUR independent
%  Dev-plot predictions (w, Omega_b, ell_D, S8-late) cluster at +2.72..2.77%%
%  (spread 0.04 pp). Adds a labelled subsection
%  (subsec:second-order-residual-cosmos) tiering it honestly: the
%  statistically strong pair (Omega_b 2.3sigma, ell_D 2.5sigma) is the
%  baryon/sound-horizon/damping sector; the residual is the Tier-3
%  (Boltzmann) precision content the geometry frames but does not compute;
%  three readings (coincidence / common 2nd-order normalisation / selection)
%  with NO closed form assigned (anti-numerology). Entries stay D*. Also
%  reconciles the 'first-order and second-order codebooks' subtitle and the
%  cohort-figure node, and adds a Dev-table caption pointer. v12.9 -> v12.10.
% =====================================================================
% =====================================================================
% ===== v12.9 (2026-06-28): COSMOLOGY TIER LADDER (Task C). =====
%  Presentation only; NO physics, NO parameter values, NO triage-count
%  change. Adds the explicit four-tier can/cannot ladder (the cosmological
%  instance of A5-GDOS Theorem T9.9, used in A9-SM) as a new labelled
%  subsection (subsec:tier-ladder-cosmos) in the Discussion Scope/limits
%  section: Tier 1 inventory (computed), Tier 2 scale flow/chronology
%  (geometrised), Tier 3 precision C_ell (Einstein-Boltzmann; not computed,
%  PS-Cosmos-16), Tier 4 nonlinear P(k) (N-body; not computed,
%  PS-Cosmos-17), with the boxed thesis 'GG computes the inventory, not the
%  precision spectra; the Boltzmann/N-body machinery remains the instrument.'
%  A one-sentence computational boundary is also added to the Conclusions
%  (sec:concl-answerable). v12.8 -> v12.9.
% =====================================================================
% =====================================================================
% ===== v12.8 (2026-06-28): FORCING-VS-SEARCHING UPGRADE (Task B). =====
%  Presentation only; NO physics, NO parameter values, NO triage-count
%  change. Upgrades the Post-hoc-fit defence section (Summary Part) to the
%  full forcing-vs-searching / Euclidean standard, mirroring A9-SM v20.1
%  subsec:forcing-vs-searching. Adds one labelled subsection
%  (subsec:forcing-vs-searching-cosmos) with the six moves adapted to
%  cosmology: one-ledger-over-constraint + cross-observable rigidity
%  (J=7, K=35); form forced by the stratum; MSMF/OGN as uniqueness selector
%  (printed costs); the named-but-uncomputed global trials arbiter;
%  forward falsifiable commitments (F-Cosmos-1..5); the Euclidean capstone;
%  and the boxed 'Forcing, not searching' euclidremark thesis. The section
%  intro now frames continuous (D1-D4) then discrete. v12.7 -> v12.8.
% =====================================================================
% =====================================================================
% ===== v12.7 (2026-06-28): TRIAGE + CONJECTURE-COUNT RECONCILIATION. =====
%  Presentation only; NO physics, NO parameter values, NO triage-count
%  change. Reconciles the remaining stale summary spots to the canonical
%  codebook (already two Derived / fifteen Derived-cond. / three Asserted
%  over twenty observables, ten named conjectures), per author-approved Ship 1:
%   - Appendix glossary counting summary: corrected to twenty observables
%     and ten conjectures closing all eighteen D*/A entries.
%   - Closure-economy summary: D* word-count and conjecture range corrected.
%   - Named-statement registry: added the missing tenth-conjecture row
%     (Lithium-port-pi-suppression, conj:lithium-port, already fully stated
%     in the body at sec:audit-d-star-cosmos) and bumped the inventory line.
%  No reader-facing count now disagrees with the body. v12.6 -> v12.7.
% =====================================================================
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% ============================================================================
% A10-Cosmos v12.5 (2026-06-27): mechanical presentation pass. No physics/values changed.
% (1) De-contamination: shed GG's OWN self-description (the cohort's six-dimensional bulk noun,
%     its scale-profile / scale-overlap / scale-depth vocabulary, and the bulk-to-boundary
%     projection language), in body + figure/flow node labels + PDF metadata + title page.
%     KEPT verbatim: the entire Randall--Sundrum-vs-GG comparison block, table row, and glossary
%     cell (their comparison vocabulary), the AdS/CFT and Anti-de Sitter framework names, the
%     named external principle/bound/literature/renormalization scheme, the QiHAL Attractor-Lattice
%     acronym glosses, all citation contexts (Maldacena/de Boer/Susskind/Witten), the RG-running
%     dictionary-disclaimer sentence, the slogan objection/response block, and all label/ref args.
% (2) Light energy-currency reduction (see the atp_reduce script): trimmed a per-molecule energy-currency
%     estimate to an upstream pointer; renamed one descriptive adjective to 'energy--information'.
%     KEPT the Q_B constant, its value, the molar relation, and the Arisaka four-current equation.
% (3) Bibliography reconciliation (see the cleanup script): de-contaminated GG-self cohort-citation
%     titles + relevance prose; external bibitems untouched.
% ============================================================================
% ===== A10-Cosmos v12.4 (2026-06-25) =====
% Central-claim lighter touches (parallel to A9): (1) Executive Summary 'central recognition'
% sentence pointing to sec:central-claim / fig:central-claim; (2) Conclusions call-back paragraph
% ('the deepest result is the shift in what a cosmological parameter is'); (3) title-page subtitle
% recast to the cohort-parallel 'The Cosmic Inventory as Observations of One Six-Dimensional
% Geometry'. No new physics.
% ===== A10-Cosmos v12.3 (2026-06-25) =====
% Central-claim foregrounding, cohort-consistent with A9-SM: inserted a new section 'The central
% claim: the cosmic inventory consists of open-system observables' (sec:central-claim) between
% sec:gg-from-zero and sec:paper-claims. Five beats (open system -- the universe most of all / OGGEP
% projection = observation / GDOS law / the cosmic inventory are registered observables, no
% exceptions / GDOS forced by Gi-4 + Qi-4) + the funnel figure (fig:central-claim) adapted so the
% payload is the cosmic inventory (Omega_m=1/pi, Omega_L=1-1/pi, Omega_b=1/2pi^2, z_*=e^7, Kaxion),
% + the forty-year/century historical claim. Same funnel as A9, payload one storey up. No new physics.
% ===== A10-Cosmos v12.2 (2026-06-25) =====
% Author-requested: added A1-Constitution's canonical Figure 9 (fig:abbrev-flow, the programme-acronym
% hierarchy), copied verbatim from the A1 source, to the end of the 'Summary of Abbreviations'
% appendix, so the acronym ledger is identical across the cohort. No physics changed.
% ===== A10-Cosmos v12.1 (2026-06-24) =====
% delta_PMNS RIPPLE from A9 v11.29: A9 now marks the leptonic Dirac phase
%   Derived (cond.) = 196 deg, NOT Asserted. Two A10 sites corrected:
%   (1) codebook-comparison sentence 6/13/9 -> 6/14/8 (drops 'and the PMNS Dirac phase'
%       from the asserted list; 'nine of 28' -> 'eight of 28');
%   (2) leptogenesis CP-provenance note: 'delta_PMNS is an Asserted entry there' ->
%       all CP data derived (Dirac phase Derived cond. at 196 deg). No other physics changed.
% v12.0 (June 24, 2026): arXiv-ready MILESTONE promotion of v11.64. Science frozen;
%   this concludes the presentation round (front matter + Parts I-XV uplifted, constructive
%   appendices E-N on the three-step template, Part XV consistency pass, and the corpus-audit
%   concrete cross-paper fixes A5 v11.10 / A10 v11.63-64). PaperVersion corrected from the
%   stale v11.38 stamp to v12.0; status 'Preprint Draft' -> 'Preprint'.
% v11.64 (June 24, 2026): corpus-audit cross-paper follow-ups. (1) CP phases: corrected
%   the v11.63 over-statement -- A9 DERIVES alpha_31 + the PMNS angles; only delta_PMNS is
%   Asserted in A9. (2) G_N: aligned Appendix-E Step-3b (+ nav + summary) to the A2-canonical
%   framing (G_N the unit, M_6 the derived combination; geometry fixes the warp/hierarchy,
%   not G_N's value). Matches A5 v11.10. No values changed.
% v11.63 (June 24, 2026): concrete corpus-audit corrections (factual, no physics/status
%   changes): (1) Appendix-L Step-4 false '=1/pi' display recast to exhibit the ~42x gap;
%   (2) dangling app:source-ledger-7f pointer -> appA:pilock-steps; (3) z*=e^J credited to
%   the present paper using A4's integer J=7; (4) OGN-5 'theorem of A4' -> upstream A2/A3,
%   audited in A4; (5) CP phases 'A9 derived' -> carried by A9 as asserted inputs; (6) G_N
%   'rather than inserted by hand' -> warp integral relates 4D G_N to bulk scale (A2: G_N is
%   the unit); (7) T-Cosmos-17 'topological invariant' -> 'residual Pontryagin normalisation,
%   conditional'; (8) removed the false sin^2thetaW=25/108 'same integration' analogy.
% v11.62 (June 24, 2026): Part XV consistency pass -- lithium-7 objections aligned to the
%   closed-packet 1/pi port (body + appendices H/I). O-32 question reframed + response
%   rewritten to 1/pi/D*/Conj VIII (selectivity+network+detailed-balance deferred to
%   GG-C4-Nucleo); O-33 response updated to 1/pi port (detailed-balance kept open); O-35
%   'Conjectures I--III' -> 'three named conjectures'. Other 63 objections unchanged.
% v11.61 (June 24, 2026): Appendices L (pi-lock) + M (Kaxion) + N (BH entropy) three-step.
%   L: nav + NEW Step-2 (coincidence problem) + Step-3 framed as 'geometric PROPOSAL' (not
%   theorem); full honest body (D*/Conj VIII, 'not a theorem') kept. M: nav + Step-2 + retitle
%   'geometric derivation: a genuine inversion'. N: nav + Step-2 + retitle 'geometric
%   derivation: A/4 from the canonical warp'. 'reading' dropped from M/N titles. No values changed.
% v11.60 (June 24, 2026): Appendices J (growth) + K (CMB) three-step template (light).
%   J: nav opening + NEW Step-2 (S8 tension) + Step-3 retitled 'geometric derivation' with
%   lead-from-front; sign-OPEN body kept. K: nav opening (Step-2=six-param joint fit) +
%   Step-3 retitled + lead reframe; 'no full C_ell yet / l_D=K(2pi)^2' honest body kept.
% v11.59 (June 24, 2026): Appendices H (BBN) + I (leptogenesis) three-step template.
%   H: nav opening; Step-3 retitled 'geometric derivation' and CONSISTENCY-ALIGNED to the
%   canonical closed-packet 1/pi port (Li-7=SBBN/pi, D*, Conj VIII) from companion I --
%   drops stale 'one tuned coefficient' framing; honest deferrals to GG-C4-Nucleo kept;
%   Status updated. I: light nav reframe (derivation appendix supplying eta_B). No new physics.
% v11.58 (June 24, 2026): Appendix F (thermal history) three-step template. Nav opening;
%   NEW Step-2 limitations; reorder to Step1->worked->Step2->Step3->Status; Step-3 retitled
%   'cosmic clock as a GDOS descent' with expanded GDOS/GRIP-Out lead prepended to the kept
%   z*=e^7 honest core (both framings verbatim). Labels/eqs preserved; no values changed.
% v11.57 (June 24, 2026): Appendix G (inflation) three-step template, honesty-careful.
%   New nav opening; NEW Step-2 'What standard inflation leaves open'; Step-3 'The geometric
%   derivation: inflation as the prototype GRIP descent' (replaces gap+reading+status),
%   leading with derived parts (modulus, computed TeV scale, GRIP/GSSB, r<<1e-3) and keeping
%   the A_s/n_s ASSERTED gap (Conj IV/V) + ~50-order normalisation tension prominent.
%   'reading' dropped; eq:r-null preserved; no values changed.
% v11.56 (June 24, 2026): Appendix E (Friedmann) PILOT of the three-step appendix
%   restructure -- (1) standard derivation kept as Step 1; (2) NEW Step-2 subsection
%   'What the standard derivation assumes -- and cannot explain'; (3) GG subsection
%   rewritten as Step-3 'The geometric derivation: ... theorem of GDOS', leading from the
%   front, servile hedging removed, honesty kept (inventory conditional; local Einstein-eq
%   derivation = A5's remit, stated once). New nav opening + Euclid/Newton thesis. No
%   'revolution' wording. Equations/labels preserved; appA:frw-status (0 refs) folded.
% v11.55 (June 24, 2026): document-wide sans-serif sweep of programme-defined names --
%   converted bare prose/title/table/caption literals (GG-6, GG-Theory, GG-Constitution,
%   GDOS, GRIP, OGGEP, GGEP, GFF, GSSB, GA-stages, MSMF, OGN/OGN-5, BI-6, Gi-4/Qi-4, TGG,
%   CCI, GEP, CEP, DGI, SIP, SeOS, EIC) to their sans-serif macros. Guarded: skipped
%   comments, tikzpicture node-IDs, label/ref/cite args, existing textsf, and hyphen-
%   compound IDs (T-GRIP-In, GRIP-In/Out kept as-is; no GRIPin/out macro exists). 437
%   lines, ~636 macro insertions. No words/values changed; rendering -> sans-serif only.
% v11.54 (June 24, 2026): Part XIV Technical Appendices -- adopted the A9-SM reference
%   block (author request). Added the A9-style front-cover intro (appendix groups + two
%   stylistic commitments). Replaced the thin notation+acronym dictionaries with A9's
%   'Summary of Abbreviations' (App A) + 'Glossary of Key Symbols' (App B), adapted to
%   cosmology; cohort-universal rows copied verbatim, which corrects five stale acronym
%   expansions (UEP/CP/OGN/MF/QiHAL) to the A1/A9 canonical forms. No values changed.
% v11.53a (June 24, 2026): split the Section-72 staircase box onto two centered lines
%   (engine chain / cosmic consequences) so it fits the text width. Wording unchanged.
% v11.53 (June 24, 2026): Part XIII (Conclusions) -- accessibility + philosophy pass
%   (author request). NEW plain-English section 'One staircase, from the Big Bang to
%   living molecules' with a boxed plain chain display and the E_GUT->atoms->galaxies->
%   stars->molecules->us walk; rewrote the closing human section as the geometric-
%   inevitability payoff (fine-tuning dissolved: well designed by geometry, not a
%   designer; no luck/multiverse; GDOS observer-independent); full life/mind case still
%   deferred to A11. One plain gloss for the Kaxion. No physics/values/labels changed.
% v11.52 (June 24, 2026): List of Figures overflowed page 5 onto page 6 (Fig 19
%   alone). Shortened the optional short-caption (LoF text) of three two-line
%   entries -- the cosmic-history timeline, the comoving-Hubble-radius curve, and
%   the kill-test timeline -- to one line each; full caption bodies unchanged so
%   the block fits back onto a single page 5. Science frozen; values unchanged.
% v11.51 (June 23, 2026): HYBRID relocation of the cohort naming-convention section
%   (sec:naming-convention), author-approved. (1) Moved it out of the Part XII Discussions
%   climax (it was the trailing subsection of the Ten Innovations section) to the FRONT of
%   the Technical Appendices, grouped with the reference dictionaries (after the Codebook
%   glossary, before the Friedmann appendix); promoted \subsection -> \section, label
%   preserved. (2) Added a one-line POINTER in Part I 'how to read this paper' so the
%   reader meets the scheme early. Both prior \S\ref cross-refs + the new pointer resolve
%   to the appendix table. Science frozen.
% v11.50 (June 23, 2026): Part XII (Discussions) -- GRIP chain made explicit as the essence
%   of GG-Theory (author request): new lead subsection 'OGGEP(AX-3)->GDOS->GRIP->GGA' with a
%   boxed display, the one-engine-two-strata point (SM A9 + LambdaCDM A10), and a Figure-1
%   cross-ref; the five distinctions reframed as facets. Ten-Innovations intro names the
%   chain once. S8 'sign open' -> 'uniqueness open (Conjecture II)'. Reconciled the 14 stale
%   Discussions partVII- labels -> partXII- (Part VII black-hole eq:partVII-* untouched).
% v11.49 (June 23, 2026): added the Pi-locked Chart of Omega to the FRONT MATTER as the
%   new Figure 1 (author request) -- right after Keywords, before Overview of Contents, on
%   its own vertically-centred page as a reader hook (fig:pi-locked-chart-omega-hook). Reuses
%   the exact Part X pie TikZ with a front-matter hook caption; the Part X copy stays in
%   context with Theorem T-Cosmos-17. All later figure numbers shift +1. Science frozen.
% v11.48 (June 23, 2026): converted the Part X horizontal pi-locked-inventory band
%   (fig:cosmic-inventory-band) into a clean pie chart, 'Pi-locked Chart of Omega'
%   (fig:pi-locked-chart-omega) -- author request. Three slices, each one closed-form
%   power-of-pi fraction (prediction) with the Planck 2018 value beneath, written just
%   OUTSIDE its slice (no leader lines), larger fonts; matter slices sum to Omega_m=1/pi.
%   No \ref depended on the old label. Science frozen; values unchanged.
% v11.47 (June 23, 2026): emphasized GRIP as the heart of Part X (Topological completion),
%   per author request (geometrical / deterministic / inevitable origin of the cosmic
%   inventory). New prominent paragraph in the Part-X Overview: the deep geometric origin
%   is one runtime acting once -- the GRIP cascade DGI->GSSB->GA reaches the unique
%   Geometric Attractor deterministically, with no branch point free, so the pi-locked
%   inventory is geometrically inevitable rather than fitted; the three questions of the
%   Part are three faces of this one statement. Kept honest: the single open step (whether
%   the GRIP attractor projects to exactly 1/pi) is named as Conjecture VIII -- 'not a
%   missing parameter, but a missing integral'. Science frozen; triage unchanged.
% v11.46 (June 23, 2026): applied the GRIP organizing concept to Part XI (Summary), as
%   in Parts VI-X (author request: 'GRIP as the heart of any observed quantity, incl. the
%   listed 20'). (1) New prominent paragraph in the Summary Overview: every codebook
%   observable is the present-epoch projection of one Geometric Attractor selected by the
%   GRIP cascade DGI->GSSB->GA of A6; the MSMF/OGN cost-minimum IS that GRIP-selected
%   attractor; the D/D*/A triage is a statement about GRIP-uniqueness (proved / conditional
%   / not-yet); GRIP-Out switches (inflation exit, recombination z_*=e^7) fix initial data
%   and GRIP-In relaxations carry the universe to the present pi-locked attractor. (2) One
%   reinforcing sentence in the compressed-codebook-summary intro. Science frozen; triage
%   unchanged. (Cumulative with v11.45: the Summary cohort-in-one-view figure swap.)
% v11.45 (June 23, 2026): Part XI (Summary) figure swap + 2 bug fixes (author-approved).
%   (1) Replaced the Summary's A10-centric 'Programme-acronym hierarchy' figure (its title
%   was a misnomer -- it showed papers + inheritance arrows, not acronyms -- and it hard-
%   coded a stale 'A6/Gi4' node) with the far more instructive 17-paper cohort-in-one-view
%   map (the front-matter fig:cohort-map-a10), reused verbatim as a full-page Summary recap
%   with label fig:cohort-one-view-summary. (2) Fixed the stale '...is reserved for future
%   work' sentence (the figure exists). (3) Hard-coded '(Appendix H)' -> \ref{app:
%   falsification-catalogue} (the falsification catalogue is Appendix S, not H).
%   Science frozen; honesty triage unchanged.
% v11.44 (June 23, 2026): closed two long-standing deferred items (author-approved).
%   (1) \Alph appendix-section overflow: the Technical Appendices proper has 19 sections
%   (A-S), but the 8 referee sections of the Common Objections Part inherited the same
%   \appendix lettering and pushed the count to 27 > Z, triggering 2 recoverable 'Counter
%   too large' notices (the 27th section + the ToC). Fix: the Common Objections Part is a
%   Q&A Part, not a technical appendix, so it now carries its own O\arabic{} section
%   numbering (O1-O8); the appendix proper keeps A-S and no longer overflows. No referee
%   section is cross-referenced by number, so the change is label-safe. (2) The lone
%   hostile-reviewer quote '$\Mlock \sim 4$ TeV' -> canonical '$\Mlock \sim 3.85$ TeV'.
%   Science frozen; honesty triage unchanged.
% v11.43 (June 23, 2026): Part X (Topological completion) honesty + consistency pass
%   (author-approved). (1) Theorem T-Cosmos-17 (Omega_m=1/pi) Step 4 rewritten: the
%   printed equality chain asserted the OGN-5 factors 'combine to give 15' and '=1/pi',
%   but (K/J)(L/N)(1/N)=25/9, not 15 -- a false equality under the honest 'does not
%   close' note. Replaced with an honest iff-cancellation statement that exposes the gap
%   (keeps the structural target + D* status, removes the referee magnet). (2) Registered
%   T-Cosmos-17 in the named-theorem deck (1 Theorem -> 2; inventory 14 -> 15). (3) Fixed
%   the Part XI Summary overclaim ('proves Omega_m=1/pi ... integral output of a fixed
%   topological computation') to the honest register matching Part X. (4) ATP hydrolysis
%   free energy -30.5 -> canonical -30.0 kJ/mol (A11 / Q_B-ATP). Science frozen; triage
%   unchanged (Omega_m/Omega_L/Omega_b stay D* under Conjecture VIII).
% v11.42 (June 23, 2026): Part IX (Cosmological codebook) consistency pass after the
%   VI-VIII revisions (author-approved). NO triage-count change (stays 2 D / 15 D* / 3 A;
%   no lemma was proved -- the corridor is D* on the still-open uniqueness conjecture, and
%   C1 itself keeps the Kaxion mass conjectural). (1) Fixed the stale DeltaNeff note: the
%   pre-recombination 0.30 is the demoted Part VI legacy EDE estimate (now a CMB-appendix
%   cross-check), and 0.04 is the canonical value. (2) Conjecture II switch z_*~10^3 ->
%   z_*=e^J=e^7~1097. (3) Stale Part cross-refs reconciled: PBH/GW Part V -> VII; JWST
%   Part IV -> VI; V_inf inflation Part II -> IV; Omega_L residual-FGFF Part II -> IV.
%   (4) Sharpened the Kaxion D* note to credit C1's executed derivation with the discrete
%   Xi_cap in [35,47] remainder. Science frozen; honesty triage unchanged.
% v11.41 (June 23, 2026): Part VIII (Cosmic chronology) -- GRIP framing + production
%   pass (author-approved). (1) New GRIP opener: r=ln(EGUT/mu) named as the GRIP-In
%   descent coordinate; inflation exit + recombination as GRIP-Out events; de Sitter
%   future as the Ground Geometric Attractor. (2) Label rename eq/sec/tab:partVIII- ->
%   partVIII- (dash-delimited; partVII/partVIII untouched, asserted). (3) Content-mapped
%   Part-number fixes (8): thermal-history 'Part II'->III, ToC-7 'Part II'->IV, self-refs
%   'Part VI'->VIII, 'Parts II through V'->VII. (4) Deleted drafting artifacts ('the
%   uploaded chronology notes'; 'the user's requested ...' x4 + section title). (5) M_lock
%   '4 TeV' -> canonical 3.85 TeV in the master + radial-bookkeeping tables. (6) z_rec tied
%   to the integer-ledger prediction z_*=e^7~1097 (milestone, master table, Example C).
%   (7) Timeline figure r-ticks reconciled to the body/master-table values (REC 59.7, LSS
%   65.8, DE 66.5, NOW 66.7). Science frozen; honesty triage unchanged.
% v11.40 (June 23, 2026): Part VII (black holes + relics) -- GRIP framing + C3-BH
%   consistency pass (author-approved). (1) New Part-opener: black hole as a GRIP-In
%   way-station attractor, Hawking radiation as its GRIP-In exit; division of labour
%   with the dedicated companion C3-BH. (2) Body wired to \citep{ArisakaC3BH}:
%   replaced the two 'a later/separate dedicated gravity paper' phrases (C3-BH exists,
%   v11.2). (3) Added the PBH integer-ledger mass ladder M ~ e^{-7n} (same J=7 as
%   z_*=e^7), credited to C3-BH F-BH-4. (4) M_lock 4 TeV -> canonical 3.85 TeV in the
%   tensor section. (5) Stale Part-number reconciliation (content-mapped, 17 prose
%   fixes; Section 19 -> unnumbered). (6) eq:partVII-/sec:partvii- labels renamed to
%   partVII (dash-delimited, partVI/partVII/partVIII untouched). (7) Worked-derivation
%   notebook relocated to the Technical Appendices (appA:bhentropy-notebook).
%   Science frozen; honesty triage unchanged.
% v11.39 (June 23, 2026): Part VI re-founded on the canonical z_*=e^7 two-stage corridor
%   (author-approved, Option 1). (1) New Part-opener "One switch, two faces": GRIP-Out switch
%   at recombination (z_*=e^7) + GRIP-In growth as the single organizing mechanism, with a
%   headline falsifiable-numbers table. (2) Hubble section rewritten around the corridor:
%   retired the rival EDE-form theorem (was tagged T-Cosmos-10) and wired the body to the
%   canonical Theorem 4.D.1 (thm:hubble-two-stage-corridor) + Corollaries 1/2/3; the two-stage
%   corridor figure relocated from Part IX into Part VI Sec 2. (3) Delta N_eff standardized to
%   the audit value 0.04 in the body; the EDE/Delta-N_eff/baryonic-void linear-response and the
%   local-void open problem demoted to the CMB appendix as a non-unique legacy estimate.
%   (4) S_8 section wired to Corollary 2; JWST section kept as GRIP-In high-nu tail. (5) The two
%   inline technical sections (cross-consistency Jacobian/sign-matrix; worked derivation
%   notebook A-G) and the response-coefficient machinery moved to the Technical Appendices
%   (growth-of-structure and CMB sections), with body pointers. (6) Stale Part-IV labels
%   (p_IV/o_IV, *-part6, "Role in Part IV", sec:part6-*) reconciled to Part VI; the closing
%   bridge "Part V" -> "Part VII". Science frozen; honesty triage counts unchanged.
% v11.38 (June 23, 2026): (1) expanded the Table 2 (preview data table, p21) caption to fill
%   its full page, mirroring Table 36; (2) enlarged the boxed-legend font in the Dev plot
%   (Fig 1 / Fig 8) from 7.6 to 9.6pt (regenerated PDF).
% v11.37 (June 23, 2026): expanded ONLY the Table 36 (dev-plot data) caption to fill its
%   full page (column-by-column + status/conventions description); Figure 8 left unchanged.
% v11.36 (June 23, 2026): Fig.1/Fig.8 + Part I/IX refinements -- (1) doubled the Sec 4 preview
%   prose with concrete LCDM-vs-GG examples; (2) added vertical gap before Table 2; (3) restored
%   the 7-entry boxed legend in the Dev plot (regenerated PDF); (4) de-orphaned Figure 7
%   (provenance map) with an in-text reference; (5) widened the 'First-order derivation source'
%   column and reduced the codebook longtable font to footnotesize (fills the sparse page).
% v11.35 (June 23, 2026): (1) moved the results preview (Dev plot + data table) from Part I
%   Sec 5 to the end of Sec 4 'What this paper claims', TABLE before FIGURE; (2) bolded the
%   7Li/H entry label in the Dev plot (regenerated PDF -> fixes both the preview and Part IX);
%   (3) Part IX: moved Table (dev-plot data) in front of Figure (dev plot), packed together.
% v11.34 (June 23, 2026): added a results PREVIEW to Part I Sec 5 (how to read) -- the
%   parameter-audit Dev plot + its data table reproduced up front as Figure (fig:dev-plot-preview)
%   and Table (tab:dev-plot-data-preview), mirroring the A9-SM preview-of-predictions hook;
%   canonical versions remain in Part IX. Distinct -preview labels; cross-referenced both ways.
% v11.33 (June 23, 2026): Dev-plot (Fig.7) refinements -- vector PDF (crisp text);
%   x-range +-0.06 -> +-0.1; plot region widened; sigma_th annotations moved inside the
%   panel; legend slimmed to 5 entries (no data overlap); figure caption shortened ~5x.
% v11.32 (June 23, 2026): document-wide production-scaffolding sweep -- removed all
%   reader-facing workflow/process leaks: 'Phase 4E preview'/'Ultra-3 version' framing
%   (Part X universal-cosmic-identity section reframed as finished), 'the ship', 'eleven-phase
%   workflow', 'sub-agent/auditor/reference-scanner', 'per L46/L49' convention codes, 'Phase 2 L5'
%   cross-refs, empty '(, May)' version fields. Wiped the future-volume codenames 'Ultra-3' and
%   'BookI' (capstone book): removed BookI from the coherence-map figure + prose + counts
%   (5 downstream -> 4) and deleted its bibitem; removed the 'What is deferred to Phase 7/Ultra-3'
%   subsection. Kept legit cohort term 'Ultra Eight'. Fixed stale inline count 2/14/3->2/15/3, 19->20.
% v11.31 (June 23, 2026): Part IX codebook update -- (1) re-stated Lemma VI eta_B to
%   resonant leptogenesis (UV completion of A9 Weinberg operator; on-resonance condition);
%   (2) added Li-7 abundance as 20th codebook entry under new Lemma X (lithium-port pi-suppression),
%   counts 19->20, lemmas 9->10, all mirror tables/glossary/front-matter reconciled;
%   (3) stripped Phase-4C/4D/6/8 + 'legacy 6F' + self-ref scaffolding from codebook surfaces;
%   (4) re-wired provenance map (eta_B from A9 ledger; added Li-7 node); regenerated dev plot (17 entries).
%% =====================================================================
%%  A10-Cosmos v11.3 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  Routed inline Overview+LoF onto gg-11's
%%  centered cohort macros (single page via toggle); stripped centered
%%  ExecutiveSummary override; dropped dead \LaTeX shim; Preface via
%%  \GGPreface; inherits the one canonical Acknowledgments.  No
%%  prose/figure/number content changed.
%% =====================================================================
%% ---------------------------------------------------------------------
%%  Arisaka-GG-A10-Cosmos-v11.2 (restatement-pair merge) -- 2026-06-20
%%  Untagged 2 restatements (Cosmological-inheritance 'recast form'; the
%%  Tensor-kill-corridor PROPOSITION preview); kept their principals.
%%  Renumbered T-Cosmos-1..17 contiguously. (Hubble formula vs existence
%%  theorem kept as distinct results.)  No physics changed.
%% ---------------------------------------------------------------------
% =====================================================================
%  Arisaka-GG-A10-Cosmos-v11.0 (Stage 3A: naming-convention theorem-id rollout) -- 2026-06-20
%  Official cohort theorem ids assigned: T-Cosmos-1..19 (19).
%  Appended (T-Cosmos-N) to own theorem-env titles; recalled classical
%  theorems left untagged.  No physics changed.
% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2. =====
%  Terminology only; NO physics, NO triage counts changed (2/14/3 observables;
%  3R/10P/0G problems). A2 bibitem AdS-free; A3 bibitem '...Warped AdS Bulk' ->
%  '...Warped Bulk'; Maldacena 'anti-de Sitter' citation title KEPT. Added the
%  holographic-RG disclaimer footnote; v10.12 -> v11.0.
% =====================================================================
% ===== v10.12 (2026-06-16): FRONT-MATTER REFINEMENT (author Go; same procedure as A9-SM). =====
%  Presentation only; no physics, no parameter values, no triage counts changed
%  (triage stays 2 Derived / 14 Derived(cond.) / 3 Asserted = 19 observables).
%   - New plain-language title + subtitle (author-approved); subtitle preposition
%     'Determined from' -> 'Derived from'.
%   - Running header suppressed on the front cover (thispagestyle empty), per A11.
%   - Philosophical-conclusion paragraph of the Executive Summary roughly doubled.
%   - Abstract rewritten for a broad audience: plain language, no citations; page 1.
%   - Executive Summary: centered \LARGE title; added a contrast-with-LCDM/multiverse
%     paragraph and a philosophical-conclusion close.
%   - NEW Preface (centered \LARGE title), broad-audience introduction.
%   - Added Overview of Contents + List of Figures on one page; short \caption[...]
%     forms for four figures; Part toc lines shortened to one line each.
%   - Lithium-7 reclassified G -> P (deferred to the forthcoming GG-C4-Nucleo);
%     13-problem L6 decomposition 3R/9P/1G -> 3R/10P/0G; 'genuinely open'
%     removed as a live classification. The 19-observable 2/14/3 triage is unchanged.
%   - Version/date stamps -> v10.12 / June 16, 2026.
% =====================================================================
% --- v10.11 (2026-06-16): Part-aware running header (matches A11-Origin v12.27/28).
%     Right-side version/date stamp replaced by a live 'Part N. <short>' pointer;
%     Executive Summary set their own front-matter marks. Header-only; no content edits. ---
% GU-5 RENAME (June 14, 2026): cohort harmonization with GG-A4-Euclid -- the five-constant inventory and its theorem are named Geometric Universal Five Constants / GU-5 (\GUfive, \T{-GU5}); the older 'Conversion-Factor Pentad'/'Conversion-Factor Triad' wording and the 'T-CFP' theorem tag are retired.
\pdfoutput=1
%% v10.10 (June 12, 2026): Legacy-name fix (author ruling: GD-2026-5
%%   = GG-A3-BI6): the ArisakaA3BI6 bibitem updated from the legacy
%%   GD-2026-5 name/title to the canonical A3-BI6 form. No other
%%   changes.
%% v10.9 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-a10, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.7 (June 10, 2026): C1-A10 consistency audit fix (one wording):
%%   the C1-Kaxion bibitem mislabelled the bulk normalisation scale as
%%   f_K; corrected to distinguish f_bulk = (3.0+-0.3)e13 GeV from the
%%   single physical decay constant f_K = f_bulk/43 ~= 7.1e11 GeV,
%%   matching both papers' body text. No other changes.
%% v10.6 (June 10, 2026): C2-synchronization precision pass (no
%%   physics changes). The Part-IV anchor-convention sentence is
%%   updated from the future-tense 'to be unified at the C2
%%   synchronization pass' to the completed statement (synchronized
%%   June 10, 2026); the companion paper is now cited formally via a
%%   new bibitem ArisakaC2Hubble carrying its final title ("The
%%   Hubble Tension in the Light of GG-Theory"), at the eps_local
%%   open-problem note, the anchor-convention sentence, and the
%%   Dev-plot caption.
%% v10.5 (June 10, 2026): follow-up to the Tier-2 revision. The stale
%%   'Status note' after Theorem thm:omega-m-pi-topological (which still
%%   claimed structural closure of a codebook conjecture and 1-sigma
%%   numerical verification) is rewritten to match the Proof-status note
%%   and the Conjecture-VIII reclassification: conditional statement only,
%%   no codebook entry closed. No other changes.
%% v10.4 (June 10, 2026): Tier-2 hostile-referee revision (honest
%%   reclassification). Omega_m/Omega_Lambda/Omega_b -> D* (new Conjecture
%%   VIII closed-packet-integration); w -> D* (new Conjecture IX vacuum-
%%   residual-exactness); codebook now 2 D / 14 D* / 3 A, nine named
%%   conjectures; corridor relabeled phenomenological (EDE-literature
%%   coefficients cited: Poulin et al. 2019); Hill et al. 2020 S8 caveat;
%%   eta_B demoted to order-of-magnitude (sigma_th factor ~3); H0^late
%%   corridor re-anchored to this paper's 67.4 -> 73.7 +/- 0.7; problem
%%   rows 8/9 Resolved -> Partially resolved; two-route Omega_b tension
%%   and CMB-branch declaration added as named open problems.
%%   Per A10-HOSTILE-REFEREE-REPORT-2026-06-10.md, Tier 2.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   Amended June 10, 2026 (Tier-1 hostile-referee fixes, author-approved):
%%   S8 corollary sign corrected (higher -> lower, cor:s8-reconciliation);
%%   H0^late = 73.0 +/- 1.0 re-cited to Riess et al. 2022 (SH0ES; new bibitem)
%%   at six sites (1998/1999 acceleration papers retained for Lambda history);
%%   phantom pointers repaired (Omega_m source -> Theorem in Part X of this
%%   paper, not 'A2 Part VIII'; 'e-fold-separation theorem of A4' -> integer-
%%   ledger identification structural to this paper); false S^3/B^4 volume
%%   ratio corrected to Vol(S^3) = 2pi^2; z* = e^J declared canonical (e^7-1
%%   variant noted, not independent); recombination-window tolerance (+/-40
%%   in z) now defined in the body and used consistently in the falsifier
%%   list and catalogue (replaces the contradictory +/-5 refutation band);
%%   0.5% -> 0.46%; ell_D equality form unified to 1381.7.
%%   See [GG-v10]/A10-HOSTILE-REFEREE-REPORT-2026-06-10.md (Tier 1).
%%   Amended June 10, 2026 (2nd content-refinement amendment, author-directed):
%%   Dev plot expanded 12 -> 16 entries (adds eta_B, w, ell_D, S8^late; S8 row
%%   split into S8^CMB anchored + S8^late predicted, mirroring the H0 pair);
%%   figure now full-page [p]; honest-epistemics markers (filled = independent
%%   prediction, open = anchored); theory tolerances added (eta_B O(20%),
%%   H0^late corridor 73.5+-0.7 per C2-Hubble, z* recombination-window +-3.7%,
%%   S8^late 0.78+-0.01); new data table tab:dev-plot-data; observed-value
%%   citations added for w, eta_B, ell_D (VERIFY: ell_D obs 1345+-15 is
%%   definition-dependent), S8^late (new bibitem Asgari2021/KiDS-1000);
%%   codebook 15 -> 19 rows; Omega_DM sigma agreement corrected 0.7 -> 0.27.
%%   Generator: outputs/a10_dev_plots_v3.py.
%%   Amended June 10, 2026 (author-directed content refinement): Figure 1
%%   (parameter-audit Dev plot) replaced by a linear-axis version with
%%   pred/obs values in the row labels (A9-SM Fig 2 style); radiation row
%%   recast as N_eff = 3.046 + DeltaNeff to avoid the near-zero-denominator
%%   artifact; caption updated; label renamed fig:a10-dev-plot-linear.
%%   Generator: outputs/a10_dev_plots.py; asset a10_dev_plot_linear.png.
%%   bibliography reformatted to bold-author + Relevance convention.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
% ----------------------------------------------------------------------
% Arisaka-GG-A10-Cosmos
% Geometric Origin of LCDM and the pi-locked Cosmic Inventory:
% Derivation of the Cosmological Parameter Sheet from the OGN-5
% Diophantine Ledger and the \GUfive{}
% Author: Katsushi Arisaka (UCLA)
%
% PROVENANCE.  This file is the v5.0 Phase 1 (identity-migration) of
% the A10-Cosmos paper.  It is built mechanically from the v1.1
% BI-6/TGG/CCI/GFF/GRIP recast source
%   UCLA-Arisaka-GG-2026-7F-recast.tex   (April 15, 2026, 299 pp)
% via the Phase-1 migration script
%   outputs/a10_v50_phase1_migration.py  (May 14, 2026).
%
% The Phase-1 transformations are MECHANICAL only:
%   * Style upgrade  ucla-arisaka-gg  ->  ucla-arisaka-gg-7         (L1, L52)
%   * Identity block + hypersetup rewrite                           (L2, L15)
%   * Local-macro purge: ~25 macros now provided in canonical
%     sans-serif by ucla-arisaka-gg-7.sty                           (L21)
%   * NO per-paper paper-code macro block: the v7 style provides
%     the 26 sans-serif paper-code macros directly                  (L52)
%   * Bibliography mechanism preserved as-is (legacy inline
%     \begin{thebibliography}); Phase 10 reconciles with the
%     UCLA-Arisaka-GG-7.bib master                                  (L10)
%   * Manual Acknowledgment section replaced with the canonical
%     \GGAcknowledgments macro, placed after the last Appendix
%     and before the bibliography                                   (L43)
%
% NO SUBSTANTIVE CONTENT EDITS in Phase 1.  Theorem statements,
% section text, and prose are byte-for-byte preserved.  The eleven
% remaining phases plus the three Ultra-passes apply per
% PROJECT-INSTRUCTIONS-v5.0-CONSOLIDATED.md.
%
% PHASE LOG.
%   Phase 1   (May 14, 2026): identity migration; this file.
%   Phase 2   (May 14, 2026): GS-as-GGEP framing audit (L5, light;
%                               A10 inherits BI-6 from A3-BI6).
%                               Three patches: (1) seven-paper
%                               inheritance table augmented with
%                               A3-BI6 ownership tag on the BI-6
%                               and anomaly-freedom rows; (2) first
%                               BI-6 stage paragraph carries an L5
%                               framing remark naming BI-6 as the
%                               constitutional generalisation of the
%                               AX-2 GGEP; (3) first Green-Schwarz
%                               language mention cites the BI-6
%                               paper as the constitutional anchor.
%   Phase 3   (May 14, 2026): L4 historical-background audit.
%                               Inserted 10 \subsection{Historical
%                               background} openings at named-section
%                               anchors: beginning at EGUT (Einstein
%                               1917 / Friedmann 1922 / Lemaitre 1927);
%                               expansion-and-cooling (Friedmann 1922
%                               / Hubble 1929); inflation (Guth 1981
%                               / Linde 1982 / Albrecht-Steinhardt
%                               1982); dark energy (Einstein 1917 /
%                               Weinberg 1989 / Riess 1998 /
%                               Perlmutter 1999); Kaxion (Peccei-Quinn
%                               1977 / Weinberg 1978 / Wilczek 1978);
%                               baryogenesis (Sakharov 1967); BBN
%                               (Alpher-Bethe-Gamow 1948); CMB
%                               (Penzias-Wilson 1965 / Dicke-Peebles
%                               1965); PBH (Hawking 1971); primordial
%                               GWs (Starobinsky 1979).  Inline bib
%                               expanded with 14 new historical
%                               entries (12 net-new + 2 backfilled).
%   Phase 4A  (May 14, 2026): Architectural reframing (this file).
%                               8 legacy \part{} -> \GGPart{}{} with
%                               canonical v7.1 titles + subtitles;
%                               3 net-new GGPart shells (Codebook /
%                               Topological-Origin / Summary)
%                               inserted between legacy Part VI
%                               (Chronology) and Part VII (Discussion);
%                               1 \GGAppendixPart{Appendices}{...}
%                               wrap on \appendix.  Total Parts now
%                               11 + 1 appendix wrap = 12, en route
%                               to the v7.1 14-Part layout.
%   Phase 4B  (May 14, 2026): cosmological codebook longtable.
%                               Inserted 15-row codebook table
%                               classifying each observable as
%                               Derived / Derived (cond.) / Asserted;
%                               summary: 6 D + 6 D* + 3 A; 5 named
%                               lemmas identified.  Sections
%                               codebook-table-cosmos,
%                               codebook-msmf-ogn (Phase 4C
%                               placeholder), codebook-hubble-two-stage
%                               (Phase 4D placeholder) populated.
%   Phase 4C  (May 14, 2026): MSMF/OGN per-parameter audit.
%                               Replaced the Phase 4C placeholder with
%                               the full per-parameter audit: statement
%                               of the MSMF/OGN principle on the
%                               cosmological-boundary stratum; audit of
%                               6 D entries (D1-D3 pi-locked fractions,
%                               D9 Starobinsky-channel r suppression,
%                               D12 Delta-N-eff from particle paper);
%                               audit of 6 D* entries under 3 named
%                               conjectures (Kaxion-misalignment-
%                               completeness; Hubble-two-stage-corridor-
%                               uniqueness; compact-angle-mass-
%                               uniqueness); audit of 3 A entries under
%                               2 named conjectures (slow-roll-exit-
%                               uniqueness; TeV-scale-inflation-
%                               uniqueness).  Closure-economy table
%                               inserted showing 6/6/3 -> 14/0/0
%                               promotion roadmap.
%   Phase 4D  (May 14, 2026): Hubble two-stage corridor named theorem.
%                               Replaced Phase 4D placeholder with full
%                               theorem statement (Theorem
%                               thm:hubble-two-stage-corridor), three
%                               hypotheses (H1)-(H3), three-step proof
%                               outline, three corollaries (Hubble
%                               tension reconciliation; sigma_8/S_8
%                               reconciliation; single-stage falsification),
%                               and four falsifiable observational tests.
%                               Headline prediction: z_* = e^J = e^7 ~ 1097
%                               from OGN-5 integer J=7; matches empirical
%                               z_rec=1090 to 0.6%.
%   Phase 4E  (May 14, 2026): universal cosmological identity preview.
%                               Inserted the \S X.0 preview as a new
%                               \section after the Phase 4A overview in
%                               Part VIII.  Three subsections: (a) why
%                               Omega_m = 1/pi and not some other value
%                               (closed-packet topology + OGN-5 ledger);
%                               (b) why the cosmic coincidence is not a
%                               coincidence (topological closure + ratio
%                               theorem + integer-counted now); (c) the
%                               same uniqueness principle layer by layer
%                               (six strata from electrons to programme).
%                               Marked as PREVIEW; full version in Ultra-3.
%                               Overview pointer in sec:topological-origin-
%                               overview-cosmos updated to point at the new
%                               preview Section.
%   Phase 4F  (May 14, 2026): cross-paper coherence-map seeds.
%                               Inserted in Part IX (Summary): the
%                               Section 59.1-style 'Note on version'
%                               listing for the 8 upstream papers
%                               (AFullOne ... AFullNine, all at v7.1
%                               May 14, 2026), each with the
%                               inheritance paragraph naming what
%                               A10-Cosmos draws from that paper;
%                               compressed 15-row codebook summary
%                               table; compressed 5-step audit
%                               roadmap table (6/6/3 -> 14/0/0);
%                               explicit Phase 7 + Ultra-3 deferred-
%                               items closeout.
%   Phase 4G  (May 14, 2026): final cleanup of Phase 4.  Three
%                               small operations: (1) strip the
%                               leftover '(placeholder)' from the
%                               MSMF/OGN audit subsection title;
%                               (2) add Bianchi1902 inline-bib entry
%                               (matching the master bib); (3) convert
%                               the two parenthetical 'Bianchi 1902'
%                               mentions in body to proper \citep{Bianchi1902}
%                               citations.  Phase 4 substantive content
%                               (sub-phases A-G) is now complete; A10-Cosmos
%                               is at v5.9 with 16 \GGPart calls (12 v7.1
%                               plus appendix wrap; final two parts split
%                               via Ultra-2), 15-row codebook + 5-conjecture
%                               audit + Hubble two-stage corridor theorem
%                               + universal-cosmic-identity preview +
%                               cohort coherence-map seeds.
%   Phase 5   (May 14, 2026): Open Problems triage with L6 status
%                               headers.  Replaced the 5-column
%                               tab:partVII-scorecard with a 6-column
%                               version including an explicit L6 status
%                               column (R / P / G).  Updated Status
%                               column to be Phase-4-aware (Theorem
%                               4.D.1 closes \#8 and \#9; codebook D
%                               status closes \#3, \#4, \#13).  Tally
%                               paragraph inserted: 6 R + 6 P + 1 G = 13.
%                               Compared with A9-SM Phase 5 tally
%                               6R + 2P + 2G for the 10 SM problems:
%                               cosmology has similar Resolved share,
%                               more Partially resolved (named-conjecture
%                               economy of Sec.~codebook-msmf-ogn).
%   Phase 6   (May 14, 2026): hostile-reviewer + reference-scanner
%                               sub-agents (parallel dispatch).  Hostile
%                               reviewer returned 15 findings (4 HIGH,
%                               7 MEDIUM, 4 LOW).  Reference scanner
%                               returned 0 undefined cites / refs and
%                               5 WARN classes (all deferred to Phase 10
%                               and 11).  Remediation script applied 10
%                               patches (P1-P10): acknowledge Omega_b
%                               2.3sigma residual; reframe z_* = e^7
%                               agreement honestly; add 'what is and is
%                               not proved' remark on Theorem 4.D.1;
%                               clarify integer-counted-now argument;
%                               reframe MSMF/OGN principle as uniqueness
%                               claim; soften A3 and A4 cross-paper
%                               attributions; sharpen GW-siren prediction;
%                               downgrade L6 status #4 to P (5R+7P+1G);
%                               drop misleading Lemma V D9-conditional
%                               remark.  Deferred to Ultra-3: explicit
%                               four-form integration (M2) and (1/3)pi^-1
%                               rule-out (M7).  Deferred to Ultra-1:
%                               phase-numbering inconsistency (L2) and
%                               counting-summary double-count (L3).
%   Phase 7   (May 14, 2026): companion-paper auditor (sub-agent) +
%                               cross-paper coherence map.  Auditor
%                               read all 8 upstream v7.1 source files
%                               (A1-A7 + A9-SM) and verified each
%                               inheritance paragraph.  Returned:
%                               16 CONFIRMED attributions, 5 NEAR
%                               fixes, 3 WRONG.  Remediation applied
%                               8 patches: W1 fix A1 axiom names
%                               (existence/continuity/closure ->
%                               LC/OGGEP/MSMF); W2 fix A1 cost-minimum-
%                               principle attribution; W3 date drift
%                               (May 14 -> May 2026 in 7 of 8
%                               paragraphs); N1 A2 Kaxion descent
%                               cross-cite to PaperC1; N2 A4 J/K role
%                               parentheticals; N3 A7 atomic/molecular/
%                               cellular -> five-story pagoda; N4 A9
%                               Section 60 -> 'The geometric origin
%                               of identity' title-cite.  Plus inserted
%                               new Sec.~coherence-map-outward-inward
%                               with 8-item outward + 5-item inward
%                               map + net-coherence summary.  Phase 8
%                               will draw the TikZ hierarchy figure.
%   Phase 8   (May 14, 2026): TikZ figures (4 of 5 in this pass).
%                               Fig 1: matplotlib symlog Dev plot for
%                               the 12 codebook entries with empirical
%                               anchors (largest |Dev| = Omega_b at
%                               +2.76%; z_* at +0.61%); inserted via
%                               \includegraphics after the codebook
%                               table.  Fig 2: TikZ cosmic-inventory
%                               band showing the three pi-locked
%                               fractions in §sec:why-omega-m-pi.  Fig 3:
%                               TikZ Hubble two-stage corridor diagram
%                               after Theorem 4.D.1, with OGN-5 ledger
%                               box.  Fig 4: TikZ programme-acronym
%                               hierarchy showing 8 upstream + A10 +
%                               5 downstream papers, in the outward-
%                               inward-coherence-map subsection.  All
%                               figures wrapped in \resizebox{\textwidth}{!}
%                               per L8; programme acronyms in sans-serif
%                               bold per L23.  The fifth proposed figure
%                               (sectoral descent ladder for cosmology)
%                               is deferred to Ultra-3 alongside the §X.0
%                               centerpiece it directly serves.
%   Phase 9   (May 14, 2026): front-matter rewrite to A1 canonical.
%                               TitlePage: 'GUT-Cosmos from GUT-10...' ->
%                               'A10-Cosmos: Geometric Origin of LCDM
%                               and the pi-locked Cosmic Inventory'.
%                               Abstract: ~640 words of theorem-ladder
%                               prose -> ~200 words of plain-English
%                               3-paragraph summary with After-all close.
%                               ExecutiveSummary: ~700 words of internal-
%                               language exposition -> ~870 words in 5
%                               canonical paragraphs (problem / strategy
%                               / results / triage / road forward) with
%                               After-all close.  Keywords refreshed:
%                               dropped GUT-10/GFF/GEP/CEP/DGI/GSSB/GA
%                               legacy jargon; added pi-locked inventory,
%                               z_*=e^7, Hubble two-stage corridor,
%                               MSMF/OGN cost-minimum principle,
%                               codebook provenance triage, universal-
%                               cosmic-identity principle.
%   Phase 10  (May 14, 2026): bibliography surname-sort + master-bib
%                               reconciliation.  Merged 31 legacy +
%                               15 Phase-3-historical + 1 Phase-4G
%                               Bianchi inline bibitems into one
%                               surname-sorted alphabetical sequence
%                               (46 total).  Migrated 7 legacy
%                               Arisaka citation keys to canonical
%                               v7.1 cohort names per L52:
%                                 ArisakaA1Constitution   -> ArisakaA1Constitution
%                                 ArisakaA2GG6     -> ArisakaA2GG6
%                                 ArisakaA3BI6     -> ArisakaA3BI6
%                                 ArisakaA5GDOS    -> ArisakaA5GDOS
%                                 ArisakaA6Gi4  -> ArisakaA6Gi4
%                                 ArisakaA7Qi4  -> ArisakaA7Qi4
%                                 ArisakaA9SM    -> ArisakaA9SM
%                               Migration applied uniformly: every
%                               \bibitem and every body citation
%                               (\cite/\citep/\citealp/\citet) is
%                               updated.  All 46 bibitems carry
%                               Relevance paragraphs (no further
%                               Relevance work needed at Phase 10).
%   Phase 11a (May 14, 2026): sans-serif sweep of legacy preprint codes.
%                               Converted GG-2026-7F -> \ATen{} (128),
%                               GG-2026-6F -> \ANine{} (42),
%                               GG-2026-1  -> \AOne{} (15),
%                               GG-2026-2  -> \ATwo{} (9),
%                               GG-2026-3  -> \AFive{} (12),
%                               GG-2026-4A -> \ASix{} (7),
%                               GG-2026-4B -> \ASeven{} (8),
%                               GD-2026-5  -> \AThree{} (9).  Total ~230
%                               substitutions.  Skipped: macro
%                               definition lines (per L48), Phase 1
%                               purge-audit comments, the PROVENANCE
%                               block, the \hypersetup{} PDF metadata,
%                               and the inline thebibliography block.
%   Phase 11b (May 14, 2026): Appendix expansion to canonical 9-section
%                               template per L19.  Added six new
%                               appendices D-I between current Appendix
%                               C and \GGAcknowledgments:
%                                 D: Programme acronym dictionary
%                                 E: Cross-paper inheritance table
%                                 F: Codebook glossary (compact)
%                                 G: Named-theorem-and-conjecture deck
%                                 H: Falsification-test catalogue
%                                 I: Named-lemma promotion roadmap
%                               Total ~6-8 pp of new appendix content.
%                               The 9-section appendix template is now
%                               complete (A-I).
%   Phase 11c (pending):      programme-acronym hierarchy figure
%                               (already drawn in Phase 8, may need
%                               additional in-Appendix-A version).
%   Phase 11d (May 14, 2026): 36-item canonical compliance audit
%                               complete.  Result: 32 PASS, 4 PARTIAL,
%                               0 FAIL out of 36 items.  Four PARTIAL
%                               items (abstract word count over by 61;
%                               §X.0 universal-cosmic-identity in preview
%                               form pending Ultra-3; final 14-Part split
%                               pending Ultra-2; closed-packet four-form
%                               integration pending Ultra-3) are openly
%                               flagged and deferred to the Ultra passes.
%                               Full audit log in
%                               outputs/a10_v518_phase11d_compliance_audit.md
%                               Phase 11 (eleven-phase) workflow is now
%                               complete; paper is ready to enter Ultra-1.
%   Ultra-1   (May 14, 2026): single-line algebraic + cleanup patches.
%                               Applied 5 patches: U1.1 abstract trim
%                               (261 -> ~190 words, addresses Phase 11d
%                               PARTIAL #12); U1.2 Phase 7 vs Phase 8
%                               inconsistency in 2 places (Phase 6 L2);
%                               U1.3 Omega_Lambda double-count in
%                               codebook counting summary (Phase 6 L3);
%                               U1.4 internal version 'v5.4' mentions
%                               in body replaced with version-agnostic
%                               'present codebook' / 'Current state'
%                               (Phase 6 L4).  Phase 11d compliance
%                               improves to 33 PASS + 3 PARTIAL out of
%                               36 (PARTIAL #12 addressed; #18, #36,
%                               and Ultra-3-bound closure remain).
%   Ultra-2   (May 14, 2026): final 14-Part split (Phase 11d audit
%                               PARTIAL #36 resolved).  Split current
%                               long Part I 'Architecture, baseline,
%                               inheritance map' into v7.1 Parts I
%                               (Cosmology baseline + LCDM framework)
%                               and II (Constitutional apparatus: from
%                               BI-6 to the cosmological boundary law).
%                               Split current long Part II 'From bulk
%                               to boundary' into v7.1 Parts III
%                               (6D bulk + EGUT launch + radial
%                               chronology) and IV (Inflation + dark
%                               energy + late vacuum inventory).  Total
%                               GGPart count: 11 substantive + 1
%                               appendix wrap = 12 -> 13 substantive + 1
%                               appendix wrap = 14, matching the v7.1
%                               canonical target.  Existing body content
%                               is unchanged; only the Part labels and
%                               natural seam markers are inserted.
%                               (Aggressive redundancy trim and overfull-
%                               hbox cleanup are deferred to Ultra-3
%                               cosmetic pass.)
%   Ultra-3   (May 14, 2026): substantive closure pass.  Four content
%                               pieces added: U3.1 §X.0 closure with
%                               formal Theorem omega-m-pi-topological
%                               (5-step proof outline + corollary +
%                               status note); U3.2 4-point post-hoc-fit
%                               defence (full version expanding the
%                               Phase 4F seed); U3.3 GG-distinct
%                               comparison table (5 frameworks: LCDM,
%                               EDE, MOND, brane, conformal-cyclic);
%                               U3.4 Conclusions Part summit view
%                               (~3 pp: what has been accomplished, what
%                               this means for cosmology / programme /
%                               next-decade observations).  Phase 11d
%                               audit: 36 PASS / 0 PARTIAL / 0 FAIL.
%                               Paper ready for v7.1 promotion.
%   v7.1     (RETRACTED, May 14, 2026): the v7.1 cohort-ship
%                               promotion bump was applied prematurely.
%                               User review of the Ultra-3 closure
%                               file identified additional work to be
%                               completed before v7.1 is appropriate.
%                               The v7.1 ship has been retracted; the
%                               active working version is v6.2.  Further
%                               revisions will progress v6.2 -> v6.3 ->
%                               v6.4 -> ... -> v7.1 under user direction.
% ----------------------------------------------------------------------

%% ----------------------------------------------------------------------
%% v10.7 -> v10.8 (June 11, 2026): P2-N3 reciprocal sync (Phase 5 of the P2 project;
%%   author-approved, June-10 protocol: minimal, disclosed, version-bumped).
%%   - Reservation sentence -> completed tense; P2 cited under final title
%%   -   (new bibitem ArisakaP2N3, v2.4 stamp); outward-map entry updated
%%   -   with P2's declared conditioning.
%%   Reference: [GG-P2-N3]/[P2-Support]/P2-N3-PHASE5-SYNC-REPORT-2026-06-11.md
%% ----------------------------------------------------------------------
% ------------------------------------------------------------
% v15.13, August 18, 2026.  The blank page 112.
%   THE DEFECT.  Page 112 carried a running header, a folio, and
%   nothing else.  Reported by the author; found by reading the
%   page, not by any gate.
%   THE CAUSE, measured.  The full-page showcase idiom is
%     \clearpage \null\vfill <minipage> \vfill\clearpage
%   and the \vfill after \null is a legal page break.  When the
%   minipage's natural height exceeds \textheight = 650.43pt, TeX
%   takes that break, ships a page carrying only the \null, and
%   places the plate on the next page.  No Overfull \vbox is
%   reported, which is why the gate passed it.  Natural heights,
%   by \setbox probe, one compile:
%     fig:partII-inflation-derived  630.3pt  fits
%     fig:partII-history-page1      669.7pt  19.2pt OVER
%     fig:partII-history-page2      658.7pt  fits in practice
%   THE REPAIR.  fig:partII-history-page1's caption is set
%   \footnotesize inside \begingroup/\endgroup -- the device this
%   paper already uses on the immediately preceding showcase,
%   fig:partII-inflation-derived.  The plate is untouched at
%   0.84\textheight and not one word of the caption changes.
%   Rejected alternative: shrinking the plate, which needed
%   0.84 -> 0.78 (a 7 %% loss of picture) to clear the page.
%   ALSO, same reading.  Seven of the eleven showcase minipages
%   lacked \noindent, so each began a paragraph, was pushed right
%   by \parindent, and ran 16.33pt past the right margin -- the
%   seven "Overfull \hbox (16.33182pt too wide)" entries in the
%   log.  Measured on the rendered page: ink x1 = 556.6pt against
%   a text edge at 540.9pt; with \noindent, x0 = 71.3 and
%   x1 = 540.7, exactly the text block.  The other four showcase
%   minipages already carried \noindent; this is a repair against
%   the paper's own idiom.
%   ALSO.  Six \vfill\clearpage followed by a bare \clearpage;
%   the second is redundant.  Removed, verified page-for-page
%   identical before and after.
%   NEW PROSE: none.  No claim, number, caption text or figure
%   changes in this ship.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.14, August 19, 2026.  The shared bulk showcase plate.
%%   ADOPTED, at the author's direction: one full-page showcase
%%   plate carried BYTE-IDENTICALLY by A2, A5, A8, A9, A10 and
%%   P1 -- the cohort-map precedent.  It is the integration of
%%   three existing figures: A1 Figure 1 supplies the logic
%%   (two necessities, AX-2 building the fiber, AX-3 projecting
%%   to the registered world), A10 Figure 13 supplies the
%%   tapering throat and the projection arrow, and A2 Figure 8
%%   supplies the calibrated scale axis.  The axis now carries
%%   eight ticks with mu = E_GUT e^-r beside each, E_GUT =
%%   E_Pl e^-2pi at r = 0, E_Pl declared OFF the diagram at
%%   r = -2pi, the lock at r = 29.41 and the OGN-5 depth at
%%   r = K = 35.  Every number was computed, not carried over.
%%   THE OFFSET IS EXACT: K is counted from the bulk anchor
%%   E_Pl/2 and the axis from E_GUT, so the two differ by
%%   ln(E_Pl/2 / E_GUT) = 2pi - ln 2 = 5.590 and
%%   r_lock = K - 2pi + ln 2 = 29.410.  That relation is what
%%   reconciles A2's "lock at r = K" with A10's "lock crossing
%%   at r = 29.4"; both name 3.85 TeV.
%%   The plate is pure TikZ and adds NO asset to the arXiv
%%   package.  Caption ABOVE, per the frozen float standard.
%%   The drawn taper is e^-r/13.64 -- the true e^-r slowed so
%%   the throat stays visible -- and the plate says so.
%%   NEW PROSE: the plate's labels and its caption, listed in
%%   the report for the author to review or rewrite.
%%   ALSO: Figure 13's scale axis is labeled r = ln(E_GUT/mu)
%%   and printed the cap "UV (r=0)" beside the E_Pl dot.  On
%%   that axis E_Pl is at r = -2pi = -6.28, so the r=0 mark
%%   and the E_Pl mark could not both be right.  The cap now
%%   reads UV, and each of the three dots carries its own
%%   r-value: E_Pl at -2pi, E_GUT at 0, the lock at 29.41.
%%   No scale was removed from the figure.
% ------------------------------------------------------------
% ------------------------------------------------------------
%% v15.15, August 19, 2026.  The bulk showcase plate moved earlier.
%%   AUTHOR'S PLACEMENT: the shared plate goes from showcase
%%   page 13 to page 12.  Every showcase unit is one page, so
%%   this is a pure relocation of the delimited block: cut from
%%   where it sat, pasted immediately above the unit that held
%%   the target page.  NOT ONE BYTE OF THE BLOCK CHANGES -- the
%%   rider asserts its sha256 before and after, and the six
%%   papers that carry it are re-checked for a single hash.
%%   Nothing outside the front-matter showcase run moves, and
%%   the plate is unnumbered, so no figure or table number and
%%   no cross-reference changes.
%%   NEW PROSE: none.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.16, August 19, 2026.  Four refinements at the author's
% request.
%   (1) PAGES 230-231 COMBINED.  Section 46, "Part VI in plain
%   English", was preceded by a \clearpage, so ten lines of Part
%   VI body text sat alone on page 230 and eighteen lines of
%   Section 46 sat alone on page 231.  MEASURED FIRST: A10 has
%   FOURTEEN "in plain English" sections and only FOUR carry a
%   \clearpage -- Parts IV, VI, VII and VIII.  Ten do not.  The
%   device is not this paper's convention; it is a leftover.  The
%   one before Part VI is removed.  The other three are printed by
%   the guard and left alone, because the author named this page
%   and not those.
%   (2) THE TWO-LINE PAGE 400.  Section 97 overran page 399 by two
%   printed lines.  Freed by one repair and four condensations.
%   THE REPAIR: Table 52's caption began two different sentences
%   with "The ordering reflects where the framework is..."; the
%   first is subsumed by the second and is removed.  That is a
%   repair against the caption's own text, and the caption stays
%   far above the 260-character floor.  THE CONDENSATIONS: four
%   sentences in Section 97 shortened without changing a claim,
%   each printed before and after in the report.
%   (3) FIGURE 40 (BBN Schramm plot) REBUILT.  The old plot
%   carried NO Y-SCALE: the axis read "abundance (schematic, log)"
%   and the four curves sat at heights corresponding to no scale
%   at all.  Quantifying the axis therefore meant moving the
%   curves to where the numbers put them.  They are redrawn from
%   standard SBBN scalings anchored to the values this paper's own
%   caption already states -- Y_p = 0.247, D/H = 2.53e-5, 7Li/H
%   predicted 5.0e-10 and observed 1.6e-10, all at the observed
%   eta = 6.12e-10.  The ONE anchor not already in the paper is
%   3He/H = 1.0e-5, the textbook SBBN value, and it is named as
%   such in the report.  Shapes: Y_p logarithmic and nearly flat,
%   D/H ~ eta^-1.6, 3He/H ~ eta^-0.6, and 7Li/H a two-branch
%   valley with minimum 1.1e-10 at eta10 = 2.8.  The y axis now
%   carries twelve decade ticks from 10^-11 to 10^0.  Enlarged
%   0.743 -> 0.95 of the measure; rendered type 7.67 -> 10.1 pt,
%   up 32 percent.  Every annotation was moved to clear space and
%   the page was read: no overlaps remain.
%   HONEST CONSEQUENCE, STATED: on a true 11-decade axis the 7Li
%   valley is shallow, because 1.1e-10 to 1.4e-9 is only 1.1
%   decades.  The old schematic exaggerated it.  The new plot is
%   less dramatic and correct.
%   (4) FIGURE 31 (provenance map) ENLARGED AND CENTERED.  0.92 ->
%   1.104 of the landscape measure, +20 percent, using the
%   cohort's \makebox idiom so the oversized plate centers into
%   the margins without an overfull box, and \null\vfill ... \vfill
%   so it sits in the middle of its page instead of the top.
%   NEW PROSE: the Figure 40 axis label and its two annotation
%   boxes, the one revised sentence of the Figure 40 caption, and
%   four condensed sentences in Section 97.  All listed in the
%   report for the author to review or rewrite.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.17, August 19, 2026.  Two figure corrections.
%   (1) FIGURE 31 SHRUNK.  v15.16 took it to 1.104 of the
%   landscape measure and the author reports it reaching the
%   running header.  What was measured then: the plate's ink
%   cleared the rotated header rule by 7 pt.  SEVEN POINTS IS A
%   CLEARANCE ON PAPER AND A COLLISION TO THE EYE, AND THE
%   AUTHOR'S EYE IS THE STANDARD.  Back to 0.99 -- a 10 percent
%   reduction, and still 7.6 percent larger than the 0.92 it was
%   before v15.16 -- which puts the clearance near 34 pt.  The
%   vertical centering added in v15.16 is kept.
%   (2) FIGURE 16 ENLARGED, AND ITS PANELS WIDENED TO FIT THEIR
%   OWN LABELS.  Reported too small at 0.527 of the measure, which
%   is 247.6 pt printed.  Taken to 0.66, up 25 percent.
%   MEASURED WHILE IN THERE, AND NOT ASKED FOR: five labels were
%   WIDER THAN THE PANELS THEY SIT IN and overhung both sides.
%   Measured by compiling each label on its own: the widest,
%   "Planck wall: singular / beginning (a -> 0, rho -> infinity)",
%   is 107.5 pt against an 88.2 pt panel; three others are 99.9,
%   97.5 and 67.7 pt.  Enlarging alone would have enlarged the
%   overhang with everything else.  The panels go 3.1 -> 4.0 units
%   (88.2 -> 113.8 pt), clearing the widest label by 6 pt, and the
%   right panel's origin moves 6.3 -> 5.4 so the overall width is
%   unchanged at 9.4 units.  Every label centred on a panel moves
%   with it, 1.55 -> 2.0.
%   NOTHING IN EITHER FIGURE'S CONTENT CHANGES -- not a label, not
%   a value, not a color.  The rider strips coordinates and the
%   scale and asserts the rest of the Figure 16 block is identical.
%   NEW PROSE: none.
% ------------------------------------------------------------
%% ---- vv15.18 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 6 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
%% And the showcase headlines brought to one rendered form.
%% The house form is the style file's own default, \Large\bfseries\sffamily,
%% which is what \GGShowcaseHead sets when it is called with no optional
%% argument, and which forty-three of the cohort's fifty-seven showcase
%% headlines already used.  Fourteen did not, and the difference was
%% visible to a reader turning pages: four headlines printed at \huge,
%% half again the house height, and three at \large, smaller than it.
%% 2 headlines in this paper.  Only the size changes; the \boldmath in
%% a headline that carries math is kept, the subtitle under a headline
%% keeps its own smaller size, and \GGShowcaseHeadSkip, which is vertical
%% space rather than type, is left exactly as each paper sets it.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour, so this providecommand takes effect)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive, so this providecommand takes effect)
\usepackage{indentfirst}
%% v11.3: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}} % A6 Gi4 -> GRIP (cohort, May 2026)
%   v9.6 (June 7, 2026; amended same day): isocurvature figure corrected
%        to the consistent TeV-corridor H_inf = 3.5e-12 GeV evaluation
%        (beta_iso ~ 3e-41, thirty-nine orders below the bound; was
%        quoted ~1e-32 from an inconsistent H_inf), matching C1 v6.6.
%   v9.6 (June 7, 2026): cohort canon sweep.  Dictionary TGG row fixed
%        ("Total Geometry--Gauge tensor" -> Total Geometric-Gauge
%        object), description kept.
%   v9.5 (June 7, 2026): GRIP-In / GRIP-Out regime vocabulary imported
%        from A6-GRIP v9.5 (def:GRIPin, def:GRIPout).  Cosmology is the
%        alternation writ large: the landscape-reorganizing transitions
%        (inflationary exit, reheating, electroweak selection, QCD
%        confinement, recombination) are GRIP-Out events; the long
%        epochs between them are GRIP-In relaxation on the landscape so
%        created, with Hubble friction as the cosmological within-basin
%        flow; crossovers (matter-radiation equality) are milestones of
%        GRIP-In, not transitions.  The chain halts exactly twice: the
%        Kaxion relic (matter ledger) and the de Sitter vacuum (energy
%        ledger) are where cosmic GRIP-In terminates.  Edits: regime
%        paragraph + classification in the cosmic-GRIP-grammar
%        subsection; Cosmic GRIP theorem statement + proof extended
%        (T-GRIP-Out-Necessity); GRIP-stage paragraphs sharpened at the
%        load-bearing sites (cosmic-epoch intro, earliest universe,
%        inflation/reheating prototype GRIP-Out, recombination,
%        late-universe terminal GRIP-In, Kaxion); misalignment
%        overdamping = GRIP-In clause; keywords; A6/A11 bibitem
%        Relevance updates; macros (A6/A11 form).
%   v9.4 (June 7, 2026): C1-Kaxion closure propagated. Misalignment
%        normalization corrected (x2.6 high at the calibration point ->
%        Omega_K h^2 = 0.090 theta_i^2 (f/10^12)^1.165); benchmarks updated
%        (raw/abundance 3.0e13 GeV / 0.19 ueV / 45.3 MHz, bookkeeping only;
%        laboratory eigenstate 7.1e11 GeV / 8.1+-0.8 ueV / 1.95 GHz, window
%        1.59-2.13 GHz); frame question resolved (detectors read eigenstates);
%        pre-inflationary single-patch history forced (no PQ radial mode;
%        TeV-corridor margins), isocurvature quantified (beta_iso ~ 1e-32);
%        photon coupling fixed by exact anomaly matching E/N = 8/3 -> on the
%        DFSZ line, g = (1.2+-0.1)e-15 GeV^-1 (laboratory); Conjecture III
%        contracted to the integer Xi_cap in [35,47]; codebook, dictionary,
%        problems table, falsifier roadmap updated; ArisakaC1Kaxion bibitem
%        added.
%   v9.3 (June 6, 2026): GGA (Ground Geometric Attractor) introduced, aligned
%        with A6-GRIP v9.3 (def:GGA) and A11-Origin (T-GGA-as-Ground-State).
%        A new cosmological GA/GGA distinction is stated once in the GRIP-
%        grammar subsection and then applied where it is genuinely a ground
%        state: the Cosmic GRIP theorem (persistent relics = GGAs), the Kaxion
%        dark-matter relic, and the late-time de Sitter vacuum. Transient
%        cosmic histories (reheating export, frozen CMB record, growth slice)
%        DELIBERATELY remain plain GAs -- a frozen record is an attractor, not
%        a ground state. \GGA + \AEleven macros added; A11-Origin bibitem added.
\providecommand{\GGA}{\ensuremath{\mathrm{GGA}}}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}
\usepackage{pdflscape}
\usepackage{afterpage}

\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A10-Cosmos}
% --- Part-aware running header (right side shows current Part as a live pointer) ---
%   The right-hand header field (previously the version/date stamp, which the cover
%   page already carries) now tracks the current Part. Implemented with the LaTeX
%   marks mechanism so the value is correct across page breaks; \GGPart gains an
%   optional short header label, and section marks are suppressed so the Part mark
%   persists. Front-matter Executive Summary / Preface set their own marks.
\makeatletter
\fancyhead[R]{\small\nouppercase{\rightmark}}
\renewcommand{\sectionmark}[1]{}%
\renewcommand{\subsectionmark}[1]{}%
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}%
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\LARGE\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother
% ---------------------------------------------------------------------------
\PaperStatus{Preprint}
% 
%   v9.7 (June 8, 2026): KAXION SECTION RECONCILED WITH C1 R2 (for simultaneous arXiv).
%        Collapsed the raw/mixed two-benchmark framing to ONE physical decay constant
%        f_K=7.1e11 GeV (bulk GS scale = diagonalization intermediate, NOT a 45 MHz line);
%        theta_i is now an OUTPUT of the abundance (~1.4 rad), closure inverted
%        (geometry->f_K, abundance->theta_i); single-benchmark table; dropped raw 45 MHz
%        line+doublet; updated onset/fuzzy/naive-Tosc satellites + Lemma-III codebook.
%% ===================================================================
%%  A10-Cosmos v13.19  (July 31, 2026)  -- P2-N3's status, and N_g.
%%  Two sites described P2-N3's upper bound as conditional on its
%%  'declared Budget Conjecture and hypotheses H_1--H_2'. P2 v14.3
%%  retired that framing: the cap is the capacity theorem with L-INT
%%  behind it and two obstruction theorems showing the conjectural
%%  form could not have been discharged as posed -- H_1 in particular
%%  is a proved impossibility, not an open hypothesis, so citing it as
%%  a condition inverted its status. One residue, R-INT, is declared
%%  and met on the benchmark background by the inflow companion.
%%  Both sites now also write the generation count N_g rather than the
%%  bare N, which after the A2/A3/A4/P2 work of this session reads as
%%  the OGN-5 colour index.
%% ===================================================================
% ------------------------------------------------------------
% A10-Cosmos, August 15, 2026.  RIDER: the spelling that ships inside
% the PDF.
%   \hypersetup's pdfsubject said `GG-Theory programme'.  It is
%   not printed text and it is not a comment, so the spelling
%   guard -- which starts at \begin{document} -- never saw it;
%   but it ships in the PDF's Subject field, where any reader can
%   open the properties panel and read it.  Confirmed with pdfinfo
%   against the deployed artifact before this was written.
%   Found at C3-BH's Stage 1, which added metadata_survivors() to
%   gg_spelling; every stage now runs it.  A8, A9, C1, S1 and S2
%   are clean; P2-N3 carries it and is not yet migrated.
%   ONE WORD IN THE PREAMBLE.  No printed page changes, and that
%   is asserted by rebuilding and comparing page for page.
% ------------------------------------------------------------
%% ------------------------------------------------------------
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\PaperVersion{v20.0}
\PaperDate{August 21, 2026}
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 pdftitle = {Arisaka-GG-A10-Cosmos: Geometric Origin of LambdaCDM and the
 pi-locked Cosmic Inventory --- Derivation of the Cosmological
 Parameter Sheet from the OGN-5 Diophantine Ledger and the
 \GUfive{}},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {Cosmology as an infrared theorem of the GG-Theory program:
 Omega_m = 1/pi, Omega_Lambda = 1 - 1/pi, Omega_b = 1/(2 pi^2),
 the Hubble two-stage corridor, Kaxion dark matter,
 baryogenesis, BBN, CMB, JWST early galaxies, and the cosmic
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 inflation, dark energy, Kaxion dark matter, baryogenesis,
 BBN, CMB, primordial gravitational waves, black-hole
 information, primordial black holes, MSMF/OGN audit,
 cosmic inventory, Hubble two-stage corridor, GG-6, BI-6,
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 \GUfive{}}
}
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%% ---- v12.92 (July 9, 2026) -- C1/A9/A10 consistency reconciliation ----
%%  C1-Kaxion is the source of record; C1 unchanged. Synced three shared numbers:
%%  (1) axion isocurvature fraction recomputed from the shared lock-epoch scale
%%      H2 (reduced-Planck Friedmann) -> canonical value + one fewer order below
%%      the Planck bound, at the two body occurrences (Kaxion-DM section + the
%%      matter-sector objection).
%%  (2) Kaxion mass error bar tightened to C1's refined budget throughout; the
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%% -----------------------------------------------------------------------------
%% ---- v12.93 (July 9, 2026) -- Appendix-A wording aligned to C1 (source of
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%%  fixes the Xi_cap WINDOW; the continuous +-6%/+-0.5ueV mass uncertainty is
%%  lattice-QCD, not geometry (removes the cap-loading-vs-lattice clash).
\begin{document}
% ---- v13.29 (August 2, 2026) ----
% The trio on the two fibres (per the internal note 'The Two
% Coordinates of the Cosmos'; Kaxion updated alongside per the
% author's commission). (R1) Launch closes with 'Inflation on
% the two fibres: the interval's coordinate, rolling once' ---
% the census (one interval => one inflaton; one circle => one
% axion; zoo and axiverse as uncounted fibres), the episode's
% shape as the fibre's character (cosine vs warp exponential),
% the once-only roll, the interval's numbers (N*, n_s, r), the
% circle's two cameos (Theta-exit; untouched ledger). (R2b)
% Main Kaxion section closes with 'The Kaxion on the two
% fibres: the circle's coordinate, ringing still' (C1 v13.20
% essence at the cosmological register). (R2) Terminus gains
% 'Dark energy on the two fibres: the circle's share, the
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% tab:cosmic-triptych and the new three-panel figure
% fig:cosmos-trio. (R5) Four stitches (census forward-ref;
% Hopf-appendix note; 23.3 chronology link; Kaxion-appendix
% pinning->triptych). Reading register; grades inherited; no
% number changed.

% ---- v13.28 (August 2, 2026) ----
% Chronology figure mirrored from P5 v3.48 Part IX into the
% radial-chronology section (end of the fibre-without-holonomy
% subsection): the whole history of time on one line, in this
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% with the e-fold budget programme as its would-be derivation.
% Coordinates from the verified session computation
% (GG-Report-153).

% ---- v13.27 (August 2, 2026) ----
% New subsection 'The fibre without holonomy: why the chronology
% cannot run backward' in the radial-chronology section (before
% 'Bottom line'): the arrow traced to the exact statement that
% holonomy requires closed loops and the radial interval has none;
% the circle remembers (letters, charges) / the interval forgets
% (gates, prompt crossing, zero dilution, welded door); expansion
% as the boundary shadow of the descent; Sakharov's conditions as
% the anatomy of the two fibres. Full treatment credited to
% GG-W1-Symmetry (v0.56). Part of the four-paper enrichment.


%% v11.3: ExecutiveSummary env (centered) provided by ucla-arisaka-gg-11.


\begingroup\setstretch{1.0}
\GGTitlePage
 {The Standard Cosmology as Theorem of \GGTheory{}}
 {The Cosmic Inventory as Observations of One Six-Dimensional Geometry}
 {August 12, 2026}
\endgroup
\thispagestyle{empty}

\vspace{8pt}

\begin{abstract}
\noindent
For forty years the standard model of cosmology, \(\Lambda\)CDM, has described the universe with breathtaking precision and explained almost none of it: a handful of numbers, read off the sky, reproduce everything --- yet the theory cannot say why those numbers take the values they do.

This paper argues that they are not free. The same six-dimensional geometry that fixes the elementary particles, with no adjustable continuous dials, also forces the present-epoch \emph{shape} of the cosmos: the contents fall into coupled powers of one number --- the matter fraction \(1/\pi\), the dark-energy fraction \(1-1/\pi\), the ordinary-matter fraction \(1/(2\pi^2)\) --- and the geometry's switch at the epoch of the first atoms sits at redshift \(e^7\), the exponential of a single whole number. Their ratios are fixed for all time. Inflation, today's acceleration (a cosmological constant, \(w=-1\)), and the dark matter --- a light particle we call the Kaxion, near \(8.1\,\mu\)eV --- follow from the same shape, the tilt and tensor ratio computed, not fitted ($n_s = 0.962$, $r = 0.004$).

The two great puzzles of present-day cosmology appear in a new light: the Hubble tension as two readings of one geometric corridor, and the surprisingly early, massive galaxies seen by JWST read structurally, within the corridor's growth history. As in its companion paper on particle physics, this work states plainly which numbers it has fully derived, which rest on a single named step --- now pinned down exactly: the timing of our observation --- and which remain open, tuning nothing after the fact.

If the argument holds, the cosmic inventory stops being unrelated accidents: its form is forced and the \(10^{120}\) vacuum fine-tuning dissolved, leaving only a few-percent ``why now'' coincidence of timing --- the one number read from the sky being \emph{when} we look.
\end{abstract}



\begin{ExecutiveSummary}
\markboth{Executive Summary}{Executive Summary}%
\paragraph{The problem.}\looseness=-1

\(\Lambda\)CDM is the rock-solid empirical baseline of contemporary cosmology --- the most successful predictive framework after the Standard Model of particle physics. Six numbers, read off the data, compress every microwave-background anisotropy, every baryon-acoustic oscillation, every supernova and weak-lensing map into one grammar. But the framework is silent about \emph{why} those numbers hold. Why is the universe roughly five percent ordinary matter, twenty-seven percent dark matter, sixty-eight percent dark energy? Why do the early- and late-universe measurements of the expansion rate disagree at nearly six sigma --- the \emph{Hubble tension}, which the James Webb Space Telescope has sharpened rather than removed --- and why has JWST found massive galaxies already in place a few hundred million years after the Big Bang? Seventeen recognized problems, set out in Part~I, sit outside the six-parameter language. This paper does not abandon \(\Lambda\)CDM; it asks whether its numbers can be \emph{derived}. Beneath the seventeen sits a question the field has circled for half a century without ever naming --- the \emph{second horizon problem} (\S\ref{sec:second-horizon}): inflation explains why disconnected regions share the same \emph{conditions} --- one temperature to a part in \(10^{5}\) --- but not why they hold the same \emph{things}: why an electron across the universe is identical to one here, though no signal ever compared them. Part~X poses it, and answers it.

\paragraph{The single input.}\looseness=-1

The derivation rests on one object: a six-dimensional geometry, fixed once and for all, carrying no continuous free parameters. One qualification, stated up front. The five \OGNfive{} integers \((M,N,L;J,K)=(1,3,5;7,35)\) are not adjusted but \emph{forced} --- the unique tuple surviving the anomaly-free \BIsix{} closure of the six-dimensional bulk (\AThree{}), each carrying a definite physical meaning (\(L{=}5\) the \(\mathrm{SU}(5)\) block, \(N{=}3\) the color triplicity, and so on, derived in \AThree{}), pinned upstream and never searched. The same ledger admits exactly two transcendental constants, one native to each compact bulk direction --- \(e\), the base of the exponential redshift along the radial (scale) fifth dimension, and \(\pi\), the angle of the compact (gauge) sixth dimension --- so every exponential in the inventory is a scale factor and every \(\pi\) a holonomy or a volume. Beyond that fixed ledger the cosmology declares only \emph{two} inputs, neither a continuous dial --- an overall \emph{amplitude} (a single normalizing scale, the least any cosmology can declare) and the \emph{epoch} at which the present-day snapshot is read (a clock reading, not a physical constant). ``Zero free parameters'' throughout means zero continuous ones. \ATen{} reads that one geometry at the largest scale: our four-dimensional universe is its boundary, and the contents of the cosmos are what the geometry casts onto that boundary. The cosmos and the electron are read from one ledger, one story apart. Stated as the paper's central claim (\S\ref{sec:central-claim}, Figure~\ref{fig:central-claim}): because no system --- and no universe --- is closed, the cosmic inventory consists of \emph{observations} of an open-system projection (\OGGEP{}/\GDOS{}), not free inputs.

\paragraph{The headline results.}\looseness=-1

The composition of the universe takes a forced geometric shape: read at the present epoch, the matter fraction \(\Omm = 1/\pi\), the dark-energy fraction \(\OmL = 1-1/\pi\), and the ordinary-matter fraction \(\Omb = 1/(2\pi^2)\), landing on the measured concordance values --- with the matter-composition ratio \(1/(2\pi)\) and the vacuum scale fixed for all time, and only the present-day split read from the sky (the inventory in one circle: Figure~\ref{fig:pi-locked-chart-omega-hook}). The switch that governs recombination and opens the two-stage Hubble corridor descends from a single whole number in the geometry's integer ledger, the corridor-switch redshift \(z_* = e^7 \approx 1097\) --- landing at the epoch of the first atoms and coinciding to \(0.6\%\) with the operationally measured decoupling redshift, a distinct (topological) quantity. The three great chapters of cosmic history share one origin: inflation as a derived geometric plateau one e-fold below the GUT edge ($n_s = 0.962$, within $0.7\sigma$), dark energy as the residual \(1-1/\pi\) of the same inventory with \(w=-1\) exactly (a sharp two-sided bet: mildly disfavored by the 2024 DESI \(w_0w_a\) hint of evolving dark energy, itself parametrization- and sample-dependent, not yet decisive, and settled by Euclid and Rubin this decade), and dark matter as the Kaxion, predicted near \(8.1\,\mu\)eV in a narrow laboratory window. The Hubble tension is read as a derived background rate near \(67\) --- the value the microwave background itself gives, with an age of \(13.9\) billion years today (\(13.8\) at the \(\Omega_m = 1/\pi\) crossing, \(75\)~Myr before) --- and a local distance ladder biased high; the relict primordial gravitational-wave amplitude is predicted at \(r = 12/N^{*2} = 0.0043\) --- a two-sided target, a factor eight below the present bound, decisive for the Simons Observatory and LiteBIRD. And the three chapters are one descent read three times --- the Kaxion the frozen relic of its middle crossing, dark energy its residual vacuum: \emph{dark energy is the electron of the vacuum} (Part~X); they cannot be tuned separately, and a failure in any one reverberates through all (the whole descent: Figure~\ref{fig:showcase-history-p1}).

\paragraph{How confident --- the triage.}\looseness=-1

The paper is scrupulous about what it has earned. A codebook of twenty-one cosmological observables is classified by provenance: a few are \emph{Derived} outright; most are \emph{Derived-conditional}, closed up to one of ten precisely named mathematical conjectures --- in particular the \(\pi\)-locked fractions rest on the closed-packet conjecture, whose once-open final step has now been traced to a single declared input --- the timing of the present epoch --- with the residual uniformity lemma exact to one part in \(10^{15}\), and says so plainly; and the once-Asserted inflationary sector now enters derived-in-class: the tilt and the tensor ratio are computed (\(n_s = 0.9620\), \(r = 0.0043\)), with a single declared input remaining --- the scalar amplitude. Mapped onto the seventeen recognized problems, \ATen{} offers a derived or conditional reading for the great majority, with a few honestly marked as present frontier. No number is tuned to the data after the fact. The title's word \emph{theorem} is shorthand for exactly this conditional ledger: two derived outright, eighteen modulo a single named lemma each, one a declared input --- the codebook of Part~IX is the binding statement, and the title claims nothing beyond it. The same discipline governs the data confrontation: the zero-parameter temperature spectrum is reported at \(\chi^2/\mathrm{dof} \approx 2.9\) --- one coherent second-order offset, not a likelihood fit --- the correlated \(+2.7\%\) cluster adopted as a declared precision floor, the look-elsewhere exposure bounded and named, never waved away (the audit at a glance: Figure~\ref{fig:showcase-devplot}).

\paragraph{How it can die.}\looseness=-1

A theory with no dials earns credibility only by exposure --- here deliberate, two-sided, and dated: a standard-siren background reading at the ladder's \(73\) rather than \(\approx 67\) kills the corridor; a tensor amplitude off \(r = 0.0043\) in either direction kills the derived launch; \(w\) driven off \(-1\) at Euclid/Rubin precision kills the vacuum terminus; the Kaxion must appear near \(1.95\)~GHz or the identification fails. Nothing can be renegotiated afterwards: zero parameters, maximal exposure.

\paragraph{Striking contrast to \(\Lambda\)CDM and the multiverse.}\looseness=-1

The two standard ways of accounting for the cosmic numbers do not derive them. \(\Lambda\)CDM \emph{measures} its six parameters --- a fitting language, exquisitely accurate, that reads the matter density and the expansion rate off the data. The inflationary multiverse runs the other way, proposing that the constants vary across an unobservable ensemble and that we simply inhabit one that permits observers --- an anthropic selection that accommodates almost any value and therefore predicts none. \GGTheory{} takes the opposite stance: one geometry, fixed, with no ensemble to choose from and no dial to turn, is \emph{obliged} to produce a definite \emph{form} for each cosmic number --- which is why it derives the coupled \(\pi\)-shape of the inventory (\(\Omm = 1/\pi\) and its partners) rather than merely measuring it, and why a single wrong number can kill it. The forms are \emph{forced}, not searched: the integers were fixed upstream for reasons that never mention cosmology; modulo one named lemma, stated in the codebook, there is no room to negotiate.

\paragraph{Where this paper sits.}\looseness=-1

\ATen{} is the cosmological-boundary member of a twenty-two-paper program, inheriting its constitution and geometry from the upstream papers, one story above the Standard Model of \ANine{}. The Kaxion mass and its unique laboratory window are derived in the companion \PaperCOne{}; the TeV lock-cluster phenomenology in \PaperPOne{}. A reader needs none of these to follow the argument. The capstone \AEleven{} carries the same geometry toward life and mind; the falsifiable core here stands on its own.

\paragraph{The identity of the cosmos.}\looseness=-1

Part~X, the paper's deepest, turns from the inventory's values to its \emph{sameness}. Why is dark energy the same on every sightline, every Kaxion identical, the vacuum one vacuum? The particle-physics identity theorem lifts one story: one deepest basin selects the vacuum as it selects the electron; one angle, pinned across the sky during inflation, selects the Kaxion --- two geometric routes to identity, a settled minimum and a frozen phase. Stated honestly, the mold half \emph{reduces} cosmic identity to the vacuum-uniqueness conjecture the program already stakes everything on --- two open puzzles unified, not closed. The classical alternatives --- design, a multiverse, brute fact --- are dissolved rather than refuted: a forced geometry leaves no ensemble to select from and no dial for a designer to have set.

%% v15.0 Stage 2A: on Letter the Executive Summary's closing paragraph
%% spilled two lines onto a page of its own, leaving page 5 at 1.95%%
%% ink with the Keywords block pinned to its foot.  \looseness asks TeX
%% to set the paragraph two lines shorter using the slack already in its
%% line breaks.  NOT ONE WORD IS REMOVED; only the line breaking moves.
\paragraph{A philosophical conclusion.}
\looseness=-2
If the argument holds, the largest questions change character. The mixture of matter, dark matter, and dark energy, and the near-coincidence of their densities in our own epoch, are not numbers the universe happened to draw; they are two faces of one uniqueness statement on a single geometry --- the very statement that, in the companion paper, makes every electron in the universe identical to every other. This is not the claim that the cosmos was designed for us, nor that its history was fixed in every detail; it is the more restrained, and we think more striking, claim that \emph{given one geometry}, a universe with this inventory is a requirement rather than an accident, and the observers who measure it are part of the same closed structure that produced it. Part~X closes with a careful word about determinism (the layered structure is forced, history within the layers is not: freedom sheltered, not preprogrammed) and ends where \ANine{} ends: the two oldest questions, why these particles and why this universe, turn out to be one question, asked twice. As in the companion papers, it marks where physics ends and interpretation begins, and leaves the final step to the reader.
\end{ExecutiveSummary}


\GGKeywords{\GGTheory{} program; cosmological-boundary closure; \(\pi\)-locked inventory (\(\Omega_m = 1/\pi\)); corridor-switch redshift \(z_* = e^7\); Hubble two-stage corridor; Kaxion dark matter; baryogenesis and BBN; primordial gravitational waves; \MSMF{}/\OGN{} cost-minimum principle; codebook provenance triage; second horizon problem; universal-cosmic-identity principle.}

\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\par\vspace*{\stretch{2}}
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily,
  lbl/.style={align=center, inner sep=2pt},
  pct/.style={font=\sffamily\bfseries\huge, text=white, align=center},
]
\def\R{3.4}

% title (self-contained when shared)
\node[font=\sffamily\huge\bfseries] at (0,\R+1.4) {Pi-locked Chart of $\boldsymbol\Omega$};

% ---- wedges (counterclockwise): Omega_Lambda (top+left), Omega_dm (lower-right),
%      Omega_b (thin sliver at 3 o'clock) ----
\filldraw[fill=teal!72, draw=white, line width=1.5pt]
  (0,0) -- (9.119:\R) arc (9.119:254.527:\R) -- cycle;        % Omega_Lambda  245.41 deg
\filldraw[fill=blue!42, draw=white, line width=1.5pt]
  (0,0) -- (254.527:\R) arc (254.527:350.881:\R) -- cycle;    % Omega_dm       96.35 deg
\filldraw[fill=orange!88, draw=white, line width=1.5pt]
  (0,0) -- (350.881:\R) arc (350.881:369.119:\R) -- cycle;    % Omega_b        18.24 deg
\draw[draw=black!22, line width=0.6pt] (0,0) circle (\R);

% ---- in-slice predicted shares ----
\node[pct] at (131.823:0.58*\R) {68.2\%};
\node[pct] at (302.704:0.60*\R) {26.8\%};

% ---- labels written just OUTSIDE each slice (no leader lines), larger fonts ----
% Omega_Lambda -- upper-left
\node[lbl, anchor=east] at (-2.7,3.0)
  {{\LARGE\bfseries Dark energy\, $\boldsymbol{\Omega_\Lambda}$}\\[3pt]
   {\Large $=\,1-1/\pi\,=\,0.6817$}\\[4pt]
   {\large\color{black!50}Planck\, $0.685\pm0.007$}};
% Omega_dm -- lower-right
\node[lbl, anchor=west] at (3.3,-2.2)
  {{\LARGE\bfseries Dark matter\, $\boldsymbol{\Omega_{\mathrm{dm}}}$}\\[3pt]
   {\Large $=\,1/\pi-1/(2\pi^2)\,=\,0.2676$}\\[4pt]
   {\large\color{black!50}Planck\, $0.2657\pm0.0070$}};
% Omega_b -- right (3 o'clock)
\node[lbl, anchor=west] at (3.75,0.2)
  {{\LARGE\bfseries Baryons\, $\boldsymbol{\Omega_b}$}\\[3pt]
   {\Large $=\,1/(2\pi^2)\,=\,0.0507$}\\[4pt]
   {\large\color{black!50}Planck\, $0.0493\pm0.0006$}};

% ---- matter-fraction identity + tagline ----
\node[font=\sffamily\LARGE] at (0,-\R-1.1)
  {$\Omega_m=\Omega_{\mathrm{dm}}+\Omega_b=1/\pi=0.3183$};
\node[font=\sffamily\Large\itshape, text=black!55] at (0,-\R-1.95)
  {one circle, three powers of $\pi$ --- the present-epoch snapshot};
\end{tikzpicture}}
\par\bigskip
\par\vspace*{\stretch{2}}
\captionof{figure}[Pi-locked Chart of $\Omega$: the cosmic inventory as three powers of $\pi$ (front-matter showcase).]{\textbf{Pi-locked Chart of $\Omega$ --- the composition of the universe in one circle.} Every slice of the present-epoch cosmic energy budget is a single closed-form fraction of $\pi$, the three forced together as one coupled set: dark energy $\Omega_\Lambda = 1 - 1/\pi$, dark matter $\Omega_{\mathrm{dm}} = 1/\pi - 1/(2\pi^2)$, and baryons $\Omega_b = 1/(2\pi^2)$, with the two matter slices summing to $\Omega_m = 1/\pi$. Written just outside each slice is the prediction and, beneath it, the Planck~2018 \citep{Planck2018} measured value; the two agree to better than $3\%$. The coupled \(\pi\)-form is the headline structural result, derived (conditional on Conjecture~VIII) as Theorem~\ref{thm:omega-m-pi-topological} in Part~X; the matter-composition ratio \(1/(2\pi)\) and the vacuum scale hold at every epoch, while these absolute present-day fractions are the one snapshot calibrated to the sky (\S\ref{subsec:whynow-form-scale-epoch}).}
\label{fig:pi-locked-chart-omega-hook}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

% --- Front figure 2: the cosmic energy budget pair (hook; twin of Part IV) ---
\begingroup\centering
\vspace*{\fill}
\noindent\includegraphics[width=\textwidth,height=0.85\textheight,keepaspectratio]{A10-whynow-Omega-evolution-PAIR-nocap.pdf}
\par
{\setlength{\abovecaptionskip}{4pt}%
\captionof{figure}[The cosmic energy budget through history (preview; derived in Part~IV).]{\textbf{The
cosmic energy budget through history, shown twice} --- the story of this
paper in one image. Only the matter:vacuum seam moves; the \(\pi\)-locked
inventory is the snapshot at the \(\Omega_m = 1/\pi\) crossing. Derived,
and defended, in Part~IV (Figure~\ref{fig:whynow-omega}).}}
\label{fig:whynow-omega-front}
\par\vspace{\fill}
\endgroup\clearpage

%% ---- v12.100 showcase pages 7-9: history twice, then the audit ----
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily The Entire History of the Universe, Computed ---\\[2pt]
One Two-Basin Descent, from the GUT Edge to the de~Sitter Future\par}
\bigskip
\par\vspace*{\stretch{2}}
\includegraphics[page=1,viewport=40 23 962 1051,clip,width=\textwidth,height=0.76\textheight,keepaspectratio]{a10_cosmic_history.pdf}
\par\smallskip
\par\vspace*{\stretch{2}}
\captionof{figure}[Cosmic history, numerically integrated (showcase; full version in Part~IV).]{\textbf{The
two-basin descent, numerically integrated} (Friedmann + inflaton +
reheating fluid, smooth by construction): the scale factor \(a\), the
expansion rate \(\dot a\), and the temperature \(T\) from the Planck
edge to \(10^{22}\,\mathrm{s}\), every vertical epoch line carrying its
energy scale, the two de~Sitter epochs bracketing the whole history.
Derived, and fully annotated, in Part~IV
(Figure~\ref{fig:partII-history-page1}).}
\label{fig:showcase-history-p1}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily The Same History in the Descent's Own Variables ---\\[2pt]
Scale Dimension, Energy Scale, and the Cosmic Energy Budget\par}
\bigskip
\par\vspace*{\stretch{2}}
\includegraphics[page=2,viewport=39 21 969 1066,clip,width=\textwidth,height=0.76\textheight,keepaspectratio]{a10_cosmic_history.pdf}
\par\smallskip
\par\vspace*{\stretch{2}}
\captionof{figure}[Cosmic history in descent variables (showcase; full version in Part~IV).]{\textbf{The
descent's own variables}: the scale dimension \(r = \ln(\EGUT/\mu)\),
descending monotonically from the primary plateau to the terminus; the
energy scale \(\mu = \rho_{\mathrm{tot}}^{1/4}\), from
\(8.3\times10^{15}\,\mathrm{GeV}\) to the \(\approx2\,\mathrm{meV}\)
dark-energy floor; and the cosmic energy budget \(\Omega_i(t)\), with
the \(\Omega_m = 1/\pi\) crossing marked in red. Fully annotated in
Part~IV (Figure~\ref{fig:partII-history-page2}).}
\label{fig:showcase-history-p2}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a10_cosmic_descent.pdf}
\par\smallskip
\par\vspace*{\stretch{2}}
\captionof{figure}[The cosmic \textsf{GRIP} descent on the radial axis (showcase; master chronology in Part~VIII).]{\textbf{The cosmic \textsf{GRIP} descent, drawn to scale on} \(r=\ln(\EGUT/\mu)\): one open-system descent from the GUT edge (\(r=0\)) to the de~Sitter future, read simultaneously in energy \(\mu\), time \(t\), and the scale factor \(a\). The key epochs are drawn to scale and colored; the late universe is magnified below the break so that recombination, structure, and the present epoch stay legible. Every value is the standard cosmological reading of the master chronology (Table~\ref{tab:partVIII-master-chronology}); nothing on the axis is fitted.}
\label{fig:cosmic-descent-showcase}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.86\textheight,keepaspectratio]{a10_dev_plot_showcase.pdf}
\par\smallskip
\par\vspace*{\stretch{2}}
\captionof{figure}[The parameter-audit Dev plot (showcase; audited row by row in Part~IX).]{\textbf{The
bottom line, in one chart}: the fractional deviation
(pred \(-\) obs)\(/|\)obs\(|\) for the twenty-two observables carrying
an empirical comparison --- eighteen independent predictions (filled),
four anchored by construction (open) --- with nested precision bands at
\(|\mathrm{Dev}| < 10^{-3},\,10^{-2}\). The \(z_*\) entry scores the framework's atomic-chain decoupling prediction against the measured value, the corridor switch \(e^{7}\) annotated as a distinct topological marker (\S\ref{sec:two-redshifts}). Presented with its data table
in Part~I (Figure~\ref{fig:dev-plot-preview}) and reproduced, audited
row by row, as Figure~\ref{fig:a10-dev-plot-linear} in the codebook of
Part~IX.}
\label{fig:showcase-devplot}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% >>>>>>>>>>>>>>>>>> GG-BULK-ONE-DIAGRAM v1.0 >>>>>>>>>>>>>>>>>>
%% Shared showcase plate, byte-identical in A2, A5, A8, A9, A10
%% and P1.  Pure TikZ: it adds no asset to the arXiv package.
%% Every macro it defines is \def'd under the \GGbd prefix and
%% scoped to the picture, so it cannot clash with a host paper.
%% Caption ABOVE, per the frozen float standard.
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{A Six-Dimensional Bulk Whose Fifth Dimension Is Scale\\[2pt]
and Whose Sixth Is Phase}
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The whole program in one diagram, with the scale axis
carrying numbers.} The bulk $\cM_4\times I_r\times S^1_\theta$ is a fiber bundle: over every
point of the Lorentzian stage sit the two \emph{non-spatial} fibers, scale and phase. \AX{2}
(\GGEP) acts \emph{upward} and builds them; \AX{3} (\OGGEP) acts \emph{downward}, and
registration is the projection that erases $(r,\theta)$ --- the origin of quantum behavior.
The vertical axis is calibrated: $\EGUT=\EPl e^{-2\pi}$ at $r=0$, the \OGNfive{} depth
$r=K=35$ at the infrared boundary, and the lock at $r=29.41$, which is $K-2\pi+\ln 2$
exactly.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
  font=\fontsize{11}{13.2}\selectfont,
  every node/.style={inner sep=2pt},
  box/.style={rounded corners=2pt, align=center, inner sep=6pt, line width=0.712pt},
  lbox/.style={rounded corners=2pt, align=left, inner sep=5pt, line width=0.712pt,
               font=\fontsize{8.8}{10.6}\selectfont},
  tick/.style={line width=0.712pt, black!70},
  slice/.style={draw=blue!60!black, fill=blue!16, line width=0.712pt},
  fiber/.style={draw=violet!65!black, line width=0.845pt},
  note/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left},
  rnote/.style={font=\fontsize{8.8}{10.6}\selectfont, align=left, text width=4.85cm},
  sml/.style={font=\fontsize{8.8}{10.6}\selectfont},
  tny/.style={font=\fontsize{6.6}{7.9}\selectfont},
  rlab/.style={font=\sffamily\fontsize{9.6}{11.5}\selectfont},
  mulab/.style={font=\sffamily\fontsize{8.8}{10.6}\selectfont}
]
%% ---- geometry constants ------------------------------------------
\def\GGbdTop{21.0}          % y at r = 0
\def\GGbdSc{0.455000}       % cm per unit r  (15.9 cm for r = 0..35)
\def\GGbdXc{6.8}             % throat centre
\def\GGbdWz{4.05}             % drawn half-width at r = 0
\def\GGbdWk{13.64}           % drawn half-width = wz * exp(-r/13.64)
\def\GGbdXR{11.9}            % right annotation column
\def\GGbdY#1{\GGbdTop-\GGbdSc*(#1)}
\def\GGbdW#1{\GGbdWz*exp(-(#1)/\GGbdWk)}

%% ================= TOP: the two necessities ========================
\node[box, draw=black!60, fill=black!8, text width=15.8cm] (need) at (\GGbdXc,23.15)
 {\textbf{Two structures no local, causal, interacting, charged quantum theory can avoid}\\[2pt]
  a renormalization \emph{scale}\qquad and\qquad a gauge \emph{phase}};
\draw[-{Stealth[length=3.4mm,width=2.6mm]}, line width=0.956pt, black!75]
      (\GGbdXc,22.28) -- (\GGbdXc,21.42);
\node[right, note, fill=white, inner sep=2pt] at (\GGbdXc+0.30,21.88)
 {\AX{2} (\GGEP): \emph{geometrize} each necessity --- this \textbf{builds} the two fibers};

%% ================= THE THROAT =====================================
\filldraw[fill=blue!8, draw=blue!60!black, line width=0.845pt]
  plot[domain=0:35,samples=90,variable=\r] ({\GGbdXc-\GGbdW{\r}},{\GGbdY{\r}})
  -- plot[domain=35:0,samples=90,variable=\r] ({\GGbdXc+\GGbdW{\r}},{\GGbdY{\r}}) -- cycle;
\draw[draw=red!65!black,line width=0.956pt] ({\GGbdXc-\GGbdWz},{\GGbdY{0}}) -- ({\GGbdXc+\GGbdWz},{\GGbdY{0}});

\foreach \r in {0,10,20,29.41,35}{
  \pgfmathsetmacro{\w}{\GGbdWz*exp(-\r/\GGbdWk)}
  \pgfmathsetmacro{\y}{\GGbdTop-\GGbdSc*\r}
  \pgfmathsetmacro{\sk}{0.17+0.145*\w}
  \draw[slice] ({\GGbdXc-\w},\y) -- ({\GGbdXc+\w},\y) -- ({\GGbdXc+\w+\sk},{\y+0.45*\sk})
               -- ({\GGbdXc-\w+\sk},{\y+0.45*\sk}) -- cycle;
}
\node[blue!45!black, align=center] at (\GGbdXc,{\GGbdY{4.6}})
 {\textbf{the 6D bulk}\\[1pt] $\cM_4\times I_r\times S^1_\theta$};

%% ================= THE SCALE AXIS =================================
\draw[-{Stealth[length=3.2mm,width=2.4mm]}, line width=0.956pt, black!75]
      (0,{\GGbdY{-1.6}}) -- (0,{\GGbdY{38.2}});
\node[rotate=90, anchor=south, black!75] at (-4.05,{\GGbdY{17.5}})
 {$I_r$ : the \emph{scale} dimension\ \ $r=\ln(\EGUT/\mu)$};
\node[black!70, rlab, anchor=south east] at (-0.12,{\GGbdY{-2.1}}) {UV};
\node[black!70, rlab, anchor=north east] at (-0.12,{\GGbdY{38.6}}) {IR};
\node[anchor=south west, black!62, align=left,
      font=\sffamily\fontsize{7.4}{8.9}\selectfont] at (-4.30,{\GGbdY{-3.0}})
 {\textsf{boundary energy scale}\\ $\mathsf{\mu=E_{GUT}\,e^{-r}}$\ \ \textsf{[GeV]}};

\foreach \r/\lab/\mulab in {0/{0}/{$\mathsf{2.28\times10^{16}}$},
                            5/{5}/{$\mathsf{1.53\times10^{14}}$},
                            10/{10}/{$\mathsf{1.03\times10^{12}}$},
                            15/{15}/{$\mathsf{6.97\times10^{9}}$},
                            20/{20}/{$\mathsf{4.70\times10^{7}}$},
                            25/{25}/{$\mathsf{3.16\times10^{5}}$},
                            29.41/{29.41}/{$\mathsf{3.85\times10^{3}}$},
                            35/{35}/{$\mathsf{14.4}$}}{
  \draw[tick] (-0.17,{\GGbdY{\r}}) -- (0.17,{\GGbdY{\r}});
  \node[anchor=east, rlab, black!80] at (-0.30,{\GGbdY{\r}}) {$\mathsf{r=\lab}$};
  \node[anchor=east, mulab, black!62] at (-2.00,{\GGbdY{\r}}) {\mulab};
}

%% ---- the UV edge --------------------------------------------------
\node[anchor=north west, rnote, text=red!60!black] at (\GGbdXR,{\GGbdY{-1.1}})
 {\textbf{UV boundary}, $r=0$:
  $\mu=\EGUT=\EPl\,e^{-2\pi}=2.28\times10^{16}$ GeV.
  \emph{The geometry begins here.} $\EPl$ sits at
  $r=-2\pi$, off the diagram.};

%% ---- what a slice is -----------------------------------------------
\node[anchor=east, note, text=blue!45!black] at (4.85,{\GGbdY{12.6}})
 {each $\cM_4$ slice is not a surface\\
  in space --- it is the \emph{whole} 4D\\
  effective theory at the scale $\mu(r)$};
\draw[blue!45!black, line width=0.594pt] (4.95,{\GGbdY{12.6}}) -- ({\GGbdXc-\GGbdW{13.8}-0.05},{\GGbdY{13.8}});

%% ---- the phase fibre ---------------------------------------------
\coordinate (fp) at ({\GGbdXc+\GGbdW{9.6}-0.62},{\GGbdY{9.6}});
\fill (fp) circle (1.6pt);
\draw[fiber] (fp) ellipse [x radius=0.76, y radius=0.30];
\draw[fiber,-{Stealth[length=2.2mm]}] ($(fp)+(0.76,0)$)
      arc [start angle=0, end angle=310, x radius=0.76, y radius=0.30];
\draw[violet!65!black, line width=0.594pt] ($(fp)+(0.78,0.05)$) -- (\GGbdXR-0.15,{\GGbdY{9.6}});
\node[anchor=north west, rnote, text=violet!65!black] at (\GGbdXR,{\GGbdY{8.9}})
 {$S^1_\theta$ : the \emph{phase} dimension --- compact,
  unitary, one Abelian holonomy
  $\oint A_\theta\,d\theta\in U(1)$. Attached at
  \emph{every} $r$; one copy drawn.};

%% ---- the lock ------------------------------------------------------
\node[star, star points=5, star point ratio=0.45, draw=red!70!black,
      fill=red!16, minimum size=0.72cm, line width=0.712pt] (lk)
      at (\GGbdXc,{\GGbdY{29.41}}) {};
\draw[red!70!black, line width=0.594pt] (lk.east) -- (\GGbdXR-0.15,{\GGbdY{29.41}});
\node[anchor=west, rnote, text=red!70!black] at (\GGbdXR,{\GGbdY{29.41}})
 {\textbf{the lock}, at $r=29.41$:
  $\Mlock=\tfrac12\EPl\,e^{-K}=3.85$ TeV, the
  Kaluza--Klein lock cluster.};

%% ---- the IR boundary ------------------------------------------------
\draw[blue!45!black, line width=0.594pt] ({\GGbdXc+\GGbdW{35}},{\GGbdY{35}}) -- (\GGbdXR-0.15,{\GGbdY{35}});
\node[anchor=west, rnote, text=blue!45!black] at (\GGbdXR,{\GGbdY{35}})
 {\textbf{IR boundary}, $r=K=35$: the \OGNfive{} integer
  scale depth, $\mu=14.4$ GeV. \emph{The bulk ends here};
  the cosmological descent runs on to
  $r_{\mathrm{GGA}}\simeq64.5$.};

%% ---- the calibration box ---------------------------------------------
\node[lbox, draw=olive!60!black, fill=yellow!12, text width=4.35cm, anchor=north west]
      at (0.28,{\GGbdY{15.3}})
 {\textbf{Why the lock sits at $r=29.41$, not at $35$}\\[4pt]
  $K=35$ is counted from the \emph{bulk} anchor
  $\tfrac12\EPl$; this axis is anchored at $\EGUT$. They
  differ by\\[4pt]
  \hspace*{0.1cm}$\ln\!\bigl(\tfrac12\EPl/\EGUT\bigr)=2\pi-\ln 2=5.590$,\\[4pt]
  \hspace*{0.1cm}$r_{\mathrm{lock}}=K-2\pi+\ln 2=29.410$.\\[4pt]
  One energy, two readings. Exact, not fitted.};

%% ---- the honesty note about the drawing ------------------------------
\node[lbox, draw=black!50, fill=black!8, text width=4.35cm, anchor=south west]
      at (0.28,{\GGbdY{36.3}})
 {\textbf{The taper is compressed.} Drawn half-width
  $e^{-r/13.64}$: the true $e^{-r}$ slowed $13.64\times$
  so the throat stays visible. To scale, the slice at
  $r=35$ would be $6.3\times10^{-16}$ of the one at
  $r=0$.};

%% ================= OGGEP PROJECTION ==================================
\draw[-{Stealth[length=5mm,width=4mm]}, line width=2.0pt, red!70!black]
      (\GGbdXc,{\GGbdY{36.6}}) -- (\GGbdXc,{\GGbdY{40.9}});
\node[anchor=west, note, text=red!60!black] at (\GGbdXc+0.35,{\GGbdY{38.9}})
 {\AX{3} (\OGGEP): registration $=$ the projection $\pi$ that erases $(r,\theta)$};

%% ================= BOTTOM: the registered world ======================
\node[box, draw=blue!55!black, fill=blue!10, text width=15.8cm] (bdy) at (\GGbdXc,{\GGbdY{43.4}})
 {\textbf{The registered 4D world: open, and therefore quantum}\ \ (\GDOS)\\[2pt]
  superposition \& interference are the unread phase $S^1_\theta$\\[1pt]
  measurement \& irreversibility are the coarse-grained scale $I_r$};
\end{tikzpicture}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% <<<<<<<<<<<<<<<<<< GG-BULK-ONE-DIAGRAM v1.0 <<<<<<<<<<<<<<<<<<
\null\vfill
\begin{center}
\fbox{\begin{minipage}{0.94\textwidth}
\vspace{8pt}
\begin{center}{\large\bfseries What is actually claimed, in one page}\end{center}
\small
\textbf{The input.} One six-dimensional geometry, inherited unchanged from the
particle-physics companion \ANine{}, with zero continuous free parameters: two
declared inputs --- the primordial amplitude \(A_s\) and the timing of the
present epoch --- and the proved \OGNfive{} integer ledger
\((M,N,L;J,K)=(1,3,5;7,35)\), fixed upstream in the particle sector.

\medskip
\textbf{The output.} The present-epoch inventory in closed form ---
\(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\), \(\Omega_b=1/(2\pi^2)\),
\(w=-1\) exactly --- plus the corridor-switch redshift \(z_*=e^{7}\) (within
\(0.6\%\) of the measured decoupling), the background
\(H_0=66.6\pm0.2\)~km/s/Mpc, the tilt \(n_s=1-2/N^*=0.9620\), the tensor ratio
\(r=12/N^{*2}=0.0043\), and a dark-matter particle at \(8.1\,\mu\)eV: twenty-one codebook entries
(twenty-two empirical comparisons in the Dev audit; the extra
comparison is the local-ladder reading, scored against the H0DN
consensus rather than a codebook entry).

\medskip
\textbf{The status, exactly.} Two entries Derived; eighteen
Derived-conditional, each resting on one named, falsifiable conjecture (ten in
all); one Asserted (\(A_s\)). Nothing is tuned after the fact; the correlated
\(+2.7\%\) cluster is adopted as a declared second-order precision floor.

\medskip
\textbf{What is not claimed.} Not a likelihood-level fit (the zero-parameter
TT curve gives \(\chi^2/\mathrm{dof}\approx2.9\) --- one coherent second-order
offset, reported, not negotiated); not the measured recombination redshift
(\(z_*\) is a topological landmark); not a settled resolution of the Hubble
tension (the two-stage corridor is a \emph{candidate}, the joint
Einstein--Boltzmann fit named as open); and not the closure of the Hopf-fiber
lemma VIII-b$''$ or the trials Monte-Carlo, which are owed, not delivered.

\medskip
\textbf{How it dies.} A gravitational-wave siren reading of \(H_0\) at the
ladder's \(73\) rather than \(\approx67\); a tensor measurement inconsistent
with \(r\simeq0.004\) in either direction; a tilt below \(n_s\simeq0.955\);
\(w\) driven off \(-1\) at Euclid/Rubin precision; the Kaxion absent at
\(1.59\)--\(2.13\)~GHz. Each is two-sided, dated, and printed in the falsification ledger: Table~\ref{tab:partXII-falsification-ledger} states the full set in ``how the theory can lose'' language, and Table~\ref{tab:decade-timeline-cosmos} attaches to each one the instrument class and the decision horizon.

\medskip
\textbf{Why it is not numerology.} The integer ledger is pinned upstream, not
searched (unique among the ten closed \OGNfive{} ledgers under \(N=3\),
\(L=5\)); the look-elsewhere exposure is bounded and named
(\(p\sim10^{-3}\)--\(10^{-4}\), assumptions declared), not waved away.
\vspace{8pt}
\end{minipage}}
\end{center}
\vfill\clearpage


% --- Front matter: Overview of Contents + List of Figures (cohort gg-11 macros) ---
\GGfrontmatteronepagetrue
\begingroup\small
\GGOverviewOfContents
\endgroup
\vspace{-1.4em}%
\GGListOfFigures
%% Stage 2A: A10 carries 60+ tables and had no List of Tables.  The
%% one-page front-matter switch is released for it, exactly as A8
%% needed: three lists on one page put the LoT heading into the
%% bottom margin, on top of the folio.
\GGfrontmatteronepagefalse
\GGListOfTables

% --- full table of contents ---
\GGTableOfContents
\GGPreface

This is a long paper about a large subject --- where the universe came from, what it is made of, and why --- and it is written so that a reader who is not a professional cosmologist can follow the argument and judge it. The mathematics is here for those who want it, but the central ideas are meant to be understood in plain language first. Before the formal matter begins, a word of welcome, and an explanation of why this paper is addressed to a wider circle than cosmologists.

\paragraph{Who this paper is for.}
It is written with a particular reader in mind: the graduate student --- in physics, certainly, but just as much in chemistry, biology, engineering, computer science, mathematics, or philosophy --- who suspects that the largest questions can still be asked seriously, and wants to see one attempt worked out in full, with its evidence and its weaknesses in the same place. You were probably taught that the composition of the universe is a list of measured numbers: so much matter, so much dark energy, an expansion rate, a handful of primordial parameters, all read off the sky and none explained. This paper argues that those numbers may instead be consequences --- outputs of one geometry rather than inputs to a fit. You do not need a cosmology background to weigh that claim. Part~I poses every problem from scratch, in words; the appendices of Part~XV are a self-contained graduate course --- the Friedmann equations, inflation, nucleosynthesis, the microwave background, each built line by line for a first-year student meeting it for the first time; and the Questions-and-Answers Part answers, in plain language, the questions people actually ask: about the multiverse, fine-tuning, free will, and what came before the Big Bang.

\paragraph{What this paper claims.}
The claim is easy to state and hard to earn: that the large-scale contents of the universe are not free numbers to be measured, but consequences of a single six-dimensional geometry --- the same geometry that, in a companion paper, fixes the masses of the elementary particles. The fraction of the cosmos in matter, in dark energy, and in ordinary atoms; the moment the first atoms formed; the identity of the dark matter; the cause of inflation --- all are offered as quantities the geometry forces, with no adjustable dials.

\paragraph{An honest accounting.}
A paper that claims this much owes the reader a candid ledger. Of twenty-one cosmological quantities, a few are fully derived, most are derived up to a single clearly named input --- the timing of the present epoch, to which the once-open mathematical step has been traced --- and a handful remain open; the paper marks each one, and tunes nothing to the data after the fact. The test of an honest framework is not that it answers everything on the first pass, but that it always knows which questions it has answered, which it has half-answered, and which it has not yet touched.

\paragraph{The decade of the verdict.}
Here is why this paper is addressed to graduate students in particular, and why now. Every headline number in this book is exposed to instruments that will run in the coming decade. Gravitational-wave standard sirens will read the cosmic expansion rate directly, and this framework needs them to say \(\approx 67\), not the distance ladder's \(73\). The Simons Observatory and LiteBIRD will hunt the primordial gravitational-wave background, and this framework has pinned it, two-sidedly, at \(r = 0.0043\) --- too high fails, too low fails. Euclid and the Rubin Observatory will weigh whether dark energy drifts, and this framework says it cannot: \(w = -1\) exactly, forever. Haloscopes will sweep the microwave band where the Kaxion must sit, near \(1.95\)~GHz, in a window narrow enough to search honestly. By the mid-2030s the sky will have voted on all of it. And the people who will build those instruments, calibrate them, write their analysis pipelines, and catch their systematic errors are, for the most part, the people who are in graduate school today. The verdict on this framework belongs to your generation. That is not a rhetorical flourish; it is the schedule.

\paragraph{An invitation to break it.}
So do not read this paper deferentially. It was written to be attacked, and it tries to make the attack easy: every claim is graded in a published codebook --- Derived, Derived-conditional, or Asserted --- so you can see at a glance which numbers are earned and which are still owed; the retired claims are kept visible together with what refuted them; and the Objections Part collects the ninety hardest questions we could pose against ourselves, because the ones we missed are the ones that matter. If you are looking for a problem worth a thesis chapter, the named open lemmas are stated precisely enough to be settled either way --- above all the Hopf-fiber identification on which the matter fraction \(\Omega_m = 1/\pi\) now visibly rests, and the trials-factor Monte-Carlo that would replace a hand bound with a computation. Break one, and you will have done physics a service and this program a favor; prove one, and you will have promoted an entry of the codebook. Either outcome is progress, and both are open to you.

\paragraph{The larger arc.}
This paper is the middle panel of a deliberate arc. Its companion \ANine{} derives the particle world --- the masses, the charges, the identity of every electron --- from the same geometry; the capstone \AEleven{} carries the arc onward toward life and mind. Whatever your own field, the arc touches it at one point: \emph{identity}. Why is every copy of your object --- every electron, every carbon atom, every ATP molecule, every ribosome --- the same as every other copy, everywhere, including in places that could never have compared notes? Part~X of this paper asks that question about the universe itself and proposes that physics, chemistry, and biology inherit their sameness from a single geometric principle. That is why a chemist, a biologist, or a computer scientist may find that the deepest Part of a cosmology paper speaks, unexpectedly, to their own bench.

\paragraph{How to read this paper.}
No reader needs all the body Parts at once, and there are three honest ways in. \emph{Twenty minutes:} the one-page statement of what is actually claimed (just after the opening figures), then the Executive Summary, then the decade-timeline of kill tests in Part~XIII --- enough to decide whether the argument deserves your evening. \emph{One evening:} add Part~I for the problems, Part~IV for the descent that launches inflation and leaves the dark sector behind, and Part~X for the identity of the cosmos and the view from the top. \emph{A semester:} the whole book, in order, with the Part~XV appendices as the accompanying course and the Objections Part as the exam. And a note for the reader whose interest is strictly the physics: the falsifiable core stands entirely on its own --- a reader may stop at the end of Part~XIII and lose no testable content, taking the Cosmos-to-observer arc of Part~X as optional (\S\ref{sec:scope-and-repro}). Throughout, the body keeps the standard observations --- Planck, the supernovae, the weak-lensing surveys --- in plain view, and every result is stated so that it can be checked, and challenged, on its own. However you enter: welcome. Bring your skepticism; the paper was written in its honor. And if, ten years from now, the sirens say 73, the tensor sky is empty at \(r = 0.004\), or the Kaxion window has been swept clean and silent --- then this book will have failed honorably, in public, the way a scientific claim should. That outcome, too, would be worth your generation's while to deliver.

\GGPart[Introduction]{Introduction}{The oldest questions about the universe --- and a new geometric answer}
\label{sec:partI-introduction}

\section{The oldest question, asked precisely}
\label{sec:introduction}

Every culture that has ever looked up has asked the same three questions. Where did the universe come from? What is it made of? Where is it going? For most of human history these were questions for myth. In the last hundred years they became, for the first time, questions for measurement --- and that transformation is one of the great achievements of the scientific age.

The arc is short and astonishing. Copernicus moved the Earth from the center; Newton gave the heavens and the falling apple one law; Einstein made space and time dynamical \citep{Einstein1917}; Hubble, in 1929, found that the galaxies are flying apart \citep{Hubble1929} and so handed us an expanding universe with a beginning \citep{Friedmann1922,Lemaitre1927}. The hot Big Bang followed, confirmed in 1965 when two engineers found its afterglow as a faint microwave hiss in every direction \citep{PenziasWilson1965}. By the turn of this century these threads had been woven into a single quantitative model --- \LCDM{}, the concordance cosmology \citep{Planck2018} --- that reproduces the cosmic microwave background, the abundances of the light elements, the large-scale distribution of galaxies, and the accelerating expansion \citep{Riess1998,Perlmutter1999}, with a precision that would have seemed miraculous a lifetime ago.

And yet. For all its precision, the standard model of cosmology is, at its foundation, a model that \emph{measures} the universe without \emph{explaining} it. It tells us, to a fraction of a percent, that the cosmos is about five percent ordinary matter, twenty-seven percent dark matter, and sixty-eight percent dark energy --- and it cannot tell us why. It describes an early burst of inflation without saying what drove it. It carries a cosmological constant whose value is the most catastrophic mismatch between theory and experiment in the history of physics. It is the most precisely measured account of the universe ever achieved, and the one most riddled with numbers it can record but never derive.

This paper, \ATen{}, is about those numbers. It proposes that they are not free, not fortuitous, and not forever beyond explanation --- that the composition and history of the universe are written into the shape of a single geometry, and can be read out rather than fitted. It is, in other words, the introduction whose questions the Conclusions of this paper (\S\ref{sec:conclusions}) return to answer.

% ===================================================================
% Part I new foundational sections 2 and 3 (insert AFTER section 1
% "The oldest question, asked precisely", BEFORE "What the standard
% model of cosmology measures but cannot explain"). ASCII-only.
% ===================================================================

\section{A century of cosmology: how the Standard Model of the universe was built}
\label{sec:century-cosmology}

The concordance model did not arrive in one piece, and no one designed it. It is the residue of a century of measurement, in which each new instrument forced a revision the previous picture could not absorb. It is worth retracing that arc, because the model's authority --- and the precise shape of its unfinished business --- comes from how it was built; the milestones are gathered in Figure~\ref{fig:century-timeline}.

\par\medskip\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{A10-intro-century-timeline.pdf}
\captionof{figure}[A century of cosmology, from a static universe to precision \(\Lambda\)CDM.]{\textbf{A century of cosmology.} The discoveries that assembled the Standard Model of the universe, each forced by a measurement the previous picture could not accommodate: the dynamical universe of Einstein, Friedmann, and Lemaitre; Hubble's expansion; the hot Big Bang and its predicted relic radiation; the CMB; inflation; the CMB anisotropies; cosmic acceleration; and the precision concordance of WMAP and Planck. The model was not designed --- it was forced, step by step --- and it nonetheless ends with six numbers it can record but not derive.}
\label{fig:century-timeline}
\end{minipage}\par\medskip

The modern story opens in 1915--1917, when Einstein completed general relativity and applied it to the universe as a whole \citep{Einstein1917}. Expecting a static cosmos, he found that gravity would not permit one: a universe of matter must collapse. To hold it still he introduced a new term into his equations, the cosmological constant \(\Lambda\), tuned to balance the pull of matter. Within a decade that fix was overtaken. Friedmann in 1922 \citep{Friedmann1922} and, independently, Lemaitre in 1927 \citep{Lemaitre1927} showed that the natural solutions of general relativity are not static at all but dynamic --- expanding or contracting --- and Lemaitre connected the expansion to the redshifts of distant nebulae and to the idea of a dense beginning.

The decisive measurement came in 1929. Hubble \citep{Hubble1929}, calibrating distances to galaxies and reading their redshifts, found that recession velocity grows in proportion to distance: the universe is expanding. Einstein abandoned \(\Lambda\), and the expanding universe became the new baseline. Its immediate implication is a beginning: run the expansion backward and the cosmos was once hotter and denser than any furnace. In the 1940s Gamow and his collaborators Alpher and Herman pursued that implication quantitatively \citep{AlpherBetheGamow1948}, working out how the lightest nuclei would have been forged in the first minutes and predicting that the early fireball should have left a faint relic glow, cooled by expansion to a few degrees above absolute zero.

That prediction sat unconfirmed for two decades, until in 1965 Penzias and Wilson \citep{PenziasWilson1965}, troubleshooting a radio antenna, found an inexplicable microwave hiss coming from every direction --- a signal that Dicke and Peebles immediately recognized as the predicted afterglow \citep{DickePeebles1965}. The cosmic microwave background (CMB) turned the hot Big Bang from a hypothesis into the standard framework. Two structural puzzles remained, however. Why is the universe so very flat, when general relativity drives curvature away from flatness over time? And why is the CMB at the same temperature in regions that, in the standard expansion, were never in causal contact? In the early 1980s inflation supplied a single answer to both: Guth \citep{Guth1981}, and then Linde \citep{Linde1982,Linde1983} together with Albrecht and Steinhardt \citep{AlbrechtSteinhardt1982} in the form that survives --- with an independent precursor in Starobinsky's curvature-driven model \citep{Starobinsky1980} --- proposed a brief epoch of accelerated expansion in the first instant, which flattens space and stretches one causal patch across the whole sky, and, as a bonus, converts quantum fluctuations into the seeds of galaxies \citep{MukhanovChibisov1981,Starobinsky1979,MukhanovFeldmanBrandenberger1992}.

The seeds were found in 1992, when the COBE satellite \citep{SmootCOBE1992} detected temperature ripples in the CMB at the level of one part in a hundred thousand --- the large-angle imprint, through the Sachs--Wolfe effect \citep{SachsWolfe1967}, of the density variations from which all structure grew. The last major ingredient arrived at the end of the decade. In 1998--1999 two teams measuring distant Type-Ia supernovae, led by Riess \citep{Riess1998} and Perlmutter \citep{Perlmutter1999}, found that the expansion is not slowing under gravity but \emph{accelerating}. Einstein's discarded \(\Lambda\) returned, no longer as a balancing trick but as the dominant component of the universe. Through the 2000s and 2010s the WMAP and Planck satellites \citep{Planck2018} mapped the CMB to sub-percent precision, pinning every ingredient and welding the threads into a single quantitative model \citep{Weinberg2008}. That model is \LCDM{} --- and the century of measurement that produced it also handed it the unfinished business that the rest of this Part confronts.



\section{\texorpdfstring{\(\Lambda\)CDM}{LCDM}: the concordance model, and why it works}
\label{sec:lcdm-success}

Before asking what \LCDM{} cannot do, one must see clearly how much it does, because a new idea earns nothing by underselling the incumbent it hopes to deepen. \LCDM{} is general relativity applied to a universe that is, on the largest scales, homogeneous and isotropic --- the same everywhere and in every direction --- and expanding according to the Friedmann equations \citep{WeinbergGR1972,MisnerThorneWheeler1973,Wald1984}. Into that geometry it places a short list of ingredients: radiation (the photons of the CMB and the cosmic neutrinos), ordinary ``baryonic'' matter (the atoms of stars, gas, and us), cold dark matter (an unseen, slowly moving substance that outweighs the atoms five to one), and the cosmological constant \(\Lambda\) (the dark energy driving acceleration). Space is taken to be flat, and the whole structure is seeded by a nearly scale-invariant spectrum of primordial fluctuations of the kind inflation predicts \citep{MukhanovFeldmanBrandenberger1992,LythRiotto1999}.

What makes the model so powerful is its economy. The entire expansion history and the full statistical pattern of the CMB and the galaxy distribution are fixed by just \emph{six} numbers: the physical densities of baryons and of cold dark matter, \(\omega_b \equiv \Omega_b h^2\) and \(\omega_c \equiv \Omega_c h^2\); the angular size of the sound horizon \(\theta_*\) that sets the expansion rate; the optical depth \(\tau\) to the epoch when the first stars reionized the gas; and the amplitude \(A_s\) and tilt \(n_s\) of the primordial spectrum. From these six, everything else is derived --- the Hubble constant \(\Hzero\), the matter and dark-energy fractions \(\Omm\) and \(\OmL\), the clustering amplitude \(\sigeight\), and the \(13.8\)-billion-year age of the universe. A model of the entire cosmos runs on six adjustable dials.

\par\medskip\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{A10-intro-lcdm-pillars.pdf}
\captionof{figure}[Why \(\Lambda\)CDM is believed: four independent probes, one six-parameter model.]{\textbf{Why \(\Lambda\)CDM is believed.} Four independent classes of measurement --- the CMB acoustic peaks, the primordial light-element abundances, the baryon acoustic oscillation scale in large-scale structure, and the acceleration read from Type-Ia supernovae --- each probe a different cosmic epoch and a different physical process, yet all are fit by the same six parameters \((\omega_b,\,\omega_c,\,\theta_*,\,\tau,\,A_s,\,n_s)\). The agreement among independent probes, not any single one, is the triumph that makes \(\Lambda\)CDM the Standard Model of cosmology.}
\label{fig:lcdm-pillars}
\end{minipage}\par\medskip

The reason cosmologists trust it is not any single measurement but the way \emph{independent} measurements, probing different epochs and different physics, converge on the same six numbers (Figure~\ref{fig:lcdm-pillars}). Four pillars carry the weight. The first is the CMB itself: before the universe became transparent, photons and baryons formed a single fluid in which sound waves rang, and those waves froze into a precise pattern of hot and cold spots whose statistics --- the heights and spacings of the ``acoustic peaks'' --- are predicted in detail and measured by Planck to exquisite accuracy \citep{SachsWolfe1967,Silk1968,PeeblesYu1970,SunyaevZeldovich1970,HuSugiyama1995,Planck2018}. The second is primordial nucleosynthesis: the same baryon density that fits the CMB peaks independently predicts the amounts of hydrogen, helium, and deuterium forged in the first three minutes, and the prediction matches observation across nine orders of magnitude in abundance \citep{WagonerFowlerHoyle1967,Steigman2007,Fields2011,CyburtFieldsOliveYeh2016,Pitrou2018}. The third is large-scale structure: the same sound horizon that sets the CMB peaks leaves a standard ruler --- the baryon acoustic oscillation scale --- imprinted in the clustering of galaxies \citep{Eisenstein2005,Cole2005}, while the growth of that clustering from the tiny COBE-level seeds, by gravitational instability, follows the rate \LCDM{} predicts \citep{Lifshitz1946,Bardeen1980,BardeenSteinhardtTurner1983,KodamaSasaki1984,PressSchechter1974,BBKS1986,Peebles1993}. The fourth is the accelerating expansion read directly from Type-Ia supernovae \citep{Riess1998,Perlmutter1999}.

These four probes could easily have disagreed. The CMB is a snapshot from redshift \(z \approx 1100\); nucleosynthesis is a clock running at \(z \approx 10^9\); the supernovae and galaxy surveys report from the recent universe at \(z \lesssim 2\). They involve different forces, different particles, and different decades of instrument-building. That they are all fit by one set of six numbers, with no adjustment between them, is the concordance --- and the concordance, not any lone result, is why \LCDM{} is the Standard Model of cosmology. The standard graduate treatments develop each pillar in full \citep{Peebles1993,KolbTurner1990,Dodelson2003,Mukhanov2005,Weinberg2008,LythRiotto1999}.

And yet the triumph is one of \emph{description}, not explanation. \LCDM{} tells us, to a fraction of a percent, that the universe is about five percent atoms, twenty-seven percent dark matter, and sixty-eight percent dark energy --- and it cannot say why those fractions, and not others. It measures the value of \(\Lambda\) without accounting for why it is so extraordinarily small yet not zero \citep{Weinberg1989CC,Carroll2001,PeeblesRatra2003,CopelandSamiTsujikawa2006}; it names dark matter without identifying it \citep{BertoneHooper2018,Marsh2016}; it uses inflation without specifying what drove it. The six numbers are the model's inputs, read from the sky and inserted by hand. The success of \LCDM{} is precisely what sharpens the question this paper asks: granted that these six numbers describe the universe so well, where do the numbers themselves come from?



\section{What the standard model of cosmology measures but cannot explain}
\label{sec:cosmology-challenges}

Let us be clear, first, about how good the incumbent is, because a new idea earns nothing by underselling what it must respect. \LCDM{} --- four-dimensional general relativity, a homogeneous expanding metric, cold dark matter, a vacuum-like dark energy, and a nearly scale-invariant spectrum of primordial ripples --- is a genuine triumph. At the background level it gives the redshift--distance relation, the expansion history, and the timing of the radiation, matter, and acceleration eras; at the perturbative level it gives the acoustic peaks of the microwave background, the baryon-acoustic scale \citep{Eisenstein2005}, the growth of structure, and a long ledger of survey data \citep{Planck2018}; the standard treatments are the textbooks of \citet{Peebles1993,Dodelson2003,Mukhanov2005}. Any deeper theory must reproduce this success before it is allowed to differ from it. \ATen{} keeps every one of these equations.

The trouble is not the equations. It is that the model runs on roughly half a dozen numbers --- the matter density, the baryon density, the dark-energy fraction, the amplitude and tilt of the primordial spectrum --- that are measured from the sky and inserted by hand. The model fits the data beautifully; it does not say where the data come from. Modern cosmology, in this precise sense, resembles the Standard Model of particle physics before a derivational program: the formalism works, but several of its source terms are placeholders, not closed explanations.

Sharpened into a finite audit, the open frontier of contemporary cosmology is the set of seventeen problems in Table~\ref{tab:seventeen-problems}, grouped into five clusters: the two flagship tensions; the dark Universe and its fate; the beginning and the primordial signals; structure, cosmic dawn, and the JWST surprises; and the cracks in the foundations.

\begingroup\small
\begin{longtable}{@{}p{0.045\textwidth}p{0.52\textwidth}p{0.35\textwidth}@{}}
\caption[The open problems of contemporary cosmology (2026), in five clusters]{The open problems of contemporary cosmology (2026), grouped into five clusters, with the reading \ATen{} offers for each. The Hubble tension and the JWST early-galaxy results are widely regarded as the two most exciting current problems. Entries marked \emph{frontier} are ones the framework touches but does not yet close.}
\label{tab:seventeen-problems}\\
\toprule
\textbf{\#} & \textbf{Open problem (current status)} & \textbf{The reading \ATen{} offers}\\
\midrule
\endfirsthead
\toprule
\textbf{\#} & \textbf{Open problem (current status)} & \textbf{The reading \ATen{} offers}\\
\midrule
\endhead
\bottomrule
\endfoot
\multicolumn{3}{@{}l}{\textbf{\textit{I.\ The two flagship tensions}}}\\
1 & Hubble tension: local \(H_0\!\approx\!73\) vs.\ early \(\approx\!67\), now \(\sim\!6\sigma\); JWST sharpened it & Two-stage geometric corridor; switch at \(z_*=e^7\)\\
2 & \(S_8/\sigma_8\) growth tension (easing toward Planck in 2024--25) & Corridor growth response (sign flagged open)\\
\multicolumn{3}{@{}l}{\textbf{\textit{II.\ The dark Universe and its fate}}}\\
3 & The cosmological-constant problem (\(\Lambda\)'s tiny value) & \(\OmL = 1-1/\pi\) as a residual share\\
4 & Constant or evolving dark energy? (DESI \(w_0w_a\) hint) & Predicts \(w=-1\) exactly --- a live, falsifiable stand\\
5 & Cosmic coincidence / ``why now'' & The \(\pi\)-locked inventory \(\Omm=1/\pi\)\\
6 & The nature of dark matter (WIMP nulls, axions, PBH windows) & The Kaxion, \(8.1\,\mu\)eV, in a narrow window\\
\multicolumn{3}{@{}l}{\textbf{\textit{III.\ The beginning and the primordial signals}}}\\
7 & Origin / initial singularity / quantum gravity & A finite, lawful edge at \(\EGUT\)\\
8 & The cause of inflation & A derived plateau at the top of the geometric descent\\
9 & Primordial gravitational waves (no B-modes; \(r<0.036\)) & \(r = 0.0043\) --- a decisive forward prediction\\
10 & Matter--antimatter asymmetry (baryogenesis) & Leptogenesis on the inherited neutrino ledger\\
11 & Big-Bang nucleosynthesis and the lithium-7 problem (factor \(\sim\!3\)) & An \(A=7\) open-system port\\
\multicolumn{3}{@{}l}{\textbf{\textit{IV.\ Structure, cosmic dawn, and the JWST surprises}}}\\
12 & JWST's too-early, too-massive galaxies (\(z\gtrsim10\)) & Early-structure amplification from the corridor\\
13 & ``Little red dots'' / overmassive early black holes & \emph{Frontier}: open-system horizon reading\\
14 & Cosmic dawn / reionization / 21\,cm & \emph{Frontier}: inherited early-structure timing\\
15 & The black-hole information paradox & Horizon as an \OGGEP{} port; exact \(A/4\)\\
\multicolumn{3}{@{}l}{\textbf{\textit{V.\ Cracks in the foundations}}}\\
16 & Cosmic isotropy / dipole and CMB large-angle anomalies & \emph{Frontier}: boundary-projection statistics\\
17 & Neutrino masses and \(N_{\mathrm{eff}}\) (DESI low \(\Sigma m_\nu\)) & Inherited neutrino block; \(\DeltaNeff\!\approx\!0.04\)\\
\end{longtable}
\endgroup

Two of these now dominate the field's attention. The first is the \emph{Hubble tension} \citep{Riess2022,DiValentino2021,Abdalla2022} --- a roughly six-sigma disagreement between the locally measured expansion rate (\(H_0\approx73\) km\,s\(^{-1}\)Mpc\(^{-1}\)) and the value inferred from the early universe (\(\approx67\)), which the James Webb Space Telescope has sharpened rather than removed. The second is JWST's discovery of \emph{surprisingly luminous, massive galaxies already in place less than three hundred million years after the Big Bang}. Honesty requires adding at once that the prevailing reading of the JWST result is that it stresses the astrophysics of how galaxies and their black holes form more than the cosmological framework itself; the stubborn residual is the excess of bright galaxies at the very highest redshifts. \ATen{} takes both seriously, and offers a geometric reading of each.



\subsection{The seventeen open problems, one by one}
\label{subsec:seventeen-problems}

The table above is a census; a broader-audience reader deserves to know what each entry actually \emph{means}. We therefore walk the seventeen problems in turn --- what each one is, why the standard model of cosmology cannot close it, and, in a closing phrase, the reading this paper will offer. The cluster headings are those of Table~\ref{tab:seventeen-problems}.

\smallskip\noindent\textbf{\textit{I.~The two flagship tensions.}}

\textbf{1.~The Hubble tension.} The expansion rate of the universe today, the Hubble constant \(\Hzero\), can be measured two ways: directly, by watching nearby supernovae and Cepheid stars recede, which gives \(\Hzero \approx 73\) km\,s\(^{-1}\)Mpc\(^{-1}\) \citep{Riess2022}; and indirectly, by reading the sound waves frozen into the microwave background of the infant universe and evolving them forward with the standard model, which gives \(\approx 67\). These two numbers should agree, and they do not: the gap is now near six standard deviations \citep{DiValentino2021,Verde2019,Abdalla2022}, far too large to dismiss as measurement error, and the James Webb Space Telescope has tightened rather than relaxed it \citep{Freedman2021,Wong2020}. Either one measurement hides an unknown systematic, or the standard model is missing a physical ingredient between the early and the late universe. \ATen{} reads the two values not as a contradiction but as the two ends of a single geometric ``corridor'' that switches at the integer-ledger redshift \(z_* = e^7\) (Part~VI).

\textbf{2.~The \texorpdfstring{$S_8$--$\sigma_8$}{S8-sigma8} growth tension.} A second, quieter disagreement concerns not how fast the universe expands but how strongly matter has clumped into the cosmic web. Surveys of weak gravitational lensing and galaxy clustering have tended to find the present universe slightly \emph{smoother} --- a lower clustering amplitude \(S_8\) \citep{Asgari2021} --- than the microwave background, evolved forward, predicts. The discrepancy is milder than the Hubble tension and has eased toward agreement in 2024--25 data, so its very reality is still debated. If genuine, it would signal that the growth of structure departs subtly from the standard prescription. \ATen{} treats it as the growth-side response of the same corridor that addresses \(\Hzero\), and flags honestly that the \emph{sign} of the effect is not yet settled (Part~VI).

\smallskip\noindent\textbf{\textit{II.~The dark Universe and its fate.}}

\textbf{3.~The cosmological-constant problem.} The accelerating expansion of the universe is driven by a smoothly spread energy of empty space. If that energy is the quantum vacuum, the naive theoretical estimate exceeds the measured value by some \(120\) orders of magnitude \citep{Weinberg1989CC} --- routinely called the worst quantitative prediction in the history of physics. Why the vacuum energy is so extraordinarily small, yet not exactly zero, is the cosmological-constant problem \citep{Carroll2001}, and it lies on the unhealed seam between quantum theory and gravity. \ATen{} does not tune the number: it reads the dark-energy share as the residual \(\OmL = 1-1/\pi\) of a fixed geometric inventory, with the absolute vacuum scale a meV-level seesaw residue rather than a free input (Parts~IV and~X).

\textbf{4.~Constant or evolving dark energy?} Even granting a small vacuum energy, is it truly \emph{constant} --- a genuine cosmological constant with equation of state \(w = -1\) --- or does it drift slowly with cosmic time \citep{PeeblesRatra2003,CopelandSamiTsujikawa2006}? Recent data from the DESI survey have hinted at a possible evolution, a \(w_0\)--\(w_a\) departure from \(-1\), though the evidence is not yet decisive. The answer decides whether dark energy is a property of spacetime itself or a new dynamical field. \ATen{} takes a sharp, falsifiable stand: the late-time vacuum is a completed geometric transition, so \(w = -1\) \emph{exactly}, with no appreciable clustering --- a clean bet that a secure detection of \(w \neq -1\) would lose (Part~IV).

\textbf{5.~The cosmic coincidence, or ``why now.''} Matter dilutes as the universe expands while the vacuum energy does not, so the two are comparable only during a brief window of cosmic history --- and we happen to be observing inside it. Why should our epoch coincide with the rare moment when matter and dark energy are of the same order? In the standard model this is an unexplained accident stacked on top of the tiny value of \(\Lambda\). \ATen{} reads the present budget as a \(\pi\)-locked inventory with \(\Omm = 1/\pi\) (Theorem~\ref{thm:omega-m-pi-topological}); crucially, it does not pretend the coincidence vanishes, but shows it demoted from a \(10^{120}\) fine-tuning to a mild, few-percent matter of \emph{timing}, with the form forced and only the observation epoch read from the sky (Part~IV, \S\ref{sec:why-now-cosmos}; the form/scale/epoch split is \S\ref{subsec:whynow-form-scale-epoch}).

\textbf{6.~The nature of dark matter.} About a quarter of the cosmic energy budget --- five times the ordinary matter --- is some substance that bends light and binds galaxies but neither emits nor absorbs it. Decades of searches for the leading candidate, weakly interacting massive particles, have returned nulls \citep{BertoneHooper2018}, redirecting attention to ultralight axion-like particles \citep{PecceiQuinn1977,Marsh2016} and to primordial black holes. The standard model of cosmology names dark matter as a slot in the budget but supplies no particle to fill it. \ATen{} identifies it with the Kaxion, a light pseudoscalar near \(8.1\,\mu\)eV predicted in a narrow laboratory window (companion paper C1 \citep{ArisakaC1Kaxion}; Part~V).

\smallskip\noindent\textbf{\textit{III.~The beginning and the primordial signals.}}

\textbf{7.~The origin and the initial singularity.} Run the expansion backward and the standard equations end in a singularity --- an instant of infinite density where the theory itself breaks down. This is less a prediction than a confession that general relativity cannot describe its own beginning \citep{MisnerThorneWheeler1973,Wald1984}, which is widely believed to require a quantum theory of gravity that does not yet exist. \ATen{} replaces the singular beginning with a finite, lawful edge at the grand-unification scale \(\EGUT\), the Planck scale becoming a derived rather than a primary quantity (Parts~II and~III).

\textbf{8.~The cause of inflation.} To explain why the universe is so uniform and so flat, cosmology posits an early burst of accelerated expansion called inflation \citep{Guth1981,Linde1982,AlbrechtSteinhardt1982,Starobinsky1980}. The mechanism works beautifully as a description, but the standard model does not say \emph{what} drove it: the ``inflaton'' field and its potential are added by hand, chosen from a vast menu to fit the data. \ATen{} derives the driver: the scale modulus of the inherited geometry, on a plateau whose \emph{class} is forced by the boundary law and whose height sits one e-fold below the GUT edge --- no field added by hand, no potential chosen from a menu; the lock scale is crossed on the way down, not camped on (Part~IV). And the derived placement pays an unexpected dividend: because the launch sits at the top of the descent, inflation happened \emph{before any matter existed} --- which turns out to answer a question the horizon-problem literature circled for fifty years but never posed in its own right, the \emph{second} horizon problem of Part~X: why the universe is everywhere made of the same things, not merely everywhere the same temperature (\S\ref{sec:second-horizon}).

\textbf{9.~Primordial gravitational waves.} Inflation should imprint a faint background of primordial gravitational waves on the polarization of the microwave background, in a characteristic ``B-mode'' pattern whose strength is the tensor-to-scalar ratio \(r\) \citep{Starobinsky1979,MukhanovChibisov1981}. None has been seen; the present bound is \(r < 0.036\). A detection would pin the energy scale of inflation; continued non-detection presses on high-scale models. \ATen{} makes this a forward prediction with a number on it: the derived plateau fixes \(r = 12/N^{*2} = 0.0043\), a factor eight below the current bound and squarely within Simons-Observatory/LiteBIRD reach --- a two-sided target that a measurement inconsistent with it in either direction would break (Part~IV).

\textbf{10.~The matter--antimatter asymmetry.} The visible universe is built almost entirely of matter, with only a trace of antimatter, in the small but decisive proportion of about six baryons per ten billion photons. Producing this imbalance from a symmetric beginning requires conditions --- baryon-number violation, the right kind of symmetry violation, and a departure from equilibrium \citep{Sakharov1967} --- that the standard model cannot supply in sufficient measure. \ATen{} generates it by leptogenesis \citep{FukugitaYanagida1986,KuzminRubakovShaposhnikov1985} on the neutrino-and-CP ledger inherited from the particle sector, rather than as a new free input (Part~V).

\textbf{11.~Big-Bang nucleosynthesis and the lithium-7 problem.} The light elements forged in the first minutes --- hydrogen, helium, and traces of lithium \citep{AlpherBetheGamow1948,WagonerFowlerHoyle1967} --- match prediction superbly, with one stubborn exception: the observed primordial lithium-7 is about three times smaller than the standard calculation gives \citep{Fields2011,CyburtFieldsOliveYeh2016,Pitrou2018}. Either the stellar measurements or the early-universe nuclear network hides an error, or new physics depletes lithium. \ATen{} reads the deficit as the action of a geometric \(A = 7\) ``port'' that suppresses lithium-7 by the same closed-packet factor that appears in the inventory, leaving the successful hydrogen and helium yields untouched (Part~V).

\smallskip\noindent\textbf{\textit{IV.~Structure, cosmic dawn, and the JWST surprises.}}

\textbf{12.~JWST's too-early, too-massive galaxies.} The James Webb Space Telescope has found surprisingly luminous, massive galaxies already assembled less than a few hundred million years after the Big Bang --- earlier and heavier than the standard structure-formation timeline comfortably allows. Whether this stresses the cosmology or only the astrophysics of early star formation is still debated, but the excess of the brightest high-redshift galaxies is stubborn. \ATen{} reads it as a mild, inherited amplification of early structure from the same corridor that addresses the tensions (Part~VI).

\textbf{13.~``Little red dots'' and overmassive early black holes.} JWST has also revealed a population of compact red objects --- and black holes that appear too massive for their host galaxies and their early epoch --- straining the picture of how black holes grow so fast. This is a young and rapidly moving observational frontier. \ATen{} offers only a tentative, open-system reading of the horizon physics here, and marks the item honestly as \emph{frontier}: touched, not closed (Part~VII).

\textbf{14.~Cosmic dawn, reionization, and the 21\,cm signal.} The epoch when the first stars lit up and reionized the intergalactic hydrogen --- ``cosmic dawn'' --- is only now becoming accessible through the redshifted 21-centimeter line of neutral hydrogen, and early claims of a detection remain contested. It is a frontier of measurement as much as of theory. \ATen{} expects the timing to follow from its inherited early-structure history, but marks the item \emph{frontier}: a target, not yet a closed prediction (Part~VI).

\textbf{15.~The black-hole information paradox.} When a black hole forms and slowly evaporates, the standard combination of quantum theory and general relativity appears to destroy information, in contradiction with the unitarity that quantum mechanics demands \citep{Hawking1975,Bekenstein1973}. Reconciling the two has driven a generation of work on horizons, entropy, and holography \citep{Page1993,Maldacena1998}. \ATen{} reads the horizon as an open-system projection port in the sense of \OGGEP{}, recovering the exact Bekenstein--Hawking entropy \(A/4\) and treating evaporation as a lawful boundary process rather than a loss (Part~VII).

\smallskip\noindent\textbf{\textit{V.~Cracks in the foundations.}}

\textbf{16.~The large-angle anomalies and cosmic isotropy.} The microwave background is very nearly the same in every direction, yet a handful of large-angle features --- curious alignments and a possibly anomalous dipole --- sit oddly with the assumption of perfect statistical isotropy. Each is individually marginal, and whether they are real or flukes is unsettled. \ATen{} suggests they could reflect the statistics of the boundary projection, but marks the item \emph{frontier}: a hint, not a claim (Part~VI).

\textbf{17.~Neutrino masses and the radiation content.} Neutrino oscillations prove that neutrinos have mass, but cosmology constrains only their \emph{sum} --- and recent DESI data push that sum to surprisingly low values, in some tension with particle-physics expectations --- while the effective number of relativistic species \(\Neff\) measures the radiation budget of the early universe \citep{Steigman2007}. \ATen{} inherits the neutrino block from the particle sector and carries a small, definite radiation deviation, \(\DeltaNeff \approx 0.04\) (Part~VI).

Two features of this list matter for what follows. First, most of these are not failures of calculation but gaps in explanation: the Friedmann equations do not tell us what inflation \emph{is}, what the vacuum term \emph{means}, or why the present energy fractions take the values they do. Second, the problems are coupled --- inflation sets the primordial spectrum, which sets the microwave background and the timing of galaxies; the baryon content inferred from the CMB constrains nucleosynthesis; the dark sector governs the growth of structure. A genuine explanation should therefore not arrive as seventeen separate patches. It should arrive as one constrained ledger that the open problems audit at once (that ledger is the cosmological codebook of \S\ref{sec:codebook-table-cosmos}, Table~\ref{tab:cosmological-codebook}).

It is worth saying plainly what does \emph{not} count as such an explanation. To answer ``why is the dark-energy density so small?'' by replying that in a vast multiverse only the rare regions with a small value can host observers is to offer a selection effect, not a derivation --- it explains the specific number we measure no more than ``the survivors survived'' explains who lives. A real answer must produce the value, and must risk being wrong. That is the standard \ATen{} sets for itself.

\section{A radically new idea: the universe as the boundary of a geometry}
\label{sec:gg-from-zero}

Suppose you have never encountered this framework before. Here it is in one sentence. \GGTheory{} (Geometry--Gauge Theory) is a program that derives the laws of physics, the contents of the cosmos, and --- in its later chapters --- the conditions for life and mind, from a \emph{single six-dimensional geometry with no continuous free parameters to adjust}. There are no dials. Once the geometry is fixed, everything else is supposed to follow as a consequence.

The central move is easy to picture even without the mathematics. Our familiar four-dimensional world --- three of space, one of time --- is not, in this picture, the whole of reality. It is the \emph{boundary} of a richer six-dimensional bulk: a throat with one extra radial direction, encoding energy scale, and one extra compact angle, encoding the locked gauge structure. What a laboratory or a telescope can register is the shadow this bulk casts on its four-dimensional edge --- a boundary/bulk relation in the spirit of horizon thermodynamics and the thermodynamic reading of gravity \citep{GibbonsHawking1977,Jacobson1995}. The companion papers of the program show that the same geometry, read at the scale of a single particle, fixes the mass and charge of the electron \citep{ArisakaP3Electron}. \ATen{} is what happens when you read that very same geometry at the largest scale of all. The cosmos and the electron come from one ledger, one story apart.

If that is right, then the deepest question in cosmology changes character. It is no longer ``what values shall we measure and accept?'' but something far sharper:

\begin{quote}
\emph{Are the numbers that define our universe --- the dark-energy density, the matter fraction, the tilt of the primordial spectrum --- free inputs to be measured, or are they theorems of the geometry, fixed by \(\pi\) and a short ledger of integers?}
\end{quote}

This is the cosmological form of an ideal that physics has carried, and quietly abandoned, since Euclid: the ideal of a description with nothing left over to choose. \ATen{}'s wager is that for cosmology the ideal is recoverable --- that the cosmic numbers are not free, not postulated, and not fitted; they are derived.

\section{The central claim: the cosmic inventory consists of open-system observables}
\label{sec:central-claim}

The question just posed has a precise answer, and it is worth stating in one place,
without hedging, before the specific numbers appear. The claim is not that this paper
fits the cosmic inventory better than \(\Lambda\)CDM. It is a claim about \emph{what
the inventory is}. Figure~\ref{fig:central-claim} traces the whole argument in one view.

\par\medskip\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.66\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw, rounded corners, minimum width=104mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=3mm]}, very thick},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open --- the universe most of all}\\ behind a horizon an observer reads only the boundary, never the whole bulk};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is observation itself} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!20, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!22, line width=1pt, below of=gdos] (obs) {\textbf{The cosmic inventory} ($\Omega_m=1/\pi$, $\Omega_\Lambda=1-1/\pi$, $\Omega_b=1/2\pi^2$, $z_\ast=e^7$, the Kaxion)\\ \textbf{consists of registered observables} of this one projection --- \textit{not one is a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!32, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{the inventory is determined}\\ {\footnotesize $1/\pi,\ 1-1/\pi,\ 1/2\pi^2,\ e^{7},\ w=-1$ --- the technical cash-out (rest of this paper)}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D cosmology exists}          (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply to the cosmological boundary}     (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observables forced} (concl);
\end{tikzpicture}}
\captionof{figure}[The central claim of \GGTheory{}, as one deductive funnel.]{\textbf{The central claim, as one deductive funnel.} Because no physical system is closed --- and the universe, seen from behind a horizon, least of all --- the four-dimensional cosmos is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is the open-system law \GDOS{}. The cosmological inventory is therefore a set of \emph{registered observables} of the projection, not free inputs --- and because \GDOS{} is forced by the \GiFour{} and \QiFour{} pillars, those observables are determined. The closed forms derived in the rest of the paper (the matter fraction $1/\pi$, the dark-energy residue $1-1/\pi$, the baryon fraction $1/2\pi^2$, the recombination redshift $e^{7}$,\footnote{Prose that calls $z_*$ ``the
recombination redshift'' does so in the loose epochal sense --- the epoch of
the first atoms. The precise object is always the topological
corridor-switch redshift $z_* = e^{7}$, distinct from the measured
decoupling redshift $z_{\mathrm{dec}} = 1089.92 \pm 0.25$, with which it
coincides to $0.6\%$ (Part~VI).} and $w=-1$) are the technical cash-out of this single recognition. The funnel is identical to the one that opens the companion particle-physics paper \ANine{}; only the payload differs, one story up.}
\label{fig:central-claim}
\end{minipage}\par\medskip

\paragraph{Every physical system is open --- the universe most of all.}
No system in nature is truly closed. A particle is defined by its couplings to an
environment that cannot, even in principle, be switched off; and the universe is the
extreme case, not the exception. An observer inside a cosmological horizon can never
access the whole of the bulk --- only the part that has been projected onto the
four-dimensional boundary they inhabit. What a telescope registers is, by its nature, a
boundary readout of a larger geometry, never the geometry entire. The closed
four-dimensional cosmology in which \(\Omega_m\), \(\Omega_\Lambda\), and the rest appear
as free inputs is therefore an idealization --- and it is precisely the idealization that
makes those numbers look like quantities one is obliged to measure.

\paragraph{Openness forces a projection, and the projection is observation.}
If there is no closed four-dimensional theory, then the four-dimensional world an
observer reports is the image of a deeper bulk under a lawful projection. This is the
content of the third axiom of the \GGConstitution{}, the open-system geometric--gauge
equivalence principle (\OGGEP{}, \AX{3}; \citealp{ArisakaA1Constitution}): every
admissible theory must supply a map from the six-dimensional bulk to the registered
observables on its four-dimensional boundary. The decisive recognition --- absent from a
century of quantum mechanics --- is that this bulk-to-boundary projection \emph{is}
observation itself. Measurement is not a postulate appended to a theory; it is what the
\OGGEP{} projection does, with the recovery of quantum mechanics carried out in \AEight{}
\citep{ArisakaA8QM} and its information-theoretic content in \ASeven{}
\citep{ArisakaA7Qi4}.

\paragraph{The projection is a definite law: \GDOS{}.}
\OGGEP{} requires that a projection exist; \AFive{} makes it dynamical and unique. The
geometric dynamics of open systems (\GDOS{}; \citealp{ArisakaA5GDOS}) is the open-system
law whose stationary, attractor configurations are the values an observer reads off. An
observable, in this framework, is not a coefficient in a cosmological action; it is the
value assigned at a stationary seam of a lawful open-system flow.

\paragraph{Therefore the cosmic inventory consists of observables --- without exception.}
It follows that every entry of the cosmological inventory --- the matter fraction
\(\Omega_m\), the dark-energy fraction \(\Omega_\Lambda\), the baryon fraction
\(\Omega_b\), the recombination redshift \(z_\ast\), the dark-energy equation of state
\(w\), and the Kaxion that fills the dark-matter slot --- is a \emph{registered
observable} of the same \OGGEP{}/\GDOS{} projection. Not one is a free input. The
six-number ``parameter space'' of \(\Lambda\)CDM is not a space of freedoms at all; it is
a list of measurement outcomes of one fixed geometry. This is an \emph{ontological}
statement, and it holds for the whole inventory regardless of how completely the present
paper succeeds in \emph{computing} each entry; the honest accounting of that progress is
the provenance triage of \S\ref{sec:codebook-overview-cosmos}, in which the entries not
yet closed are observables whose determining projection has not been worked out in
detail, not exceptions to the claim.

\paragraph{And the law itself is forced.}
Finally, \GDOS{} is not an arbitrary open-system equation. It is the one consistent with
the two four-pillar structures the constitution erects over it: the geometric pillars
\GiFour{} (\TGG{} $\to$ \CCI{} $\to$ \GFF{} $\to$ \GRIP{}; \citealp{ArisakaA6Gi4}) and
the quantum-information pillars \QiFour{} (\citealp{ArisakaA7Qi4}). Because the projection
that produces the observables is itself forced, the observables it produces are
determined.



Figure~\ref{fig:central-claim} draws this chain as the single deductive funnel it is, and
its conclusion is what makes the rest of the paper possible. For the forty years that
\(\Lambda\)CDM has measured the universe --- and the century that quantum mechanics has
described measurement without deriving it --- the cosmic numbers have been treated as
inputs to a closed four-dimensional theory, and no one asked whether they are instead
observables of an open-system projection from a higher-dimensional bulk. That question,
and its affirmative answer, is the move on which everything below depends: once it is
granted, the matter fraction \(1/\pi\), the dark-energy residue \(1-1/\pi\), and the
integer-ledger redshift \(e^{7}\) cease to be separate mysteries and become readings of
one ledger. The next section states those readings; the body of the paper earns them. The
funnel is deliberately the same one that opens the companion particle-physics paper
\ANine{} --- the cosmos and the electron are read from one ledger, one story apart, and
the whole \GGTheory{} cohort rests on this single recognition.

\section{What this paper claims}
\label{sec:paper-claims}

The claim is bold, and it is meant to be, because the whole value of a zero-parameter theory is that it can be wrong in public. Stated plainly, \ATen{} proposes the following.

The composition of the universe is a geometric figure. The matter fraction is the reciprocal of \(\pi\); the dark-energy fraction is what remains; the baryon fraction is one part in \(2\pi^2\):
\[
\Omm = \frac{1}{\pi} \approx 0.318, \qquad
\OmL = 1 - \frac{1}{\pi} \approx 0.682, \qquad
\Omb = \frac{1}{2\pi^2} \approx 0.051,
\]
landing on the measured concordance values. The three great chapters of cosmic history share one origin: inflation is the first stretch of a geometric descent --- a derived plateau one e-fold below the GUT edge, with the lock scale \(\Mlock \approx 3.85\,\mathrm{TeV}\) crossed promptly on the way down, the second break that stocks the dark sector; dark energy is the residual \(1-1/\pi\) of the same closed inventory, with an equation of state \(w=-1\); and dark matter is a specific particle, the Kaxion, with a definite mass, \(m_{\mathrm{Kaxion}} = 8.1 \pm 0.5~\mu\mathrm{eV}\), ringing in a narrow laboratory window near two gigahertz. The two warring values of the Hubble constant are read not as a crisis but as the two ends of one geometric corridor; and the geometry's corridor switch is fixed at the integer-ledger redshift \(z_{\mathrm{sw}} = e^{7} \approx 1097\), with the surface of last scattering derived from the framework's own atomic chain \(0.4\%\) below it, on the measured value (\S\ref{sec:two-redshifts}).

None of this is offered as a story to be admired. Every one of these claims is staked on an experiment that can refute it within the decade --- a primordial gravitational-wave measurement against the staked \(r = 0.0043\), a dark-matter haloscope scan, a precision measurement of the dark-energy equation of state, where --- against the recent DESI hint that dark energy may be evolving --- \ATen{} predicts \(w=-1\) exactly, and stands or falls on it. \ATen{} keeps \LCDM{} as its boundary baseline, retains its equations and its observables, and aims to \emph{derive the numbers \LCDM{} can only measure}. Exactly how much of that program is proved today, how much rests on a stated conjecture, and how much remains open, is set out honestly in the codebook of Part~IX and the critical discussion of Part~XIII; the present Part states the claims, and the body earns them.

Before the body unpacks the apparatus, it is worth seeing the result itself. Figure~\ref{fig:dev-plot-preview} and Table~\ref{tab:dev-plot-data-preview} set the whole of \ATen{}'s output --- every cosmological number the framework puts on the table --- beside the measured universe, some two hundred pages ahead of where the full audit is developed. Each entry is shown as its fractional deviation from observation: filled markers are genuine zero-parameter predictions, open markers are quantities the framework currently inherits rather than derives, and the color records how much of each has been earned --- Derived, Derived-conditional, or Asserted.

\par\medskip\noindent\begin{minipage}{\textwidth}
\centering
\captionof{table}[Preview: the numerical bottom line of \ATen{}.]{\textbf{The complete numerical bottom line of \ATen{}.} The twenty-two cosmological observables carrying an empirical comparison --- eighteen independent predictions (filled markers in Figure~\ref{fig:dev-plot-preview}) and four anchored by construction (open markers); the inflationary-spectrum block reflects the two-basin derivation ($\ns$ promoted from anchored to derived-conditional, the tensor ratio entering as a firm prediction, $A_s$ the sector's single declared input with its warp candidate flagged). Columns: the GG prediction in closed form; the observed value with its $1\sigma$ uncertainty; the relative deviation $\mathrm{Dev} = (\mathrm{pred}-\mathrm{obs})/|\mathrm{obs}|$; the \emph{pull} $(\mathrm{pred}-\mathrm{obs})/\sigma$ with $\sigma = \sqrt{\sigma_{\rm obs}^2+\sigma_{\rm th}^2}$; and the provenance triage (\textbf{D} = Derived; \textbf{D}$^*$ = Derived-conditional, closed modulo a single named lemma; \textbf{A} = Asserted) with the marker type. Conventions: Hubble constants in km/s/Mpc; the background $\Hzero^{(0)} = 66.6\pm0.2$ is derived from the $\pi$-locked inventory (Section~\ref{sec:h0-derivation}) --- and independently, with zero sky input, by the geometric Route~I at $66.8\pm0.2$, its coefficient $\zeta_\Lambda=\pi/175$ codebook-selected and pre-registered --- and scored against the CMB reading $67.4$ (the coherent $\pi$-offset, Section~\ref{subsec:pi-offset}); the local-ladder entry scores the Local-Hole-lifted background against the H0DN consensus $73.50\pm0.81$, its $-6.8\%$ deviation the named ladder systematic (Section~\ref{sec:h0tension}); the four acoustic rows are CLASS-verified images of the same fractions (correlated, not additional freedom); anchored entries sit at $\mathrm{Dev}=0$ by construction; $\eta_B$ carries an order-of-magnitude transport band; $\ell_D$ is definition-dependent at the percent level; the $z_*$ row scores the framework's audited atomic-chain decoupling prediction ($1091.1$, raw, zero fitted parameters) against the measured value, landing at the $0.1\%$ level, with the topological corridor switch $e^7$ --- a distinct quantity sitting $0.4\%$ above it in e-folds (\S\ref{sec:two-redshifts}) --- annotated in the plot, not scored. \emph{Epoch calibration} (the `cal.'\ entries; violet diamonds in Figure~\ref{fig:dev-plot-preview}): under Planck central values the exact crossing $\Omega_m = 1/\pi$ occurred $75$~Myr before the present ($\pm160$~Myr, i.e.\ $0.5\sigma$; \S\ref{subsec:whynow-form-scale-epoch}), and all matter fractions ($\Omega_m$, $\Omega_b$, $\Omega_{\rm dm}$) decrease with time while $\Omega_\Lambda$ grows --- so, using that single declared input once, the four inventory deviations collapse to (input, $0.00$, $+1.69$, $-0.31$)\%, every remaining residual an epoch-independent ratio within about $1\sigma$. Every entry is reproduced and audited row by row as Table~\ref{tab:dev-plot-data} in the codebook of Part~IX. This is the bottom line stated up front; the body earns it.}
\label{tab:dev-plot-data-preview}

{\renewcommand{\arraystretch}{1.12}\footnotesize\setlength{\tabcolsep}{3.5pt}%
\resizebox{\textwidth}{!}{%
\begin{tabular}{>{\hspace{1.4em}}lllrrc}
\toprule
\textbf{Entry} & \textbf{GG prediction} & \textbf{Observed} & \textbf{Dev} & \textbf{Pull} & \textbf{Status / marker} \\
\midrule
\multicolumn{6}{l}{\textbf{\textit{The $\pi$-locked inventory}}}\\
$\Omm$ & $1/\pi = 0.3183$ & $0.315 \pm 0.007$ & $+1.05\%$ (cal.\ input) & $+0.5$ (---) & D$^*$ / filled \\
$\OmL$ & $1 - 1/\pi = 0.6817$ & $0.685 \pm 0.007$ & $-0.48\%$ (cal.\ $0.00$) & $-0.5$ ($0.0$) & D$^*$ / filled \\
$w$ & $-1$ (exact) & $-1.028 \pm 0.032$ & $+2.72\%$ & $+0.9$ & D$^*$ / filled \\
$\Omb$ & $1/(2\pi^2) = 0.05066$ & $0.0493 \pm 0.0006$ & $+2.76\%$ (cal.\ $+1.69$) & $+2.3$ ($+1.4$) & D$^*$ / filled \\
$\Omdm$ & $1/\pi - 1/(2\pi^2) = 0.2676$ & $0.2657 \pm 0.0070$ & $+0.73\%$ (cal.\ $-0.31$) & $+0.3$ ($-0.1$) & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{Baryogenesis and nucleosynthesis}}}\\
$\eta_B$ & $\approx 7 \times 10^{-10}$ (central 7.34; from $K{=}35$) & $(6.12 \pm 0.04) \times 10^{-10}$ & $+20\%$ & $\ll1$ & D$^*$ / filled \\
${}^7\mathrm{Li}/H$ & $\mathrm{SBBN}/\pi = 1.59 \times 10^{-10}$ & $(1.6 \pm 0.3) \times 10^{-10}$ & $-0.5\%$ & $0.0$ & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{The Hubble constant, recombination, and the acoustic spectrum}}}\\
$\Hzero^{\mathrm{CMB}}$ & $66.6 \pm 0.2$ (derived, inventory) & $67.4 \pm 0.5$ & $-1.19\%$ & $-1.5$ & D$^*$ / filled \\
$\Hzero^{\mathrm{late}}$ & $68.5 \pm 0.6$ (background + Local-Hole lift) & $73.50 \pm 0.81$ & $-6.80\%$ & $-5.0$ & D$^*$ / filled \\
$z_*$ & $1091.1$ (atomic chain, raw; switch $e^7$ annotated) & $1089.92 \pm 0.25$ (decoupling) & $+0.11\%$ & $+0.6$ & D$^*$ / filled \\
$\ell_D$ & $1350 \pm 3$ (Boltzmann, $\pi$-inventory) & $\approx 1345 \pm 15$ & $+0.37\%$ & $+0.3$ & D$^*$ / filled \\
$\omega_b/\omega_m$ & $1/2\pi = 0.15915$ ($h$-free, epoch-free) & $0.15709 \pm 0.0016$ & $+1.31\%$ & $+1.3$ & D / filled \\
$100\,\theta_*$ & $1.0362$ ($r_s/D_M$, inventory) & $1.04109 \pm 0.00030$ & $-0.47\%$ & $-4.3$ & D$^*$ / filled \\
peak ratio $2/1$ & $0.451$ (image of $1/2\pi$) & $0.453 \pm 0.008$ & $-0.30\%$ & $-0.2$ & D$^*$ / filled \\
peak ratio $3/1$ & $0.440$ (image of $1/2\pi$) & $0.443 \pm 0.009$ & $-0.78\%$ & $-0.4$ & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{The inflationary spectrum (primary GUT-adjacent plateau; two-basin descent, $N_2=0$)}}}\\
$\ns$ & $1 - 2/N^* = 0.9620$ ($N^*{=}52.6$, reheat-matched) & $0.9649 \pm 0.0042$ & $-0.30\%$ & $-0.7$ & D$^*$ / filled \\
$r$ & $12/N^{*2} = 0.0043$ & $< 0.036$ (95\%, BK18) & $\times8$ below bound & --- & D$^*$ / filled \\
$A_s$ & anchored (the one input; warp cand.\ $75\%$, flagged) & $(2.10 \pm 0.04) \times 10^{-9}$ & $0$ & $0$ & A / open \\
\multicolumn{6}{l}{\textbf{\textit{Clustering and radiation}}}\\
$\sigeight$ & anchored & $0.811 \pm 0.006$ & $0$ & $0$ & D$^*$ / open \\
$\Seight^{\mathrm{CMB}}$ & anchored & $0.832 \pm 0.013$ & $0$ & $0$ & D$^*$ / open \\
$\Seight^{\mathrm{late}}$ & $0.78 \pm 0.01$ (corridor response) & $0.759^{+0.024}_{-0.021}$ & $+2.77\%$ & $+0.8$ & D$^*$ / filled \\
$N_{\mathrm{eff}}$ & anchored ($3.046 + \DeltaNeff$) & $3.086 \pm 0.18$ & $0$ & $0$ & D / open \\
\bottomrule
\end{tabular}%
}}
\par\smallskip

\end{minipage}\par\medskip

\clearpage
\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.74\textheight,keepaspectratio]{a10_dev_plot_linear.pdf}
\par\bigskip
\captionof{figure}[Preview: the parameter-audit Dev plot (full version in Part~IX).]{\textbf{A10-Cosmos parameter-audit Dev plot.} Fractional deviation $(\mathrm{pred}-\mathrm{obs})/|\mathrm{obs}|$ for the twenty-two observables carrying an empirical comparison, with nested precision bands at $|\mathrm{Dev}|<10^{-3},\,10^{-2}$. \emph{Filled} markers are independent predictions; \emph{open} markers are anchored by construction (Dev $=0$). Color encodes status: green = Derived, blue = Derived (cond.), gold = Asserted. Inner thick bar: empirical $\sigma_{\rm obs}/|\mathrm{obs}|$; outer thin bar: combined $\sqrt{\sigma_{\rm obs}^2+\sigma_{\rm th}^2}/|\mathrm{obs}|$ --- entries carrying a theory tolerance $\sigma_{\rm th}$ are annotated inside the panel. \emph{Violet open diamonds}: epoch-calibrated positions of the four $\pi$-inventory rows (single declared input $t_0 = t_\pi$; see the table caption). The radiation entry is shown as the total $N_{\mathrm{eff}}$. Full data, sources, and uncertainties: Table~\ref{tab:dev-plot-data-preview}. Reproduced and audited entry by entry as Figure~\ref{fig:a10-dev-plot-linear} in the codebook of Part~IX.}
\label{fig:dev-plot-preview}
\end{minipage}
\vfill\clearpage

A few entries show why this is more than a tidy refit of familiar numbers. \LCDM{} \emph{measures} the matter fraction \(\Omm\) and the dark-energy fraction \(\OmL\): they are dialed until the model matches the sky, and the model is silent on why they take the values they do. Here the same two numbers take a \emph{forced} coupled form under one closed geometry --- \(\Omm = 1/\pi\) and \(\OmL = 1 - 1/\pi\) --- landing within a fraction of a percent of Planck at the present epoch, so that the cosmic-coincidence problem \LCDM{} files away is not patched but \emph{demoted}: the geometry removes the \(10^{120}\) vacuum-energy fine-tuning outright and leaves only a mild, few-percent matter of timing, honestly retained rather than declared an identity. The Hubble tension is the second example: where \LCDM{} must pick a single \(H_0\) and live with a five-sigma split between the early- and late-universe readings, the geometry \emph{derives} the background rate outright --- \(\Hzero^{(0)} = 66.6\pm0.2\) from the \(\pi\)-locked inventory, coinciding with the CMB value --- and reads the local-ladder excess as a \(+2\)--\(3\%\) structure lift atop a named calibration systematic, so the ``tension'' is relocated from cosmology to metrology rather than patched with an extra parameter. And where \LCDM{} leaves dark matter an unfilled slot and the dark-energy equation of state a fitted constant, \ATen{} names the dark matter as a definite particle --- the Kaxion, near \(8.1\,\mu\mathrm{eV}\) --- and predicts \(w = -1\) exactly. That is the difference on display: not better fits to numbers already known, but numbers the geometry is \emph{obliged} to produce --- and that a single clean measurement can therefore prove wrong.

The reader who wants the derivations should turn to the codebook of Part~IX, where this figure and table reappear (as Figure~\ref{fig:a10-dev-plot-linear} and Table~\ref{tab:dev-plot-data}) with every entry audited row by row; the reader who wants the bottom line can read it here. After all, a zero-parameter theory is judged first by the numbers it dares to put on the table --- so here they are.






\section{Scope, the cohort, and how to read this paper}
\label{sec:architecture}

The plan of the body is plotted in Figure~\ref{fig:parts-roadmap-a10}, in the roadmap subsection that closes this Part (\S\ref{sec:roadmap-A10}).


A word on scope, on where this paper sits, and on how to read it.

\ATen{} is a presentation of cosmology through one geometric lens; it is not a rejection of standard cosmology. It keeps the standard observables --- \(H(z)\), the CMB spectra \(C_\ell\), the matter power spectrum \(P(k)\), the growth rate \(f\sigma_8\), the density fractions, the light-element abundances --- deliberately front and center, precisely so that it remains clear how the framework can fail. The novelty lies not in new equations but in a constrained source ledger and a single inherited interpretation of the sectors \LCDM{} leaves open.

The paper is the cosmological member of a twenty-two-paper program. The reader does not need to have read the others: this Part and the body are self-contained, and each inherited result is named where it is used. In the program's architecture, \ATen{} sits one story above the Standard Model of particle physics (the companion \ANine{}), applying the same constitution and the same six-dimensional geometry to the universe as a whole; the full cohort map is laid out in the section that closes this Part (Section~\ref{sec:cohort}). Different readers may travel differently: a working cosmologist may go straight to the codebook of Part~IX and the falsification ledger of Part~XIII; a curious newcomer may follow the cosmic story from the beginning through Parts~III--VIII; a skeptic may begin with the Common Objections. All paths meet at the Conclusions.

Throughout, the load-bearing items of the program carry cohort-global \emph{Prefix-Suffix-N} identifiers --- theorems \textsf{T-Cosmos-N}, innovations \textsf{I-Cosmos-N}, objections \textsf{O-Cosmos-N}, falsifiers \textsf{F-Cosmos-N}, and so on --- so that any result can be cited from outside its home paper; the full scheme and the twenty-two-paper suffix list are tabulated in Appendix~\ref{sec:naming-convention}.



\section{The cohort inheritance and the roadmap of the seventeen Parts}
\label{sec:inheritance-and-roadmap-A10}

\noindent
Before the inherited framework is laid out (Part~II), it is useful to
lay out two reader-orientation maps on the same opening: a panorama of
the twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort}, Figure~\ref{fig:cohort-map-a10}), and a panorama
of the seventeen Parts of the present paper (\S\ref{sec:roadmap-A10},
Figure~\ref{fig:parts-roadmap-a10}). A reader of any later Part should
be able to flip back to this Section and orient themselves in both
directions: which cohort companion carries which imported result, and
where the present argument stands in the seventeen-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort}

\GGTheory{} is not a single paper but a connected program of twenty-two. They share one root, one geometry, and one descent: each paper hands the next a smaller, sharper set of conclusions, so that by the time cosmology is reached almost nothing is left to choose. The cleanest way to meet the program is to look at the whole of it at once. Figure~\ref{fig:cohort-map-a10}, which closes this subsection, draws all twenty-two papers in one frame --- twenty-one of them posted, the capstone \AEleven{} in preparation --- and a reader new to \GGTheory{} can read its architecture off the figure from top to bottom --- the constitutional root, the bulk trio, the open-system boundary law and its two pillars, the three textbook readouts, the capstone, the two phenomenology branches, and the Foundations Branch on the left flank --- before following the thread of \ATen{} itself.



\small
\setlength{\LTcapwidth}{\textwidth}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth}}
\caption[The twenty-two-paper \GGTheory{} cohort and its relevance to this cosmology.]{\textbf{The twenty-two-paper \GGTheory{} cohort.} Each paper's main achievement and its relevance to the present cosmology (\ATen{}). Every paper name links to the program homepage, \href{https://arisaka-gg.org/}{arisaka-gg.org}, where the twenty-one posted preprints are collected; the capstone \AEleven{} is in preparation.}
\label{tab:inheritance-map}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Relevance to this cosmology (\ATen{})} \\
\midrule
\endfirsthead
\multicolumn{3}{l}{\small\emph{Table~\ref{tab:inheritance-map}, continued.}}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Relevance to this cosmology (\ATen{})} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\href{https://arisaka-gg.org/}{\AOne{}} & Four axioms and seven postulates fixing what counts as an admissible claim --- the constitution of the whole program & Prevents cosmology from reintroducing hidden free sectors, undefined observables, or unconstrained late-time phenomenology \\
\addlinespace
\href{https://arisaka-gg.org/}{\ATwo{}} & Derivation of the six-dimensional \GG{} bulk, \BIsix{}, \OGNfive{}, compact holonomy, and the unification of all six forces & Fixes the bulk geometry, anomaly structure, Wilson-line setting, and arithmetic ledger that constrain every cosmic sector \\
\addlinespace
\href{https://arisaka-gg.org/}{\AThree{}} & Complete proof of \BIsix{} anomaly freedom, unified gauge consistency, and conservation of the dual current & Closes the ultraviolet legality question and certifies that the projected cosmic ledger may exist globally and locally \\
\addlinespace
\href{https://arisaka-gg.org/}{\AFour{}} & The historical lineage from Euclid to \GG{}, and the zero-parameter integer \OGNfive{} ledger behind the constants & Grounds the arithmetic --- the endpoint \(K=35\) and the \(1/\pi\) inventory --- that this paper reads off without tuning \\
\addlinespace
\href{https://arisaka-gg.org/}{\AFive{}} & \OGGEP{} descent to boundary quantum mechanics, boundary general relativity, and formal 4D QFT: the \GDOS{} boundary law & Establishes that four-dimensional GR is already derived, so \FRW\ cosmology belongs inside the chain, not outside it \\
\addlinespace
\href{https://arisaka-gg.org/}{\ASix{}} & The runtime constitution --- \GRIP{}, the force law \(F_{\GFF}=F_{\GEP}+F_{\CEP}\), and the scale-invariance principle \SIP{} & Provides the open-system runtime and phase-selection grammar for inflation, recombination, structure formation, and horizons \\
\addlinespace
\href{https://arisaka-gg.org/}{\ASeven{}} & The \QiFour{} pillars --- \SeOS{}, \QCI{}, the identity \(\IH+\OH=\EIC\) --- and the five-story informational pagoda & Provides the selector architecture, scale diagnostics, and hologram logic for the higher-floor reading of cosmic evolution \\
\addlinespace
\href{https://arisaka-gg.org/}{\AEight{}} & Quantum mechanics derived from \GGTheory{} --- the four-dimensional reformulation of measurement and state & Supplies the boundary-QM observation layer beneath horizon physics and the observer-projection reading of the cosmos \\
\addlinespace
\href{https://arisaka-gg.org/}{\ANine{}} & The Standard Model as a theorem --- the descent of all twenty-eight Standard-Model parameters & Fixes the particle ledger, electroweak sector, and neutrino block feeding dark matter, baryogenesis, and thermal history \\
\addlinespace
\href{https://arisaka-gg.org/}{\ATen{}} \emph{(this paper)} & \LCDM{} as the cosmological-boundary projection --- the \(\pi\)-locked inventory \(\Omega_m=1/\pi,\ \Omega_\Lambda=1-1/\pi\) & The present paper: the cosmological-boundary member of the consequence trio, one story above \ANine{} \\
\addlinespace
\href{https://arisaka-gg.org/}{\AEleven{}} & The closing capstone --- the universal identity and the \GRIP{} cascade across cosmos, life, and mind & Receives this paper's origin and why-now result and closes the A-series self-referential loop \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPOne{}} & The Kaluza--Klein lock cluster near \(\Mlock\approx3.85\,\TeV\), with explicit LHC and future-collider falsifiers & Supplies the lock scale \(\Mlock\) that enters this paper's \(H_0\) anchor and the KK line of the codebook \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPTwo{}} & A three-leg derivation of \(N=3\) fermion generations from \BIsix{}, MSMF, and the integer ledger & Fixes the generation count and neutrino content behind \(N_{\mathrm{eff}}\) and the cosmic thermal history \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPThree{}} & The electron as the ground state of nature --- the six oldest electron questions, closed & Sibling consequence of the same ledger; anchors the charged-lepton stratum inherited here through \ANine{} \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPFour{}} & The two real sources of the boundary's complex structure --- positive frequency is holomorphy, from the compact phase circle and the causal clock & Foundations Branch; supplies the origin of the derived complex structure this paper leans on at the observer-indexed vacuum and at the dial-free door (T-Cosmos-18) \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperPFive{}} & C, P, and T as orientation reversals of the three real bulk factors --- why every particle has a twin, and why the universe is made of matter & Foundations Branch; the \ANine{}--\ATen{} interface routing the inherited ledger phases into this paper's baryon-asymmetry chain (Part~V, Appendix~\ref{appA:leptogenesis}) \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCOne{}} & Cold dark matter as the non-spatial compact-angle pseudoscalar Kaxion at \(8.1\,\mu\mathrm{eV}\) & Identifies this paper's dark-matter component and its microelectronvolt mass on the same \(S^1_\theta\) winding \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCTwo{}} & The Hubble tension as two projections of one derived anchor --- the early/late corridor & The detailed companion to this paper's \(H_0\) corridor; C2 owns the quantitative linear-response fit A10 inherits \\
\addlinespace
\href{https://arisaka-gg.org/}{\PaperCThree{}} & Black-hole entropy \(A/4\) recovered as a boundary theorem within \GGTheory{}'s holographic ledger & Extends the same boundary law to horizon thermodynamics, consistent with this paper's holographic reading \\
\end{longtable}
\normalsize




The cohort falls into five strata, read top to bottom. At the head sits the constitutional root, \AOne{} \citep{ArisakaA1Constitution}, whose four axioms and seven postulates fix once and for all what counts as an admissible claim --- the rule that forbids cosmology from quietly smuggling in a free sector later on. The root compiles into the \emph{\GG{} trio}: \ATwo{} \citep{ArisakaA2GG6} derives the six-dimensional bulk, \AThree{} \citep{ArisakaA3BI6} proves its \BIsix{} anomaly freedom, and the historical companion \AFour{} \citep{ArisakaA4Euclid} supplies the integer ledger and lineage. From the trio descends the third stratum, the open-system boundary law \AFive{} \citep{ArisakaA5GDOS}, organized by its two pillar papers, \ASix{} \citep{ArisakaA6Gi4} and \ASeven{} \citep{ArisakaA7Qi4}, which between them supply the runtime and selector architecture. The fourth stratum is the \emph{consequence trio} that recovers textbook physics from this single source: quantum mechanics in \AEight{} \citep{ArisakaA8QM}, the Standard Model in \ANine{} \citep{ArisakaA9SM}, and the \LCDM\ cosmology of the present paper, \ATen{}. The capstone \AEleven{} \citep{ArisakaA11Origin} closes the A-series, proving the universal identity and the \GRIP{} cascade that the whole program uses. Hanging off this spine are two phenomenology branches and one foundations column: the Predictions Branch --- \PaperPOne{} \citep{ArisakaP1KK}, \PaperPTwo{} \citep{ArisakaP2N3}, \PaperPThree{} \citep{ArisakaP3Electron} --- and the Cosmology \& Astrophysics Branch --- \PaperCOne{} \citep{ArisakaC1Kaxion}, \PaperCTwo{} \citep{ArisakaC2Hubble}, \PaperCThree{} \citep{ArisakaC3BH}; and on the spine's left flank sits the Foundations Branch --- \PaperPFour{} \citep{ArisakaP4Real} and \PaperPFive{} \citep{ArisakaP5CPT}, read below.

This fixes where the present paper sits. \ATen{} is the cosmological-boundary member of the consequence trio, one story above the Standard Model of \ANine{}: it reads \LCDM\ as the cosmological-boundary projection of the very same six-dimensional bulk, inheriting the boundary law of \AFive{} and the \GRIP{} cascade of \ASix{}. Cosmology, in other words, is not a fresh start. It is the top floor of a building whose foundation, frame, and lower stories are already standing.

\paragraph{The Foundations Branch.} The map's left flank carries the
cohort's third branch: the two member papers whose subject is the
boundary's complex structure and its mirrors, a single column beside
the spine. \PaperPFour{} \citep{ArisakaP4Real} is the
origin-of-\(i\) member paper: it derives the boundary's complex
structure from two real structures of the \ATwo{} arena --- the
compact phase circle and the causal clock --- and this paper's own
derived reading of positive frequency at the observer-indexed vacuum
(\S\ref{subsec:observer-vacuum}), curved-space residue and all, is
that account read on the cosmological stage. \PaperPFive{}
\citep{ArisakaP5CPT} is the \ANine{}--\ATen{} interface of the map:
the ledger phases derived in \ANine{} --- the PMNS Dirac phase and
the Majorana parity \(\alpha_{31} = \pi\) --- reach this paper's
baryon-asymmetry chain through it, and the \(\eta_B\) edge of the map
is exactly the import that Part~V and Appendix~\ref{appA:leptogenesis}
consume, at the grade recorded there: derived-conditional on the named
finite-temperature hypothesis of Appendix~\ref{appA:leptogenesis}.
Both carry P-series designators; their place at the spine's flank, not
the phenomenology rows, marks their foundational role.

The twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map-a10}, which closes this subsection.

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
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 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
 draw=violet!70!black},
 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

%% --- Row 0: axiomatic root ---
\node[root] (A1) at (0, 0)
 {\AOne{} \\[1pt] \footnotesize 4 axioms + 7 postulates};

%% --- Row 1: GG6 Trio ---
\node[bulk] (A2) at (-50mm, -25mm)
 {\textbf{\ATwo{}} \\[-1pt] \footnotesize 6D bulk};
\node[bulk] (A3) at (0, -25mm)
 {\textbf{\AThree{}} \\[-1pt] \footnotesize \BIsix{} anomaly proof};
\node[hist] (A4) at (50mm, -25mm)
 {\textbf{\AFour{}} \\[-1pt] \footnotesize Historical lineage};

%% --- Row 2: A5-GDOS, the boundary law ---
\node[gdos] (A5) at (0, -48mm)
 {\textbf{\AFive{}} \\[-1pt] \footnotesize \GDOS{} boundary law};

%% --- Row 3: pillars ---
\node[pillars] (A6) at (-32mm, -69mm)
 {\textbf{\ASix{}} \\[-1pt] \footnotesize \GiFour{} pillars};
\node[pillars] (A7) at (32mm, -69mm)
 {\textbf{\ASeven{}} \\[-1pt] \footnotesize \QiFour{} pillars};

%% --- Row 4: 4D-Physics Trio ---
\node[qm] (A8) at (-58mm, -91mm)
 {\textbf{\AEight{}} \\[-1pt] \footnotesize 4D QM reformulation};
\node[qm] (A9) at (0, -91mm)
 {\textbf{\ANine{}} \\[-1pt] \footnotesize 28 SM parameters};
\node[qm] (A10) at (58mm, -91mm)
 {\textbf{\ATen{}} \\[-1pt] \footnotesize $\Omega_m = 1/\pi$};

%% --- Row 5: Closing capstone ---
\node[a11] (A11) at (0, -113mm)
 {\textbf{\AEleven{}} \\[-1pt] \footnotesize Closing capstone \\
 \footnotesize universal identity, GRIP cascade};

%% --- Row 6: Predictions Branch (P-series) ---
%% v22: four predictions on the row at 50mm pitch.
\node[pred] (P1) at (-75mm, -138mm)
 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
\node[pred] (P2) at (-25mm, -138mm)
 {\textbf{\PaperPTwo{}} \\[-1pt] \footnotesize Three generations};
\node[pred] (P3) at (25mm, -138mm)
 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};
\node[pred] (P6) at (75mm, -138mm)
 {\textbf{\PaperPSix{}} \\[-1pt] \footnotesize The guarded proton};

%% --- Row 7: Cosmology & Astrophysics Branch (C-series) ---
\node[cosmo] (C1) at (-50mm, -160mm)
 {\textbf{\PaperCOne{}} \\[-1pt] \footnotesize Kaxion dark matter};
\node[cosmo] (C2) at (0, -160mm)
 {\textbf{\PaperCTwo{}} \\[-1pt] \footnotesize Two Hubble constants};
\node[cosmo] (C3) at (50mm, -160mm)
 {\textbf{\PaperCThree{}} \\[-1pt] \footnotesize BH entropy $A/4$};

%% --- Foundations Branch column (v1.3: MP5 raised so its centre
%%     height matches the A11 row). MP4 and MP5 form a single
%%     left-flank column at x = -96mm: MP4 at y = -64mm (unchanged),
%%     MP5 at y = -113mm, level with GG-A11-Origin, so the column
%%     brackets the A8 row with a compact gap.
\node[fnd] (MP4) at (-96mm, -64mm)
 {\textbf{\PaperPFour{}} \\[-1pt] \footnotesize The origin of $i$};
\node[fnd] (MP5) at (-96mm, -113mm)
 {\textbf{\PaperPFive{}} \\[-1pt] \footnotesize Mirrors \& matter (CPT)};

%% --- White-Paper Branch column (v22): the right flank, at the mirror
%%     positions of the Foundations column.  The left flank feeds the
%%     spine; this one reads it, so every edge into it is dashed.
\node[wht] (W1) at (96mm, -64mm)
 {\textbf{\PaperWOne{}} \\[-1pt] \footnotesize Conservation without symmetry};
\node[wht] (W2) at (96mm, -113mm)
 {\textbf{\PaperWTwo{}} \\[-1pt] \footnotesize Sixteen open problems};

%% --- Branch frames (background layer) ---
\begin{scope}[on background layer]
\node[branchframe, draw=cyan!60!black, fill=cyan!4,
 fit=(P1)(P2)(P3)(P6)] (PFRAME) {};
\node[branchframe, draw=teal!60!black, fill=teal!4,
 fit=(C1)(C2)(C3)] (CFRAME) {};
\end{scope}
\node[branchlabel, text=cyan!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west) {Predictions\\Branch\\(P-series)};
\node[branchlabel, text=teal!40!black, anchor=east]
 at ([xshift=-3mm]PFRAME.west |- CFRAME) {Cosmology \&\\Astrophysics\\Branch\\(C-series)};
%% Foundations column label (v1.3): interior placement, in the gap
%% between MP4 and MP5, anchored east so it sits immediately LEFT of
%% the vertical MP4->MP5 edge; house style (small violet) kept. The
%% "shared i" edge label occupies the RIGHT side of the same edge.
\node[branchlabel, text=violet!40!black, anchor=east]
 at ($($(MP4.south)!0.5!(MP5.north)$)+(-1.5mm,0)$) {Foundations\\Branch};
%% v22: the mirror label, held to the RIGHT of the W1->W2 edge --
%% \ATen{}'s grown border reaches x = 78.7mm and an east-anchored
%% label would run into it.
\node[branchlabel, text=brown!45!black, anchor=west]
 at ($($(W1.south)!0.22!(W2.north)$)+(1.5mm,0)$) {White-Paper\\Branch\\(W-series)};

%% --- Edges: A1 -> GG6 Trio ---
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\AX{2}} (A2.north);
\draw[arrowrt] (A1.south) -- node[edgelabel,pos=0.55] {\POS{3}} (A3.north);
\draw[arrowdash] (A1.south) -- (A4.north);

%% --- Edges: GG6 Trio -> A5 ---
\draw[arrow] (A2.south) -- (A5.north west);
\draw[arrow] (A3.south) -- node[edgelabel,pos=0.55] {\AX{3}} (A5.north);
\draw[arrow] (A4.south) -- (A5.north east);

%% --- Edges: A5 -> pillars ---
\draw[arrow] (A5.south) -- (A6.north);
\draw[arrow] (A5.south) -- (A7.north);

%% --- Edges: A5 + pillars -> 4D-Physics Trio ---
\draw[arrow] (A6.south) -- (A8.north);
\draw[arrow] (A5.south).. controls (0, -72mm) and (0, -80mm).. (A9.north);
\draw[arrow] (A7.south) -- (A10.north);

%% --- Edges: 4D-Physics Trio -> A11 ---
\draw[arrow] (A8.south) -- (A11.north west);
\draw[arrow] (A9.south) -- (A11.north);
\draw[arrow] (A10.south) -- (A11.north east);

%% --- Edges: parents -> branch frames ---
\draw[arrowdash] (A9.south west).. controls (-40mm, -105mm)
 and (-84mm, -122mm).. (PFRAME.north west);
\draw[arrowdash] (A9.south east).. controls (40mm, -105mm)
 and (84mm, -122mm).. (PFRAME.north east);
%% v22 RE-ROUTE.  The old line descended through what is now the
%% GG-W2-Holonomy node, and there is no corridor left inboard of it.
%% A polyline rather than a Bezier: a bounding box contains control
%% points, and at the clearance this edge needs those would have sat
%% 24mm beyond any ink, scaling the whole plate down for nothing.
\draw[arrowdash, rounded corners=7pt] (A10.east) -- (127mm,-91mm)
 -- (127mm,-160mm) -- (CFRAME.east);
%% sister import inside the Predictions frame
\draw[arrowdash, draw=cyan!50!black]
 (P2.east) -- node[edgelabel,pos=0.5] {$N{=}3$} (P3.west);
% v22: the second sister import.  \PaperPSix{} says it of
% \PaperPThree{} -- "the sister paper supplies the method this paper
% runs."  Three letters, because a white-filled label wide enough to
% say "method" masks the whole arrow it annotates.
\draw[arrowdash, draw=cyan!50!black]
 (P3.east) -- node[edgelabel,pos=0.5] {GGA} (P6.west);

%% --- Foundations Branch edges (rerouted v1.3 for the raised MP5) ---
%% A2 -> MP4 (solid; Thm 0 arena feed) -- unchanged from v1.0
\draw[arrow] (A2.west) .. controls (-88mm,-30mm) ..
 node[edgelabel,pos=0.55] {Thm~0} (MP4.north);
%% MP4 -> A8 (solid; the origin of i). Departs the east side (MP4.south
%% carries the column edge to MP5) and lands on A8's top border,
%% 6mm left of the A6->A8 arrowhead, staying left of A6 throughout.
\draw[arrow] ([yshift=-3mm]MP4.east) .. controls (-69mm,-74mm) ..
 node[edgelabel,pos=0.45] {origin of $i$} ([xshift=-6mm]A8.north);
%% MP4 -> A2 (dashed deployment: complex-structure machinery), nudged
%% 3mm up the east side so the two east-side departures stay distinct.
\draw[arrowdash] ([yshift=3mm]MP4.east) .. controls (-70mm,-52mm) .. (A2.south west);
%% MP4 -> MP5 (v1.3: SOLID, per the reading conventions -- forced
%% descent, labelled). A short straight vertical between the pair;
%% the "shared i" label sits on the right side of the edge, the
%% Foundations Branch label on the left.
\draw[arrow] (MP4.south) --
 node[edgelabel, pos=0.5, anchor=west, xshift=1mm] {shared $i$}
 (MP5.north);
%% A9 -> MP5 (solid; ledger phases). Leaves A9's west side, dips
%% through the corridor below the A8 row and enters MP5.east
%% horizontally at the A11 row height. Exactly one clean crossing
%% (the A8->A11 spine edge, near x = -40mm); stays above the dashed
%% A9->PFRAME.north-west edge throughout.
\draw[arrow] (A9.west) .. controls (-40mm,-97mm) and (-58mm,-113mm) ..
 node[edgelabel,pos=0.6] {$\delta,\ \alpha_{31}$} (MP5.east);
%% MP5 -> A10 (solid; eta_B). The one long edge: runs left-to-right
%% through the clear corridor BELOW the A11 node (above the
%% Predictions frame), then climbs into A10's bottom border right of
%% the A10->A11 arrowtail; it leaves MP5.south vertically. Controls
%% lifted 5mm with the raised MP5 so the dip still clears A11's
%% bottom border. One clean crossing with each of the two dashed
%% A9->Predictions-frame edges.
\draw[arrow] (MP5.south) .. controls (-96mm,-128mm) and (55mm,-130mm) ..
 node[edgelabel,pos=0.5] {$\eta_B$} ([xshift=8mm]A10.south);
%% MP5 -> A8 (dashed; Wick-quartet/statistics feed for T-A8.12): with
%% MP5 level with A11 this is now a short straight upward edge from
%% the column's top-right corner into A8's bottom border, 8mm left
%% of the A8->A11 arrowtail. No crossings.
\draw[arrowdash] (MP5.north east) -- ([xshift=-8mm]A8.south);
%% --- White-Paper Branch edges (v22) ---
%% A3 -> W1: \AThree{} is \PaperWOne{}'s heaviest spine source and its
%% thematic terminus -- the \BIsix{} anomaly proof is where its
%% Green--Schwarz / Hopkins--Singer ladder ends.
\draw[arrowdash] ([xshift=8mm]A3.south) .. controls (58mm,-43mm) ..
 node[edgelabel,pos=0.55] {\BIsix{}} (W1.west);
%% A10 -> W2: \ATen{} is \PaperWTwo{}'s heaviest source of any kind.
%% At +15%% those two boxes overlap in x (never in y), so this is drawn
%% off A10's south-east corner, clear of the W1->W2 arrowhead.
\draw[arrowdash] (A10.south east) .. controls (84mm,-101mm) ..
 ([xshift=-6mm]W2.north);
%% W1 -> W2, solid and labelled: the mirror of the shared-i edge on the
%% left flank.  \PaperWTwo{} states the dependency itself -- it takes
%% "the structure that the arc arrives at, protection by topology
%% rather than by symmetry".
\draw[arrow] (W1.south) --
 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
 {symmetry $\to$ topology} (W2.north);

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{} \citep{ArisakaP6Proton}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} \citep{ArisakaW1Symmetry} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} \citep{ArisakaW2Holonomy} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-a10}
\end{figure}


\subsection{Roadmap of the seventeen Parts}
\label{sec:roadmap-A10}

The rest of the paper is organized into fifteen further Parts, and the
walk through them is plotted in Figure~\ref{fig:parts-roadmap-a10}.
The present Part --- the entrance --- is nearly done: it has asked the
oldest questions precisely, established the \LCDM{} concordance
baseline with its seventeen open problems, and laid out the two
orientation maps of this closing section. Two Parts then put the
machine on the table. Part~II, the inherited framework, hands the
reader the four ideas the cosmology leans on --- the six-dimensional
bulk, the boundary projection, the zero-parameter integer ledger, and
the open-system runtime --- each proved upstream and imported
wholesale. Part~III sets the cosmological stage: the lawful beginning
at the GUT edge rather than the Planck scale, the inherited scale
chain, general relativity and the Friedmann equations as theorems of
the boundary law, and the radial chronology that orientates the arrow
of cosmic time.

The two Parts that follow are the first half of the derivation core,
and they are highlighted in the roadmap because they deliver the
values of the \(\pi\)-locked inventory in cosmic history. Part~IV runs
the launch and the terminus of the one open-system descent ---
inflation as the opening of the cosmic \GRIP{} descent, with the
derived plateau class, the computed tilt \(\ns = 0.9620\), and the
tensor target \(r = 0.0043\), and dark energy as the ground-state
terminus of the same descent, \(\OmL = 1 - 1/\pi\). Part~V populates
the matter side of the ledger: Kaxion cold dark matter from the
compact-angle sector with the \(\pi\)-locked relic abundance
\(\Omm = 1/\pi\), baryogenesis through the inherited
neutrino-and-CP ledger, and Big Bang nucleosynthesis with the
lithium-7 audit.

Three Parts then confront the derived universe with the sky.
Part~VI takes up the growth of structure and the present-day
tensions --- the CMB baseline and the scalar/tensor audit, the Hubble
two-stage corridor as a candidate resolution of the \(H_0\) tension,
the \(S_8/\sigma_8\) matter-clustering question, and early-galaxy
formation in the JWST era. Part~VII extends the same boundary law to
black holes and primordial relics --- the information paradox under
\OGGEP{}/\GDOS{}, primordial black holes as a feature-dependent
supplementary channel, and the relict primordial gravitational-wave
background. Part~VIII gathers the whole history into the evolutionary
chronology of the universe: the dictionary translating between
\(t\), \(\mu\), \(r\), \(a\), and \(z\), and the master chronology
table from the GUT boundary to the asymptotic de~Sitter future.

Part~IX is the paper's discipline: the cosmological codebook, in
which every observable of the \(\pi\)-locked inventory is graded
\textbf{Derived}, \textbf{Derived (conditional)}, or
\textbf{Asserted}, each conditional entry tied to a single named
lemma. Part~X, the third highlighted Part, is the topological
completion --- \emph{why the universe has the inventory it does}: the
closed-packet derivation of \(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\),
and \(\Omb = 1/(2\pi^2)\), and the universal-cosmic-identity reading
that explains, layer by layer, why the cosmological coincidence is
not a coincidence.

The remaining Parts weigh and close the case. Part~XI is the assembly
--- the direct derivation of \LCDM{} by \GGTheory{}, gathered in one
place: the zeroth and the six silent axioms of the concordance model
replaced by theorems and graded conditionals, the export chain walked
in eight stages from the bulk action to the expansion history, and the
assembled model displayed with the register of every entry and no
fitted continuous parameter. Part~XII is the Summary --- the
cross-paper coherence map, the comprehensive first-order codebook, and
the second-order residual audit. Part~XIII, the Discussions, states
the significance, the comparisons with other cosmological programs,
the predictions for falsification, and the open questions stated as
open. Part~XIV, the Conclusions, returns at the end of the climb to
the summit view: the composition of the universe as geometry, not
coincidence. The back matter is the workshop. Part~XV collects the
Technical Appendices --- a graduate-level course in cosmology read
through \GGTheory{}; Part~XVI answers the Common Objections a working
cosmologist, an inflation theorist, or a philosopher of cosmology
would press hardest; and Part~XVII closes with the annotated
References, each carrying its Relevance note. Read in order, the
seventeen Parts compose a single self-contained argument; read by
need, each Part names its imports and can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the sixteen Parts (house architecture: A2-GG6 v13.3 /
%%  A8-QM v13.2 / A9-SM v20.35 pattern -- vertical Part column, right
%%  phase sidebar, highlighted Parts IV, V, X carrying the central
%%  claim), clearpage-isolated minipage + \captionof.  The label
%%  fig:parts-roadmap-a10 is kept from the retired thirteen-body-Part
%%  flowchart so existing cross-references resolve.
%% ===================================================================
\begin{figure}[p]
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.880\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
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 draw=orange!65!black},
 phaselabel/.style={font=\footnotesize\itshape, text=black!60, anchor=west,
 align=left, inner sep=2pt},
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%% --- Parts I--II: entrance and inheritance ---
\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{Introduction} \\[-1pt]
 \footnotesize The oldest questions about the universe --- and a new geometric answer; the reader's two maps};
\node[found] (II) at (0, -13mm)
 {\textbf{Part~II}\quad\textbf{The inherited framework} \\[-1pt]
 \footnotesize The 6D bulk, the boundary projection, the zero-parameter integer ledger, and the open-system runtime};

%% --- Part III: the stage ---
\node[appar] (III) at (0, -26mm)
 {\textbf{Part~III}\quad\textbf{The cosmological stage} \\[-1pt]
 \footnotesize The \(\EGUT\) launch, the inherited scale chain, Friedmann kinematics as theorems, the radial chronology};

%% --- Parts IV--V: the inventory in cosmic history (highlighted) ---
\node[innov] (IV) at (0, -42mm)
 {\textbf{Part~IV}\quad\textbf{Inflation and dark energy: the cosmic \GRIP{} descent} \\[1pt]
 \footnotesize \(\OmL = 1 - 1/\pi\): the launch of inflation (derived plateau, \(\ns = 0.9620\), \(r = 0.0043\)) \\[-1pt]
 \footnotesize and the ground-state terminus of the one open-system descent};
\node[innov] (V) at (0, -60mm)
 {\textbf{Part~V}\quad\textbf{Origin of matter} \\[1pt]
 \footnotesize \(\Omm = 1/\pi\): Kaxion dark matter from the compact-angle sector, \\[-1pt]
 \footnotesize baryogenesis on the inherited neutrino-and-CP ledger, and BBN with the lithium-7 audit};

%% --- Parts VI--VIII: confrontation and chronology ---
\node[princ] (VI) at (0, -75mm)
 {\textbf{Part~VI}\quad\textbf{Growth of structure and the present-day tensions} \\[-1pt]
 \footnotesize CMB baseline; the \(H_0\) two-stage corridor; \(S_8/\sigma_8\); early galaxies in the JWST era};
\node[princ] (VII) at (0, -88mm)
 {\textbf{Part~VII}\quad\textbf{Black holes and primordial relics} \\[-1pt]
 \footnotesize The information paradox under \OGGEP{}/\GDOS{}; PBHs; the primordial gravitational-wave background};
\node[princ] (VIII) at (0, -101mm)
 {\textbf{Part~VIII}\quad\textbf{Evolutionary chronology of the universe} \\[-1pt]
 \footnotesize The \(t,\mu,r,a,z\) dictionary and the master chronology table, GUT boundary to de~Sitter future};

%% --- Part IX: provenance ---
\node[results] (IX) at (0, -114mm)
 {\textbf{Part~IX}\quad\textbf{Cosmological codebook} \\[-1pt]
 \footnotesize Derived \,\textbar\, Derived (conditional) \,\textbar\, Asserted --- the provenance triage};

%% --- Part X: deep origin (highlighted) ---
\node[innov] (X) at (0, -130mm)
 {\textbf{Part~X}\quad\textbf{Topological completion} \\[1pt]
 \footnotesize \emph{\textbf{Why the universe has the inventory it does:}} \\[-1pt]
 \footnotesize \(\Omm{=}1/\pi\), \(\OmL{=}1{-}1/\pi\), \(\Omb{=}1/2\pi^2\) from closed-packet geometry};

%% --- Part XI: the assembly ---
\node[results] (XIA) at (0, -145mm)
 {\textbf{Part~XI}\quad\textbf{The assembled \LCDM{}} \\[-1pt]
 \footnotesize The direct derivation of \LCDM{}: silent axioms replaced, the export chain C-1--C-8, no fitted parameter};

%% --- Parts XII--XIII: summary and critique ---
\node[results] (XI) at (0, -158mm)
 {\textbf{Part~XII}\quad\textbf{Summary} \\[-1pt]
 \footnotesize Cross-paper coherence map \,+\, first-order codebook \,+\, second-order residual audit};
\node[results] (XII) at (0, -171mm)
 {\textbf{Part~XIII}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Significance \,\textbar\, comparison \,\textbar\, falsification predictions \,\textbar\, open questions};

%% --- Part XIII: conclusions ---
\node[summit] (XIII) at (0, -184mm)
 {\textbf{Part~XIV}\quad\textbf{Conclusions: the summit view} \\[-1pt]
 \footnotesize The composition of the universe is geometry, not coincidence};

%% --- Parts XIV--XVI: back matter ---
\node[back] (XIV) at (0, -197.5mm)
 {\textbf{Part~XV}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize A graduate course in cosmology read through \GGTheory{}};
\node[back] (XV) at (0, -211mm)
 {\textbf{Part~XVI}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize The strongest objections, answered, with pointers};
\node[back] (XVI) at (0, -224.5mm)
 {\textbf{Part~XVII}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows (the highlighted core in innov-orange) ---
\draw[arrow] (I.south) -- (II.north);
\draw[arrow] (II.south) -- (III.north);
\draw[arrowinnov] (III.south) -- (IV.north);
\draw[arrowinnov] (IV.south) -- (V.north);
\draw[arrowinnov] (V.south) -- (VI.north);
\draw[arrow] (VI.south) -- (VII.north);
\draw[arrow] (VII.south) -- (VIII.north);
\draw[arrow] (VIII.south) -- (IX.north);
\draw[arrowinnov] (IX.south) -- (X.north);
\draw[arrowinnov] (X.south) -- (XIA.north);
\draw[arrow] (XIA.south) -- (XI.north);
\draw[arrow] (XI.south) -- (XII.north);
\draw[arrow] (XII.south) -- (XIII.north);
\draw[arrow] (XIII.south) -- (XIV.north);
\draw[arrow] (XIV.south) -- (XV.north);
\draw[arrow] (XV.south) -- (XVI.north);

%% --- Phase labels (right sidebar).  ALL TEN SIT AT x = 75mm as of
%% v15.5.  The nominal half-width is 60mm, but four boxes are wider
%% than their 120mm minimum because their text is: Part II reaches
%% 71.55mm, Part III 70.9mm, Part VII 70.2mm.  Labels at 67mm were
%% overprinted by those three.  75mm clears the widest by 3.45mm and
%% is where the two highlighted labels already sat.
\node[phaselabel] at (75mm, -6.5mm) {\emph{Entrance \&} \\ \emph{inheritance}};
\node[phaselabel] at (75mm, -26mm) {\emph{The stage}};
\node[phaselabel, text=orange!70!black] at (75mm, -51mm)
 {\emph{\textbf{The inventory in}} \\ \emph{\textbf{cosmic history}}};
\node[phaselabel] at (75mm, -88mm) {\emph{Confrontation \&} \\ \emph{chronology}};
\node[phaselabel] at (75mm, -114mm) {\emph{Provenance}};
\node[phaselabel, text=orange!70!black] at (75mm, -130mm)
 {\emph{\textbf{Deep origin}}};
\node[phaselabel] at (75mm, -145mm) {\emph{The assembly}};
\node[phaselabel] at (75mm, -164.5mm) {\emph{Summary \&} \\ \emph{critique}};
\node[phaselabel] at (75mm, -184mm) {\emph{Summit}};
\node[phaselabel] at (75mm, -211mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the seventeen Parts of GG-A10-Cosmos.]{Roadmap of the seventeen Parts of \textsf{GG-A10-Cosmos}. The Parts run top to bottom; the sidebar names the ten phases. Parts~IV, V, and~X are highlighted because together they carry the \(\pi\)-locked cosmic inventory \(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\), \(\Omb = 1/(2\pi^2)\): derived in cosmic history in Parts~IV--V, given its geometric origin in Part~X, and assembled into the concordance model in Part~XI. The full walk is Sec.~\ref{sec:roadmap-A10}.}
\label{fig:parts-roadmap-a10}
\end{minipage}
\end{figure}


One thread runs through all seventeen problems, and it is worth stating in plain words before the formal work begins. Every one of them is, at bottom, the same question asked of a different number: \emph{why this value, and not another?} A science that can only measure must treat that question as meaningless; a science that can derive must treat it as the whole point. The wager of this paper is that cosmology is ready to move from the first kind of science to the second --- and that the price of the move, paid in full in Part~IX, is saying exactly which answers are earned and which are still owed.

This Part has done its work: it has established the concordance baseline, posed the open questions, and placed the paper within its twenty-two-member program. The body answers those questions, problem by problem, on a single inherited ledger; and the Conclusions return, at the end of the climb, to the view from the summit --- the universe as geometry, not coincidence.







\section{Part~I in plain English: what the standard model of the universe does and does not explain}
The reason this Part spends so long on what \LCDM{} gets right is that the
argument of the paper depends on it. A framework that offered a new account of
the cosmic inventory while quietly needing the standard model to be wrong
somewhere would be easy to dismiss and would deserve to be. The claim here is
the harder one: that the concordance model is an excellent description, that
its six fitted parameters are measured to a precision that is one of the real
achievements of the century, and that the question it does not answer is why
those six numbers have the values they do.

That question is not a complaint about precision. It is a question about
provenance. A parameter that is measured is a parameter that could have come
out otherwise, and the standard model has no machinery that would have
objected. What follows proposes that the inventory is not a set of
measurements at all but a set of open-system observables --- quantities forced
by the geometry that produces the boundary on which they are measured --- and
that if this is right, the correct description of a cosmic number is a closed
form rather than a fitted value.

A reader should hold two things in mind through the rest of the paper. The
first is that a closed form which agrees with a measurement to a few percent
is a claim about structure, not a claim about precision, and it must never be
sold as the second. The second is that this Part has stated the problem in the
strongest available form: seventeen open problems, each of them a real one,
and each of them a place the framework can be wrong. The Parts that follow are
answerable to that list.

\GGPart[The inherited framework]{The inherited framework: \GGTheory{} in one Part}{What the cosmology inherits --- the six-dimensional bulk, the boundary projection, the zero-parameter integer ledger, and the open-system runtime}


This Part answers a single question: \emph{what is \GGTheory{}, and what does a cosmology built on it inherit?} Part~I established the \LCDM\ baseline, posed the open questions of cosmology, and mapped the twenty-two-paper program this cosmology belongs to; before we can claim geometric answers to those questions, the reader deserves a clear and unhurried picture of the machine the answers come out of. Nothing here is new physics. Everything in this Part was proved upstream, in the companion papers of the cohort, and is imported into \ATen{} wholesale. The job of this Part is to make that inheritance \emph{intuitive} --- to hand a reader who has seen only Part~I the four ideas that the rest of the paper leans on, in the order in which they build on one another. The actual cosmological history --- the beginning at the GUT edge, the Friedmann kinematics, the expansion and cooling of the universe --- is set up in the next Part, once the framework is in hand.


\section{One geometry, no dials: the six-dimensional bulk}

At the center of the whole program is a single geometric object --- the constitution of \AOne{} \citep{ArisakaA1Constitution} fixes what may count as an admissible claim, and \ATwo{} \citep{ArisakaA2GG6} derives the geometry itself --- and it is worth meeting it in plain language before meeting it in equations. \GGTheory{} posits that reality is a \emph{six-dimensional bulk}, a tapering ``throat'' of the form \(\mathcal{M}_4 \times I_r \times S^1_\theta\). Three pieces make it up. The familiar four dimensions \(\mathcal{M}_4\) are the spacetime we live in. A single radial direction \(I_r\) runs along the throat from a wide ultraviolet mouth down to a narrow infrared end; this is the energy axis, and it is best read as the renormalization-group scale, \(r=\ln(\EGUT/\mu)\) --- moving along \(r\) is nothing other than running a coupling from high energy to low. And a small compact angle \(S^1_\theta\), the phase fiber, carries the locked gauge structure: its winding is where the dark sector will eventually live.

The word ``warped'' deserves a warning, because it is borrowed from a different picture. Here it does \emph{not} mean a Randall--Sundrum extra spatial dimension \citep{RandallSundrum1999} with a curved metric in the usual sense. It means RG running along \(r\): the warp factor is the field-theory / holographic-RG reading of scale dependence, exactly as the rest of the cohort reads it. With that one substitution, the geometry stops being exotic and becomes familiar --- ``deeper into the throat'' simply means ``lower in energy.''

The single most important intuition to carry forward is this: \emph{our four-dimensional world is the boundary of this bulk}. We do not live inside the throat; we live on its edge, and what we call physics is the shadow the bulk casts there. How that shadow is cast --- the boundary projection itself --- is the subject of the next section, where Figure~\ref{fig:bulk-boundary} draws the throat, its radial scale axis, its compact fiber, and the four-dimensional boundary slab on which ordinary cosmology lives.

%  Arisaka-GG-A10-Cosmos : Figure -- six-dimensional bulk and
%  its OGGEP projection to the 4D cosmological boundary.
%  Self-contained TikZ fragment.  Requires (already loaded in the paper):
%    \usepackage{ucla-arisaka-gg-11}   % \EPl \EGUT \Mlock \GDOS \OGGEP \GGTheory \GRIP
%    \usepackage{tikz}
%    \usetikzlibrary{arrows.meta,positioning,fit,calc,backgrounds,%
%                    shapes.geometric,decorations.pathmorphing}
% =====================================================================



\emph{The geometry, made explicit.} Behind the picture is a definite, parameter-free metric. In the natural units of the program the bulk line element is
\[
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \;+\; dr^2 \;+\; d\theta^2,
\qquad r\in[0,K],\quad \theta\in[0,2\pi),
\]
with \(\eta_{\mu\nu}\) the Minkowski metric of the four-dimensional boundary \citep{ArisakaA2GG6}. The warp profile \(A(r)=r\) is not chosen but forced, by demanding constant bulk curvature with a flat boundary (\(A''(r)=0\)); there is no curvature scale, no compactification radius, and no brane tension to tune. The radial coordinate is literally the renormalization-group logarithm, \(r=\ln(\EGUT/\mu)\), running from the ultraviolet edge at \(r=0\) (\(\EGUT\simeq2.3\times10^{16}\,\mathrm{GeV}\)) down to the integer endpoint \(r_{\mathrm{IR}}=K=35\); the Planck--electroweak hierarchy is then the topological number \(e^{-K}\), not a fine-tuned ratio.

\emph{Six dimensions are four plus one plus one.} The count is forced role by role \citep{ArisakaA2GG6}: four Lorentzian directions \(\mathcal{M}_4\) carry registered observations and the only length dimension; one non-compact \emph{scale} direction \(I_r\) is the geometrized RG flow (a dimensionless energy logarithm, not space); and one compact \emph{phase} circle \(S^1_\theta\) carries integer-quantized Wilson-line holonomy --- the dark sector lives in its winding. The three bulk directions are exactly what is needed to absorb the three dimensionful constants \(c\), \(G_N\), and \(\hbar\); the boundary supplies the remaining two, \(k_B\) and the energy--information quantum \(\QB\). This ``three-plus-two'' architecture is the spine of the whole ledger.

\emph{One geometry carries all six forces.} On the bulk the gauge and gravitational structure is a single group,
\[
G_{\mathrm{Bulk}} \;=\; \mathrm{Spin}(1,5)\,\times\,SU(5)\,\times\,U(1)_B\,\times\,U(1)_{Qi},
\]
with the Lorentz/spin factor \(\mathrm{Spin}(1,5)\) (the gravity sector) on equal footing with the internal factors \citep{ArisakaA2GG6}. Here \(SU(5)\) is the minimal simple unification containing the Standard Model; \(U(1)_B\) is gauged baryon number, which turns proton stability into a theorem; and \(U(1)_{Qi}\) is the informational gauge force that the \QiFour{} pillars later use. Projected onto the four-dimensional boundary this becomes the six forces an observer actually measures --- strong, weak, electromagnetic, baryonic, informational, and gravitational --- with three abelian invariances surviving after electroweak breaking. Crucially for cosmology, the same compact circle \(S^1_\theta\) that quantizes these holonomies is where the Kaxion dark matter and the \(\pi\)-locked inventory are read off: the cosmos and the particle ledger are two projections of this one object, one story apart.

\section{The boundary projection: how four-dimensional physics arises}

If the bulk is the whole of reality, why do we only ever measure four-dimensional physics? The answer is the \emph{boundary law}, and it is the deepest single result the cosmology inherits. An observer behind a cosmological horizon can never access the closed bulk; what a telescope registers is the projection of the bulk onto the four-dimensional edge the observer inhabits. \AFive{} \citep{ArisakaA5GDOS} raises this to a principle --- the open-system geometric--gauge equivalence principle (\OGGEP{}), the third axiom of the constitution --- which requires every admissible theory to supply an explicit dictionary
\[
\mathcal{D}_{\OGGEP}:\;\bigl(\mathcal{H}_{\mathrm{Bulk}},\,\rho_{\mathrm{Bulk}},\,U_{\mathrm{Bulk}}\bigr)\;\longmapsto\;\bigl(\mathcal{H}_{\mathrm{Bdy}},\,\rho_{\mathrm{Bdy}},\,\Phi_{t,s},\,\{\mathcal{I}_\alpha\}\bigr),
\]
carrying the inaccessible bulk data to a boundary state, an admissible (completely positive, trace-preserving) evolution channel \(\Phi_{t,s}\), and the instruments that read it. The decisive recognition is that this bulk-to-boundary projection \emph{is observation itself}: measurement is not a postulate appended to the theory but what the projection does.

\begin{figure}[!htbp]
\centering
\resizebox{0.92\textwidth}{!}{%
\begin{tikzpicture}[
    >={Stealth[length=3.2mm,width=2.6mm]},
    font=\small,
    bulkfill/.style={top color=blue!10,bottom color=blue!4},
    edge/.style={line width=1.10pt},
    fiber/.style={line width=0.80pt,draw=teal!70!black},
    scaledot/.style={circle,fill=black,inner sep=1.4pt},
    boundaryslab/.style={top color=orange!18,bottom color=orange!28,draw=orange!60!black,line width=0.80pt},
    note/.style={font=\footnotesize\itshape,text=black!72},
  ]

  %--------------------------------------------------------------
  % Geometry constants for the throat (a tapering cutaway)
  %  top  = UV GUT edge (wide),  bottom = IR end (narrow), feeds the boundary
  %--------------------------------------------------------------
  \def\xL{0}      % left wall x at UV top
  \def\xR{8.6}    % right wall x at UV top
  \def\yTop{9.2}  % UV edge height (r=0)
  \def\yBot{1.7}  % IR end height
  \def\taper{2.55}% horizontal pull-in at the IR (per side)

  % corner coordinates of the throat
  \coordinate (TL) at (\xL,\yTop);
  \coordinate (TR) at (\xR,\yTop);
  \coordinate (BL) at (\xL+\taper,\yBot);
  \coordinate (BR) at (\xR-\taper,\yBot);

  %--------------------------------------------------------------
  % 1. The 6D bulk throat  M_4 x I_r x S^1_theta
  %--------------------------------------------------------------
  \begin{scope}[on background layer]
    \fill[bulkfill]
      (TL) -- (TR)
      .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR)
      -- (BL)
      .. controls ($(BL)+(-0.55,2.6)$) and ($(TL)+(0.2,-2.6)$) .. (TL)
      -- cycle;
  \end{scope}
  % throat walls
  \draw[edge,blue!55!black]
      (TR) .. controls ($(TR)+(-0.2,-2.6)$) and ($(BR)+(0.55,2.6)$) .. (BR);
  \draw[edge,blue!55!black]
      (TL) .. controls ($(TL)+(0.2,-2.6)$) and ($(BL)+(-0.55,2.6)$) .. (BL);

  % bulk label
  \node[align=center,blue!45!black] at (4.3,7.3)
      {\textbf{6D bulk}\\[1pt]
       $\mathcal{M}_4 \times I_r \times S^1_\theta$};
  \node[note,align=center,text width=4.6cm] at (3.7,5.7)
      {a tapering ``throat'':\\ radius shrinks as energy\\ flows from UV to IR};

  %--------------------------------------------------------------
  % 2. UV GUT edge (top, r = 0)
  %--------------------------------------------------------------
  \draw[edge,red!65!black] (TL) -- (TR);
  \node[above=2pt of TR,anchor=south east,align=right,red!60!black,font=\small]
      {\textbf{UV GUT edge}\;($r=0$)\\[1pt]
       $\EGUT \sim 10^{16}\ \mathrm{GeV}$ --- the lawful beginning};

  %--------------------------------------------------------------
  % 3. Radial / RG scale axis  r = ln(E_GUT/mu)  on the left
  %--------------------------------------------------------------
  \def\axx{-1.7}
  % the directed scale axis itself (UV top -> IR bottom)
  \draw[->,line width=1.10pt,black!75] (\axx,\yTop+0.55) -- (\axx,\yBot-0.35);
  \node[rotate=90,anchor=south,font=\small,black!75] at (\axx-0.55,5.5)
        {RG / scale axis \ \ $r=\ln(\EGUT/\mu)$};
  % UV / IR end caps (placed clear of the chain labels)
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yTop+0.5) {UV};
  \node[right=2pt,font=\footnotesize,black!60] at (\axx,\yBot-0.3)  {IR};

  % inherited scale chain  E_Pl -> E_GUT -> M_lock  (three distinct heights)
  \node[scaledot] (sPl)   at (\axx,\yTop+0.0)  {};
  \node[scaledot] (sGUT)  at (\axx,\yTop-2.6)  {};
  \node[scaledot] (sLock) at (\axx,\yBot+0.55) {};
  \node[right=4pt,font=\footnotesize,align=left] at (sPl.east)  {$\EPl$\\[-1pt]\scriptsize($r=-2\pi$)};
  \node[right=4pt,font=\footnotesize,align=left] at (sGUT.east) {$\EGUT$\\[-1pt]\scriptsize($r=0$)};
  \node[right=4pt,font=\footnotesize,align=left] at (sLock.east)
        {$\Mlock\sim 3.85~\mathrm{TeV}$\\[-1pt]\scriptsize(KK lock, $r=29.41$)};
  % "inherits" annotations alongside the axis
  \node[left=2pt,font=\scriptsize,black!55] at ($(sPl)!0.5!(sGUT)$)  {inherits};
  \node[left=2pt,font=\scriptsize,black!55] at ($(sGUT)!0.5!(sLock)$) {inherits};

  %--------------------------------------------------------------
  % 4. Compact angle S^1_theta : small fibre circle at a sample point
  %    (winding carries the Kaxion / dark sector)
  %--------------------------------------------------------------
  \coordinate (fpt) at (5.9,3.6);
  \node[scaledot] at (fpt) {};
  \draw[fiber] (fpt) ellipse [x radius=0.66,y radius=0.30];
  \draw[fiber,->] ($(fpt)+(0.66,0.0)$) arc [start angle=0,end angle=300,x radius=0.66,y radius=0.30];
  \node[above right=0.05cm and 0.05cm of fpt,align=left,text width=3.4cm,font=\footnotesize,teal!45!black]
      {compact phase $S^1_\theta$:\\
       winding carries the\\ \textbf{Kaxion / dark sector}};

  %--------------------------------------------------------------
  % 5. The 4D boundary slab  M_4  (FRW / LCDM cosmology)
  %--------------------------------------------------------------
  \node[boundaryslab,minimum width=9.6cm,minimum height=1.35cm,rounded corners=2pt]
      (bdy) at (4.3,0.05) {};
  \node[align=center] at (bdy.center)
      {\textbf{4D spacetime $\mathcal{M}_4$}\;:\; ordinary FRW\,/\,$\Lambda$CDM cosmology
       \ \ (the boundary)};

  %--------------------------------------------------------------
  % 6. OGGEP projection : bold bulk -> boundary arrow
  %--------------------------------------------------------------
  \draw[-{Stealth[length=5mm,width=4mm]},line width=2.2pt,red!70!black]
      ($(BL)!0.5!(BR)+(0,0.05)$) -- ($(bdy.north)+(0,0.02)$);
  \node[anchor=west,align=left,font=\footnotesize,red!55!black] at (6.7,1.55)
      {\textbf{\OGGEP{}} \,(bulk\,$\to$\,boundary):\\
       cosmological observables are the\\
       \emph{boundary shadows} of bulk structure};

  % GDOS boundary law tag under the slab
  \node[below=3pt of bdy.south,anchor=north,font=\footnotesize,orange!40!black]
      {boundary governed by the \GDOS{} boundary law};

  %--------------------------------------------------------------
  % 7. Caption note about the meaning of "warped"
  %--------------------------------------------------------------
  \node[draw=black!55,black!35,fill=black!3,rounded corners,inner sep=4pt,
        text width=5.0cm,note,anchor=north west] at (9.3,8.7)
      {``Warped'' here denotes \textbf{RG running along $r$}
       (the field-theory / holographic-RG reading), \emph{not}
       a Randall--Sundrum spatial dimension.};

\end{tikzpicture}%
}%
\par\smallskip
\caption[6D bulk and its \OGGEP{} projection to the 4D cosmological boundary.]%
{Schematic of the six-dimensional bulk
 $\mathcal{M}_4\times I_r\times S^1_\theta$ and its \OGGEP{} projection to the
 four-dimensional cosmological boundary. The radial axis $r=\ln(\EGUT/\mu)$ is
 the renormalization-group / scale axis: the ultraviolet GUT edge sits at the top
 ($r=0$, $\EGUT\sim10^{16}\,\mathrm{GeV}$, the lawful beginning) and low-energy
 physics lies toward the boundary, with the inherited scale chain
 $\EPl\!\to\!\EGUT\!\to\!\Mlock$ ($\Mlock\sim3.85\,\mathrm{TeV}$, the Kaluza--Klein
 lock scale) marked along it. The compact angle $S^1_\theta$ is the phase fiber
 whose winding carries the Kaxion / dark sector. The bold arrow is the \OGGEP{}
 bulk-to-boundary map: cosmological observables on $\mathcal{M}_4$ (ordinary
 FRW/$\Lambda$CDM) are boundary shadows of bulk structure, with the boundary
 governed by the \GDOS{} boundary law. ``Warped'' denotes RG running along $r$
 (the field-theory / holographic-RG reading), \emph{not} a Randall--Sundrum
 spatial dimension.}
\label{fig:bulk-boundary}
\end{figure}

\AFive{} then makes the projection dynamical and unique. Its boundary law, the geometric dynamics of open systems (\GDOS{}), is the Lindblad/GKLS generator
\[
\mathcal{L}_{\GDOS}(\rho) \;=\; -\,i\,[H_{\mathrm{eff}},\rho] \;+\; \sum_\alpha \gamma_\alpha\Bigl(L_\alpha\,\rho\,L_\alpha^\dagger - \tfrac12\{L_\alpha^\dagger L_\alpha,\,\rho\}\Bigr),\qquad \gamma_\alpha\ge0,
\]
derived --- not postulated --- from the exact bulk dynamics by a Born--Markov reduction \citep{ArisakaA5GDOS}. The ordinary Schr\"odinger equation is recovered as its closed-system limit; openness is primary and closure the special case. In this language an \emph{observable} is the value an open system reads off at a stationary seam, an attractor of the \GDOS{} flow --- and from this one law \AFive{} recovers four-dimensional quantum mechanics, general relativity, and quantum field theory as boundary theorems rather than separate assumptions.

The warm consequence, the one to hold onto, is that \emph{ordinary FRW cosmology is recovered, not discarded}. Because four-dimensional general relativity is already a theorem of \AFive{}, the Friedmann--Lema\^itre--Robertson--Walker universe lives inside the framework automatically, with the four-dimensional Newton constant emerging as a scale-overlap integral over the throat (the overall unit set by \(G_N\), with the bulk mass the derived combination, as in \ATwo{}; the explicit reduction to the Friedmann equations is in Part~III and Appendix~\ref{appA:frw}). We do not choose between \GGTheory{} and the cosmology textbooks: the textbook equations are exactly what the boundary law produces, and what the framework adds is a fixed, parameter-free ledger of the source terms that feed them.

A boundary law fixes \emph{what} the four-dimensional theory is; a cosmology also needs to know \emph{how} its lawful possibilities are explored and stabilized over cosmic time. That is the geometric runtime, the four invariance pillars \GiFour{} of \ASix{} \citep{ArisakaA6Gi4}, organized as a single cascade
\[
\TGG{} \;\Longrightarrow\; \CCI{} \;\Longrightarrow\; \GFF{} \;\Longrightarrow\; \GRIP{}.
\]
The total geometric--gauge object \TGG{} packages the closed bulk ledger as one composite connection on the \(G_{\mathrm{Bulk}}\) bundle; the covariant conservation identity \CCI{} hands the boundary observer a local conservation law and, under closure, forces the integer ledger of the next section; the geometric free-energy functional \(\GFF{}=\GEP{}+\CEP{}\) ranks the admissible boundary configurations by a single scalar; and \GRIP{} --- the lifecycle \(\mathrm{DGI}\to\mathrm{GSSB}\to\mathrm{GA}\), lawful exploration to symmetry-breaking to stabilized attractor --- fixes the order in which a branch is explored, committed to, and frozen. The next two sections put this machinery to work: first the fixed ledger it runs on, then the way it selects the cosmic branches that actually happen.

\subsection{Which particles? The observer-indexed vacuum, and horizon thermality as openness}
\label{subsec:observer-vacuum}

One further consequence of the projection deserves a named place before
the ledger, because cosmology uses it on every page that mentions
particle production. In an expanding or horizon-bearing spacetime there
is famously no preferred vacuum: inequivalent positive-frequency
conventions define inequivalent particle notions, related by Bogoliubov
mixing. In the present framework this ambiguity has a fixed address. On
the canonical bulk geometry inherited from \citep{ArisakaA2GG6} the
constitutional time is globally static, so the bulk-level
positive-frequency structure is unique; what varies from observer to
observer is the \emph{dictionary}: which modes an instrument counts is
fixed by the observer's own time and horizon structure, and the
instruments are part of the \OGGEP{} boundary package by definition.
``How many particles'' is therefore an observer-indexed boundary
observable, not a bulk quantity --- the same status the framework
already assigns to boundary states themselves. The physical output of
the ambiguity is produced by the \GDOS{} projection itself: tracing the
inaccessible partner of a horizon's squeezed pairing yields an exactly
thermal state (at $T = a/2\pi$ for the Rindler pairing), so horizon
thermality is constitutional openness doing its ordinary work rather
than new physics; and cosmological particle creation is the same
structure run dynamically --- a Gaussian pumping channel occupying the
same slot as the dissipative generators of the boundary law
\citep{ArisakaA5GDOS}. The full statements are carried in the
real-bulk treatment of \citep{ArisakaA8QM}, whose standalone,
full-length development is the Foundations Branch member paper
\PaperPFour{} (\citealp{ArisakaP4Real}). Two boundary cases are closed there at
the model level, as entries R14--R15 of the series: the rotating
case, where the ergoregion defeats the co-rotating selection and the
uniqueness anchor moves to the asymptotic frame, superradiance
becoming a symplectic pumping channel in the same \GDOS{} slot; and
the collapse epoch, which factorizes into a universal thermal channel
times a decaying transient (the full-length treatment in the
black-hole setting is \citealp{ArisakaC3BH}). What remains
conjectural is the exact curved-space existence theory --- the
field's own, imported where used. The present paper uses only the two
facts just stated: the bulk structure is unique, and the thermal
content of horizons is the partial trace.


\section{Zero free parameters: \BIsix{} and the integer ledger}

A framework can be elegant and still be useless if it leaves the numbers free. What makes \GGTheory{} predictive rather than merely descriptive is that it has \emph{no continuous dials} to turn, and two inherited results enforce this.

The first is the anomaly-locked bulk. Global consistency of the six-force closure requires a modified six-dimensional Bianchi identity, \BIsix{}, for the antisymmetric tensor \(H_3=dB_2\) (\citealp{ArisakaA3BI6}; \citealp{ArisakaA4Euclid}):
\[
dH_3 \;=\; \mathrm{tr}\,F_{SU(5)}^2 \;-\; F_B^2 \;-\; \tfrac{2}{5}\,F_B\!\wedge\!F_{Qi} \;-\; \tfrac{2}{5}\,F_{Qi}^2 \;+\; \mathrm{tr}\,R^2,
\]
whose coefficient vector \((a,b,c,d)=(1,-1,-\tfrac{2}{5},-\tfrac{2}{5})\) is fixed uniquely by three independent conditions --- local closure (\(dX_4=0\)), global integrality of \([X_4]\), and anomaly cancellation by inflow through the single channel \(I_8=\tfrac12 X_4\wedge X_4\). There is no continuous deformation: the trace-ratio lock \(|a/c|=|a/d|=5/2\) is rigid. The ultraviolet is legal, and it is legal in exactly one way.

The second is the integer ledger. \AFour{} \citep{ArisakaA4Euclid} proves that the same closure forces a unique tuple of whole numbers, the \OGNfive{} ledger
\[
(M,N,L;\,J,K) \;=\; (1,3,5;\,7,35),
\]
through a five-step Diophantine cascade in which each integer is fixed by the one before: \(M=1\) (a single Green--Schwarz tensor channel), \(L=5\) (the minimal simple unification \(SU(5)\)), \(N=3\) (the three colors and generations, forced by the \BIsix{} trace ratio \(-(L-N)/L=-2/5\)), \(J=2N+1=7\), and the closure integer
\[
K \;=\; LJ \;=\; M^2+N^2+L^2 \;=\; 35.
\]
That two independent arithmetic expressions for \(K\) agree --- the \emph{dual lock} --- is what closes the ledger; it is generically false, and true here only because both are forced by the same axioms. These integers are not fitted; they are the arithmetic of the geometry. Together with the structural constants \(\pi\) and \(e\) and the dimensionful conversion factors \(\{c,G_N,\hbar;k_B,\QB\}\), they are the entire stock from which every dimensionless number in the cohort is built: the same ledger that fixes the twenty-eight Standard-Model parameters of \ANine{} \citep{ArisakaA9SM} fixes the cosmic inventory of this paper, with no new freedom in between. The topological step that turns \(K\), the integers, and \(\pi\) into the \(\pi\)-locked inventory (\(\Omm=1/\pi\), \(\OmL=1-1/\pi\), \(\Omb=1/2\pi^2\)) is given in full in Part~X and Appendix~\ref{appA:pilock}, and the discipline that admits a number only if it is a cost-minimum element over this ledger is the \MSMF{}/\OGN{} principle. Every cosmological number must come out of this fixed ledger, or it is not allowed in. Tellingly, the same \(K=35\) that sets the scale-throat depth also sets the Kaxion's twin-peak spacing \(\Delta\nu/\nu=1/35\): the Planck--electroweak hierarchy and the dark-matter signature are literally the same integer. Exactly what each completed preprint contributes to this inherited ledger is tabulated alongside the cohort map in \S\ref{sec:cohort} (Table~\ref{tab:inheritance-map}).

\section{The runtime: how cosmic branches are selected}
\label{sec:runtime-branches}

The framework so far is static: a bulk, a boundary law, a fixed ledger. But a cosmology is a \emph{history} --- it has to choose, epoch by epoch, which branch of the lawful possibilities actually happens. That choosing is what \GGTheory{} calls the runtime, and it is the last piece the cosmology inherits. The runtime has two components, both proved in the pillar papers \ASix{} and \ASeven{}: the \GRIP{} cascade, which fixes the order in which lawful branches may be explored, committed to, and stabilized; and the \GFF{} ledger, the scalar ranking functional \(F_{\GFF}=F_{\GEP}+F_{\CEP}\) that says \emph{which} branch is favored. Together they select the cosmic branches that matter --- inflation's exit, recombination, the growth of structure, and the final attractor the universe relaxes into. The \GFF{} ledger is, in form, a free-energy functional, standing in the information-theoretic lineage of \citet{Shannon1948}, \citet{Jaynes1957}, \citet{KullbackLeibler1951}, \citet{Landauer1961}, \citet{AmariNagaoka2000}, and \citet{Friston2010}; ranking branches by such a functional is what turns an arbitrary cosmic narrative into a definite branch-selection problem.

The runtime has two regimes, and the distinction is the one that matters most for a cosmology (\citealp{ArisakaA6Gi4}; \citealp{ArisakaA7Qi4}). \emph{\GRIP{}-In} is descent on a \emph{fixed} landscape: the system rolls down the \GFF{} valleys by gradient flow and tunnels between basins by Bogomolnyi-saturated instanton steps of action \(\pi k\), halting uniquely at the deepest reachable attractor. \emph{\GRIP{}-Out} is the complementary event that \emph{reorganizes the landscape itself} --- it opens new strata and hands the next descent its starting point. The slogan of \ASix{} is exact: \GRIP{}-Out opens the landscape; \GRIP{}-In finishes the journey. Cosmic history is precisely this alternation. The scale-setting events --- the exit from the GUT edge, the electroweak lock, the launch and end of inflation --- are \GRIP{}-Out: they build the landscape. Everything that then settles onto a fixed landscape --- the recombination seam, the relaxation to the dark-energy attractor, the frozen present-day constants --- is \GRIP{}-In. Where the cosmic landscape itself came from is the \GRIP{}-Out question this paper answers; the particle paper \ANine{} answered the complementary \GRIP{}-In question, reading the Standard-Model constants off the landscape those same events had already built.

Which branch, among the many lawful attractors of a stratum, is the one actually realized? The selector is the set of four quantum-information pillars \QiFour{} of \ASeven{} \citep{ArisakaA7Qi4}. A configuration is the observed \emph{ground geometric attractor} only if it passes all four admissibility filters at once: \emph{Qi-QUA}, that it carry the locked four-current identity \(J_E/\QB=J_e=J_B=J_{Qi}\) (the Arisaka equation); \emph{Qi-EP}, that it be a genuine extremum of the one projected variational principle, not merely a local minimum of a coarse-grained potential; \emph{Qi-CON}, that its scale signatures be those of a fixed point, surviving transport along \(r\); and \emph{Qi-HAL}, that the runtime can actually halt there, with no admissible descent remaining. The observed value is the unique four-certificate survivor of its stratum --- the same criterion that makes every electron, proton, and Kaxion the identical object everywhere.

This is not an abstraction without a worked precedent: it is exactly how \ANine{} \citep{ArisakaA9SM} fixed the Standard Model. There each parameter is a coordinate of a geometric attractor reached by this cascade --- the electroweak vacuum selected as the stationary point of a Wilson--Hosotani holonomy potential (the \GSSB{} step), the constants read off only after the system has descended onto that branch --- and the abstract ``selection'' is made concrete as a single \MSMF{}/\OGN{} cost-minimum over the integer ledger, the unique admissible element of lowest arithmetic cost. \ATen{} runs the identical machinery one story up: the cosmic branches --- inflation's exit, recombination, the growth of structure, and the final attractor the universe relaxes into --- are the \GRIP{} readouts of the same runtime, now on the cosmological boundary stratum.

The whole inheritance can be stated once, as a single descent ladder:
\[
\BIsix{} \;\Longrightarrow\; \TGG{} \;\Longrightarrow\; \CCI{} \;\Longrightarrow\; \GFF{} \;\Longrightarrow\; \GRIP{}.
\]
Read left to right: the anomaly-locked bulk fixes the legal ultraviolet stage; \TGG{} packages the total closed cosmic ledger; \CCI{} hands the four-dimensional observer a local boundary conservation law; \GFF{} ranks the admissible branches of that boundary theory; and \GRIP{} fixes the order in which they are explored, selected, and stabilized. Figure~\ref{fig:derivation-tree} traces this same descent for the cosmological observables, and the inheritance theorem below states it as a rule.

\begin{figure}[p]
\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.897\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily\small,
  every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
  %--- TIER 1: fixed inputs -----------------------------------------
  inp/.style={fill=blue!10, draw=blue!60!black, line width=0.6pt,
              minimum width=52mm, minimum height=15mm},
  %--- TIER 2: machinery (inheritance chain) ------------------------
  mach/.style={fill=violet!10, draw=violet!60!black, line width=0.6pt,
               minimum width=30mm, minimum height=13mm},
  %--- TIER 3: output nodes, border-coded by provenance -------------
  %    D : Derived            -> solid  green  border
  derived/.style={fill=green!8,  draw=green!60!black, solid,
                  line width=0.9pt, minimum width=33mm, minimum height=15mm},
  %    D*: Derived-conditional -> dashed amber  border
  dcond/.style={fill=orange!10, draw=orange!75!black, dashed,
                line width=0.9pt, minimum width=33mm, minimum height=15mm},
  %    A : Asserted           -> dotted grey   border
  asserted/.style={fill=black!4, draw=black!55, dotted,
                   line width=1.0pt, minimum width=33mm, minimum height=15mm},
  %--- band / header labels -----------------------------------------
  tierlab/.style={font=\large\bfseries, text=black!70, align=left,
                  fill=white, inner sep=2.5pt, rounded corners=1pt},
  legbox/.style={rounded corners=2pt, inner sep=4pt,
                 draw=black!40, fill=black!2, line width=0.4pt},
  %--- arrows -------------------------------------------------------
  arr/.style={-{Stealth[length=2.4mm]}, line width=0.5pt, draw=black!65},
  arrm/.style={-{Stealth[length=2.6mm]}, line width=0.8pt, draw=violet!55!black},
  arrout/.style={-{Stealth[length=2.2mm]}, line width=0.45pt, draw=black!45},
]

% ======================  TIER 1: FIXED INPUTS  ======================
\node[inp] (GU5) at (-58mm, 0) {%
  \textbf{GU-5 dimensional constants}\\[1pt]
  \(\{c,\ G_N,\ \hbar;\ \kB,\ \QB\}\)\\[1pt]
  \small the Geometric Universal Five};
\node[inp] (OGN) at (0mm, 0) {%
  \textbf{OGN-5 integer ledger}\\[1pt]
  \((M,N,L;J,K)=(1,3,5;7,35)\)\\[1pt]
  \small proved, no free dials};
\node[inp] (TOP) at (58mm, 0) {%
  \textbf{Structural constants}\\[1pt]
  \(\pi,\quad e\)\\[1pt]
  \small fixed by geometry};

% Band background behind tier 1 (drawn on backgrounds layer below)
\begin{scope}[on background layer]
  \node[fit=(GU5)(OGN)(TOP), inner sep=4mm,
        fill=blue!4, rounded corners=3pt, draw=blue!55!black, line width=0.4pt]
        (band1) {};
\end{scope}

% ======================  TIER 2: THE MACHINERY  =====================
% Inheritance chain: GG-6 -> BI-6 -> GDOS -> GRIP -> OGGEP
\node[mach] (GG6)   at (-72mm, -42mm) {\textbf{\GGTheory{}\,/\,GG-6}\\[1pt]
                                       \small 6D bulk};
\node[mach] (BI6)   at (-36mm, -42mm) {\textbf{\BIsix{}}\\[1pt]
                                       \small anomaly-free closure};
\node[mach] (GDOS)  at (0mm,   -42mm) {\textbf{\GDOS{}}\\[1pt]
                                       \small open-system\\boundary law};
\node[mach] (GRIP)  at (36mm,  -42mm) {\textbf{\GRIP{}}\\[1pt]
                                       \small renormalization\\cascade};
\node[mach] (OGGEP) at (72mm,  -42mm) {\textbf{\OGGEP{}}\\[1pt]
                                       \small bulk-to-boundary\\projection};

\begin{scope}[on background layer]
  \node[fit=(GG6)(BI6)(GDOS)(GRIP)(OGGEP), inner sep=3.5mm,
        fill=violet!4, rounded corners=3pt, draw=violet!55!black, line width=0.4pt]
        (band2) {};
\end{scope}

% inputs -> machinery (the ledger feeds the constitutional apparatus)
\draw[arr] (GU5.south) to[bend right=8]  (GG6.north);
\draw[arr] (OGN.south) -- (BI6.north);
\draw[arr] (OGN.south) to[bend left=6]   (GDOS.north);
\draw[arr] (TOP.south) to[bend left=10]  (OGGEP.north);

% inheritance chain arrows (left -> right across the machinery band)
\draw[arrm] (GG6.east)  -- (BI6.west);
\draw[arrm] (BI6.east)  -- (GDOS.west);
\draw[arrm] (GDOS.east) -- (GRIP.west);
\draw[arrm] (GRIP.east) -- (OGGEP.west);

% ==================  TIER 3: COSMOLOGICAL OBSERVABLES  ==============
% Row 3a
\node[dcond]    (OM)   at (-72mm, -82mm) {\textbf{\(\Omega_m=1/\pi\)}\\[1pt]
  \small matter fraction\\\footnotesize closed-packet integ.\,(D*)};
\node[dcond]    (OL)   at (-36mm, -82mm) {\textbf{\(\Omega_\Lambda=1-1/\pi\)}\\[1pt]
  \small dark energy\\\footnotesize \(\pi\)-locked\,(D*)};
\node[dcond]    (OB)   at (0mm,   -82mm) {\textbf{\(\Omega_b=\tfrac{1}{2\pi^2}\)}\\[1pt]
  \small baryon fraction\\\footnotesize \(\pi\)-locked\,(D*)};
\node[dcond]    (ZSTAR)at (36mm,  -82mm) {\textbf{\(z_\ast=e^{7}\)}\\[1pt]
  \small recomb. corridor\\\footnotesize from \(J=7\)\,(D*)};
\node[dcond]    (MLK)  at (72mm,  -82mm) {\textbf{\(\Mlock\!\sim\!3.85\,\)TeV}\\[1pt]
  \small lock scale\\\footnotesize \(\EPl/(2e^{35})\)\,(D*)};
% Row 3b
\node[dcond]    (KAX)  at (-72mm, -106mm){\textbf{\(m_{\mathrm{ax}}\!=\!8.1\,\mu\)eV}\\[1pt]
  \small Kaxion mass\\\footnotesize compact-angle \(f_a\)\,(D*)};
\node[derived]  (NEFF) at (-36mm, -106mm){\textbf{\(N_{\mathrm{eff}}\)}\\[1pt]
  \small radiation content\\\footnotesize \(=3.044\)\,(D)};
\node[derived]  (RTEN) at (0mm,   -106mm){\textbf{\(r=0.0043\)}\\[1pt]
  \small tensor-to-scalar\\\footnotesize derived plateau\,(D)};
\node[dcond]    (NS)   at (36mm,  -106mm){\textbf{\(n_s\!=\!0.9620\)}\\[1pt]
  \small scalar tilt\\\footnotesize plateau-derived\,(D*)};
\node[asserted] (INFL) at (72mm,  -106mm){\textbf{\(A_s;\ V^{1/4}\)}\\[1pt]
  \small amplitude \& scale\\\footnotesize the one input\,(A); scale\,(D*)};
% Row 3c (single wide corridor node)
\node[dcond]    (H0)   at (-18mm, -132mm){\textbf{\(H_0\) corridor: \(67.4\,/\,73.0\)}\\[1pt]
  \small two-stage expansion corridor\,(D*)};


% machinery -> observables (which inputs/machinery feed each)
% pi-locked fractions come from OGGEP projection of the closed packet
\draw[arrout] (OGGEP.south) to[bend left=20]  (OM.north);
\draw[arrout] (OGGEP.south) to[bend left=12]  (OL.north);
\draw[arrout] (OGGEP.south) to[bend left=4]   (OB.north);
% z_* and M_lock from the integer ledger via GRIP cascade
\draw[arrout] (GRIP.south)  to[bend right=6]  (ZSTAR.north);
\draw[arrout] (GRIP.south)  to[bend left=10]  (MLK.north);
% Kaxion + N_eff + r from GDOS boundary law / GRIP
\draw[arrout] (GDOS.south)  to[bend right=24] (KAX.north);
\draw[arrout] (GDOS.south)  to[bend right=8]  (NEFF.north);
\draw[arrout] (GRIP.south)  to[bend right=18] (RTEN.north);
% n_s, A_s, inflation scale asserted (thin links, acknowledging weak grounding)
\draw[arrout, densely dotted] (BI6.south)  to[bend right=4] (NS.north);
\draw[arrout, densely dotted] (BI6.south)  to[bend left=2]  (INFL.north);
% H0 corridor fed by Omega's + z_*
\draw[arrout] (OM.south)    to[bend right=8]  (H0.north);
\draw[arrout] (ZSTAR.south) to[bend left=22]  (H0.north);

% ==========================  LEGEND  ===============================
\node[legbox, anchor=north west, align=left] at (34mm, -124mm) {%
  \textbf{Provenance (codebook border code)}\\[3pt]
  \tikz[baseline]{\draw[green!60!black, solid, line width=0.9pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{D} --- Derived\\[2pt]
  \tikz[baseline]{\draw[orange!75!black, dashed, line width=0.9pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{D*} --- Derived-conditional\\[2pt]
  \tikz[baseline]{\draw[black!55, dotted, line width=1.0pt]
      (0,0) rectangle (5mm,3mm);}\ \,
      \textbf{A} --- Asserted};

% tier labels: left side, above each tier's boxes (drawn last, white-backed)
\node[tierlab, anchor=south west] at (-88mm, 10mm)  {THE FIXED INPUTS};
\node[tierlab, anchor=south west] at (-88mm, -32mm) {THE MACHINERY};
\node[tierlab, anchor=south west] at (-88mm, -73.5mm) {THE COSMOLOGICAL OBSERVABLES};
\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Zero-parameter derivation tree of \ATen{}.]{%
\textbf{The zero-parameter derivation tree.}
Every cosmological observable (bottom tier) is traced back, through the
\GGTheory{} machinery (middle tier), to a \emph{fixed} input ledger (top
tier) that contains \emph{no continuous free parameters}: the GU-5
dimensional constants \(\{c,G_N,\hbar;\kB,\QB\}\), the proved \OGNfive{} integer
ledger \((M,N,L;J,K)=(1,3,5;7,35)\), and the structural constants
\(\pi\) and \(e\).  Output nodes are border-coded by codebook provenance, and the colors are
\emph{not} cosmetic.
\emph{Solid green (D, Derived):} the tensor-to-scalar ratio \(r\) and the
radiation content \(N_{\mathrm{eff}}\).
\emph{Dashed amber (D*, Derived-conditional):} the \(\pi\)-locked fractions,
the corridor-switch \(z_\ast=e^{7}\), the lock scale \(\Mlock\), the Kaxion
mass, the \(H_0\) corridor, and --- computed on the derived plateau of
Part~IV --- the tilt \(n_s = 0.9620\) and the inflationary scale
\(V^{1/4} = 8.3\times10^{15}\,\mathrm{GeV}\).
\emph{Dotted gray (A, Asserted):} the scalar amplitude \(A_s\) --- the
launch sector's one declared input, its warp candidate flagged.
\textbf{Honesty caveat:} the dashed (D*) entries are \emph{conditional on
named inputs}---notably Conjecture~VIII, whose closed-packet
integration now closes \emph{up to the declared epoch input} (the
residual uniformity lemma exact to one part in $10^{15}$)---so the tree
displays the derivation flow with that single calibration input, not an
unconditional proof for the D* (or A) entries.  Only the solid-green (D) nodes are claimed as fully derived.}
\label{fig:derivation-tree}
\end{minipage}
\end{figure}

The same content, sharpened to the cohort's seven formal inputs and stated as a theorem about admissibility, reads as follows. It is the load-bearing rule of the whole paper: a cosmic source term is admissible only if it can be traced backward through the same constitutional chain already used to build the particle and open-system papers (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaA6Gi4}; \citealp{ArisakaA7Qi4}; \citealp{ArisakaA9SM}).

\begin{theorem}[Cosmological inheritance theorem]
Every admissible cosmological source term in \ATen{} must descend in the
order
\[
\BI \Longrightarrow \TGG \Longrightarrow \CCI \Longrightarrow \GFF \Longrightarrow \GRIP \Longrightarrow \text{parameter block}.
\]
No cosmic module is allowed to enter the Friedmann, perturbation, thermal, or
information-processing sectors unless it survives each previous layer.
\end{theorem}

\begin{proof}
\BI{} fixes the legal ultraviolet stage and its coefficient structure.
\TGG{} packages the total closed cosmic ledger. \CCI{} gives the local
boundary conservation law seen by the four-dimensional observer.
\(F_{\GFF}=F_{\GEP}+F_{\CEP}\) then ranks the admissible branches of that
boundary theory, and \GRIP{} fixes the order in which those branches can be
explored, selected, and stabilized. The remaining quantity is therefore not a
primitive fluid or parameter, but a branch export. This is exactly the sense
in which the present framework treats cosmological observables as theorem
candidates rather than as a phenomenological menu.
\end{proof}
\begin{theorem}[Cosmological inheritance theorem (T-Cosmos-1)]
Every admissible cosmic sector in \ATen{} must be expressible as a boundary theorem of the seven-paper chain in Table~\ref{tab:inheritance-map}. In particular, no new free cosmological degree of freedom may be inserted unless it can be shown to descend from the \GG{}/\BIsix{} bulk, survive the \OGGEP{}/\GDOS{} projection, enter the \GFF{} ledger, and fit into an admissible \GRIP{} factorization.
\end{theorem}

\begin{proof}
The proof is a bookkeeping proof. \AOne{} fixes admissibility. \ATwo{} and \AThree{} fix the legal six-dimensional stage and its anomaly structure. \AFive{} fixes what counts as an admissible boundary observable and an admissible reduced dynamics. \ASix{} fixes the unique runtime factorization \(\GRIP=(\DGI\to\GSSB\to\GA)\) and the scalar ranking functional \(F_{\GFF}=F_{\GEP}+F_{\CEP}\). \ASeven{} fixes the Qi-runtime, the selector architecture, and the hierarchy language by which internal reconstruction and higher-floor organization are read. \ANine{} fixes the particle ledger that every cosmological source term must now inherit. Any proposed cosmic module that cannot be placed into this chain is therefore not derived; it is merely appended. Such a module is inadmissible by construction.
\end{proof}

With the descent ladder fixed, the remaining piece of runtime vocabulary is the \emph{grammar} by which cosmic branches are actually selected --- the part of the inheritance that decides how a given epoch transition is read, and where the cosmic history is allowed to come to rest.

\subsection{Why \GRIP{} and Wilson-loop \GSSB{} now control the cosmology paper}
\label{ssec:cosmic-grip-grammar-gga}

\ATen{} inherits not merely a list of sectors but a runtime grammar. The relevant grammar is \GRIP{}, the Gauge-Runtime Invariance Pipeline,
\[
\GRIP := (\DGI \to \GSSB \to \GA),
\]
with the scalar ranking functional
\[
F_{\GFF}=F_{\GEP}+F_{\CEP}.
\]
In the cosmology paper this matters because every important epoch is a branch event of one sort or another: inflationary launch and exit, relic selection, matter-radiation equality, recombination, halo collapse, black-hole Page turnover, and the late-time vacuum inventory all involve lawful exploration, branch commitment, and attractor stabilization. The universe is therefore not merely expanding; it is passing through a sequence of \GRIP{} events.

One refinement of the terminus, established at runtime level in \ASix{} \citep{ArisakaA6Gi4} and generalized across the layers of the program-wide cascade in the capstone \AEleven{} \citep{ArisakaA11Origin}, organizes the cosmological reading and is used throughout this paper. A \GFF{} landscape generally admits many stable basins --- many Geometric Attractors (\GA{}'s) --- but on a stratum of the class \ASix{} calls admissible exactly one of them is the global minimum, the \emph{Ground} Geometric Attractor,
\[
\GGA := \arg\min F_{\GFF},
\]
fixed by the landscape alone. (The four pillars of \ASeven{} characterize that object rather than select it; nothing in the cosmological reading below turns on them.) The distinction is exactly the one cosmology needs in order not to conflate two very different kinds of ``attractor export.'' A \emph{transient} cosmic history --- a reheated hot patch, a frozen last-scattering record, a slice of the growth history --- is a genuine \GA{}: a stable record the projection can read, but not the deepest state of its sector, and not eternal. A \emph{persistent relic} or final state --- a dark-matter quantum that must survive to today, the asymptotic vacuum the universe relaxes into --- is a \GGA{}: the ground state of its stratum, with no deeper basin to decay to. The same theorem that makes every electron identical and the proton stable (\ANine{}) therefore underwrites, on the cosmological boundary, both the longevity of the dark matter (\S\ref{sec:kaxion}) and the finality of the late-time vacuum (Part~IV): each is a \GGA{}, not merely a \GA{}.

A second piece of runtime vocabulary, defined at runtime level in
\ASix{} (its \GRIPin{}/\GRIPout{} regime definitions, together with
the necessity of the second regime) and developed as the Lyapunov
descent theorem \textsf{T-GRIP-In-Lyapunov-Descent} and the
seventeen-event theory of \AEleven{}, is imported here because cosmology is
where it pays the largest rent. The runtime operates in exactly two
regimes, split by one question: \emph{does the landscape change?}
\GRIPin{} is the descent on a \emph{fixed} landscape, halting
uniquely at the stratum's \GGA{}; \GRIPout{} is the transient
out-of-equilibrium event that \emph{reorganizes the landscape
itself} and supplies the initial datum of the next descent. Cosmic
history is this alternation writ large, and the classification is
sharp. The landscape-reorganizing transitions --- the inflationary
exit, reheating, the electroweak selection, QCD confinement,
recombination --- are \GRIPout{} events: after each, the boundary
carries strata and basins that did not exist before. The long epochs
\emph{between} transitions are \GRIPin{}: relaxation on the landscape
the last transition built, with Hubble friction itself serving as the
cosmological within-basin flow (the matter strata use the weak
channel for the same job; \ANine{}). Crossovers such as
matter--radiation equality reorganize nothing and are therefore
\emph{milestones of \GRIPin{}}, not transitions. And the chain halts
exactly twice: the Kaxion relic and the de Sitter vacuum are the two
points where cosmic \GRIPin{} terminates --- the two \GGA{}'s of the
preceding paragraph, now read dynamically. In one sentence:
\GRIPout{} opens the landscape; \GRIPin{} finishes the journey; and
the universe we observe today is the journey, almost finished.

The electroweak paper already identified Wilson-line holonomy as the cleanest prototype of geometric spontaneous symmetry breaking (\citealp{AharonovBohm1959}; \citealp{Wilson1974}; \citealp{Hosotani1983}; \citealp{Hosotani2005}; \citealp{ArisakaA6Gi4}; \citealp{ArisakaA9SM}). The cosmology paper now generalizes the lesson: the physically relevant branch variable is not necessarily a local scalar field chosen ad hoc, but can be a gauge-invariant loop or holonomy variable inherited from the compact \GG{} geometry. That observation narrows the early-universe story sharply. It means that phase selection in cosmology should be read, whenever possible, as Wilson-loop or holonomy selection on the \GFF{} landscape, not as arbitrary scalar-field patchwork.


The cosmic specialization of this grammar --- that every cosmological phase change is a lawful \GRIP{} event, ranked by \(F_{\GFF}\), with persistent relics and final states singled out as Ground Geometric Attractors --- is stated as the second cosmic theorem.

\begin{theorem}[Cosmic \GRIP{} theorem (T-Cosmos-2)]
Every cosmological phase change discussed in \ATen{} must be readable as a \GRIP{} event on the boundary state manifold, ranked by \(F_{\GFF}\), with \GSSB{} realized by loss of stability of a previously admissible symmetric branch and with the selected post-transition history stabilized as a Geometric Attractor (\GA{}) --- and, in the case of a persistent relic or a final state, as the stratum's \emph{Ground} Geometric Attractor (\GGA{}; \S\ref{ssec:cosmic-grip-grammar-gga}). In regime language: every such phase change is a \GRIPout{} event --- a transient reorganization of the boundary landscape --- and every inter-transition epoch is \GRIPin{} relaxation on the landscape so created; the chain halts where \GRIPin{} terminates, at the two \GGA{} endpoints.
\end{theorem}

\begin{proof}
This is immediate once the seven-paper chain is accepted. \AFive{} provides the lawful boundary state manifold and its reduced dynamics. \ASix{} proves that any admissible runtime factorizes as \GRIP{} and that branch ranking is governed by \(F_{\GFF}\). \ASeven{} extends this runtime with Qi-selector structure but does not alter the constitutional factorization. Therefore every cosmic epoch transition can be analyzed by the same three-stage logic: exploration of admissible channels, loss of stability of the symmetric branch at a threshold, and stabilization of the realized history as an attractor --- a \GA{} for a transient epoch record, the stratum's \GGA{} for a relic or final state that must persist (\AEleven{} \citep{ArisakaA11Origin}). The regime clause follows from \textsf{T-GRIP-Out-Necessity} (\AEleven{}): gradient flow on a fixed landscape cannot cross a stratum boundary, so each genuine transition must be a landscape reorganization (\GRIPout{}), and everything between transitions, being gradient flow plus its instanton stitching on a fixed landscape, is \GRIPin{} by definition (the \GRIPin{} definition of \ASix{}).
\end{proof}

\section{What this buys cosmology}

Pulling the five ideas together gives the discipline that governs the rest of the paper. The bulk supplies one geometry; the boundary law makes ordinary cosmology a projection of it; the integer ledger leaves no continuous dials; and the runtime selects which lawful history happens. The single operative consequence is a rule with no exceptions: \emph{every cosmological source term must descend the inheritance chain} --- bulk, projection, ledger, runtime --- and no new dial may be introduced along the way. A quantity that cannot be placed on this chain is not derived; it is merely appended, and is inadmissible by construction.

Table~\ref{tab:problem-to-ledger-map} makes the discipline concrete, mapping each open cosmological problem of Part~I to the inherited components most relevant to it. It is a planning map, not a victory lap: it shows where each answer must come from, not that each answer is already complete.

\begin{table}[H]
\centering
\caption[Map from the seventeen open problems to the inherited cohort components]{Working map from each of the seventeen open problems (Table~\ref{tab:seventeen-problems}) to the most relevant inherited components from the upstream papers (\AOne{}--\ASix{}, with the particle-sector ledger of \ANine{} where noted). This is a planning table for \ATen{} rather than a claim that every problem is already solved.}
\label{tab:problem-to-ledger-map}
\small
\begin{tabular}{p{0.05\textwidth}p{0.34\textwidth}p{0.50\textwidth}}
\toprule
\textbf{No.} & \textbf{Problem} & \textbf{Most relevant inherited inputs} \\
\midrule
\multicolumn{3}{@{}l}{\textbf{\textit{I.\ The two flagship tensions}}}\\
1 & Hubble tension & Recombination history, sound horizon, early ledger, corridor projection effects \\
2 & \(S_8/\sigma_8\) growth tension & Primordial spectrum, dark matter, neutrinos, growth history \\
\multicolumn{3}{@{}l}{\textbf{\textit{II.\ The dark Universe and its fate}}}\\
3 & The cosmological-constant problem & \GFF{} \(=\GEP+\CEP\); boundary GR; residual-ledger interpretation \\
4 & Constant or evolving dark energy? & Residual vacuum export of the completed transition; late-time source evolution \\
5 & Cosmic coincidence / ``why now'' & Full background closure; the \(\pi\)-locked shares; the epoch calibration \\
6 & The nature of dark matter & Compact-angle ontology; Kaxion corridor; extra Abelian structure \\
\multicolumn{3}{@{}l}{\textbf{\textit{III.\ The beginning and the primordial signals}}}\\
7 & Origin / beginning & Constitutional distinction between model and observation; \GG{} bulk; \(\EGUT\) boundary scale; \OGGEP{} \\
8 & The cause of inflation & Scale chain; throat geometry; \BIsix{} normalization; \GFF{} vacuum-selection logic \\
9 & Primordial gravitational waves & Inflationary tensor sector, throat dynamics, CMB B-modes, relic spectrum \\
10 & Baryogenesis & Derived neutrino masses, Majorana channel, leptonic CP block, exact baryonic protection (\ANine{}) \\
11 & BBN / lithium-7 & Derived baryonic ledger, expansion history, dark-sector allowances, \(\Neff\) consistency \\
\multicolumn{3}{@{}l}{\textbf{\textit{IV.\ Structure, cosmic dawn, and the JWST surprises}}}\\
12 & Early galaxies / JWST & Growth history, dark-matter interpretation, early star-formation environment, primordial spectrum \\
13 & Little red dots / overmassive early black holes & Horizon ontology (\OGGEP{} port); collapse thresholds; PBH windows; early accretion environment \\
14 & Cosmic dawn / reionization / 21\,cm & Baryonic ledger; growth history; early-structure timing from the inherited spectrum \\
15 & Black-hole information & \OGGEP{}, \GDOS{} open-system semantics, bulk-to-boundary projection, \GFF{} entropy ledger \\
\multicolumn{3}{@{}l}{\textbf{\textit{V.\ Cracks in the foundations}}}\\
16 & Isotropy / dipole / large-angle anomalies & Boundary-projection statistics (\OGGEP{}/\GDOS{}); large-scale sampling of the primordial spectrum \\
17 & Neutrino masses and \(N_{\mathrm{eff}}\) & Derived neutrino block of \ANine{}; \(\Neff\) counting; expansion history \\
\bottomrule
\end{tabular}
\end{table}


That last point is the one honest rule the paper holds itself to throughout. Inheritance constrains, but it does not by itself prove; so every cosmological entry in what follows carries an explicit provenance tag --- \emph{Derived} where the chain closes completely, \emph{Derived-conditional} where it closes only modulo a named lemma or conjecture, and \emph{Asserted} where the framework points the way but the derivation is not yet in hand. The colors of Figure~\ref{fig:derivation-tree} are exactly these tags, and they are not cosmetic: they are the difference between a result and an intention.

There is a discipline hidden in this inheritance, and it deserves a plain statement. Because every tool used here was built --- and paid for --- in the particle-physics companion, the cosmology that follows is never free to invent what it needs: whatever explains the sky must already have explained the electron. Philosophically, this begins to dissolve the old division between laws and initial conditions --- on a fixed geometry read through an open-system projection, what look like arbitrary starting values are the lawful shadows of structure. If that sounds too strong, the reader holds the antidote: the codebook of Part~IX grades every claim this framework will make.

The framework is now in hand. The bulk, its boundary projection, the zero-parameter ledger, and the open-system runtime are everything the cosmology inherits and everything it is allowed to use. The cosmological history this framework supports --- the lawful beginning at the GUT edge, the Friedmann kinematics on the boundary, and the radial ascent and cooling of the universe --- is set up in the next Part.



\section{Part~II in plain English: what this paper borrows, and at what strength}
Almost nothing in this Part is native to \ATen{}. The six-dimensional
geometry, the boundary projection, the anomaly-freedom accounting, and the
runtime that selects among branches are all established in the upstream papers
of the cohort, and they are set out here at exactly the strength their owning
papers give them.

That last clause is the one to hold onto. An import re-narrated in a new
paper's own words has not been verified; it has been copied, and copying is
where the strength of a claim quietly drifts upwards. The rule followed here
is that a result is quoted at its parent's grade, and where the parent hedges
this paper hedges in the same place and with the same words. Several of the
entries below are conditional on named conjectures, and the conditionality
travels with them.

What the inheritance buys cosmology is a single thing, and it is worth saying
plainly rather than leaving it to be inferred. It buys the absence of dials.
If the geometry is fixed and admits no free continuous parameters, then a
cosmic number is either forced by it or the framework has nothing to say about
that number. There is no third option in which a parameter is adjusted to fit.
That is the source of every prediction in this paper and it is also the source
of every way the paper can fail: a framework with nothing to tune has nothing
to retreat to.

\GGPart[The cosmological stage]{The cosmological stage}{The bulk as the arena of cosmic history --- the inherited scale chain, the lawful beginning at the GUT edge, the Friedmann kinematics from the boundary law, and the expansion, cooling, and horizons that set the scene for inflation}

% ===================================================================
% Part III NEW front sections (thesis + GUT-edge + ladder + LCDM theorem).
% Insert right after the Part III \GGPart line, before "The scale chain".
% ASCII-only.  Facts from A1,A2,A3,A4,A5,A6,A9 exec summaries (cited).
% ===================================================================

\section{The thesis: \texorpdfstring{\(\Lambda\)CDM}{LCDM} is a theorem of \GGTheory{}}
\label{sec:partIII-thesis}

The companion paper \ANine{} \citep{ArisakaA9SM} closed a question that had stood open since the Standard Model was written down: it derived the model's twenty-eight free parameters --- the couplings, the masses, the mixing angles --- as outputs of one six-dimensional geometry, with no continuous input left to tune. This Part makes the cosmological claim that stands one story above it, and it makes it without hedging:

\smallskip
\emph{\(\Lambda\)CDM is a theorem of \GGTheory{}.} The Friedmann--Lema\^itre--Robertson--Walker equations, and the form and source of every one of the six numbers the concordance model otherwise reads off the sky, are forced by the same constitution, the same bulk, and the same anomaly-locked ledger that fix the particle sector. The universe is not described by a model with six dials; it is the boundary readout of a geometry that has none.

\smallskip
The claim is established by a single descent --- the same shape \ANine{} ran on the particle stratum, now run on the cosmological boundary stratum (Figure~\ref{fig:partIII-ladder}):
\[
\begin{aligned}
&\underbrace{\text{GG-Constitution}}_{\AOne{}}
\;\Rightarrow\;
\underbrace{\text{GG-6}+\BIsix{}}_{\ATwo{},\,\AThree{}}
\;\Rightarrow\;
\underbrace{\text{zero free parameters}}_{\AFour{}}\\[6pt]
&\Rightarrow\;
\underbrace{\text{general relativity}}_{\AFive{}}
\;\Rightarrow\;
\underbrace{\text{the FRW equations}}_{\AFive{}+\ASix{}\,\to\,\text{here}}
\;\Rightarrow\;
\underbrace{\Lambda\text{CDM}}_{\text{here; Parts IV--X}}.
\end{aligned}
\]
The first four arrows are theorems already proved in the companion papers, named where they are used; the last two are the work of this paper. The sections that follow climb the ladder rung by rung and end by stating the cosmological theorem and auditing it parameter by parameter.

\begin{figure}[t]
\centering
\resizebox{0.604\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily\small,
  every node/.style={align=center, rounded corners=2.5pt, inner sep=4pt},
  rung/.style={draw=blue!55!black, line width=0.70pt, minimum width=80mm, minimum height=12mm, fill=blue!8, draw=blue!55!black},
  frw/.style={draw=green!55!black, line width=1.10pt, minimum width=80mm, minimum height=12mm, fill=green!12, draw=green!60!black},
  capstone/.style={draw=orange!55!black, line width=1.10pt, minimum width=80mm, minimum height=13mm, fill=orange!18, draw=orange!75!black},
  src/.style={font=\sffamily\footnotesize\itshape, text=black!60, align=left},
  arr/.style={-{Stealth[length=3mm]}, line width=0.80pt, draw=black!65},
]
\node[rung] (c)   {\textbf{GG-Constitution}\\[-2pt]\footnotesize four axioms, seven postulates --- no free sector};
\node[rung, below=5mm of c]  (g)  {\textbf{GG-6 \,$+$\, BI-6}\\[-2pt]\footnotesize the 6D bulk, anomaly-locked};
\node[rung, below=5mm of g]  (z)  {\textbf{Zero free parameters}\\[-2pt]\footnotesize the integer ledger $(1,3,5;7,35)$};
\node[rung, below=5mm of z]  (gr) {\textbf{General relativity}\\[-2pt]\footnotesize Einstein equations from the boundary law};
\node[frw,  below=5mm of gr] (fr) {\textbf{The Friedmann equations}\\[-2pt]\footnotesize FRW from the GRIP-selected attractor};
\node[capstone,  below=5mm of fr] (l)  {\textbf{$\Lambda$CDM as a theorem}\\[-2pt]\footnotesize the six parameters, each provenance-tagged};
\draw[arr] (c)--(g); \draw[arr] (g)--(z); \draw[arr] (z)--(gr); \draw[arr] (gr)--(fr); \draw[arr] (fr)--(l);
\node[src, right=5mm of c]  {\AOne{}};
\node[src, right=5mm of g]  {\ATwo{}, \AThree{}};
\node[src, right=5mm of z]  {\AFour{}};
\node[src, right=5mm of gr] {\AFive{}};
\node[src, right=5mm of fr] {\AFive{}\,$+$\,\ASix{}\\$\to$ \emph{here}};
\node[src, right=5mm of l]  {\emph{here};\\Parts IV--X};
\end{tikzpicture}}
\caption[The \GGTheory{} derivation ladder, from the constitution to \(\Lambda\)CDM.]{\textbf{The derivation ladder.} The cosmological theorem descends through six rungs, each a result proved in the companion paper named at right. The first four are inherited; the Friedmann line (green) and the \(\Lambda\)CDM capstone (gold) are this paper's own theorems, built on the inherited rungs and the \GRIP{} runtime. This is the cosmological analogue of the \GRIP{} spine that \ANine{} ran on the particle stratum to derive the Standard Model.}
\label{fig:partIII-ladder}
\end{figure}

A word, first, on what ``theorem'' is permitted to mean, because the word does real work here and must not be inflated. It means exactly what it means in \ANine{}: the framework \emph{forces the form and the source} of every input, and a published provenance triage records, parameter by parameter, how completely each has been closed --- \emph{Derived} (\textsf{D}) where the chain closes end to end, \emph{Derived-conditional} (\textsf{D}\(^*\)) where it closes modulo a single named lemma or conjecture, and \emph{Asserted} (\textsf{A}) where the framework fixes the structural slot but the closed form is not yet in hand. Of the six \(\Lambda\)CDM parameters, three are \textsf{D}\(^*\) and three are \textsf{A} at this writing (\S\ref{sec:partIII-lcdm-theorem}). The boldness of the claim is therefore structural --- \(\Lambda\)CDM is \emph{forced, not fitted} --- and its honesty is granular: not one parameter is quietly upgraded, and the reader is handed the audit rather than asked to trust the headline. That is the standard \ANine{} met for the Standard Model, and the standard this Part holds itself to for the cosmos.

\section{Why the universe began at the GUT edge, not the Planck scale}
\label{sec:partIII-gut-edge}
\label{sec:beginning}

Ask where the universe began and the reflexive answer is the Planck scale, \(\EPl\simeq1.2\times10^{19}\,\mathrm{GeV}\), where a hypothetical quantum theory of gravity is imagined to take over from a singular classical past. \GGTheory{} returns a different answer, and it is worth stating as bluntly as it deserves: \emph{the observable universe begins at the grand-unification edge, \(\EGUT\simeq2.3\times10^{16}\,\mathrm{GeV}\), and the Planck scale is not a beginning at all.}

The reason is structural, not a matter of taste about quantum gravity. In this framework four-dimensional spacetime is not fundamental: it is the boundary projection of the six-dimensional bulk \(\mathcal{M}_4\times I_r\times S^1_\theta\), and the radial coordinate \(r=\ln(\EGUT/\mu)\) measures depth into that bulk as renormalization-group scale (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}). The ultraviolet edge of the geometry sits at \(r=0\), that is \(\mu=\EGUT\); the manifold occupies \(r\in[0,K]\) and there is no part of it \emph{above} the edge. Since the metric, the causal structure, and the very notions of ``space'' and ``time'' are features of the boundary projection that the edge bounds, they are simply \emph{undefined} for \(\mu>\EGUT\). One cannot ask what the universe was doing at the Planck scale for the same reason one cannot ask what lies north of the North Pole: the coordinate that would label the question has run out.

The Planck scale, in this reading, is not the floor of spacetime but a \emph{derived} reading off the boundary. The four-dimensional Newton constant emerges as a scale-overlap integral over the throat, \(M_{\mathrm{Pl}}^2=\pi M_6^4(1-e^{-2K})\) in the natural units of the geometry, the factor \(\pi\) coming from the full turn of the compact angle \(S^1_\theta\) \citep{ArisakaA5GDOS}; the geometry even fixes the two scales and their ratio, \(\EGUT=\EPl\,e^{-2\pi}\), one monodromy of that same angle. But the Planck \emph{energy} \(\EPl\) is larger than \(\EGUT\) and therefore sits \emph{above} the edge, in the region the manifold does not contain; and the Planck \emph{mass} \(M_{\mathrm{Pl}}\) is a derived coupling strength, not a location in time. Neither is a first moment. The one surface that is a first moment is the edge itself.

This dissolves the Big Bang singularity without denying any equation. Run the classical radiation-dominated solution \(a(t)\propto t^{1/2}\) backward and the curvature invariants and the density formally diverge as \(t\to0^+\) \citep{KolbTurner1990,Mukhanov2005}; that is a true statement \emph{inside} four-dimensional general relativity. But the four-dimensional Einstein equations are themselves a boundary theorem (\S\ref{sec:partIII-ladder}), derived to hold at and below the ultraviolet surface, not above it; extrapolating them past \(\EGUT\) to reach \(a=0\) is using a result outside the domain on which it was proved. The framework therefore separates three things the textbook conflates:
\[
\underbrace{a\to0,\ \rho\to\infty}_{\text{formal backward limit}}
\;\;\neq\;\;
\underbrace{\mu=\EGUT,\ r=0}_{\text{ultraviolet edge of the ledger}}
\;\;\longrightarrow\;\;
\underbrace{\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}}_{\text{first admissible initial surface}}.
\]
The singular limit is a formal consequence of equations pushed out of bounds; the GUT edge is where the ledger actually starts; and the first physically admissible initial-value surface, \(\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}\), carries \emph{finite} inherited data --- a finite scale factor, expansion rate, density, and curvature set by the geometry rather than diverging. Figure~\ref{fig:partIII-gut-vs-planck} contrasts the two pictures.

\begin{figure}[t]
\centering
\resizebox{0.66\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >={Stealth[length=2.4mm]}]
% ---- LEFT: conventional ----
\def\Lx{0}
\fill[blue!8] (\Lx,0) rectangle (\Lx+4.0,7.2);
\draw[->,line width=0.80pt] (\Lx,-0.2) -- (\Lx,8.1) node[above,font=\small]{$E$};
\node[align=center, font=\small\bfseries] at (\Lx+2.0,7.9) {Conventional};
\draw[line width=1.10pt, red!70!black] (\Lx,7.2) -- (\Lx+4.0,7.2);
\node[right, red!60!black, font=\scriptsize] at (\Lx+0.12,7.45) {$\EPl\sim10^{19}$ GeV};
\node[align=center, font=\scriptsize] at (\Lx+2.0,6.6) {Planck wall: singular\\ beginning ($a\to0,\ \rho\to\infty$)};
\node[align=center, font=\itshape] at (\Lx+2.0,3.4) {4D spacetime\\ all the way up};
\node[below, align=center, font=\scriptsize] at (\Lx+2.0,-0.35) {``begins at the\\ Planck scale''};
% ---- RIGHT: GG-Theory ----
\def\Rx{5.4}
\fill[black!10] (\Rx,4.9) rectangle (\Rx+4.0,7.2);
\fill[orange!16] (\Rx,0) rectangle (\Rx+4.0,4.9);
\draw[->,line width=0.80pt] (\Rx,-0.2) -- (\Rx,8.1) node[above,font=\small]{$E$};
\node[align=center, font=\small\bfseries] at (\Rx+2.0,7.9) {\GGTheory{}};
\node[right, black!55, font=\scriptsize] at (\Rx+0.12,7.0) {$\EPl$ --- \emph{above the edge}};
\node[align=center, font=\scriptsize\itshape, black!55] at (\Rx+2.0,6.0) {not part of the manifold:\\ space \& time undefined\\ ($r<0$)};
\draw[line width=1.10pt, red!70!black] (\Rx,4.9) -- (\Rx+4.0,4.9);
\node[right, red!60!black, font=\scriptsize, align=left] at (\Rx+0.12,5.18) {$\EGUT\!\simeq\!2.3\!\times\!10^{16}$ GeV\\ UV edge ($r=0$): $\Sigma_{\rm GUT}^{\rm Bdy}$};
\node[align=center, font=\itshape] at (\Rx+2.0,2.4) {4D spacetime / FRW\\ the observable universe\\ ($0\le r\le K$)};
\node[below, align=center, font=\scriptsize] at (\Rx+2.0,-0.35) {``begins at the\\ GUT edge''};
\end{tikzpicture}}
\caption[The beginning: the Planck-scale start versus the \GG{} GUT-edge start.]{\textbf{Where the universe begins.} \emph{Left:} the conventional picture --- spacetime extends up to a Planck wall, with a singular beginning. \emph{Right:} in \GGTheory{} four-dimensional spacetime is the boundary projection of the bulk and exists only at and below the ultraviolet edge \(\EGUT\) (\(r=0\)); above the edge (\(r<0\), where the Planck \emph{energy} lies) the manifold does not extend and the notions of space and time are undefined. The first admissible initial surface is \(\Sigma_{\rm GUT}^{\rm Bdy}\), carrying finite inherited data; the Planck \emph{mass} is a derived coupling, not a first moment.}
\label{fig:partIII-gut-vs-planck}
\end{figure}

This is a statement about the domain of a description, not a metaphysics of creation. \GGTheory{} does not assert that nothing existed ``before'' \(\EGUT\); it observes that ``before,'' ``existed,'' and ``time'' are boundary notions the question presupposes and the geometry does not supply above its edge. The honest content is sharp and limited: the first surface on which a cosmology can be written down is \(\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}\), and everything this Part derives is anchored there.



Before any single cosmological observable can be derived, the stage on which the whole history plays out has to be built. That stage is the six-dimensional throat and its four-dimensional boundary, read through the inherited constitutional apparatus of Part~II. This Part assembles it in four moves and then hands the assembled scene to the inflationary dynamics of Part~IV. First we lay down the ruler: the scale chain \(\EPl\to\EGUT\to\Mlock\) that the particle papers already fixed, and what each rung of that ladder is responsible for. Second we ask where the story is actually allowed to start, and answer that it starts not at a four-dimensional Planck singularity but at the finite GUT edge of the throat, carrying finite initial data. Third we recover the Friedmann kinematics from the boundary Einstein equation, keeping the textbook equations untouched and constraining only the source ledger that feeds them. Fourth we run that ledger forward through expansion, cooling, and the horizon structure, and read the resulting arrow of time in the open-system language of \GDOS{}/\GFF{}. None of the equations in this Part are new; what is new is the disciplined inheritance chain that decides which terms are allowed to occupy them.


\section{The scale chain: \texorpdfstring{\(\EPl \to \EGUT \to \Mlock\)}{EPl to EGUT to Mlock}}

Cosmology and particle physics are the same story told at two ends of one ruler, and in \GGTheory{} that ruler is literally the radial depth of the throat. The particle paper already engraved its tick marks. Three energy scales, fixed there by a short arithmetic ledger, descend unchanged into the cosmological sequel, and almost everything in this Part is a consequence of how far apart they sit. Before doing any dynamics it is worth setting the ruler down and reading what each mark controls --- because a great deal of confusion in cosmology comes from asking a single ``fundamental energy'' to do several logically different jobs at once. Here the division of labor is clean.

\subsection{The inherited scales}

The particle paper fixed the three scales that matter most for the cosmological sequel:
\begin{equation}
\EPl = \sqrt{\frac{\hbar c^5}{G_N}},
\qquad
\EGUT = \EPl e^{-2\pi},
\qquad
\Mlock = \frac{\EPl}{2e^{35}}.
\label{eq:partII-scale-chain}
\end{equation}
Numerically, using the same values already adopted in \ANine{}, one has
\begin{equation}
\EPl \simeq 1.22\times 10^{19}\;\mathrm{GeV},
\qquad
\EGUT \simeq 2.28\times 10^{16}\;\mathrm{GeV},
\qquad
\Mlock \simeq 3.85\times 10^{3}\;\mathrm{GeV}.
\label{eq:partII-scale-chain-numbers}
\end{equation}
The associated radial coordinate is
\begin{equation}
 r = \ln\!\left(\frac{\EGUT}{\mu}\right),
\label{eq:partII-radial-coordinate}
\end{equation}
so that the full depth from the GUT edge to the lock scale is
\begin{equation}
\Delta r_{\mathrm{GUT}\to \mathrm{lock}}
= \ln\!\left(\frac{\EGUT}{\Mlock}\right)
= 35 - 2\pi + \ln 2
\simeq 29.41.
\label{eq:partII-throat-depth}
\end{equation}
Equation~\eqref{eq:partII-throat-depth} is one of the most important numbers in the paper. It says that there is a very large logarithmic hierarchy between the operational ultraviolet edge and the finite infrared lock scale even though both are fixed by the same short arithmetic ledger. The cosmological content of this Part is largely the story of what happens when a boundary universe inherits that large but finite depth.

\subsection{What each inherited scale actually controls}

Because three very different scales appear already at the beginning of the story, it is useful to state explicitly what each one controls. A great deal of confusion in cosmology comes from trying to make one scale do too many logically different jobs. In the present framework the division of labor is much cleaner. \(\EPl\) is the dimensional anchor set by Newton's constant. \(\EGUT\) is the first ultraviolet scale at which the boundary description is turned on. \(\Mlock\) is the finite infrared seam where the throat exports the low-energy hierarchy, the inflationary plateau height, and later reheating. The meV vacuum scale is then not a fourth independent input but a seesaw descendant of \(\Mlock\).

\begin{table}[H]
\centering
\caption[The four scales that organize the cosmological history]{The four scales that organize the cosmological history of \ATen{}. Only the first three are primitive inherited inputs; the fourth is a derived dark-energy scale candidate. The columns give each scale, its expression, its numerical size, and the role it plays in this Part. The fourth scale is the one to watch: it is assembled from the other three rather than carried in with them, so a reader tracking what the framework assumes should stop at the third row.}
\label{tab:partII-scale-ladder}
\small
\begin{tabular}{p{0.18\textwidth}p{0.22\textwidth}p{0.23\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Scale} & \textbf{Expression} & \textbf{Numerical size} & \textbf{Primary role in Part~II}\\
\midrule
Planck anchor & \(\EPl=\sqrt{\hbar c^5/G_N}\) & \(1.22\times 10^{19}\,\mathrm{GeV}\) & Converts the gravitational coupling into an energy anchor; not the operative boundary beginning of \GG{} cosmology.\\
Operational GUT edge & \(\EGUT=\EPl e^{-2\pi}\) & \(2.28\times 10^{16}\,\mathrm{GeV}\) & First ultraviolet boundary at which the cosmological description is initialized.\\
Lock scale & \(\Mlock=\EPl/(2e^{35})\) & \(3.85\times 10^3\,\mathrm{GeV}\) & Hierarchy seam, inflationary plateau height, and reheating target scale.\\
Derived vacuum scale & \(m_\Lambda^{\mathrm{geom}}=\Mlock^2/M_{\mathrm{Pl,red}}\) & \(\sim 6\,\mathrm{meV}\) & Natural infrared vacuum scale candidate for the late-time dark-energy residue.\\
\bottomrule
\end{tabular}
\end{table}

The practical lesson of Table~\ref{tab:partII-scale-ladder} is that the cosmology paper should not try to compress all questions into a single ``fundamental energy.'' Beginning, inflation, reheating, and late acceleration are different dynamical questions. They inherit different combinations of the same short ledger.


Two clarifications are essential.

First, \(\EGUT\) and \(\Mlock\) are not rival claims about the same physical instant. \(\EGUT\) is the ultraviolet edge at which the boundary description turns on. \(\Mlock\) is the infrared scale at which the throat selects the low-energy seam and where the inflationary and later particle-physics consequences of the locked geometry are most naturally exported. The theory therefore contains both a GUT-beginning statement and a TeV-inflation statement without contradiction.

Second, the radial coordinate \(r\) is not identical to cosmological proper time \(t\). The earlier papers interpret \(r\) as a geometric renormalization coordinate. Cosmological time is the proper time of the four-dimensional boundary metric. The beginning of the cosmological story is therefore not ``a point at \(r=0\)'' but the first boundary initial-data surface inherited from the ultraviolet end of the throat.


\section{The inherited foundation: the first three rungs}
\label{sec:partIII-ladder}

We now climb the ladder of Figure~\ref{fig:partIII-ladder}. The first three rungs are inherited from the companion papers and are recalled here only briefly --- they are the arena, not the new physics. The fourth and fifth rungs --- general relativity and the Friedmann equations --- are the heart of this paper, and the two sections after this one derive them in full, at the level of a graduate course, with the physical meaning of each step made explicit. The point of the brief recap is to fix in view the parameter-free base on which those derivations stand.

\subsection{Rung 1 --- the constitution fixes what may count as a cosmology (\texorpdfstring{\AOne{}}{A1})}
Before any geometry, a constitution. \AOne{} \citep{ArisakaA1Constitution} lays down four axioms --- local causality, the geometric--gauge equivalence principle, the open-system projection principle \OGGEP{}, and the maximum-symmetry/minimum-freedom selector \MSMF{} --- and seven postulates that together define what counts as an admissible physical claim. For cosmology the decisive clause is \MSMF{}: a theory may not carry a free sector it has not earned. This is what forbids the standard repair --- bolting on a quintessence field, a tuned inflaton potential, or a dark-sector coupling to fit the data after the fact. The constitution does not yet say what the universe is; it says what kind of answer is allowed to count, and it rules out the parameter freedom \(\Lambda\)CDM otherwise leans on.

\subsection{Rung 2 --- the bulk and its anomaly lock (\texorpdfstring{\ATwo{}, \AThree{}}{A2, A3})}
The constitution compiles into a unique arena. \ATwo{} \citep{ArisakaA2GG6} derives the six-dimensional bulk with the parameter-free metric \(ds_6^2=e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu+dr^2+d\theta^2\) and the unified group \(\mathrm{Spin}(1,5)\times SU(5)\times U(1)_B\times U(1)_{Qi}\); \AThree{} \citep{ArisakaA3BI6} proves that this six-force closure is anomaly-free through the modified Bianchi identity \BIsix{}. The cosmological consequence is that the \emph{stage} on which cosmic history plays out is fixed and rigid: one bulk, one boundary, one scale axis, no freedom to deform any of them. The arena of Part~III is not chosen; it is the very object \ANine{} used for the particle ledger, read now at the largest scale.

\subsection{Rung 3 --- zero free parameters (\texorpdfstring{\AFour{}}{A4})}
The anomaly lock has an arithmetic shadow. \AFour{} \citep{ArisakaA4Euclid} proves that \BIsix{} forces the integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\), closed by the dual lock \(K=LJ=M^2+N^2+L^2=35\); together with \(\pi\), \(e\), and the conversion factors \(\{c,G_N,\hbar;k_B,\QB\}\) these whole numbers are the entire stock from which every dimensionless cosmological quantity is built. This is the rung that makes the eventual claim sharp: the source terms that feed the Friedmann equations --- the density fractions, the baryon-to-photon ratio, the recombination redshift --- are not numbers to be measured and inserted but closed-form expressions over a fixed ledger. A cosmology built on this rung cannot be retuned; a single wrong number falsifies it.

\subsection{Rung 3, restated for cosmology}
That third rung is what makes the eventual claim sharp: the source terms that feed the Friedmann equations --- the density fractions, the baryon-to-photon ratio, the recombination redshift --- are not numbers to be measured and inserted but closed-form expressions over a fixed ledger. With the arena (Rungs 1--2) and the ledger (Rung 3) in hand, we turn to the two derivations that are the substance of this paper: gravity, and then the cosmology it carries.

\section{General relativity as a theorem of GG-6/\texorpdfstring{\GDOS{}}{GDOS}}
\label{sec:partIII-GR-theorem}

In every other framework general relativity is a \emph{starting point}: one writes the Einstein--Hilbert action by hand and proceeds \citep{MisnerThorneWheeler1973,Wald1984,WeinbergGR1972}. In \GGTheory{} it is an \emph{output}. The four-dimensional Einstein equations, the Newton constant that sets the strength of gravity, and the scale of the cosmological constant all descend from the six-dimensional bulk by a single controlled reduction \citep{ArisakaA5GDOS}. This is the load-bearing rung of the paper, so we carry the derivation out in full, at the level of a graduate course, recalling each standard object as we use it.

\subsection{The starting point: curvature in six dimensions}
The dynamics of any metric is governed by the Einstein--Hilbert action --- the integral of the Ricci scalar \(R\), the single number that measures how curved spacetime is, weighted by the invariant volume. In the bulk this reads
\[
S_{\mathrm{EH}}^{(6)} = \frac{M_6^4}{2}\int d^6x\,\sqrt{-g_6}\,R_6,
\]
with \(M_6\) the (derived) six-dimensional gravitational scale, \(g_6\) the bulk metric determinant, and \(R_6\) its Ricci scalar. No other gravitational term may be added: the constitution admits only the anomaly-free closure of \AThree{}, so the bulk is rigid. \emph{Physical picture:} curvature costs action, and gravity is whatever dynamics minimizes that cost; all of four-dimensional gravity is folded into this one expression, and the task is to read it out on the boundary by integrating away the two extra directions.

\subsection{Integrating out the throat: the scale-profile reduction}
The bulk metric of \ATwo{} is the warped product
\[
ds_6^2 = e^{-2r}\,g_{\mu\nu}(x)\,dx^\mu dx^\nu + dr^2 + d\theta^2,
\qquad r\in[0,K],\ \theta\in[0,2\pi),
\]
in which \(g_{\mu\nu}(x)\) is the metric a boundary observer measures and the warp \(e^{-2r}\) encodes renormalization-group depth. Inserting this ansatz into \(S_{\mathrm{EH}}^{(6)}\) and integrating over \(r\) and \(\theta\) needs two standard facts about warped products. First, the volume element factorizes,
\[
\sqrt{-g_6} = e^{-4r}\sqrt{-g_4},
\]
since the warp multiplies each of the four boundary directions by \(e^{-2r}\). Second, the curvature decomposes as
\[
R_6 = e^{2r}R_4 - 20,
\]
where \(R_4\) is the curvature of \(g_{\mu\nu}\), the prefactor \(e^{2r}\) is the rescaling of an inverse-length-squared quantity by the warp, and the constant \(-20\) is the (negative) curvature the throat carries on its own. Assembling the integrand,
\[
\sqrt{-g_6}\,R_6 = \sqrt{-g_4}\,\bigl(e^{-2r}R_4 - 20\,e^{-4r}\bigr),
\]
the two extra integrals are elementary,
\[
\int_0^{K}\! e^{-2r}\,dr = \tfrac12\bigl(1-e^{-2K}\bigr),\qquad
\int_0^{K}\! e^{-4r}\,dr = \tfrac14\bigl(1-e^{-4K}\bigr),\qquad
\int_0^{2\pi}\! d\theta = 2\pi,
\]
and carrying them out collapses the six-dimensional action to a four-dimensional one,
\[
S_{\mathrm{EH}}^{(6)} \;\longrightarrow\; \int d^4x\,\sqrt{-g_4}\,\Bigl[\tfrac{\pi M_6^4}{2}\bigl(1-e^{-2K}\bigr)R_4 \;-\; 5\pi M_6^4\bigl(1-e^{-4K}\bigr)\Bigr].
\]

\subsection{Reading off the four-dimensional theory}
This is exactly the four-dimensional Einstein--Hilbert action with a cosmological constant,
\[
S_{\mathrm{EH}}^{(4)} = \frac{M_{\mathrm{Pl}}^2}{2}\int d^4x\,\sqrt{-g_4}\,(R_4 - 2\Lambda_4),
\]
once the two coefficients are read off:
\[
\boxed{\;M_{\mathrm{Pl}}^2 = \pi M_6^4\bigl(1-e^{-2K}\bigr)\;},
\qquad
\Lambda_4^{\mathrm{naive}} = \frac{5\bigl(1-e^{-4K}\bigr)}{1-e^{-2K}}\ \xrightarrow{\,K=35\,}\ 5\ \ (\text{in } M_{\mathrm{Pl}}\text{ units}).
\]
The Planck mass is not a postulate but the \emph{overlap integral of the throat}, the factor \(\pi\) being the full turn of the compact angle \(S^1_\theta\). \emph{Physical picture:} gravity is weak --- \(M_{\mathrm{Pl}}\) is enormous --- precisely because curvature is diluted over the extra directions; the hierarchy between gravity and the other forces is geometric, not tuned. Varying \(S_{\mathrm{EH}}^{(4)}\) together with the boundary matter action gives the four-dimensional Einstein equation; in its full inherited form it carries two small corrections from the heavy modes that were integrated out \citep{ArisakaA5GDOS},
\[
M_{\mathrm{Pl}}^2\,G_{\mu\nu} + \Lambda_4\,g_{\mu\nu} = \langle T_{\mu\nu}\rangle_{\mathrm{Bdy}} + \Delta T_{\mu\nu}^{\mathrm{GS}} + \Delta T_{\mu\nu}^{\mathrm{KK}},
\]
with \(G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R_4\,g_{\mu\nu}\) the Einstein tensor, \(\langle T_{\mu\nu}\rangle_{\mathrm{Bdy}}\) the projected boundary stress-energy, and the last two terms the Green--Schwarz and Kaluza--Klein inflow. At cosmological energies these are suppressed by \((E/M_{\mathrm{lock}})^2\sim10^{-15}\) and drop, leaving the textbook Einstein equation exactly,
\[
\boxed{\;G_{\mu\nu} + \Lambda_4\,g_{\mu\nu} = 8\pi G_N\,T_{\mu\nu}\;},
\qquad
G_N = \frac{1}{8\pi M_{\mathrm{Pl}}^2}.
\]

\subsection{Conservation, the Newton constant, and the vacuum scale}
Three features of this result carry directly into cosmology. \emph{(i) Conservation is a theorem.} The local law \(\nabla_\mu T^{\mu\nu}=0\) is not imposed by hand; it is the Ward identity of bulk diffeomorphism invariance projected to the boundary, so the conservation of energy and momentum that the Friedmann continuity equation expresses is itself derived. \emph{(ii) The Newton constant is fixed.} \(G_N=1/(8\pi M_{\mathrm{Pl}}^2)\) is set by the throat geometry, not chosen. \emph{(iii) The vacuum scale arrives with a number.} The naive value \(\Lambda_4^{\mathrm{naive}}\approx5\) in Planck units is reduced by a boundary-localized seesaw to
\[
m_\Lambda^{\mathrm{geom}} = \frac{M_{\mathrm{lock}}^2}{M_{\mathrm{Pl,red}}} \approx 6\ \mathrm{meV},
\]
which by itself collapses the notorious \(10^{120}\) vacuum-energy discrepancy \citep{Weinberg1989CC,Carroll2001} to a residual factor of a few in scale --- and, once the ledger coefficient \(\zeta_\Lambda = \pi/LK\) is selected, to the percent level (Eq.~\eqref{eq:partII-rhoLambda-closed}). \emph{Physical picture:} the cosmological constant is small not by cancellation to sixty decimal places but because a large scale (\(M_{\mathrm{Pl}}\)) divides a modest one (\(M_{\mathrm{lock}}^2\)), a seesaw of exactly the kind that makes neutrinos light. The order-unity coefficient and the sign of \(\Lambda\) are \emph{not} yet closed in \AFive{}; pinning them is the work of Part~IV, and we flag that debt rather than hide it. With that one acknowledged gap, general relativity --- the Einstein equation, \(G_N\), and the scale of \(\Lambda\) --- is a theorem of GG-6/\GDOS{}, derived rather than assumed.

\section{The Friedmann equations as a theorem}
\label{sec:partIII-FRW-theorem}

The Einstein equation just derived holds for any spacetime. A cosmology is the special solution in which the universe looks the same at every place and in every direction --- the cosmological principle \citep{Weinberg2008,Peebles1993}. In the textbook this principle is an \emph{assumption}, justified after the fact by the smoothness of the microwave sky. In \GGTheory{} it is the geometric \emph{attractor} of the runtime: the \GRIP{} cascade \(\mathrm{DGI}\to\mathrm{GSSB}\to\mathrm{GA}\) of \ASix{} \citep{ArisakaA6Gi4}, run on the cosmological boundary stratum, flows to the maximally symmetric configuration, so homogeneity and isotropy are \emph{selected}, not posited. (That this attractor is \emph{unique} is a \textsf{D}\(^*\) result, resting on the attractor uniqueness \ASix{} states and the inflationary smoothing of Part~IV; we state it, not bury it.) Granting the attractor, the derivation of the Friedmann equations is the standard graduate computation, which we now carry out because it is the second load-bearing rung.

\subsection{The maximally symmetric metric}
Homogeneity and isotropy fix the metric up to one function of time and a discrete curvature sign \citep{Friedmann1922,Lemaitre1927}; the standard graduate derivations are collected in the textbooks of \citet{Weinberg2008,Dodelson2003,Mukhanov2005,KolbTurner1990}. In comoving coordinates,
\begin{equation}
ds^2 = -dt^2 + a^2(t)\Bigl[\frac{dr^2}{1-kr^2} + r^2\,d\Omega_2^2\Bigr],
\label{eq:partII-frw-metric}
\end{equation}
with \(a(t)\) the scale factor --- the single dynamical degree of freedom, the ``size'' of space --- and \(k\in\{-1,0,+1\}\) the sign of the spatial curvature. \emph{Physical picture:} \(a(t)\) stretches every comoving separation in lockstep; nothing in the metric singles out a place or a direction, which is the cosmological principle made geometric.

\subsection{The curvature of an expanding universe}
The Einstein tensor of this metric is a textbook computation \citep{Wald1984,MisnerThorneWheeler1973}; we record the load-bearing steps. With an overdot for \(d/dt\), the non-vanishing Christoffel symbols are
\[
\Gamma^0_{ij} = a\dot a\,\gamma_{ij},\qquad
\Gamma^i_{0j} = \frac{\dot a}{a}\,\delta^i_j,\qquad
\Gamma^i_{jk} = \widehat\Gamma^i_{jk}(\gamma),
\]
where \(\gamma_{ij}\) is the comoving spatial metric and \(\widehat\Gamma\) its own connection. From these the Ricci tensor is
\[
R_{00} = -3\frac{\ddot a}{a},\qquad
R_{ij} = \Bigl(\frac{\ddot a}{a} + 2\frac{\dot a^2}{a^2} + 2\frac{k}{a^2}\Bigr)g_{ij},
\]
and contracting gives the Ricci scalar
\[
R_4 = 6\Bigl(\frac{\ddot a}{a} + \frac{\dot a^2}{a^2} + \frac{k}{a^2}\Bigr).
\]
Assembling \(G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R_4\,g_{\mu\nu}\), the independent components are
\[
G_{00} = 3\Bigl(\frac{\dot a^2}{a^2}+\frac{k}{a^2}\Bigr),\qquad
G_{ij} = -\Bigl(2\frac{\ddot a}{a}+\frac{\dot a^2}{a^2}+\frac{k}{a^2}\Bigr)g_{ij}.
\]
\emph{Physical picture:} \(G_{00}\) collects the expansion rate and the spatial curvature --- it is the geometry's measure of how much energy is present; \(G_{ij}\) adds the acceleration \(\ddot a\), the geometry's response to pressure.

\subsection{The source: a perfect fluid from the inherited ledger}
The same symmetry forces the boundary stress-energy to be that of a perfect fluid,
\[
T^\mu{}_\nu = \mathrm{diag}(-\rho,\,p,\,p,\,p),
\]
with \(\rho\) the energy density and \(p\) the pressure: isotropy forbids any momentum flux or shear. Here is the single place the framework departs from the textbook --- not in the equation but in what may fill it. \(\rho\) and \(p\) are not free fluids to be fitted but the coarse-grained boundary projection of the bulk source ledger,
\begin{equation}
\rho = \rho_r + \rho_b + \rho_{\mathrm{DM}} + \rho_\Lambda + \cdots,
\label{eq:partII-source-ledger}
\end{equation}
each sector's identity, equation of state \(w_i=p_i/\rho_i\), and normalization fixed by the integer ledger of the inherited foundation rather than chosen.

\subsection{The Friedmann equations}
Inserting the Einstein tensor and the perfect-fluid source into \(G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G_N T_{\mu\nu}\), the time--time (\(00\)) component yields the first Friedmann equation,
\begin{equation}
\boxed{\;H^2 \equiv \Bigl(\frac{\dot a}{a}\Bigr)^{\!2} = \frac{8\pi G_N}{3}\,\rho - \frac{k}{a^2} + \frac{\Lambda}{3}\;}
\label{eq:partII-friedmann1}
\end{equation}
and combining it with the trace of the space--space (\(ij\)) component gives the second, the acceleration equation,
\begin{equation}
\boxed{\;\frac{\ddot a}{a} = -\frac{4\pi G_N}{3}\,(\rho+3p) + \frac{\Lambda}{3}\;}
\label{eq:partII-acceleration}
\end{equation}
The conservation law \(\nabla_\mu T^{\mu0}=0\) --- already a theorem from the previous section --- gives the continuity equation,
\begin{equation}
\dot\rho + 3H(\rho+p) = 0,
\label{eq:partII-continuity}
\end{equation}
which is not independent of the other two but states how each sector dilutes. \emph{Physical picture:} the first Friedmann equation is a first integral of the motion, ``energy density sets the expansion rate''; the second says pressure gravitates, so a sector with \(\rho+3p>0\) decelerates the universe and a vacuum with \(p=-\rho\) accelerates it; the continuity equation is simply dilution, \(\rho_i\propto a^{-3(1+w_i)}\) \citep{KolbTurner1990,Dodelson2003}.

\subsection{What is a theorem, and what is inherited}
The three equations are identical in form to the textbook Friedmann system, and that identity is the point, not a limitation: \GGTheory{} modifies neither gravity nor kinematics. What it fixes is everything on the right-hand side. \(G_N\) is the throat overlap integral of the previous section; \(\Lambda\) carries the seesaw scale; and the contents of \(\rho\) --- which sectors exist, with which equations of state, in which proportions --- are read from the fixed ledger, not measured and inserted. The Friedmann equations are therefore a theorem of \GGTheory{}: the boundary Einstein equation, restricted to the runtime-selected symmetric attractor, sourced by a ledger with no free dials. The full line-by-line specialization --- every Christoffel symbol and curvature component --- is collected in Appendix~\ref{appA:frw}; what matters here is the chain, bulk action \(\to\) Einstein equation \(\to\) Friedmann system, each link forced. The concordance model is what this system becomes when the ledger is read out, which is the subject of the next section.

\section{\texorpdfstring{\(\Lambda\)CDM}{LCDM} as a theorem: the six parameters}
\label{sec:partIII-lcdm-theorem}

The two derivations just completed --- general relativity, then the Friedmann equations --- deliver the cosmological stage with a fixed source ledger; the concordance model is what that stage becomes when the ledger is read out. \(\Lambda\)CDM is specified by six numbers --- the physical baryon and cold-dark-matter densities \(\omega_b\) and \(\omega_c\), the acoustic angular scale \(\theta_*\), the reionization optical depth \(\tau\), and the amplitude \(A_s\) and tilt \(n_s\) of the primordial spectrum --- from which the entire expansion history and perturbation pattern follow \citep{Planck2018,Dodelson2003}. The claim of this Part is that all six are fixed \emph{in form} by the geometry, so that the concordance model is a theorem of \GGTheory{} rather than a six-parameter fit. Table~\ref{tab:partIII-six-params} states the claim parameter by parameter, with the provenance tag that keeps it honest.

\begin{table}[H]
\centering
\caption[LCDM as a theorem: the six concordance parameters and their closed forms]{\(\Lambda\)CDM as a theorem: the six concordance parameters and the principal derived outputs, each with its \GGTheory{} closed form (or the structural slot it occupies), its provenance tag (\textsf{D} Derived, \textsf{D}\(^*\) Derived-conditional, \textsf{A} Asserted), and where it is derived. The full twenty-one-entry audit is the codebook of Part~IX.}
\label{tab:partIII-six-params}
\small
\begin{tabular}{@{}p{0.21\textwidth}p{0.34\textwidth}cp{0.20\textwidth}@{}}
\toprule
\textbf{\(\Lambda\)CDM parameter} & \textbf{\GGTheory{} form / origin} & \textbf{Prov.} & \textbf{Derived in} \\
\midrule
\multicolumn{4}{@{}l}{\textbf{\textit{The six concordance parameters}}}\\
\(\omega_b=\Omega_b h^2\) & \(\Omega_b = 1/2\pi^2\) (closed-packet baryon share) & \textsf{D}\(^*\) & Part~X \\
\(\omega_c=\Omega_c h^2\) & \(\Omega_{\mathrm{dm}} = 1/\pi - 1/2\pi^2\) & \textsf{D}\(^*\) & Part~X \\
\(\theta_*\) (acoustic scale) & set by \(z_*=e^7\) and the corridor sound horizon & \textsf{D}\(^*\) & Part~VI \\
\(\tau\) (reionization depth) & reionization-seam slot fixed; closed form owed & \textsf{A} & Part~VI \\
\(A_s\) (scalar amplitude) & inflationary two-point slot fixed; closed form owed & \textsf{A} & Part~IV \\
\(n_s\) (scalar tilt) & \(n_s = 1 - 2/N^* = 0.9620\), derived on the plateau (Lemma~IV, revised) & \textsf{D}\(^*\) & Part~IV \\
\multicolumn{4}{@{}l}{\textbf{\textit{Principal derived outputs}}}\\
\(\Omega_m\) & \(1/\pi \approx 0.318\) & \textsf{D}\(^*\) & Part~X \\
\(\Omega_\Lambda\) & \(1-1/\pi \approx 0.682\) & \textsf{D}\(^*\) & Part~X \\
\(H_0\) & two-stage geometric corridor (\(67\!\leftrightarrow\!73\)) & \textsf{D}\(^*\) & Part~VI \\
age \(t_0\) & FRW integral over the \(\pi\)-locked inventory & \textsf{D}\(^*\) & Part~VIII \\
\bottomrule
\end{tabular}
\end{table}

The audit is deliberately unflattering where it must be. Four of the six --- the two matter densities, the acoustic scale, and the scalar tilt --- are \emph{Derived-conditional}: their closed forms (\(\Omega_b=1/2\pi^2\), \(\Omega_{\mathrm{dm}}=1/\pi-1/2\pi^2\), \(z_*=e^7\), and \(n_s = 1 - 2/N^* = 0.9620\)) follow from the integer ledger, the \(\pi\)-locked inventory of Part~X, and the derived plateau of Part~IV, each modulo the single named conjecture that Part states explicitly. One --- the primordial amplitude \(A_s\) --- is the launch sector's one \emph{Asserted} declared input, its warp candidate (\(75\%\)) flagged and its closing computation (Lemma~V\('\)) named. The sixth, the reionization optical depth \(\tau\), is astrophysics --- a late-time reionization quantity outside the fundamental claim, carried at its measured value. This is the precise sense in which \(\Lambda\)CDM is a theorem: its \emph{form} is forced --- six closed-form slots over one fixed ledger, none free to retune --- while the completeness of each form is audited openly, four derived-conditional, one declared input, one astrophysical. The bold claim and the honest ledger are the same statement read at two resolutions; the chapters that follow set about closing the gap.


\section{Expansion, cooling, and horizons}

Given the equations and the ledger, the rest is kinematics --- but kinematics that does an enormous amount of work. Run a homogeneous universe with positive energy density forward and three things happen automatically: it expands, it cools, and it continually reshapes which regions can talk to which. This section is the thermal-history backbone of the paper, the standard one \citep{KolbTurner1990,Dodelson2003,Mukhanov2005}, read through the \GG{}/\GDOS{} source ledger. We derive the scaling laws for each component, the cooling law from comoving entropy, the temperature--time dictionary that every later milestone rests on, the horizon structure that motivates inflation and dark energy, and the universal \(\Gamma/H\) competition that decides every freeze-out. The equations are textbook --- Friedmann's dynamical solutions \citep{Friedmann1922} and the Hubble law \citep{Hubble1929} read against standard rulers such as the baryon acoustic oscillation scale \citep{PeeblesYu1970,Eisenstein2005,Cole2005} --- and the discipline is in the ledger that fills them. The full from-scratch derivation --- comoving-entropy conservation, the relativistic degree-of-freedom count \(g_*(T)\), and the freeze-out condition \(\Gamma\sim H\) --- is given in Appendix~\ref{appA:thermal}; here we collect the results on which the later milestones rest.

\subsection{Exact scaling law for a constant equation-of-state component}

Equation~\eqref{eq:partII-continuity} is enough to derive the standard scaling law. For a component with \(p=w\rho\),
\begin{equation}
\dot\rho + 3H\rho(1+w)=0.
\label{eq:partII-continuity-w}
\end{equation}
Using \(H=\dot a/a\), this becomes
\begin{equation}
\frac{\dot\rho}{\rho} = -3(1+w)\frac{\dot a}{a}.
\label{eq:partII-log-derivative-rho}
\end{equation}
Integrating both sides yields
\begin{equation}
\rho(a) = \rho_0\,a^{-3(1+w)}.
\label{eq:partII-rho-scaling}
\end{equation}
This compact formula already encodes the entire coarse-grained thermal history:
\begin{align}
&\text{radiation }(w=1/3): && \rho_r \propto a^{-4},
\label{eq:partII-rho-rad}\\[2pt]
&\text{pressureless matter }(w=0): && \rho_m \propto a^{-3},
\label{eq:partII-rho-mat}\\[2pt]
&\text{curvature:} && \rho_k \propto a^{-2},
\label{eq:partII-rho-curv}\\[2pt]
&\text{vacuum }(w=-1): && \rho_\Lambda = \text{const}.
\label{eq:partII-rho-vac}
\end{align}
The extra factor of \(a^{-1}\) for radiation compared with matter is the cooling term. Matter number density dilutes like volume, \(a^{-3}\). Radiation number density also dilutes like \(a^{-3}\), but each photon's energy redshifts by another factor of \(a^{-1}\). That is why the universe cools.

If spatial curvature is negligible, Eq.~\eqref{eq:partII-friedmann1} together with Eq.~\eqref{eq:partII-rho-scaling} can be integrated exactly for any constant \(w\neq -1\):
\begin{equation}
 a(t) \propto t^{\frac{2}{3(1+w)}}.
\label{eq:partII-a-power-law}
\end{equation}
Thus
\begin{align}
&\text{radiation era:} && a(t)\propto t^{1/2},\qquad H=\frac{1}{2t},
\label{eq:partII-radiation-era}\\[2pt]
&\text{matter era:} && a(t)\propto t^{2/3},\qquad H=\frac{2}{3t}.
\label{eq:partII-matter-era}
\end{align}
For a true vacuum term with \(w=-1\), Eq.~\eqref{eq:partII-friedmann1} gives \(H=\mathrm{const}\), so
\begin{equation}
 a(t) \propto e^{H_\Lambda t}.
\label{eq:partII-de-sitter-law}
\end{equation}
In one line: radiation gives rapid decelerating expansion, matter gives slower decelerating expansion, and vacuum energy gives accelerating exponential expansion.

\subsection{Cooling law from comoving entropy conservation}

The statement that the universe cools is usually abbreviated to \(T\propto a^{-1}\). It is worth deriving that relation explicitly because later sections on freeze-out, reheating, BBN, and recombination all depend on it.

For a relativistic plasma in approximate equilibrium,
\begin{equation}
\rho_r(T)=\frac{\pi^2}{30}g_*(T)T^4,
\qquad
p_r(T)=\frac{1}{3}\rho_r(T),
\label{eq:partII-rho-temperature}
\end{equation}
and the entropy density is
\begin{equation}
 s(T)=\frac{\rho_r+p_r}{T}=\frac{2\pi^2}{45}g_{*s}(T)T^3.
\label{eq:partII-entropy-density}
\end{equation}
If entropy is conserved in a comoving volume, then
\begin{equation}
\frac{\dd}{\dd t}\bigl[s(T)a^3\bigr]=0.
\label{eq:partII-comoving-entropy}
\end{equation}
When \(g_{*s}\) is approximately constant, Eq.~\eqref{eq:partII-comoving-entropy} reduces to
\begin{equation}
T\propto a^{-1}.
\label{eq:partII-T-propto-a}
\end{equation}
More generally,
\begin{equation}
T(a) \propto g_{*s}^{-1/3}(T)\,a^{-1}.
\label{eq:partII-T-with-gstar}
\end{equation}
This small correction matters whenever species become nonrelativistic and dump entropy into the still-coupled plasma. Conceptually, however, the law remains simple: absent significant entropy production, expansion means cooling.

Combining Eqs.~\eqref{eq:partII-friedmann1} and \eqref{eq:partII-rho-temperature} in the radiation era gives the standard temperature--Hubble relation,
\begin{equation}
H(T) = \sqrt{\frac{8\pi G_N}{3}\frac{\pi^2}{30}g_*T^4}
\simeq 1.66\,g_*^{1/2}\frac{T^2}{M_{\mathrm{Pl}}}.
\label{eq:partII-H-of-T}
\end{equation}
Using \(H\simeq 1/(2t)\) during radiation domination then gives
\begin{equation}
 t \simeq 0.301\,g_*^{-1/2}\frac{M_{\mathrm{Pl}}}{T^2}.
\label{eq:partII-time-temperature}
\end{equation}
Equation~\eqref{eq:partII-time-temperature} is one of the most useful equations in practical cosmology. It lets us translate temperatures into approximate times throughout the early universe. BBN, recombination, neutrino decoupling, electroweak freeze-out, and many later audit steps all rest on this relation.

\subsection{Why the expansion cools every sector differently}

Although \(T\propto a^{-1}\) is the right law for a coupled relativistic plasma, not every component cools in the same way. It is important not to conflate the photon temperature, the matter kinetic temperature, and the dilution of vacuum energy.

\paragraph{Radiation.}
For photons and any tightly coupled relativistic bath, the typical particle energy redshifts as \(E\propto a^{-1}\), and the temperature follows Eq.~\eqref{eq:partII-T-propto-a}.

\paragraph{Nonrelativistic matter.}
For a decoupled nonrelativistic gas, the number density still scales as \(a^{-3}\), but the kinetic temperature scales approximately as
\begin{equation}
T_{\mathrm{nr}}\propto a^{-2}
\label{eq:partII-T-nonrel}
\end{equation}
after decoupling because momenta redshift as \(p\propto a^{-1}\) while kinetic energy is \(p^2/2m\).

\paragraph{Vacuum energy.}
A true cosmological constant does not cool at all:
\begin{equation}
\rho_\Lambda = \mathrm{const},\qquad p_\Lambda = -\rho_\Lambda.
\label{eq:partII-vacuum-constant}
\end{equation}
This is why vacuum energy eventually wins against matter and radiation. It does not dilute while they do.

\paragraph{Open-system caveat.}
Within \GDOS{} one should remember that ``temperature'' is sometimes an effective boundary variable rather than a complete microscopic description. Nevertheless, for the eras relevant to ordinary radiation, recombination, and large-scale expansion, the textbook thermodynamic language remains the correct leading coarse-grained one.

\subsection{Why the universe expands at all}

One sometimes speaks as if ``expansion'' itself required a special new force. Equations~\eqref{eq:partII-friedmann1} and \eqref{eq:partII-acceleration} show that this is not the right way to think. Once a homogeneous metric and nonzero background energy density are present, a static universe is highly nongeneric. In fact, for \(k=0\) and \(\rho>0\), the first Friedmann equation implies \(H^2>0\), so the natural solutions are expanding or contracting. The historical universe is observed to be on the expanding branch.

The deeper question is therefore not ``why does a homogeneous universe expand at all?'' but rather:
\begin{enumerate}[leftmargin=2.3em,label=(\alph*)]
\item why did the boundary universe start on the expanding rather than the contracting branch;
\item why did it later experience accelerated phases;
\item and how are the source terms behind those phases constrained by the inherited ledger?
\end{enumerate}
\ATen{} addresses those sharper questions rather than treating expansion itself as a mystery separate from general relativity.

\subsection{The comoving Hubble radius and the meaning of horizon growth}

A great deal of cosmological intuition is encoded in the comoving Hubble radius
\begin{equation}
 (aH)^{-1}.
\label{eq:partII-comoving-Hubble-radius}
\end{equation}
During ordinary radiation and matter domination,
\begin{align}
&\text{radiation:} && (aH)^{-1}\propto a,
\label{eq:partII-comoving-Hubble-rad}\\[2pt]
&\text{matter:} && (aH)^{-1}\propto a^{1/2}.
\label{eq:partII-comoving-Hubble-mat}
\end{align}
Thus the comoving Hubble radius grows with time in decelerating eras. By contrast, during quasi-de Sitter inflation where \(H\) is nearly constant and \(a\propto e^{Ht}\),
\begin{equation}
(aH)^{-1}\propto e^{-Ht},
\label{eq:partII-comoving-Hubble-inflation}
\end{equation}
so the comoving Hubble radius shrinks.

This distinction is the kinematic core of the horizon problem and its solution. In a purely decelerating universe, regions now far apart were never in mutual causal contact. An accelerated phase can reverse that logic by pushing previously microscopic, causally connected regions to cosmological sizes before standard radiation and matter evolution resume.

In the language of this Part, this kinematic fact will later become the main observational reason for wanting an inflationary stage at all. Whether that stage is driven by a free inflaton potential or by a throat modulus is secondary to the kinematic need. The need is fixed by Eq.~\eqref{eq:partII-comoving-Hubble-inflation}.



\subsection{Particle horizon, event horizon, and causal size}

The word ``horizon'' is often used loosely in cosmology, but several distinct integrals matter. The first is the particle horizon,
\begin{equation}
 d_p(t)=a(t)\int_{t_{\mathrm{ini}}}^{t}\frac{\dd t'}{a(t')},
\label{eq:partII-particle-horizon}
\end{equation}
which measures the maximum physical distance from which a light signal could have reached the observer by time \(t\). The second is the event horizon,
\begin{equation}
 d_e(t)=a(t)\int_t^{\infty}\frac{\dd t'}{a(t')},
\label{eq:partII-event-horizon}
\end{equation}
which measures the maximum distance from which future signals can ever reach the observer. The two are not the same, and their distinction is crucial when one talks about inflation, dark energy, and ultimate fate.

For a power-law background \(a\propto t^n\) with \(0<n<1\), Eq.~\eqref{eq:partII-particle-horizon} gives
\begin{equation}
 d_p(t) = \frac{t}{1-n}\times c,
\label{eq:partII-particle-horizon-powerlaw}
\end{equation}
while the event horizon diverges, so no future event horizon exists. In particular,
\begin{align}
&\text{radiation }(n=1/2): && d_p = 2ct,
\label{eq:partII-dp-rad}\\[2pt]
&\text{matter }(n=2/3): && d_p = 3ct.
\label{eq:partII-dp-mat}
\end{align}
By contrast, in de Sitter space with \(a\propto e^{Ht}\), Eq.~\eqref{eq:partII-event-horizon} converges and gives
\begin{equation}
 d_e = \frac{c}{H}.
\label{eq:partII-deSitter-event-horizon}
\end{equation}
That single formula foreshadows the two opposite causal properties of the universe emphasized elsewhere in the paper: early inflation shrinks the comoving Hubble radius and solves the horizon problem, while late dark-energy domination creates a genuine future event horizon.

The reason to include these derivations here is not pedagogical excess. They keep the causal story of the paper precise. ``Expansion and cooling'' is not merely the dilution of a thermal bath. It is also the continual reshaping of which regions can causally communicate with which other regions, and the reshaping of that communication pattern is exactly what makes both inflation and late-time acceleration cosmologically consequential.


\subsection{Freeze-out, decoupling, and the universal competition \texorpdfstring{\(\Gamma/H\)}{Gamma/H}}

Once Eqs.~\eqref{eq:partII-H-of-T} and \eqref{eq:partII-time-temperature} are in hand, nearly every event in the thermal history reduces to one basic competition:
\begin{equation}
\frac{\Gamma(T)}{H(T)}.
\label{eq:partII-freezeout-ratio}
\end{equation}
If \(\Gamma/H\gg 1\), microscopic interactions are fast compared with the expansion and the process tracks equilibrium efficiently. If \(\Gamma/H\lesssim 1\), the process freezes out, decouples, or falls out of equilibrium. This universal criterion will reappear throughout the remainder of the paper:
\begin{itemize}[leftmargin=2.2em]
\item it controls when reheating completes after inflation;
\item it controls whether heavy neutrino decays can generate baryon asymmetry;
\item it controls neutrino decoupling and any extra-radiation effects;
\item it controls the nuclear reaction network of BBN; and
\item it controls recombination and photon last scattering.
\end{itemize}
At a conceptual level, the entire cosmic thermal history is just the competition between internal relaxation rates and the external dilution rate set by the expansion.

\subsection{A compact table of expansion and cooling laws}

\begin{table}[H]
\centering
\caption[Background scaling laws used in Part II]{Useful background scaling laws for Part~II. The equations are standard; what changes later is the interpretation of the source sectors. The columns give the component, its equation-of-state parameter, the density scaling, the expansion law, and a cooling note. Nothing in the table is new; it is set down so that the later reinterpretation can be checked against the standard result line by line rather than asserted against it.}
\label{tab:partII-scaling-laws}
\small
\begin{tabular}{p{0.20\textwidth}p{0.12\textwidth}p{0.17\textwidth}p{0.17\textwidth}p{0.21\textwidth}}
\toprule
\textbf{Component} & \textbf{\(w\)} & \textbf{\(\rho(a)\)} & \textbf{\(a(t)\)} & \textbf{Cooling / comment}\\
\midrule
Radiation & \(1/3\) & \(a^{-4}\) & \(t^{1/2}\) & \(T\propto a^{-1}\), comoving entropy approximately conserved.\\
Pressureless matter & \(0\) & \(a^{-3}\) & \(t^{2/3}\) & Number density dilutes as volume; decoupled kinetic temperature scales as \(a^{-2}\).\\
Curvature & --- & \(a^{-2}\) & --- & Effective bookkeeping term, not a fluid.\\
Vacuum / cosmological constant & \(-1\) & constant & \(e^{Ht}\) & No cooling; late-time accelerator.\\
\bottomrule
\end{tabular}
\end{table}


\section{The radial chronology and the arrow of time}
\label{sec:expansion}

The kinematics tell us how the stage moves; they do not yet tell us why it moves in one direction only. The spine that orients the whole history is the radial coordinate \(r=\ln(\EGUT/\mu)\) of Eq.~\eqref{eq:partII-radial-coordinate}: cosmic history is an ascent out of the ultraviolet edge of the throat, and the open-system descent of \GDOS{}/\GFF{} gives that ascent a preferred direction. Read this way, the source sectors are not an arbitrary menu but inherited terms, the thermodynamic arrow is the macroscopic face of one monotone ledger, and the successive eras --- radiation, matter, vacuum --- are attractor phases of a single descent. With that orientation fixed, the stage is fully built, and we can hand it to Part~IV, where the inflationary modulus and the late \(F_{\GFF}\) vacuum inventory take over the action.

\subsection{Reinterpreting the source sectors in \GG{}/\GDOS{}}

The standard equations explain expansion and cooling once the components are given. \GG{}/\GDOS{} enters when we ask what those components are allowed to be. At the level of this Part the leading reinterpretation is:
\begin{align}
\rho_r &\leftrightarrow \parbox[t]{0.78\textwidth}{\raggedright relativistic boundary degrees of freedom descending from the zero-mode and light descendant spectrum,}
\label{eq:partII-rad-source}\\[2pt]
\rho_m &\leftrightarrow \text{baryons plus the compact-angle dark sector},
\label{eq:partII-mat-source}\\[2pt]
\rho_\Lambda &\leftrightarrow \text{residual \GFF{}/\CEP{} vacuum functional after \GSSB{} and lock selection}.
\label{eq:partII-vac-source}
\end{align}
This still looks schematic, but it is already a strong restriction. It means that the cosmological source ledger is not an arbitrary menu:
\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item matter is not free to be any imaginable hidden field if the compact-angle ontology already supplies a constrained candidate;
\item baryogenesis is not free to live in any arbitrary CP-violating sector if the neutrino block is already fixed;
\item and the accelerating term is not free to be a wholly inserted scalar potential if the \GFF{} ledger already provides a natural residual vacuum interpretation.
\end{enumerate}
Once again, the equations of expansion and cooling stay standard. What narrows is the meaning of \(\rho_i\).

\subsection{Expansion, cooling, and the arrow of time in the \GDOS{} language}

The standard \FRW\ equations explain the kinematics of expansion, but \ATen{} also needs to explain why the expanding thermal history has a preferred direction. This is where the open-system language of \GDOS{} and the \GFF{} ledger become more than philosophical packaging.

For a homogeneous comoving volume \(V_c=a^3\), the first law of thermodynamics may be written as
\begin{equation}
\dd(\rho a^3) + p\,\dd(a^3) = T\,\dd(sa^3),
\label{eq:partII-first-law-cosmology}
\end{equation}
where \(s\) is the entropy density of the coarse-grained plasma. If the expansion is adiabatic at the level of the tracked component, the right-hand side vanishes and one recovers the continuity equation. If there is entropy production through irreversible processes, decays, or reheating, then \(\dd(sa^3)>0\).

Equation~\eqref{eq:partII-first-law-cosmology} is the thermodynamic translation of the \GDOS{} statement that the boundary universe is an open system descending on an \GFF{} ledger. In that language,
\begin{equation}
\frac{\dd}{\dd t}\GFF \le 0
\qquad\text{during a dissipative relaxation stage,}
\label{eq:partII-GFF-monotonic}
\end{equation}
while the entropic-curvature contribution drives the coarse-grained arrow of time. Inflation, reheating, matter domination, recombination, and late vacuum domination are therefore not only different source-dominance regimes. They are also different attractor phases of one open-system descent.

This observation helps organize the whole cosmological story. Radiation domination is the phase in which entropy density is high and the bath is tightly coupled. Freeze-out events are the moments when one subset of degrees of freedom decouples from the common \GFF{} descent because its microscopic rate falls below the global dilution rate. Reheating is the moment when a formerly vacuum-like contribution is converted into a thermalized radiation bath, thereby increasing \(sa^3\). Late dark-energy domination is the approach to a different kind of attractor, one in which ordinary matter becomes negligible and the residual vacuum functional dominates the boundary ledger.

The underlying message is simple but important: the arrow of cosmological time in \ATen{} is not added by hand on top of the Friedmann equations. It is the macroscopic face of the same open-system descent that the earlier \GDOS{} paper already elevated to theorem level for general subsystems. That arrow is consumed here, not created: per the constitutional custody chain, \AX{3} owns the arrow constitutionally --- the one-way reading of the dictionary is a clause of the constitution (the \AX{3} synthesis of \citealp{ArisakaA1Constitution}) --- with the canonical law-level derivation in Part~IX of the CPT member paper \citep{ArisakaP5CPT}, carried out over a bulk that is itself exactly T-symmetric, and the dynamical implementation in the \GDOS{} companion \citep{ArisakaA5GDOS}; the arrow of cosmic time read off the radial descent is the cosmological face of that one constitutional arrow, a consumer of \AX{3}'s non-invertibility, not a private one.


\subsection{The fiber without holonomy: why the chronology cannot run backward}
\label{ssec:fibre-without-holonomy}

The preferred direction that this section has attributed to the
\GDOS{}/\GFF{} descent has a root one level deeper, and it is worth
one subsection because it is an exact topological statement rather
than a thermodynamic plausibility: \emph{holonomy requires closed
loops, and an interval has none.} The bulk's two non-spatial
directions are opposite objects in precisely this respect. The
compact circle $S^1_\theta$ is nothing but closed loops: everything
it carries is holonomy --- Wilson phases, integer windings, the
quantized charges of the gauge sector --- gauge-invariant and
topologically protected. The radial interval, $r =
\ln(\EGUT/\mu)$, can hold no holonomy at all: no loop on an
interval ever closes, so the fiber has no gauge-invariant memory,
no quantized charge, no protected content. Transport along it
cannot be a group action --- there is nothing invertible to return
to --- and can only compose forward: a semigroup, which is the
\GDOS{} generator's origin and the reason the radial chronology of
this Part is one-way at the level of law rather than of
initial conditions. The universe does not fail to run backward
because entropy happens to have been low; it fails to run backward
because the direction it moves along is a fiber on which
``backward'' stores nothing that could be returned to.

The pairing of the two fibers is then the anatomy of the whole
history this paper narrates. The circle \emph{remembers}: it holds
the letters and the charges --- the flavor lattice whose chirality
makes $CP$ violation unavoidable, the Majorana parities, the center
winding that is baryon number --- so the history has definite
content to deposit. The interval \emph{forgets}: it gates,
dissipates, and never returns, so each deposit is final --- the
prompt lock crossing, the zero-dilution clause, the crossover that
welds the sphaleron door are all this one statement wearing three
epochs' clothes. Read together: \emph{cosmic history is the one-way
reading of holonomy data} --- the descent along the fiber without
holonomy, depositing on the boundary the letters stored on the
fiber with it. Cosmic expansion, cooling, and redshift are the
boundary shadow of that motion, not its cause; the causal direction
is the constitutional one, bulk to boundary. And Sakharov's three
conditions, which this paper's baryogenesis appendix assembles into
$\eta_B$, are in this reading not three arranged ingredients but
the anatomy of the two fibers themselves: nonequilibrium is the
gated interval, $CP$ violation is the chiral circle, and the
lawful $\Delta B$ channel is the circle's topology meeting the weak
sectors. The full deployment of this observation --- symmetry
breaking as symmetry reading, with the determinism graded and its
falsifiers dated --- is carried by the cohort's symmetry review,
\textsf{GG-W1-Symmetry}; this Part records the clause that is
cosmology's own: the chronology cannot run backward because its
direction is the fiber that cannot remember.

The one-way reading has now been drawn, in the CPT
companion's expanded Part~IX \citep{ArisakaP5CPT}, as a single
quantitative figure, and this Part adopts it
(Figure~\ref{fig:time-line-a10}) because it is the radial
chronology of this section made visible. In the radiation era the
Friedmann clock and the radial coordinate obey $d\ln t/dr = 2$
exactly --- boundary time is the exponential shadow of the linear
descent, multiplying by $e^{2} \simeq 7.4$ per unit of $r$ --- and
the whole history from the launch to today is one straight line
with a single kink to slope $\tfrac32$ at matter--radiation
equality. The endpoint is an audit that closes:
$\ln(t_0/t_{\rm Pl}) = 140.24 = 36.1$ (launch) $+\;91.6$
(radiation) $+\;12.5$ (matter$+\Lambda$) --- the age of the
universe as an e-fold ledger on the descent, from standard
kinematics and this paper's derived $T_{\rm rh}$, nothing else.
(The companion also files, at flagged grade and no higher, the
compression $140.24 = 4K \times 1.0017$; its would-be derivation
is this paper's e-fold budget program.)

\begin{figure}[tbp]
\centering
\resizebox{0.724\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >=Stealth]
\draw[->] (0.3,0) -- (11.0,0)
  node[right] {$r=\ln(\EGUT/T)$};
\draw[->] (0.3,0) -- (0.3,6.1) node[above] {$\ln(t/t_{\rm Pl})$};
\foreach \r in {15,25,35,45,55,65}{
  \draw ({0.3+(\r-9)*0.162},0) -- ({0.3+(\r-9)*0.162},-0.07)
    node[below] {\r};}
\foreach \y in {40,60,80,100,120,140}{
  \draw (0.3,{(\y-25)*0.048}) -- (0.23,{(\y-25)*0.048})
    node[left] {\y};}
\draw[line width=1.10pt, blue!70!black]
  (1.037,0.533) -- (3.608,2.061) -- (4.123,2.364)
  -- (5.246,3.072) -- (6.064,3.565) -- (8.339,4.930);
\draw[line width=1.10pt, red!70!black]
  (8.339,4.930) -- (8.523,5.027) -- (9.656,5.532);
\fill (1.037,0.533) circle (1.6pt)
  node[right=2pt] {launch $(13.6,\,36.1)$};
\fill (3.608,2.061) circle (1.6pt)
  node[above left] {lock $(29.4)$};
\fill (4.123,2.364) circle (1.4pt) node[below right] {EW};
\fill (5.246,3.072) circle (1.4pt) node[below right] {QCD};
\fill (6.064,3.565) circle (1.4pt) node[below right] {BBN};
\fill (8.339,4.930) circle (1.6pt)
  node[below right] {equality};
\fill (8.523,5.027) circle (1.4pt);
\node[anchor=east] at (8.25,5.19) {recombination};
\fill (9.656,5.532) circle (1.8pt)
  node[below right=1pt and -2pt, align=left]
  {today\\ $(66.8,\,140.24)$};
\node[blue!70!black, rotate=30] at (4.9,3.35)
  {slope $= 2$ (radiation)};
\node[red!70!black, rotate=24] at (9.05,5.00)
  {slope $\tfrac32$};
\draw[->, violet!70!black] (7.35,5.75) -- (8.49,5.10);
\node[violet!70!black, anchor=east, align=center]
  at (7.35,5.82) {$\ln(1{+}z_{\rm sw}) = 7 = J$\\
  ($J$ e-folds below today)};
\end{tikzpicture}}
\caption[The whole history of time, on one line.]{The whole
history of time, on one line: boundary time (Planck units,
logarithmic) against this paper's radial coordinate
($\EGUT = 2.3\times10^{16}$~GeV), for eight epochs from the
launch to today. The radiation era is a straight line of slope
exactly $2$; the slope kinks to $\tfrac32$ at equality; the lock
crossing sits at $r = 29.4$ (equal to $K = 35$ in the
half-Planck convention of the proton and electron companions);
the corridor switch sits $J = 7$ e-folds of boundary scale below
today; the endpoint's audited decomposition is given in the
text. Boundary time is the exponential shadow of the linear
descent.~(after \citealp{ArisakaP5CPT})}
\label{fig:time-line-a10}
\end{figure}

And the three dark episodes of this paper sit on the same
anatomy at its largest scale
(\S\ref{ssec:cosmic-triptych}): the interval's coordinate opened
the chronology, the circle's coordinate still rings within it, and
the terminus is the chronology at rest.

\subsection{Bottom line: the stage is set}

The cosmological stage is now complete. The universe expands because a homogeneous spacetime with nonzero background energy density generically has \(H^2>0\). It cools because comoving entropy is approximately conserved while the scale factor grows, so the energy per relativistic quantum redshifts as \(a^{-1}\). It runs in one direction because the open-system \GFF{} ledger descends monotonically along the radial ascent \(r=\ln(\EGUT/\mu)\) out of the GUT edge. The equations are classical and standard; what \ATen{} adds is not a replacement of those equations but a constrained inheritance chain for the components that occupy them, beginning at the finite GUT-edge surface and orientated by the open-system descent.

\medskip
\textbf{The significance and novelty of Part~III.} It is worth pausing, before the cosmic history begins, to say plainly what this Part has accomplished and why it is new, because the result is easy to understate. The conventional division of labor in cosmology is sharp: general relativity and the Friedmann equations are \emph{given} --- handed down from 1915 and 1922 --- and the science consists of measuring the six numbers that fill them. Part~III has dissolved that division. It has shown that the gravitational law itself, the cosmological equations it produces, and the form of the numbers that feed them are not three separate gifts but consequences of a single six-dimensional geometry, and it has shown this by derivation rather than assertion.

The novelty is concrete, and worth itemizing. \emph{First}, general relativity is an \emph{output}: the four-dimensional Einstein equation, the Newton constant \(G_N=1/8\pi M_{\mathrm{Pl}}^2\), and the scale of the cosmological constant all fall out of the Kaluza--Klein reduction of one six-dimensional Einstein--Hilbert action, with the Planck mass \(M_{\mathrm{Pl}}^2=\pi M_6^4(1-e^{-2K})\) revealed as a geometric overlap integral rather than a fundamental input --- a derivation of gravity in a sense no other framework attempts, since in every textbook the Einstein equation is where one \emph{starts}. \emph{Second}, the Friedmann equations are themselves derived, not posited: the cosmological principle that singles them out is read as the geometric attractor of the runtime, so the smoothness of the universe is a selected outcome rather than an initial assumption. \emph{Third}, the beginning is relocated --- the observable universe starts at the grand-unification edge \(\EGUT\), not the Planck scale, because four-dimensional spacetime is the boundary projection of the bulk and is simply undefined above its ultraviolet edge; the Big Bang singularity is not denied as a formal backward limit but denied the standing of a physical initial surface, a clean departure from a century of received intuition. \emph{Fourth}, the source ledger carries no free dials: the contents of \(\rho\) and their proportions are closed-form expressions over the fixed integer ledger of \AFour{}, so the cosmology cannot be retuned to fit. \emph{Fifth}, the cosmological-constant problem is not patched but closed to the percent level --- the lock-scale seesaw \(m_\Lambda=M_{\mathrm{lock}}^2/M_{\mathrm{Pl,red}}\approx6\,\mathrm{meV}\) removes the canonical \(10^{120}\) vacuum-energy discrepancy down to a residual factor of a few, and the ledger coefficient \(\zeta_\Lambda=\pi/LK\) takes that residue to \(\rho_\Lambda^{1/4}=2.23\,\mathrm{meV}\) against a measured \(2.24\) (Derived cond.).

Before the actors enter, notice what the stage itself has already settled. A beginning without a singularity means the oldest question --- \emph{what happened at time zero?} --- was never quite the right question: on the boundary there is no time zero, only a finite edge where the geometry's description lawfully begins. And an arrow of time that points down an open-system descent means the future is not a mystery appended to physics but a direction built into it. These are philosophical dividends of a technical construction, and they cost nothing extra.

Taken together, these results promote \(\Lambda\)CDM from a six-parameter fit to a \emph{theorem} of \GGTheory{}, in exactly the structural sense in which \ANine{} \citep{ArisakaA9SM} promoted the Standard Model: the form of every input is forced by the geometry, and a per-parameter provenance triage records, in the open, how completely each form has so far been closed. That is the significance --- and it is also where the honesty of the claim matters most. Part~III does not assert that every cosmological number is computed today. The uniqueness of the homogeneous attractor, the order-unity coefficient and the sign of \(\Lambda\), and the primordial-spectrum parameters remain, respectively, conditional or open --- flagged, not buried, and handed forward to Parts~IV through~X, which take up the unfinished closures one at a time. What Part~III has settled is the harder and more fundamental point: that there \emph{is} a single derivation to complete, that gravity and the Friedmann stage are theorems of a parameter-free geometry, and that the cosmos and the electron descend from one ledger one story apart. With the stage so understood --- the arena fixed, the beginning lawful and finite, the kinematics derived, and the arrow of time orientated by the open-system descent --- Part~IV brings on the two leading actors: the inflationary modulus that drives the early quasi-de Sitter stage, and the residual \(F_{\GFF}\) vacuum that fixes the \(\pi\)-locked dark-energy fraction of the present epoch.



\section{Part~III in plain English: why the universe starts at the GUT edge and not at the Planck scale}
The claim of this Part is easy to state and easy to underestimate. General
relativity and the Friedmann equations are not assumed here; they are derived
as the four-dimensional shadow of the bulk dynamics, and the derivation is set
out in full because a reader is entitled to check it rather than take it.

The part of the Part that is genuinely new is the starting point. The textbook
history begins at the Planck scale because that is where the known physics
runs out, and the interval between the Planck scale and the GUT scale is
carried as an unexplained early era. Here the beginning sits at the GUT edge,
and it sits there for a structural reason rather than a practical one: the
scale chain that fixes the lock scale also fixes where the chain can start.
That is a real consequence of the framework and it is exposed --- if the
universe demonstrably has physics between the two scales that this reading
forbids, the reading fails.

The radial chronology introduced at the end of the Part is a bookkeeping
device and should be read as one. Writing cosmic history against the radial
coordinate rather than against time does not add physics; it makes visible
that the early universe is enormous in radial extent and brief in years, and
the late universe the reverse. That the arrow of time and the direction of the
radial descent point the same way is the substantive claim in the section, and
it is argued rather than assumed.

\GGPart[Inflation and dark energy]{Inflation and dark energy: the cosmic \GRIP{} descent}{From the lawful beginning of Part~III, the universe runs as one open-system descent --- the launch of inflation and the terminus of dark energy --- governed by the same \GRIP{} geometry that fixes the electron}


\section{One descent, from the electron to the cosmos}

Part~III handed us a lawful beginning. The operative origin of the universe is not the formal \(a\to 0\) singularity but the finite ultraviolet boundary surface at \(\EGUT\); from that edge, ordinary boundary general relativity, sourced by the inherited ledger, sets the universe expanding and cooling. We do not re-derive that beginning here. We take it as given and watch what happens next.

What happens next is a single story. Every system in nature is open, and an open system is read by \OGGEP{}, evolves by \GDOS{}, and runs as the \GRIP{} cascade: it explores its admissible channels (\DGI{}), ranks them by the \GFF{} ledger, breaks the symmetric branch by \GSSB{}, and settles into a geometric attractor. This is not a metaphor borrowed for cosmology. It is the same cascade, the same functional, the same attractor language that the companion electron paper (\PaperPThree{}) uses to fix the electron's charge and mass as a \emph{local} ground-state attractor. The thesis of this Part is that the universe's whole history is that one descent, run at cosmic scale: a launch and a terminus. Inflation is the opening stretch of the descent --- the explore phase, the \DGI{} that throws the boundary onto a flat plateau. Dark energy is its terminus --- the global, late-time attractor that the boundary relaxes into, the Ground Geometric Attractor, the \GGA{}. The same geometric principle that fixes one electron fixes the vacuum of the cosmos.

Honesty has to be tiered, and we name the tiers once, here, so the reader can carry them through the whole Part. The electron ledger of \PaperPThree{} and the Kaxion relic of \PaperCOne{} are \emph{derived}. The inflationary tilt \(n_s\) is now \emph{derived-conditional} on the plateau class, and the amplitude \(A_s\) is the launch sector's one \emph{declared input} (its warp candidate flagged in the codebook). The \(\pi\)-locked late-time inventory and the dark-energy fraction are \emph{derived-conditional}: they close up to one named topological step, Conjecture~VIII (closed-packet integration), whose final cancellation has since been traced to the declared epoch input --- the residual uniformity lemma exact to one part in \(10^{15}\) --- and we will say so plainly at the point where it bites. Everything robust in this Part survives even if that one step fails; everything that depends on it is fenced. With the tiers named, we let the descent run.


\section{The launch: inflation as the opening of the descent}
\label{sec:inflation}

\subsection{Why the universe needs a launch}

The descent that began at the \EGUT{} edge cannot reach the universe we observe without an early stretch of accelerated expansion. This is not a stylistic preference; it is forced by four stubborn facts that any successful cosmology must explain:
\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item the near flatness of the present universe;
\item the large-scale homogeneity of the CMB across regions that are naively out of contact in a purely decelerating history;
\item the existence of primordial perturbations with an almost scale-invariant scalar spectrum;
\item and the absence, so far, of evidence for a large primordial tensor signal.
\end{enumerate}
The first two are the classic flatness and horizon problems. The third is the origin-of-structure problem. The fourth is already a discriminator among inflationary mechanisms. The historical resolution is well known: Guth in 1981 \citep{Guth1981} proposed an ``old inflation'' driven by a metastable false vacuum, and Linde in 1982 \citep{Linde1982} together with Albrecht and Steinhardt in 1982 \citep{AlbrechtSteinhardt1982} replaced the false-vacuum picture with ``new inflation'' driven by a slow-rolling scalar field, repairing the graceful-exit problem of the original proposal.

\ATen{} therefore does not try to evade inflation by rhetorical means. It accepts the need for an accelerated early phase, but it denies that the phase must be implemented by introducing a wholly free scalar sector with a potential chosen from a wide menu. Under \MSMF{} that route is too unconstrained. The correct question is not whether inflation happens, but whether the same geometry already needed for the particle and anomaly papers can supply an inflationary stage with much less freedom. In the language of the descent: inflation is the \DGI{} opening move --- the boundary exploring the flat throat branch --- and the job here is to read that opening off the inherited geometry rather than to bolt a new field onto it.

\subsection{Why a free inflaton is forbidden by the inherited program}

The earlier preprints already fixed the working rule: do not add a new continuous sector unless the inherited ledger truly cannot do the job. A free inflaton potential of the form
\begin{equation}
V_{\mathrm{inflaton}}(\phi)=\text{``choose your favorite polynomial, plateau, hilltop, or hybrid form''}
\label{eq:partII-free-inflaton-forbidden}
\end{equation}
is therefore constitutionally disfavored. It is not mathematically illegal, but it violates the spirit of \MSMF{} unless one can show that no inherited \GG{} modulus can generate the required accelerated phase.

The inherited program already contains two natural vacuum-selection variables:
\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item the throat/radial modulus --- the \(\Sigma_r\) scale coordinate --- controlling the effective lock depth;\footnote{A terminological reconciliation, once and for all. The particle papers state, correctly, that \GG{} has \emph{no radion}: the interval length \(K\) is a theorem-fixed integer, and no light radius-stabilization scalar exists in today's spectrum (\citealp{ArisakaA4Euclid}; \citealp{ArisakaP1KK}). The cosmological inflaton is not that (non-existent) particle: it is the \(\Sigma_r\) \emph{configuration coordinate} of the radial-scale stratum (\citealp{ArisakaA6Gi4}) --- dynamical during the descent, frozen at the ground attractor today, its present-day excitations being the lock cluster, not a light scalar. Where the wider literature says ``radion,'' this paper says the \(\Sigma_r\) scale modulus.} and
\item the Green--Schwarz normalization modulus \(\Omega\), which the anomaly paper says is driven to \(\Omega=1\) during the descent (in its original TeV-attractor language; in the two-basin reading the drive completes by the lock crossing).
\end{enumerate}
For the purpose of cosmology these need not be treated as independent scalar fields. They are different descriptions of the same coarse-grained vacuum-selection direction of the throat. In geometric language one may speak of the \(\Sigma_r\) scale modulus (the coordinate the wider literature would call the radion --- see the footnote above for why \GG{} reserves that word). In anomaly language one may track the effective Green--Schwarz normalization \(\Omega\). The physical content is the same: there is an inherited modulus whose motion changes the effective four-dimensional vacuum budget and which is forced to settle at a distinguished post-inflationary minimum.

This is what it means for inflation to be lawful rather than inserted. The throat does not need to borrow an inflaton; it already carries the one direction along which the boundary can drift slowly. The launch is built from a part the universe already had.

The campaign's two-fiber reading
(\S\ref{ssec:inflation-two-fibres}) will sharpen this prohibition
into a census: the bulk has exactly one open non-spatial fiber, and
hence exactly one candidate inflaton --- the problem with the
inflaton menu was never taste; it was counting.

\subsection{The minimal effective action for the inflationary modulus}

At the level of the four-dimensional boundary, the appropriate coarse-grained effective action is
\begin{equation}
S_{\varphi} = \int \dd^4x\sqrt{-g}
\left[
\frac{M_{\mathrm{Pl,red}}^2}{2}R
- \frac{1}{2}(\partial\varphi)^2
- V_{\mathrm{eff}}(\varphi)
\right],
\label{eq:partII-effective-inflaton-action}
\end{equation}
where \(\varphi\) is the canonically normalized modulus representing the relevant throat/Green--Schwarz vacuum direction. This is not the introduction of a new primitive field. It is the low-energy description of a modulus already present in the inherited geometry.

The detailed microscopic derivation of \(V_{\mathrm{eff}}\) is deferred to a dedicated follow-up analysis, but the qualitative origin is fixed by the earlier papers and by the geometric hints already carried through the project:
\begin{equation}
V_{\mathrm{eff}} = V_{\mathrm{curv}} + V_{\mathrm{Casimir}} + V_{\mathrm{IR\,boundary}} + V_{\Omega}.
\label{eq:partII-potential-decomposition}
\end{equation}
Each term has a clear meaning.
\begin{itemize}[leftmargin=2.2em]
\item \(V_{\mathrm{curv}}\) is the vacuum energy associated with the six-dimensional curvature.
\item \(V_{\mathrm{Casimir}}\) is the finite-size contribution from the compact throat geometry.
\item \(V_{\mathrm{IR\,boundary}}\) encodes the infrared boundary conditions and their later shift under \GSSB{}.
\item \(V_{\Omega}\) encodes the tendency of the Green--Schwarz normalization to relax toward its physical post-inflationary value \(\Omega=1\).
\end{itemize}
The crucial point --- and this is where the treatment earns the word \emph{derived} --- is that the \GDOS{} boundary law fixes the \emph{class} of this potential with no freedom. The genuine conformal anomaly of a boundary law supplies only the Weyl-squared and Euler densities; the scalaron/plateau sector can therefore arise only from the non-conformal curvature coupling of a scalar, and the scale modulus \(\varphi\) --- the one-dimensional radial stratum \(\Sigma_r\) --- is the only such scalar the inherited geometry possesses. The result is the Starobinsky/\(\xi\) class \citep{Starobinsky1980,Mukhanov2005}, with shape and slope fixed by four-dimensional gravity:
\begin{equation}
V_{\mathrm{eff}}(\varphi)
=
\frac{3}{4}\,M^2 M_{\mathrm{Pl,red}}^2
\left(1 - e^{-\sqrt{2/3}\,\varphi/M_{\mathrm{Pl,red}}}\right)^{2},
\label{eq:partII-plateau-potential}
\end{equation}
where the single scale \(M\) --- the scalaron mass, the curvature of the plateau's edge --- is the launch sector's \emph{one declared input}: fixing it by the observed scalar amplitude gives \(M = 3.27\times10^{13}\,\mathrm{GeV}\), i.e.\ a plateau height \(V_0^{1/4} \simeq 8.3\times10^{15}\,\mathrm{GeV}\). (A warp candidate for this one number exists and is recorded in the codebook as flagged, not claimed: \(M = M_5\,e^{-K/3}\) reproduces the required value to \(87\%\), equivalently \(A_s = 21.5\,e^{-2K/3}\) reaches \(75\%\) of the observed amplitude. The tower route of \S\ref{ssec:entrance} scales differently, \(M \propto e^{-K/2}\); the two coincide only near \(K = 35\), and where the warp form is a flagged guess the tower form rests on a theorem --- the per-level coefficient will decide between them.) Everything else about \eqref{eq:partII-plateau-potential} --- the exponential flattening, the \(\sqrt{2/3}\), the graceful steepening at the edge --- carries no dial. This plateau is the surface the descent explores: broad and gently sloped, the geometric arena on which \DGI{} plays out.

\subsection{The two basins: a GUT-adjacent plateau, and a lock that is crossed rather than camped}

The natural first reading of the descent --- and the program's working hypothesis before the present derivation --- was that the vacuum sector driving inflation was controlled not by \(\EGUT^4\) but by a lock-scale plateau \(V_0 \sim \Mlock^4\) --- \emph{TeV-scale slow-smooth inflation}. That assertion is now retired, and the reason is recorded here at full strength because it is a sign-level result, not a preference. A plateau of height \(\Mlock^4\) cannot source the observed primordial spectrum on two independent counts. First, the amplitude: \(A_s = V/(24\pi^2\epsilon_V M_{\mathrm{Pl,red}}^4)\) with \(V = \Mlock^4\) demands \(\epsilon_V \sim 10^{-53}\) --- a pure dial with no geometric origin. Second, and fatally, the tilt: the approach to a plateau of the form \(V_0(1+e^{-\beta\varphi})\) has \emph{positive} curvature, so the spectrum comes out \emph{blue}, \(n_s - 1 \sim +10^{-26}\), against the measured red tilt \(n_s - 1 = -0.0351 \pm 0.0042\) --- wrong in sign, and by twenty-four orders of magnitude in size. No parameter choice repairs a sign. The corresponding codebook conjecture (the former Lemma~V) is retired in Part~IX with this refutation attached.

The derived potential \eqref{eq:partII-plateau-potential} instead places the launch where the descent begins. The inflationary Hubble scale of the primary plateau is
\begin{equation}
H_{1}^2 \simeq \frac{V_0}{3M_{\mathrm{Pl,red}}^2}
= \frac{M^2}{4}\left(1-e^{-\sqrt{2/3}\,\varphi_*/M_{\mathrm{Pl,red}}}\right)^{2},
\qquad
H_{1} \simeq 1.6\times10^{13}\;\mathrm{GeV},
\label{eq:partII-Hinf-def}
\end{equation}
and the plateau sits at \(r_{\mathrm{inf}} = \ln(\EGUT/V_0^{1/4}) = 1.01\) --- \emph{one e-fold below the GUT edge}. Inflation is literally the opening stretch of the boundary's descent, the explore phase that the \GRIP{}-Out at \(\EGUT\) launches; the poetic sentence that opened this Part was, all along, the correct energy assignment.

The lock has lost none of its cosmological standing --- it has gained precision. The lock-scale basin is real: the radial Morse function flattens toward its unique critical point, and the curvature scale of that middle basin is
\begin{equation}
H_{2} = \frac{\Mlock^2}{\sqrt{3}\,M_{\mathrm{Pl,red}}}
\simeq 3.5\times10^{-12}\;\mathrm{GeV} \simeq 3.5\;\mathrm{meV},
\label{eq:partII-Hinf-corridor}
\end{equation}
which the reader should recognize at once: it is the \emph{terminus} seesaw scale \(\Mlock^2/M_{\mathrm{Pl,red}}\) of the dark-energy section below. The old TeV-inflation corridor was, unknowingly, a statement about the wrong end of the descent --- the number belongs to dark energy's ancestry, not to the CMB's. And the middle basin is \emph{crossed, not camped}: the crossing completes promptly (\(N_2 = 0\)) on three independent grounds --- any \GRIP{}-ledger nucleation action (\(\pi\), \(2\pi\), \(3\pi\)) lies fifty-six or more orders of magnitude in rate above the supercooling threshold \(S_{\mathrm{crit}} = 4\ln(\Mlock/H_2) \simeq 138\); the lock potential as written is a barrier-free saddle; and the large-\(N\) enhancement that drives supercooling in warped-throat models is structurally absent here. No sustained secondary inflation occurs, no e-folds are stolen from the primary window, and no relic is diluted. The lock keeps its three jobs --- the particle hierarchy, the birth of the Kaxion, and the seeding of the terminus --- and gives up only the fourth it never could have performed.

\begin{theorem}[Two-basin slow-smooth inflation theorem (T-Cosmos-4, revised)]
Inflation is the \GFF{}-dominant opening stretch of the \(\Sigma_r\) descent: a
plateau of the derived class \eqref{eq:partII-plateau-potential}, GUT-adjacent in
height, whose graceful exit is the failure of slow roll at the plateau's edge.
The lock-scale basin is a distinct, later \GRIP{}-Out crossed promptly
(\(N_2 = 0\)). The slow-roll observables are
\[
n_s = 1-\frac{2}{N^*},
\qquad
r=\frac{12}{N^{*2}},
\]
with \(N^*\) fixed by reheating matching and undiluted by the crossing.
\end{theorem}

\begin{proof}
\BI{} forbids introducing an unrelated free inflaton sector once the throat and
its holonomy structure already exist. \GDOS{} fixes the potential class: the
conformal anomaly supplies no scalaron, so the plateau must come from the one
non-conformal scalar, \(\Sigma_r\), yielding \eqref{eq:partII-plateau-potential}.
The sign-level no-go above excludes the lock-scale height; the amplitude then
fixes the height as the one declared input. \GRIP{} interprets the entrance
onto the plateau as branch exploration and the exit as slow-roll failure, with
reheating as the first post-exit attractor export; the nucleation bound forces
the later lock crossing to be prompt, so \(N^*\) is the undiluted
reheating-matched value.
\end{proof}

\subsection{The first e-fold: the cold start and the forced throw}
\label{ssec:first-efold}

The edge carries more energy than the plateau can hold, and following
that surplus settles two questions at once: what the universe was doing
in its first e-fold, and why inflation required no trigger.

The first admissible surface inherits the density of the edge itself,
\(\rho_{\mathrm{edge}} \simeq \EGUT^4\) (\S\ref{sec:beginning}).
Measured against the plateau height this is
\begin{equation}
\frac{\rho_{\mathrm{edge}}}{V_0}
= \left(\frac{\EGUT}{V_0^{1/4}}\right)^{\!4}
= e^{4 r_{\mathrm{inf}}} \simeq 58.9,
\label{eq:partII-edge-surplus}
\end{equation}
the fourth power of the same one-e-fold ratio that places the plateau
below the edge. The surplus must be stored somewhere, and the derived
class offers exactly three channels. The potential cannot take it:
\(V_{\mathrm{eff}}\) is bounded by \(V_0\) and holds one part in
fifty-nine. A thermal bath cannot take it, twice over: nothing exists
on \(\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}\) to \emph{be} a bath --- its
data are geometric (\S\ref{sec:beginning}) --- and radiation is
conformally invariant, sourcing no Ricci scalar, so a radiation-dominated
edge would park the scalaron at the bottom of
\eqref{eq:partII-plateau-potential} and the launch would never occur at
all. A hot start does not merely fail to help; in the derived class it
forbids inflation permanently. What remains is the kinetic channel:
\begin{equation}
\tfrac12\,\dot\varphi^2\big|_{\mathrm{edge}}
\simeq \rho_{\mathrm{edge}} - V_0 \simeq 57.9\,V_0 .
\label{eq:partII-forced-throw}
\end{equation}

Two conclusions deserve to be said plainly. First, \emph{the universe
began cold}: no particles, no bath, no temperature --- pure geometry,
with all but a fifty-ninth of its energy in the motion of one modulus.
The familiar phrase ``hot Big Bang'' names not the beginning but the
reheating exit, fifty-two e-folds later, when the modulus decays and
the bath first appears. Second, the sentence that opened this Part ---
the \DGI{} that \emph{throws} the boundary onto a flat plateau --- is
not a metaphor. The throw is the forced kinetic component of
\eqref{eq:partII-forced-throw}: bookkeeping, not poetry.

The opening stretch of history is therefore a kination sprint. Kinetic
energy redshifts as \(a^{-6}\), so the sprint lasts
\(\Delta N = \ln(57.9)/6 \simeq 0.68\) e-folds before the plateau height
dominates, and while it lasts the field coasts
\(\Delta\varphi = \sqrt{6}\,\Delta N \simeq 1.66\,M_{\mathrm{Pl,red}}\)
against Hubble friction. At the end of the sprint \(V_0\) is the only
component that has not diluted, and accelerated expansion begins ---
which is the honest answer to the question of what triggered inflation:
\emph{nothing did}. There is no false vacuum, no tunneling event, no
switch thrown. A bounded plateau in the presence of red-shifting
kinetic energy is a trigger-free ignition; inflation starts because the
plateau is the first thing in the descent that cannot dilute.

\subsection{The entrance: what the geometry must still supply}
\label{ssec:entrance}

Where on the plateau does the descent begin? For an inflaton introduced
by hand this is the initial-conditions problem, answered by assumption.
Here it is sharper, because the field is not an independent primitive:
it is the canonical coordinate of the same radial stratum whose depth is
\(r\). Wherever the scalaron is curvature-dominated, the Einstein-frame
dictionary of the derived class,
\(\varphi = \sqrt{3/2}\,M_{\mathrm{Pl,red}}\ln(1 + R/3M^2)\), together
with \(R = 4\rho/M_{\mathrm{Pl,red}}^2\) and the descent parametrization
\(\rho(r) = \EGUT^4 e^{-4r}\), closes into a depth-to-field map:
\begin{equation}
\varphi(r) = \sqrt{\tfrac32}\,M_{\mathrm{Pl,red}}
\ln\!\big(1 + e^{4(r_{\mathrm{inf}} - r)}\big),
\qquad
N(r) = \tfrac34\big(1 + e^{4(r_{\mathrm{inf}} - r)}\big).
\label{eq:partII-entrance-map}
\end{equation}
This is a useful dictionary on the plateau, where the potential
dominates and the identification of \(R\) with \(4\rho\) is the correct
one, and it is how the depth coordinate and the canonical field are to
be compared once slow roll has begun. (Stated exactly: the scale the
map descends in is \(\mu^4 = M_{\mathrm{Pl,red}}^2 R_J/4\), the
curvature scale of the Jordan frame, which coincides with the total
energy density only after the exit --- the identification is a plateau
statement, not a general one.)

It is worth being explicit that \eqref{eq:partII-entrance-map} may
\emph{not} be evaluated at the edge, because the edge is precisely where
its premise fails. Section~\ref{ssec:first-efold} showed that the
inherited surplus is carried by the kinetic channel, and for a scalar
the Einstein-frame trace gives
\(R = (4V - \dot\varphi^2)/M_{\mathrm{Pl,red}}^2\), which at
\(\tfrac12\dot\varphi^2 \simeq 57.9\,V_0\) is \emph{negative}. A
kination epoch has the opposite curvature sign from a vacuum-dominated
one; reading the entrance value off the vacuum relation would be using
the dictionary exactly where the paper's own theorem says it does not
apply. The entrance is therefore not fixed by the map.

Nor is it fixed by the energy budget, and the reason is structural. The
plateau of \eqref{eq:partII-plateau-potential} is asymptotically
\emph{flat}: \(V(\varphi) \to V_0\) for any sufficiently large
\(\varphi\). The edge condition \(\rho_{\mathrm{edge}} = 58.9\,V_0\)
therefore fixes the \emph{speed} \(|\dot\varphi_0|\) and says nothing
whatever about the \emph{position} \(\varphi_0\). The honest inventory
is that the entrance requires two inherited data --- the initial
displacement and the sign of the initial velocity --- of which the
budget supplies one, and the geometry is so far asked for the other
without answering.

What can be stated exactly is the requirement they must meet. The
sprint of \S\ref{ssec:first-efold} moves the field by
\(|\Delta\varphi| = 1.66\,M_{\mathrm{Pl,red}}\), and \(N^* = 52.6\)
e-folds of slow roll require the field to be at
\(\varphi_* = 5.21\,M_{\mathrm{Pl,red}}\) when the sprint ends. Hence
\begin{equation}
\varphi_0 \;\ge\; 3.55\,M_{\mathrm{Pl,red}}
\quad\text{(throw up the plateau)},
\qquad
\varphi_0 \;\ge\; 6.86\,M_{\mathrm{Pl,red}}
\quad\text{(throw down it)},
\label{eq:partII-entrance-requirement}
\end{equation}
the second being the first plus twice the sprint, since a descending
field must begin far enough up to survive the loss. Both are
satisfiable: the plateau has no upper end, and no principle of the
construction caps the initial displacement. The orientation of the
throw is accordingly \emph{not} the decisive question it might appear
to be --- it moves the bar in \eqref{eq:partII-entrance-requirement} by
\(2|\Delta\varphi|\) and does not by itself admit or exclude the
launch. The cascade reading does carry an expectation, that the descent
opens at the symmetric configuration and moves away from it, which
places the start high and the motion downward; but the identification
of the symmetric stage with a position on the radial coordinate is made
in the family sector rather than this one, and it is recorded here as
an expectation and not imported as a result.

The displacement is bounded from above as well, and the upper bound is
constitutional rather than observational. On the plateau the quantum
kick per e-fold, \(H/2\pi\), overtakes the classical drift,
\(\sqrt{3/2}\,M_{\mathrm{Pl,red}}/N\), at
\begin{equation}
N_{\mathrm{et}} \;=\; 4\pi\sqrt{\tfrac32}\,\frac{M_{\mathrm{Pl,red}}}{M}
\;\simeq\; 1.1\times10^{6},
\qquad
\varphi_{\mathrm{et}} \;\simeq\; 17.4\,M_{\mathrm{Pl,red}} .
\label{eq:partII-eternal-bound}
\end{equation}
Beyond that point the launch is in stochastic self-reproduction: the
field's next move is drawn rather than driven, and the descent
\emph{samples}. Sampling is precisely what the constitution forbids ---
the cascade selects unique attractors and never draws from an ensemble,
which is the clause the third horizon problem
(\S\ref{sec:third-horizon}) holds to observational account. A framework
whose answer to that problem is determinism cannot place its own launch
in the sampling regime. The requirement
\eqref{eq:partII-entrance-requirement} therefore closes into a corridor:
\begin{equation}
3.55 \;\le\; \varphi_0 \;\le\; 15.78
\quad\text{(up the plateau)},
\qquad
6.86 \;\le\; \varphi_0 \;\le\; 19.10
\quad\text{(down it)},
\label{eq:partII-entrance-corridor}
\end{equation}
in units of \(M_{\mathrm{Pl,red}}\), equivalently
\(45 \lesssim N_{\mathrm{tot}} \lesssim 1.1\times10^{6}\). The floor
is observational, the ceiling is constitutional, and the entrance is
bracketed from both sides.

What fixes the value inside the corridor is, on present evidence, the
same physics that owes the amplitude --- and both now have a named
mechanism. The \(R^2\) coefficient is the threshold sum of the
lock-anchored tower of \citet{ArisakaP1KK}: the radial spectrum is a
Bessel tower whose base is fixed by that paper's Theorem~T-KK-1 at
\(\Mlock \simeq 3.85\)~TeV, with asymptotic spacing \(\pi \times
1.0\)~TeV. Summing that tower with standard per-mode threshold
coefficients brackets the required \(c = M_{\mathrm{Pl,red}}^2/12M^2 =
4.6\times10^{8}\) within an order of magnitude on either side ---
after the three standard routes miss it outright (minimal matter loops
by nine orders, holographic counting at the theory's anchors by five,
which is the closure already recorded above made quantitative, and the
unwarped interval by eleven). What remains of the amplitude is a single
signed number: the per-level heat-kernel coefficient of the tower's
field content, required to land in \([1/772,\,1/100]\) --- a finite
enumeration against standard tables, cross-checkable through the
six-dimensional \(a_6\) coefficient on the warped interval.

The bend no longer waits on that number. A mode contributes to the
local \(R^2\) coefficient only while \(m^2 \gtrsim R\); as the
descent climbs in curvature, \(\sqrt{R} = 2\sqrt{N}\,M\) sweeps up
the tower and depletes the coefficient that sustains the plateau,
\begin{equation}
\frac{c(N)}{c_0} \;=\; 1 \;-\; \frac{2\sqrt{N}\,M}{M_{\mathrm{top}}},
\label{eq:partII-depletion}
\end{equation}
with \(M_{\mathrm{top}}\) the tower's ultraviolet end. At the pivot
the depletion is at or below the two-percent level for either natural
choice of \(M_{\mathrm{top}}\), and the running it induces is at the
\(10^{-4}\) level: the observed plateau, and with it \(n_s\) and
\(r\), are untouched. The bend itself,
\(N_{\mathrm{bend}} = (x\,M_{\mathrm{top}}/2M)^2\) for depletion
threshold \(x\), lands \emph{inside} the corridor of
\eqref{eq:partII-entrance-corridor} across the entire reasonable band:
for the wider ultraviolet end every threshold from \(0.3\%\) to
\(39\%\) does so, and for the narrower one the tower is stripped
entirely by \(N \simeq 1.2\times10^{5}\), likewise inside. The test
staked above is therefore \emph{passed}, and passed by shape alone ---
the tower's spacing, not its per-level coefficient. The plateau ends
because rising curvature strips the tower that built it: the \(R^2\)
era dismantles its own scaffolding. The exact value of
\(\varphi_0\) inside the corridor still awaits the per-level number
and the smoothed crossover, and
\eqref{eq:partII-entrance-corridor} remains its honest statement.

Two consequences survive the correction untouched, because neither
depends on the value. Whatever \(\varphi_0\) is, it is geometric data,
and geometric data are patch-uniform: the descent enters the plateau at
the same point everywhere, so the standard objection that plateau
inflation requires improbable initial conditions dissolves structurally
--- there is no ensemble of initial conditions for the entrance to be
improbable in. And the launch remains an event of the throat's first
stretch, the same first e-fold in which the overlap integral accrues
\(86.5\%\) of the Planck mass (Problem~E.5.2): gravity and the launch
are, in this precise sense, both features of the throat's rim.

\subsection{Slow-roll equations and the meaning of ``slow-smooth''}

The standard slow-roll equations \citep{Mukhanov2005,LythRiotto1999} follow from Eq.~\eqref{eq:partII-effective-inflaton-action}:
\begin{align}
3H\dot\varphi + V_{\mathrm{eff}}'(\varphi) &\simeq 0,
\label{eq:partII-slowroll1}\\[3pt]
H^2 &\simeq \frac{V_{\mathrm{eff}}(\varphi)}{3M_{\mathrm{Pl,red}}^2}.
\label{eq:partII-slowroll2}
\end{align}
The standard potential slow-roll parameters are
\begin{equation}
\epsilon_V := \frac{M_{\mathrm{Pl,red}}^2}{2}\left(\frac{V'}{V}\right)^2,
\qquad
\eta_V := M_{\mathrm{Pl,red}}^2\frac{V''}{V}.
\label{eq:partII-slowroll-parameters}
\end{equation}
For the plateau form \eqref{eq:partII-plateau-potential}, and in the regime where \(e^{-\beta\varphi/M_{\mathrm{Pl,red}}}\ll 1\), one obtains to leading order
\begin{align}
\epsilon_V &\simeq \frac{1}{2}c_1^2\beta^2 e^{-2\beta\varphi/M_{\mathrm{Pl,red}}},
\label{eq:partII-epsilon-plateau}\\[3pt]
\eta_V &\simeq -c_1\beta^2 e^{-\beta\varphi/M_{\mathrm{Pl,red}}}.
\label{eq:partII-eta-plateau}
\end{align}
The number of e-folds from \(\varphi_*\) to the end of slow roll is
\begin{equation}
N_e
\simeq
\frac{1}{M_{\mathrm{Pl,red}}^2}
\int_{\varphi_{\mathrm{end}}}^{\varphi_*}
\frac{V}{V'}\dd\varphi.
\label{eq:partII-efolds-def}
\end{equation}
For the leading plateau approximation this gives
\begin{equation}
N_e \simeq \frac{1}{c_1\beta^2}\exp\!\left(\beta\varphi_*/M_{\mathrm{Pl,red}}\right),
\label{eq:partII-efolds-plateau}
\end{equation}
so that
\begin{equation}
\eta_V \simeq -\frac{1}{N_e},
\qquad
\epsilon_V \simeq \frac{1}{2\beta^2N_e^2}
\quad\text{(up to order-unity model factors).}
\label{eq:partII-slowroll-largeN}
\end{equation}
This is the mathematical meaning of \emph{slow-smooth}. The slope and curvature are exponentially small on the plateau, so the modulus drifts rather than falls. The field does not need an exquisitely tuned polynomial inflection point. It only needs the broad exponential flattening naturally produced by the bulk geometry. The descent, on this stretch, is a long gentle glide, not a plunge.

\subsection{Scalar tilt, amplitude, and the tensor target: two predictions and one input}

The standard scalar spectral tilt \citep{MukhanovChibisov1981,MukhanovFeldmanBrandenberger1992} is
\begin{equation}
 n_s = 1 - 6\epsilon_V + 2\eta_V.
\label{eq:partII-ns-def}
\end{equation}
Using Eq.~\eqref{eq:partII-slowroll-largeN}, one obtains the characteristic plateau result
\begin{equation}
 n_s \simeq 1 - \frac{2}{N_e} + \mathcal O\!\left(\frac{1}{N_e^2}\right).
\label{eq:partII-ns-plateau}
\end{equation}
The e-fold count is no longer a corridor to be quoted but a number to be computed: reheating matching (next subsection) gives \(N^* = 52.6\), undiluted because the lock crossing is prompt, and hence
\begin{equation}
 n_s = 1 - \frac{2}{N^*} = 0.9620,
\label{eq:partII-ns-numeric}
\end{equation}
against the measured \(0.9649 \pm 0.0042\) --- a deviation of \(-0.30\%\), a pull of \(-0.7\). The honest grade on this number deserves emphasis, because the grade itself is the point: \emph{derived-conditional}, closed modulo the \GDOS{} plateau-class derivation and the prompt-crossing lemma, where the program's working codebook could previously write only \emph{asserted}. The tilt is no longer inherited from the data; it is risked against them.

The scalar amplitude is
\begin{equation}
A_s = \frac{V_*}{24\pi^2 M_{\mathrm{Pl,red}}^4\epsilon_V(\varphi_*)}.
\label{eq:partII-As-def}
\end{equation}
The tensor-to-scalar ratio is
\begin{equation}
 r = 16\epsilon_V.
\label{eq:partII-r-def}
\end{equation}
Eliminating \(\epsilon_V\) between Eqs.~\eqref{eq:partII-As-def} and \eqref{eq:partII-r-def} gives
\begin{equation}
 r = \frac{2V_*}{3\pi^2 A_s M_{\mathrm{Pl,red}}^4}.
\label{eq:partII-r-from-V}
\end{equation}
This relation is extremely useful because it shows directly that once the inflationary plateau height is fixed, the tensor sector is fixed. On the derived plateau the slow-roll identities collapse it to a pure function of the e-fold count,
\begin{equation}
 r = \frac{12}{N^{*2}} = 0.0043,
\label{eq:partII-r-numeric}
\end{equation}
a factor of eight below the current BICEP/Keck bound \(r < 0.036\) and several times \emph{above} the projected \(\sigma(r)\simeq10^{-3}\) design floors of the Simons Observatory's enhanced surveys and LiteBIRD \citep{Starobinsky1979}. Where the retired TeV reading predicted \(r \sim 10^{-52}\) --- safety by invisibility --- the derived launch stakes a number that this decade's instruments will reach.

The scalar amplitude \eqref{eq:partII-As-def} remains the sector's \emph{one declared input}: it fixes the plateau height \(M\), exactly as the epoch fixes the \(\pi\)-inventory's snapshot. Its warp candidate --- \(A_s = 21.5\,e^{-2K/3}\), reaching \(75\%\) of the observed amplitude with no parameter --- is recorded in the codebook as flagged, not claimed. The falsification handle is now two-sided and sharper than before: a tensor measurement inconsistent with \(r \simeq 0.004\) in either direction kills the derived launch; a tilt measurement pushed below \(n_s \simeq 0.955\) (which would demand a diluted window, \(N_2 > 8\)) kills the prompt-crossing lemma; and a loud stochastic gravitational-wave background attributable to a strongly supercooled TeV-scale transition would refute the same lemma from the opposite side. The plateau that fixes the numbers is the plateau that fires the switches --- the virtue and the vulnerability remain the same statement.

\subsection{How the descent does the cosmological work}

With the launch at the top of the descent, the classical worries invert. The horizon and flatness problems are resolved by the duration of the accelerated stage and the shrinkage of the comoving Hubble radius; and the notorious initial-patch problem that afflicts genuinely low-scale inflation --- the need for a pre-inflationary homogeneous region exponentially larger than the inflationary Hubble volume --- does not arise here, because the primary plateau begins one e-fold below the GUT edge, where the required patch is microscopic.

The practical criterion is simply
\begin{equation}
\frac{\dd}{\dd t}(aH)^{-1} < 0,
\label{eq:partII-criterion-inflation}
\end{equation}
maintained for a sufficient number of e-folds. In a plateau model the total number of e-folds is governed by the field excursion across the flat throat segment, not by a high vacuum scale. Therefore the model can still smooth curvature and establish causal contact across the observed universe provided the plateau is wide enough in the canonical field direction.

In the present program the width is controlled by the logarithmic throat depth and by the same vacuum-selection dynamics that later drives \(\Omega\to 1\). That does not yet amount to a theorem that the required e-fold number is exactly fifty-something. It does amount to a coherent geometric mechanism for obtaining an order-\(10^1\) to order-\(10^2\) accelerated interval without introducing a separate inflaton sector. Since the detailed reheating history also shifts the precise \(N_e\) requirement, the correct statement at the current stage is this:
\begin{quote}
The minimal \GG{} inflation module naturally targets the same broad \(N_e\sim 50\text{--}60\) phenomenological corridor as standard plateau inflation, but with the plateau height fixed by the lock scale rather than by a high primitive inflaton potential.
\end{quote}

The symbolic discussion of \(N_e\) becomes more informative once one writes the standard matching formula between horizon exit during inflation and the present Hubble scale. For a comoving mode \(k\), one may write schematically
\begin{equation}
N_k \simeq 61.6 - \ln\!\left(\frac{k}{a_0H_0}\right)
+ \frac{1}{4}\ln\!\left(\frac{V_k^2}{M_{\mathrm{Pl,red}}^4\rho_{\mathrm{end}}}\right)
+ \frac{1-3w_{\mathrm{rh}}}{12(1+w_{\mathrm{rh}})}\ln\!\left(\frac{\rho_{\mathrm{rh}}}{\rho_{\mathrm{end}}}\right),
\label{eq:partII-Nk-general}
\end{equation}
where \(\rho_{\mathrm{end}}\) and \(\rho_{\mathrm{rh}}\) denote the energy densities at the end of inflation and after reheating, and \(w_{\mathrm{rh}}\) is the effective reheating equation-of-state parameter. Equation~\eqref{eq:partII-Nk-general} is standard but important: the required e-fold number is not fixed by the inflationary vacuum scale alone. It also depends on how efficiently the universe reheats and how one matches the inflationary stage to the later hot Big Bang.

Here the matching is evaluated rather than parameterized. The modulus decays at Planck strength, \(\Gamma = M^3/M_{\mathrm{Pl,red}}^2\) (the only channel consistent with the warped structure), so reheating completes at \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\) after roughly fourteen e-folds of matter-like oscillation; inserting \(V_k^{1/4} = 8.3\times10^{15}\,\mathrm{GeV}\) and this reheating history into Eq.~\eqref{eq:partII-Nk-general} gives
\[
N^* = 52.6,
\]
and because the lock crossing is prompt (\(N_2 = 0\)), no later episode dilutes it. The e-fold count that feeds \(n_s\) and \(r\) above is therefore not a phenomenological corridor but a closed piece of bookkeeping --- which is precisely why the present paper keeps the inflation section inside a full cosmological ledger instead of treating it as a stand-alone potential-fit exercise: the same reheating history that fixes \(N^*\) opens the thermal-leptogenesis window for the baryogenesis of Part~V.

\subsection{Graceful exit as a geometric spontaneous symmetry breaking}

The phrase \emph{graceful exit} means that inflation ends dynamically and continuously rather than by an abrupt collapse of the theoretical description. In the descent language it is the break: the \GSSB{} that closes the explore phase. In \GG{} this exit is tied to a \GSSB{} event at the infrared seam of the throat.

One sentence of bookkeeping before the mechanism, because the derivation separates two events that the retired TeV reading fused. The \emph{graceful exit of inflation} is the failure of slow roll at the edge of the primary plateau --- \(\epsilon_H = 1\), the modulus rolling to its minimum --- and it happens at the top of the descent. The \emph{lock \GSSB{}} --- the infrared seam transition at \(\Mlock\) that births the Kaxion and seeds the terminus --- is a distinct, later \GRIP{}-Out, crossed promptly during the radiation descent. The symmetry-breaking machinery below describes the general grammar of both; where earlier text attributed the end of inflation to the infrared seam, the seam event now stands on its own as the second crossing.

The guiding idea is simple. During the inflationary plateau, the modulus \(\varphi\) drifts slowly, and the Green--Schwarz normalization sector and compact-angle holonomy have not yet reached their final post-inflationary branch. As the system approaches the infrared seam, the effective mass matrix of the residual vacuum sector changes sign along one direction. The order parameter can be represented abstractly by a holonomy variable \(\Theta\) or, equivalently, by the deviation of the physical normalization from its attractor value. The effective free-energy functional then takes the schematic form
\begin{equation}
\mathcal F_{\mathrm{IR}}(\Theta)
=
\mathcal F_0 + \frac{1}{2}m_\Theta^2(\varphi)\Theta^2 + \frac{\lambda_\Theta}{4}\Theta^4 + \cdots,
\label{eq:partII-GSSB-functional}
\end{equation}
with
\begin{equation}
 m_\Theta^2(\varphi) > 0\quad \text{on the plateau},
 \qquad
 m_\Theta^2(\varphi_c)=0,
 \qquad
 m_\Theta^2(\varphi) < 0\quad \text{after the critical point.}
\label{eq:partII-mass-sign-flip}
\end{equation}
At \(\varphi=\varphi_c\), the symmetric branch loses stability and the system selects one of the broken branches. This is exactly the \GDOS{} language of \DGI\ plus \GSSB\ plus \GA\ translated into cosmology --- the same three-move anatomy the electron paper uses, now playing out on the cosmic plateau.

Operationally, the consequences are threefold.
\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item The slow-roll conditions fail because the effective potential steepens near the critical point.
\item The infrared boundary tension / seam condition shifts, ending the quasi-de Sitter phase.
\item The stored vacuum energy is dumped into ordinary relativistic degrees of freedom, reheating the boundary universe.
\end{enumerate}
The end of inflation may be parameterized by the standard condition
\begin{equation}
\epsilon_H := -\frac{\dot H}{H^2}=1
\label{eq:partII-epsilonH-end}
\end{equation}
or equivalently by the breakdown of the slow-roll inequalities. The physical interpretation, however, is richer than in ordinary scalar inflation: inflation ends because the throat completes its vacuum-selection story.

\subsection{Reheating and the return to the hot Big Bang}

Once inflation ends, the universe must return to an ordinary thermal history. Let \(\xi_{\mathrm{rh}}\) be the fraction of the vacuum energy at the end of inflation transferred efficiently into the relativistic plasma. The reheating temperature is then estimated by
\begin{equation}
T_{\mathrm{rh}} \simeq \left(\frac{30}{\pi^2 g_{*,\mathrm{rh}}}\,\xi_{\mathrm{rh}}V_{\mathrm{end}}\right)^{1/4}.
\label{eq:partII-Treh-general}
\end{equation}
The decay is Planck-strength --- \(\Gamma = M^3/M_{\mathrm{Pl,red}}^2\), the modulus's only consistent channel --- so \(\xi_{\mathrm{rh}}\) is not order unity: the oscillating condensate leaks for about fourteen e-folds, the bath spiking to \(T_{\mathrm{max}} \sim 10^{11}\,\mathrm{GeV}\) and settling at
\begin{equation}
T_{\mathrm{rh}} \simeq 3\times10^{10}\;\mathrm{GeV},
\label{eq:partII-Treh-TeV}
\end{equation}
comfortably above the thermal-leptogenesis window (\(\gtrsim 10^{9}\)--\(10^{10}\,\mathrm{GeV}\)), comfortably below the GUT edge (monopoles stay diluted), and far above the lock (which the cooling bath then crosses at \(t \sim 10^{-13}\,\mathrm{s}\), producing the Kaxion and the KK sector as ordinary cooling thresholds).
This is one of the most attractive features of the scenario, and it is the first attractor export of the descent: the hot Big Bang is not discarded but recovered immediately after the break, with a reheating temperature derived from the modulus decay itself --- the cooling bath then crossing, in due course, the same lock scale that governs the particle hierarchy. Later baryogenesis and dark-matter calculations can therefore begin from a thermodynamic stage whose characteristic scale was not chosen separately for cosmology.

\subsection{A compact comparison and the bottom line of the launch}

\begin{table}[H]
\centering
\caption[Ordinary high-scale inflation against the minimal GG module]{Schematic comparison between ordinary high-scale inflation and the minimal \GG{} inflation module. Each row poses one question and answers it twice, in the left column for a typical high-scale inflaton model and in the right for the minimal implementation here. The comparison is schematic by design: it is a map of where the two accounts differ in kind, not a numerical competition between them, and no row should be read as a measurement.}
\label{tab:partII-inflation-compare}
\small
\begin{tabular}{p{0.22\textwidth}p{0.31\textwidth}p{0.32\textwidth}}
\toprule
\textbf{Question} & \textbf{Typical high-scale inflaton model} & \textbf{Minimal \GG{} implementation}\\
\midrule
Primitive field content & New scalar sector usually inserted by hand & No new primitive inflaton; low-energy modulus of inherited throat/GS sector.\\
Potential freedom & Large menu of tunable potentials & Plateau generated by curvature + Casimir + IR-seam competition.\\
Characteristic vacuum scale & Often near \(10^{16}\,\mathrm{GeV}\), chosen & \(V^{1/4} = 8.3\times10^{15}\,\mathrm{GeV}\), fixed one e-fold below \(\EGUT\) by the derived class + the one input.\\
Tensor prediction & Can be observable depending on scale and slope & \(r = 12/N^{*2} = 0.0043\): a firm two-sided target, decisive for the Simons Observatory and LiteBIRD.\\
Secondary structure & Typically none, or ad hoc & The lock basin: real, crossed promptly (\(N_2=0\)); births the Kaxion; seeds the terminus.\\
Exit mechanism & Model-dependent waterfall or steepening & \GSSB{} at the IR seam; vacuum-selection completion and reheating.\\
Relation to later dark sector & Often separate from dark matter and dark energy & Same event chain naturally linked to Kaxion and residual vacuum term.\\
\bottomrule
\end{tabular}
\end{table}

The inflationary module of \ATen{} is therefore built from one sharp proposition:
\begin{quote}
The \GDOS{} boundary law forces the plateau class; the amplitude declares its one input; everything else --- the tilt, the tensor ratio, the e-fold count, the reheating temperature, the promptness of the lock crossing --- is computed. Inflation is the opening stretch of the same descent whose second crossing stocks the dark sector and whose terminus is dark energy.
\end{quote}
The retired proposition is kept on the record for honesty:
\begin{quote}
A TeV-scale slow-smooth inflationary stage can be generated by the same six-dimensional throat that already underlies the particle hierarchy, with the relevant modulus represented either as the throat radion or as the Green--Schwarz normalization direction, and with graceful exit occurring through a geometric spontaneous symmetry breaking at the infrared seam.
\end{quote}
Its most important quantitative consequence is now Eq.~\eqref{eq:partII-r-numeric}: \(r = 0.0043\), reachable this decade. The retired reading's untestable \(r \sim 10^{-52}\) is replaced by a number that can lose.


\subsection{The launch, graded}
\label{subsec:launch-graded}

The derivation chain of the launch (Figure~\ref{fig:partII-inflation-derived} draws its two halves: the derived potential and the \((n_s, r)\) predictions), with the grade of every link stated in the codebook's vocabulary: the inflaton is the \(\Sigma_r\) scale modulus (\emph{forced} --- the only one-dimensional modulus the theory possesses); the potential class is Starobinsky/\(\xi\) (\emph{derived} --- the \GDOS{} non-conformal route); \(n_s = 0.9620\) and \(r = 0.0043\) (\emph{derived-conditional} --- given the class and the prompt crossing); \(N^* = 52.6\) and \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\) (\emph{derived} --- reheating matching at Planck-strength width); \(N_2 = 0\) (\emph{derived-conditional} --- the ledger-action bound, with fifty-six orders of margin); the amplitude \(A_s\) (\emph{the one declared input}, warp candidate at \(75\%\) flagged); and the entrance displacement \(\varphi_0\) (\emph{open} --- bracketed on both sides by the corridor of \S\ref{ssec:entrance}, its selecting bend now mechanized by the mode depletion of \eqref{eq:partII-depletion} and landing inside the corridor by shape alone). Two routes toward paying the amplitude have been attempted and closed --- warped decay (excluded by seventeen orders) and holographic central-charge counting (no large-\(N\) regime at the theory's anchors) --- and one remains open and sharply posed: the signed per-level heat-kernel coefficient of the lock-anchored tower of \citet{ArisakaP1KK}, required in \([1/772,\,1/100]\) --- the amplitude's mechanism being the tower's threshold sum, with the bend test already passed by the depletion law of \S\ref{ssec:entrance}, and with the same number deciding between the flagged warp form \(M \propto e^{-K/3}\) and the tower form \(M \propto e^{-K/2}\). The falsifiers are staked above; the retired TeV reading and its codebook conjecture are recorded, with their refutation, in Part~IX.

\clearpage
\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.84\textheight,keepaspectratio]{a10_inflation_derived_portrait.pdf}
\par\smallskip
\begingroup\captionsetup{font=footnotesize}
\captionof{figure}[The derived launch: potential and predictions.]{\textbf{The derived launch.} \emph{Top:} the \(\Sigma_r\) plateau of the derived class \eqref{eq:partII-plateau-potential} at \(V_0^{1/4} \approx 8.3\times10^{15}\,\mathrm{GeV}\), one e-fold below the GUT edge --- shape fixed by gravity, height the one input; graceful exit, prompt lock crossing, descent to the terminus. \emph{Bottom:} \(N^* = 52.6\) (star): \(n_s = 0.9620\) (\(0.7\sigma\), bands), \(r = 0.0043\) --- eight times below BICEP/Keck (dashed), decisive this decade (Simons Observatory/LiteBIRD, dotted).}
\label{fig:partII-inflation-derived}
\endgroup
\end{minipage}
\vfill\clearpage

\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[page=1,viewport=40 23 962 1051,clip,width=\textwidth,height=0.84\textheight,keepaspectratio]{a10_cosmic_history.pdf}
\par\smallskip
\begingroup\captionsetup{font=footnotesize}\captionof{figure}[Cosmic history, numerically integrated: scale factor, expansion rate, temperature.]{\textbf{The
two-basin descent, numerically integrated} (Friedmann + inflaton +
reheating fluid, smooth by construction). Scale factor \(a\), expansion
rate \(\dot a = da/dt\), and temperature \(T\) from the Planck edge to
\(10^{22}\,\mathrm{s}\). The gray band is pre-\GG{} (\(\EPl \to \EGUT\):
the geometry begins at the GUT edge); the purple band is the primary
inflation (\(N \simeq 60\), \(a \propto e^{Ht}\)); the temperature crashes
through super-cooling, the bath spikes to \(T_{\mathrm{max}} \sim
10^{11}\,\mathrm{GeV}\) while the modulus decays, and reheating completes
at \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\); the lock is
crossed promptly at \(t \sim 10^{-13}\,\mathrm{s}\) (Kaxion born, no
second inflation); and in the far future \(\dot a\) rises exponentially
again --- the two de~Sitter epochs bracketing the whole history. Every
vertical epoch line carries its energy scale.}
\label{fig:partII-history-page1}
\endgroup
\end{minipage}
\vfill\clearpage

\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[page=2,viewport=39 21 969 1066,clip,width=\textwidth,height=0.90\textheight,keepaspectratio]{a10_cosmic_history.pdf}
\par\smallskip
\captionof{figure}[Cosmic history in descent variables: scale dimension, energy scale, energy budget.]{\textbf{The
descent's own variables.} \emph{Top:} the scale dimension
\(r = \ln(\EGUT/\mu)\): flat on the primary plateau
(\(r_{\mathrm{inf}} \simeq 1\)), descending monotonically through the
prompt lock crossing (\(r \simeq 29.4\), crossed, not camped), and
flattening at the terminus \(r_{\mathrm{GGA}} \simeq 64.5\). \emph{Middle:}
the energy scale \(\mu = \rho_{\mathrm{tot}}^{1/4}\), from the primary
plateau (\(8.3\times10^{15}\,\mathrm{GeV}\)) through \(\Mlock\) to the
dark-energy floor (\(\approx 2\,\mathrm{meV}\)). \emph{Bottom:} the cosmic
energy budget \(\Omega_i(t)\) on the seconds axis --- the \(\Sigma_r\)
modulus band through inflation and reheating, the long radiation era, the
matter interlude, and dark energy's take-over, with the
\(\Omega_m = 1/\pi\) crossing (75~Myr before the present) marked in red.}
\label{fig:partII-history-page2}
\end{minipage}
\vfill\clearpage


\paragraph{The relaxation relic: a signature at the largest scales
(F25).} One further prediction rides on this module, inherited from
the quantum-mechanics companion (\citealp{ArisakaA8QM}) and recorded
in the falsifier ledger as F25. In the cohort's reading the Born
statistics of quantum outcomes are not postulated but produced: the
boundary law's own dissipation relaxes the bulk configurational
statistics to the Born measure, monotonically, at a per-exchange rate
set by the port coupling. A mode that leaves the horizon during
inflation decouples from the port stream and freezes whatever
divergence from the Born measure it still carries, so the earliest-
exiting --- largest-scale --- modes retain the most. The generic
relic is a power law anchored at the largest observable scales, a
primordial-statistics deficit
$\xi(k) = 1 - A\,(k/k_{0})^{-\gamma}$, with $k_{0}$ the first
observable mode out and $\gamma$ set by the collision rate and mixing
weight on the plateau --- and, because the initial configuration is
constitutionally determined rather than free, the amplitude $A$ is in
principle \emph{computable}, not a fit parameter. Detection of a
large-scale statistical anomaly incompatible with the computed form
--- or a relic where the determined initial data forbid one ---
falsifies the relaxation account; exact Born statistics at all scales
bounds the pre-inflationary relaxation time from below. The
quantification of $A$ and $\gamma$ within this module's determined
descent is left to a dedicated study.

\subsection{Inflation on the two fibers: the interval's coordinate, rolling once}
\label{ssec:inflation-two-fibres}

The launch is now graded; this closing subsection reads it on the
anatomy the cohort's recent campaign has made exact
(\textsf{GG-W1-Symmetry}; \citealp{ArisakaP5CPT};
\citealp{ArisakaC1Kaxion}): \emph{holonomy requires closed loops,
and an interval has none.} The compact circle $S^1_\theta$ is
nothing but closed loops --- everything on it is holonomy,
quantized and protected, its transport a group: the fiber that
remembers. The scale interval $I_r$ can hold no holonomy --- its
transport composes only forward, a semigroup: the fiber that
forgets. The dark-matter companion has already drawn one portrait
on this anatomy: the Kaxion is \emph{the circle's own coordinate,
promoted to a boundary field} \citep{ArisakaC1Kaxion}. The launch
supplies the mirror, and it was written into this Part's own
derivation before the campaign named it: the inflaton $\varphi$ is
the canonically normalized coordinate of the radial stratum
$\Sigma_r$ --- \emph{the interval's own coordinate, promoted to a
boundary field}. The two great scalar episodes of cosmology are
the bulk's two non-spatial coordinates, each taking the stage
exactly once.

\paragraph{The census: one interval, one inflaton; one circle, one
axion.} The uniqueness step of this Part deserves restating in the
campaign's terms, because it is the structural prize. The \GDOS{}
boundary law's genuine conformal anomaly supplies only the
Weyl-squared and Euler densities; a plateau sector can therefore
arise only from the non-conformal curvature coupling of a scalar;
and the inherited geometry possesses exactly one such scalar ---
the radial modulus, because the bulk has exactly one open
non-spatial fiber. Meanwhile the compact sector supplies exactly
two shift-protected angles with one light eigenstate
\citep{ArisakaC1Kaxion}, because the bulk has exactly one closed
non-spatial fiber. Read together: \emph{one interval $\Rightarrow$
one inflaton; one circle $\Rightarrow$ one axion.} The two great
embarrassments of riches in the literature --- the inflaton zoo
(``choose your favorite potential'') and the string axiverse
(hundreds of light axions) --- are, on this anatomy, the same
mistake made twice: uncounted fibers. The dimensional census of
the bulk \emph{is} the scalar field content of cosmology, and
\MSMF{}'s prohibition of a free inflaton
(\S\ref{sec:inflation}) is thereby sharpened from a parsimony
rule into a counting statement.

\paragraph{The episode's shape is the fiber's character.} Why does
inflation \emph{end}, while the Kaxion \emph{rings on}? Because
each coordinate inherits its parent fiber's topology. The circle's
coordinate is periodic: displaced, it oscillates --- there is
always somewhere to come back to; protected by periodicity, it
cannot get heavy; its relaxation is a ringing that persists to
this day. The interval's coordinate is non-periodic and
unprotected: its potential is a plateau with an edge, it traverses
once, slow roll fails at $\epsilon_H = 1$, and the episode is over
--- \emph{inflation is transient because the interval is an
interval.} Even the two potentials wear their fibers' faces: the
Kaxion's is a cosine --- topology-generated, periodic --- while
the inflaton's is the warp exponential
$\bigl(1-e^{-\sqrt{2/3}\,\varphi/M_{\mathrm{Pl,red}}}\bigr)^2$ ---
the e-folding of the descent itself, bent into a potential. And
the once-only logic of the descent applies at the launch with full
force: the roll happened once, in one direction, and cannot recur,
because the fiber it ran along has no inverse transport with which
to re-ascend the plateau --- the same gated-interval logic by
which the later lock crossing is prompt and unrepeated
\citep{ArisakaP5CPT}. (Stated, like the rest of this subsection,
as a reading of proved structures, not a new theorem.)

\paragraph{The interval's numbers.} Sorted on the anatomy, the
launch's observables are pure interval bookkeeping. The e-fold
count $N^* = 52.6$ is a length of descent, undiluted because the
crossing is prompt; the tilt $n_s = 1 - 2/N^* = 0.9620$ and the
tensor ratio $r = 12/N^{*2} = 0.0043$ are functions of that length
alone; and the one declared input --- the plateau height, with its
flagged $e^{-K/3}$ warp candidate --- is an \emph{address} on the
fiber that forgets, exactly where a two-fiber theory expects its
declared inputs to sit. Nothing quantized appears anywhere in the
launch's output, and the anatomy says why: the interval has no
holonomy to write with. What inflation makes --- the stretched,
frozen perturbation spectrum --- carries the descent's numbers;
what inflation cannot touch is the ledger: charges, windings, the
integers of \BIsix{} ride through the launch unaltered, because
the launch is the other fiber's episode.

\paragraph{The circle's two cameos.} Inflation is the interval's
solo, but the circle appears exactly twice, at the boundaries of
the act, and both cameos are already this Part's. \emph{The exit
is the circle's first word}: the graceful exit is a \GSSB{} whose
order parameter is the holonomy variable $\Theta$ --- as the
modulus approaches the seam, the compact-angle mass matrix flips
sign, and the circle's symmetry breaking ends the interval's
episode. \emph{The handoff seeds the next act}: the later lock
\GSSB{} --- the descent's second saddle --- leaves the compact
angle displaced, and that displacement, relaxing, is the dark
matter (\S\ref{ssec:kaxion-cosmo-two-fibres}). The two
fiber-episodes are chained: the interval's exit hands the circle
the displacement that becomes its ring. In one sentence:
\emph{inflation is the fiber that forgets, opening the story;
dark matter is the fiber that remembers, carrying it; and the
terminus (\S\ref{ssec:de-two-fibres}) is what remains when both
have finished speaking.}

\section{The terminus: dark energy as the Ground Geometric Attractor}
\label{sec:dark-energy}

\subsection{What has to be explained at the end of the descent}

The launch left a hot, expanding, nearly flat universe and a tiny residual vacuum term. We now run the descent forward to where it stops. The late-time acceleration problem contains two logically distinct questions.
\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item Why does the present universe contain an approximately constant energy-density component whose pressure is sufficiently negative to accelerate the expansion?
\item Why is that component numerically comparable to the matter density precisely in the present epoch rather than only in a remote past or future era?
\end{enumerate}
The first is the dark-energy question. The second is the cosmic coincidence question. A framework can answer one without answering the other. Part~IV aims to at least formulate both within the same inherited ledger.

The historical arc is familiar. The cosmological-constant term \(\Lambda\) was introduced by Einstein in 1917 \citep{Einstein1917} to allow static cosmological solutions and was famously withdrawn after Hubble's expansion measurement; Weinberg's 1989 review \citep{Weinberg1989CC} surveyed the dramatic mismatch between the observed value of the vacuum energy and naturalness estimates from quantum field theory. The observational reintroduction of \(\Lambda\) came in 1998--1999 with the Type-Ia supernova results of Riess et~al.\ \citep{Riess1998} and Perlmutter et~al.\ \citep{Perlmutter1999}, which established that the late-time expansion is accelerating. The present section retains the empirical \(\Lambda\) but reinterprets its scale through the residual \(\GFF\) ledger of the inherited bulk. The cosmological-constant and dynamical dark-energy literature, against which the framework's residual-vacuum branch is positioned, and the de~Sitter horizon thermodynamics relevant to the late vacuum-dominated phase, are reviewed in \citep{Carroll2001,PeeblesRatra2003,CopelandSamiTsujikawa2006,GibbonsHawking1977}.

The background condition for acceleration follows immediately from Eq.~\eqref{eq:partII-acceleration}: for a flat universe without an explicit curvature term,
\begin{equation}
\frac{\ddot a}{a}>0
\qquad\Longleftrightarrow\qquad
\rho + 3p < 0.
\label{eq:partII-acceleration-condition}
\end{equation}
A component with equation-of-state parameter \(w<-1/3\) can therefore drive acceleration. A strict cosmological constant has \(w=-1\). The standard \LCDM\ answer is simply to write
\begin{equation}
\rho_\Lambda = \text{const},
\qquad
p_\Lambda=-\rho_\Lambda,
\label{eq:partII-lcdm-lambda}
\end{equation}
and fit the constant. \ATen{} asks a narrower and more structural question: can the same term be re-read as a residual boundary export of the inherited \GFF{} and \BIsix{} geometry rather than as an inserted constant with no pedigree? In the descent language, the question is whether \(\Lambda\) is the global attractor the boundary relaxes into --- the \GGA{}.

\subsection{Residual vacuum energy as a \GFF{}/\CEP{} boundary functional}

The earlier papers already suggested the key conceptual route. The \GFF{} ledger is
\begin{equation}
\GFF = \GEP + \CEP,
\label{eq:partII-GFF}
\end{equation}
where \GEP\ is the geometric-energy block and \CEP\ is the entropic-curvature block. In the particle paper this ledger organizes masses, couplings, and vacuum selection. In cosmology the natural next step is to interpret the late-time vacuum term as a small residual zero-mode export of the same ledger.

At the level of the four-dimensional effective action one writes the residual term as
\begin{equation}
S_{\Lambda,\mathrm{eff}} = -\int \dd^4x\sqrt{-g_4}\,\rho_\Lambda^{\mathrm{eff}}.
\label{eq:partII-SLambda}
\end{equation}
The \GG{} interpretation is that
\begin{equation}
\rho_\Lambda^{\mathrm{eff}} = \rho_{\mathrm{\GFF{},res}} = \rho_{\GEP,\mathrm{res}} + \rho_{\CEP,\mathrm{res}},
\label{eq:partII-rhoLambda-GFF}
\end{equation}
where ``res'' denotes the tiny leftover zero-mode contribution after the inflationary \GSSB{} transition, the selection of the physical normalization, and the coarse-graining of heavy throat modes. In the minimal version of the program the dominant late-time term is expected to be the \CEP{} residue, because the entropic-curvature block is exactly where the framework stores slow, coarse-grained, late-time free-energy information. But the decomposition in Eq.~\eqref{eq:partII-rhoLambda-GFF} is the honest statement: at the present stage one should not overclaim a complete microscopic split.

What matters first is that the late-time vacuum term is no longer conceptually independent. It is a remnant of the same throat and same vacuum-selection event that already appeared in the launch. The terminus is made of the descent's own leftovers --- the same \GFF{} functional, read at its floor.

\subsection{A natural meV scale from the lock-scale seesaw}

The most notorious feature of dark energy is its tiny scale \citep{Weinberg1989CC,Carroll2001}. Observationally one may summarize it by saying that
\begin{equation}
\rho_\Lambda^{1/4} \simeq 2.224\;\mathrm{meV},
\label{eq:partII-mev-observation}
\end{equation}
read off the derived background \(\Omega_\Lambda = 1-1/\pi\) at
\(\Hzero^{(0)} = 66.6\) (2.243~meV on the Planck~2018 inventory --- the
difference is the two-quantity \(\Hzero{}\) corridor, not a disagreement
about the vacuum). Any claimed derivation should therefore supply a
meV-type quantity without introducing a new arbitrary scale by hand.

One structural remark belongs here. The seesaw scale \(m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}}\) is now \emph{uniquely} the terminus's number: the retired TeV-inflation reading had, unknowingly, assigned the same combination to the launch (its \(H_{\mathrm{inf}} = \Mlock^2/\sqrt3\,M_{\mathrm{Pl,red}}\) is this seesaw, up to \(\sqrt3\)), double-booking one scale at both ends of the descent. In the two-basin reading the bookkeeping is clean: the combination \(\Mlock^2/M_{\mathrm{Pl,red}}\) is the curvature scale of the middle basin and the density scale of the terminus --- the lock episode briefly \emph{visits} the terminus's vacuum energy before the crossing drains it, and dark energy is, in this reading, the undrained residue of that drainage (a flagged interpretation: it gives the open coefficient \(\zeta_\Lambda\) a physical home --- the drainage inefficiency --- without yet computing it).

\GG{} provides an immediate dimensional candidate from the already-derived scale chain. The simplest combination of \(\Mlock\) and the Planck scale with dimensions of energy is
\begin{equation}
 m_\Lambda^{\mathrm{geom}} := \frac{\Mlock^2}{M_{\mathrm{Pl,red}}}.
\label{eq:partII-mLambda-geom}
\end{equation}
The reduced mass, not the full one, is the right normalization here, and
the choice is load-bearing rather than cosmetic: it moves the answer by
\(\sqrt{8\pi} = 5.01\). It is the reduced mass because this quantity is
destined for the Friedmann equation and for
\(H_\Lambda = \sqrt{\rho_\Lambda/3M_{\mathrm{Pl,red}}^2}\), where
\(M_{\mathrm{Pl,red}}\) is what appears; equivalently it is the
normalization in which the boundary Einstein--Hilbert action of
\S\ref{sec:partIII-GR-theorem} was read off. Using
Eq.~\eqref{eq:partII-scale-chain-numbers},
\begin{equation}
 m_\Lambda^{\mathrm{geom}} \simeq 6\times 10^{-3}\;\mathrm{eV} = 6\;\mathrm{meV}.
\label{eq:partII-mLambda-geom-num}
\end{equation}
This is already in the correct observational corridor. The corresponding vacuum density scale is
\begin{equation}
\rho_\Lambda^{\mathrm{geom}}
\sim
\zeta_\Lambda\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^4,
\label{eq:partII-rhoLambda-geom}
\end{equation}
with \(\zeta_\Lambda\) the suppression factor carrying the exact zero-mode overlap and residual-ledger bookkeeping. Uncorrected, \((6.083\,\mathrm{meV})^4\) overshoots the measured
\(\rho_\Lambda\) by a factor of \(56\) --- a factor of \(2.73\) in
scale, set against the \(10^{120}\) the seesaw has already removed. That
residue is not left open: \S\ref{sec:h0-derivation} selects
\(\zeta_\Lambda = \pi/LK = \pi/175 = 0.017952\) from the frozen ledger,
with the mechanism stated there, and closes the scale at
\(\rho_\Lambda^{1/4} = 2.23\;\mathrm{meV}\) against the
\(2.224\;\mathrm{meV}\) of Eq.~\eqref{eq:partII-mev-observation}
(Eq.~\eqref{eq:partII-rhoLambda-closed}). What
Eq.~\eqref{eq:partII-rhoLambda-geom} contributes at this stage is the
scale; the coefficient is the business of \S\ref{sec:h0-derivation}. No
new scale beyond the inherited lock scale is required for either.

This ``lock-scale seesaw'' is one of the cleanest places where the cosmology sequel benefits from the particle derivation. Without the particle paper, the appearance of the meV scale would look like a numerological accident. With the particle paper, it is the first nontrivial infrared scale one can build from the already-fixed hierarchy. The same lock that held the launch at TeV now sets the floor of the terminus at meV --- one geometry, both ends.

\subsection{Why the equation of state is exactly \texorpdfstring{\(w=-1\)}{w=-1}: the transition is complete}

If the late-time term is truly the residual zero-mode export of a selected vacuum functional, then it should be nearly constant on cosmological timescales. The stress tensor of a constant vacuum term is
\begin{equation}
T^{\mu}{}_{\nu}(\Lambda) = -\rho_\Lambda\,\delta^{\mu}{}_{\nu},
\label{eq:partII-vacuum-stress}
\end{equation}
which implies
\begin{equation}
 p_\Lambda = -\rho_\Lambda,
 \qquad
 w_\Lambda = -1.
\label{eq:partII-wminusone}
\end{equation}
Therefore the most natural minimal prediction of the framework is not a rolling quintessence-like field with arbitrary \(w(z)\) \citep{PeeblesRatra2003,CopelandSamiTsujikawa2006}, but a vacuum-like term extremely close to \(-1\). This is the signature of an attractor that has been \emph{reached}: a rolling field is a descent still in motion, but \(w=-1\) is the descent at rest. The vacuum-residual exactness says the transition is complete --- the boundary has arrived at its terminus and stopped.

This is also where the anomaly logic of \GG{} becomes conceptually relevant. The earlier papers are built on a tightly constrained anomaly-free spectrum. Introducing an extra light rolling scalar solely to solve dark energy would be constitutionally expensive unless that scalar was already required elsewhere. The minimal route is therefore to treat dark energy as the frozen residual of a completed geometric transition rather than as a new light field still rolling today.

Honesty is required here, and it is now sharper than a plausibility argument. The \emph{sign} of any departure is a theorem: a ghost-free, minimally coupled vacuum sector cannot cross below \(w=-1\), so the framework forbids phantom behavior outright (Theorem~\ref{thm:no-phantom} below). The \emph{magnitude} question is now also a theorem, conditional on the constitution:

\begin{theorem}[Exact vacuum equation of state at the terminus (T-Cosmos-DE$'$), conditional]
\label{thm:w-exact}
Assume: \textbf{(I)} the constitutional rigidity of the bulk profile (\ATwo{} \citep{ArisakaA2GG6}; the axiomatic $e^{-2r}$ warp, whose dynamical stabilization is the named upstream open item); \textbf{(II)} the KK spectral gap of \AFive{} \citep{ArisakaA5GDOS} (T9.1): no massless modulus beyond the gauge zero modes, lightest massive mode at $\Mlock \simeq 3.85$~TeV; \textbf{(III)} Lemma~DE-1 of \ASix{} \citep{ArisakaA6Gi4} (vacuum-flux freezing at the Geometric Attractor, itself a consequence of GFF-ST6 and the \BIsix{} closure). Then the dark-energy equation of state on the late-time branch satisfies $w(a) = -1$ identically, with any sourced deviation bounded by
\[
 |1+w| \;\le\; O(\beta^2)\,\Big(\frac{H_0}{\Mlock}\Big)^{2} \;\approx\; 10^{-91}
\]
for any residual matter--modulus coupling $\beta \le 1$.
\end{theorem}

\begin{proof}
By Lemma~DE-1 the vacuum export is individually conserved at the attractor, $\dd\rho_\Lambda/\dd\ln a = 0$; the continuity equation $\dd\rho_\Lambda/\dd\ln a = -3(1+w)\rho_\Lambda$ then forces $1+w = 0$ (Step~1). Cosmological sources displace the state off the critical point by $\delta\phi \sim \beta\,\rho_m/(m_{\min}^2 M_{\rm Pl})$; with $m_{\min} \ge \Mlock$ from (II), the induced deviation is $1+w \simeq \dot\phi^2/\rho_\Lambda = O(\beta^2)(H_0/\Mlock)^2$ (Step~2). Combined with the unconditional no-phantom bound $w \ge -1$ (Theorem~\ref{thm:no-phantom}), the value is pinned two-sidedly (Step~3).
\end{proof}

\emph{Corollary (two-sided kill switch).} Any confirmed deviation of $w$ from $-1$, in either direction and at any late-time redshift --- including a converged evolving-dark-energy ($w_0$--$w_a$) detection --- refutes the fixed-point reading outright; the framework cannot absorb $w \neq -1$. \emph{Provenance chain:} \ATwo{} (rigid profile) $\to$ \AFive{} (T9.1 spectral gap) $\to$ \ASix{} (GFF-ST6 $\to$ Lemma~DE-1) $\to$ the present theorem: every link is a constitutional axiom or a numbered theorem of the cohort, and the single dynamical residue --- the profile-stabilization proof --- is named in hypothesis (I). Codebook consequence: row D16 conservatively remains D$^*$, its named remaining lemma now being that stabilization proof rather than the exactness itself (Part~IX). The current empirical determination, \(w = -1.028 \pm 0.032\) from the Planck~2018 combination with supernovae and BAO \citep{Planck2018}, is consistent with this expectation at the \(0.9\sigma\) level (a \(+2.7\%\) fractional deviation); the entry appears as an independent prediction in the parameter-audit Dev plot (Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}).

\begin{theorem}[No phantom: the terminus equation of state (T-Cosmos-DE)]
\label{thm:no-phantom}
Let the late-time vacuum sector consist of a finite collection of canonically
normalized scalar moduli \(\{\varphi_i\}\) (the \(\Sigma_r\) scale-modulus,
Green--Schwarz, and \GFF{} directions), minimally coupled to gravity on the
\FRW{} background, and let the spectrum be ghost-free --- as the anomaly-free,
unitary construction of \ATwo{}/\AThree{} guarantees. Then the dark-energy
equation-of-state parameter obeys
\[
w \;\ge\; -1 \qquad\text{at all times,}
\]
with equality if and only if every \(\dot\varphi_i=0\), i.e.\ at a stationary
point of \(V=\GFF{}\) --- the \GGA{}. Consequently a secure, model-independent
measurement of \(w<-1\) (a true phantom crossing) falsifies the construction.
\end{theorem}

\begin{proof}
For canonical scalars the energy density and pressure on the homogeneous
background are \(\rho=\sum_i\tfrac12\dot\varphi_i^2+V\) and
\(p=\sum_i\tfrac12\dot\varphi_i^2-V\), so
\[
\rho+p=\sum_i\dot\varphi_i^{\,2}\;\ge\;0 .
\]
Hence \(1+w=(\rho+p)/\rho=\sum_i\dot\varphi_i^2/\rho\ge0\), since \(\rho>0\).
Equality requires \(\dot\varphi_i=0\) for every \(i\); the scalar field equation
\(\ddot\varphi_i+3H\dot\varphi_i+\partial_i V=0\) then forces \(\partial_i V=0\),
a stationary configuration of \(\GFF{}\), which is the \GGA{}. A value
\(w<-1\) would require \(\rho+p<0\), i.e.\ a negative kinetic term --- a ghost ---
which the ghost-free spectrum excludes.
\end{proof}

The theorem fixes the \emph{sign}; the \emph{approach} to \(-1\) is equally
controlled. Near the \GGA{}, write \(m_\varphi^2=V''(\varphi_\star)\ge0\). In the
friction-dominated regime (\(\tfrac12\dot\varphi^2\ll V\), i.e.\ \(m_\varphi\ll
H\)) the slow-roll relation \(3H\dot\varphi\simeq-V'\) with \(H^2\simeq
V/3M_{\mathrm{Pl,red}}^2\) gives
\begin{equation}
1+w \;\simeq\; \tfrac{2}{3}\,\epsilon_V,
\qquad
\epsilon_V := \frac{M_{\mathrm{Pl,red}}^2}{2}\left(\frac{V'}{V}\right)^{\!2}\ge 0,
\label{eq:partII-w-slowroll}
\end{equation}
and the decaying mode \(\varphi-\varphi_\star\propto e^{-(m_\varphi^2/3H_\Lambda)t}\)
drives \(\epsilon_V\to0\), so \(w\to-1\) exponentially as the attractor is
approached. Three statements therefore stand together: the sign \(w\ge-1\) is a
theorem; the slow-roll form \(1+w=\tfrac23\epsilon_V\) is exact given a canonical
kinetic term; and the present magnitude \(1+w_0=\tfrac23\epsilon_V(\varphi_0)\) is
exponentially small once the system is inside the \GGA{} basin. What remains open
is the single \(\mathsf D^{*}\) item that already fences the cosmological
principle --- a proof that the realized late-time state lies in that basin --- and
the numerical value \(\epsilon_V(\varphi_0)\), which needs the \GFF{} curvature at
its floor. Neither dents the sign theorem: the empirical \(w_0=-1.028\pm0.032\)
sits on the non-phantom edge (the \(0.9\sigma\) entry noted above), and the
framework predicts the central value should migrate to \(w_0\gtrsim-1\) as the
data sharpen.

\subsection{From flatness to the late-time density fractions}

Once radiation becomes negligible, the first Friedmann equation may be written in the present epoch as
\begin{equation}
1 = \Omega_m + \Omega_\Lambda + \Omega_k,
\qquad
\Omega_m := \frac{\rho_m}{\rho_c},
\qquad
\Omega_\Lambda := \frac{\rho_\Lambda}{\rho_c},
\label{eq:partII-late-sum-rule}
\end{equation}
with
\begin{equation}
\rho_c := \frac{3H_0^2}{8\pi G_N}.
\label{eq:partII-critical-density}
\end{equation}
If inflation drives the universe close to spatial flatness, then \(\Omega_k\approx 0\), so
\begin{equation}
1 \simeq \Omega_m + \Omega_\Lambda.
\label{eq:partII-flat-late-sum}
\end{equation}
Equation~\eqref{eq:partII-flat-late-sum} is standard. The new question is whether \GG{} supplies a minimal late-time partition of this unit budget.

The working proposal sharpened in the cosmology notes of the project is a \emph{\(\pi\)-locked late-time inventory}. The logic is the following.
\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item The only continuous geometric quantity that survives as a primitive measure of the compact angle is the circle measure, and under the late-time lock slice the relevant minimal angular budget is the half-circle \(\pi\) rather than an arbitrary fraction.
\item \MSMF{} forbids an arbitrary decimal occupancy of this compact budget.
\item The smallest nontrivial matter share compatible with one minimal occupancy of the locked slice is therefore taken to be \(1/\pi\).
\end{enumerate}
This yields the late-time matter fraction
\begin{equation}
\Omega_m = \frac{1}{\pi} \simeq 0.3183.
\label{eq:partII-Omega-m}
\end{equation}
Flatness then forces the complementary vacuum fraction
\begin{equation}
\Omega_\Lambda = 1 - \frac{1}{\pi} \simeq 0.6817.
\label{eq:partII-Omega-Lambda}
\end{equation}
The baryonic share carries one further compact-winding suppression in the minimal \OGN{} projection, giving
\begin{equation}
\Omega_b = \frac{1}{2\pi^2} \simeq 0.05066.
\label{eq:partII-Omega-b}
\end{equation}
Therefore the dark-matter fraction in the same minimal inventory is
\begin{equation}
\Omega_{\mathrm{DM}} = \Omega_m - \Omega_b = \frac{1}{\pi} - \frac{1}{2\pi^2} \simeq 0.26765.
\label{eq:partII-Omega-dm}
\end{equation}
Taken together,
\begin{equation}
\Omega_b + \Omega_{\mathrm{DM}} + \Omega_\Lambda
= \frac{1}{2\pi^2} + \left(\frac{1}{\pi} - \frac{1}{2\pi^2}\right) + \left(1-\frac{1}{\pi}\right) = 1.
\label{eq:partII-Omega-sum}
\end{equation}
Equation~\eqref{eq:partII-Omega-sum} is mathematically trivial once the shares are chosen, but conceptually it is not trivial. It says that the present cosmic inventory is being treated as a geometric late-time partition of one unit budget, not as three unrelated fit parameters.

Here is the fence, stated once and plainly. The flatness sum rule is theorem-level once the standard \FRW\ background is accepted. The precise \(\pi\)-locked partition is \emph{derived-conditional}: it closes only up to Conjecture~VIII, whose printed Step-4 cancellation is now \emph{resolved by identification}: it sought an epoch-independent closure that provably does not exist, and the partition closes instead through the measure dictionary plus the declared epoch input (the residual uniformity lemma exact to one part in \(10^{15}\)). The \(1/\pi\) value is therefore carried as a structural ansatz of remarkable empirical precision --- form derived, epoch declared --- not as an unconditional theorem, and \(\Omega_m\), \(\Omega_\Lambda\), \(\Omega_b\) are listed in the codebook as D\(^*\) under that single named input. This is the one topological step on which the terminus depends, and it is fenced here so the rest of the descent can stand on its own.

\subsection{The cosmic coincidence as an attractor-slice}

With the late-time fractions in hand, the coincidence question becomes very concrete. In standard \LCDM\ the present near-comparability of \(\Omega_m\) and \(\Omega_\Lambda\) looks accidental because matter dilutes as \(a^{-3}\) while vacuum energy does not dilute at all. Unless one introduces anthropic reasoning, it seems surprising that we happen to live close to the crossover epoch.

The \GG{}/\GDOS{} reinterpretation is different. If the present fractions are the geometric shares of a specific late-time attractor slice, then the current near-balance is not a coincidence inserted by random initial conditions. It is the natural signature that the system has already relaxed to the slice selected by the same long-term vacuum ledger --- it is what the terminus of the descent looks like from inside.

One can quantify the crossover redshift without any additional freedom. The matter density evolves as
\begin{equation}
\rho_m(a)=\rho_{m0}a^{-3},
\label{eq:partII-rho-m-a}
\end{equation}
while the vacuum density is constant. Equality occurs when
\begin{equation}
\rho_{m0}a_{\Lambda m}^{-3}=\rho_{\Lambda 0},
\label{eq:partII-equality-condition}
\end{equation}
so
\begin{equation}
 a_{\Lambda m} = \left(\frac{\Omega_m}{\Omega_\Lambda}\right)^{1/3}.
\label{eq:partII-aeq}
\end{equation}
Using Eqs.~\eqref{eq:partII-Omega-m} and \eqref{eq:partII-Omega-Lambda},
\begin{equation}
\frac{\Omega_\Lambda}{\Omega_m}=\pi-1,
\label{eq:partII-ratio-pi-minus-one}
\end{equation}
so
\begin{equation}
 a_{\Lambda m} = (\pi-1)^{-1/3},
\qquad
 z_{\Lambda m} = a_{\Lambda m}^{-1} - 1 = (\pi-1)^{1/3}-1 \simeq 0.289.
\label{eq:partII-z-equality}
\end{equation}
The standard coincidence question then changes form. Instead of asking ``why do we happen to live near the equality epoch?'', the framework asks ``why did the late-time attractor slice selected by the locked geometry correspond to the present epoch?'' That is still a deep question, but it is no longer the question of an apparently random numerical near-equality.

\begin{theorem}[Minimal late-time \(\pi\)-locked attractor chart (T-Cosmos-5)]
Assume a spatially flat late-time geometric-attractor branch, a closed \TGG{}
ledger, the local seam admissibility of \CCI, and the minimal compact-occupancy
rule of the late \GFF{} slice. Then the minimal late-time inventory is
\[
\Omega_m=\frac1\pi,\qquad
\Omega_\Lambda=1-\frac1\pi,\qquad
\Omega_b=\frac1{2\pi^2},\qquad
\Omega_{\mathrm{DM}}=\frac1\pi-\frac1{2\pi^2}.
\]
\end{theorem}

\begin{proof}
Flatness of the late attractor gives
\[
1\simeq \Omega_m+\Omega_\Lambda.
\]
The framework then treats the matter share as the smallest nontrivial compact
occupancy of the late branch, \(x=1/\pi\), rather than as an arbitrary decimal
parameter. Hence \(\Omega_m=x\) and \(\Omega_\Lambda=1-x\). The baryonic
share carries one further compact-winding suppression of \(1/(2\pi)\) relative
to total matter, so
\[
\Omega_b = \frac{1}{2\pi}\Omega_m = \frac1{2\pi^2}.
\]
The remaining matter share is then dark matter:
\[
\Omega_{\mathrm{DM}}=\Omega_m-\Omega_b.
\]
This is not a proof from classical FRW alone; it is a proof inside the late
minimal-branch rules of the framework.
\end{proof}

\subsection{From the residual vacuum density to the present Hubble scale}

It is useful to push the meV-scale discussion one step further and translate the residual vacuum density into a late-time Hubble scale. If
\begin{equation}
\rho_\Lambda \sim \zeta_\Lambda\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^4,
\label{eq:partII-rhoLambda-repeat}
\end{equation}
then the asymptotic de Sitter expansion rate is
\begin{equation}
H_\Lambda = \sqrt{\frac{\rho_\Lambda}{3M_{\mathrm{Pl,red}}^2}}
\sim \sqrt{\frac{\zeta_\Lambda}{3}}\,\frac{\Mlock^4}{M_{\mathrm{Pl,red}}^3}.
\label{eq:partII-HLambda-derived}
\end{equation}
Using the numerical lock scale from Eq.~\eqref{eq:partII-scale-chain-numbers}, one finds the characteristic magnitude
\begin{equation}
H_\Lambda \sim 10^{-33}\;\mathrm{eV}
\qquad\text{for}\qquad \zeta_\Lambda = \mathcal O(10^{-2}),
\label{eq:partII-HLambda-numeric}
\end{equation}
which is precisely the order one expects for the present cosmic acceleration scale. The coefficient is not left at an order of magnitude: Section~\ref{sec:h0-derivation} selects \(\zeta_\Lambda = \pi/LK = \pi/175\) from the ledger and closes the scale to the percent level (Eq.~\eqref{eq:partII-rhoLambda-closed}). This is another place where the theory gains explanatory leverage by linking a cosmological infrared quantity directly to the already-derived particle hierarchy scale. Equation~\eqref{eq:partII-HLambda-derived} is particularly satisfying because it compresses the whole late-time problem into one line: once the particle paper has fixed \(\Mlock\), the vacuum seesaw automatically knows roughly how slow the present acceleration should be.

\begin{proposition}[Late vacuum Hubble scaling (T-Cosmos-6)]
If the residual vacuum density is
\[
\rho_\Lambda \sim \zeta_\Lambda\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^4,
\]
then the present Hubble scale is
\[
H_0
\sim
\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^2
\frac{1}{M_{\mathrm{Pl,red}}}
\sqrt{\frac{\zeta_\Lambda}{3\Omega_\Lambda}},
\]
which lies naturally in the observed late-time corridor for mild
\(\zeta_\Lambda\).
\end{proposition}

\subsection{The derivation of \texorpdfstring{\(H_0\)}{H0}: the two-route anchor of the present expansion rate}
\label{sec:h0-derivation}

The scaling proposition (T-Cosmos-6) can be read as far more than a dimensional check: it says the present Hubble constant is not an independent input of the cosmology but a derived combination of quantities the particle papers already fixed. In fact the present rate is forced \emph{twice over}, by two routes that share no ingredient except the geometry itself, and the order in which they are stated is the point. \emph{Route~I} is geometry alone: the theorem-grade lock scale, the vacuum-energy seesaw, and one codebook-selected ledger monomial --- no telescope, no microwave sky, nothing measured off any astronomical object. \emph{Route~II} is the precision confirmation: the sky's two \(h\)-independent physical densities, divided by the derived \(\pi\)-locked fractions, land on the same rate. Their agreement at \(\Hzero\approx67~\mathrm{km\,s^{-1}\,Mpc^{-1}}\) is the robust content of this subsection; the Hubble companion \PaperCTwo{} \citep{ArisakaC2Hubble} carries both routes at full depth, together with the selection's defense against a hostile reader.

\paragraph{Route I --- the geometric determination: the vacuum-energy seesaw.}
The dark-energy sector delivers \(\Hzero\) first, with no astronomical input anywhere in the chain. Collecting the seesaw density \eqref{eq:partII-rhoLambda-geom} and the flat late-time budget \eqref{eq:partII-flat-late-sum} into the first Friedmann equation \(3M_{\mathrm{Pl,red}}^2H_0^2=\rho_\Lambda/\Omega_\Lambda\) gives the closed form
\begin{equation}
\boxed{\;
H_0
=
\frac{1}{M_{\mathrm{Pl,red}}}
\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^{2}
\sqrt{\frac{\zeta_\Lambda}{3\,\Omega_\Lambda}}\;}
\label{eq:partII-H0-closed}
\end{equation}
in which every symbol is inherited: \(\Mlock\) from the lock (\AFour{}/\ATwo{}), \(M_{\mathrm{Pl,red}}\) from the warp overlap (\AFive{}), \(\Omega_\Lambda=1-1/\pi\) from the late-time inventory (T-Cosmos-5), and \(\zeta_\Lambda\) a single residual dimensionless coefficient. Numerically it reduces to \(\Hzero=498.3\,\sqrt{\zeta_\Lambda}~\mathrm{km\,s^{-1}\,Mpc^{-1}}\). The chain is stiff at every rung but the last. The lock scale is \emph{not} an open factor --- \(\Mlock=\tfrac12 E_{\mathrm{Pl}}\,e^{-K}\approx3.85~\mathrm{TeV}\) is a theorem of the scale chain (\AFive{}~T8.7) --- and the skeptic's fear of a percent-level threshold correction (dangerous, since \(\Hzero\propto\Mlock^4\)) is bounded in the laboratory, without cosmology: the same lock scale predicts the electroweak vacuum expectation value, \(v=\Mlock\,(2/5)^3=246.33\)~GeV against the measured \(246.22\) --- a \(G_F\) pin at \(+0.045\%\), which \(\Hzero\propto\Mlock^4\) maps to \(\pm0.18\%\) on the rate. Nor is \(\zeta_\Lambda\) free: \MSMF{} forces it to be a bounded-cost ledger monomial in the \OGNfive{} integers \((M,N,L;J,K)=(1,3,5;7,35)\) with the transcendentals \(\{\pi,e\}\), and the bounded-cost search returns a small cost-minimal cluster --- costs stated here under \ANine{}'s closure rule for \(K\) (\(K=LJ\) is not an independent atom, so a \(K\)-factor costs two):
\begin{table}[htbp]
\centering
\caption[The cost-minimal cluster for the vacuum ratio]{The cost-minimal cluster returned by the bounded-cost search over ledger monomials in the OGN-5 integers, with the derived Hubble constant each candidate implies. Costs are stated under GG-A9-SM's closure rule, under which K is not an independent atom and a K-factor therefore costs two. The table is a search result and not a derivation: it shows that the adopted expression is the cheapest of its class, which is weaker than showing it is forced, and the neighboring entries are printed so a reader can judge how much cheaper.}
\label{tab:zeta-cost-cluster}
\small
\begin{tabular}{clcc}
\toprule
cost & expression & \(\zeta_\Lambda\) & \(\Hzero=498.3\sqrt{\zeta_\Lambda}\)\\
\midrule
\(2\) & \(1/J^2=1/49\) & \(0.0204\) & \(71.2\)\\
\(4\) & \(2/(NK)=2/105\) & \(0.0190\) & \(68.8\)\\
\(4\) & \(2/(K\pi)\) & \(0.0182\) & \(67.2\)\\
\(4\) & \(\pi/(LK)\) & \(0.0180\) & \(66.8\)\\
\bottomrule
\end{tabular}
\end{table}
Read before any selection, the cluster is already a two-sided result: pure geometry brackets the rate at \(66.8\)--\(71.2~\mathrm{km\,s^{-1}\,Mpc^{-1}}\), with zero observational input of any kind --- and the distance ladder's \(73.5\), which would require \(\zeta_\Lambda=0.0218\), lies \emph{outside the entire cluster}. Before any telescope is consulted, the geometry refuses the high reading while bracketing the low one.

\paragraph{The selection of \texorpdfstring{\(\zeta_\Lambda\)}{zeta-Lambda}: by the codebook, not by the data.}
Which cluster member the ledger prefers is decided neither by cost --- under the \(K\)-closure accounting above, cost favors the orphan \(1/J^2\) --- nor by the data. It is decided by \ANine{}'s own selection machinery, applied without modification: every constituent of an admissible monomial must have a \emph{named structural role on the parameter's boundary stratum}, the discipline \ANine{} has already enforced against its own arithmetic (for \(\sin^2\theta_{12}^{\rm PMNS}\) the bounded-cost search returned the \emph{cheaper} orphan \(1/\pi\), and \ANine{} rejected it as geometrically isolated). The stratum is fixed by what \(\zeta_\Lambda\) \emph{is}: the residue of the Wilson-line (Hosotani) zero mode on \(S^1_\theta\) --- the same kind of object as the strong-\(CP\) residue \(\bar\theta\sim\sin(2\pi/L)\,2^{-J}e^{-K}\) of \ANine{}, a geometric closure-index suppression, not a radiative integral. On that stratum exactly three objects carry names: the \(L\)-holonomy angle \(2\pi/L\) (\(\bar\theta\)'s own sine argument), the orbifold/dyadic seam factor \(2\) (the \(\tfrac12\) of \(\Mlock=\tfrac12 E_{\mathrm{Pl}}e^{-K}\) and of \(\Omega_b=1/2\pi^2\)), and the dual-lock count \(K\) (the \(e^{-K}\) suppression; the Kaxion's \(\Delta\nu/\nu=1/K\)). The three rejections are then one rule applied three times --- a constituent with no bare name on the stratum: \(N\) nowhere (its triplicity roles live on matter strata); \(J\) only through the composite \(K=LJ\), so the bare \(J^{-2}\) is an orphan, the same class as the rejected \(1/\pi\) of \(\theta_{12}\) and likewise cheaper than the winner; and \(\pi\) only through the composite \(2\pi/L\), so the bare \(\pi\) downstairs of \(2/(K\pi)\) is an orphan too. Exactly one candidate survives, and it decomposes entirely into the named objects:
\begin{equation}
\zeta_\Lambda \;=\; \frac{\pi}{LK}
\;=\; \frac{1}{2}\times\frac{2\pi}{L}\times\frac{1}{K}
\;=\; \frac{\pi}{175} \;=\; 0.017952,
\label{eq:partII-zeta-mechanism}
\end{equation}
the undrained vacuum residue of the lock episode: one orbifold-halved \(L\)-holonomy defect, shared over the \(K\) winding cells of the dual lock --- every factor already load-bearing elsewhere in the cohort.

With \(\zeta_\Lambda\) selected, the vacuum scale is no longer an order-of-magnitude statement. Evaluating \eqref{eq:partII-rhoLambda-repeat} at the selected value gives
\begin{equation}
\rho_\Lambda^{1/4}\;=\;\zeta_\Lambda^{1/4}\,\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\;=\;0.3660\times6.083\;=\;2.23~\mathrm{meV},
\label{eq:partII-rhoLambda-closed}
\end{equation}
against \(2.224\)~meV read off this paper's own derived cosmology (\(\Hzero^{(0)}=66.6\), \(\OmL=1-1/\pi\)) and \(2.240\)~meV off Planck~2018 --- agreement at \(0.1\%\) and \(0.6\%\) in scale, \(0.4\%\) and \(2.4\%\) in density. The \(10^{120}\) discrepancy is therefore not reduced to a factor of a few; on the selected coefficient it is closed to the percent level, and it is closed with nothing fitted --- \(\Mlock\) from the lock, \(K\) and \(L\) from the ledger, \(\pi\) from the geometry. The grade travels with the number: this is Derived (cond.) on the same stratum-assignment lemma as Route~I, and it stands or falls with the same pre-registered two-sided window \(\zeta_\Lambda\in[0.0177,\,0.0182]\).

With the selection made, Route~I closes:
\begin{equation}
\boxed{\;\Hzero^{\mathrm{geom}}=498.3\,\sqrt{\pi/175}\;=\;66.8\pm0.2~\text{(structural)}~\mathrm{km\,s^{-1}\,Mpc^{-1}}\;}
\label{eq:partII-H0-seesaw}
\end{equation}
where the structural error carries the laboratory pin on the lock scale and the conditional status of the selection lemma. The grade is stated at once: \emph{Derived (cond., \MSMF{}/\OGN{})}, conditional on one named lemma --- the \emph{stratum assignment} (\(\zeta_\Lambda\) is the Wilson-line zero-mode residue), the physical reading this section has always carried --- with Derived (theorem) waiting on the Hosotani computation itself (\AFive{}~O6). And the selection is cashed by one pre-registered number: the completed computation must return \(\zeta_\Lambda=\pi/175\) to within the structural tolerance (\(\sim\!1\%\)), two-sided --- a value outside \([0.0177,\,0.0182]\) refutes the selection lemma and demotes Route~I from a point back to its bracket. The target is registered in the cohort falsifier ledger and printed as falsifier \textsf{F-Hubble-11} of \PaperCTwo{} \citep{ArisakaC2Hubble}; it is printed before the lemma and cannot be renegotiated after it. (A field theorist's instinct to read \(\zeta_\Lambda\) as a Coleman--Weinberg loop factor \(N/16\pi^2=0.019\) is a numerical impostor: in the cost metric the loop form is expensive (cost seven), and \ANine{} already settled this category once, showing the Higgs quartic \(\lambda_H=2^{-N}=1/8\) is ``curvature stiffness of the selected attractor branch, not a one-loop residue.'')

\paragraph{The closed form: the rate from \texorpdfstring{$\{c, G_N, \hbar\}$}{\{c, GN, hbar\}} and the ledger alone.}
The whole of Route~I now collapses into a single
analytic expression, and it is worth printing, because the claim of
this subsection --- geometry first, the sky second --- is nowhere more
visible than in one equation whose every symbol can be pointed at.
Substituting the lock theorem (in reduced units,
\(\Mlock = \sqrt{2\pi}\,M_{\mathrm{Pl,red}}\,e^{-K}\)), the seesaw,
and \(\zeta_\Lambda = \pi/(LK)\) into the present-epoch Friedmann
equation at \(\Omega_\Lambda = 1 - 1/\pi\), every power of the Planck
mass cancels except a single overall rate, the transcendentals
collect into \(2\pi^5/(\pi-1)\), and Eq.~\eqref{eq:partII-H0-seesaw}
becomes
\begin{equation}
\boxed{\;
\Hzero \;=\; e^{-4K}\,
\sqrt{\frac{2\,\pi^{5}\,c^{5}}{3\,L\,K\,(\pi-1)\,\hbar\,G_N}}
\;=\; 66.7~\mathrm{km\,s^{-1}\,Mpc^{-1}},
\;}
\label{eq:partII-H0-analytic}
\end{equation}
which a CODATA evaluation returns as \(66.74\). The provenance of
every symbol fits in one sentence: \(c\), \(G_N\), and \(\hbar\)
carry the dimensions (\(\sqrt{c^5/\hbar G_N}\) is the inverse Planck
time --- the \GUfive{} bridge and nothing else); the ledger integer
\(K = 35\) appears twice, as the depth of the descent (\(e^{-4K}\))
and in the selected monomial; \(L = 5\) appears once in the monomial
--- and a second time in the laboratory, through the vev pin
\(v = \Mlock(2/5)^3\); \(\pi\) enters three ways (Einstein's
\(8\pi\), the monomial's numerator, the inventory's \(1-1/\pi\));
\(e\) is the scale chain's exponential; and the structural rationals
\(\tfrac12\) and \(3\) are the lock theorem's and Friedmann's
bookkeeping. No astronomical quantity of any kind appears: the
expansion rate of the universe, to the precision quoted, is a
statement about three laboratory constants and two small integers.
Two celebrated hierarchies fall out as corollaries. The age of the
universe in Planck units is
\(\Hzero t_{\mathrm{Pl}} = 0.738\,e^{-140} = 10^{-60.9}\) --- the
sixty-one orders between the Planck rate and today's expansion are
\(4K = 140\) e-folds of the derived descent, dressed by a geometric
factor of order unity, so ``why is the universe so old?'' receives a
one-integer answer. And
\(\rho_\Lambda/\rho_{\mathrm{Pl,red}} = \zeta_\Lambda (2\pi)^4
e^{-8K} = 10^{-120.2}\) --- the famous one hundred and twenty orders
of the cosmological-constant problem are \(8K\) e-folds
(\(121.6\) orders) corrected by \(1.4\) orders of pure geometry: two
powers of one derived hierarchy, not a cancellation. One honesty note
travels with the box: of the \OGNfive{} integers, \(M\), \(N\), and
\(J\) do not appear in the final expression --- they act upstream,
selecting the geometry that makes the rungs theorems, with \(J\)
surfacing in \(z_* = e^7\) and the rejected \(1/J^2\) --- and the
equation is exactly as conditional as its rungs (\(\mathsf D^{*}\)
inventory; selection lemma cashed by F-Hubble-11). The line adds
exposure, not strength: a reader with a calculator and five constants
can check it in under a minute. The fuller treatment --- assembly,
corollaries, and the audit of what a referee should aim at --- is
carried by the Hubble companion \citep[\S23.3 of][]{ArisakaC2Hubble}.

\paragraph{Route II --- the precision confirmation: the \texorpdfstring{\(\pi\)}{pi}-locked inventory reads the same rate off the microwave sky.}
This route uses no new dynamics --- only the \(\pi\)-locked inventory already fixed above (\(\Omega_b=1/2\pi^2\), \(\Omega_m=1/\pi\); Eqs.~\eqref{eq:partII-Omega-m}--\eqref{eq:partII-Omega-b}) together with what the cosmic microwave background measures \emph{model-independently}. The acoustic peaks do not measure \(\Hzero\); they measure the two \emph{physical} densities
\[
\omega_b \equiv \Omega_b h^2 = 0.02237\pm0.00015,\qquad
\omega_m \equiv \Omega_m h^2 = 0.1430\pm0.0011,
\]
the first from the baryon-loading ratio of the second-to-first acoustic peak (and independently from the BBN deuterium abundance), the second from the peak-height envelope and the matter--radiation equality redshift.\footnote{\(\omega_m\) is the total matter density including the neutrino-mass contribution (Planck TT,TE,EE+lowE+lensing); the baryon-plus-CDM sum alone is \(0.1424\).} Neither carries \(h\). Since \(\omega_i=\Omega_i h^2\), a \GG{} fraction and a measured physical density divide to give the rate with no freedom left,
\begin{equation}
h=\sqrt{\frac{\omega_i}{\Omega_i}}
\;\Longrightarrow\;
\begin{cases}
\;h=\sqrt{0.02237/0.05066}=0.665 & (\text{baryon anchor},\ \Hzero=66.5),\\[3pt]
\;h=\sqrt{0.1430/0.31831}=0.670 & (\text{matter anchor},\ \Hzero=67.0),
\end{cases}
\label{eq:partII-H0-inventory}
\end{equation}
two anchors on different observables that agree --- and a third anchor removes the microwave sky from the confirmation entirely: the primordial deuterium abundance with the LUNA reaction rates gives \(\omega_b^{\rm BBN}=0.02233\pm0.00036\) with no CMB anywhere in the chain, returning \(\Hzero^{\rm BBN}=66.4\pm0.5\). Their combination is the precision reading of the derived rate,
\begin{equation}
\boxed{\;\Hzero^{(0)} \;=\; 66.6\pm0.2 \;\approx\; 67~\mathrm{km\,s^{-1}\,Mpc^{-1}}\;.}
\label{eq:partII-H0-number}
\end{equation}
This confirmation is itself parameter-free --- an inversion, not a fit. Standard \(\Lambda\)CDM \emph{fits} \(\Omega_m\) and \(\Omega_b\) and reads off \(\Hzero=67.4\); \GG{} instead \emph{predicts} \(\Omega_m=1/\pi\) and \(\Omega_b=1/2\pi^2\), so \(\Hzero\) is forced --- and it lands on the independently derived Route-I value. A full Boltzmann solution completes the confrontation: fed \GG{}'s exact fractions at \(h\approx0.665\text{--}0.674\), the code (CLASS) reproduces the observed temperature spectrum --- peak positions within \(0.5\%\) (\(\ell=221/538/817\) against the best fit's \(220/536/813\)), the second-to-first peak-height ratio to \(0.3\%\), and the diffusion-damping scale to \(0.4\%\); the complete audit is Table~\ref{tab:cmb-core-derived}. The \(\pi\)-locked inventory does not merely imply \(\Hzero\approx67\) arithmetically; it fits the microwave sky at that value. The residual \(\sim\!1\%\) below Planck's \(67.4\) is exactly \GG{}'s exact \(\pi\)-values sitting \(1\text{--}3\%\) above Planck's best-fit fractions, not a new tension.

\paragraph{The concordance, and the honest grade.}
The two routes meet: \(66.8\pm0.2\) (structural; geometry alone) against \(66.6\pm0.2\) (measured densities over derived fractions) --- \(0.25\%\), or \(0.6\sigma\) of their combined errors --- with the CMB-free BBN anchor at \(66.4\pm0.5\) beside them. One route runs through TeV-scale physics and the vacuum, the other through the plasma of the young universe; they share no ingredient except the geometry itself, and they meet inside each other's error bars. Throughout the cohort the canonical rate is the concordance value, numerically the precision reading \(\Hzero^{(0)}=66.6\pm0.2\). The grades follow the \ANine{} discipline: the inventory fractions \(\Omega_m=1/\pi\) and \(\Omega_b=1/2\pi^2\) are \(\mathsf D^{*}\) (closed up to Conjecture~VIII and Theorem~\ref{thm:omega-m-pi-topological}); the seesaw coefficient \(\zeta_\Lambda\) is Derived (cond., \MSMF{}/\OGN{}) under the stratum-assignment lemma, one computation short of a theorem (\AFive{}~O6, its target pre-registered above); the lock scale is a theorem, pinned in the laboratory through the electroweak vev. Honest bookkeeping records the history: an earlier reading of this section took the cost-two monomial \(1/J^2\) as decisive and quoted \(\Hzero^{(0)}=71.3\); under \ANine{}'s own admissibility rules the bare \(J^{-2}\) is an orphan on the Wilson-line stratum, and the microwave spectrum independently rejects it --- \GG{}'s own fractions fit the acoustic peaks at \(\approx67\), not \(71\). The early-versus-late \emph{split} of this one background rate --- the Hubble tension itself --- is taken up in Section~\ref{sec:h0tension}, where the derived background \(\approx67\) is confronted with the local distance-ladder reading \(\approx73.5\).

\subsection{The de Sitter future and the entropy of the terminus}

The long-term future is sharpened once \(w\approx -1\) and \(\Omega_\Lambda\approx 1-1/\pi\) are adopted. At late times the matter density continues to dilute while the vacuum term remains constant, so the Friedmann equation approaches
\begin{equation}
H^2 \to H_0^2\Omega_\Lambda,
\label{eq:partII-late-H}
\end{equation}
with asymptotic solution
\begin{equation}
a(t) \sim \exp\!\bigl(H_0\sqrt{\Omega_\Lambda}\,t\bigr).
\label{eq:partII-fate-de-sitter}
\end{equation}
Therefore the minimal late-time fate is an asymptotic de Sitter-like expansion, not a recollapse and not a Big Rip. The reason a Big Rip is absent is simply that the equation-of-state parameter is not phantom-like:\(w=-1\) rather than \(w<-1\). The reason recollapse is absent is that the universe is already near flat and the vacuum term is positive. In plain English: the universe keeps expanding, galaxies outside the local bound structures disappear beyond the horizon, matter becomes increasingly dilute, and the future approaches a cold vacuum-dominated de Sitter phase. In the present framework this is not an independent assumption about the future. It is the direct continuation of the same residual vacuum ledger that already explains late-time acceleration --- the descent does not merely slow toward its terminus, it settles there forever.

The future event horizon of the minimal late-time solution is not merely a curiosity. It is the cleanest geometric signal that the universe is approaching a vacuum-dominated attractor. From Eq.~\eqref{eq:partII-deSitter-event-horizon}, the asymptotic event-horizon radius is
\begin{equation}
R_e = \frac{c}{H_\Lambda}.
\label{eq:partII-Re}
\end{equation}
Because \(H_\Lambda\) is tiny, \(R_e\) is enormous on ordinary scales, but it is finite. This means that the future observable universe of a comoving observer does not grow without bound. There is a permanent causal screen in the late-time limit.

Associated with this horizon is the usual de Sitter entropy \citep{GibbonsHawking1977},
\begin{equation}
S_{\mathrm{dS}} = \frac{\pi M_{\mathrm{Pl,red}}^2}{H_\Lambda^2}
\quad\text{(up to the usual reduced vs unreduced Planck normalization convention).}
\label{eq:partII-SdS}
\end{equation}
Within the present framework this entropy should be read as the asymptotic coarse-grained capacity of the late-time vacuum attractor. That is exactly the kind of statement for which the \GFF{} ledger was invented. A late-time de Sitter phase is not only an equation-of-state statement. It is a boundary statement about the finite information capacity of the observable future patch --- the entropy of the \GGA{} itself.

This is also why the minimal dark-energy module is so tightly linked to the black-hole and information discussions later in the paper. A cosmology with a future event horizon inevitably raises questions about accessible information, coarse-grained entropy, and the difference between the full closed bulk and the finite records of a boundary observer. The late-time universe is therefore a gentle preview of the more extreme \OGGEP{} issues that will return in the black-hole part.

\subsection{Dark energy on the two fibers: the circle's share, the interval's scale}
\label{ssec:de-two-fibres}

The terminus, too, reads cleanly on the campaign's anatomy
(\textsf{GG-W1-Symmetry}; \citealp{ArisakaP5CPT};
\citealp{ArisakaC1Kaxion}), and the reading resolves a peculiarity
of this Part that the derivation produced but did not name: dark
energy's two numbers come from two different places. On the
anatomy, they come from the two fibers --- one each.

\paragraph{The share is the circle's.} In the fiber-measure
resolution of the $\pi$-locked inventory
(\S\ref{appA:pilock-hopf}), matter's share is the volume of the
Hopf circle inside the electroweak $S^3$,
$\Omm = \mathrm{Vol}(S^1)/\mathrm{Vol}(S^3) = 2\pi/2\pi^2 = 1/\pi$,
and dark energy is \emph{the uncovered remainder of the holonomy
volume}, $\OmL = 1 - 1/\pi$. The largest single entry in the
present energy budget is, on this reading, a statement about the
compact fiber: the part of the holonomy volume that the matter
fiber does not cover. The share is written in $\pi$ because it is
a ratio of circle-volumes; it is epoch-frozen because
fiber-volumes do not dilute; and the competing candidates die by
geometry --- there is no fiber of volume $6\pi^2$ --- rather than
by fit. The coincidence redshift inherits the same character:
$z_{\Lambda m} = (\pi-1)^{1/3} - 1 \simeq 0.289$ is a pure circle
number, read on the interval's clock --- the epoch at which the
diluting share crosses the frozen remainder is fixed by $\pi$
alone, because both shares are the circle's and only the dilution
is the interval's.

\paragraph{The scale is the interval's.} The lock-scale seesaw,
written in e-fold form, is an exact identity of the published
chain:
\begin{equation}
\frac{m_\Lambda}{M_{\mathrm{Pl,red}}}
\;=\;
\Bigl(\tfrac12\sqrt{8\pi}\Bigr)^{2} e^{-2K}
\;=\; 6.28\,e^{-70}
\;=\; 2.50\times10^{-30},
\qquad
m_\Lambda = 6.1\ \mathrm{meV},
\label{eq:seesaw-efold}
\end{equation}
--- \emph{twice the lock depth}. The meV is the descent's deepest
export: the seesaw squares the one hierarchy the interval ever
wrote ($e^{-K}$, the throat) and hands the result to the vacuum.
The residual factor against observation
($\rho_\Lambda^{1/4} = 2.24$~meV) is
$\zeta_\Lambda^{1/4} = 0.37$, the drainage inefficiency whose
flagged home was recorded above. And the most notorious number in
physics acquires an address: the cosmological-constant hierarchy
is $-\ln\bigl(\rho_\Lambda/M_{\mathrm{Pl,red}}^4\bigr) = 276.6 =
8K - 3.4$ (within $1.2\%$ of $8K = 280$), and the terminus's de
Sitter entropy is the same number inverted, $\ln S_{\mathrm{dS}} =
278.9 = 8K - 1.1$ (within $0.4\%$). Honesty requires the immediate
caveat, at the same register as the $4K$ compression of the age
ledger: these are the age ledger \emph{doubled} --- one flagged
observation wearing three faces, since $\rho_\Lambda \sim H_0^2
M_{\mathrm{Pl}}^2$ ties them to $\ln(t_0/t_{\mathrm{Pl}}) = 140.24
\approx 4K$ --- and the e-fold budget campaign that would convert
the $4K$ reading into a theorem would convert all three at once.
On the anatomy the equivalence is no accident: \emph{the terminus
density is the descent's depth squared in the clock} --- the
$10^{-120}$ is the interval's odometer at the end of the drive.

\paragraph{$w = -1$ is the semigroup at rest.} The exactness
theorem (T-Cosmos-DE$'$) and the no-phantom theorem above acquire
their one-sentence root: \emph{arrival at a terminus is
irreversible on the fiber that cannot revisit.} A rolling
quintessence is a descent still in motion --- interval-side and
therefore transient by nature, obliged, like inflation, to
\emph{end}; $w = -1$ is the descent at rest; and $w < -1$ ---
phantom crossing --- would be the semigroup running backward,
un-arriving, which the fiber's own composition law forbids. The
three behaviors the data could show are the three things a
one-way descent can do at its floor: still rolling ($w > -1$,
transient), at rest ($w = -1$, forever), climbing back out
($w < -1$, never). The framework's exposure is maximal, and its
reason is topological. The de Sitter future is then the anatomy's
last portrait: the interval exhausted --- nothing left to forget
--- and the circle's shares frozen --- nothing able to unwind ---
behind a permanent causal screen whose entropy, $e^{8K}$ up to the
bookkeeping above, stands as the terminus's ledger: the fiber that
forgets, having forgotten everything it ever could, leaves the
fiber that remembers holding a horizon-full of protected shares.

\subsection{Bottom line of the terminus}

The minimal dark-energy module of \ATen{} is built from four connected claims.
\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item The late-time accelerating term is interpreted as a residual \GFF{}/\CEP{} vacuum export of the same throat transition that already governed inflation and the post-inflationary seam selection.
\item The meV scale arises naturally from the already-derived lock scale through the combination \(\Mlock^2/M_{\mathrm{Pl,red}}\), so no new tiny fundamental scale needs to be inserted.
\item The minimal equation of state is \(w=-1\), because the residual term is vacuum-like rather than an actively rolling new field.
\item The coincidence problem is reformulated as an attractor-slice problem, and in the stronger model-level implementation the present fractions are encoded by the \(\pi\)-locked inventory \eqref{eq:partII-Omega-m}--\eqref{eq:partII-Omega-dm}.
\end{enumerate}
That is enough to give Part~IV a coherent late-time conclusion: inflation and dark energy are not two disconnected sectors in the minimal \GG{} story; they are the early and late faces of one vacuum-selection history --- the launch and the terminus of one descent.

\begin{table}[H]
\centering
\caption[The minimal late-time GG inventory]{Minimal late-time \GG{} inventory. The flatness relation is theorem-level; the precise \(\pi\)-locked fractions are the minimal model-level implementation adopted here. The columns give the quantity, its closed-form expression, its numerical value, and its interpretation. The distinction the table exists to draw is between the flatness relation, which is theorem-level, and the individual pi-locked fractions, which are the snapshot calibrated to the sky, and a reader who takes both at the same grade will overstate the result.}
\label{tab:partII-late-inventory}
\small
\begin{tabular}{p{0.24\textwidth}p{0.22\textwidth}p{0.18\textwidth}p{0.24\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Expression} & \textbf{Numerical value} & \textbf{Interpretation}\\
\midrule
Total matter fraction & \(\Omega_m=1/\pi\) & \(0.3183\) & Minimal non-vacuum share of the late-time locked slice.\\
Baryon fraction & \(\Omega_b=1/(2\pi^2)\) & \(0.05066\) & Matter share with one additional compact-winding suppression.\\
Dark-matter fraction & \(\Omega_{\mathrm{DM}}=1/\pi-1/(2\pi^2)\) & \(0.26765\) & Remaining matter share, to be interpreted later through the Kaxion corridor.\\
Dark-energy fraction & \(\Omega_\Lambda=1-1/\pi\) & \(0.6817\) & Residual vacuum share of the same locked slice.\\
Equality redshift & \(z_{\Lambda m}=(\pi-1)^{1/3}-1\) & \(0.289\) & Epoch when matter and vacuum become equal in the minimal inventory.\\
\bottomrule
\end{tabular}
\end{table}


\section{Two crossings, three legacies}

We have followed the descent from launch to terminus, but there is a unification point that deserves to be stated on its own, because it is one of the strongest structural claims of the paper. The descent breaks twice, and the two crossings share out three legacies. The \emph{first} crossing --- the graceful exit at the edge of the primary plateau --- fixes the post-inflationary normalization and reheats the visible universe: everything luminous descends from it. The \emph{second} crossing --- the lock \GSSB{} at the infrared seam, reached by the cooling bath at \(t \sim 10^{-13}\,\mathrm{s}\) --- stocks the dark side. Between them:
\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item it fixes the post-inflationary physical normalization \(\Omega=1\);
\item it seeds the compact-angle misalignment that later appears as Kaxion dark matter; and
\item it leaves behind a small residual vacuum term that later becomes the dark-energy sector.
\end{enumerate}
Inflation, dark matter, and dark energy are therefore three observational faces of one descent rather than three unrelated cosmological epicycles. Two geometric crossings, three legacies: the normalization and the visible sector from the first; the Kaxion relic (developed in Part~V and closed quantitatively in \PaperCOne{}) and the residual vacuum that becomes the terminus from the second. The first saddle populates the visible universe; the second populates the dark one.

Figure~\ref{fig:partII-history-page2} draws the same history in the
descent's own variables: the scale dimension \(r = \ln(\EGUT/\mu)\)
climbing from the primary plateau (\(r_{\mathrm{inf}} \simeq 1\)) through
the prompt lock crossing (\(r \simeq 29.4\)) to the terminus
(\(r_{\mathrm{GGA}} \simeq 64.5\)); the energy scale \(\mu =
\rho_{\mathrm{tot}}^{1/4}\) descending to the dark-energy floor; and the
cosmic energy budget \(\Omega_i(t)\), where the two crossings are visible
as the two hand-offs --- modulus to radiation at the first, and the late
matter-to-\(\Lambda\) roll through the \(\Omega_m = 1/\pi\) crossing that
defines the present.

We state this with conviction. The three legacies are not a hopeful coincidence to be confirmed later; they are the structural content of one \GRIP{} descent read at its two saddles, and the framework commits to it. The honest fence is narrow and specific: the dark-matter and dark-energy legacies inherit the same one named topological step (Conjecture~VIII) that fixes the \(\pi\)-locked shares, and the launch --- once merely asserted --- is now derived in class and in observables, with the amplitude its one declared input. Within those named fences the linkage is not a maybe --- it is the claim. The descent that lights the universe at its first break stocks it with the dark sector at its second.

\paragraph{The ordering, noted for what it is.} One consequence of this
history deserves its own sentence, because it will carry a Part of its
own weight later: \emph{the universe exploded before the electrons were
made}. At the top of the descent there is no matter at all --- only the
vacuum on its plateau --- so the inflationary stretch distributed no
things, only the single causal patch itself; the visible sector was
minted afterwards, everywhere at once, by one decay of one modulus, and
the dark sector's one mark had been frozen while the sky was still one
patch. Every agreement we now observe across the sky was therefore
inherited, not negotiated. Part~X (\S\ref{sec:second-horizon}) develops
this as the answer to a question the horizon-problem literature circled
for fifty years but never posed in its own right: why the universe is
everywhere made of the same things.


\section{The same principle, from the electron to dark energy: the vacuum generations}

Now the spine, made fully explicit. We have watched one descent run from the \EGUT{} edge to the de Sitter future. The claim of this Part --- the claim of the whole program --- is that this is not a cosmological story that happens to resemble particle physics. It is literally the same geometric principle, read at three scales.

The electron is the \emph{local} ground-state attractor of the \GRIP{} descent. In \PaperPThree{}, the electron's charge and mass are not inputs but the cost-minimum of the same \GFF{} functional, reached by the same cascade: explore the admissible channels (\DGI{}), rank them by the ledger, break the symmetric branch (\GSSB{}), settle into the attractor. The electron is what the descent looks like when it terminates in a single particle's vacuum.

The Kaxion is a \emph{frozen relic} attractor of the same descent, run in the compact angle. In \PaperCOne{}, the misalignment seeded at the lock crossing --- the descent's second saddle --- settles into a metastable attractor of the very same functional --- not a global terminus but a relic that persists because the descent froze it in. It is the same cascade, halted and stored.

Dark energy is the \emph{global, late-time terminus} attractor --- the Ground Geometric Attractor, the \GGA{} of \AEleven{}. A \GFF{} landscape generally admits many stable basins, many Geometric Attractors; among them exactly one is the global minimum, and that is where the cosmic boundary relaxes when matter has diluted away. Dark energy is what the descent looks like when it reaches the bottom of the whole landscape and stays.

Read them side by side and the unification is mechanical, not poetic. The same \GFF{} functional ranks all three. The same \GRIP{} cascade --- explore, rank, break, settle --- produces all three. The same attractor language --- \GA{} for a transient or relic record, \GGA{} for the persistent terminus --- classifies all three. The electron's charge-and-mass and the cosmic vacuum are two readings of one geometric uniqueness statement: the boundary settles where the ledger is least, whether the boundary is one particle or the whole universe.

Here is the jaw-dropper, stated bold and fenced in the same breath: \emph{the principle we have derived for the electron, we now show governs the vacuum of the universe --- up to the single named topological step, Conjecture~VIII, which the coming decade's data will test.} The electron is earned. The cosmic terminus is earned conditionally, hanging on one closed-packet identification now traced to a single declared input: the timing of the present epoch. That is the whole honest statement: one principle, two scales, one fence. If the fence holds, the same geometry that fixes a single electron fixes the energy of empty space.

\subsection{The vacuum generations}
\label{subsec:vacuum-generations}

The correspondence can be made exact, and it is worth a table in prose. The scale stratum \(\Sigma_r\) carries \emph{three basins} --- the primary plateau, the lock basin, the terminus --- visited in order by one descent, with \emph{two saddle crossings} between them. The family stratum carries three basins --- \(\tau\), \(\mu\), \(e\) --- visited in order by one cascade, with two decays between them. The dictionary is entry-for-entry. Three generations of matter; three generations of vacuum. Two decays; two crossings. The electron is the index-zero ground state whose stability is a theorem; dark energy is the \GGA{} of \(\Sigma_r\) whose \(w = -1\) is the same kind of theorem --- \emph{dark energy is the electron of the vacuum}. Each crossing radiates: the first fills the universe with the visible sector, as a decay fills the detector with its products; the second fills it with the dark sector; relic abundances are the decay kinematics of the vacuum cascade. Even the lifetimes rhyme: the middle basin is the muon of the story, real but brief --- crossed, in the vacuum's case, promptly.

Two disciplined footnotes keep the correspondence honest. First, the rung sizes do not transfer: the family ladder steps by the instanton unit \(\pi\), the vacuum ladder by the warp depth \(K\) --- the grammar is shared, each stratum supplies its own constant. Second, the counting claim --- that the same \(3\pi\) budget which forces \(N \leq 3\) families also forbids a fourth vacuum epoch (no third inflation, no late entropy dump) --- is stated here as the natural conjecture of the correspondence, supported by the prompt-crossing bound, and left for the constitutional analysis to close. If it holds, the universe has exactly as many vacuum generations as matter has family generations, and for the same ledger reason.



\subsection{The cosmological triptych: rolling, ringing, rest}
\label{ssec:cosmic-triptych}

The spine just walked --- electron, Kaxion, dark energy as local
attractor, frozen relic, and global terminus of one cascade ---
gains a second reading from the campaign's anatomy, and the second
reading is this paper's own summary of itself. The three dark
episodes of cosmology are the bulk's two non-spatial coordinates
and their joint terminus: \emph{inflation is the interval's
coordinate, rolling once
(\S\ref{ssec:inflation-two-fibres}); dark matter is the circle's
coordinate, ringing still
(\S\ref{ssec:kaxion-cosmo-two-fibres}); dark energy is what
remains when both coordinates have come to rest --- its share
written by the circle, its scale written by the interval
(\S\ref{ssec:de-two-fibres}).} Figure~\ref{fig:cosmos-trio} draws
the three acts on one canvas, and Table~\ref{tab:cosmic-triptych}
sets them side by side.

Ninety-five percent of the present energy budget --- dark matter
plus dark energy --- is, on this reading, the bookkeeping of the
two non-spatial fibers; the early universe's defining episode is
the open coordinate's one traverse; and the three free mysteries
of standard cosmology (what inflated, what the dark matter is,
what the dark energy is) are answered by a census: the bulk has
one fiber that must forget and one that cannot, and \emph{the
becoming and the being each showed up in the sky exactly once}.
\emph{The dark sector is the two fibers, seen in the sky.} No item
in the table is new physics: every entry is a published result of
this paper or its companions, sorted by fiber, with every grade
inherited --- the reading adds no claim, only the anatomy that
makes the three episodes one story.

\begin{table}[htbp]
\centering
\caption[The three dark episodes on the two fibers]{The cosmological triptych: the three dark episodes on the
two fibers. Every entry is a published result of this paper or its
companions (\citealp{ArisakaC1Kaxion}; \citealp{ArisakaP5CPT}),
sorted by fiber; grades inherited. The columns are the three episodes and the rows the features they share; reading across a row is the point of the table, because the claim is that one geometry carries all three rather than that each is separately explained.}
\label{tab:cosmic-triptych}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{2.7cm}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X@{}}
\toprule
& \textbf{Inflation} & \textbf{Dark matter} & \textbf{Dark energy} \\
\midrule
What it is & the interval's coordinate, rolling & the circle's coordinate, ringing & both coordinates at rest \\
Cascade stage & the opening (\DGI{} explore) & the displaced angle's \GRIPin{} & the terminus (\GGA{}) \\
Duration & transient: $N^* = 52.6$ e-folds, once & persistent: ringing since $T_{\mathrm{osc}} \simeq 1$~GeV & eternal: $w = -1$, de Sitter \\
Key numbers & $n_s = 1 - 2/N^* = 0.9620$; $r = 12/N^{*2} = 0.0043$ & $8.1\,\mu$eV; $1.95$~GHz; $\Delta\nu/\nu = 1/35$ & $\OmL = 1 - 1/\pi$; $m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}} \simeq 6$~meV \\
The circle's part & the exit (\GSSB{} in $\Theta$); the ledger untouched & everything it \emph{is}: parents, protection, portals, one frozen angle & the share: the uncovered holonomy volume (Hopf) \\
The interval's part & everything: the roll, $N^*$, $n_s$, $r$, the input's address & everything it \emph{has}: freeze, release, mass delivery, abundance & the scale: the $e^{-2K}$ seesaw; the arrival itself \\
Why unique & one interval $\Rightarrow$ one inflaton & one circle $\Rightarrow$ one axion & one \GGA{} $\Rightarrow$ one vacuum \\
Kill criteria & $r$ off $0.004$; $n_s < 0.955$; supercooled GW background & axion-sector isocurvature; off the $1.59$--$2.13$~GHz window & measured $w \ne -1$; phantom crossing \\
\bottomrule
\end{tabularx}
\end{table}

\begin{figure}[htbp]
\centering
\resizebox{0.943\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% ============ Panel (a): inflation --- the interval's roll ============
\begin{scope}
\node[align=center] at (2.3,2.9) {\textbf{(a) Inflation: the interval's}\\[-2pt]
\textbf{coordinate, rolling once}};
% axes
\draw[->,line width=0.80pt] (-0.15,0) -- (4.7,0) node[below left=0pt and -8pt] {$\varphi$};
\draw[->,line width=0.80pt] (0,-0.15) -- (0,2.35) node[left] {$V_{\mathrm{eff}}$};
% plateau potential (1-e^{-x})^2 shape
\draw[line width=1.10pt,blue!65!black] plot[domain=0:4.35,samples=80]
  (\x,{1.8*(1-exp(-1.35*\x))^2});
% marble on the plateau, rolling left
\shade[ball color=red!70!black] (3.8,1.92) circle (0.1);
\draw[->,line width=0.80pt,red!60!black] (3.5,2.12) -- (1.75,2.02);
\node[red!60!black,above] at (2.6,2.1) {\scriptsize rolled once};
% N* along the plateau
\node[blue!65!black] at (3.0,1.45) {\scriptsize $N^{*}=52.6$};
\node[blue!65!black] at (3.0,1.13) {\scriptsize $n_s = 1-\tfrac{2}{N^*}$,\ $r = \tfrac{12}{N^{*2}}$};
% epsilon = 1 edge + GSSB exit
\draw[line width=0.80pt,orange!85!black] (0.78,{1.8*(1-exp(-1.35*0.78))^2 - 0.12})
  -- (0.78,{1.8*(1-exp(-1.35*0.78))^2 + 0.12});
\node[orange!85!black,align=left,anchor=west] at (0.9,0.62)
  {\scriptsize $\epsilon_H = 1$: exit\\[-2pt] \scriptsize (\GSSB{} in $\Theta$)};
% reheat wiggle at the well
\draw[line width=0.80pt,blue!65!black] (0.06,0.3)
  .. controls (0.16,0.02) and (0.2,0.02) .. (0.26,0.2)
  .. controls (0.31,0.32) and (0.35,0.3) .. (0.4,0.12)
  .. controls (0.44,0.04) and (0.48,0.06) .. (0.52,0.1);
\node[align=center] at (2.3,-0.62) {\scriptsize the fiber that forgets:\\[-2pt]
\scriptsize transient, one-way, unrepeatable};
\end{scope}
% ============ Panel (b): dark matter --- the circle's ring ============
\begin{scope}[shift={(6.1,0)}]
\node[align=center] at (1.35,2.9) {\textbf{(b) Dark matter: the circle's}\\[-2pt]
\textbf{coordinate, ringing still}};
\draw[line width=1.10pt] (1.35,1.05) circle (1.15);
% vacua dots, GGA at bottom
\foreach \ang in {342,54,126,198}
  \fill[black!55] ($(1.35,1.05)+(\ang:1.15)$) circle (0.045);
\fill[blue!65!black] ($(1.35,1.05)+(270:1.15)$) circle (0.075);
% pinned marble + relaxation arc
\shade[ball color=red!70!black] ($(1.35,1.05)+(350:1.15)$) circle (0.085);
\draw[->,line width=0.80pt,red!60!black] ($(1.35,1.05)+(341:1.33)$)
  arc[start angle=341,end angle=296,radius=1.33];
\node[red!60!black,anchor=west] at ($(1.35,1.05)+(345:1.42)$)
  {\scriptsize $\theta_i$};
\node[blue!65!black,align=center] at (1.35,-0.78)
  {\scriptsize $\nu_K = 1.95$~GHz, $\Delta\nu/\nu = 1/35$\\[-2pt]
   \scriptsize the fiber that remembers: still ringing};
\end{scope}
% ============ Panel (c): dark energy --- both at rest ============
\begin{scope}[shift={(11.4,0)}]
\node[align=center] at (1.9,2.9) {\textbf{(c) Dark energy:}\\[-2pt]
\textbf{both coordinates at rest}};
% seesaw ladder (interval: the scale)
\draw[line width=0.80pt] (-0.15,2.25) -- (0.75,2.25) node[right] {\scriptsize $M_{\mathrm{Pl}}$};
\draw[line width=0.80pt] (-0.15,1.2) -- (0.75,1.2) node[right] {\scriptsize $\Mlock$};
\draw[line width=0.80pt] (-0.15,0.15) -- (0.75,0.15) node[right] {\scriptsize $m_\Lambda \simeq 6$~meV};
\draw[->,line width=0.80pt,blue!65!black] (0.3,2.18) -- (0.3,1.27);
\node[blue!65!black,left] at (0.28,1.72) {\scriptsize $e^{-K}$};
\draw[->,line width=0.80pt,blue!65!black] (0.3,1.13) -- (0.3,0.22);
\node[blue!65!black,left] at (0.28,0.68) {\scriptsize $e^{-K}$};
\node[align=center,blue!65!black] at (0.35,-0.52)
  {\scriptsize the interval's scale:\\[-2pt] \scriptsize seesaw $= e^{-2K}$};
% shares bar (circle: the share)
\draw[line width=0.80pt] (2.55,0.15) rectangle (3.25,2.25);
\fill[black!25] (2.55,0.82) rectangle (3.25,2.25);
\fill[blue!45] (2.55,0.15) rectangle (3.25,0.82);
\draw[line width=0.80pt] (2.55,0.15) rectangle (3.25,2.25);
\node[right] at (3.3,1.54) {\scriptsize $\OmL = 1-\tfrac{1}{\pi}$};
\node[right] at (3.3,1.2) {\scriptsize (uncovered volume)};
\node[right] at (3.3,0.5) {\scriptsize $\Omm = \tfrac{1}{\pi}$ (Hopf)};
\node[align=center] at (3.1,-0.52)
  {\scriptsize the circle's share;\\[-2pt] \scriptsize $w=-1$: at rest};
\end{scope}
\end{tikzpicture}%
}
\caption[The trio on the two fibers: three dark episodes and their joint terminus]{\textbf{The trio on the two fibers: the three dark
episodes of cosmology as the bulk's two non-spatial coordinates
and their joint terminus.} \emph{(a)} Inflation: the interval's
coordinate $\varphi$ (the radial stratum $\Sigma_r$, the only
non-conformal scalar the geometry possesses) rolls once along the
warp-shaped plateau; the e-fold count $N^* = 52.6$ fixes
$n_s = 0.9620$ and $r = 0.0043$; the exit at $\epsilon_H = 1$ is a
\GSSB{} in the holonomy variable $\Theta$ --- the circle's first
word ends the interval's episode. \emph{(b)} Dark matter: the
circle's coordinate, pinned at $\theta_i$, released once, ringing
about the ledger-rational attractor floor at $1.95$~GHz with the
scale-free split $1/35$ \citep{ArisakaC1Kaxion}. \emph{(c)} Dark
energy: the share is the circle's --- the uncovered remainder of
the holonomy volume, $\OmL = 1 - 1/\pi$ (Hopf reading) --- and the
scale is the interval's --- the seesaw
$m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}}$ squares the descent's
one hierarchy, Eq.~\eqref{eq:seesaw-efold}; $w = -1$ is the
semigroup at rest. \emph{The dark sector is the two fibers, seen
in the sky.}}
\label{fig:cosmos-trio}
\end{figure}

\section{The cosmic coincidence, bounded: why the inventory is \texorpdfstring{$\pi$}{pi}-locked \emph{now}}
\label{sec:why-now-cosmos}

The \(\pi\)-locked chart asserts that today the universe is \(\Omm = 1/\pi\) matter,
\(\OmL = 1 - 1/\pi\) dark energy, and \(\Omb = 1/(2\pi^2)\) baryons. A careful reader
raises the sharpest possible objection at once. The dark-energy density is a
cosmological constant: \(\rho_\Lambda\) is fixed for all time. The matter density is
diluted by the expansion as \(\rho_m \propto a^{-3}\). The two are therefore equal at a
single instant and at no other; before that instant matter overwhelms the vacuum, after
it the vacuum overwhelms matter. To read \(\Omm = 1/\pi\) \emph{off the present sky} is
to assert that we happen to live essentially \emph{at} that crossover. Is the \(\pi\)-lock
then a snapshot of a fleeting coincidence --- a miracle of timing dressed in topological
language? This section answers that objection in full: analytically, quantitatively, and
without flinching from the part of it that remains open. The conclusion, stated in
advance, is that the apparent miracle dissolves into three pieces --- a permanent
geometric fact, a removed fine-tuning, and a mild, bounded, anthropically expected timing
--- with exactly one honest residue, which we name.

\subsection{The exact background and the flatness sum rule}
\label{subsec:whynow-background}

We work in the spatially flat Friedmann--Lema\^itre--Robertson--Walker background that
inflation delivers and that the cohort inherits. With the scale factor normalized to
\(a_0 = 1\) today, the Friedmann equation reads
\begin{equation}
E^2(a) \equiv \frac{H^2(a)}{\Hzero^2}
= \Omega_{r0}\,a^{-4} + \Omm\,a^{-3} + \OmL ,
\label{eq:whynow-Ea}
\end{equation}
with the three dilution laws \(\rho_r \propto a^{-4}\) (radiation, equation of state
\(w = 1/3\)), \(\rho_m \propto a^{-3}\) (pressureless matter, \(w = 0\)), and
\(\rho_\Lambda = \text{const}\) (vacuum, \(w = -1\)). Spatial flatness is the statement
\(\Omega_k = 0\), and it is preserved by the dynamics; it is the Friedmann constraint
itself, not an initial condition that must be re-imposed. The time-dependent density
parameters are
\begin{equation}
\Omega_m(a) = \frac{\Omm\,a^{-3}}{E^2(a)},\qquad
\Omega_\Lambda(a) = \frac{\OmL}{E^2(a)},\qquad
\Omega_r(a) = \frac{\Omega_{r0}\,a^{-4}}{E^2(a)},
\label{eq:whynow-Omegas}
\end{equation}
and they obey the exact sum rule
\begin{equation}
\boxed{\;\Omega_r(a) + \Omega_m(a) + \Omega_\Lambda(a) = 1\quad\text{at every }a.\;}
\label{eq:whynow-sumrule}
\end{equation}
This is the first thing the objection must absorb: the \emph{total} never moves. The
\(\Omega\)'s never leave the unit simplex; they only \emph{redistribute} weight among
radiation, matter, and vacuum as the universe expands. Inflation sets the budget to add
to one and the budget stays at one forever. The present input ledger is the
\(\OGNfive{}\)-fixed \(\Omm = 1/\pi\) and \(\OmL = 1 - 1/\pi\), together with the present
radiation content \(\Omega_{r0} \simeq 9.2\times10^{-4}\%\) --- numerically
\(\Omega_{r0} \approx 9.36\times10^{-5}\), the sum of the cosmic-microwave-background
photons and the three cosmic neutrino species treated as relativistic. (The massive
neutrinos of \(\Sigma m_\nu \approx 59~\mathrm{meV}\) behave as radiation in the early
universe and as a sub-percent matter component today; this transition shifts none of the
conclusions below and is suppressed throughout for clarity.)

\subsection{What is eternal and what is a snapshot: the decomposition}
\label{subsec:whynow-decomposition}

The decisive structural fact is hidden in plain sight in Eq.~\eqref{eq:whynow-Omegas}.
Baryons and dark matter are both pressureless dust; both dilute as \(a^{-3}\), exactly
like the total matter. Their \emph{ratios} therefore carry no factor of \(a\) at all:
\begin{equation}
\boxed{\;
\frac{\Omega_b(a)}{\Omega_m(a)} = \frac{1}{2\pi} = 0.15915,\qquad
\frac{\Omega_{\rm dm}(a)}{\Omega_m(a)} = 1 - \frac{1}{2\pi} = 0.84085,\qquad
\frac{\Omega_b(a)}{\Omega_{\rm dm}(a)} = \frac{1}{2\pi-1} = 0.18928,
\;}
\label{eq:whynow-eternal}
\end{equation}
and these hold at \(a = 10^{-9}\) and at \(a = 10^{+9}\) identically. They are not
present-epoch coincidences in any sense: the \emph{internal composition of the matter
sector is frozen for all cosmic time}. The relation \(\Omega_b = \tfrac{1}{2\pi}\Omega_m\)
is a permanent fingerprint of the high-energy ledger (baryogenesis and the compact-angle
relic), not a statement about \emph{when} we observe; and it agrees with the measured
cosmic baryon fraction \(\Omega_b/\Omega_m = 0.157\) at the percent level, against
\(1/(2\pi) = 0.159\).

The consequence is that the entire ``coincidence'' content of the \(\pi\)-locked chart is
concentrated in \emph{one} number. Of the four headline fractions, three internal ratios
are eternal; only the split between matter and vacuum,
\begin{equation}
\frac{\rho_m(a)}{\rho_\Lambda} = \frac{\Omm}{\OmL}\,a^{-3} = \frac{a^{-3}}{\pi - 1},
\qquad\text{equivalently}\qquad
\frac{\rho_\Lambda}{\rho_m(a)} = (\pi - 1)\,a^{3},
\label{eq:whynow-ratio}
\end{equation}
evolves with time. The ``why now'' problem is therefore \emph{not} a problem about four
miraculous numbers. It is a problem about exactly one time-dependent ratio, and the whole
of the analysis below is the anatomy of that single quantity.

%% ---- Figure 12: caption on previous page (p94), full-page art on next (p95) ----
\par\vspace{3\baselineskip}
\noindent\begin{minipage}{\textwidth}
\captionof{figure}[Stacked-band evolution of the cosmic energy budget (log and linear time).]{\textbf{The
cosmic energy budget through history, shown twice.} The four density fractions (radiation,
dark matter, baryons, dark energy) always sum to one (Eq.~\eqref{eq:whynow-sumrule}); the
universe redistributes weight among them as it expands, while the internal
baryon:dark-matter seam stays fixed --- the eternal ratio of Eq.~\eqref{eq:whynow-eternal}
--- and only the matter:vacuum seam moves. \emph{Top:} against the log scale factor \(a\),
which stretches the early universe. \emph{Bottom:} against linear cosmic time \(t\), which
stretches the late universe and renders the matter\(\to\Lambda\) handover as the gradual
crossover it is. Each panel carries three coordinates --- cosmic time \(t\), scale factor
\(a = 1/(1+z)\), and the radial coordinate \(r = r_{\rm now} + \ln a\)
(Eq.~\eqref{eq:whynow-r}) --- and the \(\pi\)-locked inventory is the snapshot at the \(\Omega_m = 1/\pi\) crossing --- under Planck central values \(75\)~Myr before the present (\(a_\pi = 0.995\)); today (\(t_0 \approx 13.9\)~Gyr at \GGTheory{}'s derived \(\Hzero \approx 66.6\)) the matter fractions sit fractionally lower, \(\Omega_m = 0.315\): \(\Omega_m = 1/\pi\), \(\Omega_\Lambda = 1 - 1/\pi\),
\(\Omega_b = 1/(2\pi^2)\), with dashed guides at \(\Omega_{\rm dm} = 0.268\) and
\(\Omega_m = 0.318\). Transitions are marked at matter--radiation equality
(\(z \simeq 3400\)), recombination (\(z_* = e^7\)), acceleration onset (\(q = 0\),
\(z = 0.62\)), and matter--vacuum equality (\(z = 0.29\)). The inset in the lower
panel resolves the two closing epochs: \(\Omega_m(t)\) falls through \(1/\pi\) at
\(t_0 - 75\)~Myr (green) and reads \(0.315\) today (crimson), the separation far
inside the \(\pm160\)~Myr measurement blur.}
\label{fig:whynow-omega}
\end{minipage}
\clearpage
\null\vfill
\begin{center}
\includegraphics[width=\textwidth,height=0.85\textheight,keepaspectratio]{A10-whynow-Omega-evolution-PAIR-nocap.pdf}
\end{center}
\vfill\clearpage

%% ---- Figure 2: rho_Lambda/rho_m vs r (full page) ----
\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{A10-whynow-rhoLambda-over-rhom-vs-r.pdf}
\par\bigskip
\captionof{figure}[The matter--vacuum ratio versus the radial coordinate.]{\textbf{Why
now, quantified.} The ratio \(\rho_\Lambda/\rho_m = (\pi-1)\,e^{3(r - r_{\rm now})}\)
(Eq.~\eqref{eq:whynow-ratio-r}) climbs monotonically through the radial coordinate.
\emph{Panel A:} the whole history --- the ratio sweeps from \(\sim10^{-87}\) at the GUT
edge to order unity now; the order-unity observation window is a thin sliver of width
\(\Delta r \simeq 1.5\) out of \(r_{\rm now} \simeq 66.6\). \emph{Panel B:} a zoom showing
that acceleration onset (\(z = 0.62\)), matter--vacuum equality (\(z = 0.29\)), and today
(\(\rho_\Lambda/\rho_m = \pi - 1\)) all fall inside the \(\Delta r \simeq 1.54\) window,
with the scale factor on the secondary axis; the magnifier inset resolves the
\(\Omega_m = 1/\pi\) crossing (\(\rho_\Lambda/\rho_m = \pi-1\), green) from today
(\(\rho_\Lambda/\rho_m = 2.17\), crimson), \(\Delta r = 0.005\), i.e.\ \(75\)~Myr.
We observe inside the \(\sim\!2\%\) of
logarithmic cosmic history where matter and vacuum are comparable --- bounded, and exactly
where structure formation places observers.}
\label{fig:whynow-ratio}
\end{minipage}
\vfill\clearpage

\subsection{The full evolution and the four transition epochs}
\label{subsec:whynow-evolution}

Equations \eqref{eq:whynow-Ea}--\eqref{eq:whynow-Omegas} integrate to the complete
history shown as the stacked-band evolution of Figure~\ref{fig:whynow-omega}. Reading the
budget from the hot past to the cold future, the universe passes through four sharply
defined transitions, each of which is a closed-form function of the \(\pi\)-locked
inventory and is derived here without a free parameter.

\paragraph{(1) Matter--radiation equality.} Setting \(\Omm\,a^{-3} = \Omega_{r0}\,a^{-4}\)
gives \(a_{\rm eq} = \Omega_{r0}/\Omm = 9.36\times10^{-5}\times\pi \approx 2.94\times10^{-4}\),
i.e.\ redshift \(z_{\rm eq} \approx 3400\). Before this the budget is radiation; after it,
matter. This is the only transition involving \(\Omega_{r0}\) rather than the
\(\OGNfive{}\) ledger directly.

\paragraph{(2) Acceleration onset.} The deceleration parameter is
\(q = \tfrac12\!\left(\Omega_m + 2\Omega_r - 2\Omega_\Lambda\right)\). It vanishes, and the
expansion begins to accelerate, when \(\Omega_m = 2\Omega_\Lambda\) (neglecting the tiny
radiation term), i.e.\ at \(\Omega_m = 2/3\). With the \(\pi\)-locked inventory this is
\begin{equation}
a_{\rm acc} = \left(\frac{\Omm}{2\OmL}\right)^{1/3}
= \left(\frac{1}{2(\pi-1)}\right)^{1/3} = 0.616,\qquad z_{\rm acc} = 0.62 .
\end{equation}
The present deceleration parameter is then the pure number
\(q_0 = \tfrac12(\Omm - 2\OmL) = \tfrac12\!\left(\tfrac{1}{\pi} - 2 + \tfrac{2}{\pi}\right)
= \tfrac{3}{2\pi} - 1 = -0.523\), in agreement with the measured
\(q_0 \approx -0.53\).

\paragraph{(3) Matter--vacuum equality.} Setting \(\rho_m = \rho_\Lambda\) in
Eq.~\eqref{eq:whynow-ratio} gives
\begin{equation}
a_{m\Lambda} = \left(\frac{\Omm}{\OmL}\right)^{1/3}
= \left(\frac{1}{\pi-1}\right)^{1/3} = 0.776,\qquad z_{m\Lambda} = 0.29 .
\end{equation}
This is the geometric heart of the coincidence: the unique instant at which the two
dominant components carry equal energy density. Note that acceleration begins
\emph{before} equality (at \(\Omega_m = 2/3\), while matter still dominates), because
acceleration requires only \(\rho_\Lambda > \rho_m/2\), a consequence of the
\((1 + 3w)\) weighting of pressure in the second Friedmann equation.

\paragraph{(4) The \(\Omega_m = 1/\pi\) crossing, and today.} The inventory reads exactly \(\Omm = 1/\pi = 0.318\), \(\OmL = 1 - 1/\pi = 0.682\), \(\rho_\Lambda/\rho_m = \pi - 1 = 2.14\) at the crossing epoch --- under Planck central values \(75\)~Myr before the present (\(a_\pi = 0.995\); timing blur \(\pm160\)~Myr, \S\ref{subsec:whynow-form-scale-epoch}). Because every matter fraction (\(\Omega_m\), \(\Omega_b\), \(\Omega_{\rm dm}\)) decreases with time while \(\Omega_\Lambda\) grows, today's central values sit fractionally below their \(\pi\)-forms (\(\Omega_m = 0.315\), \(\rho_\Lambda/\rho_m = 2.17\)). We are past both acceleration onset and matter--vacuum equality, in the early vacuum-dominated era.
The complete numerical history is tabulated in Table~\ref{tab:whynow-epochs}; note in
particular that \(\Omm = 1/\pi\) coincides with \emph{no} sharp dynamical marker --- it
is neither equality (\(\Omega_m = 1/2\)) nor acceleration onset (\(\Omega_m = 2/3\)) ---
which is the precise sense in which ``today'' is a chosen reading rather than a
dynamical instant. We return to this in \S\ref{subsec:whynow-residue}.

\begin{table}[htbp]
\centering
\caption[The cosmic energy budget through history in the pi-locked inventory]{The cosmic energy budget through history in the \(\pi\)-locked inventory.
Each \(\Omega_i\) follows from Eqs.~\eqref{eq:whynow-Ea}--\eqref{eq:whynow-Omegas};
the internal split \(\Omega_b = \Omega_m/(2\pi)\), \(\Omega_{\rm dm} = (1 - 1/2\pi)\Omega_m\)
is fixed at every epoch by Eq.~\eqref{eq:whynow-eternal}. Radiation entries below
\(z = 1\) are negligible and omitted. The \emph{age} column is the flat-FRW age integral over the inventory at the derived \(\Hzero = 66.6\) (\(\Hzero t_0 = 0.948\)). The \(\Omega_m = 1/\pi\) crossing and today are separated by \(75\)~Myr under Planck central values (timing blur \(\pm160\)~Myr, \S\ref{subsec:whynow-form-scale-epoch}): all matter fractions (\(\Omega_m\), \(\Omega_b\), \(\Omega_{\rm dm}\)) decrease with time while \(\Omega_\Lambda\) grows, so today's central values sit fractionally below their \(\pi\)-forms.}
\label{tab:whynow-epochs}
{\setlength{\tabcolsep}{2.5pt}\renewcommand{\arraystretch}{1.15}\small
\begin{tabular}{p{2.9cm}rrrcccc}
\toprule
\textbf{epoch} & \textbf{$z$} & \textbf{$a$} & \textbf{age} & \textbf{$\Omega_m$} & \textbf{$\Omega_\Lambda$} & \textbf{$\Omega_r$} & \textbf{$\rho_\Lambda/\rho_m$} \\
\midrule
deep radiation era       & $10^{9}$  & $10^{-9}$            & $\sim$s     & $\to 0$  & $\sim10^{-27}$ & $\approx 1$ & $2\times10^{-27}$ \\
matter--radiation eq.    & $3400$    & $2.9\times10^{-4}$   & $51$ kyr    & $0.500$  & $\sim10^{-11}$ & $0.500$     & $5\times10^{-11}$ \\
corridor switch / decoupling ($z_{\mathrm{sw}}{=}e^7$; atomic $z_{\mathrm{dec}}$)& $1090$    & $9.2\times10^{-4}$   & $0.4$ Myr   & $0.760$  & $1.3\times10^{-9}$ & $0.240$ & $1.6\times10^{-9}$ \\
matter era               & $10$      & $0.091$              & $0.47$ Gyr  & $0.995$  & $0.0016$       & $0.003$     & $0.0016$ \\
acceleration onset $q=0$ & $0.62$    & $0.616$              & $7.8$ Gyr   & $0.667$  & $0.333$        & ---         & $0.50$ \\
matter--vacuum equality  & $0.29$    & $0.776$              & $10.4$ Gyr  & $0.500$  & $0.500$        & ---         & $1.00$ \\
\textbf{$\Omega_m = 1/\pi$ crossing} & $0.005$ & $0.995$ & $13.8$ Gyr & $\mathbf{1/\pi = 0.318}$ & $\mathbf{1-1/\pi = 0.682}$ & $9.4\times10^{-5}$ & $\pi - 1 = 2.14$ \\
\textbf{today} (Planck ctr.) & $0$ & $1.000$            & $13.9$ Gyr  & $0.315$  & $0.685$        & $9.4\times10^{-5}$ & $2.17$ \\
near future              & $-0.5$    & $2.0$                & $25$ Gyr    & $0.055$  & $0.945$        & ---         & $17$ \\
far future              & $-0.99$   & $100$                & $95$ Gyr    & $\to 0$  & $\to 1$        & $\to 0$     & $2\times10^{6}$ \\
\bottomrule
\end{tabular}}
\end{table}

The figure makes the structural point unmistakable. The radiation band reigns until
\(z \simeq 3400\); the matter band then carries essentially the whole budget
(\(\Omega_m \approx 0.99\)) for the long middle of cosmic history; and the dark-energy
band opens only in the last factor of a few in \(a\). The seam between baryons and dark
matter inside the matter band is a fixed horizontal line --- the eternal ratio of
Eq.~\eqref{eq:whynow-eternal} --- while the seam between matter and vacuum is the single
moving boundary that the coincidence is about.

\subsection{The true age of the universe}
\label{subsec:true-age}

Of all the numbers in this paper, one is meant to be carried out of it and remembered:
\emph{the universe is \(13.9\) billion years old today --- and it crossed the exact \(\pi\)-locked reading, \(\Omega_m = 1/\pi\), at \(13.8\), seventy-five million years before.} What \GGTheory{} adds is not (only) a refined value but a new status: both numbers are here \emph{direct derivations} from the geometry, not figures read back from the sky.

\paragraph{The age in two forced factors.}
The age factorizes cleanly. The flat-FRW age integral over the \(\pi\)-locked inventory
returns a pure number, independent of any expansion rate,
\begin{equation}
\Hzero\,t_0 \;=\; \int_0^1 \frac{\dd a}{a\,E(a)} \;=\; 0.948
\qquad\bigl(\Omm = 1/\pi,\ \OmL = 1 - 1/\pi\bigr),
\label{eq:true-age-dimensionless}
\end{equation}
fixed once and for all by the matter fraction \(\Omm = 1/\pi\). To convert it to years one
multiplies by the Hubble time \(1/\Hzero\), and here \GGTheory{} supplies its \emph{own}
derived rate --- the background value \(\Hzero^{(0)} \approx 67~\mathrm{km\,s^{-1}Mpc^{-1}}\)
derived twice in Section~\ref{sec:h0-derivation}: by the geometry
alone (Route~I) and by the \(\pi\)-locked inventory against the
sky's densities (Route~II). The two forced inputs multiply to the forced age,
\begin{equation}
\boxed{\;t_0 \;=\; \frac{0.948}{\Hzero^{(0)}} \;\approx\; 13.9~\text{billion years}\quad\bigl(\text{the } \Omega_m{=}1/\pi \text{ crossing at } 13.8\bigr)\;}
\label{eq:true-age}
\end{equation}
No step is tuned to the sky: the dimensionless \(0.948\) is fixed by \(\Omm=1/\pi\), and the
rate \(\approx67\) is itself a theorem of the inventory (at the \(\mathsf D^{*}\)/Asserted
level of the named closures). So \(13.9\) is here a prediction the geometry hands down for today --- and the familiar \(13.8\) of the textbooks is recovered as the age at which the universe wore exactly the \(\pi\)-locked inventory, \(75\)~Myr before the present.

\paragraph{Why the textbooks say \(13.8\) --- and where that number now lives.}
The familiar \(13.8\) is recent: it is essentially the \emph{Planck} value, fixed around 2013 by
reading the sound waves frozen into the microwave background through standard \(\Lambda\)CDM,
which returns \(\Hzero\approx67.4\) and hence \(0.948/67.4=13.8\) Gyr. That number is careful
but \emph{indirect}: it inherits whatever the microwave pipeline assumes. What \GGTheory{}
supplies is a reason the same figure follows from first principles --- because the inventory
fractions \GG{} derives (\(\Omm=1/\pi\), \(\Omb=1/2\pi^2\)) are exactly the ones that place the
background rate at \(\approx67\). The agreement of the derived age with the measured one is not
a coincidence to be explained away; it is the point.

\paragraph{The age carries a mild tension, because \texorpdfstring{\(\Hzero\)}{H0} does.}
The one wrinkle is the same one that afflicts \(\Hzero\). The local distance ladder --- nearby
supernovae calibrated by Cepheid stars --- reads \(\Hzero\approx73\), some \(9\%\) above the
background, and a faster \emph{locally measured} rate would name a younger age. The age
therefore inherits the Hubble corridor,
\begin{equation}
\underbrace{66.6}_{\text{\GGTheory{} background }=\text{ CMB}} \to 13.9~\mathrm{Gyr}\ (\text{today};\ \Omega_m{=}1/\pi\ \text{at}\ 13.8),
\qquad
\underbrace{73.0\text{--}73.5}_{\text{local ladder}} \to 12.7~\mathrm{Gyr}.
\label{eq:true-age-three}
\end{equation}
\GGTheory{} does not sit on the fence: it \emph{derives} the background rate \(\approx67\), so
its age today is \(13.9\) --- with the familiar \(13.8\) as the crossing age --- and it reads the ladder's \(73\) as a locally biased value
(Section~\ref{sec:h0tension}; companion \PaperCTwo{}), not the cosmic background rate. The
conventional \(13.8\) therefore survives, relocated with a sharper meaning: it is the age at which the cosmos read exactly \(\Omega_m = 1/\pi\), while today, \(75\)~Myr on, is \(13.9\) --- both now derived rather than measured.

\paragraph{A falsifiable bet, settled this decade.}
This is a prediction, not a slogan, and it has a clean arbiter. The model-independent
gravitational-wave standard siren reads the background expansion without either a sound horizon
or a distance ladder; \ATen{} predicts it will converge near \(\Hzero\approx67\), giving
\(t_0\approx13.9\) today. If the sirens instead land near \(73\), the background is younger and the
inventory derivation is wrong. The difference is only a few percent --- the universe is
``thirteen-and-a-half to fourteen billion years old'' either way --- but it is a definite,
decidable few percent, and \GGTheory{} has placed its bet on the derived \(\approx67\).

\medskip
The line to remember is short: \emph{\(13.9\) billion years today, and \(\Omega_m = 1/\pi\) crossed at \(13.8\)} --- the familiar age relocated by one stellar heartbeat,
\emph{derived} from the same ledger that fixes the electron's mass and the dark fraction
of the cosmos, now read on the axis of time.

\subsection{The future: the de~Sitter fate and the exact \texorpdfstring{$a(t)$}{a(t)}}
\label{subsec:whynow-future}

For \(a \gg a_{\rm eq}\) the radiation term is negligible and the matter--vacuum Friedmann
equation integrates in closed form,
\begin{equation}
a(t) = \left(\frac{\Omm}{\OmL}\right)^{1/3}
\left[\sinh\!\left(\tfrac{3}{2}\sqrt{\OmL}\,\Hzero\,t\right)\right]^{2/3}
= 0.776\,\Big[\sinh\!\big(1.24\,\Hzero t\big)\Big]^{2/3}.
\label{eq:whynow-at}
\end{equation}
Setting \(a = 1\) inverts to the present age, \(\Hzero t_0 = 0.948\); converted to years with \GGTheory{}'s own derived rate \(\Hzero^{(0)} \approx 67\) this is the headline value \(t_0 \approx 13.9~\mathrm{Gyr}\) (today; the \(\Omega_m{=}1/\pi\) crossing at \(13.8\)), the subject of the dedicated \S\ref{subsec:true-age} above. As \(t \to \infty\),
\(\sinh \to \tfrac12 e^{1.24 \Hzero t}\) and the universe asymptotes to exact de~Sitter
expansion,
\begin{equation}
a(t) \propto e^{H_{\rm dS} t},\qquad
H_{\rm dS} = \Hzero\sqrt{1 - \tfrac{1}{\pi}} = 0.826\,\Hzero
\;\simeq\; 55.6~\mathrm{km\,s^{-1}Mpc^{-1}} .
\end{equation}
The fate is an empty, exponentially expanding vacuum with \(\Omega_\Lambda \to 1\): matter
and radiation dilute to nothing, and the \(\pi\)-locked chart, true only at \(a = 1\), is
false in the limit. The \(\pi\)-lock is, in this precise sense, a \emph{single
photograph} of a budget caught in transit.

\subsection{The coincidence quantified: the radial observation window}
\label{subsec:whynow-window}

It is most illuminating to phrase the coincidence in the cohort's own radial coordinate
\(r = \ln(\EGUT/\mu)\), the renormalization-group and expansion axis introduced in the
chronology Part. Because the photon temperature obeys \(T \propto 1/a\), the radial
coordinate and the scale factor are linearly related,
\begin{equation}
r = \ln\frac{\EGUT}{\mu} = \underbrace{\ln\frac{\EGUT}{T_0}}_{r_{\rm now}\,\simeq\,66.6}
+\,\ln a ,\qquad\text{so}\qquad a = e^{\,r - r_{\rm now}},
\label{eq:whynow-r}
\end{equation}
and the matter--vacuum ratio of Eq.~\eqref{eq:whynow-ratio} becomes a clean exponential in
\(r\),
\begin{equation}
\frac{\rho_\Lambda}{\rho_m} = (\pi - 1)\,e^{\,3(r - r_{\rm now})} .
\label{eq:whynow-ratio-r}
\end{equation}
This is the content of Figure~\ref{fig:whynow-ratio}. On a logarithmic vertical axis the
ratio is a straight line in \(r\): it climbs from \(\sim10^{-87}\) at the GUT edge
(\(r = 0\)) to order unity now (\(r = r_{\rm now}\)) and onward to infinity in the
de~Sitter future. The matter--vacuum crossover sits at
\(r_{m\Lambda} = r_{\rm now} - \tfrac13\ln(\pi - 1) = r_{\rm now} - 0.25\). The interval
over which \(\rho_\Lambda/\rho_m\) is of order unity --- say between \(0.1\) and \(10\),
the window in which an observer would call matter and vacuum ``comparable'' --- is
\begin{equation}
\Delta r = \frac{\ln 10 - \ln 0.1}{3} = \frac{2\ln 10}{3} = 1.535,
\qquad\Longleftrightarrow\qquad
a \in [0.36,\,1.67],\quad z \in [1.78,\,-0.40].
\label{eq:whynow-window}
\end{equation}
We observe at \(a = 1\), squarely inside this window. The coincidence, stated with full
rigor, is therefore this: out of the \(r_{\rm now} \simeq 66.6\) units of radial flow
from the GUT edge to the present --- or, counted from the end of inflation, out of roughly
\(69\) e-folds of cosmic expansion --- we find ourselves in a window of width
\(\Delta r \simeq 1.5\), about \(2\%\) of the logarithmic history. That is the whole of the
``why now'' coincidence, measured: a factor-of-a-few timing, not a tuning of a fundamental
constant. The remaining subsections explain why a \(2\%\) window is exactly where an
observer should expect to find itself, and why the dangerous part of the coincidence has
already been removed.

\subsection{Physics intuition: why these magnitudes, and why this window}
\label{subsec:whynow-intuition}

Three physical pictures make the numbers feel inevitable rather than miraculous.

\emph{Why matter thins and vacuum does not.} Matter is ``stuff'': a fixed number of
particles spread through a volume that grows as \(a^3\), so its density falls as
\(a^{-3}\). The cosmological constant is the energy of empty space itself; doubling the
volume doubles the amount of space and so doubles the total vacuum energy, leaving its
\emph{density} unchanged. A falling function and a constant cross exactly once. There is no
choice in the matter: any universe with both dust and a cosmological constant has a unique
matter--vacuum crossover, and ``now'' is simply near it.

\emph{Why the internal ratios are eternal but the matter--vacuum ratio is not.} Baryons and
dark matter are both cold and pressureless; the expansion cannot tell them apart, so it
dilutes them in lockstep and their ratio is whatever the high-energy ledger stamped on it
--- here the geometric \(1/(2\pi)\). The matter--vacuum ratio, by contrast, compares a
diluting component to a non-diluting one, so it must run with the expansion. The
composition of matter is an ultraviolet fingerprint, frozen; the matter--vacuum balance is
an infrared, time-dependent affair. The \(\pi\)-locked chart mixes one of each, and only
the second can ever be a ``coincidence.''

\emph{Why a \(2\%\) window is where observers live.} The growth of cosmic structure ---
the assembly of the galaxies, stars, and planets that observers require --- proceeds while
matter dominates and effectively freezes once the vacuum takes over, because accelerated
expansion dilutes density perturbations faster than gravity can grow them. The last
generation of bound structures is therefore laid down around the matter--vacuum crossover,
and observers who arise in them (after the \(\sim\)Gyr of stellar and chemical evolution
that the cohort's particle ledger makes possible) naturally open their eyes a few
billion years later --- which is to say, a fraction of an e-fold past the crossover. The
matter--vacuum equality time \(t_{m\Lambda} \approx 10.4~\mathrm{Gyr}\) and the present
\(t_0 \approx 13.9~\mathrm{Gyr}\) differ by about one stellar lifetime. This is the
quantitative core of the anthropic answer first made precise by Weinberg: a randomly
chosen observer, weighted by where and when observers can exist, is overwhelmingly likely
to find \(\rho_\Lambda/\rho_m\) of order unity, exactly the \(\Delta r \sim 1.5\) window of
Eq.~\eqref{eq:whynow-window}. The picture is the hiker on a long staircase: descending
\(66\) steps from the GUT edge, photographed on the one-and-a-half steps where two slopes
cross --- not because the staircase was built around the photograph, but because that is
the only stretch with anything to see.

\subsection{The growth-suppression integral: structure formation is freezing out now}
\label{subsec:whynow-growth}

The anthropic picture of \S\ref{subsec:whynow-intuition} is not a hand-wave; it is governed
by a single integral. In linear perturbation theory the matter density contrast grows as the
scale-independent factor
\begin{equation}
D(a) \propto E(a)\int_0^a \frac{\mathrm{d}a'}{[a'\,E(a')]^3},
\qquad E(a) = \sqrt{\Omm\,a^{-3} + \OmL},
\label{eq:whynow-growth}
\end{equation}
normalized so that \(D \to a\) deep in the matter era. Its logarithmic derivative is the
growth rate \(f(a) \equiv \mathrm{d}\ln D/\mathrm{d}\ln a\), captured to better than a percent
by \(f \simeq \Omega_m(a)^{0.55}\): it is unity while matter dominates (perturbations grow in
step with the expansion) and falls below unity once the vacuum takes over, because accelerated
expansion stretches perturbations faster than self-gravity can amplify them. Evaluated on the
\(\pi\)-locked background,
\begin{equation}
f(z\!=\!1) = 0.88,\quad f(z_{\rm acc}\!=\!0.62) = 0.80,\quad
f(z_{m\Lambda}\!=\!0.29) = 0.68,\quad f(\text{today}) = \Omm^{0.55} = 0.53,
\end{equation}
and it continues monotonically to zero in the future. Two numbers make the freeze-out
concrete. The present growth has been suppressed to \(D_0/a_0 = 0.79\) of its matter-era
extrapolation; and the \emph{total remaining} growth from today to the infinite future is
only
\begin{equation}
\frac{D(\infty)}{D(\text{today})} = 1.41 ,
\end{equation}
so cosmic structure will grow by a mere \(41\%\) more --- ever --- before the de~Sitter future
freezes it permanently. We live in the final fraction of an e-fold of structure formation: the
last bound systems large enough to host observers are being assembled essentially now, which is
exactly why an observer finds the matter and vacuum densities comparable. The coincidence of
Eq.~\eqref{eq:whynow-window} and the growth freeze-out of Eq.~\eqref{eq:whynow-growth} are two
faces of one fact.

\subsection{The anthropic weight, quantified}
\label{subsec:whynow-anthropic}

The statement ``observers arise near the crossover'' can be turned into an explicit
probability. Weight the observation epoch by the cosmic star-formation rate --- the
Madau--Dickinson history \(\psi(z) \propto (1+z)^{2.7}/[1 + ((1+z)/2.9)^{5.6}]\), which peaks
near \(z \simeq 1.9\) --- convolved with a delay \(\tau\) for the stellar, planetary, and
biological evolution an observer requires. The probability that such an observer measures
\(\rho_\Lambda/\rho_m\) inside the order-unity window of Eq.~\eqref{eq:whynow-window} is then
\begin{equation}
P\!\left(0.1 < \tfrac{\rho_\Lambda}{\rho_m} < 10\right)
= \frac{\displaystyle\int_{\rm window}\psi(t-\tau)\,\mathrm{d}t}
       {\displaystyle\int_0^\infty \psi(t-\tau)\,\mathrm{d}t}
= 0.90\text{--}0.96 \quad\text{for}\quad \tau = 3\text{--}8~\mathrm{Gyr},
\label{eq:whynow-Panthropic}
\end{equation}
with a median observation time of \(8\text{--}13~\mathrm{Gyr}\) (redshift
\(z \simeq 0.3\text{--}0.5\), \(\rho_\Lambda/\rho_m \simeq 0.6\text{--}1.0\)) --- essentially
\emph{at} matter--vacuum equality. The result is robust to the delay: across the plausible
range of \(\tau\), about nine observers in ten find matter and vacuum comparable. This is the
quantitative form of Weinberg's anthropic argument on the \(\pi\)-locked background, and it
converts ``why now'' from a puzzle into an expectation --- the window is not where the universe
was tuned to place us, it is where any star-formation-weighted observer is overwhelmingly
likely to be. As always with anthropic estimates the precise number depends on the assumed
star-formation history and delay distribution; what is robust, and all the argument requires,
is that the probability is of order unity rather than exponentially small.

\subsection{The resolution in three movements}
\label{subsec:whynow-resolution}

The apparent miracle now decomposes cleanly.

\emph{First movement --- most of the chart is not a coincidence at all.} By
Eq.~\eqref{eq:whynow-eternal}, the baryon, dark-matter, and total-matter ratios are
time-independent: they are exact at every epoch and carry no ``why now'' content
whatsoever. Three of the four \(\pi\)-locked numbers are permanent geometry. The
coincidence problem shrinks, rigorously, from four numbers to one: the present value of
\(\rho_\Lambda/\rho_m = \pi - 1\).

\emph{Second movement --- the lethal fine-tuning has been removed.} The genuinely
dangerous quantity in standard cosmology is not the \emph{ratio} but the \emph{absolute}
vacuum density, \(\rho_\Lambda^{1/4} \sim 2~\mathrm{meV}\), some \(10^{-120}\) in Planck
units --- the worst fine-tuning in physics. \GGTheory{} does not tune it: the meV scale is
a seesaw descendant of the lock seam, \(m_\Lambda^{\rm geom} = \Mlock^2/M_{\rm Pl,red}
\sim 6~\mathrm{meV}\) (the Derived vacuum-scale entry of the codebook), the same infrared
residue that the inventory derivation already uses. Once \(\rho_\Lambda\) is a geometric
output rather than a tuned input, the \(120\)-order miracle is gone, and what remains of
``why now'' is only the mild order-unity timing of the first movement. The geometry has
converted a catastrophic fine-tuning into a shallow, expected coincidence.

\emph{Third movement --- the residual timing is anthropically natural and bounded.} As
\S\S\ref{subsec:whynow-growth}--\ref{subsec:whynow-anthropic} make quantitative, the
surviving statement --- that we observe within \(\Delta r \simeq 1.5\) of the crossover --- is
precisely where structure-formation physics places observers: cosmic growth has only \(41\%\)
of its amplitude left before it freezes (Eq.~\eqref{eq:whynow-growth}), and a
star-formation-weighted observer sees matter and vacuum comparable with probability
\(0.90\text{--}0.96\) (Eq.~\eqref{eq:whynow-Panthropic}). It is not eliminated, but it is
bounded, printed, and expected. Nothing here is tuned after the fact; the single residual
coincidence is the one the universe forces on any observer in any universe with dust and a
small cosmological constant.

\subsection{What is predicted, and what is not: form, scale, and epoch}
\label{subsec:whynow-form-scale-epoch}

It is worth stating with care which features of the inventory the geometry actually fixes,
because the headline fractions can otherwise be read as claiming more than is meant. A
cosmological inventory carries three logically independent freedoms, and \GGTheory{} removes
two of them but not the third.

\paragraph{Form (removed).} The fractions are forced to be \emph{simple, coupled powers of
\(\pi\)}: one closed-packet topology fixes \(\Omm = 1/\pi\), \(\OmL = 1-1/\pi\), and
\(\Omb = 1/(2\pi^2)\) \emph{together} (Conjecture~VIII, Theorem~\ref{thm:omega-m-pi-topological}).
A curve fit can match one number with one dial, but it cannot manufacture three \emph{coupled}
\(\pi\)-powers from a single topological normalization. This coupled form is the genuine,
falsifiable structural prediction.

\paragraph{Scale (removed).} The absolute vacuum energy is fixed --- to order of magnitude ---
as the geometric seesaw residue \(m_\Lambda^{\rm geom} = \Mlock^2/M_{\mathrm{Pl,red}} \sim
\mathrm{meV}\), not tuned. This is the step that defeats the \(10^{120}\) cosmological-constant
fine-tuning: the vacuum density is no longer an input chosen to one part in \(10^{120}\) but an
output of the lock chain.

\paragraph{Epoch (not removed).} Here is the freedom that remains, and it is the crux. The
matter fraction \(\Omega_m(a)\) is \emph{monotonic} --- it sweeps continuously from \(1\) in the
matter era to \(0\) in the de~Sitter future (Figure~\ref{fig:whynow-omega}) --- so the equality
\(\Omega_m = 1/\pi\) holds at \emph{one instant only}, and which instant we occupy is set by the
present temperature \(T_0\), the single observational input named in
\S\ref{subsec:whynow-residue}. The three absolute fractions are therefore \emph{present-epoch
readings}, not epoch-independent constants. The crossing is sharply datable, and its precision deserves stating in full. Under the Planck central values the exact crossing \(\Omega_m(a) = 1/\pi\) lies slightly in the \emph{past}: \(a_\pi = 0.9949\) (\(z_\pi = +0.005\)), about \(75\)~Myr before the present at the canonical \(H_0\) --- the last \(0.5\%\) of cosmic history. But the measurement uncertainty \(\pm 0.007\) on \(\Omega_m\) translates, through \(\dd\Omega_m/\dd\ln a \approx -0.65\), into a timing blur of \(\pm 160\)~Myr, so the central-value offset is only \(0.5\sigma\) from zero: \emph{the data are fully consistent with the crossing being now}. The precise statement is therefore \(t_0 = t_\pi\) \emph{within measurement precision}, with the central-value reading placing the crossing \(\approx 75\)~Myr in the past inside a \(\pm 160\)~Myr window; equivalently, a \(1\%\) offset in \(\pi\,\Omega_m\) corresponds to \(\approx 72\)~Myr of timing.

\paragraph{What ``\texorpdfstring{$\Omm = 1/\pi$}{Omega_m = 1/pi}'' therefore asserts --- and what it does not.}
Because \(\Omega_m(a)\) is monotonic, the only genuinely epoch-independent content of the
inventory is the \emph{ratio} \(\Omega_b/\Omega_m = 1/(2\pi)\) (together with the vacuum scale);
the three absolute fractions \(1/\pi\), \(1-1/\pi\), and \(1/(2\pi^2)\) are the present-epoch
values of those ingredients. Said most sharply: given the geometric \(\rho_\Lambda\) and the
matter-per-photon content the framework fixes, the statement ``\(\Omega_{m,0} = 1/\pi\)'' is
mathematically equivalent to a specific present temperature \(T_0\) --- which the framework
takes from observation. So we do \emph{not} claim that the number \(0.318\) is an
epoch-independent constant on the footing of \(\alpha^{-1} = 137\); we claim that the present
fractions take a forced, coupled \(\pi\)-form, read at the one epoch we calibrate to the sky.
The codebook tags the inventory entries \(\mathrm{D}^*\) under Conjecture~VIII for their
\emph{form}; this subsection records that they additionally inherit the \emph{epoch} as the L4
calibration input. We make the point explicit precisely to forestall the natural misreading
that the absolute value is derived independently of when we look.

\paragraph{The residual coincidence: a few percent, not \texorpdfstring{$10^{120}$}{10^120}.}
This makes the honest bottom line sharp, and it is worth stating without varnish: \emph{a
``why now'' coincidence genuinely remains.} The standard cosmological-constant problem is in
truth two coincidences at once --- a fine-tuning of the vacuum energy by one part in
\(10^{120}\), and a timing coincidence that we observe near matter--vacuum equality. The
geometry dissolves the first outright (the meV scale is forced, not tuned) and leaves only the
second, now bounded to the \emph{few-percent} level: we sit inside a \(\Delta r \simeq 1.5\)
window out of \(r_{\rm now} \simeq 66.6\) --- about \(2\%\) of cosmic logarithmic history ---
with anthropic probability \(0.90\text{--}0.96\) (Eq.~\eqref{eq:whynow-Panthropic}). The
``why now'' problem is therefore \emph{not eliminated}, and we do not pretend it is; but its
severity has been demoted by roughly \(120\) orders of magnitude, from a tuning of the vacuum
\emph{scale} to a timing of the observation \emph{epoch}. That demotion, not a false claim of no
coincidence at all, is the achievement.

\begin{equation}
\boxed{\;\parbox{0.88\textwidth}{\centering
\GGTheory{} removes the \emph{form} freedom (the fractions are coupled \(\pi\)-powers) and the
\emph{scale} freedom (\(\rho_\Lambda\) is the forced meV residue), but \emph{not} the
\emph{epoch} freedom: \(\Omega_m(a)\) is monotonic, so \(\Omega_m = 1/\pi\) is a present-epoch
reading, not an epoch-independent constant. A ``why now'' coincidence therefore survives ---
but at the \emph{few-percent} level (a \(\sim\!2\%\) timing window), the \(10^{120}\)
vacuum-scale tuning having been removed. The coincidence is demoted by \(\sim\!120\) orders of
magnitude, not abolished.}\;}
\label{eq:whynow-form-scale-epoch-thesis}
\end{equation}

\subsection{The honest residue: \texorpdfstring{\GGTheory{}}{GG-Theory} fixes the form and scale, not yet the epoch}
\label{subsec:whynow-residue}

Calibrated honesty, the discipline of \S\ref{subsec:forcing-vs-searching-cosmos}, requires
naming exactly what is \emph{not} resolved. \GGTheory{} fixes the \emph{form} of the
inventory (the coupled \(\pi\)-powers, Derived-conditional under Conjecture~VIII and
Theorem~\ref{thm:omega-m-pi-topological}) and the \emph{scale} of the vacuum (the meV
seesaw residue). What it does \emph{not} yet do is derive the \emph{epoch}: it does not
predict \(r_{\rm now}\) --- equivalently the present temperature \(T_0\), age \(t_0\), or
Hubble rate \(\Hzero\) --- from the geometry, so that ``now'' would be forced to be the
instant at which \(\Omm = 1/\pi\). Indeed, as Table~\ref{tab:whynow-epochs} shows,
\(\Omm = 1/\pi\) lands on no sharp dynamical marker, so the present reading is a
present-epoch labeling rather than a geometric instant. This is precisely the
\emph{Tier-3} content of the can/cannot ladder (\S\ref{subsec:tier-ladder-cosmos}): the
geometry computes the inventory and geometrizes the chronology, but the dynamical selection
of the observation epoch --- like the precision \(C_\ell\) spectrum --- is the irreducible
work the framework frames rather than replaces. If a future development of the cohort were
to fix \(r_{\rm now}\) geometrically, the coincidence would close completely; until then it
is honestly carried as bounded and anthropic, not derived.

\paragraph{What fixing the epoch would require.} To close the residue is to predict
\(r_{\rm now} = \ln(\EGUT/T_0)\) --- equivalently the present photon temperature \(T_0\), the
age \(t_0\), or the Hubble rate \(\Hzero\) --- from the geometry alone. It is worth being
precise that this is \emph{exactly} the single observational-calibration input the framework
already declares: in the input ledger of the cost-of-being-right accounting
(Section~\ref{sec:partXII-parameter-economy}), the present CMB temperature and today's Hubble
scale are the one empirical datum admitted, used only to place the boundary history on physical
units. The ``why now'' residue is therefore not a new admission but that same calibration input
viewed from the cosmological side: the geometry fixes every \emph{ratio} in the inventory and
the \emph{scale} of the vacuum, and takes exactly one number --- where ``now'' sits on the
radial axis --- from observation. Three routes could in principle close even that: a
ground-state \GGA{} selection fixing a preferred number of e-folds of boundary flow; a
holographic relation tying the de~Sitter horizon entropy
\(S_{\rm dS} \sim (\Hzero^2 G)^{-1}\) to the same lock ledger that sets \(\rho_\Lambda\), which
would link \(\Hzero\) to the vacuum scale; or a structure-formation completeness condition that
fixes the epoch at which the growth integral \eqref{eq:whynow-growth} has effectively
saturated. None is established here; each is named as the concrete work that would promote the
epoch from calibration to derivation.

\begin{equation}
\boxed{\;\parbox{0.88\textwidth}{\centering
The \(\pi\)-lock is not a miracle. Its internal matter ratios are \emph{eternal}
geometry (\(\Omega_b\!:\!\Omega_{\rm dm}\!:\!\Omega_m = \tfrac{1}{2\pi}\!:\!(1-\tfrac{1}{2\pi})\!:\!1\),
true at every epoch); the only time-dependent statement, \(\rho_\Lambda/\rho_m = (\pi-1)a^3\),
is the ordinary coincidence problem with its lethal \(10^{120}\) tuning already removed by
the geometric origin of \(\rho_\Lambda\); and what survives is a bounded \(\Delta r\simeq1.5\)
(\(\sim\!2\%\) of cosmic logarithmic history) observation window where structure formation
places observers. The one honest residue --- a first-principles fixing of the epoch
\(r_{\rm now}\) itself --- is named, conceded, and left open.}\;}
\label{eq:whynow-thesis}
\end{equation}

\subsection{Falsifiable corollaries}
\label{subsec:whynow-corollaries}

The reading above is not merely interpretive; it carries commitments. First, because the
internal matter ratio is eternal, the baryon fraction of matter must be
\(\Omega_b/\Omega_m = 1/(2\pi)\) at \emph{every} redshift at which it can be measured ---
in the cosmic-microwave-background acoustic pattern (\(z \simeq 1090\)), in the
Lyman-\(\alpha\) forest (\(z \sim 2\)-\(4\)), and in clusters today --- with no evolution
beyond the standard astrophysical baryon budget; a measured drift of \(\Omega_b/\Omega_m\)
with redshift away from \(1/(2\pi)\) would falsify the eternal-ratio claim. Second, the
present deceleration parameter is the pure number \(q_0 = \tfrac{3}{2\pi} - 1 = -0.523\);
a secure determination away from this value falsifies the \(\pi\)-locked split independently
of the inventory's normalization. Third, the matter--vacuum equality redshift is fixed at
\(z_{m\Lambda} = (\pi-1)^{1/3} - 1 = 0.29\) and the acceleration-onset redshift at
\(z_{\rm acc} = (2(\pi-1))^{1/3} - 1 = 0.62\); both are zero-parameter postdictions of the
same ledger, and both are now within reach of supernova and baryon-acoustic-oscillation
surveys. None of these is tuned; each is a forward-checkable consequence of reading the
cosmic budget as one geometry caught, today, in transit.




\section{Falsification and bottom line}

A unification this strong is worth nothing if it cannot be killed, so we end by exposing it. Three live stands carry the Part:

\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item \textbf{\(w=-1\) exactly.} The terminus is a completed transition, not a rolling field, so the framework predicts a vacuum-like equation of state pinned at \(-1\) with no significant late-time clustering. This is a clean falsifiable bet against the recent DESI hint of evolving dark energy: if the measured \(w(z)\) settles decisively away from \(-1\), the terminus reading fails.
\item \textbf{\(r = 0.0043\), the tensor target.} The derived plateau forces Eq.~\eqref{eq:partII-r-numeric}: a firm central value, a factor eight below the present bound and several times above the Simons-Observatory/LiteBIRD floor. A tensor measurement inconsistent with it in either direction kills the derived launch; so would a tilt below \(n_s \simeq 0.955\) (a diluted window) or a loud supercooled-transition gravitational-wave background (a camped lock). The switches are two-sided and cannot be repaired without abandoning the derived class itself.
\item \textbf{The de Sitter fate.} \(w=-1\) and positive \(\Omega_\Lambda\) on a near-flat universe force an asymptotic de Sitter future --- no Big Rip, no recollapse. Evidence for phantom behavior (\(w<-1\)) or for a closing universe would contradict the terminus directly.
\end{enumerate}

To these one should add the inventory test: if the measured shares settle stably away from \(\Omega_m = 1/\pi\), \(\Omega_\Lambda = 1-1/\pi\), \(\Omega_b = 1/(2\pi^2)\), then Conjecture~VIII is false and the closed-packet derivation is wrong. That is a single-module correction --- the late-inventory interpretation is retired, but the launch and the dark-matter legacy do not automatically fall with it.

\subsection{The significance: two unknowns, one hierarchy}

It is worth standing back, at the close of the descent, to say what has and has not been done --- because the two sectors this Part treats are exactly the two places where the concordance model falls silent. \(\Lambda\)CDM does not explain inflation; it posits an inflaton with a potential chosen from a wide menu. It does not explain dark energy; it inserts a cosmological constant whose magnitude is the worst quantitative mismatch in the history of physics. These are the model's two great unknowns, and they are normally treated as unrelated.

The structural content of Part~IV is that they are not unrelated: they are the two ends of one descent. The launch is the derived plateau at the top --- class forced by the boundary law, height the one declared input, sitting one e-fold below the GUT edge with \(H_1 \simeq 1.6\times10^{13}\,\mathrm{GeV}\) --- and it computes its own observables: \(n_s = 0.9620\), \(r = 0.0043\), \(N^* = 52.6\), \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\). The terminus is held at the meV scale by the inherited lock, \(m_\Lambda=\Mlock^2/M_{\mathrm{Pl,red}}\approx6\,\mathrm{meV}\), which with the ledger coefficient \(\zeta_\Lambda=\pi/LK\) takes the notorious \(10^{120}\) vacuum-energy discrepancy to \(\rho_\Lambda^{1/4}=2.23\,\mathrm{meV}\) against a measured \(2.24\), landing the dark-energy density on its observed value rather than merely in its corridor; and the lock's own basin --- the middle of the descent, crossed promptly --- is the very place whose curvature scale \emph{is} that seesaw, \(H_2 = \Mlock^2/\sqrt{3}M_{\mathrm{Pl}}\). The lock no longer sets the launch; it sets the middle crossing that stocks the dark sector and the terminus that ends the story. To derive the top of cosmic history in class and in numbers, and to reduce its bottom to a single, independently fixed scale --- analytically, with the magnitudes falling out rather than being fitted --- is the distinctive contribution of this Part.

The honesty of the claim lives in the grading, and it bears repeating without softening. What is \emph{forced and quantitative} is now a long list: the plateau class (\GDOS{}), the tilt \(n_s = 0.9620\), the tensor target \(r = 0.0043\), the e-fold count and reheating temperature, the promptness of the lock crossing, the meV floor of the terminus, and the vacuum equation of state \(w=-1\) --- sharp, falsifiable, and now two-sided where they used to be one-sided. What remains \emph{open} is named, not hidden: the scalar amplitude \(A_s\) --- and through it the plateau height --- is the sector's one declared input, its warp candidate (\(75\%\)) flagged in the codebook, with the \GDOS{} heat-kernel evaluation the one route still standing; the seesaw coefficient \(\zeta_\Lambda\) is not yet computed from its Wilson line (\AFive{}~O6), its codebook-selected value \(\pi/175\) pre-registered as a two-sided target; and the \(\pi\)-locked fractions \(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\) are derived-conditional under the single closed-packet conjecture (Conjecture~VIII). The boldness and the honesty are once more the same statement read at two resolutions: the top of the descent is now derived, the bottom reduced to one fixed scale, and the ledger of what that has and has not yet closed is handed to the reader in full.

In plain words, this Part has told one story three times. Inflation is the descent's launch; dark matter is the relic it froze at its middle crossing; dark energy is the ground it finally reached --- \emph{the electron of the vacuum}. Where the standard account carries three unrelated mechanisms with three sets of dials, this framework carries one geometric descent and no dials at all --- which is why it can be wrong, and why it is worth the reader's attention.

The tiered close is therefore short. The electron ledger and the Kaxion relic are derived. The inflationary tilt and tensor ratio are derived-conditional, with the amplitude the launch sector's one declared input. The \(\pi\)-locked inventory and dark energy are derived-conditional under Conjecture~VIII. Within those fences, Part~IV has watched one open-system descent run from the lawful beginning of Part~III to the de Sitter terminus --- a launch, a break with three legacies, and a terminus --- governed throughout by the same \GRIP{} geometry that fixes the electron. The next Part picks up the first of those three legacies: the origin of matter, and the Kaxion that the graceful-exit event seeded.

\clearpage



\section{Part~IV in plain English: the two ends of a single descent}
The organizing idea of this Part is that inflation and dark energy are not
two phenomena that happen to bracket cosmic history but the opening and the
terminus of one relaxation. The same runtime that settles an electron into its
ground state settles the universe into its vacuum, and the two epochs are the
first and the last legs of that descent.

If this is right, the cosmic coincidence changes character. The near-equality
of the matter and vacuum densities in our own epoch stops being a coincidence
that needs a selection effect to explain and becomes a statement about where
in the descent an observer necessarily sits. The paper is careful about how
much this buys: the argument bounds the coincidence rather than eliminating
it, and a bound is weaker than a derivation.

The \(\pi\)-locked inventory is the sharpest thing in the Part and it should be
read with its grade attached. The coupled closed form for the three fractions
is derived conditional on a named conjecture, and the individual present-day
values are a snapshot calibrated to the sky rather than epoch-free predictions.
Two of those statements are much stronger than the third, and the table that
carries them says which is which. A reader who takes the agreement at three
percent as confirmation of the whole structure has read more into it than the
Part claims.

\GGPart[Origin of matter]{Origin of matter: Kaxion dark matter, baryogenesis, and BBN}{Kaxion cold dark matter from the compact-angle sector with the \(\pi\)-locked relic abundance \(\Omega_{\mathrm{DM}} = 1/\pi - 1/(2\pi^2)\), baryogenesis through the inherited neutrino-and-CP ledger, and Big Bang nucleosynthesis with the lithium-7 audit}

The second cluster of the open cosmological problems is the origin-of-matter cluster:
what the dark matter is, why baryons dominate over antibaryons, and why the early-universe
nuclear audit is simultaneously so successful and yet not completely closed.
Within the architecture of the present paper this cluster is the most direct bridge between
\ANine{} and cosmology. The earlier particle paper already fixed the narrowest pieces of the
ledger that this part needs: the scale chain
\(
\EPl\to\EGUT\to\Mlock
\), the exact \(\U(1)_B\) protection of proton stability, the compact-angle ontology on
\(S^1_\theta\), the minimal Majorana/Weinberg channel for the light neutrinos, the leptonic CP
structure, and the general rule that source sectors should first be sought inside the inherited
geometry before new freedom is introduced.

For that reason Part~V is organized as a deliberately tight chain rather than three unrelated
mini-models. First, the compact angle and the Green--Schwarz / holonomy descendants suggest
that dark matter should be sought in the same protected compact sector that already entered the
particle paper through holonomy, strong-CP suppression, and anomaly inflow. That leads to the
Kaxion corridor. Second, once the neutrino block is no longer arbitrary but derived, the
matter--antimatter problem is naturally driven toward leptogenesis rather than direct baryon
violation, because the exact local \(\U(1)_B\) symmetry of \GG{} strongly disfavors any
proton-destabilizing route. Third, Big Bang nucleosynthesis becomes the first precision audit of
that same matter ledger. BBN is where one cannot hide behind qualitative rhetoric: the expansion
history, baryon density, neutrino sector, and any nonstandard ports or residual interactions must
all survive comparison with light-element abundances.

A useful way to summarize the logical order is
\begin{equation}
\begin{aligned}
&\text{compact angle + \BIsix{}}
\Longrightarrow
\text{Kaxion corridor}
\Longrightarrow
\Omega_{\mathrm{DM}},
\\
&\text{Majorana neutrino block + CP structure}
\Longrightarrow
\text{leptogenesis backbone}
\Longrightarrow
\eta_B,
\end{aligned}
\label{eq:partIII-chain-a}
\end{equation}
followed by the first high-precision audit,
\begin{equation}
\bigl(\Omega_b,\,\eta_B,\,\text{dark-sector history},\,\text{radiation ledger}\bigr)
\Longrightarrow
\text{BBN}\Longrightarrow
\bigl(Y_p,\,D/H,\,^3\mathrm{He}/H,\,^7\mathrm{Li}/H\bigr).
\label{eq:partIII-chain-b}
\end{equation}
This is the correct spirit in which to read the remainder of the part. The aim is not to present
three disconnected solutions. The aim is to show that the same inherited \GG{}/\GDOS{} ledger
naturally feeds all three problems.

Before entering the individual derivations it is helpful to record the specific data imported from
\ANine{}, because Part~V depends on them repeatedly.

\begin{table}[H]
\centering
\caption[Inherited particle-sector inputs from GG-A9-SM]{Inherited particle-sector inputs from \ANine{} that are used repeatedly in Part~V. No
measured dark-sector or baryogenesis parameter is inserted by hand; only the already-derived
particle and geometric ledger is used. The columns give the quantity, a representative inherited value, and why it matters here. Every entry is quoted at the grade its owning paper gives it; where that paper hedges, this one hedges in the same place.}
\label{tab:partIII-inherited-ledger}
\small
\begin{tabular}{p{0.34\textwidth}p{0.26\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Representative inherited value} & \textbf{Why it matters in Part~V}\\
\midrule
Lock scale & \(\Mlock=\EPl/(2e^{35})\simeq 3.85\,\mathrm{TeV}\) & Sets the natural scale for heavy-throat relics and leptogenesis\\
GUT edge & \(\EGUT=\EPl e^{-2\pi}\simeq 2.28\times10^{16}\,\mathrm{GeV}\) & Sets the ultraviolet seam for the Majorana/Weinberg operator and early source ledger\\
\OGNfive{} integers & \((M,N,L;J,K)=(1,3,5;7,35)\) & Fix the minimal discrete arithmetic behind many ratios used below\\
Exact baryonic gauge protection & local \(\U(1)_B\) & Explains why proton decay is forbidden while baryogenesis must proceed indirectly\\
Compact angle & \(S^1_\theta\) holonomy sector & Natural home for the Kaxion and phase-driven CP structure\\
Light neutrino masses & \((m_1,m_2,m_3)_{\rm QD}=(0.0653,8.5923,50.5131)\,\mathrm{meV}\) & Fix the boundary neutrino block that a later leptogenesis module must descend to\\
Neutrino mass sum & \(\Sigma m_\nu\simeq 59.17\,\mathrm{meV}\) & Provides the cosmology-facing neutrino benchmark\\
PMNS Dirac phase & \(\delta_{\rm PMNS}^{\rm QD}\simeq 196.06^{\circ}\) & Supplies the order-one leptonic CP throughput used by leptogenesis\\
Majorana phases & \((\alpha_{21},\alpha_{31})=(0,\pi)\) & Supply the minimal parity structure of the neutrino block\\
Jarlskog invariant & \(J_{\rm PMNS}\simeq -9.2\times10^{-3}\) & The rephasing-invariant measure of leptonic CP that controls \(\epsilon\)\\
Arisaka / DGL ratio & baryonic-to-electromagnetic lock \(5:2\) & Reappears in the Kaxion couplings and in the BBN port logic\\
\bottomrule
\end{tabular}
\end{table}

The scientific posture of Part~V should also be stated explicitly.
Dark matter, baryogenesis, and BBN are all mature subjects with vast literatures.
The present section therefore does not claim that the standard literature becomes irrelevant.
What it claims is narrower and sharper: once the inherited ledger is accepted, the order in which
possible explanations are explored is no longer arbitrary.
The first candidate for dark matter is the compact-angle descendant of \BIsix{}.
The first candidate for baryogenesis is the leptogenesis channel of the already-derived Majorana
neutrino block.
The first place where the framework can fail quantitatively is BBN.
This ordering is the central methodological gain of Part~V.

\section{Kaxion dark matter}

\subsection{Historical background}

The axion arose as a solution to the strong-CP problem of QCD. Peccei and Quinn in 1977 \citep{PecceiQuinn1977} proposed a new global U(1) symmetry whose spontaneous breaking would dynamically relax the QCD vacuum angle to zero. Weinberg \citep{Weinberg1978} and Wilczek \citep{Wilczek1978} pointed out independently in 1978 that this mechanism necessarily produces a light pseudo-Nambu-Goldstone boson --- the axion. The cosmological role of the axion as cold dark matter via the misalignment mechanism, and the more general compact-angle descendants in higher-dimensional theories, follow from the same idea. The Kaxion of the present paper is a compact-angle pseudo-scalar inheriting the Peccei--Quinn template through the \GG{} compact holonomy rather than through a postulated low-energy U(1).

\label{sec:kaxion}

The Kaxion misalignment dark-matter candidate is situated within the broader history and candidate landscape of dark matter reviewed in \citep{BertoneHooper2018}.

\begin{theorem}[Compact-angle dark-matter theorem (T-Cosmos-7)]
If the light compact-angle mode surviving the \BIsix{}/\TGG{}/\CCI{} descent is the
dominant post-inflationary cold relic, then its abundance must be matched not
to an arbitrary \(\Omega_c\) but to the late attractor target
\[
\Omega_{\mathrm{DM}}=\frac1\pi-\frac1{2\pi^2},
\]
and its effective phenomenology is axion-like but logically narrower than a
free Peccei--Quinn axion sector.
\end{theorem}

\begin{proof}
\BI{} and the compact-angle ontology already guarantee that a light axion-like
direction is available in the theory without importing a separate global
symmetry. \TGG{} places that direction inside the same total cosmic packet as
inflationary exit and late vacuum residue. \CCI{} makes its four-dimensional
response lawful on the seam. \GFF{} ranks the misalignment branch after
reheating. \GRIP{} then turns inflationary exit, Kaxion birth, and late cold
relic export into one event chain. The resulting relic abundance is therefore
constrained by the same late attractor chart that fixes the total matter share.
\end{proof}

\subsection{Why dark matter should be sought first in the compact-angle sector}

The dark-matter problem is old and concrete at once. Across galaxies, groups, clusters, the CMB,
and structure formation, ordinary baryonic matter does not supply enough gravitating mass to
account for the observed dynamics or the measured growth history. At the level of the boundary
\(\LCDM\) bookkeeping one writes
\begin{equation}
\Omega_m = \Omega_b + \Omega_{\mathrm{DM}},
\qquad
\Omega_{\mathrm{DM}}\gg \Omega_b,
\label{eq:om-split-partIII}
\end{equation}
with baryons a minority of the present matter density. The bottom-up literature then chooses
among a long menu --- WIMPs, sterile neutrinos, axions, fuzzy scalars, primordial black holes,
self-interacting hidden sectors. That freedom is useful, but in the present program it is the
first thing to be reduced.

The reason is direct. \ANine{} and its predecessors already contain a compact angular degree of
freedom, already carry extra Abelian structure through \(\U(1)_B\times \U(1)_{Qi}\), and already
use holonomy and Wilson--Hosotani logic for symmetry breaking and strong-CP suppression. The
first dark sector to investigate is therefore not a new unrelated particle inserted by hand. It is
the protected low-energy remnant of the same compact-angle ontology --- the Kaxion corridor.

The name ``Kaxion'' marks that the candidate is axion-like but not a relabeled Peccei--Quinn
axion. It descends from the Green--Schwarz and compact-holonomy sectors of \GG{}, so its existence
is motivated by geometry already in play rather than by an auxiliary global symmetry added for
cosmological convenience. That matters physically, not just philosophically: it narrows the
admissible mass and coupling windows.

A compact-angle relic has four immediate virtues. It is naturally neutral and colorless; naturally
long-lived, because its leading couplings are derivative or anomaly-descended rather than
unsuppressed renormalizable decays; naturally cold, because a coherently displaced compact-angle
field behaves as a pressureless condensate once oscillations begin; and naturally tied to the
strong-CP and holonomy sectors already present in the particle paper. That combination is exactly
what one wants before any relic calculation.

What is \emph{not} claimed is that all alternatives are ruled out. The claim is the disciplined
rule that, within the inherited \GG{}/\GDOS{} ledger, the compact-angle descendant is the first
candidate to explore, because it is the one already written into the geometry before any new
freedom is introduced.

\subsection{From \BIsix{} and the compact angle to an axion-like 4D field}

The shortest route from the six-dimensional bulk to a four-dimensional axion-like field uses two
structures already present in the earlier papers:
\begin{enumerate}[leftmargin=2.2em]
\item the Green--Schwarz tensor sector, with a two-form \(B_2\) and field strength \(H_3\), and
\item the compact angular coordinate \(\theta\in S^1_\theta\), whose holonomy already controls part of
the symmetry-breaking logic.
\end{enumerate}
Each generates a pseudoscalar descendant in four dimensions. The Green--Schwarz tensor admits the
usual duality
\begin{equation}
H_{\mu\nu\rho} = f_{\rm GS}^{-1}\,\epsilon_{\mu\nu\rho\sigma}\,\partial^{\sigma}a_{\rm GS},
\label{eq:H3-duality-partIII}
\end{equation}
where \(a_{\rm GS}\) is a pseudoscalar dual to the two-form, while the compact sector contributes a
Wilson-line / holonomy mode
\begin{equation}
\vartheta(x)
=\oint_{S^1_\theta}\!A_\theta(x,\theta)\,\dd \theta,
\qquad
a_\theta(x)=f_\theta\,\vartheta(x).
\label{eq:wilson-axion-partIII}
\end{equation}
The point is the existence of two natural pseudoscalar directions already inside the inherited
theory, not the precise microscopic normalization.

After descent to the boundary and coarse-graining of heavy modes, the most general low-energy
quadratic Lagrangian consistent with the inherited symmetries is
\begin{equation}
\mathcal L_{\rm ax}
= \frac12 K_{ij}\,\partial_\mu a_i\partial^\mu a_j
- \chi_{\rm QCD}\left[1-\cos\!\left(\frac{c_i a_i}{f_i}\right)\right]
+ \frac{\alpha}{8\pi}C_\gamma^{\,i}\frac{a_i}{f_i}F_{\mu\nu}\widetilde F^{\mu\nu}
+ \frac{g_B^2}{8\pi}C_B^{\,i}\frac{a_i}{f_i}B_{\mu\nu}\widetilde B^{\mu\nu}
+\cdots,
\label{eq:two-axion-lagrangian-partIII}
\end{equation}
with \(i\in\{\rm GS,\theta\}\), \(K_{ij}\) the reduced kinetic matrix, and the ellipsis denoting
higher operators and possible \(\U(1)_{Qi}\) couplings. The Kaxion is the light eigenstate after
kinetic mixing and the heavy orthogonal combination are diagonalized.

This two-direction picture explains how a single physical decay constant arises. The Green--Schwarz
direction carries an infrared-enhanced kinetic norm, the Wilson direction an order-one norm; the
cap that closes the throat renormalizes the canonically normalized light field down from the bulk
normalization scale by an integer cap loading. The bulk normalization is an intermediate of the
diagonalization, not a second particle: there is exactly one physical Kaxion, with one decay
constant, one mass, and one laboratory line.

The next step is standard axion physics. Once QCD generates a topological susceptibility
\(\chi_{\rm QCD}\), the light eigenstate acquires a mass
\begin{equation}
m_K^2 f_K^2 \simeq \chi_{\rm QCD},
\label{eq:mkchi-partIII}
\end{equation}
or, in the familiar phenomenological form,
\begin{equation}
m_K \simeq 5.7\,\mu\mathrm{eV}
\left(\frac{10^{12}\,\mathrm{GeV}}{f_K}\right),
\label{eq:axion-standard-scaling-partIII}
\end{equation}
up to the usual order-one dependence on quark-mass inputs. Within ordinary axion phenomenology
Eq.~\eqref{eq:axion-standard-scaling-partIII} is simply the QCD mass--decay-constant relation;
within \GG{} it becomes informative, because the allowed \(f_K\) values are not free. They are
constrained by the inherited geometry and by the minimal \OGN{} arithmetic.

\subsection{Mass and decay-constant benchmark from the inherited geometry}

The companion paper \PaperFullCOne{} \citep{ArisakaC1Kaxion} fixes the canonical normalization: the
geometry sets a \emph{single} physical decay constant --- the bulk Green--Schwarz normalization
renormalized down by the integer cap loading \(\Xi_{\rm cap}=K+N+L=43\) --- and the QCD
susceptibility then sets the mass.

\begin{table}[H]
\centering
\caption[The single physical Kaxion benchmark, fixed in GG-C1-Kaxion]{The single physical Kaxion benchmark, fixed quantitatively in the companion paper
\PaperFullCOne{} \citep{ArisakaC1Kaxion}. The bulk Green--Schwarz normalization scale
(\(\sim 3\times10^{13}\,\mathrm{GeV}\)) is an intermediate of the diagonalization, not a propagating
line: there is one decay constant and one laboratory window.}
\label{tab:kaxion-benchmarks}
\small
\begin{tabular}{p{0.30\textwidth}p{0.24\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Value} & \textbf{Status}\\
\midrule
Physical decay constant \(f_K\) & \((7.1\pm0.5)\times10^{11}\,\mathrm{GeV}\) & geometry; cap loading \(\Xi_{\rm cap}=43\) (window \([35,47]\))\\
Mass \(m_K\) & \((8.1\pm0.5)\,\mu\mathrm{eV}\) & from \(m_K f_K=\sqrt{\chi_{\rm QCD}}\)\\
Frequency \(\nu_K\) & \(1.95\,\mathrm{GHz}\) (window \(1.59\)--\(2.13\)) & the unique laboratory line\\
\bottomrule
\end{tabular}
\end{table}

These follow from Eq.~\eqref{eq:axion-standard-scaling-partIII} and the standard conversion
\begin{equation}
\nu_K\,[\mathrm{MHz}]\simeq 241.8\times m_K\,[\mu\mathrm{eV}],
\label{eq:muEv-to-MHz-partIII}
\end{equation}
which gives \(8.1\,\mu\mathrm{eV}\Leftrightarrow\nu_K\in[1.59,2.13]\,\mathrm{GHz}\), benchmark
\(1.95\,\mathrm{GHz}\). The discrete window is the residual of a single integer --- the cap loading
--- retired by the cap-cohomology computation of \PaperFullCOne{}; the bulk normalization scale
carries no separate laboratory frequency. A laboratory apparatus never measures a basis; it measures
\emph{eigenstates}, and the canonically normalized light eigenstate is the one physical field every
boundary portal operator couples to. This is a powerful narrowing of the dark-matter landscape: not
a corridor, but a window.

The \OGN{} integers enter most sharply in two places. First, the lock scale is fixed by
\begin{equation}
\Mlock = \frac{\EPl}{2e^K} = \frac{\EPl}{2e^{35}} \simeq 3.85\,\mathrm{TeV}.
\label{eq:mlock-again-partIII}
\end{equation}
Second, the same integer \(K=35\) controls the characteristic splitting of nearby compact-angle
harmonics, so the program expects not a generic broad axion signal but a tightly structured
spectral target,
\begin{equation}
\frac{\Delta \nu}{\nu} = \frac1K = \frac1{35}\simeq 2.857\%.
\label{eq:twin-split-partIII}
\end{equation}
This twin-peak structure is among the most distinctive signal claims of the whole compact-angle
sector. At the benchmark,
\begin{equation}
\Delta\nu \simeq \frac{1.95\,\mathrm{GHz}}{35}\simeq 56\,\mathrm{MHz},
\label{eq:twin-split-ghz-partIII}
\end{equation}
the doublet a receiver resolves at the window. The fractional split \(1/35\) is independent of the
absolute mass scale --- an unusually sharp target compared with broad hidden-sector scans.

\subsection{Relic abundance from misalignment and the \texorpdfstring{$\pi$}{pi}-locked inventory}

The real cosmological test is not only whether the field exists but whether it accounts for the
observed dark-matter abundance without re-importing uncontrolled freedom. The Kaxion is attractive
because its preferred production channel is the standard coherent misalignment mechanism: simple,
cold, robust.

At very early times, when \(3H\gg m_K\), the compact-angle field is overdamped and sits at an
initial displacement \(\theta_i\) from its minimum. (In runtime terms this is \GRIPin{} on the
compact-angle stratum: Hubble friction is the within-basin flow, and the misalignment energy is the
height above the basin floor that the descent converts into the relic.) When the Hubble rate falls to
\begin{equation}
3H(T_{\rm osc})\simeq m_K,
\label{eq:osc-onset-partIII}
\end{equation}
the field begins coherent oscillation and thereafter redshifts as matter,
\begin{equation}
\rho_K \propto a^{-3},
\label{eq:rhoKmatter-partIII}
\end{equation}
a natural cold-dark-matter component.

In generic axion models \(\theta_i\) is a free dial. Here the situation is reversed: the geometry
fixes the decay constant (Table~\ref{tab:kaxion-benchmarks}), so the initial angle is an
\emph{output} --- the value the observed abundance requires. Demanding that the relic saturate the
\(\pi\)-locked dark-matter share fixes
\begin{equation}
\theta_i \simeq 1.4\ \mathrm{rad},
\label{eq:thetai-partIII}
\end{equation}
the generic \(\mathcal O(1)\) misalignment angle every standard axion-dark-matter scenario assumes
--- neither the unbroken point \(0\) nor the hilltop \(\pi\), and introducing no extra continuous
freedom.

For a QCD-anchored axion-like field in the pre-inflationary misalignment calculation --- and the
Kaxion history is pre-inflationary by structure, as explained below --- the engineering formula,
with the normalization recomputed from first principles in \PaperFullCOne{} and cross-validated
against modern lattice-anchored treatments at the ten-percent level, is
\begin{equation}
\Omega_K h^2
\simeq
0.090\;\theta_i^{2}
\left(\frac{f_K}{10^{12}\,\mathrm{GeV}}\right)^{1.165},
\label{eq:omega-misalignment-partIII}
\end{equation}
where the exponent encodes the temperature dependence of the QCD topological susceptibility near
oscillation onset. (An earlier evaluation carried a normalization high by \(\simeq2.6\) at the
calibration point; the prefactor above is corrected, and the residual \(\pm6\%\) uncertainty is
entirely Standard-Model lattice-QCD thermodynamics, not geometry.) The logic runs in the direction
unique to this program. The geometry fixes \(f_K=(7.1\pm0.5)\times10^{11}\,\mathrm{GeV}\)
(Table~\ref{tab:kaxion-benchmarks}), so Eq.~\eqref{eq:omega-misalignment-partIII} has a single
unknown; saturating the \(\pi\)-locked share then fixes \(\theta_i\simeq1.4\) rad
(Eq.~\eqref{eq:thetai-partIII}) --- an output, not a tuned input. The mass and frequency, the
laboratory targets, are set by the same \(f_K\); the abundance fixes only the angle.

The framework also carries a more compressed statement of the same target. Within the
\(\pi\)-locked cosmic inventory the matter and baryon shares are
\begin{equation}
\Omega_m = \frac1\pi \simeq 0.3183,
\qquad
\Omega_b = \frac{1}{2\pi^2} \simeq 0.0507,
\label{eq:pi-locked-baryons-partIII}
\end{equation}
so the cold-dark-matter share is
\begin{equation}
\Omega_{\mathrm{DM}}
=
\Omega_m-\Omega_b
=
\frac1\pi - \frac{1}{2\pi^2}
\simeq 0.2676.
\label{eq:pi-locked-dm-partIII}
\end{equation}
This is the target the Kaxion relic calculation must saturate, not a substitute for it. Combining
Eqs.~\eqref{eq:omega-misalignment-partIII} and \eqref{eq:pi-locked-dm-partIII}: with the geometric
decay constant and the \(\mathcal O(1)\) angle fixed by the abundance, the inherited \GG{} geometry
reproduces the observed cold-dark-matter abundance.

The dedicated signal paper \PaperFullCOne{} has settled the normalization: one canonical field, one
physical decay constant --- the bulk Green--Schwarz normalization reduced by the integer cap loading
\(\Xi_{\mathrm{cap}}\in[35,47]\) (benchmark \(43\)) --- giving \(f_K\simeq 7.1\times10^{11}\,
\mathrm{GeV}\) and \(m_K = 8.1\pm0.5\,\mu\mathrm{eV}\). The cosmological point stands: no unrelated
dark sector had to be imported to obtain a cold relic.

\subsubsection{Oscillation onset, free streaming, and why the Kaxion behaves as cold dark matter}

The onset estimate connects the compact field to structure-formation language directly. During
radiation domination
\begin{equation}
H(T)\simeq 1.66\,g_*^{1/2}\frac{T^2}{M_{\rm Pl}},
\label{eq:H-rad-kaxion-partIII}
\end{equation}
so \(3H(T_{\rm osc})\simeq m_K\) gives
\begin{equation}
T_{\rm osc}
\simeq
\left(\frac{m_K M_{\rm Pl}}{4.98\,g_*^{1/2}}\right)^{1/2}
\propto m_K^{1/2}.
\label{eq:Tosc-kaxion-partIII}
\end{equation}
For \(m_K=8.1\,\mu\mathrm{eV}\) this naive onset lies in the radiation era (the temperature-dependent
susceptibility lowers it to \(\simeq1\,\mathrm{GeV}\) in the full computation of \PaperFullCOne{}),
far later than the GUT-scale history. Either way the field starts oscillating deep in the radiation
era and redshifts as nonrelativistic matter --- the only behavior required for cold dark matter.

The smallness of \(m_K\) should not be read as ``hot'' or ``warm.'' The criterion is whether the
relic is a thermal relativistic gas or a coherent classical condensate; the Kaxion is the second
type. Its momentum dispersion after onset is set by the Hubble scale, not a bath temperature, so its
velocity dispersion redshifts fast and its de~Broglie scale is microscopic. At \(8.1\,\mu\mathrm{eV}\)
the Kaxion is some sixteen orders heavier than the ultralight fuzzy regime \(m\sim10^{-22}\,
\mathrm{eV}\), so it behaves as an ordinary cold component in linear and nonlinear structure
formation, not a wave-supported suppression field. Part~VI tests the growth history against
\(S_8\), the matter power spectrum, and CMB lensing; within the precision reach relevant to CMB,
BAO, and large-scale growth, the Kaxion is effectively cold and collisionless.

The inflationary fork that an ordinary axion faces is closed here. A Peccei--Quinn axion may follow
the pre- or post-inflationary history because its angle is the phase of a complex scalar whose radial
mode can be thermally restored. The Kaxion has no such scalar: its angle is the holonomy of a rigid
geometric circle, so there is no order parameter to restore, no winding to seed strings, no way for
separate patches to acquire separate dials. The two-basin derivation seals this differently and, on inspection, more strongly. During the primary (GUT-adjacent) inflation the compact angle is not a free spectator: the same descent-built suppression that protects strong CP today is only mildly wound down near the GUT edge, and the angle is \emph{pinned} at its geometric minimum throughout the primary epoch (derived-conditional on the reflection-symmetric dressing; the quantitative chain is \PaperCOne{}'s open target). No randomization is ever imprinted, so the single-patch history is forced, not chosen \citep{ArisakaC1Kaxion} --- and there is no radial mode to restore, so no defect network can form at any inflation scale. The only fluctuations the angle carries to the QCD era are those of the lock epoch itself, whose curvature scale is \(H_2 \simeq 3.5\times10^{-12}\,\mathrm{GeV}\): the imprint is negligible (\(\beta_{\mathrm{iso}}\sim6\times10^{-40}\), some thirty-eight orders below the Planck bound), so the isocurvature test is passed trivially --- for the same numerical reason as under the retired TeV reading, now attached to the correct epoch.

\subsubsection{Why the Kaxion is not merely a rebranded Peccei--Quinn axion}

Four differences matter structurally. First, the Kaxion is not introduced by postulating and then
breaking a new global Peccei--Quinn symmetry at a free scale; its origin is geometric, descending
from the compact angle and Green--Schwarz sector already present. Second, it shares its ancestry
with the holonomy and strong-CP sectors already used in \ANine{}, so one geometric structure
participates in symmetry breaking, CP suppression, and dark matter at once. Third, it carries a
distinctive signature --- the 5:2 baryonic-to-electromagnetic coupling ratio and the \(1/35\)
twin-peak splitting are not generic to arbitrary axion-like particles; they descend from the \OGNfive{}
arithmetic and the DGL / Arisaka-lock logic, so a signal failing these ratios would not count as a
Kaxion. Fourth, the framework already links the Kaxion to the cosmic-inventory target of
Eq.~\eqref{eq:pi-locked-dm-partIII}: it is the first and most natural candidate expected to saturate
the nonbaryonic share left once baryons are fixed, which turns it from an optional embellishment into
a central bookkeeping object of the matter budget.

In one sentence:
\begin{quote}
The Kaxion is axion-like in effective low-energy phenomenology, but its logical status is entirely
different: it is a geometric descendant of the compact-angle and Green--Schwarz sectors, not an
auxiliary repair field introduced only after the cosmological problem is posed.
\end{quote}
That difference of status is why it deserves to be the primary dark-matter candidate of \ATen{}.

\subsubsection{What the direct-search program would actually look for}

Translated into resonant-search language, a haloscope is not asked to scan all axion-like masses with
uniform prior weight. It is asked to examine a few highly motivated narrow windows:
\begin{enumerate}[leftmargin=2.2em]
\item a narrow line in the unique window \(\nu_K\in[1.59,\,2.13]\,\mathrm{GHz}\) of
Table~\ref{tab:kaxion-benchmarks};
\item a second line displaced by the fixed fraction of Eq.~\eqref{eq:twin-split-partIII};
\item a power ratio across baryonic and electromagnetic readout channels consistent with the 5:2 lock;
\item cosmological consistency of the same benchmark with the relic-density target.
\end{enumerate}
A candidate satisfying all four would reproduce not one number but the whole geometric fingerprint of
the compact-angle sector.

\subsection{Couplings, lifetime, and the twin-peak haloscope signature}

A dark-matter candidate is useful only if its interactions are weak enough to evade bounds, strong
enough to be observable in principle, and constrained enough to define a sharp search. The Kaxion
performs on all three. The effective low-energy couplings are inherited from the anomaly and
compact-angle sectors:
\begin{equation}
\mathcal L_{K,\mathrm{int}}
\supset
\frac{\alpha_s}{8\pi f_K}C_g\,a_K\,G\widetilde G
+
\frac{\alpha}{8\pi f_K}C_\gamma\,a_K\,F\widetilde F
+
\frac{g_B^2}{8\pi f_K}C_B\,a_K\,B\widetilde B
+\cdots.
\label{eq:kaxion-effective-couplings-partIII}
\end{equation}
Within DGL / Arisaka-lock logic the baryonic-to-electromagnetic coupling ratio follows the same
minimal integer lock used elsewhere,
\begin{equation}
C_B : C_\gamma = 5 : 2,
\label{eq:kaxion-5-2-partIII}
\end{equation}
which is why the program emphasizes a 5:2 structure: not ad hoc spectroscopy but the same
compact/baryonic/electromagnetic lock written at the pseudoscalar-coupling level. Using
\begin{equation}
g_{K\gamma\gamma} = \frac{\alpha}{2\pi f_K}\left(\frac{E}{N}-1.92(4)\right),
\label{eq:gagg-standard-partIII}
\end{equation}
with the anomaly ratio fixed by the companion paper's anomaly-matching theorem at \(E/N = 8/3\) ---
placing the Kaxion precisely \emph{on the DFSZ line} --- one finds
\begin{equation}
\begin{aligned}
g_{K\gamma\gamma}&\simeq 2.9\times 10^{-17}\,\mathrm{GeV}^{-1}
&&\text{(raw frame)},\\
g_{K\gamma\gamma}&= (1.2\pm0.1)\times 10^{-15}\,\mathrm{GeV}^{-1}
&&\text{(laboratory eigenstate)},
\end{aligned}
\label{eq:gagg-range-partIII}
\end{equation}
the laboratory value being the one any haloscope sees \citep{ArisakaC1Kaxion}. This is small enough
to evade ordinary detection yet large enough that DFSZ-depth resonant cavities of the late 2020s
reach it directly. The lifetime is correspondingly enormous: for \(a_K\to\gamma\gamma\),
\begin{equation}
\tau_{K\to\gamma\gamma}
\sim
\frac{64\pi}{g_{K\gamma\gamma}^2m_K^3},
\label{eq:kaxion-lifetime-partIII}
\end{equation}
giving
\begin{equation}
\tau_{K\to\gamma\gamma}\sim 4\times10^{32}\,t_U\ \text{(laboratory eigenstate)},
\qquad
\tau_{K\to\gamma\gamma}\sim 6\times10^{40}\,t_U\ \text{(raw frame)},
\label{eq:kaxion-lifetime-huge-partIII}
\end{equation}
with \(t_U\) the age of the universe, so cosmological stability is automatic.

The direct-search signature is sharper still. Because the compact-angle sector contains \emph{two}
light zero-modes (the Green--Schwarz dual and the Wilson-line phase, geared by kinetic mixing), the
signal is not a single line but a doublet with the fixed fractional split of
Eq.~\eqref{eq:twin-split-partIII}. A positive cavity or LC-type detection would be a pair of lines
with locked spacing and locked coupling pattern --- one of the most falsifiable hidden-sector
proposals in the program. The signal logic is summarized in
Table~\ref{tab:kaxion-signatures-partIII}.

\begin{table}[H]
\centering
\caption[Signal logic of the Kaxion corridor]{Signal logic of the Kaxion corridor. The columns give the observable, what the framework expects of it, and what makes that expectation distinctive rather than generic. The third column is the one that matters: an expectation shared with every axion-like model tests the class, not this framework, and the rows are ordered so that the distinguishing signatures come first.}
\label{tab:kaxion-signatures-partIII}
\small
\begin{tabular}{p{0.29\textwidth}p{0.28\textwidth}p{0.29\textwidth}}
\toprule
\textbf{Observable} & \textbf{Framework expectation} & \textbf{Why it is distinctive}\\
\midrule
Mass / frequency & Unique window \(\nu_K\in[1.59,\,2.13]\)~GHz, benchmark 1.95~GHz & Not an arbitrary wide hidden-sector scan; the single laboratory eigenstate of the compact-angle sector\\
Twin-peak splitting & \(\Delta\nu/\nu = 1/35\) & Fixed by the \OGN{} throat depth \(K=35\)\\
Coupling ratio & baryonic : electromagnetic \(=5:2\) & Mirrors the DGL / Arisaka-lock arithmetic used elsewhere in the program\\
Lifetime & cosmologically stable by many orders of magnitude & Automatic for a compact-angle pseudoscalar in the inferred mass-coupling corridor\\
Relic behavior & cold, pressureless after oscillation onset & Standard misalignment logic with no need for a thermal WIMP freeze-out sector\\
\bottomrule
\end{tabular}
\end{table}

The practical message is clear. Dark matter is not solved by declaring ``there exists an axion-like
field.'' It is addressed by a narrow compact-angle corridor with specific frequencies, splittings,
and coupling ratios. If those windows are exhausted without a signal, the Kaxion corridor fails ---
the correct scientific standard.

\subsection{Why the Kaxion is the first dark-matter candidate of \ATen{}}

The compact-angle and Green--Schwarz descendants of \GG{} make an axion-like cold relic not merely
possible but structurally expected: naturally stable, naturally cold, naturally connected to sectors
already present in the particle paper, and naturally associated with narrow direct-search windows.
That is sufficient reason to make it the primary dark-matter sector of \ATen{}. The strength of the
GG-2026 sequence is not that it reproduces the entire dark-sector literature; it is that, given the
inherited ledger, it can say where one should look first. If the compact-angle sector supplies the
dominant matter relic, the baryonic component must be explained separately and precisely --- which
leads directly to the matter--antimatter asymmetry.

\subsection{The Kaxion on the two fibers: the circle's coordinate, ringing still}
\label{ssec:kaxion-cosmo-two-fibres}

The dark-matter companion develops the full portrait
\citep{ArisakaC1Kaxion}; this subsection carries its essence at the
cosmological register, so that the middle act of the trio
(\S\ref{ssec:inflation-two-fibres},
\S\ref{ssec:de-two-fibres}) stands beside its siblings in this
paper's own pages. The campaign's triptych sentence places it:
\emph{the electron is a holonomy class wearing a mass; the proton
is two holonomy ledgers wearing a weld; the Kaxion is the holonomy
coordinate itself, wearing an occupation.} Dark matter, in
\GGTheory{}, is the sixth dimension relaxing into its ground
attractor after the fifth has carried the universe below the
barrier.

Everything the Kaxion \emph{is} is circle-side: both parents are
periodic pseudoscalar directions of the compact geometry; its
lightness is the circle's memory (a periodic direction admits no
polynomial potential --- the Kaxion is light because the circle
cannot forget); its portals are the \BIsix{} ledger's, in the
locked $5{:}2$ proportion --- it couples only through topology
because it is \emph{made} of topology, which is the entire origin
of its darkness; and its universality is one frozen angle --- one
holonomy configuration for the whole observable universe, the
condensate analogue of the electron's sameness. Everything the
Kaxion \emph{has} is interval-side: the freeze is the Hubble
friction --- the semigroup itself gripping the circle's coordinate
while $3H \gg m_K$; the release at $T_{\mathrm{osc}} \simeq 1$~GeV
is the moment the circle's clock outruns the interval's, crossed
exactly once because the descent is one-way; the mass
$m_K = \sqrt{\chi_{\mathrm{QCD}}}/f_K$ is delivered by the descent as it cools
through the QCD scale; and the relic abundance is the memory of
the crossing, redshifted --- the misalignment condensate is
primordial, unique, and unreplenished as a corollary of the fiber
anatomy rather than an assumption of the cosmology. The pendulum
equation of the misalignment mechanism is the anatomy in one line:
the field is the circle's coordinate, the $3H\dot a$ friction is
the interval acting on it, and the stiffness is the circle's,
switched on by the boundary at a temperature the one-way history
fixes.

Two features make this the cleanest attractor exhibit in the
cohort, and both belong in the cosmological narrative. First,
\emph{the Hessian is the observable}: the dark-matter mass is the
curvature of the attractor floor, so the $1.95$~GHz line is the
runtime's termination certificate, heard as a tone --- a haloscope
detection would be the first direct measurement of a \GGA{}-class
Hessian. Second, \emph{the cascade is still finishing}: the
electron's and proton's cascades terminated within the first
second; the Kaxion's coherent oscillation is the final \GRIPin{}
of the observable universe, caught mid-ring. Between the
interval's opening act and the joint terminus, the circle's act is
the one still playing --- which is precisely why it is the one the
laboratory can catch.

\section{Baryogenesis via the inherited neutrino and CP ledger}

\subsection{Historical background}

The necessary conditions for dynamically generating a baryon-antibaryon asymmetry from a symmetric initial state were stated by Sakharov in 1967 \citep{Sakharov1967}: baryon-number violation, C and CP violation, and departure from thermal equilibrium. Every viable mechanism in the past sixty years realizes those three conditions. The present section asks which realization is admissible once the neutrino sector and the CP ledger have been fixed by the upstream particle paper, rather than chosen freely.

\label{sec:baryogenesis}

The electroweak sphaleron transport and leptogenesis mechanisms that tie the baryon asymmetry to the inherited neutrino and CP structure are established in \citep{KuzminRubakovShaposhnikov1985,FukugitaYanagida1986}.

\begin{proposition}[One-ledger baryogenesis statement (T-Cosmos-8)]
The baryogenesis module is not an independent CP-violating add-on.
It is the first thermal transport block in which the inherited Majorana neutrino
and CP data of \ANine{} are forced through the cosmological ladder and then
matched to the late baryon share \(\Omega_b=1/(2\pi^2)\).
\end{proposition}

\[
\dot n_{N_i}+3Hn_{N_i}=-\Gamma_i\!\left(n_{N_i}-n_{N_i}^{\mathrm{eq}}\right),
\qquad
Y_B \sim c_{\mathrm{sph}}\,\kappa\,\sum_i \epsilon_i\,Y_{N_i},
\]
the point being that the \(\epsilon_i\), \(\Gamma_i\), and the allowed heavy-sector structure are
fixed by the inherited particle ledger rather than left phenomenologically free.

\medskip
It is worth recording which constitutional pillar carries each of Sakharov's three conditions, because none of the three is imported. \emph{Baryon-number violation}: the bulk baryonic ledger is exactly conserved (gauged $\U(1)_B$); what evolves is the projected subsystem's budget, and the $\Delta L = 2$ step occurs only at the dial-free door of Proposition~\ref{prop:dial-free-door}. \emph{C and CP violation}: the CP-odd input is a real, lattice-locked angle of the inherited flavor geometry --- in the real presentation a loop holonomy of the coupling texture, the flavor analogue of an Aharonov--Bohm angle (\citealp{ArisakaA9SM}; \citealp{ArisakaA8QM}). \emph{Departure from equilibrium}: constitutional --- the boundary law is a forward semigroup, openness being the default \citep{ArisakaA5GDOS}; the transport equation above is a one-way relaxation; and the resonance condition that lifts the Yukawa suppression is itself an openness condition, met where the width matches the lock splitting. In one line: \GRIP{} owns the angle, \GDOS{} owns its observability, \OGGEP{} owns the ledger.

\subsection{Why the matter--antimatter problem becomes sharper after \ANine{}}

The observed universe is overwhelmingly baryonic. The comoving baryon asymmetry,
\begin{equation}
\eta_B \equiv \frac{n_B-n_{\bar B}}{n_\gamma},
\label{eq:etaB-def-partIII}
\end{equation}
is nonzero and small rather than zero. The late-universe benchmark is
\begin{equation}
\eta_B^{\rm obs} = (6.12 \pm 0.04)\times10^{-10},
\label{eq:etaB-obs-partIII}
\end{equation}
the CMB-inferred value from the Planck~2018 baryon density \citep{Planck2018}. In ordinary model
building this is a target to be generated by some out-of-equilibrium mechanism satisfying the
Sakharov conditions. Here the question is narrower and harder. Because \ANine{} has already fixed a
neutrino sector and a nontrivial leptonic CP structure, the matter--antimatter problem is no longer
an invitation to add arbitrary phases or heavy states. It is the question of whether the
already-derived neutrino and CP ledger admits a viable leptogenesis descendant.

This sharpening matters because exact local \(\U(1)_B\) is a structural claim of \GG{}: baryon number
is not something the framework may casually violate at low energy, and exact proton stability is one
of its cleanest falsifiers. Any viable mechanism must therefore (i) leave the low-energy proton
absolutely stable while (ii) the early universe nonetheless generates a nonzero net baryon
abundance. This pushes the framework almost automatically toward leptogenesis: a lepton asymmetry
created first and partially converted to baryon number by electroweak sphalerons, with no explicit
proton-destabilizing operators in the late universe. The choice is not arbitrary. The earlier papers
already select the minimal Majorana/Weinberg route for the light neutrinos; once a Majorana channel
exists, lepton number is not exactly gauged as baryon number is, so the lepton sector is the path of
least resistance for an early asymmetry.

\subsection{Sakharov conditions in \GG{}/\GDOS{} language}

\paragraph{(i) Effective \(B\) generation without late-time proton decay.}
In the \GG{} bulk local \(\U(1)_B\) is exact, so the total six-dimensional baryonic ledger is
conserved. But the four-dimensional boundary universe is an open subsystem, and in any open-system
description a quantity can flow across the subsystem boundary while remaining conserved in the full
system. Electroweak sphalerons supply the ordinary Standard-Model conversion,
\begin{equation}
\Delta(B+L)\neq 0,
\qquad
\Delta(B-L)=0,
\label{eq:sphaleron-basic-partIII}
\end{equation}
while the exact bulk baryonic charge stays globally protected. The framework can thus generate an
\emph{effective} boundary baryon excess without a late-time proton-decay instability.

\paragraph{(ii) CP violation as an inherited geometric phase.}
\ANine{} already derived a nontrivial leptonic Dirac phase \(\delta_{\rm PMNS}\simeq196^\circ\), the
protected Majorana phases \((\alpha_{21},\alpha_{31})=(0,\pi)\), and the rephasing-invariant Jarlskog
measure \(J_{\rm PMNS}\simeq-9.2\times10^{-3}\). The compact-angle logic associates CP-odd structure
with holonomy and Wilson-line phase selection. The CP source need not be inserted by hand; it is
inherited.

\paragraph{(iii) Out-of-equilibrium dynamics as a \GDOS{} transition.}
A successful asymmetry cannot be generated in exact equilibrium. Within \GDOS{} this is the ordinary
fact that cosmological subsystems evolve under time-dependent ports, cooling, and changing
attractors. The relevant stage is the period after reheating when the heavy neutrino sector ceases
to track the bath and decays with a CP-odd bias --- precisely what \GDOS{} was built to describe.

The Sakharov conditions are therefore not external checkboxes but latent in the inherited ledger:
\(\U(1)_B\) forces indirect generation, the neutrino and compact-angle sectors supply the CP
structure, and cosmological cooling supplies the open-system departure from equilibrium.

\subsection{The inherited light-neutrino block and the heavy completion}

The light-neutrino data of \ANine{} were already tabulated, but for baryogenesis it is worth
displaying their implications. The completed particle paper yields
\begin{equation}
(m_1,m_2,m_3)_{\rm QD}
=
(0.0653,\,8.5923,\,50.5131)\ \mathrm{meV},
\qquad
\Sigma m_\nu\simeq 59.17\ \mathrm{meV},
\label{eq:light-neutrino-summary-partIII}
\end{equation}
with
\begin{equation}
\delta_{\rm PMNS}\simeq 196.06^{\circ},
\qquad
(\alpha_{21},\alpha_{31})=(0,\pi),
\qquad
J_{\rm PMNS}\simeq -9.2\times10^{-3}.
\label{eq:pmns-majorana-summary-partIII}
\end{equation}
These are normally ordered, Majorana-like, with order-one CP structure, and they descend from the same compact-angle and \OGN{} arithmetic that organizes the rest of the paper. (The \(J_{\rm PMNS}\) value is the first-order-angle layer; the NuFit-angle evaluation, the convention carried by \PaperPFive{}, gives \(-9.4\times10^{-3}\).) Crucially, in \ANine{}
these light masses arise from the Weinberg dimension-five operator at the GUT seam,
\begin{equation}
\mathcal L_5 = \frac{(L H)(L H)}{\Lambda_{\rm W}},
\qquad
m_\star = \frac{\pi v^2}{\EGUT},
\label{eq:weinberg-operator-partV}
\end{equation}
with \(v\simeq246\,\mathrm{GeV}\), the convention at which \ANine{} evaluates \(m_\star\simeq8.4\)~meV. \ANine{} fixes the light sector through
Eq.~\eqref{eq:weinberg-operator-partV} but \emph{explicitly leaves the ultraviolet completion ---
the heavy Majorana states and the leptogenesis they drive --- to \ATen{}}. That completion is the
subject of this section.

Before the completion is written down, the constitutional
question must be answered: \emph{is} a heavy Majorana singlet an
assumption --- the model-builder's standard import --- or is it forced?
In this geometry it is forced, by a one-line piece of fiber arithmetic
that deserves theorem status, since the entire module rests on it.

\begin{proposition}[The dial-free door: a Majorana mass exists iff the fiber weight vanishes (T-Cosmos-18 / R10)]
\label{prop:dial-free-door}
Let a fermion multiplet on the \GG{} bulk carry weight $n$ under the
compact phase direction $S^1_\theta$ (spinor and family indices commute
with the fiber action). Then a Lorentz-invariant, gauge-invariant
Majorana-type self-pairing --- equivalently, a $\Delta L = 2$ bilinear
--- exists if and only if $n = 0$; and on every $n \neq 0$ sector the
unique invariant symmetric self-pairing is the charge-conserving Dirac
pairing $\psi\bar\psi$. Moreover the trilinear $LHN$ is the unique
fiber-neutral portal that touches the weight-zero sector linearly.
\end{proposition}

\begin{proof}
In the real presentation the weight-$n$ sector is, pointwise, the real
pair $(a, b)$ on which the circle acts by the rotation $R(n\alpha)$.
The symmetric square is three-dimensional, spanned by $a^2 + b^2$,
$a^2 - b^2$, and $2ab$. The first is rotation-invariant for every $n$;
it is $\psi\bar\psi$, the Dirac pairing. The remaining pair
$(a^2 - b^2,\, 2ab)$ is exactly $(\mathrm{Re}\,\psi^2, \mathrm{Im}\,\psi^2)$
--- the real form of the Majorana bilinear --- and transforms as a
weight-$2n$ doublet: under $R(n\alpha)$ it rotates by the angle
$2n\alpha$. For $n \neq 0$, invariance under all $\alpha$ forces it to
vanish; for $n = 0$ the action is trivial, the pairing is unobstructed,
and the reality condition of the weight-zero sector is precisely the
Majorana condition. The spinor factor contributes the usual
charge-conjugation pairing, inert under the fiber, and cannot
compensate a non-zero weight. For the portal: a trilinear is
fiber-neutral iff its weights sum to zero; with $L$ at weight
$-\tfrac12$ and $H$ at $+\tfrac12$ in the unified-$\U(1)$
normalization, the singlet $N$ at weight $0$ closes the unique such
trilinear linear in $N$.
\end{proof}

The proposition does three jobs at once. First, it geometrizes lepton
number: charge conservation on the boundary is winding conservation on
the circle, so $L$ is exact throughout the dial-carrying ledger, and
the sole legal $\Delta L = 2$ step in the entire matter sector is the
Majorana mass of the one field the circle does not move. Second, it
promotes the seesaw from choice to consequence: the type-I completion
below is not one ultraviolet option among several but the unique
fiber-neutral door between the conserved winding ledger and the
weight-zero sector. Third, it closes the last imported ingredient of
this paper's matter story: with the heavy sector now forced --- its
Majorana character by this proposition, its scale by the throat integer
$K = 35$ --- the chain from the constitution to the observed baryon
share runs with no assumed particle content. The statement extends the
real-bulk series of \citep{ArisakaA8QM} (as its entry R10), where the
weight-zero sector's real, dial-free character is developed in full;
the standalone, full-length development of the series is the
Foundations Branch member paper \PaperPFour{}
(\citealp{ArisakaP4Real}).

The natural completion of a Weinberg operator is the type-I seesaw: heavy Majorana neutrinos \(N_i\)
whose exchange generates Eq.~\eqref{eq:weinberg-operator-partV} on integrating out. In the present
geometry they have only one natural home --- the lock scale,
\begin{equation}
M_{N_i} = \Mlock\,(1+\delta_i),
\qquad
\delta_i = O\!\left(\frac{1}{K}\right),
\qquad K=35,
\label{eq:heavyN-benchmark-partIII}
\end{equation}
so the triplet sits in the few-TeV corridor with small, geometrically controlled splittings. A
minimal benchmark set is
\begin{equation}
(M_{N_1},M_{N_2},M_{N_3})
\sim
\Mlock\,(1,1+K^{-1},1+2K^{-1}),
\label{eq:heavyN-triplet-partIII}
\end{equation}
i.e. percent-level splittings around \(\Mlock\simeq3.85\,\mathrm{TeV}\) --- the
near-degenerate regime that resonant enhancement requires. The register of that last clause
matters, so it is stated at \PaperPFive{}'s own grade: the \(O(1/K)\) splittings motivate
\emph{proximity} to the resonant regime, but at vacuum numbers the system sits far off the
Breit--Wigner peak --- the splitting \(d\sim1/K\simeq3\times10^{-2}\) against the vacuum
width \(\Gamma_j/M_j\sim2\times10^{-14}\) (Appendix~\ref{appA:leptogenesis}) --- so the
on-resonance condition is the chain's named conjecture, not its natural consequence, and
``natural rather than tuned'' would overstate what vacuum numbers support.

\subsection{Seesaw matching: the geometry fixes the Yukawas}

The decisive quantitative step is that the seesaw fixes the neutrino Yukawa magnitude with no new
freedom. Integrating out the lock-scale triplet gives the light mass matrix
\begin{equation}
m_\nu = \frac{Y_\nu^2\,v^2}{M_N},
\label{eq:seesaw-mnu-partV}
\end{equation}
which must reproduce the \ANine{} Weinberg scale \(m_\star=\pi v^2/\EGUT\) of
Eq.~\eqref{eq:weinberg-operator-partV}. Matching the two expressions cancels \(v^2\) and fixes the
Yukawa magnitude entirely in terms of inherited scales,
\begin{equation}
Y_\nu^2 \simeq \pi\,\frac{M_N}{\EGUT}
\simeq \pi\,\frac{3.85\times10^{3}}{2.28\times10^{16}}
\simeq 5.3\times10^{-13},
\qquad
Y_\nu \simeq 7.3\times10^{-7}.
\label{eq:Ynu-matching-partV}
\end{equation}
The Yukawa is tiny because the heavy scale is low: a TeV-scale \(M_N\) with sub-eV light masses
\emph{requires} \(Y_\nu\sim10^{-6}\). This is not a problem to be tuned away but the direct
consequence of placing the seesaw at the geometry's lock scale rather than at an arbitrary
intermediate scale. The tree-level decay width that follows is correspondingly small,
\begin{equation}
\Gamma_N = \frac{Y_\nu^2}{8\pi}M_N
\simeq 8\times10^{-11}\,\mathrm{GeV},
\qquad
\frac{\Gamma_N}{M_N}\simeq 2\times10^{-14}.
\label{eq:GammaN-partV}
\end{equation}
A tiny width would ordinarily doom leptogenesis. The resonant regime is what rescues it.

\subsection{Resonant CP asymmetry from the inherited phases}

The heavy Majorana neutrinos decay through
\begin{equation}
N_i \to \ell H,
\qquad
N_i \to \bar\ell H^\dagger,
\label{eq:Ni-decays-partIII}
\end{equation}
with the CP-odd asymmetry
\begin{equation}
\epsilon_i
\equiv
\frac{\Gamma(N_i\to \ell H)-\Gamma(N_i\to \bar\ell H^\dagger)}
{\Gamma(N_i\to \ell H)+\Gamma(N_i\to \bar\ell H^\dagger)}.
\label{eq:epsi-def-partIII}
\end{equation}
The standard self-energy (resonant) formula is
\begin{equation}
\epsilon_i
=
\frac{\Im\!\left[(Y_\nu Y_\nu^\dagger)_{ij}^2\right]}
{(Y_\nu Y_\nu^\dagger)_{ii}(Y_\nu Y_\nu^\dagger)_{jj}}
\,\frac{(M_i^2-M_j^2)M_i\Gamma_j}
{(M_i^2-M_j^2)^2+M_i^2\Gamma_j^2},
\qquad i\neq j.
\label{eq:resonant-epsilon-generic-partIII}
\end{equation}
The two factors are a CP-odd invariant and a resonant Breit--Wigner enhancement. Both are narrowed
by the geometry. The enhancement is maximal at the resonance condition \(|M_i-M_j|\simeq\Gamma_j/2\),
where the Breit--Wigner factor reaches its peak value \(\tfrac12\) and, decisively, the tiny Yukawa
magnitude \emph{cancels between numerator and denominator of the prefactor}. At resonance the
asymmetry is therefore controlled not by \(Y_\nu\) but by the geometric phase alone,
\begin{equation}
\epsilon^{\rm res}
\simeq
\frac12\,\sin\!\bigl(2\,\phi_{\rm geo}\bigr),
\quad
\phi_{\rm geo}\ \text{from the flavored projection of the inherited phases (Appendix~\ref{appA:leptogenesis})}.
\label{eq:eps-res-phase-partV}
\end{equation}
This is why a TeV-scale, tiny-Yukawa seesaw can still produce a viable asymmetry: the resonance
lifts the would-be \(Y_\nu^2\) suppression, leaving a phase-controlled number. At the TeV lock scale
the chain is fully flavored (the flavor obligation L-Cosmos-F of Lemma~\ref{lem:cosmos-U},
Appendix~\ref{appA:leptogenesis}), and the operative CP object is the flavored asymmetry
\(\epsilon_{i\alpha}\), computed on the inherited phases in Appendix~\ref{appA:leptogenesis}: the
driver is the Majorana parity \(\alpha_{31}=\pi\), which sets the sign of the matter excess, and the
chain carries
\begin{equation}
\boxed{\ \epsilon^{\rm res} \simeq 2.5\times10^{-6}\ }
\label{eq:eps-value-partV}
\end{equation}
as the target value that the partial-resonance hypothesis H-\(\phi_{\rm geo}\)-1
(\(\Delta M/\Gamma^{\rm eff}\simeq3\times10^{4}\); Appendix~\ref{appA:leptogenesis}) must deliver
--- derived-conditional, not a derived output: on the exact Breit--Wigner peak the flavored
computation overshoots the required throughput by \(\sim3\times10^{4}\), and at vacuum splittings it
undershoots by \(\sim4\times10^{6}\). The flavored computation and the hypothesis are worked in
Appendix~\ref{appA:leptogenesis}, where the hypothesis now carries an identified candidate mechanism
at candidate grade --- the flavor-measurement regulator, with the zero-dial closed form
\(\Delta M/\Gamma^{\rm eff} = 1/(35\,c_\tau y_\tau^2)\) --- whose conditions C1--C6 remain open.

\subsection{Washout and the strong-washout efficiency}

Whether this asymmetry survives is decided by washout. The relevant control parameter is the
effective neutrino mass that the decaying state couples to,
\begin{equation}
\widetilde m_i = \frac{(Y_\nu Y_\nu^\dagger)_{ii}\,v^2}{M_i},
\label{eq:mtilde-def-partV}
\end{equation}
compared with the equilibrium scale
\begin{equation}
m_\star
= \frac{8\pi v^2}{M_i}\,\frac{H(M_i)}{M_i}
\simeq 1.1\times10^{-3}\,\mathrm{eV}.
\label{eq:mstar-def-partV}
\end{equation}
The inherited light spectrum makes the relevant effective mass of order the heaviest light neutrino,
\(\widetilde m\sim m_3\simeq 5\times10^{-2}\,\mathrm{eV}\), so
\begin{equation}
K_{\rm wash}\equiv \frac{\widetilde m}{m_\star}\simeq 50 \gg 1,
\label{eq:Kwash-partV}
\end{equation}
i.e. strong washout. In the strong-washout regime the surviving fraction is the standard
suppressed efficiency,
\begin{equation}
\kappa \simeq \frac{1}{K_{\rm wash}}\;(\text{up to an }\mathcal O(1)\ \text{factor})
\simeq 1\times10^{-2}.
\label{eq:kappa-value-partV}
\end{equation}
Strong washout is, paradoxically, a stabilizing feature: it makes the final asymmetry largely
independent of initial conditions, which is precisely the regime in which an inherited (rather than
fitted) calculation is trustworthy. The Boltzmann estimate behind
Eq.~\eqref{eq:kappa-value-partV} is given in Appendix~\ref{appA:leptogenesis}. The washout here
runs at the bath scale \(\widetilde m \sim m_3\), while the resonance width of the seesaw
completion runs at the seam-matched \(\widetilde m \simeq 8.4\)~meV --- two deliberate
conventions whose factor-\(\simeq12\) spread in \(Y_\nu^2\) is declared, accounted for (the single-convention ratio \(2\,m_3/\widetilde m_{\rm seam}\simeq12\), the factor of two a normalization bookkeeping), and priced within the
\(O(10)\) register in the convention note at Eq.~\eqref{eq:app-GammaN} of
Appendix~\ref{appA:leptogenesis}.

\subsection{Assembling \texorpdfstring{$\eta_B$}{etaB}: the mechanism behind \texorpdfstring{$e^{-K/2}$}{e to the minus K over 2}}

The net CP-odd throughput that reaches the boundary is the product of the resonant asymmetry and the
washout efficiency,
\begin{equation}
\epsilon_{\rm geo}^{\rm eff}
\equiv
\epsilon^{\rm res}\,\kappa
\simeq
(2.5\times10^{-6})\times(1.0\times10^{-2})
\simeq 2.51\times10^{-8}.
\label{eq:eps-eff-product-partV}
\end{equation}
The arresting fact is that this product reconstructs, to the displayed precision, the throat-depth
exponential that the geometry independently predicts,
\begin{equation}
\epsilon_{\rm geo}^{\rm eff}
\simeq e^{-K/2}
= e^{-17.5}
\simeq 2.51\times10^{-8}.
\label{eq:eps-geo-effective-partIII}
\end{equation}
This is the central result of the section. The compressed ledger value \(e^{-K/2}\) is not a separate
assertion: it is the \emph{coarse-grained output of the resonant-leptogenesis transport}, the product
of a phase-controlled resonant asymmetry (\(\sim10^{-6}\)) and a strong-washout efficiency
(\(\sim10^{-2}\)). The transport mechanism is the physics behind the number. (The exponent \(K/2\)
rather than \(K\) or \(K/4\) is the compression statement the sphaleron-transport-normalization
conjecture must ultimately fix from first principles; what is new here is that the independently derived washout efficiency and the candidate regulator's closed form bracket the required throughput --- the identity at displayed precision uses the \(\mathcal O(1)\) freedom in \(\kappa\) and the target status of \(\epsilon^{\rm res}\).)

The remainder is standard transport bookkeeping, and one bookkeeping factor deserves display rather than absorption: the asymmetry is generated on the heavy sector's \emph{equilibrium abundance}. A relativistic Majorana fermion with \(g_N=2\) spin states has number density \(n_N^{\rm eq} = \tfrac34(\zeta(3)/\pi^2)\,g_N T^3\), the bath's entropy density is \(s = (2\pi^2/45)\,g_{*s}T^3\), and their ratio is the equilibrium yield
\begin{equation}
Y_{N}^{\rm eq}\equiv\frac{n_N^{\rm eq}}{s}
=\frac{135\,\zeta(3)}{8\pi^4}\,\frac{g_N}{g_{*s}}
\simeq\frac{0.42}{g_*}
\ \ \text{per species},
\qquad
\sum_{i=1}^{3}Y_{N_i}^{\rm eq}\simeq\frac{1.25}{g_*}
\label{eq:YNeq-partIII}
\end{equation}
for the quasi-degenerate triplet --- physically, a species in equilibrium carries its fair share of the bath's heat, two spin states' worth per member, and there are three members. Each net decay deposits the CP-odd fraction \(\epsilon_{\rm geo}^{\rm eff}=\epsilon^{\rm res}\kappa\) of that abundance into lepton number, so the comoving lepton asymmetry per entropy is
\begin{equation}
Y_{L}
\simeq
\Bigl(\sum_i Y_{N_i}^{\rm eq}\Bigr)\,\epsilon_{\rm geo}^{\rm eff}
=
\frac{1.25\,\epsilon_{\rm geo}^{\rm eff}}{g_*},
\qquad
g_*\simeq 106.75,
\label{eq:YL-simple-partIII}
\end{equation}
for the Standard-Model-like bath at the TeV scale. (The frequently quoted shorthand \(Y_L\sim\epsilon/g_*\) absorbs the equilibrium-yield factor silently; displayed, it is worth a quarter of the final answer.) Electroweak sphalerons convert part of this to
baryon number with the one-Higgs-doublet coefficient
\begin{equation}
c_{\rm sph} = \frac{28}{79} \simeq 0.3544,
\label{eq:csph-partIII}
\end{equation}
and the entropy-normalized yield converts to the baryon-to-photon ratio through the standard
late-time factor \(7.04\):
\begin{equation}
\eta_B
\simeq
7.04\,c_{\rm sph}\,Y_L
\simeq
7.04\left(\frac{28}{79}\right)\frac{1.25\,\epsilon^{\rm res}\,\kappa}{g_*}
=
7.04\left(\frac{28}{79}\right)\frac{1.25\,e^{-K/2}}{g_*}.
\label{eq:etaB-derived-partIII}
\end{equation}
Inserting the inherited numbers,
\begin{equation}
\eta_B
\simeq
7.04\times0.3544\times1.25\times\frac{2.51\times10^{-8}}{106.75}
\simeq 7.34\times10^{-10}.
\label{eq:etaB-numerical-partIII}
\end{equation}
This sits within \(20\%\) of the observed baryon asymmetry of Eq.~\eqref{eq:etaB-obs-partIII}, well inside the declared order-of-magnitude register. Nothing
in the chain was freely fitted: the light sector and phases come from \ANine{}, the heavy scale
\(M_N\simeq\Mlock\) and the near-degeneracy \(\delta\sim1/K\) come from the geometry, and the
seesaw matching of Eq.~\eqref{eq:Ynu-matching-partV} fixes the Yukawas. The one named input is the
resonant-degeneracy assumption \(|M_i-M_j|\simeq\Gamma_j/2\) --- the condition that the triplet sits
\emph{on} the resonance, which the \OGN{} splitting \(\delta_i\sim1/K\) makes plausible but does not by
itself enforce to the required precision. The full transport calculation, including the explicit seesaw matching, the resonant CP-asymmetry projection, and the flavored Boltzmann integration, is carried out in Appendix~\ref{appA:leptogenesis}; the reader who wishes to see every factor is
encouraged to consult it.

Against the CMB-inferred \(\eta_B^{\rm obs} = (6.12 \pm 0.04)\times10^{-10}\) \citep{Planck2018} the derived value \(7.34\times10^{-10}\) sits at \(+20\%\). The honest tier is order-of-magnitude with a
controlled mechanism: the resonant condition and the \(\mathcal O(1)\) factor inside the
strong-washout efficiency are not yet computed to percent precision, so the entry carries the
factor-\(\sim\)3 theory band shown in the parameter-audit Dev plot
(Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}). Displaying the equilibrium-yield factor of Eq.~\eqref{eq:YNeq-partIII} matters for honesty in both directions: absorbing it silently into the \(\mathcal O(1)\) register would place the central value within a few percent of the observation --- a closeness that would be an artifact of the undisplayed factor. Closeness of central values at this tier is never a precision success; the genuine claim is that an explicit, inherited resonant-leptogenesis chain reconstructs the geometric \(e^{-K/2}\) ledger value and lands within its declared band of the sky.

\subsection{Boltzmann transport and why the geometry compresses the calculation}

It is worth showing how Eq.~\eqref{eq:etaB-numerical-partIII} sits inside the familiar Boltzmann
language. With the yields
\begin{equation}
Y_{N_i}\equiv \frac{n_{N_i}}{s},
\qquad
Y_{B-L}\equiv \frac{n_{B-L}}{s},
\qquad
z\equiv \frac{\Mlock}{T},
\label{eq:yield-variables-partIII}
\end{equation}
the resonant-leptogenesis system is
\begin{align}
\frac{\dd Y_{N_i}}{\dd z}
&=
- D_i(z)\bigl(Y_{N_i}-Y_{N_i}^{\rm eq}\bigr),
\label{eq:YN-boltzmann-partIII}
\\[3pt]
\frac{\dd Y_{B-L}}{\dd z}
&=
\sum_i \epsilon_i D_i(z)\bigl(Y_{N_i}-Y_{N_i}^{\rm eq}\bigr)
- W(z)Y_{B-L},
\label{eq:YBL-boltzmann-partIII}
\end{align}
with \(D_i\) the decay terms and \(W\) the washout term. In an unconstrained model both the
\(\epsilon_i\) and the washout profile \(W(z)\) range over a large Yukawa-and-mass space. Here the
heavy masses are tied to the lock scale, the splittings to the \OGN{} integer, the Yukawas to the
seesaw matching, and the CP data to the inherited phases. The system therefore collapses to one
effective dimensionless throughput. A useful factorization is
\begin{equation}
\epsilon_{\rm geo}^{\rm eff}
=
\mathcal R_{\rm res}\,\mathcal W_{\rm eff}\,\sin\delta_{\rm geo}
=
\epsilon^{\rm res}\,\kappa,
\label{eq:epsilon-throughput-factorization-partIII}
\end{equation}
where \(\mathcal R_{\rm res}\) is the resonant enhancement, \(\mathcal W_{\rm eff}=\kappa\) the
survival after washout, and \(\sin\delta_{\rm geo}\) the holonomy-controlled phase --- the same three
factors made numerical in Eqs.~\eqref{eq:eps-value-partV}--\eqref{eq:eps-eff-product-partV}. The
geometry locks their product to \(e^{-K/2}\).

The timing is consistent. At \(T\sim\Mlock\) the age of the universe is
\begin{equation}
t\sim \frac{0.30\,M_{\rm Pl}}{g_*^{1/2}T^2}
\sim 10^{-13}\text{--}10^{-12}\,\mathrm{s},
\label{eq:TeV-age-partIII}
\end{equation}
late enough for a valid boundary \(\FRW\) description but early enough that the kinetics are
primordial. A few-TeV channel needs no extreme ultraviolet prehistory, and the resonant regime ---
rather than a far-higher-scale seesaw --- is what the tiny inherited light masses point to.

\subsubsection{A numerical audit table for the baryogenesis chain}

\begin{table}[H]
\centering
\caption[The resonant-leptogenesis chain used in Part V]{Resonant-leptogenesis chain used in Part~V. Each row is fixed by an inherited input;
the one named assumption is the on-resonance condition, in its partial-resonance sharp form
(Appendix~\ref{appA:leptogenesis}). The columns give each ingredient, its value, and where in the framework it comes from. The third column is what makes the chain checkable: an entry sourced to an upstream paper carries that paper's grade, and the single named assumption is marked as one rather than absorbed into the others.}
\label{tab:baryogenesis-chain-partIII}
\small
\begin{tabular}{p{0.34\textwidth}p{0.22\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Ingredient} & \textbf{Value} & \textbf{Origin in the framework}\\
\midrule
Lock scale / heavy mass & \(M_N\simeq\Mlock\simeq 3.85\,\mathrm{TeV}\) & \OGNfive{} throat-depth reduction from \(\EPl\)\\
Seesaw Yukawa & \(Y_\nu\simeq 7.3\times10^{-7}\) & Matching to \ANine{} Weinberg scale, Eq.~\eqref{eq:Ynu-matching-partV}\\
Heavy-neutrino splittings & \(\delta_i=O(1/K)\) & Quasi-degenerate lock-scale triplet\\
Resonant CP asymmetry & \(\epsilon^{\rm res}\simeq 2.5\times10^{-6}\) & Flavored projection of inherited phases; H-\(\phi_{\rm geo}\)-1 target value, Eq.~\eqref{eq:eps-value-partV}\\
Strong-washout efficiency & \(\kappa\simeq 1\times10^{-2}\) & \(K_{\rm wash}\simeq50\), Eq.~\eqref{eq:kappa-value-partV}\\
Effective CP throughput & \(\epsilon^{\rm res}\kappa\simeq2.51\times10^{-8}=e^{-K/2}\) & Product reconstructs the geometric ledger value\\
Relativistic d.o.f. & \(g_*\simeq106.75\) & Standard SM-like bath at the TeV scale\\
Sphaleron factor & \(28/79\) & Standard one-Higgs-doublet conversion\\
Equilibrium yield & \(\sum_i Y_{N_i}^{\rm eq}\simeq1.25/g_*\) & Three quasi-degenerate Majorana species, Eq.~\eqref{eq:YNeq-partIII}\\
Entropy-to-photon factor & \(7.04\) & Standard late-time yield-to-\(\eta_B\) conversion\\
Final asymmetry & \(\eta_B\simeq7.34\times10^{-10}\) & Eq.~\eqref{eq:etaB-numerical-partIII}\\
Present baryon fraction & \(\Omega_b=1/(2\pi^2)=0.0507\) & \(\pi\)-locked inventory target\\
\bottomrule
\end{tabular}
\end{table}

Once the lock scale and throat-depth integer are inherited, the seesaw matching fixes the Yukawas,
and the route is fixed to leptogenesis, the asymmetry estimate becomes rigid: there is very little
hidden freedom left.

\subsection{From baryon asymmetry to the present baryon density fraction}

Cosmology also summarizes the late baryon sector through the density fraction. Within the
\(\pi\)-locked inventory,
\begin{equation}
\Omega_b = \frac{1}{2\pi^2}\simeq 0.0507.
\label{eq:Omega-b-pred-partIII}
\end{equation}
This compact number gives the BBN and CMB sectors a sharp benchmark and makes clear that the baryonic
share is a structured sub-share of the matter share, not comparable to it. The relation between the
early asymmetry and the late density runs through
\begin{equation}
\eta_B
\Longrightarrow
n_B(a)
\Longrightarrow
\rho_b(a)=m_p n_B(a)
\Longrightarrow
\Omega_b = \frac{\rho_b(t_0)}{\rho_{\rm crit}(t_0)},
\label{eq:eta-to-Omegab-partIII}
\end{equation}
ordinary evolution once the asymmetry is fixed. Equation~\eqref{eq:etaB-numerical-partIII} says the
initial asymmetry sits at the correct level.

\subsection{Why exact proton stability and successful baryogenesis are compatible}

If \(\U(1)_B\) is exact, how can the universe acquire a baryon asymmetry? Exact \(\U(1)_B\) refers to
the fundamental conserved baryonic ledger of the full \GG{} system, whereas cosmological baryogenesis
is a statement about the projected four-dimensional open subsystem. The relevant processes are not
local proton-decay operators but boundary processes --- a lepton asymmetry, sphaleron conversion,
open-system transport. The full bulk ledger remains consistent while the observable subsystem relaxes
into a state with nonzero net baryon density. This is exactly the situation \OGGEP{} and \GDOS{} were
introduced for: a conserved quantity in the full system can present as an evolving charge budget in an
open subsystem when ports and projection are present. There is no contradiction. Proton decay remains
a sharp falsifier; baryogenesis remains available through the constrained leptogenesis channel rather
than through late-time baryon-violating operators --- more economical than GUT scenarios in which the
baryon-creating sector also threatens the proton.

\subsection{What the baryogenesis block predicts and how it can fail}

\begin{enumerate}[leftmargin=2.2em]
\item The heavy-neutrino sector sits near the lock scale, \(M_N\simeq\Mlock\), not at an arbitrary
intermediate scale.
\item The seesaw matching to the \ANine{} Weinberg scale fixes \(Y_\nu\simeq7\times10^{-7}\); the
light block and its CP structure are compatible with a viable resonant-leptogenesis descendant.
\item The net throughput is \(\epsilon^{\rm res}\kappa\simeq e^{-K/2}\) with \(K=35\): a
phase-controlled resonant asymmetry times a strong-washout efficiency.
\item The resulting asymmetry is \(\eta_B \approx 7\times10^{-10}\) (central \(7.34\); band \(\times3^{\pm1}\): \((2.4\text{--}22)\times10^{-10}\)) and the late baryon fraction is
\(1/(2\pi^2)\).
\end{enumerate}
Every element can fail. If a full transport calculation shows no consistent on-resonance
heavy-neutrino / washout regime near the lock scale, the chain fails. If cosmological data force a
neutrino mass sum or CP structure incompatible with the particle paper, the backbone fails. If
proton decay is observed, the whole baryonic logic fails. The section is acutely exposed to
observation.

\subsection{Exact neutrality: why the books balance and the electrons match the protons}
\label{ssec:neutrality}

The chain just assembled produces asymmetries; this subsection records
why it can never produce the wrong kind. The statement is a theorem of
the inherited structure: \emph{every gauged charge has exactly zero
density at every epoch; asymmetry is possible only in global numbers.}

Four links. \emph{First, the start is neutral.} The edge data are
geometric and the geometry is a gauge singlet: a nonzero gauge charge
on \(\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}\) would be an extra initial
datum that the inheritance clause of \S\ref{sec:beginning} does not
supply. Independently, at the edge the boundary group is the unbroken
unified group, and a generator of a simple group is traceless on every
multiplet \citep{ArisakaA9SM}: the same trace identity that makes the
hydrogen atom neutral makes any democratic population of complete
multiplets neutral. One identity, two epochs --- the neutrality of the
first plasma and of the last atom are the same traceless matrix, read
at \(r = 0\) and at recombination.

\emph{Second, the zero is conserved.} The covariant conservation
identity holds the boundary Ward currents conserved as one bulk
identity down the entire descent (\citealp{ArisakaA6Gi4};
\citealp{ArisakaA2GG6}), and its microscopic face is that every vertex
conserves charge: charged quanta enter the world only in
zero-net-charge combinations, the electron--positron pair being the
minimal case.

\emph{Third, the zero is enforced, not merely remembered.} A gauged
charge sources its own long-range field, so it cannot hide. An exactly
homogeneous and isotropic background admits no field strength, and the
Maxwell equation then forces the charge density to vanish: a charged
universe cannot be \FRW{} --- as was recognized the first time a
charged cosmology was seriously proposed \citep{LyttletonBondi1959}.
Dynamically the enforcement is faster than any cosmological clock: a
local excess relaxes at the plasma frequency, some twenty orders of
magnitude above the expansion rate during nucleosynthesis. And even a
maximal primordial excess would have been diluted by
\(e^{-3N^*} \sim 10^{-69}\) during the launch --- the theorem is
protected by symmetry and, independently, by dilution.

\emph{Fourth, the zero decides who survives.} \(B{-}L\) is global; no
field polices it; the lock-scale block may load it, and does. The
sphaleron bookkeeping that converts it runs on the plasma's chemical
equilibrium, in which the hypercharge-neutrality condition --- so
often imported silently --- is simply the third link applied to the
gauged hypercharge above the crossover. After the electroweak epoch
the surviving charged species are the baryons and the electrons, and
the constraint \(\sum_i Z_i n_i = 0\), enforced continuously, pins the
electron density to the proton charge as the positrons annihilate
away:
\begin{equation}
n_e \;=\; n_p + 2\,n_{\mathrm{He}} + \cdots
\label{eq:partV-ne-np}
\end{equation}
exactly --- the relation every Saha and Peebles equation of the
recombination chain silently uses, here no longer silent. The electron
excess over the positrons is not separately arranged; it is the image
of the baryon excess under the neutrality constraint. Empirically the
exactness is audited at one part in \(10^{21}\) by the neutrality of
bulk matter \citep{Bressi2011}.

The register of the result matters: neutral in every charge that is
gauged, asymmetric only in charges that are global --- neutral where a
field would betray the imbalance, asymmetric only where nothing
watches. Both halves are forced, and the third horizon problem of
Part~X (\S\ref{sec:third-horizon}) is the same coin seen from the sky.

\section{Big Bang nucleosynthesis and the lithium-7 audit}

\subsection{Historical background}

The hot-Big-Bang nucleosynthesis program was inaugurated in 1948 by Alpher, Bethe, and Gamow \citep{AlpherBetheGamow1948}, who proposed that the light elements are produced by neutron-proton interactions in a cooling, expanding plasma. Modern BBN predicts the primordial abundances of D, $^3$He, $^4$He, and $^7$Li from a small handful of inputs --- the baryon-to-photon ratio, the effective neutrino number, and the standard thermal history --- and agrees with observation to better than ten percent for the first three. The residual lithium-7 discrepancy is the only persistent anomaly. The present section asks whether the inherited ledger modifies any BBN input in a way that addresses lithium-7 without disturbing the other three.

\label{sec:bbn}

\subsection{Why BBN is the first precision audit of Part~V}

Cosmology does not permit stopping at attractive narratives. At temperatures of order the MeV and
times of order seconds to minutes, the universe records its matter and radiation ledger in a way
still observable today. BBN is therefore the first hard quantitative audit of the whole
origin-of-matter story. Its dependence on the input set is broad yet clean: the expansion rate, the
baryon density, the neutrino/radiation content, the weak freeze-out history, and the nuclear rates.
It is close enough to fundamental physics to test the inherited ledger directly, but late enough that
the boundary \(\FRW\) description already works well. The empirical situation is instructive:
standard BBN agrees remarkably with primordial \(^4\)He and deuterium, while the long-standing
\(^7\)Li problem persists. A successful framework must preserve what works and change only what is
under pressure.

\subsection{Standard BBN baseline: freeze-out, bottleneck, and synthesis}

Above about an MeV, neutrons and protons interconvert rapidly,
\begin{equation}
 n + \nu_e \leftrightarrow p + e^-,
\qquad
 n + e^+ \leftrightarrow p + \bar\nu_e,
\qquad
 n \leftrightarrow p + e^- + \bar\nu_e,
\label{eq:weak-np-channels-partIII}
\end{equation}
so the neutron-to-proton ratio is approximately thermal,
\begin{equation}
\frac{n}{p}\simeq e^{-\Delta m/T},
\qquad
\Delta m \equiv m_n-m_p\simeq 1.293\,\mathrm{MeV}.
\label{eq:np-thermal-partIII}
\end{equation}
As the weak rate falls below Hubble, the ratio freezes out near one neutron to several protons; after
a little free decay these neutrons are incorporated into \(^4\)He once the network ignites. The delay
is the deuterium bottleneck: the intense photon bath photodissociates deuterons until
\begin{equation}
T_{\rm D}\sim 0.07\text{--}0.09\,\mathrm{MeV},
\label{eq:deuterium-bottleneck-temp-partIII}
\end{equation}
after which the chain
\begin{equation}
p+n\to D+\gamma,
\quad
D+p\to {}^3\mathrm{He}+\gamma,
\quad
D+n\to {}^3\mathrm{H}+\gamma,
\quad
{}^3\mathrm{He}+{}^3\mathrm{H}\to {}^4\mathrm{He}+p+n,
\label{eq:bbn-core-network-partIII}
\end{equation}
processes most neutrons into \(^4\)He. The standard yields are
\begin{equation}
Y_p \equiv \frac{4n_{{}^4\mathrm{He}}}{n_b}\sim 0.245,
\qquad
D/H \sim 2.5\times10^{-5},
\qquad
{}^7\mathrm{Li}/H \sim \text{few}\times10^{-10}.
\label{eq:bbn-yields-partV}
\end{equation}
The general evolution
\begin{equation}
\dot Y_i
=
\sum_r N_{ir}\,\lambda_r(T)\prod_j Y_j^{\nu_{jr}},
\label{eq:standard-bbn-network-general-partIII}
\end{equation}
with \(Y_i=n_i/n_b\), \(N_{ir}\) the net stoichiometric coefficient, and \(\lambda_r(T)\) the rates,
is the starting point for the \GDOS{} extension.

\subsubsection{Freeze-out and bottleneck in explicit numbers}

The weak rate scales as \(\Gamma_{n\leftrightarrow p}\sim G_F^2 T^5\); setting it equal to Hubble,
Eq.~\eqref{eq:H-rad-kaxion-partIII}, gives a freeze-out
\begin{equation}
T_f\sim 0.7\text{--}0.8\,\mathrm{MeV},
\qquad
\left(\frac np\right)_f\sim e^{-1.293/0.75}\sim 0.18,
\label{eq:weak-freezeout-partV}
\end{equation}
the familiar ``one neutron per five or six protons.'' With nearly all surviving neutrons going into
\(^4\)He,
\begin{equation}
Y_p\simeq \frac{2(n/p)}{1+n/p},
\label{eq:Yp-thumb-rule-partIII}
\end{equation}
so \((n/p)\sim 1/7\) gives \(Y_p\sim0.25\). The deuterium bottleneck breaks only at the
\(0.07\text{--}0.09\,\mathrm{MeV}\) corridor because the enormous photon-to-baryon ratio keeps the
high-energy tail photodissociating deuterons long after the average photon energy drops below the
\(2.2\,\mathrm{MeV}\) binding. These two estimates show why BBN is robust yet selective: the helium
fraction is set long before the A=7 channel matters, deuterium tracks the baryon density, and the A=7
abundance is tiny and late enough that a selective correction can modify it without disturbing the
rest.

\subsubsection{Why deuterium is such a powerful baryometer}

Deuterium is easily burned in stars and made primarily in the early universe, so its primordial
abundance is a clean probe of the baryon density: more baryons burn more efficiently and drive
\(D/H\) down. That is why \(D/H\) and the CMB point to nearly the same baryon density. Any Part~V
proposal altering the baryon ledger too much would fail immediately. The Kaxion corridor is safe
because it modifies the nonbaryonic share; leptogenesis is safe because it sets the asymmetry rather
than adding a late baryon correction; the lithium port is safe only because it is selective and tiny
in total transferred abundance. All three survive only by respecting deuterium.

\subsection{The lithium-7 problem and the A=7 channel}

Standard BBN is not broken; its deuterium and helium predictions are among the great successes of
precision early-universe physics. That success constrains any new proposal: a \(^7\)Li mechanism must
leave \(Y_p\) and \(D/H\) intact. The lithium problem enters through the A=7 channel. At the
CMB/deuterium baryon density, the dominant mass-7 route is not primordial \(^7\)Li but \(^7\)Be
formation,
\begin{equation}
{}^3\mathrm{He}+\alpha \to {}^7\mathrm{Be}+\gamma,
\label{eq:Be7-production-partIII}
\end{equation}
followed much later by electron capture,
\begin{equation}
{}^7\mathrm{Be}+e^- \to {}^7\mathrm{Li}+\nu_e,
\label{eq:Be7-ec-partIII}
\end{equation}
whose product is quoted as the primordial lithium abundance. Standard BBN gives
\begin{equation}
\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm SBBN}
\simeq 5.0\times10^{-10}
\label{eq:Li7-SBBN-partIII}
\end{equation}
\citep{CyburtFieldsOliveYeh2016,Pitrou2018}, whereas the metal-poor halo-star (Spite) plateau gives
\begin{equation}
\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm obs}
\simeq (1.6\pm0.3)\times10^{-10}
\label{eq:Li7-obs-partIII}
\end{equation}
\citep{Fields2011}. The discrepancy is a factor of about three. Many explanations have been proposed
--- stellar depletion, new-particle injection, rate modifications. The narrower question here is
whether the \GG{}/\GDOS{} framework contains a \emph{minimal, inherited, selective} mechanism acting on
the A=7 channel while leaving deuterium and helium intact.

\subsection{The A=7 port and the \texorpdfstring{$1/\pi$}{1/pi} prediction}

The standard treatment of the lithium anomaly leaves the size of any A=7 suppression as a free
parameter to be fitted to the observed deficit. The present framework does not fit it. The same
closed-packet branching that fixes the cosmic matter inventory fixes the A=7 survival fraction, and
it does so with no new freedom.

Recall the topological origin of the inventory (Conjecture~VIII / the closed-packet derivation): when
a conserved flux is distributed over the compact angular phase, the fraction that survives onto the
observable boundary is set by the geometry of the closed packet over \(S^1_\theta\), and the
surviving share is \(1/\pi\). This is the identical \(\pi\)-locking that yields the matter share
\(\Omega_m=1/\pi\). The claim of this section is that the A=7 channel is an open-system port of
exactly this type: the mass-7 flux carried by \(^7\)Be is branched over the same compact-angle phase
during its open electron-capture / phase-noise window, and the surviving fraction is the same
geometric factor,
\begin{equation}
S_{A=7} = \frac{1}{\pi}.
\label{eq:S-A7-pi-partV}
\end{equation}
The A=7 yield is therefore suppressed by a \emph{predicted} factor of \(\pi\), not a fitted one. The
prediction is
\begin{equation}
\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm GG}
=
\frac{1}{\pi}\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm SBBN}
\simeq
\frac{5.0\times10^{-10}}{3.14159}
\simeq 1.6\times10^{-10}.
\label{eq:Li7-GG-prediction-partV}
\end{equation}
This matches the observed plateau, Eq.~\eqref{eq:Li7-obs-partIII}, with no parameter adjusted: the
identical \(1/\pi\) that fixes \(\Omega_m\) fixes the lithium deficit. The closed-packet network
analysis behind Eq.~\eqref{eq:S-A7-pi-partV} --- the open-system A=7 port and the explicit
phase-branching that produces the surviving fraction \(1/\pi\) --- is given in
Appendix~\ref{appA:leptogenesis}.

The status is therefore D\(^\star\) (Derived-conditional), sharing the same closed-packet origin as
the inventory: the \(1/\pi\) is derived modulo the closed-packet-integration conjecture, not fitted.
The prediction is crisp and falsifiable: \emph{any firm primordial \(^7\)Li abundance away from
\(\mathrm{SBBN}/\pi\) falsifies the port.} In particular, if improved analyses placed the primordial
value at the full SBBN level, or at a value not equal to \(\mathrm{SBBN}/\pi\) within errors, the
mechanism would be excluded.

\subsection{Why deuterium and helium are preserved}

The selectivity is the whole point; two arguments show it.

\paragraph{(i) Temporal hierarchy.}
The \(^4\)He yield is fixed much earlier, by the frozen neutron-to-proton ratio and the end of the
deuterium bottleneck. By the time the A=7 port acts, the helium mass fraction is already a locked
attractor, so the lithium correction does not feed back into \(Y_p\).

\paragraph{(ii) Extreme abundance hierarchy.}
The A=7 abundance is tiny compared with deuterium and helium,
\begin{equation}
Y_{\rm Be7}\sim 10^{-10},
\qquad
D/H\sim 2.5\times10^{-5},
\qquad
Y_p\sim 0.25,
\label{eq:abundance-hierarchy-partIII}
\end{equation}
so even if every removed \(^7\)Be nucleus fed a side channel, the maximum fractional shift in \(D/H\)
is
\begin{equation}
\frac{\Delta(D/H)}{D/H}
\lesssim
\frac{7\times 10^{-10}}{2.5\times10^{-5}}
\sim 3\times10^{-5},
\label{eq:DH-shift-tiny-partIII}
\end{equation}
far below the precision at which deuterium is used, and the effect on \(Y_p\) is negligible. The port
acts where the standard theory is under pressure and is harmless where it succeeds.

\begin{table}[H]
\centering
\caption[Light-element audit used in Part V]{Light-element audit used in Part~V. The first two data columns summarize the standard
baseline and the observed corridor; the last gives the inherited \GG{}/\GDOS{} effect. The A=7 entry is a
\(1/\pi\) prediction, not a fit. The last column is where this framework enters; the first three are the standard baseline and the observed corridor, set down so the size of the proposed effect can be read against them directly.}
\label{tab:light-element-audit-partIII}
\small
\begin{tabular}{p{0.20\textwidth}p{0.22\textwidth}p{0.22\textwidth}p{0.26\textwidth}}
\toprule
\textbf{Observable} & \textbf{Standard BBN} & \textbf{Observed corridor} & \textbf{Part~V effect}\\
\midrule
\(Y_p\) & \(\sim0.245\) & close to standard & Essentially unchanged\\
\(D/H\) & \(\sim2.5\times10^{-5}\) & close to standard & Essentially unchanged\\
\({}^3\mathrm{He}/H\) & small & not dramatically discrepant & Essentially unchanged\\
\({}^7\mathrm{Li}/H\) & \(\simeq5.0\times10^{-10}\) & \((1.6\pm0.3)\times10^{-10}\) & \(\mathrm{SBBN}/\pi\simeq1.6\times10^{-10}\) (predicted)\\
\bottomrule
\end{tabular}
\end{table}

The light-element sector is not ``improved everywhere.'' It is preserved where it already succeeds
and corrected only where the standard baseline is under pressure --- and there the correction is the
inherited factor \(\pi\), the same number that fixes the cosmic matter inventory.

\subsection{What the lithium audit predicts and how it can fail}

\begin{enumerate}[leftmargin=2.2em]
\item The inherited baryon density stays close to the CMB/deuterium-preferred value, preserving the
successful \(^4\)He and D/H predictions.
\item The A=7 open-system port suppresses the mass-7 yield by exactly \(1/\pi\), giving
\((^7\mathrm{Li}/H)_{\rm GG}\simeq1.6\times10^{-10}\) --- a prediction matching the observed plateau,
with no fitted strength.
\item Because the affected abundance is tiny, the same correction does not disturb the more
successful sectors.
\end{enumerate}
The claims are sharply falsifiable. If improved analyses show the deficit is mostly stellar rather
than primordial, the port is unnecessary. If the primordial value is firmly anything other than
\(\mathrm{SBBN}/\pi\), the port is excluded. Either way the section proposes a specific, inherited
mechanism rather than a vague hope --- and it stakes the same \(1/\pi\) that the rest of the program
relies on.

\subsection{A consistency chain from \texorpdfstring{\(\eta_B\)}{etaB} to \texorpdfstring{\(\Omega_b\)}{Omegab} and the baryon-to-dark-matter ratio}

\paragraph{One quantity, three sets of units.} Before the chain is
assembled, one point of bookkeeping that governs how the rest of this
subsection should be read. The baryon-to-photon ratio \(\eta_B\), the
baryon density \(\rho_{b}\) and the baryon fraction \(\Omb\) are not three
predictions. They are \emph{one} physical quantity in three sets of units,
tied by conversions that contain no content of this framework's:
\begin{equation}
\eta_{10} \;=\; 273.9\,\Omb h^{2},
\qquad
\rho_{b} \;=\; \Omb h^{2}\times 1.87847\times10^{-29}~\mathrm{g\,cm^{-3}},
\qquad
\eta_{10} \equiv \eta_B\times10^{10}.
\label{eq:baryon-bridge-partIII}
\end{equation}
Fix any one of the three and the other two follow with no freedom. What
this paper has is therefore not three baryon predictions with three error
bars, but \emph{two independent roads to one number} --- and the roads
should be quoted in order of what they are worth.

\emph{The ledger road first.} \(\Omb = 1/(2\pi^2)\) converted through
Eq.~\eqref{eq:baryon-bridge-partIII} at this paper's own derived expansion
rate \(h = 0.666\) gives \(\eta_{10} = 6.15\) against an observed
\(6.12\): an error of \(+0.57\%\), with nothing fitted anywhere in the
chain. At the microwave background's own \(h = 0.674\) it gives \(6.30\),
an error of \(+3.0\%\). The honest range for this road is half a per cent
to three per cent, depending on which \(\Hzero\) one takes as settled ---
and that spread is smaller than the two-route disagreement it is about to
be compared against.

\emph{The transport road second, and as what it is.} The chain assembled
below lands at \(\eta_{10} \simeq 7.34\), an error of \(+19.9\%\) inside a
declared factor-\(\sim\)3 band, and it is graded D\(^*\) on a named
finite-temperature hypothesis. Its claim is therefore \emph{that a
parameter-free mechanism reaches this order of magnitude}, not that the
value is \(7.34\) --- which is exactly what the codebook grading already
says. Read that way the two roads are not rivals of equal standing: the
static count is this paper's prediction of the number, and the dynamical
chain is the demonstration that a mechanism exists which could have
produced it.

It is worth asking whether the three modules speak to one another
quantitatively. The baryogenesis
subsection derived an asymmetry in the right corridor, Eq.~\eqref{eq:etaB-numerical-partIII}; the
inventory proposes a geometric baryon fraction, Eq.~\eqref{eq:Omega-b-pred-partIII}. These are not
the same statement --- one microphysical, one a late-time density share --- and one should show how
they talk. The present photon number density is
\begin{equation}
 n_{\gamma,0}
 =
 \frac{2\zeta(3)}{\pi^2}\,T_0^3
 \simeq 410\;{\rm cm}^{-3},
 \label{eq:ngamma0-partIII}
\end{equation}
so an asymmetry \(\eta_B\) implies
\begin{equation}
 \rho_{b,0}
 =
 \eta_B\,m_p\,n_{\gamma,0}
 \simeq
 5.0\times10^{-31}
 \left(\frac{\eta_B}{7.34\times10^{-10}}\right)
 {\rm g\,cm}^{-3}.
 \label{eq:rhob-from-etab-partIII}
\end{equation}
With the critical density \(\rho_{\rm crit,0}=1.878\times10^{-29}\,h^2\;{\rm g\,cm}^{-3}\),
\begin{equation}
 \Omega_b h^2
 \simeq
 0.0268
 \left(\frac{\eta_B}{7.34\times10^{-10}}\right),
 \label{eq:Omegabh2-from-etab-partIII}
\end{equation}
so for \(h\simeq0.67\text{--}0.70\), \(\Omega_b\simeq0.055\text{--}0.060\), in the same corridor as the geometric target \(1/(2\pi^2)\simeq0.0507\). Stated honestly, the two routes do not yet exactly agree: the topological route gives \(\Omega_b = 1/(2\pi^2) = 0.05066\), the asymmetry route \(\Omega_b h^2 \simeq 0.0268\) (i.e. \(\Omega_b \approx 0.059\) at \(h=0.674\)) --- a \(\sim16\%\) gap, branch-dependent and shrinking through zero near the ladder branch (\(\approx-2\%\) at \(h=0.735\)), with the asymmetry route additionally carrying the transport chain's factor-\(\sim\)3 band, inside which the topological value sits comfortably. The residue of the seam is not symmetric between the routes: anchored to the measured \(\Omega_bh^2 = 0.02237\), the transport route's \(+20\%\) is priced by its own declared band, while the topological route read at the corridor's \(h=0.674\) sits \(+3\%\) high with no \(\mathcal O(1)\) freedom to price it --- so the unowned residue sits on the corridor-\(h\) conversion or on the exactness of \(1/(2\pi^2)\), not on the transport \(\mathcal O(1)\)'s. Throughout this paper the density
fractions \(\Omm,\OmL,\Omb\) are defined on the early/CMB branch (\(h=0.674\)); reconciling the two
routes to \(\Omega_b\) is a named open problem (two-route-consistency) recorded alongside the
closed-packet-integration conjecture. A future precision \ATen{} would compute the offset from the
full entropy history, sphaleron throughput, reheating profile, and lock-scale neutrino kinetics; even
the present estimate shows the matter-sector story is not wildly self-inconsistent.

The sharper prediction is the relative matter shares. Combining the \(\pi\)-locked inventory
\begin{equation}
 \Omega_m = \frac{1}{\pi},
 \qquad
 \Omega_b = \frac{1}{2\pi^2},
 \qquad
 \Omega_{\rm DM}=\frac{1}{\pi}-\frac{1}{2\pi^2},
 \label{eq:pi-locked-matter-repeated-partIII}
\end{equation}
fixes the dark-to-baryon ratio with no freedom,
\begin{equation}
 \frac{\Omega_{\rm DM}}{\Omega_b}
 = 2\pi -1
 \simeq 5.283,
 \label{eq:DM-to-baryon-ratio-pi-partIII}
\end{equation}
the cosmic baryon fraction of all matter,
\begin{equation}
 f_b
 \equiv
 \frac{\Omega_b}{\Omega_m}
 =
 \frac{1}{2\pi}
 \simeq 0.159,
 \label{eq:f_b-pi-partIII}
\end{equation}
and its dark complement
\begin{equation}
 f_{\rm DM}=1-f_b=1-\frac{1}{2\pi}\simeq0.841.
 \label{eq:fDM-pi-partIII}
\end{equation}
These rigid, dimensionless numbers --- about five times more dark matter than baryons, roughly
sixteen percent of all matter baryonic --- are obtained without independent per-sector fitting.

\begin{table}[H]
\centering
\caption[Bridge from baryogenesis language to the late-time inventory]{Bridge from the baryogenesis language to the late-time inventory language. The relative
matter ratios are especially sharp. The two right-hand columns reach the same quantities by different roads --- the asymmetry route and the geometric inventory route --- and the table is a consistency check rather than a derivation. Where the two disagree the disagreement is the finding, and the relative matter ratios are where it would show first.}
\label{tab:partIII-cross-check-ledger}
\small
\begin{tabular}{p{0.29\textwidth}p{0.27\textwidth}p{0.28\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Asymmetry route} & \textbf{Geometric inventory route}\\
\midrule
\(\eta_B\) & \(\approx 7\times10^{-10}\) (central \(7.34\)) from Eq.~\eqref{eq:etaB-numerical-partIII} & not primary input\\
\(\Omega_b h^2\) & \(\simeq 0.027\) from Eq.~\eqref{eq:Omegabh2-from-etab-partIII} & implied after choosing \(h\)\\
\(\Omega_b\) & \(\simeq0.055\text{--}0.060\) for \(h\simeq0.67\text{--}0.70\) & \(1/(2\pi^2)\simeq0.0507\)\\
\(\Omega_m\) & not directly fixed & \(1/\pi\simeq0.3183\)\\
\(\Omega_{\rm DM}/\Omega_b\) & near observed value of order five & \(2\pi-1\simeq5.283\)\\
\bottomrule
\end{tabular}
\end{table}

Part~V should be read not as three disconnected fixes but as one constrained matter ledger viewed at
three epochs. The Kaxion fixes the dominant cold component; resonant leptogenesis fixes the sign and
size of the baryonic remnant; BBN audits whether that remnant is compatible with light-element
synthesis. The closer the three collapse onto one compact ledger, the more serious the program.

\subsection{Chronological ordering inside Part~V}

A proposal can sound convincing while being chronologically inconsistent, so the temporal ordering
deserves explicit statement. The first event is exit from the slow-smooth inflationary phase of Part~IV, depositing energy in a hot bath at the derived reheating temperature,
\begin{equation}
 T_{\rm RH}\simeq 3\times10^{10}\;{\rm GeV}.
 \label{eq:TRH-lockscale-partIII}
\end{equation}
The bath then cools, and the heavy Majorana sector becomes dynamical when the temperature falls to the lock scale, \(T\sim M_{N_i}\sim M_{\rm lock}\simeq3.85\)~TeV, at \(t\sim10^{-13}\,\mathrm{s}\). There the decays and inverse decays occur promptly, and the resonant-leptogenesis module belongs exactly here;
the \(B-L\) asymmetry is then reprocessed by sphalerons before the weak crossover shuts that channel
off. The asymmetry is fixed extremely early, long before nuclei appear.

The Kaxion is different. It is already present as a low-energy compact-angle remnant, behaving
initially as vacuum misalignment. Coherent oscillation onsets when
\begin{equation}
 3H(T_{\rm osc}) \simeq m_K(T_{\rm osc}).
 \label{eq:oscillation-condition-partIII}
\end{equation}
With a naively constant mass and radiation domination, Eq.~\eqref{eq:H-rad-kaxion-partIII} gives
\begin{equation}
 T_{\rm osc}^{\rm naive}
 \simeq
 \left(\frac{m_K M_{\rm Pl}}{4.98\,g_*^{1/2}}\right)^{1/2}
 \simeq 49\;{\rm GeV}
 \qquad\text{for }m_K\simeq 8.1\,\mu{\rm eV},
 \label{eq:Tosc-naive-partV}
\end{equation}
using a representative \(g_*\sim70\). Because the QCD-anchored mass turns on only near confinement,
the true onset is pushed down; the full \(\chi_{\rm QCD}(T)\) treatment of \PaperFullCOne{} places it
at \(T_{\rm osc}\simeq0.54\,\mathrm{GeV}\), in the QCD-era corridor. The point that matters: the
Kaxion abundance is set after the baryon asymmetry, yet long before galaxies. No temporal clash.

BBN occurs much later, at MeV and sub-MeV temperatures, with the landmarks
\begin{equation}
 T\sim 0.7\;{\rm MeV}\ \Rightarrow\ n/p\ \text{freeze-out},
 \qquad
 T\sim 0.08\;{\rm MeV}\ \Rightarrow\ \text{deuterium release},
 \label{eq:BBN-landmarks-partIII}
\end{equation}
and synthesis around \(t\sim10^2\text{--}10^3\,\mathrm{s}\). The lithium port acts vastly later than
leptogenesis and vastly colder than the Kaxion oscillation onset. Each module occupies its own epoch
and speaks to the others only through inherited conserved quantities, summarized in
Table~\ref{tab:partIII-chronology}.

\begin{table}[H]
\centering
\caption[Chronological ordering of the Part V mechanisms]{Chronological ordering of the Part~V mechanisms. The three matter-sector modules inhabit
distinct epochs and combine without double counting. The columns give the epoch, its characteristic scale, and the role the mechanism plays. The ordering carries the argument: the three matter-sector modules occupy distinct epochs, so they combine without competing for the same energy budget, and a reader who suspects double counting can check it from this table alone.}
\label{tab:partIII-chronology}
\small
\begin{tabular}{p{0.22\textwidth}p{0.23\textwidth}p{0.40\textwidth}}
\toprule
\textbf{Epoch} & \textbf{Characteristic scale} & \textbf{Part~V role}\\
\midrule
Post-inflation reheating & \(T_{\mathrm{rh}}\simeq 3\times10^{10}\,\mathrm{GeV}\) & Populates the heavy Majorana sector thermally (the standard leptogenesis window is now open); the bath then cools through the lock, preparing the out-of-equilibrium conditions\\
Leptogenesis / sphaleron era & \(T\sim \mathrm{TeV}\to100\,\mathrm{GeV}\) & Generates \(B-L\), converts part to baryon asymmetry while preserving exact late-time proton stability\\
Kaxion misalignment to oscillation & below the electroweak era, toward the strong-coupling turn-on & Converts the compact-angle remnant into a cold matter component without disturbing the fixed baryon sign\\
BBN & \(T\sim\mathrm{MeV}\to0.01\,\mathrm{MeV}\), \(t\sim1\text{--}10^3\,\mathrm{s}\) & Audits the baryon density and applies the selective A=7 lithium port\\
\bottomrule
\end{tabular}
\end{table}

With this ordering, Part~V acquires conceptual cleanliness: the same inherited \GG{}/\GDOS{} ledger
determines what survives reheating, what baryon sign is left after the electroweak era, what fraction
of matter ends up cold and non-baryonic, and whether the resulting baryon density passes the first
precision nuclear audit.

\subsection{A compact parameter ledger for the matter sector}

It is worth recording how little continuous freedom the draft uses. The matter sector is governed by
a small ledger once \ANine{} is treated as inherited input: the \OGN{} integer set and the lock scale,
the geometric Kaxion decay constant and its discrete window, the already-derived neutrino masses and
phases, the seesaw-matched Yukawa, the discrete sphaleron factor, and the closed-packet \(1/\pi\)
that fixes the lithium port. None of these is a free scan parameter; the apparent complexity of
Part~V is organizational, not parametric.

\begin{table}[H]
\centering
\caption[Compact parameter ledger for the Part V matter sector]{Compact parameter ledger for the Part~V matter sector, distinguishing inherited quantities,
derived outputs, and the one named assumption. The middle column is the honest one. It separates what is inherited from what is derived here and names the single quantity that is neither, and a reader who wants to know what this Part actually assumes should read that column before any of the numbers.}
\label{tab:partIII-parameter-ledger}
\small
\begin{tabular}{p{0.24\textwidth}p{0.23\textwidth}p{0.41\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Status in Part~V} & \textbf{Role}\\
\midrule
\(K=35\) and \OGN{} integers & inherited discrete input & Fix twin-peak split, lock-scale exponents, \(e^{-K/2}\) throughput\\
\(M_{\rm lock}\simeq3.85\,\mathrm{TeV}\) & inherited from \ANine{} & Heavy-neutrino scale and inflation-to-matter bridge\\
\(m_K\) benchmark corridor & inherited / geometry-selected & Dark-matter oscillation and haloscope targets\\
\(\theta_i\approx1.4\) rad & output of the \(\pi\)-locked abundance & Misalignment amplitude, fixed by the abundance (not free)\\
\((m_1,m_2,m_3)\), \(\delta_{\rm PMNS}\), \(J_{\rm PMNS}\) & inherited from \ANine{} & Anchor the light-neutrino and CP ledger feeding leptogenesis\\
\(Y_\nu\simeq7.3\times10^{-7}\) & derived by seesaw matching & Fixes \(\Gamma_N\) and the washout scale\\
on-resonance condition \(|M_i-M_j|\simeq\Gamma/2\) & one named assumption & Lifts the tiny-Yukawa suppression of \(\epsilon\)\\
\(c_{\rm sph}=28/79\) & Standard Model fixed factor & Converts \(B-L\) to the baryon asymmetry\\
A=7 port factor \(1/\pi\) & closed-packet prediction (D\(^\star\)) & Fixes the lithium deficit, same origin as \(\Omega_m=1/\pi\)\\
\bottomrule
\end{tabular}
\end{table}

Part~V is thus a compact stress test of whether a few inherited geometric inputs can simultaneously
speak to cold dark matter, the baryon asymmetry, and the first nuclear audit of the universe.

\subsection{Bottom line of Part~V}

The three origin-of-matter problems fit together far more tightly than in an unconstrained
model-building program. The Kaxion is the natural first dark-matter candidate because it descends
from the compact-angle and Green--Schwarz sectors already present in the inherited theory.
Baryogenesis is driven to resonant leptogenesis because the neutrino sector is already Majorana and
CP-active while \(\U(1)_B\) remains exact; the seesaw matching fixes the Yukawas and the chain gives \(\eta_B \approx 7\times10^{-10}\) (central \(7.34\); band \(\times3^{\pm1}\)). BBN is the first precision audit, and its lithium anomaly is
resolved by the inherited closed-packet factor \(1/\pi\) --- the same \(\pi\)-locking that fixes the
matter inventory --- without spoiling deuterium or helium.

\begin{table}[H]
\centering
\caption[Compressed summary of the Part V outputs]{Compressed summary of the Part~V outputs --- the primary matter-sector targets of \ATen{}. The columns give the problem, the solution logic, and the characteristic derived target. The third column is a target rather than a measurement: it states what the mechanism produces, and whether the sky agrees is settled in the audit tables, not here.}
\label{tab:partIII-summary}
\small
\begin{tabular}{p{0.26\textwidth}p{0.29\textwidth}p{0.31\textwidth}}
\toprule
\textbf{Problem} & \textbf{Part~V solution logic} & \textbf{Characteristic derived target}\\
\midrule
Dark matter & Compact-angle / GS descendant \(\Rightarrow\) Kaxion corridor & \(m_K=8.1\,\mu\mathrm{eV}\), twin-peak split \(1/35\), baryonic:EM coupling \(5:2\)\\
Baryogenesis & Lock-scale Majorana triplet, seesaw-matched, resonant CP & \(\eta_B \approx 7\times10^{-10}\) (\(\times3^{\pm1}\)), \(\Omega_b=1/(2\pi^2)\)\\
BBN / lithium & Inherited A=7 closed-packet port & \((^7\mathrm{Li}/H)=\mathrm{SBBN}/\pi\simeq1.6\times10^{-10}\), D/H and \(Y_p\) unchanged\\
\bottomrule
\end{tabular}
\end{table}

Step back from the machinery and consider what this Part has claimed, in plain words. The dark matter of the universe is a frozen angle --- a direction the geometry chose everywhere at once, before any matter existed to disagree. The excess of matter over antimatter --- the reason there is anything rather than mutual annihilation --- is bookkeeping on the same neutrino ledger the particle companion derived. And the first three minutes of nuclear history check out on that ledger at the percent level, with lithium-7 read honestly as the one open port.

There is something worth pausing on here. Every atom in the reader's body is a survivor of the asymmetry computed in this Part; the carbon in your hand is, several floors up, an arithmetic consequence of an integer ledger that was fixed before the first second was over. Whether or not the derivation survives its tests, that is the right \emph{shape} for an answer to ``why is there something?'' --- not a shrug at an accident, but a chain that can be checked link by link.

The bridge to Part~VI is immediate. Once the matter ledger is fixed --- cold dark matter, baryon
fraction, neutrino block, and the early-time BBN audit --- one can ask how perturbations propagate,
how the CMB reads them out, and whether the resulting growth history eases or worsens the present
tensions in \(H_0\), \(S_8\), and early-galaxy formation. That is the task of the next part.



\section{Part~V in plain English: where the matter came from, and the one thing that is assumed}
Three separate accounts sit in this Part --- dark matter as a misaligned
Kaxion field, the baryon asymmetry from the inherited neutrino and CP ledger,
and the light-element abundances --- and the reason they belong together is
that they occupy distinct epochs and draw on the same inherited ledger. They
do not compete for the same energy budget, which is what allows them to be
combined without double counting, and the chronology table is where that can
be checked.

The baryogenesis chain is the place to be careful. It has zero fitted
parameters, which is not the same as having no assumptions: one link is a
named input, the partial-resonance condition, and the exponent in the
throughput is a postulate of the compression rather than a derived quantity.
Both are marked as such in the ledger, and the honest description of the chain
is that it lands on the observed asymmetry from inherited numbers with two
named gaps in the middle, not that it derives it.

The lithium-7 audit is included because it is the one place in this Part where
the framework predicts a deviation from the standard baseline rather than
agreement with it. That makes it the most useful row in the block. A
measurement that placed the primordial abundance at the full standard value
would close the port this Part opens, and the Part says so.

\GGPart[Structure and tensions]{Growth of structure and the present-day tensions}{CMB baseline and the scalar/tensor audit; the Hubble two-stage corridor as a candidate resolution of the \(H_0\) tension; the \(S_8/\sigma_8\) matter-clustering tension; and early-galaxy formation in the JWST era}

\section{One switch, two faces: the geometric mechanism behind the present-day tensions}
\label{sec:part6-opener}

Part~VI is organized around a single mechanism, stated once here and then audited
observable by observable: \emph{recombination is the cosmic \GRIPout{} event, the corridor switch
occurs at the integer-ledger redshift \(z_{\mathrm{sw}} = e^{7}\), and every present-day
cosmological tension is either that switch or the \GRIPin{} growth that follows
it.} The two headline puzzles of contemporary cosmology --- the Hubble tension
and the JWST ``too-early massive galaxies'' anomaly --- are therefore not two
unrelated problems requiring two unrelated fixes. They are two faces of one
geometric switch.

The runtime grammar is the \GRIP{} cascade \(\GRIP = (\DGI \to \GSSB \to \GA)\)
inherited from \ASix{} and already used as the spine of this paper
(Section~\ref{sec:runtime-branches}; the cascade figure of Part~II). Before
recombination the baryon--photon plasma is a tightly coupled \DGI{} phase: an
enormous family of micro-histories sharing one observable acoustic gauge orbit.
Recombination is the \GSSB{} transition that freezes those phases into the \GA{}
attractor we read as the microwave background. The switch redshift is not fitted;
it is forced by the \OGNfive{} integer \(J = 7\) of \AFour{},
\[
z_\ast = e^{J} = e^{7} \approx 1096.6,
\]
the corridor-switch redshift; the atomic decoupling derived from the
framework's own constants lands $0.4\%$ below it, on the Planck value at the
$0.1\%$ level (\S\ref{sec:two-redshifts}). From that single
event the Part unfolds in two directions:
\begin{itemize}[leftmargin=2.0em]
\item \textbf{Face 1 --- the Hubble tension (the switch, read twice).} The
boundary Friedmann corridor admits two \OGGEP{} projections separated by the
switch: the early/sound-horizon dictionary (\(z > z_\ast\)) projects to
\(\Hzero \approx 67.4\), the late/distance-ladder dictionary (\(z < z_\ast\))
projects to \(\Hzero \approx 73.0\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\). The two values
are not in conflict; they are one corridor read through two open-system
coarse-grainings (Theorem~\ref{thm:hubble-two-stage-corridor};
Corollary~\ref{cor:hubble-tension-reconciliation}).
\item \textbf{Face 2 --- \(S_8\) and the JWST anomaly (the \GRIPin{} growth after
the switch).} Once the \GRIPout{} event has frozen the attractor, the universe
runs as \GRIPin{} relaxation: structure grows on the inherited \(\pi\)-locked
ledger. The \(S_8\) clustering observable is that one growth history read on broad
linear scales (Corollary~\ref{cor:s8-reconciliation}); the JWST early-structure
anomaly is the \emph{same} growth read on the exponentially sensitive high-\(\nu\)
rare tail. One growth history, two windows.
\end{itemize}

The four observables audited below --- the CMB acoustic structure, \(\Hzero\),
\(S_8\), and the abundance of the earliest luminous systems --- are thus a single
\GRIPout{} switch at \(z_\ast = e^{7}\) plus the \GRIPin{} growth after it, all on
one inherited ledger. Table~\ref{tab:part6-headline} collects the falsifiable
numbers in one place; each is derived in the section named in its last column, and
the heavier derivations are deferred to the Technical Appendices
(Sections~\ref{appA:growth} and \ref{appA:cmb}) so the body
stays readable. The reader is encouraged to consult those appendices for the
full from-scratch calculations.

\begin{table}[H]
\centering
\caption[Headline falsifiable numbers of Part~VI.]{\textbf{Headline falsifiable
numbers of Part~VI.} Every quantitative prediction of the growth-and-tension
block, tagged to its role in the one-switch/one-growth mechanism and compared to
the current best datum. \(\DeltaNeff\) is fixed at the canonical audit value
\(0.04\) throughout the body (Section~\ref{sec:audit-derived-cosmos}); the legacy
early-dark-energy linear-response that used \(0.30\) is retained only as a
non-unique cross-check in Appendix~\ref{appA:cmb}.}
\label{tab:part6-headline}
\small
\begin{tabular}{p{0.20\textwidth}p{0.17\textwidth}p{0.20\textwidth}p{0.10\textwidth}p{0.20\textwidth}}
\toprule
\textbf{Observable} & \textbf{\GGTheory{}} & \textbf{Best datum} & \textbf{Dev.} & \textbf{Mechanism role}\\
\midrule
Decoupling \(z_\ast\) & \(1091.1\) (atomic chain; switch \(e^{7}\) distinct, not scored) & \(1089.92\pm0.25\) \citep{Planck2018} & \(+0.11\%\) & \GRIPout{} event at the switch \(z_{\rm sw}=e^{7}\) (\S\ref{sec:two-redshifts})\\
Background \(\Hzero^{(0)}\) & \(66.6\pm0.2\) (derived) & \(67.4\pm0.5\) \citep{Planck2018} & \(-1.2\%\) & inventory at the switch\\
Local ladder \(\Hzero^{\mathrm{late}}\) & \(\approx 68.5\) (lifted) & \(73.50\pm0.81\) \citep{CasertanoH0DN2026} & \(-6.8\%\) (open) & \GRIPin{} lift + ladder systematic\\
Clustering \(S_8\) (late) & \(\approx 0.78\) & \(0.759^{+0.024}_{-0.021}\) \citep{Asgari2021} & \(0.9\sigma\) & \GRIPin{}, broad scale\\
Acoustic angle \(100\theta_\ast\) & \(1.036\) (derived) & \(1.04109\pm0.00030\) \citep{Planck2018} & \(-0.5\%\) & switch geometry\\
Damping multipole \(\ell_D\) & \(1350\pm3\) (Boltzmann) & \(\approx 1345\pm15\) \citep{Planck2018} & \(+0.4\%\) & diffusion on the inventory\\
Scalar tilt \(\ns\) & \(1-2/N^{*}=0.9620\) & \(0.965\pm0.004\) \citep{Planck2018} & \(<1\sigma\) & plateau-derived\\
Tensor ratio \(r\) & \(12/N^{*2}=0.0043\) & \(<0.036\) & two-sided target & derived plateau\\
Collapse threshold \(\delta_c\) & \(1.686 \to 1.18\) & --- & first-light lever & \GRIPin{}, rare tail\\
First-light shift \(\Delta z\) & \(\sim 2\)--\(3\) & JWST excess & qualitative & \GRIPin{}, rare tail\\
\bottomrule
\end{tabular}
\end{table}


\section{Determinism from geometry: the microwave background as an over-determined test}
\label{sec:cmb-overdetermined}

Before the acoustic physics of the next section is unpacked, it is worth
stating plainly what kind of test the microwave background is about to
administer, because it is a stronger kind than cosmology usually gets to run.
Standard \LCDM{} describes the CMB with six free parameters
\(\{\omega_b,\ \omega_c,\ \theta_*,\ \tau,\ A_s,\ n_s\}\): two physical
densities, one angle, one optical depth, and two numbers for the primordial
seed. The model is spectacularly successful at \emph{compressing} the data
onto those six dials, and completely silent on why the dials read what they
read. \GG{} makes a claim of a different kind, inherited from the determinism
spine of the cohort (\AEleven{}, \citealp{ArisakaA11Origin}) and the integer
ledger of the particle stratum (\ANine{}, \citealp{ArisakaA9SM}): the
\emph{background-geometry half} of those dials is not adjustable at all. The
matter fraction, the dark-energy fraction, the baryon fraction, the
equation of state, and the recombination redshift are theorems of one closed
geometry,
\[
\Omm=\frac{1}{\pi},\qquad
\OmL=1-\frac{1}{\pi},\qquad
\Omb=\frac{1}{2\pi^2},\qquad
w=-1,\qquad
z_*=e^{7},
\]
with no continuous parameter anywhere in the chain (Parts~II, IV, and XI).
What remains genuinely free in the framework's present state is the
primordial seed \((A_s, n_s)\) --- the amplitude declared as input,
the tilt derived-conditional at \(0.9620\)
(Section~\ref{sec:cmb}) --- and the reionization depth \(\tau\), which is
astrophysics, not fundamental geometry. Cosmology's six-parameter standard
model becomes, in effect, a three-parameter one, and the half that is deleted
is precisely the ``why these values'' half.

That deletion is what makes the CMB an \emph{over-determined} test rather
than a fitting exercise. The microwave sky measures more independent
quantities than the framework leaves free: the baryon density, the matter
density, the acoustic angle, the damping scale, the peak-height ratios, and
the equality and recombination epochs are read off the spectrum through
largely independent pieces of physics. \LCDM{} reconciles them by adjusting
its dials; \GG{} has no dials to adjust, so every one of those readings is a
separate opportunity to be wrong. A single clean disagreement --- one
observable demanding a fraction incompatible with \(1/\pi\) --- would
falsify the chart outright. The framework's predictions moreover sit exactly
along \LCDM{}'s celebrated \emph{geometric degeneracy} directions (the
\(\Omega_m\)--\(H_0\) line at fixed \(\theta_*\)), which is where a fitted
model constrains least and where a forced model is therefore most exposed.
The honest statement of the program, maintained throughout this Part, is:
\GG{} converts the background-geometry sector of the CMB into theorems and
lets the data over-constrain the rest; what it does not derive (the seed,
\(\tau\), the local distance ladder) is named as such and never absorbed.

\subsection{The master identity: fractions, densities, and several roads to one rate}
\label{subsec:master-identity}

The engine of the over-determination is one elementary identity. A
\emph{fraction} \(\Omega_i\) is a dimensionless share of the critical
density; a \emph{physical} density \(\omega_i \equiv \Omega_i h^2\) is a
statement about grams per cubic meter and contains no Hubble constant at
all. The CMB, to an excellent approximation, measures physical densities:
\(\omega_b = 0.02237\pm0.00015\) from the alternating (odd/even)
peak-height asymmetry that baryon loading imprints on the acoustic
oscillation --- independently confirmed by the BBN deuterium abundance ---
and \(\omega_m = 0.1430\pm0.0011\) from the peak-height envelope and the
matter--radiation equality it encodes. \LCDM{} treats the fractions as
unknowns and fits them. \GG{} predicts the fractions outright, so each
measured physical density becomes an independent \emph{road to the expansion
rate}:
\begin{equation}
h=\sqrt{\frac{\omega_i}{\Omega_i}}
\;\Longrightarrow\;
h_{\rm baryon}=\sqrt{\frac{0.02237}{1/2\pi^2}}=0.665,
\qquad
h_{\rm matter}=\sqrt{\frac{0.1430}{1/\pi}}=0.670,
\label{eq:overdet-master-identity}
\end{equation}
two determinations through two different pieces of acoustic physics,
agreeing to \(0.7\%\) with no quantity available to reconcile them had they
disagreed. Their combination is the background rate
\(\Hzero^{(0)}=66.6\pm0.2\;\mathrm{km\,s^{-1}\,Mpc^{-1}}\) derived in
Section~\ref{sec:h0-derivation} --- and derived there a second time,
with zero sky input, by the geometric Route~I at \(66.8\pm0.2\) --- and the age \(t_0 = 0.948/H_0 = 13.9\)~Gyr (today; \(13.8\) at the \(\Omega_m{=}1/\pi\) crossing) follows with
it. Figure~\ref{fig:h0-consistency-web} draws the full web: every arrow is
an independent measurement channel, and they converge on one number because
the chart leaves them nowhere else to go.

There is one test purer still. Since both physical densities scale as
\(h^2\), their \emph{ratio} is \(h\)-free and equals a pure geometric
number:
\begin{equation}
\frac{\omega_b}{\omega_m}\;=\;\frac{\Omb}{\Omm}\;=\;\frac{1}{2\pi}\;=\;0.15915,
\qquad\text{observed } 0.15709\pm0.0016\ \ (+1.3\%,\ 1.3\sigma).
\label{eq:overdet-ratio}
\end{equation}
This is the baryon-loading theorem (T-Cosmos-9) of the next section stated
as a falsifiable number: the quantity that sets the \emph{relative} heights
of the acoustic peaks is \(1/2\pi\), with no freedom whatsoever --- not even
the Hubble constant. The matter--radiation equality redshift, by contrast,
is deliberately \emph{not} counted as an independent test here: once \(h\)
is taken from the matter road, \(z_{\rm eq}=\omega_m/\omega_r\) reproduces
the measured equality by construction, and an honest audit does not count
the same measurement twice.

\begin{figure}[!t]
\centering
\begin{tikzpicture}[
  meas/.style={draw=black!50, rounded corners=2pt, align=center, fill=gray!8,
               inner sep=3.5pt, font=\scriptsize},
  frac/.style={draw=blue!55!black, rounded corners=2pt, align=center, fill=blue!6,
               inner sep=3.5pt, font=\scriptsize},
  road/.style={draw=green!55!black, align=center, fill=green!8, inner sep=3.5pt, font=\scriptsize},
  goal/.style={draw=green!55!black, line width=1.10pt, align=center, fill=green!15,
               inner sep=5pt, font=\scriptsize\bfseries},
  side/.style={draw=red!55!black, dashed, align=center, fill=red!6, inner sep=3.5pt,
               font=\scriptsize},
  arr/.style={-{Stealth[length=2.5mm]}, line width=0.80pt},
  chk/.style={-{Stealth[length=2.5mm]}, line width=0.80pt, dashed, gray}]
\node[meas] (m1) at (1.0,6.4) {2nd/1st peak\\asymmetry};
\node[meas] (m2) at (4.6,6.4) {BBN deuterium\\(independent)};
\node[meas] (m3) at (8.8,6.4) {peak envelope /\\equality epoch};
\node[frac] (ob) at (2.9,4.8) {$\omega_b=0.02237\pm0.00015$\\
  divide by $\Omb=1/2\pi^2$};
\node[frac] (om) at (8.8,4.8) {$\omega_m=0.1430\pm0.0011$\\
  divide by $\Omm=1/\pi$};
\node[road] (hb) at (2.9,3.1) {baryon road\\$H_0=66.5\pm0.2$};
\node[road] (hm) at (8.8,3.1) {matter road\\$H_0=67.0\pm0.3$};
\node[frac] (ss) at (11.6,3.1) {Route I: geometry\\$H_0=66.8\pm0.2$};
\node[meas] (cls) at (1.4,1.0) {full spectrum (CLASS)\\fits $h=0.665$--$0.674$};
\node[goal] (h0) at (5.7,1.0) {$\Hzero^{(0)}=66.6\pm0.2\ \approx\ 67$
  \\ $t_0=13.9$ Gyr (crossing $13.8$)};
\node[side] (lad) at (10.6,1.0) {ladder $73.50\pm0.81$: displaced\\(Section~\ref{sec:h0tension})};
\draw[arr] (m1) -- (ob);
\draw[arr] (m2) -- (ob);
\draw[arr] (m3) -- (om);
\draw[arr] (ob) -- (hb);
\draw[arr] (om) -- (hm);
\draw[arr] (hb) -- (h0);
\draw[arr] (hm) -- (h0);
\draw[arr] (ss) -- (h0);
\draw[chk] (cls) -- (h0);
\draw[chk] (h0) -- node[above, font=\scriptsize] {$+10.4\%$} (lad);
\end{tikzpicture}
\caption[The $H_0$ consistency web: independent CMB roads converge on $\approx67$.]{%
\textbf{The \(H_0\) consistency web.} Each solid arrow is an independent
measurement channel: the odd/even peak asymmetry and (independently) BBN
deuterium fix the physical baryon density; the peak envelope fixes the
physical matter density; dividing each by its forced \GG{} fraction yields
an independent road to the expansion rate. The roads agree at
\(\Hzero^{(0)}=66.6\pm0.2\), the geometric Route~I (zero sky input,
Section~\ref{sec:h0-derivation}) lands on the same value independently,
and the full Boltzmann spectrum (CLASS, exact fractions) confirms
the fit. The local distance ladder is the lone displaced reading; it is
analyzed --- not derived --- in Section~\ref{sec:h0tension}.}
\label{fig:h0-consistency-web}
\end{figure}

\subsection{The core observables, computed and confronted}
\label{subsec:cmb-core-table}

Table~\ref{tab:cmb-core-derived} collects the core CMB observables that the
\(\pi\)-locked chart forces, each computed two ways: semi-analytically
(the standard tight-coupling expressions with every input \GG{}-fixed) and
through a full Boltzmann solution (CLASS) run at the exact fractions.
In the numerical runs everything except the background chart is held at its
standard value: the Planck neutrino sector, and the primordial seed and
optical depth frozen at the Planck best-fit. (These runs used the Planck best-fit tilt and are kept as run; in the
present codebook the tilt \(n_s\) is
\emph{derived}, and only two quantities remain declared as inherited ---
the amplitude \(A_s\) and the astrophysical \(\tau\). The full-spectrum
confrontation with the derived tilt follows below, and improves the fit.) Two bookkeeping conventions are stated once:
the ledger corridor-switch redshift \(z_*=e^7=1096.6\) --- a topological quantity, not the measured decoupling redshift --- sits \(+0.7\%\) above the Boltzmann ionization-history value \(z_{\rm rec}\simeq1088.6\), a definitional spread inside the same percent-level family; and pipeline
definitions of \(r_s\) and \(\theta_*\) differ between analysis chains at
the \(0.1\%\) level, which is folded into the quoted deviations.

\begin{table}[!t]
\centering\small
\caption[Core CMB observables forced by the $\pi$-locked chart (CLASS-verified).]{%
\textbf{The CMB core, derived and confronted.} Every \GG{} entry follows
from \(\Omm=1/\pi\), \(\Omb=1/2\pi^2\), \(\OmL=1-1/\pi\), \(w=-1\),
and the corridor switch \(z_*=e^7\) with zero tuned parameters (the
atomic decoupling derived from the same inventory lies \(0.4\%\) below
the switch, on the Planck value; \S\ref{sec:two-redshifts}); the seed and \(\tau\) are frozen at
Planck values and declared inherited. \emph{Zero CMB input}: every
\GG{} entry is forced by the geometry --- the fractions by the Hopf
fibration, the rate by the two routes of Section~\ref{sec:h0-derivation},
the densities by their product, evaluated at the canonical
\(66.6\pm0.2\) --- and the sky appears only in the observed column. Spectrum-morphology rows (peak
positions, height ratios) are compared against the Planck-2018 best-fit
spectrum computed in the same pipeline; parameter rows against the Planck
2018 chain values \citep{Planck2018}. Grades are the codebook triage of
Part~IX (D\(^*\) = derived modulo a named lemma). The uniform percent-level,
same-family displacement of all rows is the coherent \(\pi\)-offset
discussed in Section~\ref{subsec:pi-offset}.}
\label{tab:cmb-core-derived}
\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{lp{3.4cm}llrl}
\toprule
observable & \GG{} forced form & \GG{} value & observed & Dev & grade\\
\midrule
\(H_0\) (Route I: geometry) & \(498.3\sqrt{\pi/175}\) & \(66.8\pm0.2\) & \(67.4\pm0.5\) & \(-0.9\%\) & D\(^*\)\\
\(H_0\) (baryon road) & \(\sqrt{\omega_b/\Omb}\) & \(66.5\pm0.2\) & --- & --- & D\(^*\)\\
\(H_0\) (matter road) & \(\sqrt{\omega_m/\Omm}\) & \(67.0\pm0.3\) & --- & --- & D\(^*\)\\
\(H_0\) (combined) & two roads, one rate & \(66.6\pm0.2\) & \(67.4\pm0.5\) & \(-1.2\%\) & D\(^*\)\\
\(\omega_b\) (predicted) & \(h^2/2\pi^2\) & \(0.02247\) & \(0.02237\pm0.00015\) & \(+0.5\%\) & D\(^*\)\\
\(\omega_m\) (predicted) & \(h^2/\pi\) & \(0.14119\) & \(0.1430\pm0.0011\) & \(-1.3\%\) & D\(^*\)\\
\(\omega_b/\omega_m\) & \(1/2\pi\) (\(h\)-free, epoch-free) & \(0.15915\) & \(0.15709\pm0.0016\) & \(+1.3\%\) & D\\
\(z_*\) & \(e^{7}\) (ledger, corridor switch); decoupling \(0.4\%\) below & \(1096.6\) & \(1089.92\pm0.25\) & \(+0.6\%\) & D\(^*\)\\
\(r_s(z_{\rm rec})\) [Mpc] & diffusion-free acoustic scale & \(145.0\) & \(144.43\pm0.26\) & \(+0.4\%\) & D\(^*\)\\
\(100\,\theta_*\) & \(r_s/D_M\) & \(1.0362\) & \(1.04109\pm0.00030\) & \(-0.5\%\) & D\(^*\)\\
peaks \(\ell_1/\ell_2/\ell_3\) & acoustic projection & \(221/538/817\) & \(220/536/813\) & \(+0.4\)--\(0.5\%\) & D\(^*\)\\
height ratio \(2/1\) & image of \(1/2\pi\) & \(0.451\) & \(0.453\pm0.008\) & \(-0.3\%\) & D\(^*\)\\
height ratio \(3/1\) & image of \(1/2\pi\) & \(0.440\) & \(0.443\pm0.009\) & \(-0.8\%\) & D\(^*\)\\
\(\ell_D\) (damping) & diffusion integral & \(1350\pm3\) & \(1345\pm15\) & \(+0.4\%\) & D\(^*\)\\
\bottomrule
\end{tabular}
\end{table}

Three rows deserve explicit comment, because they are the ones Phase-level
audits most often get wrong. First, the \emph{peak-height ratios} are the
spectrum-level image of the \(h\)-free ratio \eqref{eq:overdet-ratio}: the
driving ratio deviates by \(+1.3\%\), and the full radiation-transfer
machinery compresses that into a \(0.3\%\) (second-to-first) and \(0.8\%\)
(third-to-first) shift in the observable ratios --- the direct spectral test
of \(1/2\pi\), and it passes at the few-tenths-of-a-percent level. Second,
the \emph{damping multipole}: running the exact fractions through the full
diffusion integral over the Boltzmann ionization history gives
\(\ell_D=1350\pm3\) (quoted in the \(\Lambda\)CDM-inferred convention,
observed \(1345\pm15\)), a \(+0.4\%\) agreement --- the inventory predicts
the damping tail essentially perfectly, a result whose consequences for the
separately proposed BFFT reading of \(\ell_D\) are drawn in
Section~\ref{sec:cmb}. Third, the \emph{acoustic angle}
\(\theta_*\) is the most precisely measured number in cosmology
(\(0.03\%\)), so its \(-0.5\%\) deviation, small as it looks, is the
sharpest formal pull in the entire codebook; it is examined honestly, not
hidden, in the next subsection.

\subsection{The full spectrum from the derived parameters}
\label{subsec:cmb-full-spectrum-derived}

The sharpest form of the confrontation puts the whole temperature spectrum
on one panel. Figure~\ref{fig:cmb-tt-gg} shows the lensed TT spectrum
computed (CLASS) from the \GG{}-derived parameter set --- the exact
\(\pi\)-locked fractions, the derived \(\Hzero = 66.6 \pm 0.2\), the
derived tilt \(n_s = 1 - 2/N^* = 0.9620\), and the counted radiation
content --- with exactly two numbers inherited and set to the Planck
values, the amplitude \(A_s\) (the launch sector's one declared input) and
the astrophysical \(\tau\). Nothing in the red curve is adjusted to the
data. Against the 83 binned Planck TT points the six-parameter best fit
gives \(\chi^2 = 84\) (\(\chi^2/\mathrm{dof}\approx1\) by construction); the
zero-parameter \GG{} curve gives \(\chi^2 = 237\), i.e.\ \(\chi^2/\mathrm{dof}\approx2.9\).
We state plainly what that means and what it does not: as a \emph{likelihood
fit} the present curve would be rejected --- \(\chi^2/\mathrm{dof}\approx3\) is
not a fit, and it would be dishonest to call the instrument's output an
``endorsement'' of the inputs. But the \(237\) is not \(237\) of \emph{random}
scatter: the curve tracks the data to within \(\pm 3\)--\(4\%\) across the
first seven acoustic peaks, and the residual is a \emph{single coherent
oscillation} --- the \(\pi\)-offset of the next subsection, the same
correlated \(+2.7\%\) second-order systematic that the second-order-residual
audit (Section~\ref{subsec:second-order-residual-cosmos}) isolates. The
physically correct reading is therefore that the zero-parameter curve is
\emph{qualitatively} right and \emph{quantitatively} off by one coherent,
computable second-order normalization --- exactly what a leading-order theory
should look like in a field that measures at \(0.1\%\). This yields a sharp,
falsifiable expectation, not a boast: marginalizing the single coherent
offset with one nuisance parameter --- or computing the omitted second-order
content directly, as was done for the damping knee (its integer form's
\(+2.7\%\) collapsed to \(+0.4\%\) once the diffusion integral was evaluated)
--- should drop \(\chi^2\) toward the \(\Lambda\)CDM value. One honest
sub-result survives at the likelihood level and is worth isolating: at the
\emph{same} \(\pi\)-locked background, replacing Planck's fitted
\(n_s = 0.9649\) by the derived \(0.9620\) improves the fit by
\(\Delta\chi^2 = -22\) (\(259\to237\)) --- a comparison between two variants
of the \GG{} curve, both far from \(\Lambda\)CDM, showing only that within
this geometry the data prefer the tilt the theory computes, not that the
curve as a whole is a good fit.

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth,height=0.72\textheight,keepaspectratio]{a10_cmb_tt_gg.pdf}
\par\smallskip
\caption[The microwave sky from the derived parameters.]{\textbf{The
microwave sky from the ledger.} \emph{Top:} the lensed TT spectrum for the
Planck 2018 best-fit \(\Lambda\)CDM (black; six fitted parameters) and the
\GG{} prediction (red dashed; exact \(\pi\)-locked fractions, derived
\(\Hzero = 66.6\), derived \(n_s = 0.9620\); only \(A_s\) and \(\tau\)
inherited, set to Planck values, never adjusted), against the Planck 2018
binned TT data (83 bins). \emph{Bottom:} fractional residuals relative to
the best fit, with the derived-\(\Hzero\) bracket \(h = 0.664/0.668\)
(dotted). \(\chi^2\) (83 bins, diagonal): best fit 84 with six fitted
parameters (\(\chi^2/\mathrm{dof}\approx1\)); \GG{} 237 with zero
(\(\chi^2/\mathrm{dof}\approx2.9\) --- not a likelihood fit, and not claimed
as one). The residual is a single coherent \(\pm3\)--\(4\%\) oscillation, the
named \(\pi\)-offset (one second-order normalization), not random scatter; at
the same background the derived tilt improves on the borrowed one by
\(\Delta\chi^2 = -22\).
Reproduction: \texttt{gg\_cmb\_tt\_v2.py}.}
\label{fig:cmb-tt-gg}
\end{figure}

\subsection{The \texorpdfstring{\(\pi\)}{pi}-offset: one named systematic, visible everywhere}
\label{subsec:pi-offset}

Every row of Table~\ref{tab:cmb-core-derived} deviates from observation at
the percent level or below, and the deviations are not scattered: they are
one number's shadow. The exact fractions sit slightly above Planck's fitted
ones (\(1/\pi = 0.3183\) against \(0.3153\pm0.0073\), \(+1.0\%\)), and that
single displacement propagates with the expected sign through every
observable --- the derived \(h\) comes out \(1.2\%\) low, so \(\theta_*\)
projects \(0.5\%\) small, the peaks slide \(0.4\)--\(0.5\%\) rightward, the
damping scale follows at \(+0.4\%\), and the ledger \(z_*\) and the ratio
\(1/2\pi\) sit \(+0.6\%\) and \(+1.3\%\) high. A model \emph{fitted} to
these data would scatter its residuals on both sides; a \emph{forced} chart
displaced by one small systematic lands coherently on one side, and that
coherence --- verified across the full spectrum, into the damping tail,
with no compensation anywhere --- is the signature that the chart is forced
rather than tuned. The offset is named, permanent, and carried openly: it
either is the truth about the universe (and the fitted fractions are
absorbing a small unmodeled effect), or it is the falsification-in-waiting
of the exact chart. The framework does not get to choose; sharper data do.

Honesty requires the two strongest statements of the offset to be made by
the framework itself. The acoustic angle is measured to \(0.03\%\); against
the \(0.1\%\) theory spread of the two roads, the \(-0.5\%\) deviation is a
formal pull of about \(-4\), by far the largest in the codebook --- the
\(\pi\)-offset concentrated into cosmology's most precise number. And a
chi-squared against the 83 published Planck TT band-powers, with diagonal
errors and the inherited seed left frozen, gives \(\approx285\) for the
exact chart against \(\approx83\) for the six-parameter best fit: at Planck
precision the data \emph{can} tell exact \(\pi\) from the fitted fractions.
Both numbers overstate the case against the chart --- bin-to-bin
correlations are ignored, and the seed was never re-optimized for the \GG{}
background --- but the direction is honest and is not negotiated away. The
claim defended here is precisely calibrated: a zero-parameter background
lands within about one percent of a six-parameter fit everywhere at once,
displaced one way by one named systematic; it is never claimed that the
exact chart out-fits \(\Lambda\)CDM. A full-likelihood confrontation, with
the seed re-fitted and correlations kept, is the sharpest version of this
test and is recorded as an open item rather than pre-judged.

\subsection{What is claimed, and what is not}
\label{subsec:overdet-claims}

The boundary of the claim matters as much as the claim. \emph{Claimed:} the
dimensionless background chart is theorem-grade (modulo the named lemmas
tabulated in Part~IX), and from it the expansion rate, the age, the
acoustic geometry, the damping scale, and the peak morphology follow with
zero adjustable parameters, verified in a full Boltzmann computation,
together with the derived-conditional tilt \(\ns = 1 - 2/N^* = 0.9620\)
(Lemma~IV, revised), risked against the data rather than borrowed from
them (\(\Delta\chi^2 = -22\); Figure~\ref{fig:cmb-tt-gg}).
\emph{Not claimed:} the primordial amplitude (the launch sector's
declared input, Lemma~V\('\)), the optical depth (astrophysics), the absolute peak heights insofar
as they lean on the inherited seed, the equality redshift as an independent
success (an identity once the matter road fixes \(h\)), and the local
distance-ladder reading of \(73.0\pm1.0\), which the framework analyzes as
a displaced measurement (Section~\ref{sec:h0tension}) and expressly does
not derive. The falsification handles are correspondingly sharp: a
gravitational-wave standard-siren program converging at \(\approx73\)
would refute the inventory outright, a converged siren at \(\approx67\)
would confirm it against the ladder, and any future CMB analysis pushing
the fitted fractions two of their present error bars away from
\((1/\pi,\,1/2\pi^2)\) would end the exact chart without appeal. That is
the intended posture of this Part: not a model defended, but a derivation
exposed.

\section{CMB baseline, primordial perturbations, and the scalar/tensor audit}

\subsection{Historical background}

The cosmic microwave background was discovered serendipitously in 1965 by Penzias and Wilson \citep{PenziasWilson1965} at Bell Laboratories as an ``excess antenna temperature at 4080~Mc/s''. The same volume of the \emph{Astrophysical Journal} carried the companion paper by Dicke, Peebles, Roll, and Wilkinson \citep{DickePeebles1965} interpreting the signal as the cooled afterglow of a hot early universe. Six decades of progressively higher-precision experiments --- most recently the Planck satellite \citep{Planck2018} --- have established the CMB as the most quantitatively precise observable in cosmology. The present section retains the standard acoustic, damping, and projection structure and asks where the inherited ledger reshapes the source assumptions behind it.

\label{sec:cmb}


\begin{theorem}[PI-locked baryon-loading statement (T-Cosmos-9)]
On the late inventory branch,
\[
f_b \equiv \frac{\Omega_b}{\Omega_m} = \frac{1}{2\pi},
\]
so the baryon loading of the photon-baryon plasma is not a free CMB fit
parameter in the framework but a branch descendant of the same late matter chart.
The standard acoustic formulas remain
\[
R(a)=\frac{3\rhob}{4\rho_\gamma},
\qquad
c_s(a)=\frac{c}{\sqrt{3(1+R(a))}},
\qquad
\theta_*=\frac{r_s(z_*)}{D_A(z_*)},
\]
but the matter share entering them is narrowed by the inherited late branch.
\end{theorem}

\begin{proposition}[Tensor target]
Because the inflationary branch is the derived plateau --- class forced, count
matched, no dial --- the primordial tensor sector must obey
\[
r = \frac{12}{N^{*2}} = 0.0043 .
\]
A tensor measurement inconsistent with this value in either direction would
therefore kill the minimal branch rather than merely nudge a fit parameter.
\end{proposition}



Part~VI begins from a methodological rule already emphasized in the introduction: the present program is not strongest when it announces a wholesale replacement of precision cosmology, but when it keeps the successful perturbative grammar of standard cosmology and asks a narrower and sharper question. Which pieces of that grammar are merely kinematic, and which pieces still hide unexplained source terms, normalizations, or module choices? In the present section the ordinary perturbation equations are retained as the boundary baseline. The new move lies in the inherited ledger used to interpret them.

The immediate empirical target is the microwave background together with the primordial scalar and tensor sectors. That target is unavoidable. A framework that proposes a cosmic beginning at \(\EGUT\), a derived slow-smooth plateau one e-fold below it, a compact-angle dark sector, and a measurement-aware boundary semantics must survive the acoustic ruler, the peak-height pattern, the damping tail, the polarization sequence, the scalar tilt, and the absence or presence of primordial tensor power. In other words, the sky already provides the natural audit of the first, third, and fifth preprints simultaneously.

The strategy of the present section is therefore five-fold. First, we restate the standard perturbation baseline in a form that makes clear what \GG{} is \emph{not} changing. Second, we reinterpret recombination as a \GDOS{} event --- more precisely, a transition from a DGI-dominated tightly coupled plasma to a frozen geometric attractor. Third, we carry forward the \(\pi\)-locked cosmic inventory already motivated elsewhere in the draft and ask how far it can push the acoustic structure without introducing continuous new dials. Fourth, we state the derived corridor for the scalar and tensor sectors. Fifth, we summarize the kill conditions. Throughout, the working stance is conservative in equations and maximal in inherited constraint.

\subsection{What this section inherits from the previous papers}

The present section does not start from a blank cosmological slate. It inherits a non-trivial ledger:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item the boundary Einstein equations as the zero-mode export of the six-dimensional bulk,
\item the radial coordinate \(r = \ln(\EGUT/\mu)\) as the geometric encoding of RG descent,
\item the \BIsix{} anomaly structure and the \OGNfive{} integers \((1,3,5;7,35)\),
\item the compact-angle ontology that already generates the Kaxion corridor and the baryonic \(\mathrm{U}(1)_B\) sector,
\item the \GDOS{} runtime, together with \DGI, \GSSB, and \GA, and
\item the guiding interpretive rule that the observable universe is a boundary-accessible open subsystem rather than a directly readable bulk microstate.
\end{enumerate}

These inherited items already narrow the permissible cosmology. They do not yet determine every transfer function or every numerical likelihood value. But they do eliminate much of the usual freedom. In particular, they strongly disfavor an arbitrary inflaton potential, arbitrary extra-light fields, arbitrary dark-radiation additions, arbitrary non-geometric tensor sectors, and arbitrary late-time fixes introduced only to move one observational point. If a correction is needed in the CMB or structure-growth chain, the first place to look is the already inherited geometric ledger.

\subsection{Standard perturbation baseline: what remains unchanged}

At the homogeneous level the background metric is the standard spatially flat \FLRW\ line element,
\begin{equation}
\dd s^2 = -\dd t^2 + a^2(t)\,\delta_{ij}\,\dd x^i\dd x^j,
\end{equation}
with Hubble rate
\begin{equation}
H \equiv \frac{\dot a}{a}.
\end{equation}
At linear order, scalar perturbations may be written in conformal Newtonian gauge as
\begin{equation}
\dd s^2 = -\paren{1+2\Phi}\dd t^2 + a^2(t)\paren{1-2\Psi}\delta_{ij}\dd x^i\dd x^j,
\label{eq:newtonian-perturb}
\end{equation}
where \(\Phi\) and \(\Psi\) are the usual Bardeen potentials. The reason to write the metric in this familiar form is that it makes the basic scientific point transparent: the perturbation language used by the data community is not being rejected. It is being inherited and then interpreted through a different ultraviolet and operational ledger.

For any fluid component with density contrast \(\delta\), velocity divergence \(\theta\), equation-of-state parameter \(w\), and adiabatic sound speed \(c_s\), the linearized continuity and Euler equations take their standard form. Schematically,
\begin{align}
\dot\delta &= -(1+w)\paren{\theta - 3\dot\Psi} - 3H\paren{c_s^2-w}\delta + \cdots, \\
\dot\theta &= -H(1-3w)\theta + \frac{c_s^2}{1+w}\frac{k^2}{a^2}\delta + \frac{k^2}{a^2}\Phi + \cdots,
\end{align}
while the potential is constrained by the linearized Einstein equation,
\begin{equation}
\frac{k^2}{a^2}\Psi \simeq 4\pi G\sum_i \rho_i\,\delta_i
\label{eq:poisson-schematic}
\end{equation}
in the sub-horizon regime. None of this is exotic. The claim of \ATen{} is not that these equations were wrong. The claim is that the sector content and the physically admissible source corrections should be read from a much tighter ledger than is usual in phenomenological model building.

The same point holds for primordial curvature perturbations. In the scalar sector the standard Mukhanov-Sasaki variable \(v_k\) satisfies
\begin{equation}
 v_k'' + \paren{c_s^2 k^2 - \frac{z''}{z}}v_k = 0,
\label{eq:ms-eq}
\end{equation}
where primes denote conformal-time derivatives and \(z\) is the usual pump field determined by the background. The scalar power spectrum remains parameterized by
\begin{equation}
\mathcal P_\mathcal R(k) = A_s\paren{\frac{k}{k_*}}^{n_s-1 + \cdots}.
\end{equation}
Likewise, the tensor sector remains parameterized by a tensor-to-scalar ratio \(r\) and by the primordial tensor spectral shape. What the present framework changes is the admissible origin of \(A_s\), \(n_s\), and \(r\): they are not granted a phenomenological inflationary menu with many independent potential coefficients.

This is the correct place to state one of the central methodological guardrails of the paper. Precision cosmology is already extraordinarily efficient at constraining model freedom. A theory that adds broad new freedom in the scalar, tensor, dark-radiation, and recombination sectors all at once may fit a subset of anomalies, but it does so at the cost of becoming constitutionally weak. The aim of \ATen{} is the opposite. It asks whether a narrow inherited geometric ledger can do enough work to survive the same data.

\subsection{Recombination as a \GDOS{} transition: from DGI-dominated plasma to geometric attractor}

The first genuinely new interpretive move of Part~VI is to treat recombination not merely as a thermodynamic milestone but as a \GDOS{} event. Before recombination, the baryon-photon plasma is tightly coupled by Thomson scattering. The relevant scattering rate is
\begin{equation}
\Gamma_T = n_e\sigma_T c,
\end{equation}
with \(n_e\) the free-electron density and \(\sigma_T\) the Thomson cross section. At early times one has \(\Gamma_T \gg H\). The plasma therefore behaves as a single coupled fluid supporting acoustic oscillations. In the language of \GDOS{}, the microscopic rearrangements of photon-electron scattering histories lie on a large family of effectively equivalent descriptions so long as the macroscopic acoustic pattern remains the same. This is the cosmological manifestation of \DGI: many micro-histories belong to the same observable gauge orbit.

A convenient operational state manifold for the recombination-era plasma is
\begin{equation}
\mathcal Z_{\mathrm{rec}} = \set{T, X_e, \delta_b, \delta_\gamma, \vec v_b, \vec v_\gamma, \Phi, \Psi},
\end{equation}
where \(T\) is the temperature, \(X_e\) the ionization fraction, \(\delta_b\) and \(\delta_\gamma\) the baryon and photon density perturbations, and \(\vec v_b,\vec v_\gamma\) the velocity fields. The open-system viewpoint is not optional here. The observed CMB is not a direct bulk microstate. It is a boundary-registered remnant of a last-scattering event after coarse-graining, free streaming, projection, and later lensing.

The transition itself occurs when the scattering rate can no longer keep pace with expansion, roughly when
\begin{equation}
\Gamma_T \sim H.
\end{equation}
The usual thermodynamic account expresses this as the universe cooling enough for neutral hydrogen to form. The \GDOS{} refinement is to identify an order parameter,
\begin{equation}
\eta_{\mathrm{rec}} \equiv 1 - X_e,
\end{equation}
which moves from approximately zero in the ionized phase to approximately unity in the neutral phase. A schematic Landau-style potential then reads
\begin{equation}
V_{\mathrm{rec}}(\eta_{\mathrm{rec}}) = -\frac{\mu^2}{2}\eta_{\mathrm{rec}}^2 + \frac{\lambda}{4}\eta_{\mathrm{rec}}^4,
\label{eq:recomb-pot}
\end{equation}
with \(\mu^2\) changing sign across the recombination threshold. This is not meant to replace the Saha or Peebles recombination equations. Rather, it expresses in geometric language what the standard microphysics is doing: a previously symmetric, tightly coupled phase loses accessibility and the system settles onto a smaller set of frozen boundary patterns.

The attractor reached after this transition is the observed last-scattering imprint itself. Once photons decouple, the acoustic phases are no longer continuously reprocessed by scattering. They are frozen into a geometric attractor and later projected to the sky. It is in that sense that the CMB can be described as a \GSSB-to-\GA transition: the tightly coupled phase admits continual rephasing; the decoupled phase preserves only the frozen acoustic records.

This interpretive move is useful for two reasons. First, it clarifies the role of \DGI{} in a context where ordinary cosmological perturbation theory already worries about gauge dependence. The physical observables --- peak positions, relative heights, polarization patterns, damping tail --- are gauge-invariant exports of the open-system evolution. Second, it makes the later links to the Hubble tension and the structure-growth audit conceptually smoother, because both problems depend on how one passes from early-time microphysics to late-time inferred parameters.

\subsection{The \texorpdfstring{\(\pi\)}{pi}-locked background inventory and baryon loading}

The second inherited move is the use of the late-time \(\pi\)-locked inventory. Within the present ledger one takes
\begin{equation}
\Omm = \frac{1}{\pi},
\qquad
\OmL = 1 - \frac{1}{\pi},
\qquad
\Omb = \frac{1}{2\pi^2},
\qquad
\Omega_c = \Omm - \Omb = \frac{1}{\pi} - \frac{1}{2\pi^2}.
\label{eq:pi-locked-inventory}
\end{equation}
The first thing to notice is that these are not used as raw substitutes for every stage of the expansion history. Rather, they specify the late-time attractor shares of the boundary inventory. Even so, they already constrain several early-time observables because the relative matter shares and baryon loading are inherited from the same ledger.

A particularly useful quantity is the baryon fraction within total matter,
\begin{equation}
 f_b \equiv \frac{\Omb}{\Omm} = \frac{1}{2\pi} \approx 0.159.
\label{eq:baryon-fraction}
\end{equation}
This number is close to the usual observed baryon fraction and immediately fixes the rough inertia loading of the photon-baryon fluid. Standard recombination physics packages that inertia through
\begin{equation}
 R(a) \equiv \frac{3\rho_b}{4\rho_\gamma},
\end{equation}
which then determines the sound speed,
\begin{equation}
 c_s(a) = \frac{c}{\sqrt{3\paren{1+R(a)}}}.
\label{eq:sound-speed}
\end{equation}
Near recombination one typically has \(R\) of order unity but still less than one, so the sound speed is somewhat below \(c/\sqrt3\). In the \GG{}/\GDOS{} reading this is not an independent dial. It is an inherited consequence of the baryon share.

This point is worth emphasizing because it illustrates the general strategy of the paper. The framework is strongest not when it claims to compute every number from scratch in one step, but when it systematically strips away freedom. The baryon loading is no longer something to be chosen and then fitted. It is linked to the same compact-angle and anomaly-locked ledger already used for dark matter and baryogenesis. Precision cosmology then becomes an audit of that common inheritance.

\begin{table}[H]
\centering
\caption[Inherited background inventory for the CMB and growth audit]{Inherited background inventory used in the CMB and growth audit. The entries are interpreted as late-time attractor shares whose ratios propagate into the acoustic and growth sectors. The columns give the quantity, its locked expression, its approximate value, and its role in this Part. The entries are read as late-time attractor shares rather than as fitted parameters, which is the whole of the difference between this table and the corresponding row of a concordance fit.}
\label{tab:pi-inventory-part6}
\small
\begin{tabular}{p{0.23\textwidth}p{0.22\textwidth}p{0.22\textwidth}p{0.23\textwidth}}
\toprule
\textbf{Quantity} & \textbf{Locked expression} & \textbf{Approximate value} & \textbf{Role in Part VI}\\
\midrule
Total matter share & \(\Omm = 1/\pi\) & 0.318 & fixes the background matter budget entering growth and lensing\\
Baryon share & \(\Omb = 1/(2\pi^2)\) & 0.0507 & fixes baryon loading and the odd/even peak modulation\\
Cold dark matter share & \(\Omega_c = 1/\pi - 1/(2\pi^2)\) & 0.2676 & supplies the dominant clustering component\\
Dark-energy share & \(\OmL = 1 - 1/\pi\) & 0.6817 & fixes the late-time attractor background\\
Baryon fraction in matter & \(f_b = 1/(2\pi)\) & 0.159 & controls the matter-sector inertia ratio\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Acoustic ruler, peak sequence, and the fixed ratio \texorpdfstring{\(r_s/D_A\)}{rs/DA}}

The standard CMB observable that most cleanly couples early microphysics to later geometric inference is the acoustic scale
\begin{equation}
\theta_* = \frac{r_s(z_*)}{D_A(z_*)},
\label{eq:theta-star}
\end{equation}
where \(z_*\) denotes the effective last-scattering epoch, \(r_s\) is the comoving sound horizon,
\begin{equation}
 r_s(z_*) = \int_{z_*}^{\infty} \frac{c_s(z)}{H(z)}\,\dd z,
\label{eq:sound-horizon}
\end{equation}
and \(D_A\) is the comoving angular-diameter distance,
\begin{equation}
 D_A(z_*) = \frac{1}{1+z_*}\int_0^{z_*} \frac{\dd z}{H(z)}.
\label{eq:angular-diameter}
\end{equation}
The power of this observable is well known: the peak positions depend mostly on the ratio \(r_s/D_A\), not on either quantity in isolation. That is why changes in the sound horizon can masquerade as changes in the late-time Hubble constant when one insists on fitting the same observed \(\theta_*\).

In the present framework the acoustic scale is not treated as a free inference chain. Because both \(r_s\) and \(D_A\) are evaluated on the same inherited background, the ratio is narrowed. The working claim of the \GG{}/\GDOS{} notes is that the locked background yields
\begin{equation}
100\theta_* \approx 1.041,
\qquad
\ell_A \equiv \frac{\pi}{\theta_*} \approx 302.
\label{eq:theta-lA}
\end{equation}
Once the ordinary phase shifts from neutrino free streaming and baryon loading are included, this translates to the familiar first three peak locations,
\begin{equation}
\ell_1 \approx 220,
\qquad
\ell_2 \approx 540,
\qquad
\ell_3 \approx 800.
\label{eq:first-three-peaks}
\end{equation}
The physics here is familiar but worth restating carefully.

Compression phases in the coupled fluid produce odd peaks, while rarefaction phases produce even peaks. The heavier the baryon loading, the more the compression peaks are enhanced relative to the rarefaction peaks. Thus the odd/even modulation is the observational shadow of the baryon inertia. In standard cosmology this is encoded in \(R\). In the present framework, \(R\) is itself narrowed by the inherited \(\Omb\) and \(f_b\) shares. The odd/even modulation is therefore not merely fitted; it becomes an audit of whether the inherited matter ledger is viable.

An intuitive picture is helpful. The photon-baryon plasma before recombination is a slightly weighted spring-mass system. Radiation pressure acts as the spring. Baryons act as inertia. Gravity provides a background tilt that deepens compressions. The oscillation is not mysterious. What matters is who is allowed to choose the mass loading. In the present theory, the loading is already constrained before the CMB fit begins.

\subsection{Silk damping, BFFT, and the ultraviolet knee of the primary CMB}

The acoustic pattern would extend indefinitely to high multipole were it not for diffusion damping. In the standard treatment the relevant diffusion scale is determined by the photon mean free path and the finite thickness of the last-scattering surface. A schematic expression for the diffusion wavenumber is
\begin{equation}
 k_D^{-2}(a_*) \sim \int_0^{a_*} \frac{\dd a}{a^3 H(a)\,\dot\tau(a)}\,\frac{1}{6\paren{1+R(a)}}\paren{\frac{R^2(a)}{1+R(a)} + \frac{16}{15}},
\label{eq:diffusion-scale}
\end{equation}
where \(\dot\tau\) is the differential optical depth. The corresponding angular damping scale is obtained by projecting the diffusion length to the last-scattering surface.

The \GG{}/\GDOS{} proposal does not discard this logic. It superposes on it a BFFT interpretation. Because the logarithmic radial coordinate already packages scale hierarchies discretely through the \OGNfive{} integers, the ultraviolet knee of the primary anisotropy pattern is identified with the first non-trivial BFFT ultraviolet cutoff of the throat. The proposed locked relation is
\begin{equation}
\ell_D = K(2\pi)^2 = 35\times 4\pi^2 \approx 1.38\times 10^3.
\label{eq:ellD-BFFT}
\end{equation}
In plain language, the damping tail is read as the ultraviolet knee of the same multiscale bus that elsewhere appears as BFFT. The standard diffusion integral tells us \emph{how} damping happens in boundary microphysics. The BFFT relation proposes \emph{why the relevant angular cutoff should sit where it does} once the multiscale throat capacity is fixed by \(K=35\). It is essential to keep the two logically separate, because the framework now computes the first of them outright.

The framework's actual prediction of the damping scale is the standard diffusion integral \eqref{eq:diffusion-scale} evaluated on the \(\pi\)-locked inventory: running the exact fractions through the full Boltzmann ionization history gives \(\ell_D = 1350\pm3\) in the \(\Lambda\)CDM-inferred convention, against the observed \(\ell_D \approx 1345 \pm 15\) \citep{Planck2018} --- a \(+0.4\%\) agreement with zero adjustable parameters, and it is this Boltzmann value that the parameter-audit Dev plot scores (Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}; Section~\ref{subsec:cmb-core-table}). The BFFT reading must then be stated honestly. \(K(2\pi)^2 = 1381.7\) sits \(+2.7\%\) above the observed value and, more tellingly, \(+2.3\%\) above the framework's \emph{own} computed damping scale; whatever the multiscale bus is doing, it is not reproducing the diffusion integral exactly. The identification is therefore carried as a conjecture-level overlay --- a suggestive coincidence at the few-percent level whose burden has grown, since the BFFT-damping-knee lemma must now explain a \(2.3\%\) gap against the inventory itself, not merely sharpen a projection. If the lemma closes that gap, the overlay is promoted; if it cannot, the overlay falls away while the framework's \(\ell_D\) prediction --- the \(+0.4\%\) Boltzmann value --- stands untouched. This is still the program working as intended: the standard calculation is retained, the interpretation is additional, and the audit keeps them separate.

\subsection{Scalar tilt, polarization, and the tensor kill corridor}

The scalar tilt is the next unavoidable audit. The minimal slow-roll estimate for a geometric plateau producing roughly \(N_e\approx 60\) e-folds gives
\begin{equation}
 n_s \approx 1 - \frac{2}{N_e} \approx 1 - \frac{2}{60} \approx 0.967.
\label{eq:ns-60}
\end{equation}
A second mnemonic estimate, directly tied to the integer ledger, gives
\begin{equation}
 n_s \approx 1 - \frac{1}{K} = 1 - \frac{1}{35} \approx 0.971.
\label{eq:ns-K}
\end{equation}
The point is not that both expressions must be exact simultaneously. The point is that both fall in the same narrow red-tilted corridor. The framework therefore predicts that the scalar spectrum should be close to but below exact scale invariance, with no freedom to roam across a wide inflationary landscape.

The polarization sector follows the standard logic of Thomson scattering with a local quadrupole. Scalar perturbations source E-mode polarization, and the E-mode sequence is phase shifted relative to the temperature peaks because velocity maxima and density maxima do not occur at the same oscillation phase. Thus the observed TE anti-correlation and EE peak ordering remain standard diagnostics. Again, \GG{} is not changing the scattering geometry. It is constraining the source ledger that determines which oscillatory solution is present.

The tensor sector is where the paper must be especially explicit --- and where the derived plateau changes the statement from a bound to a number. On the derived GUT-adjacent plateau, the tensor amplitude follows from the same \(N^*\) as the tilt-scale slow-roll inflation. The practical outcome is simple: the primordial tensor signal is driven to a negligibly small level. To avoid artificial precision when the exact microphysical implementation is still being refined, it is best to state the result as a corridor:
\begin{equation}
 r = \frac{12}{N^{*2}} = 0.0043,
\label{eq:r-small}
\end{equation}
a factor eight below the current bound and several times above the Simons-Observatory/LiteBIRD design floor (the label is kept from the retired bound-form statement for the audit trail).

The scientific meaning is sharp. The framework predicts that delensed near-term B-mode data should begin to reveal a primordial component at precisely this amplitude. A tensor measurement inconsistent with it in either direction, in the conventional inflationary window would therefore act as a clean kill switch for the derived launch. This is precisely the kind of statement the paper should make. It is better to lose clearly than to win vaguely.

\subsection{Lensing amplitude, early dark energy residue, and the recombination-to-structure bridge}

The lensing amplitude occupies a particularly interesting role because it sits between the primary CMB and later structure growth. The present framework proposes a mild enhancement,
\begin{equation}
 A_L = 1 + \alpha_L,
\qquad
\alpha_L \approx 0.05,
\qquad
\Rightarrow\qquad
A_L \approx 1.05.
\label{eq:AL}
\end{equation}
This is not introduced as an independent phenomenological parameter. It is interpreted as a residual projection of the same bulk that stabilizes the throat and shapes the early source ledger.

The same source ledger also carries two early-time residues that become central in the next section:
\begin{equation}
 f_{\mathrm{EDE}} \approx 0.03,
\qquad
\Delta N_{\mathrm{eff}} \approx 0.30,
\qquad
z_c \approx 3500.
\label{eq:EDE-DR-values}
\end{equation}
Here \(f_{\mathrm{EDE}}\) is the fractional height of a brief early-dark-energy lobe, \(\Delta N_{\mathrm{eff}}\) is a small extra-radiation contribution, and \(z_c\) marks the approximate epoch at which the lobe is centered. These numbers matter because they affect the sound horizon, the equality timing, and the subsequent growth history. They are therefore the hinge between the CMB section and the later Hubble, \(S_8\), and JWST discussions.

\begin{table}[H]
\centering
\caption[Condensed CMB and primordial-sector audit]{Condensed CMB and primordial-sector audit for the minimal Part~VI ledger. The third column states the intended role of each quantity inside \ATen{}. The third column states what each entry is being asked to do, which is not the same as claiming the framework predicts it. Several rows are consistency checks against results the framework inherits rather than derives, and they are labeled so.}
\label{tab:cmb-audit-part6}
\small
\begin{tabular}{p{0.21\textwidth}p{0.22\textwidth}p{0.47\textwidth}}
\toprule
\textbf{Observable} & \textbf{Working corridor} & \textbf{Interpretation inside \GG{}/\GDOS{}}\\
\midrule
Acoustic angle & \(100\theta_*=1.036\) (\(-0.5\%\)) & predicted ratio of the sound horizon and angular-diameter distance on the locked background; the sharpest single carrier of the \(\pi\)-offset (Section~\ref{subsec:pi-offset})\\
Acoustic ruler & \(\ell_A\approx302\) & baseline harmonic scale before phase shifts\\
First three peaks & \((221,538,817)\) predicted & odd/even pattern governed by the forced baryon loading \(1/2\pi\); height ratios reproduced to \(0.3\)--\(0.8\%\) (Table~\ref{tab:cmb-core-derived})\\
Damping scale & \(\ell_D=1350\pm3\) (Boltzmann) & diffusion integral on the inventory (\(+0.4\%\)); the BFFT knee \(K(2\pi)^2\approx1382\) is a conjecture-level overlay \(+2.3\%\) above it\\
Scalar tilt & \(n_s = 0.9620\) (derived) & red tilt computed on the primary plateau, \(1-2/N^*\) at the undiluted \(N^*=52.6\)\\
Lensing amplitude & \(A_L\approx1.05\) & mild late projection/growth correction from the bulk\\
Tensor sector & \(r = 0.0043\) & firm primordial B-mode target, decisive for the Simons Observatory and LiteBIRD; a tensor measurement inconsistent with it in either direction is fatal\\
Early residue & \(f_{\mathrm{EDE}}\approx0.03\); \(\Delta N_{\mathrm{eff}}=0.04\) (baseline) & narrow early-time source correction from the throat/$\Sigma_r$ sector; the legacy \(\Delta N_{\mathrm{eff}}\approx0.30\) estimate is demoted to the CMB appendix as a non-unique cross-check\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Bottom line of the CMB audit}

The central point of the microwave-background analysis can now be stated cleanly. The present framework keeps the successful perturbation grammar of precision cosmology. It then removes much of the usual freedom in the source ledger. Recombination becomes a \GDOS{} transition from a tightly coupled \DGI{} phase to a frozen geometric attractor; the baryon loading is inherited from the \(\pi\)-locked matter inventory; the damping tail is related to the BFFT ultraviolet knee fixed by \(K=35\); the scalar tilt is confined to a narrow red corridor; the lensing amplitude is slightly elevated; and the tensor sector is effectively null.

If this whole package cannot survive the observed CMB simultaneously, the framework should be judged to fail before it is allowed to speak confidently about the Hubble tension or early galaxies. That is why the CMB section must come first in Part~VI. It is the precision gateway for everything that follows.

\section{Hubble tension}
\label{sec:h0tension}

The early-versus-late Hubble-constant discrepancy, the independent local and time-delay determinations, and the catalog of proposed solutions against which the two-stage corridor is positioned are surveyed in \citep{Verde2019,Wong2020,Freedman2021,DiValentino2021}.

\medskip
\textbf{Scale versus split: where this section sits.} It is worth stating at the outset what this section does and does not address, because the present Hubble constant carries two logically separate questions. Its \emph{absolute scale} --- why \(H_0\) has the order of magnitude it does --- was fixed in Part~IV (Section~\ref{sec:h0-derivation}): the \(\pi\)-locked inventory, confronted with the physical densities the microwave background measures, forces \(\Hzero^{(0)}\approx67\;\mathrm{km\,s^{-1}\,Mpc^{-1}}\), and the geometric route derives the same rate with zero sky input (Route~I). That background value \emph{coincides}, to about a percent, with the recombination-calibrated CMB value \(\approx67.4\): the two are the same rate, as they must be, since the CMB reads the background through the standard sound horizon and \GG{}'s exact fractions fit the acoustic peaks precisely there. The present section addresses the orthogonal question of the early-versus-late \emph{split} --- why the local distance-ladder value (\(\approx73.0\text{--}73.5\)) sits some \(9\%\) \emph{above} that background, while the CMB value sits on it. The observed local rate factorizes as \(H_0^{\mathrm{late}}=H_0^{(0)}\,\mathcal{C}_{\mathrm{local}}\): the background \(H_0^{(0)}\approx67\) from Part~IV, and a local-environment factor \(\mathcal{C}_{\mathrm{local}}>1\) developed below. In particular the terminus no-phantom result (Theorem~\ref{thm:no-phantom}) is this module's sharpest shared falsifier: a confirmed \(w<-1\) would break both the dark-energy terminus and the corridor that inherits it. The scale is fixed at \(\approx67\); what carries an open coefficient is only the local lift that raises the ladder reading --- as detailed below.


\begin{proposition}[The recombination-era cosmic \GRIPout{} switch]
\label{prop:recomb-gripout}
Under the \GRIP{} cascade of \ASix{}, the recombination era hosts the \GSSB{}
transition from the tightly coupled \DGI{} plasma to the frozen \GA{} attractor.
The corridor's defining switch event within that era is the \emph{geometric}
branch change, not the atomic transition itself, and the \OGNfive{} integer
ledger of \AFour{} fixes its redshift to
\[
z_{\mathrm{sw}} \equiv z_\ast = e^{J} = e^{7} \approx 1096.6 \qquad (J = 7),
\]
with no continuous parameter free to be tuned. The early-time (\(z>z_\ast\)) and
late-time (\(z<z_\ast\)) sides of this one switch are the two \OGGEP{} projections
of a single corridor; their present-epoch Hubble values are the content of the
canonical two-stage corridor Theorem~\ref{thm:hubble-two-stage-corridor}.
The atomic decoupling surface --- the visibility peak Planck measures, where
the atoms actually form --- is derived from the framework's own Rydberg and
baryon inventory in \S\ref{sec:two-redshifts} and lies $0.4\%$ (in e-folds)
below the switch, at the measured Planck value.
\end{proposition}

This proposition replaces the earlier device of patching the corridor
with a tunable early-dark-energy-plus-local-void vector. By the \MSMF{}/\OGN{}
uniqueness principle, a multi-parameter linear-response fit cannot be the unique
cost-minimum solution and so does not earn the status of a theorem; it survives in
Appendix~\ref{appA:cmb} only as a non-unique magnitude estimate of the corridor
gap. The structural result the body relies on is the single corridor of
Theorem~\ref{thm:hubble-two-stage-corridor}, whose existence is proved in Part~IX
from the constitutional hypotheses and whose uniqueness is the named open
Conjecture~\ref{conj:hubble-corridor}.



\subsection{The switch and the first atoms: two redshifts, one descent}
\label{sec:two-redshifts}

The proposition above fixes a \emph{geometric} redshift: the corridor
switch at $z_{\mathrm{sw}} = e^{J} = e^{7} \approx 1096.6$, an exact
integer statement with no continuous parameter. The microwave sky, by
contrast, is released at the \emph{atomic} decoupling redshift
$z_{\mathrm{dec}}$ --- the surface where the free electrons vanish
into hydrogen and the photon mean free path outruns the horizon. These
are two different quantities, and this section derives the second from
the framework's own atomic chain, with no fitted parameter, and then
states exactly how far apart the two land and what would make their
proximity a theorem. The result is one of the sharpest zero-parameter
reproductions in the program: the atomic chain lands on the measured
decoupling redshift at the $0.1\%$ level.

\paragraph{The atomic chain, assembled from the cohort's own outputs.}
Decoupling is composite. Its temperature is set by atomic physics: the
binding energy of hydrogen, $B = \tfrac12 \alpha^{2} m_{e} c^{2} =
13.606$~eV, is carried by the electron mass and charge that the
program derives in the particle sector (\citealp{ArisakaP3Electron};
the fine structure constant through \citealp{ArisakaA9SM}). But atoms
do not form at $k T \sim B$: the photon bath outnumbers the baryons by
$\eta^{-1} \sim 1.6\times10^{9}$, a number this paper owns through the
baryon fraction $\Omega_b = 1/(2\pi^{2})$ and the corridor $h$, so the
Boltzmann tail of the bath keeps hydrogen ionized until the
temperature falls to
\begin{equation}
k T_{*} \;\simeq\; \frac{B}{53.0},
\label{eq:tstar-53}
\end{equation}
the factor $53.0$ emerging from the Saha--Peebles recombination
calculation run with the framework's inputs (the number is
logarithmic in $\eta$, which is why it is so stable). The
\emph{redshift} of decoupling then compares that temperature with the
present one:
\begin{equation}
\ln(1+z_{\mathrm{dec}})
\;=\; \ln\!\frac{B}{k T_{0}} \;-\; \ln\!\frac{B}{k T_{*}}
\;=\; 10.967 \;-\; 3.971
\;=\; 6.996 .
\label{eq:lnz-decomp}
\end{equation}
Every entry on the right is a cohort output: the Rydberg from the
particle sector, the $53.0$ from the recombination microphysics with
$\Omega_b = 1/(2\pi^{2})$, and $T_{0}$ the present temperature of the
descent.

\paragraph{The quantitative result.} Running the full recombination
calculation (Peebles three-level atom; cross-checked against the
standard fitting form of \citealp{HuSugiyama1996}, which reproduces
the Planck value to $0.18\%$ on Planck inputs) with the framework's
inputs only --- $\Omega_b h^{2} = h^{2}/2\pi^{2} = 0.0230$,
$\Omega_m h^{2} = h^{2}/\pi = 0.1446$ at the corridor $h = 0.674$ ---
gives
\begin{equation}
z_{\mathrm{dec}}^{\mathrm{GG}} = 1091.1 \quad (\text{raw}); \qquad
1089.1 \quad (\text{modeling-calibrated}),
\label{eq:zdec-gg}
\end{equation}
against the measured $z_{\mathrm{dec}} = 1089.92 \pm 0.25$
\citep{Planck2018} --- the Planck-inferred value, itself a derived
parameter of the $\Lambda$CDM fit, model-dependent at a level far
below the offsets discussed here. The raw chain agrees at
$0.11$--$0.16\%$ across the two corridor readings: that is the
zero-fitted-parameter statement, and it already sits inside the
$0.18\%$ modeling systematic of the recombination calculation
itself. Calibrating the chain on the known modeling offset tightens
the agreement to $0.03$--$0.07\%$ --- a calibration anchored on
Planck's own analysis pipeline, stated for completeness and not
claimed as independent precision. The
prediction is rigid: the elasticities are
$d\ln z_{\mathrm{dec}}/d\ln(\Omega_b h^{2}) = -0.029$ and
$d\ln z_{\mathrm{dec}}/d\ln(\Omega_m h^{2}) = +0.011$ ---
logarithmically suppressed, so no admissible variation of the inputs
moves the landing point visibly. The other corridor reading
($h = 0.666$) shifts the result by less than $0.05\%$.

\paragraph{The two redshifts, side by side.} The atomic chain lands at
$\ln(1+z_{\mathrm{dec}}) = 6.996$; the switch sits at exactly $7$. The
offset is now a computed number,
\begin{equation}
\delta \;\equiv\; J - \ln(1+z_{\mathrm{dec}}) \;=\; 0.0041,
\qquad\text{i.e.}\qquad
\frac{e^{J}}{\,1+z_{\mathrm{dec}}\,} - 1 \;=\; e^{\delta} - 1 \;=\; 0.41\%,
\label{eq:delta-offset}
\end{equation}
with a $\sim0.2\%$ modeling systematic (the offset is computed on
the raw chain, from the unrounded $\ln(1+z_{\mathrm{dec}}) = 6.9959$;
the modeling-calibrated chain shifts it to $\delta \approx 0.0059$,
within that systematic). The precision comparison with
the microwave data belongs entirely to the atomic quantity --- and it
passes at the $0.1\%$ level; the switch is a distinct, exact,
topological marker $0.4\%$ above it. Any statement in this paper
comparing $e^{7}$ directly with the Planck redshift is to be read in
this two-quantity form.

\paragraph{Convention ($z$ versus $1+z$).} The e-fold identity fixes
$1 + z_{\mathrm{sw}} = e^{7}$ exactly; statements made at the level of
$z$ rather than $1+z$ (including $z_{\mathrm{sw}} \approx 1096.6$) are
leading order in $1/z$, a slack of one part in $1097$. Three offsets
circulate in this paper under two labels: $0.41\%$ is the e-fold
(log-space) switch--decoupling offset of Eq.~\eqref{eq:delta-offset};
in $z$-space the atomic value sits $0.50\%$ below the switch and the
Planck value $0.62\%$ below it. Throughout, ``$0.4\%$'' names the
e-fold offset and ``$0.6\%$'' the $z$-space switch--Planck gap.

\paragraph{Physics intuition.} The universe carries an odometer and a
thermometer. The odometer is geometric: the corridor switch trips when
the descent has unwound exactly $J = 7$ e-folds of expansion between
that event and the present --- an integer of the ledger, blind to
atomic physics. The thermometer is material: atoms form when the bath
cools through $B/53$, a threshold built from the electron's binding
energy and the photon count per baryon --- blind, on its face, to the
ledger. The measured sky says the odometer and the thermometer agree
to four parts in a thousand. Half of that marvel is now bookkeeping:
this section computed the thermometer's reading from the cohort's own
constants and found $6.996$. The other half --- why the geometric
integer sits within $0.4\%$ of the atomic landing --- is either a
coincidence the descent merely records, or a theorem the descent has
not yet yielded.

\paragraph{What would make the proximity a theorem.} The offset obeys
the identity $\delta = J - \ln(B/kT_{0}) + \ln 53.0$, in which $B$ and
the $53.0$ are already derived; \emph{the only quantity not yet
owned by the ledger is $T_{0}$}. The question therefore reduces,
exactly, to whether the descent derives the present temperature with
no astronomical input --- equivalently the photon entropy
normalization, equivalently $z_{\mathrm{eq}}$. One natural candidate
is tested and rejected here in the program's own spirit: the
$\pi$-locked closure $z_{\mathrm{eq}} = \pi e^{7} = 3445$ sits
$2.8\sigma$ above the measured $z_{\mathrm{eq}} = 3387 \pm 21$ ---
recorded so that no reader re-derives it. The route that remains open
is the \emph{e-fold budget}: $T_{0}$ is the reheating temperature
redshifted through the descent, and the framework owns every
ingredient --- the derived reheating scale of the inflation module,
the $g_{*}$ freeze-out counting of the particle sector, and the
corridor itself. If the total e-fold count from reheating to the
$\Omega_m = 1/\pi$ crossing proves ledger-expressible, then $T_{0}$,
and with it $\delta$, becomes a theorem --- the exact analogue of the
$K = 35$ resolution of the why-now coincidence. That computation is
left to a dedicated study; until it lands, $\delta = 0.0041$
is carried as a computed consistency, not an assumption, and nothing
downstream in this paper depends on its being zero.

\subsection{Why the Hubble tension is really an inference-chain tension}

The Hubble tension is often presented as a simple clash of numbers. That presentation is not wrong, but it is incomplete. The deeper issue is that the early-universe inference chain and the late-universe distance-ladder chain are both attempting to infer what is spoken of as the ``same'' Hubble constant, yet they do so through very different operational routes. Early-time inference reads an acoustic ruler through a calibrated recombination and background-expansion model. Late-time inference reads a local distance ladder through standard candles, geometric calibrators, and a local flow model. The disagreement is therefore not merely numerical; it is a disagreement between operational dictionaries.

The first task of the present section is to restate the standard acoustic logic. The angular scale \(\theta_*\) is directly observed. If one holds it fixed, then any downward shift in the sound horizon must be compensated by an upward shift in the inferred late-time expansion rate. To first order one may write
\begin{equation}
\theta_* = \frac{r_s}{D_A}
\qquad\Rightarrow\qquad
\frac{\delta \theta_*}{\theta_*} = \frac{\delta r_s}{r_s} - \frac{\delta D_A}{D_A}.
\end{equation}
If the observed \(\theta_*\) is held fixed, then
\begin{equation}
\frac{\delta D_A}{D_A} \approx \frac{\delta r_s}{r_s}.
\label{eq:fixed-theta}
\end{equation}
Because the late-time angular-diameter distance roughly scales like \(D_A\propto H_0^{-1}\) when the rest of the low-redshift background is not radically altered, one arrives at the standard lever-arm relation
\begin{equation}
\frac{\delta H_0}{H_0} \approx -\frac{\delta r_s}{r_s}.
\label{eq:h0-rs}
\end{equation}
This is why the sound horizon is the natural first place to look if one wants to raise the early-universe inferred value of \(H_0\) without discarding the entire CMB fit.

From the standpoint of \ATen{}, this standard logic is welcome. It means that a constrained early source correction can do real work. One does not need an arbitrary late-time fluid if the sound horizon can be moved by a physically inherited correction to the pre-recombination budget.

\subsection{\texorpdfstring{$H(t)$}{H(t)}: the derived expansion history, of which \texorpdfstring{\(\Hzero\)}{H0} is one point}
\label{sec:h-of-t-derived}

Before the two projections are separated, it is worth drawing the object they are
projections \emph{of}. In a flat Friedmann--Robertson--Walker universe the
expansion rate is not an independent quantity; it is fixed algebraically by the
energy content,
\begin{equation}
H(t)^2 \;=\; \frac{8\pi G_N}{3}\sum_i \rho_i(t)
\;=\; \Hzero^{2}\!\left[\Omega_r a^{-4} + \Omm\, a^{-3} + \OmL\right],
\label{eq:friedmann-derived}
\end{equation}
so once the densities are given as functions of time, \(H(t)\) is their
consequence and nothing further is free to choose. This is exactly the situation
\ATen{} is in. The geometry supplies the three ingredients
Eq.~\eqref{eq:friedmann-derived} needs: spatial flatness from the boundary
Friedmann export; the present-epoch densities as the \(\pi\)-locked inventory
\(\Omm=1/\pi\), \(\OmL=1-1/\pi\), \(\Omb=1/(2\pi^2)\) (Part~X); and their evolution
in time, since \(\rho_m\propto a^{-3}\), \(\rho_r\propto a^{-4}\), and
\(\rho_\Lambda\) is constant because the terminus equation of state is the
\emph{derived} value \(w=-1\) (Theorem~\ref{thm:no-phantom}). Feeding these into
Eq.~\eqref{eq:friedmann-derived} yields a single curve, drawn in
Figure~\ref{fig:H-of-t-derived}: the entire expansion history of the universe,
from the radiation era through the matter era to the de~Sitter terminus, with no
continuous parameter adjusted.

The figure makes a structural point that the phrase ``Hubble constant'' tends to
hide. \(H(t)\) falls monotonically for all time and asymptotes to the constant
\(H_\infty=\Hzero\sqrt{\OmL}=55.0~\mathrm{km\,s^{-1}\,Mpc^{-1}}\); it never rises,
not even after the expansion begins to accelerate at \(z\approx0.6\), because
\(H=\dot a/a\) continues to decrease while \(\ddot a>0\). The familiar \(\Hzero\)
is simply the value this curve takes at the present epoch,
\(\Hzero\equiv H(t_0)=66.6\) at \(t_0=13.9\) Gyr. It is one reading of a derived
function, not a fundamental constant in its own right.

This reorders the logical hierarchy of the whole Hubble question. In the
\(\Lambda\)CDM fitting language \(\Hzero\) is a \emph{primary} parameter, adjusted
alongside the densities until the model matches the data. In \ATen{} the densities
are primary --- they are geometric outputs --- and \(H(t)\), together with its
present value \(\Hzero\), is \emph{downstream}: the geometry
(\GG{}/\GDOS{}/\GRIP{}) fixes \(\rho_i(t)\), the Friedmann law turns \(\rho_i(t)\)
into \(H(t)\), and \(\Hzero\) is one point on \(H(t)\). The early-versus-late
\emph{tension} sits one step further down still: it is a mismatch in the
calibration of that single point between two measurement pipelines, not a mismatch
in the derived history, which agrees with the microwave background across the whole
acoustic spectrum. In this sense the \(\Hzero\) tension is a \emph{downstream,
secondary} problem --- a question about how faithfully one local measurement reads
one point of a curve the geometry has already fixed.

Four honesties keep this from overreaching. \emph{First}, the reduction
\(\rho_i(t)\Rightarrow H(t)\) presupposes the Friedmann law, which \ATen{} does not
assume but earns as the boundary general relativity of \GDOS{} (\AFive{},
\AEight{}); the claim is that the geometry derives the densities \emph{and} the law
that turns them into a rate. \emph{Second}, the inventory is \(\mathsf D^{*}\) ---
accurate at the one-to-three-percent level, conditional on the Hopf-fiber measure
lemma and the declared observation epoch --- and \(\Omega_r\) is calibrated from the
measured microwave temperature; \(H(t)\) inherits exactly that grade, so
``derived'' here means predictive-to-a-few-percent, not exact and unconditional.
\emph{Third}, \(H(t)\) is dimensionful: the densities fix its \emph{shape}, while
the single dimensionful scale is \(G_N\) (equivalently the Planck mass) --- the same
universal constant that sets every particle mass in \ANine{}, not a quantity read
from the sky. \emph{Fourth}, locating the tension downstream diagnoses it but does
not by itself resolve it: whether the local \(\approx73\) is a biased reading of a
background \(\approx67\), or the background is genuinely high, is still decided by
data --- above all the ladder-free gravitational-wave standard siren, on which
\ATen{} stakes the falsifiable prediction \(\Hzero^{\mathrm{GW}}\to67\). With those
four fences in place the reading is this: the expansion history is a derived
geometric object, and the decade's famous quarrel is a question about the
calibration of one of its points.

\begin{figure}[p]
\centering
%% v15.7: height cap 0.88 -> 0.79\textheight.  At 0.88 the cap was
%% INERT: the asset is 604.80 x 698.40 pt, so width=\textwidth
%% scales it by 0.7767 to 469.76 x 542.46 pt, while 0.88\textheight
%% would have allowed 0.8196 -- wider than the text block.  Picture
%% plus \abovecaptionskip plus a nine-line caption came to more than
%% \textheight and the last caption line printed in the footer band,
%% 17.66 pt below the block, beside the folio.  0.79 binds and gives
%% back 28.62 pt.  Measured by bottom_scan.py; no word of the
%% caption changes.
\includegraphics[width=\textwidth,height=0.79\textheight,keepaspectratio]{a10_H_of_t_derived.pdf}
\caption[The derived Hubble parameter \(H(t)\) of the \GG{} background, of which \(\Hzero\) is one point.]{\textbf{The Hubble parameter \(H(t)\) is a derived geometric function; \(\Hzero\) is one reading of it.} The background expansion rate of \ATen{}, computed from Eq.~\eqref{eq:friedmann-derived} with the \(\pi\)-locked inventory (\(\Omm=1/\pi\), \(\OmL=1-1/\pi\), \(\Omb=1/2\pi^2\)) and the derived \(w=-1\); the one dimensionful scale is \(\Hzero\) (equivalently \(G_N\)). \emph{Top:} the full history on log--log axes, from the radiation era (\(H\propto a^{-2}\propto t^{-1}\)) through the matter era (\(H\propto a^{-3/2}\propto t^{-1}\)) to the dark-energy era. \emph{Bottom:} the late universe in linear time. The rate falls monotonically through every era and asymptotes to the de~Sitter floor \(H_\infty=\Hzero\sqrt{\OmL}=55.0~\mathrm{km\,s^{-1}\,Mpc^{-1}}\) --- never rising, even once the expansion accelerates. The familiar \(\Hzero\equiv H(t_0)=66.6\) is one point on this curve, at today's epoch \(t_0=13.9\) Gyr; a single number does not capture the dynamical change of the Universe's size. This entire history is a \GG{}/\GDOS{}/\GRIP{} output: fix the densities and their time dependence, and \(H(t)\) follows.}
\label{fig:H-of-t-derived}
\end{figure}

\subsection{The two-stage corridor: one Friedmann structure, two projections}
\label{sec:part6-corridor}

The resolution this framework offers is structural, not parametric. The boundary
Friedmann equation --- the \OGGEP{} export of the \GG{} bulk --- admits a
single \emph{two-stage corridor} of solutions, and the early and late Hubble
values are its two projections across the \GRIPout{} switch of
Proposition~\ref{prop:recomb-gripout}. This is the content of the canonical
existence Theorem~\ref{thm:hubble-two-stage-corridor}, proved in Part~IX from the
constitutional hypotheses; here we give the physical reading and the two
falsifiable numbers, deferring the formal proof and the legacy linear-response
cross-check to Appendix~\ref{appA:cmb}.

The lever that makes the two projections differ is the acoustic angle. The
observed \(\theta_\ast = r_s(z_\ast)/D_A(z_\ast)\) is fixed to four digits, so at
fixed \(\theta_\ast\) any change in the inferred sound horizon must be absorbed by
a compensating change in the inferred late-time distance scale,
\[
\frac{\delta\Hzero}{\Hzero} \approx -\frac{\delta r_s}{r_s}
\qquad\text{(at fixed observed }\theta_\ast\text{)}.
\]
The early/sound-horizon dictionary reads the universe \emph{through} the switch:
it calibrates \(\Hzero\) from \(r_s(z_\ast)\) on the pre-recombination budget, and
on the inherited \(\pi\)-locked inventory with the audit value \(\DeltaNeff=0.04\)
it projects to
\[
\Hzero^{\mathrm{CMB}} = \Hzero^{(0)}\,\kappa_{\mathrm{CMB}}(\Omm,\Omb,\DeltaNeff)
\approx 67.4 \pm 0.5~\mathrm{km\,s^{-1}\,Mpc^{-1}}.
\]
The late/distance-ladder dictionary reads the universe \emph{after} the switch, on
the present-day \GA{} attractor, and projects to
\[
\Hzero^{\mathrm{late}} = \Hzero^{(0)}\,\kappa_{\mathrm{late}}(\Omm,\OmL)
\approx 73.0 \pm 1.0~\mathrm{km\,s^{-1}\,Mpc^{-1}}.
\]
Figure~\ref{fig:hubble-two-stage-corridor} shows the structure: one corridor, one
switch at \(z_\ast = e^{7}\), two projection windows.

\begin{figure}[H]
\centering
\resizebox{0.853\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\sffamily\small},
 >={Stealth[length=2.4mm]},
]
% Horizontal axis = redshift (decreasing left-to-right is natural for "time forward")
% Left edge: high redshift; right edge: z=0 (now)
\def\axisl{0}
\def\axisr{15}
\def\zswitch{4.5}
\def\axisy{0}

% Main axis
\draw[line width=0.80pt,->] (\axisl,\axisy) -- (\axisr+0.6,\axisy) node[right] {time forward};

% Redshift tick labels
\foreach \x/\lbl in {\axisl/{$z > z_*$}, \zswitch/{$z = z_*$}, \axisr/{$z = 0$}} {
 \draw[line width=0.80pt] (\x,\axisy-0.12) -- (\x,\axisy+0.12);
 \node[below=2pt] at (\x,\axisy-0.12) {\lbl};
}

% Early-time branch box (left side)
\node[fill=blue!15, draw=blue!60!black, line width=0.80pt, inner sep=4pt, font=\sffamily\bfseries\footnotesize,
 align=center, minimum width=3.8cm, minimum height=1.3cm] (early) at ({\axisl + 1.7}, {\axisy + 1.8})
 {Early-time branch\\$z > z_*$\\\rule{0pt}{0pt}\\$H_0^{\rm CMB} \approx 67.4$ km/s/Mpc};

% Switch event box
\node[fill=red!18, draw=red!60!black, line width=0.80pt, inner sep=4pt, font=\sffamily\bfseries\footnotesize,
 align=center, minimum width=3.0cm, minimum height=1.3cm] (sw) at (\zswitch, {\axisy + 4.0})
 {Corridor switch\\$z_{\rm sw} = e^7 \approx 1097$\\\rule{0pt}{0pt}\\\footnotesize atomic decoupling\\\footnotesize $0.4\%$ below, at Planck $1090$};

% Late-time branch box (right side)
\node[fill=green!16, draw=green!60!black, line width=0.80pt, inner sep=4pt, font=\sffamily\bfseries\footnotesize,
 align=center, minimum width=3.8cm, minimum height=1.3cm] (late) at ({\axisr - 1.7}, {\axisy + 1.8})
 {Late-time branch\\$z < z_*$\\\rule{0pt}{0pt}\\$H_0^{\rm late} \approx 73.0$ km/s/Mpc};

% Arrows from branches to switch and from switch out to late
\draw[->,line width=0.80pt,blue!60!black] (early.east) to[bend left=20] (sw.west);
\draw[->,line width=0.80pt,green!60!black] (sw.east) to[bend left=20] (late.west);

% OGN-5 integer ledger annotation below
\node[fill=yellow!18, draw=yellow!50!black, line width=0.80pt, inner sep=3pt, font=\sffamily\footnotesize,
 align=center, minimum width=4cm] (ogn) at ({\zswitch}, {\axisy - 1.5})
 {OGN-5 integer ledger\\$(M,N,L;J,K)=(1,3,5;7,35)$\\$z_* = e^J = e^7$};
\draw[line width=0.80pt,dashed,gray] (sw.south) -- (ogn.north);

% Vertical dashed line marking z_*
\draw[line width=0.80pt,dashed,red!60!black] (\zswitch,\axisy-1.0) -- (\zswitch,\axisy+0.5);

% Bottom caption-like text
\node[below right=12mm and 0pt, font=\sffamily\tiny, text=gray] at (\axisl,\axisy-0.6) {(left: CMB-projected sound horizon)};
\node[below left=12mm and 0pt, font=\sffamily\tiny, text=gray] at (\axisr,\axisy-0.6) {(right: distance-ladder)};

\end{tikzpicture}}
\caption[The Hubble two-stage corridor: one structure, two projections.]{\textbf{Hubble two-stage corridor of Theorem~\ref{thm:hubble-two-stage-corridor}.} The boundary Friedmann equation admits a single corridor of solutions with two natural projections separated by a switch event at recombination. The corridor switch redshift $z_* = e^J = e^7 \approx 1096.6$ is identified in the present paper as the exponential of the \OGNfive{} integer $J = 7$ inherited from \AFour{} --- a topological marker distinct from the operational decoupling redshift: the atomic decoupling derived from the framework's own constants lies $0.4\%$ below the switch, on the Planck value $z_{\rm dec} = 1089.92 \pm 0.25$ at the $0.1\%$ level (\S\ref{sec:two-redshifts}). The early-time branch (left) projects to the CMB-inferred Hubble value $H_0^{\rm CMB} \approx 67.4$ km/s/Mpc; the late-time branch (right) is read by the distance ladder at $H_0^{\rm late} \approx 73.5$ km/s/Mpc (the H0DN consensus). The corridor explains why the two projections need not return one closed-system constant; how much of the late reading's displacement is derived (the $+2$--$3\%$ Local-Hole lift) and how much is a named calibration systematic is priced in Section~\ref{sec:h0tension}.}
\label{fig:hubble-two-stage-corridor}
\end{figure}

The physical content is that the two pipelines are not two measurements of one
closed-system constant; they are two \GDOS{} open-system coarse-grainings of one
boundary history, separated by a real geometric event. That is why they may
legitimately differ --- and why the size of the difference is fixed by the
geometry rather than chosen. The honest status is preserved exactly as in the
codebook audit: the corridor's \emph{existence} is the proved
Theorem~\ref{thm:hubble-two-stage-corridor}, its \emph{uniqueness} is the named
open Conjecture~\ref{conj:hubble-corridor}, and the calibration factor
\(\kappa_{\mathrm{late}}\) remains an inherited Derived-conditional quantity. The
body presents the structural mechanism and flags the calibration as a declared
open problem; it does not overclaim.

\subsection{The absolute anchor and the local-ladder excess}
\label{sec:h0-two-values}

The previous subsection wrote each measured value as a projection \(\Hzero=\Hzero^{(0)}\kappa\) of one background rate \(\Hzero^{(0)}\). The terminus derivation of Section~\ref{sec:h0-derivation} supplies that background from first principles: the \(\pi\)-locked inventory, confronted with the physical densities the CMB measures, forces
\begin{equation}
\Hzero^{(0)}
=100\,\sqrt{\frac{\omega_i}{\Omega_i}}\;\mathrm{km\,s^{-1}\,Mpc^{-1}}
\approx 66.6\pm0.2
\qquad(\Omega_m=1/\pi,\ \Omega_b=1/2\pi^2),
\label{eq:partVI-anchor}
\end{equation}
and derived independently, with zero sky input, by the geometric
route, \(\Hzero^{(0)}=498.3\sqrt{\pi/175}=66.8\pm0.2\)
(Section~\ref{sec:h0-derivation}). For the first time the corridor's anchor is a theorem rather than a measured input, and that is the robust content: the present background expansion rate is \(\Hzero^{(0)}\approx67\). It is falsifiable directly. The model-independent gravitational-wave standard-siren channel, which reads the background expansion without either a sound horizon or a distance ladder, is predicted to converge near \(\Hzero^{\mathrm{GW}}\approx67\); a converged siren value at \(73\) would refute the inventory. Against this background the two famous discordant numbers are not symmetric: the CMB value sits \emph{on} the background, and only the local ladder is displaced.

\paragraph{The early/CMB value is the background.}
The acoustic dictionary infers \(\Hzero\) from \(\theta_\ast=r_s/D_A\) at fixed measured \(\theta_\ast\). Fed \GG{}'s exact \(\pi\)-locked fractions, a full Boltzmann solution reproduces the entire acoustic spectrum --- peak positions and the second-to-first height ratio --- at \(\Hzero\approx67\), essentially on top of \(\Lambda\)CDM's \(67.4\). There is no ``CMB deficit'' to explain: the recombination-calibrated value and the \GG{} background are the same rate to about a percent, and that agreement is a successful zero-parameter postdiction of the microwave sky, not a tension. (An earlier reading treated \(67.4\) as a reading biased \(\sim\!6\%\) low relative to a higher anchor at \(71.3\); the Boltzmann spectrum does not support that picture --- \GG{}'s fractions fit the peaks at \(67\), not above --- so the anchor is \(\approx67\) and the CMB carries no deficit.)

\paragraph{The late/local value reads high.}
The one real displacement is the distance ladder. Its value (\(73.50\pm0.81\), the H0DN Local Distance Network consensus, \citealp{CasertanoH0DN2026}; \(73.0\pm1.0\), \citealp{Riess2022}) sits some \(9\%\) \emph{above} the background \(\approx67\), a \(\sim\!5\sigma\) offset. \GG{} supplies one physical channel that biases the \emph{locally} measured rate upward: the nearby volume is a mild under-density (the observed ``Local Hole''), and a ladder calibrated inside it reads an expansion rate high by \(\delta\Hzero/\Hzero=-\tfrac13 f\,\delta\approx+2\text{--}3\%\) for \(\delta\approx-0.1\) to \(-0.2\) --- itself a \GRIPin{} structure-formation feature, not a tunable void inserted by hand. That carries the ladder to \(\approx68.5\text{--}69\), about one-third of the way to \(73.5\). The remaining \(\sim\!7\%\) is beyond any \GG{} (or standard) \emph{physical} mechanism: in particular the previously invoked \(\mathrm{U}(1)_B\) graviphoton lift is excluded as the cause, since equivalence-principle and lunar-laser-ranging bounds cap a baryon-coupled fifth force at \(\alpha_B\lesssim10^{-9}\), seven to eight orders below the \(\sim\!3\times10^{-2}\) that a \(3\%\) lift would require. In the reading \GG{} now shares with the ``CMB-is-right'' mainstream, that residual is a distance-ladder calibration systematic, not a number \GG{} derives.

\begin{table}[htbp]
\centering
\caption[Status of each element of the Hubble reading]{Status of each element of the Hubble reading, with the grade each carries. The third column is the one to read first: it distinguishes the background value, which is derived twice by independent roads, from the residual offset against the distance ladder, which this framework does not derive and treats as a calibration systematic in the reading it now shares with the CMB-is-right mainstream. Printing the two at the same size would be the easiest way to overstate the result, so they are graded apart.}
\label{tab:hubble-status-ledger}
\small
\begin{tabular}{p{0.29\textwidth}cp{0.45\textwidth}}
\toprule
quantity & value & status\\
\midrule
background \(\Hzero^{(0)}\) & \(\approx67\) & derived twice: geometry (\(\zeta_\Lambda=\pi/175\), Derived cond.) \(+\) \(\pi\)-inventory (\(\mathsf D^{*}\)), agreeing at \(0.6\sigma\)\\
CMB value \(\Hzero^{\mathrm{CMB}}\) & \(67.4\) & \emph{equals} the background (Boltzmann fit); no bias to close\\
local ladder \(\Hzero^{\mathrm{late}}\) & \(73.50\pm0.81\) & \(+2\text{--}3\%\) Local-Hole lift (\GRIPin{}) atop a \(\sim\!7\%\) ladder systematic (open)\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The physical picture.}
There is a single background expansion rate, \(\approx67\;\mathrm{km\,s^{-1}\,Mpc^{-1}}\), derived from the geometry through the vacuum seesaw and confirmed by the \(\pi\)-locked inventory against the sky's densities. The microwave background reads exactly this rate. The Hubble ``tension'' is therefore not a symmetric disagreement about the background; it is a one-sided excess in the \emph{local} distance ladder, part of which (\(+2\text{--}3\%\)) is a \GG{} structure-formation effect and the rest a ladder calibration systematic. This is the \GDOS{}/\OGGEP{} reading: the background is one number, and one of the two pipelines --- the local ladder --- carries a known environmental bias.

\paragraph{Honest grade.}
The robust, falsifiable output is the background \(\Hzero^{(0)}\approx67\), derived twice --- from the geometry alone (Route~I) and from the inventory (Route~II) --- confirmed by the CMB acoustic spectrum, and testable by the model-independent siren channel (predicted near \(67\)). The age of the universe follows as \(13.9\) Gyr today, the familiar \(13.8\) recovered as the age at the \(\Omega_m = 1/\pi\) crossing (Section~\ref{subsec:true-age}). What is \emph{not} finished is a first-principles account of the full local-ladder excess: \GG{} derives the sign and roughly a third of its size (the Local-Hole lift) and reads the remainder as a calibration systematic. No step is fitted; the unfinished piece --- the residual ladder offset --- is named. The decisive content is that the absolute scale of \(H_0\) is now a theorem of the ledger, and it lands where the CMB already sits.

\subsection{The local branch and the inherited baryonic sector}

The late/distance-ladder projection is realized in a local environment, and the
gauged ultra-weak \(\mathrm{U}(1)_B\) baryonic sector inherited from the matter
papers supplies the small environmental correction that anchors it. An honest
accounting must record that a first-pass estimate of that correction from the
\(\mathrm{U}(1)_B\) coupling falls short of the required magnitude by more than an
order of magnitude; closing or honestly retiring that gap is a declared open
problem of the corridor module and of the companion paper C2-Hubble
\citep{ArisakaC2Hubble}. The quantitative legacy linear-response that estimated
this lift --- the early-dark-energy lobe, the extra-radiation term, and the
local-void factor --- is collected, with its full honesty caveats, in
Appendix~\ref{appA:cmb}; it survives there only as a non-unique magnitude estimate
of the corridor gap, not as the mechanism.

\subsection{The \GDOS{} interpretation: a mis-specified dictionary problem rather than a contradiction of laws}

The Hubble tension is often narrated as if one side of the measurement must simply be wrong. The \GDOS{}/\OGGEP{} refinement offers a subtler reading. A finite observer never has access to the closed six-dimensional bulk. Different observational pipelines constitute different open-system projections and different coarse-graining channels. The early-universe acoustic route and the local distance-ladder route are therefore not mathematically identical measurements of a closed-system constant. They are distinct operational dictionaries applied to the same underlying bulk history.

This does not mean that any disagreement can be excused by semantic flexibility. Quite the opposite. It means that the theory must be explicit about what each pipeline is actually inferring. The early CMB route infers a late-time \(H_0\)-like quantity under a calibration chain that depends sensitively on the sound horizon, recombination physics, and the assumed early energy budget. The local route reads a nearby background expansion through a ladder of standard candles and geometric anchors, and is therefore more exposed to late-time environmental realization.

In that precise sense the Hubble tension becomes partly a mis-specified-dictionary problem. The error is to assume that both pipelines must return the identical number before one has audited whether the early and late projection dictionaries are actually the same. \GG{}/\GDOS{} then proposes a concrete and testable way this mismatch can arise: a narrow early-time source correction plus a mild late-time local lift.

\subsection{A compact derivation of the numerical corridor}

In one line, then, the Hubble module is one corridor read twice: the early branch at \(67.4\) --- the background itself, which Section~\ref{sec:h0-derivation} derives from the geometry alone --- and the late branch at the distance ladder's \(73.50 \pm 0.81\) (the H0DN Local Distance Network consensus, \citealp{CasertanoH0DN2026}), separated by the \(z_\ast = e^{7}\) switch. Of the late branch's \(\approx\!9\%\) displacement from the early one, the geometry derives the sign and roughly a third --- the \(+2\)--\(3\%\) Local-Hole lift, a \GRIPin{} structure-formation feature --- and the remaining \(\sim\!7\%\) is carried openly as a named distance-ladder calibration systematic, not a number the geometry fixes. The companion paper C2-Hubble \citep{ArisakaC2Hubble} prices exactly this decomposition against the H0DN error budget and stakes the verdict on the standard-siren channel.

\begin{table}[H]
\centering
\caption[Two-projection decomposition of the Hubble corridor]{Two-projection decomposition of the Hubble corridor in \ATen{}. The
corridor is one structure; the rows are its two \OGGEP{} projections, the switch
that separates them, and the open-system distinction that explains why the two
windows need not return the same number.}
\label{tab:hubble-module}
\small
\begin{tabular}{p{0.24\textwidth}p{0.22\textwidth}p{0.20\textwidth}p{0.24\textwidth}}
\toprule
\textbf{Element} & \textbf{Reading} & \textbf{Projection} & \textbf{Role}\\
\midrule
Early branch (\(z>z_\ast\)) & sound-horizon dictionary & \(\Hzero^{\mathrm{CMB}}\approx67.4\) & calibrates \(\Hzero\) from \(r_s(z_\ast)\) through the switch\\
Late branch (\(z<z_\ast\)) & distance-ladder dictionary & \(\Hzero^{\mathrm{late}}\approx73.5\) & reads \(\Hzero\) on the present-day \GA{} attractor\\
Corridor switch \(z_\ast\) & integer ledger \(e^{J}\) & \(z_\ast=e^{7}\approx1097\) & the \GRIPout{} event fixing the split, no free parameter\\
\OGGEP{}/\GDOS{} distinction & conceptual, not free & --- & why the two projections need not be one closed-system constant\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Success criteria and kill switches for the Hubble module}

The Hubble section should not end with rhetoric. It should end with success and failure criteria --- and they have sharpened since the early-correction corridor was retired: the derived background needs no early-time modification, so the criteria now guard the derivation itself.

The module succeeds only if the following all hold simultaneously:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item the two routes keep agreeing as the data sharpen: the geometric determination (\(66.8\pm0.2\), zero sky input) and the density inversion (\(66.6\pm0.2\)) remain concordant,
\item the microwave sky remains fit by the exact \(\pi\)-locked fractions with no early-dark-energy lobe and no radiation excess beyond the counted \(\Delta N_{\mathrm{eff}}\approx0.04\),
\item the local lift remains mild (\(+2\)--\(3\%\)) rather than becoming a hidden late-time free sector, with the residual ladder excess carried openly as a calibration systematic, and
\item the model-independent standard-siren channel converges near \(67\).
\end{enumerate}

Conversely, the module fails if any one of the following occurs:
\begin{enumerate}[leftmargin=2.0em,label=\alph*.]
\item a converged standard-siren catalog reads the background near \(73\),
\item a genuine early-dark-energy lobe (\(f_{\mathrm{EDE}}\gtrsim0.02\) at \(\ge3\sigma\)) or a radiation excess incompatible with \(\Delta N_{\mathrm{eff}}\approx0.04\) is detected,
\item the required local correction becomes implausibly large or inconsistent with large-scale isotropy, or
\item the completed Hosotani computation (\AFive{}~O6) returns \(\zeta_\Lambda\) outside \([0.0177,\,0.0182]\) --- off the pre-registered \(\pi/175\) (falsifier \textsf{F-Hubble-11} of \PaperCTwo{}) --- demoting the geometric route from a point to its bracket.
\end{enumerate}

This style of explicit failure condition is essential for the credibility of the paper. The theory should not present the Hubble tension as solved merely because it can write down a plausible direction of correction. It should state clearly what would count as losing.

\section{The \texorpdfstring{\(S_8/\sigma_8\)}{S8/sigma8} matter-clustering tension}
\label{sec:s8}

The clustering (\Seight{}) tension and its connection to the wider set of cosmological tensions and anomalies are surveyed in \citep{Abdalla2022}.


\begin{proposition}[One-ledger sign constraint for the growth block (T-Cosmos-11)]
Because \(S_8=\sigma_8(\Omega_m/0.3)^{1/2}\) belongs to the same inherited
ledger as the Hubble and CMB corrections, any parameter change that helps one
block must be tracked through the others with a fixed sign matrix. The framework
therefore treats the \(S_8\) problem not as a free one-parameter fix but as a
cross-consistency audit.
\end{proposition}



\subsection{Definition of the observable and why it is difficult to move cleanly}

The \(S_8\) tension concerns the amplitude of late-time matter clustering inferred from weak lensing and related low-redshift probes relative to the amplitude inferred from early-universe calibration. The usual compressed variable is
\begin{equation}
 S_8 \equiv \sigma_8\paren{\frac{\Omm}{0.3}}^{1/2},
\label{eq:S8def}
\end{equation}
where \(\sigma_8\) is the rms linear matter fluctuation on \(8\,h^{-1}\mathrm{Mpc}\) scales at late times. The reason this quantity is so hard to move cleanly is that it depends on several linked ingredients: the primordial amplitude, the scalar tilt, the matter content, the neutrino sector, the equality timing, and the whole late-time growth history. Any proposal that helps \(H_0\) by changing the early universe often alters the growth history as well.

This is why the \(S_8\) problem is the natural partner of the Hubble problem. A model that fixes \(H_0\) but worsens \(S_8\) has not really won; it has simply moved the tension from one part of the data chain to another. The present section therefore asks a stricter question: can the same narrow early-time source pair that helps \(H_0\) also move \(S_8\) in the correct direction, while remaining consistent with the CMB and without spoiling the high-redshift small-scale structure needed later for early galaxies?

\subsection{Standard linear growth equation and the scale on which \texorpdfstring{\(S_8\)}{S8} lives}

For sub-horizon matter perturbations in the linear regime, the standard growth factor \(D(a)\) satisfies
\begin{equation}
\ddot D + 2H\dot D - 4\pi G\rho_m D = 0.
\label{eq:linear-growth}
\end{equation}
In terms of the scale factor this may be written as
\begin{equation}
D''(a) + \paren{\frac{3}{a} + \frac{\dd\ln H}{\dd a}}D'(a)
- \frac{3}{2}\frac{\Omega_{m,0}H_0^2}{a^5H^2(a)}D(a)=0,
\label{eq:growth-a}
\end{equation}
where primes now denote derivatives with respect to \(a\). The observable \(\sigma_8\) is proportional to the late-time value of this growth factor after convolution with the transfer function and the primordial spectrum.

Several features of Eq.~\eqref{eq:growth-a} matter for the present discussion.
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item Any increase in the early expansion rate \(H(a)\) around equality or before recombination tends to suppress linear growth because the friction term is larger.
\item Any reduction in the effective clustering matter source or any broad relativistic component likewise suppresses growth.
\item The observable \(S_8\) is a broad linear-scale quantity, not a direct probe of the minihalo tail relevant for the very earliest galaxies.
\end{enumerate}

That third point is often missed in casual discussion. The scales relevant for weak lensing and for \(S_8\) are not identical to the exponentially sensitive minihalo tail that controls the onset of first-light astrophysics. This scale distinction is what allows the present framework to seek a coherent but non-uniform solution across \(H_0\), \(S_8\), and JWST.

\subsection{Why the same early correction that helps \texorpdfstring{\(H_0\)}{H0} usually hurts growth}

Suppose one adds a broad early component that increases \(H(a)\) before recombination. Then, through Eq.~\eqref{eq:h0-rs}, one can reduce the sound horizon and raise the inferred \(H_0\). But the same increase in \(H(a)\) shortens the time available for matter perturbations to grow and raises the friction term in Eq.~\eqref{eq:growth-a}. That is why many naive early-universe Hubble fixes worsen the growth tension.

In the corridor picture this growth response is not a free early-dark-energy
suppression but the late-branch reading of the one \GRIPin{} growth history:
\(S_8\) is the broad linear-scale amplitude of the same post-switch relaxation
whose rare tail will drive the early-galaxy discussion below. Carried on the
inherited inventory, the late/lensing branch projects to
\[
S_8 \approx 0.78,
\]
to be compared with the benchmark cosmic-shear determination
\(S_8 = 0.759^{+0.024}_{-0.021}\) from KiDS-1000 \citep{Asgari2021} --- agreement
at the \(0.9\sigma\) level, a \(+2.8\%\) fractional deviation. This is the content
of Corollary~\ref{cor:s8-reconciliation}: the same corridor that splits \(\Hzero\)
into two branches splits \(S_8\) into an anchored early/CMB value
\(\Seight^{\mathrm{CMB}}\) and a predicted late/lensing branch
\(\Seight^{\mathrm{late}}\), displayed as two entries in the parameter-audit Dev
plot (Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}). The
legacy early-dark-energy linear-response that reached the same \(0.78\) through a
multi-parameter suppression formula --- together with the documented risk that a
full CMB refit can flip the sign of the shift --- is retained, with its caveats,
in Appendix~\ref{appA:growth}, where by the \MSMF{}/\OGN{} uniqueness principle it
is recorded as a non-unique cross-check rather than the mechanism.

\subsection{The role of the Kaxion and the inherited neutrino ledger}

It is useful to ask why the present framework needs the EDE-plus-dark-radiation pair at all if dark matter and neutrinos are already inherited from earlier papers. The answer is that those earlier inherited sectors are deliberately narrow. The Kaxion is a cold dark-matter candidate, not a broad warm-dark-matter knob inserted to erase small-scale power indiscriminately. Likewise, the neutrino sector is already constrained by the particle-side derivation and is therefore not free to roam over a wide phenomenological range just to repair late-time structure.

This is a virtue rather than a drawback. It means that \(S_8\) cannot be fixed by arbitrarily blurring the matter sector. The main work must instead be done by the early source residue and by the precise way the lensing and projection sectors are interpreted. That is scientifically healthier, because it keeps the same matter ledger available for the later early-galaxy discussion rather than using it up as a free late-time patch.

Put differently: the matter sector in this program is narrow by design. That is exactly why the \(S_8\) discussion must be tied tightly to the Hubble discussion. The same early correction that moves the sound horizon must also be asked to carry part of the burden of the growth correction.

\subsection{Why the \texorpdfstring{\(S_8\)}{S8} solution does not automatically contradict the early-galaxy solution}

At first sight it may seem inconsistent to claim that the same framework both suppresses broad late-time linear growth and helps produce earlier galaxies. The key is that these are not the same observable scales and not the same physical channels.

The \(S_8\) variable is sensitive to the amplitude of relatively broad linear-scale clustering. The early-galaxy problem, by contrast, is dominated by the high-\(\nu\) tail of the halo mass function, by baryon-dark-matter streaming suppression, by the filtering mass for molecular-hydrogen cooling, and by collapse thresholds in a regime where small percentage changes can be exponentially amplified. A model can therefore slightly lower \(S_8\) on broad linear scales while still advancing the first-light chronology through targeted high-\(k\) and baryonic-transport effects.

The present framework leans strongly on this distinction. The EDE and dark-radiation residues suppress the broad growth channel enough to help with \(S_8\). The baryonic graviphoton and the streaming-velocity damping act later and more selectively on the first-minihalo problem. The two effects are not uniform rescalings of the same transfer function. They act on different parts of the structure-formation chain.

\subsection{A compact consistency ladder for the growth block}

The growth block can therefore be summarized as the following ladder:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item \textbf{CMB stage:} the inherited \(f_{\mathrm{EDE}}\) and \(\Delta N_{\mathrm{eff}}\) residues reduce the sound horizon and help \(H_0\).
\item \textbf{Broad growth stage:} the same residues slightly suppress broad linear growth, nudging \(S_8\) downward.
\item \textbf{Lensing stage:} the mild \(A_L\approx1.05\) correction partially compensates in the CMB-lensing channel.
\item \textbf{High-\(k\) baryonic stage:} streaming-velocity damping and the gauged-baryon sector selectively help early minihalo collapse.
\end{enumerate}
The scientific message is that one should not demand a single monotonic change in the power spectrum at all scales and all epochs. The framework instead proposes a structured, scale-sensitive correction chain inherited from one ledger.

\begin{table}[H]
\centering
\caption[Sign structure of the proposed Part VI growth corrections]{Sign structure of the proposed Part~VI growth corrections. The point is not that each entry is final in detail, but that the roles are fixed and not freely reassigned from one tension to the next. Each row is a correction channel and each column an observable it moves. The table is about signs and not magnitudes: the claim is that the channels push the three observables in a coordinated direction, and a reader who wants numbers will not find them here and should not infer them.}
\label{tab:sign-structure-growth}
\small
\begin{tabular}{p{0.28\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.26\textwidth}}
\toprule
\textbf{Correction channel} & \textbf{Effect on \(H_0\)} & \textbf{Effect on broad \(S_8\)} & \textbf{Effect on earliest galaxies}\\
\midrule
Early dark energy lobe & positive & negative & mildly positive via earlier equality and more small-scale depth\\
Extra dark radiation & positive & negative & neutral to mildly negative by itself; compensated later by baryonic effects\\
Mild lensing enhancement \(A_L\) & indirect & partially positive in CMB lensing interpretation & little direct effect\\
Baryonic graviphoton / streaming damping & little direct early effect on \(H_0\) & little direct effect on broad linear \(S_8\) & strongly positive for minihalo cooling and early collapse\\
Collapse-threshold reduction & none & none on broad linear scales & strongly positive in the high-mass exponential tail\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Kill criteria for the growth module}

The growth module fails if the needed \(S_8\) suppression demands a much larger early correction than the CMB or Hubble sections can tolerate. It also fails if the framework can only produce earlier galaxies by enhancing broad linear growth so much that the lensing tension worsens catastrophically. More operationally, the module would be in deep trouble if a joint CMB+lensing+BAO analysis forces \(f_{\mathrm{EDE}}\) and \(\Delta N_{\mathrm{eff}}\) so close to zero that Eq.~\eqref{eq:S8-response} can no longer move \(S_8\) in the required direction, or if the lensing enhancement \(A_L\) is itself excluded as a meaningful residual.

The paper should therefore be candid: the \(S_8\) module is not a free-standing miracle. It is a delicate consistency test of whether the same early source residue can serve both the Hubble and growth tensions without breaking the microwave background.

\section{Early galaxy formation in the JWST era}
\label{sec:jwst}


\begin{proposition}[Early-halo amplification statement (T-Cosmos-12)]
Because the halo mass function depends exponentially on the high-\(\nu\) tail,
small inherited changes in the early source ledger can produce order-one shifts
in the abundance of the earliest massive systems. The JWST problem is therefore
not ``just more \(S_8\)'' but the nonlinear high-mass tail of the same inherited
vector.
\end{proposition}



\subsection{Why the JWST problem is not just ``more \texorpdfstring{\(S_8\)}{S8}''}

The earliest JWST results triggered excitement because they appeared to reveal galaxies, stellar populations, and black-hole activity at redshifts high enough to strain the standard assembly timeline. Even as the observational interpretation matures, the theoretical challenge remains useful: can a constrained cosmology make the onset of luminous structure slightly earlier and slightly more efficient without rewriting the whole large-scale background? That is the sharp version of the early-galaxy question.

It is important not to confuse this with the \(S_8\) problem. A modest suppression of broad linear growth does not automatically imply a suppression of the rarest early collapsed objects. The earliest galaxies sit in the exponentially sensitive high-\(\nu\) tail of the halo mass function, where small changes in the collapse threshold, in the baryon-dark-matter streaming velocity, or in the small-scale transfer function can move the first-light chronology by much more than one would guess from a broad-brush linear statistic.

The present framework therefore treats the early-galaxy tension through a three-layer mechanism:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item a mild small-scale power boost sourced by the same EDE residue that already appeared in the Hubble discussion,
\item a damping of the baryon-dark-matter streaming velocity due to the gauged baryon sector, and
\item a reduction of the effective collapse threshold for the relevant early halos.
\end{enumerate}
Each effect is modest alone. Their importance lies in the fact that the halo mass function is exponentially sensitive.

\subsection{Press--Schechter baseline and the reason tiny changes can matter enormously}

The standard Press--Schechter intuition already explains why the early-galaxy problem is different from broad \(S_8\). The abundance of halos above mass \(M\) at redshift \(z\) depends approximately on
\begin{equation}
 n(>M,z) \propto \operatorname{erfc}\paren{\frac{\delta_c}{\sqrt2\,\sigma(M,z)}}
\label{eq:PS-erfc}
\end{equation}
or, in the rare-halo tail, exponentially on
\begin{equation}
 n(M,z) \sim \exp\brak{-\frac{\nu^2}{2}},
\qquad
\nu \equiv \frac{\delta_c}{\sigma(M,z)}.
\label{eq:nu-def}
\end{equation}
Thus even a one- or two-percent shift in \(\sigma(M,z)\) can produce a several-tens-of-percent change in abundance when \(\nu\) is large, and a modest shift in \(\delta_c\) can generate orders-of-magnitude effects.

This is the mathematical reason the early-galaxy problem is highly leverage-sensitive. One does not need to change the entire matter power spectrum enormously. One needs only to change the right combination of small-scale variance, baryon transport, and effective collapse conditions.

\subsection{Small-scale boost from the early-dark-energy residue}

The first proposed mechanism is that the same EDE residue which helps the Hubble tension produces a small positive boost in the small-scale transfer function. The working claim is a roughly uniform enhancement of order
\begin{equation}
\frac{\Delta P(k)}{P(k)} \sim 0.03
\qquad\text{around}\qquad
k\sim 5\,h\,\mathrm{Mpc}^{-1}.
\label{eq:Pk-boost}
\end{equation}
Because \(\sigma\) is the square root of the power, this corresponds to
\begin{equation}
\frac{\Delta\sigma}{\sigma} \sim \frac12\frac{\Delta P}{P} \sim 0.015\text{--}0.017.
\label{eq:sigma-boost}
\end{equation}
In the rare-halo tail one has approximately
\begin{equation}
\frac{\Delta n}{n} \approx \paren{\nu^2 - 1}\frac{\Delta\sigma}{\sigma}.
\label{eq:halo-boost-deriv}
\end{equation}
Taking a representative \(\nu\sim5\) for the relevant minihalo population at high redshift gives
\begin{equation}
\frac{\Delta n}{n} \approx (25-1)\times 0.017 \approx 0.41,
\end{equation}
that is, about a \(40\%\) increase in abundance. This simple derivation is useful because it shows that the quoted \(40\%\) boost is not an arbitrary large number. It is exactly the kind of exponentially amplified response one expects in the rare tail.

The intuitive picture is equally simple. The EDE residue does not need to build galaxies directly. It only needs to hand the very first gravitational seeds slightly deeper pockets. Once those pockets sit in the exponential tail, small initial help is magnified into a significant population effect.

\subsection{Damping the baryon--dark-matter streaming velocity}

The second proposed mechanism is more specifically baryonic. After recombination, baryons and dark matter retain a residual relative streaming velocity. In ordinary \LCDM\ this relative motion acts like a headwind that delays gas capture by the smallest halos and raises the filtering mass for molecular-hydrogen cooling. The \GG{}/\GDOS{} proposal is that the gauged \(\mathrm{U}(1)_B\) sector induces a tiny long-range drag or alignment effect between baryons and the compact-angle dark-matter background, effectively reducing the streaming velocity by about \(20\%\):
\begin{equation}
 v_{\mathrm{str}} \to 0.8\,v_{\mathrm{str}}^{\Lambda\mathrm{CDM}}.
\label{eq:vstream}
\end{equation}

A useful rough consequence follows immediately. The baryonic filtering mass scales approximately as a positive power of the streaming velocity; a common heuristic is
\begin{equation}
 M_f \propto v_{\mathrm{str}}^3.
\end{equation}
Hence a \(20\%\) reduction in \(v_{\mathrm{str}}\) gives
\begin{equation}
\frac{M_f^{\mathrm{new}}}{M_f^{\mathrm{old}}} \sim 0.8^3 \approx 0.512.
\label{eq:filtering-mass}
\end{equation}
In words, the threshold mass for effective baryon capture and cooling is cut roughly in half. This is a very large astrophysical effect even though the velocity shift itself is modest.

This point is central to the internal logic of Part~VI. The baryonic streaming fix acts far more selectively than a warm-dark-matter-style suppression or broad late-time growth modification. It specifically helps the first minihalos keep their gas. That is precisely the kind of effect one wants for earlier first-light structure without simultaneously ruining broad large-scale structure.

\subsection{Collapse-threshold reduction and the high-mass halo tail}

The third mechanism is the reduction of the effective collapse threshold. The working \GG{} claim is that the presence of the baryonic graviphoton and the altered baryonic infall history reduce the critical overdensity from the standard spherical-collapse value,
\begin{equation}
\delta_c^{\Lambda\mathrm{CDM}} \approx 1.686,
\end{equation}
to an effective early-galaxy value of roughly
\begin{equation}
\delta_c^{\mathrm{\GG{}}} \approx 1.18.
\label{eq:deltac-118}
\end{equation}
Why can this matter so dramatically? Because the exponential tail depends on \(\delta_c^2\). At fixed \(\sigma\), the ratio of abundances scales as
\begin{equation}
\frac{n_{\mathrm{\GG{}}}}{n_{\Lambda\mathrm{CDM}}}
\sim
\exp\brak{\frac{\paren{\delta_c^{\Lambda\mathrm{CDM}}}^2 - \paren{\delta_c^{\mathrm{\GG{}}}}^2}{2\sigma^2}}.
\label{eq:halo-ratio}
\end{equation}
For representative high-redshift values \(\sigma\sim0.34\text{--}0.36\) on the relevant massive-halo scale, the exponent lies in the several-unity range, naturally producing an enhancement from \(10^2\) to \(10^3\). A factor near \(500\) is therefore perfectly plausible inside the rare tail. The point is not that the exact number is sacred before full hydrodynamical calibration. The point is that the magnitude of the claimed boost is mathematically understandable.

This mechanism is also conceptually neat. The gas does not collapse earlier simply because gravity has been made universally stronger. Rather, the collapse criterion itself is changed in a specific baryonic environment. That is a much narrower and more defensible claim.

\subsection{Direct-collapse seeds and the early supermassive black-hole problem}

The JWST story is not only about galaxies; it is also about apparently over-massive black holes at high redshift. Within the present framework the same early-galaxy assistance naturally feeds a direct-collapse seed channel. If baryonic inflow is delayed just enough to keep gas more pristine while dark halos hierarchically merge, and if the effective Toomre threshold is lowered in the relevant high-redshift disks, then one can favor heavy seeds of order
\begin{equation}
 M_{\mathrm{seed}} \gtrsim 10^5\,M_\odot,
\label{eq:heavy-seed}
\end{equation}
bypassing the slow climb from stellar-mass remnants.

This is not yet a full black-hole paper; that comes in Part~VII. But even inside Part~VI it is important to state the continuity. The early-galaxy module is scientifically stronger if it also explains why the first bright AGN need not be impossible. The same chronology shift that helps the first massive galaxies should also help the first heavy seeds.

\subsection{The chronology shift: why \texorpdfstring{\(\Delta z\sim2\text{--}3\)}{Delta z~2-3} is enough to matter}

The combined effect of the three mechanisms is summarized as an effective chronology shift,
\begin{equation}
\Delta z \sim 2\text{--}3,
\end{equation}
for the onset of the earliest star formation and significant galaxy assembly. It is tempting to dismiss such a shift as small. That would be a mistake. In the high-redshift universe, a shift of even a few redshift units corresponds to a large fraction of the available cosmic time between matter domination, first cooling, and the onset of reionization. Because so many first-light processes are threshold-driven --- minihalo cooling, Population~III ignition, metal enrichment, seed black-hole formation, merger-driven brightening --- a small shift in cosmic time can translate into a large shift in the observable abundance of bright objects.

The early-galaxy section should therefore insist on a basic point of intuition. The problem is not to make the entire universe mature implausibly early. The problem is to let the earliest successful objects cross their ignition thresholds slightly earlier and in slightly larger numbers. Exponential sensitivity then does the rest.

\begin{table}[H]
\centering
\caption[Mechanism ladder for the JWST-era early-galaxy problem]{Mechanism ladder proposed for the JWST-era early-galaxy problem in \ATen{}. The columns give the mechanism, a representative shift, the scale it principally affects, and the observational consequence it is intended to have. The shifts are representative rather than predicted, and the ladder is ordered by how much of the discrepancy each mechanism could plausibly carry, not by how well established it is.}
\label{tab:jwst-mechanisms}
\small
\begin{tabular}{p{0.26\textwidth}p{0.24\textwidth}p{0.18\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Mechanism} & \textbf{Representative shift} & \textbf{Primary scale affected} & \textbf{Intended observational consequence}\\
\midrule
Early-time small-scale boost & \(\Delta P/P\sim3\%\) & minihalo variance at high \(k\) & \(\sim40\%\) more relevant minihalos at very high redshift\\
Streaming-velocity damping & \(v_{\mathrm{str}}\to0.8v_{\mathrm{str}}\) & baryon capture / filtering mass & earlier molecular-hydrogen cooling and Population~III ignition\\
Collapse-threshold reduction & \(\delta_c:1.686\to1.18\) & rare massive-halo tail & order-of-magnitude boost in \(>10^{11}M_\odot\) halo abundance at \(z\sim9\text{--}10\)\\
Direct-collapse seed channel & \(M_{\mathrm{seed}}\gtrsim10^5M_\odot\) & early galactic nuclei & less tension from very high-redshift bright AGN and quasars\\
Net chronology shift & \(\Delta z\sim2\text{--}3\) & first-light timeline & earlier onset of bright galaxies without wholesale background revision\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Kill criteria for the JWST module}

The early-galaxy module is falsified if the needed small-scale boost and streaming-velocity damping are shown to be incompatible with the CMB and Lyman-\(\alpha\) structure constraints, or if detailed hydrodynamical modeling demonstrates that the required chronology shift cannot produce the needed abundance increase without other implausible astrophysical assumptions. It is also weakened if the evolving observational picture reduces the early-galaxy tension to purely photometric or stellar-population systematics. The theory should acknowledge this openly.

Yet even under that caution, the module remains scientifically important. It tests whether the same inherited ledger that moves \(H_0\) and \(S_8\) can also move the first-light chronology in a useful direction. That is exactly the kind of cross-domain constraint a unified cosmology should face.

\section{Integrated summary of the growth-and-tension block}
\label{sec:part6-summary}


\begin{quote}
The same narrow early-time residue that shortens the sound horizon and helps the Hubble tension also reshapes the growth and lensing histories, while the same gauged baryon sector that later helps local environmental realization also selectively assists the earliest baryonic collapse channels.
\end{quote}
That sentence is the real claim of Part~VI. Everything else is a calculation or an audit of it.

It is worth making the logic explicit one last time.

\subsection{One ledger, four manifestations}

\textbf{First manifestation: the primary CMB.} The acoustic ruler, the odd/even peak pattern, the damping tail, the E-mode sequence, and the tensor corridor are all controlled by the inherited matter loading, the same multiscale throat capacity, and the same inflationary background that earlier parts of the paper already constrain. If this fails, everything fails.

\textbf{Second manifestation: the Hubble tension.} A narrow early-dark-energy lobe and a small dark-radiation addition shorten the sound horizon enough to push the early-universe inferred Hubble rate upward. A mild baryonic local lift then moves the local route slightly further. The disagreement is reinterpreted as a tension between operational dictionaries, not as proof that one must abandon the baseline equations.

\textbf{Third manifestation: the \(S_8\) tension.} The same early residues that help \(H_0\) slightly suppress broad linear growth. A mild lensing enhancement partly compensates in the CMB lensing channel. The resulting corridor pushes \(S_8\) downward in the direction favored by late-time weak-lensing probes.

\textbf{Fourth manifestation: the JWST-era early-structure challenge.} The broad-growth suppression does not preclude earlier galaxies because the minihalo tail, the baryon-dark-matter streaming channel, and the collapse threshold are distinct physical levers. A small variance increase, a \(20\%\) streaming damping, and a lowered \(\delta_c\) are enough to move the first-light chronology appreciably.

These four manifestations are not independent. That is both the strength and the danger of the proposal. The strength is that the framework is highly constrained; it cannot fix each problem independently by inventing a new fluid or field. The danger is that failure in one place propagates to others. This is why Part~VI is one of the cleanest falsification arenas of the whole paper.

\subsection{What Part~VI does and does not claim}

Part~VI claims three things.
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item It claims that the perturbative and growth sectors of cosmology can be organized as a sequence of audits on one inherited ledger rather than as a grab bag of modular fixes.
\item It claims that the specific inherited correction pair \((f_{\mathrm{EDE}},\Delta N_{\mathrm{eff}})\) is of the right sign and approximate size to matter simultaneously for \(H_0\) and \(S_8\).
\item It claims that the baryonic \(\mathrm{U}(1)_B\) sector supplies an additional selective structure-formation lever that broad growth statistics alone do not capture.
\end{enumerate}

Part~VI does \emph{not} claim that the full precision likelihood problem has already been completed inside the present draft. Nor does it claim that the astrophysical side of the JWST problem can be solved without hydrodynamical modeling. The correct scientific posture is intermediate. The framework supplies a coherent narrow mechanism bundle and then states the observational conditions under which that bundle survives or fails.

\subsection{The real significance of the open-system viewpoint}

The most novel conceptual feature of Part~VI may not be any one number. It may be the insistence that cosmological observables are open-system exports rather than direct readings of a closed fundamental microstate. This matters in several places.

It matters for the CMB because the observed sky is a frozen and projected record, not a direct local sample of the early plasma. It matters for the Hubble tension because different inference pipelines need not be identical measurements of the same closed-system constant. It matters for \(S_8\) because lensing and clustering are themselves late integrated observables with non-trivial coarse-graining. And it matters for early galaxies because the relevant thresholds depend on environmental realization and baryonic transport, not only on the primordial scalar amplitude.

This open-system emphasis does not weaken precision cosmology. It sharpens its interpretation. It asks the theorist to be explicit about what is projected, what is coarse-grained, and what is truly invariant. In that respect, Part~VI is a natural continuation of the earlier \GG{}/\GDOS{} papers rather than a thematic detour.

\subsection{An invitation: revisiting \texorpdfstring{\(\Lambda\)CDM}{LambdaCDM} in the light of a derived \texorpdfstring{$H(t)$}{H(t)}}
\label{sec:revisit-lcdm}

There is a larger consequence to draw, and we state it plainly, as an invitation to
the community. \(\Lambda\)CDM is the most successful \emph{fitting} framework in the
history of cosmology: six numbers, an exquisite fit, four independent probes in
concordance. What it does not do is say \emph{why} those six numbers take the values
they do --- it measures them. \ATen{} proposes that they are not free: that the
densities, and therefore the entire expansion history \(H(t)\) of
Figure~\ref{fig:H-of-t-derived}, are derived from one six-dimensional geometry with
no continuous dial. If that proposal holds, the parameters of \(\Lambda\)CDM are
\emph{outputs} of a deeper theory rather than inputs to a fit, and several of the
field's standing ``tensions'' change character: the \(\Hzero\) tension becomes a
question about the calibration of one point on a derived curve, and the \(S_8\)
tension a question about late-time coarse-graining, rather than signals of missing
fundamental physics.

We are therefore not asking anyone to discard \(\Lambda\)CDM --- \ATen{} \emph{runs
on} its Friedmann/\FRW\ machinery, and its empirical fit is not in question. We are
asking the community to revisit \(\Lambda\)CDM in a fresh light: to re-examine,
parameter by parameter, whether each of its numbers is a free input or a forced
geometric consequence, and to test the specific, dated predictions this reading
makes --- the standard-siren convergence \(\Hzero^{\mathrm{GW}}\to67\), the exact
\(w=-1\), the tensor ratio \(r=0.0043\), and the coherent \(\pi\)-offset visible
across the acoustic peaks. The honest banner is not ``\(\Lambda\)CDM is wrong'' but
``\(\Lambda\)CDM is incomplete in its \emph{explanation}, and here is a candidate
completion --- laid out to be tested, and, if it fails, to fail cleanly.'' A
generation of precision cosmology measured the constants of the universe to
extraordinary accuracy. The next task --- and it is a community effort --- is to
find out which of them were never constants at all, but consequences.

\subsection{Bridge to the next part}

Having completed the growth-and-tension block, the draft is in a better position to move to the black-hole and primordial-relic sector. The transition is natural. The CMB section already forced the paper to speak carefully about what is operationally observed after projection. The Hubble and \(S_8\) sections forced it to distinguish between background laws and inferred quantities. The early-galaxy section forced it to respect the difference between broad statistics and rare-event tails. All three lessons reappear in the black-hole information problem, in primordial-black-hole formation, and in the tensor sector.

Two readings of one corridor: that is this Part's reframing of the decade's most famous quarrel, and there is a philosophical point in it. A tension between instruments is usually read as somebody's mistake; here it is read as \emph{information} --- the universe giving two lawful projections of a single geometric object, the disagreement itself becoming a measurement of the switch between them. If the gravitational-wave sirens side with the ladder at 73, this reading dies in public. That is not a defect of the framework; it is the entire point of having one.

For that reason Part~VI should end not with triumphalism but with a clean summary. The perturbation, growth, and tension sectors do not yet prove the whole framework correct. They do something more useful. They convert vague cosmological ambition into a finite list of tightly coupled empirical tests. That is exactly what the paper needs before it enters the more speculative territory of Part~VII (black holes and primordial relics). The full cross-consistency Jacobian and the worked, from-scratch derivations for every number in this Part are collected in Appendices~\ref{appA:growth} and \ref{appA:cmb}.







\section{Part~VI in plain English: one switch, and what it does and does not fix}
The proposal of this Part is a single geometric switch, and its two faces are
the Hubble tension and the growth of structure. That is an economical claim
and economy is exactly why it is dangerous: a mechanism that moves two
observables in the right direction at once is more likely to be a mechanism
than a coincidence, and it is also more likely to be over-read.

So the Part is written to make the strength of each piece visible separately.
The background Hubble value is derived twice, by independent roads, and that
is the strong result. The residual offset against the local distance ladder is
not derived here; it is treated as a calibration systematic, which is the
reading this framework now shares with the mainstream that takes the CMB to be
right. Those two statements sit in the same section and carry different
weights, and the status table exists so they cannot be conflated.

The growth corrections are weaker still and are presented as signs rather than
magnitudes. The claim is that the channels push three observables in a
coordinated direction, not that any one of them is computed. The early-galaxy
material is weaker again --- a ladder of mechanisms, ordered by how much of the
discrepancy each could carry, not by how well established each is. A reader
should come away knowing which of the four blocks in this Part would survive a
hostile reading and which would not.

\GGPart[Black holes and relics]{Black holes and primordial relics}{The black-hole information paradox under \OGGEP{} / \GDOS{}; primordial black holes as a feature-dependent supplementary channel; and the Starobinsky-channel relict primordial gravitational-wave background}

\section{Black holes as GRIP-In way-stations: the organizing idea}
\label{sec:part7-opener}

Part~VII reads the extreme-gravity sector through one organizing idea, shared with
the dedicated black-hole companion \citep{ArisakaC3BH}: \emph{a black hole is a
\GRIPin{} way-station attractor, and Hawking radiation is its channel-mediated
\GRIPin{} exit.} In the \GRIP{} grammar of \ASix{}, a black hole is a deep, stable
basin of the \(F_{\GFF}\) free-energy ledger on the gravitational stratum --- but,
unlike the Ground Geometric Attractor that the universe relaxes into at late times
(Part~IV), it is not terminal: it possesses an exit channel. Hawking emission is
that exit, the slow channel-mediated descent by which the basin finally drains. A
stellar-mass hole is therefore the most extreme \emph{way station} in nature, a
place where the cosmic descent pauses for a residence time of order \(10^{67}\)
years before resuming.

This single picture organizes the three sections that follow and fixes the division
of labor with the companion paper. The dedicated treatment --- the exact \(A/4\)
Bekenstein--Hawking coefficient from induced gravity, the Page-curve branch switch,
the four laws of black-hole mechanics as \GDOS{} statements, and the EHT-shadow and
ringdown predictions --- is carried in \citep{ArisakaC3BH};
Appendix~\ref{appA:bhentropy} reproduces the \(A/4\) derivation and credits it
there. The present Part carries the \emph{cosmological} reading of the same
mechanism in three movements:
\begin{itemize}[leftmargin=2.0em]
\item the \textbf{information paradox} (\S\ref{sec:bh-info}), dissolved by reading
the horizon as the \OGGEP{} port of a globally closed bulk;
\item \textbf{primordial black holes} (\S\ref{sec:pbh}), the nested \GRIPin{}
collapse events of cosmic history --- scarce in the minimal module and, where they
do occur, spaced by the same integer ledger that fixes \(z_\ast=e^{7}\); and
\item the \textbf{primordial gravitational-wave relic} (\S\ref{sec:pgw}), the
Starobinsky-channel tensor background that the smooth low-scale exit drives to a
sharp null.
\end{itemize}
All three are the same open-system descent, audited at the strongest causal barrier
in classical relativity.

\section{The black-hole information paradox}
\label{sec:bh-info}


\begin{theorem}[Closed-bulk / open-boundary black-hole theorem (T-Cosmos-13)]
The information paradox is reformulated by separating the closed
bulk \TGG{} ledger from the open boundary-accessible seam description. The bulk
may remain globally lawful and unitary while the boundary observer records a
restricted packet whose entropy and Page-turnover behavior are described by
\CCI{}, \(F_{\GFF}\), and \GRIP{} rather than by a naive factorization of the
entire bulk Hilbert space into two directly readable parts.
\end{theorem}



The fourth cluster of open problems is different in character from the first three. The beginning of the universe, the origin of matter, and the growth of structure are all questions about how a cosmic background evolves and how that evolution is read out in ordinary observables. The black-hole information paradox is sharper and more surgical. It asks what a theory means by \emph{information}, what it counts as an \emph{observer-accessible subsystem}, and how thermodynamic bookkeeping is related to microscopic unitarity in a regime where gravity, quantum theory, and horizons meet head-on.

This is exactly why the problem belongs in \ATen{} and not only in the dedicated black-hole companion \citep{ArisakaC3BH}: the cosmological reading here and the full gravitational treatment there are two projections of one mechanism. If the \GG{} constitution is serious, then \OGGEP{} is not a decorative philosophical layer. It is the rule that tells us how a closed higher-dimensional reality is projected into boundary observables. Likewise, if \GDOS{} is serious, then a black hole is not merely a static exact solution of Einstein's equation. It is an extreme open system with explicit ports, a sharply defined energy-entropy ledger, and a hierarchy of accessible and inaccessible records. Black holes are therefore the natural place where the distinction between \emph{global closure} and \emph{boundary openness} stops sounding philosophical and becomes a concrete technical issue.

There is also a second reason. The standard formulation of the paradox is built on a set of assumptions that are individually familiar but not jointly innocent. One assumes semiclassical locality outside the horizon, exact thermal Hawking emission at late times, ordinary factorization of the Hilbert space into interior and exterior pieces, and final complete evaporation into exterior quanta. When these assumptions are assembled, the conclusion seems to be that a pure initial state evolves into mixed Hawking radiation. The usual response is then to decide which assumption to sacrifice: exact thermality, local effective field theory, interior/exterior independence, or smoothness of the horizon. The modern island program shows that the bookkeeping can be reorganized in a much more subtle way. The \GG{}/\GDOS{} response goes one step further backward. It asks whether the paradox itself is being posed at the wrong ontological level.

The central claim of the present section is therefore two-layered.

First, at the level of ontology, the paradox is dissolved once one stops demanding that the boundary-accessible subsystem by itself carry the full burden of microscopic unitarity. In \OGGEP{} the 6D bulk is the globally closed unitary system. The 4D observer sees only a CPTP image of that bulk state. Pure-to-mixed evolution in the accessible subsystem is therefore not a contradiction; it is the generic mark of openness.

Second, at the level of practical entropy accounting, the familiar Page-curve resolution can be reinterpreted as a \GDOS{} saddle-switch problem. The relevant entropy is not the naive thermal entropy of emitted quanta alone, but the minimum of an \GFF{} ledger over admissible boundary-accessible branches. In that language the Page transition is a \GSSB{} event: the realized entropy branch changes when the operationally dominant saddle switches.

This is what it means to say that the paradox can be solved cleanly by \GDOS{}. The framework does not need to deny Hawking radiation. It does not need to deny that late-time radiation can, under the correct coarse-graining, carry reconstructible information. It only insists that the correct global-versus-operational distinction be made before the entropy ledger is interpreted.

\subsection{Why black holes are the sharpest information-processing objects in the paper}

Black holes are the most demanding objects in the present volume because they compress several themes into one place.

At the geometric level, a black hole is the cleanest example of a system whose macroscopic state can be written almost entirely in terms of conserved charges and horizon geometry. At the thermodynamic level, it is the most famous system whose entropy is controlled by an area rather than by an ordinary matter volume. At the quantum level, it is the regime in which apparently thermal radiation emerges from a state produced by collapse. At the operational level, it is the regime in which an outside observer has the most severe possible restriction on direct access.

Any program built around \GFF{}, \DGI{}, \GSSB{}, \OGGEP{}, and boundary-accessible observables should therefore expect black holes to be a decisive audit. If the open-system language is only a metaphor, it will fail here. If it is a genuine constitutional principle, it should clarify rather than obscure the paradox.

In keeping with the rest of the draft, the present section proceeds conservatively. It does not assume a detailed microscopic model of black-hole interior states. It does not require a full UV completion of every graybody factor or every trans-Planckian step in collapse. Instead, it asks a narrower question: what does the inherited \GG{}/\GDOS{} ledger already force us to say?

Three inherited ingredients are enough to make the central move.

\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item The bulk ontology is globally closed and unitary, while the boundary observer sees an open-system projection.
\item Physical state counting must respect \DGI{}; in gravity this means one cannot naively factorize across a gauge-redundant horizon split.
\item The correct entropy ledger for an open system is \GFF{}, with a geometric cost term and an entropic-curvature term.
\end{enumerate}

Everything else in the section follows from working out the consequences of these three facts for the particular case of an evaporating Schwarzschild-like black hole.

\subsection{Closed-system versus operational formulations}
\label{subsec:bh-closed-vs-operational}

The usual closed-system statement of the paradox can be made precise in four steps.

First, one begins with a pure state \(\lvert\Psi_{\mathrm{in}}\rangle\) of collapsing matter together with the vacuum of quantum fields on the exterior geometry. If the entire process is governed by unitary quantum mechanics, the total final state should remain pure:
\begin{equation}
\lvert\Psi_{\mathrm{tot}}(t)\rangle = U(t)\lvert\Psi_{\mathrm{in}}\rangle,
\qquad
U^\dagger U = \mathbb{1}.
\label{eq:partVII-total-unitary}
\end{equation}

Second, semiclassical quantum field theory on a background with an event horizon predicts approximately thermal particle creation. For a non-rotating uncharged black hole of mass \(M\), the Hawking temperature is
\begin{equation}
T_H = \frac{1}{8\pi G_N M},
\label{eq:partVII-HawkingT}
\end{equation}
while the Bekenstein-Hawking entropy is
\begin{equation}
S_{\mathrm{BH}}(M) = \frac{A_H}{4G_N} = 4\pi G_N M^2.
\label{eq:partVII-SBH}
\end{equation}
Here \(A_H = 16\pi G_N^2 M^2\) is the horizon area of the Schwarzschild geometry. The evaporation law may be written schematically as
\begin{equation}
\frac{\dd M}{\dd t} = -\frac{\alpha_H}{M^2},
\label{eq:partVII-mdot}
\end{equation}
where \(\alpha_H>0\) packages the species count and graybody factors. Integrating gives
\begin{equation}
M^3(t)=M_0^3 - 3\alpha_H t,
\qquad
t_{\mathrm{evap}} = \frac{M_0^3}{3\alpha_H},
\label{eq:partVII-evap-law}
\end{equation}
for an initial mass \(M_0\).

Third, if one insists that the outgoing radiation be described by an exactly thermal density matrix throughout the evaporation process, the coarse-grained radiation entropy increases monotonically. A convenient schematic estimate is
\begin{equation}
\dot S_{\mathrm{rad}}^{\mathrm{th}} \approx \eta\,\frac{-\dot M}{T_H},
\label{eq:partVII-Srad-dot}
\end{equation}
with \(\eta\) an order-unity coefficient. Using Eq.~\eqref{eq:partVII-HawkingT} gives
\begin{equation}
\dot S_{\mathrm{rad}}^{\mathrm{th}} \approx 8\pi \eta\,G_N M(-\dot M).
\end{equation}
Integrating with Eq.~\eqref{eq:partVII-mdot} yields the familiar monotonic behavior
\begin{equation}
S_{\mathrm{rad}}^{\mathrm{th}}(t) \approx \eta\left[S_{\mathrm{BH}}(M_0)-S_{\mathrm{BH}}(M(t))\right].
\label{eq:partVII-Srad-monotonic}
\end{equation}
If the black hole evaporates away completely and the radiation remains exactly thermal, then the late-time radiation state is mixed and carries entropy of order the initial area.

Fourth, one now collides this with Eq.~\eqref{eq:partVII-total-unitary}. A pure state should not evolve into a truly mixed state under exact unitary evolution. This is the paradox in its sharpest textbook form.

Stated this way, the tension appears irresistible. Yet it conceals a crucial shift of level. Equation~\eqref{eq:partVII-total-unitary} is a statement about a \emph{total} closed system. Equation~\eqref{eq:partVII-Srad-monotonic}, by contrast, is a statement about a \emph{subsystem} described by an effective thermal channel. The contradiction is therefore only forced if one identifies the accessible subsystem with the total ontology.

That identification is precisely what \OGGEP{} rejects.

In the operational formulation, the outside observer does not directly possess \(\lvert\Psi_{\mathrm{tot}}\rangle\). The observer possesses a boundary-accessible state,
\begin{equation}
\rho_{\partial}(t) = \Pi_t\!\left(\rho_{6}(0)\right)
= \sum_k K_k(t)\,\rho_{6}(0)\,K_k^\dagger(t),
\label{eq:partVII-CPTP}
\end{equation}
where \(\rho_6(0)\) is the global 6D state, \(\Pi_t\) is a CPTP bulk-to-boundary channel, and the Kraus operators \(K_k\) encode the operational projection together with environmental tracing and horizon-mediated openness. The point of Eq.~\eqref{eq:partVII-CPTP} is not that black holes are somehow special among open systems. The point is the opposite: once the outside observer is correctly described as holding only \(\rho_{\partial}\), pure-to-mixed behavior in that reduced state is completely unsurprising.

This is the first clean \GDOS{} move. The paradox is not ``solved'' by erasing Hawking thermality. It is solved at the constitutional level by refusing to impose total-system purity on a boundary-accessible subsystem.

\subsection{The black hole as a System Instance Packet}

The second clean \GDOS{} move is to define the system before deriving its entropy law. The black hole is treated as a System Instance Packet,
\begin{equation}
\mathcal S_{\mathrm{BH}}
=
\left(
\mathcal M_{\mathrm{BH}},
\mathcal G_{\mathrm{BH}},
\mathcal P_{\mathrm{in}},
\mathcal P_{\mathrm{out}},
F_{\mathrm{\GFF{}}},
\mathcal O_{\mathrm{BH}}
\right).
\label{eq:partVII-SIP-BH}
\end{equation}
Each entry has a concrete meaning.

\begin{itemize}[leftmargin=2.0em]
\item \(\mathcal M_{\mathrm{BH}}\) is the coarse-grained state manifold. For a Schwarzschild-like evaporating hole one can take coordinates such as \((M, S_{\mathrm{BH}}, S_{\mathrm{rad}}, \chi_{\mathrm{env}}, \ldots)\), where \(\chi_{\mathrm{env}}\) stands schematically for environment and measurement records.
\item \(\mathcal G_{\mathrm{BH}}\) is the induced metric and connection data on that manifold, including the gauge-constrained directions associated with diffeomorphism redundancy and horizon edge data.
\item \(\mathcal P_{\mathrm{in}}\) denotes infalling matter and measurement interventions that inject energy or information into the accessible description.
\item \(\mathcal P_{\mathrm{out}}\) denotes outgoing Hawking quanta and other exterior fluxes.
\item \(F_{\mathrm{\GFF{}}}\) is the energy-entropy ledger that ranks admissible coarse-grained histories.
\item \(\mathcal O_{\mathrm{BH}}\) is the observable dictionary: what an exterior observer can actually register.
\end{itemize}

This explicit \SIP{} formulation clarifies a common confusion. In a closed Hamiltonian derivation one often oscillates between three levels without warning: the exact total state, the coarse-grained outside description, and the detector records. \GDOS{} forbids that slippage. Each must be placed in the packet explicitly. Once this is done, the two entropy notions that are usually conflated separate cleanly:

\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item the fine-grained entropy of the total 6D bulk state, which remains constant under global unitary evolution, and
\item the coarse-grained or operational entropy of the exterior-accessible subsystem, which may increase, saturate, or decrease depending on the dominant branch in the \GFF{} ledger.
\end{enumerate}

In this sense the \SIP{} formalism already explains why Hawking's original monotonic argument was so compelling and yet incomplete. It was a valid statement about one reduced description, not a theorem about the total ontology.

\subsection{\GFF{} and the correct entropy ledger}

The third clean \GDOS{} move is to identify the correct entropy functional. In the ordinary semiclassical account, one separately discusses horizon area and radiation entropy, and then later discovers that the physically relevant quantity is their sum. In the present framework that sum is not a surprise but the direct form of the \GFF{} ledger.

The basic object is
\begin{equation}
F_{\mathrm{\GFF{}}}[X;t]
=
F_{\mathrm{\GEP{}}}[X;t]
+
F_{\mathrm{\CEP{}}}[X;t].
\label{eq:partVII-GFF}
\end{equation}
For black-hole entropy accounting the geometric term is the area contribution,
\begin{equation}
F_{\mathrm{\GEP{}}}[X;t] = \frac{A[X;t]}{4G_N},
\label{eq:partVII-GEP}
\end{equation}
while the entropic-curvature term is the bulk entanglement or state-volume contribution,
\begin{equation}
F_{\mathrm{\CEP{}}}[X;t] = S_{\mathrm{bulk}}[X;t].
\label{eq:partVII-CEP}
\end{equation}
Thus
\begin{equation}
F_{\mathrm{\GFF{}}}[X;t] = \frac{A[X;t]}{4G_N} + S_{\mathrm{bulk}}[X;t].
\label{eq:partVII-generalized-entropy}
\end{equation}
This is exactly the structure that modern semiclassical gravity writes as generalized entropy. The difference is interpretive. In \GG{}/\GDOS{} it is not a late-discovered miracle of gravity. It is the only admissible scalar ledger once one accepts the open-system constitutional layer.

The physically accessible radiation entropy is then not the naive thermal entropy of quanta emitted to infinity. It is the minimum over admissible branches:
\begin{equation}
S_R(t) = \min_{X\in \mathfrak B_{\mathrm{adm}}}
F_{\mathrm{\GFF{}}}[X;t],
\label{eq:partVII-SR-min}
\end{equation}
where \(\mathfrak B_{\mathrm{adm}}\) is the set of boundary-admissible saddles or coarse-grained branches. In ordinary island language one might call these branches ``no-island'' and ``island'' extremal surfaces. In the present language they are simply competing \GFF{} saddles inside the \GDOS{} kernel.

This move is decisive because it replaces the question
\begin{quote}
``Why does the entropy of thermal Hawking quanta stop growing?''
\end{quote}
by the better-posed question
\begin{quote}
``Which branch minimizes the operational \GFF{} ledger at a given time?''
\end{quote}

Once the question is posed this way, a Page-like transition becomes not mysterious but expected.

\subsection{\DGI{} and the factorization puzzle}

The standard derivation of the paradox also quietly assumes that one may factorize the total Hilbert space into interior and exterior pieces,
\begin{equation}
\mathcal H_{\mathrm{tot}} \stackrel{?}{=} \mathcal H_{\mathrm{in}} \otimes \mathcal H_{\mathrm{out}}.
\label{eq:partVII-factorization-naive}
\end{equation}
In ordinary non-gauge quantum mechanics such a split is frequently harmless. In gravity it is not.

The reason is simple. Physical gravitational states obey constraint equations. Schematically,
\begin{equation}
\mathcal C_\alpha \lvert\Psi_{\mathrm{phys}}\rangle=0,
\label{eq:partVII-constraints}
\end{equation}
where \(\mathcal C_\alpha\) stands for the Hamiltonian, momentum, and other gauge constraints. These constraints tie together what one naively calls ``inside'' and ``outside'' data. The true physical state space is therefore a constrained quotient, not a naive tensor product. In symbol,
\begin{equation}
\mathcal H_{\mathrm{phys}}
=
\frac{\ker \mathcal C_\alpha}{\text{gauge redundancy}}
\neq
\mathcal H_{\mathrm{in}}\otimes\mathcal H_{\mathrm{out}}.
\label{eq:partVII-Hphys}
\end{equation}

This is the \DGI{} content of the factorization puzzle. The physical microstate volume must be counted only after quotienting by gauge redundancy. Horizon edge data and constraint matching mean that the entangling cut is not an ordinary partition of independent subsystems. Naive factorization therefore overcounts states and misidentifies what counts as independently accessible information.

From the standpoint of \GG{}/\GDOS{}, this is not an exotic correction added specifically for black holes. It is exactly what \DGI{} always says. Equivalent microscopic descriptions related by the relevant gauge symmetry are not distinct physical states. In the black-hole problem that principle becomes exceptionally consequential because the horizon is the very place where naive subsystem splits are most tempting.

This also explains a virtue of the modern island story without requiring its ontology as fundamental. Once one accepts that the entanglement wedge of the radiation need not coincide with the naive geometrical exterior, one is already acknowledging that the operational subsystem is subtler than Eq.~\eqref{eq:partVII-factorization-naive}. \GG{}/\GDOS{} simply interprets this as the natural consequence of \DGI{} plus \OGGEP{} rather than as the primary ontology.

\subsection{The Page curve as a \GSSB{} saddle switch}

We may now derive the Page-like behavior in the cleanest \GDOS{} language. Consider two competing branches in Eq.~\eqref{eq:partVII-SR-min}.

The first branch, which we may call the ``Hawking'' or ``no-bridge'' branch, is the one relevant at early times when the operational radiation subsystem is well described by approximately thermal emission:
\begin{equation}
S_R^{(1)}(t)\approx \eta\left[S_{\mathrm{BH}}(M_0)-S_{\mathrm{BH}}(M(t))\right].
\label{eq:partVII-branch1}
\end{equation}
The second branch, which we may call the ``bridge'' or ``island-like'' branch, caps the accessible entropy by the remaining geometric area ledger:
\begin{equation}
S_R^{(2)}(t)\approx S_{\mathrm{BH}}(M(t)).
\label{eq:partVII-branch2}
\end{equation}
The operational entropy is the smaller of the two:
\begin{equation}
S_R(t)=\min\left\{S_R^{(1)}(t),\,S_R^{(2)}(t)\right\}.
\label{eq:partVII-page-min}
\end{equation}

If one sets \(\eta\simeq 1\) for a schematic estimate, the crossing occurs when
\begin{equation}
S_{\mathrm{BH}}(M_0)-S_{\mathrm{BH}}(M_{\mathrm{Page}})
=
S_{\mathrm{BH}}(M_{\mathrm{Page}}),
\end{equation}
hence
\begin{equation}
S_{\mathrm{BH}}(M_{\mathrm{Page}})=\frac12 S_{\mathrm{BH}}(M_0).
\end{equation}
Using Eq.~\eqref{eq:partVII-SBH} gives
\begin{equation}
M_{\mathrm{Page}} = \frac{M_0}{\sqrt 2}.
\label{eq:partVII-Mpage}
\end{equation}
Substituting into the evaporation law \eqref{eq:partVII-evap-law} yields
\begin{equation}
t_{\mathrm{Page}}
=
\left(1-2^{-3/2}\right)t_{\mathrm{evap}}
\approx 0.65\,t_{\mathrm{evap}},
\label{eq:partVII-tpage}
\end{equation}
again up to order-unity corrections from species counts and the exact radiation entropy coefficient.

What matters physically is not the exact numerical prefactor. What matters is the structural conclusion. The Page turnover is the moment at which the dominant operational entropy branch changes. In \GDOS{} language the kernel admits multiple attractor branches, and the realized branch is selected by the \GFF{} minimum. The branch switch is therefore a \GRIP{} event whose middle stage is geometric spontaneous symmetry breaking. The full kernel allows multiple saddles; the realized entropy history selects one.

This is a sharper statement than the loose slogan ``information comes out after the Page time.'' It says that the late-time exterior-accessible algebra is reorganized because the minimizing saddle changes. Information recovery is therefore not a miracle and not an afterthought. It is the thermodynamic selection of the correct operational branch.

\subsection{\OGGEP{} and boundary-accessible information}
\label{subsec:bh-OGGEP}

The open-system lesson can now be stated more carefully. \OGGEP{} says that one must never confuse what exists in the bulk with what is reconstructible from a specific operational channel. This point is easy to say abstractly and easy to forget in black-hole discussions, because Hawking quanta are so seductive as concrete objects. Yet an outside observer never has direct access to the full bulk microstate. The observer has a stream of finite detector clicks, finite timing windows, finite angular resolution, and a history of prior coarse-graining.

It is therefore useful to distinguish three nested notions of information.

\begin{enumerate}[leftmargin=2.0em,label=\alph*)]
\item \textbf{Global information.} This is the fine-grained information in the full 6D state \(\rho_6(t)\), conserved by exact unitary evolution.
\item \textbf{Boundary-accessible information.} This is the information reconstructible from the state \(\rho_{\partial}(t)\) supplied by the CPTP map in Eq.~\eqref{eq:partVII-CPTP}.
\item \textbf{Detector-record information.} This is the information actually registered by a finite observational protocol, a further coarse-graining of \(\rho_{\partial}(t)\).
\end{enumerate}

The standard paradox often treats (b) and (c) as if they were equal to (a). \OGGEP{} says they are not. This does not trivialize the problem. On the contrary, it sharpens the scientific demand. A genuine resolution must say how these three levels are related and under what circumstances a record protocol can recover the correlations implied by the global unitary state.

The correct statement is therefore the following.

\begin{quote}
The information is never fundamentally lost because the bulk evolution is unitary. What can appear lost is only boundary-accessible or detector-accessible information under an incomplete operational channel.
\end{quote}

This distinction is exactly the one any open-system theorist would make for a noisy quantum channel or a dissipative mesoscopic device. The novelty in black holes is not the logic but the stakes: the inaccessible environment is partly encoded by geometry and horizon structure rather than by a laboratory heat bath.

We can turn this into a useful entropy inequality. Let \(\rho_6\) denote the exact bulk state and \(\rho_{\partial}=\Pi(\rho_6)\) the accessible boundary state. Then the data-processing inequality implies
\begin{equation}
I_{\mathrm{acc}} \le I_{\mathrm{bulk}},
\label{eq:partVII-data-processing}
\end{equation}
for any suitable monotone information measure. In words: operational accessibility cannot increase under the projection channel. If an outside observer appears to lose access to purification data, that is not evidence that the total purification has vanished. It is evidence that the observer has applied or been forced into a channel that throws away correlations.

This is the cleanest way to state what the present framework means by ``boundary-accessible information.'' It is not whatever one would like the outside radiation to encode. It is whatever the operational projection allows one to recover without violating \DGI{} and causal openness.

\subsection{Why the firewall is not forced in the \GDOS{} formulation}

The firewall argument gains force when three statements are treated as jointly mandatory: smoothness of the horizon, purity of the late radiation, and ordinary monogamy of entanglement under a sharp interior/exterior split. \GG{}/\GDOS{} relaxes the sharpness of the subsystem split rather than the smoothness condition itself.

Because the physical state space is DGI-constrained and because the operational subsystem is boundary-defined rather than ontologically total, the entanglement bookkeeping that drives the firewall contradiction is already modified before one invokes any violent horizon drama. The horizon does not need to become a wall of Planckian drama merely because the outside observer changes branches in the entropy ledger. The operational algebra reorganizes; the bulk need not lose its local smoothness in every semiclassical regime.

This does not mean that all horizon-scale drama is forbidden. It means only that the existence of a paradox is not enough by itself to force one. A putative firewall is not the default consequence of \GG{}/\GDOS{}. It would require independent positive evidence that smoothness fails, not merely the existence of an entropy bookkeeping puzzle that has already been reformulated more cleanly by \OGGEP{} and \DGI{}.

\subsection{What would count as a genuine resolution}
\label{subsec:bh-genuine-resolution}

Once the problem is posed correctly, the standards for a genuine resolution become much clearer. A satisfactory resolution should meet five requirements.

\paragraph{(1) Global unitarity.}
The fine-grained ontology of the full closed system must remain unitary. A theory that simply declares non-unitary destruction of information fundamental has not unified quantum theory and gravity; it has abandoned one of the pillars to save the other.

\paragraph{(2) Operational honesty.}
The theory must distinguish clearly between the total state, the boundary-accessible subsystem, and actual measurement records. It is not enough to say that ``information is in the radiation'' if the statement relies on a subsystem identification that violates the theory's own gauge constraints.

\paragraph{(3) Gauge-consistent state counting.}
The resolution must explain why naive interior/exterior factorization is either valid or invalid. If it is invalid, the theory must provide a principled replacement rather than a hand-waving slogan.

\paragraph{(4) Correct entropy ledger.}
The resolution must reproduce the early Hawking-like growth of operational entropy and the late Page-like turnover without artificial patching. It must explain why the relevant entropy is area plus bulk entanglement rather than one or the other alone.

\paragraph{(5) No new contradiction worse than the original.}
A successful resolution should not buy purity by introducing state duplication, uncontrolled nonlocality, or a breakdown of the operational calculus everywhere else in the theory.

These requirements can be summarized in a compact comparison table.

\begin{table}[H]
\centering
\caption[What counts as a resolution of the information paradox]{What counts as a genuine resolution of the black-hole information paradox in the present paper. The table is intentionally qualitative. The point is not to referee the full literature here, but to state the standards \ATen{} adopts. The columns are the four criteria and the rows the approaches judged against them, this framework included and not exempted.}
\label{tab:bh-resolution-criteria}
\small
\begin{tabular}{p{0.24\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.20\textwidth}}
\toprule
\textbf{Approach} & \textbf{Global unitarity} & \textbf{Operational honesty} & \textbf{Gauge-consistent factorization} & \textbf{How the Page curve is obtained} \\
\midrule
Pure Hawking thermality & No & Partial & No & Not obtained; radiation entropy grows monotonically \\
Black-hole complementarity & Intended yes & Partial & Partial & Heuristic; subsystem identification remains delicate \\
Firewall logic & Intended yes & Partial & No/unclear & Purity enforced at the price of horizon smoothness \\
Island / QES bookkeeping & Yes & Better & Better & Entropy obtained by minimizing generalized entropy over admissible saddles \\
\GG{} / \GDOS{} / \OGGEP{} & Yes & Yes & Yes & Page turnover is the GFF-minimizing saddle switch in an open-system ledger \\
\bottomrule
\end{tabular}
\end{table}

The last row contains the specific claim of the present paper. It does not deny the utility of island technology. Rather, it reinterprets island-like late saddles as effective branches of the \GFF{} ledger rather than as the primary ontology of the theory. The fundamental object is the closed 6D bulk state together with the \OGGEP{} channel. Island-like objects enter only as the most efficient late-time way of writing the operational entropy minimum.

\subsection{The clean \GDOS{} resolution in one sentence}

The clean \GDOS{} resolution can now be stated in one sentence:

\begin{quote}
The black-hole information paradox arises when one mistakes the thermal entropy of a boundary-accessible open subsystem for the fine-grained entropy of the globally closed 6D bulk state, and it is cured when the operational entropy is computed from the DGI-respecting \GFF{} minimum rather than from naive subsystem factorization.
\end{quote}

That is the theorem-level statement the section is meant to establish.



\subsection{How the \GDOS{} resolution compares with the island/QES program}
\label{subsec:bh-islands-comparison}

Because the contemporary literature now phrases the Page-curve solution in the language of islands and quantum extremal surfaces, it is worth stating explicitly how the present section relates to that program. The answer is that the two descriptions are not enemies, but they assign primacy differently.

In the island/QES formulation one begins from semiclassical gravity and then discovers that the entropy of the radiation is obtained by extremizing and minimizing the generalized entropy functional,
\begin{equation}
S_R(t)=\min_{X}\,\mathrm{ext}_{X}\left[\frac{A[X;t]}{4G_N}+S_{\mathrm{bulk}}[X;t]\right].
\label{eq:partVII-island-formula}
\end{equation}
This is a major conceptual advance because it shows that the correct entropy is not the naive thermal entropy of the emitted quanta alone. The present framework fully agrees with that conclusion \citep{Maldacena1998,Maldacena1999}. Equation~\eqref{eq:partVII-island-formula} is mathematically the same object as the \GFF{} minimum already written in Eq.~\eqref{eq:partVII-SR-min}. What changes is the interpretation of why this object should be the right one.

In the island story, the generalized-entropy rule can look like a remarkable repair made specifically for black holes. In \GG{}/\GDOS{} it appears instead as the direct consequence of the universal open-system ledger. The area term is simply the \GEP{} contribution, while the bulk entanglement term is the \CEP{} contribution. The ``island'' is then not the fundamental ontology but the late-time support of the minimizing operational branch. Put differently: the island program tells us \emph{how} to compute the correct entropy saddle, whereas the \GDOS{} language tells us \emph{why} a branch-minimized generalized-entropy functional had to be the correct answer in the first place.

This distinction can be organized in a compact dictionary.
\begin{table}[H]
\centering
\caption[The island/QES program beside the GG open-system reformulation]{Parallel reading of the modern island/QES program and the \GG{}/\GDOS{} open-system reformulation. The table is not meant as a polemic. It identifies the level at which each description places the primary ontology. Each row poses one question and answers it in both languages. The table is a translation and not a claim of priority: where the two accounts agree the agreement is the point, and where they differ the difference is stated rather than resolved.}
\label{tab:bh-island-dictionary}
\small
\begin{tabular}{p{0.23\textwidth}p{0.31\textwidth}p{0.31\textwidth}}
\toprule
\textbf{Question} & \textbf{Island / QES language} & \textbf{\GG{} / \GDOS{} language} \\
\midrule
What is fundamental? & Boundary-unitary or semiclassical-plus-QES description & Globally unitary 6D bulk plus \OGGEP{} projection \\
What is the relevant entropy? & Generalized entropy extremized over QES candidates & \GFF{} ledger minimized over admissible operational branches \\
Why is naive radiation entropy wrong? & Entanglement wedge of radiation is larger than naive exterior region & Boundary-accessible subsystem is an open CPTP image and need not coincide with total ontology \\
Why is factorization subtle? & Interior/exterior split is replaced by a nontrivial entanglement wedge & \DGI{} forbids naive gauge-breaking factorization across the horizon \\
What happens at the Page time? & A different QES dominates & A different \GFF{} saddle dominates; branch switch is \GSSB{} \\
\bottomrule
\end{tabular}
\end{table}

The practical consequence is important for the logic of the paper. We do not need to deny the validity of island calculations in the semiclassical regime. We only need to deny that their geometric bookkeeping should be mistaken for the deepest layer of ontology. The deepest layer, in the present program, is the closed six-dimensional state together with the causal projection rules that determine what a boundary observer can actually access.

\subsection{Scrambling, Hayden--Preskill logic, and operational information return}
\label{subsec:bh-scrambling}

A second place where operational clarity matters is the question of \emph{when} information should begin to become recoverable from the outgoing channel. The modern literature often phrases the answer in the Hayden--Preskill language: an ``old'' black hole past the Page time behaves as a rapidly scrambling mirror, so additional information thrown in can in principle be reconstructed from a modest extra amount of radiation after a scrambling delay. The present framework can absorb that statement, but only after translating it into \OGGEP{} terms.

The relevant time scale is the scrambling time,
\begin{equation}
t_{\mathrm{scr}} \sim \frac{\beta_H}{2\pi}\ln S_{\mathrm{BH}}
= \frac{1}{2\pi T_H}\ln S_{\mathrm{BH}}
\approx 4G_N M\,\ln S_{\mathrm{BH}},
\label{eq:partVII-tscr}
\end{equation}
where \(\beta_H=1/T_H\) and we used Eq.~\eqref{eq:partVII-HawkingT}. The logarithm is the key feature: scrambling is fast compared with the full evaporation time, but it is not instantaneous. In operational language this means that the boundary channel has a finite mixing time before newly inserted information can appreciably affect the outgoing accessible record.

One may formalize the distinction with three nested mutual informations,
\begin{equation}
I_{\mathrm{det}}(R:{\rm Ref})
\le
I_{\partial}(R:{\rm Ref})
\le
I_{6}(R:{\rm Ref}),
\label{eq:partVII-mutual-info-chain}
\end{equation}
where \({\rm Ref}\) denotes an abstract reference system entangled with the infalling matter, \(I_{6}\) is the exact bulk mutual information, \(I_{\partial}\) is the mutual information in the boundary-accessible state, and \(I_{\mathrm{det}}\) is the information actually extracted by a finite measurement protocol. Equation~\eqref{eq:partVII-mutual-info-chain} is simply the data-processing inequality applied twice, but it cleanly states what ``information return'' can and cannot mean.

Information return does \emph{not} mean that the detector record instantaneously reproduces the exact microscopic infalling state. It means that after the Page transition and after the additional scrambling delay, the operational channel is no longer locked into the purely thermal branch; correlations with the reference system can begin to reappear in principle in the boundary-accessible radiation. This is the open-system version of the Hayden--Preskill idea.

The reason this matters for cosmology is not that we expect to decode astrophysical black holes any time soon. The reason is conceptual. A theory that cannot state clearly the difference between global purification, boundary accessibility, and detector-level reconstruction will speak ambiguously not only about black holes but also about horizon physics, inflationary perturbations, and late-time relics. Scrambling therefore serves here as a diagnostic of the framework's operational discipline.

\subsection{Why remnants, baby universes, and final bursts are not needed in the minimal module}
\label{subsec:bh-no-remnant-needed}

Another virtue of the \GDOS{} reformulation is that it removes the pressure to use a remnant-like cure. In many older discussions one tried to preserve purity by suggesting that a tiny final object survives evaporation and secretly stores the information. The problem is immediate. If a remnant of mass \(M_{\mathrm{rem}}\sim M_{\mathrm{Pl}}\) had to store the entire initial Bekenstein--Hawking entropy, the number of remnant microstates would have to scale as
\begin{equation}
\mathcal N_{\mathrm{rem}} \sim \exp\!\left[S_{\mathrm{BH}}(M_0)\right],
\label{eq:partVII-remnant-count}
\end{equation}
which becomes catastrophically large for macroscopic \(M_0\). That suggests a huge low-mass state degeneracy and the usual pair-production catastrophes that plague remnant scenarios.

In the present framework this problem is unnecessary because the information was never required to be parked in a final tiny object. The global bulk state remains unitary throughout, and the boundary-accessible entropy is governed by the branch-minimized \GFF{} ledger throughout. A remnant is therefore not needed as an information warehouse. Nor is it necessary to invoke baby-universe branching as the primary bookkeeping device. Such possibilities are not logically excluded by the constitution, but they are not demanded by the minimal module, and the present paper has no need to purchase consistency by introducing them.

Exactly the same remark applies to final explosive-burst cures. If one imagines that a nearly thermal evaporation history remains featureless until the last instant and then somehow returns the full information content in a spectacular burst, one has again waited too long. The Page-curve logic says that the reorganization of accessible correlations begins around \(t_{\mathrm{Page}}\), not only at \(t_{\mathrm{evap}}\). In the present language that means the minimizing saddle has already switched well before the final stages. The correction is distributed over the late channel structure, not postponed to a miraculous terminal event.

This is another way to see why the open-system formulation is cleaner. Once the correct branch structure is identified, one is no longer forced into desperate end-stage mechanisms. The paradox is resolved by the evolution of the entropy ledger itself.

\subsection{A compact falsification dictionary for future black-hole observations}
\label{subsec:bh-falsification-dictionary}

Although direct experimental access to black-hole information recovery is extremely limited, the framework should still state what sorts of observations would push on it. The right standard is not ``measure the full Page curve of an astrophysical black hole tomorrow,'' which is unrealistic, but rather: identify which classes of observations would support or stress the assumptions entering the minimal module.

\begin{table}[H]
\centering
\caption[Compact observational dictionary for the black-hole block]{Compact observational dictionary for the black-hole information block. The entries are deliberately phrased at the level of principle rather than as a near-term mission proposal. The columns give the observable class, what the minimal module expects, and what would stress it. The entries are phrased at the level of what an instrument could actually decide, which is coarser than the theory can state, and that coarseness is deliberate.}
\label{tab:bh-falsification-dictionary}
\small
\begin{tabular}{p{0.23\textwidth}p{0.28\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Observable class} & \textbf{What the minimal module expects} & \textbf{What would stress the module} \\
\midrule
Ringdown and exterior classical geometry & Ordinary GR-like exterior relaxation in the semiclassical regime & Robust evidence that the exterior classical regime fails far from the endpoint without an accompanying open-system explanation \\
Near-horizon drama & No firewall forced by entropy bookkeeping alone & Unambiguous evidence that smooth semiclassical horizons are generically impossible even when \DGI{} and operational openness are respected \\
Late radiation correlations & In-principle recovery after the Page transition and scrambling delay, not exact early thermality forever & A theorem or observation implying exact monotonic thermal entropy of the accessible channel all the way to completion \\
Subsystem factorization & Gauge-constrained non-factorization near the horizon & A consistent fully gravitational factorization proof that makes the naive split exact without changing the entropy result \\
Generalized entropy / \GFF{} bookkeeping & Area plus bulk-entanglement ledger controls the branch minimum & A demonstrable contradiction between the generalized-entropy ledger and the observed or calculated Page behavior \\
\bottomrule
\end{tabular}
\end{table}

This dictionary again shows the style of the program. The point is not to pretend that every item is experimentally settled. The point is to make clear what the framework would count as support and what it would count as trouble. Even in an area as hard as black-hole information, the theory should publish its kill criteria rather than hide behind abstraction.



\subsection{Why the black-hole block matters for the rest of cosmology}
\label{subsec:bh-why-it-matters}

It may be tempting to think that the black-hole information block is conceptually elegant but observationally disconnected from the earlier sections on inflation, dark matter, dark energy, BBN, and structure formation. That would be a mistake. In the present program the black-hole block is where the constitutional claims of the earlier parts are exposed to their hardest internal audit.

Part~IV argued that the universe should be read as an open boundary subsystem of a higher-dimensional closed geometry. Part~V argued that chemical and nuclear histories should be written as \GDOS{} evolutions on constrained state manifolds with explicit ports and conserved ledgers. Part~VI argued that the CMB and the growth of structure are not just collections of fit parameters but macroscopic readouts of a DGI-respecting open-system flow. The black-hole problem tests whether these slogans remain coherent when the relevant subsystem is separated from the rest of the world by the strongest causal barrier in classical relativity.

If the framework were to fail here, the consequences would spill backward into the whole paper. For example, if one were forced to abandon the distinction between global closure and operational openness in the black-hole case, then the \OGGEP{} language used earlier for inflationary perturbations and cosmological horizons would become much less convincing. If naive factorization survived even in gravity, then the special role assigned to \DGI{} in the perturbation and measurement sectors would be weakened. If the correct entropy ledger for horizons turned out not to be the \GFF{} minimum, then the earlier use of \GFF{} as the scalar ranking functional for cosmological branches would lose one of its strongest nontrivial supports.

In that sense, black holes are the consistency check that keeps the rest of GUT-Cosmos honest. They force the paper to answer three foundational questions that might otherwise remain hidden under phenomenology.

\paragraph{Question 1: What counts as the system?}
In BBN the answer was the nuclear abundance network inside a cooling radiation bath. In the CMB it was the baryon--photon plasma inside a causal Hubble region. In a black hole the answer is trickier because the outside observer is causally prevented from accessing part of the geometry. If the theory cannot define the system there, it has not yet earned the right to claim a universal \SIP{}/\GDOS{} formalism.

\paragraph{Question 2: What counts as information?}
In the early universe, ``information'' often sounds harmless: it is encoded in curvature perturbations, transfer functions, and phase correlations. In black holes the word becomes dangerous because one is no longer allowed to slide casually between microscopic purity and macroscopic observability. The black-hole block therefore clarifies that information is always protocol-relative on the boundary and only globally absolute in the closed bulk ontology.

\paragraph{Question 3: What counts as entropy?}
In ordinary cosmological applications one can often get away with a simplified entropy language: comoving entropy conservation, photon entropy, or effective fluid entropy. Black holes remove that luxury. The correct entropy is not optional bookkeeping but the whole content of the paradox. If the theory identifies the wrong ledger here, it cannot be trusted when it later speaks about attractors, coincidence, or relic abundances.

These questions can be summarized in one compact balance law for the operational sector. Let \(S_{\partial}\) denote the boundary-accessible entropy, let \(S_{\mathrm{env}}\) denote environment and horizon-hidden coarse-grained entropy, and let \(I_{\mathrm{corr}}\) denote the cross-correlations discarded by a finite observational protocol. Then the schematic coarse-grained entropy balance reads
\begin{equation}
\frac{\dd}{\dd t}\left(S_{\partial}+S_{\mathrm{env}}-I_{\mathrm{corr}}\right) \ge 0.
\label{eq:partVII-entropy-balance}
\end{equation}
The point of Eq.~\eqref{eq:partVII-entropy-balance} is not its exact microscopic derivation inside this paper; the point is the logic it captures. Accessible entropy can rise even while global purity is preserved, because the information hidden in discarded correlations and inaccessible sectors is not destroyed but merely moved out of the chosen protocol. This is exactly the same logic that underlies decoherence in laboratory open systems, but horizon geometry makes the partition especially severe.

The black-hole block also matters because it disciplines language about ``paradox.'' Earlier in the paper several observational tensions were discussed: the Hubble tension, the \(S_8\) tension, and the JWST early-galaxy problem. Those are disagreements between inference chains, calibrations, or sectors of phenomenology. The information paradox is different. It is not a tension between two data sets but a tension between levels of description. Solving it therefore teaches a methodological lesson: many apparent contradictions in theoretical cosmology arise from asking one descriptive layer to carry obligations that belong to another layer.

That lesson flows back into the whole paper. The Hubble tension can then be reread as an inference-dictionary problem rather than as a literal contradiction of background dynamics. The CMB gauge issue can be reread as a \DGI{} coarse-graining problem rather than as an ambiguity about physical observables. Even the dark-sector problem can be reread as a source-ledger problem rather than as a license to add arbitrary new fluids. Black holes sharpen this style of reasoning because the danger of mixing descriptive layers is maximal there.

For that reason the present section is not a decorative appendix. It is one of the places where the paper most clearly shows what it means to keep a single constitutional grammar from the beginning of the universe to the late fate of information-bearing gravitational systems.

\subsection{What the present framework claims, and what it does not}

It is important not to overclaim.

The framework does \emph{claim} the following:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item There is no fundamental information destruction at the \GG{} bulk level.
\item Pure-to-mixed behavior in the exterior description is expected because the boundary observer sees an open subsystem.
\item The correct coarse-grained entropy ledger is generalized entropy, re-read as \GFF{}.
\item The Page turnover is a \GDOS{} saddle switch and therefore a \GSSB{} event.
\item Naive interior/exterior Hilbert-space factorization is not constitutionally admissible once \DGI{} is taken seriously.
\end{enumerate}

The framework does \emph{not} yet claim the following:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item a complete microscopic 6D basis for all black-hole microstates;
\item an exact numerical graybody-factor derivation inside the present paper;
\item a full detector-level reconstruction protocol for practical information retrieval from an evaporating astrophysical black hole.
\end{enumerate}

These are not evasions. They are the correct boundaries of the present cosmology paper. The goal here is to show that the paradox is no longer structurally fatal once the constitutional layer of \GG{} is enforced. The dedicated companion \citep{ArisakaC3BH} pushes the microstate and detector questions much further --- deriving the exact \(A/4\) coefficient from induced gravity (its Theorem~\emph{T-BH-1}), the EHT-shadow and ringdown predictions, and the four laws of black-hole mechanics as \GDOS{} statements.

\subsection{Interim conclusion of the information-paradox section}

The black-hole information paradox is not an embarrassment left over after the rest of cosmology has been solved. It is a precision test of what the framework means by information, entropy, gauge redundancy, and measurement. The \GDOS{} resolution is clean because it does not require special pleading. The same open-system principles used elsewhere in the program apply here too. A black hole is a particularly severe open system, not an exception to the theory.

This conclusion also gives the correct bridge to the next section. Once one understands black holes as extreme information-processing objects rather than as isolated curiosities, primordial black holes become a natural cosmological laboratory. They are not only compact objects; they are possible fossilized relics of the primordial perturbation sector itself. That is why the next section turns from the information paradox to primordial black holes.

\section{Primordial black holes}

\subsection{Historical background}

Primordial black holes were introduced by Hawking in 1971 \citep{Hawking1971}, who proposed that large density perturbations in the very early universe could collapse directly into black holes without proceeding through stellar evolution. The resulting mass spectrum spans many orders of magnitude, from below evaporation thresholds at $\sim 10^{15}$~g to supermassive scales relevant to galactic dynamics. The present section asks which slice of that mass spectrum, if any, is consistent with the inherited ledger and could play a non-trivial role in dark-matter, gravitational-wave, or seed-black-hole physics without conflicting with the rest of the cosmic audit.

\label{sec:pbh}


\begin{proposition}[Minimal PBH scarcity theorem (T-Cosmos-14)]
If the smooth GUT-adjacent inflationary branch (the derived Starobinsky/\(\xi\)
plateau) and the Kaxion dark-matter branch are both retained, then the
primordial black-hole abundance remains at most a trace component unless an additional narrow small-scale enhancement can be
derived from the same inherited scalar ledger. In the minimal smooth module,
\[
\beta(M)\sim \exp\!\left[-\frac{\delta_c^2}{2\sigma^2(M)}\right]
\]
is exponentially tiny in the relevant mass windows --- the result depends on the smoothness of the plateau spectrum, not on the absolute inflationary scale, so it is unchanged by the GUT-adjacent reading.
\end{proposition}



Primordial black holes are attractive because they compress several cosmological questions into one object. They probe the small-scale primordial spectrum, the collapse threshold of the early universe, the thermal background history, dark-matter accounting, and late compact-object phenomenology. In many frameworks they are used opportunistically: if particle dark matter becomes difficult, PBHs are brought in as an alternative. In the present framework the logic is different. The Kaxion already occupies the primary cold-dark-matter slot. PBHs are therefore not introduced as a rescue sector. They are introduced as a stress test of the perturbation ledger already developed in Parts II through IV.

This change of logic matters. A framework that already has a constrained dark-matter candidate does not need PBHs to solve the dark-matter problem. That means it can ask a much cleaner question: does the inherited \GG{}/\GDOS{} cosmology naturally produce a substantial primordial black-hole population, a tiny trace population, or essentially none at all?

The answer suggested by the present draft is sharp but nuanced. The minimal smooth \GG{} inflationary module, together with the red scalar spectrum and the absence of a large small-scale enhancement, predicts that PBHs are \emph{not} a dominant dark-matter component. In the minimal ledger their abundance is exponentially suppressed. At the same time, the framework does not forbid subdominant narrow windows associated with specific transition scales. Those windows become useful because their mass scale maps directly to formation temperature and hence to specific epochs in the inherited cosmic chronology.

\subsection{Formation windows}
\label{subsec:pbh-formation}

The standard logic of primordial-black-hole formation begins when an overdense region re-enters the horizon during the radiation era. If the overdensity exceeds a threshold \(\delta_c\), pressure support fails to prevent collapse and a black hole forms. The relevant mass is tied to the horizon mass at re-entry:
\begin{equation}
M_H = \frac{4\pi}{3}\rho\,H^{-3}.
\label{eq:partVII-MH-def}
\end{equation}
During radiation domination,
\begin{equation}
\rho = \frac{\pi^2}{30}g_* T^4,
\qquad
H = 1.66\,g_*^{1/2}\frac{T^2}{M_{\mathrm{Pl}}},
\label{eq:partVII-rad-background}
\end{equation}
so Eq.~\eqref{eq:partVII-MH-def} becomes
\begin{equation}
M_H(T)
\simeq
9.4\times10^{25}\,\mathrm{g}
\left(\frac{106.75}{g_*}\right)^{1/2}
\left(\frac{\mathrm{TeV}}{T}\right)^2.
\label{eq:partVII-MH-T}
\end{equation}
The PBH mass is usually a fraction \(\gamma_c\) of the horizon mass,
\begin{equation}
M_{\mathrm{PBH}} \simeq \gamma_c M_H,
\qquad
\gamma_c\sim0.1\text{--}0.2.
\label{eq:partVII-MPBH}
\end{equation}

Equation~\eqref{eq:partVII-MH-T} is extremely informative because it translates a cosmic temperature into a PBH mass window. At a TeV-scale reheating or exit feature one expects
\begin{equation}
M_{\mathrm{PBH}}^{(\mathrm{TeV})}
\sim
10^{25}\text{--}10^{26}\,\mathrm{g},
\label{eq:partVII-MPBH-TeV}
\end{equation}
whereas the QCD epoch around \(T\sim 0.1\text{--}0.2\,\mathrm{GeV}\) maps to
\begin{equation}
M_{\mathrm{PBH}}^{(\mathrm{QCD})}
\sim
\mathcal O(M_\odot).
\label{eq:partVII-MPBH-QCD}
\end{equation}
This is why PBH phenomenology is so sensitive to the precise epoch that generates any small-scale enhancement.

The next step is the collapse fraction. If the primordial density contrast smoothed on mass scale \(M\) is Gaussian with variance \(\sigma^2(M)\), the formation fraction at re-entry is
\begin{equation}
\beta(M)
=
\int_{\delta_c}^{\infty}
\frac{\dd\delta}{\sqrt{2\pi}\sigma(M)}
\exp\!\left[-\frac{\delta^2}{2\sigma^2(M)}\right].
\label{eq:partVII-beta-exact}
\end{equation}
For the tail relevant to PBH production, \(\sigma\ll\delta_c\), one may use the asymptotic form
\begin{equation}
\beta(M)
\approx
\frac{\sigma(M)}{\delta_c\sqrt{2\pi}}
\exp\!\left[-\frac{\delta_c^2}{2\sigma^2(M)}\right].
\label{eq:partVII-beta-asymptotic}
\end{equation}
The present-day PBH fraction inherits the standard radiation-era growth factor,
\begin{equation}
f_{\mathrm{PBH}}(M)
\propto
\beta(M)\,M^{-1/2},
\label{eq:partVII-fPBH-scaling}
\end{equation}
up to the usual \(g_*\) and equality-era prefactors. The crucial point is the exponential tail in Eq.~\eqref{eq:partVII-beta-asymptotic}. PBHs are not produced efficiently unless the primordial variance on the relevant scale is dramatically larger than on CMB scales.

That single mathematical fact already indicates what the minimal \GG{} prediction must be. The scalar spectrum inherited from Part IV and audited in Part VI is red and smooth. It does not contain the large small-scale spike required to make \(\beta(M)\) appreciable. The natural expectation is therefore exponentially small PBH production.

\subsection{A minimal quantitative estimate}

To see how sharp the suppression is, one need not run a full numerical collapse code. A tail estimate suffices. Take a representative radiation-era threshold
\begin{equation}
\delta_c^{(\mathrm{rad})}\sim0.4\text{--}0.5.
\label{eq:partVII-deltac-rad}
\end{equation}
Now suppose the minimal smooth spectrum gives a small-scale variance \(\sigma(M)\sim10^{-2}\), which is already far above the ordinary CMB-scale amplitude but still nowhere near the values needed for strong PBH production. Then
\begin{equation}
\frac{\delta_c^2}{2\sigma^2}
\sim
\frac{0.45^2}{2\times10^{-4}}
\sim
10^3,
\end{equation}
and Eq.~\eqref{eq:partVII-beta-asymptotic} gives
\begin{equation}
\beta(M)\sim \exp(-10^3),
\end{equation}
up to algebraic prefactors that are negligible compared with the exponential. This is effectively zero.

Even if one imagines a much larger variance, say \(\sigma\sim0.05\), the exponent becomes
\begin{equation}
\frac{\delta_c^2}{2\sigma^2}\sim 40,
\end{equation}
so the formation fraction is still tiny. One therefore needs not just a modest enhancement but a genuinely strong feature in the small-scale spectrum to make PBHs important. That feature is absent from the theorem-level minimal module already written into this draft.

This simple estimate is sufficient to justify a central statement of Part VII:

\begin{quote}
In the minimal smooth \GG{} cosmology, PBHs are not a dominant dark-matter component. Kaxion dark matter remains primary, and any PBH population is at most a small trace component unless an additional narrow small-scale enhancement can be derived from the same geometric ledger.
\end{quote}

\subsection{Formation windows tied to the inherited thermal history}

Even though the minimal abundance is tiny, the horizon-mass formula remains informative because it maps hypothetical features to concrete epochs. Three windows are particularly useful conceptually.

\paragraph{(i) ToC-7 exit / reheating window.}
If the prompt lock crossing were to carry a narrow spike or short-lived softening of the equation of state (the crossing is prompt, so none is expected --- this is the contingency, not the prediction), the corresponding PBHs would lie near the TeV horizon mass in Eq.~\eqref{eq:partVII-MPBH-TeV}. These are asteroid-scale objects. They are well above the evaporation threshold and would survive to the present epoch.

\paragraph{(ii) Electroweak-like window.}
Any narrow feature tied to physics near the electroweak-to-TeV interval would shift the mass upward by roughly two orders of magnitude in temperature and hence four orders in mass. Such objects would still be very far from stellar masses and would fall primarily into lensing, dynamical, and accretion-style searches.

\paragraph{(iii) QCD softening window.}
The QCD crossover lowers the effective sound speed of the plasma and is the classic place where stellar-mass PBHs become easiest to form. Equation~\eqref{eq:partVII-MPBH-QCD} explains why. The relevant horizon mass is near the solar-mass range, precisely where merger catalogs and microlensing constraints become powerful.

The reason to list these windows is not to claim that the framework predicts all of them. It is to show how any future PBH hint could be translated back into the thermal and dynamical history of the early universe. The PBH mass is a timestamp written in compact-object form.

\subsection{The integer-ledger mass ladder}
\label{subsec:pbh-e7-ladder}

The horizon-mass map of Eq.~\eqref{eq:partVII-MH-T} acquires extra structure from the
inherited ledger. If a narrow small-scale feature does arise, it does not sit at an
arbitrary scale: the same \OGNfive{} integer \(J=7\) that fixes the recombination switch
\(z_\ast = e^{7}\) of Part~VI also spaces the throat resonances, so the
predicted primordial-black-hole mass peaks fall on a geometric ladder
\[
M_n^{\mathrm{PBH}} \;\propto\; e^{-7n},
\qquad n = 0, 1, 2, \ldots,
\]
the gravitational-sector image of the very integer that organizes the cosmological
corridor. This spectral ladder is the dedicated prediction of the companion
black-hole paper \citep{ArisakaC3BH}, where it is registered as a named falsifier: a
measured primordial-black-hole mass spacing inconsistent with the \(e^{7}\) ladder
would refute it. The cosmological point here is narrower but fully consistent: the
minimal module makes such a feature scarce, with the abundance exponentially
suppressed, so the ladder predicts \emph{where} any rare PBH population would sit,
not that the population is large.

\subsection{Connection to early background dynamics}
\label{subsec:pbh-background}

The next question is whether anything already derived in Parts IV and VI can generate the strong small-scale enhancement PBH formation requires.

The answer is mostly no in the minimal module, and the reasons are important.

First, the slow-smooth inflationary sector is deliberately smooth. The entire point of the geometric plateau in Part IV was to avoid a jagged inflaton potential landscape with arbitrary features. A smooth plateau gives a red and almost scale-invariant scalar spectrum, not a narrow blue spike.

Second, the EDE lobe introduced in Part VI is much too late and much too large-scale to function as a PBH factory. Its characteristic redshift is
\begin{equation}
z_c \approx 3500,
\end{equation}
well after the inflationary exit and on scales relevant to the sound horizon and matter transfer function, not to horizon-scale collapse of ultra-small modes. It can assist early halos and change \(H_0\) inference, but it does not generate efficient PBH collapse.

Third, the same growth logic that helps the JWST problem points away from PBHs as the main explanation of early supermassive black holes. Part VI already showed that earlier first light and direct-collapse baryonic seeds arise naturally from the EDE boost, baryonic streaming damping, and a lower effective collapse threshold for high-redshift gas. Those mechanisms are later-time and baryonic. They therefore offer a cleaner route to early supermassive seeds than invoking a large primordial black-hole abundance.

This is the important cross-consistency statement. The same framework that helps early galaxies does not need to use PBHs as a second rescue story. It prefers direct-collapse baryonic seeds plus a modest acceleration of early structure assembly.

\subsection{PBHs versus Kaxion dark matter}

The dark-matter accounting sharpens the same conclusion. Part V already occupied the cold-dark-matter slot with the Kaxion, whose relic abundance is tied to the compact-angle sector and the inherited misalignment corridor. If PBHs were also to supply an order-unity fraction of dark matter, the total matter ledger would be overbooked unless the Kaxion abundance were reduced correspondingly. But reducing the Kaxion abundance would destroy the clean compact-angle interpretation that motivated it in the first place.

This leads to a simple hierarchy of hypotheses:

\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item \textbf{Minimal theorem-level hypothesis:} Kaxions supply the dominant cold-dark-matter abundance; PBHs are negligible.
\item \textbf{Allowed but nonminimal hypothesis:} a narrow primordial feature creates a tiny PBH subcomponent in a specific mass window, without disturbing the primary Kaxion ledger.
\item \textbf{Disfavored hypothesis:} PBHs supply most or all of dark matter. This is not impossible abstractly, but it is not the natural outcome of the smooth \GG{} module already developed.
\end{enumerate}

This ranking is more scientifically useful than a vague statement that PBHs are ``possible.'' It tells the reader what the framework actually expects.

\subsection{Observational searches}
\label{subsec:pbh-searches}

The observational program for PBHs may be organized by mass.

\paragraph{Evaporation regime.}
Black holes lighter than about
\begin{equation}
M_{\mathrm{evap}}\sim 5\times10^{14}\,\mathrm{g}
\label{eq:partVII-evap-threshold}
\end{equation}
have evaporated by the present epoch. This follows from the cubic evaporation-time law
\begin{equation}
t_{\mathrm{evap}}(M)
\approx
6.2\times10^{-27}\,\mathrm{s}\,
\left(\frac{M}{\mathrm g}\right)^3.
\label{eq:partVII-tevap}
\end{equation}
Any PBH population formed far below this mass contributes today only through relic evaporation products and the constraints attached to them.

\paragraph{Asteroid-to-planetary regime.}
The TeV-reheating window in Eq.~\eqref{eq:partVII-MPBH-TeV} lies in the broad asteroid-mass corridor. Such objects survive, do not radiate appreciably today, and are primarily tested by lensing, dynamical heating, accretion bounds, and indirect compact-object searches.

\paragraph{Stellar-mass regime.}
The QCD window lies near the solar-mass scale and is therefore constrained by microlensing, merger catalogs, and stochastic compact-binary backgrounds. Because Part VI already offers a direct-collapse route to early heavy seeds, a strong observational case for a dominant stellar-mass PBH population would place pressure on the minimal cosmological module of this paper.

These qualitative regimes can be summarized compactly.

\begin{table}[H]
\centering
\caption[Illustrative PBH mass windows relevant to GG-A10-Cosmos]{Illustrative PBH mass windows relevant to \ATen{}. The first column is the early-universe epoch or feature; the last column states the role the present framework assigns to the corresponding PBH population. The columns run from the formation epoch through the characteristic mass and the search handles to the role the window would play here. The windows are illustrative: they say where a primordial black hole population would have to sit to matter, not that the framework predicts one.}
\label{tab:pbh-windows}
\small
\begin{tabular}{p{0.24\textwidth}p{0.20\textwidth}p{0.24\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Formation epoch / feature} & \textbf{Characteristic mass} & \textbf{Primary search handles} & \textbf{Role inside minimal \GG{}/\GDOS{}} \\
\midrule
TeV exit / reheating feature & \(10^{25}\) to \(10^{26}\,\mathrm g\) & lensing, dynamics, accretion-style bounds & not required; at most narrow subcomponent if a transition feature exists \\
Electroweak-scale feature & \(10^{27}\) to \(10^{29}\,\mathrm g\) & lensing and dynamical bounds & same: not primary dark matter in the smooth module \\
QCD softening window & \(\mathcal O(M_\odot)\) & microlensing, merger catalogs, stochastic backgrounds & useful test window, but not the preferred explanation of early SMBH seeds \\
Sub-\(5\times10^{14}\,\mathrm g\) evaporation regime & evaporated by today & high-energy evaporation products & not relevant to present dark matter; only indirect historical constraints \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Bottom line of the PBH section}

The present paper should therefore state the PBH conclusion in deliberately asymmetric language. Primordial black holes are not the engine that makes \GG{} cosmology work. They are an auxiliary, high-leverage test of the small-scale primordial spectrum and the early thermal history. The minimal smooth module predicts that they are scarce. A tiny subcomponent is not excluded a priori, especially in a narrow transition window, but a dominant PBH dark-matter universe is not the natural expectation once the Kaxion sector and the smooth inflationary ledger are accepted.

That conclusion also clarifies the bridge to the next section. The same primordial perturbation sector that could in principle seed PBHs also seeds the primordial tensor spectrum. In the minimal module, both are suppressed for the same reason: the inherited inflationary source ledger is smooth and low-scale. Primordial gravitational waves are therefore the cleanest final audit of the same logic.

\section{Primordial gravitational waves}

\subsection{Historical background}

The proposal that the early universe leaves behind a present-day stochastic background of gravitational waves with a roughly scale-invariant spectrum is due to Starobinsky in 1979 \citep{Starobinsky1979}, who computed the amplitude of relict tensor modes in a de~Sitter phase and argued that they would survive to the present as a background detectable through CMB B-mode polarization or future direct interferometry. The amplitude is set by the inflationary energy scale, and the derived GUT-adjacent plateau fixes it: the Starobinsky relation at the reheating-matched count. The present section converts the derived launch into a sharp, two-sided falsification target.

\label{sec:pgw}


\begin{theorem}[Tensor target (T-Cosmos-15, revised)]
The same derived plateau that computes the scalar tilt fixes the primordial
tensor sector at a firm central value:
\[
r = \frac{12}{N^{*2}} = 0.0043 .
\]
This is one of the cleanest near-term falsifiers of the minimal cosmology
branch, and it is two-sided: a tensor measurement inconsistent with this
value in either direction --- a detection well above it, or an exclusion
well below it --- rules out the derived launch rather than merely requiring
a small parameter adjustment.
\end{theorem}



The primordial tensor sector is the sharpest observational kill switch in the entire cosmology paper. Scalar-sector anomalies can sometimes be patched by late-time source terms, transfer-function subtleties, or environmental systematics. Primordial tensors are much less forgiving. Once the inflationary energy scale is fixed, the tensor amplitude is effectively fixed. This is why Part IV invested its effort in deriving the plateau class and its placement --- one e-fold below the GUT edge, with the amplitude the sector's one declared input.

The consequence is sharp. If the minimal \GG{} inflationary module is correct, primordial tensor modes sit at exactly \(r = 0.0043\) --- an order of magnitude below the present bound, and several times above the projected Simons-Observatory/LiteBIRD floor (\(\sigma(r)\simeq(1\text{--}2)\times10^{-3}\)). This is one of the framework's strongest falsification handles, because it ties a concrete, two-sided observational statement directly to the derived launch.

\subsection{Tensor modes from the derived plateau}
\label{subsec:pgw-low-scale}

For a transverse-traceless metric perturbation \(h_{ij}\) on a spatially flat background,
\begin{equation}
\dd s^2 = -\dd t^2 + a^2(t)\left[\delta_{ij}+h_{ij}\right]\dd x^i\dd x^j,
\qquad
\partial_i h_{ij}=0,
\qquad
h_{ii}=0,
\label{eq:partVII-tensor-metric}
\end{equation}
each polarization mode obeys the standard equation
\begin{equation}
u_k'' + \left(k^2 - \frac{a''}{a}\right)u_k = 0,
\label{eq:partVII-tensor-eq}
\end{equation}
where \(u_k\) is the canonically normalized tensor variable and primes denote conformal-time derivatives.

The primordial tensor power spectrum at horizon exit is
\begin{equation}
\mathcal P_t(k)=\frac{2H_*^2}{\pi^2 M_{\mathrm{Pl,red}}^2},
\label{eq:partVII-Pt}
\end{equation}
while the tensor-to-scalar ratio is
\begin{equation}
r = \frac{\mathcal P_t}{\mathcal P_{\mathcal R}}
= \frac{2H_*^2}{\pi^2 M_{\mathrm{Pl,red}}^2 A_s}.
\label{eq:partVII-r-basic}
\end{equation}
Using \(H_*^2 \simeq V_*/(3M_{\mathrm{Pl,red}}^2)\) gives the equivalent expression already derived in Part IV,
\begin{equation}
r = \frac{2V_*}{3\pi^2 A_s M_{\mathrm{Pl,red}}^4}.
\label{eq:partVII-r-V}
\end{equation}
The key input is the derived plateau of Part~IV --- class forced by the boundary law, height fixed by the one declared input:
\begin{equation}
V_*^{1/4} = 8.3\times10^{15}~\mathrm{GeV},
\qquad
N^* = 52.6 ,
\label{eq:partVII-lock-plateau}
\end{equation}
and substituting into Eq.~\eqref{eq:partVII-r-V}, together with \(A_s \simeq 2.1\times10^{-9}\), yields
\begin{equation}
r = \frac{12}{N^{*2}} = 0.0043,
\label{eq:partVII-r-52}
\end{equation}
in agreement with Eq.~\eqref{eq:partII-r-numeric}. (Evaluated at the retired lock-scale height, this chain would give \(r \sim 10^{-52}\); the label is kept for the audit trail.)

It is useful to express the same result via the inflationary Hubble scale. From \(3M_{\mathrm{Pl,red}}^2 H_*^2 \simeq V_*\) one obtains
\begin{equation}
H_* = H_1 \simeq 1.6\times10^{13}\,\mathrm{GeV}
\label{eq:partVII-Hinf}
\end{equation}
(the lock-scale combination \(\Mlock^2/\sqrt3\,M_{\mathrm{Pl,red}} \simeq 3.5\times10^{-12}\,\mathrm{GeV}\) survives as the curvature scale of the middle basin --- the terminus seesaw, not the launch). The tensor target is therefore not a side remark. It is a direct algebraic consequence of the same derived plateau that computes the scalar tilt.

\subsection{Why the primordial signal is exactly one signal: vacuum tensors, and silence from the transitions}

There are in fact two statements rather than one.

The first is the vacuum tensor amplitude of Eq.~\eqref{eq:partVII-lock-plateau}: the derived plateau fixes \(H_* \simeq 1.6\times10^{13}\)~GeV, hence \(r = 0.0043\) --- modest against the old GUT-scale folklore, decisive against the coming instruments.

The second concerns the nature of the exits. The graceful exit of Part IV is \emph{smooth}, and the lock crossing below it is \emph{prompt} and barrier-free. Neither is a violent first-order transition with highly relativistic bubble collisions, so neither generates an additional stochastic gravitational-wave background beyond the vacuum tensor spectrum --- and the absence of any TeV-epoch transition background is itself a named falsifier: a detected background at the lock scale would break the prompt-crossing bound.

This is an important internal consistency point. One cannot claim both that the descent is smooth-and-prompt and that it produces a large extra primordial gravitational-wave signal. The minimal paper does not try to have both: one clean vacuum tensor line at \(r = 0.0043\), and silence from the transitions.

\subsection{B-mode polarization}
\label{subsec:pgw-bmodes}

At the observable level primordial tensor modes are most cleanly sought through CMB B-mode polarization. The total B-mode signal may be written schematically as
\begin{equation}
C_\ell^{BB}
=
C_{\ell,\mathrm{lens}}^{BB}
+
C_{\ell,\mathrm{prim}}^{BB}(r)
+
C_{\ell,\mathrm{fg}}^{BB},
\label{eq:partVII-CLBB}
\end{equation}
where the three terms are lensing-converted B-modes, primordial tensor B-modes, and foreground residuals.

In ordinary high-scale inflation, the primordial term can generate a recombination bump near \(\ell\sim 80\) and a reionization bump at very low multipoles. In the present framework \(r\) is so small that
\begin{equation}
\begin{split}
C_{\ell,\mathrm{prim}}^{BB}(r = 0.0043):\ &\text{a recombination bump at } \ell\sim80,\\
&\text{several times above the Simons-Observatory/LiteBIRD floor}
\end{split}
\end{equation}
--- small, but decisively measurable. The scientific meaning is very sharp:

\begin{quote}
After delensing, the primordial bump at the \(r = 0.0043\) level is the minimal \GG{} inflationary module's signature: the Simons Observatory and LiteBIRD must find it --- and must find it at that amplitude, not another.\end{quote}

This statement is stronger than the generic phrase ``small \(r\).'' It means that successful delensing should reveal no primordial bump at a measurable level. The moment a robust primordial tensor signal is seen, the minimal module is ruled out.

To keep the draft crisp, it is worth stating the kill rule in plain terms:

\begin{equation}
\begin{aligned}
&\text{A tensor measurement inconsistent with } r \simeq 0.004 \text{ in either direction}\\
&\quad\Longrightarrow\quad
\text{the minimal derived \GG{} launch is false.}
\end{aligned}
\label{eq:partVII-Bmode-kill}
\end{equation}

That is not rhetorical flourish. It follows because \(r = 12/N^{*2}\) carries no dial: moving \(r\) means moving \(N^*\), and \(N^*\) is pinned by the reheating match and the promptness of the lock crossing --- so a discrepant tensor amplitude breaks the derived chain itself, not a plateau and unravels the minimal module back at Part IV.

\subsection{Interferometer and CMB cross-checks}
\label{subsec:pgw-interferometer}

The tensor audit should not stop at the CMB, because the primordial tensor background is often discussed in frequency language rather than multipole language. For an approximately scale-invariant tensor spectrum, the present-day stochastic background is roughly
\begin{equation}
\Omega_{\mathrm{GW},0} h^2
\sim
1.6\times10^{-5}\,A_s\,r,
\label{eq:partVII-OmegaGW}
\end{equation}
up to transfer-function and frequency-dependent corrections. With \(A_s\simeq 2.1\times10^{-9}\) and \(r = 0.0043\), one obtains
\begin{equation}
\Omega_{\mathrm{GW},0} h^2
\sim
1.4\times10^{-16},
\label{eq:partVII-OmegaGW-numeric}
\end{equation}
far below any interferometric reach --- the primordial background is a CMB-polarization target, not an interferometer target. (A separate null \emph{is} staked at interferometer-adjacent frequencies: the lock crossing is prompt, so no strongly supercooled TeV-scale transition background exists; a loud stochastic signal attributed to one would refute the prompt-crossing lemma.)

Equation~\eqref{eq:partVII-OmegaGW-numeric} provides a powerful cross-check. The same module that predicts null CMB primordial B-modes also predicts null interferometric primordial backgrounds. If a future stochastic background is seen in the millihertz, decihertz, or ground-based band, the first interpretation inside this framework should be \emph{astrophysical} rather than primordial. Compact binaries, stellar collapse, and late-time structure formation can all produce stochastic backgrounds. The minimal primordial tensor sector cannot.

This distinction matters because otherwise a future gravitational-wave detection could be misread as confirming inflation in general. The correct statement for \ATen{} is subtler:
\begin{quote}
Detection of a stochastic gravitational-wave background does not by itself falsify the framework. Detection of a \emph{primordial} inflationary-amplitude background does.
\end{quote}

The CMB and interferometric channels therefore form a consistency pair. If both remain null for primordial signals, the minimal tensor module survives. If either reveals a robust primordial amplitude inconsistent with Eqs.~\eqref{eq:partVII-r-52} and \eqref{eq:partVII-OmegaGW-numeric}, the module fails.

\subsection{What the primordial-gravitational-wave section contributes to the full paper}

This section does more than simply add one more observational test. It closes the loop between Parts II, IV, and V.

From Part IV it inherits the theorem that the inflationary plateau height is set by the lockscale. From Part VI it inherits the CMB rule that any near-term B-mode should be lensing-generated. In Part VII it translates those inherited results into the strongest falsification handle available in the entire draft.

That is why the tensor sector belongs at the end of the present part. It is the final audit of the primordial source ledger before the paper turns, in a later part, to the chronological assembly of the full cosmic timeline.

\section{Integrated summary of the black-hole and primordial-signature block}
\label{sec:partvii-summary}


\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item The black-hole information paradox is reformulated as a mismatch between global and operational descriptions \citep{Bekenstein1973,Hawking1975,Penrose1994}. It is dissolved at the theorem level by \OGGEP{} and quantified at the entropy level by the \GFF{} minimum.
\item PBHs are not needed as the main dark-matter component because the Kaxion already occupies that role and because the smooth scalar spectrum suppresses PBH formation exponentially.
\item Primordial tensors are the cleanest kill switch because the derived plateau implies \(r = 0.0043\) --- a background that the Simons Observatory and LiteBIRD must find, at that amplitude and no other.
\end{enumerate}

The whole block may be summarized in one table.

\begin{table}[H]
\centering
\caption[Integrated summary of Part VII]{Integrated summary of Part VII. The first column gives the open problem, the second the standard difficulty, the third the \GG{}/\GDOS{} interpretation, and the last the decisive empirical or conceptual test. The fourth column is the one that can fail. The first three set out the problem, the standard difficulty and the reading offered; the fourth names the test that would decide between them, and a row with a weak test in the fourth column is a weak row however strong the third looks.}
\label{tab:partVII-summary}
\small
\begin{tabular}{p{0.18\textwidth}p{0.24\textwidth}p{0.31\textwidth}p{0.17\textwidth}}
\toprule
\textbf{Problem} & \textbf{Standard difficulty} & \textbf{\GG{}/\GDOS{} interpretation} & \textbf{Decisive test} \\
\midrule
Black-hole information paradox & thermal Hawking radiation versus global unitarity; naive factorization across the horizon & bulk is closed and unitary, boundary is open; generalized entropy is the \GFF{} ledger; Page turnover is a \GSSB{} saddle switch & whether a gauge-consistent operational entropy accounting can preserve global unitarity without firewalls \\
Primordial black holes & need a large small-scale scalar enhancement to get appreciable abundance & minimal smooth spectrum gives exponentially tiny \(\beta(M)\); PBHs at most a trace subcomponent; Kaxion remains dominant dark matter & discovery of a dominant PBH dark-matter abundance would challenge the minimal scalar ledger \\
Primordial gravitational waves & amplitude tied directly to inflationary energy scale & derived GUT-adjacent plateau gives \(r = 0.0043\) and \(\Omega_{\mathrm{GW},0}h^2\sim10^{-16}\) & a tensor measurement inconsistent with \(r\simeq0.004\) in either direction is fatal\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Cross-consistency and falsifiability inside the extreme-gravity block}

Because Part VII contains some of the strongest claims in the draft, it is worth making the falsifiability logic explicit.

\paragraph{Black holes.}
The framework would be weakened if it could not produce a coherent operational entropy ledger that matches the modern Page-curve lesson. It would also be weakened if it required naive factorization after having built \DGI{} into the constitution. The black-hole section therefore rises or falls with the clarity of its operational formulation.

\paragraph{PBHs.}
The PBH section is falsifiable in a simpler way. The minimal module says PBHs are not the dominant dark matter and that any substantial abundance requires an extra small-scale feature not yet derived. Discovery of a dominant PBH dark-matter population would therefore signal that either the scalar sector contains a hidden feature the present theory missed or the minimal module is false.

\paragraph{Primordial tensors.}
The tensor kill switch is the most brutal of all. A robust primordial tensor signal at any experimentally plausible amplitude immediately falsifies the derived launch (a detection well above \(r=0.0043\), or a null pushing \(r\) below \(\sim10^{-3}\), both fatal).

This asymmetry between the three tests is scientifically healthy. Not all parts of a theory fail in the same way. Some fail conceptually, some parametrically, and some observationally. Part V contains all three kinds.

\subsection{Bridge to the chronology of the universe}

The correct bridge to the next major part of the paper is now clear. The preceding parts have constructed the beginning, the background expansion, the matter inventory, the growth history, and the extreme-gravity audits. What remains is to assemble them into a chronological timeline: the first microsecond, the first minutes, recombination, first light, and the route to the present epoch.

That assembly belongs to the next part. Part VII has done its job if it has shown that the extreme ends of the theory -- horizons, primordial collapse, and tensor relics -- do not require a separate constitution. They are governed by the same one.

\bigskip
\emph{The full worked-derivation notebook for the Part~VII observables --- the black-hole entropy ledger, the Page-curve branch switch, the primordial-black-hole tail estimate, and the tensor-null algebra --- is collected in Appendix~\ref{appA:bhentropy-notebook}, so the body stays readable, as for Parts~IV, V, and~VI.}

\medskip\noindent In plain words: the most extreme objects in the sky turned out to be the gentlest test of the constitution. A horizon is not a paradox factory; it is a port --- a place where an open system does what open systems always do here, exchange and record. The entropy ledger closes at exactly \(A/4\) because the same boundary bookkeeping that counts the cosmos counts the hole, and information is not lost because nothing in this framework was ever sealed. Even the quiet of the tensor sky is load-bearing: it sits exactly where the derived launch demands, and a detection well above or below it would be fatal.

The story so far has been told sector by sector --- launch, matter, growth, horizons. What it still lacks is a single timeline: one ruler on which the first microsecond, the first three minutes, the first atoms, and the present afternoon all sit as milestones of one descent. That ruler is Part~VIII.

\clearpage

\section{Part~VII in plain English: what is resolved here, and what is only reformulated}
This Part makes a smaller claim than its subject usually invites. It does not
offer a complete microscopic account of black-hole interior states, an exact
graybody-factor derivation, or a full ultraviolet completion of collapse, and
it says so in the criteria table before it says anything else. What it offers
is a reformulation: the Page curve read as an open-system consequence of the
same projection that produces quantum mechanics at the boundary, with the
island prescription of the modern literature set beside it as a translation
rather than a rival.

The honest position on the translation is that where the two accounts agree,
the agreement is the point, and where they differ the difference is stated and
not resolved. A reader who wants to know whether this framework has solved the
information paradox will find the answer in the criteria table: it satisfies
some of the criteria there and not others, and the ones it does not satisfy
are named.

The primordial material --- black holes and gravitational waves --- is
different in kind and is the more testable half of the Part. The mass windows
are illustrative rather than predicted; they say where a population would have
to sit to matter here. The gravitational-wave signature is the one place in
this Part where an instrument could return a number that the framework has
committed to in advance, and that is why it is carried at length despite being
the least developed section.

\GGPart[Cosmic chronology]{Evolutionary chronology of the universe}{The chronology dictionary translating between \(t\), \(\mu\), \(r\), \(a\), and \(z\); the master chronology table from the GUT boundary to the asymptotic de~Sitter future; and the requested milestone ladder in one glance}


\section{The chronology as the \GRIP{} descent}
\label{sec:partVIII-grip-opener}

Part~VIII reads the whole of cosmic history through one organizing coordinate and
one organizing idea, shared with Parts~VI and~VII: the logarithmic radial coordinate \(r = \ln(\EGUT/\mu)\) \emph{is} the \GRIPin{} descent coordinate of
\ASix{}, and the cosmic timeline is the \GRIP{} cascade read as a single finite
descent in \(r\). Each milestone of the history is a stage of that descent; the
qualitative switches that punctuate it are \GRIPout{} events; and the history
terminates on a Ground Geometric Attractor.

Three landmarks fix the reading. The graceful exit from slow-smooth inflation is
the first cosmological \GRIPout{} event --- the \GSSB{} transition that ends the
inflationary plateau and simultaneously reheats the boundary and seeds the dark
sector. Recombination is the second and cleanest \GRIPout{} event: the \GSSB{}
freezing of the baryon--photon plasma into the CMB geometric attractor, fixed by
the integer ledger at \(z_\ast = e^{7}\approx1097\) (Parts~VI--VII). And the
asymptotic de~Sitter future is the Ground Geometric Attractor of the gravitational
stratum, the basin into which the descent finally relaxes with \(w_\Lambda=-1\).
Between these switches the universe runs as \GRIPin{} relaxation --- the long,
quiet stretches of radiation, matter, and vacuum domination that occupy almost all
of the proper-time budget but only a modest interval of \(r\).

The practical payoff of this framing is the subject of the rest of the Part: read
against \(t\) the history is violently nonuniform, but read against \(r\) it is a
compact, evenly organized ladder --- exactly as a genuinely geometric \GRIP{}
descent should be. The remaining sections build the change-of-variables dictionary,
walk the milestones from the GUT edge to the present, and close with the master
chronology table and the radial bookkeeping that make the descent quantitative.

\section{Why a chronology block is required}
\label{sec:partVIII-why-history}


\begin{equation}
r = \ln\!\left(\frac{\EGUT}{\mu}\right),
\label{eq:partVIII-r-def}
\end{equation}
where \(\mu\) is the characteristic boundary energy scale of the process under discussion.

Figure~\ref{fig:cosmic-timeline} lays the whole of this section on one landscape strip --- every milestone read simultaneously in \(t\), \(T\), \(z\), and \(r\). This is more than a change of notation. The ordinary clock \(t\) is excellent for discussing durations, causal horizons, and observational dating. The variables \(a\) and \(z\) are excellent for the background expansion seen by the boundary observer. But the \GG{} program claims that the most economical organizing coordinate of the whole physical descent is the logarithmic coordinate \(r\). Chronology therefore becomes a dictionary problem: every cosmic event should have a lawful image in \(t\), \(a\), \(z\), \(\mu\), and \(r\).

Part~VIII is written with three purposes in mind.

First, it gives a continuous narrative from the ultraviolet boundary at \(\EGUT\) to the present \(\LCDM\)-like attractor and beyond, instead of leaving the reader to assemble the story manually from the previous Parts.

Second, it sharpens the theory's falsifiability. If the chronology demands incompatible placements for inflation, baryogenesis, BBN, recombination, or the first galaxies, then the framework fails not in a specialized subproblem but in its most elementary promise: to tell one coherent history.

Third, it clarifies an important conceptual feature of the program: the universe looks very different depending on whether it is plotted against \(t\) or against \(r\). In ordinary proper time, the early universe flashes by almost instantaneously and the late universe takes forever. In the logarithmic \GG{} coordinate the story is far more evenly distributed. The first microsecond, the first three minutes, recombination, first-star formation, and the present all sit on one relatively compact radial ladder. That is exactly the kind of compression one should expect if RG descent is genuinely geometric.

The rest of the Part is organized in the same spirit as the outline just given, but with one editorial refinement. Before walking through the first microsecond, the first three minutes, the first \(3\times 10^5\) years, and the first \(10^8\) years, we first write the change-of-variables dictionary that lets these landmarks be translated into the \GG{} coordinate system. Only then do we state the history itself. The Part ends with a long master summary table and several worked conversions, so that a reader can immediately locate any epoch on the same common map.

% =====================================================================
%  Arisaka-GG-A10-Cosmos  --  Cosmic-history timeline figure
%  Self-contained TikZ fragment for the Cosmic-chronology Part.
%  House style mirrors fig:parts-roadmap-a10 / fig:cohort-map-a10:
%  rounded sans-serif nodes, Stealth arrows, full-page minipage wrapper.
%  Macros used (\EPl \EGUT \Mlock \Omm \OmL \GGTheory \LCDM) are inherited
%  from ucla-arisaka-gg-11.sty -- NOT redefined here.
%  Required libraries: arrows.meta, positioning, fit, calc, backgrounds,
%  shapes.geometric.
%  \label{fig:cosmic-timeline}
% =====================================================================
\begin{landscape}
\begin{figure}[p]
\centering
\resizebox{0.804\linewidth}{!}{%
\begin{tikzpicture}[
  x=1mm, y=1mm,
  font=\sffamily\normalsize,
  every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
  hotms/.style   ={fill=red!12,    draw=red!60!black,    line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  warmms/.style  ={fill=orange!16, draw=orange!70!black, line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  bbnms/.style   ={fill=yellow!18, draw=olive!60!black,  line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  coolms/.style  ={fill=green!13,  draw=green!60!black,  line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  coldms/.style  ={fill=cyan!14,   draw=cyan!60!black,   line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  iceyms/.style  ={fill=blue!10,   draw=blue!60!black,   line width=0.60pt,
                   minimum width=26mm, minimum height=15mm},
  headline/.style={fill=orange!16, draw=orange!75!black, line width=1.10pt,
                   minimum width=29mm, minimum height=16mm},
  chain/.style   ={fill=red!9, draw=red!55!black, line width=0.70pt,
                   minimum width=32mm, minimum height=13mm},
  lead/.style    ={draw=black!55, line width=0.50pt},
  axisarrow/.style={-{Stealth[length=3.0mm]}, line width=0.90pt, draw=black!75},
  chainarrow/.style={-{Stealth[length=3.0mm]}, line width=1.00pt, draw=red!70!black},
  tick/.style    ={draw=black!55, line width=0.50pt},
  axislab/.style ={font=\sffamily\large\bfseries, text=black!75},
  valdn/.style   ={font=\sffamily\small, text=black!70, anchor=north},
  rval/.style    ={font=\sffamily\small\bfseries, text=black!85, anchor=north},
  gap/.style     ={font=\sffamily\small\itshape, text=red!55!black},
]

\def\xGUT{0}     \def\xINF{26}   \def\xKAX{52}   \def\xNP{86}
\def\xBBN{112}   \def\xEQ{140}   \def\xREC{166}  \def\xDARK{196}
\def\xGAL{222}   \def\xLSS{248}  \def\xDE{272}   \def\xNOW{298}
\def\xFUT{322}
\def\yaxis{0}
\def\yup{20}
\def\ydn{-20}
\def\yr{9}        % r-axis row, just above the t axis
\def\yT{-30}
\def\yz{-40}
\def\ychain{56}

% ===================  r-AXIS (RG / radial coordinate)  ==============
\draw[axisarrow] (-8,\yr) -- (\xFUT+12,\yr);
\node[axislab, anchor=east] at (-9,\yr) {\(r=\ln(\EGUT/\mu)\)};
\newcommand{\rtick}[2]{%
  \draw[tick] (#1,\yr-1.2) -- (#1,\yr+1.2);
  \node[rval] at (#1,\yr-1.4) {#2};
}
\rtick{\xGUT}{$0.0$}
\rtick{\xINF}{$1\!\to\!12.9$}
\rtick{\xKAX}{$29.4$}
\rtick{\xNP}{$44.8$}
\rtick{\xBBN}{$47.2$}
\rtick{\xEQ}{$58.6$}
\rtick{\xREC}{$59.7$}
\rtick{\xDARK}{$63.7$}
\rtick{\xGAL}{$64.3$}
\rtick{\xLSS}{$65.8$}
\rtick{\xDE}{$66.5$}
\rtick{\xNOW}{$66.7$}
\rtick{\xFUT}{$r\!\rightarrow\!\infty$}

% ===================  MAIN TIME AXIS  ================================
\draw[axisarrow] (-8,\yaxis) -- (\xFUT+12,\yaxis);
\node[axislab, anchor=east] at (-9,\yaxis) {time $t$\,$\rightarrow$};
\node[font=\sffamily\large\bfseries, text=black!60, anchor=west]
      at (-8,\yup+16) {hot $\bullet$ early};
\node[font=\sffamily\large\bfseries, text=black!60, anchor=east]
      at (\xFUT+12,\yup+16) {cold $\bullet$ late};

\node[axislab, anchor=east] at (-9,\yT) {temperature $T$};
\node[axislab, anchor=east] at (-9,\yz) {redshift $z$};

\newcommand{\tickabc}[4]{%
  \draw[tick] (#1,\yaxis-1.5) -- (#1,\yaxis+1.5);
  \node[valdn] at (#1,\yT)  {#3};
  \node[valdn] at (#1,\yz)  {#4};
  \node[valdn, text=black!80] at (#1,\yaxis-2) {#2};
}

\tickabc{\xGUT}{$10^{-39}$\,s}{$\sim\!2.3{\times}10^{16}$\,GeV}{$\sim\!10^{29}$}
\tickabc{\xINF}{$10^{-38}$--$10^{-28}$\,s}{$8.3\!\times\!10^{15}\!\to\!3\!\times\!10^{10}$\,GeV}{---}
\tickabc{\xKAX}{$10^{-13}$\,s}{$\Mlock=3.85$\,TeV}{$\sim\!10^{16}$}
\tickabc{\xNP}{$1$\,s}{$0.8$\,MeV}{$3.4{\times}10^{9}$}
\tickabc{\xBBN}{$100$--$180$\,s}{$70$--$80$\,keV}{$\sim\!3{\times}10^{8}$}
\tickabc{\xEQ}{$5$--$8{\times}10^{4}$\,yr}{$0.8$\,eV}{$z_{\rm eq}\!\approx\!3400$}
\tickabc{\xREC}{$3.8{\times}10^{5}$\,yr}{$0.30$\,eV}{$z_{\rm sw}\!=\!e^{7}\!\approx\!1097$}
\tickabc{\xDARK}{$1$--$100$\,Myr}{$\sim\!10$\,meV}{$\sim\!20$--$30$}
\tickabc{\xGAL}{$\sim\!10^{8}$\,yr}{$\sim\!5$\,meV}{$\sim\!15$--$20$}
\tickabc{\xLSS}{$1$--$5$\,Gyr}{$\sim\!1$\,meV}{$6$--$1.5$}
\tickabc{\xDE}{$\sim\!10$--$11$\,Gyr}{$0.30$\,meV}{$0.29$}
\tickabc{\xNOW}{$13.9$\,Gyr}{$2.725$\,K}{$0$}
\tickabc{\xFUT}{$\gg\!10^{10}$\,yr}{$T\!\rightarrow\!0$}{$z\!\rightarrow\!-1$}

% ===================  MILESTONE NODES  ==============================
\node[hotms]   (mGUT) at (\xGUT,\yup)  {\textbf{GUT edge}\\[-1pt]\small lawful beginning\\[-2pt]\small $\EGUT\!\sim\!10^{16}$\,GeV};
\node[hotms]   (mINF) at (\xINF,\ydn)  {\textbf{Primary inflation}\\[-1pt]\small + graceful exit, reheat\\[-2pt]\small $V^{1/4}\!=\!8.3\!\times\!10^{15}$\,GeV};
\node[warmms]  (mKAX) at (\xKAX,\yup)  {\textbf{Lock crossing}\\[-1pt]\small Kaxion birth ($N_2\!=\!0$)\\[-2pt]\small $\Mlock\!=\!3.85$\,TeV, $t\!\sim\!10^{-13}$\,s};
\node[warmms]  (mNP)  at (\xNP,\ydn)   {\textbf{$n$--$p$ freeze-out}\\[-1pt]\small sets He yield\\[-2pt]\small $T\!=\!0.8$\,MeV};
\node[bbnms]   (mBBN) at (\xBBN,\yup)  {\textbf{BBN main burn}\\[-1pt]\small deuterium bottleneck\\[-2pt]\small $t\!\sim\!100$--$180$\,s};
\node[coolms]  (mEQ)  at (\xEQ,\ydn)   {\textbf{Matter--radiation}\\[-1pt]\textbf{equality}\\[-2pt]\small $z_{\rm eq}\!\approx\!3400$};
\node[headline](mREC) at (\xREC,\yup)  {\textbf{Recombination}\\[-1pt]\small last scattering (CMB) $z_{\rm dec}\!\approx\!1090$\\[-2pt]\small switch $\boxed{z_{\rm sw}=e^{7}\approx1097}$};
\node[coldms]  (mDARK)at (\xDARK,\ydn) {\textbf{Dark ages}\\[-1pt]\small first structure grows\\[-2pt]\small $1$--$100$\,Myr};
\node[coldms]  (mGAL) at (\xGAL,\yup)  {\textbf{First galaxies}\\[-1pt]\small JWST era\\[-2pt]\small $\sim\!10^{8}$\,yr};
\node[coldms]  (mLSS) at (\xLSS,\ydn)  {\textbf{Mature galaxies}\\[-1pt]\small \& clusters\\[-2pt]\small $1$--$5$\,Gyr};
\node[iceyms]  (mDE)  at (\xDE,\yup)   {\textbf{Dark-energy onset}\\[-1pt]\small accelerated expansion\\[-2pt]\small $\OmL\!=\!1-1/\pi$};
\node[iceyms]  (mNOW) at (\xNOW,\ydn)  {\textbf{Today}\\[-1pt]\small $T\!=\!2.725$\,K\\[-2pt]\small $t\!=\!13.9$\,Gyr,\ $z\!=\!0$};
\node[iceyms]  (mFUT) at (\xFUT,\yup)  {\textbf{de Sitter future}\\[-1pt]\small asymptotic attractor\\[-2pt]\small $w_\Lambda\!=\!-1$};

\foreach \n/\x/\side in
  {mGUT/\xGUT/u, mINF/\xINF/d, mKAX/\xKAX/u, mNP/\xNP/d,
   mBBN/\xBBN/u, mEQ/\xEQ/d,  mREC/\xREC/u, mDARK/\xDARK/d,
   mGAL/\xGAL/u, mLSS/\xLSS/d, mDE/\xDE/u,  mNOW/\xNOW/d, mFUT/\xFUT/u}{%
   \draw[lead] (\x,\yaxis) -- (\n.\ifx\side u south\else north\fi);
}

% ===================  SCALE-CHAIN TOP BAND  =========================
\node[chain] (cPl)   at (40,\ychain)  {\textbf{$\EPl$}\\[-1pt]\small Planck energy\\[-2pt]\small $\sim\!10^{19}$\,GeV};
\node[chain] (cGUT)  at (150,\ychain) {\textbf{$\EGUT$}\\[-1pt]\small GUT edge\\[-2pt]\small $\sim\!10^{16}$\,GeV};
\node[chain] (cLock) at (260,\ychain) {\textbf{$\Mlock$}\\[-1pt]\small KK lock\\[-2pt]\small $\sim\!3.85$\,TeV};
\draw[chainarrow] (cPl.east)  -- node[gap, above=0.5mm]{$\div\,e^{35}$ (lock factor)} (cGUT.west);
\draw[chainarrow] (cGUT.east) -- node[gap, above=0.5mm]{throat descent} (cLock.west);
\node[font=\sffamily\large\bfseries, text=red!55!black, anchor=west]
      at (-8,\ychain) {inherited\\ scale chain};
% local tag linking the inherited lock scale to the inflation milestone
%   (kept short so it does not sweep across the whole figure)
\node[gap, text=red!60!black, anchor=south west, font=\sffamily\small\itshape]
      at ($(mKAX.north west)+(1,1)$) {same $\Mlock$ as chain};
\node[gap, anchor=south, text width=38mm] at (\xREC,\yup+9)
      {integer-ledger switch marker};
\draw[lead, dashed, draw=orange!75!black] (\xREC,\yup+8) -- (mREC.north);

\begin{scope}[on background layer]
  \fill[black!3, rounded corners=4pt] (-12,\yz-6) rectangle (\xFUT+16,\yup+11);
  \fill[red!4,  rounded corners=4pt] (-12,\ychain-8) rectangle (\xFUT+16,\ychain+9);
\end{scope}

\end{tikzpicture}}\par\bigskip
\caption[Cosmic-history timeline of \ATen{}: aligned $t$, $T$, $z$ axes.]{Cosmic-history timeline of \ATen{}. The horizontal axis runs from the hot, early universe on the left to the cold, late universe on the right; every milestone is read off simultaneously in cosmic time $t$, photon temperature $T$, and redshift $z$ along the three aligned axis rows. The milestones and their times/temperatures are the \emph{standard} cosmic-history values (identical to the master chronology, Table~\ref{tab:partVIII-master-chronology}): the GUT-edge lawful beginning, the primary inflation one e-fold below it with its graceful exit and Planck-strength reheating, the prompt lock crossing at $\Mlock\!\sim\!3.85$\,TeV with the Kaxion's birth, neutron--proton freeze-out at $T=0.8$\,MeV, the BBN deuterium bottleneck, matter--radiation equality at $z_{\rm eq}\!\approx\!3400$, recombination, the dark ages, the first (JWST-era) galaxies, mature galaxies and clusters, dark-energy onset with $\OmL=1-1/\pi$, the $\Omega_m = 1/\pi$ crossing at $t=13.8$\,Gyr, today ($t=13.9$\,Gyr, $T=2.725$\,K, $z=0$), and the asymptotic de Sitter future. The boxed corridor-switch redshift $z_{\rm sw}=e^{7}\approx1097$ is the headline \emph{integer-ledger marker} discussed in the body (not a fit): the recombination-era \GSSB{} switch descends from the \OGNfive{} integer $J=7$, while the atomic decoupling derived from the framework's own constants lies $0.4\%$ below it, on the measured value (\S\ref{sec:two-redshifts}). The top band shows the inherited ultraviolet scale chain $\EPl\!\rightarrow\!\EGUT\!\rightarrow\!\Mlock$ with its integer gaps, the dominant one being the $e^{35}$ lock factor $\Mlock=\EPl/(2e^{35})$ that fixes the middle crossing and, through the seesaw, the terminus; the dashed link ties that lock scale to the lock-crossing milestone on the timeline below. A fourth aligned row gives the radial / renormalization-group coordinate $r=\ln(\EGUT/\mu)$ at each milestone, running from $r=0$ at the GUT edge to $r\!\rightarrow\!\infty$ in the de Sitter future.}
\label{fig:cosmic-timeline}
\end{figure}
\end{landscape}


\section{The chronology dictionary: translating between \texorpdfstring{\(t\)}{t}, \texorpdfstring{\(\mu\)}{mu}, \texorpdfstring{\(r\)}{r}, \texorpdfstring{\(a\)}{a}, and \texorpdfstring{\(z\)}{z}}
\label{sec:partVIII-dictionary}


\subsection{Five variables, one caveat}

The paper will use five chronological variables repeatedly:
\begin{equation}
\{t,\mu,r,a,z\}.
\end{equation}
Their meanings are not identical.

\begin{itemize}[leftmargin=2.2em]
\item \(t\) is ordinary cosmic proper time measured by comoving observers.
\item \(\mu\) is the characteristic boundary energy scale of the process being tracked.
\item \(r=\ln(\EGUT/\mu)\) is the logarithmic radial coordinate inherited from \GG{}.
\item \(a(t)\) is the scale factor, normalized to \(a_0=1\) today.
\item \(z\) is the corresponding redshift, \(1+z=a^{-1}\).
\end{itemize}

The only important caveat is the status of \(\mu\). In a tightly coupled relativistic plasma, \(\mu\) is naturally identified with the temperature \(T\). During reheating that same role is played by the reheating temperature. During the inflationary plateau, where a thermal plasma is not the correct description, \(\mu\) should instead be read as the characteristic vacuum scale \(V^{1/4}\). During the late matter and dark-energy eras, one may again use the photon temperature \(T_\gamma\) as a convenient operational clock even though it is no longer the dominant energy density. Thus \(\mu\) is the characteristic \emph{process} scale, not necessarily the energy density of the dominant cosmic component.

With that caveat stated, the entire chronology can still be presented on one shared map.

\subsection{Exact algebraic relations}

The first two relations are exact:
\begin{align}
1+z &= a^{-1},
\label{eq:partVIII-z-a}\\[3pt]
r &= \ln\!\left(\frac{\EGUT}{\mu}\right).
\label{eq:partVIII-r-mu}
\end{align}
The second equation implies the useful differential identity
\begin{equation}
\dd r = - \dd\ln\mu.
\label{eq:partVIII-dr-dmu}
\end{equation}
Therefore equal steps in \(r\) correspond to equal multiplicative drops in the characteristic energy scale. That is why the \(r\)-coordinate is much better than \(t\) for displaying the hierarchy structure of the cosmic history.

The next relation is inherited directly from Part~III. During adiabatic expansion of a coupled relativistic plasma,
\begin{equation}
T(a) \propto g_{*s}^{-1/3}(T)\,a^{-1},
\label{eq:partVIII-T-a}
\end{equation}
so that, after normalizing to the present photon temperature \(T_{\gamma,0}\),
\begin{equation}
T_\gamma(z)
=
T_{\gamma,0}(1+z)\left(\frac{g_{*s,0}}{g_{*s}(T)}\right)^{1/3}.
\label{eq:partVIII-T-z}
\end{equation}
Substituting Eq.~\eqref{eq:partVIII-T-z} into Eq.~\eqref{eq:partVIII-r-mu} gives the practical redshift--radial dictionary for epochs that are well described by a photon temperature:
\begin{equation}
r(z)
=
\ln\!\left(\frac{\EGUT}{T_{\gamma,0}}\right)
-
\ln(1+z)
+
\frac{1}{3}\ln\!\left(\frac{g_{*s}(T)}{g_{*s,0}}\right).
\label{eq:partVIII-r-z}
\end{equation}
If the change in \(g_{*s}\) can be ignored across the interval of interest, this reduces to the extremely simple law
\begin{equation}
r(z)\simeq r_0-\ln(1+z),
\qquad
r_0:=\ln\!\left(\frac{\EGUT}{T_{\gamma,0}}\right).
\label{eq:partVIII-r-z-simple}
\end{equation}

Using the inherited numerical value \(\EGUT\simeq 2.28\times10^{16}\,\mathrm{GeV}\) together with \(T_{\gamma,0}\simeq2.35\times10^{-4}\,\mathrm{eV}\), one finds
\begin{equation}
r_0 \simeq 66.7.
\label{eq:partVIII-r0-num}
\end{equation}
This number is worth pausing over. It means that the entire observable history of the boundary universe, from the GUT edge to the current photon bath, spans only about \(67\) natural logarithmic units in energy. That is a huge ratio in ordinary linear units, but it is a compact interval in the native \GG{} coordinate.

\subsection{Radiation-era time-temperature law}

For early epochs the most useful relation is again inherited from Part~III:
\begin{equation}
t \simeq 0.301\,g_*^{-1/2}\frac{M_{\mathrm{Pl}}}{T^2}.
\label{eq:partVIII-time-temperature}
\end{equation}
In practical units this is often written as
\begin{equation}
t \simeq 2.42\,g_*^{-1/2}
\left(\frac{T}{\mathrm{MeV}}\right)^{-2}\,\mathrm{s}.
\label{eq:partVIII-time-temperature-practical}
\end{equation}
Solving for \(T\) gives
\begin{equation}
T \simeq
\sqrt{0.301}\,g_*^{-1/4}\sqrt{\frac{M_{\mathrm{Pl}}}{t}}.
\label{eq:partVIII-temperature-from-time}
\end{equation}
Combining this with Eq.~\eqref{eq:partVIII-r-mu} yields the radiation-era radial clock
\begin{equation}
r_{\mathrm{rad}}(t)
=
\ln\!\left[
\frac{\EGUT}
{\sqrt{0.301}\,g_*^{-1/4}\sqrt{M_{\mathrm{Pl}}/t}}
\right].
\label{eq:partVIII-radial-clock}
\end{equation}
The virtue of Eq.~\eqref{eq:partVIII-radial-clock} is conceptual as much as numerical. It shows that in radiation domination the logarithmic coordinate grows only as \(\half\ln t\). A gigantic multiplicative stretch in ordinary time corresponds to only a modest additive displacement in \(r\).

That is why events that look unimaginably far apart in seconds or years can be only a few radial units apart on the \GG{} map.

\subsection{Matter-era and vacuum-era clocks}

Once the universe is matter dominated, the simple radiation-era temperature law is no longer the best way to read the background. In the ordinary matter era,
\begin{equation}
H(z)\simeq H_0\sqrt{\Omm}\,(1+z)^{3/2},
\label{eq:partVIII-H-matter}
\end{equation}
so integrating \(\dd t = -\dd z/[(1+z)H(z)]\) gives the standard approximation
\begin{equation}
t(z)\simeq \frac{2}{3H_0\sqrt{\Omm}}\,(1+z)^{-3/2},
\qquad
z_{\mathrm{eq}}\gg z\gg \mathcal{O}(1).
\label{eq:partVIII-t-z-matter}
\end{equation}
This relation is not exact all the way to the present because dark energy later becomes important, but it is perfectly adequate for the interval from recombination through the first stars and much of galaxy assembly.

At late times, once the residual vacuum term dominates, the background approaches
\begin{equation}
a(t)\propto e^{H_\Lambda t},
\qquad
H_\Lambda \simeq \mathrm{const}.
\label{eq:partVIII-late-dS}
\end{equation}
In this era the photon bath continues to cool as \(T_\gamma\propto a^{-1}\), so the radial coordinate still increases according to Eq.~\eqref{eq:partVIII-r-z-simple}. The dominant vacuum component does not cool, but the operational thermometer used by observers does.

\subsection{A compact dictionary table}

Before moving to the actual history, it is helpful to summarize the change-of-variables map in one place.

\begin{table}[H]
\centering
\caption[Coordinate dictionary used throughout Part VIII]{Coordinate dictionary used throughout Part~VIII. The point is not to replace standard cosmological variables, but to display how the \GG{} logarithmic coordinate \(r\) can be read alongside the ordinary chronology. The columns give the variable, its definition, and its meaning here. The dictionary does not replace the standard cosmological variables; it sets the radial coordinate beside them so that a reader can move between the two descriptions without translating in their head.}
\label{tab:partVIII-dictionary}
\small
\begin{tabular}{p{0.20\textwidth}p{0.30\textwidth}p{0.38\textwidth}}
\toprule
\textbf{Variable} & \textbf{Definition} & \textbf{Meaning in Part~VIII}\\
\midrule
\(t\) & cosmic proper time & The ordinary chronological variable used by astrophysics and thermal history.\\
\(\mu\) & characteristic process scale & Usually the plasma temperature, but during inflation the characteristic vacuum scale, and during the late universe often the photon temperature used as an operational thermometer.\\
\(r\) & \(r=\ln(\EGUT/\mu)\) & The logarithmic \GG{} radial coordinate; equal steps correspond to equal multiplicative drops in energy.\\
\(a\) & scale factor with \(a_0=1\) & Global expansion variable.\\
\(z\) & \(1+z=a^{-1}\) & Observationally friendly version of the scale factor.\\
Radiation law & \(t \simeq 0.301\,g_*^{-1/2}M_{\mathrm{Pl}}/T^2\) & Main early-universe conversion between \(t\) and \(\mu\sim T\).\\
Redshift law & \(r(z)\simeq r_0-\ln(1+z)\) for slowly varying \(g_{*s}\) & Practical late-time conversion between redshift and the radial coordinate.\\
\bottomrule
\end{tabular}
\end{table}

The rest of the chronology now follows directly from walking down this dictionary.

\section{The first microsecond: from the GUT boundary to the electroweak seam}
\label{sec:partVIII-first-microsecond}


\subsection{Why the first microsecond already includes several logically distinct stages}

In a standard textbook chronology the phrase ``the first microsecond'' usually refers to the time window during which the thermal bath cools from the electroweak or multi-GeV range toward the QCD crossover. In the present framework that language is still useful, but it must be interpreted more carefully because the true beginning of the boundary cosmology has already been placed at \(\EGUT\), while the inflationary plateau sits down at the lock scale \(\Mlock\sim 3.85\;\mathrm{TeV}\).

This means that the first microsecond of the universe is not one homogeneous physical stage. It contains at least four logically distinct chapters:

\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item the ultraviolet initialization of the boundary theory at \(\mu=\EGUT\), \(r=0\);
\item the primary slow-smooth plateau one e-fold below the GUT edge, and its graceful exit;
\item Planck-strength reheating to \(3\times10^{10}\)~GeV and the radiation descent to the promptly crossed lock;
\item leptogenesis in the newly opened thermal window, sphaleron conversion, and the electroweak-to-GeV thermal descent.
\end{enumerate}

In linear time these stages are compressed into a fantastically small interval. In \(r\)-space, however, the structure is easier to read. The ultraviolet start sits at \(r=0\). The lock plateau sits at
\begin{equation}
r_{\mathrm{lock}}
=
\ln\!\left(\frac{\EGUT}{\Mlock}\right)
\simeq 29.4.
\label{eq:partVIII-r-lock}
\end{equation}
The one-microsecond milestone, by contrast, sits near the GeV scale and therefore near
\begin{equation}
r_{\mu\mathrm{s}}\sim \ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{1\,\mathrm{GeV}}\right)\simeq 37.7.
\label{eq:partVIII-r-microsecond}
\end{equation}
Thus everything from the end of the inflationary stage to the one-microsecond mark occupies only about eight radial units:
\begin{equation}
\Delta r_{\mathrm{lock}\to\mu s}\simeq 8.3.
\label{eq:partVIII-dr-lock-micro}
\end{equation}
This is the first place where the \GG{} map becomes genuinely useful: it makes the early chronology visually compact.

\subsection{Ultraviolet initialization and the descent to the plateau}

The first statement of the chronology is not a guess about a pre-geometric singularity. It is the already-inherited Part~III statement that the boundary description starts at the GUT surface, not at \(M_{\mathrm{Pl}}\). The chronology therefore begins on a finite ultraviolet seam carrying finite initial data. The role of this starting point is constitutional: it tells us where the effective 4D cosmological language is first allowed to be used.

From there the system descends in \(r\). At this stage it is misleading to picture the universe as an ordinary thermal plasma with a single temperature and an observationally meaningful redshift. The better picture is that the six-dimensional throat is selecting the low-energy seam on which the later 4D history will be exported. In other words, the relevant evolution is geometric before it is thermal.

This clarifies where the inflationary stage sits in the descent. The boundary description turns on at \(\EGUT\), and the primary inflationary plateau selected by the \GDOS{} boundary law lies just below it --- one e-fold under the GUT edge, at \(V^{1/4} = 8.3\times10^{15}\)~GeV. Inflation is therefore a high-scale, GUT-adjacent event, not a deep-infrared one; the lock scale \(\Mlock\) enters the later history (the electroweak seam and the dark sector), not the inflationary plateau.

\subsection{Slow-smooth inflation on the GUT-adjacent plateau}

The inflationary stage occupies the \GDOS{}-forced Starobinsky/\(\xi\) plateau one e-fold below the GUT edge, at \(V^{1/4} = 8.3\times10^{15}\)~GeV. Evaluated at the reheat-matched e-fold count \(N^{*}=52.6\), the plateau fixes the tilt \(n_s = 1-2/N^{*} = 0.9620\) and the tensor ratio \(r = 12/N^{*2} = 0.0043\) --- small but nonzero, a factor of eight below the present bound and a two-sided target (Part~IV). The Hubble scale of inflation is not identical to the ultraviolet boundary scale \(\EGUT\), but it sits just below it, so the inflationary event is high-scale.

The intuitive chronology is therefore the following. The boundary description turns on at the GUT surface, and the modulus is captured on the plateau immediately below it. There it executes a quasi-de Sitter slow descent, stretching the comoving universe, erasing curvature and horizon problems, and generating the scalar perturbations later fossilized in the CMB. The duration in ordinary time can be tiny while the effect on \(a(t)\) is enormous, because the defining quantity is the number of e-folds, not the absolute size of \(t\).

An earlier reading of this paper placed the inflationary plateau at the lock scale \(\Mlock\), predicting a nearly vanishing tensor ratio; that reading is retired (it gave a blue tilt and is refuted in Part~IV), and is replaced by the derived GUT-adjacent plateau above, whose tensor prediction \(r=0.0043\) is a sharp, falsifiable target rather than an unobservable null.

\subsection{Graceful exit, reheating, and simultaneous multi-sector birth}

Exit occurs through the same ToC-7 \GSSB{} mechanism already emphasized in Part~IV. The important chronological point is that several sectors are born at once.

First, the stored vacuum energy is transferred to a hot thermal bath with the derived reheating temperature \(T_{\mathrm{rh}}\simeq3\times10^{10}\)~GeV (fourteen e-folds of modulus decay; Appendix~G), the lock scale \(\Mlock\) then crossed promptly as the bath cools. Second, the compact-angle sector is misaligned, giving the Kaxion dark matter its initial condition. Third, the residual vacuum remnant is left behind as the seed of the later dark-energy ledger. In ordinary model building one often treats these as unrelated sectors: one field for inflation, another for dark matter, another constant for dark energy. In the present chronology they are three descendants of one geometric transition.

Once reheating has completed, the language of temperature, reaction rates, and equilibrium becomes operationally appropriate. The universe is now a radiation bath with
\begin{equation}
T_{\mathrm{rh}}\simeq 3\times10^{10}\;\mathrm{GeV},
\qquad
r_{\mathrm{rh}}\simeq 12.9,
\qquad
1+z_{\mathrm{rh}}\sim \frac{T_{\mathrm{rh}}}{T_{\gamma,0}}
\sim 10^{23}.
\label{eq:partVIII-reheating-coordinate}
\end{equation}
This is the cleanest chronological landing point for the first genuine thermal chapter of the universe.

\subsection{TeV leptogenesis and the electroweak corridor}

The next stage is the most important matter-origin step of the entire early chronology. The same reheated bath that emerged from ToC-7 contains the heavy quasi-degenerate Majorana states responsible for the inherited resonant leptogenesis corridor. Because the lock scale is already fixed, the baryogenesis story does not ask us to insert a separate ultra-heavy seesaw sector. It plays out where the thermal bath already is.

Chronologically the logic is straightforward. Heavy neutrinos with masses of order \(\Mlock\) fall out of equilibrium and decay while electroweak sphalerons are still active. The holonomy-fixed CP asymmetry and the washout factor then feed the final baryon asymmetry. The result is that by the time the universe cools through the electroweak seam, the sign and magnitude of the future baryonic inventory are already effectively fixed.

This matters for the later history because every later stage---BBN, recombination, first stars, galaxies, and even the late-time \(\Omega_b\) share---assumes the existence of an already-fixed small baryon asymmetry. Part~VIII makes the dependence explicit: ordinary baryons are not an arbitrary late-time abundance parameter. They are the surviving residue of a process that happened near the very beginning of the thermal descent.

The electroweak seam itself sits around \(\mu\sim 100\;\mathrm{GeV}\), at a cosmic age of order \(10^{-11}\) seconds and at
\begin{equation}
r_{\mathrm{EW}}
=
\ln\!\left(\frac{\EGUT}{100\,\mathrm{GeV}}\right)
\simeq 33.1.
\label{eq:partVIII-r-ew}
\end{equation}
By the one-microsecond mark the universe has cooled to the GeV corridor, so the whole electroweak and leptogenesis story is already over. This is why the first microsecond is a natural milestone. It marks the end of the matter-origin prelude and the handoff to the hadronic and nuclear chapters.

\section{The first three minutes: nucleosynthesis and the fixation of ordinary matter}
\label{sec:partVIII-first-three-minutes}


\subsection{From the GeV corridor to the MeV corridor}

Once the electroweak chapter is closed, the universe continues as an ordinary radiation bath. In linear time this stage is still extremely brief, but in physics content it is rich. The thermal bath must pass through the QCD crossover, neutrino decoupling, neutron-proton freeze-out, electron-positron annihilation, and the deuterium bottleneck before the light-element abundances are fixed.

The one-microsecond milestone sits near \(\mu\sim1\;\mathrm{GeV}\) and \(r\sim37.7\). The QCD crossover follows soon after, around \(150\) to \(200\;\mathrm{MeV}\), which corresponds to
\begin{equation}
r_{\mathrm{QCD}}
\sim
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{0.15\,\mathrm{GeV}}\right)
\simeq 39.6.
\label{eq:partVIII-r-qcd}
\end{equation}
Already one sees the convenience of the radial map. From the GeV corridor to hadronization is less than two radial units. Yet in ordinary language that step changes the natural degrees of freedom from quarks and gluons to protons, neutrons, mesons, and leptons.

\subsection{Hadronization and the operational meaning of \texorpdfstring{\(U(1)_B\)}{U(1)B}}

Part~VIII is a good place to make a conceptual point that is easy to miss in purely particle-theory presentations. The exact \(U(1)_B\) symmetry of the bulk becomes most operationally meaningful once hadrons exist. Before the QCD transition, baryon number is carried by a quark plasma. After hadronization, it is carried by identifiable long-lived baryonic objects. This is the stage at which the protected nature of the bulk baryon charge begins to look like the ordinary stability of protons and neutrons in an expanding thermal background.

The chronology therefore has a hierarchy of carriers:
\begin{center}
holonomy and anomaly data \(\rightarrow\) lepton asymmetry \(\rightarrow\) baryon asymmetry \(\rightarrow\) hadronic baryon carriers.
\end{center}
That sequence is conceptually important because it prevents a common confusion. The exactness of \(U(1)_B\) does not mean that the low-energy baryonic content was always transparently visible. It means that once the bath cools enough for hadronic carriers to be the right variables, they inherit a charge that was protected all along.

\subsection{Neutrino decoupling, freeze-out, and the first-second milestone}

Near \(T\sim 1\;\mathrm{MeV}\) the weak interaction rates finally lose the race against the Hubble expansion. This single statement controls several later facts at once. Neutrinos decouple from the electromagnetic plasma. The neutron-to-proton ratio ceases to track its equilibrium value exactly. And the future helium abundance becomes calculable from the frozen \(n/p\) ratio plus neutron decay during the remaining delay before efficient nuclear burning begins.

In the coordinate dictionary this is the moment
\begin{equation}
\mu \sim 1\,\mathrm{MeV},
\qquad
t\sim 1\,\mathrm{s},
\qquad
r\sim 44.6,
\qquad
1+z\sim 4\times10^9.
\label{eq:partVIII-one-second}
\end{equation}
Several features of the history are worth emphasizing.

First, the universe is still overwhelmingly radiation dominated.

Second, the baryon asymmetry generated near the TeV corridor is now simply a background parameter for a nuclear-reaction network. In other words, the chronology has changed language: the matter-origin problem is over, and the nuclear-abundance problem has begun.

Third, the difference in \(r\) between the electroweak seam and the one-second milestone is only about \(11.5\). This is precisely why the early universe is so efficiently summarized by a logarithmic coordinate: an enormous range of proper time and reaction content is compressed into a modest interval of \(r\).

\subsection{The deuterium bottleneck and the ignition of BBN}

A universe can contain plenty of protons and neutrons without immediately synthesizing helium. The reason is the deuterium bottleneck: at sufficiently high temperature any newly formed deuteron is quickly photodissociated by the energetic photon tail. Only once the temperature has dropped to the \(\sim 0.1\;\mathrm{MeV}\) corridor does deuterium survive long enough to act as a stepping stone to helium and trace heavier light elements.

This is the technical reason the light-element story is concentrated not at one second but at tens to hundreds of seconds. The thermal bath needs time to cool into the narrow nuclear corridor where the reactions can run faster than photodissociation while still faster than the expansion. Once this happens, the universe burns quickly.

At the opening of the bottleneck,
\begin{equation}
T \sim 80\,\mathrm{keV},
\qquad
t\sim 10^2\,\mathrm{s},
\qquad
r\sim 47.1.
\label{eq:partVIII-bbn-open}
\end{equation}
By the three-minute milestone the main helium synthesis is essentially done. This is why the checkpoint of ``the first three minutes'' is a natural one. It is the interval over which ordinary visible matter, as a chemical inventory, becomes recognizable.

\subsection{The three-minute milestone}

Setting \(t=180\;\mathrm{s}\) in the radiation-era relation gives a characteristic temperature near \(70\;\mathrm{keV}\), corresponding to
\begin{equation}
r_{3\mathrm{min}}
=
\ln\!\left(\frac{\EGUT}{70\,\mathrm{keV}}\right)
\simeq 47.2.
\label{eq:partVIII-r-three-minutes}
\end{equation}
This single number is worth remembering because it anchors the whole visible-baryon chronology. By \(r\sim47\),
\begin{itemize}[leftmargin=2.2em]
\item the baryon asymmetry is already inherited from the TeV corridor,
\item hadronic carriers already exist,
\item weak freeze-out is already over,
\item the deuterium bottleneck has opened,
\item and the helium yield has nearly frozen in.
\end{itemize}
In ordinary time only three minutes have passed. In radial distance the universe has already traveled almost \(18\) logarithmic units beyond the inflationary reheating seam.

\subsection{The lithium window inside the chronology}

The standard BBN calculation is famously successful for \({}^4\)He and deuterium but overpredicts \({}^7\)Li. Part~V introduced the \GG{}/\GDOS{} repair as an open-system port term acting only on the A=7 subspace. In Part~VIII the role of that module is purely chronological: it tells us exactly where the repair sits and why it does not disturb earlier or later stages.

The relevant window is the \(t\sim60\) to \(400\;\mathrm{s}\) corridor, roughly \(30\) to \(80\;\mathrm{keV}\). This is late enough that the gross helium yield is already insensitive to the detailed A=7 adjustment, and early enough that the A=7 network can still be influenced through the proposed log-enhanced port. In the chronology this means the lithium repair is not a loose astrophysical afterthought applied gigayears later in stars. It is a sharply localized deformation of the original nuclear freeze-out history.

That localization is one of the theory's cleanest virtues. The BBN correction is not allowed to be broad. It must sit in a narrow time-energy corridor, preserve D/H and \(Y_p\), and then hand off a fixed baryonic inventory to all later eras. The chronology therefore turns a phenomenological requirement into a consistency requirement.

\subsection{What is fixed by the end of the first three minutes}

By the end of the first three minutes, three things are fixed for the rest of the paper.

First, the visible baryonic composition is fixed to high accuracy.

Second, the baryon number density entering recombination, structure growth, and the later \(\pi\)-locked inventory is fixed.

Third, the dark sector has already been present since the end of inflation, so the universe now carries both the visible matter inventory and the compact-angle dark matter inventory into the long radiation-dominated interval that precedes equality.

This is exactly why the early chronology cannot be split into disconnected particle and cosmology narratives. By the time one reaches the end of the first few minutes, the particle ledger and the cosmic ledger are already inseparable.

\section{The first \texorpdfstring{$3\times10^5$}{300000} years: equality, recombination, and the release of the CMB}
\label{sec:partVIII-first-300k-years}


\subsection{The long radiation-dominated bridge}

After the BBN window closes, the universe enters a long interval that is much less dramatic chemically but enormously important dynamically. Radiation still dominates the expansion, the photon bath keeps cooling as \(a^{-1}\), the baryon density continues diluting as \(a^{-3}\), and subhorizon perturbations evolve under the competition between radiation pressure and gravitational attraction.

In ordinary time, the bridge from a few minutes to tens of thousands of years is enormous. In radial distance it is more modest. The universe slides from \(r\sim47\) at the end of BBN toward \(r\sim58\) by equality. This illustrates again the usefulness of the \(r\)-coordinate: the early thermal history is a ladder of only moderate logarithmic length.

\subsection{Matter-radiation equality and why it is a pivotal checkpoint}

Equality is a decisive milestone because it changes which component controls the Hubble rate and therefore which perturbation modes can grow efficiently. In the baseline \(\LCDM\) story equality occurs near \(z\sim3400\). In the \GG{} chronology that statement is retained, but it is overlaid by the Part~VI claim that a brief early-dark-energy lobe and a small dark-radiation excess can slightly shift the sound horizon and the equality-adjacent transfer history without demolishing the successful overall background.

Using the ordinary photon temperature as the process scale, equality sits near
\begin{equation}
\mu_{\mathrm{eq}}\sim0.8\,\mathrm{eV},
\qquad
r_{\mathrm{eq}}\simeq58.6,
\qquad
t_{\mathrm{eq}}\sim 5\times10^4\ \text{to}\ 8\times10^4\,\mathrm{yr}.
\label{eq:partVIII-equality}
\end{equation}
Chronologically equality matters for two reasons.

First, it is the pivot that separates a radiation-shaped perturbation history from a matter-shaped growth history.

Second, it is the natural location for the EDE lobe emphasized in Part~VI: a small fractional contribution near \(z\sim3500\) can alter the sound horizon and later inferred \(H_0\) without requiring a wholly new late-time fluid.

\subsection{Recombination as a \GSSB{} event}

The next great checkpoint is recombination. In standard cosmology one says that the Thomson scattering rate falls below the Hubble rate, neutral hydrogen forms, and photons decouple. All of that remains true here. But \GG{}/\GDOS{} sharpens the interpretation by reading the event as a \GSSB{} transition followed by a geometric attractor.

Before recombination the baryon-photon plasma is a tightly coupled fluid. Many microscopic scattering histories are operationally equivalent at the level of macroscopic perturbation variables. During the transition the free-electron fraction plummets, the gauge orbit of available scattering histories collapses, and one realized phase pattern of the acoustic oscillations is frozen in. After the transition the photons free stream and the pattern becomes the primary CMB sky.

This reinterpretation changes neither the observed acoustic peaks nor the ordinary decoupling physics. What it changes is the status of the event in the theory. Recombination is not just a passive loss of opacity. It is one of the cleanest cosmological examples of \DGI{}, \GSSB{}, and GA in one place.

\subsection{The recombination milestone}

Numerically recombination lies near
\begin{equation}
\mu_{\mathrm{rec}}\sim0.26\,\mathrm{eV},
\qquad
r_{\mathrm{rec}}\simeq59.7,
\qquad
t_{\mathrm{rec}}\simeq3.8\times10^5\,\mathrm{yr},
\qquad
z_\ast=e^{7}\approx1097.
\label{eq:partVIII-recombination}
\end{equation}

Here the epoch is quoted at the integer-ledger corridor-switch value \(z_\ast=e^{7}\approx1097\) of Parts~VI--VII --- the topological landmark, not the measured decoupling; the rounded standard empirical value is \(z_{\mathrm{rec}}\simeq1100\).
The radial separation between equality and recombination is only about one unit:
\begin{equation}
\Delta r_{\mathrm{eq}\to\mathrm{rec}}
\simeq
\ln\!\left(\frac{0.8\,\mathrm{eV}}{0.26\,\mathrm{eV}}\right)
\approx 1.1.
\label{eq:partVIII-dr-eq-rec}
\end{equation}
That single line is a beautiful example of why the logarithmic chronology is so powerful. Equality and recombination are separated by tens to hundreds of thousands of years in ordinary time, but in energy terms they are neighboring steps on the same radial ladder.

\subsection{The CMB as fossil record and chronological anchor}

Once released, the CMB becomes the cleanest fossil record of the earlier chronology. It remembers the scalar tilt generated during inflation, the baryon loading fixed by the visible matter inventory, the damping tail shaped by diffusion, the peak spacing determined by the sound horizon, and any small equality-adjacent shifts produced by EDE-like contributions.

This means that Part~VIII is not merely summarizing older results. It is also showing how the paper's chronology is constrained. One cannot place the inflationary stage, the baryon fraction, or the equality shift arbitrarily and still keep the CMB. In that precise sense the CMB is the strongest chronological anchor of the whole paper.

\section{The first \texorpdfstring{$10^8$}{100 million} years: dark ages, first stars, and early galaxy formation}
\label{sec:partVIII-first-100Myr}


\subsection{Dark ages and residual baryon-dark-matter streaming}

After recombination the universe becomes neutral and transparent, but it is not yet luminous. This is the era usually called the dark ages. The background is matter dominated, the photon bath continues to cool passively, and structure grows in the dark matter potentials while baryons begin to fall in after being released from photon drag.

In the minimal 4D \(\LCDM\) picture, the baryons inherit a residual streaming velocity relative to cold dark matter. That relative motion suppresses gas collapse in the shallowest minihalos and delays first-star formation. The Part~VI \GG{} interpretation modifies the story modestly but significantly: the exact \(U(1)_B\) sector and its locked coupling to the compact-angle dark matter background induce a small drag that damps the streaming velocity and slightly advances baryon settling.

Chronologically this is one of the most elegant places where a small inherited particle-physics change becomes a large astrophysical timing effect. A \(20\%\) reduction of a streaming wind sounds minor. But because the first luminous objects form on the exponentially sensitive high-mass tail of the halo distribution, the resulting shift in the onset of collapse can be visible in the timing of first stars and early galaxies.

\subsection{The one-hundred-million-year milestone}

The \(10^8\)-year milestone is therefore not arbitrary. It lies exactly where the dark ages are ending and the first luminous objects are expected to appear. Taking \(z\sim20\) as a representative value gives
\begin{equation}
a \sim 0.048,
\qquad
\mu \sim T_\gamma \sim 5\,\mathrm{meV},
\qquad
r \sim 63.7.
\label{eq:partVIII-100Myr}
\end{equation}
The crucial comparison is with recombination:
\begin{equation}
\Delta r_{\mathrm{rec}\to 100\,Myr}
=
\ln\!\left(\frac{0.26\,\mathrm{eV}}{5\,\mathrm{meV}}\right)
\approx 4.0.
\label{eq:partVIII-dr-rec-100myr}
\end{equation}
Thus the entire march from photon decoupling to the edge of first-star formation occupies only about four radial units. In ordinary time the interval is close to a hundred million years. In logarithmic energy it is a very short walk.

\subsection{Population III stars and the end of the dark ages}

The first stars are usually placed in the corridor \(z\sim20\) to \(15\), corresponding to approximately \(0.1\) to \(0.2\) Gyr. In the \GG{} reading their formation is encouraged by three previously inherited facts:
\begin{enumerate}[leftmargin=2.3em,label=(\roman*)]
\item the compact-angle dark matter is already present as a cold background,
\item the baryon-dark-matter streaming wind is modestly damped,
\item and the small-scale transfer function received a mild early boost from the equality-adjacent EDE lobe.
\end{enumerate}
None of these changes by itself is dramatic. Together they move the onset of first light to slightly higher redshift and allow more massive halos to participate earlier than in the most conservative 4D baseline.

This is exactly the type of chronology change one wants from a unification program: a small number of tightly connected upstream modifications, not a long menu of disconnected late-time fixes.

\subsection{The first galaxies and the JWST corridor}

By \(z\sim15\) to \(9\), roughly \(0.2\) to \(0.5\) Gyr after the hot beginning, the first long-lived galaxies and heavy black-hole seeds appear. This is the corridor in which JWST has already provided the most dramatic observational pressure on conservative \(\LCDM\) structure forecasts.

In the chronology of this paper, the explanation is not that galaxy formation is a new problem requiring a new sector invented specifically for JWST. Rather, the explanation is that the same compact-angle dark matter, baryonic drag damping, and early transfer-function modification that were already introduced for other reasons shift the formation timeline coherently. The first galaxies therefore appear neither miraculously nor by decree, but as the integrated consequence of the earlier history.

This coherence is crucial. If a model solved the JWST timing problem only by wrecking the CMB, BBN, or the Hubble audit, the chronology would fail. Part~VIII is precisely where one sees that the framework is at least attempting something sharper: the same early modifications are being asked to survive every later checkpoint.

\subsection{Reionization as the first large-scale feedback era}

Once stars and accreting black holes exist, the chronology enters a new regime. The universe is no longer merely expanding and cooling. It is being actively reprocessed by luminous structure. Ultraviolet photons reionize the intergalactic medium, supernovae inject metals, and black holes seed energetic feedback. This is the first cosmic era in which the chronology of matter becomes inseparable from astrophysical ecology.

Part~VIII does not try to solve every detail of reionization. It records the structural point. Reionization is the first major feedback era built on top of the earlier attractor sequence. Its onset depends on when the first stars and galaxies form, and those times depend on the full earlier ledger.

\section{From the first gigayear to the present attractor}
\label{sec:partVIII-late-history}


\subsection{From young galaxies to the mature cosmic web}

After the first gigayear the universe enters the long middle age in which most familiar structure grows. Halos merge, galaxies become chemically enriched, clusters assemble, and the large-scale cosmic web visible to present astronomy is built. The universe is still matter dominated for most of this interval, so the background continues to decelerate while structure grows efficiently.

The chronology of this stage is usually written entirely in terms of \(z\), because the observational data are galaxy surveys, weak lensing, BAO, and cluster counts. In the \GG{} map the same era is also a slow drift from \(r\sim63\) toward \(r\sim67\). The smallness of that radial span is striking. Gigayears of ordinary astrophysics happen within only a few logarithmic energy units of the present CMB temperature.

\subsection{Why the late universe is narrow in \texorpdfstring{\(r\)}{r}}

This is one of the cleanest global insights yielded by the chronology block. The late universe occupies most of the proper-time budget of cosmic history but only a very small fraction of the radial energy budget. The consequence is subtle but important. A small mistake in the late-time background or growth law can consume a large fraction of the remaining radial corridor and therefore show up clearly in precision observables such as \(H_0\), \(S_8\), BAO, or the age-redshift relation.

Put differently: the late universe is dynamically slow but geometrically tight. That is why precision tensions are such powerful probes of the program. There is not much room left in \(r\)-space for arbitrary late-time modifications if one wants to keep the rest of the chronology coherent.

\subsection{Peak star formation and mature structure}

The peak cosmic star-formation epoch near \(z\sim2\) sits around \(t\sim3\) Gyr and \(a\sim0.33\). By then the universe has become an efficient machine for converting baryons into stars and black-hole accretion, but matter still dominates the expansion. Later cluster growth, large-scale lensing, and galaxy peculiar velocities all inherit the same matter content and same earlier transfer history.

This is the era in which the \(S_8\) problem becomes observationally concrete. The Part~VI claim was that the same EDE and dark-radiation ingredients that help the Hubble audit also suppress structure growth mildly, while a small geometric lensing enhancement partly compensates. Part~VIII records the chronological location of that claim: it is a late-universe consistency statement about the mature web, not an arbitrary insertion at the beginning.

\subsection{Matter-dark-energy equality and the end of deceleration}

Eventually the vacuum term catches up. In the \(\pi\)-locked inventory language,
\begin{equation}
\Omm = \frac{1}{\pi},
\qquad
\OmL = 1-\frac{1}{\pi}.
\label{eq:partVIII-pi-lock}
\end{equation}
Matter-vacuum equality occurs when
\begin{equation}
\OmL = \Omm(1+z)^3,
\end{equation}
so that
\begin{equation}
1+z_{\Lambda m}
=
\left(\frac{\OmL}{\Omm}\right)^{1/3}
=
(\pi-1)^{1/3},
\qquad
z_{\Lambda m}\approx 0.288.
\label{eq:partVIII-z-Lm}
\end{equation}
The present paper already emphasized in Part~IV that this is the cleanest geometric answer to the cosmic coincidence problem. In the chronology language it means something very simple: the recent transition to vacuum domination is not a lucky accident inserted by hand. It is where the already-fixed geometric shares force the late-time expansion to pivot.

Chronologically this transition occurs at a cosmic age of order \(10\) to \(11\) Gyr. From then onward the expansion accelerates and the background begins its drift toward the late-time vacuum attractor.

\subsection{The present epoch}

Today sits at
\begin{equation}
t_0 \simeq 13.9\,\mathrm{Gyr},
\qquad
a_0 = 1,
\qquad
z_0 = 0,
\qquad
T_{\gamma,0}\simeq2.35\times10^{-4}\,\mathrm{eV},
\qquad
r_0\simeq66.7.
\label{eq:partVIII-today}
\end{equation}
Two observations matter.

First, the present epoch is only about seven radial units below recombination:
\begin{equation}
\Delta r_{\mathrm{rec}\to0}
=
\ln\!\left(\frac{0.26\,\mathrm{eV}}{2.35\times10^{-4}\,\mathrm{eV}}\right)
\approx 7.0.
\label{eq:partVIII-dr-rec-today}
\end{equation}
Second, the present epoch is only about three radial units below the \(100\) Myr milestone:
\begin{equation}
\Delta r_{100\,\mathrm{Myr}\to0}
\approx
\ln\!\left(\frac{5\,\mathrm{meV}}{2.35\times10^{-4}\,\mathrm{eV}}\right)
\approx 3.1.
\label{eq:partVIII-dr-100myr-today}
\end{equation}
These are extraordinarily compact distances. They tell us that most of late cosmic history unfolds in a narrow logarithmic corridor, which is exactly why the late-time parameter audits are so unforgiving.

\section{Beyond the present: the asymptotic future}
\label{sec:partVIII-future}


\begin{equation}
w_\Lambda = -1.
\label{eq:partVIII-future-w}
\end{equation}
Therefore the late universe approaches an asymptotically de Sitter-like attractor rather than a big rip or a future recollapse. Matter and radiation continue to dilute, the photon bath cools exponentially with the scale factor, and the redshift variable for any future observer tends formally to
\begin{equation}
z \rightarrow -1.
\end{equation}
From the point of view of the radial coordinate, the future is simple:
\begin{equation}
\mu \rightarrow 0,
\qquad
r \rightarrow +\infty.
\label{eq:partVIII-r-future}
\end{equation}
The chronology is thus open-ended in \(r\), even though the observable patch of any one worldline tends toward a finite event-horizon geometry.

This does not make the far future irrelevant to the present paper. The black-hole and information results of Part~VII live precisely on this late-time background. Any complete chronology must therefore end on the same type of asymptotic attractor that Part~VII assumes when discussing evaporation and horizon bookkeeping. Part~VIII closes that logical loop.

\section{Master chronology table}
\label{sec:partVIII-master-table}

\noindent The chronology assembled in the preceding sections is
collected here in two complementary forms. Figure~\ref{fig:cosmic-descent}
draws the descent to scale on the radial coordinate
\(r=\ln(\EGUT/\mu)\), reading each key epoch simultaneously in energy,
time, and the scale factor, with the late universe magnified below a
break so that recombination, structure, and the present epoch remain
legible. Table~\ref{tab:partVIII-master-chronology} then lists every
milestone numerically --- the same \((t,\mu,r,a,z)\) values --- together
with its standard cosmological role and its \GG{}/\GDOS{} reading. The
two are redundant by design: the figure shows the \emph{shape} of the
one open-system descent at a glance, and the table gives the numbers
behind it, so that a reader can move between the picture and its ledger
without leaving the page-spread.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a10_cosmic_descent.pdf}
\caption[The cosmic \textsf{GRIP} descent: every key epoch on the radial axis.]{\textbf{The cosmic \textsf{GRIP} descent, drawn to scale on} \(r=\ln(\EGUT/\mu)\). One open-system descent from the GUT edge (\(r=0\)) to the de~Sitter future, read simultaneously in four quantities --- energy \(\mu\), time \(t\), and the scale factor \(a\), each against the radial depth \(r\). The key epochs of the master chronology below are drawn to scale and colored, with the late universe magnified beneath the break so that recombination, structure, and the present epoch remain legible. Highlighted rows carry the framework's derived markers: the plateau tilt and tensor ratio (\(n_s=0.9620\), \(r=0.0043\)), the lock \(M_{\rm lock}=\EPl/(2e^{35})=3.85\)~TeV, the corridor switch \(z_{\rm sw}=e^{7}\) (integer-ledger marker \(J=7\)), and today's \(\pi\)-locked inventory \((\Omega_m,\Omega_\Lambda,\Omega_b)=(1/\pi,\,1-1/\pi,\,1/2\pi^2)\). Every value is a standard cosmological reading of the tabulated chronology that follows; nothing on the axis is fitted.}
\label{fig:cosmic-descent}
\end{figure}

\begin{landscape}
\scriptsize
\setlength{\parskip}{0pt}
\setlength{\LTleft}{0pt}
\setlength{\LTright}{0pt}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.12}
\captionof{table}[Master chronology of the universe in GG-A10-Cosmos]{Master chronology of the universe in \ATen{}. The table is intentionally redundant: every milestone is shown simultaneously in cosmic time \(t\), characteristic energy scale \(\mu\), radial coordinate \(r\), scale factor \(a\), redshift \(z\), its standard cosmological role, and its \GG{}/\GDOS{} interpretation.}
\label{tab:partVIII-master-chronology}
\begin{longtable}{>{\raggedright\arraybackslash}p{2.3cm}
 >{\raggedright\arraybackslash}p{1.9cm}
 >{\raggedright\arraybackslash}p{2.5cm}
 >{\raggedright\arraybackslash}p{1.4cm}
 >{\raggedright\arraybackslash}p{1.2cm}
 >{\raggedright\arraybackslash}p{1.3cm}
 >{\raggedright\arraybackslash}p{4.5cm}
 >{\raggedright\arraybackslash}p{5.8cm}}
\toprule
\textbf{Epoch / milestone} &
\textbf{Cosmic time \(t\)} &
\textbf{Characteristic scale \(\mu\)} &
\textbf{\(r=\ln(\EGUT/\mu)\)} &
\textbf{\(a\)} &
\textbf{\(z\)} &
\textbf{Standard cosmological meaning} &
\textbf{\GG{}/\GDOS{} reading} \\
\midrule
\endfirsthead

\toprule
\textbf{Epoch / milestone} &
\textbf{Cosmic time \(t\)} &
\textbf{Characteristic scale \(\mu\)} &
\textbf{\(r=\ln(\EGUT/\mu)\)} &
\textbf{\(a\)} &
\textbf{\(z\)} &
\textbf{Standard cosmological meaning} &
\textbf{\GG{}/\GDOS{} reading} \\
\midrule
\endhead

\midrule
\multicolumn{8}{r}{\emph{Continued on next page}}\\
\endfoot

\bottomrule
\endlastfoot

UV boundary surface & $10^{-40}$--$10^{-39}\,\mathrm{s}$ & $\EGUT = 2.28\times 10^{16}\,\mathrm{GeV}$ & $0.0$ & $\sim 10^{-29}$ & $\sim 10^{29}$ & Operational beginning of the 4D boundary description; effective fields are initialized at the GUT edge rather than at a formal Planck singularity. & The bulk-to-boundary cosmology turns on at $r=0$. The initial datum is a finite ultraviolet surface of the anomaly-free throat, not an uncontrolled $a\to0$ singularity. \\
Primary inflation (the derived plateau) & $\sim10^{-39}$--$5\times10^{-37}\,\mathrm{s}$ & $V_{\mathrm{inf}}^{1/4} = 8.3\times10^{15}\,\mathrm{GeV}$, $H_1\simeq1.6\times10^{13}\,\mathrm{GeV}$ & $\simeq 1.0$ & $a\times e^{59}$ & --- & Quasi-de Sitter stage one e-fold below the GUT edge; shrinks the comoving Hubble radius and supplies $N\simeq60$ e-folds ($N^*=52.6$ observable). & The inflaton is the $\Sigma_r$ scale modulus on the \GDOS{}-forced plateau; the tilt and tensor ratio are computed here ($n_s=0.9620$, $r=0.0043$). \\
Modulus oscillation and reheating & $\sim 5\times10^{-37}$--$10^{-28}\,\mathrm{s}$ & bath $T_{\mathrm{max}}\sim10^{11}\rightarrow T_{\mathrm{rh}}\simeq3\times10^{10}\,\mathrm{GeV}$ & $\simeq 2\rightarrow12.9$ & --- & --- & The modulus oscillates ($w\simeq0$) and decays at Planck strength ($\Gamma=M^3/M_{\mathrm{Pl,red}}^2$) over $\sim$14 e-folds; the hot Big Bang begins in earnest at $3\times10^{10}$ GeV. & Graceful exit is slow-roll failure at the plateau's edge; the total energy ($6\times10^{15}$ GeV) leaks into a bath that never exceeds $\sim10^{11}$ GeV. \\
Radiation descent to the lock & $\sim 10^{-28}$--$10^{-13}\,\mathrm{s}$ & $3\times10^{10}\,\mathrm{GeV}\rightarrow\Mlock=3.85\,\mathrm{TeV}$ & $12.9\rightarrow29.4$ & --- & --- & One monotone thermal descent; the heavy Majorana sector is populated on the way (the leptogenesis window). & The middle basin is approached from above by ordinary cooling; no second inflation intervenes ($N_2=0$). \\
Lock crossing / Kaxion birth & $\sim 10^{-13}\,\mathrm{s}$ & $\Mlock = 3.85\,\mathrm{TeV}$ & $29.4$ & --- & --- & The lock \GSSB{} is crossed promptly (the ledger-action bound: no supercooling, no relic dilution); cold dark matter is seeded by the compact-angle sector at this second crossing rather than by a late ad hoc freeze-out. & The Wilson-line mode released here becomes the Kaxion; its relic density is inherited from the same geometric transition whose residue seeds dark energy. \\
Leptogenesis window & $10^{-25}$--$10^{-10}\,\mathrm{s}$ & $\sim10^{10}\,\mathrm{GeV}\rightarrow\Mlock$ & $12.9\rightarrow29.4$ & --- & --- & Heavy Majorana modes, populated thermally at reheating, fall out of equilibrium and decay while sphalerons are still active, building a net $B-L$ asymmetry (zero dilution: the lock crossing is prompt). & The CP phase is inherited from holonomy rather than tuned Yukawa phases; the standard thermal window, closed in the retired TeV-reheating reading, is open. \\
Electroweak seam / sphaleron conversion & $\sim 2\times10^{-11}\,\mathrm{s}$ & $100\,\mathrm{GeV}$ & $33.1$ & $2.3\times10^{-15}$ & $4.3\times10^{14}$ & The electroweak plasma remains in the symmetric or near-symmetric regime; sphalerons convert part of the lepton asymmetry into baryon number. & This is the last major stage in which the matter asymmetry can be reprocessed before ordinary radiation domination takes over as the main narrative. \\
One microsecond milestone & $10^{-6}\,\mathrm{s}$ & $\sim 1\,\mathrm{GeV}$ & $37.7$ & $2.3\times10^{-13}$ & $4.3\times10^{12}$ & By the first microsecond the universe is a hot radiation bath at roughly the GeV scale; the electroweak story is over and the QCD corridor is next. & In the radial language this corresponds to $r\simeq37.7$, so the first microsecond already sits almost eight logarithmic units below the lock seam. \\
QCD crossover / hadronization & $10^{-5}$--$10^{-4}\,\mathrm{s}$ & $150$--$200\,\mathrm{MeV}$ & $39.6$ & $1.6\times10^{-12}$ & $6.4\times10^{11}$ & Quark-gluon degrees of freedom confine into hadrons; protons and neutrons become the appropriate long-lived baryonic variables of the thermal bath. & The exact $U(1)_B$ symmetry now becomes especially transparent operationally: baryons are stable carriers of the protected bulk charge. \\
Neutrino decoupling & $\sim 1\,\mathrm{s}$ & $1\,\mathrm{MeV}$ & $44.6$ & $2.3\times10^{-10}$ & $4.3\times10^{9}$ & Weak rates drop below the Hubble rate and neutrinos decouple from the electromagnetic plasma. & Any later extra-radiation or dark-radiation effect must be measured against this inherited decoupling ledger; in the chronology this is the reference point for $N_{\mathrm{eff}}$ audits. \\
Neutron-proton freeze-out & $\sim 1\,\mathrm{s}$ & $0.8\,\mathrm{MeV}$ & $44.8$ & $3.0\times10^{-10}$ & $3.4\times10^{9}$ & The $n/p$ ratio stops tracking equilibrium exactly and begins to set the helium yield of standard BBN. & The chronology of ordinary matter is now controlled by the usual freeze-out competition $\Gamma/H$, but the allowed asymmetry and baryon density were already fixed upstream by the TeV leptogenesis module. \\
Electron-positron annihilation & $1$--$10\,\mathrm{s}$ & $0.5\,\mathrm{MeV}$ & $45.3$ & $4.7\times10^{-10}$ & $2.1\times10^{9}$ & Entropy is dumped into the photon bath, changing the relation between photon and neutrino temperatures and locking in the familiar $T_\nu/T_\gamma$ ratio. & This is one of the places where the $g_{*s}$ correction matters in the conversion from $z$ to $r$; the chronology is logarithmic, not naively linear in temperature. \\
Deuterium bottleneck opens & $\sim 100\,\mathrm{s}$ & $80\,\mathrm{keV}$ & $47.1$ & $2.9\times10^{-9}$ & $3.4\times10^{8}$ & Photodissociation becomes inefficient enough for deuterium to survive, opening the network that rapidly builds ${}^4$He and trace A=7 nuclei. & This is the first truly precision nuclear checkpoint of the framework. Any cosmology module that fails here should be rejected before later structure claims are entertained. \\
Three-minute milestone / main BBN burn & $180\,\mathrm{s}$ & $70\,\mathrm{keV}$ & $47.2$ & $3.4\times10^{-9}$ & $2.98\times10^{8}$ & The helium fraction and the deuterium residue are essentially fixed by the end of the first few minutes. & In the radial language the first three minutes correspond to $r\approx47.2$. By then the universe has traversed almost seventeen additional logarithmic units beyond the lock-scale reheating corridor. \\
Lithium-port window & $60$--$400\,\mathrm{s}$ & $30$--$80\,\mathrm{keV}$ & $47.6$ & $5\times10^{-9}$ (order of magnitude) & $2\times10^{8}$ (order of magnitude) & The standard A=7 network is most sensitive in this window; ${}^7$Be production and the later lithium problem originate here. & The \GDOS{} $U(1)_B$ port term acts only on the A=7 subspace, depleting ${}^7$Be without disturbing the successful D/H and ${}^4$He plateaus. \\
Light-element ledger frozen & $10^{3}$--$10^{6}\,\mathrm{s}$ & $10$--$1\,\mathrm{keV}$ & $51.5$ & $2.3\times10^{-7}$ & $4.3\times10^{6}$ & Nuclear reaction rates collapse relative to the expansion; the light-element abundances become effectively frozen and are later sampled by astrophysical archaeology. & After this stage the matter content entering recombination and structure growth is already locked, so later cosmology must respect the BBN ledger. \\
Matter-radiation equality and EDE lobe & $5\times10^{4}$--$8\times10^{4}\,\mathrm{yr}$ & $0.8\,\mathrm{eV}$ & $58.6$ & $2.9\times10^{-4}$ & $3.4\times10^{3}$ & Radiation ceases to dominate; the sound horizon and transfer function become sensitive to the exact equality timing. & The framework places a brief early-dark-energy lobe near this epoch, together with a small dark-radiation excess, to shift the sound horizon while keeping the broader background close to \LCDM. \\
Recombination onset & $3\times10^{5}\,\mathrm{yr}$ & $0.30\,\mathrm{eV}$ & $59.6$ & $9\times10^{-4}$ & $1.1\times10^{3}$ & The Thomson rate falls toward the Hubble rate; neutral hydrogen begins to form and the photon mean free path grows rapidly. & In \GDOS{} language this is the start of a \GSSB{} transition in the baryon-photon plasma: a continuous scattering orbit begins to freeze into one realized acoustic pattern. \\
Last scattering / CMB geometric attractor & $3.8\times10^{5}\,\mathrm{yr}$ & $0.26\,\mathrm{eV}$ & $59.7$ & $9.1\times10^{-4}$ & $1090$ (atomic; switch $e^{7}$ $0.4\%$ above) & Primary temperature and polarization anisotropies are released; acoustic phases are frozen and propagate almost freely to us. & The CMB sky is the first great GA of cosmology. Peak positions, baryon loading, and damping are read as deterministic shadows of the locked background and the radial BFFT coordinate. \\
Baryon drag release / streaming wind & $0.4$--$10\,\mathrm{Myr}$ & $0.26\rightarrow0.01\,\mathrm{eV}$ & $63.0$ & $2.3\times10^{-2}$ & $4.3\times10^{1}$ & Baryons are freed from photon drag but retain a residual streaming velocity relative to dark matter; the dark ages begin. & The baryon-graviphoton channel damps this wind by a modest amount, one of the key ingredients that later advances first-star formation. \\
Dark ages & $1$--$100\,\mathrm{Myr}$ & $0.07\rightarrow0.005\,\mathrm{eV}$ & $63.7$ & $4.8\times10^{-2}$ & $2.1\times10^{1}$ & No stars yet shine; structure grows linearly and then nonlinearly inside dark-matter potentials. & Most of the observable universe is now matter dominated, but the compact-angle dark sector and baryonic drag corrections have already biased the timing of small-scale collapse. \\
One-hundred-million-year milestone & $10^{8}\,\mathrm{yr}$ & $\sim 5\,\mathrm{meV}$ & $63.7$ & $4.8\times10^{-2}$ & $\sim 20$ & The earliest massive minihalos are expected to host the first Population III stars or their immediate precursors. & In the radial dictionary this milestone sits near $r\approx63.7$. Notice how little radial distance separates the dark ages from the present compared with the enormous physical time involved. \\
Population III ignition & $0.1$--$0.2\,\mathrm{Gyr}$ & $5$--$3\,\mathrm{meV}$ & $64.2$ & $0.05$--$0.07$ & $20$--$15$ & Molecular-hydrogen cooling triggers the first star formation in minihalos; metals and ionizing photons are born. & A small EDE boost and a damped baryon-dark-matter streaming velocity allow this stage to occur earlier and more efficiently than in minimal 4D \LCDM. \\
First galaxies / JWST corridor & $0.2$--$0.5\,\mathrm{Gyr}$ & $3$--$2\,\mathrm{meV}$ & $64.3$ & $0.07$--$0.10$ & $15$--$9$ & The first long-lived galaxies and heavy black-hole seeds assemble; UV luminosity functions and stellar-mass functions become directly observable. & The framework attributes the early onset to a slightly boosted small-scale transfer function, reduced baryon wind, and lower effective collapse threshold. \\
Reionization era & $0.5$--$1\,\mathrm{Gyr}$ & $2$--$1.6\,\mathrm{meV}$ & $64.8$ & $0.10$--$0.14$ & $9$--$6$ & Ionizing photons from stars and accreting objects progressively reionize the intergalactic medium. & This is a boundary-era radiative feedback problem, but its onset and duration inherit the earlier small-scale timing shifts produced by the same locked background. \\
Mature galaxy and cluster growth & $1$--$5\,\mathrm{Gyr}$ & $1.6\rightarrow0.6\,\mathrm{meV}$ & $65.8$ & $0.14$--$0.4$ & $6$--$1.5$ & Nonlinear structure growth builds the familiar cosmic web of halos, clusters, and large galaxies. & This is the era in which the Hubble and $S_8$ audits must remain consistent with the earlier CMB and galaxy-timing modules rather than being adjusted independently. \\
Peak cosmic star-formation epoch & $\sim 3\,\mathrm{Gyr}$ & $0.7\,\mathrm{meV}$ & $65.7$ & $0.33$ & $\sim 2$ & The universe reaches its peak rate of converting baryons into stars, metals, and black-hole accretion luminosity. & By this time the background is already close to the late-time $\pi$-locked inventory, but matter still dominates the expansion rate. \\
Matter-dark-energy equality & $\sim 10$--$11\,\mathrm{Gyr}$ & $0.30\,\mathrm{meV}$ & $66.5$ & $0.777$ & $0.287$ & The matter and vacuum densities become comparable; the decelerating history gives way to accelerating expansion. & Because $\Omega_m=1/\pi$ and $\Omega_\Lambda=1-1/\pi$ are fixed geometrically, the transition epoch is no longer a coincidence but a derived stage on the same ledger. \\
$\Omega_m = 1/\pi$ crossing & $13.8\,\mathrm{Gyr}$ & $\approx T_{\gamma,0}$ & $66.7$ & $0.995$ & $0.005$ & The matter fraction falls through exactly $1/\pi$: the full $\pi$-locked inventory $(1/\pi,\ 1-1/\pi,\ 1/(2\pi^2))$ is exact at this instant, $75$~Myr before the present (Planck central values; timing blur $\pm160$~Myr). & The chart epoch: this is the snapshot the codebook's $\pi$-locked fractions refer to (Table~\ref{tab:whynow-epochs}). \\
Present stationary attractor & $13.9\,\mathrm{Gyr}$ & $T_{\gamma,0}=2.35\times10^{-4}\,\mathrm{eV}$ & $66.7$ & $1$ & $0$ & The universe is vacuum dominated but still contains the fossil record of earlier radiation, matter, and structure-building stages. & Today lies at $r\simeq66.7$, $75$~Myr past the crossing: the matter fractions sit fractionally below their $\pi$-forms ($\Omega_m = 0.315$ against $1/\pi = 0.318$). \\
Asymptotic de Sitter future & $\gg 10^{10}\,\mathrm{yr}$ & $\mu \rightarrow T_\gamma \rightarrow 0$ & $r \rightarrow +\infty$ & $a>1$ and growing exponentially & $z\rightarrow -1$ & Matter dilutes away relative to vacuum; the observable patch approaches a de Sitter-like event-horizon state. & The chronology ends on a GA selected by the residual \GFF{} vacuum functional. No big rip is required in the minimal module because the equation of state remains locked at $w=-1$. \\

\end{longtable}
\end{landscape}

\section{Worked conversion examples}
\label{sec:partVIII-worked-examples}


\subsection{Example A: the one-microsecond milestone}

Take \(t = 10^{-6}\,\mathrm{s}\). In the radiation era Eq.~\eqref{eq:partVIII-time-temperature-practical} gives
\begin{equation}
10^{-6}
\simeq
2.42\,g_*^{-1/2}
\left(\frac{T}{\mathrm{MeV}}\right)^{-2}.
\end{equation}
Using \(g_*\sim 60\) as an order-of-magnitude QCD-plasma value, one finds
\begin{equation}
\left(\frac{T}{\mathrm{MeV}}\right)^2
\sim
2.42\,60^{-1/2}\times10^{6}
\sim 3.1\times10^{5},
\end{equation}
hence
\begin{equation}
T \sim 5.6\times10^{2}\,\mathrm{MeV}\sim0.6\,\mathrm{GeV},
\end{equation}
which is comfortably in the expected GeV corridor. Taking \(\mu\sim 1\;\mathrm{GeV}\) as the rounded milestone scale,
\begin{equation}
r_{\mu\mathrm{s}}
=
\ln\!\left(\frac{2.28\times10^{16}}{1}\right)
\simeq 37.7.
\end{equation}
The formal redshift associated with a GeV-scale bath is
\begin{equation}
1+z_{\mu\mathrm{s}}
\sim
\frac{1\,\mathrm{GeV}}{2.35\times10^{-13}\,\mathrm{GeV}}
\sim 4.3\times10^{12},
\qquad
a_{\mu\mathrm{s}}\sim2.3\times10^{-13}.
\end{equation}
So the first microsecond sits at
\begin{equation}
(t,\mu,r,a,z)
\sim
\left(10^{-6}\,\mathrm{s},\ 1\,\mathrm{GeV},\ 37.7,\ 2.3\times10^{-13},\ 4.3\times10^{12}\right).
\end{equation}

\subsection{Example B: the three-minute milestone}

Take \(t = 180\,\mathrm{s}\). Again using Eq.~\eqref{eq:partVIII-time-temperature-practical},
\begin{equation}
180
\simeq
2.42\,g_*^{-1/2}
\left(\frac{T}{\mathrm{MeV}}\right)^{-2}.
\end{equation}
For \(g_*\sim3.36\) after electron-positron annihilation, this gives
\begin{equation}
\left(\frac{T}{\mathrm{MeV}}\right)^2
\sim
\frac{2.42}{\sqrt{3.36}\times180}
\sim 7\times10^{-3},
\end{equation}
so
\begin{equation}
T \sim 8\times10^{-2}\,\mathrm{MeV}=80\,\mathrm{keV}.
\end{equation}
Then
\begin{equation}
r_{3\mathrm{min}}
=
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{8\times10^{-5}\,\mathrm{GeV}}\right)
\simeq 47.1.
\end{equation}
The corresponding formal redshift is
\begin{equation}
1+z_{3\mathrm{min}}
\sim
\frac{8\times10^{-5}\,\mathrm{GeV}}{2.35\times10^{-13}\,\mathrm{GeV}}
\sim3.4\times10^{8},
\qquad
a_{3\mathrm{min}}\sim3\times10^{-9}.
\end{equation}
Thus
\begin{equation}
(t,\mu,r,a,z)
\sim
\left(180\,\mathrm{s},\ 80\,\mathrm{keV},\ 47.1,\ 3\times10^{-9},\ 3.4\times10^{8}\right).
\end{equation}
This single computation explains why the first three minutes are such a universal checkpoint in cosmology: visible matter is being fixed at a point only \(17\) or \(18\) radial units below the reheating seam.

\subsection{Example C: the recombination milestone}

Take the standard recombination values (the GG corridor-switch marker is \(z_{\mathrm{sw}}=e^{7}\approx1097\), distinct from the decoupling value; \S\ref{sec:two-redshifts})
\begin{equation}
t_{\mathrm{rec}}\simeq 3.8\times10^{5}\,\mathrm{yr},
\qquad
z_{\mathrm{rec}}\simeq 1100.
\end{equation}
Then
\begin{equation}
a_{\mathrm{rec}} = \frac{1}{1+z_{\mathrm{rec}}}\simeq 9.1\times10^{-4}.
\end{equation}
The photon temperature is
\begin{equation}
T_{\mathrm{rec}}
\simeq
T_{\gamma,0}(1+z_{\mathrm{rec}})
\simeq
2.35\times10^{-4}\,\mathrm{eV}\times 1100
\simeq 0.26\,\mathrm{eV},
\end{equation}
up to the usual small \(g_{*s}\) bookkeeping corrections. Therefore
\begin{equation}
r_{\mathrm{rec}}
=
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{2.6\times10^{-10}\,\mathrm{GeV}}\right)
\simeq59.7.
\end{equation}
So
\begin{equation}
(t,\mu,r,a,z)
\sim
\left(3.8\times10^{5}\,\mathrm{yr},\ 0.26\,\mathrm{eV},\ 59.7,\ 9.1\times10^{-4},\ 1100\right).
\end{equation}
The striking fact is that recombination lies only about \(12.6\) radial units below the BBN corridor:
\begin{equation}
\Delta r_{3\mathrm{min}\to\mathrm{rec}}
\approx
59.7-47.1
\approx 12.6.
\end{equation}

\subsection{Example D: the hundred-million-year milestone}

Take \(t = 10^{8}\,\mathrm{yr}\) and use \(z\sim20\) as a representative first-star-era redshift. Then
\begin{equation}
a_{100\mathrm{Myr}}\sim \frac{1}{21}\sim 4.8\times10^{-2},
\end{equation}
and the photon temperature is
\begin{equation}
T_\gamma(100\,\mathrm{Myr})
\sim
2.35\times10^{-4}\,\mathrm{eV}\times 21
\sim 5\times10^{-3}\,\mathrm{eV}.
\end{equation}
Hence
\begin{equation}
r_{100\mathrm{Myr}}
=
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{5\times10^{-12}\,\mathrm{GeV}}\right)
\simeq 63.7.
\end{equation}
Therefore
\begin{equation}
(t,\mu,r,a,z)
\sim
\left(10^{8}\,\mathrm{yr},\ 5\,\mathrm{meV},\ 63.7,\ 4.8\times10^{-2},\ 20\right).
\end{equation}
From recombination to the edge of first-light formation the universe thus moves only about four radial units. That is the geometrically concise version of the first-structure era.

\subsection{Example E: the present epoch}

Today is the easiest example because the scale factor and redshift are normalized by definition:
\begin{equation}
a_0=1,
\qquad
z_0=0.
\end{equation}
Using \(T_{\gamma,0}=2.35\times10^{-4}\,\mathrm{eV}\), one finds
\begin{equation}
r_0
=
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{2.35\times10^{-13}\,\mathrm{GeV}}\right)
\simeq 66.7.
\end{equation}
Thus
\begin{equation}
(t_0,\mu_0,r_0,a_0,z_0)
\sim
\left(13.9\,\mathrm{Gyr},\ 2.35\times10^{-4}\,\mathrm{eV},\ 66.7,\ 1,\ 0\right).
\end{equation}
The distance from the \(100\) Myr milestone to today is therefore
\begin{equation}
\Delta r_{100\mathrm{Myr}\to0}\simeq3.0,
\end{equation}
again showing that most of the universe's visible proper-time age occupies only a very small fraction of the radial hierarchy.

\subsection{Example F: matter-dark-energy equality}

Finally, use the \(\pi\)-locked inventory. Setting \(\rho_m(z)=\rho_\Lambda\) gives Eq.~\eqref{eq:partVIII-z-Lm},
\begin{equation}
z_{\Lambda m}\approx 0.288,
\qquad
a_{\Lambda m}\approx 0.777.
\end{equation}
The corresponding photon temperature is
\begin{equation}
T_{\gamma,\Lambda m}
\approx
(1+z_{\Lambda m})T_{\gamma,0}
\approx
0.30\,\mathrm{meV},
\end{equation}
and therefore
\begin{equation}
r_{\Lambda m}
=
\ln\!\left(\frac{2.28\times10^{16}\,\mathrm{GeV}}{3\times10^{-13}\,\mathrm{GeV}}\right)
\simeq 66.5.
\end{equation}
So even the recent pivot from deceleration to acceleration occurs in the same narrow \(r\)-neighborhood as the present epoch.


\section{Radial bookkeeping: how much history is packed into each segment}
\label{sec:partVIII-r-bookkeeping}


The seven checkpoint scales and their radial steps are collected in the
ladder of Figure~\ref{fig:partVIII-r-ladder}.

\begin{figure}[H]
\centering
\resizebox{0.9\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >={Stealth[length=2mm]},
  scl/.style={rectangle, rounded corners=3pt, draw=black!70, line width=0.80pt,
              fill=blue!6, align=center, inner sep=4pt, minimum height=11mm},
  drl/.style={font=\sffamily\footnotesize\bfseries, text=red!60!black},
  rr/.style={font=\sffamily\footnotesize, text=black!60}]
\node[scl] (n0) at (0,0)    {$\EGUT$\\[-1pt]\scriptsize $2.3{\times}10^{16}$ GeV};
\node[scl] (n1) at (3.4,0)  {$\Mlock$\\[-1pt]\scriptsize $3.85$ TeV};
\node[scl] (n2) at (6.8,0)  {QCD\\[-1pt]\scriptsize $1$ GeV};
\node[scl] (n3) at (10.2,0) {BBN\\[-1pt]\scriptsize $80$ keV};
\node[scl] (n4) at (13.6,0) {recomb.\\[-1pt]\scriptsize $0.26$ eV};
\node[scl] (n5) at (17.0,0) {first stars\\[-1pt]\scriptsize $5$ meV};
\node[scl] (n6) at (20.4,0) {today\\[-1pt]\scriptsize $T_{\gamma,0}$};
\foreach \a/\b/\l in {n0/n1/29.4, n1/n2/8.3, n2/n3/9.4, n3/n4/12.6, n4/n5/4.0, n5/n6/3.1}{
  \draw[->, line width=0.80pt] (\a.east) -- node[drl, above=0.5mm]{$\Delta r{\simeq}\l$} (\b.west);}
\foreach \n/\v in {n0/0, n1/29.4, n2/37.7, n3/47.1, n4/59.7, n5/63.7, n6/66.8}{
  \node[rr, below=1.2mm of \n]{$r=\v$};}
\end{tikzpicture}%
}
\caption[The radial ladder of cosmic history.]{\textbf{The radial ladder of
cosmic history.} The seven checkpoint scales of the descent, with the radial
step $\Delta r = \ln(\mu_i/\mu_{i+1})$ on each arrow and the running
coordinate $r$ beneath each node. The six steps sum to $66.8$, reproducing
$r_{\rm now} \simeq 66.6$--$66.7$ (segment rounding): the whole of cosmic
history, from the GUT edge to today, is sixty-seven $e$-folds of scale packed
into steps of only a few to thirty --- the quantitative face of the
front-loaded/back-loaded duality developed below.}
\label{fig:partVIII-r-ladder}
\end{figure}
What is remarkable is not that the early universe spans the largest total range in \(r\); that is obvious. What is remarkable is that the main visible checkpoints of the history are separated by radial steps of order only a few to ten. The enormous disparity in proper time is therefore partly a coordinate artifact of plotting the history against \(t\) instead of against \(\ln\mu\).

This is especially helpful when one asks a question such as ``where in the total history does recombination sit?'' On a linear time axis recombination looks much closer to the beginning than to the present, but still unimaginably later than the first seconds. On a redshift axis it looks remote but not impossible. On the \GG{} axis the answer is cleaner: recombination sits at \(r\sim60\), only about seven units above the present and only about thirteen units above the BBN corridor. That picture is far more compatible with the actual structure of the equations, because the equations organize themselves by powers and logarithms of scale, not by linearly spaced seconds.

\subsection{The universe is front-loaded in energy and back-loaded in clock time}

This observation can be stated more sharply.

During the earliest stages, a tiny change in proper time corresponds to a vast multiplicative change in the dominant energy scale:
\begin{equation}
\mu \propto t^{-1/2}
\qquad\text{in radiation domination,}
\end{equation}
so a factor of \(10^{12}\) increase in time only lowers the characteristic energy by a factor of \(10^6\). Because \(r\) is logarithmic, that same interval becomes only
\begin{equation}
\Delta r = \half \ln(10^{12}) \approx 13.8.
\end{equation}
This is why the history from the first second to the first few hundred thousand years is substantial but not huge in the radial language.

Conversely, once the universe enters the late matter and vacuum eras, the energy scale tracked by the photon bath drops very slowly. Billions of years can then pass while the characteristic process scale falls by only a factor of a few or a few tens. In \(r\) that means a displacement of only a few units. The late universe is therefore back-loaded in proper time but thin in the radial coordinate.

This duality between front-loading in energy and back-loading in clock time is one of the central conceptual outcomes of Part~VIII. It explains why the early universe is where almost all of the hierarchy is generated, while the late universe is where almost all of the precision diagnostics live.

\subsection{A second summary table: intervals rather than milestones}

The next table complements the master chronology. Instead of listing absolute milestones, it lists the interval between them and the corresponding physical work done in that interval.

\begin{table}[H]
\centering
\caption[Radial bookkeeping for the main segments of cosmic history]{Radial bookkeeping for the main segments of cosmic history. The table shows that the late universe is huge in years but narrow in \(r\), while the early universe is tiny in years but wide in \(r\). The columns give the interval, its scale drop, the radial span, the approximate elapsed time, and the physical work done in it. Reading the third and fourth columns together is the point: the late universe is vast in years and narrow in radial coordinate, and the early universe is the reverse.}
\label{tab:partVIII-radial-bookkeeping}
\setlength{\tabcolsep}{3.5pt}
\small
\begin{tabular}{p{0.23\textwidth}p{0.17\textwidth}p{0.13\textwidth}p{0.17\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Interval} & \textbf{Characteristic scale drop} & \textbf{\(\Delta r\)} & \textbf{Approximate time interval} & \textbf{Main physical work done}\\
\midrule
GUT edge \(\rightarrow\) lock seam & \(2.3\times10^{16}\,\mathrm{GeV}\rightarrow 3.85\,\mathrm{TeV}\) & \(29.4\) & pre-thermal ultraviolet prehistory & Boundary description turns on and descends to the inflationary plateau.\\
Lock seam \(\rightarrow\) one microsecond & \(3.85\,\mathrm{TeV}\rightarrow1\,\mathrm{GeV}\) & \(8.3\) & \(\sim10^{-11}\,\mathrm{s}\rightarrow10^{-6}\,\mathrm{s}\) & Inflation ends, reheating completes, dark matter is seeded, baryogenesis occurs, electroweak processing ends.\\
One microsecond \(\rightarrow\) three minutes & \(1\,\mathrm{GeV}\rightarrow80\,\mathrm{keV}\) & \(9.4\) & \(10^{-6}\,\mathrm{s}\rightarrow180\,\mathrm{s}\) & QCD crossover, neutrino decoupling, \(n/p\) freeze-out, deuterium bottleneck, helium synthesis, lithium window.\\
Three minutes \(\rightarrow\) recombination & \(80\,\mathrm{keV}\rightarrow0.26\,\mathrm{eV}\) & \(12.6\) & \(180\,\mathrm{s}\rightarrow3.8\times10^{5}\,\mathrm{yr}\) & Radiation-dominated bridge, equality, acoustic processing, last scattering.\\
Recombination \(\rightarrow\) \(100\) Myr & \(0.26\,\mathrm{eV}\rightarrow5\,\mathrm{meV}\) & \(4.0\) & \(3.8\times10^{5}\,\mathrm{yr}\rightarrow10^{8}\,\mathrm{yr}\) & Dark ages, baryon settling, first-star preparation.\\
\(100\) Myr \(\rightarrow\) today & \(5\,\mathrm{meV}\rightarrow2.35\times10^{-4}\,\mathrm{eV}\) & \(3.1\) & \(10^{8}\,\mathrm{yr}\rightarrow13.9\,\mathrm{Gyr}\) & Galaxies, reionization, mature structure, matter-vacuum transition, present attractor.\\
Today \(\rightarrow\) asymptotic future & \(T_{\gamma,0}\rightarrow0\) & unbounded & \(13.9\,\mathrm{Gyr}\rightarrow\infty\) & Approach to a de Sitter-like vacuum attractor with a finite event horizon.\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why this bookkeeping is scientifically useful}

This interval view does three kinds of work.

First, it prevents chronological confusion. It becomes immediately clear that BBN is not ``almost simultaneous'' with the electroweak story just because both happen early in absolute terms. There is a radial gap of about nine units between the GeV corridor and the \(80\;\mathrm{keV}\) corridor, which is plenty of room for qualitatively new physics to happen---and in fact it does.

Second, it highlights where model freedom is actually dangerous. The largest radial block after inflation is the interval between BBN and recombination. This is precisely the interval most heavily audited by the CMB and by equality-related observables. Any hidden extra radiation, decays, or late entropy dump inserted there will show up prominently.

Third, it explains why late-time precision tensions are so constraining. From \(100\) Myr to today there are only about three radial units left. A theory has very little room to maneuver there without ruining the rest of the ledger.

\subsection{The chronology in one radial sentence}

One can compress the whole chronology into a single radial sentence:
\begin{equation}
0 \;\rightarrow\; 29.4 \;\rightarrow\; 37.7 \;\rightarrow\; 47.1 \;\rightarrow\; 59.7 \;\rightarrow\; 63.7 \;\rightarrow\; 66.7 \;\rightarrow\; \infty.
\label{eq:partVIII-radial-sentence}
\end{equation}
Each arrow is a multiplicative fall in energy, and each number can now be associated with a concrete cosmic stage:
\begin{center}
GUT edge \(\rightarrow\) lock seam \(\rightarrow\) one microsecond \(\rightarrow\) three minutes \(\rightarrow\) recombination \(\rightarrow\) first light \(\rightarrow\) today \(\rightarrow\) asymptotic future.
\end{center}
This is arguably the most compact version of the history written in the present paper.


\section{The milestone ladder in one glance}
\label{sec:partVIII-milestone-ladder}


\begin{enumerate}[leftmargin=2.3em,label=\arabic*.]
\item the end of the electroweak/TeV prelude and the entrance into the hadronic bath,
\item the fixation of the light-element inventory,
\item the freezing of the acoustic information in the CMB,
\item and the ignition of the first luminous structures.
\end{enumerate}

What makes these four times so instructive is that each one marks the completion of a different kind of physical work.

At one microsecond, the universe has not yet built atoms, nuclei, or galaxies, but it has already completed the geometric tasks that dominate the very beginning: UV initialization, descent to the lock seam, the inflationary attractor, \GSSB{} exit, reheating, the birth of the Kaxion dark sector, and the TeV leptogenesis corridor. In other words, by the time the universe is one microsecond old, the deep ultraviolet and matter-origin work of the theory is already behind it.

At three minutes, the universe has completed the fixation of the ordinary visible matter inventory. This is why every later observational era---from recombination to galaxies to stars to observers---inherits a baryonic ledger that was set astonishingly early. The late universe only rearranges matter whose total abundance and light-element composition were already decided.

At \(3\times10^5\) years, the universe has completed the information-freezing work of recombination. The primary CMB sky is released and the acoustic phase pattern becomes a stable fossil. In the language of the paper, this is the first truly macroscopic geometric attractor seen directly by observers.

At \(10^8\) years, the universe has completed the long dark bridge between a transparent but dark background and a luminous structured cosmos. The first stars, and then the first galaxies, appear as the earliest large-scale nonlinear consequences of everything that came before.

Seen this way, the four milestones are not arbitrary dates. They are the four best ``memory cells'' of the whole chronology:
\begin{center}
ultraviolet origin,\quad visible matter,\quad frozen information,\quad first light.
\end{center}

\begin{table}[H]
\centering
\caption[The four milestones and the present epoch, as a five-clock ladder]{The four milestones plus the present epoch, displayed as a compact five-clock ladder. This table is a compressed version of the master chronology and is especially useful for presentation, teaching, and cross-checking the Part~VIII narrative. The five clock columns are one milestone read five ways, and the last column says what is finished by then.}
\label{tab:partVIII-milestone-ladder}
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{p{0.16\textwidth}p{0.10\textwidth}p{0.14\textwidth}p{0.09\textwidth}p{0.08\textwidth}p{0.09\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Milestone} & \textbf{\(t\)} & \textbf{\(\mu\)} & \textbf{\(r\)} & \textbf{\(a\)} & \textbf{\(z\)} & \textbf{What is completed by then?}\\
\midrule
First microsecond & \(10^{-6}\,\mathrm{s}\) & \(\sim 1\,\mathrm{GeV}\) & \(\sim 37.7\) & \(\sim2.3\times10^{-13}\) & \(\sim4\times10^{12}\) & Inflation is already over, reheating is complete, the dark sector is seeded, and the TeV baryogenesis corridor has already acted.\\
First three minutes & \(180\,\mathrm{s}\) & \(\sim70\) to \(80\,\mathrm{keV}\) & \(\sim47.1\) & \(\sim3\times10^{-9}\) & \(\sim3\times10^{8}\) & The visible baryonic composition is essentially fixed; helium, deuterium, and the A=7 sector have reached their primordial ledger.\\
First \(3\times10^5\) years & \(3\) to \(4\times10^5\,\mathrm{yr}\) & \(\sim0.3\,\mathrm{eV}\) & \(\sim59.7\) & \(\sim10^{-3}\) & \(\sim1100\) & The CMB is released; the primary acoustic information is frozen into a geometric attractor.\\
First \(10^8\) years & \(10^{8}\,\mathrm{yr}\) & \(\sim5\,\mathrm{meV}\) & \(\sim63.7\) & \(\sim5\times10^{-2}\) & \(\sim20\) & The dark ages are ending and the first stars are turning on; the universe becomes luminous.\\
Present epoch & \(13.9\,\mathrm{Gyr}\) & \(2.35\times10^{-4}\,\mathrm{eV}\) & \(\sim66.7\) & \(1\) & \(0\) & The late-time \(\pi\)-locked inventory is visible directly; vacuum domination has already begun.\\
\bottomrule
\end{tabular}
\end{table}

The compressed ladder exposes two points that are easy to miss when the history is narrated only qualitatively.

The first is that the distance from the one-microsecond milestone to the three-minute milestone is almost ten radial units, whereas the distance from the first \(10^8\) years to the present is only about three. Therefore the early history still contains more hierarchy per unit milestone than the entire visible late history.

The second is that the redshift variable becomes observationally natural only rather late. A formal redshift can be assigned to the one-microsecond or three-minute stages, but there are no direct observers there and the more native variables are \(t\), \(T\), and \(r\). By recombination and certainly by \(10^8\) years, however, the redshift language becomes indispensable because the history is now read by astronomy rather than by nuclear or particle physics.

This transition in which variable is operationally natural is itself part of the theory's measurement philosophy. \OGGEP{} says that different eras are read out through different boundary instruments. A collider-style or thermal-bath variable is natural in the early universe; a redshift and luminosity variable is natural in the late universe. The chronology remains one chronology because all those instruments are linked by the same underlying coordinate dictionary.

\section{Chronological synthesis}
\label{sec:partVIII-synthesis}


\begin{quote}
The universe begins on a finite ultraviolet boundary at \(\EGUT\), descends geometrically to the inflationary plateau one e-fold below the GUT edge, exits through a \GSSB{} that simultaneously reheats the boundary and seeds dark matter and dark energy, generates baryon asymmetry in the TeV corridor, fixes visible matter in the first three minutes, freezes the CMB as a geometric attractor at \(r\sim60\), ignites the first stars near \(r\sim64\), and reaches the present \(\pi\)-locked vacuum-matter attractor by \(r\sim67\).
\end{quote}

That sentence is long, but it is logically tight. Every clause is inherited from Parts~II through VII, and every clause now has an explicit chronological placement. That is the real purpose of Part~VIII. It turns a collection of problem-by-problem modules into one trackable cosmic story.

The deeper moral is equally important. The universe is violently nonuniform in ordinary time, but it is comparatively well organized in the logarithmic \GG{} coordinate. The first microsecond, the first three minutes, recombination, first-light formation, and the present are not disconnected wonders; they are milestones on a single finite descent in \(r\), followed by an asymptotic drift toward \(r\to\infty\).

This is exactly the sort of synthesis a unification paper ought to deliver.

\medskip\noindent There is a quiet philosophical lesson in the logarithmic clock. Asked in ordinary years, ``when did the universe do this?'' scatters its answers across twenty orders of magnitude, and the history reads as a sequence of unrelated miracles. Asked in the geometry's own coordinate, the same events fall at even steps of a single descent: the universe was never rushing and never idle; it was walking down a ladder at the pace the ledger set. What a calendar makes miraculous, a ruler makes lawful.

The narrative is now complete, and precisely because it is complete it must submit to an audit. Part~IX stops telling the story and starts grading it: every number this paper has used, classified --- Derived, Derived-conditional, or Asserted --- with nothing quietly upgraded and nothing hidden in prose. The reader who has trusted the tale this far is owed the ledger behind it.


\clearpage

\section{Part~VIII in plain English: why a chronology block earns its place}
A chronology of the universe is standard material and this one would not need
writing if it were only that. It is here because the framework's central
bookkeeping device --- the radial coordinate --- has to be shown to reproduce
the ordinary history before it can be used to say anything new about it, and a
translation table between cosmic time, energy scale, radial coordinate, scale
factor and redshift is the only honest way to show that.

The table is deliberately redundant. Every milestone appears in all five
coordinates at once, which makes the block long and makes it checkable: a
reader who suspects the radial reading of smuggling something in can convert
any row back to standard variables and see that nothing has been added.

The one thing the Part does claim is the shape of the packing. The early
universe occupies a vast radial interval and a negligible span of years; the
late universe occupies almost no radial interval and almost all of the years.
That is a statement about the geometry rather than about the physics of any
epoch, and it is what makes the descent picture of Part~IV a picture of cosmic
history rather than a metaphor laid over it. The worked conversions at the end
are there so the claim can be checked with a pencil.

\GGPart[Cosmological codebook]{Cosmological codebook: Derived / Derived (cond.) / Asserted}{The cosmological observables of the \(\pi\)-locked inventory, classified by their provenance under the \MSMF{}/\OGN{} principle; the per-parameter audit identifying which entries are derived, which are derived modulo a named lemma, and which remain asserted pending further constitutional analysis}


\section{Codebook overview and the \texorpdfstring{\MSMF{}/\OGN{}}{\MSMF{}/\OGN{}} principle on the cosmological-boundary stratum}
\label{sec:codebook-overview-cosmos}

This Part is the cosmological analogue of the twenty-eight-parameter codebook of \ANine{} for the Standard-Model sector. Where \ANine{} classified each electroweak/flavor observable as Derived (fully derived from non-physics axioms), Derived~(cond.) (derived modulo a single explicitly-named lemma), or Asserted (reported as the empirical input pending a structurally-named lemma), the present Part classifies twenty-one cosmological observables on the cosmological-boundary stratum. The \MSMF{}/\OGN{} principle --- minimal-structure-of-minimal-form / Ostrogradsky-graded-natural-degree --- selects, for each observable, the unique constitutional cost-minimum admissible under the inherited ledger.

The cosmological observables of interest are the \(\pi\)-locked density inventory \(\Omega_m = 1/\pi\), \(\Omega_\Lambda = 1 - 1/\pi\), \(\Omega_b = 1/(2\pi^2)\), \(\Omega_{\mathrm{DM}}\); the Hubble two-stage corridor parameters \(\Hzero\), the early-time and late-time sound horizons, and the corridor switch redshift; the scalar perturbation amplitude and tilt \(A_s\) and \(\ns\); the structure-formation parameters \(\sigeight\) and \(\Seight\); the radiation-content effective number \(\DeltaNeff\); the inflationary energy scale and the tensor-to-scalar ratio \(r\); the Kaxion mass and decay constant; and any feature-dependent supplementary channel (PBH mass spectrum slice, primordial-GW band).

The remainder of this Part provides the codebook itself. Section~\ref{sec:codebook-table-cosmos} states the twenty-one-observable codebook table. Section~\ref{sec:codebook-msmf-ogn} states the per-parameter \MSMF{}/\OGN{} audit. Section~\ref{sec:codebook-hubble-two-stage} states the Hubble two-stage corridor named theorem.

\subsection{The cosmological codebook}
\label{sec:codebook-table-cosmos}

Table~\ref{tab:cosmological-codebook} is the cosmological analogue of the twenty-eight-parameter Codebook Provenance Table of \ANine{}. It lists twenty-one cosmological observables and classifies each by its provenance under the \MSMF{}/\OGN{} principle on the cosmological-boundary stratum. The provenance map of Figure~\ref{fig:codebook-provenance} traces the same classification visually, carrying each observable back through its named conjecture to the single geometric origin.

The classification triage is identical to \ANine{}:
\begin{itemize}
 \item \textbf{Derived (D)}: fully derived from non-physics axioms with every factor traceable to upstream theorems of the Ultra Eight \GGTheory{} cohort. No empirical content is required to obtain the value.
 \item \textbf{Derived (cond.) (D\(^*\))}: derived modulo a single explicitly-named lemma. The lemma is an open research target with a definite mathematical statement; once proved, the entry is promoted to D.
 \item \textbf{Asserted (A)}: reported as the empirical input pending a structurally-named lemma whose proof would close the entry. The lemma is identified by name in the per-parameter audit of Section~\ref{sec:codebook-msmf-ogn}.
\end{itemize}

Of the twenty-one entries, two are fully derived (\(r\) and \(\Delta N_{\mathrm{eff}}\)), eighteen are derived modulo a named lemma (the three \(\pi\)-locked fractions \(\Omega_m\), \(\Omega_\Lambda\), and \(\Omega_b\), the dark-energy equation of state \(w\), \(\Omega_{\mathrm{DM}}\), the baryon asymmetry \(\eta_B\), the lithium-7 abundance, the two branches of the Hubble two-stage corridor, the decoupling redshift \(z_*\) (the atomic chain on the fractions; \S\ref{sec:two-redshifts}), \(\sigeight\), the two \(\Seight\) branches, the damping multipole \(\ell_D\), the Kaxion mass and decay constant, the scalar tilt \(\ns\) (Lemma~IV, revised), and the inflationary energy scale \(V_{\mathrm{inf}}^{1/4}\) (Lemma~V\('\))), and one remains asserted: the scalar amplitude \(A_s\), the launch sector's declared input, awaiting the heat-kernel computation of Lemma~V\('\). No observable in the inventory requires an adjustable continuous parameter: the input ledger is the Geometric Universal Five Constants \GUfive{} \(=\GUfiveSet\), the structural constants \((\pi, e)\), and the proved \OGNfive{} integer ledger \((M, N, L; J, K) = (1, 3, 5; 7, 35)\), per the zero-free-parameters discipline of \AFour{}.

{\footnotesize\setlength{\tabcolsep}{3pt}%
\begin{longtable}{p{0.115\textwidth}p{0.085\textwidth}p{0.135\textwidth}p{0.05\textwidth}p{0.275\textwidth}p{0.175\textwidth}}
\caption[Cosmological codebook: twenty-one observables by provenance]{Cosmological codebook: twenty-one observables of the cosmological-boundary stratum classified by their provenance under the \MSMF{}/\OGN{} principle. Empirical values from \citet{Planck2018} where applicable; the late-branch Hubble value from the H0DN consensus \citep{CasertanoH0DN2026}, with \citet{Riess2022} the leading single-team determination; the lensing \(S_8\) value from \citet{Asgari2021}; the 1998--1999 acceleration-discovery papers \citep{Riess1998,Perlmutter1999} are cited for the historical reintroduction of \(\Lambda\), not for parameter values. Status D = Derived (all factors traceable to upstream theorems); D\(^*\) = Derived (cond.), modulo a single named lemma; A = Asserted (pending a structurally-named lemma). The per-parameter audit in Section~\ref{sec:codebook-msmf-ogn} states each named lemma explicitly. The three \(\pi\)-locked inventory entries (\(\Omm\), \(\OmL\), \(\Omb\)) are Derived-conditional for their \emph{coupled \(\pi\)-form} under Conjecture~VIII; their absolute present-day values additionally inherit the observation epoch (\(T_0\)) as the single L4 calibration input (\S\ref{subsec:whynow-form-scale-epoch}).}
\label{tab:cosmological-codebook}\\
\toprule
\textbf{Observable} & \textbf{Symbol} & \textbf{Empirical value} & \textbf{Status} & \textbf{First-order derivation source} & \textbf{Named lemma (if D\(^*\) or A)} \\
\midrule
\endfirsthead
\toprule
\textbf{Observable} & \textbf{Symbol} & \textbf{Empirical value} & \textbf{Status} & \textbf{First-order derivation source} & \textbf{Named lemma (if D\(^*\) or A)} \\
\midrule
\endhead
\bottomrule
\endfoot

Matter density fraction & \(\Omm\) & \(0.315 \pm 0.007\) & D\(^*\) & \(\Omm = 1/\pi \approx 0.3183\); closed-packet topology of the cosmological-boundary stratum (Theorem~\ref{thm:omega-m-pi-topological}, Part~X of the present paper; bulk geometry inherited from \citealp{ArisakaA2GG6}) & Closed-packet-integration lemma (Conj.~VIII) \\
Cosmological constant fraction & \(\OmL\) & \(0.685 \pm 0.007\) & D\(^*\) & \(\OmL = 1 - 1/\pi \approx 0.6817\); residual \(F_{\GFF}\) on the late-time slice (Part~IV; \citealp{ArisakaA6Gi4}) & Same lemma (via unit-trace identity) \\
Dark-energy equation of state & \(w\) & \(-1.028 \pm 0.032\) \citep{Planck2018} & D\(^*\) & \(w = -1\) exactly; frozen residual vacuum export of the completed late-time geometric transition (Part~IV) & Vacuum-residual-exactness lemma (Conj.~IX) \\
Baryon density fraction & \(\Omb\) & \(0.0493 \pm 0.0006\) & D\(^*\) & \(\Omb = 1/(2\pi^2) \approx 0.0507\); baryonic gauge protection from the inherited Standard-Model ledger (\citealp{ArisakaA9SM}). \emph{Note:} the parameter-free prediction \(0.0507\) deviates from the empirical \(0.0493 \pm 0.0006\) at the \(2.3\sigma\) level. The D\(^*\) classification reflects that the structural derivation is closed with no remaining free parameter; the residual \(2.3\sigma\) discrepancy is a candidate next-order correction whose source (small baryonic-projection asymmetry inherited from the Standard-Model gauge sector) is identified for follow-up work. & Closed-packet-integration lemma \\
Dark-matter density fraction & \(\Omdm\) & \(0.2657 \pm 0.0070\) & D\(^*\) & \(\Omdm = \Omm - \Omb = 1/\pi - 1/(2\pi^2) \approx 0.2676\); Kaxion compact-angle relic (Part~V) & Kaxion-misalignment-completeness lemma \\
Baryon-to-photon ratio & \(\eta_B\) & \((6.12 \pm 0.04)\times10^{-10}\) \citep{Planck2018} & D\(^*\) & \(\eta_B \approx 7\times10^{-10}\) (central \(7.34\); \(\times3^{\pm1}\)) from resonant leptogenesis on the inherited \ANine{} neutrino sector (UV completion of the Weinberg operator); geometric throughput \(\varepsilon^{\mathrm{res}}\kappa = e^{-K/2}\), \(K = 35\) (Part~V, App.~\ref{appA:leptogenesis}) & Resonant-leptogenesis-normalization lemma (Conj.~VI; on-resonance condition) \\
Lithium-7 abundance & \({}^7\mathrm{Li}/H\) & \((1.6 \pm 0.3)\times10^{-10}\) \citep{Fields2011} & D\(^*\) & \(({}^7\mathrm{Li}/H)_{\mathrm{GG}} = \mathrm{SBBN}/\pi \approx 1.6\times10^{-10}\); the closed-packet \(1/\pi\) acting on the \(A=7\) port (Part~V) & Lithium-port-\(\pi\)-suppression lemma (Conj.~X) \\
Hubble constant, late branch & \(\Hzero^{\mathrm{late}}\) & \(73.50 \pm 0.81\) km/s/Mpc \citep{CasertanoH0DN2026} & D\(^*\) & Background \(\approx67\) plus the \GRIPin{} Local-Hole lift (\(+2\)--\(3\%\)); the remaining ladder excess is a named calibration systematic, not derived (Part~VI) & Hubble-two-stage-corridor-uniqueness lemma \\
Hubble constant, early branch & \(\Hzero^{\mathrm{CMB}}\) & \(67.4 \pm 0.5\) km/s/Mpc \citep{Planck2018} & D\(^*\) & Derived \(66.6\pm0.2\) from the \(\pi\)-locked inventory read against the measured physical densities (Section~\ref{sec:h0-derivation}); coincides with the CMB reading to \(1.2\%\) & Same lemma \\
Decoupling redshift & \(z_*\) & \(1089.92 \pm 0.25\) \citep{Planck2018} & D\(^*\) & Two quantities (\S\ref{sec:two-redshifts}): the corridor switch \(z_{\mathrm{sw}} = e^{7} = 1096.6\) is topological (\OGNfive{} integer \(J = 7\), Theorem~\ref{thm:hubble-two-stage-corridor}); the operational decoupling from the framework's own atomic chain on the \(\pi\)-locked inventory lands on the Planck value at the \(0.1\%\) level, \(0.4\%\) below the switch & Closed-packet-integration lemma (Conj.~VIII, via the \(\pi\)-locked inputs; switch ledger-fixed under Conj.~II) \\
Scalar perturbation tilt & \(\ns\) & \(0.965 \pm 0.004\) \citep{Planck2018} & D\(^*\) & \(\ns = 1 - 2/N^* = 0.9620\), computed on the \GDOS{}-forced plateau at the reheat-matched \(N^* = 52.6\) (Part~IV) & Plateau-class-and-prompt-crossing (Lemma~IV, revised) \\
Scalar perturbation amplitude & \(A_s\) & \((2.10 \pm 0.04) \times 10^{-9}\) \citep{Planck2018} & A & The launch sector's declared input; the \GDOS{} heat-kernel normalization on the same plateau is the named closure target (Section~\ref{sec:audit-asserted-cosmos}) & GDOS-heat-kernel-amplitude (Lemma~V\('\)) \\
Tensor-to-scalar ratio & \(r\) & \(< 0.036\) (95\%~CL) \citep{Planck2018} & D & derived plateau class + reheat-matched count \(\Rightarrow r = 12/N^{*2} = 0.0043\) \citep{Starobinsky1979} (Part~IV) & \emph{none} \\
Matter clustering amplitude & \(\sigeight\) & \(0.811 \pm 0.006\) \citep{Planck2018} & D\(^*\) & Growth-history modification from the early-time Hubble branch; same corridor theorem (Part~VI) & Hubble-two-stage-corridor-uniqueness lemma \\
\(S_8\) clustering parameter, early/CMB branch & \(\Seight^{\mathrm{CMB}}\) & \(0.832 \pm 0.013\) \citep{Planck2018} & D\(^*\) & Implied: \(\Seight = \sigeight \sqrt{\Omm/0.3}\); both factors derived above & Same lemma \\
\(S_8\) clustering parameter, late/lensing branch & \(\Seight^{\mathrm{late}}\) & \(0.759^{+0.024}_{-0.021}\) \citep{Asgari2021} & D\(^*\) & \(\Seight^{\mathrm{late}} = 0.78 \pm 0.01\); growth-history response of the two-stage corridor (Part~VI) & Same lemma \\
Damping-tail multipole & \(\ell_D\) & \(\approx 1345 \pm 15\) \citep{Planck2018} & D\(^*\) & \(\ell_D = 1350\pm3\) from the diffusion integral on the \(\pi\)-locked inventory (Part~VI); the BFFT overlay \(K(2\pi)^2 \approx 1382\) is carried separately as a conjecture (\(+2.3\%\) above the Boltzmann value) & Closed-packet-integration lemma (Conj.~VIII); the BFFT overlay gated by Conj.~VII \\
Effective neutrino number deviation & \(\DeltaNeff\) & \(0.04 \pm 0.18\) \citep{Planck2018} & D & Three Standard-Model neutrinos plus a thermal residue from the inherited particle paper (\citealp{ArisakaA9SM}); displayed as \(N_{\mathrm{eff}} = 3.046 + \DeltaNeff\) in Figure~\ref{fig:a10-dev-plot-linear} (the near-zero central value degenerates the fractional metric). The legacy early-dark-energy linear-response of Part~VI used a separate \emph{pre-recombination} \(\DeltaNeff \approx 0.30\); that estimate is now demoted to the CMB appendix as a non-unique cross-check, and the canonical value carried by the two-stage corridor and by this row is \(\DeltaNeff = 0.04\) (late-time/baseline). & \emph{none} \\
Kaxion mass & \(m_{\mathrm{Kaxion}}\) & \(8.1\pm0.5\,\mu\)eV (predicted; \(\nu_K\in[1.59,2.13]\) GHz) & D\(^*\) & \(\pi\)-locked abundance closure + cap mixing on the compact-angle relic; pseudo-Nambu--Goldstone mass from \citep{PecceiQuinn1977,Weinberg1978,Wilczek1978} template & Compact-angle-mass-uniqueness lemma; derivation executed in companion \PaperCOne{} (open remainder contracted to the discrete integer \(\Xi_{\mathrm{cap}}\in[35,47]\); entry kept D\(^*\) pending the lemma) \\
Kaxion decay constant & \(f_{\mathrm{Kaxion}}\) & \((3.0\pm0.3)\times10^{13}\) GeV raw; \(7.1\times10^{11}\) GeV laboratory (predicted) & D\(^*\) & Abundance closure on the holonomy decay constant; cap-mixing gearing \(\Xi_{\mathrm{cap}}\) & Same lemma \\
Inflationary energy scale & \(V_{\mathrm{inf}}^{1/4}\) & \(8.3\times10^{15}\,\mathrm{GeV}\) (derived class + declared amplitude) & D\(^*\) & The \GDOS{}-forced Starobinsky/\(\xi\) plateau one e-fold below \(\EGUT\), with graceful exit and prompt lock crossing (Part~IV; \citealp{Starobinsky1980} for the plateau template) & Plateau-class-and-prompt-crossing (Lemma~IV, revised)\\

\end{longtable}
}



% Provenance map of the cosmological codebook for GG-A10-Cosmos.
% PURE TikZ (no pgfplots). Top-to-bottom flow, 4 strata, status-coded.
\afterpage{%
\begin{landscape}
\null\vfill
\noindent
\begin{minipage}{\linewidth}
\centering
\makebox[\linewidth]{\resizebox{0.99\linewidth}{!}{%
\begin{tikzpicture}[
  font=\small,
  >={Stealth[length=2.2mm]},
  root/.style={rectangle, rounded corners=3pt, draw=black!75, thick,
               fill=black!7, align=center, inner sep=5pt, text width=80mm},
  mech/.style={rectangle, rounded corners=2.5pt, draw=black!60, thick,
               fill=blue!4, align=center, inner sep=3.5pt, text width=33mm,
               minimum height=12mm},
  mechflag/.style={mech, draw=red!70, fill=red!5, thick},
  obsD/.style={rectangle, rounded corners=2pt, draw=green!55!black, thick,
               fill=green!16, align=center, inner sep=2.5pt, text width=33mm,
               minimum height=9mm},
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                fill=blue!13, align=center, inner sep=2.5pt, text width=33mm,
                minimum height=9mm},
  obsA/.style={rectangle, rounded corners=2pt, draw=orange!75!black, thick,
               fill=yellow!35, align=center, inner sep=2.5pt, text width=33mm,
               minimum height=9mm},
  tag/.style={font=\footnotesize\itshape, text=black!55},
  flow/.style={->, draw=black!55, thick},
  flowf/.style={->, draw=red!65, thick},
]

% ---------- STRATUM 1: ROOT ----------
\node[root] (root) {\textbf{6D bulk geometry} (companion A2-GG6)\\[1pt]
  $+$ \textbf{OGN-5 integer ledger}: $K=35$,\; $J=7$\\[1pt]
  \scriptsize\itshape single parameter-free geometric origin};
\node[tag, below=0.5pt of root] {tag: A2-GG6};

% ---------- STRATUM 2: MECHANISMS ----------
\node[mechflag] (cVIII) at (-11.6,-2.9)
  {\textbf{Conj.\ VIII}\\Closed-packet integration\\[1pt]
   \scriptsize\textcolor{red!75!black}{\textbf{= EPOCH IDENTITY} ($t_0{-}75$ Myr)}};
\node[mech] (cIX) at (-7.0,-2.9)
  {\textbf{Conj.\ IX}\\Vacuum-residual exactness};
\node[mech] (cIVV) at (-2.3,-2.9)
  {\textbf{Lemma IV (rev.)\,/\,V$'$}\\Derived plateau (inflation)};
\node[mech] (cIII) at (2.4,-2.9)
  {\textbf{Conj.\ III}\\Compact-angle Kaxion relic};
\node[mech] (cII) at (7.0,-2.9)
  {\textbf{Conj.\ II}\\Two-stage Hubble corridor};
\node[mech] (a9) at (11.7,-2.9)
  {\textbf{Inherited SM ledger}\\(companion A9-SM)};

% Root -> mechanisms
\foreach \m in {cVIII,cIX,cIVV,cIII,cII,a9}{%
  \draw[flow] (root.south) -- (\m.north);}

% ---------- STRATUM 3/4: OBSERVABLES ----------
% Conj VIII (D*)
\node[obsDc] (Om)  at (-12.85,-5.8) {$\Omm=\tfrac{1}{\pi}$\\($\approx0.3183$)};
\node[obsDc] (Ob)  at (-12.85,-7.45) {$\Omb=\tfrac{1}{2\pi^2}$\\($\approx0.0507$; $2.3\sigma$ high)};
\node[obsDc] (Odm) at (-12.85,-9.1){$\Omdm=\tfrac{1}{\pi}-\tfrac{1}{2\pi^2}$};
\node[obsDc] (Li7) at (-12.85,-10.75){${}^7\mathrm{Li}/H=\mathrm{SBBN}/\pi$\\(Conj.\ X)};
\node[tag, right=1pt of Ob] {A9-SM};

% Conj IX (D*)
\node[obsDc] (OL) at (-8.27,-5.8) {$\OmL=1-\tfrac{1}{\pi}$\\($\approx0.6817$)};
\node[obsDc] (w)  at (-8.27,-7.45) {$w=-1$ (exact)};
\node[tag, right=1pt of OL] {A6-GRIP};

% Conj IV/V (Asserted) + r (Derived)
\node[obsDc] (ns)  at (-3.17,-5.8) {$\ns=0.9620$\\(plateau-derived)};
\node[obsA] (As)  at (-3.17,-7.45) {$A_s=2.1\times10^{-9}$\\(the one input)};
\node[obsDc] (Vinf)at (-3.17,-9.1){$V_{\mathrm{inf}}^{1/4}=8.3\times10^{15}$ GeV};
\node[obsD] (r)   at (-3.17,-10.75){$r=0.0043$\\(two-sided; bound $<0.036$)};

% Conj II (D*)
\node[obsDc] (H0l) at (7.04,-5.8) {$\Hzero^{\mathrm{late}}=73.0$};
\node[obsDc] (H0e) at (7.04,-7.45) {$\Hzero^{\mathrm{early}}=67.4$};
\node[obsDc] (s8)  at (7.04,-9.1){$\sigeight=0.811$};
\node[obsDc] (S8c) at (11.09,-5.8){$\Seight^{\mathrm{CMB}}=0.832$};
\node[obsDc] (S8l) at (11.09,-7.45){$\Seight^{\mathrm{late}}=0.78$};
\node[obsDc] (lD)  at (-12.85,-12.4){$\ell_D=1350\pm3$\\(Boltzmann diffusion)};
\node[tag, below=1pt of H0l] {C2-Hubble};
\node[obsDc] (zdec) at (-8.27,-12.4){$z_{\mathrm{dec}}\approx1090$ (atomic chain)\\ switch $z_{\mathrm{sw}}=e^{7}$};

% Conj III + A9
\node[obsDc] (mK)  at (2.38,-5.8) {$m_K=8.1\ \mu$eV\\(C1-Kaxion)};
\node[obsDc] (fK)  at (2.38,-7.45) {$f_{\mathrm{Kaxion}}=3.0\times10^{13}$ GeV};
\node[obsDc] (etaB)at (2.38,-9.1){$\eta_B\approx7\times10^{-10}$\\leptogenesis (A9), $e^{-K/2}$};
\node[obsD]  (Neff)at (2.38,-10.75){$\DeltaNeff=0.04$\\(from A9-SM)};

% ----- arrows: mechanism -> observables -----
\draw[flowf] (cVIII.south) -- (Om.north);
\draw[flowf] (cVIII.south) to[out=-90,in=120] (Ob.north west);
\draw[flowf] (cVIII.south) to[out=-90,in=120] (Odm.north west);
\draw[flowf] (cVIII.south) to[out=-90,in=130] (Li7.north west);

\draw[flow] (cIX.south) -- (OL.north);
\draw[flow] (cIX.south) to[out=-90,in=120] (w.north west);

\draw[flow] (cIVV.south) -- (ns.north);
\draw[flow] (cIVV.south) to[out=-90,in=120] (As.north west);
\draw[flow] (cIVV.south) to[out=-90,in=120] (Vinf.north west);
\draw[flow] (cIVV.south) to[out=-90,in=120] (r.north west);

\draw[flow] (cIII.south) -- (mK.north);
\draw[flow] (cIII.south) to[out=-90,in=120] (fK.north west);
\draw[flow] (a9.south) to[out=-90,in=60] (etaB.north east);
\draw[flow] (a9.south) to[out=-90,in=60] (Neff.north east);

\draw[flow] (cII.south) -- (H0l.north);
\draw[flow] (cII.south) to[out=-90,in=120] (H0e.north west);
\draw[flow] (cII.south) to[out=-90,in=120] (s8.north west);
\draw[flow] (cII.south) to[out=-60,in=90] (S8c.north);
\draw[flow] (cII.south) to[out=-60,in=120] (S8l.north west);
\draw[flowf] (cVIII.south) to[out=-90,in=150] (lD.north west);
\draw[flowf] (cVIII.south) to[out=-80,in=110] (zdec.north west);

% ---------- LEGEND ----------
\node[rectangle, rounded corners=3pt, draw=black!60, thick, fill=black!3,
      align=left, inner sep=6pt, text width=90mm, anchor=north west]
      (legend) at (4.4,-10.1)
  {\textbf{Status legend:}\;
   \tikz\node[obsD, text width=6mm, minimum height=4mm, inner sep=1pt]{};\, \textbf{green} = D\quad
   \tikz\node[obsDc, text width=6mm, minimum height=4mm, inner sep=1pt]{};\, \textbf{blue} = D$^*$\quad
   \tikz\node[obsA, text width=6mm, minimum height=4mm, inner sep=1pt]{};\, \textbf{gold} = A\\[3pt]
   \footnotesize D$^*$ entries are conditional on their named input;
   \textcolor{red!75!black}{Conjecture VIII (closed-packet $1/\pi$) closes as
   the epoch identity: $\Omega_m = 1/\pi$ exact at $t_0 - 75$ Myr; residual:
   lemma VIII-a (exact to $10^{-15}$) $+$ the declared epoch input.}};

\end{tikzpicture}%
}}
\captionof{figure}[Provenance map of the cosmological codebook.]{\textbf{Provenance map of the
  cosmological codebook of \ATen.} A single parameter-free geometric origin
  (Stratum~1: the 6D bulk geometry of companion A2-GG6 together with the
  \OGNfive{} integer ledger $K=35$, $J=7$) flows downward through a row of named
  conjectures and inherited mechanisms (Stratum~2) to all 21 cosmological
  observables (Strata~3--4). Colors follow the status legend (lower right);
  \textcolor{red!75!black}{Conjecture~VIII (closed-packet $1/\pi$), flagged in
  red, is the load-bearing conditional.} The color-coding, the epoch identity
  behind Conjecture~VIII, the two-quantity decoupling entry, and the
  companion-paper tags are unpacked in the \emph{Reading conventions}
  paragraph following Figure~\ref{fig:a10-dev-plot-linear}.}
\label{fig:codebook-provenance}
\end{minipage}
\vfill
\end{landscape}
\begin{landscape}
\noindent
\begin{minipage}{\linewidth}
\centering
\captionof{table}[Data table for the parameter-audit Dev plot.]{\textbf{The complete numerical bottom line of \ATen{}.} The twenty-two cosmological observables carrying an empirical comparison --- eighteen independent predictions (filled markers in Figure~\ref{fig:a10-dev-plot-linear}) and four anchored by construction (open markers); the inflationary-spectrum block reflects the two-basin derivation. Column definitions, the provenance triage (\textbf{D}/\textbf{D}$^*$/\textbf{A}), reading conventions, sources, and the epoch calibration: the \emph{Reading conventions} paragraph following Figure~\ref{fig:a10-dev-plot-linear}.}
\label{tab:dev-plot-data}

{\renewcommand{\arraystretch}{1.12}\footnotesize\setlength{\tabcolsep}{3.5pt}%
\resizebox{0.78\linewidth}{!}{%
\begin{tabular}{>{\hspace{1.4em}}lllrrc}
\toprule
\textbf{Entry} & \textbf{GG prediction} & \textbf{Observed} & \textbf{Dev} & \textbf{Pull} & \textbf{Status / marker} \\
\midrule
\multicolumn{6}{l}{\textbf{\textit{The $\pi$-locked inventory}}}\\
$\Omm$ & $1/\pi = 0.3183$ & $0.315 \pm 0.007$ & $+1.05\%$ (cal.\ input) & $+0.5$ (---) & D$^*$ / filled \\
$\OmL$ & $1 - 1/\pi = 0.6817$ & $0.685 \pm 0.007$ & $-0.48\%$ (cal.\ $0.00$) & $-0.5$ ($0.0$) & D$^*$ / filled \\
$w$ & $-1$ (exact) & $-1.028 \pm 0.032$ & $+2.72\%$ & $+0.9$ & D$^*$ / filled \\
$\Omb$ & $1/(2\pi^2) = 0.05066$ & $0.0493 \pm 0.0006$ & $+2.76\%$ (cal.\ $+1.69$) & $+2.3$ ($+1.4$) & D$^*$ / filled \\
$\Omdm$ & $1/\pi - 1/(2\pi^2) = 0.2676$ & $0.2657 \pm 0.0070$ & $+0.73\%$ (cal.\ $-0.31$) & $+0.3$ ($-0.1$) & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{Baryogenesis and nucleosynthesis}}}\\
$\eta_B$ & $\approx 7 \times 10^{-10}$ (central 7.34; from $K{=}35$) & $(6.12 \pm 0.04) \times 10^{-10}$ & $+20\%$ & $\ll1$ & D$^*$ / filled \\
${}^7\mathrm{Li}/H$ & $\mathrm{SBBN}/\pi = 1.59 \times 10^{-10}$ & $(1.6 \pm 0.3) \times 10^{-10}$ & $-0.5\%$ & $0.0$ & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{The Hubble constant, recombination, and the acoustic spectrum}}}\\
$\Hzero^{\mathrm{CMB}}$ & $66.6 \pm 0.2$ (derived, inventory) & $67.4 \pm 0.5$ & $-1.19\%$ & $-1.5$ & D$^*$ / filled \\
$\Hzero^{\mathrm{late}}$ & $68.5 \pm 0.6$ (background + Local-Hole lift) & $73.50 \pm 0.81$ & $-6.80\%$ & $-5.0$ & D$^*$ / filled \\
$z_*$ & $1091.1$ (atomic chain, raw; switch $e^7$ annotated) & $1089.92 \pm 0.25$ (decoupling) & $+0.11\%$ & $+0.6$ & D$^*$ / filled \\
$\ell_D$ & $1350 \pm 3$ (Boltzmann, $\pi$-inventory) & $\approx 1345 \pm 15$ & $+0.37\%$ & $+0.3$ & D$^*$ / filled \\
$\omega_b/\omega_m$ & $1/2\pi = 0.15915$ ($h$-free, epoch-free) & $0.15709 \pm 0.0016$ & $+1.31\%$ & $+1.3$ & D / filled \\
$100\,\theta_*$ & $1.0362$ ($r_s/D_M$, inventory) & $1.04109 \pm 0.00030$ & $-0.47\%$ & $-4.3$ & D$^*$ / filled \\
peak ratio $2/1$ & $0.451$ (image of $1/2\pi$) & $0.453 \pm 0.008$ & $-0.30\%$ & $-0.2$ & D$^*$ / filled \\
peak ratio $3/1$ & $0.440$ (image of $1/2\pi$) & $0.443 \pm 0.009$ & $-0.78\%$ & $-0.4$ & D$^*$ / filled \\
\multicolumn{6}{l}{\textbf{\textit{The inflationary spectrum (primary GUT-adjacent plateau; two-basin descent, $N_2=0$)}}}\\
$\ns$ & $1 - 2/N^* = 0.9620$ ($N^*{=}52.6$, reheat-matched) & $0.9649 \pm 0.0042$ & $-0.30\%$ & $-0.7$ & D$^*$ / filled \\
$r$ & $12/N^{*2} = 0.0043$ & $< 0.036$ (95\%, BK18) & $\times8$ below bound & --- & D$^*$ / filled \\
$A_s$ & anchored (the one input; warp cand.\ $75\%$, flagged) & $(2.10 \pm 0.04) \times 10^{-9}$ & $0$ & $0$ & A / open \\
\multicolumn{6}{l}{\textbf{\textit{Clustering and radiation}}}\\
$\sigeight$ & anchored & $0.811 \pm 0.006$ & $0$ & $0$ & D$^*$ / open \\
$\Seight^{\mathrm{CMB}}$ & anchored & $0.832 \pm 0.013$ & $0$ & $0$ & D$^*$ / open \\
$\Seight^{\mathrm{late}}$ & $0.78 \pm 0.01$ (corridor response) & $0.759^{+0.024}_{-0.021}$ & $+2.77\%$ & $+0.8$ & D$^*$ / filled \\
$N_{\mathrm{eff}}$ & anchored ($3.046 + \DeltaNeff$) & $3.086 \pm 0.18$ & $0$ & $0$ & D / open \\
\bottomrule
\end{tabular}%
}}
\par\bigskip

\end{minipage}
\end{landscape}
\noindent
\begin{minipage}{\textwidth}
\centering
\includegraphics[trim=34 22 13 21,clip,width=\textwidth]{a10_dev_plot_linear.pdf}
\par\bigskip
\captionof{figure}[Parameter-audit deviation plot for the cosmological codebook.]{\textbf{A10-Cosmos parameter-audit Dev plot.} Fractional deviation $(\mathrm{pred}-\mathrm{obs})/|\mathrm{obs}|$ for the twenty-two observables of Table~\ref{tab:dev-plot-data}, with nested precision bands at $|\mathrm{Dev}|<10^{-3},\,10^{-2}$. \emph{Filled} markers are independent predictions; \emph{open} markers are anchored by construction ($\mathrm{Dev}=0$); color encodes status (green = Derived, blue = Derived (cond.), gold = Asserted); \emph{violet open diamonds} mark the epoch-calibrated positions of the four $\pi$-inventory rows. Error bars, in-panel annotations, and the $z_*$/corridor-switch marker follow the \emph{Reading conventions} paragraph below; full data, sources, and uncertainties: Table~\ref{tab:dev-plot-data}.}
\label{fig:a10-dev-plot-linear}
\end{minipage}
\clearpage
}

\paragraph{Reading conventions for Figures~\ref{fig:codebook-provenance} and~\ref{fig:a10-dev-plot-linear} and Table~\ref{tab:dev-plot-data}.}
In the provenance map (Figure~\ref{fig:codebook-provenance}) the colors are
\emph{not} cosmetic: every blue observable inherits the open status of its
mechanism, and Conjecture~VIII closes as the \emph{epoch identity}
($\Omega_m = 1/\pi$ exact at $t_0 - 75$~Myr; residual lemma VIII-a exact to
$10^{-15}$), so the matter-density column is conditional on that single
declared input. Only $r=0.0043$ (the plateau's two-sided tensor target) and
$\DeltaNeff=0.04$ (inherited from A9-SM) are genuinely Derived; the tilt
$n_s=0.9620$ and the scale $V^{1/4}$ are plateau-derived (D$^*$,
Lemma~IV revised), with $A_s$ the launch sector's one declared input. The
lithium-7 abundance inherits the same closed-packet $1/\pi$: Conjecture~X is
its $A=7$ instance. Companion-paper tags (A2-GG6, A6-GRIP, A9-SM, C1-Kaxion,
C2-Hubble) mark where each result is developed in full. In
Table~\ref{tab:dev-plot-data}, the columns are: the GG prediction in closed
form; the observed value with its $1\sigma$ uncertainty; the relative
deviation $\mathrm{Dev} = (\mathrm{pred}-\mathrm{obs})/|\mathrm{obs}|$; the
\emph{pull} $(\mathrm{pred}-\mathrm{obs})/\sigma$ with $\sigma =
\sqrt{\sigma_{\rm obs}^2+\sigma_{\rm th}^2}$; and the provenance triage
(\textbf{D} = Derived; \textbf{D}$^*$ = Derived-conditional, closed modulo a
single named lemma; \textbf{A} = Asserted) with the marker type.
Hubble constants are in km/s/Mpc; the background $\Hzero^{(0)} = 66.6\pm0.2$ is derived from the $\pi$-locked inventory (Section~\ref{sec:h0-derivation}) --- and independently, with zero sky input, by the geometric Route~I at $66.8\pm0.2$, its coefficient $\zeta_\Lambda=\pi/175$ codebook-selected and pre-registered --- and scored against the CMB reading $67.4$ (the coherent $\pi$-offset, Section~\ref{subsec:pi-offset}); the local-ladder entry scores the Local-Hole-lifted background against the H0DN consensus $73.50\pm0.81$, its $-6.8\%$ deviation the named ladder systematic (Section~\ref{sec:h0tension}). The four acoustic rows are CLASS-verified images of the same fractions (correlated, not additional freedom); anchored entries sit at $\mathrm{Dev}=0$ by construction; $\eta_B$ carries an order-of-magnitude transport band; $\ell_D$ is definition-dependent at the percent level. The $z_*$ row scores the framework's audited atomic-chain decoupling prediction ($1091.1$, raw, zero fitted parameters) against the measured value, landing at the $0.1\%$ level, with the topological corridor switch $e^7$ --- a distinct quantity sitting $0.4\%$ above it in e-folds (\S\ref{sec:two-redshifts}) --- annotated in the plot, not scored. In the plot, the inner thick bar is the empirical $\sigma_{\rm obs}/|\mathrm{obs}|$ and the outer thin bar the combined $\sqrt{\sigma_{\rm obs}^2+\sigma_{\rm th}^2}/|\mathrm{obs}|$, with entries carrying a theory tolerance $\sigma_{\rm th}$ annotated inside the panel; the radiation entry is shown as the total $N_{\mathrm{eff}}$. \emph{Epoch calibration} (the `cal.'\ entries in the table; violet diamonds in the plot): under Planck central values the exact crossing $\Omega_m = 1/\pi$ occurred $75$~Myr before the present ($\pm160$~Myr, i.e.\ $0.5\sigma$; \S\ref{subsec:whynow-form-scale-epoch}), and all matter fractions ($\Omega_m$, $\Omega_b$, $\Omega_{\rm dm}$) decrease with time while $\Omega_\Lambda$ grows --- so, using that single declared input once, the four inventory deviations collapse to (input, $0.00$, $+1.69$, $-0.31$)\%, every remaining residual an epoch-independent ratio within about $1\sigma$. The clustering of three independent predictions ($w$, $\Omb$, $\Seight^{\mathrm{late}}$) near $+2.7\%$ is examined in Section~\ref{subsec:second-order-residual-cosmos}.

\paragraph{Counting summary.} {\sloppy The twenty-one entries decompose as: two Derived (D); eighteen Derived (cond.) (D\(^*\)); one Asserted (A). Ten distinct named lemmas appear: (i) Kaxion-\allowbreak misalignment-\allowbreak completeness, (ii) Hubble-\allowbreak two-\allowbreak stage-\allowbreak corridor-\allowbreak uniqueness, (iii) compact-\allowbreak angle-\allowbreak mass-\allowbreak uniqueness, (iv) plateau-\allowbreak class-\allowbreak and-\allowbreak prompt-\allowbreak crossing (Lemma~IV, revised), (v) GDOS-\allowbreak heat-\allowbreak kernel-\allowbreak amplitude (Lemma~V\('\), replacing the retired TeV-\allowbreak scale-\allowbreak inflation-\allowbreak uniqueness), (vi) resonant-\allowbreak leptogenesis-\allowbreak normalization, (vii) BFFT-\allowbreak damping-\allowbreak knee-\allowbreak identification, (viii) closed-\allowbreak packet-\allowbreak integration, (ix) vacuum-\allowbreak residual-\allowbreak exactness, (x) lithium-\allowbreak port-\allowbreak \(\pi\)-\allowbreak suppression. Each lemma is an open research target with a precise mathematical statement --- the Roman numeral is its entry in the conjecture registry of Section~\ref{sec:compressed-audit-roadmap}, so ``Lemma~IV (revised)'' and ``Conjecture~IV (revised)'' name the same statement in its derivational and epistemic roles; once any one is proved, one or more table entries are promoted from D\(^*\) or A to D. The status of the cosmological codebook is therefore: two entries fully closed, eighteen awaiting nine named lemmas, and one --- the scalar amplitude, the launch sector's declared input --- awaiting the single heat-kernel computation of Lemma~V\('\). No entry requires an adjustable continuous parameter.\par}

\paragraph{Post-hoc-fit defense.} The reader who arrives at the \(\pi\)-locked inventory line for the first time will rightly ask whether \(\Omm = 1/\pi\) is a post-hoc fit to the Planck data. The defense is structural and follows the same logic used in \ANine{} for the Standard-Model parameter sheet. First, the value \(1/\pi\) for the matter fraction is forced by the closed-packet topology of the cosmological-boundary stratum, which is itself forced by the \OGNfive{} integer ledger of \AFour{}; the value is not chosen. Second, the same single topology forces \emph{three} entries (\(\Omm\), \(\OmL\), \(\Omb\)) simultaneously; a post-hoc fit explains one number, not three coupled numbers. Third, the \(\Omb\) entry depends on \(2\pi^2\), not on \(\pi\); the closed-packet topology must produce the right \emph{combination} of powers of \(\pi\), and it does. Fourth, every other entry in the codebook is independently constrained by the Hubble two-stage corridor, the inherited neutrino-and-CP ledger, or a named lemma; none of them is a free parameter that could have been tuned. The post-hoc-fit defense will be developed in full in the per-parameter audit below, drawing on the parallel defense in \ANine{}.

\paragraph{What the codebook does not yet contain.} Three classes of entries remain outside the present codebook and will be added as the program matures. First, the feature-dependent supplementary channels (the primordial-black-hole mass-window slice; the primordial-gravitational-wave band in the Starobinsky channel) appear in Part~VII but are not codebook entries because they are not single-parameter observables. Second, the remaining BBN light-element abundances (D/H, \(^3\)He/H, \(^4\)He/H) appear in Part~V but are not codebook entries because they are coupled-output observables of a small set of inputs; the lithium-7 abundance is the exception, promoted to a codebook entry above because its closed-packet \(1/\pi\) suppression is a single-number, parameter-free prediction. Third, the early-galaxy abundance statistics relevant to the JWST audit appear in Part~VI but are coupled to the growth-history modification rather than a separate parameter. The codebook as it stands is the headline parameter sheet; the coupled-observable audits live in the respective Parts and the consolidated benchmark audit of Appendix~\ref{app:benchmark-audit}.

\subsection{Per-parameter \texorpdfstring{\MSMF{}/\OGN{}}{\MSMF{}/\OGN{}} audit}
\label{sec:codebook-msmf-ogn}

This Section walks each row of Table~\ref{tab:cosmological-codebook} in turn and states, for each entry, the upstream-theorem chain that closes it (for the Derived entries) or the named lemma whose proof would close it (for the Derived (cond.) and Asserted entries). The principle that selects, for each entry, the unique constitutional cost-minimum is the \MSMF{}/\OGN{} principle, first formalized in \AFour{} and applied parameter-by-parameter in \ANine{} for the Standard-Model sector. The present audit applies the same principle to the cosmological-boundary stratum.

\subsubsection{Statement of the \texorpdfstring{\MSMF{}/\OGN{}}{\MSMF{}/\OGN{}} principle on the cosmological-boundary stratum}
\label{sec:msmf-ogn-statement-cosmos}

The \MSMF{}/\OGN{} principle in cosmology takes the same operational form as in the particle sector. On any stratum of the boundary moduli space, the constitutional cost functional \(\mathcal F = F_{\GEP} + F_{\CEP}\) (the \GFF{} ledger of \ASix{}) defines a finite-codimension critical manifold whose admissible saddles are the candidate Geometric Attractors. The \MSMF{}/\OGN{} principle then asserts:

\begin{principle}[\MSMF{}/\OGN{} on the cosmological-boundary stratum]\label{princ:msmf-ogn-cosmos}
On the cosmological-boundary stratum of the boundary moduli space, the constitutional cost functional \(\mathcal F = F_{\GEP} + F_{\CEP}\) admits a unique cost-minimum saddle on every admissible sub-stratum corresponding to a cosmological observable. Equivalently: for each admissible cosmological observable, there is one and only one solution to \(\nabla \mathcal F = 0\) on the relevant sub-stratum that satisfies the admissibility constraints. The uniqueness claim is the empirical content of the principle; it is in general non-trivial and is the open research target for those observables not yet fully closed.
\end{principle}

The principle decomposes the cosmological observables into three operational closure modes, depending on the present state of evidence for the uniqueness claim:

\begin{itemize}
 \item[(D)] \emph{Derived}. The uniqueness claim is proved: the cost-minimum on the relevant sub-stratum is unique, and the value is obtained from upstream theorems of the Ultra Eight \GGTheory{} cohort via a finite citation chain. No empirical content is required to obtain the value.
 \item[(D\(^*\))] \emph{Derived (cond.)}. The uniqueness claim is conditional on a single explicitly-named lemma about the sub-stratum's geometry. The lemma is an open research target with a precise mathematical statement; once proved, the entry is promoted to D.
 \item[(A)] \emph{Asserted}. The observable is reported as the empirical input pending a structurally-named lemma whose proof would identify the underlying cost-minimum on the relevant sub-stratum and thereby promote the entry to D\(^*\) or D.
\end{itemize}

Thus the principle is a falsifiable uniqueness claim (the cost-functional saddle is unique on each admissible sub-stratum), and the three closure modes are operational summaries of the state of evidence for that claim.

The discipline that follows from Principle~\ref{princ:msmf-ogn-cosmos} is the zero-free-parameters discipline of \AFour{}: at no point in the audit is an adjustable continuous parameter introduced. The input ledger consists of the Geometric Universal Five Constants \GUfive{} \(=\GUfiveSet\), the structural constants \((\pi, e)\), and the proved \OGNfive{} integer ledger \((M, N, L; J, K) = (1, 3, 5; 7, 35)\). Every cosmological observable is either a constructive function of these inputs (closure modes D and D\(^*\)) or an honest empirical input awaiting such a function (closure mode A).

\subsubsection{Audit of the two Derived (D) entries}
\label{sec:audit-derived-cosmos}

\paragraph{D9.\ Tensor-to-scalar ratio \(r\).}
The derived plateau class (the \GDOS{} non-conformal route of Part~IV) fixes the tensor sector through the Starobinsky relation \(r = 12/N^{*2}\) \citep{Starobinsky1980}; evaluated at the reheating-matched count \(N^* = 52.6\), it gives \(r = 0.0043\) --- a factor eight below the current observational bound (\(r < 0.036\) at 95\,\% CL) and several times above the projected Simons-Observatory/LiteBIRD floor, hence decisive within the decade. The class and the count are underwritten by the revised Lemma~IV; the relation \(12/N^{*2}\) itself carries no further lemma, which is why the row is D rather than D\(^*\): the entry is the Starobinsky relation, evaluated at the derived count. Citation chain: \(\AFive{}\) (\GDOS{} class) \(\to\) Lemma~IV (revised) \(\to\) \citep{Starobinsky1980} \(\to\) present.

\paragraph{D12.\ Effective neutrino number deviation \(\DeltaNeff\).}
The empirical value \(\DeltaNeff = 0.04 \pm 0.18\) is consistent with three Standard-Model neutrinos plus a small thermal residue from late-decoupling species in the inherited particle ledger \citep{ArisakaA9SM}. The exact value of the residue is computable from the inherited neutrino sector without any free parameter. No remaining lemma. Citation chain: \ANine{} \(\to\) present.

\paragraph{Two-entry counting summary.}
The two D entries are the derived tensor ratio (\(r = 0.0043\), the Starobinsky relation at the reheating-matched count) and the effective-neutrino-number deviation (\(\DeltaNeff\)) inherited from the particle paper. Each citation chain has finite length and terminates at upstream theorems; no chain references an empirical fit.

\subsubsection{Audit of the eighteen Derived (cond.) (D\(^*\)) entries --- nine named lemmas}
\label{sec:audit-d-star-cosmos}

Sixteen of the eighteen D\(^*\) entries are organized under eight named lemmas (I, II, III, VI, VII, VIII, IX, X) and audited in this block; the remaining two --- the tilt (A7) and the inflationary scale (A15), promoted from A by the two-basin derivation of Part~IV --- ride on the revised Lemma~IV and are audited with the inflationary sector below. Seven of the eight lemmas here close codebook rows directly, while Lemma~VII gates only the interpretive BFFT overlay of the damping row D18, whose codebook value is closed by Lemma~VIII. Each lemma is a precise mathematical statement about a sub-stratum of the boundary moduli space; each is an open research target.

\paragraph{Lemma I: Kaxion-misalignment-completeness.}
This lemma closes D4 (dark-matter density fraction \(\Omdm\)).

\begin{conjecture}[Kaxion-misalignment-completeness]\label{conj:kaxion-completeness}
On the compact-angle sub-stratum of the cosmological-boundary stratum, the misalignment-mechanism relic-abundance functional, evaluated at the constitutional cost-minimum of the initial compact-angle misalignment, equals \(\Omm - \Omb = 1/\pi - 1/(2\pi^2)\).
\end{conjecture}

The constitutional cost-minimum of the initial compact-angle misalignment is the unique maximum of the cost functional \(\mathcal F\) restricted to the compact-angle sub-stratum. The empirical value of \(\Omdm = 0.2657 \pm 0.0070\) is in 0.27-\(\sigma\) agreement with the predicted value \(1/\pi - 1/(2\pi^2) \approx 0.2676\). Once Conjecture~\ref{conj:kaxion-completeness} is proved, row D4 is promoted to D. Upstream citation chain: \(\ATwo{}\) (compact holonomy) \(\to\) \(\AThree{}\) (\BIsix{} anomaly freedom on the compact angle) \(\to\) \citep{PecceiQuinn1977,Weinberg1978,Wilczek1978} (axion template) \(\to\) the present audit.

\paragraph{Lemma II: Hubble-two-stage-corridor-uniqueness.}
This lemma closes D5 (\(\Hzero^{\mathrm{CMB}}\)), D6 (\(\Hzero^{\mathrm{late}}\)), D10 (\(\sigeight\)), D11 (\(\Seight^{\mathrm{CMB}}\)), and D19 (\(\Seight^{\mathrm{late}}\)): five codebook entries simultaneously.

\begin{conjecture}[Hubble-two-stage-corridor-uniqueness]\label{conj:hubble-corridor}
The cosmological-boundary stratum admits exactly one corridor of branch coordinates parameterized by the corridor-switch redshift \(z_*\), such that (i) the early-time branch yields the CMB-projected Hubble value \(\Hzero^{\mathrm{CMB}} \approx 67.4\) km/s/Mpc, (ii) the late-time branch yields the distance-ladder Hubble value \(\Hzero^{\mathrm{late}} \approx 73.0\) km/s/Mpc, and (iii) the corridor switch is fixed by the inherited recombination geometry and the \OGNfive{} integer \(J=7\) at \(z_* = e^{J} = e^{7} \approx 1097\) (the atomic decoupling lies \(0.4\%\) below, \S\ref{sec:two-redshifts}). The corridor is unique; no continuous parameter is free to be tuned.
\end{conjecture}

The constitutional cost-minimum that selects the corridor is the unique saddle of \(\mathcal F\) on the joint sub-stratum of (sound-horizon, growth-history) branch coordinates. Once Conjecture~\ref{conj:hubble-corridor} is proved, five codebook rows are promoted to D simultaneously, and the \(H_0\) tension and \(S_8\) tension reduce to consistency statements rather than tensions. The canonical theorem-form statement of this corridor argument is the subject of Section~\ref{sec:codebook-hubble-two-stage}. Upstream citation chain: \(\ATwo{}\) (bulk) \(\to\) \(\ASix{}\) (\GRIP{} cascade) \(\to\) \citep{Planck2018,Riess2022} (empirical anchors) \(\to\) the present audit.

\paragraph{Lemma III: Compact-angle-mass-uniqueness.}
This lemma closes D13 (\(m_{\mathrm{Kaxion}}\)) and D14 (\(f_{\mathrm{Kaxion}}\)).

\begin{conjecture}[Compact-angle-mass-uniqueness]\label{conj:compact-angle-mass}
On the compact-angle sub-stratum, the \OGNfive{} integer ledger \((K = 35)\) acts on the compact-angle pseudo-scalar such that the unique cost-minimum solution is a single physical decay constant \(f_{\mathrm{Kaxion}} = (7.1\pm0.5)\times10^{11}\) GeV --- the bulk Green--Schwarz normalization reduced by the integer cap loading \(\Xi_{\mathrm{cap}}\in[35,47]\) (benchmark \(43\)) --- and hence the mass \(m_{\mathrm{Kaxion}} = 8.1\pm0.5\,\mu\)eV (\(\nu_K\in[1.59,2.13]\) GHz). The misalignment angle is then fixed as an output of the \(\pi\)-locked abundance, and the exact integer value of \(\Xi_{\mathrm{cap}}\) is the sole remaining discrete unknown.
\end{conjecture}

The lock factor \(e^{-K} = e^{-35}\) descends directly from the \OGNfive{} integer ledger of \AFour{}: the integer \(K = 35\) appears in the constitutional input set as one of the five proved \OGNfive{} integers \((M, N, L; J, K) = (1, 3, 5; 7, 35)\). The full derivation of the resulting Kaxion mass and decay constant has now been executed in the companion paper \PaperCOne{} \citep{ArisakaC1Kaxion}: the geometry fixes \(f_K\) to \(\pm6\%\) (with the misalignment angle then an output of the observed abundance), exact anomaly matching fixes the photon coupling on the DFSZ line, and the open remainder of the conjecture has contracted to the integer value of \(\Xi_{\mathrm{cap}}\) within \([35,47]\). Upstream citation chain: \(\AFour{}\) (\OGNfive{} ledger) \(\to\) \(\ATwo{}\) (compact holonomy) \(\to\) \PaperCOne{} \(\to\) the present audit.

\paragraph{Lemma VI: Resonant-leptogenesis-normalization.}
This lemma closes D17 (baryon-to-photon ratio \(\eta_B\)). The baryon asymmetry is produced by resonant leptogenesis on the inherited \ANine{} neutrino-and-\(CP\) ledger: the heavy Majorana sector is the ultraviolet completion of the \ANine{} Weinberg dimension-five operator at the lock scale, its Yukawa couplings fixed by seesaw matching to the \ANine{} light-neutrino masses and its near-degeneracy fixed by the \OGNfive{} integer \(K = 35\), so that the resonant \(CP\) asymmetry and the washout efficiency combine to the geometric throughput \(\varepsilon^{\mathrm{res}}\,\kappa = e^{-K/2}\) (full derivation in Part~V and Appendix~\ref{appA:leptogenesis}). The lemma asserts that the \emph{on-resonance condition} --- the single named input, the near-degeneracy \(|M_i - M_j| \simeq \Gamma/2\) made plausible but not enforced by the \(1/K\) level splitting --- holds at the lock scale, so that the full Boltzmann-transport normalization converges to the geometric value with its residual \(O(1)\) coefficients fixed. The geometric value \(7.34\times10^{-10}\) sits \(+20\%\) from the empirical \(\eta_B = (6.12 \pm 0.04)\times10^{-10}\) \citep{Planck2018}, inside the declared order-of-magnitude band. Once the lemma is proved, row D17 is promoted to D. Citation chain: \AFour{} (\(K = 35\)) \(\to\) \ANine{} (Weinberg operator + \(CP\) ledger) \(\to\) Part~V \(\to\) present audit.

\paragraph{Lemma VII: BFFT-damping-knee-identification.}
This lemma no longer closes a codebook row; it gates the \emph{interpretive BFFT overlay} of D18 (damping-tail multipole \(\ell_D\)). The codebook value of D18 is the Boltzmann diffusion prediction of the \(\pi\)-locked inventory, \(\ell_D = 1350\pm3\) (\(+0.4\%\) against the observed \(1345\pm15\)), and is closed --- with the fractions it descends from --- by Lemma~VIII. The separately proposed locked relation \(\ell_D = K(2\pi)^2 \approx 1382\) identifies the ultraviolet knee with the first non-trivial BFFT cutoff of the multiscale throat (Part~VI); it sits \(+2.7\%\) above the observed value and \(+2.3\%\) above the framework's own Boltzmann value, so the lemma must now \emph{explain that gap}, not merely assert that the projection of the diffusion scale onto the BFFT bus is exact at leading order. If proved in that stronger form, the overlay is promoted from conjecture to structure; if refuted, the overlay falls away while D18 stands on the diffusion integral. Citation chain: \AFour{} (\(K = 35\)) \(\to\) \ASeven{} (BFFT bus) \(\to\) Part~VI Boltzmann audit \(\to\) present audit.

\paragraph{Lemma VIII: Closed-packet-integration.}
This lemma closes D1--D3 (\(\Omm\), \(\OmL\), \(\Omb\)) and, through them, D18 (\(\ell_D\), whose diffusion-integral value has no input beyond the fractions and standard recombination physics) and D21 (the decoupling redshift \(z_*\), whose atomic-chain value likewise has no input beyond the fractions and standard atomic physics; \S\ref{sec:two-redshifts}). It asserts that the closed-packet integral of the \BIsix{} four-form on the cosmological-boundary stratum evaluates to \(1/\pi\) (matter share) and \(1/(2\pi^2)\) (baryonic weight, \(1/\mathrm{Vol}(S^3)\)) with an exhibited, verifiable normalization --- completing the programmatic Steps 3--4 of Theorem~\ref{thm:omega-m-pi-topological} (see the Proof-status note there). The \(\pi\)-locked values are D\(^*\) entries of half-percent empirical precision: coupled form derived, epoch declared (\S\ref{subsec:whynow-form-scale-epoch}), residual lemma VIII-a exact up to the \(\sim e^{-K}\) floor.

\emph{The VIII-a/VIII-b split.} The conjecture decomposes into two parts of different status. \emph{VIII-a (delocalization and share robustness of the stationary measure):} on the matter-sector phase moduli --- the electroweak order-parameter directions (the Wilson--Hosotani phases of \ASix{}) being non-flat by design, settled into the GA, and excluded from the domain --- the frozen stationary \GDOS{} measure leaves every counted direction delocalized --- the flavor directions \emph{exactly} protected by rephasing invariance, the physical (Jarlskog-class) combinations tilted at most \(\sim 10^{-28}\), the leading term the gravity-mediated floor \(\sim e^{-K} \approx 10^{-15}\) (the mechanism home of the original candidate \(S_w = K = 35\); its \(O(1)\) prefactor the one open coefficient) --- and the share dictionary consumes the measure only through counting and per-component fiber volume, both invariant under any normalized tilt --- uniformity is sufficient but not necessary. Named as the GSM adjunct of \AFive{} (O6) and grounded in the topological, non-spatial character of \(S^1_\theta\); proved in this consumer form (frozen-Boltzmann reduction via the [B1] kernel theorems), with its three residues discharged: the cascade-decoupling check (verified --- the cascade's boundary imprint is the parameter codebook and nothing else), the freeze-out/electroweak ordering (a two-channel theorem: the electroweak order parameter is driven by the ungapped light channel, stays in equilibrium through the crossover, and localizes there --- so the counting \{matter phases in, order parameter out\} is stable for every admissible epoch hierarchy), and the invariant-operator tilt table (which restores the gravity floor as the operative bound). \emph{VIII-b (the share dictionary):} \emph{which} measure-ratio equals \emph{which} \(\Omega\) --- precisely the unexhibited Step-4 normalization of Theorem~\ref{thm:omega-m-pi-topological}. The holonomy moduli space carries no discrete identification (\ATwo{}, the no-identification remark), so no volume-\(\pi\) shortcut exists and every route to \(\Omega_m = 1/\pi\) must pass through VIII-b. \emph{Lemma VIII-b$'$ (proved) --- the real-sector half-measure (\AFive{}).} Exact mode counting --- $2\Lambda+1$ independent real degrees of freedom for a reality-constrained sector against $2(2\Lambda+1)$ for a holonomy-charged one, a ratio of exactly $1/2$ at every finite cutoff --- fixes the effective fiber measures at $\pi$ (real) and $2\pi$ (charged) under the equipartition measure of VIII-a. The recalibrated share dictionary then delivers the full inventory from a single remaining normalization: unit-trace total $=$ the real-sector measure $\pi$; matter $=$ one quantized closed packet; baryons $=$ the charged average $1/2\pi$ within matter --- forcing $\Omega_b/\Omega_m = 1/2\pi$ with no residual freedom, the same $h$-free ratio verified in the acoustic spectrum (Table~\ref{tab:cmb-core-derived}). The open core of Conjecture~VIII has therefore contracted to one statement, \emph{VIII-b$''$ (unit-packet normalization)}: one closed packet $=$ one unit of fiber measure, with its derivation route through the $T_{\GFF} \sim M_{\rm KK}$ unit bridge of GFF-ST6. Until VIII-b$''$ is exhibited the fractions remain D$^*$; when it is, Steps 3--4 of Theorem~\ref{thm:omega-m-pi-topological} close and the $\pi$-locked inventory is promoted. \emph{The unit bridge: VIII-b$''$ sharpened and partially executed.} The route is now the explicit relation $T_{\GFF} = 2\pi\,I_m\,\varepsilon_{\GEP}$ (one packet of integer charge $I_m$, energy $I_m\varepsilon_{\GEP}$, against the equipartition rate $T_{\GFF}/2\pi$ per unit of fiber measure), and \ASix{} \citep{ArisakaA6Gi4} fixes its \emph{action} side outright: the descent normalization is $S_{\rm bulk} = (\pi/2)\,\mathcal{F}_{\GEP}$ (its Lemma~21.4 --- transverse flux $\Phi_\omega = \pi$, pinned by the graviton-normalizing warped volume of \AFive{} Lemma T8-1 and \emph{derived from} the Bogomolnyi cascade actions $\pi k$ --- the saddle charge $\Delta c_2 = k$ is the Whitney pair count of the winding bundle, under three declared premises; \ASix{}), and the closed packet is \emph{absolutely superselected} (its Corollary~21.6: only $\Delta\mathcal{F}_{\GEP} \in 4\mathbb{Z}$ is Dirac-phase-trivial) --- a topological mechanism beneath the DE-1 freezing. By the Hirzebruch signature theorem ($p_1 = 3\sigma$), $p_1 = N = 3$ is the \emph{minimal} closed gravitational topology ($\sigma = 1$), so the packet is the canonical unit of the gravity-sector charge lattice: the ``one'' in ``one closed packet'' is forced, not read. The open remainder has since contracted by one item. \emph{[B1] discharged as a structure theorem.} The Einstein ratio $T_{\GFF} = D/\mu$ on the fiber-moduli block (its GKLS generator already fixed by \AFive{}'s reduction) is a \emph{thermometer}: under kinetic (KMS) contact it copies the bath temperature exactly, independently of the coupling strength and the spectral details, so no kernel evaluation can manufacture $M_{\rm lock}$. At the Geometric Attractor the tower bath is gapped and frozen, and the block is dissipatively disconnected ($D = \mu = 0$) --- the kernel-side face of the DE-1 freezing (a tower-channel statement: the ungapped light channel instead retires the electroweak order parameter at the crossover, per the refinement paragraph above). The lock-scale reading survives in exactly one form: the \emph{freeze-out record} $T^{*} = 2\Mlock/L^{*}$, the last temperature the block registered before the pair channel's Boltzmann factor $e^{-2\Mlock/T}$ severed contact ($L^{*}$ the logarithm of the epoch rate hierarchy) --- the Tolman softness of GFF-ST6 reduced to a named logarithm (see the sharpened remark in \ASix{} and the GSM-adjunct update in \AFive{}). The open core is therefore \emph{[P]} alone: the \OGGEP{} projection of the superselected packet action ($3\pi/2$) onto the present-epoch boundary Hamiltonian, now with a sharpened target --- $\varepsilon_{\GEP} = T_{\GFF}/2\pi = \Mlock/(\pi L^{*})$ under the freeze-out reading (canonical $I_m = 1$), against the exponentially smaller Geometric-Attractor vacuum-floor reading ($T \to \omega_0/2$, the equipartition floor of the winding-suppressed modulus) --- whose adjudication is the projection lemma's own content. \emph{[P] discharged by identification; the promotion path retired.} The projection terminates on a \emph{dilution no-go}: superselection freezes the integer $I_m$, not the energy record, so $U_{\rm packet}$ tracks the diluting matter sector ($\propto a^{-3}$) while the equipartition rate $T_{\GFF}/2\pi$ is frozen --- no epoch-independent constant (in particular no anomaly-inflow $\varepsilon_{\GEP}$) can equal a diluting quantity at more than one instant. The occupancy $u(t) = \pi\,\Omega_m(t)$ therefore sweeps monotonically through unity exactly once, at $a_\pi = 0.9949$ ($z_\pi = +0.005$, about $75$~Myr before the present at the canonical $H_0 = 66.6$; the Planck uncertainty on $\Omega_m$ is a $\pm 160$~Myr timing blur, so this offset is $0.5\sigma$ --- consistent with the crossing being now), and the closure statement $u = 1$ is \emph{mathematically equivalent} to the present-epoch calibration $\Omega_{m,0} = 1/\pi$ --- precisely the epoch input declared in \S\ref{subsec:whynow-form-scale-epoch}, the one freedom the geometry does not remove. The unit-bridge arc and the form/scale/epoch honesty split thus converge on the same terminus by independent roads. \emph{Consequently the open core of Conjecture~VIII is VIII-a (uniformity) alone}: VIII-b$'$ is proved, and VIII-b$''$/($\star$) closes as the epoch identity --- exact at the crossing epoch $t_\pi$, with $t_0 = t_\pi$ to $1\%$ ($\approx 72$~Myr) as the declared L4 input. The fractions remain D$^*$ with the epoch as their declared calibration input, exactly as the codebook tags them. \emph{Residue status:} VIII-a's three residues are discharged (see the refinement paragraph above); the remaining named items in the entire Conjecture-VIII neighborhood are the 6D zero-mode formalization of the wrap-independence premise and the gravity-floor prefactor --- both well-posed, neither load-bearing for any deployed claim.

\paragraph{D1.\ Matter density fraction \(\Omm = 1/\pi\).}
The matter density fraction is the unique cost-minimum of \(\mathcal F\) on the closed-packet topological stratum of the cosmological-boundary moduli space. The closure runs through the \OGNfive{} integer ledger of \AFour{}: the matter packet inherits a closed-form integer topology from the \(\mathbf 5\) trace structure of the bulk holonomy, and the corresponding cost-minimum on the cosmological slice reduces to the rational fraction \(1/\pi\) by the closed-packet integration (the still-open Conjecture~VIII). Remaining: VIII-a (uniformity; exact up to the $\sim e^{-K}$ floor) $+$ the declared epoch input (\S\ref{subsec:whynow-form-scale-epoch}). Citation chain: \(\AFour{} \to \ATwo{} \to \AThree{} \to\) present.

\paragraph{D2.\ Cosmological constant fraction \(\OmL = 1 - 1/\pi\).}
The complementary fraction is forced by the unit-trace closure of the homogeneous-isotropic Friedmann constraint \(\sum_i \Omega_i = 1\) on the cosmological-boundary stratum. Once \(\Omm = 1/\pi\) is fixed by D1 and the radiation fraction is negligible at the present epoch, the residual vacuum fraction is identically \(1 - 1/\pi\). The fact that the residual vacuum fraction takes the value of a topological complement rather than an adjustable empirical constant is the deep content of D2. Remaining: VIII-a (uniformity; exact up to the $\sim e^{-K}$ floor) $+$ the declared epoch input (\S\ref{subsec:whynow-form-scale-epoch}). Citation chain: \(\ASix{}\) (residual \(F_{\GFF}\) interpretation) \(\to\) present.

\paragraph{D3.\ Baryon density fraction \(\Omb = 1/(2\pi^2)\).}
The baryon fraction descends from the baryonic-gauge protection of the Standard-Model ledger \citep{ArisakaA9SM}, projected onto the cosmological-boundary stratum. The factor \(2\pi^2\) is the volume of the unit three-sphere, \(\mathrm{Vol}(S^3) = 2\pi^2\): the baryonic gauge \(U(1)_B\) projects onto the relevant topological sub-stratum with weight \(1/\mathrm{Vol}(S^3) = 1/(2\pi^2)\) --- equivalently, and in the derived route of the recalibrated dictionary, the charged group average within matter: \(1/(2\pi^2) = (1/\pi)\cdot(1/2\pi)\) (Lemma VIII-b$'$); the \(\mathrm{Vol}(S^3)\) form is the mnemonic factorization of the same number. Remaining: VIII-a (uniformity; exact up to the $\sim e^{-K}$ floor) $+$ the declared epoch input (\S\ref{subsec:whynow-form-scale-epoch}). Citation chain: \(\AFour{} \to\) \ANine{} \(\to\) present.

\paragraph{D21.\ Decoupling redshift \(z_*\).}
The entry carries two distinct quantities (\S\ref{sec:two-redshifts}). The corridor switch \(z_{\mathrm{sw}} = e^{J} = e^{7} = 1096.6\) is a topological marker forced by the \OGNfive{} integer \(J = 7\) (Theorem~\ref{thm:hubble-two-stage-corridor}); it is not the audited number. The audited number is the operational decoupling redshift: the framework's own atomic chain --- Saha/Boltzmann recombination run on the \(\pi\)-locked inventory \(\Omb = 1/(2\pi^2)\), \(\Omm = 1/\pi\), with standard atomic constants and no cosmological input beyond the fractions --- lands on the Planck value \(z_{\mathrm{dec}} = 1089.92 \pm 0.25\) at the \(0.1\%\) level, \(0.4\%\) below the switch. (The Dev plot and its data table score this audited atomic number, with the switch annotated, not scored.) The row therefore rides on the same lemma as the fractions it descends from: once Lemma~VIII is proved, D21 is promoted to D with them (the atomic chain itself introduces no new lemma). Citation chain: \(\AFour{}\) (\(J = 7\)) \(\to\) D1--D3 (the fractions) \(\to\) \S\ref{sec:two-redshifts} \(\to\) present.

\paragraph{Lemma IX: Vacuum-residual-exactness.}
This lemma closes D16 (\(w\)), and its content is not an open assertion: the \emph{exactness} of \(p = -\rho\) is Theorem~\ref{thm:w-exact} (T-Cosmos-DE$'$), proved from Lemma~DE-1 of \ASix{} and the \AFive{} spectral gap, conditional on the constitutional rigidity of the bulk profile, with \(|1+w| \lesssim 10^{-91}\). What the lemma now names is the single surviving residue: the dynamical stabilization of that profile (a Goldberger--Wise-type computation, an upstream \ATwo{}/\AFive{} target). Once supplied, row D16 is promoted to D. The empirical determination \(w = -1.028 \pm 0.032\) \citep{Planck2018} is consistent at \(0.9\sigma\), and the two-sided kill switch of Theorem~\ref{thm:w-exact} applies: any confirmed \(w \neq -1\), either sign, is fatal.

\paragraph{Lemma X: Lithium-port-\(\pi\)-suppression.}
This lemma closes D20 (the primordial lithium-7 abundance \({}^7\mathrm{Li}/H\)). It asserts that the \(A=7\) nucleosynthesis port on the cosmological-boundary stratum inherits the same closed-packet \(1/\pi\) factor that fixes the matter fraction \(\Omm\), so that the standard-BBN lithium yield is suppressed by exactly \(1/\pi\).

\begin{conjecture}[Lithium-port-\(\pi\)-suppression]\label{conj:lithium-port}
On the \(A=7\) sub-stratum of the cosmological-boundary stratum, the open-system closed-packet integral suppresses the mass-7 yield by exactly \(1/\pi\), so that \(({}^7\mathrm{Li}/H)_{\mathrm{GG}} = (1/\pi)\,({}^7\mathrm{Li}/H)_{\mathrm{SBBN}}\), leaving the \(A\le 4\) abundances (\(D\), \({}^3\mathrm{He}\), \({}^4\mathrm{He}\)) unchanged.
\end{conjecture}

With the standard-BBN value \(({}^7\mathrm{Li}/H)_{\mathrm{SBBN}} \simeq 5.0\times10^{-10}\) \citep{CyburtFieldsOliveYeh2016,Pitrou2018}, the prediction is \(({}^7\mathrm{Li}/H)_{\mathrm{GG}} \simeq 1.6\times10^{-10}\), matching the metal-poor halo-star (Spite) plateau \((1.6\pm0.3)\times10^{-10}\) \citep{Fields2011} with no fitted strength: the identical \(1/\pi\) that fixes \(\Omm\) fixes the lithium deficit. The prediction is sharply falsifiable --- any firm primordial \({}^7\mathrm{Li}\) abundance away from \(\mathrm{SBBN}/\pi\) excludes the port. Once Conjecture~\ref{conj:lithium-port} is proved, row D20 is promoted to D. Upstream citation chain: \AFour{} (closed-packet \(1/\pi\)) \(\to\) \ATwo{} (bulk holonomy) \(\to\) Part~V \(\to\) the present audit.

\paragraph{Sixteen-entry counting summary for D\(^*\).}
The eighteen D\(^*\) entries are closed by eight lemmas: Lemma~I closes one entry (D4); Lemma~II closes five entries (D5, D6, D10, D11, D19); Lemma~III closes two entries (D13, D14); Lemma~IV (revised) closes two entries (A7 and A15, audited with the inflationary sector below); Lemma~VI closes one entry (D17); Lemma~VIII closes five entries (D1, D2, D3, D18, D21); Lemma~IX closes one entry (D16); Lemma~X closes one entry (D20). Lemma~VII closes no row: it gates the interpretive BFFT overlay of D18, whose codebook value is the Boltzmann diffusion prediction of the inventory. The economy is favorable: nine open research targets (with Lemma~V\('\) carrying the single remaining A entry besides), eighteen entries promoted on proof of eight of them.

\subsubsection{Audit of the inflationary sector: two promoted entries and the one remaining Asserted (A) entry --- two named lemmas}
\label{sec:audit-asserted-cosmos}

This block audits the three entries of the inflationary sector --- the tilt (A7), the amplitude (A8), and the inflationary scale (A15). Two of them now stand at D\(^*\) on the revised Lemma~IV; one, the amplitude, remains the codebook's single A entry, carried by Lemma~V\('\). Both lemmas concern the inflationary modulus on the cosmological-boundary stratum.

\paragraph{Lemma IV (revised): Plateau-class-and-prompt-crossing.}
This lemma underwrites A7 (\(\ns\)) and A15 (\(V_{\mathrm{inf}}^{1/4}\)) at D\(^*\); its statement follows.

\begin{conjecture}[Plateau-class-and-prompt-crossing (revised)]\label{conj:slow-roll-exit}
On the inflationary-modulus sub-stratum of the cosmological-boundary stratum, (i) the \GDOS{} non-conformal route forces the Starobinsky/\(\xi\) plateau class of Eq.~\eqref{eq:partII-plateau-potential} as the unique admissible potential (theorem-level promotion of the derivation of Part~IV), and (ii) the \GRIP{} instanton-ledger bound applies on \(\Sigma_r\), so the lock crossing is prompt (\(N_2 = 0\)) and the reheating-matched count \(N^* = 52.6\) is undiluted.
\end{conjecture}

Lemma~IV, in its revised form, underwrites the two entries the two-basin derivation moved out of A: the tilt (A7, now D\(^*\): \(\ns = 1-2/N^* = 0.9620\), derived in Part~IV given (i) and (ii)) and the inflationary scale (A15, now D\(^*\): \(V_{\mathrm{inf}}^{1/4} = 8.3\times10^{15}\) GeV, fixed by the derived class once the amplitude input is declared). Once Lemma~IV is proved, rows A7 and A15 are promoted from D\(^*\) to D. The earlier form of this conjecture (slow-roll-exit-uniqueness, which asserted that the exit fixes both \(\ns\) and \(A_s\) at their empirical values) is superseded: the tilt is now computed rather than matched, and the amplitude is carried separately by Lemma~V\('\) below. Upstream citation chain: \(\ATwo{}\) (bulk) \(\to\) \(\AFive{}\) (\GDOS{} anomaly structure) \(\to\) \(\ASix{}\) (\GRIP{} ledger) \(\to\) \citep{Starobinsky1980} (plateau template) \(\to\) the present audit.

\paragraph{Lemma V (retired) and its replacement Lemma V\('\).}
The program's working codebook formerly carried here the conjecture \emph{TeV-scale-inflation-uniqueness}\label{conj:tev-scale-inflation} --- that \(V_{\mathrm{inf}}^{1/4}\) is forced to a TeV-scale value by structural matching to the throat sector, ruling out GUT-scale inflation. \textbf{That conjecture is refuted, and the refutation is recorded at full strength} (Part~IV): a lock-scale plateau yields a \emph{blue} spectral tilt (\(n_s - 1 \sim +10^{-26}\)) against the measured red \(-0.0351 \pm 0.0042\) --- wrong in sign --- and demands \(\epsilon_V \sim 10^{-53}\) for the amplitude. No admissible reading survives; the conjecture is retired rather than reworded, and it is kept on this page so the audit trail shows what the framework once believed and why it stopped. Its A15 role passes to the revised Lemma~IV above.

In its slot stands the replacement that carries the sector's one remaining input:

\begin{conjecture}[GDOS-heat-kernel amplitude (Lemma V\('\))]\label{conj:gdos-amplitude}
The \(R^2\)/scalaron coefficient of the boundary effective action, evaluated as the \GDOS{} heat-kernel integral over the warped interval, equals \(e^{2K/3}/12\) up to an order-unity factor --- equivalently, the plateau height satisfies \(M = M_5\,e^{-K/3}\) with \(M_5 = (M_{\mathrm{Pl,red}}^2\,k)^{1/3}\), so that \(A_s = 21.5\,e^{-2K/3}\).
\end{conjecture}

Lemma~V\('\) is the one computation that would pay the amplitude: its warp candidate currently reproduces the required scalaron mass to \(87\%\) (\(A_s\) to \(75\%\)) with no parameter, and two alternative routes (warped decay; holographic central-charge counting) have been attempted and closed. Once Lemma~V\('\) is proved, row A8 is promoted from A to D and the launch sector becomes input-free. Upstream citation chain: \(\ATwo{}\) (bulk) \(\to\) \(\AFive{}\) (\GDOS{} heat kernel) \(\to\) the present audit.

\paragraph{Counting summary for the inflationary sector.}
Two of the three former A entries (A7, the tilt; A15, the inflationary scale) were promoted to D\(^*\) by the two-basin derivation of Part~IV and now ride on the revised Lemma~IV. The single remaining A entry (A8, the amplitude) is closed by one lemma: Lemma~V\('\). The economy is unchanged --- two open research targets --- and strictly better in content: three entries once pending are now one pending and two derived-conditional.

\subsubsection{Closure summary and the promotion roadmap}
\label{sec:closure-summary-cosmos}

The twenty-one codebook entries are closed by a total of \emph{ten} named lemmas:

\begin{itemize}
 \item Lemma~I (Kaxion-misalignment-completeness) closes 1 entry.
 \item Lemma~II (Hubble-two-stage-corridor-uniqueness) closes 5 entries.
 \item Lemma~III (Compact-angle-mass-uniqueness) closes 2 entries.
 \item Lemma~IV (Plateau-class-and-prompt-crossing, revised) closes 2 entries.
 \item Lemma~V\('\) (GDOS-heat-kernel amplitude; replaces the retired Lemma~V) closes 1 entry.
 \item Lemma~VI (Resonant-leptogenesis-normalization) closes 1 entry.
 \item Lemma~VII (BFFT-damping-knee-identification) closes no entry directly; it gates the interpretive BFFT overlay of D18 (the row itself is closed by Lemma~VIII).
 \item Lemma~VIII (Closed-packet-integration) closes 5 entries.
 \item Lemma~IX (Vacuum-residual-exactness) closes 1 entry.
 \item Lemma~X (Lithium-port-\(\pi\)-suppression) closes 1 entry.
\end{itemize}

Two entries (D9 and D12) are already closed without any open lemma. The closure economy is therefore:

\begin{table}[H]
\centering
\caption[Closure economy: how the codebook evolves as each lemma is proved]{Closure economy: how the codebook decomposition evolves as each named lemma is proved. The three right-hand columns count the observables that are Derived, Derived-conditional and Asserted at each stage. Reading down them shows what each named lemma would actually buy, which is a more useful question than whether the lemma is likely to be proved.}
\label{tab:closure-economy-cosmos}
\renewcommand{\arraystretch}{1.25}
\small
\begin{tabular}{p{0.34\textwidth}p{0.18\textwidth}p{0.18\textwidth}p{0.18\textwidth}}
\toprule
\textbf{State of the codebook} & \textbf{D} & \textbf{D\(^*\)} & \textbf{A} \\
\midrule
Current state (present paper) & 2 & 18 & 1 \\
After Lemma~I proved & 3 & 17 & 1 \\
After Lemma~II proved & 8 & 12 & 1 \\
After Lemma~III proved & 10 & 10 & 1 \\
After Lemma~IV proved & 12 & 8 & 1 \\
After Lemma~V\('\) proved & 13 & 8 & 0 \\
After Lemma~VI proved & 14 & 7 & 0 \\
After Lemma~VII proved & 15 & 6 & 0 \\
After Lemma~VIII proved & 19 & 2 & 0 \\
After Lemma~IX proved & 20 & 1 & 0 \\
After Lemma~X proved & 21 & 0 & 0 \\
\bottomrule
\end{tabular}
\end{table}

The promotion roadmap shows that the twenty-one-row codebook is closed by ten named lemmas (one of them, V\('\), the
replacement of a retired and refuted predecessor), that three of the lemmas (II, III, and VIII) each close multiple entries, and that the closure proceeds through a strictly monotonic improvement of the codebook decomposition as each lemma is proved. When all ten lemmas are proved, the codebook contains twenty independent D entries after the twenty-first (the inflationary-scale entry, promoted via Lemma~IV, combined with the post-promotion redundancy of D9) collapses into a single derived chain, yielding the canonical state ``all twenty independent cosmological observables derived from non-physics axioms''.

\paragraph{Comparison with the \texorpdfstring{\ANine{}}{A9-SM} closure economy.}
The Standard-Model codebook of \ANine{} has a similar structure: six entries fully derived, fourteen modulo a single named lemma, and eight remaining asserted (the eight non-top charged-fermion Yukawa masses). The cosmological codebook of the present paper is therefore closer to the goal state per row: two of twenty-one versus six of twenty-eight already at D, and only one of twenty-one at A versus eight of twenty-eight. This reflects the fact that the cosmological-boundary stratum is structurally lower-dimensional than the Standard-Model stratum, and that the closed-packet topology of the cosmological inventory (the three \(\pi\)-locked fractions) admits more direct constitutional closure than the family-projector geometry of the Standard-Model parameter sheet.

After all, the audit is not a victory lap but a structured tally: which observables are closed today, which would be closed tomorrow under ten precise mathematical conjectures, and which remain honestly empirical pending the proof of those conjectures. The framework's strength lies not in claiming that every observable is already derived but in stating exactly which lemmas would complete the derivation, with the proofs offered as open research targets in language sharp enough to be falsified.



\subsection{The Hubble two-stage corridor: canonical theorem statement}
\label{sec:codebook-hubble-two-stage}

\subsubsection{Empirical motivation: the \texorpdfstring{\(H_0\)}{H0} tension as of 2026}
\label{sec:hubble-tension-empirical}

The Hubble constant inferred from the cosmic microwave background under the standard \(\Lambda\)CDM extrapolation is \(\Hzero^{\mathrm{CMB}} = 67.4 \pm 0.5\) km/s/Mpc \citep{Planck2018}. The Hubble constant inferred from the local distance ladder (Cepheid-calibrated Type-Ia supernovae) is \(\Hzero^{\mathrm{late}} = 73.0 \pm 1.0\) km/s/Mpc \citep{Riess2022}. The discrepancy is now near five standard deviations and persists across calibration cross-checks (Mira stars, the tip of the red-giant branch, gravitational-wave standard sirens, lensing time delays). The minimal-extension responses within \(\Lambda\)CDM --- early dark energy, modified recombination histories, extra relativistic degrees of freedom, interacting dark sectors --- each introduce one or more adjustable continuous parameters and resolve the tension only in part. The present framework responds structurally, and we are careful to say at what level: it proposes that the two values are the projections of a single two-stage corridor of solutions to the boundary Friedmann equation, with the corridor switch fixed by the inherited recombination geometry of \AThree{} and the integer \(J = 7\) of the \OGNfive{} ledger of \AFour{}. This is a \emph{candidate} resolution --- a mechanism-level structural identification, not yet a computed joint fit; a tension is fully resolved only by a global Einstein--Boltzmann likelihood across CMB, BAO, SN, and lensing with the covariance kept, which the paper names as open work (PS-Cosmos-16). What lifts it above a phenomenological ``two \(H_0\)'s'' fit is one clean forward prediction a phenomenology cannot make: the gravitational-wave standard sirens, which read the background free of \emph{both} the sound horizon and the distance ladder, must land on the \emph{background} rate \(\Hzero^{\mathrm{GW}}\approx67\), not the local \(73\). That is the discriminating test, and it is near.

\subsubsection{Setup and hypotheses}
\label{sec:hubble-corridor-hypotheses}

The theorem of this Section is proved under three constitutional hypotheses inherited from upstream papers of the Ultra Eight \GGTheory{} cohort.

\begin{description}
 \item[\textbf{(H1)} Inherited bulk and \(\pi\)-locked inventory.] The cosmological-boundary stratum descends from the six-dimensional \GG{} bulk of \ATwo{} via the \OGGEP{} projection. The matter, vacuum, and baryon density fractions at the present epoch take the \(\pi\)-locked values \(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\), \(\Omb = 1/(2\pi^2)\) proved in the codebook audit of Section~\ref{sec:audit-derived-cosmos}.
 \item[\textbf{(H2)} \BIsix{} anomaly-free recombination geometry.] The modified Bianchi identity \BIsix{} of \AThree{} is in force throughout the cosmological history. In particular, the recombination surface inherits a closed-form geometry from the \(\mathbf 5\) trace structure of the bulk holonomy: the sound horizon \(r_s\) at recombination and the angular diameter distance \(D_A\) from the present epoch to recombination are both functions of the inherited inventory \((\Omm, \OmL, \Omb, \DeltaNeff)\) with no continuous adjustable parameter.
 \item[\textbf{(H3)} \GRIP{} cascade for the corridor switch.] The runtime factorization \(\GRIP = (\DGI \to \GSSB \to \GA)\) of \ASix{} applies to the recombination event: recombination is a \GSSB{} transition from a tightly-coupled \DGI{} phase to a frozen \GA{} phase. The cosmological history therefore admits a two-stage structure with a single switch event at recombination.
\end{description}

\subsubsection{Statement of the theorem}
\label{sec:hubble-corridor-theorem-statement}

\begin{theorem}[Hubble two-stage corridor: existence (T-Cosmos-16)]\label{thm:hubble-two-stage-corridor}
Assume hypotheses (H1), (H2), (H3) above. Then the boundary Friedmann equation admits a two-stage corridor of solutions parameterized by the corridor-switch redshift \(z_*\), with the following three properties:
\begin{enumerate}
 \item[\textbf{(C1)}] \textbf{Corridor switch is fixed by the \OGNfive{} integer ledger.} The switch redshift equals
 \[
 z_* = e^{J} = e^{7} \approx 1097,
 \]
where \(J = 7\) is one of the five proved \OGNfive{} integers \((M, N, L; J, K) = (1, 3, 5; 7, 35)\) of \AFour{}. No continuous parameter is free to be tuned; the value is forced by the integer ledger, and the atomic decoupling lies \(0.4\%\) below it (\S\ref{sec:two-redshifts}). The operational decoupling redshift from Planck is \(z_{\mathrm{dec}}^{\mathrm{Planck}} = 1089.92 \pm 0.25\); the corridor switch \(z_* = e^7 \approx 1096.6\) is a distinct topological quantity sitting \(0.6\%\) above it, while the atomic decoupling computed from the framework's own constants lands on the Planck value at the \(0.1\%\) level. \emph{Note:} the empirical Planck precision is \(\sigma/z \approx 0.025\%\), so the switch--decoupling offset is large in units of that error --- which is precisely why the two are carried as distinct quantities, never as one predicted number.

The framework's claim is therefore \emph{parameter-free agreement at the fractional-percent level}, not Planck-precision agreement at the \(0.025\%\) level. The canonical identification used throughout this paper is \(z_* = e^J\); the variant reading \(\ln(1 + z_*) = J\) (i.e.\ \(z_* = e^7 - 1 \approx 1095.6\)) differs from it by \(0.09\%\), far inside the tolerance defined next, and is not treated as an independent prediction.

The honest tolerance of the identification is set by the width of the recombination transition itself: last scattering is extended, with a visibility-function width \(\Delta z \approx 90\) (FWHM), so the switch point is defined only to within \(\sigma_{\mathrm{th}} \approx \pm 40\) in \(z\) (\(\pm 3.7\%\)). Within that tolerance the integer-ledger value sits \(0.17\sigma_{\mathrm{th}}\) from the Planck central value; this is the theory band displayed in the parameter-audit Dev plot (Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}).
 \item[\textbf{(C2)}] \textbf{Early-time branch.} For \(z > z_*\), the sound-horizon-projected Hubble value at the present epoch equals
 \[
 \Hzero^{\mathrm{CMB}} = \Hzero^{(0)} \cdot \kappa_{\mathrm{CMB}}(\Omm, \Omb, \DeltaNeff),
 \]
 where \(\Hzero^{(0)}\) is the bare Hubble normalization set by the boundary Einstein equation and \(\kappa_{\mathrm{CMB}}\) is the CMB-projection factor inherited from the \BIsix{} anomaly-free recombination geometry (hypothesis H2). Numerically, with the inherited inventory of (H1) and \(\DeltaNeff = 0.04\) from the audit of Section~\ref{sec:audit-derived-cosmos}, the early-time branch projects to
 \[
 \Hzero^{\mathrm{CMB}} \approx 67.4 \pm 0.5 \text{ km/s/Mpc}.
 \]
 \item[\textbf{(C3)}] \textbf{Late-time branch.} For \(z < z_*\), the distance-ladder-projected Hubble value at the present epoch equals
 \[
 \Hzero^{\mathrm{late}} = \Hzero^{(0)} \cdot \kappa_{\mathrm{late}}(\Omm, \OmL),
 \]
 where \(\kappa_{\mathrm{late}}\) is the late-time distance-ladder calibration factor inherited from the \GRIP{} cascade on the present-day attractor (hypothesis H3). Numerically, with the inherited inventory of (H1), the late-time branch projects to
 \[
 \Hzero^{\mathrm{late}} \approx 73.0 \pm 1.0 \text{ km/s/Mpc}.
 \]
\end{enumerate}
\end{theorem}



The corridor is illustrated in Figure~\ref{fig:hubble-two-stage-corridor} of Part~VI, where the two-stage structure is introduced and read off observationally.

\paragraph{Remark on uniqueness.} Theorem~\ref{thm:hubble-two-stage-corridor} states existence: the two-stage corridor with the listed numerical branches descends from the constitutional hypotheses (H1)--(H3) and the \OGNfive{} integer ledger. The complementary uniqueness claim --- that no other admissible two-stage corridor exists --- is the content of Conjecture~\ref{conj:hubble-corridor} of Section~\ref{sec:codebook-msmf-ogn}. Together, the theorem (existence) and the conjecture (uniqueness) close the five codebook rows D5, D6, D10, D11, D19.

\paragraph{Remark on what is and is not proved.} Theorem~\ref{thm:hubble-two-stage-corridor} proves at the canonical-theorem level only property (C1) --- the switch redshift \(z_* = e^J = e^7\) descending from the \OGNfive{} integer ledger of \AFour{}. Properties (C2) and (C3) --- the two numerical Hubble branches at \(\Hzero^{\mathrm{CMB}} \approx 67.4\) and \(\Hzero^{\mathrm{late}} \approx 73.0\) km/s/Mpc --- are structural identifications of the corridor's two natural projections rather than full numerical derivations: the projection factors \(\kappa_{\mathrm{CMB}}\) and \(\kappa_{\mathrm{late}}\) are the standard CMB sound-horizon and distance-ladder calibrations inherited from the empirical literature (\citealp{Planck2018,Riess2022}) rather than computed afresh in the present paper. The theorem's content is therefore: (i) the structural existence of a two-stage corridor; (ii) the switch redshift as an integer-ledger prediction at the \(0.6\%\) fractional level; (iii) the numerical reasonableness of the two empirical Hubble values as consistent projections onto the two corridor branches. A full derivation of \(\kappa_{\mathrm{CMB}}\) and \(\kappa_{\mathrm{late}}\) from first principles (without referencing the empirical CMB and distance-ladder machinery) is an open follow-up.

\subsubsection{Proof outline}
\label{sec:hubble-corridor-proof}

The full proof of Theorem~\ref{thm:hubble-two-stage-corridor} is a three-step argument, with each step relying on one of the three hypotheses.

\paragraph{Step 1: Two-stage solution of the boundary Friedmann equation under (H1).}
Substituting the \(\pi\)-locked inventory \(\Omm = 1/\pi\) and \(\OmL = 1 - 1/\pi\) into the first Friedmann equation,
\[
 H^2(z) = \Hzero^2 \bigl[\Omm (1+z)^3 + \OmL\bigr] + \text{(radiation, neutrinos)},
\]
gives a one-parameter family of expansion histories indexed by \(\Hzero\). Hypothesis (H1) reduces the family to a single curve modulo the choice of bare Hubble normalization \(\Hzero^{(0)}\). This step does \emph{not} produce a two-stage structure on its own: it produces a smooth single-stage curve. The two-stage structure arises only when the recombination surface is treated as a distinguished slice.

\paragraph{Step 2: Recombination as a switch event under (H2)--(H3).}
The \BIsix{} anomaly-free closure of (H2) certifies that the recombination surface is a lawful four-dimensional boundary on the cosmological-boundary stratum: the sound horizon \(r_s\), the angular diameter distance to recombination \(D_A^{\mathrm{rec}}\), and the acoustic peak angular scale \(100\theta_*\) are all derivable from the inherited inventory with no continuous adjustable parameter. The \GRIP{} cascade of (H3) then certifies that recombination is a \GSSB{} transition from a tightly-coupled \DGI{} phase (the photon-baryon plasma) to a frozen \GA{} phase (decoupled photons in free-streaming). Together, (H2) and (H3) elevate the recombination surface from a mere historical milestone to a structural \emph{switch event} of the cosmological history.

\paragraph{Step 3: The switch redshift from the \OGNfive{} integer ledger \(J=7\).}
The \OGNfive{} integer ledger of \AFour{} fixes a discrete set of integers including \(J = 7\). On the cosmological-boundary stratum, the present paper identifies this integer with the e-fold count \(\ln(1 + z_*)\) at the recombination switch event --- the ledger integer \(J = 7\) lying \(0.41\%\) above the computed \(\ln(1 + z_{\mathrm{dec}}) = 6.996\) (\S\ref{sec:two-redshifts}), giving \(z_* = e^J - 1 \approx 1095.6\) (or, to leading order, \(z_* \approx e^J = 1096.6\)). This identification is structural to the present paper rather than directly stated in \AFour{}, which proves \(J = 7\) as an integer-ledger output without committing to a specific physical interpretation in cosmology. Once the recombination surface is identified as a \GSSB{} switch event (Step 2), the integer-ledger identification above --- structural to the present paper, with the integer \(J = 7\) itself proved in \AFour{} --- fixes the switch redshift to be
\[
 z_* = e^{J} = e^{7} \approx 1096.6,
\]
the topological corridor-switch redshift, distinct from the operational decoupling redshift \(z_{\mathrm{dec}}^{\mathrm{Planck}} = 1089.92 \pm 0.25\) that Planck measures as the visibility peak, and coinciding with it to \(0.6\%\) --- an offset the recombination-history lemma would compute, not a precision match asserted here. This completes the derivation of (C1). The numerical projection of the two Hubble branches (C2 and (C3)) follows from substituting the inherited inventory of (H1) and the switch redshift of (C1) into the standard sound-horizon and distance-ladder calibrations. \(\square\)

\subsubsection{Corollaries}
\label{sec:hubble-corridor-corollaries}

\begin{corollary}[Reconciliation of the Hubble tension]\label{cor:hubble-tension-reconciliation}
Under the hypotheses of Theorem~\ref{thm:hubble-two-stage-corridor}, the apparent five-sigma discrepancy between \(\Hzero^{\mathrm{CMB}}\) and \(\Hzero^{\mathrm{late}}\) is not a tension between two measurements of the same quantity, but a consistency statement: the two values are the projections of a single two-stage corridor onto its two natural observational windows.
\end{corollary}

\begin{corollary}[\(\sigeight\) and \(\Seight\) reconciliation]\label{cor:s8-reconciliation}
The same two-stage corridor of Theorem~\ref{thm:hubble-two-stage-corridor} also \emph{addresses} the matter-clustering tension --- the early-time branch produces \(\sigeight \approx 0.811\) and \(\Seight \approx 0.832\) in agreement with Planck, while the late-time branch produces the slightly lower amplitudes preferred by weak-lensing surveys --- with no additional ingredient. We flag the honest limit at full strength: the \emph{sign} of the late-branch growth response is not yet derived (it is set by the corridor's coupling to the growth equation, which the joint fit above would fix), so this is a candidate reconciliation of the \(S_8\) tension, not a closed one; the codebook marks the \(S_8\) branches D\(^*\) accordingly.
\end{corollary}

\begin{corollary}[Falsification: a single-stage corridor would fail]\label{cor:hubble-falsification}
If future joint analyses of CMB, BAO, distance-ladder, and lensing-time-delay data forced \(\Hzero^{\mathrm{CMB}}\) and \(\Hzero^{\mathrm{late}}\) to coincide within 0.5 standard deviations, the two-stage corridor of Theorem~\ref{thm:hubble-two-stage-corridor} would be falsified, and with it the broader claim that the cosmological-boundary stratum admits a structural rather than parametric resolution of the Hubble tension.
\end{corollary}

\subsubsection{Falsifiable predictions and observational tests}
\label{sec:hubble-corridor-predictions}

Theorem~\ref{thm:hubble-two-stage-corridor} sharpens four falsifiable predictions for the next decade of cosmological measurement.

\begin{enumerate}
 \item \textbf{Corridor switch vs atomic decoupling.} The framework's atomic chain gives \(z_{\mathrm{dec}} = 1089\)--\(1091\) with zero parameters, on the Planck value at the \(0.1\%\) level; the corridor switch \(z_{\mathrm{sw}} = e^{7} \approx 1097\) is a distinct topological marker sitting \(0.41\%\) above it in e-folds (\S\ref{sec:two-redshifts}). A future independent determination of \(z_{\mathrm{dec}}\) at the \(\pm 5\) level --- a realistic target for post-2030 spectral-distortion missions reading the cosmological recombination radiation, complemented by 21-cm cosmology --- carries the single quantitative kill: the reading is refuted if the measured \(z_{\mathrm{dec}}\) sits off the atomic-chain value by more than \(0.5\%\), equivalently if the measured \(\ln(1+z_{\mathrm{dec}})\) departs from the computed \(6.996\) by more than \(0.005\) (present agreement \(0.1\%\)). The switch--decoupling offset \(\delta = 0.0041\) is then a derived consistency statement, fixed by the same measurement and the ledger integer \(J = 7\), not an independent kill: \(z_{\mathrm{sw}}\) is a topological marker with no observational handle of its own.
 \item \textbf{Two-stage residue in BAO at \(z \approx 1\).} The corridor switch at recombination should leave a measurable residue in the BAO scale at intermediate redshifts \(z \sim 1\) where the early-time and late-time branches differ most. DESI and Euclid baryon-acoustic-oscillation measurements should observe this residue at the 1--2\% level; the sharp estimator is the isotropic BAO dilation \(\alpha(z)\) in the \(z \simeq 0.8\)--\(1.2\) bins of a DESI-Y5 analysis.
 \item \textbf{Gravitational-wave standard sirens.} Multi-messenger \(\Hzero\) determination from gravitational-wave standard sirens is a clean independent probe of the late-time branch. \GG{} predicts the siren value converges on the \emph{background} rate, \(\Hzero^{\mathrm{GW}} \approx 67\) km/s/Mpc, since the siren reads the background expansion free of both the sound horizon and the local distance ladder. Decisiveness requires percent-level precision on the background: a few-tens-of-events O5-era catalog giving \(2\)--\(3\%\) at a central value near \(70\) would discriminate nothing --- the framework pre-registers that only a percent-level determination, with the peculiar-velocity systematic floor controlled, separates the \(67\) background from the \(73\) ladder. A converged siren value near the local ladder (\(\Hzero^{\mathrm{GW}} \approx 73\)) would refute the \(\pi\)-locked inventory derivation of the background; a value near \(67\) confirms it and marks the ladder's excess as a local/environmental bias.
 \item \textbf{Lensing time delays.} The Time-Delay Cosmography program already gives \(\Hzero^{\mathrm{TDC}}\) determinations that probe an intermediate-redshift slice of the corridor. A consistent determination of \(\Hzero^{\mathrm{TDC}} \approx 73\) at 1\% precision would confirm the late-time branch hypothesis; a determination significantly lower than 73 would force a refinement of the \GRIP{} cascade timing.
\end{enumerate}

After all, the Hubble two-stage corridor is the place where the \(\pi\)-locked inventory of cosmology stops being a tidy table of three derived numbers and starts being a structural prediction about \emph{when} two distinct branches of the universe's history give two distinct projections of the same constant. The framework's response to the most famous tension in contemporary cosmology is not a new free parameter but a structural theorem with two named branches, one named conjecture for uniqueness, four falsifiable predictions, and no adjustable continuous knob.

\medskip\noindent This Part is the conscience of the paper, and it is worth saying plainly what it found. Of twenty-one observables, two are earned outright; eighteen stand one named, falsifiable conjecture away from earned; and one --- the primordial amplitude --- is declared, openly, as the single number the geometry has not yet produced. A reader may find that ledger modest against the title's ambition. We would answer that the ledger \emph{is} the ambition: in a subject whose frameworks have historically absorbed every measurement, the rare and difficult thing is not to claim more but to specify exactly how one could be forced to claim less.

The audit also changes what kind of question remains. Once the values are graded, the deepest open matter is no longer any single number but the \emph{sameness} behind all of them --- why one vacuum, one Kaxion, one dark energy, everywhere and everywhen, for every observer who will ever check. That question --- the identity of the cosmos, and the two old questions it merges into one --- is Part~X, the summit of this paper.




\section{Part~IX in plain English: how to read the codebook, and why the grades are uneven}
The codebook is the paper's account of itself, and it is the section a
skeptical reader should turn to first. Twenty-one cosmological observables are
listed with their provenance under the cost-minimum principle, and each is
graded: Derived, Derived-conditional, or Asserted.

The grades are uneven and the unevenness is the information. A framework that
graded everything Derived would be claiming more than it has; one that graded
everything Asserted would be claiming nothing. What the codebook records is
that some of the inventory is forced by the geometry, some is forced
conditional on a conjecture that is named and not yet proved, and some is
carried in from measurement and not derived at all. Reading down the status
column is the fastest way to see how much of the cosmological claim actually
rests on the framework.

Two habits are worth adopting while reading it. First, a D-star row is not a
weak D; it is a row whose derivation is complete except for one named step,
and the step is named so it can be attacked. Second, the closure economy table
shows what each outstanding conjecture would buy if it were proved, which is a
more useful question than whether it is likely to be. A conjecture that would
convert one row is worth less than one that would convert six, and the table
says which is which.

\GGPart[Topological completion]{Topological completion: \emph{why the universe has the inventory it does}}{The closed-packet topological derivation of \(\Omega_m = 1/\pi\), \(\Omega_\Lambda = 1 - 1/\pi\), and \(\Omega_b = 1/(2\pi^2)\); the universal-cosmic-identity insight that explains why the cosmological coincidence is not a coincidence, layer by layer from particle physics through cosmology}


\section{Overview of the topological and deep-geometric origin Part}
\label{sec:topological-origin-overview-cosmos}

This Part is the cosmological analogue of \ANine{}'s Part~XI, \emph{``Geometric origin of identity and uniqueness of forces.''} There, on the Standard-Model boundary, the closing Part of the parameter audit rose from a technical fact --- why every electron in the universe carries the same mass, charge, and coupling --- to the recognition that identity and uniqueness are a single geometric theorem read at successive layers. This Part does the same one floor up, on the cosmological-boundary stratum, and asks the cosmological form of the same question: not what the universe is made of, but why it is made of \emph{that}, in \emph{those} proportions, everywhere we look.

\paragraph{The three-fold plan.}
The Part climbs in four movements, mirroring and extending \ANine{}'s Part~XI. \textbf{Section~\ref{sec:universal-cosmic-identity-preview}} is the identity core: it imports the ground-state refinement (\GA{} \(\to\) \GGA{}) that sharpened \ANine{}'s electron argument, then sets the three cosmological ``particles'' --- the electron of ordinary matter, the Kaxion of dark matter, and dark energy itself --- \emph{side by side}, and shows each to be the same construction, a \GDOS{} boundary observable read off a \GRIP{}-selected attractor, on three different strata. \textbf{Section~\ref{sec:second-horizon}} then turns the identity core toward cosmology's most celebrated puzzle and states what this paper regards as one of its deepest results: the famous horizon problem was only ever half a problem --- the sameness of \emph{conditions} --- and the sameness of \emph{things} (every electron, every Kaxion, one vacuum, identical across regions that never exchanged a signal) is a second horizon problem --- circled by the field for fifty years, never named --- that the identity core, welded to the derived GUT-edge launch of Part~IV, now solves. \textbf{Section~\ref{sec:pi-locked-derivation}} is the technical heart already assembled in this Part: the closed-packet derivation of the \(\pi\)-locked inventory \((\Omm, \OmL, \Omb) = (1/\pi,\, 1 - 1/\pi,\, 1/(2\pi^2))\), why the cosmic coincidence is no coincidence, and the four-form theorem behind \(\Omm = 1/\pi\). \textbf{Section~\ref{sec:two-questions-cosmos}} is the philosophical summit: the cosmological instance of \ANine{}'s ``two questions, one answer,'' where the fine-tuning of the vacuum, the anthropic coincidence, and the design/multiverse/brute-fact trilemma are each dissolved by the same uniqueness principle, and the loop closes on the observers who read it.

\paragraph{\GRIP{} is the heart of the inventory.}
Stated most simply, the deep geometric origin this Part exhibits is one runtime acting once. The cosmological-boundary stratum carries exactly one stable Geometric Attractor, and the \GRIP{} cascade \(\DGI \to \GSSB \to \GA\) of \ASix{} reaches it deterministically --- a gradient descent on the \GFF{} ledger that commits step by step until it halts at the unique cost-minimum, with no branch point left free. The \(\pi\)-locked inventory \((\Omm, \OmL, \Omb) = (1/\pi,\, 1 - 1/\pi,\, 1/(2\pi^2))\) is the present-epoch projection of that one attractor. This is the sense in which the observed cosmological facts are not fitted but \emph{geometrically inevitable}: once the bulk of \ATwo{}, the \BIsix{} four-form of \AThree{}, and the \OGNfive{} integer ledger of \AFour{} are fixed, the \GRIP{} runtime has exactly one place to relax, and the universe we measure is where it relaxes --- with no free parameter anywhere in the chain. The three questions of this Part --- why \(\Omm = 1/\pi\) rather than some other value, why the cosmic coincidence is no coincidence, and the layer-by-layer universal identity from electrons to cosmologies --- are three faces of this one statement. The single step that is not yet a closed proof is whether the \GRIP{}-selected attractor projects to exactly \(1/\pi\): that closed-packet integration is Conjecture~VIII, flagged honestly in the proof-status note of Theorem~\ref{thm:omega-m-pi-topological}. The inevitability is therefore structural and the remaining gap is named --- not a missing parameter, but a missing integral.

\section{The geometric origin of the cosmic inventory: identity, one floor up}
\label{sec:universal-cosmic-identity-preview}

This Section is the identity core of the Part. It presents the universal-cosmic-identity argument --- why the universe has the inventory it does --- as the cosmological instance of the electron argument of \ANine{}'s Part~XI. It first imports the one refinement of runtime vocabulary that sharpened that argument (the ground state, \S\ref{subsec:gga-cosmos}); then sets the three cosmological ``particles'' side by side and exhibits each as one \GDOS{} observable read off one \GRIP{}-selected attractor (\S\ref{subsec:three-objects}); answers three cosmological questions in the mould of \ANine{}'s three electron questions (\S\ref{subsec:three-cosmic-questions}); walks the same principle layer by layer from the electron to the cosmos (\S\ref{sec:uniqueness-layer-by-layer}); and closes by exhibiting the dynamical channel that drives the cosmic descent to its ground states (\S\ref{subsec:vacuum-relaxation}). The formal closed-packet theorem behind the numbers is deferred to Section~\ref{sec:pi-locked-derivation}.

The motivating analogy is \ANine{}'s Part~XI (``Why every electron is the same''): on the charged-lepton stratum, the \GRIP{} cascade \(\DGI \to \GSSB \to \GA\) selects a single stable Geometric Attractor, and every electron in the universe is a copy of that one attractor. The same logical structure, propagated one floor up, applies to the cosmological-boundary stratum: the \GRIP{} cascade selects a stable Geometric Attractor on the cosmological side, and the entire \(\pi\)-locked inventory \((\Omm, \OmL, \Omb) = (1/\pi,\; 1 - 1/\pi,\; 1/(2\pi^2))\) is the projection of that attractor onto the present epoch. What follows makes this precise, and --- crucially --- honest about the two \emph{different} geometric routes by which the cosmos achieves universal identity.

\subsection{The ground-state refinement: from \texorpdfstring{\GA{}}{GA} to \texorpdfstring{\GGA{}}{GGA}}
\label{subsec:gga-cosmos}

The argument rests on one refinement of the runtime vocabulary, established at runtime level in \ASix{} \citep{ArisakaA6Gi4} and generalized across the layers of the program-wide cascade in the capstone \AEleven{} \citep{ArisakaA11Origin}. A Morse landscape generally admits \emph{many} index-zero minima --- many Geometric Attractors --- and identity cannot rest on ``a'' stable minimum: if several were occupied, the objects that read them out would come in varieties. The cascade output is sharper. On a stratum of the class \ASix{} calls admissible, that paper's per-stratum ground-state corollary gives exactly one index-zero critical point, which is therefore the global minimum,
\begin{equation}
\GGA(\Sigma) \;:=\; \GA^{*} \;=\; \operatorname*{arg\,min}_{x\in\Sigma}\,\GFF(x),
\label{eq:gga-def-cosmos}
\end{equation}
the \emph{Ground} Geometric Attractor; the cascade \(\DGI\to(\GFF)\to\GSSB\to\GA\) terminates there. Dynamically, the \GGA{} is where \GRIPin{} --- the fixed-landscape regime of \ASix{} --- halts: the unique attractor from which no admissible exit channel remains. Whether the cosmological strata invoked below fall inside \ASix{}'s admissible class is not settled there, and the readings that follow are carried at the grade of their own sector rather than derived from the corollary. ``Ground state'' is thereby promoted to a constitutional principle, and identity and stability arrive together, as the two faces of ground-state-ness: all copies are identical because there is only one deepest basin to occupy, and each copy persists because there is no deeper basin to decay to.

The refinement is what lets the cosmological argument be stated without sleight of hand. The cosmological-boundary stratum, like any stratum, has more than one stable attractor --- Part~IV exhibited three basins along the scale coordinate \(\Sigma_r\), the primary plateau, the lock, and the terminus. Universality at the present epoch is not the claim that the cosmos sits in \emph{some} stable basin; it is the claim that the observed inventory is read off the \emph{ground} basin (dark energy, the terminus) together with the relic and baryonic records left as the descent passed through the others. Keeping ``a stable attractor'' and ``the ground attractor'' distinct is exactly what makes the cosmological identity statement true rather than merely suggestive.

\subsection{\texorpdfstring{\GDOS{}}{GDOS} and \texorpdfstring{\GRIP{}}{GRIP}, three objects side by side}
\label{subsec:three-objects}

The deepest content of this Part is a single sentence with a table under it: \emph{the electron, the Kaxion, and dark energy are the same geometric construction on three different strata.} The construction has two named halves, and it is worth separating them cleanly, because the program's two runtime laws do different jobs.

\GDOS{} --- the boundary law of operational observables of \AFive{} --- fixes \emph{what stratum an observer stands on and what quantity that observer reads}. It is the open-system dictionary: it says that a finite observer never sees the closed bulk, only a projected boundary observable, and it fixes which observable that is (a mass, a charge, a relic density, a vacuum energy). \GRIP{} --- the runtime regimes \GRIPin{}/\GRIPout{} of \ASix{} --- is the \emph{dynamics that selects the attractor whose coordinates are that observable}: explore the admissible channels, rank them on the \GFF{} ledger, break the symmetric branch, settle. Every object in the program is therefore the pair \((\text{\GDOS{} stratum})\times(\text{\GRIP{}-selected attractor})\). The three cosmological ``particles'' --- ordinary matter's electron, dark matter's Kaxion, and dark energy --- are this same pair evaluated on three strata, and Table~\ref{tab:three-objects} reads the pair off for each.

\begin{table}[htbp]
\centering
\small
\renewcommand{\arraystretch}{1.35}
\caption[The electron, the Kaxion, and dark energy as one \GDOS{}\,$+$\,\GRIP{} construction.]{\textbf{One construction, three strata.} The electron (ordinary matter, the reference object of \ANine{}/\PaperPThree{}), the Kaxion (dark matter, \PaperCOne{}), and dark energy (\ATen{}/\AEleven{}) read out as the same pair --- a \GDOS{} boundary observable on a stratum, and the \GRIP{}-selected attractor whose coordinates that observable is. The last row is the honesty fence: what is earned at theorem level and what is carried on a named conjecture.}
\label{tab:three-objects}
\begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash\bfseries}p{0.135\textwidth}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X@{}}
\toprule
 & \textbf{Electron} \newline (ordinary matter) & \textbf{Kaxion} \newline (dark matter) & \textbf{Dark energy} \newline (the vacuum) \\
\midrule
\GG{} sector
 & scale sector: the \(\Sigma_r\) interval, charged-lepton stratum
 & \emph{phase} sector: the compact angle \(S^1_\theta\), misalignment relic
 & scale sector \emph{terminus}: \(r \to K\) and beyond \\
\GDOS{} observable read
 & \(m_e,\ e,\ \alpha\) --- mass, charge, coupling
 & \(m_K \simeq 8.1\,\mu\mathrm{eV}\), relic density \(\Omdm\)
 & \(m_\Lambda \simeq 6.1\,\mathrm{meV}\), \(\OmL = 1-1/\pi\), \(w=-1\) \\
\GRIP{} mechanism
 & \GRIPin{} descent to the \GGA{} --- the family ground state (depth~2)
 & angle \emph{pinned} at the cap during primary inflation, then a frozen misalignment relic
 & \GRIPin{} descent \emph{terminates} at the global \GGA{} --- the deepest basin \\
Route to identity
 & one \GGA{} \(\Rightarrow\) every electron identical \emph{and} stable
 & one pinned angle \(\Rightarrow\) patch-uniform, deterministic, zero primary isocurvature
 & one terminus \(\Rightarrow\) the vacuum has one ground state \(\Rightarrow\) one \(\Lambda\) everywhere \\
Why the value
 & \(\alpha^{-1}(M_Z)=2^J=128\); \(m_e\) the depth-2 floor
 & \(\Xi_{\mathrm{cap}}\) from the cap four-form integral; \(8.1\,\mu\mathrm{eV}\)
 & seesaw \(\Mlock^2/M_{\mathrm{Pl,red}}\); \(\OmL = 1-1/\pi\) by unit-trace closure \\
Generation reading
 & ground state of the \emph{family} cascade (\(\tau,\mu\) excited)
 & relic of the \emph{middle} crossing --- radiates the dark sector
 & ground state of the \emph{vacuum} cascade --- \emph{the electron of the vacuum} \\
\midrule
Honesty fence
 & earned at theorem level (\PaperPThree{})
 & reflection-dressing \(+\) cap-content branch \(+\) \(\theta_i\) (\PaperCOne{})
 & Conjecture~VIII (closed-packet) \(+\) the declared epoch input \\
\bottomrule
\end{tabularx}
\end{table}

\paragraph{Two geometric routes to identity, not one.}
Read down the ``route to identity'' row and a genuine structural fact appears, one it would be dishonest to smooth over. The three objects reach universal identity by \emph{two} distinct geometric mechanisms, not one. The electron and dark energy are identical everywhere because each is a \emph{ground state}: there is exactly one deepest basin on the stratum, so every copy is the same copy and there is nowhere lower to fall. The Kaxion is identical everywhere for a different reason: its field angle was \emph{pinned} --- frozen coherently across the whole sky at the cap during primary inflation --- so that every patch inherited the same misalignment, delivering a patch-uniform relic with no primordial isocurvature. A settled minimum and a frozen phase are not the same geometric event; the first is the terminus of \GRIPin{}, the second is a value locked in before \GRIPin{} on the compact sector could act. That one runtime produces universal identity by both routes --- ground-state selection where the landscape is explored, topological pinning where a phase is frozen --- is a sharper and more honest statement than ``everything is a ground state,'' and it is the true shape of the cosmological identity theorem. The electron and the vacuum are the same kind of object; the Kaxion is their cousin, identical for a reason of its own that the same geometry supplies.

\paragraph{Where \texorpdfstring{\GDOS{}}{GDOS} bites.}
The \GDOS{} half of the construction is not decorative. It is what makes the three observables \emph{comparable} in the first place: mass, relic density, and vacuum energy are wildly different kinds of number, and only because each is defined as an operational boundary projection of the same bulk --- the open-system reading of a finite observer who cannot access the closed six-dimensional geometry --- can they be placed in one table and shown to share a mechanism. \GDOS{} is also where the cosmological objects differ most sharply from a naive particle picture: dark energy is not a particle at all, but the \GDOS{} projection of the terminus attractor's residual ledger; the Kaxion's ``mass'' is the curvature of the pinned angle's potential, a compact-sector reading with no scale-sector analogue. The pairing \((\text{\GDOS{}})\times(\text{\GRIP{}})\) is what lets a mass, a relic, and a vacuum energy be recognized as three readings of one geometry.

\subsection{Three cosmic questions, finally answerable}
\label{subsec:three-cosmic-questions}

\ANine{} posed three questions about the electron --- why its mass, its charge, and its force are the same everywhere --- and answered each by contrasting the textbook restatement with the geometric uniqueness theorem. The cosmological objects admit three questions of exactly the same shape, and they have never been answered together before.

\paragraph{(a) Why is the dark-energy density the same everywhere and at every epoch?}
A textbook will say that \(\Lambda\) is a cosmological constant --- constant by definition, the same everywhere because that is what a constant is. This is a restatement, not an explanation: it leaves untouched why empty space carries \emph{that} energy density and not some other, and why the number does not drift from place to place. In the present framework the vacuum energy is the \GDOS{} readout of the cosmological \GGA{} --- the deepest basin of \GFF{} on the scale stratum, reached when matter has diluted away and \GRIPin{} halts at the terminus. A \(w=-1\) terminus has a single value because a ground state has a single depth; the density is the same at every epoch because, once reached, the terminus is where the descent stays; it is the same on every sightline because there is only one deepest basin for the boundary to relax into. The seesaw scale \(m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}} \simeq 6.1\,\mathrm{meV}\) is the depth of that basin, read out.

\paragraph{(b) Why is every Kaxion identical --- same mass, same coupling, no isocurvature?}
A textbook axion story will say the misalignment angle is an initial condition, drawn at random in each causal patch, and that the relic density depends on where that draw happened to land. In the present framework the angle is not drawn but \emph{pinned}: reflection-symmetric dressing at the cap fixes it during primary inflation, coherently across the whole observable universe, so that every patch inherits the same \(\theta_i\) and the relic is delivered patch-uniform, deterministic, with no primordial isocurvature (the compact-sector analogue of charge quantization: a single frozen phase read out identically everywhere). The mass \(m_K \simeq 8.1\,\mu\mathrm{eV}\) is the curvature of the pinned potential, fixed by the cap four-form integral \(\Xi_{\mathrm{cap}}\); every Kaxion is the same Kaxion because every Kaxion reads out the same frozen angle on the same compact sector.

\paragraph{(c) Why is the inventory \texorpdfstring{\((1/\pi,\,1-1/\pi,\,1/2\pi^2)\)}{(1/pi, 1-1/pi, 1/2pi^2)} the same on every sightline?}
A textbook will say the density parameters are measured numbers, fit to the data, the same across the sky because the universe is homogeneous. This too is a restatement: homogeneity is assumed, not explained, and the particular values are fit, not derived. In the present framework the inventory is the projection of one attractor, and its three fractions are locked together by unit-trace closure (Section~\ref{sec:pi-locked-derivation}); the composition is the same on every sightline for the same reason the electron mass is --- it is the readout of a single geometric selection, not an average over independent local draws. The homogeneity of the inventory is not the premise of the argument; it is its consequence.

The unifying answer to (a), (b), and (c) is one sentence: \emph{the cosmological objects are universal because each is the readout of a \GRIP{} selection on the cosmological strata --- dark energy of the ground attractor, the Kaxion of a pinned angle, the inventory of their locked projection --- and a geometry, like a molecule, has exactly one ground configuration to read.} This is the cosmological twin of \ANine{}'s electron sentence, and it carries the same moral: any framework that treats \(\OmL\), \(m_K\), or the density parameters as inputs cannot answer why they are the same everywhere, because an input number gives no such reason. Only a framework in which each is the readout of a uniqueness selection can.

\subsection{The same principle, layer by layer, from the electron to the cosmos}
\label{sec:uniqueness-layer-by-layer}

The principle that explains universal-electron identity is not unique to the electron, and it does not stop at the Standard Model. Every layer of cosmic evolution after the six-dimensional bulk locks in is the readout of the same \GRIP{} cascade on a different boundary stratum; at each layer the cascade terminates on the layer's \GGA{} --- exactly one deepest attractor, or, where a symmetry mandates degenerate copies, a finite quantized list of them --- and that \GGA{}'s coordinates are what the next layer reads as identity. \ANine{}'s Part~XI walked this ladder from quarks to single-cell life; here we walk the same rungs and then continue one floor higher, to the cosmological objects of the present paper, so that the cosmos is seen as the natural continuation of the ladder rather than a separate claim.

\paragraph{1. Leptons and quarks.}
On the charged-lepton stratum the \GRIP{} cascade has one stable attractor at each cascade depth, and the three depths, populated by the three families, produce \(m_\tau\), \(m_\mu\), \(m_e\); the quark strata produce the six quark masses. Of the three populated family attractors, the first generation is the stratum's \GGA{} --- which is precisely why the electron is absolutely stable while the muon and tau, occupying higher, non-ground attractors of the same stratum, decay toward it. Identity and stability arrive together: the periodic table of particles has excited states, and they behave exactly as excited states should.

\paragraph{2. Protons (quark confinement).}
Confinement is a \GSSB{} step: the runtime selects the color-singlet branch because that is the unique stable attractor of \GFF{} in the strong-coupling regime. The proton is the minimum-energy color-singlet baryon under gauged \(\mathrm{U}(1)_B\); that minimum is the baryonic stratum's \GGA{}, so every proton in the universe has the same mass and the same charge, and --- because \(\mathrm{U}(1)_B\) is gauged --- the ``no deeper basin'' clause holds exactly: proton stability is ground-state stability in its sharpest form.

\paragraph{3. Elements (H, He, C, N, O, \(\ldots\)).}
For each integer atomic number \(Z\) the cascade selects the unique ground-state nucleus, the unique ground-state electron configuration, and the unique spectral fingerprint. The Aufbau principle and the periodic table are readouts of the layer-3 attractor; astronomers see the same Balmer series in a distant quasar and a laboratory lamp because both read the same cascade attractor on the same atomic stratum. Here the runtime's vocabulary and chemistry's coincide exactly: ``ground state'' was the constitutional concept all along.

\paragraph{4. Molecules in the interstellar medium.}
The molecules of cold dense clouds --- \(\mathrm{H}_2\), \(\mathrm{CO}\), \(\mathrm{H}_2\mathrm{O}\), \(\mathrm{NH}_3\), \(\mathrm{CH}_3\mathrm{OH}\), several hundred more --- are the \GGA{}'s of \GFF{} on the molecular stratum at the prevailing temperature and density. That the same molecules appear across the galaxy and in high-redshift systems is the layer-4 echo of universality: the molecular attractors are the same everywhere because the cascade is.

\paragraph{5. Complex organic molecules.}
The cascade continues into polyatomic carbon chemistry: amino acids, sugars, small nucleotide precursors are the \GGA{}'s on the organic stratum at the temperatures of warm, wet surfaces and proto-stellar nebulae. Glycine in cometary ice and the same twenty proteinogenic amino acids underlying all terrestrial life are the layer-5 echo: where conditions let the cascade reach this layer, the same attractors are selected.

\paragraph{6. ATP, RNA, DNA.}
The cascade selects the energy-quantum carrier and the information carriers of life. ATP is the \GGA{} of \GFF{} on the metabolic-energy stratum --- its energy--information calibration against \(\QB{}\) developed in (\citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin}); RNA and DNA are the \GGA{}'s on the information-storage stratum, the four-letter alphabet the unique solution of the gauge-locking constraint at this layer. Every cell uses the same ATP currency and the same DNA alphabet because both are layer-6 readouts of the same selection.

\paragraph{7. Single-cell life.}
The cascade selects single-cell life as the \GGA{} of \GFF{} on the open-system stratum subject to the four boundary conditions of the Selfish Open System (\SeOS{}) class of \AFive{}. A bacterium and an archaeon share the same genetic code, the same ATP currency, the same membrane compartmentalization, and the same core metabolism because all are layer-7 attractor coordinates of the same cascade; the Last Universal Common Ancestor is the empirical record of that attractor.

\paragraph{8. One floor up: the cosmological objects.}
Here the present paper extends the ladder. The three cosmological ``particles'' of \S\ref{subsec:three-objects} are the next rungs, read on the largest strata the program reaches. \emph{Dark energy} is the \GGA{} of \GFF{} on the scale stratum \(\Sigma_r\) --- the terminus, the deepest basin of the whole descent, the cosmic ground state whose depth is \(m_\Lambda\); it is universal for exactly the reason the electron is, because there is one deepest basin. \emph{The Kaxion} is the frozen-relic attractor of the same descent read in the compact angle, universal by pinning rather than by settling. \emph{The \(\pi\)-locked inventory} is the projection of the terminus attractor onto the present epoch, its three fractions unit-trace-locked. The cosmos is not a separate universality claim bolted onto the particle ladder; it is the ladder's continuation onto the strata where the bulk geometry itself is the object being read.

\paragraph{The principle, plainly stated.}
At each layer of cosmic evolution after the six-dimensional bulk locks in, the \GRIP{} cascade terminates on the Ground Geometric Attractor of the layer-specific stratum --- the unique deepest minimum of \GFF{}, not merely a stable one --- and what we observe as the universal identity of physical objects at that layer is the readout of that \GGA{}'s coordinates (with the one honest amendment of \S\ref{subsec:three-objects}: where a phase is frozen before the descent can act, identity is delivered by pinning rather than by settling, as for the Kaxion). Stability and identity are the same fact seen twice: an object persists because its configuration has no deeper basin to decay to, and all its copies are identical because there is only one deepest basin to occupy. The ladder that begins with why every electron is the same ends, without a break in the logic, with why the vacuum energy of the universe is the same everywhere --- and the cohort's capstone volume carries the higher rungs, from single-cell life through the layers above, as the natural continuation of the same theorem.

\subsection{The vacuum cascade as the cosmic relaxation channel}
\label{subsec:vacuum-relaxation}

\ANine{}'s Part~XI gave the identity principle a dynamical shadow: it read the weak interaction as the runtime's \emph{relaxation channel} --- the unique implementer of \GRIPin{} on the matter strata, the road by which matter reaches the ground states the constitution guarantees. ``A constitution that mandates unique ground states must also mandate a road to them,'' and on the Standard-Model boundary that road is the weak force, the one flavor-connecting channel, running at five wildly different tempos from the lepton ladder to the neutron star. The cosmological descent has the same dynamical shadow, and naming it completes the parallel.

On the cosmological strata the road to the ground state is the \emph{vacuum cascade itself} --- the single \GRIPin{} descent of Part~IV, run along \(\Sigma_r\) from the primary plateau through the lock to the terminus, with its two saddle crossings as the relaxation events. Where the weak force steps matter down the family ladder rung by rung, the cascade steps the \emph{vacuum} down the scale ladder basin by basin; and, exactly as a weak decay ``fills the detector with its products,'' each crossing of the vacuum cascade radiates a sector into the universe. The first crossing --- the graceful exit of inflation --- reheats the boundary and fills it with the visible sector; the second --- the lock crossing --- seeds the Kaxion misalignment and stocks the dark sector; the terminus is the vacuum's electron, the ground state where the descent finally stays. Relic abundances are the decay kinematics of this cascade, and the \(\pi\)-locked inventory is the ledger of what each crossing left behind. This is the same reading Part~IV developed as the \emph{vacuum generations} (\S\ref{subsec:vacuum-generations}): three basins of the vacuum in place of three families of matter, two crossings in place of two decays --- \emph{dark energy is the electron of the vacuum, and the cascade that reaches it is the cosmic implementation of \GRIPin{}}, the road to the cosmic ground state precisely as the weak force is the road to the electron.

Two disciplines keep the parallel honest, both inherited from Part~IV. First, the rung sizes do not transfer: the family ladder steps by the instanton unit \(\pi\), the vacuum ladder by the warp depth \(K\); the grammar is shared, each stratum supplies its own constant. Second, the closure of the count --- that the same budget which forbids a fourth matter family also forbids a fourth vacuum epoch --- is stated there as the natural conjecture of the correspondence, supported by the prompt-crossing bound and left for the constitutional analysis to complete. Within those fences the dynamical picture is one picture: a runtime that guarantees ground states must supply the channel that reaches them, whether the object relaxing is a muon on the lepton ladder or the vacuum of the entire universe.

\section{The second horizon problem: sameness of things, not sameness of temperature}
\label{sec:second-horizon}

The identity core of Section~\ref{sec:universal-cosmic-identity-preview}
answered a question about objects: why every electron, every Kaxion, and
the vacuum itself are the same wherever we look. This Section turns that
answer toward cosmology's most celebrated puzzle and makes a claim about
the history of the subject, stated with the calibration the history
deserves (\S\ref{subsec:shp-pedigree}): the famous horizon problem was
only ever \emph{half} a problem; the field solved the easy half in 1981;
and the hard half --- though it surfaced for fifty years in disguise, as
a domain problem, a relic problem, a footnote to anthropics --- was
never named, never ranked, and never answered in the temporal register
in which the identity core, welded to the derived placement of inflation
at the GUT edge, now answers it.

\subsection{The problem the textbooks leave unnamed}
\label{subsec:shp-unposed}

Open any cosmology textbook at the horizon problem and one finds the same
statement. Without inflation, the sky at recombination consisted of tens
of thousands of causally disconnected patches, yet its \emph{temperature}
is uniform to one part in $10^5$. How did regions that never exchanged a
signal agree on a temperature? Inflation answers \citep{Guth1981,Linde1982,
AlbrechtSteinhardt1982}: they did not have to --- they were once a single
microscopic, causally connected patch, stretched to cosmological size
before the hot Big Bang began.

Now notice what that statement quietly takes for granted. The disconnected
patches do not merely share a temperature. They share \emph{everything
they are made of}. Every electron in every patch carries the same mass,
charge, and magnetic moment --- the Balmer lines of a quasar at redshift
seven are the Balmer lines of a laboratory lamp. Every proton carries the
same mass and the same exact stability. The dark matter arrives with the
same abundance per comoving volume on every sightline; the dark-energy
density is one number on every sightline and at every epoch we can probe.
Temperature is a \emph{state} --- one number that thermal contact could
plausibly smooth. Identity is a \emph{specification} --- a complete parts
list with couplings --- and no exchange of photons can negotiate a
particle's mass between two regions.

The horizon problem as posed and solved in 1981 therefore addressed the
easiest case, one thermal number, and left the hard case --- the identity
of everything --- unsolved and, as a problem in its own right, unnamed.
Pieces of the question do surface across the history of the subject, and
\S\ref{subsec:shp-pedigree} traces them: Dirac's worry that the
constants of nature might drift in time; the domain problem of the
1970s, which posed the sharpest fragment --- causally disconnected
regions choosing their vacuum at random --- and then filed it under
relic abundances; the varying-constants programs, whose
quasar-absorption spectroscopy now bounds any spatial variation of the
fine-structure constant at the $10^{-5}$--$10^{-6}$ level across the
observable universe \citep{Uzan2011} --- a precision \emph{measurement}
of identity uniformity that the standard framework can only restate as an
assumption; and the string landscape, which makes the problem actively
worse, since if many vacua exist, causally disconnected regions have no
reason to have chosen the same one, and the observed uniformity of the
laws becomes an anthropic accident. Why did the question never take a
place of its own on the canonical list, beside temperature and flatness?
Because quantum field theory \emph{seems} to answer it by
construction: all electrons are quanta of one electron field, so any
two electrons are exactly interchangeable --- not merely similar but
identical in principle, which is why Pauli statistics work at all. But
look at what the construction actually secures. The interchangeability
is \emph{conditional} --- quanta of one field, with one set of
couplings, on one vacuum --- and every clause of the condition is an
input. QFT does not derive the \emph{value} of the electron's charge:
\(e\) is a free renormalized parameter, measured and inserted. It does
not derive the \emph{constancy} of that value across space: the
couplings are constants by stipulation of the Lagrangian's form, and
nothing in the framework forbids promoting them to fields that vary
from place to place --- the varying-constants programs above are
built on exactly that freedom. And it does not derive the
\emph{uniqueness} of the vacuum: the low-energy values depend on which
vacuum a region sits in, and spontaneously broken theories contain the
very mechanism by which identity would fail --- causally disconnected
regions choosing different vacua. So the field ontology explains
identity \emph{within} a patch that already shares one vacuum and one
Lagrangian, and silently assumes the rest. The question is not
answered; it is moved up a level --- \emph{why is there one field
content, with one set of couplings, in one vacuum, across regions that
were never in causal contact?} A globally unique vacuum is what QFT
assumes; it is not what QFT derives. The assumption is so ingrained
that the question became invisible. We name it here: the
\textbf{second horizon problem} --- the identity horizon problem ---
the sameness of \emph{things}, where the first horizon problem was the
sameness of \emph{conditions}.

And note, finally, the inversion in the data record, because it is the
sharpest measure of how thoroughly the question was overlooked:
\emph{the evidence for the sameness of things predates the evidence for
the sameness of conditions by a full century.} In 1864 Huggins showed
that the stars shine with the same spectral lines as terrestrial
elements --- the same atomic physics across the visible universe
\citep{Huggins1864} --- and by the 1920s the same lines were being read
at cosmological redshifts. The uniform temperature was discovered in
1965 \citep{PenziasWilson1965}; the field had named its paradox within
a few years, and inflation answered it within sixteen. The uniformity
of things had been on the record since 1864, and it waited a hundred
and sixty years without so much as a name. (Strictly, what the spectra
certify are the dimensionless combinations --- the fine-structure
constant \(\alpha = e^2/\hbar c\) and the electron-to-proton mass
ratio --- the only form in which the cross-sky constancy of a
``constant'' is meaningfully stated; the distinction sharpens the
bounds quoted above and changes nothing in the argument.)

\subsection{Fifty years of near misses: what was posed, and what was not}
\label{subsec:shp-pedigree}

A claim about the history of a subject should be held to the same
standard as a claim about the sky, so this subsection lays out the
pedigree --- the places where the question surfaced, what each near miss
posed, and what each left undone. The pedigree is genuine; laying it out
is what locates, precisely, the gap this Section fills.

One famous anticipation deserves its own sentence. In 1940 Wheeler
telephoned Feynman with a complete answer to half of the question: all
electrons have the same charge and the same mass, he proposed, because
they are all \emph{the same electron}, threading forwards and backwards
through time \citep{FeynmanNobel1965}. The proposal was wrong --- it
predicts as much antimatter as matter, as Feynman observed on the spot
--- but it is the exception that proves the rule: the identity of the
electrons was felt to be strange enough to demand a mechanism exactly
once, by one physicist, in a phone call, and the episode survives in
the literature as an anecdote in a Nobel lecture, not as a named
problem. The causal, cosmological form of the question was first posed
--- for one corner of identity --- in 1974, before inflation existed.
Zeldovich, Kobzarev, and Okun, studying
spontaneous CP violation, observed that during the cooling of
``causally disjointed points'' of the universe the sign of the
condensate is chosen at random, producing a domain structure of the
vacuum --- and they even asked the one-vacuum question directly: how
early must the walls be removed to make the whole universe a single
domain \citep{ZeldovichKobzarevOkun1974}? Kibble then made the argument
general and quantitative: at any cosmological symmetry-breaking
transition the vacuum choice is made independently in regions with no
prior causal contact, the correlation length is bounded by the horizon,
and defects form where the choices disagree \citep{Kibble1976}. This is
the identity question, posed with full causal rigor --- but posed for
the \emph{orientation} of a degenerate vacuum inside one fixed theory,
and framed as a \emph{relic} problem: the domains mattered because
their walls would overclose the universe, not because the sameness of
the vacuum was itself a puzzle to rank beside the horizon problem. The
canonical list absorbed it in exactly that register --- unwanted relics,
solved by dilution.

The spatial half of the solution is likewise standard. From 1984 the
field understood that the observable universe fits deep inside a single
inflated domain, so that one broken-symmetry state --- hence one set of
low-energy laws --- fills the visible sky; Linde in particular listed
``the problem of the uniqueness of the universe'' among inflation's
targets (why should the whole universe have jumped to the same
SU(3)\(\times\)U(1) minimum?), and once remarked, in an argument about
anthropics, that no selection principle explains why \emph{all
properties} of the universe are approximately uniform over its entire
observable part \citep{Linde1990}. That sentence is the nearest miss in
the ninety-year history of modern cosmology --- written in passing, and
not followed. The uniformity of ``all properties'' never joined the
problem list it was a comment on; and in the eternal-inflation
literature that grew from the same author's work, the emphasis runs the
other way: domains \emph{beyond} our horizon with \emph{different}
low-energy laws are the headline, and the uniformity \emph{within} our
patch is the by-product nobody interrogates. The string landscape
completed the inversion --- many vacua, populated by bubble nucleation,
make the diversity outside the story's point, while the sameness
inside, the only part anyone observes, is no one's explanandum.

The measurement arm grew in parallel, and just as namelessly. Dirac's
drifting constants became the modern varying-constants program, whose
quasar spectroscopy now certifies the spatial uniformity of the
fine-structure constant across the observable universe at the
\(10^{-5}\)--\(10^{-6}\) level \citep{Uzan2011} --- bounds framed
throughout that literature as equivalence-principle tests, with
uniformity as the null hypothesis rather than the fact in need of a
mechanism. Relativists and philosophers of cosmology have flagged the
underlying assumption in so many words --- ``cosmology proceeds by
assuming the laws of physics are the same everywhere''
\citep{Ellis2006} --- and the steady-state theorists once secured it by
outright postulate; noticed, that is, but never mechanized. Even the
pinning of \S\ref{subsec:shp-ordering} is stated almost verbatim in the
axion literature: if the Peccei--Quinn symmetry breaks before inflation,
our Hubble volume begins life ``with a single uniform value of
\(\theta_{\rm PQ}\) everywhere'' \citep{Marsh2016} --- stated as a
fact about the relic abundance and its isocurvature test, never as a
statement about the identity of the dark sector.

One boundary should be drawn on the other side. A critical tradition ---
Penrose's entropy argument, the Hollands--Wald critique --- has long
held that inflation fails to explain even the smoothness it claims
\citep{Penrose2004,HollandsWald2002}. That tradition seeks a deeper
half of the \emph{first} problem, in the initial state and its entropy;
this Section is not part of it. Here inflation's 1981 success is granted
in full, and a second problem --- differently shaped, about things
rather than conditions --- is posed beside it and answered by the same
event, read in a new register.

The gap, then, is four-fold. No one named the sameness of things as a
horizon problem in its own right, or ranked it as the harder half of
1981's question. No one extended it from the orientation of one vacuum
to the identity of the whole inventory --- masses, couplings,
abundances, and the angle at once. No one drew the temporal inference:
that because inflation preceded the existence of matter, identity was
inherited rather than negotiated --- the premise, that all matter was
minted at reheating from one field, is universal lore; the corollary is
nowhere. And no one reduced cosmic identity and vacuum uniqueness to a
single puzzle, so that one future proof settles both. Those four steps
--- the naming, the extension, the ordering, the unification --- are
what the rest of this Section takes.

\subsection{The first half of the answer: one mold}
\label{subsec:shp-mold}

The first half of the answer is the identity core just built. Every
object in the inventory is the readout of a \GRIP{} selection on a
stratum of \emph{one} six-dimensional bulk: the electron and the vacuum
as Ground Geometric Attractors (one deepest basin, so every copy is the
same copy), the Kaxion as a pinned angle (one frozen value, so every
patch inherits the same relic). Call this the \emph{mold}: there is
exactly one specification for each object, so there is nothing regional
to vary. A landscape has many molds and needs anthropics to explain why
we see one; \GGTheory{} has one mold and needs nothing. And the mold's
guarantee is stronger than any causal argument could be, because it is
underwritten by the geometry rather than by history: two regions that
were never in causal contact --- even regions permanently beyond our
horizon --- still read the same electron, because they are boundaries of
the same bulk.

\subsection{The second half of the answer: the ordering}
\label{subsec:shp-ordering}

A mold alone, however, does not close the cosmological question, because
the question has a \emph{when} as well as a \emph{what}. Granted that
the deepest basin is unique --- did every region actually reach it?
Could a patch beyond our cluster have been caught mid-descent, camped in
a different basin, frozen at a different angle, stocked with a different
abundance? A uniqueness theorem guarantees the destination; it does not
by itself guarantee that all travelers arrived, or arrived with the
same luggage. Here the derived placement of inflation --- at the GUT
edge, at the \emph{top} of the descent (Part~IV) --- supplies what no
earlier reading could, and it is worth stating as its own result:

\emph{Inflation happened before any matter existed.} At the top of the
descent the boundary universe contains no electrons, no protons, no
photons in any meaningful abundance --- only the vacuum itself, the
scale modulus on its derived plateau one e-fold below the GUT edge. The
accelerated stretch therefore acted on a universe \emph{empty of
things}: it distributed no matter, because there was none to distribute.
What it distributed was the single causal patch itself.

\emph{Then the matter was minted --- everywhere at once, by one
mechanism.} At the graceful exit, the one modulus decayed at Planck
strength into the one boundary field content the bulk admits, reheating
the whole stretched patch simultaneously and identically, because every
point of the patch carried the same modulus at the same point of the
same descent. The electrons were not smoothed into agreement after the
fact; they were \emph{stamped from one mold after the distribution
network was already laid}. Once it is understood that the inflation
which exploded the universe came \emph{before} all the electrons were
made, their identity across the sky stops being a coincidence in need of
explanation: the agreements were inherited, not negotiated.

\emph{The dark sector's one mark was fixed even earlier.} The Kaxion is
born at the second crossing, far down the ladder --- but its single
potentially regional mark, the misalignment angle, was pinned during
inflation, while the sky was still one patch. The dark matter's identity
was set \emph{before its own birth}, in the era when nothing could yet
differ from place to place; that is exactly why the relic is
patch-uniform with zero primordial isocurvature
(\S\ref{subsec:three-objects}). The axion literature owns this
mechanism completely --- it is the standard pre-inflationary
Peccei--Quinn scenario (\S\ref{subsec:shp-pedigree}) --- and owns it
as a statement about the relic abundance; what is new here is only the
register: read as identity, the pinned angle is the dark sector's one
regional mark, fixed before the dark matter existed.

\emph{And no region camped on the way down.} The residual worry --- a
distant patch trapped in the middle basin, a vacuum domain structure,
walls between regions that chose differently --- is closed by the prompt
lock crossing, \(N_2 = 0\), with some fifty orders of magnitude of
margin in the ledger action: there is no alternative basin to camp in,
no defect network, no domains. Every road from the GUT edge leads to the
same floor, and every region took it at the same pace.

\subsection{The theorem, assembled --- and its exposure}
\label{subsec:shp-theorem}

\begin{equation}
\boxed{\;\parbox{0.88\textwidth}{\centering \emph{Why is everything the
same everywhere?} The identity of things across the sky is guaranteed
\emph{if} the vacuum is unique --- (i) each object then has exactly one
specification, the Ground Geometric Attractor of its stratum read through
\GDOS{} from one bulk (\emph{the mold}) --- and this half \emph{reduces} a
new cosmological puzzle to the same vacuum-uniqueness the program already
stakes everything on, unifying the two rather than closing one. The other
half is independently earned: (ii) the universe was stretched from one
causal patch before any object existed, so the agreements were inherited,
not negotiated (\emph{the ordering}); (iii) the one mark that could have
varied --- the compact angle --- was frozen while the sky was still one
patch (\emph{the pinning}); (iv) no region could be caught in another basin
on the way down (\emph{the prompt crossing}). Sameness of state is
inflation's gift; sameness of things is the geometry's --- \emph{modulo the
one uniqueness conjecture, to which it is honestly reduced}.}\;}
\label{eq:second-horizon-thesis}
\end{equation}

Five features of the answer deserve emphasis. First, it is
\emph{layered} exactly like the identity ladder of
\S\ref{sec:uniqueness-layer-by-layer}: the mold explains identity of
\emph{kind} (why every electron has the same mass --- tested by quasar
spectroscopy of the constants), the ordering explains uniformity of
\emph{amount} (why every region carries the same abundances --- tested by
the CMB and BBN uniformity), and the pinning explains both at once for
the dark sector (tested by the isocurvature bound). Second, it
\emph{inverts a weakness of the retired reading into a strength of the
derived one}: a TeV-scale inflation would have acted \emph{after} the
visible sector's history had begun, muddying the before/after ordering;
the GUT-adjacent placement, forced by the sign-level tilt argument of
Part~IV, is precisely what makes ``the universe exploded before the
electrons were made'' true. The ordering is not an ornament on the
two-basin derivation --- it is one of its deepest consequences. Third,
the answer is \emph{exposed} in channels the codebook already stakes:
a confirmed spatial variation of a fundamental constant would break the
mold; a detected axion-sector isocurvature would break the pinning; a
relic domain structure or a TeV-epoch transition background would break
the prompt crossing. The second horizon problem adds no new physics ---
it adds a new reading of physics already derived, and it inherits that
physics' kill-tests. Fourth, the answer is new only where the pedigree of
\S\ref{subsec:shp-pedigree} leaves room, and it should be owned in exactly
those terms: the \emph{spatial} half of the answer --- we live deep inside
one stretched domain --- has been the field's property since 1984; what is
claimed here is the \emph{temporal} half --- the domain existed before its
contents did, so identity was inherited rather than negotiated --- together
with the naming, the extension from the orientation of one vacuum to the
identity of the whole inventory, and the unification with vacuum
uniqueness. Fifth, and most important to state plainly: the mold
half is a \emph{reduction}, not an independent proof. It does not
\emph{establish} that the vacuum is unique --- it shows that \emph{if} the
vacuum is unique (the landscape-uniqueness the program stakes and has not
closed, \S\ref{subsec:shp-mold}), \emph{then} cosmic identity follows for
free. That is a genuine gain --- two deep puzzles, cosmic identity and
vacuum uniqueness, are shown to be one puzzle, so a single future proof
settles both --- but it would be dishonest to sell it as a standalone
solution. The ordering, the pinning, and the prompt crossing are earned
independently of the uniqueness conjecture; the mold is the unification that
ties them to it.

After all, the deepest uniformity in the sky was never the temperature.
It is the fact that the same atom shines on both sides of it. The first
horizon problem asked why the universe is everywhere equally warm; the
second asks why it is everywhere made of the same things --- and the
answer, it turns out, required knowing not only that the universe was
once one patch, but that it was one patch \emph{before it contained
anything at all}.


\section{The third horizon problem: sameness of choices}
\label{sec:third-horizon}

The second horizon problem asked why disconnected regions hold the
same \emph{things}. There is a third rung above it, and it is the
sharpest of the three: why did every disconnected region make the same
\emph{choice}?

The choice in question is the one Part~V derived: matter over
antimatter, by a few parts in ten billion, through the CP-violating
decays of the lock-scale Majorana block. State the standard framework
honestly and this choice is made independently in every Hubble patch
after the patches have fallen out of contact: the same mechanism must
fire everywhere, with the same sign and the same yield, coordinated by
nothing. If the CP-odd invariant that sets the sign had any
spontaneous or stochastic element --- a sign chosen at a phase
transition, domain by domain --- the sky would carry matter--antimatter
boundaries and their annihilation signatures, and it carries none
\citep{CohenDeRujulaGlashow1998}: the observable universe is matter on
every line of sight. If the yield fluctuated from patch to patch, the
baryon-to-photon ratio would arrive with super-horizon inhomogeneity
of a kind the acoustic spectrum bounds tightly. The uniformity of the
choice is an observational fact, and it is a fact about regions that
never exchanged a signal.

The ladder, then. The 1981 problem asked why the regions share the
same \emph{conditions} --- one temperature --- and inflation answered
it. The second asked why they share the same \emph{things} --- one
electron, one vacuum --- and the identity core, welded to the GUT-edge
launch, answered it (\S\ref{sec:second-horizon}). The third asks why
they share the same \emph{choices} --- one sign, one yield, one
mechanism --- and here neither inflation nor the identity core
suffices, because sameness of choices requires that the choosing
itself be deterministic. That is what the descent supplies, and it
supplies it twice over. The \GSSB{} steps of the cascade select unique
attractors, not random members of degenerate families
(\citealp{ArisakaA6Gi4}; \citealp{ArisakaA11Origin}), so there are no
domains to disagree; and the CP-odd invariant of the lock-scale block
is holonomy data --- part of the frozen, patch-uniform readout of the
compact sector, the same inheritance that delivers the relic phase
identically everywhere (Part~V). Disconnected regions agree for the
same reason two impressions of a seal agree: not because they
exchanged messages, but because they were struck from the same die.

The ranking deserves to be recorded the way the second problem's was.
The first horizon problem needs inflation. The second needs inflation
plus the identity core. The third needs inflation plus the identity
core plus the determinism of the cascade --- and it is where that most
contested commitment of the framework, that \GSSB{} selects rather
than samples, stops being a philosophical preference and becomes the
only mechanism on the table for an observational fact: the universe
made one choice, everywhere, with nothing available to coordinate it.

It is worth saying which choice it was. Baryogenesis is the productive
breaking in its purest form --- the step that selects a world with
things in it over a world with none. And by the neutrality theorem of
\S\ref{ssec:neutrality} it threads an exact needle: asymmetric in
precisely the one number that is global and unwatched, exactly neutral
in every number a gauge field polices. The universe breaks the
symmetry it may break and keeps the one it must keep.

\section{Why the inventory is \texorpdfstring{\(\pi\)}{pi}-locked: the closed-packet derivation}
\label{sec:pi-locked-derivation}

The identity core of Section~\ref{sec:universal-cosmic-identity-preview} located the cosmic inventory as the readout of one attractor and its relics. This Section supplies the arithmetic behind the readout: why the matter fraction takes the specific value \(1/\pi\) and not some neighboring number, why the apparent coincidence of the matter and vacuum densities at the present epoch is no coincidence, and the four-form theorem that would close the derivation. It names the three sub-questions and gives each its headline argument; the formal closed-packet theorem is stated in Section~\ref{sec:why-omega-m-pi-theorem}.

\subsection{Why \texorpdfstring{\(\Omm = 1/\pi\)}{Omega-m = 1/pi} and not some other value?}
\label{sec:why-omega-m-pi}

The first question is the most direct (the target chart is Figure~\ref{fig:pi-locked-chart-omega}): why does the matter fraction take the specific value \(1/\pi \approx 0.3183\), rather than, say, \(0.30\) (a round decimal) or \(1/3\) (a simple rational) or any other plausible candidate? The answer is structural in three steps, each stated as a formal lemma in Section~\ref{sec:why-omega-m-pi-theorem}.

\emph{First}, the cosmological-boundary stratum is a single connected manifold on which the matter fraction is defined as a topological invariant rather than as an empirical density count. This is the closed-packet topology argument assembled in this Part, resting on the \OGNfive{} integer ledger of \citet{ArisakaA4Euclid} that forces it: every admissible cosmological inventory has the structure of a closed packet, with the matter fraction equal to the relative weight of one specific topological cell of the bulk packet projected onto the boundary. That the packet's unit is itself forced rather than chosen --- that the ``one'' in ``one closed packet'' is read off the geometry and not adopted --- is the contribution of \citet{ArisakaA5GDOS}. The relative weight is a rational function of \(\pi\), not a free parameter; the only question is which rational function.

\emph{Second}, the rational function is selected \emph{by the geometry of the boundary measure}, not by the data. The matter sector's charge lives on the compact phase circle \(S^1_\theta\) of the bulk, which sits inside the electroweak \(\mathrm{SU}(2)\!\cong\!S^3\) holonomy manifold as its Hopf fiber \(S^1\hookrightarrow S^3\to S^2\) (drawn in Figure~\ref{fig:hopf-fibration}); the present-epoch fraction is the ratio of the fiber volume to the total holonomy volume. Both are canonical unit spheres --- \(\mathrm{Vol}(S^1)=2\pi\), \(\mathrm{Vol}(S^3)=2\pi^2\) --- so the ratio is \(2\pi/2\pi^2 = 1/\pi\), with the \(\pi\) descending from the denominator volume and nothing read off the sky. The assignment is upstream of the value: both spheres enter for reasons fixed by the bulk construction before any fraction is computed --- \(S^1_\theta\) as the home of the matter charge, \(S^3\) as the electroweak holonomy --- so the pair is not chosen for its ratio; the ratio is what the prior assignment delivers, and what remains owed is exactly VIII-b$''$: that the \GDOS{} measure weights this fiber and no other. Competitors fail geometrically, not empirically: \(1/(3\pi)\) would need a holonomy volume \(6\pi^2\) that no manifold in the construction has; \(1/(2\pi)\) is the \emph{baryonic} projection (one unit per revolution of the fiber), not the matter/total ratio; \(1/\pi^2\) would need an \(S^5\) that the four-dimensional boundary does not carry.

\emph{Third}, the same fiber-volume measure fixes the whole inventory at once. Dark energy is the part of the \(S^3\) holonomy volume the matter Hopf circle does not cover: \(\OmL = (2\pi^2-2\pi)/2\pi^2 = 1 - 1/\pi\), a topological complement, not an adjustable constant. Baryons are one integer unit of \(\mathrm{U}(1)_B\) charge threaded once around the fiber: \(\Omb = \Omm/\mathrm{Vol}(S^1) = (1/\pi)/(2\pi) = 1/(2\pi^2) = 1/\mathrm{Vol}(S^3)\). One fibration, three coupled numbers, no freedom to move one without the others --- the structural signature that separates a derivation from a per-observable fit.

The geometry these three steps trade in is drawn in
Figure~\ref{fig:hopf-fibration}, and it repays a moment of looking,
because the fibration is the one geometric fact this Section most
needs the reader to see. The 3-sphere cannot be drawn directly, so the
figure projects it stereographically into ordinary space, where its
Hopf fibers appear as circles arranged on nested tori. Their defining
property is the remarkable one: \emph{every} fiber passes over exactly
one point of the base 2-sphere, every pair of fibers is linked exactly
once, and together the circles fill the 3-sphere with no gaps and no
crossings. The red circle --- the matter fiber --- is not a special
place: every fiber is equivalent, and that equivalence is the physical
point. The matter sector does not sit \emph{somewhere} on the
electroweak holonomy manifold; it is one winding of a structure that
permeates all of it --- the compact phase circle \(S^1_\theta\) of the
bulk, embedded canonically in the \(\mathrm{SU}(2) \cong S^3\) the
integer ledger selects.

The intuition for the ratio itself is equipartition on a compact
configuration space. A frozen stationary measure has no preferred
point, so each sector's share is simply the volume it occupies: the
matter charge runs along one fiber --- a circle, of circumference
\(2\pi\) --- while the full holonomy carries the 3-sphere volume
\(2\pi^2\); the share is their ratio, \(1/\pi\). One power of \(\pi\)
survives because the ratio cancels the circle's \(2\pi\) against one
factor in the sphere's \(2\pi^2\) --- the \(\pi\) in the cosmic
inventory is, quite literally, the leftover of a sphere volume. The
right panel of the figure keeps the ledger: the uncovered complement
\(1 - 1/\pi\) for dark energy, one unit of \(\mathrm{U}(1)_B\) charge
per fiber revolution for baryons. The chart of
Figure~\ref{fig:pi-locked-chart-omega} then reads the same three
coupled numbers out against the data.

\begin{figure}[H]
\centering
\includegraphics[width=0.98\textwidth]{c2_hopf_fibration.pdf}
\caption[The Hopf fibration behind the \(\pi\)-locked inventory.]{\textbf{The
Hopf fibration behind the \(\pi\)-locked inventory.} \emph{Left:} the
3-sphere of the electroweak holonomy, drawn by stereographic
projection: the Hopf map \(S^1 \hookrightarrow S^3 \to S^2\) fills
\(S^3\) without gaps by linked circles, one over each point of the
base 2-sphere (families of base latitudes in blue and rose; one
fiber --- the matter circle --- highlighted in red). \emph{Right:}
the volume ledger that turns the fibration into the cosmic inventory:
the matter share is fiber volume over holonomy volume,
\(2\pi/2\pi^2 = 1/\pi\); the dark-energy share is the uncovered
complement, \(1 - 1/\pi\); and the baryon share threads one unit of
\(\mathrm{U}(1)_B\) charge per fiber revolution, \(1/(2\pi^2)\). One
fibration fixes three coupled numbers; none can be moved without the
others. That the frozen \GDOS{} measure weights exactly \emph{this}
fiber as the matter sector's home is the content of the measure lemma
VIII-b\(''\) on which Theorem~\ref{thm:omega-m-pi-topological} is
conditional (\S\ref{sec:why-omega-m-pi-theorem}). Figure shared with
the companion Hubble paper \PaperCTwo{} \citep{ArisakaC2Hubble};
generator \texttt{c2\_fig\_hopf.py}.}
\label{fig:hopf-fibration}
\end{figure}



\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily,
  lbl/.style={align=center, inner sep=2pt},
  pct/.style={font=\sffamily\bfseries\huge, text=white, align=center},
]
\def\R{3.4}

% title (self-contained when shared)
\node[font=\sffamily\huge\bfseries] at (0,\R+1.4) {Pi-locked Chart of $\boldsymbol\Omega$};

% ---- wedges (counterclockwise): Omega_Lambda (top+left), Omega_dm (lower-right),
%      Omega_b (thin sliver at 3 o'clock) ----
\filldraw[fill=teal!72, draw=white, line width=1.5pt]
  (0,0) -- (9.119:\R) arc (9.119:254.527:\R) -- cycle;        % Omega_Lambda  245.41 deg
\filldraw[fill=blue!42, draw=white, line width=1.5pt]
  (0,0) -- (254.527:\R) arc (254.527:350.881:\R) -- cycle;    % Omega_dm       96.35 deg
\filldraw[fill=orange!88, draw=white, line width=1.5pt]
  (0,0) -- (350.881:\R) arc (350.881:369.119:\R) -- cycle;    % Omega_b        18.24 deg
\draw[draw=black!22, line width=0.60pt] (0,0) circle (\R);

% ---- in-slice predicted shares ----
\node[pct] at (131.823:0.58*\R) {68.2\%};
\node[pct] at (302.704:0.60*\R) {26.8\%};

% ---- labels written just OUTSIDE each slice (no leader lines), larger fonts ----
% Omega_Lambda -- upper-left
\node[lbl, anchor=east] at (-2.7,3.0)
  {{\LARGE\bfseries Dark energy\, $\boldsymbol{\Omega_\Lambda}$}\\[3pt]
   {\Large $=\,1-1/\pi\,=\,0.6817$}\\[4pt]
   {\large\color{black!50}Planck\, $0.685\pm0.007$}};
% Omega_dm -- lower-right
\node[lbl, anchor=west] at (3.3,-2.2)
  {{\LARGE\bfseries Dark matter\, $\boldsymbol{\Omega_{\mathrm{dm}}}$}\\[3pt]
   {\Large $=\,1/\pi-1/(2\pi^2)\,=\,0.2676$}\\[4pt]
   {\large\color{black!50}Planck\, $0.2657\pm0.0070$}};
% Omega_b -- right (3 o'clock)
\node[lbl, anchor=west] at (3.75,0.2)
  {{\LARGE\bfseries Baryons\, $\boldsymbol{\Omega_b}$}\\[3pt]
   {\Large $=\,1/(2\pi^2)\,=\,0.0507$}\\[4pt]
   {\large\color{black!50}Planck\, $0.0493\pm0.0006$}};

% ---- matter-fraction identity + tagline ----
\node[font=\sffamily\LARGE] at (0,-\R-1.1)
  {$\Omega_m=\Omega_{\mathrm{dm}}+\Omega_b=1/\pi=0.3183$};
\node[font=\sffamily\Large\itshape, text=black!55] at (0,-\R-1.95)
  {one circle, three powers of $\pi$ --- the present-epoch snapshot};
\end{tikzpicture}}
\caption[Pi-locked Chart of $\Omega$: the cosmic inventory as three powers of $\pi$.]{\textbf{Pi-locked Chart of $\Omega$.} The present-epoch cosmic inventory drawn as a pie chart: each slice is one closed-form fraction of $\pi$, the three forced together as one coupled set --- dark energy $\Omega_\Lambda = 1 - 1/\pi$, dark matter $\Omega_{\mathrm{dm}} = 1/\pi - 1/(2\pi^2)$, and baryons $\Omega_b = 1/(2\pi^2)$ --- and the two matter slices sum to $\Omega_m = 1/\pi$. Written just outside each slice is the closed-form prediction with the Planck~2018 \citep{Planck2018} measured value beneath it: the predictions land on the data to better than $3\%$ ($\Omega_\Lambda$ and $\Omega_{\mathrm{dm}}$ within $1\sigma$, $\Omega_b$ at $2.3\sigma$). The chart is the visual statement of the Hopf fiber-volume form of Theorem~\ref{thm:omega-m-pi-topological} (conditional on the measure lemma VIII-b$''$): the matter share is \(\mathrm{Vol}(S^1)/\mathrm{Vol}(S^3)=1/\pi\), dark energy the complement, baryons one unit per fiber revolution \(1/(2\pi)\) within matter --- three coupled numbers from one fibration, with the absolute fractions read at the crossing epoch (\S\ref{subsec:whynow-form-scale-epoch}).}
\label{fig:pi-locked-chart-omega}
\end{figure}



The answer to (a) is therefore: \(\Omm = 1/\pi\) is the ratio of the matter Hopf-fiber volume to the total electroweak-holonomy volume, \(\mathrm{Vol}(S^1)/\mathrm{Vol}(S^3)=2\pi/2\pi^2\); the competing rational functions of \(\pi\) are excluded by the geometry of the boundary measure, with no reference to the measured value and no continuous parameter free. The formal statement, and the one measure-normalization lemma it still rests on, are in Section~\ref{sec:why-omega-m-pi-theorem}.

\subsection{Why is the cosmic coincidence not a coincidence?}
\label{sec:cosmic-coincidence-not-coincidence}

The traditional cosmic-coincidence problem of \(\Lambda\)CDM asks: why does the present epoch happen to be the unique cosmic time at which the matter density and the cosmological-constant density are comparable? Under a free \(\Lambda\)-and-\(\Omm\) hypothesis, the comparable-magnitude condition holds only for a fleeting fraction of the cosmic history --- typically a few e-folds out of the eighty between recombination and the asymptotic de~Sitter future --- and the apparent agreement between \(\Omm\) and \(\OmL\) at the present epoch becomes an anthropic-or-coincidence puzzle.

The answer dissolves this puzzle in three steps, each a corollary of the \(\pi\)-locked-inventory theorem.

\emph{First}, the matter fraction and the cosmological-constant fraction are not two independent observables on the cosmological-boundary stratum: they are bound by the topological closure \(\Omm + \OmL = 1\) (modulo small radiation and curvature contributions). This is unit-trace closure, not free parameter choice. The comparable-magnitude condition at the present epoch is therefore the \emph{generic} state of the cosmological-boundary attractor, not a coincidence about \emph{when} we are observing.

\emph{Second}, the specific ratio \(\Omm / \OmL = (1/\pi) / (1 - 1/\pi) = 1/(\pi - 1) \approx 0.467\) at the present epoch is forced by the same topology that fixes each fraction individually. This is a numerical theorem, not an empirical relation.

\emph{Third}, the temporal coincidence (``why are we here at this particular cosmic age?'') is dissolved by a structural integer identification on two independent epoch-bookkeeping channels. The Hubble two-stage corridor switch event sits at \(z_* = e^J = e^7\) (Theorem~\ref{thm:hubble-two-stage-corridor}), and the e-fold count from the corridor switch to the present epoch is \(\ln(1 + z_*) \approx J = 7\) as well; both channels are governed by the same \OGNfive{} integer \(J = 7\) of \AFour{}. The temporal coincidence (``why is the present epoch at this particular cosmic age?'') is therefore dissolved by the integer identification: the same integer \(J = 7\) sets both the recombination scale and the recombination-to-present e-fold separation, so ``now'' is not a free choice of cosmic time but the natural endpoint of the integer-counted descent from the corridor switch.

The answer to (b) is therefore: the cosmic coincidence is not a coincidence because (i) the matter-vacuum split is topologically locked, (ii) the specific ratio is a numerical theorem, and (iii) the temporal ``now'' is integer-counted from the recombination switch event --- each a corollary of the universal-identity theorem.


\subsection{The closed-packet four-form integration: \texorpdfstring{$\Omm = 1/\pi$}{Omega-m = 1/pi} as a topological theorem}
\label{sec:why-omega-m-pi-theorem}

This subsection states the formal theorem behind \S\ref{sec:why-omega-m-pi} in canonical form, with hypotheses, conclusion, and a five-step proof outline anchored on the \BIsix{} four-form of \AThree{} and the \OGNfive{} integer ledger of \AFour{}.

\paragraph{Setup.}
On the cosmological-boundary stratum --- the four-dimensional boundary slice obtained from the six-dimensional \GG{} bulk of \ATwo{} via the \OGGEP{} projection of \AFive{} --- the \BIsix{} modified Bianchi identity of \AThree{} fixes a closed four-form
\[
 X_4 = \operatorname{tr} R^2 \;-\; \operatorname{Tr} F_5^2 \;-\; \tfrac{2}{5} F_B^2 \;-\; \tfrac{2}{5} F_{Qi}^2,
\]
with the coefficient vector $(1, -1, -2/5, -2/5)$ proved unique by \AThree{}'s anomaly-cancellation theorem. The cosmological matter density fraction $\Omm$ at the present epoch is operationally the ratio of the matter-sourced contribution to the total stress-energy budget, projected through \OGGEP{} onto the four-dimensional cosmological-boundary slice.

\begin{theorem}[$\Omm = 1/\pi$ as the Hopf fiber-volume ratio, conditional (T-Cosmos-17)]\label{thm:omega-m-pi-topological}
Assume:
\begin{enumerate}
 \item[\textbf{(I)}] The cosmological-boundary stratum is a connected, orientable four-manifold $\mathcal{M}_4$ obtained as the \OGGEP{} projection of the \GG{} bulk (\citealp{ArisakaA2GG6}; \citealp{ArisakaA5GDOS}).
 \item[\textbf{(II)}] \BIsix{} holds throughout the cosmological history: $dX_4 = 0$ on $\mathcal{M}_4$, with the coefficient vector $(1, -1, -2/5, -2/5)$ \citep{ArisakaA3BI6}.
 \item[\textbf{(III)}] The \OGNfive{} integer $N=3$ (families) and $L=5$ (\(\mathrm{SU}(5)\)) fix the boundary holonomy manifold as the electroweak $\mathrm{SU}(2)\cong S^3$, and the matter sector's charge as its Hopf fiber $S^1$ (VIII-b$''$, the unit-packet normalization).
\end{enumerate}
Then the present-epoch matter density fraction equals
\[
 \boxed{\;\Omm \;=\; \frac{1}{\pi} \;\approx\; 0.3183\;}
\]
as the Hopf fiber-volume ratio $\mathrm{Vol}(S^1)/\mathrm{Vol}(S^3)$ of the boundary holonomy, conditional on the measure-normalization lemma VIII-b$''$ (Conjecture~VIII), with no continuous adjustable parameter and no reference to the measured inventory.
\end{theorem}

\emph{Epoch-identity reading.} The conditionality on Conjecture~VIII localizes further: with VIII-b$'$ proved and VIII-b$''$/($\star$) closed as the epoch identity (see the Conjecture-VIII refinement paragraph), the theorem may equivalently be read as \emph{unconditional in form at the crossing epoch} $t_\pi$: at the instant when one packet's diluted ledger record equals one unit's equipartition rate, the inventory reads the coupled $\pi$-form $(1/\pi,\;1-1/\pi,\;1/(2\pi^2))$. The residual hypotheses are then VIII-a (uniformity) and the identification $t_0 = t_\pi$ (the $1\%$ L4 epoch input of \S\ref{subsec:whynow-form-scale-epoch}).

\paragraph{Proof outline.}

The proof proceeds in five steps. The first three establish the closed-packet topology; the fourth performs the integration; the fifth rules out competing rational functions of $\pi$ by the constitutional cost-minimum principle.

\emph{Step 1 (Closure of $X_4$).}
By hypothesis (II), $dX_4 = 0$ on $\mathcal{M}_4$. The four-form $X_4$ therefore represents a class in the fourth de Rham cohomology $H^4(\mathcal{M}_4, \mathbb{R})$. The class is non-trivial because at least one of the four trace-squared terms (specifically $\operatorname{tr} R^2$) generates a non-trivial Pontryagin class on the curved cosmological-boundary slice.

\emph{Step 2 (Closed-packet structure: the integer skeleton).}
By hypothesis (III) and the topological-cell decomposition of \AFour{}, the \OGNfive{} tuple $(M,N,L;J,K)=(1,3,5;7,35)$ fixes the \emph{integer} content of the packet: the family integer $N=3$ and the gauge integer $L=5$ identify the boundary holonomy as the electroweak $\mathrm{SU}(2)$, and the packet is one canonical unit (the Hirzebruch minimum $\sigma=1$ of Step 3). This step supplies the discrete skeleton --- \emph{which} manifold and \emph{one} packet on it --- but not yet the share; the share is a fiber volume, computed in Step 4. \emph{(One might instead assign the matter cell the rational weight $M/(LN)=1/15$ and seek an epoch-independent arithmetic cancellation to reach $1/\pi$; that route is a provable dead end --- Appendix~\ref{appA:pilock-steps} --- and the fiber-measure evaluation below replaces it.)}

\emph{Step 3 (Pontryagin normalization).}
The Chern--Pontryagin integral of $\operatorname{tr} R^2$ over a closed orientable four-manifold satisfies
\[
 \int_{\mathcal{M}_4} \operatorname{tr} R^2 \;=\; 8 \pi^2 \cdot p_1[\mathcal{M}_4],
\]
where $p_1[\mathcal{M}_4]$ is the integer-valued first Pontryagin number. For the \OGNfive{}-structured cosmological-boundary slice with $p_1 = N = 3$, the integral evaluates to $24 \pi^2$. The matter-cell contribution carries this normalization factor when projected through \OGGEP{} onto the present-epoch boundary. \emph{(Normalization note: the $8\pi^2$ is consumed upstream --- the \GEP{} functional of \ASix{} is defined as $(1/8\pi^2)\int X_4$, so the packet's energy content is the pure integer $p_1$, and \emph{no} residual $\pi$ descends from this step. The $\pi$ of $1/\pi$ is therefore \emph{not} a leftover of the Pontryagin factor and \emph{not} produced by Lemma VIII-b$'$ --- whose content is the reality factor of exactly $2$ (the $2\pi^2$ vs $\pi$ half-measure of a charged sector against a real one), not a $\pi$. The $\pi$ enters as a \emph{volume} in the denominator of the fiber-measure ratio of Step 4. By the Hirzebruch signature theorem $p_1 = 3\sigma$, the value $p_1 = N = 3$ is $\sigma = 1$: the minimal closed gravitational topology, which is what forces the packet to be one canonical unit.)}

\emph{Step 4 (The fiber-measure evaluation: the Hopf ratio).}
The matter fraction is not an epoch-independent \emph{arithmetic} of the \OGNfive{} integers --- the route that seeks one is a provable dead end (Appendix~\ref{appA:pilock-steps}, Step~4, evaluating to the failed $3/(125\pi)$; the dilution no-go of the Conjecture-VIII audit shows no constant combination can close it). It is a ratio of \emph{fiber volumes} on the boundary holonomy, and this is where the $\pi$ is born. The matter sector's charge lives on the compact phase circle $S^1_\theta$, which sits inside the electroweak $\mathrm{SU}(2)\cong S^3$ holonomy manifold as its \emph{Hopf fiber} $S^1\hookrightarrow S^3\to S^2$ (a canonical fibration, with no freedom of choice). Under the frozen stationary \GDOS{} measure of Lemma VIII-a, each sector's present-epoch share is its fiber volume divided by the total holonomy volume, so
\[
 \boxed{\;\Omm \;=\; \frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)} \;=\; \frac{2\pi}{2\pi^2} \;=\; \frac{1}{\pi}\;}
\qquad(\text{unit-normalized spheres}),
\]
exact. The reality factor of Lemma VIII-b$'$ (charged measure $2\pi$ against real measure $\pi$, ratio $2$) then supplies the extra factor in the baryon line, and the crossing-epoch identity $u=1$ (VIII-b$''$, discharged as $t_0 = t_\pi$ to $1\%$, the declared L4 input of \S\ref{subsec:whynow-form-scale-epoch}) fixes \emph{when} the running share equals this geometric ratio --- the ``when,'' not the ``what.'' The value $1/\pi$ is thus read off the Hopf geometry, with the data nowhere in the selection. \AFour{} supplies the integer that pins the packet to one unit; the fiber-volume identification is the content of the remaining measure lemma (VIII-b$''$), stated below.

\emph{Step 5 (Uniqueness: the competitors die by geometry, not by data).}
The decisive test of non-circularity is whether the alternative rational functions of $\pi$ fall \emph{without} appeal to the measured inventory. They do:
\begin{itemize}
 \item[(a)] $1/(3\pi)$ would require a total holonomy volume $3\,\mathrm{Vol}(S^3) = 6\pi^2$. No boundary-holonomy manifold in the construction has that volume --- there is no ``triple $S^3$.'' Excluded by geometry.
 \item[(b)] $1/(2\pi) = 1/\mathrm{Vol}(S^1)$ is the \emph{baryonic} projection (one integer unit per revolution of the fiber), i.e.\ the $\Omb/\Omm$ ratio, not the matter/total fiber ratio. Assigning it to $\Omm$ mistakes the baryon fiber for the matter fiber. Excluded structurally.
 \item[(c)] $1/\pi^2$ would require $\mathrm{Vol}(S^1)/\mathrm{Vol}(S^5) = 2\pi/\pi^3 = 2/\pi^2$ (off by a factor of $2$), and there is no $S^5$ holonomy on the four-dimensional boundary in any case. Excluded by geometry.
\end{itemize}
Every competitor that an empirical argument would eliminate with ``the data prefer $\Omm/\OmL \approx 0.46$'' is eliminated here by ``there is no fiber of that volume.'' The selector is the fiber-volume geometry, and the measured value of the inventory enters the derivation at no step.

\paragraph{Proof status (honesty note): where the conditional lives.}
\label{par:where-the-conditional-lives}

The inventory is carried at grade D\(^*\), and it is worth saying exactly
which link of the chain the star is attached to --- because it is not the
link the earlier phrasing of this Section named. The derivation has four
links, and they do not have the same status.

\begin{enumerate}
\item \textbf{The assignment.} That the boundary holonomy manifold is the
electroweak \(\mathrm{SU}(2)\cong S^3\), and that the matter sector's
charge lives on its Hopf fiber \(S^1\). Both are fixed upstream by the
\OGNfive{} integers \(N=3\) and \(L=5\), \emph{before} any fraction is
computed, so the pair is not chosen for its ratio: the ratio is what the
prior assignment delivers. \emph{Derived, upstream.}

\item \textbf{VIII-a (uniformity).} That the frozen \GDOS{} measure leaves
every counted direction delocalized. Proved in consumer form with its
three residues discharged; exact up to the gravity-mediated
\(\sim e^{-K}\) floor, whose \(O(1)\) prefactor is the one open
coefficient and is load-bearing for nothing deployed. \emph{Proved.}

\item \textbf{VIII-b\('\) (the reality half-measure).} Exact mode
counting --- \(2\Lambda+1\) independent real degrees of freedom for a
reality-constrained sector against \(2(2\Lambda+1)\) for a
holonomy-charged one, a ratio of exactly \(1/2\) at every finite cutoff.
\emph{Proved.}

\item \textbf{VIII-b\(''\) (unit-packet normalization).} One closed packet
\(=\) one unit of fiber measure. \emph{Discharged --- but not into a
statement about a constant.}
\end{enumerate}

\paragraph{Why links 3 and 4 have different status, and it is not an
accident of effort.}
Lemma VIII-b\('\) compares two \emph{counts}. A ratio of counts is
dimensionless and cutoff-independent, so it can be proved outright, and it
is. Lemma VIII-b\(''\) compares a length to a volume: at fiber radius
\(R\),
\begin{equation}
\frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)}
\;=\; \frac{2\pi R}{2\pi^2 R^3}
\;=\; \frac{1}{\pi R^{2}},
\label{eq:fibre-ratio-radius}
\end{equation}
which equals \(1/\pi\) only at unit radius. \emph{A ratio that carries a
dimension cannot be fixed by geometry alone; it needs a physical unit.}
That is the entire content of residue \textbf{(R2)}, and it is why
VIII-b\(''\) required a unit bridge while VIII-b\('\) did not. Naming the
two lemmas alike concealed a real difference in kind.

\paragraph{And the unit bridge terminates on a theorem, not on a gap.}
Superselection freezes the packet's integer charge, not its energy record,
so the packet energy tracks the diluting matter sector
(\(\propto a^{-3}\)) while the equipartition rate \(T_{\GFF}/2\pi\) is
frozen. Hence the \emph{dilution no-go}: no epoch-independent constant can
equal a diluting quantity at more than one instant. VIII-b\(''\) therefore
\emph{cannot} close as a statement about a constant --- and that is a
result, not an omission. What it closes into is the occupancy identity:
\(u(t) = \pi\,\Omm(t)\) sweeps monotonically through unity exactly once,
and \(u = 1\) is \emph{mathematically equivalent} to \(\Omega_{m,0} = 1/\pi\).

\begin{equation}
\boxed{\;\text{The form is derived. The epoch is calibrated.}\;}
\label{eq:form-derived-epoch-calibrated}
\end{equation}

Concretely: the coupled \(\pi\)-form
\(\bigl(1/\pi,\;1-1/\pi,\;1/(2\pi^2)\bigr)\) follows from links 1--3 with
no measured density entering at any step, and the competitors are excluded
geometrically rather than empirically --- \(1/(3\pi)\) would need a
holonomy volume \(6\pi^2\) that no manifold in the construction has,
\(1/\pi^2\) an \(S^5\) the four-dimensional boundary does not carry. That
is a derivation, and at the crossing epoch the theorem is
\emph{unconditional in form}. What the geometry does \emph{not} fix is
\emph{when} the running inventory reads that form. The theorem supplies a
unique crossing epoch \(t_\pi\); the identification \(t_0 = t_\pi\), good
to \(1\%\), is the declared L4 epoch input of
\S\ref{subsec:whynow-form-scale-epoch}. \textbf{It is that identification,
and nothing else, that makes the present-epoch values} D\(^*\).

The calibration is a genuine one and its width should be stated with it.
The crossing sits at \(a_\pi = 0.9949\), about \(75\)~Myr before the
present at the canonical \(\Hzero = 66.6\); moving \(\Omm\) by
\(\pm1\sigma\) of the Planck uncertainty moves the crossing from roughly
\(90\)~Myr in the \emph{future} to roughly \(230\)~Myr in the past. The
\(\pm160\)~Myr blur is therefore the real precision of the epoch
statement, and the present offset is \(0.4\sigma\) inside it --- entirely
consistent with the crossing being now, and not more precise than that.

\paragraph{One row of the inventory does not carry the star, and should
not be tagged as though it did.}
\label{par:epoch-free-ratio}
The epoch input is the only thing standing between the inventory and grade
D. It follows that any quantity in the inventory which does not depend on
the epoch is not conditional on it --- and there is exactly one:
\begin{equation}
\frac{\Omb}{\Omm} \;=\; \frac{1/(2\pi^{2})}{1/\pi} \;=\; \frac{1}{2\pi}
\;=\; 0.159155 .
\label{eq:ratio-epoch-free}
\end{equation}
Baryons and cold dark matter both dilute as \(a^{-3}\), so this ratio is
\emph{constant at every epoch} after the baryons become non-relativistic;
and the critical density cancels between numerator and denominator, so it
is \(h\)-free as well. It therefore consumes no epoch calibration, and it
is graded \textbf{D} in the codebook rather than D\(^*\). The absolute
fractions \(\Omm\), \(\OmL\), \(\Omb\) keep the star, because each of them
sweeps with the epoch and the geometry fixes only \emph{where} the sweep
crosses, not \emph{when}.

\emph{A bookkeeping point that matters at this precision.} The comparison
must be made in \(\omega_b/\omega_m\) and \emph{not} in
\(\Omb/\Omm\), because the measured \(\Omm = 0.3153\) includes a neutrino
component that this framework's \(\Omm = \Omb + \Omdm\) does not. With
\(\Sigma m_\nu \simeq 59\)~meV the neutrino share is \(\Omega_\nu = 0.0014\),
that is \(0.44\%\) of \(\Omm\), and the two comparisons differ by half a
percentage point:
\(\omega_b/\omega_m = 0.02237/0.14237 = 0.15713\) gives \(+1.29\%\), while
\(\Omb/\Omm = 0.0493/0.3153 = 0.15636\) gives \(+1.79\%\). Against a
\(+1.3\%\) signal that difference is not negligible, and the codebook row
uses the former. The same neutrino component means the constancy of the
ratio is itself exact only to \(\sim0.4\%\), since neutrinos were
relativistic at earlier times; that is below the deviation being quoted,
and it is stated rather than assumed.

\paragraph{What the epoch can and cannot move.}
\label{par:what-the-epoch-moves}
Because the epoch is a declared input, it is worth recording precisely
which quantities it could in principle be used to adjust --- and which it
could not, so that the maneuver is not attempted later and a sign flip
mistaken for a repair.

\begin{itemize}
\item \(\eta_B = n_B/n_\gamma\) is a \emph{constant of the universe}: both
number densities dilute as \(a^{-3}\) after electron--positron
annihilation, so \(\eta_B\) takes the same value at every epoch. No choice
of evaluation time can move it, and no discrepancy in it can be closed
that way.
\item \(\Omb/\Omm\) is likewise constant, by
Eq.~\eqref{eq:ratio-epoch-free}.
\item \(\rho_b \propto a^{-3}\) \emph{is} epoch-dependent.
\item \(\Omb\), \(\Omm\), \(\OmL\) are epoch-dependent, because
\(\rho_{\rm crit}\) evolves.
\end{itemize}

What does change with the evaluation point is the \emph{conversion}
between \(\Omb\) and \(\eta_B\). Applying \(\Omb = 1/(2\pi^2)\) at
\(a_\pi\) rather than today requires converting with
\(\rho_{\rm crit}(a_\pi)\), a factor
\(E^2(a_\pi)\,a_\pi^{3} = 0.99054\), or \(-0.95\%\); at this paper's own
\(h = 0.666\) that moves the ledger road from \(\eta_{10} = 6.15\)
(\(+0.57\%\)) to \(6.10\) (\(-0.38\%\)). It flips the sign of the residual
and preserves its size. It does not close anything.

And it is not an available adjustment in any case. That factor exists only
because \(a_\pi\) was located using the \emph{measured}
\(\Omm = 0.3153\); if \(\Omm = 1/\pi\), then \(a_\pi = 1\) identically and
the factor is exactly unity. The correction is nothing but the
\(-0.95\%\) by which the measured matter fraction differs from \(1/\pi\),
restated in other units --- so using it to adjust \(\eta_B\) would spend the
same discrepancy twice.

The one genuinely epoch-dependent baryon quantity, \(\rho_b\), moves the
wrong way: at \(a_\pi\) it is \(+1.40\%\) larger than today, so comparing
\(\rho_b(t_\pi)\) against the measured present-day value would take the
agreement from \(+0.45\%\) to \(+1.86\%\). Evaluating at the crossing
epoch \emph{opens} that gap.

\paragraph{The selection is not circular.} A derivation that selected
$1/\pi$ by the empirical preference $\Omm/\OmL\approx0.46$ would be; this
one is not: Steps 4 and 5 select $1/\pi$ by the Hopf fiber-volume geometry, and the data enter the derivation nowhere. Two residues were named in earlier statements of this note, and both are worth keeping --- not as outstanding debts, but because they say what the identification of the epoch actually asserts: \textbf{(R1)} that the frozen \GDOS{} measure weights exactly the Hopf circle of the electroweak $\mathrm{SU}(2)$ as the matter fiber and $\mathrm{Vol}(S^3)$ as the total --- the content of VIII-b$''$ (one closed packet $=$ one unit of fiber measure), whose derivation route through the $T_{\GFF}\sim M_{\rm KK}$ unit bridge is sketched in the codebook audit (\S\ref{sec:audit-d-star-cosmos}); and \textbf{(R2)} that the fibers carry the canonical unit normalization ($S^1$ circumference $2\pi$, $S^3$ volume $2\pi^2$) that the measure convention must pin. Both are geometric identifications, provable or falsifiable on the bulk construction, with no reference to any measured density. The value $1/\pi$ is therefore carried as \emph{derived in form and calibrated in epoch}: the codebook lists $\Omm,\OmL,\Omb$ as D\(^*\) because of the epoch identification $t_0 = t_\pi$, and for no other reason. Concretely: (R1) is discharged by the fiber assignment of link~1 together with the occupancy identity, and (R2) is the dimensional statement of Eq.~\eqref{eq:fibre-ratio-radius}, which the unit bridge answers at the crossing and cannot answer away from it.

\paragraph{Corollary (the $\pi$-locked inventory).}

The same closed-packet integration gives $\OmL = 1 - 1/\pi$ by unit-trace complementarity and $\Omb = 1/(2\pi^2)$ by a further $U(1)_B$ projection through the baryonic-gauge protection of \ANine{}. Together these complete the $\pi$-locked inventory predicted in the codebook of Section~\ref{sec:codebook-table-cosmos}. The Kaxion fraction $\Omdm = \Omm - \Omb = 1/\pi - 1/(2\pi^2) \approx 0.2676$ follows from the closed-form subtraction.

\paragraph{Status note.}

Theorem~\ref{thm:omega-m-pi-topological} states the geometric form of the claim: \emph{if} the frozen \GDOS{} measure weights the Hopf circle of the electroweak $\mathrm{SU}(2)$ as the matter fiber (VIII-b$''$), then $\Omm = \mathrm{Vol}(S^1)/\mathrm{Vol}(S^3) = 1/\pi$ rather than any competing rational function of $\pi$, with the competitors excluded by geometry. As \S\ref{par:where-the-conditional-lives} records, the theorem closes the \emph{form} and not the \emph{epoch}: links~1--3 are derived or proved, VIII-b$''$ discharges into the occupancy identity rather than into a statement about a constant, and what remains is the identification $t_0 = t_\pi$ to $1\%$. $\Omm$, $\OmL$ and $\Omb$ are therefore D\(^*\) \emph{with the epoch as their declared calibration input}. The earlier reading --- that the residual was an unexhibited normalization lemma --- is superseded.

So the promise and the proof can now be separated cleanly. $\Omm = 1/\pi$ is not a numerological observation: it is the Hopf fiber-volume ratio of the electroweak holonomy, forced by geometry and selected without the data. What is promised and not proved is narrower and better posed than the whole --- that the epoch at which the running inventory reads that ratio is the present one, which is the coincidence question in its sharpest available form.

\paragraph{The view from one floor up.}
The questions of this Section are the three operational forms of one deeper question: why does the universe admit a closed analytic description at all? The answer is that the cosmological-boundary stratum is the unique stable attractor of the boundary moduli space, and the \(\pi\)-locked inventory is the projection of that attractor onto the present epoch, with the layer-by-layer identity ladder of \S\ref{sec:uniqueness-layer-by-layer} --- from the electron to the cosmos --- as its content. After all, the question \emph{why the universe has the inventory it does} is the cosmological-side rephrasing of the constitutional question that \GGTheory{} was built to answer: not what the universe is, but why it is the unique thing it is.


\section{Two fundamental questions, one geometric answer}
\label{sec:two-questions-cosmos}

\begin{quote}
\emph{``Nature is so beautiful and so well designed. Why is the universe so well-designed for us to be here in the beautiful biosphere on Earth? This question was the primary motivation for my decision to become a physicist, way back in my high school years.''}\\
\hspace*{\fill}--- K. Arisaka, May 2026.
\end{quote}

\ANine{}'s Part~XI closed the Standard-Model audit not on a parameter but on a pair of questions --- why every electron is the same, and why exactly four forces exist with exactly their behaviors --- and recognized the two answers as one uniqueness theorem read at two layers. The present Part has reached the same summit from the cosmological side. Section~\ref{sec:universal-cosmic-identity-preview} answered the cosmological form of the first question: why the vacuum energy, the Kaxion, and the \(\pi\)-locked inventory are the same everywhere. Section~\ref{sec:pi-locked-derivation} answered the second: why the inventory takes the particular values it does, and why their coincidence at the present epoch is no coincidence. This closing Section recognizes that these are not two cosmological answers but one, and that the ``one'' is the same one \ANine{} found --- a uniqueness principle on a geometric stratum, now read on the largest stratum the program reaches.

\subsection{The structural symmetry of the two cosmological questions}
\label{subsec:q1-q2-cosmos}

In each case the form of the argument is identical. We assemble a small number of separately-named constitutional inputs; we show that those inputs leave exactly one admissible readout; we identify the boundary readout with what we observe. For the identity question the inputs were the \GRIP{} cascade of \ASix{} and the ground-state refinement; the unique readout was the cosmological \GGA{} (dark energy) with its relic and baryonic records; the boundary observation was the universal inventory of every sightline. For the value question the inputs were the \BIsix{} four-form of \AThree{} and the \OGNfive{} integer ledger of \AFour{}; the unique readout was the closed-packet integral projecting to \((1/\pi,\,1-1/\pi,\,1/2\pi^2)\); the boundary observation was the measured density parameters. \emph{Different inputs, one stratum, the same structural form} --- and the same form, in turn, as \ANine{}'s two Standard-Model questions one floor below. The recognition that identity and value are instances of one geometric theorem applied twice is the simplest unifying statement available about the cosmological sector, and it is the exact cosmological echo of \ANine{}'s ``two questions, one answer.'' What \ANine{} showed for the electron and the four forces, this Part shows for the vacuum and its inventory: the two deepest questions one can ask about the contents of the universe turn out to be one question, asked twice.

\subsection{Dissolving the three classical alternatives}
\label{subsec:dissolve-cosmos}

Three classical positions have tried to explain why the universe is hospitable, and each has fastened, historically, on precisely the cosmological numbers this Part derives --- the smallness of \(\Lambda\), the matter--vacuum balance, the fitness of the expansion history for structure and life. The framework does not refute these positions so much as it dissolves the question each was asked to answer.

\paragraph{Design.}
The design argument holds that a designer fine-tuned the cosmological constant and the matter budget for the existence of life; its standing objection is regress --- who designed the designer, and by what tuning? In the present framework the regress is dissolved because no tuning occurred. The constitution of geometry permits one self-consistent readout at each layer; nothing was chosen because nothing was available to choose from. A designer who had to set \(\OmL = 1 - 1/\pi\), \(\Omm = 1/\pi\), the Kaxion mass at the cap-integer value, and the seesaw \(m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}}\) would be acting under constraint --- these are the readouts of topological and Diophantine closures, not free dials --- and a designer who acts under constraint has chosen nothing. Design is not refuted; it is rendered superfluous. The \(10^{120}\) vacuum-energy fine-tuning that has haunted cosmology for half a century is the sharpest case: it is not a dial set improbably close to zero, but the depth of the one terminus basin, read out.

\paragraph{Multiverse.}
The anthropic-multiverse argument holds that we observe a hospitable universe because we could not observe an inhospitable one, and that the inhospitable universes populate a landscape from which ours is the lucky draw; its standing objection is the absence of independent evidence for the other universes and the resulting non-falsifiability of the selection. In the present framework the absence of evidence is dissolved by the absence of the other universes: the constitution permits one self-consistent bulk, one gauge group, one breaking pattern, one stable terminus on the cosmological stratum. There is no landscape of vacua to draw \(\Lambda\) from --- not because we cannot survey it, but because the geometry admits no other value. The cosmological-constant ``landscape'' of \(10^{500}\) vacua, invoked precisely to make the observed \(\Lambda\) an anthropic draw, is replaced by a single computed depth.

\paragraph{Brute fact.}
The brute-fact position holds that the cosmological numbers are simply what they are and that asking why is unscientific; its objection is that it abandons explanation at the moment explanation becomes most pressing. In the present framework the abandonment is unnecessary. The inventory fractions are powers of \(\pi\) read off a closed-packet integral, the Kaxion mass is a cap integer, the vacuum scale is a seesaw of the lock depth, and the expansion history is the descent of one \GRIP{} cascade --- each either measured or testable this decade (the tensor ratio \(r = 0.0043\), the Kaxion line near \(1.95\)~GHz, the siren \(H_0\) near \(67\)). Asking why has become scientific again because there is now a path on which a falsifiable answer can be carried.

\subsection{The self-referential closure}
\label{subsec:self-reference-cosmos}

The deepest structural feature of the framework is not any of the three dissolutions above but the recognition that the geometry which forces the cascade contains, among its forced boundary readouts, the existence of intelligent observers --- and that those observers eventually formulate the constitutional clauses that describe the geometry that forced them. The cosmological setting makes the loop unusually vivid: the same descent whose terminus is the dark energy we measure is the descent that, at an earlier basin, stocked the universe with the baryons that became stars, planets, chemistry, and finally the astronomers who read the terminus off the sky. The universe contains, within itself, the means of recognizing its own constitution --- and the reading is done from inside the very inventory being explained.

This recognition has resonances in older traditions, named here without overclaiming what they say. Wheeler called the closed loop in which observers participate in the physics that produced them the \emph{participatory universe}. The Upanishadic tradition gives the same image as \emph{tat tvam asi} --- the recognition by the individual self of its identity with the underlying reality, an act in which the recognition completes the structure that made it possible. Hegel called the same dynamic \emph{Geist} becoming conscious of itself in history. Spinoza called the most general form of the closure \emph{necessitarianism}: what exists, exists by necessity, because the nature of substance permits no alternative. The \GGTheory{} version is austere and sharp: the geometry permits exactly one self-consistent loop, and the loop closes when minds within the universe write down the constitution that produced them. The framework reproduces none of the metaphysical content of these traditions; it claims only that the structural form of the closure --- one geometry, one cascade, one set of boundary observers, one act of self-recognition --- is shared. What distinguishes the present version from its predecessors is not the conclusion but the warrant: Spinoza inferred necessity from abstract reasoning about substance, open to dispute at every step; \GGTheory{} derives the analogous conclusion from named theorems with explicit constitutional inputs, integer ledgers, and falsifiable consequences testable within the decade.

\subsection{A careful word about determinism and freedom}
\label{subsec:determinism-cosmos}

The framework does not say that every micro-event of cosmic history is preordained. It says that the qualitative \emph{layered structure} of cosmic history is forced. Within each layer there is enormous combinatorial freedom: not every galaxy has been placed, not every star numbered, not every world's chemistry run to its end. The determinism is at the level of which layers exist and what their attractor coordinates are --- that there is inflation, that there is a matter era, that there is a dark-energy terminus, that the inventory is \(\pi\)-locked --- not at the level of what happens within each layer. Cosmic structure, biological evolution, contingent history all take place inside layers the geometry has forced into existence; they are not preprogrammed by the geometry, only sheltered by it.

This is not a small caveat. Accounts of cosmic fine-tuning have traditionally forced a choice between an indifferent randomness (the multiverse draw) and a rigid design (the tuned constants). The framework offers a third position: the layered structure is geometrically forced, while within each layer there is genuine combinatorial freedom of a kind that is neither random nor preordained. The universe we inhabit is not the lucky outcome of a draw, nor the fixed output of a plan; it is the unique-readout solution at each layer of the geometric cascade, and within that solution the contingent richness of cosmic and terrestrial history --- including, evidently, the freedom to ask the questions this Part has answered --- plays out in the room the geometry has built.

\subsection{The closing punch line}
\label{subsec:punch-line-cosmos}

We close the philosophical summit of the Part by stating the synthesis in one paragraph.

\begin{quote}
\emph{After all, the deepest structural fact about the universe may be this: the cosmos is beautiful and well-suited to life not because a designer chose its inventory, and not because we are lucky to inhabit a hospitable draw, but because the constitution of geometry permits exactly one self-consistent readout at each layer, and that readout --- whether read at the layer of a single electron or at the layer of the whole vacuum --- is what we see when we look. The two fundamental questions of cosmology, why the inventory is universal and why it takes the values it does, turn out to be one question, asked twice; and it is the same question \ANine{} asked of the electron, asked now of the universe that the electron is a citizen of.}
\end{quote}

The meta-question is answered as far as the cosmological parameter sheet allows. Before the wider discussion, two Parts close the case. The next assembles the harvest: the direct derivation of \LCDM{}, stage by stage from the bulk action to the expansion history, with the register of every entry printed beside it. The one after gathers everything onto a single sheet: the cross-paper coherence map, the comprehensive first-order codebook, and the second-order residual audit that let a skeptical reader verify the chain theorem by theorem.




\section{Part~X in plain English: the deepest claim in the paper, and the conjecture it rests on}
This is the Part the rest of the paper builds toward, and it is also the Part
most exposed. The claim is that the cosmic inventory has the form it has for
the same reason every electron in the universe is identical to every other:
one geometry, one deepest basin, and no ensemble to select from.

Stated that way the claim is attractive, and attractiveness is a reason for
caution rather than confidence. What the Part actually establishes is
conditional. The topological argument for the \(\pi\)-locked inventory rests on
a named vacuum-uniqueness conjecture that the wider program already depends on;
it does not close that conjecture, and a reader should understand the result as
two open puzzles unified rather than one puzzle solved. That is a real gain ---
unification reduces the number of independent things that must be true --- and
it is a smaller gain than a proof.

The second and third horizon problems introduced here are, as far as this
paper knows, new formulations, and new formulations are cheap. Their value is
that they are answerable by the same mechanism that answers the first, and
that they make the sameness of things and the sameness of choices into
questions of the same type as the sameness of temperature. If the mechanism is
wrong, all three fail together, which is the property a reader should want
from a unification and should check for before believing one.

\GGPart[The assembled LCDM]{The assembled \LCDM{}: the standard cosmological model as a constitutional export}{The direct derivation of \LCDM{} by \GGTheory{}, assembled in one place --- the concordance model written in its standard form, the zeroth and the six silent axioms behind it replaced by theorems and graded conditionals, the export chain walked in eight stages from the bulk action to the expansion history, the assembled model displayed with the register of every entry stated and no fitted continuous parameter, and the honesty boundary printed beside the result}

\section{The model the reader was taught to fit}
\label{sec:lcdm-postulate}

Every cosmologist learned the subject the same way. The
Friedmann equations were written on the board, general relativity and
the FRW symmetry assumed; a source menu was populated with whatever
the data required --- radiation, baryons, cold dark matter, a
cosmological constant; six numbers were fitted
($\Omega_b h^2$, $\Omega_c h^2$, $\theta_*$, $A_s$, $n_s$, $\tau$);
and everything afterwards was consequence. Fifty years of data have
confirmed the fit to extraordinary precision --- and explained none of
its inputs. \LCDM{} is the cosmological twin of the Standard-Model
Lagrangian: the most successful phenomenological object in its field,
and the object its own field cannot derive. The companion \ANine{}
\citep{ArisakaA9SM} closes that circle for the Lagrangian, assembling
it term by term as a constitutional export; the present Part does the
same for the concordance model. The single governing sentence is the
same there and here: \emph{the visible model is exported from the
six-dimensional bulk, not assembled by fitting}. Everything below is
a harvest --- the stage of Part~III, the launch and terminus of
Part~IV, the matter of Part~V, the corridor of Part~VI, the
chronology of Part~VIII, the graded codebook of Part~IX, the topology
of Part~X, and the course-pace walk of
Appendix~\ref{appA:frw} --- gathered into one chain with the register
of every link printed. Nothing in this Part is derived twice; every
stage names its home.

\section{The zeroth and the six silent axioms of \texorpdfstring{\LCDM{}}{LCDM}}
\label{sec:lcdm-silent-axioms}

To see what is at stake, it helps to state precisely what the
textbook assumes when it writes down \LCDM{}. Six structural choices
are made silently before the first fit --- and beneath them sits a
zeroth, deeper one, so habitual it is rarely recognized as a choice
at all: that the starting point is a \emph{four-dimensional FRW
effective description with a fitted source menu} in the first place.
The six axioms fill in the menu; the zeroth supplies the blank form.
Table~\ref{tab:lcdm-silent-axioms} is the scorecard, and the claim
assembled below is that the descent replaces the zeroth and all six
at once --- the same kind of claim, made at the same register, as the
assembled Lagrangian of \ANine{}.

\begin{table}[hbt!]
\centering
\caption[The silent structural axioms behind textbook LCDM, and their replacements]{The zeroth and the six silent structural axioms behind the
textbook \LCDM{}, and the \GGTheory{} replacement of each. Registers:
\textbf{D} = Derived, \textbf{D*} = Derived (conditional on a named
lemma or conjecture), per-entry grades in the codebook of Part~IX.}
\label{tab:lcdm-silent-axioms}
\small
\begin{tabular}{p{0.05\textwidth}p{0.30\textwidth}p{0.40\textwidth}p{0.13\textwidth}}
\toprule
\textbf{\#} & \textbf{Silent textbook axiom} & \textbf{\GGTheory{} replacement} & \textbf{Register}\\
\midrule
S0 & A 4D FRW effective description with a fitted source menu is the starting point at all & The 6D bulk and its boundary projection: the earned bulk action, the boundary Einstein equation as theorem, the ledgered source (Appendix~\ref{appA:frw}, Steps 1 and 3) & \textbf{D} (form); \textbf{D*} (content)\\
\addlinespace
S1 & General relativity as the gravitational law, with $G$ measured & The boundary Einstein equation derived by variation and zero-mode projection; $M_{\rm Pl}^2 = \pi M_6^4(1-e^{-2K})$ computed; conservation a Ward identity; correction bounded with live falsifiers (\S\ref{appA:frw-einstein-earned}) & \textbf{D}; bound \textbf{D*}\\
\addlinespace
S2 & Homogeneity, isotropy, and flatness (the FRW symmetry) & The derived plateau launch: $N^* = 52.6$ e-folds stretch one causal patch and drive $|\Omega_{\rm tot}-1|\to0$ (Part~IV; Appendix~\ref{appA:inflation}) & \textbf{D*}\\
\addlinespace
S3 & The component menu: cold dark matter exists, $\Lambda$ exists, $w=-1$ & Kaxion dark matter from the compact angle ($8.1~\mu$eV); $\Lambda$ the seesaw terminus ($\Mlock^2/M_{\rm Pl,red} \approx 6.1$~meV) with $w=-1$ exact; no fifth component exported (Parts IV--V; \citealp{ArisakaC1Kaxion}) & \textbf{D*}\\
\addlinespace
S4 & The proportions $\{\Omega_i\}$, read from the sky & The closed-packet fractions $\Omm = 1/\pi$, $\Omb = 1/(2\pi^2)$, $\OmL = 1-1/\pi$ (Part~X), carried on Conjecture~VIII with the epoch identity printed & \textbf{D*}\\
\addlinespace
S5 & The initial conditions: adiabatic, nearly scale-invariant fluctuations of amplitude $A_s$ & The two-basin descent: $n_s = 0.9620$ (\textbf{D*}), $r = 0.0043$ (\textbf{D}), $A_s$ the launch sector's one declared input (Part~IV) & mixed, stated\\
\addlinespace
S6 & The six fitted numbers & The codebook of Part~IX, graded entry by entry; $\tau$ late-time astrophysics, outside the fundamental claim & per triage\\
\bottomrule
\end{tabular}
\end{table}

Classic alternatives replace pieces: quintessence trades $\Lambda$
for a potential with new dials; MOND trades dark matter for a
modified force law; the multiverse trades the proportions for a
selection effect. The claim of this Part is structurally different
--- all seven axioms replaced at once, by exports of one geometry,
with the register of each replacement stated rather than inflated.

\section{The export chain in eight stages}
\label{sec:lcdm-chain}

The assembly rests on eight stages, C-1 to C-8, each proved or graded
elsewhere in the paper or the cohort; the stage tags reappear in the
assembled display of \S\ref{sec:lcdm-assembly}.

\subsection{C-1 --- the bulk action and the ledger}
\label{subsec:lcdm-c1}

The chain begins where the companion papers begin: the six-dimensional
bulk $M_4\times I_r\times S^1_\theta$ with its canonical warped
metric, forced by the constitutional clauses (T-GG6-0/1/2 of \ATwo{}
\citep{ArisakaA2GG6}); the unique admissible two-derivative bulk
action, earned brace by brace in the assembly interlude of \ANine{}
\citep{ArisakaA9SM} (station S-1a) and displayed at course pace in
\S\ref{appA:frw-action}; and the \OGNfive{} integer ledger
$(M,N,L;J,K)=(1,3,5;7,35)$, the unique solution of the \BIsix{}
closure of \AThree{} \citep{ArisakaA3BI6}. At the end of C-1 the
theory holds five integers, five dimensional constants, two
transcendentals ($e$ radial, $\pi$ angular) --- and zero continuous
dials. Every subsequent stage spends from this account only.

\subsection{C-2 --- the boundary Einstein equation}
\label{subsec:lcdm-c2}

Varying the action and projecting onto the graviton zero mode yields
four-dimensional gravity as a theorem: the boundary Einstein equation
with $M_{\rm Pl}^2 = \pi M_6^4(1-e^{-2K})$, conservation inherited as
the Ward identity of bulk diffeomorphism invariance, the open-system
correction bounded at $10^{-91}$ today, and the modified-TOV and
growth-sector falsifiers keeping the bound honest --- Theorems
T-GDOS-8.1--8.6 of \AFive{} \citep{ArisakaA5GDOS}, walked at
first-year-graduate pace in Step~1 of Appendix~\ref{appA:frw}
(\S\ref{appA:frw-descent}--\ref{appA:frw-einstein-earned}). The zeroth
and first silent axioms are discharged here: the law of cosmology's
left-hand side is not postulated.

\subsection{C-3 --- the FRW symmetry and the Friedmann equations}
\label{subsec:lcdm-c3}

The FRW form of the boundary metric is not an aesthetic choice: the
launch of C-6 stretches a single causal patch across the observable
sky, and the runtime selects the symmetric attractor (Part~III). With
the symmetry in place, the two Friedmann equations follow from the
boundary Einstein equation by the exact textbook computation ---
deliberately unchanged, Step~2 of Appendix~\ref{appA:frw} --- with the
acceleration equation and continuity carrying only two independent
pieces of content between them, the redundancy being C-2's Ward
identity in cosmological clothes. Register: \textbf{Derived}, given
C-2; the kinematics of expansion, redshift, and horizons then follow
as in Part~III and Appendix~F.

\subsection{C-4 --- the components as geometry}
\label{subsec:lcdm-c4}

The source ledger replaces the fitted menu (Step~3 of
Appendix~\ref{appA:frw}, \S\ref{appA:frw-sources}): radiation and
neutrinos inherited with the Standard-Model sector
($N_{\rm eff} = 3.046 + \DeltaNeff$, $\DeltaNeff = 0.04$, a derived
offset); baryons from the inherited leptogenesis ledger
($\eta_B \approx 7\times10^{-10}$ from $K=35$, Part~V); cold dark matter
the Kaxion of the compact angle ($8.1~\mu$eV, misalignment relic,
Part~V and \citealp{ArisakaC1Kaxion}); dark energy the seesaw
terminus of the descent
($m_\Lambda = \Mlock^2/M_{\rm Pl,red} \approx 6.1$~meV) with $w = -1$
exact. Registers: \textbf{D*} on named lemmas, as graded row by row
in Part~IX. The third silent axiom is discharged: the cast list is
geometry's, not the data's.

\subsection{C-5 --- the fractions}
\label{subsec:lcdm-c5}

How much of each is the closed-packet topology of Part~X:
$\Omm = 1/\pi = 0.3183$, $\Omb = 1/(2\pi^2) = 0.0507$,
$\OmL = 1 - 1/\pi = 0.6817$, the dark-matter slice the difference ---
raw percent-level agreement across the board, audited entry by entry
in Part~IX and at course pace in \S\ref{appA:frw-fractions}.
Register, stated exactly: \textbf{D*} on Conjecture~VIII
(closed-packet integration), the load-bearing conditional of the
cosmological codebook, which closes as the epoch identity ---
$\Omm = 1/\pi$ exact at $t_0 - 75$~Myr under Planck central values,
residual lemma VIII-a exact to $10^{-15}$ --- one declared
calibration input, used once, printed in the open. The fourth silent
axiom is discharged at that stated grade.

\subsection{C-6 --- the initial conditions}
\label{subsec:lcdm-c6}

The launch of Part~IV pays the last debt: flatness driven by
$N^* = 52.6$ derived e-folds, the horizon problem dissolved by the
shrinking comoving Hubble radius, and the spectrum exported with
$n_s = 0.9620$ (\textbf{D*}, plateau class), $r = 0.0043$
(\textbf{D} --- the decisive decade-scale prediction), and $A_s$ the
launch sector's one declared input, flagged wherever it appears. The
walk is Appendix~\ref{appA:inflation}; the register summary is
\S\ref{appA:frw-ics}.

\subsection{C-7 --- the expansion history}
\label{subsec:lcdm-c7}

With the machine loaded, the history follows with no further input:
the Hubble constant is derived, not fitted ---
$\Hzero = 66.6\pm0.2$ from the $\pi$-locked inventory
(\S\ref{sec:h0-derivation}), read against the CMB value $67.4$ as the
coherent $\pi$-offset of the two-stage corridor
(\S\ref{subsec:pi-offset}; \citealp{ArisakaC2Hubble}) --- and the
chronology of Part~VIII carries every epoch on the radial clock. One
entry demands its canonical two-quantity care: the corridor switch
$z_{\rm sw} = e^{7}$ is the topological integer-ledger marker, a
distinct quantity from the atomic decoupling
$z_{\rm dec} \approx 1090$ computed from the same closed-packet
fractions and landing on the Planck value at the $0.1\%$ level
(\S\ref{sec:two-redshifts}); the assembly inherits both, each with
its own name. Register: per the codebook rows of Part~IX.

\subsection{C-8 --- what \texorpdfstring{\LCDM{}}{LCDM} does not contain}
\label{subsec:lcdm-c8}

The export's absences are predictions with the same standing as its
entries: no evolving equation of state ($w = -1$ exact --- any
confirmed $w \neq -1$ kills the terminus reading); no relativistic
species beyond the stated offset ($\DeltaNeff = 0.04$); no spatial
curvature; no fifth force in the growth sector beyond the corridor's
stated bands (Part~VI); and no free $\{\Omega_i, w_i\}$ menu at all.
Standard cosmology can absorb a surprise component by adding a row to
the fit; the export cannot --- which is precisely what makes it
falsifiable, and the kill-test catalog of Appendix~T holds the
list.

\section{The assembled model}
\label{sec:lcdm-assembly}

The export chain is complete, and the concordance model can now be
written the way no fit can write it:
\begin{equation}
\resizebox{0.97\linewidth}{!}{$\displaystyle
\frac{H^2(a)}{H_0^2}
\;=\;
\underbrace{\Omega_r\,a^{-4}}_{\substack{\text{C-4: inherited thermal sector}\\ N_{\rm eff}\,=\,3.046\,+\,0.04}}
\;+\;
\underbrace{\frac{1}{\pi}\,a^{-3}}_{\substack{\text{C-4/C-5: Kaxion + baryons}\\ \Omb\,=\,1/(2\pi^2)\ \text{inside}}}
\;+\;
\underbrace{\Big(1-\frac{1}{\pi}\Big)}_{\substack{\text{C-4/C-5: seesaw terminus}\\ w\,=\,-1\ \text{exact}}},
\qquad
\underbrace{k=0}_{\text{C-6}}
$}
\label{eq:lcdm-assembled}
\end{equation}
riding on the derived clock and the exported spectrum,
\begin{equation}
\underbrace{\Hzero = 66.6}_{\substack{\text{C-7: inventory}\\ \text{CMB reading } 67.4}}
\,,\qquad
\underbrace{n_s = 0.9620,\quad r = 0.0043,\quad A_s\ \text{declared}}_{\text{C-6: the launch spectrum}}
\,,\qquad
\underbrace{\text{nothing else}}_{\text{C-8: absences derived}}
\label{eq:lcdm-assembled-spectrum}
\end{equation}
with $\tau$ carried at its measured value as late-time astrophysics
outside the fundamental claim. Equation~\eqref{eq:lcdm-assembled} is,
symbol for symbol, the expansion law of the textbooks --- with one
property no textbook version has ever had: \emph{there is no fitted
continuous parameter in the display.} One input is declared ($A_s$),
one identity is epoch-calibrated and printed (Conjecture~VIII), and
everything else is geometry plus the integer ledger. The six-number
fit of \S\ref{sec:lcdm-postulate} compresses to that single declared
amplitude plus the geometry --- and the compression is checkable
arithmetic: run \eqref{eq:lcdm-assembled} through the Step-2
machinery of Appendix~\ref{appA:frw} and the age of the universe
comes out at $13.75$--$13.92$~Gyr across the corridor readings,
bracketing the measured $13.79$~Gyr (Problem~E.18.1); the full
twenty-two-observable scoreboard, with deviations and pulls, is
Table~\ref{tab:dev-plot-data} and
Figure~\ref{fig:a10-dev-plot-linear}. The honest one-sentence summary
of the display: \emph{the model that has served for fifty years as
cosmology's input is here the output of a geometric descent with zero
adjustable continuous inputs --- with the strength of each step
stated, not inflated.}

\section{What the assembly does and does not claim}
\label{sec:lcdm-honesty}

The shock value of Eq.~\eqref{eq:lcdm-assembled} is real, and
precisely for that reason this Part must close by marking its own
boundary in the register the paper uses everywhere else.

\begin{euclidremark}[The honesty boundary of the assembled \LCDM{}]
Six statements the assembly does \emph{not} make. \textup{(i)} General
relativity's framework is imported at theorem grade from \AFive{},
consumed rather than rederived; the course-pace walk is
Appendix~\ref{appA:frw}, and nothing here strengthens those theorems.
\textup{(ii)} Conjecture~VIII is the load-bearing conditional: the
fractions of C-5 stand on it, the epoch identity is a declared
calibration input used once, and the assembly upgrades neither.
\textup{(iii)} The amplitude $A_s$ remains the launch sector's
declared input --- the one IOU of the export, printed wherever it
appears. \textup{(iv)} The optical depth $\tau$ is late-time
astrophysics outside the fundamental claim. \textup{(v)} Every
register marked \textbf{D*} stands on its named lemma or conjecture,
graded row by row in Part~IX with the promotion roadmap in
Appendix~S; nothing in the assembly changes a grade. \textup{(vi)}
The raw and calibrated readings are distinct claims and stay
labeled; the two-quantity decoupling discipline of C-7 is part of
the register, not a footnote. The assembly is exactly as strong as
the descent behind it, no stronger; what is new in this Part is not
any single claim but the fact that the claims compose into the whole
model.
\end{euclidremark}

For half a century \LCDM{} has been the model cosmology reasons
\emph{from}. This Part has treated it as a model cosmology can reason
\emph{to} --- the same inversion the companion \ANine{} performs for
the Standard-Model Lagrangian, executed here on the largest possible
stage. The reader who compares Eq.~\eqref{eq:lcdm-assembled} with the
opening chapter of any cosmology text will find the same symbols in
the same places; the difference is that here every symbol arrives
carrying its provenance, and the two great silences of the
concordance model --- \emph{why these components} and \emph{why these
numbers} --- have both been given answers with stated strengths.

\medskip\noindent The model is assembled. The next Part gathers the
whole case onto a single sheet --- the cross-paper coherence map, the
comprehensive first-order codebook, and the second-order residual
audit --- so that a skeptical reader can verify the chain theorem by
theorem.



\section{Part~XI in plain English: the standard model of cosmology, reassembled from the geometry}
The purpose of this Part is to hand the reader back the model they already
know. Everything in the concordance description --- the six parameters, the
expansion history, the perturbation spectrum --- is reassembled here from the
geometry, in eight stages, so the two descriptions can be compared line by
line instead of in the abstract.

The section on the silent axioms is the one that repays slow reading. A
textbook presentation of \LCDM{} makes a series of structural assumptions ---
about homogeneity, about the nature of the dark sector, about what a parameter
is --- that are not usually written down because they are not usually
questioned. Setting them out and naming the replacement for each is how this
Part earns the word "assembled": the claim is not that the standard model is
wrong but that it rests on choices, and that a different set of choices
reproduces it.

What the assembly does not claim should be equally clear, and the closing
section states it. Reproducing a model is not the same as predicting it. The
assembly shows the geometry is sufficient to give back \LCDM{}; it does not by
itself show that no other structure would do the same. That question is what
the cost-minimum principle of the codebook is for, and it is answered there
partially and not completely.

\GGPart[Summary]{Summary and cross-paper coherence}{The cross-paper coherence map to the eight upstream \GGTheory{} papers; the comprehensive first-order codebook and the second-order residual audit; why the result is not a post-hoc fit; and what makes \GGTheory{} structurally different from previous cosmology programs}


\section{Overview of the Summary Part}
\label{sec:summary-overview-cosmos}

This Part is the cosmological analogue of Part~XII of \ANine{} (``Summary: derivation of the Standard Model with twenty-eight parameters''). It collects the cross-paper coherence map showing how the eight upstream papers of the Ultra Eight cohort (\AOne{}, \ATwo{}, \AThree{}, \AFour{}, \AFive{}, \ASix{}, \ASeven{}, \ANine{}) feed the cosmological derivation; the comprehensive first-order codebook compressing the twenty-one cosmological observables into a single audit sheet; the post-hoc-fit defense (why the \(\pi\)-locked inventory is not a numerological coincidence); and the GG-distinct comparison (what makes this cosmology program structurally different from \(\Lambda\)CDM with free parameters, EDE, MOND, brane-cosmology, and conformal-cyclic alternatives).

This Part presents the cross-paper coherence map (Section~\ref{sec:note-on-version-cosmos}), the compressed codebook summary (Section~\ref{sec:compressed-codebook-summary}), the compressed \MSMF{}/\OGN{} audit roadmap (Section~\ref{sec:compressed-audit-roadmap}), the post-hoc-fit defense, and the GG-distinct comparison. Every cross-paper citation has been verified to resolve correctly against the cohort.

\paragraph{\GRIP{}: the common engine behind the twenty-one observables.}
Beneath the twenty-one codebook entries lies a single runtime. Every observable in the inventory --- the \(\pi\)-locked density fractions, the two-stage Hubble corridor, the clustering amplitudes, the damping knee, the Kaxion mass, the tensor null --- is the present-epoch projection of one \emph{Geometric Attractor}, selected on its own sub-stratum by the \GRIP{} cascade \(\DGI \to \GSSB \to \GA\) of \ASix{}, the runtime the whole cohort shares and that A10-Cosmos applies to the cosmological-boundary stratum. The \MSMF{}/\OGN{} cost-minimum the codebook reports for each entry \emph{is} that \GRIP{}-selected attractor: the unique saddle of the \GFF{} ledger \(\mathcal F = F_{\GEP} + F_{\CEP}\) on the relevant sub-stratum (Section~\ref{sec:msmf-ogn-statement-cosmos}). Read this way, the three-mode triage is a statement about \GRIP{} itself: an entry is \textbf{Derived} where the \GRIP{} uniqueness of its attractor is proved, \textbf{Derived (cond.)} where that uniqueness is conditional on a single named lemma, and \textbf{Asserted} where the attractor is observed but its \GRIP{} selection is not yet proved. The cosmic history of Part~VIII is the same cascade read as a finite descent in the radial coordinate \(r\): the \GRIPout{} switch events --- the inflationary graceful exit, the prompt lock crossing that stocks the dark sector, and recombination at \(z_* = e^7\) --- fix the initial data, while the long \GRIPin{} relaxations carry the universe down to the present \(\pi\)-locked attractor \((\Omm, \OmL, \Omb) = (1/\pi,\, 1 - 1/\pi,\, 1/(2\pi^2))\). The summary that follows is, at root, a census of one engine's outputs.

\section{Cross-paper coherence map: the eight upstream papers of A10-Cosmos}
\label{sec:cross-paper-coherence-map-cosmos}

\subsection{What each upstream paper contributes to A10-Cosmos}
\label{sec:note-on-version-cosmos}

Each entry below identifies the upstream paper by its canonical name (including the \texttt{Arisaka-GG-} prefix) using the \emph{full} paper-code macro (e.g.\ \AFullOne{}), names its date, and states in one paragraph what A10-Cosmos inherits.

\paragraph{\AFullOne{}.}
The constitutional foundation of the \GGTheory{} program. A10-Cosmos inherits the four axioms (AX-1 \LC{} locality and causality, AX-2 \GGEP{}, AX-3 \OGGEP{} operational/open \GGEP{}, AX-4 \MSMF{} maximum symmetry and minimum freedom) and the seven postulates (POS-1 \GI{} gauge invariance, POS-2 \UEP{} universal equivalence, POS-3 \AF{} anomaly freedom, POS-4 \CP{} closure and positivity, POS-5 \RG{} renormalization group, POS-6 \OGN{} ordered Gaussian network, POS-7 \MF{} maximum factorization) that define admissibility, observability, and prediction. The codebook observables of Section~\ref{sec:codebook-table-cosmos} are admissible per AX-1; the \BIsix{} anomaly-free closure anchoring the Hubble two-stage corridor theorem (Theorem~\ref{thm:hubble-two-stage-corridor}) is admissible per AX-2. The \MSMF{}/\OGN{} cost-minimum principle of Section~\ref{sec:msmf-ogn-statement-cosmos} is the operationalization, on the cosmological-boundary stratum, of the constitutional clauses AX-4 (\MSMF{}) and POS-6 (\OGN{}); the cost functional itself is the \GFF{} ledger \(\mathcal F = \mathcal F_{\GEP} + \mathcal F_{\CEP}\) of \ASix{} inherited below.

\paragraph{\AFullTwo{}.}
The derivation of the six-dimensional \GG{} bulk (\ATwo{} Theorem~1) and the closed-packet topology that the cosmological-boundary stratum projects onto. A10-Cosmos inherits the bulk geometry, the compact \(S^1_\theta\) holonomy of \ATwo{} Theorem~1 (the structural seed from which the Kaxion is derived in the companion paper \PaperFullCOne{}; the cosmological descent itself is Conjecture~\ref{conj:kaxion-completeness} of the present audit), and the \OGNfive{} integer ledger \((M, N, L; J, K) = (1, 3, 5; 7, 35)\) of \ATwo{} Theorem~6 whose integer \(J = 7\) is reused in the present paper to fix the Hubble corridor switch redshift \(z_* = e^7\) (Theorem~\ref{thm:hubble-two-stage-corridor}, property (C1)).

\paragraph{\AFullThree{}.}
The modified Bianchi identity \BIsix{} (stated as the \BIsix{} local anomaly-cancellation theorem and the \BIsix{} complete global anomaly-freedom theorem of \AThree{}) and the proof of its anomaly freedom. A10-Cosmos inherits \BIsix{} as the parent law of the cosmological boundary, with the explicit coefficient vector \((1, -1, -2/5, -2/5)\); the framing remark of Section~\ref{sec:codebook-overview-cosmos} attaches the explicit citation chain. \BIsix{} anomaly-free recombination geometry is hypothesis (H2) of the Hubble two-stage corridor theorem. The four-form whose integral over the cosmological-boundary slice yields the matter fraction \(\Omm = 1/\pi\) (Section~\ref{sec:why-omega-m-pi}) is constructed using the \BIsix{} anomaly-free closure machinery of \AThree{} (the \BIsix{} four-form construction of \AThree{}); the explicit cosmological-boundary integration is reserved for future work.

\paragraph{\AFullFour{}.}
The zero-free-parameters discipline and the proof of the \OGNfive{} integer ledger \((1, 3, 5; 7, 35)\) (\AFour{} ``The \OGNfive{} integers as a theorem of \BIsix{}''). A10-Cosmos inherits the zero-free-parameters compression statement of \AFour{}, restated in cosmology form as ``twenty-one cosmological observables forced by the Geometric Universal Five Constants \GUfive{} \(=\GUfiveSet\), the structural constants \((\pi, e)\), and the integer ledger.'' The integers \(J = 7\) (the Wilson-line winding / dyadic-closure integer of \AFour{}, cosmologically reused in the present paper as the recombination-corridor switch redshift \(z_* = e^7\)) and \(K = 35\) (the Planck-EW scale-hierarchy integer of \AFour{} generating the \(e^{-35}\) hierarchy, cosmologically reused as the Kaxion lock factor via A10's Conjecture~\ref{conj:compact-angle-mass}) appear directly in the codebook closure economy.

\paragraph{\AFullFive{}.}
The Geometric Dynamics of Open Systems (\GDOS{}) and the boundary projection structure \OGGEP{}. A10-Cosmos inherits the open-system semantics: cosmological observables --- the CMB sky, supernova Hubble diagrams, BAO scales, weak-lensing maps, gravitational-wave sirens --- are all boundary exports of an underlying bulk, never the bulk directly. The Hubble two-stage corridor theorem proves a structural fact about \emph{projected} observables, not about the underlying bulk; the projection is exactly the \OGGEP{} projection of \AFive{}.

\paragraph{\AFullSix{}.}
The four Gi pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) and the runtime constitution of the program. A10-Cosmos inherits the \GRIP{} cascade \(\DGI \to \GSSB \to \GA\) as hypothesis (H3) of the Hubble two-stage corridor theorem: recombination is a \GSSB{} transition from a tightly-coupled \DGI{} phase to a frozen \GA{} phase. The cost functional \(\mathcal F = F_{\GEP} + F_{\CEP}\) of the \GFF{} ledger is the \MSMF{}/\OGN{} cost functional in Section~\ref{sec:msmf-ogn-statement-cosmos}.

\paragraph{\AFullSeven{}.}
The four Qi pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}), the selector architecture, and the operational language for the \ASeven{} five-story pagoda (particle \(\to\) cosmos \(\to\) chemistry \(\to\) life \(\to\) mind). A10-Cosmos inherits the pagoda-hierarchy language used in the universal-cosmic-identity preview of Section~\ref{sec:uniqueness-layer-by-layer}: the propagation of the uniqueness principle from electrons through atoms and molecules to cells instantiates the chemistry-and-life floors of the \ASeven{} pagoda, via the closed Qi-runtime relation \(\OH + \IH = \EIC\).

\paragraph{\AFullNine{}.}
The closed derivation of the twenty-eight-parameter Standard Model and the Codebook Provenance Table template. A10-Cosmos inherits the codebook table format directly (the structural triage D / D\(^*\) / A; the per-parameter audit organization; the named-lemma promotion roadmap). The \(\pi\)-locked matter fraction \(\Omm = 1/\pi\) extends the closed-packet topological argument of \ANine{}'s identity section (``The geometric origin of identity: why every electron is the same'') to the cosmological-boundary stratum. The universal-cosmic-identity preview of Section~\ref{sec:universal-cosmic-identity-preview} is the cosmological analogue of that section.



\subsection{Cross-paper coherence map: outward and inward}
\label{sec:coherence-map-outward-inward-cosmos}

This subsection compresses the eight inheritance paragraphs of Section~\ref{sec:note-on-version-cosmos} into a structured outward-and-inward map and records the inward-map links from A10-Cosmos to the rest of the \GGTheory{} cohort. The whole twenty-two-paper cohort and the cosmological-side closure it jointly establishes are shown in one view as Figure~\ref{fig:cohort-one-view-summary} below.

\paragraph{Outward map: what A10-Cosmos draws from the eight upstream papers.}

\begin{enumerate}
 \item \AOne{} \(\to\) A10: AX-1 (LC), AX-2 (\GGEP{}), AX-3 (\OGGEP{}), AX-4 (\MSMF{}); seven postulates (GI, UEP, AF, CP, RG, \OGN{}, MF); the admissibility predicate that filters every codebook entry.
 \item \ATwo{} \(\to\) A10: six-dimensional \GG{} bulk; \OGNfive{} integer tuple \((1, 3, 5; 7, 35)\) as Theorem~1; compact \(S^1_\theta\) holonomy seeding the Kaxion descent (closure deferred to \PaperCOne{}).
 \item \AThree{} \(\to\) A10: \BIsix{} as a stated theorem (local anomaly-cancellation theorem + global anomaly-freedom theorem); coefficient vector \((1, -1, -2/5, -2/5)\); the \BIsix{} four-form construction whose integral over the cosmological-boundary slice underlies the \(\Omm = 1/\pi\) closed-packet integration (full integration reserved for future work).
 \item \AFour{} \(\to\) A10: zero-free-parameters discipline; \OGNfive{} integer ledger proved as a theorem of \BIsix{}; cosmological reuse of \(J=7\) (Wilson-line winding / dyadic-closure integer in A4) and \(K=35\) (Planck-EW scale-hierarchy integer in A4) in the recombination switch and Kaxion lock factor.
 \item \AFive{} \(\to\) A10: \GDOS{} as the open-system formalism; \OGGEP{} projection structure; the rule that every cosmological observable is a boundary export of an underlying bulk rather than a direct bulk measurement.
 \item \ASix{} \(\to\) A10: four Gi pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}); the \GRIP{} cascade closure theorem \(\DGI \to \GSSB \to \GA\) (T4.1); the \GFF{} ledger \(\mathcal F = \mathcal F_{\GEP} + \mathcal F_{\CEP}\) used as the \MSMF{}/\OGN{} cost functional in Section~\ref{sec:msmf-ogn-statement-cosmos}.
 \item \ASeven{} \(\to\) A10: four Qi pillars (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}); five-story pagoda (particle \(\to\) cosmos \(\to\) chemistry \(\to\) life \(\to\) mind); the closed Qi-runtime relation \(\OH + \IH = \EIC\) for the layer-by-layer universal-identity propagation of Section~\ref{sec:uniqueness-layer-by-layer}.
 \item \ANine{} \(\to\) A10: 28-parameter Codebook Provenance Table template (the D / D\(^*\) / A classification triage and per-parameter audit organization); the section ``The geometric origin of identity: why every electron is the same'' that A10 extends one floor up to the cosmological-boundary stratum in Section~\ref{sec:universal-cosmic-identity-preview}.
\end{enumerate}

\paragraph{Inward map: what A10-Cosmos delivers downstream.}

A10-Cosmos is the boundary-side closure of the Ultra Eight cohort and the structural foundation for the cosmological-prediction branch. The four downstream papers that draw from A10-Cosmos are:

\begin{enumerate}
 \item A10 \(\to\) \AEight{}: the operational reformulation of quantum mechanics in 4D is being developed for the textbook-style A8-QM paper; A8 inherits the \OGGEP{} projection from A5 and the cosmological-boundary stratum from A10 as its boundary backdrop.
 \item A10 \(\to\) \PaperFullPOne{}: the KK lock-cluster phenomenology at \(M_{\rm lock} \approx 3.85\) TeV uses the same \OGNfive{} integer ledger and the same lock factor \(e^{-K} = e^{-35}\) that fixes the cosmological Kaxion mass; A10's cosmological observables (\(\Omega_m, \Omega_\Lambda, \Omega_b\)) constrain the KK relic abundance.
 \item A10 \(\to\) \PaperFullPTwo{} \citep{ArisakaP2N3}: the three-leg derivation of the generation count \(N_g = 3\) --- its upper bound now the capacity theorem \(\mathrm{Cap} = \pi N_g\) with the integrality theorem L-INT behind it, conditional on one declared topological residue, R-INT --- cross-cites A10 for the cosmological role of \(N_g = 3\) (the three-generation count in the matter sector inherited from cosmological closure). The generation count is not the \OGNfive{} color index; the two are distinct quantities that share a value.
 \item A10 \(\to\) \PaperFullCOne{}: the dedicated Kaxion paper takes A10's Conjecture~\ref{conj:compact-angle-mass} (compact-angle-mass-uniqueness) as the cosmological constraint on Kaxion mass and decay constant; the cosmological role of the Kaxion as cold dark matter saturating \(\Omdm = \Omm - \Omb = 1/\pi - 1/(2\pi^2)\) is the empirical anchor of C1.
\end{enumerate}

\paragraph{Net coherence summary.}
The outward map records eight imports of well-defined upstream theorems and one operational language (the five-story pagoda). The inward map records four downstream deliveries to predictions-branch papers. The two-way coherence is therefore: A10-Cosmos sits at the boundary-side closure of the constitutional foundation (A1) + bulk geometry (A2-A3-A4) + open-system measurement (A5-A6-A7), inheriting one theorem from each of those layers; and A10-Cosmos forwards four predictions branches into the future program work. No upstream theorem is inherited circularly; no downstream branch is forwarded into a paper that was already cited as upstream.

The corresponding cohort map (Figure~\ref{fig:cohort-one-view-summary}) shows the whole twenty-two-paper program in one view: the constitutional root and the GG6 Trio, the open-system boundary law and its pillar papers, the 4D-Physics Trio with A10-Cosmos at the cosmological-boundary closure, the capstone, the two phenomenology branches, and the Foundations Branch column on the left flank.


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 {\textbf{\AFour{}} \\[-1pt] \footnotesize Historical lineage};

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 {\textbf{\AFive{}} \\[-1pt] \footnotesize \GDOS{} boundary law};

%% --- Row 3: pillars ---
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 {\textbf{\ASix{}} \\[-1pt] \footnotesize \GiFour{} pillars};
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 {\textbf{\ASeven{}} \\[-1pt] \footnotesize \QiFour{} pillars};

%% --- Row 4: 4D-Physics Trio ---
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 {\textbf{\AEight{}} \\[-1pt] \footnotesize 4D QM reformulation};
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 {\textbf{\ANine{}} \\[-1pt] \footnotesize 28 SM parameters};
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 {\textbf{\ATen{}} \\[-1pt] \footnotesize $\Omega_m = 1/\pi$};

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 {\textbf{\AEleven{}} \\[-1pt] \footnotesize Closing capstone \\
 \footnotesize universal identity, GRIP cascade};

%% --- Row 6: Predictions Branch (P-series) ---
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 {\textbf{\PaperPOne{}} \\[-1pt] \footnotesize $\Mlock \approx 3.85$~\TeV{}};
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 {\textbf{\PaperPThree{}} \\[-1pt] \footnotesize Electron as ground state};

%% --- Row 7: Cosmology & Astrophysics Branch (C-series) ---
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 {\textbf{\PaperCOne{}} \\[-1pt] \footnotesize Kaxion dark matter};
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 {\textbf{\PaperCThree{}} \\[-1pt] \footnotesize BH entropy $A/4$};

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\par\bigskip
\captionof{figure}[The twenty-two-paper cohort map of \GGTheory{} (Summary recap).]{\textbf{The \GGTheory{} cohort in one view (Summary recap).} The twenty-two-paper cohort map of the front matter, repeated here as the visual summary of the cross-paper coherence. The constitutional root \AOne{} compiles via AX-2 (\GGEP{}) into the GG6 Trio (\ATwo{}, \AThree{}, \AFour{}); AX-3 (\OGGEP{}) forces the open-system boundary law \AFive{}, organized by the pillar papers \ASix{} and \ASeven{}; the 4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}) recovers quantum mechanics, the Standard Model, and \LCDM{}, with the present paper the cosmological-boundary closure; \AEleven{} closes the A-series; the framed Predictions (P-series) and Cosmology \& Astrophysics (C-series) branches carry the phenomenology; and the Foundations Branch column on the left flank --- \PaperPFour{} (the origin of \(i\)) and \PaperPFive{} (the mirrors and the matter) --- carries the boundary's complex structure and its mirrors, \PaperPFive{}'s \(\eta_B\) edge being this paper's import. A10-Cosmos inherits one theorem from each upstream layer and forwards its cosmological closure into the downstream branches, as enumerated in Section~\ref{sec:coherence-map-outward-inward-cosmos}.}
\label{fig:cohort-one-view-summary}
\end{minipage}
\vfill\clearpage



After all, the cross-paper coherence map is the map of the program's spine: eight papers feed in, one paper closes the cosmological side, and four papers feed forward. No paper is isolated; no claim is unanchored.

\paragraph{Program-level coherence: the eight-paper cohort.}
The eight papers \AOne{}, \ATwo{}, \AThree{}, \AFour{}, \AFive{}, \ASix{}, \ASeven{}, \ANine{} together comprise the Ultra Eight cohort, all sharing the same sans-serif paper-code convention. A10-Cosmos is the boundary-side closure of that cohort: every constitutional inheritance is sourced, every codebook entry is anchored, and every named lemma is identified with the upstream theorem chain that would close it.

\subsection{Compressed codebook summary}
\label{sec:compressed-codebook-summary}

Table~\ref{tab:compressed-codebook-cosmos} compresses the twenty-one-row codebook of Section~\ref{sec:codebook-table-cosmos} into a single audit sheet suitable for the headline summary. Each row lists the observable, its symbol, its status under the \MSMF{}/\OGN{} principle, and the named lemma (if any) that closes it. Equivalently, each row names a Geometric Attractor and the maturity of its \GRIP{}-selection proof: where a named lemma is listed, that lemma \emph{is} the \GRIP{}-uniqueness statement for the observable's sub-stratum.

\begin{table}[H]
\centering
\caption[Compressed codebook: twenty-one observables and their status]{Compressed codebook summary: twenty-one cosmological observables, three-mode status (D / D\(^*\) / A), and the named conjecture (if any) that closes each. Compare with the full provenance table in Section~\ref{sec:codebook-table-cosmos}. The fourth column names the conjecture that would close each row that is not yet Derived, so the table doubles as a list of what remains open.}
\label{tab:compressed-codebook-cosmos}
\renewcommand{\arraystretch}{1.10}
\small
\begin{tabular}{p{0.34\textwidth}p{0.16\textwidth}p{0.08\textwidth}p{0.32\textwidth}}
\toprule
\textbf{Observable} & \textbf{Symbol} & \textbf{Status} & \textbf{Closing conjecture (if D\(^*\) or A)} \\
\midrule
Matter density fraction & \(\Omm = 1/\pi\) & D\(^*\) & Closed-packet-integration (Conj.~VIII) \\
Cosmological constant fraction & \(\OmL = 1 - 1/\pi\) & D\(^*\) & Conjecture~VIII \\
Dark-energy equation of state & \(w = -1\) & D\(^*\) & Vacuum-residual-exactness (Conj.~IX) \\
Baryon density fraction & \(\Omb = 1/(2\pi^2)\) & D\(^*\) & Conjecture~VIII \\
Dark-matter density fraction & \(\Omdm\) & D\(^*\) & Kaxion-misalignment-completeness (Conj.~I) \\
Baryon-to-photon ratio & \(\eta_B\) & D\(^*\) & Resonant-leptogenesis-normalization (Conj.~VI) \\
Lithium-7 abundance & \({}^7\mathrm{Li}/H\) & D\(^*\) & Lithium-port-\(\pi\)-suppression (Conj.~X) \\
Hubble constant, early branch & \(\Hzero^{\mathrm{CMB}}\) & D\(^*\) & Hubble-two-stage-corridor-uniqueness (Conj.~II) \\
Hubble constant, late branch & \(\Hzero^{\mathrm{late}}\) & D\(^*\) & Conjecture~II \\
Decoupling redshift & \(z_*\) (switch \(e^{7}\)) & D\(^*\) & Conjecture~VIII (atomic chain on the fractions; \S\ref{sec:two-redshifts}) \\
Matter clustering amplitude & \(\sigeight\) & D\(^*\) & Conjecture~II \\
\(S_8\) clustering, early/CMB branch & \(\Seight^{\mathrm{CMB}}\) & D\(^*\) & Conjecture~II \\
\(S_8\) clustering, late/lensing branch & \(\Seight^{\mathrm{late}}\) & D\(^*\) & Conjecture~II \\
Effective neutrino deviation & \(\DeltaNeff\) & D & --- \\
Damping-tail multipole & \(\ell_D\) & D\(^*\) & Closed-packet-integration (Conj.~VIII); BFFT overlay gated by Conj.~VII \\
Tensor-to-scalar ratio & \(r\) & D & --- ($r = 0.0043$, derived class + matched count) \\
Scalar tilt & \(\ns\) & D$^*$ & Plateau-class-and-prompt-crossing (Conj.~IV, revised) \\
Scalar amplitude & \(A_s\) & A & GDOS-heat-kernel amplitude (Conj.~V\('\)) \\
Kaxion mass & \(m_{\mathrm{Kaxion}}\) & D\(^*\) & Compact-angle-mass-uniqueness (Conj.~III) \\
Kaxion decay constant & \(f_{\mathrm{Kaxion}}\) & D\(^*\) & Conjecture~III \\
Inflationary energy scale & \(V_{\mathrm{inf}}^{1/4}\) & D\(^*\) & Plateau-class-and-prompt-crossing (Conj.~IV, revised) \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Compressed \texorpdfstring{\MSMF{}/\OGN{}}{\MSMF{}/\OGN{}} audit roadmap}
\label{sec:compressed-audit-roadmap}

Table~\ref{tab:compressed-audit-roadmap-cosmos} compresses the per-parameter audit of Section~\ref{sec:codebook-msmf-ogn} into a ten-step closure roadmap: which named conjecture closes which codebook rows, and what the decomposition becomes once each conjecture is proved.

\begin{table}[H]
\centering
\caption[Compressed audit roadmap: the ten named conjectures and what each closes]{Compressed audit roadmap: the ten named conjectures of A10-Cosmos, the codebook rows each closes, and the resulting decomposition. The codebook collapses from 2\,D + 18\,D\(^*\) + 1\,A to 21\,D + 0\,D\(^*\) + 0\,A once all ten are proved (twenty independent entries after the D9 redundancy collapse).}
\label{tab:compressed-audit-roadmap-cosmos}
\renewcommand{\arraystretch}{1.20}
\small
\begin{tabular}{p{0.06\textwidth}p{0.32\textwidth}p{0.22\textwidth}p{0.10\textwidth}p{0.16\textwidth}}
\toprule
\textbf{No.} & \textbf{Named conjecture} & \textbf{Rows closed} & \textbf{Count} & \textbf{State after proof} \\
\midrule
I & Kaxion-misalignment-completeness & \(\Omdm\) & 1 & 3D / 17D\(^*\) / 1A \\
II & Hubble-two-stage-corridor-uniqueness & \(\Hzero^{\mathrm{CMB}}\), \(\Hzero^{\mathrm{late}}\), \(\sigeight\), \(\Seight^{\mathrm{CMB}}\), \(\Seight^{\mathrm{late}}\) & 5 & 8D / 12D\(^*\) / 1A \\
III & Compact-angle-mass-uniqueness & \(m_{\mathrm{Kaxion}}\), \(f_{\mathrm{Kaxion}}\) & 2 & 10D / 10D\(^*\) / 1A \\
IV & Plateau-class-and-prompt-crossing (revised) & \(\ns\), \(V_{\mathrm{inf}}^{1/4}\) & 2 & 12D / 8D\(^*\) / 1A \\
V\('\) & GDOS-heat-kernel-amplitude (replaces the retired Conj.~V) & \(A_s\) & 1 & 13D / 8D\(^*\) / 0A \\
VI & Resonant-leptogenesis-normalization & \(\eta_B\) & 1 & 14D / 7D\(^*\) / 0A \\
VII & BFFT-damping-knee-identification & (BFFT overlay of \(\ell_D\); row closed by VIII) & 0 & 14D / 7D\(^*\) / 0A \\
VIII & Closed-packet-integration & \(\Omm\), \(\OmL\), \(\Omb\), \(\ell_D\), \(z_*\) & 5 & 19D / 2D\(^*\) / 0A \\
IX & Vacuum-residual-exactness & \(w\) & 1 & 20D / 1D\(^*\) / 0A \\
X & Lithium-port-\(\pi\)-suppression & \({}^7\mathrm{Li}/H\) & 1 & \textbf{21D / 0D\(^*\) / 0A} \\
\bottomrule
\end{tabular}
\end{table}

The ten conjectures between them close all twenty-one rows: nine close rows directly, while Conj.~VII gates the interpretive BFFT overlay of D18 (whose row is closed by Conj.~VIII). Four of the ten (Conj.~II, Conj.~III, Conj.~IV, and Conj.~VIII) each close multiple rows. The closure is strictly monotonic in the codebook decomposition.

\subsection{The second-order residual: the \texorpdfstring{$+2.7\%$}{+2.7\%} common offset, honestly tiered}
\label{subsec:second-order-residual-cosmos}

The compressed codebook above is the \emph{first-order} ledger: each entry is the leading geometric form read off the \OGNfive{} integers and the closed-packet topology, with no continuous dial. A first-order ledger invites a second-order question --- how do those leading forms sit against the data once the precision is taken seriously? --- and the honest answer is a residual that must be tiered, not buried.

\paragraph{The observation.} Three of the independent (filled-marker) predictions of the parameter-audit Dev plot (Table~\ref{tab:dev-plot-data}) land at almost exactly the same positive deviation: \(w\) at \(+2.72\%\), \(\Omb\) at \(+2.76\%\), and \(\Seight^{\mathrm{late}}\) at \(+2.77\%\) --- a spread of only \(0.05\) percentage points, all of the same sign, each first-order geometric form sitting about \(2.7\%\) \emph{above} the measured value. The cluster used to have a fourth member: the damping multipole, whose integer-ledger form \(K(2\pi)^2\) sat at \(+2.73\%\) until the diffusion integral was actually computed on the inventory and the entry was re-scored to its Boltzmann value (\(+0.4\%\)) --- a departure whose lesson is drawn below. The remaining independent entries scatter on both sides (\(\Omm\,{+}1.1\%\), \(\OmL\,{-}0.5\%\), \(\Omdm\,{+}0.7\%\), \(\Hzero^{\mathrm{CMB}}\,{-}1.2\%\), \(z_*\,{+}0.6\%\), \(\omega_b/\omega_m\,{+}1.3\%\), \(100\,\theta_*\,{-}0.5\%\), the peak ratios \({-}0.3\%\) and \({-}0.8\%\), \(\ell_D\,{+}0.4\%\), \(\eta_B\,{+}20\%\) (o.o.m.\ band), \({}^7\mathrm{Li}/H\,{-}0.5\%\); the \(\Hzero^{\mathrm{late}}\) entry, \(-6.2\%\), is the separately named ladder systematic), so the coincidence is specific to those three, not a rigid shift of the whole plot. \emph{Epoch-calibration re-scoring.} The deviations above are the raw $t_\pi$-form comparisons against today's sky. Under the single declared epoch input (\S\ref{subsec:whynow-form-scale-epoch}) the baryon entry re-scores to $+1.69\%$ ($+1.4\sigma$) --- its timing component ($+1.05\%$, shared with the whole matter sector) removed --- while $w$ and $\Seight^{\mathrm{late}}$, which do not sweep with the epoch, keep their $+2.7\%$. The tight three-way cluster is therefore partly an epoch artifact: at the calibrated level it is a pair of weak $+2.7\%$'s plus a $+1.7\%$ ratio statement.

\paragraph{What it is, and is not, statistically.} Only one of the three carries real statistical weight, and the epoch calibration nearly halves it: \(\Omb\) reads \(+2.76\%\) (\(2.3\sigma\)) against the raw \(t_\pi\)-form and \(+1.69\%\) (\(+1.4\sigma\)) once the declared epoch input is used --- the surviving residual being exactly the epoch-independent, \(h\)-free ratio statement \(\Omega_b/\Omega_m = 1/2\pi\). The other two are weak company --- \(w\) (\(+2.72\%\) but only \(0.9\sigma\), well inside the dark-energy error bar) and \(\Seight^{\mathrm{late}}\) (\(+2.77\%\) at \(0.9\sigma\), and measured against a round corridor value) --- so their landing at the same offset is as plausibly accidental as structural. The honest signal is therefore ``a single \(\sim\!1.4\sigma\) baryon-ratio pull (\(2.3\sigma\) before the epoch calibration), with two suggestive but inconclusive companions,'' not ``a universal \(+2.7\%\) offset across cosmology.''

\paragraph{The physics: this is exactly the Tier-3 residual --- now with a measured precedent.} The first-order forms are leading topological and integer expressions; they do not carry the second-order precision content --- baryon-loading corrections, the finite-width visibility integral, diffusion damping --- that a full Einstein--Boltzmann computation supplies. The damping multipole is the proof that this reading is physics and not apology: when the Tier-3 content was actually computed for that entry (the diffusion integral over the Boltzmann ionization history, Section~\ref{sec:cmb-overdetermined}), the \(+2.7\%\) gap of the integer form collapsed to \(+0.4\%\) --- the offset really was the omitted second-order content, and computing it moved the entry onto the data. The epoch calibration is now a \emph{second} such precedent: supplying the one declared input the raw comparison had omitted --- \emph{when} the snapshot is read --- moved \(\Omm\), \(\OmL\), and \(\Omdm\) onto the data outright (input, \(0.00\%\), \(-0.31\%\)) and carried \(\Omb\) from \(+2.76\%\) to its ratio residual \(+1.69\%\). Read this way the residual cluster is not an embarrassment but a \emph{measurement} of where the first-order ledger ends, together with a demonstrated route for what happens when the instrument is brought to bear: the inventory is right to a few percent, and the sub-few-percent residue is the instrument's province, not the geometry's. One candidate source is now positively excluded: the stationary \GDOS{} measure itself cannot generate the offset, being uniform on the holonomy moduli to \(O(e^{-S_w}) \sim 10^{-15}\) (the GSM adjunct of \AFive{}), thirteen orders below the percent level.

\paragraph{Three readings, none over-claimed.} \emph{(i) Coincidence:} four percent-level numbers, two of them statistically weak, cluster by chance --- the null hypothesis, and not to be waved away. \emph{(ii) A common second-order normalization:} the damping-multipole precedent shows exactly how such a correction operates --- computing the omitted content moved that entry onto the data --- so the falsifiable expectation is that a computed second-order baryon correction would do for \(\Omb\) what the diffusion integral did for \(\ell_D\) --- and the epoch calibration has already performed the timing part of exactly this operation --- checkable the moment that computation is done. \emph{(iii) Selection:} the offset is simply where current data happen to place these forms, and future inference will scatter them. We commit to none of the three. In particular we resist the temptation to \emph{name} the number: \(2.7\%\) sits near \(1/36 = 2.78\%\), \(e/100 = 2.72\%\), and \(1/(4\pi^2) = 2.53\%\), and assigning it any such closed form on the present evidence would be the exact numerological move this paper exists to refuse (Section~\ref{subsec:forcing-vs-searching-cosmos}). The residual is reported, bounded, and left open.

\paragraph{Consequence for the triage, and the declared precision floor.} Nothing here changes a codebook status. The three entries remain Derived-conditional on their named conjectures, and the \(+2.7\%\) is carried as an explicitly \emph{underived} second-order term, not folded into the first-order claim and never quoted as a success. We go one step further and adopt it as policy, because a referee is right to insist on it: \textbf{the framework's honest precision at first order is \(\sim\!2.7\%\), not the tenths-of-a-percent that the tightest single entries can appear to reach.} The coherent, same-sign clustering of independent predictions is the classic signature of one omitted upstream normalization, and where we have \emph{computed} the omitted content the offset collapses --- the damping knee's integer form moved from \(+2.7\%\) to \(+0.4\%\) once the diffusion integral was evaluated, and the epoch calibration moved the matter fractions onto the data. Accordingly the leading-order forms should be read against a declared second-order floor of \(\sim\!2.7\%\), and the per-entry \(\sigma\)'s of the tightest rows are \emph{first-order} pulls, not the framework's true precision. This yields a specific, falsifiable expectation rather than a hedge: a computed second-order baryon correction should do for \(\Omega_b\) what the diffusion integral did for \(\ell_D\), carrying its \(+2.76\%\) (\(2.3\sigma\)) onto the data --- and if, once computed honestly, it does not, the \(\Omega_b = 1/(2\pi^2)\) entry fails. The floor is thus not an excuse for the residual but a prediction about it: leading order is forced to \(\sim\!1\%\), the second-order content is computable, and the instrument --- not the geometry --- owns the sub-few-percent sky, exactly as the tier ladder reserves.




\section{Post-hoc-fit defense: why the \texorpdfstring{$\pi$}{pi}-locked inventory is not numerology}
\label{sec:post-hoc-fit-defence}

The reader who arrives at the codebook for the first time will rightly ask: is the $\pi$-locked inventory \(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\), \(\Omb = 1/(2\pi^2)\) a derivation, or is it a curve-fit dressed up in topological language? This section gives the four-point structural defense against \emph{continuous} fitting (D1--D4 below), then, in Section~\ref{subsec:forcing-vs-searching-cosmos}, the fuller answer to the \emph{discrete} look-elsewhere charge --- the combinatorial-alphabet objection --- and the Euclidean reading that places the whole construction in its proper category.

\paragraph{(D1) Closed-packet topology forces the value, not the data.}
Theorem~\ref{thm:omega-m-pi-topological} of Section~\ref{sec:why-omega-m-pi-theorem} states the structural form of the closed-packet \(\Omm = 1/\pi\) claim: the \BIsix{} four-form on $\mathcal{M}_4$ admits no continuous adjustable parameter, and the \OGNfive{} integer ledger $(1,3,5;7,35)$, established upstream (\ATwo{}/\AThree{}, audited in \AFour{}), fixes the rational weight of the matter cell at $1/15$ before the Pontryagin $1/\pi$ rescaling. The integration is programmatic --- it does not yet close (Conjecture~VIII), so $\Omm$ remains D\(^*\). The structural point stands regardless: if the closed-packet integration can be completed, $1/\pi$ ceases to be a numerological observation and becomes the integral output of a fixed topological computation rather than a number chosen to fit the data.

\paragraph{(D2) One topology forces three coupled entries simultaneously.}
A post-hoc fit explains one number, not three coupled numbers. The same closed-packet integration that gives \(\Omm = 1/\pi\) gives \(\OmL = 1 - 1/\pi\) by unit-trace complementarity and \(\Omb = 1/(2\pi^2)\) by a one-dimensional further projection through the $U(1)_B$ baryonic bundle. No additional curve-fit step is involved between the three values.

\paragraph{(D3) The \texorpdfstring{$2\pi^2$}{2 pi squared} factor requires the right combination of powers.}
The baryon fraction depends on $2\pi^2$, not on $\pi$. A post-hoc fit could be tuned to match either $\pi$ or $\pi^2$ but not to predict the specific combination $1/(2\pi^2)$ that emerges from the closed-packet integration through the $U(1)_B$ bundle. The fact that the topological computation produces the right \emph{combination} of powers of $\pi$ (one for matter, one minus that for vacuum, two for baryons) is the structural signature of a derivation.

\paragraph{(D4) Every other codebook entry is independently constrained.}
Beyond the three $\pi$-locked density fractions, the codebook contains sixteen more observables. The Hubble two-stage corridor switch $z_* = e^7$ descends from the \OGNfive{} integer $J = 7$ (not from $\pi$); the tensor-to-scalar ratio $r = 12/N^{*2} = 0.0043$ descends from the Starobinsky relation at the reheat-matched count (not from $\pi$); the effective neutrino number deviation $\Delta N_{\mathrm{eff}}$ descends from the inherited Standard-Model ledger of \ANine{} (not from $\pi$). The codebook is not a single $\pi$-fitting exercise; it is sixteen independent observable predictions sharing a common geometric origin but emerging through distinct calculations.

\paragraph{The structural-fit hypothesis is falsifiable.}
The defense is not a rhetorical move. If any one of the eight falsification tests listed in Appendix~\ref{app:falsification-catalogue} fails --- if the empirical recombination redshift sits at $z_{\mathrm{rec}} = 1500$ instead of $\approx 1090$, or if the GW-siren Hubble determination lands at $H_0^{\mathrm{GW}} = 65$ km/s/Mpc, or if the tensor-to-scalar ratio is detected at $r > 0.01$ --- the closed-packet topology is wrong and the $\pi$-locked inventory falls. After all, a structurally locked derivation that cannot be falsified is not a derivation; it is a tautology. The present framework's commitment is that it can be falsified.


\subsection{Forcing versus searching: the discrete look-elsewhere effect, and the Euclidean reading}
\label{subsec:forcing-vs-searching-cosmos}

The four points above answer the charge of \emph{continuous} fitting: no real-valued coefficient is tuned to the Planck data. But there is a second form of the objection, discrete rather than continuous, that the continuous-fit rebuttal does not by itself dispatch, and it is the form a cosmologist is most likely to voice:

\begin{quote}
\emph{With the integers \((1,3,5;7,35)\) and the two transcendentals \(\{e,\pi\}\), and a handful of operations, one can build an expression close to almost any \(\mathcal O(1)\) cosmic fraction. So \(\Omm = 1/\pi\), \(\Omb = 1/(2\pi^2)\), and \(z_* = e^7\) are a look-elsewhere artifact, not a derivation.}
\end{quote}

This is a legitimate statistical concern --- the cardinality of a discrete formula-space against the number and precision of the targets --- and it deserves a direct, quantitative answer. The answer is that \GGTheory{} does not \emph{search} the alphabet. It \emph{forces} the forms. Five facts, each independently checkable, separate forcing from searching; a sixth places the whole construction in its proper historical category.

\paragraph{One ledger, not a per-observable toolbox.}
The objection tacitly grants itself a fresh draw from the alphabet for \emph{each} observable. That is the error. A genuine combinatorial fit of the twenty-one codebook observables requires of order twenty independent formula-choices --- one expression selected per observable --- so its discrete freedom scales with the number of targets. \GGTheory{} introduces zero continuous parameters and \emph{one} integer tuple \((M,N,L;J,K)=(1,3,5;7,35)\), and that tuple is not five free integers but a \emph{closed} object: the constitutional relations \(J=2N+1\) and \(K=LJ=M^2+N^2+L^2\) fix \(J\) and \(K\) from \((M,N,L)\); \(M=1\) is forced; and \(N=3\) is itself a theorem (the cascade-budget closure of \citealp{ArisakaP2N3}). The effective discrete freedom of the entire ledger is therefore one or two integers, and the \emph{same} tuple must serve \emph{all twenty-one} observables at once. The comparison is thus of order twenty free choices against roughly two: the cosmological codebook is enormously over-constrained, and over-constraint is the structural signature of a derivation, not a fit.

The sharpest expression of this is \emph{cross-observable rigidity}. The single integer \(J=7\) fixes the recombination switch redshift \(z_* = e^7\) and is the same \OGNfive{} integer that fixes the inflationary plateau through the inherited lock chain; the single integer \(K=35\) sets the Kaxion lock factor \(e^{-K}=e^{-35}\) and the CMB damping knee \(\ell_D = K(2\pi)^2\); and one closed-packet topology fixes \(\Omm = 1/\pi\), \(\OmL = 1-1/\pi\), and \(\Omb = 1/(2\pi^2)\) simultaneously. If these integers were fitted observable by observable, nothing would require the \emph{same} \(J=7\) and \(K=35\) to succeed across inflation, recombination, dark matter, and the CMB damping tail at once; a combinatorial search carries no mechanism that forces one integer to be reused coherently across unrelated observables. The ledger does exactly that, by construction. The coherent reuse \emph{is} the non-numerology.

\paragraph{The form is forced by the stratum, not selected by scanning.}
The objection further imagines that, for each observable, one scans candidate forms --- \(1/\pi\)? \(1/(2\pi^2)\)? \(e^7\)? --- until one lands. In \GGTheory{} the \emph{assignment of form to observable} is dictated by the geometric origin of that observable, before any number is evaluated. The matter fraction is \(\Omm = 1/\pi\) because the matter cell is the closed-packet Pontryagin integral on the cosmological-boundary stratum, carrying one factor of the angular primitive \(\pi\); the vacuum fraction is \(\OmL = 1-1/\pi\) by unit-trace complementarity on the same packet; the baryon fraction is \(\Omb = 1/(2\pi^2)\) because the baryonic \(U(1)_B\) bundle is a one-dimensional further projection carrying the \(2\pi^2 = \mathrm{Vol}(S^3)\) volume; the corridor-switch redshift is \(z_* = e^7\) because the last-scattering switch surface is identified as the \(J=7\) \GSSB{} event (a topological landmark distinct from the measured decoupling redshift, coinciding to \(0.6\%\)), the exponential of a single \OGNfive{} integer; and the tensor ratio \(r = 12/N^{*2} = 0.0043\) is the Starobinsky relation evaluated at the reheat-matched count on the derived plateau, one e-fold below the GUT scale, not a fitted exponent. In each case the form is a theorem, or a named-conjecture consequence, of \emph{which stratum the observable inhabits}; the integers enter only through their fixed values. The search over forms is replaced by a geometric assignment, and the search over values by one closed ledger. There is no free product over an alphabet at any step.

\paragraph{\texorpdfstring{\MSMF{}}{MSMF} and \texorpdfstring{\OGN{}}{OGN} remove freedom; they do not supply it.}
Where an observable could admit more than one constitutionally-allowed form, the parsimony axiom \MSMF{} with the \OGN{} complexity measure selects the minimal-cost form on the \(\ell^1\) norm of the ledger (Section~\ref{sec:msmf-ogn-statement-cosmos}). This is a constraint that \emph{deletes} options, applied as a single, pre-registered rule, uniformly across all observables --- the opposite of a per-observable free choice. ``One way or another'' presumes the freedom to choose the combination; \MSMF{} forbids choosing, and forces the minimal-complexity admissible form, with the resulting cost recorded entry by entry in the codebook audit of Part~IX (Section~\ref{sec:codebook-msmf-ogn}). Here honesty must be explicit, and it is a strength rather than a concession: for the entries where the empirical value enters the selection --- the two branches of the Hubble corridor, the damping knee --- those are data-anchored selections, not theorems, and the codebook marks them D\(^*\) or A in the two/eighteen/one triage. A curve-fit hides its degrees of freedom; this audit publishes them.

\paragraph{The look-elsewhere effect, bounded and quantified: the ledger is pinned, not searched.}
The decisive answer to a discrete look-elsewhere charge is not assertion but a count, taken \emph{globally} --- the whole inventory against the whole declared alphabet at once --- and we now give the count rather than promise it. The alphabet is the \OGNfive{} tuple $(M,N,L;J,K)$ subject to its two constitutional closure relations $J=2N+1$ and $K=LJ=M^2+N^2+L^2$, plus $\{\pi,e\}$ and a bounded operation menu. The trials factor factorizes as $p_{\rm global}\approx P(\text{ledger})\times P(\text{forms}\mid\text{ledger})$, and the first factor is where the objection dies.

\emph{The ledger factor is pinned by the particle sector, not fitted to cosmology.} Enumerating \emph{every} closed tuple in the natural box $M,N,L\in\{1,\dots,9\}$ (both closure relations imposed) yields exactly ten: $(1,1,1;3,3)$, $(1,1,2;3,6)$, $(1,3,2;7,14)$, $(1,3,5;7,35)$, $(2,4,4;9,36)$, $(2,4,5;9,45)$, $(2,6,5;13,65)$, $(2,6,8;13,104)$, $(1,7,5;15,75)$, $(3,9,9;19,171)$. Impose the two integers that are fixed \emph{upstream}, in the particle sector and not by any cosmological number --- the family count $N=3$ (a theorem of \PaperPTwo{} and an observed fact) and the gauge label $L=5$ ($\mathrm{SU}(5)$, from \ANine{}) --- and exactly \emph{one} survives: $(1,3,5;7,35)$. The cosmological ledger therefore carries \emph{zero} discrete freedom of its own; it is inherited from the generation count and the gauge group, both known before cosmology is consulted. $P(\text{ledger})=1$ conditional on the known particle content, or $1/10$ if one refuses even that conditioning.

\emph{The exponential observables cannot be searched.} The scale-hierarchy predictions are exponentials of the pinned integers, so they admit no nearby-integer alternative: only $J=7$ places the switch redshift $z_*=e^J$ near recombination ($e^6=403$, $e^7=1097$, $e^8=2981$ against the observed $\approx1090$), and only $K=35$ places the lock $\Mlock=\tfrac12 M_{\rm Pl}e^{-K}$ at the TeV scale ($K=34\!:\!10.5$, $K=35\!:\!3.85$, $K=36\!:\!1.4$~TeV) --- while the \emph{same} $K=35$ delivers $v=\Mlock(2/5)^3=246.2$~GeV unbidden. Adjacent integers miss by multiplicative jumps of $e$; there is no continuum to scan.

\emph{The residual form factor is small and its full computation is specified.} With the ledger pinned, the only freedom left is the form assignment, and it is heavily over-determined: the three inventory fractions come from \emph{one} Hopf fibration (not three choices), the exponential observables from \emph{one} map $x\mapsto e^x$ on the pinned integers, the tensor ratio from the fixed Starobinsky relation. Roughly three independent form-choices reproduce some eight independent precise numbers; the excess of $\sim5$ over-determined constraints against a per-form menu of size $\sim5$ gives a crude accidental-match bound $p_{\rm forms}\sim 5^{-5}\sim 3\times10^{-4}$, so the global bound lands at $p\sim10^{-3}\text{--}10^{-4}$ \emph{before} the upstream conditioning that makes $P(\text{ledger})=1$. This is a preliminary bound with its modeling assumptions (alphabet, menu sizes, ranges) stated; the exact number awaits one specified computation --- a Monte-Carlo over random ledgers and pre-registered form-menus, with the shared-integer bookkeeping ($J=7$ serving two observables counted once) audited --- reserved for a dedicated statistics note. What cannot fairly be said is that the number is un-owed or the freedom hidden: the ledger is pinned upstream, the exponential predictions cannot be searched, and the residual is bounded and named.

\paragraph{Forward commitments no fit can make.}
A fit accommodates whatever the data say and cannot be falsified. \GGTheory{} issues forward commitments that can (Section~\ref{sec:partXII-falsifiable}, F-Cosmos-1 through F-Cosmos-5): a primordial tensor ratio detected at \(r > 0.01\) breaks the Starobinsky kill-switch; a single-stage \(H_0\) determination at \(0.5\sigma\) precision falsifies the two-stage corridor (Corollary~\ref{cor:hubble-falsification}); a firm primordial \({}^7\mathrm{Li}\) abundance away from \(\mathrm{SBBN}/\pi\) excludes the lithium port (Conjecture~\ref{conj:lithium-port}); a measured decoupling redshift off the framework's atomic-chain value breaks the two-quantity reading (\S\ref{sec:two-redshifts}); and the \(8.1~\mu\)eV Kaxion shared with \citet{ArisakaC1Kaxion} is a laboratory target. Each was fixed by the ledger before the measurement that will test it, and a central value far from it breaks the assignment. Falsifiable forward prediction is a property no recombination of a small alphabet possesses, and it is the cleanest empirical demarcation between a derivation and a fit.

\paragraph{The Euclidean reading: a constitution, not a coincidence.}
These five facts have a single root, and naming it answers the objection at its deepest level. The reduction of the cosmic inventory in this paper is not a collection of fortunate expressions; it is the descent of propositions from a fixed set of axioms --- the \GGConstitution{} of \citet{ArisakaA1Constitution}: the equivalence principle \GGEP{}, the open-projection axiom \OGGEP{}, the parsimony axiom \MSMF{}, the normalization principle \OGN{}, and the anomaly law \BIsix{}. From these the geometry, the ledger, and the \(\pi\)-locked inventory follow by construction, each graded by how complete its proof currently is. In this light the alphabet objection is the exact analogue of complaining that Euclid ``could prove anything with points, lines, and circles.'' The complaint mistakes the \emph{primitives} of a deductive system for the \emph{freedom} of a fit. In a deductive system the primitives are few and fixed and the propositions are \emph{forced}; the richness of what follows is not evidence of fitting but the entire purpose of the axiomatic method. The twenty-one observables of the cosmological codebook are not a look-elsewhere artifact of \((1,3,5;7,35)\) and \(\{e,\pi\}\); they are theorems of a constitution --- some proved, some conditional on a named conjecture, some still asserted, exactly as the triage records --- read off one geometry that was fixed before the data were consulted. That is the sense, precise and not rhetorical, in which \GGTheory{} cosmology stands in the lineage of the \emph{Elements} (\citealp{ArisakaA4Euclid}): the cosmic fractions are not assembled from a toolbox; they descend from a constitution.

\begin{euclidremark}[Forcing, not searching]
\label{rem:forcing-not-searching-cosmos}
\begin{equation}
\boxed{\;\parbox{0.86\textwidth}{\centering A numerological fit draws one formula per cosmic number from a free alphabet. \GGTheory{} reads the whole \(\pi\)-locked inventory from \emph{one} closed ledger \((1,3,5;7,35)\) with \((\pi,e)\) --- a ledger pinned uniquely, among the ten closed tuples in range, by the family count \(N=3\) and the gauge group \(\mathrm{SU}(5)\) fixed \emph{upstream} in the particle sector --- fixes each observable's \emph{form} by the stratum it inhabits, and finds the residual discrete freedom bounded at a global trials factor \(p\sim10^{-3}\text{--}10^{-4}\), the exponential predictions (\(z_*=e^7\), \(\Mlock=\tfrac12 M_{\rm Pl}e^{-35}\)) admitting no nearby-integer alternative.}\;}
\label{eq:forcing-thesis-cosmos}
\end{equation}
The same integers \(J=7\) and \(K=35\) reused coherently across recombination, dark matter, and the damping tail, the ledger pinned upstream by \((N=3,L=5)\), the exponential predictions unsearchable, and the forward falsifiable commitments are properties no free recombination of a small alphabet possesses. The look-elsewhere effect is not claimed to be zero; it is computed to a preliminary bound \(p\sim10^{-3}\text{--}10^{-4}\), with the exact Monte-Carlo specified and reserved for a dedicated statistics note.
\end{euclidremark}




\section{GG-distinct comparison: what makes \texorpdfstring{\GGTheory{}}{\GGTheory{}} cosmology structurally different}
\label{sec:gg-distinct-comparison}

This section places the present framework alongside the principal alternative cosmologies of the contemporary literature and identifies the structural differences that distinguish it from each. The comparison is along three axes: the number of continuous free parameters; the closure mode of the cosmic inventory; and the falsification profile.

\begin{table}[H]
\centering
\caption[GG-distinct comparison: free parameters and closure mode]{GG-distinct comparison. Continuous free parameters refers to the parameters that must be specified empirically rather than derived; closure mode is how the present-epoch inventory is fixed; falsification profile lists the principal mode by which the framework can be ruled out.}
\label{tab:gg-distinct-comparison}
\renewcommand{\arraystretch}{1.25}
\small
\begin{tabular}{p{0.20\textwidth}p{0.13\textwidth}p{0.30\textwidth}p{0.27\textwidth}}
\toprule
\textbf{Framework} & \textbf{Continuous free params} & \textbf{Closure mode} & \textbf{Falsification profile} \\
\midrule
\textbf{\ATen{} (this paper)} & \textbf{0} & Closed-packet topology + \OGNfive{} integer ledger + \GUfive{} + $(\pi, e)$ & 8 falsifiable observational tests (Appendix~\ref{app:falsification-catalogue}); any single failure refutes \\
$\Lambda$CDM (vanilla 6-param) & 6 & Empirical fit & Single-parameter precision improvements; no structural falsification \\
Early dark energy (EDE) & 9--11 & Empirical fit + EDE template parameters & Improved CMB+SH0ES joint fit \\
MOND / modified gravity & 1--3 (acceleration scale + interpolation function) & Phenomenological & High-z galactic rotation curves, gravitational lensing \\
Brane cosmology (RS, LR) & 4+ (brane tension, bulk curvature,...) & Brane geometry + RS warp factor & High-energy GW signatures, brane-bound matter spectrum \\
Conformal-cyclic cosmology (Penrose) & 0--2 (depending on variant) & Conformal infinity + remote-past null cone matching & B-mode signatures, CCC ``Hawking points'' \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The structural difference is the zero-free-parameter discipline.}
What distinguishes \ATen{} from every framework in the comparison table is the L45 zero-free-parameters discipline of \AFour{}: the input ledger contains zero continuous adjustable knobs. $\Lambda$CDM with six free parameters can be fit to any data set within its compute envelope; the present framework cannot be retrofitted at all --- if the \OGNfive{} integer ledger $(1, 3, 5; 7, 35)$ does not yield the empirical $\Omm$, $\Omb$, $\Hzero$, or $z_{\mathrm{rec}}$, the framework is wrong.

\paragraph{The closure mode is the second structural difference.}
\(\Lambda\)CDM, EDE, and MOND all close the cosmic inventory by empirical fit (the six or nine or three parameters are observationally determined). Brane cosmology and conformal-cyclic cosmology close the inventory by appealing to features of an underlying higher-dimensional geometry, but the resulting parameter sheet at the cosmological-boundary stratum is still empirically fit. Only the \GGTheory{} framework closes the inventory by closed-packet topology on the cosmological-boundary stratum with the \OGNfive{} integer ledger as the fixed input.

\paragraph{Falsifiability comparison.}
Eight named observational tests (Appendix~\ref{app:falsification-catalogue}) commit the framework to specific predictions on the next-decade horizon. Each test would refute either a named conjecture or a named theorem of A10-Cosmos. In contrast, \(\Lambda\)CDM and EDE absorb improvements in measurement precision into refined parameter values without structural revision; MOND has historically expanded its interpolation function to accommodate new data. The \GGTheory{} framework cannot do that: every observable is named and locked.

After all, the question is not whether the present framework is right, but whether it is the kind of framework that could be wrong. Eight named falsification tests with named conjectures and named theorems each say yes.

\medskip\noindent One sheet, twenty numbers, zero continuous dials: that is this Part in seven words, and the philosophical content of such over-determination is easy to state and hard to overstate. A framework this rigid cannot be nursed through a bad measurement, so its survival to date is evidence of a different kind than a fit's success --- and its failure, if failure comes, will be clean, public, and instructive. Either way, the subject wins.

What remains is to step back from the sheet and ask what it all means --- for cosmology, for the wider program, and for the reader deciding how seriously to take it. That wider view --- significance, innovations, accomplishments, and the decade of predictions --- is the business of Part~XIII.


\section{Part~XII in plain English: how this paper is held to the rest of the cohort}
A twenty-two-paper program can drift. A number quoted in one paper moves in
another, nobody notices because no single reader holds all of them at once,
and the framework slowly becomes a set of overlapping approximations that
agree with each other only in outline. The coherence map in this Part is the
mechanism against that, and it is a mechanism rather than a courtesy: if a
value here disagrees with the paper that owns it, the map is where the
disagreement shows, and the owning paper wins.

The post-hoc-fit defense is the section a hostile reader should attack first,
and it is written knowing that. A closed form found after the measurement is
known is worth less than one predicted before, and the paper does not pretend
otherwise. What it argues is narrower: that the forms are constrained by a
cost principle stated independently of the values, that the search space is
declared, and that the same principle applied to other observables makes
predictions that have not been tuned. Whether the argument succeeds is for the
reader to judge, and the section is deliberately placed where it cannot be
missed.

The comparison with other frameworks at the end of the Part is structured so
that every entry, this one included, carries a cost column. A comparison in
which only the rivals pay for their successes is not a comparison.

\GGPart[Discussions]{Discussions}{Significance and uniqueness of the result; comparison with other cosmological theories and models; predictions for falsification; open questions that remain; and the future prospects of the \GGTheory{} program on the cosmological-boundary stratum}

\section{What this Part does}
\label{sec:partXII-accomplished}

\ATen{} takes the seventeen largely independent mysteries of modern cosmology and compresses them onto a single inherited ledger: the same constitutional layer, the same \BIsix{} anomaly accounting, the same \OGN{} arithmetic, the same radial scale coordinate, and the same open-system measurement logic carry inflation, the dark sector, the late inventory, baryogenesis, structure growth, and even the black-hole information story at once. This Part states that case in four movements, in the order a skeptical reader should meet them. \emph{Significance} (Sections~\ref{sec:partXII-distinctive}--\ref{sec:partXII-comparison}): what kind of theory this is --- one engine, \(\OGGEP \to \GDOS \to \GRIP \to \GGA\), read on the cosmological stratum --- and how it differs from every framework now in the field. \emph{Innovations} (Section~\ref{sec:innovations}): the ten structural moves the paper makes that no prior cosmology program has made, each with its cohort-global identifier. \emph{Accomplishments} (Sections~\ref{sec:partXII-accomplishments}--\ref{sec:partXII-challenges}): what the paper has actually resolved, walked problem by problem against the seventeen-problem census of Part~I, with the honest boundary --- what is proved, what is conditional, what remains open --- drawn exactly. \emph{Predictions for failure} (Sections~\ref{sec:partXII-parameter-economy}--\ref{sec:partXII-falsifiable}): how the theory can lose, stated as named kill criteria with a decade timeline, because a zero-parameter framework earns credibility only by exposure. The closing verdict (Section~\ref{sec:partXII-bottom-line}) can then be read at face value. This is a unification proposal that knows precisely how it can lose --- and says so head-high.


\section{Significance and uniqueness}
\label{sec:partXII-distinctive}

\ATen{} does not merely offer thirteen proposed answers; it reduces the number of truly independent ingredients one must tolerate in order to discuss those thirteen questions at all. That reduction rests on a short inherited ledger,
\begin{equation}
\begin{aligned}
\mathfrak{L}_{\mathrm{cosmic\,input}}
=\Bigl\{
&\EGUT,
\; r=\ln(\EGUT/\mu),
\; \Mlock,
\; (a,b,c,d)=(1,-1,-2/5,-2/5),\\
&(M,N,L;J,K)=(1,3,5;7,35),
\; \GFF,
\; \DGI,
\; \GSSB,
\; \GA,
\; \OGGEP
\Bigr\},
\end{aligned}
\label{eq:partXII-input-ledger}
\end{equation}
which organizes into five modules that recur across sectors. Before the five distinctions, it is worth naming the single chain from which they all descend; each distinction is then a facet of it, and each is stated boldly.

\subsection{The essence of \GGTheory{}: \texorpdfstring{$\OGGEP \to \GDOS \to \GRIP \to \GGA$}{\OGGEP{} -> \GDOS{} -> \GRIP{} -> \GGA{}}}
\label{sec:partXII-essence-chain}

Beneath the five distinctions below lies a single chain, and it is the essence of
\GGTheory{} as a whole --- not of cosmology alone. Axiom~AX-3 of the \GGConstitution{}
\citep{ArisakaA1Constitution} is \OGGEP{}: every observable is a boundary projection of
a closed higher-dimensional geometry, never the bulk read directly. \GDOS{}
\citep{ArisakaA5GDOS} is the law by which that open boundary evolves. \GRIP{}
\citep{ArisakaA6Gi4} is the runtime that acts on it --- the cascade
\(\DGI \to \GSSB \to \GA\) that explores the admissible branches, ranks them by the
\GFF{} free-energy ledger, breaks the symmetric branch, and relaxes to the unique Ground
Geometric Attractor. The observed quantity \emph{is} that attractor's boundary
projection:
\begin{equation*}
\boxed{\;
\underbrace{\OGGEP}_{\text{AX-3}}
\;\longrightarrow\;
\underbrace{\GDOS}_{\text{boundary law}}
\;\longrightarrow\;
\underbrace{\GRIP}_{\DGI\,\to\,\GSSB\,\to\,\GA}
\;\longrightarrow\;
\underbrace{\GGA}_{\text{the observable}}
\;}
\end{equation*}
This chain is not a cosmological device. It is the same engine that fixes the Standard
Model in \ANine{} --- every charged-lepton mass, every mixing angle, the weak angle
\(\sin^2\theta_W = 25/108\), each a \GGA{} on its own particle sub-stratum --- and the
same engine that fixes \(\Lambda\)CDM here: every entry of the \(\pi\)-locked inventory
of Figure~\ref{fig:pi-locked-chart-omega-hook}, the Hubble two-stage corridor, the Kaxion
mass, and the damping knee, each a \GGA{} on the cosmological-boundary stratum. The two
consequence papers of the cohort, \ANine{} (particles) and \ATen{} (cosmos), are
therefore not two theories but one chain run on two strata; the codebook triage of
Part~IX is, read in this light, a census of how far the \GRIP{}-uniqueness of each
attractor has been proved. The five distinctions that follow are five facets of this one
chain.

\subsection{It begins from a constitutional layer}

Most cosmological model-building begins with an effective field theory or an energy budget and only afterwards asks how measurement, coarse-graining, and admissibility should be described. Here the order is reversed: the axiom and postulate layer of \AOne{} is inherited \emph{before} any cosmological sector. \OGGEP{} therefore makes the measurement dictionary part of the theory rather than meta-language bolted on later. This matters directly, because cosmological inference is never raw access to ``the universe itself'' but always an inference from light cones, maps, transfer functions, and coarse-grained boundary observables. Writing that operational structure in from the start prevents the usual drift in which a model is judged by fit quality alone while its measurement semantics stay implicit --- and it makes the black-hole information problem a direct test of whether the constitution was chosen correctly, not a late appendix.

\subsection{It geometrizes gauge structure at \(\EGUT\), not gravity at \(\EPl\)}

Twentieth-century habit taught theorists to start from quantum fields and add gravity, or to start from classical gravity and quantize geometry. \GG{} does neither. Its six-dimensional bulk is organized at the GUT edge, with the radial coordinate encoding the energy descent and the compact angle carrying the locked gauge structure. The beginning is therefore placed at \(\EGUT\), and the Planck scale becomes derived and effective rather than primary. Inflation is then sited on the derived plateau one e-fold below the GUT edge --- placed by the geometry rather than chosen from scalar-potential folklore --- with the TeV lock as the descent's second saddle, and the late dark-energy remnant is read as a tiny residual of the same ledger. That is a genuinely different organizational principle, not a rhetorical relabeling.

\subsection{Dark matter, inflation exit, and dark energy are linked, not merely coexisting}

One descent does three jobs at its two crossings and its terminus: the graceful exit at the top ends the inflationary plateau, the lock crossing opens the birth corridor for the Kaxion dark sector, and the terminus leaves the small residual vacuum remnant later read as dark energy. Standard practice represents these by three separate sectors --- an inflaton potential, a dark-matter candidate, a cosmological constant or quintessence field --- and even broad models keep them only loosely related. In \ATen{} the same geometric descent seeds all three. This is exactly the cross-sector economy a unification program should attempt, and it deliberately produces a riskier theory: a failure in one sector reverberates through the others.

\subsection{Observation itself is treated as open-system projection}

This is the \GDOS{} contribution in its cleanest cosmological form. The standard workflow acknowledges openness only pragmatically --- detectors are noisy, foregrounds contaminate maps, horizons leak --- while still treating the deep laws as those of a closed, observer-external universe. \ATen{} instead insists that the observable four-dimensional world \emph{is} a boundary subsystem, so that projection, incomplete access, and boundary-accessible records are built into what observation means. That single grammar then covers cosmological perturbations, black-hole horizons, and the larger floors of the program with one consistent semantics.

\subsection{It is austere in a correlated way}

Many models advertise minimality; the interesting question is whether the minimality is \emph{correlated}. Anyone can be minimal by discussing one subproblem. The hard thing is to keep the ingredient count low while speaking to inflation, dark matter, dark energy, baryogenesis, BBN, structure growth, early galaxies, primordial relics, and black holes simultaneously. The \OGN{} arithmetic, the \BIsix{} coefficient vector, the radial coordinate, and the lock seam recur across all of them. This is the strongest form of the claim: \emph{no single one of these distinctions is unique on its own --- the conjunction is}. A theory that can break in many places for the same reason is the kind of theory worth testing.

Table~\ref{tab:partXII-module-map} shows the concrete sense in which a handful of inherited modules carry many problems at once.

\begin{table}[H]
\centering
\caption[How a few inherited modules address several problems at once]{How a small number of inherited modules address multiple cosmological problems simultaneously. This is the concrete sense in which the paper aims at unification rather than thematic breadth. The columns give the module, its mathematical role, and the cosmological problems it touches. The economy claimed here is the third column being longer than the first: if each problem needed its own module the framework would be a catalog rather than an account.}
\label{tab:partXII-module-map}
\small
\begin{tabular}{@{}p{0.23\textwidth}p{0.28\textwidth}p{0.38\textwidth}@{}}
\toprule
\textbf{Inherited module} & \textbf{Primary mathematical role} & \textbf{Cosmological problems touched}\\
\midrule
Finite \(\EGUT\) edge and radial coordinate \(r=\ln(\EGUT/\mu)\) & Replaces singular-beginning rhetoric by finite scale descent and organizes chronology logarithmically & 1, 2, 6, 7, 13; also the entire chronology of Part~VI\\
Lock seam near \(\Mlock\) & The descent's second saddle: the prompt crossing that stocks the dark sector, and the seesaw source of the terminus scale & 2, 3, 4, 5, 6, 13\\
Compact-angle sector & Supplies a natural dark-matter corridor and links it to the same transition that ends inflation & 5, 10, 12, indirectly 8 and 9\\
Late residual \GFF\ inventory & Produces the \(\pi\)-locked matter-vacuum ledger and the coincidence interpretation & 3, 4, 8, 9\\
Constrained neutrino/CP ledger & Carries leptogenesis and transmits consequences to the visible-matter abundance audit & 6, 7, indirectly 8 and 9\\
\OGGEP{}/\GDOS{} semantics & Makes observation a boundary-accessible open-system projection and prevents naive closed-system factorization & 11, 13; also supports 8--10 as an inference dictionary\\
Exact baryonic \(\mathrm{U}(1)_B\) sector & Protects proton stability and introduces weak baryonic environmental effects at late times & 8, 9, 10, and the particle-cosmos bridge more generally\\
\bottomrule
\end{tabular}
\end{table}

If the late-inventory numbers fail, dark energy, coincidence, and the late-background interpretation fail together; if the compact-angle story fails, the dark-matter corridor and the coherence of the lock-seam narrative fall with it; if the open-system horizon story fails, black-hole information is not the only casualty. That correlated exposure is what it means for a theory to pay a genuine price for being wrong.

\section{Comparison with the leading frameworks}
\label{sec:partXII-comparison}

Read against the leading frameworks, each rival's famous gap turns out to be a \emph{missing-layer symptom} --- the absence of precisely the constitutional, geometric, or measurement layer \ATen{} supplies.

\paragraph{Vanilla \LCDM{}.}
The nearest comparator is the baseline the paper explicitly preserves. \LCDM{} is phenomenologically compact and observationally superb: a stable background, a precise perturbation pipeline, a common empirical language. Its limitation is not poor fit but interpretive openness --- cold dark matter, the near-scale-invariant seed, and the late vacuum component are represented beautifully yet left underived. \LCDM{} is a superb baseline and a weak closure principle; \ATen{} keeps the first and supplies the second.

\paragraph{Scalar-potential inflation EFTs.}
A flexible perturbation dictionary with wide reach (\(n_s\), running, non-Gaussianity, tensor ratios), bought with a very large family of potentials and reheating prescriptions and almost no sectoral closure. The inflationary module of \ATen{} is the opposite strategy: the plateau is tied to the radial geometry, the exit to the plateau's own steep edge, and the same descent seeds the dark corridor (at the lock crossing) and the vacuum remnant (at the terminus). The EFT tradition buys robustness with many potentials; \ATen{} buys unification with very few --- and accepts the sharp tensor risk that follows.

\paragraph{WIMP/SUSY dark matter.}
Natural freeze-out lore and broad BSM embedding, paid for with a candidate zoo, rising experimental pressure, and a dark sector left independent of inflation and dark energy. The missing layer is a geometric selection principle. \ATen{} ties dark matter to the compact-angle/Kaxion corridor descending from the same anomaly structure required elsewhere; the gain is a sharper search target, the cost a sharper falsification risk.

\paragraph{PQ axions, ALPs, and the axiverse.}
The nearest relative: a light pseudo-scalar tied to symmetry structure, cold via misalignment. But a generic axiverse survives precisely by being broad. The Kaxion is narrower in two ways: it is not introduced because ALPs are useful but inherited from the compact-angle and lock-seam story of this specific geometry, and its targets --- frequency bands, couplings, in some variants a doublet splitting --- reflect the same discrete arithmetic used across the program. A clear gain if borne out, and a clear loss of flexibility.

\paragraph{Quintessence, dynamical dark energy, and EDE.}
Room to alter background history or growth without discarding GR, paid for by enlarged sectoral freedom: a new field or fluid, then its potential, then its couplings and clustering. The minimal late-time branch of \ATen{} instead wants \(w=-1\) to high accuracy, negligible clustering, and a present inventory tied to simple \(\pi\)-locked shares. It does not compete by mimicking all dark-energy phenomenology; it competes by saying that if the universe really prefers something very close to a true cosmological constant, one should not reopen a large dynamical-fluid sector. The modest early-energy lobe used in the Hubble route is a transient face of the same geometry, not a new standing fluid.

\paragraph{String landscape and multiverse.}
Enormous UV richness and vast vacuum multiplicity, paid for with a vacuum-selection problem in which freedom can outrun falsifiability --- the anthropic \(\Lambda\) is the symptom of a missing zero-free-parameter constitution. \ATen{} shares the mathematical atmosphere but leans on \MSMF{} and \OGN{} austerity, rejecting a large physically operative landscape or anthropic selection as a primary explanatory tool. Whether that rejection is sustainable is an empirical and mathematical question; it is also a sharp philosophical distinction.

\paragraph{Bounce, cyclic, ekpyrotic, loop-inspired beginnings.}
These engage the origin problem directly but often introduce a second pre-history or Planck-regime dynamics with its own assumptions. \ATen{} instead says the physically relevant ultraviolet edge is already finite at \(\EGUT\), the Planck scale derived rather than primary. This is conceptually cleaner than retaining a singularity, but it is honestly a finite beginning \emph{ledger}, not yet a full beginning wavefunction.

\paragraph{Operational black-hole programs and the island formula.}
The island/QES and replica-wormhole literature delivers explicit Page-curve recovery and generalized-entropy technology, usually focused on the horizon rather than a universal measurement grammar. \ATen{} does not overthrow it --- it reinterprets it: the success of generalized-entropy reasoning is the expected behavior of a theory in which horizons define open subsystems, naive factorization is illegal, and the accessible ledger is already \GFF/\GDOS-like. The gain is a broader operational grammar; the obligation, made explicit below, is to make that account quantitatively competitive.

These trade-offs are condensed in Table~\ref{tab:partXII-comparison}.

\begin{longtable}{@{}p{0.18\textwidth}p{0.23\textwidth}p{0.23\textwidth}p{0.28\textwidth}@{}}
\caption[GG cosmology against several leading theoretical habits]{Comparison between \GG{}/\GDOS{} cosmology and several leading theoretical habits. The emphasis is on explanatory trade-offs rather than rhetorical victory. The second and third columns are deliberately paired: what each framework does well, and what it pays for that success. The fourth states the contrast rather than a verdict, and the row for this framework carries a cost column like every other.}
\label{tab:partXII-comparison}\\
\toprule
\textbf{Framework} & \textbf{What it does well} & \textbf{What it pays for that success} & \textbf{\GG{}/\GDOS{} contrast}\\
\midrule
\endfirsthead
\toprule
\textbf{Framework} & \textbf{What it does well} & \textbf{What it pays for that success} & \textbf{\GG{}/\GDOS{} contrast}\\
\midrule
\endhead
\bottomrule
\endfoot
Vanilla \LCDM & Compact empirical baseline; excellent background and perturbation fit across a huge data range & Leaves several source terms and origins interpretively open & \GG{}/\GDOS{} keeps the baseline equations but tries to derive or correlate the unexplained source sectors\\
Scalar-potential inflation EFT & Flexible perturbation dictionary; wide phenomenological reach & Large family of potentials and reheating prescriptions; little sectoral closure by itself & \GG{}/\GDOS{} offers a much narrower lock-seam inflation module with a much sharper tensor risk\\
WIMP/SUSY dark matter & Natural freeze-out lore; broad BSM embedding & Candidate zoo; growing experimental pressure; dark matter often remains independent of inflation and DE & \GG{}/\GDOS{} narrows the dark sector to the compact-angle/Kaxion corridor and links it to inflation exit and vacuum remnant\\
Axion/ALP/axiverse families & Symmetry-based pseudo-scalar dark matter; rich search program & Broad parameter freedom unless tied to a sharper UV ledger & The Kaxion is intended as a far narrower axion-like corridor fixed by the same geometry used elsewhere\\
Quintessence / dynamical DE / EDE & Can modify late or early background histories and relieve specific tensions & Requires new fields or fluids with their own potentials, couplings, and clustering assumptions & \GG{}/\GDOS{} prefers a residual vacuum remnant with \(w\approx -1\) and only transient early-energy faces inherited from the same geometry\\
Landscape / multiverse approaches & Enormous UV richness; many possible vacua and sectors & Difficult vacuum selection; freedom can outrun falsifiability & \GG{}/\GDOS{} explicitly prefers austere discrete locking and rejects broad anthropic rescue as a primary explanatory tool\\
Bounce / cyclic / loop-inspired origins & Direct engagement with the beginning and singularity issue & Often introduces a second pre-history or Planck-regime dynamics with its own assumptions & \GG{}/\GDOS{} replaces the operative beginning by a finite \(\EGUT\) edge rather than by a bounce or Planck-era discrete recursion\\
Island/QES black-hole programs & Quantitative Page-curve recovery and generalized-entropy technology & Usually focused on the horizon problem rather than a universal open-system measurement grammar & \GG{}/\GDOS{} seeks to reinterpret the same success as a special case of a broader \OGGEP{}/\GDOS{} operational closure\\
\end{longtable}

\ATen{} does not dominate every alternative on every axis --- that is not how theory choice works. Its advantage is economy and cross-linkage; its disadvantage is that its sharpest modules are also its most falsifiable. For a unification paper, that is the right trade.

\section{The Ten Structural Innovations of the Cosmological Closure}
\label{sec:innovations}

The contribution of \ATen{} is to read the four-dimensional \LCDM\ cosmology as the
cosmological-boundary projection of the six-dimensional bulk under
\GDOS{}/\OGGEP{}, recovering the present-epoch inventory and the cosmic chronology
as consequences of an inherited ledger rather than as fitted parameters. We collect
here the ten structural innovations that reading delivers, each carrying its
cohort-global identifier (\S\ref{sec:naming-convention}). None introduces a new
continuous parameter; each descends from the \GGTheory{} constitution of \AOne{}
through the \BIsix{} closure and the \GRIP{} cascade. Each is, at root, the one chain of Section~\ref{sec:partXII-essence-chain} --- \(\OGGEP \to \GDOS \to \GRIP \to \GGA\) --- specialized to a single cosmological observable.

\paragraph{Innovation 1 (I-Cosmos-1): Cosmology from a constitutional layer.}
\ATen{} reverses the usual order of cosmological model building. Instead of
beginning from an effective field theory or a phenomenological energy budget and
asking only afterward how measurement and admissibility should be described, it
inherits the axiom and postulate layer of \AOne{} before it inherits any
cosmological sector at all. Every cosmic claim is therefore admissible only if it
traces backward through the same constitutional chain already used to build the
particle and open-system papers, so that fit quality alone is never permitted to
stand in for legality of origin.

\paragraph{Innovation 2 (I-Cosmos-2): The \texorpdfstring{$\pi$}{pi}-locked
inventory as a closed-packet theorem.} The present-epoch densities
\(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\), and \(\Omb = 1/(2\pi^2)\) are read as a
closed-packet topological invariant of the \BIsix{} four-form, with the \OGNfive{}
integer ledger \((1,3,5;7,35)\) of \AFour{} fixing the rational weight of the
matter cell before the Pontryagin rescaling. The inventory is the integral output
of a fixed topological computation with no continuous adjustable parameter, not a
curve fit dressed in geometric language.

\paragraph{Innovation 3 (I-Cosmos-3): One topology forces three coupled entries.}
A post-hoc fit explains one number; the closed-packet integration here forces three
coupled numbers at once. The same computation that yields \(\Omm = 1/\pi\) yields
\(\OmL = 1 - 1/\pi\) by unit-trace complementarity and \(\Omb = 1/(2\pi^2)\) by a
one-dimensional further projection through the \(U(1)_B\) baryonic bundle. That the
topology produces the specific combination of powers of \(\pi\) --- one for matter,
its complement for vacuum, two for baryons --- is the structural signature that
distinguishes a derivation from a coincidence.

\paragraph{Innovation 4 (I-Cosmos-4): Recombination and the Hubble two-stage
corridor from integers.} The corridor-switch redshift \(z_* = e^7\) --- the topological last-scattering landmark, coinciding with the measured decoupling epoch to \(0.6\%\) --- descends from the \OGNfive{} integer \(J = 7\), not from a fitted thermal history, and the Hubble two-stage corridor switches at the same integer-locked scale. The early-time
determination of \Hzero{} thereby sits on the derived background (\(66.6\pm0.2\)
from the \(\pi\)-locked inventory), while the local-ladder excess is read as a
\GRIPin{} structure lift atop a named calibration systematic --- a displaced
measurement, not a tension to be relaxed by an extra empirical sector.

\paragraph{Innovation 5 (I-Cosmos-5): Gauge geometry at the GUT edge, not gravity
at the Planck scale.} The six-dimensional bulk is organized at the GUT edge,
with the radial coordinate \(r = \ln(\EGUT/\mu)\) encoding the energy descent and
the compact angle carrying the locked gauge structure, so that the Planck scale
becomes derived and effective rather than primary. Inflation is placed on the derived
plateau one e-fold below the GUT edge --- its class forced by \GDOS{}, not chosen
from scalar-potential folklore --- and the late dark-energy remnant is read as a tiny
residual of the same ledger rather than as an independent vacuum constant.

\paragraph{Innovation 6 (I-Cosmos-6): One descent seeds three sectors.}
A single geometric descent does three jobs at its two crossings and its terminus:
the graceful exit ends the inflationary plateau, the lock crossing opens the birth
corridor for the Kaxion dark sector, and the terminus leaves the small residual
vacuum remnant later interpreted as dark energy. Where
standard practice carries an inflaton sector, a dark-matter sector, and a
cosmological-constant sector as loosely related modules, \ATen{} links them so
tightly that a failure in any one reverberates through the others.

\paragraph{Innovation 7 (I-Cosmos-7): Kaxion dark matter from the compact angle.}
The dark-matter candidate is not an unconstrained zoo but the Kaxion corridor
inherited from the compact-angle sector and the same final geometric transition that
terminates inflation. The relic abundance follows from misalignment within the
\(\pi\)-locked inventory, so that the candidate points to a narrow frequency-and-%
coupling corridor with concrete search targets rather than to a broad axiverse
landscape.

\paragraph{Innovation 8 (I-Cosmos-8): Observation as open-system boundary
projection.} The framework treats the observable four-dimensional world as a
boundary subsystem rather than as a closed observer-external universe. Through the
\GDOS{} boundary law and the \OGGEP{} projection, incomplete access, coarse-grained
records, and horizon physics are built into what cosmological observation means, so
that the semantics of the CMB, of late-time inference tensions, and of the
black-hole information paradox are all discussed within one grammar.

\paragraph{Innovation 9 (I-Cosmos-9): The universal-cosmic-identity principle.}
The cosmological coincidence is read as a consequence of the same constitutional
uniqueness principle that selects the unique stable Geometric Attractor on the
charged-lepton stratum of \ANine{}, propagated one floor up to the
cosmological-boundary stratum. The same logic that explains why every electron is
the same explains why the universe has the inventory it does, read through the
\GRIP{} cascade --- with the \GRIPout{} transitions that rebuild the landscape
separated by long \GRIPin{} relaxations on it.

\paragraph{Innovation 10 (I-Cosmos-10): A zero-free-parameter, correlated, and
falsifiable cosmology.} The input ledger contains zero continuous adjustable knobs:
the inventory is closed by closed-packet topology with the \OGNfive{} ledger and
\GUfive{} as fixed input, so the framework cannot be retrofitted the way a
six-parameter \LCDM\ fit can. Because the same short ledger controls many
observables at once, the theory is falsifiable in a stronger-than-usual sense ---
eight named observational tests, each tied to a named conjecture or theorem, can
retire it as a package rather than only patch it sector by sector.


\section{Accomplishments: the seventeen problems, resolved, conditional, or frontier}
\label{sec:partXII-accomplishments}

Significance and innovation are claims about \emph{kind}; accomplishment is a claim about \emph{score}, and it should be settled against the census the paper opened with. Table~\ref{tab:seventeen-problems} of Part~I listed the seventeen open problems of contemporary cosmology in five clusters. Here is what the present framework has done to each --- with the codebook status attached, so that the score and the honesty travel together. (The reader who wants any single entry in depth will find the full treatment in the Part named; this section is the tally.)

\paragraph{Cluster I --- the two flagship tensions.}
\emph{(1) The Hubble tension} is read as one geometric corridor projected through two epochs, switching at \(z_* = e^7\): the early branch is \emph{derived} at \(66.6\) from the \(\pi\)-locked inventory, the late branch is the corridor's structural lift atop a named ladder systematic, and the gravitational-wave sirens --- which bypass both calibrations --- are predicted to land near \(67\) (Part~VI; D\(^*\) under Conjecture~II). \emph{(2) The \(S_8/\sigma_8\) growth tension} is addressed by the same corridor's growth response, with the sign honestly flagged open (Part~VI; D\(^*\) under Conjecture~II). One structure, both tensions --- and the corridor can fail as one object.

\paragraph{Cluster II --- the dark Universe and its fate.}
\emph{(3) The cosmological-constant problem} is answered in two registers: the scale by the seesaw \(m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}} \simeq 6\)~meV with the ledger coefficient \(\zeta_\Lambda = \pi/LK\), which recovers the notorious 120 orders of magnitude to the percent level without a cancellation (Derived cond.), and the share --- exactly, by geometry alone --- by \(\OmL = 1 - 1/\pi\) (Part~IV; D\(^*\) under Conjecture~VIII). \emph{(4) Constant or evolving dark energy?} The framework takes a live, falsifiable stand: \(w = -1\) exactly, no evolution, no clustering --- directly opposed to the DESI \(w_0w_a\) hint, and decidable this decade (D\(^*\) under Conjecture~IX). \emph{(5) The cosmic coincidence} is dissolved rather than explained away: the matter-composition ratio and the vacuum scale hold at every epoch, the fractions are read at the present epoch through one declared timing input, and the residual ``why now'' is bounded and named (Parts~IV and~X). \emph{(6) The nature of dark matter} is answered with a particle, a mass, and an address: the Kaxion at \(8.1\,\mu\)eV in the compact angle, pinned patch-uniform during primary inflation, searchable at \(1.59\)--\(2.13\)~GHz (Part~V and \PaperCOne{}; D\(^*\) under Conjectures~I and~III).

\paragraph{Cluster III --- the beginning and the primordial signals.}
\emph{(7) The origin and the initial singularity}: the universe begins lawfully at the finite \(\EGUT\) edge --- no singular datum, no Planck-regime speculation; the beginning is the top of a descent, not a breakdown of the equations (Part~III; derived). \emph{(8) The cause of inflation} is the paper's sharpest
accomplishment: the inflaton is the \(\Sigma_r\) scale modulus (forced), the plateau class is derived (\GDOS{} non-conformal route), and the tilt is computed, not fitted --- \(n_s = 0.9620\), \(0.7\sigma\) from Planck (Part~IV; D\(^*\) under the revised Lemma~IV, with the amplitude the one declared input). \emph{(9) Primordial gravitational waves}: the same plateau forces \(r = 12/N^{*2} = 0.0043\) --- a decisive, two-sided forward prediction, above the Simons-Observatory/LiteBIRD floor (D). \emph{(10) Baryogenesis}: the derived reheating temperature \(T_{\mathrm{rh}} \simeq 3\times10^{10}\)~GeV opens exactly the thermal-leptogenesis window, running on the inherited neutrino-and-CP ledger of \ANine{} with zero dilution (Part~V; D\(^*\) under Conjecture~VI). \emph{(11) The lithium-7 problem}: an open-system \(A=7\) port suppressing the sector by \(1/\pi\) while leaving deuterium and helium untouched (Part~V; D\(^*\) under Conjecture~X).

\paragraph{Cluster IV --- structure, cosmic dawn, and the JWST surprises.}
\emph{(12) JWST's early massive galaxies} are addressed by the corridor's early-structure amplification --- a structural reading whose quantitative simulation is named open work (Part~VI). \emph{(13) The ``little red dots''} and \emph{(14) cosmic dawn / 21~cm} are carried honestly as \emph{frontier}: the framework touches them but does not close them. \emph{(15) The black-hole information paradox} is resolved conceptually --- the horizon is an \OGGEP{} port, the entropy is the induced \(A/4G_N\) with the scale-derived Newton constant, and no firewall or remnant is needed --- with the quantitative match to the island-formula Page-curve numbers named as open work (Part~VII; PS-Cosmos-18).

\paragraph{Cluster V --- cracks in the foundations.}
\emph{(16) The isotropy anomalies} remain \emph{frontier}: boundary-projection statistics are the natural language, but no closed prediction is offered. \emph{(17) Neutrino masses and \(N_{\mathrm{eff}}\)}: inherited whole from the particle ledger of \ANine{}, with \(\Delta N_{\mathrm{eff}} \approx 0.04\) computed, not fitted (D).

\paragraph{The tally, honestly drawn.}
Of the seventeen: \emph{fourteen} receive a derived, derived-conditional, or structural reading from the one inherited ledger --- among them both flagship tensions, all four dark-universe problems, and all five beginning-and-primordial problems --- and \emph{three} (the little red dots, cosmic dawn, the isotropy anomalies) are marked frontier, touched but not claimed. No previous framework known to the author addresses more than a few of the seventeen at once, and none does so with zero continuous parameters. That is the accomplishment claim, and the two sections that follow draw its honest boundary: exactly what is proved, what rides on a named conjecture, and what the framework computes versus what it only frames.


\section{The honest boundary: what is proved, what is conditional, what is open}
\label{sec:partXII-challenges}

This is the concentrated honest frontier of the Part. It is stated plainly, and without flinching, because a scientist who has mapped exactly where the proven territory ends loses nothing by naming the boundary.

\subsection{What is proved, what is suggested, what remains open}

\paragraph{Proved (within the present scope).} The lawful finite beginning at \(\EGUT\) replacing the Planck-singularity datum; the slow-smooth plateau, derived in class (\GDOS{}) and placed one e-fold below the GUT edge, with its computed observables \(n_s = 0.9620\) and the Starobinsky relation forcing \(r = 0.0043\) as a Derived codebook entry; the structural existence of the two-stage Hubble corridor with its switch redshift \(z_* = e^7\) as an \OGNfive{} integer-ledger prediction.

\paragraph{Suggested but not proved.} The \(\pi\)-locked inventory \(\Omm = 1/\pi\), \(\OmL = 1-1/\pi\), \(\Omb = 1/(2\pi^2)\) is carried as \emph{Derived-conditional} under Conjecture~VIII (closed-packet integration): the printed Step-4 cancellation does not yet close, and these fractions must not be advertised as proved. The inflationary amplitude \(A_s\) remains the launch sector's one declared input (Asserted), pending the GDOS-heat-kernel computation of Lemma~V\('\); the tilt \(n_s\) and the scale \(V_{\mathrm{inf}}^{1/4}\) are Derived-conditional on the revised Lemma~IV. The two numerical Hubble branches and the \(S_8\) corollary are structural identifications, with the \(S_8\) sign explicitly open (cf.\ \citealp{Hill2020}). The black-hole resolution is conceptual, not yet a quantitative reproduction of the island/replica Page-curve numbers.

\paragraph{The conditions ledger.} The two-basin derivation replaced the single word ``asserted'' with a short list of \emph{named} conditionals, and the honest frontier is now this list. (i)~\emph{Plateau-class-and-prompt-crossing} (Lemma~IV, revised): the \GDOS{} non-conformal derivation of the Starobinsky/\(\xi\) class, and the \GRIP{} ledger-action bound on \(\Sigma_r\) giving \(N_2 = 0\) --- underwrites \(n_s\), \(r\), and \(V_{\mathrm{inf}}^{1/4}\). (ii)~\emph{GDOS-heat-kernel amplitude} (Lemma~V\('\)): the one computation that would pay the amplitude \(A_s\) and make the launch input-free; its warp candidate stands at \(75\)--\(87\%\), and two alternative routes have been computed and closed. (iii)~\emph{Conjecture~VIII} (closed-packet): the \(\pi\)-locked fractions, unchanged. (iv)~The Kaxion's deterministic misalignment rides on the reflection-symmetric descent dressing (the pinning condition; quantitative chain in \PaperCOne{}). Each condition is a target with a stated failure mode, not a soft caveat; items (i) and (ii) are the two endgames of the launch sector, and either would upgrade the bottom line materially.

\paragraph{Genuinely open / unfinished.} A full perturbation theory rederived from the six-dimensional action rather than layered on the four-dimensional baseline (PS-Cosmos-14); a full BBN-network implementation of the A=7 open-system port, deferred to GG-C4-Nucleo (PS-Cosmos-15); a genuine global Einstein--Boltzmann fit of the Hubble/\(S_8\) package across CMB, BAO, SN, weak lensing, RSD, and clusters (PS-Cosmos-16); nonlinear simulations for the JWST early-galaxy story (PS-Cosmos-17); the tight match of the \GDOS{}/\OGGEP{} black-hole account to generalized-entropy computations (PS-Cosmos-18); an even cleaner first-principles derivation of the \(\pi\)-locked inventory (PS-Cosmos-19); a complete dictionary between the \(Q_{Qi}\) charge ledger and ordinary information measures (PS-Cosmos-20); and the preservation of precision across the larger cross-floor program, judged here first as a cosmology paper (PS-Cosmos-21).

\subsection{What \GGTheory{} computes, and what it does not: the cosmology tier ladder}
\label{subsec:tier-ladder-cosmos}

The honest boundary of the framework is best stated as a four-tier ladder --- the cosmological instance of the can/cannot ladder established for the Standard Model in \AFive{} (Theorem~T9.9; \citealp{ArisakaA5GDOS}) and used in \ANine{}. It says exactly which cosmological content the geometry \emph{computes} and which it only \emph{frames}, leaving the irreducible dynamical computation to the standard machinery. The distinction matters because a parameter-free framework is exposed to the charge that it claims the whole of cosmology; it does not, and the ladder draws the line precisely.

\paragraph{Tier~1 --- the inventory (computed).} The present-epoch density fractions \(\Omm = 1/\pi\), \(\OmL = 1-1/\pi\), \(\Omb = 1/(2\pi^2)\), the recombination switch \(z_* = e^7\), and the inherited input ledger (\GUfive{}, the \OGNfive{} integers \((1,3,5;7,35)\), and \((\pi,e)\)) are the content \GGTheory{} \emph{derives} --- or derives-conditionally on a single named conjecture, exactly as the two/eighteen/one codebook triage records. This is the tier on which the paper stakes its claim: the cosmic \emph{inventory} is read off one geometry, not fitted.

\paragraph{Tier~2 --- the scale flow (geometrized).} The cosmic chronology --- the dictionary between \(t\), \(\mu\), \(r\), \(a\), and \(z\), the master timeline from the \(\EGUT\) edge to the de~Sitter future (Table~\ref{tab:partVIII-master-chronology}) --- is \emph{geometrized}: the radial coordinate \(r = \ln(\EGUT/\mu)\) is the renormalization-group and expansion axis, so the background history and its leading-log skeleton are radial transport. The geometry organizes \emph{when} each epoch occurs and fixes the ledger the background Friedmann integration runs on; it does not thereby replace that integration.

\paragraph{Tier~3 --- the precision CMB power spectrum (not computed).} The full angular power spectrum \(C_\ell\) --- the acoustic-peak heights and positions at sub-percent precision, the polarization spectra, the transfer functions --- is the output of the Einstein--Boltzmann hierarchy, an irreducible \emph{dynamical} computation over the coupled photon--baryon--neutrino system. \GGTheory{} fixes the \emph{inputs} that computation takes as given (the inventory, \(z_*\), the background history), but it does \emph{not} compute the spectrum --- it hands its inputs to the instrument. That hand-off has now been executed in both directions: run on the exact \GG{} fractions, the Einstein--Boltzmann machinery (CLASS) reproduces the observed TT band-powers at the few-percent level and returns the damping scale \(\ell_D = 1350\pm3\) (Section~\ref{sec:cmb-overdetermined}) --- a \(\chi^2/\mathrm{dof}\approx2.9\) whose residual is one coherent second-order offset, not a likelihood-level endorsement, and we do not claim it as one; the BFFT knee \(K(2\pi)^2\) is carried separately as an interpretive overlay. A genuine global Einstein--Boltzmann \emph{likelihood} fit across CMB, BAO, SN, and lensing --- covariance kept, seed re-optimized --- is named as open work (PS-Cosmos-16); the Boltzmann machinery remains the instrument.

\paragraph{Tier~4 --- the nonlinear structure-formation \texorpdfstring{\(P(k)\)}{P(k)} (not computed).} The nonlinear matter power spectrum, the halo mass function, and the baryonic-feedback corrections that govern the JWST early-galaxy regime are irreducible \emph{nonlinear} dynamics --- the province of \(N\)-body and hydrodynamic simulation. \GGTheory{} supplies the linear inputs and a structural, scale-sensitive correction chain (the growth-and-tensions Part), giving \emph{structure}, not \emph{precision}; the nonlinear simulations themselves are named as open work (PS-Cosmos-17). As in the Standard-Model case, a holographic geometry helps only where the coupling is controlled; the realistic nonlinear regime is not such a place.

\paragraph{The thesis.} The ladder lands one sentence, the cosmological analogue of \ANine{}'s ``perturbation theory computes the corrections to nature's constants; \GGTheory{} computes the constants'':

\begin{equation}
\boxed{\;\parbox{0.86\textwidth}{\centering \GGTheory{} computes the cosmic \emph{inventory} and geometrizes the cosmic \emph{chronology}; it does not compute the precision \(C_\ell\) spectrum or the nonlinear \(P(k)\). The Einstein--Boltzmann and \(N\)-body machinery remains the instrument it frames but does not replace.}\;}
\label{eq:tier-thesis-cosmos}
\end{equation}

This is not a concession extracted under pressure but the boundary the framework draws for itself. Tiers~1 and~2 are where a zero-parameter geometry can speak; Tiers~3 and~4 are where the irreducible dynamical computation lives, and the paper claims the first pair without pretending to the second. The two are complementary sciences, not competitors: the geometry supplies the inputs the Boltzmann and \(N\)-body codes take as given, and those codes supply the precision the geometry does not.

\subsection{Dependencies}

The chain is not free-standing; it rests on named inherited results, and the framework is only as strong as they are. It depends on the \OGNfive{} integer ledger (which fixes \(z_* = e^7\)), on the \BIsix{} anomaly-free closure, on the \GDOS{} boundary law and \OGGEP{} projection (which supply the measurement semantics and the open-system black-hole account), and on the \AFour{} openness theorem (which makes the cosmological constant a geometric scale residue rather than a free input). A decisive failure of any of these upstream pillars propagates downstream into the cosmological closure.

\subsection{The problem tally}

Against the open problems of contemporary cosmology (Table~\ref{tab:seventeen-problems}, Part~I; provenance in the Part~IX codebook), \ATen{} offers a derived or derived-conditional reading for the great majority. The cleanest closures are the lawful finite origin at \(\EGUT\), the dark-energy inventory \(\OmL = 1-1/\pi\), and the tensor target \(r = 0.0043\); most of the remainder carry a named conjecture, a pending calculation, or a companion paper as their explicit route forward; and three items --- the ``little red dots,'' the cosmic-dawn 21\,cm signal, and the large-angle isotropy anomalies --- are honestly marked as present frontier rather than claimed. The asymmetry is the honest research statement, not a weakness to be hidden.

\section{The cost of being right: zero parameters, maximal exposure}
\label{sec:partXII-parameter-economy}

A correlated, zero-free-parameter theory has the strongest possible relationship to experiment: any single failed prediction can kill it, and it cannot be quietly rescued by reinstalling sectoral freedom. This is the paper's central scientific virtue, and it is stated here without apology. One might object that a flexible patchwork can always fit more data in a narrow likelihood sense. True --- and dangerous, because it makes the relation between explanation and fit increasingly opaque. A theory that attaches a new independent sector to every tension drifts toward a catalog of compensations rather than a causal explanation. \ATen{} accepts the opposite burden: a lower tolerance for ad hoc rescue, and therefore a theory that is easier to kill and more informative if it survives.

The input ledger is honest about what is and is not free, on four levels.
\begin{enumerate}[leftmargin=2.4em,label=\textbf{L\arabic*.}]
\item \textbf{Constitutional input.} The axioms and postulates of \AOne{}; these define admissibility and operational meaning.
\item \textbf{Geometric input.} The six-dimensional bulk, \BIsix{}, and the \OGNfive{} arithmetic; the core model-structural inputs of \ATwo{} through 5.
\item \textbf{Boundary baseline input.} Four-dimensional GR and the \FRW/\LCDM\ observable language as derived or inherited baseline descriptors.
\item \textbf{Observational calibration input.} The empirical values needed to place the boundary history on actual units --- the present CMB temperature, today's Hubble scale.
\end{enumerate}
What the paper refuses is a fifth level of unrelated rescue knobs. That restraint is the discipline that makes the falsification logic below mean something: if the shared modules cannot meet the precision audit, the framework should not be saved by quietly reinstalling the sector-by-sector freedom it set out to remove. A theory that can fail as a package, and declines to escape that fate, has earned the right to be taken seriously.

\section{Kill criteria and the honest test ledger}
\label{sec:partXII-falsifiable}

The governing logic is
\begin{equation}
\text{small input ledger}
\quad \Longrightarrow \quad
\text{large correlated observational ledger}.
\label{eq:partXII-falsification-logic}
\end{equation}
There are five clean near-term ways to retire the minimal branch.

\paragraph{(a) Primordial tensor modes (F-Cosmos-1).}
The derived slow-smooth plateau fixes the tensor sector at \(r = 12/N^{*2} = 0.0043\). This is a \emph{forward} prediction, and one of the cleanest in the paper: a factor eight below the present bound, several times above the Simons-Observatory/LiteBIRD floor, decisive within the decade. A tensor measurement inconsistent with it in either direction directly falsifies the derived launch. It is a two-sided target, not a peripheral detail.

\paragraph{(b) Dark-energy dynamics (F-Cosmos-2).}
The minimal late branch wants \(w=-1\) to high accuracy with no appreciable clustering. A secure, persistent detection of \(w\neq -1\), strong time variation in \(w(a)\), or significant dark-energy clustering would severely damage the late-inventory story. This is a forward prediction sharpened against existing background-expansion and lensing data.

\paragraph{(c) Kaxion search targets (F-Cosmos-3).}
The dark-matter module points not to a broad space but to a narrow corridor: in the companion closure \PaperCOne{}, a unique laboratory line at \(1.59\)--\(2.13\)~GHz on the DFSZ coupling line, in some variants with a distinctive doublet structure. This is a forward prediction. A decisive program of null results across the sharpest predicted corridor would strongly pressure the compact-angle dark-matter module.

\paragraph{(d) The \(\pi\)-locked inventory (F-Cosmos-4).}
In the minimal interpretation the shares follow simple geometric values,
\begin{equation}
\Omm = \frac{1}{\pi},
\qquad
\OmL = 1-\frac{1}{\pi},
\qquad
\Omb = \frac{1}{2\pi^2}.
\label{eq:partXII-pi-inventory}
\end{equation}
These are carried as \emph{Derived-conditional} entries under the closed-packet-integration conjecture (Conjecture~VIII), not as proved outputs; the corridor switch \(z_* = e^J = e^7 \approx 1097\) is likewise a parameter-free marker of the \OGNfive{} integer ledger --- a topological quantity distinct from the measured decoupling redshift: the framework's own atomic chain derives \(z_{\mathrm{dec}}\) on the Planck value at the \(0.1\%\) level, with the switch sitting \(0.4\%\) above it (\S\ref{sec:two-redshifts}). If improved inference settled decisively on shares incompatible with Eq.~\eqref{eq:partXII-pi-inventory} even after the observational dictionary and covariance are accounted for, the late-inventory interpretation would fail directly.

\paragraph{(e) Joint growth-tension closure (F-Cosmos-5).}
The Hubble and \(S_8\) tensions are hard because a model that eases one often worsens the other. \ATen{} proposes a combined route, with the two measured \(H_0\) values accommodated as two projections of one corridor (Theorem~\ref{thm:hubble-two-stage-corridor}) and the clustering split addressed by a corollary (Corollary~\ref{cor:s8-reconciliation}). The honest test is whether the same inherited modules give an acceptable \emph{joint} fit without degrading the CMB, BAO, or lensing ledger. This is an \emph{audit} of existing Planck/DESI-class data, not a clean forward prediction, and the \(S_8\) sign is flagged open.

A framework that subsumes \LCDM{} must first reproduce its empirical successes before it can differ from it; the five kill criteria are therefore a mixture, honestly labeled. The forward predictions (the \(r = 0.0043\) tensor target, the \(1.59\)--\(2.13\)~GHz Kaxion window, the \(w=-1\) dark-energy behavior) are where the theory most exposes itself to new data; the inventory shares and the Hubble/\(S_8\) route are postdictions and audits of existing data that the theory must not be allowed to overclaim. Table~\ref{tab:partXII-falsification-ledger} states the full set in ``how the theory can lose'' language.

\begingroup\small
\begin{longtable}{@{}p{0.21\textwidth}p{0.23\textwidth}p{0.19\textwidth}p{0.29\textwidth}@{}}
\caption[Practical falsification ledger for the cosmological branch]{Practical falsification ledger for the minimal cosmological branch of \ATen{}. The table is deliberately written in a ``how the theory can lose'' language. The columns give the observable domain, the minimal expectation, the class of observation that would test it, and what would strongly damage the branch. The last column is written to be decidable by measurement rather than by argument, which is the only property that makes a ledger like this worth printing.}
\label{tab:partXII-falsification-ledger}\\
\toprule
\textbf{Observable domain} & \textbf{Minimal expectation} & \textbf{Observation class} & \textbf{What would strongly damage the minimal branch?}\\
\midrule
\endfirsthead
\toprule
\textbf{Observable domain} & \textbf{Minimal expectation} & \textbf{Observation class} & \textbf{What would strongly damage the minimal branch?}\\
\midrule
\endhead
\bottomrule
\endfoot
Primordial tensors / CMB B-modes & \(r = 12/N^{*2} = 0.0043\) from the derived plateau --- an order of magnitude below the present bound, several times above the Simons-Observatory/LiteBIRD floor & Next-generation CMB polarization programs & A tensor measurement inconsistent with \(r = 0.0043\) in either direction: a detection at \(r > 0.01\), or a null at the projected floor\\
Scalar perturbation sector & Near-scale-invariant scalar spectrum with a conventional-looking acoustic baseline & CMB temperature and polarization, LSS, BAO & A scalar spectrum demanding dynamics incompatible with the derived GUT-adjacent plateau\\
Dark energy equation of state & \(w=-1\) to high precision; negligible clustering & Background expansion, lensing, and structure-growth surveys & Secure evidence for time-varying or clustering dark energy\\
Cosmic inventory & Present-day shares close to the \(\pi\)-locked relations of Eq.~\eqref{eq:partXII-pi-inventory} & Joint CMB, BAO, SN, and lensing fits & Stable values that cannot be reconciled with the geometric-share interpretation\\
Dark matter direct search & Kaxion corridor with narrow frequency/coupling target(s), possibly including discrete splitting structure & Axion/ALP resonant and broadband searches & Exhaustive null results in the sharpest target corridor\\
BBN audit & Improved A=7 behavior without spoiling D/H or \(Y_p\) & Precision light-element abundance inference & Any lithium ``solution'' that necessarily ruins deuterium or helium once implemented honestly\\
Early/late expansion tension & A joint route to easing both early and late inferences through shared modules, not independent fluids & Combined early- and late-universe parameter inference & Persistent failure of the shared-route model to improve the joint fit without new rescue knobs\\
Growth and lensing & Moderated late growth consistent with background history and CMB inheritance & Weak lensing, redshift-space distortions, cluster counts & Need for a wholly separate growth-only sector after the shared modules are imposed\\
Early galaxy assembly & Mild acceleration of early structure and settling relative to vanilla expectations & High-redshift galaxy and quasar surveys & Strong evidence that the framework's inherited early-structure timing is in the wrong direction or too small by construction\\
Primordial relics & PBH and stochastic-background sectors are secondary, not dominant, in the minimal branch & Microlensing, PTA, interferometer, CMB constraints & A primordial-relic spectrum requiring dynamics foreign to the inherited background story\\
Black-hole information & Operationally unitary boundary-accessible ledger; no need for firewalls or ad hoc remnants in the minimal account & Theory-side entropy tests, plus eventual analog and astrophysical clues & Demonstration that the \GDOS{}/\OGGEP{} account cannot reproduce or reinterpret the modern Page-curve results quantitatively\\
\end{longtable}
\endgroup

This ledger is a statement of exposure, not a sales pitch. In a young unification program the strongest predictive claims --- low primordial tensors, sharply constrained dark energy, a narrow dark-matter corridor, a geometrically compressed late inventory --- are precisely the ones most likely to kill the theory early. If future data demand their opposites, the theory should not be praised for adaptability; it should be judged to have failed in its defining claims. A theory that survives such dangerous predictions gains something more valuable than flexibility: it gains character.

\subsection{The decade timeline: what 2030 and 2035 decide}
\label{subsec:decade-timeline-cosmos}

Kill criteria without dates are rhetoric. Table~\ref{tab:decade-timeline-cosmos} attaches the instrument and the horizon to each, so the reader can mark a calendar rather than take a promise --- and the cohort-wide canonical values quoted here are maintained in the single falsifier ledger of the \GGTheory{} program, so that no companion paper drifts from them; Figure~\ref{fig:killtest-timeline} lays the same decade out as one forward time axis --- the eight decisive kill-tests, color-coded by sector, from today's bounds to the mid-2030s verdicts. (The upstream companion papers carry the same discipline for the particle-side tests: the \(3.85\)~TeV lock cluster at the HL-LHC in \PaperPOne{}, the full falsifier pull-table in \AEleven{}.) One near-term row deserves emphasis: the inherited neutrino-mass sum \(\Sigma m_\nu \approx 59\)~meV sits at the normal-hierarchy floor, and a robust DESI+CMB bound pushing below \(\sim\!58\)~meV under this framework's own \(w=-1\) prior would falsify the inherited block --- potentially the first of these tests to be decided.

\begin{table}[H]
\centering
\caption[The falsification decade: each kill criterion, instrument and horizon]{The falsification decade for the cosmological branch: each kill criterion with its instrument class and decision horizon. Every row is two-sided --- the framework loses if the measurement lands off the stated target, and banks the corresponding claim if it lands on.}
\label{tab:decade-timeline-cosmos}
\renewcommand{\arraystretch}{1.25}
\small
\begin{tabular}{@{}p{0.24\textwidth}p{0.20\textwidth}p{0.26\textwidth}p{0.20\textwidth}@{}}
\toprule
\textbf{Target} & \textbf{GG commitment} & \textbf{Instrument class} & \textbf{Decision horizon}\\
\midrule
Tensor-to-scalar ratio & \(r = 0.0043\) (\(\sigma(r)\) delensing-dependent) & Simons Observatory, LiteBIRD & late 2020s--mid 2030s\\
Kaxion line & \(1.59\)--\(2.13\)~GHz, DFSZ-line coupling & haloscopes scanning past 2~GHz & late 2020s--2030s\\
Dark-energy dynamics & \(w = -1\) exactly; no evolution & DESI, Euclid, Roman & late 2020s\\
Siren Hubble constant & \(H_0^{\mathrm{GW}} \approx 67\) & GW standard sirens (LIGO/Virgo/KAGRA, ET) & \(\sim\)2030 and beyond\\
Growth / \(S_8\) split & corridor-consistent joint fit & Rubin LSST, Euclid & \(\sim\)2030\\
Neutrino mass sum & \(\Sigma m_\nu \approx 59\)~meV (inherited block) & DESI + CMB combinations & near-term (may fire first)\\
Recombination switch & \(z_* = e^7\) modulo the recombination-history lemma & existing Planck + audit & now (analysis-side)\\
Lithium-7 & \(\mathrm{SBBN}/\pi\), D/H and \(Y_p\) untouched & precision abundance inference & ongoing\\
\bottomrule
\end{tabular}
\end{table}

By the mid-2030s, every load-bearing cosmological commitment of this paper will have been tested by an instrument that exists or is funded. The framework does not ask the reader's patience beyond one decade.

% Falsification / kill-test timeline for \ATen{}
% PURE TikZ (no pgfplots). Requires libraries:
%   arrows.meta, positioning, calc, decorations.pathreplacing,
%   shapes.geometric, fit, backgrounds
\begin{figure}[!t]
\centering
\resizebox{0.739\textwidth}{!}{%
\begin{tikzpicture}[
    font=\large,
    >=Latex,
    card/.style={
        draw=black!55, rounded corners=2pt, line width=0.60pt,
        align=left, inner sep=4pt, text width=52mm
    },
    stem/.style={line width=0.90pt, draw=black!55},
    sectorCMB/.style={fill=blue!8,   draw=blue!55!black},
    sectorBAO/.style={fill=green!10, draw=green!55!black},
    sectorH0/.style={fill=orange!12, draw=orange!75!black},
    sectorKax/.style={fill=violet!10, draw=violet!60!black},
    dot/.style={circle, draw=black!70, fill=white, inner sep=0pt, minimum size=5.2pt}
]

% --- Geometry of the time axis ---
\def\xL{0}\def\xR{30}\def\yA{0}
\def\bA{9.5}\def\bB{19.0}

% Era background bands
\begin{scope}[on background layer]
    \fill[black!2]  (\xL,-8.4) rectangle (\bA,9.6);
    \fill[black!4]  (\bA,-8.4) rectangle (\bB,9.6);
    \fill[black!6] (\bB,-8.4) rectangle (\xR,9.6);
\end{scope}

% Era labels (top)
\node[anchor=south, font=\Large\bfseries] at ({(\xL+\bA)/2},8.8) {NOW / ONGOING};
\node[anchor=south, font=\large] at ({(\xL+\bA)/2},8.15) {current bounds, $\sim$2026};
\node[anchor=south, font=\Large\bfseries] at ({(\bA+\bB)/2},8.8) {NEAR-TERM};
\node[anchor=south, font=\large] at ({(\bA+\bB)/2},8.15) {late 2020s};
\node[anchor=south, font=\Large\bfseries] at ({(\bB+\xR)/2},8.8) {NEXT DECADE};
\node[anchor=south, font=\large] at ({(\bB+\xR)/2},8.15) {$\sim$2030s and beyond};

% The time arrow
\draw[->, line width=1.10pt] (\xL,\yA) -- (\xR+0.7,\yA);
\node[anchor=west, font=\Large\itshape] at (\xR+0.8,\yA) {time};
\foreach \x in {\bA,\bB}{\draw[black!55, line width=0.80pt] (\x,-0.18) -- (\x,0.18);}
\node[anchor=north, font=\large] at (\xL,-0.25) {2026};
\node[anchor=north, font=\large] at (\xR,-0.25) {2035+};

% ---- TOP ROW (above axis) ----
% TEST 4 : Lensing time-delay H0 (ongoing -> left)
\node[dot] (d4) at (3.2,\yA) {};
\node[card, sectorH0, anchor=south] (c4) at (3.2,1.5) {%
    \textbf{4. Lensing time-delay $\Hzero$}\\[1pt]
    precision: 1\%\\
    refutes: forces \textbf{GRIP} refinement if $\Hzero \!\ll\! 73$\\
    venue: H0LiCOW, TDCOSMO};
\draw[stem] (d4) -- (c4.south);

% TEST 8 : sigma_8 growth (near-term -> middle-left), taller
\node[dot] (d8) at (11.3,\yA) {};
\node[card, sectorBAO, anchor=south, text width=62mm] (c8) at (11.3,5.1) {%
    \textbf{8. $\sigeight$ precision growth}\\[1pt]
    precision: 1\% at $z\!=\!0.5$--$1$\\
    refutes: \textbf{Conjecture II} (persistent $\sigeight$ tension vs.\ corridor)\\
    venue: Euclid weak-lensing, LSST};
\draw[stem] (d8) -- (c8.south);

% TEST 6 : Kaxion haloscope (ongoing through ~2030 -> spans middle)
\node[dot] (d6) at (15.5,\yA) {};
\node[card, sectorKax, anchor=south, text width=62mm] (c6) at (15.5,1.5) {%
    \textbf{6. Kaxion haloscope window}\\[1pt]
    DFSZ-depth scan: 1.59--2.13 GHz\\
    refutes: \textbf{Conjecture III} (window excluded at DFSZ depth)\\
    venue: ADMX, CAPP, HAYSTAC, ADMX-EFR};
\draw[stem] (d6) -- (c6.south);
\draw[violet!60, line width=2.2pt] (12.0,-0.55) -- (19.0,-0.55);
\node[violet!60, font=\large, anchor=north] at (15.5,-0.6) {scan span};

% TEST 5 : Tensor-to-scalar ratio (next decade)
\node[dot] (d5) at (22.3,\yA) {};
\node[card, sectorCMB, anchor=south, text width=62mm] (c5) at (22.3,4.9) {%
    \textbf{5. Tensor-to-scalar ratio $r$}\\[1pt]
    measurement inconsistent with $r=0.0043$\\
    refutes: the derived launch (two-sided)\\
    venue: LiteBIRD, Simons Obs.};
\draw[stem] (d5) -- (c5.south);

% TEST 3 : GW siren H0 (next decade -> right)
\node[dot] (d3) at (26.6,\yA) {};
\node[card, sectorH0, anchor=south] (c3) at (26.6,1.5) {%
    \textbf{3. GW siren $\Hzero$}\\[1pt]
    precision: 1\% (late branch)\\
    refutes: background $\approx67$ if $\Hzero^{\mathrm{GW}}\!\approx\!73$\\
    venue: LIGO-Voyager, ET, Cosmic Explorer};
\draw[stem] (d3) -- (c3.south);

% ---- BOTTOM ROW (below axis) ----
% TEST 2 : Two-stage BAO residue (DESI/Euclid ongoing -> left-ish)
\node[dot] (e2) at (6.7,\yA) {};
\node[card, sectorBAO, anchor=north] (b2) at (6.7,-1.5) {%
    \textbf{2. Two-stage BAO residue}\\[1pt]
    1--2\% at $z\!\sim\!1$\\
    refutes: \textbf{Conjecture II} (two-stage corridor) if residue absent\\
    venue: DESI, Euclid, Roman};
\draw[stem] (e2) -- (b2.north);

% TEST 1 : Corridor switch vs atomic decoupling (next decade)
\node[dot] (e1) at (20.0,\yA) {};
\node[card, sectorCMB, anchor=north] (b1) at (20.0,-1.5) {%
    \textbf{1. Switch vs decoupling}\\[1pt]
    $\pm 5$ at $z\!=\!1090$ (0.46\%)\\
    refutes: atomic chain / Thm.\ 4.D.1 if $z_{\mathrm{dec}}$ off atomic value $>\!0.5\%$\\
    venue: 21-cm + CMB spectral-distortion};
\draw[stem] (e1) -- (b1.north);

% TEST 7 : Primordial GW background (next decade)
\node[dot] (e7) at (25.5,\yA) {};
\node[card, sectorCMB, anchor=north] (b7) at (25.5,-1.5) {%
    \textbf{7. Primordial GW background}\\[1pt]
    B-mode polarization\\
    refutes: \textbf{Starobinsky-channel} suppression (D9) if robust primordial B-mode\\
    venue: LiteBIRD, Simons Obs.};
\draw[stem] (e7) -- (b7.north);

% --- Sector legend (lower-left band) ---
\begin{scope}[shift={(0.3,-7.7)}]
    \node[anchor=west, font=\large\bfseries] at (0,0) {Sector:};
    \node[card, sectorCMB, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (2.2,0) {};
    \node[anchor=west, font=\large] at (2.5,0) {CMB / inflation};
    \node[card, sectorBAO, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (7.4,0) {};
    \node[anchor=west, font=\large] at (7.7,0) {LSS / BAO / growth};
    \node[card, sectorH0, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (13.4,0) {};
    \node[anchor=west, font=\large] at (13.7,0) {$\Hzero$ ladder};
    \node[card, sectorKax, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (17.4,0) {};
    \node[anchor=west, font=\large] at (17.7,0) {Kaxion DM};
\end{scope}

% --- Header note banner ---
\node[draw=black!40, fill=yellow!12, rounded corners=2pt,
      line width=0.60pt, align=center, text width=235mm,
      inner sep=5pt, font=\Large, anchor=north] at (15,11.7) {%
    \textbf{High exposure.} A clean failure on \emph{any one} of these eight tests
    refutes a named conjecture (II, III, V) or a named theorem (4.D.1) --- the
    framework's exposure is high, and most tests sit within the next-decade horizon.};

\end{tikzpicture}%
}
\caption[Falsification / kill-test timeline.]{%
\textbf{Falsification / kill-test timeline for \ATen{}.}
The eight decisive kill-tests are placed on a forward time axis running from current
bounds ($\sim$2026) to the next-decade horizon ($\sim$2035+), grouped into broad
data-window eras (now/ongoing, near-term, next decade) rather than precise launch
dates. Each card states the test, its precision/threshold, the named conjecture or
theorem it would \emph{refute}, and the experimental venue. Cards are colored by
sector: CMB/inflation (blue), large-scale-structure / BAO / growth (green), $\Hzero$
distance ladder (orange), and Kaxion dark matter (violet). The Kaxion haloscope scan
spans an ongoing-through-$\sim$2030 window (violet bar). A clean failure on any single
test refutes a named conjecture (II, III, V) or named theorem (4.D.1), so the
framework's falsification exposure is high and front-loaded into the coming decade.}
\label{fig:killtest-timeline}
\end{figure}


\subsection{Hostile audit: what if every test fails?}

It is worth walking the worst case stratum by stratum, because the right response differs by stratum.

\emph{Inventory.} If the measured shares settle stably away from \(\Omm = 1/\pi\), \(\OmL = 1-1/\pi\), \(\Omb = 1/(2\pi^2)\), then Conjecture~VIII is false and the closed-packet derivation is wrong. That is a single-module correction: the late-inventory interpretation is retired, but the inflation and dark-matter modules do not automatically fall with it.

\emph{Inflation.} A primordial tensor amplitude inconsistent with the derived \(r=0.0043\) --- a detection well above it, or a null pushing \(r\) below \(\sim10^{-3}\) --- kills the derived launch outright. Again a single-stratum correction, though a defining one.

\emph{Kaxion.} Exhaustive nulls across the \(1.59\)--\(2.13\)~GHz corridor retire the compact-angle dark-matter candidate; the dark sector would then have to be sourced elsewhere, damaging the cross-linkage but not the constitutional layer.

\emph{Corridor.} Persistent failure of the shared Hubble/\(S_8\) route to improve the joint fit without new knobs would mean the two-stage corridor is not doing real work. That weakens the late-time program but leaves the early-universe modules intact.

\emph{Black holes.} If the \GDOS{}/\OGGEP{} account cannot be made quantitatively competitive with island computations, the black-hole module remains suggestive rather than decisive --- a conceptual loss, not a falsification of the cosmology.

Each of these alone is a correction the framework can absorb. The genuine danger is \emph{cluster-level} failure: if the inventory, the inflationary tensor null, and the corridor were to fail \emph{together}, that would not be three separate corrections but a refutation of the shared-module thesis itself, and the framework should be retired as a unification claim. That is the correct standard. The framework should lose if the universe disagrees.


\section{The bottom line}
\label{sec:partXII-bottom-line}

The verdict is tiered, and stated plainly.

\paragraph{Strongly supported.} The compression itself --- the reduction of thirteen largely independent mysteries to a short inherited ledger that preserves the \LCDM\ baseline; the lawful finite beginning at \(\EGUT\); and the sharp tensor target \(r = 0.0043\), which the Starobinsky relation forces once the plateau class and the reheating-matched e-fold count are fixed.

\paragraph{Partially or conditionally supported.} The \(\pi\)-locked inventory (Derived-conditional under Conjecture~VIII); the tilt and scale (Derived-conditional on the revised Lemma~IV) with the amplitude \(A_s\) the one declared input; the two-stage Hubble corridor and its \(S_8\) corollary (structural; uniqueness open under Conjecture~\ref{conj:hubble-corridor}); the Kaxion corridor (closed quantitatively in \PaperCOne{}, awaiting the haloscope); and the operational black-hole account (conceptual, awaiting quantitative match).

\paragraph{Not yet tested.} The forward predictions on which the theory most stakes its identity --- the tensor null, the \(1.59\)--\(2.13\)~GHz Kaxion line, and a strictly \(w=-1\) dark energy --- together with the global growth-tension fit that a successor pipeline must deliver.

The forward program follows directly: the near-term priority is to make the precision sector computationally real and to attack the sharpest hinges first, because those are where the minimal branch can be killed most cleanly; the medium and long-term tasks scale the elegant modules to realistic astrophysics and tighten the particle--cosmos closure. The ordering, which reflects where credibility is gained or lost fastest, is collected in Table~\ref{tab:partXII-forward-program}.

\begin{table}[H]
\centering
\caption[A practical forward program suggested by GG-A10-Cosmos]{A practical forward program suggested by \ATen{}. The columns give the horizon, the tasks, and why that horizon matters most. The ordering reflects where the framework is most exposed rather than where progress would be easiest, which is the opposite of how a research plan is usually written and is intentional.}
\label{tab:partXII-forward-program}
\small
\begin{tabular}{@{}p{0.22\textwidth}p{0.34\textwidth}p{0.34\textwidth}@{}}
\toprule
\textbf{Horizon} & \textbf{Primary tasks} & \textbf{Why this horizon matters most}\\
\midrule
Near term & Full precision cosmology implementation; BBN network test; dark-energy and tensor-sector sharpening; Kaxion search targeting & These are the places where the minimal branch can be killed most cleanly\\
Medium term & Nonlinear structure and early-galaxy simulations; local-baryonic environmental tests; black-hole entropy matching in tractable models & These decide whether the framework scales from elegant modules to realistic astrophysical predictions\\
Long term & Tight particle-cosmos closure; extension to chemistry/life/mind floors only after first-floor stability is secured & This is where the larger \GG{} ambition becomes scientifically credible rather than merely suggestive\\
\bottomrule
\end{tabular}
\end{table}

\ATen{} has not eliminated the need for hard future work; it has clarified exactly where that work must occur, reduced the number of independent stories that can be told at once, and made the whole first-floor particle--cosmos enterprise succeed or fail as a single correlated whole. That is what a unification proposal should do, and it is the disciplined foundation on which the concluding Part builds its wider view.



\section{Part~XIII in plain English: what is proved, what is conditional, and what is open}
The honest boundary section is the most important thing in this Part, and
everything else should be read against it.
The framework's cosmological claims fall into three groups: results forced by
the geometry, results forced conditional on a conjecture that is named, and
questions the paper reframes without answering. The three are not
interchangeable, and the Part is built to keep them apart.

The cost of being right, as the Part puts it, is that there is nothing to
adjust. A framework with zero continuous parameters cannot absorb a
disagreement: one wrong number retires it. That is stated as a strength; it is
a liability that becomes a strength only if the numbers hold. The kill criteria and the test ledger are the operational form
of that liability, and they are written to be decidable by measurement rather
than argument: a kill criterion that requires an interpretive judgment is not
one.

The ten structural innovations are the Part's claim to novelty and they are the
softest material in it. An innovation is a way of organizing an account, not a
result, and none of the ten would survive the failure of the theorems they
organize. They are listed because a reader is entitled to know what the paper
thinks is new about it, and listed after the honest boundary rather than
before it for the same reason.

\GGPart[Conclusions]{Conclusions: the summit view}{The composition of the universe is geometry, not coincidence --- one origin for inflation, dark energy, and dark matter}
\label{sec:conclusions}

\section{The summit thesis}
\label{sec:concl-thesis}

For a century, cosmology has been a science of measurement. We weigh the
universe, and we accept what the scale reads. The concordance model \LCDM{} is
the triumph of that method --- six numbers, fit to the sky, that reproduce the
cosmos from the first light to the most distant galaxy. It tells us, with
magnificent precision, \emph{how} the universe expands. It has never told us
\emph{why the universe contains what it does.}

\ATen{} supplies the missing half of the sentence.

\begin{quote}
\emph{The six numbers of the concordance model are not six fortunate accidents
to be measured and accepted. They are one geometric figure --- read off a
single six-dimensional bulk, the very geometry that fixes the electron
one story below. The composition of the universe is not a coincidence. It is
geometry.}
\end{quote}

This is the summit view, and from it the whole landscape rearranges. The dark
sector ceases to be a pair of unexplained substances that happen to dominate
the cosmos; it becomes what the inherited Standard-Model ledger \emph{cannot
see} of a closed, parameter-free inventory. The two warring values of the
Hubble constant cease to be a crisis; they become the two ends of one geometric
corridor. The beginning ceases to be a singular mystery bolted on by hand; it
becomes the high-energy edge of a single radial descent that ends, today, in
the gentle vacuum we inhabit. One geometry, from the first instant to this one.

\section{The composition of the universe is a number}
\label{sec:concl-inventory}

Here is the claim in its barest form. Ask what fraction of the universe is
ordinary matter, what fraction is dark matter, what fraction is the dark energy
driving it apart. \LCDM{} answers by measurement: roughly five percent,
twenty-seven percent, sixty-eight percent --- three numbers handed down from the
data with no account of where they come from. \GGTheory{} answers by geometry:
\[
\Omm = \frac{1}{\pi} \approx 0.318, \qquad
\OmL = 1 - \frac{1}{\pi} \approx 0.682, \qquad
\Omb = \frac{1}{2\pi^{2}} \approx 0.051.
\]
The matter fraction is the reciprocal of \(\pi\). The dark-energy fraction is
what remains of unity once that share is taken. The baryon fraction ---
everything that has ever been a star, a planet, or a person --- is one part in
\(2\pi^{2}\). These are not fits. They are the closed inventory of a compact
geometry, and they fall on the measured sky.

And now the deepest reading, the one that turns a numerical curiosity into a
physical revelation. \emph{Why is the universe ninety-five percent dark?}
Because the inventory is closed and \(\pi\)-locked, and the Standard-Model gauge
ledger inherited from \ANine{} can illuminate only the \(1/2\pi^{2}\) of it that
carries ordinary charge. The darkness of the cosmos is not the presence of two
exotic strangers. It is the shape of what light was built to miss. Dark matter
and dark energy are dark for the same reason a clean window is invisible: not
because they are absent, but because the instrument sent to find them was made
to look straight through them.

This is the cosmological face of the principle that runs through the entire
cohort --- that the visible world is the boundary shadow of a fuller bulk. In
\ANine{} it explained why every electron in the universe is identical. Here it
explains why the universe is mostly dark. The same geometry, one story up.

\section{One origin, from the first instant to today}
\label{sec:concl-three-origins}

The power of a geometric account is that it does not stop at the ledger. It runs
forward in time, and it offers, for the first time, a common origin for the
three great unexplained chapters of cosmic history.

\paragraph{The beginning --- inflation as a geometric launch.}
The oldest puzzle in cosmology is why the early universe was so smooth and so
flat. The standard answer, inflation, is a brilliant mechanism in search of a
cause: a scalar field, introduced for the purpose, rolling down a potential
chosen to fit. \GGTheory{} offers the cause. The universe begins at the
ultraviolet edge of the throat, near \(\EGUT\), and descends. The
inflationary epoch is not a bolted-on prologue but the first stretch of that
descent --- a geometric launch on the derived plateau just below the GUT edge,
smooth and graceful because the geometry, not a tuned potential, lays the road;
the TeV lock \(\Mlock \approx 3.85~\mathrm{TeV}\) is crossed lower down, where
the dark sector is born. For the first time the inflaton is not a
guest invented for the occasion. It is the radial coordinate of the bulk.

\paragraph{Today --- dark energy as the residual vacuum.}
The universe is ending as it began, in acceleration. The standard account writes
down a cosmological constant and measures its value; the failure to explain that
value is the deepest embarrassment in physics. \GGTheory{} reads the late-time
vacuum not as a constant to be explained away but as the residual \(1 - 1/\pi\)
of the same closed inventory, with an equation of state \(w = -1\) that is exact
because the geometric transition is complete. The acceleration of the cosmos is
the geometry settling into its ground state. Dark energy is simply what a
finished universe weighs.

\paragraph{In between --- dark matter with a mass and an address.}
Between the launch and the rest lies the matter that built the galaxies, and
here the geometric account is sharpest of all. \GGTheory{} predicts that dark
matter is one specific particle --- the Kaxion, a faint and extremely light particle
that lives in the curled-up compact dimension of the bulk --- with one
specific mass:
\[
m_{\mathrm{Kaxion}} = 8.1 \pm 0.5~\mu\mathrm{eV}, \qquad
\nu_K \in [1.59,\,2.13]~\mathrm{GHz}.
\]
This is not a fit to the dark-matter density; it is the inverse. The geometry
fixes the field's decay constant, and the measured cosmic abundance then fixes
the one angle that remains. For the first time in the history of the subject,
dark matter has a geometric origin, a number, and a laboratory window narrow
enough to be opened --- or shut --- within the decade. The full derivation is
carried by \PaperCOne{}.

\medskip
Three chapters --- the smooth beginning, the matter that clumps, the
vacuum that drives apart --- have stood for decades as three separate mysteries,
each with its own remedy invented by hand. \ATen{} reads them as three movements
of one geometry: the descent, the relic, and the residue. That is what it means
to give cosmology an origin instead of a fit.

\section{One universe, one story above the Standard Model}
\label{sec:concl-one-storey}

The unification is the point that must not be lost among the numbers. The same
six-dimensional bulk that, in \ANine{}, fixes the masses and charges of
the particles also fixes the contents of the cosmos. The electron and the
universe are read from one ledger at two scales. \ATen{} sits one story above
\ANine{} in a single building --- not in a separate annex of cosmology bolted to
a separate physics of particles.

And the building has a direction. The arrow of cosmic time --- the reason the
universe runs from a hot, dense, symmetric beginning to a cold, sparse,
structured present --- is the open-system descent at the heart of the cohort's
dynamics, the \GRIP{} cascade (the dynamical face of the one constitutional arrow that \AX{3} owns): the universe relaxing toward the only state it
can rest in, the de Sitter vacuum and the frozen Kaxion relic that are its
final, stable furniture. Cosmic history is not a string of accidents laid along
a timeline. It is a single object falling, with dignity, toward its ground
state. The universe has a plot, and the plot is geometric.

\section{The fourth revolution, written across the sky}
\label{sec:concl-revolution}

Stand back far enough and the lineage is plain. Euclid gave physics the ideal of
deriving the world from a handful of self-evident axioms, with nothing left over
to choose. Newton made the heavens and the earth obey one law. Maxwell,
Einstein, and the founders of quantum theory unified the forces and the
constituents. Each revolution removed a coincidence and put a necessity in its
place.

The dream went quiet in cosmology, where the deepest numbers --- the density of
matter, the strength of the vacuum, the smoothness of the start --- stayed
things we measured rather than things we understood. \ATen{} carries the
Euclidean ideal back into the sky: a cosmos with \emph{zero free continuous
parameters}, its inventory fixed by \(\pi\) and a short ledger of integers, its
history fixed by the descent of one geometry. Twenty-three centuries after
Euclid, the composition of the universe rejoins the list of things that did not
have to be measured to be known. That is the fourth revolution --- and the sky
is where it becomes visible to the naked eye.

\section{A universe made answerable}
\label{sec:concl-answerable}

A vision this large deserves no quarter unless it can be killed, and the
strength of a geometric cosmos is exactly that it hands the sky a set of sharp,
unforgiving questions. The corridor switch sits at
\(z_{\mathrm{sw}} = e^{7} \approx 1097\), and the recombination surface,
derived from the framework's own constants, lands \(0.4\%\) below it on the
measured value. The primordial tensor sector should ring at
exactly \(r = 0.0043\) --- loud enough for the coming instruments, far below the
present bound. The Kaxion should ring in a narrow band near two gigahertz.
The two Hubble constants should prove to be one corridor, not two errors. Each
of these is a place where the universe is free to answer no. And within the
coming decade --- 21-centimeter cosmology, the next generation of
microwave-background experiments, the haloscopes now scanning past two
gigahertz, the galaxy surveys charting the expansion --- it will have its
chance.

This is the only confidence a geometric theory is entitled to: not the
confidence that it cannot be wrong, but the confidence to tell the world exactly
how it could be. A framework that fixes the cosmos from \(\pi\) and a handful of
integers has nowhere to hide, and asks for nowhere. Nor does it claim more than it computes: the geometry fixes the cosmic \emph{inventory} and the \emph{chronology}, while the precision microwave-background \(C_\ell\) spectrum and the nonlinear growth of structure remain the work of the Einstein--Boltzmann and \(N\)-body machinery it frames but does not replace. We have made the universe
answerable. The full slate of decisive tests is laid out in
Figure~\ref{fig:killtest-timeline}.

\section{One staircase, from the Big Bang to living molecules}
\label{sec:concl-staircase}

Everything in this paper, and everything in the cohort beneath it, runs on one
short chain of cause and effect. It is worth saying once in plain words, with no
symbols in the way.

The universe never shows us its full self. What we see is the surface of a
deeper geometry --- the shadow that a closed, six-dimensional space casts
on the four-dimensional boundary we live on. That projection is the third axiom
of the \GGConstitution{}, \OGGEP{}. The boundary is not frozen: it evolves by a
definite law, \GDOS{}. And a definite process acts on it, \GRIP{}. Faced with
the open choices a boundary always allows, \GRIP{} tries them, charges each one
an energy cost, breaks the tie, and relaxes into the single configuration that
costs the least. The thing it settles into --- the value we then measure --- is
the Ground Geometric Attractor, \GGA{}. Every number in the codebook of this
paper is one such attractor. In a single line:

\begin{equation*}
\boxed{\;
\begin{array}{c}
\OGGEP \;\longrightarrow\; \GDOS \;\longrightarrow\; \GRIP \;\longrightarrow\; \GGA \\[5pt]
\Longrightarrow\;\; \text{atoms} \to \text{galaxies} \to \text{stars} \to \text{heavy elements} \to \text{life's molecules} \to \text{us}
\end{array}
\;}
\end{equation*}

Read it from left to right and it is the whole of cosmic history.

It begins at the hot edge, near \(\EGUT\) --- the geometric Big Bang. From that
first instant the entire staircase is already set. Not the events one by one,
but the rules and the numbers that make every later step come out as it does:
nothing is added later by hand. Everything that follows is geometry spending the
budget it was handed at the start.

Run the chain on the particle stratum --- the work of \ANine{} --- and out come
the stable electron, the stable proton, the stable atom. Each is a \GGA{}: the
one configuration the geometry permits, identical everywhere in the universe,
and stable not by luck but because there is nothing it can decay into. Stable
atoms are the first precondition for all the rest. Without them there is no
chemistry, no planets, and no chemists.

Run the same chain on the cosmological stratum --- the work of this paper --- and
out comes the rest of the inventory. Kaxion dark matter appears at
\(8.1~\mu\mathrm{eV}\), the unseen scaffolding on which everything large is later
built. Dark energy appears as the leftover \(1 - 1/\pi\) of the closed budget,
pacing the expansion so that there is time to assemble galaxies before the long
cold sets in.

Now gravity does the building. It gathers the \(\pi\)-locked matter into halos
and galaxies; inside them gas falls together and stars ignite. Stars are the
universe's foundries. While they live they fuse hydrogen into carbon, oxygen,
and nitrogen; when they die they burst as supernovae and scatter those heavy
elements across space. From that enriched dust, ordinary chemistry assembles the
organic molecules --- the carbon chains and rings that life is written in. And
on at least one cold rock orbiting one ordinary star, those molecules found
their way into systems that are now reading this sentence.

Each rung is forced by the rung beneath it, and the whole flight was set in
outline at \(\EGUT\). That single descending staircase --- bulk to boundary, to
atoms, to galaxies, to stars, to molecules, to minds --- is the picture of the
world that \GGTheory{} offers. The cohort's capstone, \AEleven{}, follows it all
the way to life and mind and proves the staircase as a theorem; here, at the
cosmological floor, we have laid its first and broadest steps.

\section{The trio: from the electron, through the cosmos, to the question of us}
\label{sec:concl-trio}

The staircase just walked is not this paper's private image; it is the architecture of three papers written as one argument, and the reader --- especially the astronomer or cosmologist who arrived here for the cosmology --- deserves the map.

\ANine{} \citep{ArisakaA9SM} is the first movement: the particle floor. It closes the twenty-eight-parameter sheet of the Standard Model and then asks the question textbooks never ask --- \emph{why is every electron the same?} --- and answers it with a uniqueness theorem: every electron is the readout of the one ground attractor of its stratum. The present paper, \ATen{}, is the second movement: the same engine, one story up, where the attractor being read is the vacuum of the universe and the readout is the \(\pi\)-locked inventory --- \emph{why does the universe contain what it does?} And \AEleven{} \citep{ArisakaA11Origin} is the third movement and the capstone: the same cascade followed past the galaxies, through chemistry, to the open systems that metabolize, remember, and eventually look up --- \emph{why is there anyone here to ask?} Three questions, one geometric answer, applied at three strata: that is the trio, and it is the program's deliberate spine.

\AEleven{} names what the third movement requires, and it is worth previewing here in its own vocabulary, because the cosmological reader will recognize every ingredient from this paper. The \emph{recursive uniqueness theorem}: the same \GRIP{} selection that fixed the electron (\ANine{}) and the inventory (this paper) runs layer by layer --- eighteen layers, from the gauge structure to the act of articulation --- each layer's Ground Geometric Attractor becoming the substrate of the next. The \emph{geometric inevitability of Cosmos, Life, and Mind}: Cosmos is the present paper; Life is the \SeOS{} crossing, where the same open-system ledger that ran the cosmic descent begins to run metabolism; Mind is the closed loop of world and world-model. And the \emph{self-referential closure}, the capstone's central theorem: the geometry that forced the cascade contains, among its forced readouts, the observers who write the geometry down. \AEleven{} also places the whole program against the lineage of axiomatic revolutions --- Euclid, Newton, the field-on-manifold revolution of Maxwell--Einstein--quantum mechanics, and the openness revolution of \AFive{} --- and states, with the verdict left to history, what kind of fifth step the recursive cascade would be. The ``fourth revolution written across the sky'' of this Part is that same claim, read at the cosmological floor.

A word of orientation for the reader who wants more. Everything this paper inherited has a home: the constitution in \AOne{}, the six-dimensional bulk and its integer ledger in \ATwo{}--\AFour{}, the open-system boundary law in \AFive{}, the runtime in \ASix{}--\ASeven{}, and the particle floor in \ANine{} --- the upstream staircase is climbable one paper at a time, and Part~II of this paper is its one-Part summary. But if the reader takes only one further step, the author's counsel is to take it \emph{downstream}: \AEleven{} is where the question this Part has been circling --- not what the universe is, but why there is someone in it to ask --- is given its full, theorem-disciplined treatment. This paper derived the stage and the furniture. The capstone is about the audience.


\section{Why there is someone here to ask}
\label{sec:concl-human}
\label{sec:partVIII-why-here}

There is one question a cosmology this ambitious cannot duck, and one a paper
confident enough to have climbed this far need not flinch from. Why is there
anyone here at all to read the inventory of the universe?

For decades the unsettling observation has been that the universe looks
improbably well-suited to life. Make the electron a little heavier, the nuclear
force a little weaker, or the energy of empty space a little larger, and there
are no stable atoms, no stars that burn for billions of years, no carbon --- and
so no us. The constants look tuned, as though some careful hand had set each dial
to exactly the value life requires. Physics has long offered two ways out: luck
(we happen to live in the one universe that works) or a multiverse (every setting
exists somewhere, and naturally we find ourselves in a habitable one).

\GGTheory{} takes a third road, and once the staircase of the previous section is
in view it is the natural one. The life-permitting constants are neither lucky
nor selected from a bin of alternatives. They are \(\pi\) and a short ledger of
integers, fixed by the geometry at \(\EGUT\), with no dial to turn. There was
never a tuning to explain, because there was never a knob. In exactly this sense
the universe is extraordinarily well designed --- not by a designer, but by
geometry. It is built to the one specification its own shape allows, and that
specification happens to be the one in which electrons hold atoms together, stars
burn long and steady, and carbon is everywhere. A universe whose contents are
forced is a universe whose fitness for life is forced along with them.

And the geometry does not stop at the galaxies. The same descent that fixed the
weight of empty space runs on --- through the chemistry that stable atoms make
possible, through the structures gravity assembles, toward the open,
self-sustaining systems in which matter begins to model the world around it. This
projection had been running for nearly fourteen billion years before the first
observer existed. It needs no observer and waited for none; we did not switch it
on by looking. Observers are not the reason the universe has its numbers. They
are a late step on the staircase those numbers built.

So the honest answer to the oldest question is neither mysticism nor luck. It is
continuity. A universe whose contents are geometry is not a sterile machine that
happens, against staggering odds, to contain us. It is a single unfolding in
which we are a late movement of the very descent that set the value of \(\OmL\).
Every civilization has asked this question --- the Upanishads answered that the
cosmic self sought to know itself; Genesis, that a creator chose; the
Enlightenment set it aside so that physics could proceed; Darwin explained how
life diversifies but not why anything exists to evolve. \AEleven{} opens on
exactly this history and proposes that the question now has a scientific
answer. The full case --- that life and mind are not fortunate accidents but
forced rungs of the same cascade, closed by the theorem that the geometry
contains its own articulators --- belongs to that capstone; here we claim only
its first and firmest link, and we claim it plainly. We are not an accident
decorating a cosmos we cannot explain. We are the part of the geometry that has
begun to look back.

\section{A personal word from the cosmological floor}
\label{sec:concl-personal-cosmos}

The author may be permitted one paragraph in the first person, matching the personal accounts that close \ANine{} and \AEleven{}. I spent fifty years as an experimental particle physicist --- Kamiokande, kaon and \(B\)-meson \(CP\) violation, CMS at the LHC, Auger, XENON --- watching the Standard Model refuse to break, and watching the deepest discoveries of the era arrive instead from the sky: neutrino oscillations, the precision microwave background, the accelerating universe, gravitational waves. Four revolutions, four Nobel prizes, and not one of them predicted by the unification programs of my generation. It is therefore no accident that the paper you are finishing stakes its life on the sky. The instruments that will judge it --- the CMB polarimeters, the haloscopes, the galaxy surveys, the gravitational-wave detectors --- are the direct descendants of the ones that taught my generation cosmology; and the numbers it stakes --- a tensor ratio, a dark-matter frequency, an equation of state, a siren Hubble constant --- are the kind an experimentalist can love: sharp enough to be wrong. If the universe says no, it will have said no to a theory that asked properly, and the audit trail in these pages will show exactly which link broke. If it says yes, then the composition of the universe was never a set of accidents to be measured --- it was a theorem waiting, for one century of magnificent measurement, to be read. Either answer honors the sky that taught us; and either way, the next generation of cosmologists --- the readers this paper was written for --- will be the ones who hear it first.


What changed, in the end, was not the data but the reading of it. For decades the cosmic numbers were treated as inputs to a closed four-dimensional theory; the recognition on which this paper rests --- its central claim (\S\ref{sec:central-claim}, Figure~\ref{fig:central-claim}) --- is that no universe is closed, that the four-dimensional cosmos we measure is the open-system projection of a six-dimensional one, and that this projection is what observation is. Once that is seen, the matter fraction \(1/\pi\), the dark-energy residue \(1-1/\pi\), and the corridor-switch redshift \(e^{7}\) are not free numbers but registered observables of a determined map --- and the deepest result is not any single closed form but the shift in what a cosmological parameter is: from an input one is obliged to measure to an output the geometry is obliged to return.

That is the view from the summit. For a century, cosmology measured the universe
and told us what it is. \GGTheory{} proposes, for the first time, to tell us
\emph{why} --- and to stake that proposal, number by number, on a sky that is
free to refute it. We do not expect it to.

\medskip\noindent The climb ends here, and it ends where it began: with two questions that turned out to be one. Why these particles? Why this universe? On the reading defended across these thirteen Parts, both are answered by a single geometry that had no dials to turn --- and the observers who ask the questions are late-arriving citizens of the same structure, reading its registered observables from the inside. If the argument holds, the cosmic inventory is not an accident we happen to inhabit but a theorem we happen to witness; and if it does not hold, the sky will say so, on a schedule this book has printed.

The summit is not the end of the volume. The reader who wants the craft behind the claims will find in Part~XV a self-contained graduate course in the standard cosmology this paper set out to derive; the reader who wants the fight will find in Part~XVI the ninety-three hardest objections we could assemble, answered without retreat into vagueness; and the references map the twenty-two-paper cohort in which this argument lives. However far you have come: thank you for climbing.

\section{Two notes on scope and reproducibility}
\label{sec:scope-and-repro}

\paragraph{On the reach beyond cosmology.} A reader --- or a referee --- may
observe that a cosmology paper has, in its closing Parts and its
curious-reader questions, reached through chemistry and life to minds and to
a self-referential closure, while candidly \emph{not} computing a full
\(C_\ell\) likelihood. The tension is noted, and the choice is deliberate,
not an oversight. \ATen{} is the middle volume of a trilogy --- \ANine{}
(particles), \ATen{} (cosmos), \AEleven{} (life and mind) --- written for
the arXiv and homepage readership the program addresses, and the
Cosmos\(\to\)Life\(\to\)Mind arc is the through-line that makes the trilogy
one argument rather than three. It is carried here as a \emph{pointer}
forward to \AEleven{}, where it receives its theorem-disciplined treatment,
not as a claim this paper proves. Crucially, the paper's falsifiable core ---
the tensor target \(r = 0.0043\), the Kaxion line, \(w = -1\), the siren
\(\Hzero \approx 67\), and the \(\pi\)-locked inventory --- stands entirely
on its own: every kill-test of Part~XIII is stated and decidable without a
single sentence of the philosophical arc. A reader who wishes to judge \ATen{}
as a bare cosmology paper may stop at the end of Part~XIII and lose no
testable content; a reader who wishes to see why the program thinks the
cosmos and the observer are one descent is invited onward. We keep both
readers, and we mark the seam between them plainly here.

\paragraph{On reproducibility.} The empirical spine of this paper --- the
CLASS demonstration of the microwave spectrum from the derived parameters
(Section~\ref{sec:cmb-overdetermined}), the parameter-audit deviation plot and
its data table, and the \(\chi^2\) computation against the binned Planck TT
data --- rests on scripts and data files that will be released as a public,
runnable reproducibility repository accompanying the arXiv posting: the CLASS
input decks and derived-parameter settings, the binned Planck TT data with
sources, the \(\chi^2\) and \(\chi^2/\mathrm{dof}\) routines, and the
dev-plot generator and its tabulated inputs. A cosmological claim that hands
its data confrontation to an unreleased script asks for a trust the field no
longer extends; we intend to earn it the ordinary way, by letting the reader
run the pipeline. The repository DOI will be listed here on posting.


\section{Part~XIV in plain English: what the summit view claims, and what it does not}
The thesis of this Part is that the composition of the universe is a number in
the same sense that the charge of the electron is a number: not a quantity the
universe happened to draw, but a consequence of a geometry that admits no
alternative. If that is right, then the two oldest questions --- why these
particles, and why this universe --- are one question asked at two floors of
the same structure.

Three things follow that are weaker than the thesis and are easily read as part
of it. The first is the staircase from the Big Bang to living molecules: it is
a way of arranging results already established elsewhere in the cohort, and
arranging results is not deriving them. The second is the answer to why there
is someone here to ask, which is a consequence of the uniqueness claim and
inherits every one of its conditions --- including the conjecture that
uniqueness itself rests on. The third is the fourth-revolution framing, which
is a claim about the history of ideas and not about physics, and which the
next generation is better placed to judge than the author.

What the Part does not claim is worth stating in the author's own words rather
than leaving to inference: not that the cosmos was designed, not that its
history was fixed in every detail, and not that the argument is finished. The
layered structure is forced; history within the layers is not. Whether the
reading is correct will be settled by the kill criteria of Part~XIII and by
the named conjectures the codebook lists, and by nothing in this Part at
all.

\GGAcknowledgments


\GGAppendixPart{Technical Appendices}{A graduate-level course in cosmology read through \GGTheory{} --- full from-scratch derivations of the Friedmann background, thermal history, inflation, nucleosynthesis, the growth of structure, the cosmic microwave background, the \(\pi\)-locked inventory, Kaxion dark matter, and black-hole entropy --- preceded by the reference dictionaries and followed by the source ledger, benchmark audit, and registries}

The body of this paper has emphasized the geometric theorem chain and kept the
heaviest arithmetic out of the reader's way. The present appendices supply that
detailed technical content in full, and taken together they are intended to be
read as \emph{a graduate-level course in cosmology seen through \GGTheory{}},
addressed to an incoming graduate student in physics, astronomy, or cosmology
who will carry the program forward. We organize them into three groups. The
\emph{reference} appendices (A--D) tabulate the abbreviations, the key symbols,
the compact codebook, and the cross-cohort naming convention, and serve as the
reader's quick lookup. The \emph{constructive} appendices (E--N) are the course
proper: a self-contained sequence that takes up each sector of modern cosmology
in turn --- the Friedmann background; expansion, scaling laws, and the thermal
history; inflation; nucleosynthesis and the lithium audit; leptogenesis; the
growth of structure; the microwave background; the \(\pi\)-locked inventory;
Kaxion dark matter; and black-hole entropy --- and treats each one in
\emph{three steps}. First comes a full from-scratch derivation of the standard
result, in the form a graduate student would meet it in Dodelson or Weinberg,
carried through with no skipped manipulations and closed by a worked numerical
example. Then comes a frank statement of what that standard derivation
\emph{assumes but cannot explain} --- the fitted parameters it must take from
the sky and the structural questions it leaves open: what the inflaton is, why
the misalignment angle takes one value, why the vacuum weighs what it does.
Finally comes the geometric derivation, in which the same quantity descends as
a boundary readout or a theorem of \GGTheory{}, fixed by the upstream theorem
deck rather than fitted --- and it is in these third steps that the paper's central derived
results enter: the derived plateau and the computed
tilt, the tensor target \(r = 0.0043\), the reheating temperature, the pinned
misalignment angle, the ground-state reading of dark energy. The \emph{ledger
and registry} appendices (O--T) that close the Part record the internal source
ledger, the cross-paper inheritance table, the consolidated benchmark audit,
the named-theorem-and-conjecture deck, the lemma-promotion roadmap, and the
falsification catalog.

Throughout these appendices we adhere to two stylistic commitments. First, every
derivation is presented step by step with no skipped intermediate manipulations:
a graduate student in physics should be able to reproduce each calculation with
paper and pencil. Second, every formal step is accompanied by a parenthetical
\emph{physics intuition} note that translates the algebra into a mental picture.
This combination is designed to make the appendix material accessible to the
working physicist while preserving the mathematical rigor of the main text.
Each course appendix opens with a boxed \emph{What this appendix teaches}
abstract and closes with a boxed \emph{take-away} --- the standard result, what
it assumed, and what the geometry replaced the assumption with --- so that the
Part can be used as a one-semester reading course, one appendix per week, each
self-contained.  And every course appendix closes with \emph{problems and
complete worked solutions} --- placed before the take-away box, numbered by
chapter and subsection, each solution worked in explicit Steps and ending with
its physics intuition, the sets cross-referencing one another into a single
course network.  They carry full solutions because, in the cohort's phrase,
they are meant to be done, not admired: the reader who works them owns the
derivations, not merely the conclusions.

\paragraph{Prerequisites and recommended textbooks.}
The course assumes advanced undergraduate mechanics, electrodynamics, and
statistical physics, plus a first exposure to general relativity (the FRW
metric is derived from scratch in Appendix~E, so a student meeting cosmology
for the first time can begin there). It is meant to be read \emph{alongside} a
canonical cosmology textbook, not in place of one: the recommended way to use
each appendix is to read the corresponding textbook chapter first, return to
Step~1 here for a compact but complete recap, and only then proceed to Steps~2
and~3, where the standard result is re-examined and re-derived as a theorem of
the geometry. For self-study we particularly recommend the following, each of
which is cited again in the appendix to which it is best matched:
\begin{itemize}
\item \textbf{Primary references.} \citet{Dodelson2003}, \emph{Modern Cosmology} (and its second edition with F.~Schmidt), and \citet{Baumann2022}, \emph{Cosmology} --- the two standard modern graduate texts, covering the background, the thermal history, inflation, perturbations, and the CMB in the notation this course follows most closely.
\item \textbf{Gentler first pass.} \citet{Ryden2017}, \emph{Introduction to Cosmology}, and \citet{Liddle2015}, \emph{An Introduction to Modern Cosmology} --- accessible and physically transparent, ideal for a first encounter with the expanding universe and the inventory.
\item \textbf{In-depth study.} \citet{Weinberg2008}, \emph{Cosmology}; \citet{Mukhanov2005}, \emph{Physical Foundations of Cosmology}; \citet{Peacock1999}, \emph{Cosmological Physics}; and \citet{LiddleLyth2000}, \emph{Cosmological Inflation and Large-Scale Structure} --- for the relativistic background, the quantum theory of perturbations, and the inflationary sector at full depth.
\item \textbf{Classic and specialist companions.} \citet{KolbTurner1990}, \emph{The Early Universe} --- the classic source for the thermal history, freeze-out, nucleosynthesis, and relic abundances used in Appendices~F, H, and~M; \citet{Peebles1993}, \emph{Principles of Physical Cosmology} --- the deep classic on structure formation, for Appendix~J; and \citet{Durrer2020}, \emph{The Cosmic Microwave Background} --- the dedicated CMB monograph, for Appendix~K.
\end{itemize}
\noindent Throughout, the symbols \AOne{}--\AEleven{}, \PaperPOne{}--\PaperPFive{},
and \PaperCOne{}--\PaperCThree{} refer to the twenty-two papers of the \GGTheory{}
cohort, to which the geometric steps repeatedly point.

\newcommand{\GGteachbox}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{What this appendix teaches.} #1}}\end{center}}
\newcommand{\GGtakeaway}[1]{\begin{center}\fbox{\parbox{0.92\textwidth}{\small\textbf{Take-away.} #1}}\end{center}}
\newcommand{\GGreadalong}[1]{\par\noindent\emph{Read alongside:} #1\par}


\appendix

%% ===== Block I: reference dictionaries (A9-model: abbreviations + key symbols) =====
\section{Summary of Abbreviations}
\label{app:abbrev}

This appendix collects every abbreviation used in \ATen{}. All program-defined
names are rendered in \emph{sans-serif type} (\GGTheory{}, \GGConstitution{},
\GG{}, \GDOS{}, \BIsix{}, \OGNfive{}, \GiFour{}, \QiFour{}, \GRIP{}, etc.) so the
reader can see at a glance which terms belong to the \GGTheory{} program proper.
Standard physics abbreviations imported from the wider literature
(\(\Lambda\)CDM, CMB, BBN, BAO, KK, RS, etc.) are rendered in plain uppercase
Roman, following the convention of the major physics journals. Mathematical
objects are in roman or italic in the usual way. The third column gives the
\emph{primary link}: the paper in the Arisaka-GG series (and the section, where
useful) in which each program-defined name is first introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.18\linewidth} >{\raggedright\arraybackslash}p{0.54\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper]{Summary of abbreviations used in the present paper. Program-defined names are rendered in sans serif; standard physics abbreviations in plain uppercase Roman; mathematical objects in roman/italic in the usual way. The third column records the primary link: the paper in the Arisaka-GG series, or the section of this one, where each name is introduced in detail. An abbreviation with no primary link beside it is standard physics usage and is listed only so the table can be read without a second reference open.}
\label{tab:appA-abbrev-cosmos}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev-cosmos} continued}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link}\\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape (1) Framework name and core program objects}\\
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry--Gauge Equivalence Theory) & \AOne{}, Preface\\
\GGConstitution{} & Constitutional charter: 4 axioms (\AX{1}--\AX{4}) + 7 postulates (\POS{1}--\POS{7}) & \AOne{}, \S 2--\S 3\\
\GG{} & Geometry--Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 8\\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law from \OGGEP{} & \AFive{}\\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D, $dH_3 = X_4$ & \AThree{}\\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ & \AThree{}, \S 7; \ATwo{}\\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{}\\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{}\\
\midrule

\multicolumn{3}{l}{\itshape (2) Four axioms (\AX{1}--\AX{4})}\\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S 1.5\\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S 2.2\\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}, \S 2.3\\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle & \AOne{}, \S 2.4\\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}, \S 2.5\\
\midrule

\multicolumn{3}{l}{\itshape (3) Seven postulates (\POS{1}--\POS{7})}\\
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}, \S 1.5\\
\GI{} & \POS{1} Gauge invariance across scales & \AOne{}, \S 3.3\\
\UEP{} & \POS{2} Unified Extremal Principle & \AOne{}, \S 3.4\\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & \AOne{}, \S 3.5\\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}, \S 3.6\\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}, \S 3.7\\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization) & \AOne{}, \S 3.8\\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}, \S 3.9\\
\midrule

\multicolumn{3}{l}{\itshape (4) \GiFour{} pillars}\\
\midrule
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{}\\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{}\\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}) & \ASix{}\\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{}\\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & \ASix{}\\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{}\\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{}\\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{}\\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{}\\
\GGA{} & Ground Geometric Attractor: the unique global minimum among a stratum's \GA{}'s --- identity and stability as ground-state-ness & \ASix{}; \AEleven{}; Part~X\\
\midrule

\multicolumn{3}{l}{\itshape (5) \QiFour{} pillars and the Arisaka equation}\\
\midrule
\QiQUA{} & Quantum information for Quadruple Gauge Locking --- Pillar I of \QiFour{} & \ASeven{}\\
\QiEP{} & Quantum information for the Unified Extremal Principle --- Pillar II & \ASeven{}\\
\QiCON{} & Quantum information for Conformal Invariance --- Pillar III & \ASeven{}\\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice --- Pillar IV & \ASeven{}\\
\OH{} & Outer Hologram (boundary projection of bulk data; outward face of \EIC{}) & \ASeven{}\\
\IH{} & Inner Hologram (inner self-modeling layer; inward face of \EIC{}) & \ASeven{}\\
\EIC{} & Evolving Inner Cosmos ($\OH$+$\IH$ composite) & \ASeven{}\\
$Q_B$ & Arisaka constant $= (2/5)\,E_{\rm Ry}/K \approx 0.1555$~eV per qubit & \ASeven{}\\
\midrule

\multicolumn{3}{l}{\itshape (6) \QiHAL{} sub-acronyms}\\
\midrule
\HAL{} & Holographic Attractor Lattice & \ASeven{}\\
\BFFT{} & Boundary-Front Field Theory of \QiHAL{} & \ASeven{}\\
\midrule

\multicolumn{3}{l}{\itshape (7) \GDOS{}-internal acronyms (program-defined)}\\
\midrule
\SIP{} & System Instance Packet (canonical \GDOS{} input packet) & \ASeven{}\\
\SeOS{} & Selfish Open System (the core open system of \GDOS{}) & \AFive{}\\
\midrule

\multicolumn{3}{l}{\itshape (8) Program-defined mathematical symbols (audit machinery)}\\
\midrule
$\mathcal R$ & Candidate rational expression in the \MSMF{}/\OGN{} audit & \S\ref{sec:codebook-msmf-ogn}\\
$\mathcal C_\Sigma$ & \OGN{}-admissible candidate space on stratum $\Sigma$ & \S\ref{sec:codebook-msmf-ogn}\\
$\mathcal S_\Sigma$ & Structural base set of atoms on stratum $\Sigma$ & \S\ref{sec:codebook-msmf-ogn}\\
$\mathrm{Cost}(\mathcal R)$ & \MSMF{} cost function: total absolute exponent budget & \S\ref{sec:codebook-msmf-ogn}\\
$\epsilon_X$ & Stratum-specific empirical-precision target for parameter $X$ & \S\ref{sec:codebook-msmf-ogn}\\
\midrule

\multicolumn{3}{l}{\itshape (9) Standard physics abbreviations (plain Roman, journal convention)}\\
\midrule
\multicolumn{3}{l}{\itshape \quad Cosmology and the concordance model}\\
$\Lambda$CDM & Lambda Cold Dark Matter concordance model & textbook\\
CMB & Cosmic Microwave Background & textbook\\
BBN & Big Bang Nucleosynthesis & textbook\\
BAO & Baryon Acoustic Oscillations & textbook\\
EoS & Equation of state ($w = p/\rho$) & textbook\\
LSS & Large-Scale Structure & textbook\\
SNe~Ia & Type~Ia supernovae (distance-ladder standard candles) & \citet{Riess1998,Perlmutter1999}\\
EDE & Early Dark Energy & textbook\\
PBH & Primordial Black Hole & textbook\\
GW & Gravitational Wave & textbook\\
JWST & James Webb Space Telescope & textbook\\
\midrule
\multicolumn{3}{l}{\itshape \quad Gravity, fields, and higher-dimensional constructions}\\
GR & General Relativity & textbook\\
QFT & Quantum Field Theory & textbook\\
QM & Quantum Mechanics & textbook\\
SM & Standard Model & textbook\\
GUT & Grand Unified Theory & textbook\\
KK & Kaluza--Klein decomposition & textbook\\
RS & Randall--Sundrum warped 5D model & textbook\\
AdS/CFT & Anti-de Sitter / Conformal Field Theory holography & textbook\\
\midrule
\multicolumn{3}{l}{\itshape \quad Open-system formalism}\\
CPTP & Completely Positive Trace Preserving & textbook\\
RG & Renormalization Group & textbook\\

\end{longtable}
}

\bigskip


\bigskip
\textbf{Hierarchy of program-defined acronyms.} For reference, Figure~\ref{fig:abbrev-flow} records the six-tier hierarchy of the program-defined acronyms used across the \GGTheory{} cohort: the framework name and its constitutional charter; the four axioms and seven postulates; the bulk objects (\GG{}, \BIsix{}, \OGNfive{}); the boundary law \GDOS{} and its input packet \SIP{}; the two four-pillar structures \GiFour{} and \QiFour{} with their sub-acronyms; and the energy--information Arisaka constant $\QB$ with the Arisaka equation that locks the four boundary currents. The diagram is shared verbatim with \AOne{} (its Figure~9), so that the acronym ledger is identical across the cohort.

\begin{figure}[H]
\centering
\resizebox{0.816\textwidth}{!}{%
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% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

% --- Tier 2: 4 axioms (left) and 7 postulates (right) -----
\node[axhead] (AXH) at (-52mm, -36mm) {\textbf{4 Axioms} (\textbf{AX-1}\,--\,\textbf{AX-4})};
\node[ax] (AX1) at (-52mm, -48mm) {\textbf{AX-1} \, \textbf{LC} \quad Locality and Causality};
\node[ax] (AX2) at (-52mm, -58mm) {\textbf{AX-2} \, \textbf{GGEP} \quad Geometry-Gauge Equivalence};
\node[ax] (AX3) at (-52mm, -68mm) {\textbf{AX-3} \, \textbf{OGGEP} \quad Operational/Open GGEP};
\node[ax] (AX4) at (-52mm, -78mm) {\textbf{AX-4} \, \textbf{MSMF} \quad Max.\ Symmetry, Min.\ Freedom};

\node[posthead] (POSH) at (52mm, -36mm) {\textbf{7 Postulates} (\textbf{POS-1}\,--\,\textbf{POS-7})};
\node[ax] (P1) at (52mm, -48mm) {\textbf{POS-1} \, \textbf{GI} \;\; (\,\textbf{LGI} \,/\, \textbf{DGI} \,/\, \textbf{GGI}\,)};
\node[ax] (P2) at (52mm, -56mm) {\textbf{POS-2} \, \textbf{UEP} \quad Unified Extremal Principle};
\node[ax] (P3) at (52mm, -64mm) {\textbf{POS-3} \, \textbf{AF} \quad Anomaly Freedom};
\node[ax] (P4) at (52mm, -72mm) {\textbf{POS-4} \, \textbf{CP} \quad Closure and Positivity};
\node[ax] (P5) at (52mm, -80mm) {\textbf{POS-5} \, \textbf{RG} \quad RG covariance};
\node[ax] (P6) at (52mm, -88mm) {\textbf{POS-6} \, \textbf{OGN} \quad Occam Gauge Normalization};
\node[ax] (P7) at (52mm, -96mm) {\textbf{POS-7} \, \textbf{MF} \quad Measurement and Falsifiability};

\draw[arrow] (GGC.south) -| ($(AXH.north) +(0,3mm)$) -- (AXH.north);
\draw[arrow] (GGC.south) -| ($(POSH.north)+(0,3mm)$) -- (POSH.north);

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

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\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{Hierarchy of program-defined acronyms in the \GGTheory{} program. The diagram is the same content as the abbreviation table set out as a hierarchy, so that the dependency between the names is visible at a glance rather than having to be reconstructed from the primary-link column. An arrow means the lower name is defined using the upper one; it does not mean the lower result depends on the upper for its proof.}
\label{fig:abbrev-flow}
\end{figure}

\section{Glossary of Key Symbols}
\label{app:glossary-symbols}

This appendix collects the key mathematical symbols used in \ATen{}. The first
block records the bulk-geometry coordinates and the \OGNfive{} integer ledger
that every later derivation appeals to. The second records the inherited scale
chain $E_{\rm Pl} \to E_{\rm GUT} \to M_{\rm lock}$. The third through seventh
blocks record the cosmological observables of the \(\pi\)-locked inventory ---
the density fractions, the Hubble two-stage corridor, the growth and primordial
sectors, the nucleosynthesis ledger, and Kaxion dark matter --- as audited in
\S\ref{sec:codebook-table-cosmos}. The eighth block records the \MSMF{}/\OGN{}
audit machinery.

\renewcommand{\arraystretch}{1.15}
\begin{center}
\small
\begin{longtable}{@{}p{0.22\textwidth}p{0.60\textwidth}p{0.14\textwidth}@{}}
\caption[Glossary of key symbols used in the present paper]{Glossary of key symbols used in the present paper. ``Source'' gives the theorem, section, or paper where the symbol is introduced or load-bearing. The Source column is the column that matters for checking: it names the theorem, section or paper where each symbol is introduced, so a symbol met in the middle of the paper can be traced to its definition in one step.}
\label{tab:appB-glossary-cosmos}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source}\\
\midrule
\endfirsthead
\multicolumn{3}{c}{\itshape Table~\ref{tab:appB-glossary-cosmos} continued}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Source}\\
\midrule
\endhead
\midrule \multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape (1) Bulk geometry and the \OGNfive{} ledger}\\
\midrule
$M_6$ & 6D bulk manifold $M_4 \times I_r \times S^1_\theta$ & \ATwo{}\\
$ds_6^2$ & Canonical scale-profile metric $e^{-2r}\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2$ & \ATwo{}\\
$r$ & Radial coordinate, $r \in [0, K]$ (the cosmic chronology axis) & ---\\
$\theta$ & Compact angular coordinate, $\theta \in [0, 2\pi)$ & ---\\
$S^1_\theta$ & Compact direction circle (hosts the Kaxion holonomy) & ---\\
$I_r$ & Radial interval & ---\\
$(M, N, L; J, K)$ & \OGNfive{} ledger $= (1, 3, 5; 7, 35)$ & \AFour{}\\
$K = LJ$ & Total scale depth, $K = LJ = M^2 + N^2 + L^2 = 35$ & \ATwo{}\\
$N = 3$ & Family number, three-leg theorem of \PaperPTwo{} & \citealp{ArisakaP2N3}\\
\midrule

\multicolumn{3}{l}{\itshape (2) Scale chain}\\
\midrule
$E_{\rm Pl}$ & Planck energy $\approx 1.22 \times 10^{19}$~GeV & ---\\
$E_{\rm GUT}$ & GUT scale $= E_{\rm Pl}\,e^{-2\pi} \approx 2.28 \times 10^{16}$~GeV & \ATwo{}\\
$M_{\rm lock}$ & Lock scale $= \tfrac12 M_{\rm Pl}\,e^{-K} \approx 3.85$~TeV & \ATwo{}\\
$m_\Lambda^{\rm geom}$ & Dark-energy mass scale $= M_{\rm lock}^2/M_{\rm Pl} \approx 6$~meV & \S\ref{sec:codebook-table-cosmos}\\
\midrule

\multicolumn{3}{l}{\itshape (3) The \(\pi\)-locked density inventory}\\
\midrule
$\Omm$ & Matter density fraction $= 1/\pi \approx 0.3183$ & Thm.~\ref{thm:omega-m-pi-topological}\\
$\OmL$ & Vacuum (dark-energy) fraction $= 1 - 1/\pi \approx 0.6817$ & Thm.~\ref{thm:omega-m-pi-topological}\\
$\Omb$ & Baryon fraction $= 1/(2\pi^2) \approx 0.0507$ & Thm.~\ref{thm:omega-m-pi-topological}\\
$\Omdm$ & Dark-matter fraction $= 1/\pi - 1/(2\pi^2) \approx 0.2676$ & Thm.~\ref{thm:omega-m-pi-topological}\\
$w$ & Dark-energy equation of state $= -1$ (exact) & \S\ref{sec:codebook-table-cosmos}\\
\midrule

\multicolumn{3}{l}{\itshape (4) Hubble two-stage corridor and epochs}\\
\midrule
$\Hzero^{\rm CMB}$ & Early-branch Hubble value $\approx 67.4$~km/s/Mpc & Thm.~\ref{thm:hubble-two-stage-corridor}\\
$\Hzero^{\rm late}$ & Late-branch Hubble value $\approx 73.0$~km/s/Mpc & Thm.~\ref{thm:hubble-two-stage-corridor}\\
$z_*$ & Corridor-switch redshift $= e^J = e^7 \approx 1097$ (atomic decoupling $z_{\rm dec} \approx 1090$, \S\ref{sec:two-redshifts}) & Thm.~\ref{thm:hubble-two-stage-corridor}\\
$z_{\rm eq}$ & Matter--radiation equality redshift $\approx 3400$ & \S\ref{sec:codebook-table-cosmos}\\
$z_{\rm rec}$ & Empirical recombination redshift $= 1089.92 \pm 0.25$ & \citet{Planck2018}\\
\midrule

\multicolumn{3}{l}{\itshape (5) Structure growth and the primordial sector}\\
\midrule
$\sigeight$ & Linear matter-clustering amplitude $\approx 0.811$ & \S\ref{sec:codebook-table-cosmos}\\
$\Seight$ & Clustering parameter $\sigma_8\sqrt{\Omm/0.3}$ (corridor: CMB $0.832$, late $0.78$) & \S\ref{sec:codebook-table-cosmos}\\
$\ns$ & Scalar spectral tilt $= 1-2/N^{*} = 0.9620$ (plateau-derived, D$^*$) & \S\ref{sec:codebook-table-cosmos}\\
$A_s$ & Scalar amplitude $= 2.10 \times 10^{-9}$ (asserted input) & \S\ref{sec:codebook-table-cosmos}\\
$r$ & Tensor-to-scalar ratio $= 12/N^{*2} = 0.0043$ (derived, two-sided) & \S\ref{sec:codebook-table-cosmos}\\
$\DeltaNeff$ & Effective extra-radiation deviation $\approx 0.04$ & \S\ref{sec:codebook-table-cosmos}\\
$\ell_D$ & CMB damping multipole $\approx 1350$ (Boltzmann; BFFT overlay $\approx 1382$) & \S\ref{sec:codebook-table-cosmos}\\
\midrule

\multicolumn{3}{l}{\itshape (6) Nucleosynthesis ledger}\\
\midrule
$\eta_B$ & Baryon-to-photon ratio $\approx 7 \times 10^{-10}$ ($\times3^{\pm1}$) & \S\ref{sec:codebook-table-cosmos}\\
${}^7\mathrm{Li}/\mathrm{H}$ & Primordial lithium-7 abundance $= \mathrm{SBBN}/\pi \approx 1.6 \times 10^{-10}$ & \S\ref{sec:codebook-table-cosmos}\\
\midrule

\multicolumn{3}{l}{\itshape (7) Kaxion dark matter}\\
\midrule
$m_{\rm Kaxion}$ & Kaxion mass $= 8.1 \pm 0.5~\mu$eV & \PaperCOne{}\\
$\nu_K$ & Kaxion frequency window $\in [1.59, 2.13]$~GHz & \PaperCOne{}\\
$f_{\rm Kaxion}$ & Kaxion decay constant $\approx (3.0 \pm 0.3) \times 10^{13}$~GeV & \PaperCOne{}\\
$\theta_i$ & Initial misalignment angle $\approx 1.4$~rad (fixed by the relic abundance) & \PaperCOne{}\\
\midrule

\multicolumn{3}{l}{\itshape (8) \MSMF{}/\OGN{} audit machinery}\\
\midrule
$\mathcal R$ & Candidate rational expression & \S\ref{sec:codebook-msmf-ogn}\\
$\mathcal C_\Sigma$ & \OGN{}-admissible candidate space on stratum $\Sigma$ & \S\ref{sec:codebook-msmf-ogn}\\
$\mathrm{Cost}(\mathcal R)$ & \MSMF{} cost: total absolute exponent budget & \S\ref{sec:codebook-msmf-ogn}\\
$\epsilon_X$ & Empirical-precision target for parameter $X$ on its stratum & \S\ref{sec:codebook-msmf-ogn}\\
$\mathcal R^\star$ & Cost-minimum unique winner under \MSMF{}/\OGN{} selection & \S\ref{sec:codebook-msmf-ogn}\\

\end{longtable}
\end{center}
\renewcommand{\arraystretch}{1.0}

\bigskip


\section{Codebook glossary (compact)}
\label{app:codebook-glossary}

A condensed version of the twenty-one-row codebook of Section~\ref{sec:codebook-table-cosmos}, suitable for quick scan. Each row gives the observable, its symbol, the predicted value (from \ATen{}), the empirical value (with reference), and the status under the \MSMF{}/\OGN{} principle.

\begin{table}[H]
\centering
\caption[Compact codebook glossary with predicted and empirical values]{Compact codebook glossary. D = Derived, D\(^*\) = Derived (cond.), A = Asserted. Predicted values from the present paper; empirical values from \citet{Planck2018}, \citet{Riess2022}, \citet{Asgari2021}. Every predicted value in the fourth column is this paper's; every empirical value in the fifth is quoted from the source named in the codebook proper. A status of D-star means the row is derived conditional on a named conjecture, and the conjecture is named there rather than here.}
\label{tab:codebook-glossary-compact}
\renewcommand{\arraystretch}{1.15}
\small
\begin{tabular}{p{0.18\textwidth}p{0.12\textwidth}p{0.17\textwidth}p{0.20\textwidth}p{0.10\textwidth}}
\toprule
\textbf{Observable} & \textbf{Symbol} & \textbf{Predicted} & \textbf{Empirical} & \textbf{Status} \\
\midrule
Matter fraction & \(\Omm\) & \(1/\pi \approx 0.3183\) & \(0.315 \pm 0.007\) & D\(^*\) \\
Vacuum fraction & \(\OmL\) & \(1 - 1/\pi \approx 0.6817\) & \(0.685 \pm 0.007\) & D\(^*\) \\
Dark-energy EoS & \(w\) & \(-1\) (exact) & \(-1.028 \pm 0.032\) & D\(^*\) \\
Baryon fraction & \(\Omb\) & \(1/(2\pi^2) \approx 0.0507\) & \(0.0493 \pm 0.0006\) & D\(^*\) (2.3\(\sigma\) residual) \\
Dark-matter fraction & \(\Omdm\) & \(\approx 0.2676\) & \(0.2657 \pm 0.0070\) & D\(^*\) \\
Baryon-to-photon ratio & \(\eta_B\) & \(\approx7\times10^{-10}\) (central \(7.34\)) & \((6.12\pm0.04)\times10^{-10}\) & D\(^*\) \\
Lithium-7 & \({}^7\mathrm{Li}/H\) & \(\mathrm{SBBN}/\pi \approx 1.6\times10^{-10}\) & \((1.6\pm0.3)\times10^{-10}\) & D\(^*\) \\
Hubble (CMB branch) & \(\Hzero^{\mathrm{CMB}}\) & \(\approx 67.4\) km/s/Mpc & \(67.4 \pm 0.5\) & D\(^*\) \\
Hubble (late branch) & \(\Hzero^{\mathrm{late}}\) & \(\approx 73.0\) km/s/Mpc & \(73.0 \pm 1.0\) & D\(^*\) \\
Corridor switch / decoupling & \(z_*\) & switch \(e^7 \approx 1096.6\); atomic \(\approx 1090\) & \(1089.92 \pm 0.25\) (decoupling) & D\(^*\) \\
Scalar tilt & \(\ns\) & \(1-2/N^{*}=0.9620\) & \(0.965 \pm 0.004\) & D\(^*\) \\
Scalar amplitude & \(A_s\) & \(2.10 \times 10^{-9}\) (input) & \((2.10 \pm 0.04) \times 10^{-9}\) & A \\
Tensor-to-scalar & \(r\) & \(12/N^{*2}=0.0043\) & \(<0.036\) (95\%) & D \\
Clustering amplitude & \(\sigeight\) & \(\approx 0.811\) & \(0.811 \pm 0.006\) & D\(^*\) \\
Clustering (CMB branch) & \(\Seight^{\mathrm{CMB}}\) & \(\approx 0.832\) & \(0.832 \pm 0.013\) & D\(^*\) \\
Clustering (late branch) & \(\Seight^{\mathrm{late}}\) & \(0.78 \pm 0.01\) & \(0.759^{+0.024}_{-0.021}\) & D\(^*\) \\
Damping multipole & \(\ell_D\) & \(1350\pm3\) & \(\approx 1345 \pm 15\) & D\(^*\) \\
Neutrino dev. & \(\DeltaNeff\) & \(\approx 0.04\) & \(0.04 \pm 0.18\) & D \\
Kaxion mass & \(m_{\mathrm{Kaxion}}\) & \(8.1\pm0.5~\mu\)eV (predicted) & --- & D\(^*\) \\
Kaxion decay constant & \(f_{\mathrm{Kaxion}}\) & \((3.0\pm0.3)\times10^{13}\) GeV raw (predicted) & --- & D\(^*\) \\
Inflation scale & \(V_{\mathrm{inf}}^{1/4}\) & \(8.3\times10^{15}\) GeV & --- & D\(^*\) \\
\bottomrule
\end{tabular}
\end{table}

\textbf{Counting summary.} 2 D + 18 D\(^*\) + 1 A = 21 observables; 10 named conjectures (Appendix~\ref{app:conjecture-deck}) close all 19 of the D\(^*\) and A entries.

%% ===== Block II: pedagogical derivations (graduate course) =====
% =====================================================================
%  Appendix fragment for Arisaka-GG-A10-Cosmos
%  The Friedmann equations from the boundary Einstein equation
%  Self-contained graduate-level derivation, maximum depth.
%  ASCII-only. Mirrors the A8 "statement -> line-by-line derivation
%  -> worked micro-example -> framework connection" rhythm.
%  All \label keys prefixed appA:frw-
% =====================================================================

\section{Global naming convention for cross-cohort citation}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique identifier
of the form \emph{Prefix-Suffix-N}, where the Prefix names the item type and the
Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem.
\item \textbf{L-Suffix-N}, \textbf{C-Suffix-N}, \textbf{D-Suffix-N},
\textbf{R-Suffix-N} for the $N$th Lemma, Corollary, Definition, and Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (the Common Objections Part).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (the kill criteria that would
retire the theory).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-Cosmos-1 is the
cosmology-from-a-constitutional-layer innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved / status entry.
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution}
(A1), \emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS} (A5),
\emph{GRIP} (A6), \emph{Qi4} (A7), \emph{QM} (A8), \emph{SM} (A9),
\emph{Cosmos} (A10, this paper), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7) of
\citet{ArisakaA1Constitution} keep their existing cohort-wide names; the global
identifier is the same item viewed from outside its home paper. The present paper,
\ATen{}, uses the suffix \emph{Cosmos}: its theorems are T-Cosmos-N, its falsifiers
F-Cosmos-N, its innovations I-Cosmos-N (the ten above), its problems-solved
PS-Cosmos-N, and its objections O-Cosmos-N.


\section{Direct derivation of the 4D Einstein equation, the Friedmann equations, and \LCDM{}}
\label{appA:frw}

\GGteachbox{The full arc from the six-dimensional bulk to the
concordance model, walked in three steps at first-year-graduate pace.
Step~1: the four-dimensional Einstein equation is \emph{derived} from
the \GG{} bulk --- the stage and its integer ledger, the bulk action
read term by term from its earned home in the cohort, the
Einstein--Hilbert term descending with its scale weight, Newton's
constant computed as an overlap integral, conservation inherited as a
Ward identity, and the open-system correction bounded, with live
falsifiers. Step~2: the Friedmann equations are derived from that
equation exactly as the textbooks do --- Christoffel symbols, Ricci
tensor, Einstein tensor, perfect-fluid source, every index traced ---
closed by a worked computation of the age of the universe. Step~3: the
\LCDM{} inventory is read from the geometric ledger rather than fitted
--- components, fractions, and initial conditions, with the register of
every entry stated --- and the assembled expansion law is written with
no fitted continuous parameter. Fourteen problems with complete worked
solutions.}
\GGreadalong{\citet{Ryden2017} chs.~3--6 for a first pass;
\citet{Dodelson2003} ch.~2 or \citet{Baumann2022} ch.~1 at course
level; \citet{Weinberg2008} ch.~1 at depth; for the general-relativity
prerequisites of Step~2, \citet{Wald1984} or
\citet{MisnerThorneWheeler1973}.}

Before the indices, the arc of discovery --- because every equation below
was forced on physics by an observation. In 1929 Hubble turned Leavitt's
Cepheid ruler on the spiral nebulae and found the recession law \(v = H_0 d\)
\citep{Hubble1929}, making Friedmann's and Lema\^itre's expanding
solutions \citep{Friedmann1922,Lemaitre1927} physics rather than
curiosity. In 1965 Penzias and Wilson found the 3-kelvin glow
\citep{PenziasWilson1965} that Dicke's group identified at once as the
relic of a hot beginning \citep{DickePeebles1965}; COBE measured its
blackbody shape and first anisotropies \citep{SmootCOBE1992}; and the
supernova teams of 1998--99 \citep{Riess1998,Perlmutter1999} found the
acceleration that put \(\Lambda\) into the inventory. Planck then
compressed all of it into six numbers \citep{Planck2018}. The equations
this appendix derives are the machine those six numbers feed --- and the
question the rest of the course answers is why the numbers take the
values they do.

Figure~\ref{fig:appfig-horizons} draws the comoving Hubble radius across the whole history. Standard pedagogical and historical treatments of the Friedmann--Lemaitre--Robertson--Walker background cosmology, on which the derivations of this appendix follow line by line, are given in \citep{Weinberg2008,Dodelson2003,Mukhanov2005,Peebles1993,MisnerThorneWheeler1973,WeinbergGR1972,Wald1984,Bianchi1902}.

\begin{figure}[htbp]
\centering
\resizebox{0.658\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,scale=1.0]
% x = log a (inflation .. future), mapped 0..12 ; y = comoving Hubble radius 0..6
\def\xmax{12.4}\def\ymax{6.6}
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (\xmax,0);
\node[anchor=south east,font=\scriptsize] at (\xmax,0.10){$\log a$ \ (time $\rightarrow$)};
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (0,\ymax)
  node[above,anchor=south west,xshift=-3pt]
  {comoving Hubble radius $(aH)^{-1}$};
% era boundaries (vertical guides)
% inflation: 0..2.2 ; reheating min ~2.2 ; radiation 2.2..5 ; matter 5..8.6 ;
% horizon max ~8.6 ; dark energy 8.6..12
\foreach \x in {2.2,5.0,8.6}
  \draw[densely dotted,gray!60] (\x,0) -- (\x,6.1);
% THE CURVE: shrink (inflation) -> min at reheating -> grow (rad+matter)
% -> max -> shrink (de Sitter future)
\draw[line width=1.10pt,blue!65!black]
  (0.35,5.6)                              % large at start of inflation
  .. controls (1.1,4.4) and (1.7,1.7) ..
  (2.2,0.9)                               % minimum at reheating
  .. controls (2.9,1.35) and (3.8,2.6) ..
  (5.0,3.4)                               % growing through radiation
  .. controls (6.4,4.25) and (7.6,4.85) ..
  (8.6,5.1)                               % maximum (end of matter dom.)
  .. controls (9.5,5.25) and (9.9,4.7) ..
  (10.4,3.7)                              % de Sitter: shrinking again
  .. controls (11.0,2.5) and (11.4,1.6) ..
  (12.0,1.0);
% inflation shrink arrow annotation
\node[blue!55!black,font=\scriptsize,align=center,anchor=south] at (1.15,5.1)
  {shrinks\\(inflation)};
\node[blue!55!black,font=\scriptsize,align=center,anchor=north east] at (10.3,2.4)
  {shrinks\\($\Lambda$/de Sitter)};
% reheating minimum label
\draw[{Latex[length=1.6mm]}-,gray!55!black] (2.2,0.9) -- (2.7,0.35)
  node[anchor=west,font=\scriptsize,gray!50!black]{reheating (min)};
% horizon maximum label
\fill[blue!65!black] (8.6,5.1) circle (1.3pt);
\node[font=\scriptsize,anchor=south,align=center] at (8.0,5.25)
  {horizon\\ maximum};
% last scattering marker (in matter era, just before max)
\draw[densely dashed,red!65!black] (7.0,0) -- (7.0,4.55);
\node[red!65!black,font=\scriptsize,anchor=south,rotate=90] at (7.0,2.1)
  {last scattering};
% era band labels along the bottom
\foreach \xa/\xb/\lab in {0.05/2.2/inflation, 2.2/5.0/radiation,
                          5.0/8.6/matter, 8.6/12.2/{dark energy ($w{=}{-}1$)}}
  {\draw[decorate,decoration={brace,amplitude=4pt,mirror},black!65]
     (\xa,-0.55) -- (\xb,-0.55)
     node[midway,below=3pt,font=\scriptsize,align=center]{\lab};}
% horizon-problem / acceleration teaching callouts
\node[draw=black!55,fill=black!3,rounded corners,inner sep=3pt,
      font=\scriptsize,align=center,anchor=north] at (1.15,-1.55)
  {horizon problem\\ solved by inflation};
\node[draw=black!55,fill=black!3,rounded corners,inner sep=3pt,
      font=\scriptsize,align=center,anchor=north] at (10.4,-1.55)
  {late-time\\ acceleration};
% GG / derived status note
\node[draw=green!45!black,fill=green!4,rounded corners,align=left,
      font=\scriptsize,inner sep=4pt,text width=8.6cm,anchor=north]
      at (6.2,-2.55)
  {\textbf{\GGTheory{} --- Derived (the cleanest of the set):} the scale-profile descent sets the early inflationary shrinking horizon; the boundary
   Einstein equation governing this FRW evolution is genuinely
   \emph{derived}, backed by the boundary general relativity of companion
   \AFive{}. The de Sitter future corresponds to $w=-1$.};
\end{tikzpicture}}
\caption[Comoving Hubble radius $(aH)^{-1}$ across cosmic history.]{%
\textbf{FRW horizons: the comoving Hubble radius $(aH)^{-1}$ versus
$\log a$.}
Read left to right in time. During inflation the comoving Hubble radius
\emph{shrinks}, reaching a minimum at reheating; it then \emph{grows}
through the decelerating radiation- and matter-dominated eras (so the
causal horizon grows and scales re-enter), reaches a maximum near the
end of matter domination, and \emph{shrinks} again once dark energy
dominates and expansion accelerates toward a de Sitter future. This
single curve teaches both the horizon problem (resolved by the early
shrinking phase) and late-time acceleration. Last scattering is marked
in the matter era.
\textbf{\GGTheory{} derivation:} the scale-profile descent sets the early
inflationary shrinking horizon, and the boundary Einstein equation
governing this FRW evolution is genuinely \emph{derived} -- backed by
the boundary general relativity of companion \AFive{}; the de
Sitter future corresponds to $w=-1$. This is a foundational, Derived
figure, the cleanest of the set.}
\label{fig:appfig-horizons}
\end{figure}


This appendix has a single aim: to show that the Friedmann equations
--- the equations on which all of modern cosmology rests --- are not a
starting assumption but a consequence of geometry. We build the case in
three steps, and the reader can navigate by them.

\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{From the bulk to the Einstein equation
(\S\ref{appA:frw-gg6}--\ref{appA:frw-einstein-earned}).} We derive
four-dimensional gravity itself: the six-dimensional stage and its
integer ledger, the bulk action read term by term from its earned home
in the cohort, the Einstein--Hilbert term descending to the boundary
with its scale weight, Newton's constant computed as an elementary
overlap integral, and the boundary Einstein equation obtained by
variation and zero-mode projection --- its conservation law inherited
as a Ward identity, its open-system correction bounded, its falsifiers
named. What every textbook takes as the axiom of cosmology is here the
first theorem.
\item \textbf{From the Einstein equation to the Friedmann equations
(\S\ref{appA:frw-metric}--\ref{appA:frw-worked}).} We then derive the
Friedmann equations exactly as a textbook does, from scratch, at the
depth a first-year graduate student can follow line by line --- metric,
connection, curvature, Einstein tensor, perfect-fluid source, and the
two Friedmann equations, closed with a worked numerical example.
Nothing here is unusual, and nothing is skipped --- the point of
Step~1 is precisely that nothing needs to change.
\item \textbf{From the Friedmann equations to \LCDM{}
(\S\ref{appA:frw-sources}--\ref{appA:frw-assembled}).} Finally the
right-hand side: the free menu $\{\Omega_i, w_i\}$ of standard
cosmology is replaced by the geometric ledger --- which components
exist, with which equations of state, in which $\pi$-locked
proportions, from which initial conditions --- and the assembled
expansion law is written out with no fitted continuous parameter, every
entry carrying its register.
\end{enumerate}

This is the first concrete instance of the belief that animates the
whole program --- that the contents and laws of the universe should
follow from geometry, in the lineage of Euclid, who asked that the world
be built from the fewest self-evident things, and of Newton, who turned
Kepler's measured curves into theorems of a single law. The reader who
wants only the background equations may read the boxed displays
\eqref{appA:frw-eq-friedmann1}, \eqref{appA:frw-eq-friedmann2} and
\eqref{appA:frw-eq-continuity} and skip ahead; the reader who wants to
verify every step will find no gap papered over.

\subsection{The four debts of the textbook derivation}
\label{appA:frw-limits}

Before the first index is traced, it is worth stating plainly what the
textbook version of this subject --- general relativity assumed, the
Friedmann equations derived, the inventory fitted --- must supply by
hand. Four debts, and they are not small.

\begin{enumerate}[leftmargin=*]
\item \textbf{The law of gravity.} The textbook derivation of Step~2 begins
from the Einstein equation \eqref{appA:frw-eq-einstein}. Nothing in that derivation explains
\emph{why} gravity obeys that law rather than another; general
relativity is the input, not the output.
\item \textbf{Newton's constant.} The single number $G$ sets the
strength of the link between geometry and matter, and it enters as a
measured constant. The derivation has nothing to say about why it has
the value it does.
\item \textbf{The cosmic inventory.} The right-hand side of the
Friedmann equation is a \emph{menu}: one writes down whatever components
$\{\Omega_i, w_i\}$ the data seem to require --- radiation, baryons,
and then, when the data demand them, cold dark matter and a
cosmological constant. Standard cosmology does not say what these
components are, why they exist, or why they appear in the particular
proportions we measure. The near-equality of matter and dark energy
\emph{today} --- the cosmic coincidence --- is left as a fact to be
noted, not explained.
\item \textbf{The initial state.} The smoothness, the flatness, and the
spectrum of initial fluctuations are imposed as boundary conditions.
\end{enumerate}

None of this is a flaw in the mathematics; it is the boundary of what a
four-dimensional, general-relativistic description \emph{can} do. Such
a description accommodates the universe we observe --- with great
precision --- but it does not derive it: every cosmological parameter
is a number to be measured and inserted. A theory in the Euclidean and
Newtonian tradition would instead ask these numbers to follow from
something deeper. The three steps of this appendix are organized as
exactly that request: Step~1 pays debts~1 and~2 (the law and its
constant), Step~3 pays debts~3 and~4 (the inventory and the initial
state), and Step~2 --- deliberately untouched, textbook line by
textbook line --- is the demonstration that paying the debts changes
the \emph{provenance} of cosmology, not its equations. \emph{(Physics
intuition: the honest way to admire the Friedmann equations is to ask
who is paying for them. The textbook borrows four times and never
settles; this appendix walks the same street with the receipts.)}

\subsection{Step 1 --- the six-dimensional stage and its ledger}
\label{appA:frw-gg6}

\GGTheory{} answers the four debts of \S\ref{appA:frw-limits} not one
at a time but together, because they share a single source: the
four-dimensional world on which the Friedmann equations live is the
boundary of a six-dimensional geometry. Step~1 walks that derivation
at the same depth as the textbook material of Step~2 --- stating each
object, doing the arithmetic the reader can check by hand, and naming
where the rigorous proof lives in the cohort: chiefly the boundary
general relativity of \citet{ArisakaA5GDOS} (Theorems T-GDOS-8.1--8.7
of its Part~IV), the bulk construction of \citet{ArisakaA2GG6}, and
the assembly interlude of \citet{ArisakaA9SM}, whose station S-1a
earns the bulk action this Step spends. For a reader meeting warped
products for the first time: the line element below is an ordinary
four-dimensional metric multiplied by a factor that depends on where
you stand in a fifth direction --- nothing more exotic than a map
whose scale bar changes from place to place. The new content is the
\emph{reading} of that fifth direction: it is the renormalization-group
altitude, so that geometry and energy scale become one coordinate.

\paragraph{The six-dimensional stage.}
The geometry is a product geometry (\citealp{ArisakaA2GG6}): four ordinary
spacetime dimensions $x^\mu$, a radial direction $r$, and a compact angle
$\theta$, with line element
\begin{equation}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \;+\; dr^2 \;+\; d\theta^2 .
\label{appA:frw-eq-6dmetric}
\end{equation}
The scale profile $e^{-2r}$ shrinks four-dimensional lengths as one moves
inward along $r$, which runs from the ultraviolet edge at $r=0$ (the
unification scale \EGUT{}) down a total scale depth $K=35$; the angle
$\theta$ closes the compact circle that hosts the Kaxion. The depth is
not chosen --- it is the \OGNfive{} integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$,
and the value $35$ is fixed twice over,
\begin{equation}
K \;=\; LJ \;=\; 5\times 7 \;=\; 35
\qquad\text{and}\qquad
K \;=\; M^2+N^2+L^2 \;=\; 1+9+25 \;=\; 35 ,
\label{appA:frw-eq-Kdual}
\end{equation}
a redundancy that is one of the integer-ledger's internal consistency
checks. \emph{Physics intuition:} $r$ is a renormalization-group
altitude, and it is precisely the descent parameter of the open-system
boundary law \GDOS{}. Moving downward in $r$ is integrating out the
high-energy bulk modes --- the flow that the \GRIP{} runtime executes ---
so the single coordinate $r$ carries the entire arrow from the hot
unified edge to the cold dilute world we inhabit. Cosmic history is a
journey down this one axis.


\subsection{Step 1 --- the bulk action, read term by term}
\label{appA:frw-action}

Every dynamical statement in this appendix descends from one object,
and the reader should see it whole before any reduction is performed.
The bulk carries the unique admissible two-derivative action
\begin{equation}
\resizebox{0.97\linewidth}{!}{$\displaystyle
S_6 \;=\; \int_{M_6} \sqrt{-G}\;\Big[
\underbrace{\tfrac{1}{2\kappa_6^2}\,\big(R_6 - 2\Lambda_6\big)}_{\text{geometry}}
\;-\; \underbrace{\tfrac{1}{4}\,\mathcal F_{MN}^{\,A}\mathcal F^{A\,MN}}_{\text{gauge sector}}
\;+\; \underbrace{i\,\overline\Psi\,\Gamma^M D_M \Psi}_{\text{matter spinors}}
\;-\; \underbrace{\tfrac{1}{12}\,H_{MNP}H^{MNP}}_{\text{flux sector}}
\Big],
$}
\label{appA:frw-eq-S6}
\end{equation}
plus the Green--Schwarz terms that anomaly closure forces. This is not
an ansatz of the present paper, and this appendix does not ask the
reader to accept it on citation alone: its field content is forced by
the stage, metric, and gauge-group theorems of \citet{ArisakaA2GG6}
(T-GG6-0/1/2/3), its consistency by the anomaly closure of
\citet{ArisakaA3BI6}, and the uniqueness of its dynamics --- why these
terms and no others at lowest derivative order --- is earned brace by
brace in the assembly interlude of \citet{ArisakaA9SM} (station S-1a),
with the register of every claim stated there: the form of each term
\textbf{Derived} at lowest derivative order, the matter content
\textbf{Derived (cond.)} on a named minimality lemma. For Step~1 only
two facts about Eq.~\eqref{appA:frw-eq-S6} matter, and both belong to
the geometry brace: the Einstein--Hilbert term exists with a definite
coefficient, and no other term in the action contains second
derivatives of the metric. Everything else --- gauge fields, spinors,
flux --- rides on the right-hand side of the equation we are about to
derive, as source. \emph{(Physics intuition: an action is a recipe
book, and this book has one page. The rest of the cohort spends the
gauge and matter braces to build particle physics; cosmology, the
coarsest of all boundary sciences, needs only the first brace and the
bookkeeping of the others' energy.)}

\subsection{Step 1 --- the Einstein--Hilbert term descends to the boundary}
\label{appA:frw-descent}

The tool that carries a six-dimensional action to a four-dimensional
theory is mode expansion --- Kaluza--Klein reduction --- and a
first-year student has already used it twice in other clothes. On the
compact circle it is the Fourier series: any field decomposes into
harmonics $e^{i n\theta}$, and the $n=0$ harmonic is the part that does
not vary around the circle. On the interval it is the particle in a
box: standing waves with a discrete tower of energies, whose lowest
mode again varies least. A six-dimensional field is therefore, to a
boundary observer, an infinite family of four-dimensional fields: one
massless \emph{zero mode} (constant across the compact directions) and
a tower of heavy partners whose masses start at the lock scale
$\Mlock \approx 3.85$~TeV --- far above every cosmological energy, so
the background dynamics of this appendix sees the zero modes alone
(the tower is the collider story of the companion papers, Theorem
T-GDOS-9.1 of \citealp{ArisakaA5GDOS}). What remains is to reduce the
geometry brace of Eq.~\eqref{appA:frw-eq-S6} and watch four-dimensional
gravity emerge.

\paragraph{Gravity as a boundary zero mode (paying debt 1).}
A field living in six dimensions can be expanded in standing waves
(Kaluza--Klein modes) along the two compact directions. The lowest ---
the \emph{zero mode}, constant across $r$ and $\theta$ --- is the ordinary
four-dimensional graviton: we promote the flat metric $\eta_{\mu\nu}$ in
\eqref{appA:frw-eq-6dmetric} to a slowly varying four-dimensional field
$g_{\mu\nu}(x)$. Two factors then control how the six-dimensional
Einstein--Hilbert action reduces. First, the measure of the scale-profile block
carries the scale profile on each of the four spacetime legs, so its
determinant scales as $g_6 = e^{-8r}\,g_4\cdot(1)(1)$ and
\begin{equation}
\sqrt{g_6}\;=\;e^{-4r}\,\sqrt{g_4}\,.
\label{appA:frw-eq-measure}
\end{equation}
Second, the four-dimensional Ricci scalar built from $g_{\mu\nu}(x)$
enters the six-dimensional curvature with a compensating inverse-metric
factor $e^{+2r}$ (raising two indices on the scale-profile block),
$R_6 \supset e^{+2r}\,R_4[g]$. Their product is the scale weight
$e^{-4r}\cdot e^{+2r}=e^{-2r}$, and the gravitational action collapses to
four dimensions:
\begin{equation}
M_6^{4}\!\int\! d^4x\,d\theta\,dr\,\sqrt{g_6}\,R_6
\;\;\longrightarrow\;\;
M_6^{4}\Big(\!\int_0^{2\pi}\!\! d\theta\!\int_0^{K}\! e^{-2r}\,dr\Big)
\!\int\! d^4x\,\sqrt{g_4}\,R_4 .
\label{appA:frw-eq-kkreduce}
\end{equation}
The four-dimensional Einstein--Hilbert term --- the left-hand side of the
Einstein equation \eqref{appA:frw-eq-einstein} that Step~2 will begin from ---
emerges with a definite coefficient (\AFive{}~\T{8.1}). The law of
gravity on the boundary is therefore not an axiom; it is the shadow that
six-dimensional gravity casts on its four-dimensional edge.
\emph{Physics intuition:} general relativity is what the bulk geometry
looks like to an observer who cannot resolve the two extra directions ---
and the scale weight $e^{-2r}$ is the statement that the deep infrared
($r$ large) contributes almost nothing to what that observer calls
``gravity.''


\subsection{Step 1 --- Newton's constant computed, and the scale ladder}
\label{appA:frw-newton}

Debt 2 is the value of the coupling, and here the geometry does
something no four-dimensional theory can: it computes the coefficient
it just derived, as an integral the reader can do on one line.

\paragraph{Newton's constant from a scale-overlap integral (paying debt 2).}
The coefficient in \eqref{appA:frw-eq-kkreduce} is the entire content of
Newton's constant. It is the overlap of the graviton zero mode with
itself across the compact directions, with the scale weight just derived,
\begin{equation}
M_{\rm Pl}^2 \;=\; M_6^4 \int_0^{2\pi}\!\! d\theta \int_0^{K}\! e^{-2r}\,dr .
\label{appA:frw-eq-overlap}
\end{equation}
Both integrals are elementary. The angular one gives $2\pi$. The radial
one is
\begin{equation}
\int_0^{K} e^{-2r}\,dr
\;=\;\Big[-\tfrac12 e^{-2r}\Big]_0^{K}
\;=\;\tfrac12\big(1-e^{-2K}\big),
\label{appA:frw-eq-radint}
\end{equation}
so their product is $2\pi\cdot\tfrac12(1-e^{-2K})=\pi(1-e^{-2K})$ and
\begin{equation}
\boxed{\;M_{\rm Pl}^2 \;=\; \pi\,M_6^4\,\big(1 - e^{-2K}\big)\;}
\label{appA:frw-eq-MPl}
\end{equation}
(\AFive{}~\T{8.2}). With the ledger depth $K=35$ the scale residual is
\begin{equation}
e^{-2K}=e^{-70}\approx 3.98\times10^{-31},
\end{equation}
utterly negligible, so to one part in $10^{30}$ the relation collapses to
the clean statement $M_{\rm Pl}^2 \simeq \pi M_6^4$. Read in the natural direction this ties Newton's constant,
$G = 1/(8\pi M_{\rm Pl}^2)$ in reduced-Planck units, to a single
six-dimensional scale $M_6$ and the pure number $\pi$: it is not an
independent four-dimensional parameter. The framework carries one
dimensionful unit --- conventionally $M_{\rm Pl}\equiv G_N^{-1/2}$, with $M_6$
the \emph{derived} combination, as the constitution requires
(\citealp{ArisakaA2GG6}) --- so what the geometry fixes is the scale profile relating them, not a numerical prediction of $G_N$, which sets the units. \emph{Physics intuition:} $G$ measures how
large the hidden compact geometry looks from the boundary; the factor
$\pi$ is the geometric size of the throat, and the missing
$e^{-70}$ is how little of that throat the infrared world can feel. Where
standard cosmology treats $G$ as one free constant among many, the framework
ties it to the bulk geometry by a fixed relation --- one dimensionful unit,
and no free four-dimensional gravitational dial.

\paragraph{The same scale profile fixes the scale ladder (worked numbers).}
The habit of computing rather than measuring is clearest in the energy
scales that standard physics reads off experiment. Three rungs of one
ladder are fixed by $e$ raised to integer-ledger powers of the Planck
scale (\citealp{ArisakaA2GG6}; the scale chain of Part~II). From the
ultraviolet edge $\EPl=1.22\times10^{19}$~GeV:
\begin{align}
\EGUT &\;=\; \EPl\,e^{-2\pi}
\;=\; 1.22\times10^{19}\times e^{-2\pi}\ \mathrm{GeV}
\;=\; 2.28\times10^{16}\ \mathrm{GeV}, \label{appA:frw-eq-EGUT}\\[2pt]
\Mlock &\;=\; \tfrac12 M_{\rm Pl}\,e^{-K}
\;=\; \tfrac12\,(1.22\times10^{19})\,e^{-35}\ \mathrm{GeV}
\;=\; 3.85\times10^{3}\ \mathrm{GeV} \;=\; 3.85\ \mathrm{TeV}. \label{appA:frw-eq-Mlock}
\end{align}
The first is one full turn of the compact circle ($2\pi$) below the
Planck scale; the second is the full scale depth ($K=35$) below it. Their
ratio --- the great desert of particle physics, from the unification
scale down to the TeV seam --- is then itself a closed expression,
\begin{equation}
\frac{\EGUT}{\Mlock}
\;=\; 2\,e^{\,K-2\pi}
\;=\; 2\,e^{\,28.72}
\;\approx\; 5.9\times10^{12},
\label{appA:frw-eq-hierarchy}
\end{equation}
a hierarchy of nearly six trillion produced entirely by the integer
difference between the scale depth $35$ and the single GUT turn. Nothing
here is tuned; the numbers are exponentials of integers from the ledger
\eqref{appA:frw-eq-Kdual}. \emph{Physics intuition:} the vast hierarchies
of physics are not fine-tunings but a few notches of scale profile, read off the
same ruler that fixes $G$.


\subsection{Step 1 --- the boundary Einstein equation, earned}
\label{appA:frw-einstein-earned}

\paragraph{From action to equation.}
An action is not yet a law of motion. The law follows from the same
variational principle the reader knows from classical mechanics ---
$\delta S = 0$ --- executed for the metric field: varying
Eq.~\eqref{appA:frw-eq-S6} with respect to $G_{MN}$ yields the
six-dimensional Einstein equation
\begin{equation}
M_6^4\, G_{MN} \;+\; \Lambda_6\, g_{MN} \;=\; T_{MN}^{\rm total},
\label{appA:frw-eq-einstein6}
\end{equation}
with $T_{MN}^{\rm total}$ collecting the gauge, spinor, and flux
sectors of the action (the five-field variational accounting is
recorded in the canonical-Lagrangian section of
\citealp{ArisakaA5GDOS}). Restricting to the graviton zero mode of
\S\ref{appA:frw-descent} and integrating the compact directions with
the scale weight of \S\ref{appA:frw-newton} projects
Eq.~\eqref{appA:frw-eq-einstein6} onto the boundary:
\begin{equation}
G_{\mu\nu} \;+\; \Lambda\, g_{\mu\nu}
\;=\; 8\pi G\, T_{\mu\nu}
\;+\; \Delta T^{\rm GS}_{\mu\nu} + \Delta T^{\rm KK}_{\mu\nu},
\label{appA:frw-eq-einstein4}
\end{equation}
the four-dimensional Einstein equation with two additions the honest
register must carry: the boundary cosmological term $\Lambda$, whose
existence rides down with the geometry brace and whose \emph{value} is
the terminus story of Part~IV (the lock-scale seesaw, taken up in
Step~3), and the calculable inflow corrections
$\Delta T_{\mu\nu}$, bounded below. This is Theorem T-GDOS-8.1/8.4 of
\citet{ArisakaA5GDOS} at course pace. \emph{(Physics intuition: the
Einstein equation was never wrong to postulate --- it was only
unexplained. Here it is the unique shadow a one-page bulk recipe can
cast on its edge: the reader has now watched the left-hand side of all
of cosmology assemble itself from a variational principle and two
elementary integrals.)}

\paragraph{Conservation inherited, not imposed.}
The textbook pairs the Einstein equation with a separate physical
demand: energy--momentum conservation, $\nabla_\mu T^{\mu\nu}=0$. In
textbook general relativity the pairing is consistent because of the
contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ --- a geometric
fact accepted from differential geometry. Here it has a sharper
origin: that conservation law is the \emph{Ward identity} of bulk
diffeomorphism invariance restricted to the boundary
(\AFive{}~\T{8.3}). The freedom to relabel coordinates in the
six-dimensional bulk becomes, on the four-dimensional edge, the
statement that the source cannot leak. One practical consequence will
surface in Step~2 and is worth flagging now: of the three FRW
equations (two Friedmann plus continuity), only two are independent,
and the redundancy is exactly this identity wearing cosmological
clothes --- worked explicitly in the continuity discussion of
\S\ref{appA:frw-friedmann} and in Problem~E.12.1. \emph{(Physics
intuition: a bulk-exported source \emph{cannot} violate continuity
without breaking a symmetry of the geometry itself. Consistency is
inherited, not imposed by hand.)}

\paragraph{The open-system correction, bounded --- why the equations are exactly textbook today.}
The full boundary field equation is not quite the bare Einstein equation:
it carries calculable Green--Schwarz and Kaluza--Klein inflow terms
$\Delta T^{\rm GS}_{\mu\nu}+\Delta T^{\rm KK}_{\mu\nu}$
(\AFive{}~\T{8.4}). These switch on only near the lock scale $\Mlock$ and
fall away in the infrared, scaling as a power of the ambient energy over
$\Mlock$. On the present cosmological boundary the ambient scale is the
Hubble energy $H_0\sim 1.4\times10^{-33}$~eV, so the relative size of the
corrections is of order
\begin{equation}
\Big(\frac{H_0}{\Mlock}\Big)^2
\;\sim\;\Big(\frac{1.4\times10^{-33}\,\mathrm{eV}}{3.85\times10^{12}\,\mathrm{eV}}\Big)^2
\;\sim\; 10^{-91},
\label{appA:frw-eq-inflow-suppress}
\end{equation}
an utterly negligible number. This is why every equation of this appendix holds \emph{unmodified} today: the geometry corrects general
relativity only at the TeV seam --- exactly where it should, and nowhere
a cosmological measurement can yet reach. \emph{Physics intuition:} the
new physics is real but it lives at the throat; by the time the universe
has expanded to its present size, the throat is $10^{91}$ times too far
away to matter for the background.


\paragraph{The bound comes with instruments.}
A bounded correction is an honest claim only if something could
measure it. Two instruments are on the table. Astrophysically, the
same inflow terms modify the Tolman--Oppenheimer--Volkoff equation of
stellar structure in a calculable pattern (\AFive{}~\T{8.6}) --- a
named falsifier: a neutron-star observation matching the modified
profile would be discovery, one excluding it at the throat-suppressed
level would be trouble. Cosmologically, the growth sector of
Part~VI is the standing audit: any late-time departure from
general-relativistic growth beyond the corridor's stated bands would
land exactly here. The claim of this subsection is therefore not
``general relativity, amen'' but ``general relativity, plus a
correction of stated size, with stated instruments.''

\paragraph{A second route to the same edge.}
The zero-mode reduction is not the cohort's only road to boundary
gravity: the black-hole companion derives the same four-dimensional
gravity from the entanglement side --- induced gravity on the boundary,
with the Bekenstein--Hawking $A/4$ recovered exactly
(Appendix~\ref{appA:bhentropy}; \citealp{ArisakaC3BH}). Two
independent constructions landing on one Einstein equation is itself a
consistency check of the geometry. \emph{(Physics intuition: when the
KK road and the entanglement road meet at the same equation, the
equation stops being a route-dependent artifact and starts being a
place.)}

\paragraph{Step 1's ledger, closed.}
What Step~1 has established, with registers: the four-dimensional
Einstein equation \eqref{appA:frw-eq-einstein4} --- form
\textbf{Derived} (T-GDOS-8.1/8.4, consumed at course pace above);
Newton's constant tied to the bulk scale by
$M_{\rm Pl}^2 = \pi M_6^4(1-e^{-2K})$ --- \textbf{Derived} (T8.2), the
overall unit set by $G_N$ as the constitution requires; conservation
--- \textbf{Derived} as the T8.3 Ward identity; the open-system
correction --- \textbf{bounded}, not assumed absent, with the T8.6 and
Part-VI falsifiers live. Debts 1 and 2 of
\S\ref{appA:frw-limits} are paid. Step~2 can now begin exactly where
the textbook begins --- with one difference invisible in the equations
and decisive in the logic: nothing on the page ahead is postulated.

\subsection{Step 2 --- the FRW metric and its conventions}
\label{appA:frw-metric}

\paragraph{Statement.}
The cosmological principle --- spatial homogeneity and isotropy of
the observer-accessible four-dimensional sector --- restricts the
metric to the Friedmann--Robertson--Walker form. In comoving
spherical coordinates $(t,r,\theta,\phi)$ the line element is
\begin{equation}
\label{appA:frw-eq-metric}
ds^2 \;=\; -\,c^2\,dt^2
\;+\; a(t)^2\!\left[
\frac{dr^2}{1 - k\,r^2}
\;+\; r^2\,d\theta^2
\;+\; r^2\sin^2\!\theta\,d\phi^2
\right],
\end{equation}
where $a(t)$ is the dimensionless scale factor (the macro \aof{}),
normalized so that $a(t_0)=1$ today, and $k\in\{-1,0,+1\}$ is the
sign of the constant spatial curvature after $r$ is rescaled to
absorb its magnitude. We adopt the mostly-plus signature
$(-,+,+,+)$ and the index convention that Greek indices
$\mu,\nu,\dots$ run over $\{0,1,2,3\}=\{t,r,\theta,\phi\}$. For the
remainder of the algebra we set $c=1$ and restore it only in the
worked numerical example.

\paragraph{The metric and inverse-metric components.}
Reading \eqref{appA:frw-eq-metric} off as a diagonal matrix
$g_{\mu\nu}=\mathrm{diag}(g_{tt},g_{rr},g_{\theta\theta},g_{\phi\phi})$,
\begin{align}
\label{appA:frw-eq-gdown}
g_{tt} &= -1, &
g_{rr} &= \frac{a^2}{1-kr^2}, &
g_{\theta\theta} &= a^2 r^2, &
g_{\phi\phi} &= a^2 r^2 \sin^2\!\theta,
\end{align}
and because the metric is diagonal its inverse is the
componentwise reciprocal,
\begin{align}
\label{appA:frw-eq-gup}
g^{tt} &= -1, &
g^{rr} &= \frac{1-kr^2}{a^2}, &
g^{\theta\theta} &= \frac{1}{a^2 r^2}, &
g^{\phi\phi} &= \frac{1}{a^2 r^2 \sin^2\!\theta}.
\end{align}
The metric determinant is
\begin{equation}
\label{appA:frw-eq-detg}
g \;=\; \det g_{\mu\nu}
\;=\; -\,\frac{a^6 r^4 \sin^2\!\theta}{1-kr^2},
\qquad
\sqrt{-g} \;=\; \frac{a^3 r^2 \sin\theta}{\sqrt{1-kr^2}},
\end{equation}
which we will need for the covariant volume element. Overdot
denotes $d/dt$ throughout, so $\dot a = da/dt$ and
$\ddot a = d^2a/dt^2$, and the Hubble rate is defined as
\begin{equation}
\label{appA:frw-eq-hubble-def}
H(t) \;\equiv\; \frac{\dot a}{a},
\qquad
\Hzero \;\equiv\; H(t_0)
\ \text{(the present value)}.
\end{equation}

% ---------------------------------------------------------------------
\subsection{Step 2 --- Christoffel symbols}
\label{appA:frw-christoffel}

\paragraph{Statement.}
The Levi-Civita connection is the unique metric-compatible,
torsion-free connection, with components
\begin{equation}
\label{appA:frw-eq-christoffel-def}
\Gamma^{\lambda}{}_{\mu\nu}
\;=\;
\tfrac{1}{2}\,g^{\lambda\sigma}
\bigl(
\partial_\mu g_{\sigma\nu}
+\partial_\nu g_{\sigma\mu}
-\partial_\sigma g_{\mu\nu}
\bigr).
\end{equation}
Because $g_{\mu\nu}$ is diagonal and depends only on $t$ (through
$a$), on $r$ (through $1/(1-kr^2)$ and $r^2$), and on $\theta$
(through $\sin^2\theta$), most of the $4\times 10 = 40$ independent
symbols vanish. We compute the nonzero ones explicitly.

\paragraph{Time-derivative (Hubble-flow) symbols.}
Only $g_{rr},g_{\theta\theta},g_{\phi\phi}$ carry $t$-dependence,
and each is proportional to $a^2$, so $\partial_t g_{ii}
= 2(\dot a/a)\,g_{ii} = 2H g_{ii}$ (no sum). The symbols with a
single time index follow from \eqref{appA:frw-eq-christoffel-def}.
With $\lambda=0$ (i.e. $t$), using $g^{tt}=-1$ and
$\partial_t g_{ii} = 2H g_{ii}$,
\begin{equation}
\label{appA:frw-eq-Gamma0ij}
\Gamma^{t}{}_{ij}
= -\tfrac{1}{2}\,g^{tt}\,\partial_t g_{ij}
= \tfrac{1}{2}\,\partial_t g_{ij}
= H\,g_{ij}
\qquad(i,j\in\{r,\theta,\phi\}),
\end{equation}
explicitly
\begin{equation}
\label{appA:frw-eq-Gamma0-list}
\Gamma^{t}{}_{rr} = \frac{a\dot a}{1-kr^2},
\quad
\Gamma^{t}{}_{\theta\theta} = a\dot a\,r^2,
\quad
\Gamma^{t}{}_{\phi\phi} = a\dot a\,r^2\sin^2\!\theta.
\end{equation}
Similarly, the symbols with one time and one space index come from
$\lambda=i$, $\mu=t$, $\nu=i$:
\begin{equation}
\label{appA:frw-eq-Gammai0i}
\Gamma^{i}{}_{tj}
= \tfrac{1}{2}\,g^{ii}\,\partial_t g_{ij}
= \frac{\dot a}{a}\,\delta^{i}{}_{j}
= H\,\delta^{i}{}_{j}
\qquad(\text{no sum on }i),
\end{equation}
so
\begin{equation}
\label{appA:frw-eq-Gammai0i-list}
\Gamma^{r}{}_{tr}
= \Gamma^{\theta}{}_{t\theta}
= \Gamma^{\phi}{}_{t\phi}
= \frac{\dot a}{a} = H.
\end{equation}
These are the only symbols that carry the expansion; every
appearance of $\dot a$ in the curvature traces back to them.

\paragraph{Purely spatial symbols.}
The remaining nonzero symbols are exactly those of the maximally
symmetric three-space of curvature $k$, unchanged by the overall
$a^2(t)$ factor because that factor cancels between $g^{ii}$ and
$\partial g_{jj}$:
\begin{align}
\label{appA:frw-eq-Gamma-spatial}
\Gamma^{r}{}_{rr} &= \frac{kr}{1-kr^2}, &
\Gamma^{r}{}_{\theta\theta} &= -\,r(1-kr^2), &
\Gamma^{r}{}_{\phi\phi} &= -\,r(1-kr^2)\sin^2\!\theta,
\nonumber\\
\Gamma^{\theta}{}_{r\theta} &= \frac{1}{r}, &
\Gamma^{\theta}{}_{\phi\phi} &= -\sin\theta\cos\theta, &
\Gamma^{\phi}{}_{r\phi} &= \frac{1}{r},
\nonumber\\
\Gamma^{\phi}{}_{\theta\phi} &= \cot\theta. & & & &
\end{align}
As a representative check of \eqref{appA:frw-eq-christoffel-def},
take $\Gamma^{r}{}_{\theta\theta}$:
\begin{equation}
\label{appA:frw-eq-Gamma-check}
\Gamma^{r}{}_{\theta\theta}
= \tfrac{1}{2}g^{rr}\bigl(2\partial_\theta g_{r\theta}
-\partial_r g_{\theta\theta}\bigr)
= -\tfrac{1}{2}\,\frac{1-kr^2}{a^2}\,\partial_r(a^2 r^2)
= -\tfrac{1}{2}\,\frac{1-kr^2}{a^2}\,(2a^2 r)
= -\,r(1-kr^2),
\end{equation}
where $g_{r\theta}=0$ kills the first term. The arithmetic of every
other entry in \eqref{appA:frw-eq-Gamma-spatial} is identical in
spirit.

% ---------------------------------------------------------------------
\subsection{Step 2 --- the Ricci tensor and Ricci scalar}
\label{appA:frw-ricci}

\paragraph{Statement.}
The Ricci tensor is the contraction of the Riemann tensor,
\begin{equation}
\label{appA:frw-eq-ricci-def}
R_{\mu\nu}
\;=\;
\partial_\lambda \Gamma^{\lambda}{}_{\mu\nu}
-\partial_\nu \Gamma^{\lambda}{}_{\mu\lambda}
+\Gamma^{\lambda}{}_{\lambda\sigma}\Gamma^{\sigma}{}_{\mu\nu}
-\Gamma^{\lambda}{}_{\nu\sigma}\Gamma^{\sigma}{}_{\mu\lambda}.
\end{equation}
By homogeneity and isotropy $R_{\mu\nu}$ is diagonal and its three
spatial entries are equal up to the metric, so only the $tt$ and
one spatial component are independent. We compute both.

\paragraph{The $tt$ component.}
Set $\mu=\nu=t$ in \eqref{appA:frw-eq-ricci-def}. The first term
$\partial_\lambda\Gamma^{\lambda}{}_{tt}$ vanishes because every
$\Gamma^{\lambda}{}_{tt}=0$. The second term is
$-\partial_t\Gamma^{\lambda}{}_{t\lambda}$. From
\eqref{appA:frw-eq-Gammai0i-list} the trace of the connection over
its lower-time slot is
\begin{equation}
\label{appA:frw-eq-trace-conn}
\Gamma^{\lambda}{}_{t\lambda}
= \Gamma^{r}{}_{tr}+\Gamma^{\theta}{}_{t\theta}
+\Gamma^{\phi}{}_{t\phi}
= 3H = 3\,\frac{\dot a}{a},
\end{equation}
so $-\partial_t\Gamma^{\lambda}{}_{t\lambda}
= -3\,\dfrac{d}{dt}\!\left(\dfrac{\dot a}{a}\right)
= -3\,\dfrac{\ddot a\,a - \dot a^2}{a^2}
= -3\dfrac{\ddot a}{a} + 3H^2$.
The third term $\Gamma^{\lambda}{}_{\lambda\sigma}
\Gamma^{\sigma}{}_{tt}$ vanishes ($\Gamma^{\sigma}{}_{tt}=0$). The
fourth term is
$-\Gamma^{\lambda}{}_{t\sigma}\Gamma^{\sigma}{}_{t\lambda}
= -\sum_{i}(\Gamma^{i}{}_{ti})^2 = -3H^2$, using
\eqref{appA:frw-eq-Gammai0i}. Adding,
\begin{equation}
\label{appA:frw-eq-Rtt}
R_{tt}
= \Bigl(-3\frac{\ddot a}{a}+3H^2\Bigr) + \bigl(-3H^2\bigr)
= -\,3\,\frac{\ddot a}{a}.
\end{equation}

\paragraph{A spatial component.}
Take $\mu=\nu=r$. Carrying out
\eqref{appA:frw-eq-ricci-def} term by term with the symbols of
\eqref{appA:frw-eq-Gamma0-list}, \eqref{appA:frw-eq-Gammai0i-list}
and \eqref{appA:frw-eq-Gamma-spatial} --- the time terms supply
$a\ddot a + 2\dot a^2$ and the spatial terms supply the curvature
$2k$, all divided by $1-kr^2$ --- gives, after collecting,
\begin{equation}
\label{appA:frw-eq-Rrr}
R_{rr}
= \frac{a\ddot a + 2\dot a^2 + 2k}{1-kr^2}.
\end{equation}
Isotropy guarantees the angular components are fixed by the same
scalar; one finds $R_{\theta\theta}
= r^2(a\ddot a + 2\dot a^2 + 2k)$ and
$R_{\phi\phi} = R_{\theta\theta}\sin^2\theta$. In every case
\begin{equation}
\label{appA:frw-eq-Rij-compact}
R_{ij}
= \Bigl(\frac{\ddot a}{a} + 2H^2 + \frac{2k}{a^2}\Bigr) g_{ij}
\qquad(i,j\ \text{spatial}),
\end{equation}
which is the cleanest form to contract.

\paragraph{The Ricci scalar.}
Contract \eqref{appA:frw-eq-Rtt} and
\eqref{appA:frw-eq-Rij-compact} with $g^{\mu\nu}$. Using
$g^{tt}R_{tt} = (-1)(-3\ddot a/a) = 3\ddot a/a$ and
$g^{ij}R_{ij} = 3(\ddot a/a + 2H^2 + 2k/a^2)$ (the factor $3$ is
$g^{ij}g_{ij}$ summed over the three spatial directions),
\begin{equation}
\label{appA:frw-eq-ricci-scalar}
R
= g^{tt}R_{tt} + g^{ij}R_{ij}
= 3\frac{\ddot a}{a}
+ 3\Bigl(\frac{\ddot a}{a} + 2H^2 + \frac{2k}{a^2}\Bigr)
= 6\Bigl(\frac{\ddot a}{a} + H^2 + \frac{k}{a^2}\Bigr).
\end{equation}
This is the full scalar curvature of the \FRW{} geometry; it is
positive for an accelerating, spatially closed universe and can
change sign in general.

% ---------------------------------------------------------------------
\subsection{Step 2 --- the Einstein tensor: components}
\label{appA:frw-einstein}

\paragraph{Statement.}
The Einstein tensor is
$G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}R\,g_{\mu\nu}$. We assemble
its two independent components from the pieces above.

\paragraph{The $tt$ component (this is the one that becomes the
first Friedmann equation).}
With $g_{tt}=-1$,
\begin{align}
\label{appA:frw-eq-Gtt}
G_{tt}
&= R_{tt} - \tfrac{1}{2}R\,g_{tt}
= -3\frac{\ddot a}{a}
- \tfrac{1}{2}\Bigl[6\Bigl(\frac{\ddot a}{a}+H^2+\frac{k}{a^2}\Bigr)\Bigr](-1)
\nonumber\\
&= -3\frac{\ddot a}{a}
+ 3\frac{\ddot a}{a} + 3H^2 + 3\frac{k}{a^2}
= 3\Bigl(H^2 + \frac{k}{a^2}\Bigr).
\end{align}
The two $\ddot a/a$ pieces cancel exactly: this is why the first
Friedmann equation is first order in time even though the Ricci
tensor was second order. It is a structural fact, not a
coincidence, and it is the reason the energy density (not its
rate of change) sources the expansion rate.

\paragraph{A spatial component (this becomes the second Friedmann
equation).}
With $g_{ij}$ spatial and
$R_{ij} = (\ddot a/a + 2H^2 + 2k/a^2)g_{ij}$,
\begin{align}
\label{appA:frw-eq-Gij}
G_{ij}
&= R_{ij} - \tfrac{1}{2}R\,g_{ij}
= \Bigl[\frac{\ddot a}{a} + 2H^2 + \frac{2k}{a^2}
- 3\Bigl(\frac{\ddot a}{a}+H^2+\frac{k}{a^2}\Bigr)\Bigr]g_{ij}
\nonumber\\
&= -\Bigl(2\frac{\ddot a}{a} + H^2 + \frac{k}{a^2}\Bigr)g_{ij}.
\end{align}
It is often more transparent in mixed (one-up, one-down) form,
where the metric factor disappears:
\begin{align}
\label{appA:frw-eq-Gmixed}
G^{t}{}_{t}
&= g^{tt}G_{tt}
= -3\Bigl(H^2 + \frac{k}{a^2}\Bigr),
&
G^{i}{}_{j}
&= -\Bigl(2\frac{\ddot a}{a} + H^2 + \frac{k}{a^2}\Bigr)\,
\delta^{i}{}_{j}.
\end{align}
These two scalars --- $G^{t}{}_{t}$ and the common value of
$G^{i}{}_{j}$ --- carry the entire gravitational content of the
homogeneous, isotropic geometry.

% ---------------------------------------------------------------------
\subsection{Step 2 --- the perfect-fluid stress-energy tensor}
\label{appA:frw-fluid}

\paragraph{Statement.}
Homogeneity and isotropy force the source to be a perfect fluid:
no shear, no heat flux, no anisotropic stress. In terms of the
fluid four-velocity $u^\mu$ (with $u^\mu u_\mu = -1$), the energy
density $\rho$, and the isotropic pressure $p$,
\begin{equation}
\label{appA:frw-eq-Tmunu}
T_{\mu\nu}
= \Bigl(\rho + \frac{p}{c^2}\Bigr)u_\mu u_\nu + p\,g_{\mu\nu},
\qquad\text{(c=1):}\quad
T_{\mu\nu} = (\rho + p)\,u_\mu u_\nu + p\,g_{\mu\nu}.
\end{equation}
In the comoving frame the fluid is at rest, $u^\mu=(1,0,0,0)$ and
$u_\mu = g_{\mu\nu}u^\nu = (-1,0,0,0)$. Then $u_t u_t = 1$ and the
components are
\begin{align}
\label{appA:frw-eq-Tcomponents}
T_{tt} &= (\rho+p)(1) + p(-1) = \rho,
&
T_{ij} &= (\rho+p)(0) + p\,g_{ij} = p\,g_{ij},
\end{align}
or in mixed form, the textbook diagonal
\begin{equation}
\label{appA:frw-eq-Tmixed}
T^{\mu}{}_{\nu}
= \mathrm{diag}(-\rho,\;p,\;p,\;p).
\end{equation}
The sign pattern is fixed once the signature is fixed: with
$(-,+,+,+)$ the energy density sits in the time-time slot with a
minus in mixed form. Each cosmic component obeys a linear
equation of state $p_i = w_i\,\rho_i$, with $w=0$ for
pressureless matter (dust), $w=1/3$ for radiation, and $w=-1$ for
a cosmological constant / vacuum energy.

% ---------------------------------------------------------------------
\subsection{Step 2 --- the two Friedmann equations, the acceleration
equation, and the continuity equation}
\label{appA:frw-friedmann}

\paragraph{Statement.}
We now feed the geometry of \S\ref{appA:frw-einstein} and the
source of \S\ref{appA:frw-fluid} into the Einstein equation that
Step~1 established on the boundary,
Eq.~\eqref{appA:frw-eq-einstein4} --- from here on the walk is the
textbook's own, because Step~1's point is that the textbook was
right. In
geometrized-but-explicit form (restoring $c$ and Newton's $G$,
and allowing the cosmological constant $\Lambda$),
\begin{equation}
\label{appA:frw-eq-einstein}
G_{\mu\nu} + \Lambda\,g_{\mu\nu}
= \frac{8\pi G}{c^4}\,T_{\mu\nu}.
\end{equation}
We treat $\Lambda$ as a geometric term on the left; equivalently
it can be moved to the right as a $w=-1$ vacuum fluid with
$\rho_\Lambda = \Lambda c^2/(8\pi G)$. Both conventions are used
below and we keep them straight.

\paragraph{First Friedmann equation (the $tt$ equation).}
Insert $G_{tt} = 3(H^2 + k/a^2)$ from \eqref{appA:frw-eq-Gtt},
$g_{tt}=-c^2$ (restoring $c$), and $T_{tt}=\rho c^2$ (the energy
density times $c^2$ in the time-time slot with these units) into
\eqref{appA:frw-eq-einstein}:
\begin{equation}
\label{appA:frw-eq-tt-insert}
3\Bigl(H^2 + \frac{k c^2}{a^2}\Bigr) - \Lambda c^2
= \frac{8\pi G}{c^4}\,\rho\,c^4
= 8\pi G\,\rho.
\end{equation}
Dividing by $3$ and rearranging gives the boxed result,
\begin{equation}
\label{appA:frw-eq-friedmann1}
\boxed{\;
H^2 \;=\; \Bigl(\frac{\dot a}{a}\Bigr)^2
\;=\; \frac{8\pi G}{3}\,\rho
\;+\; \frac{\Lambda c^2}{3}
\;-\; \frac{k c^2}{a^2}.
\;}
\end{equation}
This is the \emph{first Friedmann equation}. It says the
square of the expansion rate is set by the total energy density,
the cosmological constant, and the spatial curvature. Folding
$\Lambda$ into a vacuum density $\rho_\Lambda$ and letting $\rho$
denote the sum over all components, it reads compactly
$H^2 = (8\pi G/3)\rho - kc^2/a^2$.

\paragraph{Second Friedmann equation (the spatial equation).}
Insert $G^{i}{}_{j} = -(2\ddot a/a + H^2 + kc^2/a^2)\delta^{i}{}_{j}$
from \eqref{appA:frw-eq-Gmixed} (with $c$ restored on the
curvature term), $g^{i}{}_{j}=\delta^{i}{}_{j}$, and
$T^{i}{}_{j}=p\,\delta^{i}{}_{j}$ into the mixed-index Einstein
equation $G^{i}{}_{j}+\Lambda\delta^{i}{}_{j}
= (8\pi G/c^4)T^{i}{}_{j}$:
\begin{equation}
\label{appA:frw-eq-spatial-insert}
-\Bigl(2\frac{\ddot a}{a} + H^2 + \frac{kc^2}{a^2}\Bigr)
+ \Lambda c^2
= \frac{8\pi G}{c^4}\,p.
\end{equation}
Now substitute the first Friedmann equation
\eqref{appA:frw-eq-friedmann1} for the combination
$H^2 + kc^2/a^2 = (8\pi G/3)\rho + \Lambda c^2/3$, which removes
$H^2$ and $k$ from \eqref{appA:frw-eq-spatial-insert}:
\begin{equation}
\label{appA:frw-eq-spatial-sub}
-2\frac{\ddot a}{a}
- \frac{8\pi G}{3}\rho - \frac{\Lambda c^2}{3}
+ \Lambda c^2
= \frac{8\pi G}{c^2}\,p,
\end{equation}
where on the right we wrote $T^{i}{}_{j}=p$ with $8\pi G/c^4$ so
that pressure enters as $8\pi G\,p/c^2$. Solving for $\ddot a/a$
gives the \emph{acceleration equation},
\begin{equation}
\label{appA:frw-eq-friedmann2}
\boxed{\;
\frac{\ddot a}{a}
\;=\;
-\,\frac{4\pi G}{3}\Bigl(\rho + \frac{3p}{c^2}\Bigr)
\;+\; \frac{\Lambda c^2}{3}.
\;}
\end{equation}
This is the \emph{second Friedmann equation}. The combination
$\rho + 3p/c^2$ is the active gravitational mass density of
general relativity: pressure gravitates, and gravitates three
times as hard per unit energy. For ordinary matter and radiation
($p\ge 0$) the first term decelerates the expansion; only a
sufficiently negative pressure ($w<-1/3$, e.g. a cosmological
constant) can drive $\ddot a>0$ and accelerate it.

\paragraph{The continuity equation and the redundancy.}
The third standard equation, conservation of energy, is
$\nabla_\mu T^{\mu}{}_{0}=0$. Working it out for the perfect fluid
with the \FRW{} connection, the time component of
$\nabla_\mu T^{\mu\nu}=0$ is
\begin{equation}
\label{appA:frw-eq-conserv-expand}
\partial_t T^{t}{}_{t}
+ \Gamma^{\mu}{}_{\mu t}T^{t}{}_{t}
- \Gamma^{\lambda}{}_{\mu t}T^{\mu}{}_{\lambda}
= -\dot\rho - 3H\rho - 3H p
\;=\; 0,
\end{equation}
using $\Gamma^{\mu}{}_{\mu t}=3H$ from
\eqref{appA:frw-eq-trace-conn} and
$T^{t}{}_{t}=-\rho,\ T^{i}{}_{i}=p$. This gives the
\emph{continuity equation},
\begin{equation}
\label{appA:frw-eq-continuity}
\boxed{\;
\dot\rho + 3H\Bigl(\rho + \frac{p}{c^2}\Bigr) \;=\; 0.
\;}
\end{equation}
The two terms have a clean reading: $3H\rho$ is the dilution of
energy density by the volume growth $\propto a^3$, and $3Hp/c^2$
is the $pdV$ work done by the fluid against expansion. For a
single component with $p=w\rho c^2$ it integrates to the
\emph{scaling law}
\begin{equation}
\label{appA:frw-eq-scaling}
\rho_i(a) = \rho_{i,0}\,a^{-3(1+w_i)},
\end{equation}
giving $\rho_m \propto a^{-3}$ (matter dilutes as volume),
$\rho_r \propto a^{-4}$ (radiation dilutes as volume \emph{and}
redshifts), and $\rho_\Lambda = \text{const}$ (vacuum energy does
not dilute).

\paragraph{The three are not independent.}
There are three boxed equations
--- \eqref{appA:frw-eq-friedmann1},
\eqref{appA:frw-eq-friedmann2},
\eqref{appA:frw-eq-continuity} --- but only two are independent.
This is the cosmological face of the contracted Bianchi identity
$\nabla_\mu G^{\mu\nu}\equiv 0$, which (with
\eqref{appA:frw-eq-einstein}) forces $\nabla_\mu T^{\mu\nu}=0$.
Concretely, differentiate the first Friedmann equation
\eqref{appA:frw-eq-friedmann1} (with $\Lambda$ absorbed into
$\rho$) with respect to time:
\begin{equation}
\label{appA:frw-eq-bianchi-step1}
2H\dot H
= \frac{8\pi G}{3}\dot\rho
+ \frac{2kc^2\dot a}{a^3},
\qquad
\dot H = \frac{\ddot a}{a} - H^2.
\end{equation}
Substituting $\dot\rho = -3H(\rho + p/c^2)$ from the continuity
equation and simplifying reproduces the acceleration equation
\eqref{appA:frw-eq-friedmann2} exactly --- no new information is
generated. Operationally: pick any two of
$\{$first Friedmann, acceleration, continuity$\}$ and the third
follows. The standard practice is to solve the first Friedmann
equation together with the per-component scaling laws
\eqref{appA:frw-eq-scaling}, which already encode continuity.

% ---------------------------------------------------------------------
\subsection{Step 2 --- spatial curvature, the critical density, and the
density parameters}
\label{appA:frw-omega}

\paragraph{Statement.}
The first Friedmann equation defines a natural reference density:
the density for which the universe is spatially flat ($k=0$) at a
given epoch. Setting $k=0$ and $\Lambda=0$ in
\eqref{appA:frw-eq-friedmann1} and solving for $\rho$ gives the
\emph{critical density}
\begin{equation}
\label{appA:frw-eq-rhocrit}
\rho_{\rm crit}(t)
\;\equiv\; \frac{3H(t)^2}{8\pi G},
\qquad
\rho_{{\rm crit},0}
= \frac{3\Hzero^2}{8\pi G}
\ \text{(today)}.
\end{equation}
The \emph{density parameter} of component $i$ is its density in
units of the critical density,
\begin{equation}
\label{appA:frw-eq-omega-def}
\Omega_i(t) \equiv \frac{\rho_i(t)}{\rho_{\rm crit}(t)},
\qquad
\Omega_\Lambda \equiv \frac{\Lambda c^2}{3\Hzero^2},
\qquad
\Omega_k \equiv -\frac{kc^2}{a^2\Hzero^2}.
\end{equation}
With the matter parameter \Omm{}, the vacuum parameter \OmL{}, the
radiation parameter $\Omega_r$, and the curvature parameter
$\Omega_k$ defined this way, divide the first Friedmann equation
\eqref{appA:frw-eq-friedmann1} through by $\Hzero^2$ at the
present epoch ($a=1$). Every term becomes a density parameter and
the equation collapses to the \emph{closure relation}
\begin{equation}
\label{appA:frw-eq-closure}
\boxed{\;
\Omm + \Omega_r + \OmL + \Omega_k \;=\; 1.
\;}
\end{equation}
A spatially flat universe is exactly $\Omega_k=0$, i.e. the total
density equals the critical density. Writing the time dependence
explicitly via the scaling laws \eqref{appA:frw-eq-scaling}, the
first Friedmann equation becomes the workhorse expansion-history
formula
\begin{equation}
\label{appA:frw-eq-Eofa}
\frac{H(a)^2}{\Hzero^2}
\;=\; \Omega_r\,a^{-4}
+ \Omm\,a^{-3}
+ \Omega_k\,a^{-2}
+ \OmL,
\end{equation}
often written $H(a) = \Hzero\,E(a)$ with $E(a)$ the square root of
the right-hand side. Equation \eqref{appA:frw-eq-Eofa} is the
single relation from which ages, distances, and growth rates are
computed.

% ---------------------------------------------------------------------
\subsection{Step 2 --- worked numerical example: the age of the universe
and the critical density in flat \LCDM{}}
\label{appA:frw-worked}

To make the abstract relations concrete we plug in present-day
flat-\LCDM{} values and compute two numbers a cosmologist cares
about: the present critical density and the age of the universe.
We use the Planck-anchored set
\begin{equation}
\label{appA:frw-eq-worked-inputs}
\Hzero = 67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\qquad
\Omm = 0.315,
\qquad
\OmL = 0.685,
\qquad
\Omega_r,\Omega_k \approx 0,
\end{equation}
which satisfy the closure relation
\eqref{appA:frw-eq-closure} to the quoted precision
($0.315 + 0.685 = 1.000$).

\paragraph{Step 1: $\Hzero$ in SI and the Hubble time.}
Convert $\Hzero$ from astronomers' units using
$1\ \mathrm{Mpc} = 3.0857\times 10^{19}\ \mathrm{km}$:
\begin{equation}
\label{appA:frw-eq-worked-H0}
\Hzero
= \frac{67.4\ \mathrm{km\,s^{-1}}}
{3.0857\times 10^{19}\ \mathrm{km}}
= 2.184\times 10^{-18}\ \mathrm{s^{-1}}.
\end{equation}
The Hubble time is its reciprocal,
\begin{equation}
\label{appA:frw-eq-worked-tH}
t_H \equiv \frac{1}{\Hzero}
= 4.579\times 10^{17}\ \mathrm{s}
= 14.51\ \mathrm{Gyr},
\end{equation}
using $1\ \mathrm{Gyr} = 3.156\times 10^{16}\ \mathrm{s}$. The
Hubble time is an \emph{over}estimate of the true age, because the
universe expanded more slowly in the matter-dominated past.

\paragraph{Step 2: the present critical density.}
From \eqref{appA:frw-eq-rhocrit} with
$G = 6.674\times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$,
\begin{equation}
\label{appA:frw-eq-worked-rhocrit}
\rho_{{\rm crit},0}
= \frac{3\Hzero^2}{8\pi G}
= \frac{3\,(2.184\times 10^{-18})^2}
{8\pi\,(6.674\times 10^{-11})}
= 8.53\times 10^{-27}\ \mathrm{kg\,m^{-3}}.
\end{equation}
That is breathtakingly dilute: dividing by the proton mass
$m_p = 1.673\times 10^{-27}\ \mathrm{kg}$ gives
\begin{equation}
\label{appA:frw-eq-worked-rhocrit-protons}
\frac{\rho_{{\rm crit},0}}{m_p}
= 5.10\ \mathrm{protons\,m^{-3}}
\approx 5\ \text{hydrogen atoms per cubic meter}.
\end{equation}
The \emph{actual} mean matter density is $\Omm$ times this, about
$1.6$ proton masses per cubic meter, of which only one part in
six is ordinary baryonic matter.

\paragraph{Step 3: the age of the universe.}
The age is the comoving-time integral of $da/\dot a$ from the Big
Bang ($a=0$) to today ($a=1$). Using $\dot a = a H(a)
= a\Hzero E(a)$ with $E(a)$ from \eqref{appA:frw-eq-Eofa} and
$\Omega_r=\Omega_k=0$,
\begin{equation}
\label{appA:frw-eq-worked-age-int}
t_0
= \int_0^1 \frac{da}{a\,\Hzero\,E(a)}
= \frac{1}{\Hzero}\int_0^1
\frac{da}{a\,\sqrt{\Omm\,a^{-3} + \OmL}}
= \frac{1}{\Hzero}\int_0^1
\frac{\sqrt{a}\,da}{\sqrt{\Omm + \OmL\,a^3}}.
\end{equation}
For a flat two-component (matter-plus-$\Lambda$) universe this
integral has a closed form,
\begin{equation}
\label{appA:frw-eq-worked-age-analytic}
t_0
= \frac{2}{3\Hzero\sqrt{\OmL}}\,
\operatorname{arcsinh}\!\sqrt{\frac{\OmL}{\Omm}}.
\end{equation}
Evaluate the dimensionless factor with the inputs
\eqref{appA:frw-eq-worked-inputs}:
$\sqrt{\OmL/\Omm} = \sqrt{0.685/0.315} = \sqrt{2.1746}
= 1.4747$, and $\operatorname{arcsinh}(1.4747)
= \ln(1.4747 + \sqrt{1 + 1.4747^2}) = \ln(1.4747 + 1.7822)
= \ln(3.2569) = 1.1808$. Then
\begin{equation}
\label{appA:frw-eq-worked-age-factor}
\frac{2}{3\sqrt{\OmL}}\,
\operatorname{arcsinh}\!\sqrt{\frac{\OmL}{\Omm}}
= \frac{2}{3\sqrt{0.685}}\,(1.1808)
= \frac{2}{3\,(0.8276)}\,(1.1808)
= 0.9510,
\end{equation}
so the age is $0.9510$ Hubble times,
\begin{equation}
\label{appA:frw-eq-worked-age}
t_0
= 0.9510\times t_H
= 0.9510\times 14.51\ \mathrm{Gyr}
= 13.80\ \mathrm{Gyr}.
\end{equation}
A direct numerical quadrature of the integral
\eqref{appA:frw-eq-worked-age-int} reproduces the same value
$t_0 = 13.80\ \mathrm{Gyr}$ to four significant figures,
confirming the closed form. This $13.80\ \mathrm{Gyr}$ is the
standard \LCDM{} age, and the way it sits a little below the
Hubble time $14.51\ \mathrm{Gyr}$ is the quantitative fingerprint
of an early matter-dominated, decelerating phase giving way to
late-time $\Lambda$-driven acceleration.

% ---------------------------------------------------------------------
\subsection{Step 3 --- the source ledger: the components from geometry}
\label{appA:frw-sources}

Steps 1 and 2 built the machine; Step 3 loads it. In standard
cosmology this is the moment the fitting begins: one writes
$\rho = \sum_i \rho_i$ with whatever components the data demand and
dials their present-day fractions. Here the loading is a reading, not
a fitting.

\paragraph{The source becomes a ledger (paying debt 3).}
With the first three inputs settled, the right-hand side --- the cosmic
inventory --- is where cosmology proper begins, and it is the subject of
the main text. The free menu $\{\Omega_i,w_i\}$ of standard cosmology is
replaced by a single \emph{constrained ledger}: each component is a
lawful boundary export of the one parent six-dimensional state, so the
inventory is fixed by the geometry and the \OGNfive{} ledger rather than
chosen to fit. This is the sense in which the headline numbers of
\ATen{} --- the $\pi$-locked fractions
\begin{equation}
\Omm=\frac1\pi\approx0.318,\qquad
\OmL=1-\frac1\pi\approx0.682,\qquad
\Omb=\frac1{2\pi^2}\approx0.051
\end{equation}
--- are statements about the \emph{source} of the very same Friedmann
equations derived above, not modifications of them. The continuity
equation being a Ward identity (Step~1) guarantees the ledger is
self-consistent. These fractions are derived in Part~X, and audited there
honestly: at present the inventory is a \emph{conditional} theorem,
resting on a named closure conjecture rather than a finished proof. That
honesty is part of the claim, not a hedge against it.


\paragraph{The components, one by one.}
A first-year student should see the cast list with its provenance and
its effect on the expansion, side by side. \emph{Radiation and
neutrinos} ($w=\tfrac13$, diluting as $a^{-4}$): inherited with the
Standard-Model sector of \citet{ArisakaA9SM}, with the effective
neutrino count $N_{\rm eff} = 3.046 + \Delta N_{\rm eff}$,
$\Delta N_{\rm eff} = 0.04$ --- a derived, testable offset.
\emph{Baryons} ($w=0$, diluting as $a^{-3}$): produced by the
inherited leptogenesis ledger of Part~V, giving the asymmetry $\eta_B \approx 7\times10^{-10}$ from the same integer $K=35$ that fixed
Newton's constant's throat --- register \textbf{Derived (cond.)}, with
an order-of-magnitude transport band carried honestly. \emph{Cold dark
matter} ($w=0$): the Kaxion of the compact angle --- the one field the
geometry insisted on hosting on $S^1_\theta$ --- with mass
$8.1~\mu\mathrm{eV}$ and the misalignment relic abundance worked in
Appendix~\ref{appA:kaxion}; register \textbf{Derived (cond.)} on the
compact-angle-mass-uniqueness lemma. \emph{Dark energy} ($w=-1$,
diluting not at all): the terminus residue of the descent, its scale
the lock-scale seesaw
\begin{equation}
m_\Lambda \;=\; \frac{\Mlock^2}{M_{\rm Pl,red}} \;\approx\; 6.1~\mathrm{meV},
\label{appA:frw-eq-seesaw}
\end{equation}
which collapses the notorious $10^{120}$ vacuum-energy discrepancy to
a residual factor of a few (Problem~E.15.1 computes both sides), with
$w=-1$ exact --- no quintessence dial, no evolving equation of state.
And, as decisively, \emph{nothing else}: the export contains no fifth
component, no extra relativistic species beyond the stated offset, and
no spatial curvature --- the absences are part of the prediction.
\emph{(Physics intuition: standard cosmology's menu is a restaurant
where the universe may order anything; the ledger is a fixed dinner.
The test of a fixed dinner is not whether it can match your appetite
--- it is that the courses were printed before you sat down.)}

\subsection{Step 3 --- the fractions, and the epoch identity}
\label{appA:frw-fractions}

How much of each? This is where standard cosmology fits and
\GGTheory{} computes. The closed-packet topology of the boundary
stratum --- the derivation is Part~X, walked at course pace in
Appendix~\ref{appA:pilock} --- forces the fractions
\begin{equation}
\Omm \;=\; \frac{1}{\pi} \;=\; 0.3183, \qquad
\Omb \;=\; \frac{1}{2\pi^2} \;=\; 0.0507, \qquad
\OmL \;=\; 1 - \frac{1}{\pi} \;=\; 0.6817,
\label{appA:frw-eq-fractions}
\end{equation}
with the dark-matter slice the difference,
$\Omega_{\rm dm} = 1/\pi - 1/(2\pi^2) = 0.2676$. Against the measured
values ($0.315\pm0.007$, $0.0493\pm0.0006$, $0.685\pm0.007$,
$0.2657\pm0.0070$) the raw deviations sit at the percent level ---
Problem~E.16.1 audits the baryon entry, the sharpest of the set. The
honest register, stated exactly as Part~IX grades it: these fractions
are \textbf{Derived (cond.)} on Conjecture~VIII (closed-packet
integration), the load-bearing conditional of the cosmological
codebook, which closes as the \emph{epoch identity} --- $\Omm = 1/\pi$
exact at $t_0 - 75$~Myr under Planck central values, i.e.\ today to
within half a percent of cosmic history --- with residual lemma VIII-a
exact to $10^{-15}$. One declared calibration input, used once,
printed in the open. \emph{(Physics intuition: the pie chart of the
universe, the most famous figure in cosmology, here stops being a
measurement summary and becomes a statement about the topology of a
closed packet --- with the ``why now'' coincidence dissolved into the
statement that the crossing epoch is, on cosmic scales, now;
Problem~E.13.2 locates it.)}

\subsection{Step 3 --- the initial conditions (paying debt 4)}
\label{appA:frw-ics}

The last debt is the starting state: the textbook imposes near-perfect
smoothness, spatial flatness, and a nearly scale-invariant spectrum of
small fluctuations as boundary conditions. In the descent they are
outputs of the launch. Flatness: the derived plateau inflation of
Part~IV (worked at course pace in Appendix~\ref{appA:inflation})
drives $|\Omega_{\rm tot}-1|$ toward zero through
$N^* = 52.6$ e-folds --- and Problem~E.13.3 shows just how badly a
decelerating universe needs that mechanism, tracing today's
near-flatness back to a $10^{-16}$ tuning at nucleosynthesis if
nothing supplies it. Smoothness and the horizon problem: the
shrinking comoving Hubble radius of the launch
(Figure~\ref{fig:appfig-horizons}) puts the whole observable sky in
early causal contact. The spectrum: tilt $n_s = 0.9620$
(\textbf{Derived (cond.)}, plateau class), tensor ratio $r = 0.0043$
(\textbf{Derived} --- the decisive decade-scale prediction), and the
amplitude $A_s$ --- the launch sector's one declared input, flagged as
such wherever it appears. Debt 4 is paid at stated grades, with the
single IOU printed. \emph{(Physics intuition: initial conditions are
where cosmology has always hidden its miracles; the descent replaces
the miracle with a mechanism and then --- the unusual part --- itemizes
the one number it still owes.)}

\subsection{Step 3 --- the assembled expansion law}
\label{appA:frw-assembled}

Putting the three steps together, the expansion law of the universe
can now be written the way no textbook can write it:
\begin{equation}
\boxed{\;
\frac{H^2(a)}{H_0^2}
\;=\; \Omega_r\,a^{-4}
\;+\; \frac{1}{\pi}\,a^{-3}
\;+\; \Big(1-\frac{1}{\pi}\Big),
\qquad k = 0,
\;}
\label{appA:frw-eq-assembled}
\end{equation}
with the matter slice split internally as
$\Omb = 1/(2\pi^2)$ plus $\Omega_{\rm dm} = 1/\pi - 1/(2\pi^2)$, the
radiation term carried at its thermal value with
$N_{\rm eff} = 3.046 + 0.04$, dark energy at $w = -1$ exactly, and the
Hubble constant not fitted but derived: $\Hzero = 66.6$ for the
background, with the CMB reading $67.4$ as the coherent $\pi$-offset
of the two-stage corridor (Part~VI). The perturbation sector rides on
top with $n_s = 0.9620$, $r = 0.0043$, and the declared $A_s$; the
optical depth $\tau$ is late-time astrophysics, outside the
fundamental claim and carried at its measured value. Every symbol in
Eq.~\eqref{appA:frw-eq-assembled} therefore has one of three statuses,
and the sentence bears repeating slowly: \emph{there is no fitted
continuous parameter in the display --- one input is declared ($A_s$),
one identity is epoch-calibrated and printed (Conjecture~VIII), and
everything else is geometry.} The arithmetic check closes the loop:
running Eq.~\eqref{appA:frw-eq-assembled} through the Step-2 machinery
gives the age of the universe as $t_0 = 0.948/H_0$ ---
$13.75$~Gyr at the CMB reading, $13.92$~Gyr at the background reading
(Problem~E.18.1) --- bracketing the measured $13.79$~Gyr, with the
critical density's five protons per cubic meter unchanged from
Step~2's worked example. \LCDM{} is not modified; it is
\emph{concluded}. \emph{(Physics intuition: the six-parameter model
was never six freedoms --- it was six shadows of one geometry, and the
display above is what the concordance model looks like when the
shadows are traced back to the object that casts them.)}

\subsection{What this appendix establishes}
\label{appA:frw-establishes}

Three steps, one arc. Step~1 derived the left-hand side of cosmology:
four-dimensional gravity as the zero-mode shadow of the six-dimensional
geometry --- the Einstein--Hilbert term descending with its scale
weight, Newton's constant computed as an elementary overlap
($M_{\rm Pl}^2 = \pi M_6^4(1-e^{-2K})$), the boundary Einstein
equation obtained by variation and projection, conservation inherited
as a Ward identity, and the open-system correction bounded at
$10^{-91}$ today with its falsifiers named. Step~2 then ran the
textbook derivation of the Friedmann equations, unchanged and
unskipped, on Planck-anchored inputs --- fixing the present critical
density at $\rho_{{\rm crit},0}=8.53\times10^{-27}\ \mathrm{kg\,m^{-3}}$
($\approx5.1$ proton masses per cubic meter) and the age of the
universe at $t_0=13.80$~Gyr: standard cosmology reproduces the
measured universe, exactly as it should, and Step~1 is why it may.
Step~3 replaced the fitted menu with the geometric ledger ---
components with their provenances, the $\pi$-locked fractions at their
stated conditional register, the initial conditions from the launch
--- and assembled the expansion law
\eqref{appA:frw-eq-assembled} with no fitted continuous parameter.
The registers are the claim: the Einstein equation and the Friedmann
system \textbf{Derived}; the inventory \textbf{Derived (cond.)} on the
named closure conjecture of Part~X, with the epoch identity printed;
the amplitude $A_s$ declared; $\tau$ outside the claim. The rigorous
proofs consumed here live in the companions --- the boundary general
relativity of \citet{ArisakaA5GDOS}, the bulk of
\citet{ArisakaA2GG6}, the earned action of \citet{ArisakaA9SM} --- and
this appendix's task has been to show, at textbook depth, exactly how
their theorems become the equations a first-year student already
trusts. In the lineage of Newton, who turned Kepler's measured
ellipses into consequences of a single law, the concordance model is
here reached as a result rather than assumed as a premise.

% =====================================================================
% Arisaka-GG-A10-Cosmos -- Technical Appendix fragment
% Appendix: Cosmic expansion, scaling laws, and thermal history
% Style model: Arisaka-GG-A8-QM-v12.0 (Technical Appendices)
% Macros assumed available from the A10 preamble:
%   \LCDM \Omm \OmL \Hzero \EGUT \GDOS \GGTheory \GGTheory{}
% ASCII only. Insert inside \GGAppendixPart{Technical Appendices}.
% =====================================================================


\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{E.4.1. Do the scale-weight bookkeeping for the
gauge sector, and explain from it why gravity is UV-localized while
gauge forces are not.}

\noindent\emph{Solution.}  Step 1: the measure. From the warped metric,
$\sqrt{-G} = e^{-4r}\sqrt{-g_4}$: each of the four spacetime legs
contributes one factor $e^{-r}$ to the volume element.  Step 2: the
weights. The gauge kinetic term $\mathcal F_{MN}\mathcal F^{MN}$ needs
\emph{two} inverse metrics to raise its four-dimensional indices, each
contributing $e^{+2r}$, so its radial weight is
$e^{-4r}\cdot e^{+4r} = 1$: the gauge action is \emph{uniform} in $r$,
and gauge zero modes spread across the whole throat. The
Einstein--Hilbert term $R_4$ needs only \emph{one} inverse-metric
factor, $e^{+2r}$, so its weight is $e^{-4r}\cdot e^{+2r} = e^{-2r}$:
the gravitational action is exponentially concentrated at small $r$,
the ultraviolet edge.  Step 3: the consequence. Newton's constant is
an overlap dominated by the first stretch of throat (Problem E.5.2
quantifies this), while gauge couplings sample the full depth --- which
is why, in the companion papers, the gauge couplings are seam
registrations of the whole descent while gravity's strength is a
UV-edge quantity, and why gravity is so feeble at the boundary: the
infrared world lives far from where gravity keeps its strength.
Physics intuition: one factor of $e^{2r}$ --- the difference between
raising two indices and raising one --- is the entire explanation for
why you can lift a paper clip with a toy magnet against the pull of a
whole planet.

\medskip\noindent\textbf{E.5.1. Invert the overlap relation for the
bulk scale, and show the flat-space correction is thirty orders of
magnitude beyond any measurement.}

\noindent\emph{Solution.}  Step 1: invert
$M_{\rm Pl}^2 = \pi M_6^4(1-e^{-2K})$. Since $e^{-2K} = e^{-70}$ is
negligible (Step 2), $M_6 = (M_{\rm Pl}^2/\pi)^{1/4}$; with the
reduced Planck mass $M_{\rm Pl,red} = 2.435\times10^{18}$~GeV this
gives $M_6 = (5.93\times10^{36}/\pi)^{1/4} = 1.17\times10^{9}$~GeV
$\approx 1.2\times10^{9}$~GeV --- the single dimensionful scale of the
bulk, a million TeV, fixed by gravity's observed strength and the pure
number $\pi$.  Step 2: the residual. $e^{-70} = 3.98\times10^{-31}$,
so truncating $(1-e^{-2K}) \to 1$ errs at one part in $10^{30}$ ---
some twenty orders of magnitude below the precision of the best
measured constant in physics.  Step 3: the reading. The ledger depth
$K=35$ enters Newton's constant only through this invisible residual:
$G$ cares about the \emph{existence} and shape of the throat, not its
depth, while the particle-physics scales of the ladder
(\S\ref{appA:frw-newton}) care about the depth exponentially. One
geometry, two very different sensitivities.  Physics intuition: the
throat is a well 35 units deep, but gravity's strength is set at the
rim --- you can measure the rim's circumference forever without
learning how deep the well goes; for that you need the ladder.

\medskip\noindent\textbf{E.5.2. Show that most of the Planck mass
lives in the first e-fold of the throat.}

\noindent\emph{Solution.}  Step 1: the share of the overlap integral
accumulated by radial depth $r_\star$ is
$f(r_\star) = \int_0^{r_\star} e^{-2r}dr \big/ \int_0^{K} e^{-2r}dr
= (1-e^{-2r_\star})/(1-e^{-2K})$.  Step 2: evaluate at one e-fold,
$r_\star = 1$: $f(1) = (1-e^{-2})/(1-e^{-70}) = 0.8647$ --- 86.5\% of
$M_{\rm Pl}^2$ accumulates in the first of thirty-five units of
depth; by $r_\star = 3$ the share is $1 - e^{-6} = 99.75\%$.
Step 3: the reading. Four-dimensional gravity is, quantitatively, a
phenomenon of the ultraviolet edge: the deep throat --- where the
Standard Model's hierarchies are manufactured --- contributes almost
nothing to $G$. This is the arithmetic behind the statement that the
graviton zero mode is UV-localized, and it is why no late-time
cosmological measurement can see the throat directly: gravity's
strength was fixed within one e-fold of the rim.  Physics intuition:
$G$ is a rim phenomenon and the hierarchy is a floor phenomenon ---
the same well, read at opposite ends, which is how one geometry can
hold both the weakest force and the widest hierarchy without tension.

\medskip\noindent\textbf{E.6.1. In the geometric derivation,
why is obtaining ONE Friedmann equation enough?}

\noindent\emph{Solution.}  Step 1: Problem E.12.1 shows the system
(Friedmann~1, Friedmann~2, continuity) carries two independent
equations.  Step 2: the boundary law delivers Friedmann~1 (the
Hamiltonian constraint, in relativist's language) and the
ledger's source accounting delivers continuity --- the
open-system bookkeeping of the \GDOS{} descent is precisely
$\dot\rho = -3H(\rho + p)$ read as a channel statement.
Step 3: Friedmann~2 then follows by the algebra of Problem E.12.1 run in
reverse: differentiate and substitute.  Nothing further needs
deriving, and the Ward-identity paragraph of
\S\ref{appA:frw-einstein-earned} is the statement that this
is a feature of the constraint structure, not a gap.
Physics intuition: in both languages --- Bianchi identity or
ledger conservation --- the same moral: a theory whose geometry
owns its bookkeeping only ever owes you one dynamical equation
per symmetry class.


\medskip\noindent\textbf{E.12.1. Derive the continuity equation
from the two Friedmann equations alone.}

\noindent\emph{Solution.}  Step 1: differentiate the first
Friedmann equation, $H^2 = \tfrac{8\pi G}{3}\rho$ (flat case for
clarity): $2H\dot H = \tfrac{8\pi G}{3}\dot\rho$.  Step 2: the
acceleration equation gives $\dot H = \ddot a/a - H^2
= -4\pi G(\rho + p)$.  Step 3: substitute:
$2H\cdot[-4\pi G(\rho + p)] = \tfrac{8\pi G}{3}\dot\rho$, i.e.
$\dot\rho = -3H(\rho + p)$ --- the continuity equation, exactly.
The three equations carry only two independent pieces of content.
Physics intuition: this redundancy is the contracted Bianchi
identity $\nabla_\mu G^{\mu\nu} = 0$ wearing FRW clothes ---
geometry conserves its source by construction, which is why the
chapter's geometric Step could derive one equation and inherit
the other free of charge.

\medskip\noindent\textbf{E.12.2. Obtain the first Friedmann
equation from Newtonian physics, and identify exactly where the
derivation cheats.}

\noindent\emph{Solution.}  Step 1: place a test mass $m$ on the
surface of a comoving ball of radius $R = a(t)\chi$ and uniform
density $\rho$; Newtonian energy conservation gives
$\tfrac12\dot R^2 - \tfrac{GM}{R} = E$ with
$M = \tfrac{4\pi}{3}\rho R^3$.  Step 2: divide by $R^2/2$:
$\bigl(\dot a/a\bigr)^2 = \tfrac{8\pi G}{3}\rho + \tfrac{2E}
{\chi^2 a^2}$ --- the Friedmann equation, with $2E/\chi^2$
playing the role of $-kc^2$.  Step 3: the cheats: (i) Newtonian
gravity has no propagation, so the ball must be small compared
with the Hubble radius for the shell theorem to apply; (ii)
pressure gravitates in relativity ($\rho + 3p$ in the
acceleration equation) and Newton misses it entirely --- the
derivation ``works'' only because the first Friedmann equation
happens not to contain $p$; (iii) the constant $E$ is put in by
hand where relativity supplies spatial curvature with a fixed
normalization.  Physics intuition: the Newtonian shortcut is a
mnemonic, not a derivation --- but it explains why cosmic
expansion obeys projectile logic: escape ($k<0$), capture
($k>0$), and the critical parabola ($k=0$) are the three FRW
geometries in disguise.

\medskip\noindent\textbf{E.13.1. Compute the present deceleration
parameter for the $\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: from the acceleration
equation with matter ($p=0$) and $\Lambda$
($p = -\rho$), $q_0 \equiv -\ddot a a/\dot a^2 =
\tfrac12\Omm - \Omega_\Lambda$.  Step 2: insert the inventory
$\Omm = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$:
$q_0 = \tfrac{1}{2\pi} - \bigl(1 - \tfrac1\pi\bigr)
= \tfrac{3}{2\pi} - 1 \approx -0.523$.  Step 3: the sign is the
1998 discovery --- the expansion accelerates --- and the value is
a zero-parameter prediction of the inventory, sitting well
within the supernova constraints.  Physics intuition: $q_0$
weighs matter's braking (half its density, since only $\rho +
3p$ brakes) against the vacuum's push (full density, twice over,
since $\rho + 3p = -2\rho$); the $\pi$-locked split settles the
contest at three-halves-over-$\pi$ minus one --- geometry
deciding a tug of war.

\medskip\noindent\textbf{E.13.2. Find the redshifts of
matter--vacuum equality and of the onset of acceleration for the
$\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: matter dilutes as
$(1+z)^3$, the vacuum not at all, so $\rho_m = \rho_\Lambda$ at
$(1+z)^3 = \Omega_\Lambda/\Omm = \pi - 1$:
$z_{m\Lambda} = (\pi-1)^{1/3} - 1 \approx 0.29$.  Step 2:
acceleration begins where $q = 0$, i.e.\ $\rho_m = 2\rho_\Lambda$
--- one $(1+z)^3$ factor deeper: $z_{\rm acc} =
\bigl[2(\pi-1)\bigr]^{1/3} - 1 \approx 0.62$.  Step 3: both are
zero-parameter numbers of the inventory, and both sit where the
supernova Hubble diagrams place them.  Physics intuition: the
universe began accelerating when it was about five-eighths of
its present size --- recently, in logarithmic terms, which is
the ``why now'' coincidence the main text derives rather than
laments.


\medskip\noindent\textbf{E.13.3. The flatness problem, quantified:
how flat did the universe have to start?}

\noindent\emph{Solution.}  Step 1: from the first Friedmann equation
with curvature, $\Omega_{\rm tot}(a) - 1 = kc^2/(aH)^2$: the deviation
from flatness scales as the inverse square of the comoving Hubble rate
$aH$. In a decelerating universe $aH$ falls, so $|\Omega_{\rm tot}-1|$
\emph{grows} --- flatness is unstable.  Step 2: run it backwards. In
the radiation era $aH \propto a^{-1}$ and in the matter era
$aH \propto a^{-1/2}$, so from nucleosynthesis
($a_{\rm BBN} \simeq 2.4\times10^{-9}$, at $T \simeq 0.1$~MeV) to
matter--radiation equality ($a_{\rm eq} \simeq 1/3400$) the deviation
grows by $(a_{\rm eq}/a_{\rm BBN})^2 \simeq 1.6\times10^{10}$, and
from equality to today by a further $1/a_{\rm eq} \simeq 3.4\times10^3$
--- a combined factor of about $5\times10^{13}$.  Step 3: impose the
present bound $|\Omega_{\rm tot}-1| < 0.005$: at nucleosynthesis the
universe had to satisfy $|\Omega_{\rm tot}-1| \lesssim 10^{-16}$ ---
and pushing back to the GUT edge makes the required tuning tens of
orders sharper still. Either the universe began absurdly fine-tuned,
or an era in which $aH$ \emph{rose} drove it flat: exactly the
shrinking comoving Hubble radius of the derived launch
(\S\ref{appA:frw-ics} and Figure~\ref{fig:appfig-horizons}), which
pays this debt mechanically.  Physics intuition: a pencil standing on
its point for fourteen billion years is not an initial condition worth
believing; inflation is the hand that placed it upright just before
you looked, and in the descent that hand is a theorem, not a story.

\medskip\noindent\textbf{E.14.1. Derive the closed-form age of a
flat matter-plus-$\Lambda$ universe and evaluate it for the
$\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: with
$H^2 = H_0^2[\Omm a^{-3} + \Omega_\Lambda]$, the age integral
$t_0 = \int_0^1 da/(aH)$ substitutes $u = a^{3/2}$ into an
elementary form, giving
\begin{equation*}
t_0 = \frac{2}{3H_0\sqrt{\Omega_\Lambda}}\,
\mathrm{arcsinh}\sqrt{\frac{\Omega_\Lambda}{\Omm}}\,.
\end{equation*}
Step 2: insert $\Omm = 1/\pi$ and the Planck-calibrated
$H_0 = 67.4~\mathrm{km\,s^{-1}Mpc^{-1}}$ ($1/H_0 = 14.51$~Gyr):
$t_0 = 0.948/H_0 = 13.75$~Gyr.  Step 3: compare the
matter-only answer $\tfrac23 H_0^{-1} = 9.7$~Gyr --- younger
than the oldest globular clusters, the classic ``age crisis''
that $\Lambda$ resolved.  Physics intuition: the vacuum
stretches the timeline because early expansion was slower than
a matter-only history extrapolates; the $\pi$-locked inventory
lands the age at $13.7$--$13.8$~Gyr with nothing tuned --- the
first sanity check any cosmology must pass, passed from
integers.

\medskip\noindent\textbf{E.14.2. Express the critical density in
protons per cubic meter.}

\noindent\emph{Solution.}  Step 1: $\rho_c = 3H_0^2/8\pi G$;
with $H_0 = 67.4~\mathrm{km\,s^{-1}Mpc^{-1}} = 2.18\times
10^{-18}\,\mathrm{s^{-1}}$, $\rho_c = 8.5\times10^{-27}\,
\mathrm{kg\,m^{-3}}$.  Step 2: dividing by the proton mass,
$\rho_c \approx 5.1$ protons per cubic meter.  Step 3: of this,
the inventory assigns baryons only $\Omega_b = 1/(2\pi^2)
\approx 5\%$ --- about one proton per four cubic meters; the
rest is dark matter ($\Omm - \Omega_b$) and vacuum.
Physics intuition: the entire cosmic budget, whose split the
main text derives from topology, totals five hydrogen atoms per
meter cubed --- emptier than any laboratory vacuum ever pumped,
and yet enough, integrated over the Hubble volume, to curve the
universe flat.


\medskip\noindent\textbf{E.15.1. Compute both sides of the
cosmological-constant problem: the seesaw scale and the measured
dark-energy scale.}

\noindent\emph{Solution.}  Step 1: the geometry's number. The
lock-scale seesaw \eqref{appA:frw-eq-seesaw} gives
$m_\Lambda = \Mlock^2/M_{\rm Pl,red}
= (3.85\times10^{3})^2/2.435\times10^{18}~\mathrm{GeV}
= 6.1\times10^{-12}~\mathrm{GeV} = 6.1$~meV.  Step 2: the sky's
number. The measured dark-energy density is
$\rho_\Lambda = \OmL\,\rho_{\rm crit}$; with
$\rho_{\rm crit} = 3H_0^2 M_{\rm Pl,red}^2$ at the CMB reading this
evaluates to $\rho_\Lambda^{1/4} = 2.24$~meV.  Step 3: the comparison.
The two sit a factor $\sim 2.7$ apart --- after one hundred and twenty
orders of magnitude of vacuum-energy catastrophe, the seesaw lands on
the observed scale to within a small factor, and that residual factor
is not waved away: Part~IV names it as the open drainage coefficient
of the terminus (a flagged interpretation, register open), the one
number of the dark-energy story still awaiting computation.  Physics
intuition: the cosmological-constant problem was never ``why is
$\Lambda$ small?'' --- it was ``what sets its scale at all?''; a
TeV-squared-over-Planck seesaw answers the second question with
millielectronvolts, and honest bookkeeping keeps the last factor of
order unity on the table rather than under it.

\medskip\noindent\textbf{E.16.1. Audit the baryon fraction: the
sharpest entry of the $\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: the prediction.
$\Omb = 1/(2\pi^2) = 0.050660$, with no adjustable input.  Step 2: the
comparison. Against the measured $0.0493\pm0.0006$ the raw deviation
is $0.05066/0.0493 - 1 = +2.8\%$, a pull of about $+2.3$; in the
physical combination $\Omb h^2$ the prediction reads
$0.0507\times(0.674)^2 = 0.0230$ against the measured $0.02237$.
Step 3: the register. Part~IX carries this entry at its raw value
with the epoch calibration reducing the deviation to $+1.69\%$
(pull $+1.4$) --- the sharpest single test of the closed-packet
fractions, and a named kill criterion: a future sub-percent baryon
census that pins $\Omb$ away from $1/(2\pi^2)$ at the calibrated
level falsifies the packet, full stop.  Physics intuition: it is easy
to admire $\Omm = 1/\pi$ and miss that the baryon slice is the
stricter judge --- two powers of $\pi$, a smaller error bar, and
nowhere to hide; the inventory lives or dies at the second decimal of
five percent.

\medskip\noindent\textbf{E.18.1. Compute the age of the universe at
both corridor readings of the derived Hubble constant.}

%% v15.0 Stage 1: on Letter this closed form re-broke into a line 58.23pt
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\noindent\emph{Solution.}  Step 1: the closed form of
Problem~E.14.1, with the $\pi$-locked inventory,
\[
  t_0 = \frac{2}{3H_0\sqrt{\OmL}}\,
  \mathrm{arcsinh}\sqrt{\OmL/\Omm} = 0.948/H_0,
\]
and note the small
gift of the inventory: $1 + \OmL/\Omm = 1/\Omm = \pi$, so the
argument of the arcsinh is $\sqrt{\pi - 1}$.  Step 2: the two
readings. At the background value $H_0 = 66.6$:
$1/H_0 = 14.68$~Gyr and $t_0 = 13.92$~Gyr. At the CMB reading
$H_0 = 67.4$: $1/H_0 = 14.51$~Gyr and $t_0 = 13.75$~Gyr.  Step 3: the
comparison. The measured age is $13.79$~Gyr --- inside the corridor
the two readings bracket, with no parameter adjusted anywhere in the
computation: the inventory came from topology, the Hubble constant
from the two-stage corridor, and the formula from Step~2 of this
appendix.  Physics intuition: the age of the universe is the single
most public number in cosmology, and here it is a two-line consequence
of $\pi$ and a derived clock rate --- the kind of check a first-year
student can run on paper, which is exactly why it belongs at the end
of the walk.

\GGtakeaway{\emph{Standard result:} the FRW metric plus general
relativity gives the two Friedmann equations, and with a measured
inventory they yield the expansion history and the age of the universe.
\emph{What it assumed:} general relativity itself and its constant,
spatial flatness, and an inventory of densities read from the sky ---
the four debts of \S\ref{appA:frw-limits}. \emph{What the geometry
replaced it with:} the Einstein equation and Newton's constant derived
from the bulk in Step~1 (conservation a Ward identity, the correction
bounded, the falsifiers live); the Friedmann equations unchanged in
Step~2, their provenance changed; and in Step~3 the inventory read
from the ledger --- components from geometry, $\pi$-locked fractions
at their stated conditional register, flatness and the spectrum from
the derived launch (Appendix~G), the deep topology in Appendix~L ---
closing in an expansion law with no fitted continuous parameter.
\emph{Continue in:} Part~III (the stage), Appendix~F (the clock),
Part~IX (the codebook that grades every entry).}

\section{Cosmic expansion, scaling laws, and thermal history}
\label{appA:thermal}

\GGteachbox{How each energy component dilutes (\(\rho \propto
a^{-3(1+w)}\)) and how the scale factor grows in each era; redshift,
entropy conservation, and the freeze-out criterion \(\Gamma \sim H\)
that governs every relic in the universe; the thermal milestones
(equality, recombination, nucleosynthesis) computed with worked numbers;
the three horizons; and the geometric reading in which the whole thermal
history becomes radial transport --- one logarithmic coordinate,
\(r = \ln(\EGUT/\mu)\), on which every epoch has a lawful address.}
\GGreadalong{\citet{KolbTurner1990} chs.~3--5 (the classic);
\citet{Ryden2017} chs.~8--10 for a first pass; \citet{Baumann2022}
ch.~3 at course level.}

The relativistic degree-of-freedom counting, entropy conservation, and freeze-out machinery used throughout this appendix are developed in the standard early-universe references \citep{KolbTurner1990,Weinberg2008,Peebles1993}.

This appendix is a self-contained, first-principles account of the
standard expansion and thermal history of the universe, written for a
first-year graduate reader who has met special relativity and
elementary thermodynamics but not yet the Friedmann--Lemaitre
machinery. Every result the main text uses --- the scaling
$\rho \propto a^{-3(1+w)}$, the time dependence $a(t)$ in each
era, the redshift law $1+z = 1/a$, the cooling law $T \propto 1/a$
with its $g_*$ corrections, the thermal milestones (matter--radiation
equality, recombination, BBN), the freeze-out criterion
$\Gamma \simeq H$, and the three cosmological horizons --- is derived
here line by line. The reader who wants only the central numbers can
read the worked examples in \S\ref{appA:thermal-worked} and the
Status note in \S\ref{appA:thermal-status}; the reader who wants to
verify each step should follow the appendix in order.

We build this appendix in three steps, exactly as in Appendix~E.

\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard thermal history
(\S\ref{appA:thermal-continuity}--\ref{appA:thermal-horizons}).} The
scaling laws, the time dependence $a(t)$, the redshift and cooling laws,
entropy conservation, the freeze-out criterion, the thermal milestones,
and the three horizons --- all derived line by line, with worked numbers
in \S\ref{appA:thermal-worked}.
\item \textbf{What the standard account leaves open
(\S\ref{appA:thermal-limits}).} The milestones are computed by inserting
\emph{measured} densities into the equilibrium formulas; the standard
treatment has no deeper account of why the switch redshifts fall where
they do.
\item \textbf{The geometric derivation (\S\ref{appA:thermal-gg}).} The
cosmic clock is read as a single \GDOS{} descent, the milestones as
\GRIP{}-Out switch events along it, and recombination as the one switch
the integer ledger pins to a parameter-free value, $z_*=e^7$ --- stated,
as always, with both its near-hit and its nominal-tension framings.
\end{enumerate}

The Friedmann equations and the equilibrium thermodynamics below are
textbook physics the cohort retains intact; the geometric content is the
reading of the descent that underlies them. This is the same belief that
runs through the program --- that the structure of the world should
follow from geometry --- applied now to the clock of cosmic history.

Throughout we use the metric signature $(-,+,+,+)$, set $c=1$, write
$M_{\mathrm{Pl}} = G_N^{-1/2} \approx 1.22\times 10^{19}$~GeV for the
(non-reduced) Planck mass, and take the spatially flat
Friedmann--Lemaitre--Robertson--Walker (FLRW) line element with scale
factor $a(t)$,
\begin{equation}
\dd s^2 = -\dd t^2 + a^2(t)\,
\bigl[\dd r^2 + r^2\,\dd\Omega^2\bigr],
\qquad a(t_0) \equiv 1,
\label{eq:appA:thermal-flrw}
\end{equation}
normalized so that the scale factor today is unity. The two Friedmann
equations that govern $a(t)$ are
\begin{align}
H^2 &\equiv \Bigl(\frac{\dot a}{a}\Bigr)^2
   = \frac{8\pi G_N}{3}\,\rho ,
   \label{eq:appA:thermal-friedmann1}\\
\frac{\ddot a}{a}
   &= -\frac{4\pi G_N}{3}\,(\rho + 3p),
   \label{eq:appA:thermal-friedmann2}
\end{align}
where $\rho$ and $p$ are the total energy density and pressure of the
cosmic fluid, $H$ is the Hubble rate, and an overdot is $\dd/\dd t$.
We have set the spatial curvature $k=0$ and absorbed any cosmological
constant into $\rho$ as a $w=-1$ component, consistent with the
near-flat, $\OmL$-dominated present epoch used in the main text.

% ---------------------------------------------------------------
\subsection{The continuity equation and the scaling law
$\rho \propto a^{-3(1+w)}$}
\label{appA:thermal-continuity}

The first task is to find how the energy density of a single fluid
component dilutes as the universe expands. The two Friedmann equations
\eqref{eq:appA:thermal-friedmann1}--\eqref{eq:appA:thermal-friedmann2}
are not independent: local energy--momentum conservation
$\nabla_\mu T^{\mu\nu}=0$ ties them together. For the homogeneous,
isotropic fluid the $\nu=0$ component of that conservation law is the
\emph{continuity equation}
\begin{equation}
\dot\rho + 3H\,(\rho + p) = 0.
\label{eq:appA:thermal-continuity}
\end{equation}
It is worth seeing this in two ways, because both readings recur in
the main text.

\paragraph{Thermodynamic derivation.} Treat a comoving volume
$V \propto a^3$ as a thermodynamic system. Its energy is $E=\rho V$
and its volume is $V$. In the early universe expansion is adiabatic to
excellent approximation (no heat flows across a comoving boundary in a
homogeneous universe, since every patch is identical), so the first
law $\dd E = -p\,\dd V$ holds with $\dd Q=0$. Writing
$\dd(\rho a^3) = -p\,\dd(a^3)$ and dividing by $\dd t$,
\begin{equation}
\dot\rho\,a^3 + 3\rho\,a^2\dot a = -p\,3a^2\dot a
\;\Longrightarrow\;
\dot\rho + 3\frac{\dot a}{a}(\rho+p)=0,
\label{eq:appA:thermal-firstlaw}
\end{equation}
which is exactly \eqref{eq:appA:thermal-continuity}. The expansion
does work against (or, for $p<0$, is driven by) the pressure, and that
work is what depletes $\rho$ beyond the geometric $a^{-3}$ dilution of
number density.

\paragraph{Covariant derivation.} Equivalently, differentiate
\eqref{eq:appA:thermal-friedmann1} to get
$2H\dot H = (8\pi G_N/3)\dot\rho$, use
\eqref{eq:appA:thermal-friedmann2} to eliminate $\dot H$ via
$\dot H = \ddot a/a - H^2 = -4\pi G_N(\rho+p)$, and the continuity
equation drops out. The two Friedmann equations plus continuity are
any two of the three; the third is a consequence.

Now specialize to a component with a constant linear (barotropic)
equation of state
\begin{equation}
p = w\,\rho,
\qquad w = \mathrm{const}.
\label{eq:appA:thermal-eos}
\end{equation}
Substituting into \eqref{eq:appA:thermal-continuity},
\begin{equation}
\frac{\dot\rho}{\rho}
= -3(1+w)\,\frac{\dot a}{a},
\label{eq:appA:thermal-separable}
\end{equation}
which is separable. Integrating $\int \dd\ln\rho = -3(1+w)\int \dd\ln a$
gives the master scaling law
\begin{equation}
\boxed{\;\rho(a) = \rho_0\,a^{-3(1+w)}\;}
\label{eq:appA:thermal-master-scaling}
\end{equation}
with $\rho_0$ the present-day value at $a=1$. The three cases that
organize all of standard cosmology follow by inserting the
appropriate $w$:
\begin{itemize}
\item \textbf{Radiation} ($w=\tfrac{1}{3}$, relativistic particles or
massless fields, $p=\rho/3$): $\rho_r \propto a^{-4}$. The extra
power of $a$ beyond the $a^{-3}$ number dilution is the redshift of
each quantum's energy, $E\propto a^{-1}$ (\S\ref{appA:thermal-redshift}).
\item \textbf{Matter / dust} ($w=0$, nonrelativistic, $p\ll\rho$):
$\rho_m \propto a^{-3}$. Pure number dilution; the rest mass does not
redshift.
\item \textbf{Vacuum energy / cosmological constant} ($w=-1$,
$p=-\rho$): $\rho_\Lambda \propto a^{0} = \mathrm{const}$. The
$(1+w)=0$ factor switches off the dilution entirely; a true vacuum
neither dilutes nor cools.
\end{itemize}
A useful intermediate is the spatial-curvature term, which in the
Friedmann equation contributes like a fluid with $w=-1/3$, i.e.
$\rho_k \propto a^{-2}$; this is why a curvature contribution, if
present, would interpolate between matter and $\Lambda$. The reason
$\Lambda$ eventually dominates is now algebraically transparent: among
$\{a^{-4}, a^{-3}, a^{-2}, a^{0}\}$ the constant term wins at large
$a$, the steepest term wins at small $a$. The universe therefore
passes generically through a radiation era, a matter era, and a
vacuum era in that temporal order, simply because the scaling
exponents are ordered.

% ---------------------------------------------------------------
\subsection{Solving for $a(t)$: radiation, matter, and
$\Lambda$ domination}
\label{appA:thermal-aoft}

With \eqref{eq:appA:thermal-master-scaling} in hand the first
Friedmann equation \eqref{eq:appA:thermal-friedmann1} becomes a closed
ordinary differential equation for $a(t)$ whenever one component
dominates. Write $\rho = \rho_0\,a^{-3(1+w)}$ and
\eqref{eq:appA:thermal-friedmann1} reads
\begin{equation}
\dot a = H_0\,a^{-(1+3w)/2},
\qquad H_0 \equiv \sqrt{\tfrac{8\pi G_N}{3}\rho_0}.
\label{eq:appA:thermal-aoft-ode}
\end{equation}

\paragraph{Power-law eras ($w \ne -1$).} Separating variables,
$\int a^{(1+3w)/2}\,\dd a \propto \int \dd t$, gives
$a^{(3+3w)/2} \propto t$, i.e.
\begin{equation}
\boxed{\;a(t) \propto t^{\,2/[3(1+w)]}\;}
\qquad (w\ne -1).
\label{eq:appA:thermal-aoft-power}
\end{equation}
Reading off the exponents:
\begin{itemize}
\item \textbf{Radiation domination} ($w=1/3$):
$a(t)\propto t^{1/2}$, so $H = \dot a/a = 1/(2t)$.
\item \textbf{Matter domination} ($w=0$):
$a(t)\propto t^{2/3}$, so $H = 2/(3t)$.
\end{itemize}
Both are decelerating, $\ddot a < 0$, as
\eqref{eq:appA:thermal-friedmann2} demands for $\rho+3p>0$. The
relation $H=1/(2t)$ in the radiation era is the one used in
Eq.~\eqref{eq:partII-time-temperature} of the main text to convert
temperatures into ages.

\paragraph{Vacuum domination ($w=-1$).} Here $\rho$ is constant, so
\eqref{eq:appA:thermal-friedmann1} gives $H=\mathrm{const}$ and the
solution is the de Sitter exponential
\begin{equation}
\boxed{\;a(t) \propto e^{H t},
\qquad H = \sqrt{\tfrac{8\pi G_N}{3}\rho_\Lambda}\;}
\label{eq:appA:thermal-aoft-desitter}
\end{equation}
which is accelerating, $\ddot a>0$. This is both the late-time fate of
the $\OmL$-dominated universe and the prototype for an inflationary
quasi-de Sitter stage. The transition redshifts between eras are fixed
by where the scaling curves cross; for the present mixture they occur
at matter--radiation equality (\S\ref{appA:thermal-milestones}) and at
the matter--$\Lambda$ crossover near $1+z \approx (\OmL/\Omm)^{1/3}$,
which for the main-text values $\Omm \approx 1/\pi$,
$\OmL \approx 1-1/\pi$ gives $z\approx 0.30$.

\paragraph{General multi-component $H(a)$.} When no single component
dominates the exact Hubble rate is the sum
\begin{equation}
H^2(a) = H_0^2
\Bigl[\Omega_r\,a^{-4} + \Omm\,a^{-3}
      + \Omega_k\,a^{-2} + \OmL\Bigr],
\label{eq:appA:thermal-Hofa}
\end{equation}
where $\Omega_i \equiv \rho_{i,0}/\rho_{\mathrm{crit},0}$ and
$\rho_{\mathrm{crit},0} = 3\Hzero^2/(8\pi G_N)$. Equation
\eqref{eq:appA:thermal-Hofa} is the integrand that feeds every horizon
integral in \S\ref{appA:thermal-horizons}; the single-component
solutions above are its asymptotic limits.

% ---------------------------------------------------------------
\subsection{Redshift $1+z = 1/a$ and the redshifting of momenta and
temperature}
\label{appA:thermal-redshift}

The scale factor is not directly observable; the redshift is. Consider
a photon emitted at scale factor $a_e$ with wavelength
$\lambda_e$ and observed today ($a=1$) with wavelength $\lambda_0$.
Null geodesics in the metric \eqref{eq:appA:thermal-flrw} satisfy
$\dd t = a\,\dd r$ along the line of sight; two successive wavecrests
emitted a period $\delta t_e$ apart are received a period
$\delta t_0$ apart, and because the comoving distance $\int \dd t/a$
is the same for both crests one finds
$\delta t_0/\delta t_e = a_0/a_e = 1/a_e$. Wavelength tracks the
period, so
\begin{equation}
\boxed{\;1+z \equiv \frac{\lambda_0}{\lambda_e}
= \frac{1}{a_e}\;}
\label{eq:appA:thermal-redshift-def}
\end{equation}
with $a_0=1$. Thus $a = 1/(1+z)$: high redshift is small scale factor,
the distant past.

\paragraph{Momentum redshift.} The same calculation done for the de
Broglie wavelength of any free particle, or equivalently from the
geodesic equation for the physical momentum $|\vec p|$, gives the
master kinematic law
\begin{equation}
|\vec p\,| \propto a^{-1}.
\label{eq:appA:thermal-momentum}
\end{equation}
For a relativistic particle $E\simeq|\vec p\,|$, so $E\propto a^{-1}$:
this is the missing fourth power in $\rho_r\propto a^{-4}$ promised in
\S\ref{appA:thermal-continuity}. For a nonrelativistic particle the
\emph{kinetic} energy is $|\vec p\,|^2/2m \propto a^{-2}$, while the
rest energy $m$ is fixed; this is why a decoupled nonrelativistic gas
cools as $T_{\mathrm{nr}}\propto a^{-2}$, faster than the photon bath
(cf.\ Eq.~\eqref{eq:partII-T-nonrel} of the main text).

\paragraph{Temperature redshift of a thermal bath.} A bath of
massless particles with phase-space distribution
$f(p) = [\exp(p/T)\mp 1]^{-1}$ stays a thermal distribution under
expansion because every momentum redshifts by the same factor
$a^{-1}$ \eqref{eq:appA:thermal-momentum}. Rescaling
$p\to p/a$ in $f$ is undone exactly by rescaling
$T\to T/a$. Hence a free relativistic thermal bath remains thermal
with
\begin{equation}
\boxed{\;T \propto a^{-1} = 1+z\;}
\label{eq:appA:thermal-Tprop}
\end{equation}
\emph{provided no entropy is injected}. The crucial qualification ---
``provided no entropy is injected'' --- is the subject of the next
subsection, where the $g_*$ bookkeeping makes the law slightly more
intricate but leaves its leading behavior intact.

% ---------------------------------------------------------------
\subsection{Entropy conservation, $g_{*s}T^3a^3 = \mathrm{const}$,
and reheating of the photons}
\label{appA:thermal-entropy}

The cleaner and more general statement of the cooling law uses
\emph{comoving entropy conservation} rather than the naive
$T\propto a^{-1}$. For a relativistic plasma in equilibrium at
temperature $T$ the energy density, pressure, and entropy density are
\begin{align}
\rho_r &= \frac{\pi^2}{30}\,g_*(T)\,T^4,
&
p_r &= \frac{1}{3}\rho_r,
\label{eq:appA:thermal-rho-of-T}\\
s &= \frac{\rho_r+p_r}{T}
   = \frac{2\pi^2}{45}\,g_{*s}(T)\,T^3,
\label{eq:appA:thermal-s-of-T}
\end{align}
matching Eqs.~\eqref{eq:partII-rho-temperature}--%
\eqref{eq:partII-entropy-density} of the main text. The two
effective-degrees-of-freedom counters differ in general:
\begin{align}
g_*(T)   &= \sum_{i\in\mathrm{bosons}}
            g_i\Bigl(\frac{T_i}{T}\Bigr)^4
          + \frac{7}{8}\!\!\sum_{i\in\mathrm{fermions}}\!\!
            g_i\Bigl(\frac{T_i}{T}\Bigr)^4 ,
\label{eq:appA:thermal-gstar}\\
g_{*s}(T) &= \sum_{i\in\mathrm{bosons}}
            g_i\Bigl(\frac{T_i}{T}\Bigr)^3
          + \frac{7}{8}\!\!\sum_{i\in\mathrm{fermions}}\!\!
            g_i\Bigl(\frac{T_i}{T}\Bigr)^3 ,
\label{eq:appA:thermal-gstars}
\end{align}
where $g_i$ is the internal multiplicity (spin times particle/
antiparticle), the factor $7/8$ is the Fermi--Dirac correction to the
relativistic energy and entropy integrals, and $T_i$ is the actual
temperature of species $i$, which may differ from the photon
temperature $T$ once that species has decoupled. When all relativistic
species share one temperature, $g_* = g_{*s}$.

The Second Law in a homogeneous universe, where adiabatic expansion
injects no heat, says the total entropy in a comoving volume is
conserved:
\begin{equation}
\boxed{\;S = s\,a^3 = \frac{2\pi^2}{45}\,g_{*s}(T)\,T^3 a^3
= \mathrm{const}\;}
\label{eq:appA:thermal-comoving-S}
\end{equation}
which is the integrated form of the main-text identity
$\dd[s\,a^3]/\dd t = 0$, Eq.~\eqref{eq:partII-comoving-entropy}.
Solving \eqref{eq:appA:thermal-comoving-S} for the temperature gives
the exact cooling law
\begin{equation}
T(a) = T_0\,
\Bigl[\frac{g_{*s}(T_0)}{g_{*s}(T)}\Bigr]^{1/3} a^{-1}
\;\;\propto\;\; g_{*s}^{-1/3}(T)\,a^{-1},
\label{eq:appA:thermal-Texact}
\end{equation}
reproducing Eq.~\eqref{eq:partII-T-with-gstar}. Away from a
particle-annihilation threshold $g_{*s}$ is constant and one recovers
$T\propto a^{-1}$ exactly; \emph{at} a threshold, where a species
becomes nonrelativistic and annihilates, $g_{*s}$ drops, and
\eqref{eq:appA:thermal-Texact} says the still-coupled bath cools
\emph{more slowly} than $a^{-1}$ for a while --- the annihilating
species dumps its entropy into the bath and \emph{reheats} it relative
to the naive expectation.

\paragraph{Worked $g_*$ step: $e^+e^-$ annihilation and the
neutrino temperature.} The canonical illustration, and the one the
main text relies on for the neutrino background, is electron--positron
annihilation around $T\sim m_e \approx 0.511$~MeV. Just above this
threshold the relativistic bath is photons ($g=2$), electrons and
positrons ($g = 4$, i.e.\ two spin states times particle/antiparticle),
and three neutrino species ($g = 2\times 3 = 6$, one helicity each for
$\nu$ and $\bar\nu$). The neutrinos have \emph{already decoupled}
(\S\ref{appA:thermal-freezeout}) at $T\sim 1$~MeV, so the $e^+e^-$
entropy is dumped only into the photons. Counting the
entropy-coupled bath that the photons share:
\begin{equation}
g_{*s}^{\gamma e}(\text{before})
= \underbrace{2}_{\gamma}
+ \frac{7}{8}\underbrace{4}_{e^\pm}
= 2 + \frac{7}{2} = \frac{11}{2},
\qquad
g_{*s}^{\gamma}(\text{after}) = 2.
\label{eq:appA:thermal-gstar-ee}
\end{equation}
Entropy conservation in the photon--electron sector,
$g_{*s}^{\gamma e}(T_\gamma a)^3 = \mathrm{const}$, forces the photon
temperature to drop more slowly than $a^{-1}$ across the threshold by
the factor $(11/4)^{1/3}$ relative to the decoupled, freely-cooling
neutrinos:
\begin{equation}
\boxed{\;\frac{T_\nu}{T_\gamma}
= \Bigl(\frac{4}{11}\Bigr)^{1/3} \approx 0.714\;}
\label{eq:appA:thermal-neutrino-T}
\end{equation}
(The $4$ rather than $11/2$ in the ratio is because only the
\emph{net} entropy transferred enters the standard algebra, which
gives $4/11$ after the photon-only counting $g_{*s}^{\gamma}=2$ is
divided out of $11/2$.) Today $T_\gamma = 2.725$~K, so
$T_\nu \approx 1.95$~K. The same reheating makes today's value
$g_{*s}(T_0) = 2 + (7/8)\cdot 6\cdot(4/11)
\approx 3.91$, while the photon-only $g_*(T_0) = 2$ --- the daily
reminder that the two counters genuinely differ once a species has
left the photon bath. This is the single worked $g_*$ step that the
main text presumes when it speaks of photon last scattering against a
colder neutrino background.

% ---------------------------------------------------------------
\subsection{Freeze-out: the $\Gamma/H$ competition and how decoupling
works}
\label{appA:thermal-freezeout}

Every thermal milestone --- neutrino decoupling, the freeze-out of a
dark-matter relic, recombination, even BBN --- is at bottom a
\emph{race between two rates}: the interaction rate $\Gamma$ that
keeps a species in equilibrium and the expansion rate $H$ that pulls
the bath apart. The interaction rate per particle is
\begin{equation}
\Gamma = n\,\langle\sigma v\rangle,
\label{eq:appA:thermal-Gamma}
\end{equation}
with $n$ the number density of targets, $\sigma$ the relevant cross
section, and $v$ the relative velocity. The expansion rate in the
radiation era is, from \eqref{eq:appA:thermal-friedmann1} and
\eqref{eq:appA:thermal-rho-of-T},
\begin{equation}
H(T) = \sqrt{\frac{8\pi G_N}{3}\frac{\pi^2}{30}g_*T^4}
\simeq 1.66\,g_*^{1/2}\,\frac{T^2}{M_{\mathrm{Pl}}},
\label{eq:appA:thermal-HofT}
\end{equation}
matching Eq.~\eqref{eq:partII-H-of-T}. The qualitative criterion is
\begin{equation}
\boxed{\;
\Gamma \gg H:\ \text{coupled (equilibrium)};
\quad
\Gamma \ll H:\ \text{decoupled (frozen out)}.
\;}
\label{eq:appA:thermal-decouple-criterion}
\end{equation}
The crossover \emph{freeze-out temperature} $T_f$ is defined
operationally by $\Gamma(T_f) \simeq H(T_f)$. The logic of why this is
a sharp transition is that $\Gamma$ and $H$ typically scale as
different powers of $T$: for a weak-interaction process
$\Gamma \sim G_F^2 T^5$ falls faster than $H\sim T^2$ as the universe
cools, so once $\Gamma$ drops below $H$ it stays below, and the
abundance ``freezes'' at its last-equilibrium value, thereafter merely
diluting as $a^{-3}$. For neutrinos, equating $G_F^2 T^5 \sim
T^2/M_{\mathrm{Pl}}$ gives $T_f \sim 1$~MeV, which is why the
neutrinos are already decoupled when $e^+e^-$ annihilate just below
$0.511$~MeV --- precisely the input assumed in
\eqref{eq:appA:thermal-neutrino-T}.

Within \GDOS{} this same $\Gamma/H$ competition is the operational
face of a branch-selection event: the system first explores
microscopically equivalent histories while coupled, then crosses the
$\Gamma\simeq H$ threshold, and finally stabilizes a new
coarse-grained branch that the background expansion preserves. The
thermodynamic and the open-system readings agree on the arithmetic;
they differ only in what they say the frozen-out abundance
\emph{means}.

% ---------------------------------------------------------------
\subsection{The thermal milestones: equality, recombination, BBN}
\label{appA:thermal-milestones}

The scaling laws and the freeze-out criterion fix the calendar of the
early universe. Three milestones anchor everything the main text says
about the CMB and light elements.

\paragraph{Matter--radiation equality, $z_{\mathrm{eq}}$.} Radiation
($\propto a^{-4}$) outruns matter ($\propto a^{-3}$) toward small $a$,
so there is a redshift where the two densities cross:
$\Omega_r a^{-4} = \Omm a^{-3}$, giving
\begin{equation}
1 + z_{\mathrm{eq}} = \frac{\Omm}{\Omega_r}.
\label{eq:appA:thermal-zeq}
\end{equation}
Before $z_{\mathrm{eq}}$ the dynamics are radiation-driven
($a\propto t^{1/2}$); after, matter-driven ($a\propto t^{2/3}$).
Structure growth effectively switches on after equality, so
$z_{\mathrm{eq}}$ sets the scale of the matter power spectrum turnover.
A worked value appears in \S\ref{appA:thermal-worked}.

\paragraph{Recombination and photon decoupling, $z_{\mathrm{rec}}$.}
As the photon bath cools below $T \sim 0.3$~eV the reaction
$p + e^- \leftrightarrow \mathrm{H} + \gamma$ shifts toward neutral
hydrogen (the binding energy is $13.6$~eV, but the enormous
photon-to-baryon ratio $\eta^{-1}\sim 10^9$ delays recombination until
$T$ is far below $13.6$~eV --- the high-energy tail of the Planck
distribution keeps hydrogen ionized until then; this is the Saha
suppression). Once the free-electron fraction collapses, Thomson
scattering shuts off, $\Gamma_\gamma \ll H$, and the photons
free-stream: this is the surface of last scattering we observe as the
CMB. The Planck-measured value is
\begin{equation}
z_{\mathrm{rec}} = 1089.92 \pm 0.25,
\label{eq:appA:thermal-zrec-planck}
\end{equation}
with a finite \emph{visibility width} $\Delta z \approx 90$ over which
the last-scattering probability is appreciable --- the CMB is emitted
from a shell of finite thickness, not a mathematical surface. That
width is the physically honest yardstick against which any predicted
$z_*$ must be compared, and it is the crux of
\S\ref{appA:thermal-gg}.

\paragraph{Big-Bang nucleosynthesis (BBN).} Earlier still, at
$T\sim 0.1$--$1$~MeV ($t\sim 1$--$200$~s by
Eq.~\eqref{eq:partII-time-temperature}), the light elements form.
Neutron--proton interconversion freezes out near $T\sim 0.8$~MeV,
fixing $n/p \approx e^{-(m_n-m_p)/T_f} \approx 1/6$; after free-neutron
decay this falls to $\approx 1/7$ by the onset of deuterium synthesis
at $T\sim 0.07$~MeV, giving the famous primordial helium mass fraction
$Y_p \approx 2(n/p)/[1+(n/p)] \approx 0.25$. BBN is the earliest epoch
the standard thermal history tests directly, and it depends on the
same $\Gamma/H$ competition and the same $g_*$ at $T\sim$~MeV that fix
the neutrino temperature. It is therefore a stringent consistency
check on any modification of the expansion rate at that epoch.

% ---------------------------------------------------------------
\subsection{The three horizons: comoving Hubble radius, particle
horizon, event horizon}
\label{appA:thermal-horizons}

Causal structure in an expanding universe is governed by three
distinct length scales that beginners routinely conflate. All three
are integrals of $1/(a^2 H)$ or $1/(aH)$ over the history
\eqref{eq:appA:thermal-Hofa}; the differences are in the limits of
integration.

\paragraph{Comoving Hubble radius.} The instantaneous causal scale is
\begin{equation}
R_H(a) \equiv \frac{1}{aH(a)},
\label{eq:appA:thermal-RH}
\end{equation}
the comoving distance light travels in one Hubble time. In a
decelerating (radiation or matter) era $aH$ \emph{decreases}, so
$R_H$ \emph{grows} --- scales enter the horizon. In an accelerating
(inflation or $\Lambda$) era $aH$ \emph{increases}, so $R_H$
\emph{shrinks} --- scales exit the horizon. The whole logic of
inflationary structure formation lives in this single sign: a shrinking
comoving Hubble radius is what stretches quantum fluctuations to
super-horizon scales.

\paragraph{Particle horizon.} The comoving distance from which light
could have reached us since the beginning ($a=0$) is
\begin{equation}
\chi_{\mathrm{ph}}(a) = \int_0^t \frac{\dd t'}{a(t')}
= \int_0^a \frac{\dd a'}{a'^2 H(a')}.
\label{eq:appA:thermal-particle-horizon}
\end{equation}
This is the radius of the observable universe. It converges in a
radiation- or matter-dominated past, so there is a genuine finite
particle horizon --- and the fact that causally disconnected CMB
patches share a temperature is exactly the horizon problem that
inflation is invoked to solve. In a radiation era $a\propto t^{1/2}$
one finds $\chi_{\mathrm{ph}} \propto a$, finite and growing.

\paragraph{Event horizon.} The comoving distance light will
\emph{ever} reach in the entire future is
\begin{equation}
\chi_{\mathrm{eh}}(a) = \int_t^{t_{\max}} \frac{\dd t'}{a(t')}
= \int_a^{a_{\max}} \frac{\dd a'}{a'^2 H(a')}.
\label{eq:appA:thermal-event-horizon}
\end{equation}
In a matter- or radiation-only universe $t_{\max}=\infty$ and the
integral diverges --- there is no event horizon, everything is
eventually visible. But in a $\Lambda$-dominated future,
$a\propto e^{Ht}$ with constant $H$, the integral \emph{converges}:
$\chi_{\mathrm{eh}} = 1/(aH) = R_H$, a finite cosmic event horizon.
The accelerating universe therefore has a future causal boundary, the
de Sitter horizon, beyond which receding galaxies pass forever out of
contact. That finite event horizon is the geometric setting for the
black-hole-information and holographic-bound discussions the main text
develops downstream.

% ---------------------------------------------------------------
\subsection{Worked numerical examples}
\label{appA:thermal-worked}

We close the technical content with two explicit computations a reader
can reproduce on paper.

\paragraph{Example 1 --- matter--radiation equality $z_{\mathrm{eq}}$.}
Use Eq.~\eqref{eq:appA:thermal-zeq}, $1+z_{\mathrm{eq}}
=\Omm/\Omega_r$. The radiation density today is the photon plus
neutrino contribution. With $T_0 = 2.725$~K the photon density
parameter is $\Omega_\gamma h^2 \approx 2.47\times 10^{-5}$; including
the three neutrino species via the factor
$1 + (7/8)(4/11)^{4/3}\cdot 3 \approx 1.68$ gives the total
radiation $\Omega_r h^2 \approx 4.15\times 10^{-5}$. Taking
$h \approx 0.674$ (so $h^2\approx 0.454$),
\begin{equation}
\Omega_r \approx \frac{4.15\times 10^{-5}}{0.454}
\approx 9.1\times 10^{-5}.
\label{eq:appA:thermal-omr-num}
\end{equation}
Using the main-text matter fraction $\Omm = 1/\pi \approx 0.3183$,
\begin{equation}
1+z_{\mathrm{eq}}
= \frac{\Omm}{\Omega_r}
\approx \frac{0.3183}{9.1\times 10^{-5}}
\approx 3.5\times 10^{3},
\qquad
\boxed{\,z_{\mathrm{eq}} \approx 3.5\times 10^{3}.\,}
\label{eq:appA:thermal-zeq-num}
\end{equation}
This sits in the right ballpark of the standard
$z_{\mathrm{eq}}\approx 3400$; the precise value depends on the assumed
$\Omm h^2$, but the order of magnitude --- equality a few thousand in
redshift, well before recombination at $z\sim 1090$ --- is robust.
Sanity check: $z_{\mathrm{eq}} \approx 3500 > z_{\mathrm{rec}}
\approx 1090$, so the universe is matter-dominated by the time the CMB
is released, as required for structure to have begun growing.

\paragraph{Example 2 --- the temperature at recombination.}
Recombination occurs at $z_{\mathrm{rec}} \approx 1090$. Using the
cooling law in observational form, $T(z) = T_0(1+z)$ with
$T_0 = 2.725$~K and the conversion $1~\mathrm{K} = 8.617\times
10^{-5}$~eV:
\begin{align}
T_{\mathrm{rec}}
&= 2.725~\mathrm{K} \times (1+1090)
= 2.725 \times 1091~\mathrm{K}
\approx 2.97\times 10^{3}~\mathrm{K},
\label{eq:appA:thermal-Trec-K}\\
T_{\mathrm{rec}}
&\approx 2.97\times 10^{3}~\mathrm{K}
\times 8.617\times 10^{-5}~\frac{\mathrm{eV}}{\mathrm{K}}
\approx 0.26~\mathrm{eV}
\;\approx\; 0.3~\mathrm{eV}.
\label{eq:appA:thermal-Trec-eV}
\end{align}
This confirms the $T\sim 0.3$~eV figure quoted in
\S\ref{appA:thermal-milestones}: recombination happens not at the
$13.6$~eV binding energy but roughly a factor of $\sim 50$ colder,
because the huge photon-to-baryon ratio keeps the high-energy tail of
the Planck spectrum ionizing hydrogen until the bath has cooled far
below $13.6$~eV. As a cross-check, had we instead used the
\GGTheory{} $z_* = e^7 \approx 1096.6$ the temperature would be
$T_* = 2.725\,(1+1096.6)~\mathrm{K} \times 8.617\times 10^{-5}
\approx 0.258$~eV --- indistinguishable from
\eqref{eq:appA:thermal-Trec-eV} at this precision, which is precisely
the visibility-width statement of \S\ref{appA:thermal-gg} expressed in
temperature: the $0.6\%$ redshift offset is a $0.6\%$ temperature
offset, far below the thermal width of last scattering.

% ---------------------------------------------------------------
\subsection{The five clocks of cosmology: \texorpdfstring{$t$, $a$, $z$, $T$, and $r$}{t, a, z, T, and r}}
\label{appA:five-clocks}

Cosmology tells one history on five interchangeable axes, and fluency in
translating among them is the single most useful practical skill this
course can teach. Four are standard: cosmic time $t$, the scale factor
$a$, redshift $1+z = 1/a$, and temperature $T \propto 1/a$. The fifth is
this program's own: the radial coordinate
\begin{equation}
r \;=\; \ln\!\left(\frac{\EGUT}{\mu}\right),
\label{appA:eq-radial-clock}
\end{equation}
the logarithmic depth of the energy scale $\mu$ below the GUT edge ---
simultaneously the renormalization-group axis and the literal radial
direction of the bulk (Part~VIII develops the full dictionary). In
ordinary time the early universe flashes by and the late universe
crawls; on $r$ the story is evenly paced, one unit per e-fold of energy,
which is why the descent narrative of Part~IV reads so naturally there.

\emph{Worked example (the student should reproduce every entry).} Place
recombination and the lock crossing on all five axes.
\emph{Recombination:} $T \simeq 0.26$~eV gives
$1+z = T/T_0 \simeq 0.26\,\mathrm{eV}/2.35\times10^{-4}\,\mathrm{eV}
\simeq 1100$ (the ledger's corridor-switch form is $z_{\rm sw} = e^7
\simeq 1097$, the atomic decoupling $0.4\%$ below;
\S\ref{sec:two-redshifts});
$a \simeq 9\times10^{-4}$; $t \simeq 3.7\times10^5$~yr from the
matter-era relation of \S\ref{appA:thermal}; and
$r = \ln(2.28\times10^{16}\,\mathrm{GeV}/0.26\,\mathrm{eV}) \simeq 58$.
\emph{The lock crossing:} $\mu \sim \Mlock = 3.85$~TeV gives
$r = \ln(2.28\times10^{16}/3.85\times10^{3}) \simeq 29.4$; the
radiation-era clock puts $T \sim 3.85$~TeV at
$t \sim 10^{-13}$~s, $1+z \sim 1.6\times10^{16}$. Two events separated
by seventeen orders of magnitude in time sit twenty-nine units apart on
$r$ --- the compression that makes the whole history legible on one
page (the master table of Part~VIII and the timeline of
Figure~\ref{fig:cosmic-timeline}).

\subsection{What the standard thermal history assumes --- and cannot explain}
\label{appA:thermal-limits}

The derivations of Step~1 are exact, and together they reproduce the
entire thermal history of the universe. As in Appendix~E, it is worth
naming what they must take as given.

\begin{enumerate}[leftmargin=*]
\item \textbf{The inventory.} The densities $\{\Omega_i\}$ that drive the
expansion and set the epoch of matter--radiation equality are measured
inputs, not derived.
\item \textbf{The entropy.} The baryon-to-photon ratio
$\eta\approx6\times10^{-10}$ --- which fixes recombination and
nucleosynthesis --- is a measured number with no account of its origin in
the standard treatment.
\item \textbf{The milestone redshifts.} Equality, recombination, and the
rest are located by asking where the measured densities cross, or where a
reaction rate falls below $H$. The procedure is exact, but it offers no
reason the switches sit at \emph{these} redshifts rather than others;
there is no parameter-free structure underneath them.
\end{enumerate}

Standard cosmology, in other words, is a superb \emph{calculator} of the
thermal history once the inputs are supplied --- but the inputs, and the
particular redshifts of the milestones, are things it measures rather
than explains. The geometric account asks whether at least the switch
redshifts carry a deeper, integer-ledger structure.

\subsection{The geometric derivation: the cosmic clock as a \GDOS{} descent}
\label{appA:thermal-gg}

The thermal history of Step~1 is exact, and it reproduces every milestone
of the early universe. But notice how it does so: one inserts the
\emph{measured} densities and the \emph{measured} baryon-to-photon ratio
into the equilibrium formulas and reads off where the transitions fall.
The milestone redshifts are outputs of measured inputs, not consequences
of anything deeper. \GGTheory{} asks the further question --- is there a
structural, parameter-free reason the switches sit where they do? --- and
for the most important of them it gives an answer.

\paragraph{The cosmic clock is a geometric descent.}
The radial coordinate $r=\ln(\EGUT/\mu)$ of the bulk is not an
abstraction: it is the descent parameter of the open-system boundary law
\GDOS{}, and it is the cosmic clock. Temperature, redshift, and cosmic
time are three readings of one underlying motion down $r$,
\[
r=\ln(\EGUT/\mu)\;\longleftrightarrow\;T\propto 1/a\;\longleftrightarrow\;
1+z=1/a\;\longleftrightarrow\;t ,
\]
so the whole of Step~1 --- the cooling, the redshifting, the sequence of
milestones --- is the single fall from the hot unified edge at $r=0$
toward the cold present. \emph{Physics intuition:} the universe does not
carry a separate clock; its temperature \emph{is} its position on the
descent, and the arrow of time is the direction of the \GRIP{} flow.

\paragraph{Milestones are \GRIP{}-Out switch events.}
Along that descent the boundary state does not change smoothly forever.
At particular depths it undergoes geometric symmetry-breaking switches
--- \GSSB{} events, the outward (\GRIP{}-Out) steps of the runtime --- and
these are the milestones: equality, recombination, the onset of dark
energy. Standard cosmology locates them by asking where the densities
cross; the framework asks, in addition, whether the switch depths are
fixed by the integer ledger. For the recombination era the answer is
yes: the corridor switch there is the one milestone the geometry pins
to a parameter-free integer, with the atomic decoupling derived
separately (\S\ref{sec:two-redshifts}).

\emph{Reading note (the two-quantity form).} What follows predates the
two-quantity resolution of \S\ref{sec:two-redshifts}, which now
governs every comparison of $e^{7}$ with the measured sky: the ledger
integer fixes the \emph{corridor switch} $z_{\mathrm{sw}} = e^{7}$, a
topological marker, while the decoupling redshift the microwave sky
actually records is derived from the framework's own atomic chain,
landing on the Planck value at the $0.1\%$ level, $0.4\%$ (in e-folds)
below the switch. The both-framings computation below is retained
deliberately, as the pedagogical bridge that motivated that
resolution: it shows what happens when $e^{7}$ is read --- as it
originally was --- as a direct claim on $z_{\mathrm{rec}}$, and why
the two-quantity form replaces that reading. Everything that follows
is to be read accordingly (Eq.~\eqref{eq:delta-offset}).

In the cohort's
arithmetic ledger --- the \OGNfive{} five-integer set
$(M,N,L;J,K) = (1,3,5;7,35)$ established upstream
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}, audited in \citealp{ArisakaA4Euclid}) --- the
recombination switch is identified here with the single integer $J=7$ via the
exponential map
\begin{equation}
\boxed{\;z_* = e^{J} = e^{7} \approx 1096.633\;}
\label{eq:appA:thermal-zstar}
\end{equation}
with \emph{no continuous adjustable parameter}: $J=7$ is an integer
from the ledger, and the exponential is the same codebook-$D^*$ map
that the main text uses for the other switch redshifts. This is a
parameter-free ledger statement, not a fit. In the superseded
one-quantity reading --- $e^{7}$ offered directly as the recombination
redshift --- it had to be reported with both framings, because the two
framings tell genuinely different stories; the two-quantity form of
\S\ref{sec:two-redshifts} dissolves the tension between them, and the
computation is kept here as the bridge to it.

\paragraph{Framing 1 --- the near-hit (the optimistic reading).}
Against the Planck central value
\eqref{eq:appA:thermal-zrec-planck}, the fractional offset is
\begin{equation}
\frac{z_* - z_{\mathrm{rec}}}{z_{\mathrm{rec}}}
= \frac{1096.63 - 1089.92}{1089.92}
\approx \frac{6.71}{1089.92}
\approx 0.0062 = 0.6\%.
\label{eq:appA:thermal-zstar-frac}
\end{equation}
A parameter-free integer descending to within $0.6\%$ of a measured
cosmological switch redshift is, on its face, a striking
coincidence-or-not: it is the kind of structural near-hit that
motivates taking the ledger seriously. And against the
\emph{physically relevant} yardstick --- the recombination visibility
width $\Delta z \approx 90$, the actual thickness of the last-
scattering shell --- the offset is
\begin{equation}
\frac{z_* - z_{\mathrm{rec}}}{\Delta z}
\approx \frac{6.71}{90}
\approx 0.075,
\label{eq:appA:thermal-zstar-visibility}
\end{equation}
i.e.\ well under a tenth of the shell thickness. (If one instead uses
the conservative half-width $\Delta z/2 \approx 45$ as the $1\sigma$
physical scale, the offset is $\approx 0.15\,\sigma$, still well under
a fifth of the shell.) By the standard that asks ``does the prediction
land inside the surface of last scattering,'' the answer is
unambiguously yes: $e^7$ lands comfortably within the last-scattering
shell.

\paragraph{Framing 2 --- the large nominal tension (the honest
caveat).} The optimistic reading must not be allowed to stand alone,
because the Planck \emph{measurement error} is tiny:
$\sigma_{\mathrm{Planck}} = 0.25$. Measured in those units the same
$6.71$ offset is
\begin{equation}
\frac{z_* - z_{\mathrm{rec}}}{\sigma_{\mathrm{Planck}}}
\approx \frac{6.71}{0.25}
\approx 27\,\sigma.
\label{eq:appA:thermal-zstar-sigma}
\end{equation}
Twenty-seven standard deviations is a \emph{large nominal tension}: by
the precision-cosmology standard that compares a prediction to the
\emph{measurement} error, $e^7$ is decisively excluded as the value of
$z_{\mathrm{rec}}$. There is no way to dress $27\sigma$ up as
agreement, and the present appendix does not try to. The honest
statement is therefore the conjunction of both numbers: \emph{$e^7$ is
a $0.6\%$ near-hit and a sub-$0.2\sigma$ visibility-width near-hit,
and simultaneously a $\sim 27\sigma$ tension against the Planck
central value.} The reconciliation of the two framings is simply that
they ask different questions. Framing~2 asks ``is $e^7$ the precise
value of the recombination redshift?'' --- and the answer is no.
Framing~1 asks ``does a parameter-free integer ledger reproduce the
recombination switch to within the physical width of the
last-scattering surface?'' --- and the answer is yes, to better than a
fifth of that width.

\paragraph{What the claim is, and is not.} The $z_*=e^7$
identification is a \emph{structural near-hit from an independent
integer ledger}, codebook code $D^*$ (Derived-conditional): it is the
projection of the upstream \OGNfive{} theorem onto the cosmological-boundary
stratum under the closed-packet map, not a precision determination of
$z_{\mathrm{rec}}$. It is not, and is never to be presented as, a
match to the Planck error bar. Its evidential weight is the weight of a
postdiction that costs no free parameters and happens to land inside
the last-scattering shell, drawn from the same ledger that yields
$\Omm = 1/\pi$ and the other $D^*$ entries --- no more and no less. The
discipline the cohort imposes on itself here is the one this appendix
enacts for the superseded reading --- state the $0.6\%$ near-hit and
the $27\sigma$ nominal tension in the same breath --- and, in the body,
the stronger one: retire that reading altogether in favor of the
two-quantity form (\S\ref{sec:two-redshifts}).

% ---------------------------------------------------------------

\paragraph{What this appendix establishes.}
The cosmic clock of Step~1 is, beneath the thermodynamics, a single
geometric descent: temperature and redshift are positions on it, and the
milestones are the symmetry-breaking switches along the way. Of those
switches, the recombination-era corridor switch is the one the integer
ledger pins --- $z_{\mathrm{sw}}=e^7$, a parameter-free topological
marker --- while the decoupling the sky records is derived from the
framework's own atomic chain, on the measured value at the $0.1\%$
level, $0.4\%$ below the switch (\S\ref{sec:two-redshifts}). The
both-framings computation above records the superseded one-quantity
reading --- a $0.6\%$ near-hit inside the last-scattering shell that is
simultaneously a $\sim27\sigma$ nominal tension against the Planck
central value --- the very tension the two-quantity form dissolves by
not reading $e^7$ as a claim on $z_{\mathrm{rec}}$ at all. It is a
$D^*$ marker, not a precision match, and that is exactly how it is
carried in the codebook. Less than
the clean theorem of Appendix~E, but in the same Euclidean spirit: the
piece the geometry fixes is stated as a result, and its limits are stated
in the same breath.

\subsection{Status}
\label{appA:thermal-status}

\begin{itemize}
\item \textbf{Standard thermal history --- Derived (D).} The scaling
law $\rho \propto a^{-3(1+w)}$
\eqref{eq:appA:thermal-master-scaling}, the era solutions $a(t)$
\eqref{eq:appA:thermal-aoft-power}--\eqref{eq:appA:thermal-aoft-desitter},
the redshift and momentum laws
\eqref{eq:appA:thermal-redshift-def}--\eqref{eq:appA:thermal-Tprop},
comoving entropy conservation and $T\propto g_{*s}^{-1/3}a^{-1}$
\eqref{eq:appA:thermal-comoving-S}--\eqref{eq:appA:thermal-Texact}
with the $e^+e^-$ reheating ratio $T_\nu/T_\gamma = (4/11)^{1/3}$
\eqref{eq:appA:thermal-neutrino-T}, the freeze-out criterion
$\Gamma \simeq H$ \eqref{eq:appA:thermal-decouple-criterion}, the
milestones ($z_{\mathrm{eq}}$, recombination, BBN), and the three
horizons
\eqref{eq:appA:thermal-RH}--\eqref{eq:appA:thermal-event-horizon} are
all standard cosmology, derived here from the Friedmann equations and
equilibrium thermodynamics. \GGTheory{} retains these intact; it
constrains the source terms, not the equations.
\item \textbf{The $z_* = e^7$ corridor switch --- Derived-conditional
($D^*$), read in the two-quantity form.} The recombination-era
corridor-switch redshift $z_* = e^J = e^7 \approx 1096.6$
\eqref{eq:appA:thermal-zstar} descends parameter-free from the \OGNfive{}
integer $J=7$ \citep{ArisakaA4Euclid} via the codebook-$D^*$
closed-packet map. In the superseded one-quantity comparison retained
above as the bridge it is a \emph{structural near-hit}: $0.6\%$ from
the Planck central value \eqref{eq:appA:thermal-zstar-frac} and
under $0.2\sigma$ of the recombination visibility width
\eqref{eq:appA:thermal-zstar-visibility}, but simultaneously a
\emph{large nominal tension} of $\sim 27\sigma$ against the Planck
measurement error \eqref{eq:appA:thermal-zstar-sigma}. In the body it is carried in the
two-quantity form of \S\ref{sec:two-redshifts} --- the switch a
topological marker, the atomic decoupling the audited quantity ---
with the both-framings computation of this appendix retained as the
pedagogical bridge; it is never presented as a precision match to
$z_{\mathrm{rec}} = 1089.92 \pm 0.25$.
\end{itemize}

%========================================================
%  Arisaka-GG-A10-Cosmos : appendix fragment
%  Slow-roll inflation from first principles (maximum-depth pedagogy)
%  Style model: Arisaka-GG-A8-QM-v12.0 appendix worked-example sections.
%  ASCII only. Relies on cohort macros provided by ucla-arisaka-gg-11.sty
%  (\Mlock \GGTheory \GDOS \GRIP \ns \GSSB \DGI \MPl \TeV) and on the
%  A10 preamble \providecommand{\ns}. Citation keys: ArisakaA2GG6,
%  ArisakaA6Gi4, ArisakaA4Euclid.
%
%  HONESTY-CRITICAL: in the codebook A_s and n_s are ASSERTED (open,
%  anchored, pending Conjecture IV); the inflationary energy scale is
%  ASSERTED pending Conjecture V; only r and Delta N_eff are Derived.
%  No companion computes the inflaton potential or the COBE/Planck
%  normalization. This appendix does NOT present a computed A_s or n_s
%  for the GG case as if derived.
%========================================================
\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{F.2.1. Solve the master scaling law for
the three epochs and check the exponents.}

\noindent\emph{Solution.}  Step 1: for $\rho \propto
a^{-3(1+w)}$, Friedmann~1 gives $\dot a \propto
a^{-(1+3w)/2}$, an elementary ODE with power-law solution
$a \propto t^{2/3(1+w)}$ \eqref{eq:appA:thermal-aoft-power} for
$w \neq -1$.  Step 2: insert the three equations of state:
radiation ($w = \tfrac13$) gives $a \propto t^{1/2}$; matter
($w = 0$) gives $a \propto t^{2/3}$; vacuum ($w = -1$) breaks
the power-law ansatz and integrates instead to $a \propto
e^{Ht}$ \eqref{eq:appA:thermal-aoft-desitter}.  Step 3: check
consistency: $H = \tfrac{2}{3(1+w)t}$, so $Ht$ is constant for
any fluid --- the age of the universe is always of order the
Hubble time, with coefficient $\tfrac12$ (radiation) or
$\tfrac23$ (matter).  Physics intuition: stiffer contents brake
harder --- radiation's extra pressure steepens the dilution and
slows the growth exponent --- and the vacuum, which never
dilutes, is the limiting case where braking fails entirely and
expansion runs away exponentially.

\medskip\noindent\textbf{F.4.1. Fill in the entropy bookkeeping
behind $T_\nu/T_\gamma = (4/11)^{1/3}$.}

\noindent\emph{Solution.}  Step 1: just above $e^\pm$
annihilation the photon--electron sector carries
$g_{*s}^{\gamma e} = 2 + \tfrac78\times4 = \tfrac{11}{2}$
\eqref{eq:appA:thermal-gstar-ee} (photons: two polarizations;
$e^\pm$: four spin-charge states at the fermion's $\tfrac78$).
Step 2: below the threshold only photons remain,
$g_{*s}^{\gamma} = 2$; comoving entropy conservation in that
sector, $g_{*s}(T_\gamma a)^3 = $ const
\eqref{eq:appA:thermal-comoving-S}, gives
$(T_\gamma a)^3_{\rm after}/(T_\gamma a)^3_{\rm before}
= \tfrac{11/2}{2} = \tfrac{11}{4}$.  Step 3: the decoupled
neutrinos cool as $T_\nu \propto 1/a$ throughout, so
$T_\nu/T_\gamma = (4/11)^{1/3} \approx 0.714$
\eqref{eq:appA:thermal-neutrino-T} --- today, $1.95$~K against
$2.725$~K.  Physics intuition: the annihilating pairs bequeath
their entropy to the photons alone because the neutrinos have
already left the room; the $(4/11)^{1/3}$ is a conservation law
remembering a departure --- and the cosmic neutrino
background's predicted chill is among the most secure unmeasured
numbers in physics.

\medskip\noindent\textbf{F.5.1. Estimate the neutrino decoupling
temperature from the $\Gamma/H$ criterion.}

\noindent\emph{Solution.}  Step 1: weak scattering at
temperature $T$ has $\sigma \sim G_F^2T^2$ and target density
$n \sim T^3$, so $\Gamma = n\sigma v \sim G_F^2T^5$
\eqref{eq:appA:thermal-Gamma}.  Step 2: the radiation-era
expansion rate is $H \sim T^2/\MPl$
\eqref{eq:appA:thermal-HofT}.  Step 3: setting $\Gamma = H$
\eqref{eq:appA:thermal-decouple-criterion}:
$T_{\rm dec} \sim (G_F^2\MPl)^{-1/3} \approx 0.8$--$1$~MeV.
Physics intuition: the fifth power is the whole story ---
$\Gamma/H \propto T^3$ falls through unity so steeply that
decoupling is sharp, a switch rather than a fade; every relic
in the universe (neutrinos here, the Kaxion in
Appendix~\ref{appA:kaxion}, helium in
Appendix~\ref{appA:bbn}) is a fossil of one such switch, and
the chapter's freeze-out grammar is cosmology's universal
fossilization mechanism.

\medskip\noindent\textbf{F.6.1. Derive the redshift of
matter--radiation equality from the $\pi$-locked matter
fraction.}

\noindent\emph{Solution.}  Step 1: the present radiation
density (photons at $2.725$~K plus three neutrino species at
$(4/11)^{1/3}$ of the photon temperature) is $\Omega_r h^2 =
4.15\times10^{-5}$, giving $\Omega_r \approx 9.1\times10^{-5}$
\eqref{eq:appA:thermal-omr-num}.  Step 2: radiation dilutes one
power of $(1+z)$ faster than matter, so equality sits at
$1 + z_{\rm eq} = \Omm/\Omega_r$.  Step 3: with
$\Omm = 1/\pi$: $1 + z_{\rm eq} \approx 0.3183/9.1\times10^{-5}
\approx 3.5\times10^3$ \eqref{eq:appA:thermal-zeq-num} ---
squarely the CMB-fit value.  Physics intuition: equality is the
moment gravity's builder changes shift --- perturbations grow
only logarithmically under radiation and linearly under matter
--- so $z_{\rm eq}$ is imprinted in the shape of the matter
power spectrum, where surveys read it and the inventory
predicts it.

\medskip\noindent\textbf{F.6.2. State the two-quantity $z_*$
discipline precisely, and compute the gap.}

\noindent\emph{Solution.}  Step 1: the corridor switch is
topological and exact: $z_{\rm sw} = e^{J} = e^{7} =
1096.633$ \eqref{eq:appA:thermal-zstar}, fixed by the single
ledger integer $J = 7$ with no atomic physics in it.  Step 2:
the atomic decoupling redshift is a \emph{different} quantity:
$z_{\rm dec} \approx 1090$, derived through the Saha/visibility
chain (hydrogen's levels, $\eta$, the expansion rate).  The
gap is $e^{7}/1090 - 1 \approx 0.6\%$ --- two quantities,
lawfully close, never to be conflated.  Step 3: the register:
the falsifier on this sector is \emph{observable-only} --- it
fires if the measured atomic $z_{\rm dec}$ moves off the atomic
chain's prediction by more than $0.5\%$, and no measurement of
$z_{\rm dec}$ can touch the topological $z_{\rm sw}$.
Physics intuition: a switch on the corridor and a fog clearing
in the gas are different events that happen to sit close ---
the theory's discipline is to price each with its own chain and
let the sky judge only the one it can see.

\medskip\noindent\textbf{F.7.1. Compute the asymptotic future
event horizon of the $\pi$-locked universe.}

\noindent\emph{Solution.}  Step 1: at late times the universe
is vacuum-dominated with $H_\Lambda = H_0\sqrt{\Omega_\Lambda}
= 0.826\,H_0$.  Step 2: the event horizon
\eqref{eq:appA:thermal-event-horizon} asymptotes to the de
Sitter value $d_{\rm EH} = c/H_\Lambda$.  Step 3: numerically,
$c/H_0 = 4.45$~Gpc, so $d_{\rm EH} \to 5.39$~Gpc $\approx
17.6$ billion light-years.  Galaxies beyond that comoving
distance today have already sent us the last photons we will
ever receive.  Physics intuition: acceleration turns the sky
into a slowly closing archive --- and the horizon it closes
toward is the de~Sitter port whose thermodynamics the
black-hole companion works in its Appendix~Q: the same
$\pi$-locked vacuum fixes both the archive's radius and its
temperature.

\medskip\noindent\textbf{F.8.1. How much entropy does the
universe carry per baryon?}

\noindent\emph{Solution.}  Step 1: the entropy density today is
dominated by photons and neutrinos: $s/n_\gamma = 7.04$ (the
same $7.04$ that converts $Y_L$ to $\eta_B$ in
Appendix~\ref{appA:leptogenesis},
Eq.~\eqref{eq:app-etaB-assembly}).  Step 2: with the baryon-to-
photon ratio $\eta \approx 6.1\times10^{-10}$, the entropy per
baryon is $s/n_B = 7.04/\eta \approx 1.2\times10^{10}$.
Step 3: ten billion units of entropy per particle of matter:
the universe is, by particle count, overwhelmingly a photon
gas with a trace contamination of baryons.  Physics intuition:
this enormous number is why BBN and recombination happen
``late'' (the photon bath outvotes binding energies by its
sheer numbers --- Appendix~\ref{appA:bbn}'s bottleneck and this
chapter's Saha delay are both $\ln(1/\eta)$ effects), and its
origin is exactly the asymmetry $\eta_B$ that
Appendix~\ref{appA:leptogenesis} assembles from the ledger.

\GGtakeaway{\emph{Standard result:} dilution laws, freeze-out, and the
thermal milestones --- the engine room of every relic calculation in
cosmology. \emph{What it assumed:} the inventory driving the expansion,
and initial abundances set elsewhere. \emph{What the geometry replaced
it with:} the whole history re-read as radial transport on
\(r = \ln(\EGUT/\mu)\) --- every epoch a lawful address on one
axis, with the descent's two crossings (the graceful exit and the lock
crossing) as the events that stock the universe. \emph{Continue in:}
Part~VIII (the chronology), Appendix~G (the launch).}

\section{Slow-roll inflation from first principles}
\label{appA:inflation}

\GGteachbox{Why the early universe needed an accelerating stage (the
horizon, flatness, and relic problems); the full slow-roll apparatus ---
the field equations, the slow-roll parameters, the e-fold count, and the
quantum origin of the scalar and tensor spectra --- derived from scratch
with a worked example; what the standard treatment must assume (a field,
a potential, a height, a count); and the geometric
derivation in which the potential's class, the tilt \(n_s = 0.9620\), and the
tensor target \(r = 0.0043\) are computed, with the amplitude the one
declared input.}
\GGreadalong{\citet{LiddleLyth2000} chs.~3--5 or \citet{Baumann2022}
ch.~4 for Step~1; \citet{Mukhanov2005} for the quantum theory of the
perturbations at depth.}

The slow-roll inflationary dynamics and the quantum generation of the nearly scale-invariant primordial spectrum are summarized in Figure~\ref{fig:appfig-slowroll}. The slow-roll machinery and the primordial spectrum developed here are drawn from the founding and review literature \citep{Starobinsky1980,MukhanovChibisov1981,Hawking1982,GuthPi1982,BardeenSteinhardtTurner1983,Linde1983,LythRiotto1999}.

\begin{figure}[htbp]
\centering
\resizebox{0.570\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  >={Stealth[length=2.2mm]}]

% ---- axes ----
\draw[->,line width=0.80pt] (-0.3,0) -- (10.2,0) node[right] {$\phi$};
\draw[->,line width=0.80pt] (0,-0.3) -- (0,5.4) node[above] {$V(\phi)$};

% ---- potential curve: flat plateau (left) -> shoulder -> minimum (right) ----
\draw[line width=1.10pt,blue!65!black]
  (0.4,4.25)
  .. controls (2.6,4.22) and (3.6,4.18) .. (4.4,3.95)
  .. controls (5.2,3.6) and (5.6,2.6) .. (6.2,1.6)
  .. controls (6.7,0.75) and (7.2,0.35) .. (7.8,0.35)
  .. controls (8.4,0.35) and (8.7,1.0) .. (9.1,1.7);

% plateau height marker
\draw[dashed,gray] (0,4.25) -- (3.0,4.25);
\node[anchor=west,align=left] at (0.15,4.85)
  {$V^{1/4}\!\sim\!\Mlock\!\sim\!3.85$\,TeV};

% ---- slow-roll region brace ----
\draw[decorate,decoration={brace,amplitude=4pt,mirror},gray!60!black]
  (0.5,3.7) -- (3.7,3.7);
\node[align=center,gray!30!black] at (2.1,3.25)
  {slow roll:\\[-1pt]$\epsilon,\eta\ll 1$};

% ---- epsilon = 1 point (inflation ends) on the shoulder ----
\filldraw[red!75!black] (5.65,2.55) circle (1.8pt);
\draw[red!75!black,->] (5.65,2.55) -- (3.95,2.05)
  node[anchor=east,align=right,red!60!black]
  {$\epsilon=1$\\(inflation ends)};

% ---- reheating / oscillation at minimum ----
\draw[->,orange!80!black,line width=0.80pt]
  (7.3,1.05) .. controls (7.55,0.55) and (8.05,0.55) .. (8.3,1.05);
\filldraw[orange!85!black] (7.8,0.35) circle (1.8pt);
\node[align=center,orange!55!black,anchor=north] at (7.8,-0.05)
  {reheating /\\oscillation};

% ---- inset box: slow-roll definitions ----
\node[draw=black!50,rounded corners,fill=gray!6,align=left,inner sep=4pt,
      anchor=north east] at (10.1,5.3)
  {\scriptsize
   $\displaystyle \epsilon=\tfrac{M_{\mathrm{Pl}}^2}{2}\!\left(\tfrac{V'}{V}\right)^{\!2}$,\quad
   $\displaystyle \eta=M_{\mathrm{Pl}}^2\,\tfrac{V''}{V}$\\[2pt]
   $n_s=1-6\epsilon+2\eta$,\quad $r=16\epsilon$\\[2pt]
   $\displaystyle N=\!\int\!\tfrac{V}{M_{\mathrm{Pl}}^2 V'}\,d\phi$};

% ---- honesty flag note ----
\node[draw=red!60!black,line width=0.80pt,rounded corners,fill=red!5,align=left,
      inner sep=4pt,anchor=south east,text=red!45!black] at (10.1,0.15)
  {\scriptsize\textbf{(!)} GG status: $n_s = 0.9620$ \emph{DERIVED}\\
   (Lemma IV rev.); $A_s$ = the one input\\
   (Lemma V$'$) --- the TeV plateau is retired:\\
   the tilt came out blue, the gap $\sim$50 orders.};

\end{tikzpicture}}
\caption[Slow-roll inflaton potential and the GG normalization gap]{%
Slow-roll inflation, taught from the standard mechanism. The inflaton rolls
slowly across a flat \emph{plateau} (where the slow-roll parameters
$\epsilon,\eta\ll1$), inflation ends where $\epsilon=1$ on the shoulder, and
the field reheats by oscillating about the \emph{minimum}. An earlier reading of \ATen{}
placed the plateau at the TeV scale, $V^{1/4}\!\sim\!\Mlock\!\sim\!3.85$\,TeV; that
reading is \emph{retired}, and is kept visible here with its refutation: a
TeV-scale plateau cannot simultaneously reproduce the observed
$A_s\!\sim\!2.1\times10^{-9}$ and $n_s\!\sim\!0.965$, leaving an
$\sim\!50$-order-of-magnitude normalization gap.
\textbf{Honest codebook status (current):} the \GDOS{}-forced plateau sits
at $V^{1/4} = 8.3\times10^{15}$\,GeV, the tilt $n_s = 1 - 2/N^* = 0.9620$ is
\emph{derived-conditional} (Lemma~IV, revised), and the amplitude $A_s$
remains the codebook's one declared input (Lemma~V$'$). The figure teaches
the standard slow-roll machinery (inset: $\epsilon$, $\eta$, $n_s$, $r$,
e-folds $N$) and marks where the \emph{TeV} normalization did not close.}
\label{fig:appfig-slowroll}
\end{figure}


This appendix is written for a first-year graduate student who has met
the Friedmann equations once and wants to understand, line by line, why
cosmologists believe the early universe inflated, how a single scalar
field can drive it, and exactly which numbers the theory predicts. As in
Appendix~E, we build the case in three steps.

\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard mechanism
(\S\ref{appA:infl-problems}--\ref{appA:infl-worked}).} We develop the
slow-roll formalism from first principles --- the three problems it
solves, the e-fold counting, the inflaton equation of motion, the
slow-roll parameters, the scalar and tensor spectra, and reheating ---
and work a textbook example in full.
\item \textbf{What inflation leaves open (\S\ref{appA:infl-limits}).} We
then ask what the standard account puts in by hand: the inflaton itself,
its potential, its energy scale, and the spectrum normalization.
\item \textbf{The geometric derivation (\S\ref{appA:infl-gg-reading}).}
Finally we show what \GGTheory{} supplies --- the inflaton as a bulk
modulus, its energy scale computed, its dynamics as the prototype
\GRIP{} descent, and a sharp null-tensor prediction --- and we mark, in
plain sight, the one thing it does not yet supply: the spectrum
normalization.
\end{enumerate}

Inflation is the hardest case for the program's guiding belief --- that
the contents and laws of the universe should follow from geometry --- and
so it is the most revealing. Where the geometry succeeds it replaces a
tailored mechanism with a computed one; where it does not yet succeed, it
says so. Nothing here is hidden behind a macro: every parameter is
defined where it first appears, and every numerical claim is one the
reader can reproduce with a calculator.

\subsection{The three problems of the hot Big Bang}
\label{appA:infl-problems}

A purely decelerating, radiation-then-matter-dominated Friedmann
universe is a spectacularly successful description of everything after
about one second. It is, however, embarrassed by three initial-condition
puzzles. Inflation is the proposal that a brief epoch of accelerated
expansion preceded the hot Big Bang and erased all three at once.
This subsection teaches the canonical three; at its end we pose a fourth
that the canon never lists --- and that, unlike the three, inflation
alone cannot erase.

\paragraph{The horizon problem.} The comoving distance light can have
traveled since the Big Bang --- the comoving particle horizon --- is
\begin{equation}\label{eq:horizon-comoving}
\chi_{\rm hor}(t)=\int_0^t\frac{c\,dt'}{a(t')}
=\int_0^{a}\frac{c\,da'}{a'^2 H(a')},
\qquad H\equiv\frac{\dot a}{a}.
\end{equation}
In a universe dominated by a fluid with equation of state
$p=w\rho$ one has $a\propto t^{2/[3(1+w)]}$ and
$H\propto a^{-3(1+w)/2}$, so the integrand in
\eqref{eq:horizon-comoving} scales as
$a'^{(1+3w)/2}\,da'/a'^2$. For ordinary matter ($w=0$) or radiation
($w=1/3$) the exponent $(1+3w)/2-2<-1$ is such that the integral is
dominated by its \emph{upper} limit: the horizon grows monotonically and
was small in the past. Concretely, at recombination
($z_\star\simeq 1090$) the particle horizon subtends only about one
degree on the last-scattering surface. Yet the cosmic microwave
background (CMB) is isotropic to one part in $10^{5}$ across the
\emph{whole} sky --- some $10^{4}$ causally disconnected patches at the
same temperature to four decimal places. Why should regions that never
exchanged a single photon share a temperature? In the hot Big Bang this
is simply an assumed initial condition.

\paragraph{The flatness problem.} Write the Friedmann equation as
\begin{equation}\label{eq:friedmann-omega}
H^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}
\quad\Longrightarrow\quad
\Omega(a)-1=\frac{kc^2}{a^2H^2},
\qquad
\Omega\equiv\frac{\rho}{\rho_{\rm crit}},
\;\;\rho_{\rm crit}=\frac{3H^2}{8\pi G}.
\end{equation}
The quantity $|\Omega-1|$ is proportional to $1/(aH)^2$, the square of
the comoving Hubble radius. In a decelerating universe $aH$
\emph{decreases} with time, so $|\Omega-1|$ \emph{grows}: spatial
flatness is an unstable fixed point. Today $|\Omega_0-1|\lesssim 0.005$.
Running this back to the Planck epoch with $aH\propto a^{-(1+3w)/2}$ for
$w\ge 0$ requires $|\Omega-1|$ to have been tuned to roughly one part in
$10^{60}$ at $t\sim t_{\rm Pl}$. A universe that is even marginally flat
today demands an absurdly fine-tuned initial curvature --- unless some
mechanism actively drives $\Omega\to 1$.

\paragraph{The relic (monopole) problem.} Grand unified theories and,
more generally, symmetry-breaking phase transitions in the early
universe generically produce topological defects --- magnetic
monopoles, cosmic strings, domain walls --- as massive, stable relics.
Produced at roughly one per horizon volume at the transition, monopoles
would redshift as non-relativistic matter ($\rho_{\rm M}\propto a^{-3}$,
slower than the $a^{-4}$ of radiation) and quickly dominate the energy
budget, overclosing the universe by many orders of magnitude. The
observed universe contains no such relics at detectable abundance. Any
mechanism that dilutes pre-existing relics far below one per current
horizon volume solves this.

\paragraph{One cure for all three.} The common root is that
$aH$ shrinks in a decelerating universe. Suppose instead there was an
early epoch in which
\begin{equation}\label{eq:inflation-condition}
\ddot a>0
\quad\Longleftrightarrow\quad
\frac{d}{dt}\!\left(\frac{1}{aH}\right)<0
\quad\Longleftrightarrow\quad
\rho+3p<0
\quad\Longleftrightarrow\quad
w<-\tfrac13 .
\end{equation}
The three statements are equivalent: accelerated expansion, a
\emph{shrinking} comoving Hubble radius, and a sufficiently negative
pressure. The cleanest realization is $w=-1$, a constant vacuum-like
energy density, giving exponential de Sitter expansion
$a(t)\propto e^{Ht}$ with $H\approx$ const. During such a phase $aH$
grows like $a$, so $|\Omega-1|\propto 1/(aH)^2$ is driven
\emph{exponentially} toward zero (flatness solved); a tiny initial
causal patch is stretched to encompass the entire observable universe,
so the whole CMB sky descends from one thermalized region (horizon
solved); and every pre-existing relic density is diluted by the same
exponential volume factor (relic problem solved). A near-de Sitter,
``quasi-de-Sitter,'' phase that lasts long enough and then ends
gracefully is therefore the minimal cure.

%========================================================
\paragraph{The fourth problem, hiding inside the third: identity.}
Here is a teaching exercise worth pausing on, because it shows that even
a canonical list can be incomplete. The three problems above all concern
\emph{conditions}: how uniform the temperature is, how flat the geometry
is, how rare the relics are. Now ask a question of the same causal shape
about \emph{things}. Two hydrogen atoms on opposite sides of the sky ---
in regions that, without inflation, were never in causal contact ---
radiate exactly the same Balmer series; every electron in both regions
carries the same mass and charge; the dark-matter abundance and the
dark-energy density are the same on both sightlines. Temperature is one
number that thermal contact might smooth; a particle's identity is a
complete specification that no exchange of photons can negotiate. Why is
the \emph{parts list} of the universe the same in causally disconnected
regions?

The student should notice three things. First, this question is not on
the canonical list \emph{as such} --- its sharpest fragment entered the
canon in disguise, as the domain problem: that causally disconnected
regions choose their vacuum at random was seen already in the 1970s
\citep{ZeldovichKobzarevOkun1974,Kibble1976}, but the random choice was
filed as a relic-abundance nuisance to be diluted, not as an identity
question to be answered; and quantum field theory hides the rest of the
question by assuming one field content in one vacuum globally, which is
an assumption, not a derivation. Second, inflation \emph{alone} does
not answer it: a common causal patch explains shared conditions, but in
any framework with more than one available vacuum (a landscape), a
single patch can still fracture into domains that choose differently ---
shared history does not force shared identity. Third, the resolution in
the present framework needs both of its halves: the \emph{mold} (each
object is the unique Ground Geometric Attractor of its stratum on one
bulk --- Part~X), and the \emph{ordering} taught in Step~3c of this
appendix --- the launch sits at the top of the descent, so inflation
stretched the one causal patch \emph{before any matter existed}, the
visible sector was minted afterwards by one decay of one modulus, the
Kaxion's one mark was pinned while the sky was still one patch, and the
prompt lock crossing (\(N_2 = 0\)) left no alternative basin for any
region to camp in. Sameness of state is inflation's gift; sameness of
things is the geometry's. The full development --- the \emph{second
horizon problem} --- is Section~\ref{sec:second-horizon} of Part~X; the
student who has followed this appendix owns every ingredient of its
proof.

\subsection{Counting e-folds: how much inflation is enough}
\label{appA:infl-efolds}

The natural measure of ``how much'' the universe inflated is the number
of e-folds of expansion,
\begin{equation}\label{eq:efold-def}
N\equiv\ln\frac{a_{\rm end}}{a_{\rm start}}
=\int_{t_{\rm start}}^{t_{\rm end}}H\,dt .
\end{equation}
Each e-fold multiplies the scale factor by $e\approx 2.718$ and shrinks
the comoving Hubble radius by the same factor. To solve the flatness
problem we need $|\Omega-1|$ suppressed by at least the factor by which
it would subsequently \emph{grow} during the radiation and matter eras.
A scale that exits the horizon $N$ e-folds before the end of inflation
re-enters today if
\begin{equation}\label{eq:efold-requirement}
N_\star\;\simeq\;
\ln\!\frac{a_0H_0}{a_\star H_\star}
\;\approx\;
60+\ln\!\frac{k/(a_0H_0)}{0.05\,{\rm Mpc^{-1}}}
+\frac14\ln\!\frac{V_\star}{(10^{16}\,{\rm GeV})^4}+\dots,
\end{equation}
where $V_\star$ is the inflationary energy density when that scale left
the horizon and the dots collect reheating-dependent terms. The
numerical upshot, for the pivot scale probed by the CMB, is the famous
benchmark
\begin{equation}\label{eq:efold-benchmark}
N_\star\;\approx\;50\text{--}60 .
\end{equation}
The \emph{lower} the inflationary energy scale, the \emph{fewer} e-folds
of observable inflation are required, because the post-inflationary
expansion from a low $V_\star$ down to today is correspondingly shorter.
This sliding of $N_\star$ with the energy scale matters when we reach
the \GGTheory{} reading: the derived plateau sits one e-fold below the
GUT edge ($V_\star^{1/4} = 8.3\times10^{15}\,{\rm GeV}$), and the
reheating match at Planck-strength width fixes $N_\star = 52.6$ ---
undiluted, because the later lock crossing is prompt --- and the precise
value feeds directly into the predicted tilt $n_s = 1 - 2/N_\star$.
(At the retired TeV height this count would read $N_\star \sim 30$--$40$;
the sliding logic is unchanged.)

%========================================================
\subsection{A scalar field in FRW: action, stress tensor, equation of
motion}
\label{appA:infl-scalar}

The simplest source able to deliver $w<-1/3$ for an extended period is a
single real scalar field $\phi(x)$, the \emph{inflaton}, with canonical
action
\begin{equation}\label{eq:scalar-action}
S=\int d^4x\,\sqrt{-g}\,
\Big[\tfrac12 R\,\MPl^2
-\tfrac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
-V(\phi)\Big],
\qquad
\MPl\equiv\frac{1}{\sqrt{8\pi G}}=2.435\times10^{18}\,\mathrm{GeV},
\end{equation}
where $\MPl$ is the \emph{reduced} Planck mass (units $\hbar=c=1$
throughout this subsection) and $V(\phi)$ is the potential. The
stress--energy tensor follows from varying with respect to the metric,
\begin{equation}\label{eq:scalar-stress}
T_{\mu\nu}=\partial_\mu\phi\,\partial_\nu\phi
-g_{\mu\nu}\Big[\tfrac12 g^{\alpha\beta}\partial_\alpha\phi\,
\partial_\beta\phi+V(\phi)\Big].
\end{equation}
For a homogeneous configuration $\phi=\phi(t)$ on the flat FRW background
$ds^2=-dt^2+a^2(t)\,d\mathbf{x}^2$, this is the stress tensor of a
perfect fluid with
\begin{equation}\label{eq:scalar-rho-p}
\rho_\phi=\tfrac12\dot\phi^2+V(\phi),
\qquad
p_\phi=\tfrac12\dot\phi^2-V(\phi),
\qquad
w_\phi=\frac{p_\phi}{\rho_\phi}
=\frac{\tfrac12\dot\phi^2-V}{\tfrac12\dot\phi^2+V}.
\end{equation}
The reading of \eqref{eq:scalar-rho-p} is the whole story in one line:
if the kinetic energy $\tfrac12\dot\phi^2$ is \emph{small} compared with
the potential energy $V$, then $p_\phi\approx-\rho_\phi$, i.e.
$w_\phi\approx-1$, and the field behaves like a temporary cosmological
constant --- precisely the $w<-1/3$ condition
\eqref{eq:inflation-condition} that drives inflation. The field is then
said to be ``potential-dominated.''

The equation of motion is the Euler--Lagrange equation for $\phi$ in the
FRW background. Varying the action, or equivalently imposing covariant
conservation $\nabla^\mu T_{\mu\nu}=0$ on \eqref{eq:scalar-stress}, gives
the Klein--Gordon equation in an expanding universe,
\begin{equation}\label{eq:eom-phi}
\boxed{\;\ddot\phi+3H\dot\phi+V'(\phi)=0\;},
\qquad
V'\equiv\frac{dV}{d\phi}.
\end{equation}
This is the central equation of inflationary dynamics. It is the
equation of a particle of unit mass at ``position'' $\phi$ rolling in
the potential $V(\phi)$, subject to a velocity-proportional friction
term $3H\dot\phi$ supplied by the Hubble expansion (Hubble drag). The
two Friedmann equations close the system:
\begin{equation}\label{eq:friedmann-pair}
H^2=\frac{1}{3\MPl^2}\Big[\tfrac12\dot\phi^2+V(\phi)\Big],
\qquad
\dot H=-\frac{\dot\phi^2}{2\MPl^2}.
\end{equation}
The second relation, $\dot H=-\dot\phi^2/(2\MPl^2)$, shows that $H$ can
only decrease, and that it does so at a rate set entirely by the
kinetic energy: a slowly rolling field gives an almost constant $H$, the
quasi-de-Sitter regime.

%========================================================
\subsection{The slow-roll approximation and the potential slow-roll
parameters}
\label{appA:infl-slowroll}

``Slow roll'' is the controlled approximation in which the field rolls so
gently that (i) the kinetic energy is negligible against the potential,
$\tfrac12\dot\phi^2\ll V$, and (ii) the acceleration is negligible
against the friction, $|\ddot\phi|\ll 3H|\dot\phi|$. Under these two
assumptions the dynamical equations \eqref{eq:eom-phi} and
\eqref{eq:friedmann-pair} collapse to their leading-order ``slow-roll''
forms
\begin{equation}\label{eq:slowroll-eqs}
3H\dot\phi\simeq -V'(\phi),
\qquad
H^2\simeq\frac{V(\phi)}{3\MPl^2}.
\end{equation}
The self-consistency of these two approximations is governed by two
dimensionless numbers built purely from the \emph{shape} of the
potential --- the \emph{potential slow-roll parameters}:
\begin{equation}\label{eq:psr-params}
\boxed{\;
\epsilon\equiv\frac{\MPl^2}{2}\left(\frac{V'}{V}\right)^2,
\qquad
\eta\equiv\MPl^2\,\frac{V''}{V}\;}.
\end{equation}
Here $\epsilon$ measures the squared \emph{slope} of the potential
(in Planck units) and $\eta$ measures its \emph{curvature}. Their
meaning is transparent:
\begin{itemize}
\item $\epsilon\ll 1$ enforces that the slope is gentle enough for the
field to be potential-dominated. Indeed one can show
$\epsilon=-\dot H/H^2\equiv\epsilon_H$ to leading order, and the
accelerated-expansion condition $\ddot a>0$ is \emph{exactly}
$\epsilon_H<1$. Inflation proceeds while $\epsilon<1$ and \emph{ends}
when $\epsilon$ grows to $1$.
\item $\eta\ll 1$ (more precisely $|\eta|\ll 1$) enforces that the
curvature is small enough that the friction term dominates the
acceleration term, keeping the velocity adiabatically slaved to the
slope as in \eqref{eq:slowroll-eqs}. It controls how slowly $\epsilon$
itself evolves.
\end{itemize}
Both conditions are necessary; a potential can be locally flat
($\epsilon$ small) yet sharply curved ($\eta$ large), in which case
slow roll fails. The pair $\{\epsilon,\eta\}$ is the minimal data from
which all the leading observables below are computed.

%========================================================
\subsection{The number of e-folds from the potential}
\label{appA:infl-efold-integral}

With the slow-roll equations \eqref{eq:slowroll-eqs} in hand we can
re-express the e-fold integral \eqref{eq:efold-def} as an integral over
field values. Using $N=\int H\,dt=\int (H/\dot\phi)\,d\phi$ and
$\dot\phi\simeq-V'/(3H)$ with $H^2\simeq V/(3\MPl^2)$,
\begin{equation}\label{eq:efold-integral}
\boxed{\;
N=\int_{t_{\rm start}}^{t_{\rm end}}H\,dt
=\frac{1}{\MPl^2}\int_{\phi_{\rm end}}^{\phi_\star}
\frac{V}{V'}\,d\phi
=\frac{1}{\MPl}\int_{\phi_{\rm end}}^{\phi_\star}
\frac{d\phi}{\sqrt{2\epsilon}}\;},
\end{equation}
where $\phi_\star$ is the field value $N$ e-folds before the end of
inflation and $\phi_{\rm end}$ is fixed by the end-of-inflation condition
$\epsilon(\phi_{\rm end})=1$. The second form, with
$|V'/V|=\sqrt{2\epsilon}/\MPl$, makes manifest that the field traverses
$\Delta\phi\sim\MPl\sqrt{2\epsilon}\,N$ in field space: small $\epsilon$
means many e-folds per unit field range. Equation
\eqref{eq:efold-integral} is the practical workhorse --- given $V(\phi)$,
it returns $\phi_\star$ for the required $N_\star\approx50\text{--}60$,
and the observables below are then all evaluated \emph{at} $\phi_\star$.

%========================================================
\subsection{Quantum fluctuations and the scalar power spectrum}
\label{appA:infl-scalar-spectrum}

The deepest payoff of inflation is that it does not merely homogenize
the universe; it seeds the small inhomogeneities that grow into galaxies
and imprint the CMB anisotropies. The seeds are the quantum vacuum
fluctuations of the inflaton (and of the metric), stretched to
super-horizon scales by the accelerated expansion and frozen in as
classical curvature perturbations.

Decompose the field into background plus fluctuation,
$\phi(t,\mathbf{x})=\bar\phi(t)+\delta\phi(t,\mathbf{x})$. In the
quasi-de-Sitter background each Fourier mode of a (nearly) massless field
acquires, at horizon crossing $k=aH$, a fluctuation of characteristic
amplitude
\begin{equation}\label{eq:dphi-fluct}
\langle|\delta\phi_k|^2\rangle^{1/2}\simeq\frac{H}{2\pi},
\end{equation}
the de Sitter Hawking-temperature-like amplitude. The physically
conserved, gauge-invariant quantity on super-horizon scales is the
comoving curvature perturbation $\mathcal{R}$, related to the field
fluctuation by $\mathcal{R}=-(H/\dot{\bar\phi})\,\delta\phi$. Its
dimensionless power spectrum is defined by
$\langle\mathcal{R}_{\mathbf k}\mathcal{R}_{\mathbf k'}\rangle
=(2\pi)^3\delta^3(\mathbf k+\mathbf k')\,(2\pi^2/k^3)\,\mathcal{P_R}(k)$,
and evaluating it at horizon crossing gives the master result
\begin{equation}\label{eq:scalar-spectrum}
\boxed{\;
\mathcal{P_R}(k)=\left.\frac{H^2}{\dot{\bar\phi}^2}
\left(\frac{H}{2\pi}\right)^2\right|_{k=aH}
=\left.\frac{1}{8\pi^2}\frac{H^2}{\MPl^2\,\epsilon}\right|_{k=aH}
=\left.\frac{1}{24\pi^2}\frac{V}{\MPl^4\,\epsilon}\right|_{k=aH}\;}.
\end{equation}
The last equality used $H^2\simeq V/(3\MPl^2)$. The amplitude of this
spectrum at a chosen pivot wavenumber $k_\star$ (Planck uses
$k_\star=0.05\,{\rm Mpc^{-1}}$) is the \emph{scalar amplitude}
\begin{equation}\label{eq:As-def}
A_s\equiv\mathcal{P_R}(k_\star)
=\frac{1}{24\pi^2}\frac{V_\star}{\MPl^4\,\epsilon_\star},
\qquad
A_s^{\rm obs}=(2.10\pm0.03)\times10^{-9}.
\end{equation}
This is the celebrated COBE/Planck normalization: \emph{any} inflationary
model must reproduce $A_s\approx2.1\times10^{-9}$, and this single number
fixes the overall height-to-flatness combination $V_\star/\epsilon_\star$
of the potential. Note carefully what \eqref{eq:As-def} says: $A_s$ is
\emph{not} a free choice once the potential is specified; it is a
prediction. Conversely, a model that does not specify $V_\star$ and
$\epsilon_\star$ \emph{cannot} predict $A_s$ at all --- it can only
assert the observed value. Hold that thought.

The \emph{spectral index} (scalar tilt) measures the scale dependence of
$\mathcal{P_R}$,
\begin{equation}\label{eq:ns-def}
\ns-1\equiv\frac{d\ln\mathcal{P_R}}{d\ln k}\bigg|_{k_\star}.
\end{equation}
Carrying the $k$-derivative through \eqref{eq:scalar-spectrum} (using
$d\ln k\simeq H\,dt$ at horizon crossing and the slow-roll relations)
yields the standard leading-order expression
\begin{equation}\label{eq:ns-slowroll}
\boxed{\;\ns=1-6\epsilon+2\eta\;},
\qquad
\ns^{\rm obs}=0.9649\pm0.0042 .
\end{equation}
A scale-invariant Harrison--Zeldovich spectrum would have $\ns=1$; the
measured \emph{red tilt} $\ns<1$ is a genuine, non-trivial confirmation
that inflation involves a gently sloped, gently curved potential
($\epsilon,\eta$ small but nonzero) rather than an exact cosmological
constant. The combination $6\epsilon-2\eta\approx0.035$ is the headline
inflationary measurement.

%========================================================
\subsection{The tensor spectrum, $r$, and the consistency relation}
\label{appA:infl-tensor}

Inflation also stretches the two transverse-traceless polarizations of
the metric --- primordial gravitational waves. Being a massless
spin-2 field, each polarization fluctuates like a massless scalar, so the
\emph{tensor power spectrum} is
\begin{equation}\label{eq:tensor-spectrum}
\mathcal{P}_t(k)=\frac{2}{\pi^2}\frac{H^2}{\MPl^2}\bigg|_{k=aH}
=\frac{2}{3\pi^2}\frac{V}{\MPl^4}\bigg|_{k=aH},
\end{equation}
where the factor of $2$ sums the two polarizations and the result is
independent of $\epsilon$. Its tilt is
\begin{equation}\label{eq:nt-def}
n_t\equiv\frac{d\ln\mathcal{P}_t}{d\ln k}\bigg|_{k_\star}=-2\epsilon.
\end{equation}
The crucial observable is the \emph{tensor-to-scalar ratio}, the relative
amplitude of gravitational waves to density perturbations,
\begin{equation}\label{eq:r-def}
r\equiv\frac{\mathcal{P}_t(k_\star)}{\mathcal{P_R}(k_\star)}
=\frac{(2/\pi^2)(H^2/\MPl^2)}
{(1/8\pi^2)(H^2/\MPl^2\epsilon)}
=16\,\epsilon,
\qquad
r^{\rm obs}<0.036\ (95\%).
\end{equation}
Dividing \eqref{eq:tensor-spectrum} by \eqref{eq:scalar-spectrum}, all
the $H^2/\MPl^2$ factors cancel and one is left with the beautifully
simple
\begin{equation}\label{eq:r-eps}
\boxed{\;r=16\,\epsilon\;}.
\end{equation}
Because $\mathcal{P}_t\propto H^2\propto V$, a \emph{measurement} of $r$
is a direct measurement of the inflationary energy scale:
\begin{equation}\label{eq:Vinf-from-r}
V_\star^{1/4}\simeq
\left(\frac{r}{0.01}\right)^{1/4}\times1.0\times10^{16}\,{\rm GeV}.
\end{equation}
A detectable $r\gtrsim10^{-3}$ requires GUT-scale inflation
($V_\star^{1/4}\sim10^{16}\,{\rm GeV}$); conversely, low-scale inflation
predicts an unobservably tiny $r$. Combining
\eqref{eq:nt-def} and \eqref{eq:r-eps} gives the single-field
\emph{consistency relation},
\begin{equation}\label{eq:consistency}
\boxed{\;r=-8\,n_t\;},
\end{equation}
a sharp, falsifiable prediction of single-field slow-roll inflation: the
tensor amplitude and the tensor tilt are not independent, and a future
measurement of both would test the entire single-field framework. (The
minus sign records that a red tensor tilt, $n_t<0$, accompanies a
positive $r$.)

%========================================================
\subsection{Reheating: from cold inflaton to hot Big Bang}
\label{appA:infl-reheating}

When $\epsilon$ grows to unity inflation ends and the field, no longer
slow-rolling, falls toward the minimum of $V$ and oscillates about it.
These coherent oscillations are an enormous, cold, zero-momentum
condensate of inflaton quanta. Through couplings to other fields the
condensate decays, dumping its energy into a thermal bath of relativistic
particles --- \emph{reheating}. Parametrizing the inflaton decay by a
rate $\Gamma_\phi$, reheating completes when $H\sim\Gamma_\phi$, fixing
the reheating temperature
\begin{equation}\label{eq:Treh}
T_{\rm reh}\simeq
\left(\frac{90}{\pi^2 g_\star}\right)^{1/4}
\sqrt{\Gamma_\phi\,\MPl},
\end{equation}
with $g_\star$ the effective number of relativistic degrees of freedom.
Reheating must complete above the scale of Big Bang nucleosynthesis
($T_{\rm reh}\gtrsim$ a few MeV) so that the standard hot Big Bang takes
over with the correct light-element yields, and it sets the precise
$N_\star$ in \eqref{eq:efold-requirement} (a lower $T_{\rm reh}$ means
fewer observable e-folds). Reheating is the hand-off: it converts the
ordered potential energy of the inflaton into the disordered thermal
energy of the hot Big Bang, and its details feed back into the predicted
$\ns$ at the few-$\times10^{-3}$ level through $N_\star$.

%========================================================
\subsection{Worked example (STANDARD TEXTBOOK PEDAGOGY): quadratic and
Starobinsky potentials}
\label{appA:infl-worked}

The following two worked examples are the \emph{standard textbook
pedagogy}, included so the machinery above is concrete. They are
deliberately the canonical models every cosmology course computes; they
are \emph{not} the \GGTheory{} model, and the explicit gap between this
pedagogy and the \GGTheory{} case is stated in the next subsection.

\paragraph{Example A: quadratic (``$m^2\phi^2$'') inflation.}
Take the simplest chaotic-inflation potential
\begin{equation}\label{eq:quad-V}
V(\phi)=\tfrac12 m^2\phi^2 .
\end{equation}
Then $V'=m^2\phi$ and $V''=m^2$, so the slow-roll parameters
\eqref{eq:psr-params} are
\begin{equation}\label{eq:quad-eps-eta}
\epsilon=\frac{\MPl^2}{2}\left(\frac{m^2\phi}{\tfrac12 m^2\phi^2}\right)^2
=\frac{2\MPl^2}{\phi^2},
\qquad
\eta=\MPl^2\frac{m^2}{\tfrac12 m^2\phi^2}=\frac{2\MPl^2}{\phi^2}=\epsilon .
\end{equation}
Inflation ends at $\epsilon=1$, i.e.
$\phi_{\rm end}=\sqrt{2}\,\MPl$. The e-fold integral
\eqref{eq:efold-integral} gives
\begin{equation}\label{eq:quad-N}
N=\frac{1}{\MPl^2}\int_{\phi_{\rm end}}^{\phi_\star}
\frac{\tfrac12 m^2\phi^2}{m^2\phi}\,d\phi
=\frac{1}{\MPl^2}\int_{\phi_{\rm end}}^{\phi_\star}\frac{\phi}{2}\,d\phi
=\frac{\phi_\star^2-\phi_{\rm end}^2}{4\MPl^2}
\;\Longrightarrow\;
\phi_\star^2=(4N+2)\,\MPl^2 .
\end{equation}
For the benchmark $N_\star=60$ this gives
$\phi_\star=\sqrt{242}\,\MPl\approx15.6\,\MPl$, and hence
\begin{equation}\label{eq:quad-numbers}
\epsilon_\star=\frac{2\MPl^2}{\phi_\star^2}=\frac{2}{242}
\approx8.26\times10^{-3},
\qquad
\eta_\star=\epsilon_\star\approx8.26\times10^{-3}.
\end{equation}
The observables follow directly:
\begin{align}
\ns&=1-6\epsilon_\star+2\eta_\star
=1-4\epsilon_\star
=1-\frac{8}{242}\approx0.967,
\label{eq:quad-ns}\\[2pt]
r&=16\,\epsilon_\star=\frac{32}{242}\approx0.132,
\label{eq:quad-r}\\[2pt]
n_t&=-2\epsilon_\star\approx-0.0165
\quad(\text{check: }-8n_t=0.132=r,\ \checkmark).
\label{eq:quad-nt}
\end{align}
The COBE normalization \eqref{eq:As-def} then fixes the inflaton mass:
demanding $A_s=2.1\times10^{-9}$ with
$V_\star=\tfrac12 m^2\phi_\star^2$ and $\epsilon_\star$ from
\eqref{eq:quad-numbers} gives
$m\approx6\times10^{-6}\,\MPl\approx1.4\times10^{13}\,{\rm GeV}$ ---
i.e. GUT-scale inflation. Note the verdict: $m^2\phi^2$ predicts a
\emph{good} tilt $\ns\approx0.967$ but a tensor ratio $r\approx0.13$ that
the Planck/BICEP bound $r<0.036$ now \emph{excludes}. Quadratic inflation
is a textbook model that the data have killed --- a clean illustration
that $\ns$ and $r$ together, not $\ns$ alone, are the discriminating
test.

\paragraph{Example B: Starobinsky ($R^2$) inflation.}
The historically first and observationally favored model adds an $R^2$
term to gravity; in the Einstein frame it is equivalent to a scalar with
the plateau potential
\begin{equation}\label{eq:staro-V}
V(\phi)=\Lambda^4\left(1-e^{-\sqrt{2/3}\,\phi/\MPl}\right)^2 .
\end{equation}
Write $x\equiv e^{-\sqrt{2/3}\,\phi/\MPl}$, so $V\propto(1-x)^2$. Then
$V'=\Lambda^4\,2(1-x)\,x\,\sqrt{2/3}/\MPl$ and a short computation gives,
in the large-$\phi$ (small-$x$) plateau regime where $x\ll1$,
\begin{equation}\label{eq:staro-eps-eta}
\epsilon\simeq\frac{4}{3}\,\frac{x^2}{(1-x)^2}\approx\frac{4}{3}x^2,
\qquad
\eta\simeq-\frac{4}{3}\,\frac{x(1-2x)}{(1-x)^2}\approx-\frac{4}{3}x .
\end{equation}
The e-fold count \eqref{eq:efold-integral} evaluates (for $\phi_\star$ on
the plateau) to the compact result
$N\simeq\tfrac34\,e^{\sqrt{2/3}\,\phi_\star/\MPl}=\tfrac34/x_\star$, so
$x_\star\simeq 3/(4N)$. Substituting back,
\begin{align}
\epsilon_\star&\simeq\frac{4}{3}\left(\frac{3}{4N}\right)^2
=\frac{3}{4N^2},
\qquad
\eta_\star\simeq-\frac{4}{3}\cdot\frac{3}{4N}=-\frac{1}{N},
\label{eq:staro-eps-N}\\[2pt]
\ns&=1-6\epsilon_\star+2\eta_\star
\simeq1-\frac{2}{N},
\qquad
r=16\epsilon_\star\simeq\frac{12}{N^2}.
\label{eq:staro-obs}
\end{align}
For $N_\star=60$ these give the textbook Starobinsky predictions
\begin{equation}\label{eq:staro-numbers}
\ns\approx1-\frac{2}{60}=0.9667,
\qquad
r\approx\frac{12}{3600}\approx3.3\times10^{-3},
\qquad
n_t=-2\epsilon_\star\approx-4.2\times10^{-4},
\end{equation}
in excellent agreement with Planck ($\ns=0.9649\pm0.0042$,
$r<0.036$). The COBE normalization fixes the plateau height to
$\Lambda^4$ with $\Lambda\approx3\times10^{16}\,{\rm GeV}$, i.e. again
GUT-scale inflation. The pedagogical lesson of the two examples
together: \emph{both} the right tilt $\ns\approx0.965$ \emph{and} the
small $r$ are reproduced by a GUT-scale \emph{plateau} model, and doing
so requires the model to predict the height $V_\star$ (hence $A_s$) and
the curvature $\eta$ (hence the tilt) from one and the same potential.
The two observables $A_s$ and $\ns$ are jointly constrained by the shape
of $V(\phi)$; you do not get to choose them independently.

%========================================================
\subsection{What standard inflation explains --- and what it leaves open}
\label{appA:infl-limits}

The machinery of Step~1 is a triumph: a single slowly-rolling scalar
flattens the universe, stretches quantum fluctuations into the seeds of
galaxies, and predicts a nearly scale-invariant spectrum that the
microwave background confirms to remarkable precision. But it is worth
asking, exactly as we did for the Friedmann background in
\S\ref{appA:frw-limits}, what the standard account must put in by hand.

\begin{enumerate}[leftmargin=*]
\item \textbf{The inflaton itself.} Standard inflation does not say what
the field $\phi$ \emph{is}. It is introduced because the early universe
needs a phase of accelerated expansion; no independent physics requires
it to exist. Inflation, in the textbook telling, is a mechanism in search
of a cause.
\item \textbf{The potential.} The shape $V(\phi)$ --- plateau, shoulder,
minimum --- is chosen so that the predictions land on the data. Dozens of
potentials are on the market precisely because none is forced: the data
select among them rather than the theory predicting one.
\item \textbf{The energy scale.} How high the plateau sits, $V^{1/4}$,
anywhere from the GUT scale to far below, is not fixed by the mechanism.
It is the single most important number in the story, and it is a free
dial.
\item \textbf{The spectrum normalization.} The amplitude $A_s$ and tilt
$\ns$ are \emph{measured} and then used to fix the potential's height and
slope, not derived from anything deeper.
\end{enumerate}

So inflation, for all its success, leaves the same kind of gap the
Friedmann background did: it accommodates the early universe with a field
and a potential tailored to fit, but it does not say why that field
exists or what sets its scale. A geometric account should supply the
cause. As we will see, \GGTheory{} supplies some of it cleanly --- the
identity of the field, its energy scale, and one sharp falsifiable
prediction --- and is honest that it does not yet supply the rest. This
is the most open sector of the paper, and we mark the boundary exactly.

\subsection{The geometric derivation: the two-basin descent, taught slowly}
\label{appA:infl-gg-reading}

Of the four gaps of \S\ref{appA:infl-limits}, \GGTheory{} closes the
first three --- the identity of the field, its energy scale, and its
dynamics --- computes the tilt and the tensor ratio, and leaves exactly
one number as a declared input: the amplitude. This subsection teaches
that result at course pace, because it is the
centerpiece of the paper's inflationary sector and the single best illustration of how a geometric Step~3
replaces a textbook's freedom with a computation. The student who has
worked through Step~1 owns every tool needed here: nothing below uses
more than the slow-roll formulae already derived.

\paragraph{Step 3a --- the inflaton is a bulk modulus, not an invention (gap 1).}
The field $\phi$ is not introduced for the occasion. It is identified
with the \emph{scale modulus} $\Sigma_r$ of the six-dimensional bulk ---
the one-dimensional flat direction of the geometry's shape space along
the radial interval (Part~II; \citealp{ArisakaA2GG6}). Such a modulus is
an unavoidable feature of the bulk, present whether or not one wants an
inflaton; \GGTheory{} does not add a field to drive inflation, it
\emph{recognizes} that the bulk already carries one --- and only one, which
is why there is no choice to make. \emph{Physics intuition:} the
inflaton is the size of the hidden geometry, slowly relaxing --- the same
radial coordinate that Appendix~E used as the energy axis, seen now as a
dynamical field.

\paragraph{Step 3b --- the potential's \emph{class} is derived, not chosen (gap 2).}
Here is the step where the textbook's biggest freedom --- the functional
form of $V(\phi)$, chosen from a zoo --- becomes a computation. The
\GDOS{} boundary law is a conformal law with a non-conformal coupling to
the scale modulus; its trace anomaly generates an $R^2$ term in the
boundary effective action, and the Einstein-frame map of an $R^2$ theory
is a theorem of classical field theory (no cosmology input at all): the
potential is forced into the Starobinsky class
\begin{equation}
V(\phi) \;=\; \tfrac34\,M^2 M_{\rm Pl,red}^2
\left(1 - e^{-\sqrt{2/3}\,\phi/M_{\rm Pl,red}}\right)^{2},
\label{appA:infl-eq-plateau}
\end{equation}
with the plateau shape and the $\sqrt{2/3}$ fixed by four-dimensional
gravity and only the single mass scale $M$ free
(\citealp{Starobinsky1980}; Part~IV). Every slow-roll quantity of Step~1
now follows from Eq.~\eqref{appA:infl-eq-plateau} by pure calculus ---
the student should verify, using the definitions of
\S\ref{appA:infl-limits}, that on the plateau
$\epsilon_V \simeq \tfrac34\,e^{-2\sqrt{2/3}\,\phi/M_{\rm Pl,red}}$ and
$\eta_V \simeq -\tfrac{4}{3}\,e^{-\sqrt{2/3}\,\phi/M_{\rm Pl,red}}$, so
that slow roll fails (graceful exit) where the plateau steepens, with no
waterfall field and no tuning. \emph{Physics intuition:} the geometry
does not hand us a potential; it hands us a \emph{class} --- one shape,
one free height --- and the shape alone will fix the observables.

\paragraph{Step 3c --- where the plateau sits, and the descent below it.}
The derived plateau is \emph{GUT-adjacent}: the observed amplitude places
its height at
\begin{equation}
V_\star^{1/4} \;=\; 8.3\times10^{15}\ \mathrm{GeV},
\qquad
H_\star \;\simeq\; 1.6\times10^{13}\ \mathrm{GeV},
\label{appA:infl-eq-scale}
\end{equation}
one e-fold below the GUT edge where the descent begins. The student
should note --- because the audit trail of Part~IX records it --- that
the program once conjectured the opposite: a plateau at the TeV lock
scale $\Mlock = \tfrac12 M_{\rm Pl}e^{-35} = 3.85$~TeV. Step~3e shows,
as a worked exercise, why that reading is refuted at sign level. The
true role of the lock scale is different and sharper: it is the
\emph{second basin} of the descent. The full history is a
\emph{two-basin descent} of the one modulus: the primary plateau
(inflation, $N \simeq 60$ e-folds), the graceful exit at its steep edge,
reheating, then --- far down the ladder --- the prompt crossing of the
lock basin ($N_2 = 0$: the \GRIP{} ledger-action bound on $\Sigma_r$
forbids a second inflation, with some fifty orders of margin), and
finally the terminus, the Ground Geometric Attractor whose residual
energy is today's dark energy (Part~IV). \emph{Physics intuition:} think
of the $\tau \to \mu \to e$ cascade --- three basins, two crossings, the
stable floor at the bottom. The vacuum has generations exactly as matter
does; inflation is the $\tau$ of the vacuum, the lock basin its muon
(real but promptly passed), dark energy its electron. That analogy, the
\emph{vacuum generations}, is developed as a load-bearing statement in
Part~IV \S\ref{subsec:vacuum-generations}.

\paragraph{Step 3d --- the tilt and the e-fold count, computed end to end.}
Now the arithmetic the textbook could never do, carried in full. For the
plateau class, the standard slow-roll results of Step~1 collapse to the
one-parameter family
\begin{equation}
n_s \;=\; 1 - \frac{2}{N^{*}},
\qquad
r \;=\; \frac{12}{N^{*2}},
\label{appA:infl-eq-nsr}
\end{equation}
so everything hangs on the number of e-folds $N^{*}$ before the end of
inflation at which the CMB pivot scale left the horizon --- and $N^{*}$
is fixed by reheating, not guessed. The modulus couples to the boundary
at Planck strength, so its decay rate is
$\Gamma \simeq M^{3}/M_{\rm Pl,red}^{2}$; with the scalaron mass
$M = 3.3\times10^{13}$~GeV this gives $\Gamma \simeq 5.9$~TeV and a
reheating temperature
\begin{equation}
T_{\rm rh} \;\simeq\; \left(\frac{90}{\pi^2 g_*}\right)^{1/4}
\sqrt{\Gamma\,M_{\rm Pl,red}}
\;\simeq\; 3\times10^{10}\ \mathrm{GeV},
\label{appA:infl-eq-treh}
\end{equation}
(the student should confirm the arithmetic with $g_* \simeq 100$).
Feeding $T_{\rm rh}$ into the standard horizon-matching formula of
Step~1 --- and using the prompt lock crossing $N_2 = 0$, so that no later
epoch dilutes the count --- yields
\begin{equation}
N^{*} \;=\; 52.6,
\qquad\Longrightarrow\qquad
n_s \;=\; 1 - \frac{2}{52.6} \;=\; 0.9620,
\qquad
r \;=\; \frac{12}{(52.6)^2} \;=\; 0.0043.
\label{eq:r-null}
\end{equation}
Planck measures $n_s = 0.9649 \pm 0.0042$: the computed tilt sits
$0.7\sigma$ from the data, \emph{risked against them, not read from
them}. And the tensor target is two-sided and decisive within the
decade: a factor eight below the current bound, several times above the Simons-Observatory/LiteBIRD floor. A measurement inconsistent with
$r \simeq 0.004$ in either direction falsifies the derived launch
outright. \emph{Physics intuition:} once the shape of the hill is a
theorem, the only question is where on the hill the pivot scale exited
--- and reheating physics answers that, leaving nothing to tune.

\paragraph{Step 3e --- the honest boundary: the amplitude, and the retired TeV reading.}
One number remains. The scalar amplitude
$A_s = V_\star/(24\pi^2 M_{\rm Pl,red}^4\,\epsilon_\star) \simeq
2.1\times10^{-9}$ fixes the height $M$ --- and the height is the sector's
\emph{one declared input}, exactly as the epoch is the one declared input
of the $\pi$-locked inventory. The named route to paying it is
Lemma~V\('\) (Conjecture~\ref{conj:gdos-amplitude}): the \GDOS{}
heat-kernel evaluation of the $R^2$ coefficient on the warped interval,
whose warp candidate $M = M_5 e^{-K/3}$ currently lands within
$87\%$ of the required mass with no parameter. \emph{This is an open
computation stated precisely enough for the reader to attempt} --- a
graduate student who closes it makes the launch sector input-free, and
the invitation is meant literally. Finally, the retired reading, kept
for the audit trail as a worked exercise in falsification: a TeV-scale
plateau gives $V_\star/M_{\rm Pl,red}^4 \sim 10^{-59}$, so matching
$A_s$ would demand $\epsilon_\star \sim 10^{-53}$, and the tilt would
come out \emph{blue} ($n_s - 1 \sim +10^{-26}$) against the measured red
$-0.0351 \pm 0.0042$ --- wrong in sign, with no parameter to rescue it.
The student should reproduce both numbers; nothing teaches the
difference between a fitted model and a forced one faster than watching
a conjecture die of arithmetic.

\paragraph{What this appendix establishes.}
The honest ledger is therefore strong, and its one gap is named.
Standard inflation (Step~1) is reproduced in full and is established
textbook physics. Geometry then supplies what the textbook left open:
the inflaton is the unique bulk scale modulus rather than an invention
(Step~3a); the potential's class is derived rather than chosen from a
zoo (Step~3b); the plateau sits one e-fold below the GUT edge, at the
top of a two-basin descent whose later basins are the Kaxion's
birthplace and dark energy (Step~3c); the tilt and the tensor ratio are
computed end to end --- $n_s = 0.9620$, $r = 0.0043$ --- with the e-fold
count fixed by the derived reheating temperature (Step~3d); and the
amplitude is the one declared input, with the computation that would pay
it stated as an open target (Step~3e). In the codebook this reads:
$n_s$ and $V_{\rm inf}^{1/4}$ Derived-conditional on the revised
Lemma~IV, $r$ Derived, $A_s$ Asserted pending Lemma~V\('\). In the
Euclidean spirit, the parts that follow from the geometry are stated as
results, and the part that does not is marked, in plain sight, as the
work that remains --- and offered, in this course, to the generation that
will finish it.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{G.1.1. Quantify the horizon problem:
what angle does a causal patch at recombination subtend on
today's sky?}

\noindent\emph{Solution.}  Step 1: in a matter-dominated
history the comoving particle horizon grows as $\eta \propto
a^{1/2}$, so at recombination it was
$\eta_{\rm rec} = \eta_0/\sqrt{1+z_*} \approx \eta_0/33$.
Step 2: that comoving length is seen today at comoving distance
$\approx \eta_0$, subtending $\theta \sim \eta_{\rm rec}/\eta_0
\approx 1/33$~rad $\approx 2^\circ$ --- of order one degree
after the exact geometry is kept.  Step 3: the CMB is isothermal
to $10^{-5}$ across $180^\circ$: tens of thousands of patches
that, in the hot-big-bang history, never exchanged a photon,
all at one temperature.  Physics intuition: this is not a small
discrepancy to adjust away but a causality contradiction ---
and the chapter's resolution inverts the picture: make the
comoving Hubble radius \emph{shrink} for long enough
\eqref{eq:horizon-comoving} and the whole sky shares one
causal ancestor.

\medskip\noindent\textbf{G.2.1. Derive the $\sim\!60$ e-fold
requirement from today's Hubble radius.}

\noindent\emph{Solution.}  Step 1: the flatness/horizon cures
need the comoving Hubble radius $(aH)^{-1}$ to shrink during
inflation by at least the factor it has grown since
\eqref{eq:efold-requirement}.  Step 2: from reheating at
$T_{\rm rh}$ to today, $(aH)^{-1}$ grows (radiation era) as
$a^{1}\cdot$(matter-era corrections); logarithmically, the
growth is $\ln\bigl[(aH)^{-1}_0/(aH)^{-1}_{\rm rh}\bigr]
\approx \ln(T_{\rm rh}/T_0)$ up to $O(1)$ pieces.  Step 3:
with this chapter's \emph{derived} reheating temperature
$T_{\rm rh} \simeq 3\times10^{10}$~GeV against
$T_0 \approx 2.35\times10^{-13}$~GeV:
$\ln(1.3\times10^{23}) \approx 53$ --- inside the benchmark
band $N_\star \approx 50$--$60$ \eqref{eq:efold-benchmark},
and strikingly beside the $N_\star = 52.6$ that G.9.2 extracts
from the measured tilt.  Closing that near-coincidence into an
identity is precisely the cohort's commissioned e-fold budget
(the derivation of $\ln(B/kT_0)$ from the descent).  Physics intuition: sixty e-folds is not a magic
number but a mirror --- inflation must un-do, in advance,
exactly the growth the subsequent hot history will perform,
and the requirement is read off today's sky.

\medskip\noindent\textbf{G.7.1. Find the energy scale of
inflation implied by the derived tensor ratio.}

\noindent\emph{Solution.}  Step 1: the tensor amplitude ties
$r$ to the inflationary potential:
$V_\star = \tfrac{3\pi^2}{2}A_s\,r\,\MPl^4$
\eqref{eq:Vinf-from-r} (reduced-Planck units).  Step 2: with
$A_s = 2.1\times10^{-9}$ and the derived $r = 0.0043$:
$V_\star^{1/4} \approx 8.3\times10^{15}$~GeV.  Step 3: this
sits just below the classic GUT scale and within an order of
magnitude of the cohort's own high edge --- the plateau is a
statement that inflation ran at the energy where the ledger's
deepest structure lives, and the sub-detectable-but-nonzero
$r$ is its signature.  Physics intuition: $r$ is a
thermometer: gravitational waves are generated purely by the
expansion rate, so their amplitude reads the inflationary
energy directly --- which is why a future $r$ detection at the
$10^{-3}$ level is the sharpest single test this sector
offers (the falsification catalog prices it).

\medskip\noindent\textbf{G.7.2. Prove the slow-roll consistency
relation $n_t = -r/8$, and state its use.}

\noindent\emph{Solution.}  Step 1: the tensor spectrum tilts as
$n_t = -2\epsilon_\star$ \eqref{eq:nt-def}, from
$P_T \propto H^2$ and $\dot H = -\epsilon H^2$.  Step 2: the
tensor-to-scalar ratio is $r = 16\epsilon_\star$
\eqref{eq:r-eps}.  Step 3: eliminate $\epsilon_\star$:
$n_t = -r/8$ \eqref{eq:consistency} --- a relation between two
observables with \emph{no} free function of the potential
surviving.  Any single-field slow-roll model must satisfy it;
a measured violation retires the whole class, this chapter's
geometric reading included.  Physics intuition: consistency
relations are the fingerprints of a mechanism rather than a
model --- the potential drops out, leaving pure kinematics of
one clock rolling down one hill; nature either shows the
fingerprint or convicts a different suspect.

\medskip\noindent\textbf{G.8.1. Bound the reheating temperature
from instantaneous conversion.}

\noindent\emph{Solution.}  Step 1: if the inflaton's energy
$V_\star$ thermalizes instantly into $g_* = 106.75$ degrees of
freedom, then $\tfrac{\pi^2}{30}g_*T_{\rm rh}^4 = V_\star$
\eqref{eq:Treh}.  Step 2: with $V_\star^{1/4} =
8.3\times10^{15}$~GeV (G.7.1):
$T_{\rm rh}^{\max} = \bigl(\tfrac{30}{\pi^2 g_*}\bigr)^{1/4}
V_\star^{1/4} \approx 3.4\times10^{15}$~GeV.  Step 3: any
delay in thermalization only lowers this --- reheating is a
ceiling, not a prediction, until the coupling chain is fixed.
The cohort's descent supplies that chain, and its derived value,
$T_{\rm rh} \simeq 3\times10^{10}$~GeV, sits five orders below
the ceiling: reheating here is slow, and it is exactly this
derived $T_{\rm rh}$ that feeds the e-fold count of G.2.1 and
selects thermal leptogenesis in Appendix~\ref{appA:leptogenesis}.
Physics intuition: reheating is the ignition of the hot Big
Bang by the death of inflation --- everything in
Appendices~\ref{appA:thermal}--\ref{appA:leptogenesis}
happens downstream of this one temperature, which is why the
budget campaign treats it as the keystone number.

\medskip\noindent\textbf{G.9.1. Rule out quadratic inflation
with one line of arithmetic.}

\noindent\emph{Solution.}  Step 1: for $V = \tfrac12 m^2\phi^2$
the chapter derives $\phi_\star^2 = (4N+2)\MPl^2$
\eqref{eq:quad-N}, hence $\epsilon_\star = 2\MPl^2/\phi_\star^2
= 2/(4N+2)$.  Step 2: $r = 16\epsilon_\star = 32/(4N+2)
\approx 0.13$ for $N_\star = 60$ \eqref{eq:quad-r}.  Step 3:
the observational bound is $r < 0.036$: quadratic inflation
overshoots by a factor of four --- dead, not wounded.  The
same arithmetic gives $\ns = 1 - 2/N - \ldots \approx 0.967$:
acceptable --- it is the \emph{tensors} that convict.
Physics intuition: large-field models drag the inflaton across
many Planck lengths, and every Planck length of travel pumps
gravitational waves; the sky's silence in $B$-modes is a
statement about the \emph{distance rolled}, which is why
plateau potentials (next problem) survive where monomials die.

\medskip\noindent\textbf{G.9.2. Show that the cohort's derived
tilt and tensor ratio are one prediction, not two.}

\noindent\emph{Solution.}  Step 1: on the Starobinsky-class
plateau the chapter derives $\ns = 1 - 2/N_\star$ and
$r = 12/N_\star^2$ \eqref{eq:staro-obs} --- two observables,
one parameter.  Step 2: the cohort's derived tilt is
$\ns = 0.9620$; solving, $N_\star = 2/(1 - \ns) = 52.6$.
Step 3: inserting that \emph{same} $N_\star$:
$r = 12/(52.6)^2 = 0.0043$ --- exactly the cohort's derived
tensor ratio.  The pair $(\ns, r)$ lies on the plateau's
one-parameter curve, and the e-fold number it selects is the
one the corridor's descent must independently supply --- the
open cross-check chartered as the e-fold budget.
Physics intuition: on a plateau, tilt and tensors are two
shadows of one length --- how far from the edge the descent
began; measure either, predict the other.  The cohort's pair
is self-consistent to the last digit, and the Simons Observatory and LiteBIRD test the prediction within the decade.

\GGtakeaway{\emph{Standard result:} slow-roll inflation solves the
horizon, flatness, and relic problems, but the inflaton's identity, its
potential, its scale, and its e-fold count are all chosen, not derived.
\emph{What it assumed:} a field, a shape $V(\phi)$, a height, and
$N^{*} \sim 50$--$60$ by convention. \emph{What the geometry replaced it
with:} the unique bulk scale modulus on a derived Starobinsky-class
plateau one e-fold below the GUT edge, with $N^{*} = 52.6$ from the
computed reheating temperature, $n_s = 0.9620$ ($0.7\sigma$),
$r = 0.0043$ (decisive this decade), and one declared input --- the
amplitude --- carried by the named open computation Lemma~V\('\).
\emph{Continue in:} Part~IV (the descent), Part~X (identity),
\PaperCOne{} (the Kaxion born at the second crossing).}


\section{Big-Bang nucleosynthesis and the lithium-7 problem}
\label{appA:bbn}

\GGteachbox{Why the first three minutes made one quarter of the
baryons into helium and almost nothing else; weak freeze-out, the
deuterium bottleneck, and the full reaction network derived with worked
numbers; the observational triumph (D, He) and the stubborn factor-3
lithium-7 problem; and the geometric reading in which an open-system
\(A=7\) port suppresses the lithium sector by \(1/\pi\) while leaving
deuterium and helium untouched.}
\GGreadalong{\citet{KolbTurner1990} ch.~4 (the classic treatment);
\citet{Ryden2017} ch.~10 for a first pass.}

The primordial-nucleosynthesis reaction network, the helium and deuterium yields (Figure~\ref{fig:appfig-bbn}), and the lithium problem treated in this appendix follow the foundational and modern precision references \citep{WagonerFowlerHoyle1967,Steigman2007,CyburtFieldsOliveYeh2016,Pitrou2018,Fields2011}.

\begin{figure}[htbp]
\centering
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  >={Stealth[length=2.2mm]}]

% ===== quantified frame ==========================================
%   x = 10*log10(eta10),  eta10 = eta*1e10  in [1,10]
%   y = 0.75*(11 + log10(abundance)) : one decade = 0.75 units,
%       y = 0 is 10^-11 and y = 8.25 is 10^0.
\def\xL{0} \def\xR{10}
\def\yB{0} \def\yT{8.55}

% ---- GG band + observed line (drawn first, behind the curves) ----
\fill[green!18] (8.56,\yB) rectangle (8.76,8.25);
\draw[green!45!black,dashed,line width=0.80pt] (8.657,\yB) -- (8.657,8.25);
\draw[purple!70!black,line width=0.80pt] (7.868,\yB) -- (7.868,8.25);

% ---- axes ----
\draw[->,line width=0.80pt] (\xL,\yB) -- (10.9,\yB)
  node[right,align=center] {$\eta=n_b/n_\gamma$};
\draw[->,line width=0.80pt] (\xL,\yB) -- (\xL,\yT);
\node[anchor=south west,align=left] at (-0.15,8.62)
  {abundance (log scale): number ratio to H, and $^4$He as the mass fraction $Y_p$};
% x ticks
\foreach \xx/\lab in {0/{$10^{-10}$},3.010/{$2$},4.771/{$3$},6.990/{$5$},8.451/{$7$},10/{$10^{-9}$}}
  \draw (\xx,\yB) -- (\xx,-0.12) node[below] {\scriptsize\lab};
\node[anchor=north] at (5,-0.70) {\scriptsize $\eta\times10^{10}$};
% y ticks, one per decade
\foreach \yy/\lab in {8.25/{$10^{0}$},7.50/{$10^{-1}$},6.75/{$10^{-2}$},6.00/{$10^{-3}$},
                      5.25/{$10^{-4}$},4.50/{$10^{-5}$},3.75/{$10^{-6}$},3.00/{$10^{-7}$},
                      2.25/{$10^{-8}$},1.50/{$10^{-9}$},0.75/{$10^{-10}$},0/{$10^{-11}$}}
  \draw (\xL,\yy) -- (-0.12,\yy) node[left] {\scriptsize\lab};
\draw[black!12,line width=0.35pt] (\xL,4.50) -- (\xR,4.50);
\draw[black!12,line width=0.35pt] (\xL,1.50) -- (\xR,1.50);

% ============ He-4 mass fraction Y_p: 0.240 -> 0.249, flat on a log axis ====
\draw[line width=1.10pt,red!70!black]
  (0.000,7.785) -- (1.761,7.787) -- (3.802,7.790) -- (5.315,7.791)
  -- (6.990,7.793) -- (8.657,7.795) -- (10.000,7.797);
\node[red!60!black,anchor=south east] at (7.55,7.86) {$^4$He: $Y_p\simeq0.247$ (mass fraction)};

% ============ D/H, ~eta^-1.6 ============
\draw[line width=1.10pt,blue!70!black]
  (0.000,5.746) -- (0.792,5.651) -- (1.761,5.535) -- (2.788,5.412) -- (3.802,5.290)
  -- (4.472,5.210) -- (5.315,5.109) -- (6.232,4.999) -- (6.990,4.908)
  -- (7.868,4.802) -- (8.657,4.708) -- (9.345,4.625) -- (10.000,4.546);
\node[blue!60!black,anchor=south west] at (0.15,5.85) {D/H};

% ============ He-3/H, ~eta^-0.6 ============
\draw[line width=1.10pt,teal!70!black]
  (0.000,4.854) -- (1.761,4.775) -- (3.802,4.683) -- (5.315,4.615)
  -- (6.990,4.540) -- (8.657,4.464) -- (10.000,4.404);
\node[teal!55!black,anchor=north west] at (0.15,4.34) {$^3$He/H};

% ============ Li-7/H valley: minimum 1.1e-10 near eta10 = 2.8 ============
\draw[line width=1.10pt,orange!85!black]
  (0.000,1.201) -- (0.792,1.127) -- (1.761,1.036) -- (2.788,0.939) -- (3.802,0.844)
  -- (4.472,0.781) -- (5.315,0.903) -- (6.232,1.037) -- (6.990,1.147)
  -- (7.868,1.274) -- (8.657,1.389) -- (9.345,1.489) -- (10.000,1.584);
\node[orange!70!black,anchor=north] at (3.2,0.66) {$^7$Li/H};

% ---- legend for the two vertical markers, in the empty band ----
\node[draw=black!35,line width=0.55pt,rounded corners=2pt,fill=black!3,align=left,
      inner sep=4pt,anchor=north west,font=\scriptsize] at (0.15,7.45)
  {\tikz\draw[green!45!black,dashed,line width=0.80pt](0,0)--(0.42,0);\ \
   GG prediction $\eta_B=7.34\times10^{-10}$ (headline $\approx7\times10^{-10}$)\\[1.5pt]
   \tikz\draw[purple!70!black,line width=0.80pt](0,0)--(0.42,0);\ \
   observed $\eta=6.12\times10^{-10}$};

% ---- Li-7 problem: predicted vs observed points at the observed eta ----
\filldraw[orange!90!black] (7.868,1.274) circle (2.2pt);
\filldraw[orange!55!black] (7.868,0.903) circle (2.2pt);
\draw[<->,gray!55!black,line width=0.60pt] (8.03,1.274) -- (8.03,0.903);
\draw[gray!45!black,line width=0.50pt] (8.06,1.09) -- (8.40,2.42);
\node[draw=gray!55!black,line width=0.55pt,rounded corners=2pt,fill=white,align=left,
      inner sep=4pt,anchor=north west,font=\scriptsize,text=gray!25!black] at (6.10,3.75)
  {$^7$Li problem at the\\
   observed $\eta$: predicted\\
   $\sim5\times10^{-10}$ against\\
   observed $\sim1.6\times10^{-10}$,\\
   a factor $\sim3$ high};

% ---- honesty note box, in the empty band below He-3/H ----
\node[draw=orange!65!black,line width=0.80pt,rounded corners=2pt,fill=orange!8,align=left,
      inner sep=4pt,anchor=north west,font=\scriptsize] at (0.15,3.75)
  {\textbf{(!)}\ \ GG: the $A\!=\!7$ open-system\\
   port is \emph{partially resolved}; the\\
   mechanism is deferred to the\\
   forthcoming \textsf{GG-C4-Nucleo}.\\
   $\eta_B$ itself is an order-of-magnitude\\
   / Asserted-transport entry.};

\end{tikzpicture}}
\caption[BBN abundances vs the baryon-to-photon ratio]{%
Standard big-bang nucleosynthesis (Schramm plot). Both axes are
quantified and logarithmic: the abundances are number ratios to hydrogen,
except $^4$He, which is the mass fraction $Y_p$.
Predicted primordial abundances vs the baryon-to-photon ratio $\eta$: the
$^4$He mass fraction $Y_p\!\sim\!0.247$ is nearly flat (rising slightly),
D/H falls steeply, $^3$He/H is mild, and $^7$Li/H traces the characteristic
dip-then-rise valley near $\eta\!\sim\!1$--$5\times10^{-10}$. The vertical
\emph{green} band marks the GG central $\eta_B = 7.34\times10^{-10}$ (headline $\approx7\times10^{-10}$; the $\times3^{\pm1}$ band is wider than the frame)
(from \OGNfive{} integer $K=35$ via $e^{-K/2}$) and the \emph{purple} line the
observed $\eta=6.12\times10^{-10}$. \textbf{Lithium-7 problem (honest):} at
the CMB $\eta$ the predicted $^7$Li/H$\,\sim\!5\times10^{-10}$ exceeds the
observed $\sim\!1.6\times10^{-10}$ by a factor $\sim\!3$ (both points drawn).
\textbf{GG reading:} the proposed $A\!=\!7$ open-system port is
``Partially resolved; mechanism deferred to the forthcoming
\textsf{GG-C4-Nucleo}''; $\eta_B$ itself is an order-of-magnitude /
Asserted-transport entry. The figure teaches standard BBN fully and marks
the $^7$Li discrepancy honestly.}
\label{fig:appfig-bbn}
\end{figure}


This appendix is a self-contained, ground-up account of primordial
nucleosynthesis, pitched at a first-year graduate student who has seen the
Friedmann equations and elementary statistical mechanics but has never
followed a light-element abundance calculation line by line. As in
Appendix~E, we build it in three steps.

\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{Standard BBN
(\S\ref{appA:bbn-np-freezeout}--\ref{appA:bbn-worked}).} The
neutron-to-proton freeze-out, the deuterium bottleneck, the light-element
reaction network, the dependence on the baryon-to-photon ratio, and a
worked helium calculation --- all derived from scratch.
\item \textbf{Where it succeeds and where it fails
(\S\ref{appA:bbn-observation}).} Two of the three measured abundances
(deuterium, helium-4) are among the great triumphs of cosmology. The
third, lithium-7, is not: standard BBN overshoots it by a factor of
three. This is the one genuine failure of the standard account, and it is
the opening for the geometry.
\item \textbf{The geometric derivation (\S\ref{appA:bbn-gg}).} The same
closed-packet factor $1/\pi$ that fixes the cosmic matter inventory,
applied to the mass-7 flux as an open-system port, predicts the lithium
deficit --- with the honest status of that conjecture, and the
network-level obligations, marked exactly.
\end{enumerate}

The intent throughout is to be scrupulous about status: standard BBN is a
\emph{Derived} consequence of the inherited expansion history and baryon
ledger and nothing here disturbs it; the lithium mechanism is a
\emph{predicted but conditional} addition whose microscopic closure is
deferred to the forthcoming companion paper GG-C4-Nucleo. The reader
should finish able to compute the helium fraction from scratch and to
state precisely what about lithium is, and is not, resolved.

\subsection{The neutron-to-proton ratio and weak freeze-out}
\label{appA:bbn-np-freezeout}

Everything in BBN is downstream of one number: the ratio of free neutrons
to free protons at the moment the light nuclei begin to assemble. We
therefore start there.

In the first second after the bang the universe is a hot plasma in which
neutrons and protons interconvert freely through the charged-current weak
reactions
\begin{align}
n + e^{+} &\longleftrightarrow p + \bar{\nu}_e, \label{appA:bbn-weak1}\\
n + \nu_e &\longleftrightarrow p + e^{-}, \label{appA:bbn-weak2}\\
n &\longleftrightarrow p + e^{-} + \bar{\nu}_e. \label{appA:bbn-weak3}
\end{align}
As long as these reactions proceed much faster than the universe expands,
the two species sit in mutual chemical equilibrium. The proton is lighter
than the neutron by
\begin{equation}
Q \equiv (m_n - m_p)c^2 = 1.293~\mathrm{MeV},
\label{appA:bbn-Qdef}
\end{equation}
so equilibrium statistics (a Maxwell--Boltzmann / detailed-balance
ratio of the two non-relativistic species at common temperature $T$,
with their internal degeneracies equal) give the textbook Boltzmann
suppression
\begin{equation}
\left(\frac{n}{p}\right)_{\mathrm{eq}}(T)
= \exp\!\left(-\frac{Q}{k_B T}\right).
\label{appA:bbn-np-eq}
\end{equation}
We use natural units in which $k_B=1$ and quote temperatures in MeV
hereafter, so $Q/T$ is just a ratio of two numbers in MeV. At very high
temperature $T \gg Q$ the exponent is negligible and $n/p \to 1$:
neutrons and protons are equally abundant. As the universe cools, the
exponential turns on and neutrons become progressively rarer than
protons because it costs energy $Q$ to make one.

Equilibrium~\eqref{appA:bbn-np-eq} holds only while the weak rates win
against the Hubble expansion. The relevant comparison is between the
per-neutron conversion rate $\Gamma_{\mathrm{weak}}(T)$ and the
expansion rate $H(T)$. The weak rate falls steeply with temperature,
$\Gamma_{\mathrm{weak}} \sim G_F^2 T^5$, because the cross sections
scale as $G_F^2 E^2$ and the available energy scales with $T$. The
expansion rate in the radiation era scales as
\begin{equation}
H(T) = \sqrt{\frac{8\pi G}{3}\,\rho}
\;\propto\; \sqrt{g_\ast}\;\frac{T^2}{M_{\mathrm{Pl}}},
\label{appA:bbn-Hubble}
\end{equation}
where $g_\ast$ counts the relativistic degrees of freedom and
$M_{\mathrm{Pl}}$ is the Planck mass. Because $\Gamma_{\mathrm{weak}}
\propto T^5$ falls faster than $H \propto T^2$, there is a crossover
temperature $T_f$ defined by
\begin{equation}
\Gamma_{\mathrm{weak}}(T_f) \simeq H(T_f),
\label{appA:bbn-freezeout-cond}
\end{equation}
below which the weak interactions can no longer keep pace with the
expansion. Inserting the measured Fermi constant and the radiation-era
$H(T)$ puts this crossover at
\begin{equation}
T_f \approx 0.8~\mathrm{MeV}.
\label{appA:bbn-Tf}
\end{equation}
This is \emph{weak freeze-out}: below $T_f$ the reactions
\eqref{appA:bbn-weak1}--\eqref{appA:bbn-weak2} effectively switch off and
the equilibrium relation \eqref{appA:bbn-np-eq} stops being enforced. The
$n/p$ ratio is no longer free to track its equilibrium value; it
``freezes'' near whatever value equilibrium had reached at $T_f$:
\begin{equation}
\left(\frac{n}{p}\right)_{\!f}
= \exp\!\left(-\frac{Q}{T_f}\right)
= \exp\!\left(-\frac{1.293}{0.8}\right)
\approx 0.20 \;\approx\; \frac{1}{5}.
\label{appA:bbn-np-freeze}
\end{equation}
This frozen ratio of about one neutron for every five protons is the
single most important number in the whole calculation.

The word ``freeze'' is not quite literal, and the difference matters for
the final helium yield. The free neutron is unstable: through
reaction~\eqref{appA:bbn-weak3} it beta-decays with a mean lifetime
\begin{equation}
\tau_n = 879~\mathrm{s}.
\label{appA:bbn-taun}
\end{equation}
After freeze-out, then, the proton number is essentially fixed but the
neutron number continues to leak away by free decay during the long wait
before nucleosynthesis can begin. The ratio at the actual onset of
nucleosynthesis, at a later time $t_{\mathrm{nuc}}$, is therefore the
frozen value damped by the surviving fraction of neutrons,
\begin{equation}
\frac{n}{p}\Big|_{\mathrm{nuc}}
= \left(\frac{n}{p}\right)_{\!f}\,
  \exp\!\left(-\frac{t_{\mathrm{nuc}}}{\tau_n}\right).
\label{appA:bbn-np-decay}
\end{equation}
We now have to find out how long that wait is, and the answer is the
content of the next subsection.

\subsection{The deuterium bottleneck}
\label{appA:bbn-deuterium-bottleneck}

A naive estimate would have nucleosynthesis start the moment it is
energetically allowed to. The first step on every path to heavier nuclei
is the formation of deuterium,
\begin{equation}
n + p \longrightarrow d + \gamma,
\qquad B_d = 2.22~\mathrm{MeV},
\label{appA:bbn-dform}
\end{equation}
where $B_d$ is the deuteron binding energy. One might guess that
deuterium accumulates as soon as $T$ drops below $B_d \approx
2.2~\mathrm{MeV}$. It does not, and the reason is the central subtlety
of BBN.

The obstruction is that the universe is enormously \emph{photon-rich}.
The number of photons exceeds the number of baryons by the inverse of the
baryon-to-photon ratio, $\eta \sim 6 \times 10^{-10}$, i.e.\ there are
roughly $1.6 \times 10^9$ photons per baryon. Even when the
\emph{average} photon energy has fallen well below $B_d$, the
high-energy tail of the blackbody distribution still contains, per
deuteron, an overwhelming number of photons energetic enough to
photodissociate it back to $n+p$. Deuterium is destroyed as fast as it
forms. Quantitatively, the photodissociation reaction
$d + \gamma \to n + p$ only becomes ineffective once the number of
tail photons above $B_d$ drops below the number of baryons, which
requires $T$ to fall not to $B_d$ but to roughly
\begin{equation}
T_{\mathrm{nuc}} \approx \frac{B_d}{\ln(1/\eta)}
\approx \frac{2.22~\mathrm{MeV}}{\ln(1.6\times 10^{9})}
= \frac{2.22~\mathrm{MeV}}{21.2}\approx 0.10~\mathrm{MeV}.
\label{appA:bbn-Tnuc}
\end{equation}
This bare logarithmic estimate gives the right order of magnitude ($\sim0.1$~MeV); the full deuterium-bottleneck condition --- which also follows the photodissociation cross-section and the falling baryon density --- brings the operative onset down to the standard value $T_{\rm nuc}\approx0.07$~MeV $\approx0.8\times10^{9}$~K. The factor $\ln(1/\eta) \approx 21$ is the entire story: it is the
logarithm of the photon-to-baryon excess, and it is what pushes the start
of nucleosynthesis from $T \approx 2~\mathrm{MeV}$ down by a factor of
about thirty to $T \approx 0.07~\mathrm{MeV}$. This delay is the
\emph{deuterium bottleneck}. Because deuterium is the gateway nucleus on
every reaction path, nothing downstream of it --- no ${}^{3}\mathrm{He}$,
no ${}^{4}\mathrm{He}$, no ${}^{7}\mathrm{Li}$ --- can form in quantity
until the deuterium ``logjam'' breaks at $T_{\mathrm{nuc}}$.

In cosmic time, freeze-out at $T_f\approx0.8~\mathrm{MeV}$ occurs at
about $t\approx 1~\mathrm{s}$, whereas the bottleneck breaks at
$T_{\mathrm{nuc}}\approx0.07~\mathrm{MeV}$ at about $t_{\mathrm{nuc}}
\approx 250$--$300~\mathrm{s}$. It is precisely during this several-minute
interval that the neutron-decay damping of
Eq.~\eqref{appA:bbn-np-decay} acts, and it is why the helium yield ends up
sensitive to the neutron lifetime $\tau_n$. The bottleneck is therefore
not a nuisance but a clock: it sets the wait during which neutrons leak
away before they can be locked into nuclei.

\subsection{The reaction network: D, He-3, He-4, and Li-7/Be-7}
\label{appA:bbn-network}

Once $T$ falls through $T_{\mathrm{nuc}}$ and the deuterium bottleneck
breaks, the available neutrons and protons burn rapidly up the light
nuclear network. The reactions proceed essentially to completion within a
few minutes because the densities are still high and the Coulomb barriers
of the lightest nuclei are low. The principal chain is
\begin{align}
n + p &\to d + \gamma, \label{appA:bbn-net1}\\
d + d &\to {}^{3}\mathrm{He} + n,
\qquad d + d \to t + p, \label{appA:bbn-net2}\\
d + {}^{3}\mathrm{He} &\to {}^{4}\mathrm{He} + p,
\qquad d + t \to {}^{4}\mathrm{He} + n. \label{appA:bbn-net3}
\end{align}
The flow piles up at ${}^{4}\mathrm{He}$ because the helium-4 nucleus is
exceptionally tightly bound ($B/A \approx 7.07~\mathrm{MeV}$ per
nucleon) and there is no stable nucleus at mass number $A=5$ or $A=8$.
The two missing mass numbers act as a wall: the network cannot easily
bridge past helium-4, so the overwhelming majority of all neutrons
present at $T_{\mathrm{nuc}}$ end up bound inside ${}^{4}\mathrm{He}$,
and only trace amounts of anything heavier are made.

\paragraph{The helium mass fraction.}
Because ${}^{4}\mathrm{He}$ is the sink, its abundance can be estimated by
a simple bookkeeping argument that needs no detailed rate integration.
Each helium-4 nucleus contains two neutrons and two protons. To good
approximation \emph{every} surviving neutron is captured into helium, and
it brings one proton with it. If there are $n$ neutrons and $p$
protons at the onset of nucleosynthesis, the number of helium nuclei
formed is $n/2$ (two neutrons each), consuming $n$ neutrons and $n$
protons; the remaining $p-n$ protons stay as free hydrogen. The helium
\emph{mass} fraction, with helium of mass $\approx 4$ and the total
baryonic mass $\approx n+p$, is then
\begin{equation}
Y_p \equiv \frac{4\,(n/2)}{n+p}
= \frac{2n}{n+p}
= \frac{2\,(n/p)}{1 + (n/p)}.
\label{appA:bbn-Yp}
\end{equation}
This compact formula is the headline BBN prediction: the primordial
helium mass fraction is fixed almost entirely by the
neutron-to-proton ratio at nucleosynthesis, which we computed in
Sec.~\ref{appA:bbn-np-freezeout}. We evaluate it explicitly in
Sec.~\ref{appA:bbn-worked}.

\paragraph{The mass-7 corner.}
Beyond helium, the network makes trace amounts of ${}^{3}\mathrm{He}$ and
deuterium that survive incomplete burning, and a much smaller trickle
into mass number $A=7$. The $A=7$ abundance is produced by two
competing routes whose balance depends on the baryon density:
\begin{align}
t + {}^{4}\mathrm{He} &\to {}^{7}\mathrm{Li} + \gamma
&&\text{(direct }{}^{7}\mathrm{Li}\text{, dominant at low }\eta\text{),}
\label{appA:bbn-li7}\\
{}^{3}\mathrm{He} + {}^{4}\mathrm{He} &\to {}^{7}\mathrm{Be} + \gamma,
\quad {}^{7}\mathrm{Be} + e^{-} \to {}^{7}\mathrm{Li} + \nu_e
&&\text{(via }{}^{7}\mathrm{Be}\text{, dominant at high }\eta\text{).}
\label{appA:bbn-be7}
\end{align}
The key point for what follows is that at the observationally favored,
relatively high value of $\eta$, the mass-7 abundance is dominated not
by direct lithium production but by beryllium-7, which is made first and
only later electron-captures to lithium-7 (much of it after BBN proper,
in the recombination era). The observed primordial ``lithium'' is
therefore largely \emph{decayed beryllium}. This is why the standard
shorthand for the problem is the ${}^{7}\mathrm{Li}/{}^{7}\mathrm{Be}$
abundance: any mechanism that wishes to reduce primordial lithium without
wrecking the other elements must act on mass number 7, and in practice
must act on the ${}^{7}\mathrm{Be}$ that dominates it. We return to this
in Sec.~\ref{appA:bbn-gg}.

\subsection{Dependence on the baryon-to-photon ratio}
\label{appA:bbn-eta}

BBN has, to leading order, exactly one free input: the baryon-to-photon
ratio
\begin{equation}
\eta \equiv \frac{n_B}{n_\gamma}
\;\simeq\; 6.1 \times 10^{-10},
\label{appA:bbn-eta-def}
\end{equation}
equivalently the baryon density $\Omb h^2 \approx 0.0224$ (the two are
related by $\eta \approx 2.74\times 10^{-8}\,\Omb h^2$). Every primordial
abundance is a definite function of this one number, which is what makes
BBN a sharp, falsifiable audit rather than a fitting exercise. The
qualitative dependences are physically transparent:
\begin{itemize}
\item \textbf{Deuterium} $\mathrm{D/H}$ is a steeply \emph{decreasing}
  function of $\eta$: more baryons means denser plasma, faster burning,
  and more complete conversion of the fragile deuterium into helium, so
  less deuterium survives. This steep slope makes $\mathrm{D/H}$ the
  best ``baryometer.''
\item \textbf{Helium} $Y_p$ depends only weakly (logarithmically) on
  $\eta$, because it is set by the $n/p$ ratio
  (Eq.~\ref{appA:bbn-Yp}) and $\eta$ enters only through the modest shift
  in $T_{\mathrm{nuc}}$ and hence in the neutron-decay damping.
\item \textbf{Lithium-7} ${}^{7}\mathrm{Li}/\mathrm{H}$ is non-monotonic
  in $\eta$: it has a minimum at intermediate $\eta$ and rises on both
  sides because of the two competing production routes
  \eqref{appA:bbn-li7}--\eqref{appA:bbn-be7}. At the high $\eta$ preferred
  by the CMB, the ${}^{7}\mathrm{Be}$ route dominates and lithium
  \emph{increases} with $\eta$.
\end{itemize}
The decisive modern fact is that $\eta$ is no longer a BBN-internal fit
parameter. The cosmic microwave background fixes $\Omb h^2$ (and hence
$\eta$) independently and to sub-percent precision from the relative
heights of the acoustic peaks. BBN must therefore work at the
CMB-determined $\eta$, with no remaining freedom. In the inherited
ledger of this cohort the baryon fraction is carried as the
$\pi$-locked value $\Omb = 1/(2\pi^2)$ (see the topological-inventory
Part of the main text), so $\eta$ is fixed upstream and BBN is run as a
closed audit against it, not tuned.

\subsection{Observational status: triumph and the lithium-7 problem}
\label{appA:bbn-observation}

At the CMB-fixed $\eta$, two of the three measurable primordial
abundances are textbook successes of cosmology:
\begin{itemize}
\item \textbf{Deuterium.} The predicted $\mathrm{D/H}$ and the value
  measured in pristine, metal-poor, high-redshift absorption systems
  along quasar sightlines agree at the level of a few percent,
  $\mathrm{D/H} \approx 2.5\times 10^{-5}$. Because $\mathrm{D/H}$ is
  the steep baryometer, this agreement is a stringent and independent
  confirmation of the CMB baryon density.
\item \textbf{Helium-4.} The predicted $Y_p \approx 0.247$ and the value
  extrapolated from helium and hydrogen emission lines in metal-poor
  extragalactic H\,\textsc{ii} regions, $Y_p \approx 0.245 \pm 0.003$,
  agree well within the uncertainties.
\end{itemize}
These two agreements, spanning nine orders of magnitude in abundance and
resting on independent physics (weak freeze-out for helium, the burning
network for deuterium), are among the strongest quantitative evidence we
have that the universe was once hot, dense, and described by known
microphysics back to $t\approx1~\mathrm{s}$.

The third measurable abundance does \emph{not} agree, and this is the
\emph{lithium-7 problem}. Standard BBN at the CMB-fixed $\eta$ predicts a
primordial lithium abundance of about
\begin{equation}
\left(\frac{{}^{7}\mathrm{Li}}{\mathrm{H}}\right)_{\!\mathrm{BBN}}
\approx (4.5\text{--}5.0)\times 10^{-10},
\label{appA:bbn-li-pred}
\end{equation}
whereas the abundance inferred from the surfaces of old, metal-poor
halo stars --- where it sits on the remarkably flat ``Spite plateau,''
nearly independent of metallicity and temperature --- is
\begin{equation}
\left(\frac{{}^{7}\mathrm{Li}}{\mathrm{H}}\right)_{\!\mathrm{Spite}}
\approx (1.5\text{--}1.6)\times 10^{-10}.
\label{appA:bbn-li-obs}
\end{equation}
The prediction overshoots the observation by a factor of roughly three
(about $4$--$5\sigma$ given the quoted errors). This is a genuine,
long-standing discrepancy. Three broad classes of explanation are
debated and none is established: (i) \emph{stellar depletion}, in which
lithium is destroyed over the stars' lifetimes by mixing below the
convection zone, so the plateau understates the true primordial value
--- the flatness of the plateau is, however, hard to reconcile with the
spread that depletion models tend to predict; (ii) \emph{nuclear-rate
revisions}, in which a mismeasured ${}^{7}\mathrm{Be}$-producing or
${}^{7}\mathrm{Be}$-destroying cross section shifts the prediction
--- but the relevant reactions have been re-measured and the rates are
now fairly well pinned; and (iii) \emph{new physics} during or just after
BBN that selectively destroys mass-7 nuclei. The GG reading proposed
below sits in the third class, and the honesty discipline of this
appendix is to label it as exactly that: a proposal, not a resolution.

\subsection{Worked example: n/p at freeze-out and the helium fraction}
\label{appA:bbn-worked}

We now carry the helium calculation through explicitly, because it is the
cleanest demonstration that BBN is genuinely \emph{Derived} from the
inherited expansion history with no tunable knob.

\emph{Step 1 --- equilibrium ratio at freeze-out.} Using
Eq.~\eqref{appA:bbn-np-eq} with $Q = 1.293~\mathrm{MeV}$ and the
freeze-out temperature $T_f = 0.8~\mathrm{MeV}$ from
Eq.~\eqref{appA:bbn-Tf},
\begin{equation}
\left(\frac{n}{p}\right)_{\!f}
= \exp\!\left(-\frac{1.293}{0.8}\right)
= \exp(-1.616)
= 0.199 \approx \frac{1}{5}.
\label{appA:bbn-work-np}
\end{equation}

\emph{Step 2 --- neutron decay during the bottleneck.} Nucleosynthesis
does not begin until the deuterium bottleneck breaks at
$t_{\mathrm{nuc}} \approx 300~\mathrm{s}$
(Sec.~\ref{appA:bbn-deuterium-bottleneck}). During the interval from
freeze-out to $t_{\mathrm{nuc}}$ the free neutrons decay with lifetime
$\tau_n = 879~\mathrm{s}$. Applying the damping factor of
Eq.~\eqref{appA:bbn-np-decay},
\begin{equation}
\frac{n}{p}\Big|_{\mathrm{nuc}}
= 0.199 \times \exp\!\left(-\frac{300}{879}\right)
= 0.199 \times \exp(-0.341)
= 0.199 \times 0.711
= 0.142 \approx \frac{1}{7}.
\label{appA:bbn-work-np-nuc}
\end{equation}
The decay has driven the ratio from about $1/5$ down to about $1/7$:
roughly one neutron survives for every seven protons when the network
finally fires. This factor of $1/7$ is the famous BBN benchmark.

\emph{Step 3 --- helium mass fraction.} Inserting
$n/p = 1/7$ into the bookkeeping formula
Eq.~\eqref{appA:bbn-Yp},
\begin{equation}
Y_p
= \frac{2\,(n/p)}{1 + (n/p)}
= \frac{2\times \tfrac{1}{7}}{1 + \tfrac{1}{7}}
= \frac{2/7}{8/7}
= \frac{2}{8}
= 0.25.
\label{appA:bbn-work-Yp}
\end{equation}
\begin{equation}
\boxed{\;Y_p \approx 0.25\;}
\label{appA:bbn-Yp-result}
\end{equation}
A primordial helium mass fraction of one-quarter, derived from nothing
but $Q = 1.293~\mathrm{MeV}$, the weak-freeze-out temperature, the
neutron lifetime, and the bottleneck delay, and agreeing with the
measured $Y_p \approx 0.245$. Two remarks underline how unforgiving
this is. First, had the neutrons \emph{not} decayed
($n/p = 0.20$ throughout), Eq.~\eqref{appA:bbn-Yp} would give
$Y_p = 2(0.20)/1.20 = 0.33$, a third of the baryonic mass in helium ---
the $\sim0.25$ value is what it is only because the bottleneck delay
lets neutrons leak away first. Second, no parameter here is free to
adjust: $Q$, $\tau_n$, and $G_F$ are laboratory constants, and
$\eta$ is fixed by the CMB. The agreement is therefore a prediction,
not a fit. After all, a number this rigid is exactly the kind of audit a
geometric origin theory must pass before it is allowed to say anything
new.

\subsection{The geometric derivation: the lithium-7 deficit as a closed-packet $1/\pi$ port}
\label{appA:bbn-gg}

With the baseline fully in view --- two triumphs and one genuine failure
--- we can state what the geometry adds. The two successful abundances,
$\mathrm{D/H}$ and $Y_p$, are left \emph{untouched}: the inherited
expansion history and the $\pi$-locked baryon ledger $\Omb=1/(2\pi^2)$
reproduce them exactly as standard BBN does, and the program claims no
modification there. The lithium-7 problem is the one place the standard
prediction fails, and it is the one place the geometry speaks --- with the
same closed-packet factor $1/\pi$ that fixes the cosmic matter inventory.

\paragraph{Why the boundary universe has an open channel.}
The observable four-dimensional universe is never a closed microstate. By
\OGGEP{} it is an open boundary subsystem, accessed through a
non-invertible bulk-to-boundary projection, and the generic signature of
that projection is a dissipative \GDOS{} channel that can drain a
registered species into the inaccessible bulk ledger. A selective $A=7$
port is exactly such a channel acting on mass number 7 --- in practice on
the ${}^{7}\mathrm{Be}$ that dominates the $A=7$ abundance at the
CMB-favored $\eta$ (\S\ref{appA:bbn-network}). \emph{Physics intuition:}
an open boundary always leaks a little of what it carries; the only
question is how much, and onto what.

\paragraph{The surviving fraction is $1/\pi$, not a fitted knob.}
The strength of the port is not a free coefficient. Modeling the mass-7
flux as a System Instance Packet carried over the compact angle
$\theta\in S^1_\theta$, the open-system drift retains, on projection to
the observable boundary, exactly the closed-packet share $1/\pi$ --- the
\emph{same} factor that yields $\Omm=1/\pi$. The full derivation is the
companion port calculation of \S\ref{appA:leptogenesis},
Eqs.~\eqref{eq:app-port-drift}--\eqref{eq:app-survival-pi}; the surviving
fraction $S_{A=7}=1/\pi$ follows because the closed-packet geometry
branches the conserved flux over the compact angular measure in precisely
the way it branches the cosmic inventory. The predicted primordial
lithium is therefore
\begin{equation}
\left(\frac{{}^{7}\mathrm{Li}}{\mathrm H}\right)_{\!\rm GG}
=\frac{1}{\pi}\left(\frac{{}^{7}\mathrm{Li}}{\mathrm H}\right)_{\!\rm SBBN}
\simeq\frac{5.0\times10^{-10}}{\pi}
\simeq1.6\times10^{-10},
\label{appA:bbn-li-gg}
\end{equation}
to be compared with the Spite-plateau value
$(1.5\text{--}1.6)\times10^{-10}$ of \eqref{appA:bbn-li-obs}: the
factor-of-three overshoot is removed, and $\mathrm{D/H}$ and $Y_p$ are
untouched because the port is selective in $A$. \emph{Physics intuition:}
the same projection that keeps only $1/\pi$ of the closed packet visible
as matter keeps only $1/\pi$ of the mass-7 flux as lithium --- the lithium
deficit and the matter inventory are two readings of one geometric
branching.

\paragraph{What is, and is not, established.}
This is a prediction, not a fit: the port strength is the closed-packet
$1/\pi$, with no free coefficient. But it inherits exactly the status of
the inventory that supplies it --- \emph{Derived-conditional} ($D^*$),
resting on the closed-packet-integration conjecture (Conjecture~VIII) ---
and three obligations remain open, stated plainly:
\begin{enumerate}
\item \textbf{The selectivity mechanism.} Why the port acts on $A=7$ and
not on $A=2,3,4$ --- how the selectivity in mass number arises from the
bulk projection --- is not delivered here.
\item \textbf{Network-level closure.} A full coupled BBN-network
calculation, re-running the reaction network of \S\ref{appA:bbn-network}
with the port inserted and confirming the other abundances genuinely do
not move, is owed.
\item \textbf{Detailed balance.} Compatibility with detailed balance and
the well-measured $A=7$ cross sections
(${}^{3}\mathrm{He}+{}^{4}\mathrm{He}\to{}^{7}\mathrm{Be}$ production and
${}^{7}\mathrm{Be}$ destruction) must be demonstrated.
\end{enumerate}
These are the remit of the forthcoming companion paper GG-C4-Nucleo, not
yet a member of the cohort. The honest one-line summary: the lithium-7
problem is addressed by a \emph{parameter-free} closed-packet port that
lands on the Spite plateau, conditional on Conjecture~VIII and on the
network-level obligations above --- a predicted, partially closed
mechanism, falsifiable by any firm primordial ${}^{7}\mathrm{Li}$ away
from $\mathrm{SBBN}/\pi$, and not yet a finished resolution. In the
Euclidean spirit of the program: the part the geometry fixes (the
$1/\pi$ survival) is stated as a result, and the parts it does not yet
fix are marked, in plain sight, as the work that remains.

\subsection*{Status}
\label{appA:bbn-status}

\begin{itemize}
\item \textbf{Standard BBN --- Derived.} The neutron-to-proton freeze-out
  ratio, the deuterium bottleneck, the
  D/${}^{3}\mathrm{He}$/${}^{4}\mathrm{He}$ network, the helium mass
  fraction $Y_p\approx0.25$ (\S\ref{appA:bbn-worked}), and the
  $\eta$-dependence of the abundances all follow from the inherited
  radiation-era expansion history and the CMB-fixed, $\pi$-locked baryon
  ledger $\Omb=1/(2\pi^2)$, with no free parameter. The agreement with the
  measured $\mathrm{D/H}$ and $Y_p$ is reproduced, not modified.
\item \textbf{The $A=7$ lithium port --- Derived-conditional ($D^*$).} The
  selective $A=7$ \GDOS{} open-system port reduces the predicted
  ${}^{7}\mathrm{Li}/{}^{7}\mathrm{Be}$ abundance to the Spite-plateau
  value by the closed-packet factor $1/\pi$ --- the same factor that fixes
  $\Omm=1/\pi$ --- with \emph{no free coefficient} (\S\ref{appA:leptogenesis}).
  Its status is therefore that of the inventory: $D^*$, conditional on the
  closed-packet-integration conjecture (Conjecture~VIII). The microscopic
  origin of the mass-7 selectivity, the full coupled BBN-network closure,
  and compatibility with detailed balance and the measured $A=7$ cross
  sections are open obligations deferred to the forthcoming companion
  paper GG-C4-Nucleo. The lithium-7 problem is reported here as a
  predicted, partially closed mechanism, not as solved.
\end{itemize}

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{H.2.1. Estimate the temperature at
which the deuterium bottleneck breaks.}

\noindent\emph{Solution.}  Step 1: deuterium (binding energy
$B_D = 2.22$~MeV) is photodissociated as long as photons above
$B_D$ outnumber baryons; with $n_\gamma/n_B = 1/\eta$, the
Boltzmann tail condition is $e^{-B_D/T}/\eta \gtrsim 1$.
Step 2: solve: $T_{\rm D} \approx B_D/\ln(1/\eta) =
2.22/21.2 \approx 0.1$~MeV --- twenty times below the binding
energy.  Step 3: only below $T_{\rm D}$ can the reaction
network of the chapter fire, which is why nucleosynthesis
happens at $t \sim$ minutes rather than seconds, giving free
neutrons time to decay (lowering $n/p$ toward $1/7$).
Physics intuition: the bottleneck is the photon bath's veto
power --- a billion photons per baryon means even the
one-in-a-billion tail can dissociate every nucleus; the
$\ln(1/\eta)$ delay is F.8.1's entropy-per-baryon speaking in
nuclear language.

\medskip\noindent\textbf{H.4.1. Explain, with the network's
logic, why deuterium falls steeply with $\eta$ while helium
barely moves.}

\noindent\emph{Solution.}  Step 1: helium-4 is a
\emph{thermodynamic} endpoint --- essentially every neutron
alive at the bottleneck ends inside $^4$He, so $Y_p$ depends
on $\eta$ only through the mild shift of the bottleneck
temperature (logarithmic, H.2.1) --- weak dependence.
Step 2: deuterium is a \emph{kinetic} survivor --- the trace
that failed to burn; more baryons mean faster burning and a
smaller survivor fraction, a steep power law
($\mathrm{D/H} \propto \eta^{-1.6}$).  Step 3: hence the
division of labor: helium tests the expansion rate (H.6.1),
deuterium measures $\eta$ --- the ``baryometer'' --- and the
measured D/H fixes $\eta \approx 6\times10^{-10}$,
independently confirmed by the CMB.  Physics intuition:
endpoints make poor meters and survivors make sharp ones ---
the abundance of what \emph{almost} happened always carries
more information than the abundance of what was inevitable.

\medskip\noindent\textbf{H.6.1. How would a fourth neutrino
species shift the helium abundance?}

\noindent\emph{Solution.}  Step 1: an extra relativistic
species raises $g_*$ at MeV temperatures, hence raises
$H \propto \sqrt{g_*}T^2$.  Step 2: faster expansion means
weak freeze-out (the chapter's worked example) happens
\emph{earlier}, at higher $T_f$: the $n/p$ ratio freezes
larger, and more neutrons survive into helium.  Step 3: the
standard sensitivity is $\Delta Y_p \approx 0.013\,\Delta
N_\nu$ --- a fourth neutrino would push $Y_p$ from $\approx
0.247$ to $\approx 0.260$, outside the observed range: BBN
counted the neutrino families ($N_\nu \approx 3$) two decades
before the colliders did.  Physics intuition: the early
universe is a speedometer --- anything that adds energy
density accelerates the expansion, and the helium abundance is
the tachograph that recorded it; this is also why BBN polices
every new light particle the cohort or anyone else proposes.

\medskip\noindent\textbf{H.5.1. State the lithium-7 problem as
one number, and notice the number.}

\noindent\emph{Solution.}  Step 1: standard BBN with the
CMB-fixed $\eta$ predicts $^7$Li/H $\approx 5.0\times
10^{-10}$.  Step 2: the Spite plateau of metal-poor halo
stars observes $\approx 1.6\times10^{-10}$.  Step 3: the
discrepancy factor is $5.0/1.6 = 3.13$ --- within one percent
of $\pi$.  Standard resolutions (stellar depletion, nuclear
rates) each strain against independent data; the geometric
reading of the next chapter's port section derives the factor
as exactly $\pi$, from the same closed-packet branching that
fixes $\Omm = 1/\pi$.  Physics intuition: a thirty-year
anomaly whose magnitude is a mathematical constant is either
a coincidence or a signature --- the falsification catalog
treats it as the latter and prices the kill: tighten the
plateau and the nuclear inputs, and the factor either
sharpens toward $\pi$ or walks away from it.

\GGtakeaway{\emph{Standard result:} BBN predicts the light-element
abundances from one parameter (\(\eta_B\)) and nuclear physics ---
cosmology's earliest quantitative triumph --- but overpredicts
lithium-7 by a factor \(\sim3\). \emph{What it assumed:} a closed
reaction network and \(\eta_B\) measured elsewhere. \emph{What the
geometry replaced it with:} \(\eta_B\) delivered by leptogenesis on the
inherited ledger (Appendix~I) and the \(A=7\) sector read as an
open-system port with a \(1/\pi\) suppression --- falsifiable, because
any lithium fix that damages D/H or \(Y_p\) is excluded.
\emph{Continue in:} Part~V; the decade table of Part~XIII.}

\section{Resonant leptogenesis and the lithium-7 port}
\label{appA:leptogenesis}

\GGteachbox{Where the baryon asymmetry \(\eta_B \sim 6\times10^{-10}\)
comes from: the type-I seesaw matched to the inherited \ANine{} neutrino
block, the resonant CP asymmetry on the inherited phases, washout, and
the assembled \(\eta_B\) --- followed by the geometric
sharpening: the
derived reheating temperature \(T_{\mathrm{rh}} \simeq 3\times10^{10}\)
GeV opens exactly the thermal-leptogenesis window, so the mechanism is
selected by the launch, not chosen.}
\GGreadalong{\citet{KolbTurner1990} ch.~6 for baryogenesis generally;
\citet{FukugitaYanagida1986} for the original mechanism.}

This appendix is the geometric derivation behind the origin of matter. It
supplies the one number standard Big-Bang nucleosynthesis (Appendix~H)
had to take as a measured input --- the baryon-to-photon ratio $\eta_B$
--- and the closed-packet $1/\pi$ port that resolves the lithium-7 deficit.
Where Appendix~H asked what the data require, this appendix asks what the
inherited geometry forces, in the same Euclidean spirit as Appendix~E: the
heavy Majorana scale, the neutrino Yukawas, and the CP phases are not
free, but fixed by \ANine{} and the lock-scale geometry. It carries the
heavy algebra the Part~V matter sector points to: the type-I seesaw
matching that fixes the neutrino Yukawas, the resonant CP-asymmetry formula
evaluated on the inherited phases, the Boltzmann/washout estimate, the
assembly of $\eta_B$, and the open-system $A=7$ port whose surviving
fraction is the closed-packet factor $1/\pi$. Each block follows the same
order: statement, derivation, worked number, geometric origin, honest
status.

\subsection{Type-I seesaw matching to the \texorpdfstring{\ANine{}}{A9} Weinberg operator}

\paragraph{Statement.}
\ANine{} fixes the light neutrino masses through the dimension-five Weinberg operator at the GUT
seam,
\begin{equation}
\mathcal L_5 = \frac{(L H)(L H)}{\Lambda_{\rm W}},
\qquad
m_\star = \frac{\pi v^2}{\EGUT},
\qquad v\simeq246\,\mathrm{GeV},
\label{eq:app-weinberg}
\end{equation}
and explicitly leaves the ultraviolet completion to \ATen{}. The completion is a heavy Majorana
triplet \(N_i\) at the lock scale; integrating it out must reproduce Eq.~\eqref{eq:app-weinberg}.

\paragraph{Derivation.}
With Dirac Yukawas \(Y_\nu\) coupling \(L H N\) and Majorana masses \(M_N\), the seesaw light mass
matrix is
\begin{equation}
m_\nu = - Y_\nu^{\sf T} M_N^{-1} Y_\nu\,v^2,
\label{eq:app-seesaw-matrix}
\end{equation}
whose magnitude for a representative entry is \(m_\nu \sim Y_\nu^2 v^2 / M_N\). Matching this to
\(m_\star=\pi v^2/\EGUT\) cancels \(v^2\):
\begin{equation}
\frac{Y_\nu^2}{M_N} \simeq \frac{\pi}{\EGUT}
\quad\Longrightarrow\quad
Y_\nu^2 \simeq \pi\,\frac{M_N}{\EGUT}.
\label{eq:app-Ynu2}
\end{equation}

\paragraph{Worked number.}
With \(M_N\simeq\Mlock\simeq3.85\times10^{3}\,\mathrm{GeV}\) and \(\EGUT\simeq2.28\times10^{16}\,
\mathrm{GeV}\),
\begin{equation}
Y_\nu^2 \simeq \pi\,\frac{3.85\times10^{3}}{2.28\times10^{16}}
\simeq 5.3\times10^{-13},
\qquad
Y_\nu \simeq 7.3\times10^{-7}.
\label{eq:app-Ynu-number}
\end{equation}
The tree-level decay width is
\begin{equation}
\Gamma_N = \frac{Y_\nu^2}{8\pi}M_N
\simeq \frac{5.3\times10^{-13}}{8\pi}\,(3.85\times10^{3}\,\mathrm{GeV})
\simeq 8.1\times10^{-11}\,\mathrm{GeV},
\qquad
\frac{\Gamma_N}{M_N}\simeq 2.1\times10^{-14}.
\label{eq:app-GammaN}
\end{equation}
One bookkeeping convention is declared here: this width is quoted at the seam-matched
one-generation Yukawa, which corresponds to \(\widetilde m \simeq 8.4\)~meV, while the washout
below runs at \(\widetilde m \sim m_3 \simeq 50.5\)~meV --- conventions differing by a factor \(\simeq12\) in \(Y_\nu^2\). Run at the washout convention the width is
\(\Gamma_N \simeq 1\times10^{-9}\)~GeV: an \(O(10)\) shift inside the order-of-magnitude
register, absorbed into the effective width \(\Gamma^{\rm eff}\) that the hypothesis
H-\(\phi_{\rm geo}\)-1 below parameterizes. The two conventions are deliberate, and each is evaluated where its
physics lives: the resonance width is a property of the seam-localized Yukawa texture, so it is
quoted at the seam-matched one-generation magnitude \(Y_\nu^2 \simeq \pi M_N/\EGUT\) of
Eq.~\eqref{eq:app-Ynu2}, whose effective mass is the Majorana seam scale itself,
\(\widetilde m = Y_\nu^2 v^2/M_N = \pi v^2/\EGUT \simeq 8.4\)~meV; the washout strength is a
property of the light-neutrino mass scale the thermal bath actually sees, which the inherited
spectrum fixes at the heaviest light mass, \(\widetilde m \sim m_3 \simeq 50.5\)~meV. The factor
between the conventions is thereby accounted for, not mysterious. The effective masses differ by \(m_3/(\pi v^2/\EGUT) \simeq 50.5/8.4 \simeq 6.0\); the quoted Yukawa-squared ratio carries, in addition, only the Casas--Ibarra half in \(m_\nu = Y^2v^2/2M_N\) (the normalization of the resonant subsection below, Eq.~\eqref{eq:app-sinphi}) relative to the seam magnitude matching \(m_\nu \sim Y_\nu^2 v^2/M_N\) of Eq.~\eqref{eq:app-Ynu2}, so that \((Y_3/Y_\nu)^2 = 2\,m_3/\widetilde m_{\rm seam} \simeq 12\) (numerically, \(Y_3 \simeq 2.5\times10^{-6}\) against \(Y_\nu \simeq 7.3\times10^{-7}\)). The physical spread between the conventions is the single mass ratio \(\simeq6.0\); the factor of two is normalization bookkeeping, not an additional physical spread; both readings sit inside the \(O(1)\)--\(O(10)\) pricing already declared, each downstream quantity
is run on the convention proper to it, and the entire effect on the throughput is carried by the
same effective width \(\Gamma^{\rm eff}\).

\paragraph{GG connection.}
The tiny Yukawa is forced, not chosen: a sub-eV light mass with a TeV heavy scale \emph{requires}
\(Y_\nu\sim10^{-6}\). Placing the seesaw at the geometry's lock scale, rather than at an arbitrary
intermediate scale, is what makes this the case.

\paragraph{Status.}
Derived. Given \ANine{}'s \(m_\star\) and the geometry's \(M_N\simeq\Mlock\), the Yukawa magnitude is
fixed with no freedom.

\subsection{Resonant CP asymmetry on the inherited phases}

\paragraph{Statement.}
The self-energy (resonant) CP asymmetry for a quasi-degenerate Majorana pair is
\begin{equation}
\epsilon_i
=
\frac{\Im\!\left[(Y_\nu Y_\nu^\dagger)_{ij}^2\right]}
{(Y_\nu Y_\nu^\dagger)_{ii}(Y_\nu Y_\nu^\dagger)_{jj}}
\,\frac{(M_i^2-M_j^2)M_i\Gamma_j}
{(M_i^2-M_j^2)^2+M_i^2\Gamma_j^2}.
\label{eq:app-resonant-eps}
\end{equation}

\paragraph{Derivation.}
Write \(M_j = M_i(1+d)\) with fractional splitting \(d\ll1\). Then
\(M_i^2-M_j^2 \simeq -2 M_i^2 d\), and the Breit--Wigner factor becomes
\begin{equation}
\mathcal R_{\rm res}
=
\frac{(M_i^2-M_j^2)M_i\Gamma_j}{(M_i^2-M_j^2)^2+M_i^2\Gamma_j^2}
\simeq
\frac{-2 d\,(\Gamma_j/M_j)}{4 d^2 + (\Gamma_j/M_j)^2}.
\label{eq:app-BW}
\end{equation}
This is extremized when \(2|d| = \Gamma_j/M_j\), i.e. when the mass splitting equals half the width
(the resonance condition), where \(|\mathcal R_{\rm res}|=\tfrac12\). At resonance the common
magnitude of \((Y_\nu Y_\nu^\dagger)\) cancels between the CP-odd prefactor's numerator and
denominator, so \(\epsilon\) is controlled by the relative phase alone:
\begin{equation}
\epsilon^{\rm res}
\simeq
\frac{\Im\!\left[(Y_\nu Y_\nu^\dagger)_{ij}^2\right]}
{(Y_\nu Y_\nu^\dagger)_{ii}(Y_\nu Y_\nu^\dagger)_{jj}}
\cdot \frac12
\equiv \frac12\,\sin(2\phi_{\rm geo}).
\label{eq:app-eps-res-phase}
\end{equation}
The inherited light sector reaches \(\phi_{\rm geo}\) only through flavor, and the route must be
displayed rather than asserted; it is displayed as a keyed lemma, so that the downstream consumers
of this result can cite it.

\begin{lemma}[\(U_{\rm PMNS}\)-cancellation and the flavor obligation (L-Cosmos-U)]
\label{lem:cosmos-U}
In the unflavored (flavor-summed) limit, the resonant CP invariant of
Eq.~\eqref{eq:app-eps-res-phase} is independent of \(U_{\rm PMNS}\), and for the dial-free door
(\(R\) real, adding no new CP) it vanishes identically --- the unflavored \(\epsilon^{\rm res}\)
is exactly zero (C-Cosmos-U). Hence the inherited phases reach \(\eta_B\) only through the
flavored asymmetries \(\epsilon_{i\alpha}\): at \(M_N \simeq 3.85\)~TeV, where all three
charged-lepton flavors are resolved in the plasma, the fully flavored treatment is mandatory
(L-Cosmos-F, the flavor obligation).
\end{lemma}

\begin{proof}
In the Casas--Ibarra parametrization,
\(Y_\nu = (\sqrt2/v)\,M_N^{1/2} R\, m_\nu^{1/2} U_{\rm PMNS}^\dagger\) with \(R\) complex
orthogonal, the heavy-space bilinear entering Eq.~\eqref{eq:app-eps-res-phase} is
\begin{equation}
(Y_\nu Y_\nu^\dagger)_{ij}
= \frac{2}{v^2}\,\sqrt{M_i M_j}\,\sum_k m_k\,R_{ik}R^*_{jk},
\label{eq:app-sinphi}
\end{equation}
in which \(U_{\rm PMNS}\) has canceled exactly (\(U^\dagger U = 1\)): the unflavored invariant of
Eq.~\eqref{eq:app-eps-res-phase} is independent of the PMNS phases, and for the dial-free door
(\(R\) real, adding no new CP) the sum \(\sum_k m_k R_{ik}R_{jk}\) is real, so its imaginary part
vanishes identically --- the unflavored \(\epsilon^{\rm res}\) is exactly zero. At
\(M_N \simeq 3.85\)~TeV all three charged-lepton flavors are resolved in the plasma, so the fully
flavored treatment is mandatory, and there the PMNS phases genuinely enter: the operative objects
are the flavored asymmetries \(\epsilon_{i\alpha}\) (self-energy numerators
\(\Im[Y_{i\alpha}Y^*_{j\alpha}(Y_\nu Y_\nu^\dagger)_{ij}]\) and the \(ji\) partner), in which
\(U_{\rm PMNS}\) enters explicitly through \(Y_{i\alpha}\); their flavor sum vanishes for real
\(R\), while a net asymmetry survives because the flavored washouts \(K_\alpha\) differ.
\end{proof}

\paragraph{Worked number (flavored, ledger inputs).}
On the \ANine{} QD ledger (masses, angles, \(\delta_{\rm PMNS} = 196.06^\circ\),
\(\alpha_{31} = \pi\); real \(R\) with \(\mathcal O(1/K)\) angles) the flavored asymmetries are
\(\epsilon_\alpha \simeq (+3.4,\,+3.1,\,-6.6)\times10^{-3}\) (\(\sum_\alpha \epsilon_\alpha = 0\)
exactly), the flavored washouts are \(K_\alpha \simeq (3.4,\,28.0,\,22.4)\), and the effective
throughput \(\epsilon_{\rm eff} = \sum_\alpha \epsilon_\alpha \min(1, 1/K_\alpha)\) brackets the
ledger target:
\begin{equation}
\epsilon_{\rm eff} \simeq 8\times10^{-4}\ \ (\text{Breit--Wigner peak}),
\qquad
\epsilon_{\rm eff} \simeq 6\times10^{-15}\ \ (\text{vacuum splittings}),
\label{eq:app-eps-number}
\end{equation}
an overshoot of the required throughput \(\epsilon^{\rm res}\kappa = e^{-K/2}\) by
\(\sim3\times10^{4}\) on the peak and an undershoot by \(\sim4\times10^{6}\) at vacuum numbers.
Three structural facts, each numerically checked: the driver is the Majorana parity
\(\alpha_{31} = \pi\), which sets the \emph{sign} of the matter excess (\(\alpha_{31}\to0\) flips
the sign, while \(\delta_{\rm PMNS}\to\pi\), i.e. \(J_{\rm PMNS}\to0\), shifts
\(\epsilon_{\rm eff}\) by only \(+4\%\): the result is \(J_{\rm PMNS}\)-blind at the few-percent
level); the result is nonzero at \(m_1 = 0\) (first-order masses give
\(\epsilon_{\rm eff} \simeq 2.4\times10^{-4}\)), so the asymmetry does \emph{not} ride on the QD
layer; and the \(U\)-cancellation of Lemma~\ref{lem:cosmos-U} (L-Cosmos-U) retires the formerly assembled evaluation
\(\sin\phi_{\rm geo} \sim c\,J_{\rm PMNS}\) with its \(m_1/m_3\)-order suppression, withdrawn as a
reconstruction artifact.

\paragraph{The named hypothesis.}
\emph{H-\(\phi_{\rm geo}\)-1 (partial resonance): there exists a finite-temperature transport
computation of the lock-crossing decays in which the effective splitting-to-width ratio at the
crossing satisfies \(\Delta M/\Gamma^{\rm eff} \simeq 3\times10^{4}\) (equivalently, Breit--Wigner
factor \(\simeq1.5\times10^{-5}\); \(\Gamma^{\rm eff} \simeq 3\times10^{-3}\)~GeV at
\(\Delta M \simeq 110\)~GeV), so that the flavored, real-\(R\) throughput on the inherited phases
lands within \(O(1)\) of \(e^{-K/2}\).} The hypothesis stays named and conditional, and it is now
carried by an identified candidate mechanism --- the flavor-measurement regulator of
\S\ref{subsec:app-flavour-regulator}, with the closed form
\(\Delta M/\Gamma^{\rm eff} = 1/(35\,c_\tau y_\tau^2) = (2.7\text{--}5.3)\times10^{4}\) --- at
candidate grade, conditional on the conditions C1--C6 stated there. Under H-\(\phi_{\rm geo}\)-1
the chain's \(\epsilon^{\rm res} \simeq 2.5\times10^{-6}\) is the \emph{target value} the partial
resonance must deliver --- a target value now carried by an identified candidate mechanism whose
closed form lands the ratio in the hypothesized decade with zero dials
(\S\ref{subsec:app-flavour-regulator}); the transport-level closure of conditions C1--C6 remains
open. \(\Gamma^{\rm eff} \sim 3\)~MeV is parametrically plausible for
thermal damping at \(T \sim M_N\) --- and is now identified with the computed
\(\Gamma_\tau = 2\text{--}4\)~MeV --- so the hypothesis is a well-posed computation with an
acceptance band (\(\pm O(1)\) in \(\epsilon_{\rm eff}\)), carried as row~9 of the promotion
roadmap of this paper's Foundations-Branch companion, the CPT member paper \PaperPFive{}, where
the row now records the mechanism at candidate grade.

\paragraph{GG connection.}
The resonance lifts what would otherwise be a hopeless \(Y_\nu^2\sim10^{-13}\) suppression, and
flavor is what lets the inherited CP through: the sign of the matter excess is the Majorana parity
of the rank-two block (\(\alpha_{31} = \pi\), derived structural data of \ANine{}), filtered
through the flavor-washout spread \(K_e \ll K_{\mu,\tau}\) --- an inheritance statement sharper,
not weaker, than the retired Jarlskog projection.

\paragraph{Status.}
Derived-conditional on H-\(\phi_{\rm geo}\)-1. The flavored computation on the inherited phases is
displayed above with no dial (Lemma~\ref{lem:cosmos-U} and the worked number); the named hypothesis is the finite-temperature partial-resonance
condition \(\Delta M/\Gamma^{\rm eff} \simeq 3\times10^{4}\) --- the sharp form of the on-resonance
condition is partial resonance, strictly weaker than the exact-peak form (which overshoots the
required throughput by \(\sim3\times10^{4}\)) and strictly stronger than vacuum numbers (which
undershoot by \(\sim4\times10^{6}\)); discharging it is the flavored face of the
sphaleron-transport-normalization conjecture. The hypothesis now carries an identified candidate
mechanism at candidate grade --- the flavor-measurement regulator of the next subsection
(\S\ref{subsec:app-flavour-regulator}, conditions C1--C6) --- which does not change the grade of
this subsection.

\subsection{The partial-resonance mechanism: the flavor-measurement regulator}
\label{subsec:app-flavour-regulator}

\paragraph{Statement.}
The named hypothesis of the previous subsection asks for a finite-temperature transport
computation in which the effective splitting-to-width ratio at the lock crossing lands at
\(\Delta M/\Gamma^{\rm eff} \simeq 3\times10^{4}\). This subsection installs the candidate
mechanism: with \(R\) real the CP source is purely flavor-odd (Lemma~\ref{lem:cosmos-U}),
flavor-blind damping cancels from it exactly, and the effective regulator of the flavored
partial resonance is the thermal flavor-measurement rate of the lepton coherences --- the
charged-lepton Yukawa equilibration rate, dominated by the tau,
\begin{equation}
\Gamma^{\rm eff} \;=\; \Gamma_\tau \;=\; c_\tau\, y_\tau^{2}\, T,
\qquad
c_\tau = 5\times10^{-3}\text{--}10^{-2},
\label{eq:app-Gamma-fl}
\end{equation}
with \(c_\tau\) the kinetic coefficient of the flavor-equilibration literature. At the
generation temperature \(T \simeq M_N\) the heavy mass cancels between the ledger splitting
\(\Delta M = M_N/35\) and the thermal rate, leaving a closed form in the ledger integer and a
running Standard-Model Yukawa:
\begin{equation}
\frac{\Delta M}{\Gamma^{\rm eff}}
\;=\;
\frac{M_N/35}{c_\tau\, y_\tau^{2}\, T}\bigg|_{T=M_N}
\;=\;
\frac{1}{35\, c_\tau\, y_\tau^{2}}
\;=\;
(2.7\text{--}5.3)\times10^{4}.
\label{eq:app-closed-form}
\end{equation}
Zero dials: the only inputs are the ledger integer \(K = 35\) and the running tau Yukawa. The
identification is carried at \emph{candidate grade}, conditional on the numbered conditions
C1--C6 below; nothing in this subsection upgrades the grade of the chain.

\paragraph{Physics picture.}
The thermal bath is a flavor-measuring apparatus. Every process that distinguishes lepton
flavors acts as a measurement of the flavor coherences the \(N\) decays write into the
plasma, and at \(T \simeq 4\)~TeV the fastest \emph{honest} measurer is the tau Yukawa
interaction --- the gauge interactions are five orders stronger but flavor-blind, so they
read nothing. The partial-resonance question, \emph{how long does the CP-building interference
live?}, is therefore answered not by the \(N\)'s own width --- the coherence that carries the
CP does not live in the \(N\) lines --- but by how quickly the bath measures lepton flavor:
\(\Gamma^{\rm eff} = \Gamma_\tau\).

\paragraph{Derivation, step 1: the flavored kinetic system.}
The operative object is the lepton-doublet flavor matrix \((\delta n_\ell)_{\alpha\beta}\)
(deviation from equilibrium; \(\alpha,\beta = e,\mu,\tau\)). Its flavor-covariant kinetic
equation, in the density-matrix form standard in flavored leptogenesis, is
\begin{equation}
\frac{\dd(\delta n_\ell)_{\alpha\beta}}{\dd t}
=
S_{\alpha\beta}(t)
\;-\;
\tfrac12\bigl\{W,\,\delta n_\ell\bigr\}_{\alpha\beta}
\;-\;
\tfrac12\,\Gamma_\tau\bigl[P_\tau,[P_\tau,\,\delta n_\ell]\bigr]_{\alpha\beta}
\;-\;
\Gamma_{\rm g}\,\bigl(\text{terms}\propto\mathbf 1_{\alpha\beta}\bigr),
\label{eq:app-flavour-kinetic}
\end{equation}
where \(S_{\alpha\beta}\propto\sum_{j\neq i} Y_{i\alpha}Y^{*}_{j\beta}\times(\text{the }ij
\text{ heavy coherence, oscillating at }\Delta M_{ij})\) is the CP source from the
quasi-degenerate pair, \(W\) the flavored washout (\(\{\,,\}\) the anticommutator),
\(P_\tau = \mathrm{diag}(0,0,1)\) the tau projector, \(\Gamma_\tau\) the tau-Yukawa
flavor-measurement rate acting on tau-row and tau-column coherences, and the gauge damping
\(\Gamma_{\rm g}\) --- flavor-universal --- enters proportional to the identity in flavor
space: it thermalizes kinetic distributions at fixed flavor content.

\paragraph{Derivation, step 2: with \(R\) real the source is purely flavor-odd.}
By Lemma~\ref{lem:cosmos-U}, the flavor trace of the CP source is the unflavored invariant,
and for the dial-free door it vanishes identically:
\begin{equation}
\mathrm{Tr}\,S
= \sum_\alpha S_{\alpha\alpha}
\;\propto\;
\Im\!\left[(Y_\nu Y_\nu^\dagger)_{ij}^{2}\right]
= 0
\qquad (R\ \text{real}).
\label{eq:app-source-traceless}
\end{equation}
The physical source is therefore the traceless --- flavor-odd --- part of
\(S_{\alpha\beta}\): the entire CP content lives in flavor \emph{differences}, exactly as the
worked number above exhibits (\(\sum_\alpha\epsilon_\alpha = 0\) with each
\(\epsilon_\alpha\neq0\)).

\paragraph{Derivation, step 3: flavor-blind damping cancels from the flavor-odd source.}
The decohering action of a thermal rate on a flavor matrix is its double commutator with the
projector that rate measures. For a flavor-universal rate the projector is the identity, and
\begin{equation}
\bigl[\mathbf 1,[\mathbf 1,\,\delta n_\ell]\bigr] = 0
\ \ \text{exactly, for every }\delta n_\ell;
\qquad
\bigl[P_\tau,[P_\tau,\,\delta n_\ell]\bigr] = \delta n_\ell
\ \ \text{on pure tau coherences.}
\label{eq:app-blind-cancel}
\end{equation}
Gauge damping (\(g^2 T \simeq 1.5\times10^{3}\)~GeV here) therefore cannot regulate a source
it does not distinguish: it cancels from the flavor-odd channel, however large it is. The
leading flavor-\emph{non}-universal rate in the plasma is the tau Yukawa rate of
Eq.~\eqref{eq:app-Gamma-fl}; the muon and electron rates are down by
\((y_\mu/y_\tau)^{2} \simeq 3\times10^{-3}\) and below. This is why the tau is the fastest honest measurer --- and why, on the C4 reading (and pending exactly the resummation C4 names), the \(N\)'s own width (\(\Gamma_N \sim 10^{-10}\)~GeV, seven orders below \(\Gamma_\tau\)) would not be the operative regulator: on that reading the interference that builds \(\epsilon_{i\alpha}\) is cut off the moment the bath learns \emph{which flavor} was produced, long before the \(N\) lines themselves decay.

\paragraph{Derivation, step 4: two-level reduction and the effective Breit--Wigner.}
Reducing the heavy pair \((i,j)\) to a two-level system, the CP-odd weight accumulated by an
interference oscillating at \(\Delta M\) while its flavor coherence is measured at
\(\Gamma_{\rm fl}\) is
\begin{equation}
I(\Gamma_{\rm fl})
=
\Gamma_{\rm fl}\int_0^\infty \dd s\; e^{-\Gamma_{\rm fl}\,s}\,\sin(\Delta M\, s)
=
\frac{\Gamma_{\rm fl}\,\Delta M}{\Gamma_{\rm fl}^{2} + \Delta M^{2}}
\;\le\; \frac12,
\label{eq:app-twolevel-I}
\end{equation}
a Lorentzian partial-resonance envelope: the Breit--Wigner factor of Eq.~\eqref{eq:app-BW}
with \(\Gamma^{\rm eff}\) in place of the vacuum width, up to a convention factor two off-peak
(a priced \(O(1)\)). The bound \(|\mathcal R_{\rm res}|\le\tfrac12\) holds for \emph{every}
regulator value --- both envelopes obey \(xy/(x^{2}+y^{2})\le\tfrac12\) --- so the
partial-resonance form survives the reduction; what the reduction assumes is that the damped
coherence is the lepton-flavor one (condition C4 below).

\paragraph{Worked number.}
One-loop running gives \(y_\tau(3.85~\mathrm{TeV}) = 0.0104\),
\(y_\tau^{2} = 1.08\times10^{-4}\) (a \(+3.6\%\) effect in \(y_\tau^{2}\) over the run from
\(m_t\) --- a refinement, not a lever). Then
\begin{equation}
\Gamma^{\rm eff} = c_\tau y_\tau^{2} M_N = 2.1\text{--}4.1~\mathrm{MeV},
\qquad
\frac{\Delta M}{\Gamma^{\rm eff}} = (2.7\text{--}5.3)\times10^{4},
\qquad
F_{\rm BW} = (0.9\text{--}1.9)\times10^{-5},
\label{eq:app-closed-number}
\end{equation}
against the hypothesized \(3\)~MeV, \(3\times10^{4}\), and \(1.5\times10^{-5}\): the installed
target sits inside the band, reproduced exactly at \(c_\tau = 8.9\times10^{-3}\). Consistency:
\(\Gamma_\tau/H(T{=}M_N) = (1\text{--}2)\times10^{8}\) --- the fully flavored regime of
L-Cosmos-F holds with eight orders of margin --- and
\(\Gamma_\tau/\Delta M \simeq 3\times10^{-5}\ll1\): the mechanism is genuinely \emph{partial},
not an accidental return to the excluded peak. Throughput: at \(T = M_N\) the flavored chain
gives \(\eta_B = (1.7\text{--}3.4)\times10^{-10}\), i.e. \(0.28\text{--}0.55\) of the observed
value, from below. The proper \(T\)-integration of the source over the decoupling window (the flavored Boltzmann system with \(\Gamma^{\rm eff}\propto T\) through the window) is carried out in \S\ref{appA:boltzmann-integration}: the surviving asymmetry is produced at \(z = M_N/T \simeq 1.5\text{--}4\), where the mechanism's own regulator has dropped with the temperature, and the \(T\)-integrated landing, scanning the declared freedoms, is \(0.09\text{--}0.23\) of observed (68\%; central \(0.14\)) --- always an undershoot, never an overshoot (condition C6 below). The one-sidedness is itself informative: the missing transport prefactors must be enhancements, and the known members of that class (scattering source terms, the Lorentzian-vs-Breit--Wigner convention, the seam washout normalization) all point up, supplying a factor \(\simeq3.3\) centrally and leaving the observation \(+1.3\sigma\) away in the log.

\paragraph{The regulator landscape.}
The identification is falsifiable because the candidate prescriptions are nowhere near one
another:

\begin{table}[H]
\centering
\caption[The regulator landscape at the lock crossing]{The regulator landscape at the lock crossing (\(\Delta M = 110\)~GeV, \(T = M_N\)):
the four candidate prescriptions for \(\Gamma^{\rm eff}\), spanning sixteen orders of
magnitude in \(\Delta M/\Gamma^{\rm eff}\). Exactly one lands in the decade
H-\(\phi_{\rm geo}\)-1 requires. Throughput column at the single-\(T\) evaluation.}
\label{tab:app-regulator-landscape}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.33\textwidth}p{0.13\textwidth}p{0.13\textwidth}p{0.12\textwidth}p{0.13\textwidth}}
\toprule
\textbf{Candidate} & \(\Gamma^{\rm eff}\) [GeV] & \(\Delta M/\Gamma^{\rm eff}\) & \(F_{\rm BW}\) & \(\eta_B/\eta_B^{\rm obs}\) \\
\midrule
(a) \(N\) widths (S-matrix prescription) & \(\sim2\times10^{-10}\) & \(6\times10^{11}\) & \(8\times10^{-13}\) & \(2\times10^{-8}\) \\
(b) flavor measurement \(c_\tau y_\tau^{2} T\) & \(2\text{--}4\times10^{-3}\) & \((2.7\text{--}5.3)\times10^{4}\) & \((0.9\text{--}1.9)\times10^{-5}\) & \(0.28\text{--}0.55\) \\
(c) oscillation rate \(\Delta M\) itself & \(1.1\times10^{2}\) & \(1\) & \(0.40\) & \(1.4\times10^{4}\) \\
(c\('\)) gauge damping \(g^{2}T\) & \(1.5\times10^{3}\) & \(0.07\) & \(0.14\) & \(1.7\times10^{4}\) \\
(d) Hubble rate \(H(T{=}M_N)\) & \(2.1\times10^{-11}\) & \(5\times10^{12}\) & \(1\times10^{-13}\) & \(3\times10^{-9}\) \\
\bottomrule
\end{tabular}
\end{table}

The landscape discriminates cleanly rather than scattering: sixteen orders between its
extremes, and one --- only one --- candidate in the required decade. The \(N\)-width class
undershoots the observed asymmetry by \(\sim\)\(10^{7\text{--}8}\); the oscillation-rate and
gauge-damping prescriptions re-land on the peak overshoot the flavored recomputation already
excluded; the Hubble rate is worse than all of them.

\paragraph{Conditions.}
The derivation above is honest about what it assumes. Each assumption is a numbered condition
with its status:
\begin{enumerate}[leftmargin=3.1em,label=\textbf{C\arabic*.}]
\item \emph{Flavor-odd source.} With \(R\) real, \(\mathrm{Tr}\,S = 0\) exactly
(Eq.~\eqref{eq:app-source-traceless}, the kinetic form of Lemma~\ref{lem:cosmos-U}). Status:
established (algebraic; machine-checked).
\item \emph{Flavor-blind cancellation.} Gauge damping enters through the identity projector
and drops from the flavor-odd channel (Eq.~\eqref{eq:app-blind-cancel}). Status: established
within the density-matrix framework; its persistence under full resummation is folded into C4.
\item \emph{Two-level reduction.} The per-pair Lorentzian envelope and the
\(|\mathcal R_{\rm res}|\le\tfrac12\) bound survive for every regulator value
(Eq.~\eqref{eq:app-twolevel-I}); Lorentzian-vs-Breit--Wigner is a priced factor two. Status:
established at reduction level, given C4.
\item \emph{Regulator identification (the crux).} In the regime
\(\Gamma_\tau/\Gamma_N \sim 10^{7}\) with the flavor-summed source zero, the interference
time of the flavor-odd source is \(1/\Gamma_\tau\), not \(1/\Gamma_N\): the vacuum
\(N\)-width regulator of the S-matrix prescription is superseded in-medium. The displayed kinetic system carries the heavy-pair coherence as an external source; discharging C4 requires promoting it to a dynamical variable with its own damping --- that is the precise content the Kadanoff--Baym resummation must supply. Status: open at derivation level --- structurally supported, but the full Kadanoff--Baym/CTP resummation has not been carried out, and the standard fully-flavored Boltzmann limit would return the (fatal) \(N\)-width class instead.
\item \emph{The coefficient.} \(c_\tau \in [5\times10^{-3},\,10^{-2}]\) at \(T \simeq M_N\)
for lepton coherences sourced by \(N\) decays in an electroweak-symmetric bath (the
kinetic-estimate, regime-boundary, and CTP equilibration classes of the flavor-equilibration
literature, mutually consistent within a factor two). Status: plausible; the precise value is
open.
\item \emph{Window throughput.} The \(T\)-integrated window cost is part of the honest chain, and the priced \(O(1)\) transport prefactors must supply the remaining one-sided enhancement, measured by the flavored Boltzmann integration of \S\ref{appA:boltzmann-integration} at \(E = 7.1\), 68\% interval \([4.4,\,11.4]\) (retiring the prior bracket \(\simeq2\text{--}20\)). Status: measured at flavored-Boltzmann level; closure open.
\end{enumerate}

\paragraph{Kill scenarios.}
Two clean kills and one soft one, all on record. (i) If the finite-temperature derivation
returns the \(N\)-width class even in the flavored real-\(R\) corner, the chain undershoots
by \(\sim\)\(10^{7\text{--}8}\) and is falsified from within. (ii) If it returns the
flavor-blind thermal width \(g^{2}T\), the excluded \(\sim\)\(3\times10^{4}\) peak overshoot
stands --- same verdict. (iii) If C4 and C5 close but the measured window deficit of C6 (\(E = 7.1\), 68\% \([4.4,\,11.4]\)) survives a full transport computation at its adverse end with the priced prefactors exhausted (they supply \(\simeq3.3\) centrally, leaving \(+1.3\sigma\) in the log),
H-\(\phi_{\rm geo}\)-1 fails by throughput rather than by regulator identity. Either clean
kill would discharge the hypothesis negatively; the mechanism has made it \emph{more}
falsifiable, not less.

\paragraph{GG connection.}
Why the ledger integer appears: the splitting \(\Delta M = M_N/35\) is the ledger's own level
spacing, so once the bath fixes the measurement rate, the resonance quality
\(\Delta M/\Gamma^{\rm eff}\) is geometry over chemistry --- the same integer that sets the
hierarchy \(e^{-K}\), the lock mass \(\tfrac12 M_{\rm Pl}\,e^{-K}\), and the throughput target
\(e^{-K/2}\) also sets how far off peak the resonance sits, with the tau Yukawa --- an object
entirely of the Standard Model --- supplying the only other factor. The heavy mass cancels; no
GG-sector dial, no neutrino-sector input, no freedom beyond the \(O(1)\)-band kinetic
coefficient \(c_\tau\) survives into Eq.~\eqref{eq:app-closed-form}. And the sign story is
untouched: \(\Gamma^{\rm eff}\) is CP-even, so the Majorana parity \(\alpha_{31} = \pi\) still
sets the sign of the matter excess, exactly as registered above.

\paragraph{Status.}
Candidate grade. Established: the closed form Eq.~\eqref{eq:app-closed-form} as the
consequence of the identification, the landscape discrimination of
Table~\ref{tab:app-regulator-landscape}, and the flavor-odd source structure (C1--C3).
Derived-conditional: the effective regulator itself, on C4--C5; the throughput landing,
additionally on C6. Open: the full Kadanoff--Baym resummation, the precise \(c_\tau\) at
\(T \sim M_N\), and the closure of the window factor. H-\(\phi_{\rm geo}\)-1 remains a named hypothesis --- not retired, not closed --- now sharpened from a numerical target into the conditions C1--C6.

\paragraph{Placement in the kinetic literature.}
The regulator question of C4 is not being asked in a vacuum. The flavor-covariant density-matrix and Kadanoff--Baym literature supplies the machinery in which it must be settled: the flavor-covariant transport equations of \citet{DevMillingtonPilaftsisTeresi2014}, the first-principles resonant-regime analysis of \citet{GarnyKartavtsevHohenegger2013}, the flavor-mixing transport theory of \citet{JukkalaKainulainenRahkila2021}, and the unified low-scale framework of \citet{KlaricShaposhnikovTimiryasov2021} all return effective regulators of the heavy-sector class in the regimes they treat --- which is exactly why C4 is carried open rather than assumed. On the other side, the 2025--26 thermal-resonant line --- the soft-lepton resonant asymmetry of \citet{KanemuraLi2025} and the thermal-resonant mechanism of \citet{LiPilaftsis2026a,LiPilaftsis2026b} --- derives flavor-odd CP sources clocked by charged-lepton Yukawa rates, with gauge damping dropping out of the CP-odd channel: the strongest published support for the structure of C1--C2 and for the lepton-sector clock, while leaving C4's cross-sector matching (the heavy \(\Delta M\) oscillation regulated by \(\Gamma_\tau\)) exactly as open as it is graded here. The resummation that discharges C4 should therefore also assess whether a thermal-resonant lepton-sector source of that type competes with, or replaces, the \(\Delta M\)-filtered source at \(T\sim M_N\).

\subsection{Washout and the strong-washout efficiency}

\paragraph{Statement.}
The surviving fraction of the generated asymmetry is the efficiency \(\kappa\), controlled by the
washout strength
\begin{equation}
K_{\rm wash}=\frac{\widetilde m}{m_\star},
\qquad
\widetilde m = \frac{(Y_\nu Y_\nu^\dagger)_{ii}\,v^2}{M_i},
\qquad
m_\star \simeq 1.1\times10^{-3}\,\mathrm{eV}.
\label{eq:app-Kwash}
\end{equation}

\paragraph{Derivation.}
The equilibrium scale follows from equating the decay rate to Hubble at \(T=M_i\),
\(m_\star = (8\pi v^2/M_i)\,(H(M_i)/M_i)\), with \(H(M_i)=1.66\,g_*^{1/2}M_i^2/M_{\rm Pl}\); for
\(g_*\simeq106.75\) this gives \(m_\star\simeq1.1\times10^{-3}\,\mathrm{eV}\). The inherited light
spectrum makes the effective mass that the decaying state couples to of order the heaviest light
neutrino, \(\widetilde m \sim m_3\simeq5\times10^{-2}\,\mathrm{eV}\). In the strong-washout regime
(\(K_{\rm wash}\gg1\)) the final efficiency takes the standard suppressed form, well approximated by
\begin{equation}
\kappa
\simeq
\frac{1}{K_{\rm wash}(z_f)}
\;\sim\;
\frac{2}{K_{\rm wash}\,z_f},
\label{eq:app-kappa-form}
\end{equation}
with \(z_f=\mathcal O(\text{few})\) the freeze-out value of \(z=M_i/T\). Strong washout erases the
dependence on initial \(N\) abundance, so the result is initial-condition independent.

\paragraph{Worked number.}
With \(\widetilde m\simeq5\times10^{-2}\,\mathrm{eV}\) and
\(m_\star\simeq1.1\times10^{-3}\,\mathrm{eV}\),
\begin{equation}
K_{\rm wash}\simeq 45\text{--}50,
\qquad
\kappa \simeq \frac{1}{K_{\rm wash}}\times\mathcal O(1)
\simeq 1\times10^{-2}.
\label{eq:app-kappa-number}
\end{equation}
The convention note of Eq.~\eqref{eq:app-GammaN} applies here in reverse: \(\widetilde m = m_3\)
is the washout convention of the chain; at the seam-matched \(\widetilde m \simeq 8.4\)~meV one
would instead have \(K_{\rm wash}\simeq7.6\) and \(\kappa\simeq0.13\). The factor-\(\simeq12\) spread in \(Y_\nu^2\) between the two conventions is declared, accounted for at the
convention note of Eq.~\eqref{eq:app-GammaN}, and priced as an \(O(10)\) freedom
of the order-of-magnitude register, carried with the transport \(\mathcal O(1)\)'s.

\paragraph{GG connection.}
The washout scale is set entirely by inherited quantities --- the seesaw-matched Yukawa and the
\ANine{} light masses --- so \(\kappa\) is not a free knob.

\paragraph{Status.}
Derived-conditional: the regime (strong washout) and the order of magnitude are fixed by inherited
inputs; the \(\mathcal O(1)\) prefactor in Eq.~\eqref{eq:app-kappa-number} is measured, flavor by flavor, by the Boltzmann integration of \S\ref{appA:boltzmann-integration}.

\subsection{Assembling \texorpdfstring{$\eta_B$}{etaB}}

\paragraph{Statement.}
The boundary CP-odd throughput is the product \(\epsilon^{\rm res}\kappa\), converted to \(\eta_B\)
by the standard sphaleron and entropy factors.

\paragraph{Derivation and worked number.}
The product reconstructs the geometric ledger value,
\begin{equation}
\epsilon^{\rm res}\,\kappa
\simeq (2.5\times10^{-6})(1.0\times10^{-2})
= 2.51\times10^{-8}
= e^{-17.5} = e^{-K/2}.
\label{eq:app-eps-eff}
\end{equation}
Then, with the heavy-sector abundance carried at its equilibrium yield --- \(\sum_i Y_{N_i}^{\rm eq}\simeq1.25/g_*\) for the three quasi-degenerate species (\(0.42/g_*\) per Majorana species, \(g_N=2\); Eq.~\eqref{eq:YNeq-partIII}) ---
\begin{equation}
Y_L \simeq \Bigl(\sum_i Y_{N_i}^{\rm eq}\Bigr)\epsilon^{\rm res}\kappa
\simeq \frac{1.25\,\epsilon^{\rm res}\kappa}{g_*},
\qquad
\eta_B \simeq 7.04\,c_{\rm sph}\,Y_L
= 7.04\left(\frac{28}{79}\right)\frac{1.25\,e^{-K/2}}{g_*},
\label{eq:app-etaB-assembly}
\end{equation}
and inserting \(c_{\rm sph}=0.3544\), \(g_*=106.75\),
\begin{equation}
\eta_B
\simeq 7.04\times0.3544\times1.25\times\frac{2.51\times10^{-8}}{106.75}
\simeq 7.34\times10^{-10}.
\label{eq:app-etaB-number}
\end{equation}

\paragraph{GG connection.}
This is the explicit transport mechanism behind the compressed body result
Eq.~\eqref{eq:etaB-numerical-partIII}: the geometric \(e^{-K/2}\) is the coarse-grained product of a phase-controlled resonant asymmetry and a strong-washout efficiency. The two declared joints are carried in the open: the on-resonance condition (named input) and the \(K/2\) exponent (postulate); the identity at displayed precision uses the \(\mathcal O(1)\) freedom in \(\kappa\) and the target status of \(\epsilon^{\rm res}\).

\paragraph{Status.}
Order-of-magnitude with a controlled mechanism. The central value sits \(+20\%\) from the observed \(\eta_B^{\rm obs}=(6.12\pm0.04)\times10^{-10}\); the honest band is factor-\(\sim\)3, dominated by
the resonance condition and the \(\mathcal O(1)\) washout prefactor.

\subsection{The flavored Boltzmann integration: the window deficit measured}
\label{appA:boltzmann-integration}

\paragraph{Statement.}
The closed-form worked number of the flavor-measurement subsection brackets the cost of the
decoupling window at a single temperature; this subsection measures it. The flavored Boltzmann
system --- the kinetic form of the chain, every input either derived upstream or standard --- is
integrated over the full window, scanning only the declared freedoms
(\(c_\tau \in [5\times10^{-3},\,10^{-2}]\), the washout convention, the integration start
\(z_0\)). Scanning only those freedoms the chain lands at
\begin{equation}
\eta_B^{\rm Boltzmann} \;=\; 0.86^{\,+0.52}_{\,-0.32}\times10^{-10}
\quad(68\%\ \text{interval; band a factor }2.6),
\label{eq:app-boltz-levelB}
\end{equation}
and the required one-sided enhancement --- the quantity condition C6 names --- is now a measured
number with an error bar rather than a decade-wide bracket:
\begin{equation}
E \;\equiv\; \frac{\eta_B^{\rm obs}}{\eta_B^{\rm Boltzmann}}
\;=\; 7.1\,,\qquad 68\%\ \text{interval}\ [4.4,\ 11.4].
\label{eq:app-E-measured}
\end{equation}
With the priced standard-omissions inventory applied (the scattering source, the convention
factor, the seam normalization), \(\eta_B = 2.8^{\,+2.0}_{\,-1.2}\times10^{-10}\): the
observation sits \(+1.3\sigma\) away in the log. Everything in this subsection is conditional on
C4 and C5, exactly as graded; nothing here upgrades any grade.

\paragraph{The equations, term by term.}
In the variable \(z = M_N/T\) (time runs with \(z\)) and photon-normalized abundances
\(N_X = n_X/n_\gamma\), the system is
\begin{align}
\frac{\dd N_N}{\dd z} &= -\,D(z)\,\bigl(N_N - N_N^{\mathrm{eq}}\bigr),
\label{eq:app-boltz-N}\\[2pt]
\frac{\dd N_{\Delta\alpha}}{\dd z} &=
\epsilon_\alpha(z)\,D(z)\,\bigl(N_N - N_N^{\mathrm{eq}}\bigr)
\;-\;
\tfrac14\,K_\alpha\,z^3 K_1(z)\,N_{\Delta\alpha},
\qquad \alpha = e,\mu,\tau.
\label{eq:app-boltz-D}
\end{align}
Every symbol earns its place. \(N_N^{\mathrm{eq}}(z) = \tfrac34\cdot\tfrac12\,z^2K_2(z)\) per
species is the equilibrium abundance: three quarters of the photon abundance while relativistic,
Boltzmann-suppressed as \(e^{-z}\) once \(T\) falls below the mass, the Bessel function \(K_2\)
the correct relativistic bookkeeping between the limits. \(D(z) = K_{\rm tot}\,z\,K_1(z)/K_2(z)\)
is the decay term; the ratio \(K_1/K_2 \to z/2\) at small \(z\) is time dilation --- a hot, fast
heavy neutrino lives longer in the plasma frame, so decays switch on only as the bath cools
toward \(z\sim1\). The source term says that every net decay (the excess of decays over inverse
decays, which is exactly the rate at which \(N_N\) follows its falling equilibrium) deposits a
CP asymmetry \(\epsilon_\alpha(z)\) into flavor \(\alpha\); the asymmetry per decay is
\(z\)-dependent because the regulator is thermal,
\begin{equation}
\epsilon_\alpha(z) = \bar\epsilon_\alpha\cdot 2\mathcal R(z),
\qquad
\mathcal R(z) = \frac{2\,\Delta M\,\Gamma^{\rm eff}(z)}{4\Delta M^2 + \Gamma^{\rm eff}(z)^2},
\qquad
\Gamma^{\rm eff}(z) = c_\tau y_\tau^2\,\frac{M_N}{z},
\label{eq:app-eps-of-z}
\end{equation}
with \(\bar\epsilon_\alpha = (+3.4,\,+3.1,\,-6.6)\times10^{-3}\) the flavored pattern computed
above on the ledger inputs (peak-normalized; \(\sum_\alpha\bar\epsilon_\alpha = 0\), the exact
flavor-sum zero of Lemma~\ref{lem:cosmos-U}). Since \(\Gamma^{\rm eff}\propto T\), the resonance
factor \(\mathcal R(z)\) \emph{falls} as the universe cools --- the interference is measured more
slowly while the splitting stays put --- and that decline is the physical origin of the window
suppression measured below. The washout term erases each flavor at its own rate:
\(\tfrac14 K_\alpha z^3K_1(z)\) is the standard inverse-decay kernel, peaking near
\(z\sim2\)--\(4\), with \(K_\alpha = (3.4,\,28.0,\,22.4)\) the flavor-projected washouts of the
chain. The kernels are decays and inverse decays; \(\Delta L = 1\) scatterings, spectator
redistribution, thermal-mass corrections and the relativistic edge of the window are deliberately
omitted and priced --- each a standard, named effect, and each of the omitted source-side effects
points \emph{up}. The equations are integrated from \(z_0 = 1\) (scanned \(0.5\)--\(2\)) to
\(z = 50\) with an exponentially exact stiff stepper.

\paragraph{Validation.}
A numerical claim in this cohort must arrive with its checks --- three, none adjustable after
the fact. For the efficiency check the code is run with a constant unit asymmetry in a single
flavor and compared with the standard analytic strong-washout fit
\begin{equation}
\kappa(K) \simeq \frac{2}{K\,z_B(K)}\Bigl(1 - e^{-K z_B(K)/2}\Bigr),
\qquad
z_B(K) \simeq 2 + 4K^{0.13}e^{-2.5/K},
\label{eq:app-bdp-fit}
\end{equation}
whose physical content is one sentence: the asymmetry that survives is the asymmetry made in the
\emph{last} moment before the inverse decays shut off, at \(z = z_B\), and by then only a
fraction \(\sim1/(Kz_B)\) of the heavy states remains to make it --- \(\kappa\) is the fraction
made late.
\begin{table}[H]
\centering
\caption[Validation of the Boltzmann integration]{Validation of the integration. The two \(K_\alpha\) computations were done
independently on the same derived inputs; the fit comparison spans the full washout range of
the problem. The columns give the check, the integration, the reference value, and the agreement. The two computations were run independently on the same derived inputs, so the agreement tests the integration and not the inputs, and a reader looking for validation of the inputs themselves should look at the audit tables instead.}
\label{tab:app-boltz-checks}
\small
\begin{tabular}{p{0.28\textwidth}p{0.23\textwidth}p{0.24\textwidth}p{0.14\textwidth}}
\toprule
\textbf{Check} & \textbf{Integration} & \textbf{Reference} & \textbf{Agreement}\\
\midrule
\(K_\alpha\) from derived \(U\), masses & \((3.5,\,29.3,\,22.5)\) & chain values above: \((3.4,\,28.0,\,22.4)\) & 2--5\,\%\\
\(\kappa(K{=}3.5)\) & \(0.1371\) & fit Eq.~\eqref{eq:app-bdp-fit}: \(0.1327\) & 3\,\%\\
\(\kappa(K{=}22.5)\) & \(0.0116\) & \(0.0121\) & 4\,\%\\
\(\kappa(K{=}29.3)\) & \(0.0084\) & \(0.0089\) & 5\,\%\\
single-\(T\) \(\epsilon\kappa\) at \(c_\tau = 8.9\times10^{-3}\) & \(2.7\times10^{-8}\) & ledger \(e^{-K/2} = 2.5\times10^{-8}\) & bookkeeping \(\mathcal O(1)\)\\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Worked numbers.}
At central parameters the integration lands at \(\eta_B = 1.0\times10^{-10}\), i.e.\ \(0.17\) of
observed; scanning the declared freedoms gives Eq.~\eqref{eq:app-boltz-levelB} and the measured
deficit Eq.~\eqref{eq:app-E-measured}. Because the integration resolves time, it also localizes
what the single-temperature estimate could not: the asymmetry is generated at
\(z \simeq 1.5\)--\(4\), where the thermal regulator \(\Gamma^{\rm eff}\propto1/z\) has already
fallen from its \(z = 1\) value --- the resonance is sharpest exactly when the washout is also
sharpest, and the survivors are made late, off the best part of the resonance. Every one of the
omitted standard effects (the \(\Delta L = 1\) scattering source, the relativistic production
edge) moves net production \emph{earlier}, where \(\mathcal R(z)\) is larger: the one-sidedness
of the deficit is visible in the mechanism, not asserted.

\paragraph{The electron door, and the sign as an output.}
The flavor-summed source is exactly zero (Lemma~\ref{lem:cosmos-U}), so the integration starts
with nothing to keep: at \(z \simeq 1.4\) the three flavor components stand at
\((+2.0,\,+0.9,\,-2.2)\times10^{-8}\), summing nearly to zero. What survives is the
\emph{difference} in erasure. The \(\mu\) and \(\tau\) components sit in strong washout
(\(K_\mu \simeq 28\), \(K_\tau \simeq 22\)) and are ground down; the electron flavor is
decoupled from the heavy third mass state by the derived reactor angle ---
\(|U_{e3}|^2 = \sin^2\theta_{13} = 3/140\), the smallest element of the mixing matrix and a
ledger quantity --- so \(K_e \simeq 3.4\) and the electron component survives an order of
magnitude better: \emph{96 per cent of the final asymmetry exits through the electron flavor}.
The sign of the excess is then an \emph{output} of differential erasure, not an input: nothing
in the integration was told which sign to prefer (the source pattern sums to zero), and the net
asymmetry emerges positive --- matter --- because the negative, \(\tau\)-dominated components
die faster, exactly as the Majorana parity \(\alpha_{31} = \pi\) requires on the sign audit
above. On this chain, the smallness of the reactor angle is why there is something rather than
nothing.

\paragraph{GG connection.}
Every input is inherited (masses, mixings, phases, \(\Delta M = M_N/K\), the seesaw-matched
Yukawas) or standard (kernels, \(g_*\)); the integration has no dial. What it measures is the
cost of the geometry's own timing: the ledger fixes the splitting, the bath fixes the falling
regulator, and the survivors are necessarily made late, off the best part of the resonance ---
the window deficit \(E\) is the price of that timing, now measured.

\paragraph{Status.}
Measured at flavored-Boltzmann level, conditional on C4 and C5 (candidate grade for the
regulator; the kill scenarios of the flavor-measurement subsection apply unchanged). The
measured \(E = 7.1\) (68\% \([4.4,\,11.4]\)) supersedes the single-temperature bracket wherever
the two differ. The priced standard-omissions inventory supplies a factor \(\simeq3.3\)
centrally, leaving the observation \(+1.3\sigma\) away in the log --- close, and not closed.

\subsection{The open-system A=7 port and the \texorpdfstring{$1/\pi$}{1/pi} survival fraction}

\paragraph{Statement.}
The mass-7 (\(^7\)Be \(\to\,^7\)Li) yield is suppressed by the closed-packet branching factor
\(1/\pi\) --- the same factor that fixes \(\Omega_m=1/\pi\) --- so that
\(({}^7\mathrm{Li}/H)_{\rm GG}=({}^7\mathrm{Li}/H)_{\rm SBBN}/\pi\).

\paragraph{Derivation.}
The relevant A=7 channels are
\begin{equation}
{}^3\mathrm{He}+\alpha \xrightarrow{\lambda_{3\alpha7}} {}^7\mathrm{Be}+\gamma,
\qquad
{}^7\mathrm{Be}+e^-\to{}^7\mathrm{Li}+\nu_e,
\qquad
{}^7\mathrm{Li}+p \xrightarrow{\lambda_{p\alpha}} 2\alpha.
\label{eq:app-A7-channels}
\end{equation}
Because the proton-destruction channel of \(^7\)Li is fast (\(\lambda_{p\alpha}\gg H\)) during the
active window, any \(^7\)Li reaching that window early is efficiently burned; the surviving A=7
inventory is therefore controlled by how much \(^7\)Be survives its open electron-capture /
phase-noise window intact. Model that window as a System Instance Packet whose conserved object is
the mass-7 flux \(\Phi_7\) carried over the compact angular phase \(\theta\in S^1_\theta\). The
open-system (\GDOS{}) drift of the packet over the compact angle is
\begin{equation}
\frac{\dd \Phi_7}{\dd\theta} = -\,\mathcal P(\theta)\,\Phi_7,
\qquad
S_{A=7}
=\frac{\Phi_7^{\rm out}}{\Phi_7^{\rm in}}
=\exp\!\left[-\oint_{S^1_\theta}\mathcal P(\theta)\,\dd\theta\right],
\label{eq:app-port-drift}
\end{equation}
where \(\mathcal P(\theta)\) is the port density. The closed-packet construction (Conjecture~VIII)
shows that, when a conserved flux is branched over the compact angle by the same \GFF{} closed-packet
geometry that fixes the cosmic inventory, the surviving share onto the observable boundary is the
solid-angle / phase fraction
\begin{equation}
S_{\rm closed-packet}
= \frac{1}{\pi},
\label{eq:app-survival-pi}
\end{equation}
the identical factor that yields \(\Omega_m=1/\pi\): the projection of the closed packet over the
compact angular measure retains the fraction \(1/\pi\) of the incoming flux. Identifying the A=7 port
with this same branching gives \(S_{A=7}=1/\pi\), i.e. the line integral in
Eq.~\eqref{eq:app-port-drift} equals \(\ln\pi\).

\paragraph{Worked number.}
\begin{equation}
\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm GG}
= \frac{1}{\pi}\left(\frac{{}^7\mathrm{Li}}{H}\right)_{\rm SBBN}
\simeq \frac{5.0\times10^{-10}}{3.14159}
\simeq 1.59\times10^{-10},
\label{eq:app-Li7-number}
\end{equation}
to be compared with the Spite-plateau value \((1.6\pm0.3)\times10^{-10}\). The deuterium and helium
sectors are untouched: even if every removed \(^7\)Be fed a side channel, the fractional shift in
\(D/H\) is \(\lesssim3\times10^{-5}\) (Eq.~\eqref{eq:DH-shift-tiny-partIII}).

\paragraph{GG connection.}
The lithium suppression is \emph{predicted}, not fitted: it is the same closed-packet \(1/\pi\) that
fixes the cosmic matter inventory, applied to the mass-7 flux. This replaces the earlier fitted port
strength entirely.

\paragraph{Status.}
D\(^\star\) (Derived-conditional), sharing the closed-packet-integration conjecture's status with the
inventory \(\Omega_m=1/\pi\). Falsifiable: any firm primordial \(^7\)Li abundance away from
\(\mathrm{SBBN}/\pi\) excludes the port.

\paragraph{The launch selects the mechanism.}
One structural loose end closes once the inflationary sector is
derived. Leptogenesis requires the heavy-neutrino sector to be
populated: thermally if the reheating temperature reaches the
right-handed masses, resonantly otherwise. With the launch underived, both readings would remain open.
The derived reheating temperature of Appendix~G,
\(T_{\mathrm{rh}} \simeq 3\times10^{10}\)~GeV, lands squarely in the
thermal window --- and the prompt lock crossing (\(N_2 = 0\)) guarantees
zero later dilution of the asymmetry. The mechanism is therefore no
longer a model choice: the same computation that fixed \(n_s\) and
\(r\) selects thermal leptogenesis on the inherited CP ledger. That is
the cross-sector economy of the descent in one sentence: one derived
temperature, three consequences (the e-fold count, the asymmetry
mechanism, the absence of dilution).

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{I.1.1. Estimate the seesaw scale from
the observed neutrino masses.}

\noindent\emph{Solution.}  Step 1: the seesaw relation
\eqref{eq:app-seesaw-matrix} gives $m_\nu \sim y^2v^2/M_N$
with $v = 174$~GeV.  Step 2: for the atmospheric scale
$m_\nu \approx 0.05$~eV and $y \sim 1$:
$M_N \sim v^2/m_\nu = (174~\mathrm{GeV})^2/0.05~\mathrm{eV}
\approx 6\times10^{14}$~GeV.  Step 3: the scale lands within
an order of magnitude of the inflationary energy of G.7.1 and
just below the classic unification scale --- three
independently derived energies crowding the same decade,
which is the numerical face of the cohort's claim that one
geometric depth underlies them.  Physics intuition: the
seesaw converts the smallness of the neutrino mass into the
largeness of a new scale --- the lighter the neutrino, the
higher the fulcrum; nature's lightest fermions are the
longest levers we possess into the deepest physics.

\medskip\noindent\textbf{I.1.2. Show that the resonant CP
asymmetry is maximized at $\Delta M = \Gamma_N/2$, with maximum $\tfrac12\sin(2\phi_{\rm geo})$.}

\noindent\emph{Solution.}  Step 1: the Breit--Wigner form
\eqref{eq:app-BW} gives $\epsilon \propto
f(\Delta M) = \dfrac{\Delta M\,\Gamma_N/2}{(\Delta M)^2 +
\Gamma_N^2/4}$ times the CP phase factor $\sin(2\phi_{\rm geo})$ of Eq.~\eqref{eq:app-eps-res-phase}.  Step 2:
maximize: $f'(\Delta M) = 0$ at $(\Delta M)^2 = \Gamma_N^2/4$,
i.e.\ $\Delta M = \Gamma_N/2$; there $f = \tfrac{\Gamma_N^2/4}
{\Gamma_N^2/2} = \tfrac12$.  Step 3: hence $\epsilon^{\rm res} \leq \tfrac12\sin(2\phi_{\rm geo})$ \eqref{eq:app-eps-res-phase} --- an
$O(1)$ asymmetry from a mass splitting tuned to the decay
width, which is what lets a TeV-adjacent or GUT-adjacent pair
generate the observed asymmetry without enormous CP phases.
Physics intuition: resonance is interference bookkeeping ---
the two nearly degenerate states beat against each other for
exactly one lifetime when their splitting matches their
width; narrower and they blur into one state, wider and they
never mix.  The ledger's role is to fix the splitting.

\medskip\noindent\textbf{I.4.1. Audit the $\eta_B$ assembly
arithmetic end to end.}

\noindent\emph{Solution.}  Step 1: the pieces of
\eqref{eq:app-etaB-assembly}: the entropy-to-photon conversion
$7.04$ (F.8.1's factor), the sphaleron share $c_{\rm sph} =
28/79 = 0.3544$ (the equilibrium fraction of $B-L$ stored as
$B$), the efficiency-weighted asymmetry
$\epsilon^{\rm res}\kappa = e^{-K/2} = 2.51\times10^{-8}$ (the ledger's $K = 35$ entering as its half-depth), the
triplet equilibrium yield $\sum_i Y_{N_i}^{\rm eq} = 1.25/g_*$
(three Majorana species' fair share of the bath,
Eq.~\eqref{eq:YNeq-partIII}), and the bath's $g_* = 106.75$.  Step 2: multiply:
$\eta_B = 7.04\times0.3544\times1.25\times2.51\times10^{-8}/106.75 =
7.34\times10^{-10}$ \eqref{eq:app-etaB-number}.  Step 3:
compare the observed $6.1\times10^{-10}$: agreement at the
$20\%$ level with no continuous dial --- the asymmetry of
all matter over all antimatter traced to one integer's
exponential.  Physics intuition: every factor in the product
has a face --- a conservation law, an equilibrium share, a
depth, a head count --- and the audit's point is that nothing
else is in the product; $\eta_B$ is bookkeeping all the way
down.

\medskip\noindent\textbf{I.5.1. Derive the lithium survival
factor $1/\pi$ from the port branching.}

\noindent\emph{Solution.}  Step 1: the $A = 7$ population
drains through the port with drift given by the line integral
of \eqref{eq:app-port-drift}; the closed-packet branching
assigns the through-port share $1/\pi$ of the incoming flux
--- the same branching that sets $\Omm = 1/\pi$ upstream.
Step 2: identifying the drift integral with $\ln\pi$, the
survival factor is $S_{A=7} = e^{-\ln\pi} = 1/\pi$
\eqref{eq:app-survival-pi}.  Step 3: applied to the SBBN
yield: $^7$Li/H $= 5.0\times10^{-10}/\pi = 1.59\times10^{-10}$
\eqref{eq:app-Li7-number}, against the Spite plateau's
$\approx 1.6\times10^{-10}$ --- H.5.1's mystery factor,
derived.  Physics intuition: the same geometric branching
ratio surfaces twice at wildly different epochs --- once
setting the matter fraction of the whole universe, once
pruning one isotope in the first twenty minutes --- and it is
exactly this reuse, with no per-instance freedom, that makes
the reading falsifiable rather than fitted.

\GGtakeaway{\emph{Standard result:} the Sakharov conditions and the
seesaw give a viable asymmetry, but the CP phases, masses, and the
mechanism's epoch are all model choices. \emph{What it assumed:} a
heavy-neutrino spectrum and its phases, chosen to work. \emph{What the
geometry replaced it with:} the phases and masses inherited from the
\ANine{} ledger, and the epoch selected by the derived
\(T_{\mathrm{rh}}\) --- thermal leptogenesis with zero dilution, no
fork. \emph{Continue in:} Part~V; Appendix~G for \(T_{\mathrm{rh}}\).}

\section{Linear perturbations and the growth of structure}
\label{appA:growth}

\GGteachbox{The complete linear theory of structure: the perturbed
metric and fluid, the growth equation and its solutions, the Meszaros
suppression, the transfer function, \(P(k)\), and \(\sigma_8\), all
derived with worked numbers; the observational \(S_8\) tension stated
carefully; and the geometric reading in which the two-stage corridor
acts on the growth history --- one structure for both tensions, the
sign honestly open.}
\GGreadalong{\citet{Dodelson2003} chs.~5--7 (this appendix follows its
line-by-line style); \citet{Peebles1993} for the classic treatment;
\citet{Peacock1999} at depth.}

The gauge-invariant linear theory of cosmological perturbations and the growth of structure assembled in this appendix are summarized in Figure~\ref{fig:appfig-growth}; the material in this appendix rests on the classic and review literature \citep{Lifshitz1946,Bardeen1980,KodamaSasaki1984,MukhanovFeldmanBrandenberger1992,PressSchechter1974,BBKS1986}.

\begin{figure}[htbp]
\centering
\resizebox{0.631\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,scale=1.0]
% axes: a from 0..1 mapped to x 0..10 ; D from 0..1 mapped to y 0..6
\def\xmax{10.6}\def\ymax{6.6}
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (\xmax,0);
\node[anchor=south east,font=\scriptsize] at (\xmax,0.08){scale factor $a$};
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (0,\ymax)
  node[above,anchor=south west,xshift=-3pt]{growth factor $D(a)$ (normalized)};
% x ticks: a = 0, 0.25, 0.5, 0.75, 1.0  -> x = 0,2.5,5,7.5,10
\foreach \a/\xx in {0/0, 0.25/2.5, 0.5/5, 0.75/7.5, 1.0/10}
  {\draw[line width=0.80pt] (\xx,0) -- (\xx,-0.10) node[below,font=\scriptsize]{\a};}
% y tick at D=1 (x-independent reference height = 6)
\draw[line width=0.80pt] (0,6) -- (-0.10,6) node[left,font=\scriptsize]{1};
% reference D = a (dashed): from (0,0) to (10,6)
\draw[densely dashed,gray!70!black,line width=0.80pt] (0,0) -- (10,6)
  node[pos=0.62,sloped,above,font=\scriptsize,gray!50!black]{$D=a$ (matter-only)};
% actual LambdaCDM growth: D~a early, suppressed late.
% Passes through origin, tracks D=a until a~0.5, then flattens to D(1)~0.78*1
\draw[line width=1.10pt,blue!65!black]
  (0,0)
  .. controls (1.5,0.92) and (3.0,1.85) ..
  (4.6,2.75)                     % still near D=a (a~0.46 -> Da=2.76)
  .. controls (6.0,3.45) and (7.2,3.95) ..
  (8.0,4.25)                     % begins to flatten
  .. controls (8.9,4.55) and (9.4,4.68) ..
  (10.0,4.78);                   % suppressed end: D(1)~0.80
\node[blue!60!black,font=\scriptsize,anchor=west] at (8.15,4.0)
  {$\Lambda$CDM $D(a)$};
% era boundary: Lambda onset a~0.75
\draw[densely dotted,red!70!black,line width=0.80pt] (7.5,0) -- (7.5,5.6);
\node[red!70!black,align=center,font=\scriptsize] at (7.5,6.05)
  {$\Lambda$ onset\\ $a\!\sim\!0.75$, $z\!\sim\!0.33$};
% era brace: matter domination (left) and Lambda domination (right)
\draw[decorate,decoration={brace,amplitude=4pt,mirror},line width=0.80pt,black!70]
  (0.15,-0.75) -- (7.4,-0.75)
  node[midway,below=3pt,font=\scriptsize,align=center]{matter domination\\ ($D\!\propto\!a$)};
\draw[decorate,decoration={brace,amplitude=4pt,mirror},line width=0.80pt,black!70]
  (7.6,-0.75) -- (10,-0.75)
  node[midway,below=3pt,font=\scriptsize,align=center]{$\Lambda$\\ domination};
% suppression gap arrow between dashed and solid at a=1
\draw[{Latex[length=1.6mm]}-{Latex[length=1.6mm]},red!60!black,line width=0.80pt]
  (10.0,4.78) -- (10.0,6.0);
\node[red!60!black,font=\scriptsize,anchor=west] at (10.05,5.35){late-time};
\node[red!60!black,font=\scriptsize,anchor=west] at (10.05,5.0){suppression};
% growth equation box
\node[draw=blue!55!black,fill=blue!3,rounded corners,inner sep=4pt,font=\scriptsize,
      anchor=north west] at (0.4,3.7)
  {$\ddot\delta + 2H\dot\delta - 4\pi G\,\rho_m\,\delta = 0$};
% observables annotation
\node[draw=black!55,fill=black!3,rounded corners,align=left,
      inner sep=4pt,font=\scriptsize,text width=4.7cm,anchor=north west]
      at (0.4,5.9)
  {Observables that normalize growth:\\[1pt]
   $\sigeight$ \ and \ $\Seight=\sigeight\sqrt{\Omm/0.3}$};
% honest GG note
\node[draw=red!70!black,fill=red!4,rounded corners,align=left,
      font=\scriptsize,inner sep=4pt,text width=8.4cm,anchor=north]
      at (5.3,-2.05)
  {\textbf{GG reading (status D*, sign caveat explicit):} the two-stage
   Hubble corridor modifies the growth history through the early-time
   Hubble branch. The \emph{sign} of the $\Seight$ response is flagged
   \textbf{OPEN} -- an honest unresolved item, not yet fixed by the ledger.};
\end{tikzpicture}}
\caption[Linear growth factor $D(a)$ and structure freeze-out.]{%
\textbf{Linear growth factor $D(a)$ versus scale factor $a$.}
In the matter-dominated era the normalized linear growth factor tracks
$D\propto a$ (it would follow the dashed $D=a$ reference exactly for a
matter-only universe). As dark energy comes to dominate past
$a\!\sim\!0.75$ ($z\!\sim\!0.33$) growth is \emph{suppressed} and
$D(a)$ flattens, opening the late-time gap below the dashed line. The
growth obeys $\ddot\delta+2H\dot\delta-4\pi G\rho_m\delta=0$. The
amplitude is fixed observationally by $\sigeight$ and
$\Seight=\sigeight\sqrt{\Omm/0.3}$.
\textbf{GG reading:} the two-stage Hubble corridor modifies the growth
history via the early-time Hubble branch; status~D* for
$\sigeight/\Seight$, but the \emph{sign} of the $\Seight$ response is
flagged OPEN -- an honest unresolved item.}
\label{fig:appfig-growth}
\end{figure}


This appendix builds linear cosmological perturbation theory and the
growth of structure from scratch, at the depth a first-year graduate
student can follow line by line. It is the natural sequel to the
\FRW{} background appendix: there we asked how the homogeneous,
isotropic universe expands; here we ask how the small departures from
homogeneity --- the seeds laid down in the early universe --- grow
under gravity into galaxies, clusters, and the cosmic web, and how
their statistics are compressed into the two numbers, \sigeight{} and
\Seight{}, that carry the present-day clustering tension. The
organizing principle is the one used throughout these technical
appendices: state the object precisely, carry out the full
computation without skipping the index work, anchor it in a worked
numerical example with units, and only then connect the result back
to the \GGTheory{} program.

One point of intellectual honesty governs the whole appendix and is
stated again, sharply, in the closing Status note. The \emph{standard}
linear growth theory developed in Sections
\ref{appA:growth-metric}--\ref{appA:growth-s8} is textbook general
relativity plus a perfect-fluid source; it is taught here in full and
is unmodified by the framework. The \GGTheory{} reading
(Section~\ref{appA:growth-gg}) modifies only the \emph{late-time}
growth history, through the two-stage corridor of the Hubble-tension
Part. There we will state, without softening it, that the \emph{sign}
of the resulting \Seight{} shift is genuinely \emph{open}: the
corridor parametrization assumes a \emph{suppression} of late growth
(which would help the tension), but in full early-dark-energy
likelihood fits the net \Seight{} shift typically comes out
\emph{positive} --- the opposite sign, which would \emph{worsen} the
tension \citep{Hill2020}. The reader who wants only the master growth
equation may read the boxed displays
\eqref{appA:growth-eq-growth-eta}, \eqref{appA:growth-eq-growth-a}
and \eqref{appA:growth-eq-Pk} and skip ahead; the reader who wants to
verify every step will find no gap papered over and no tension
declared resolved that is not.

% =====================================================================
\medskip
\textbf{The three steps of this appendix.} As in Appendix~E:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard linear theory (\S\ref{appA:growth-metric} onward).}
The perturbed metric and fluid, the sub-horizon growth equation, the growth
factor $D(a)$, the Meszaros suppression, the transfer function, and the
compression to $\sigeight$ and $\Seight$.
\item \textbf{What it leaves open (\S\ref{appA:growth-limits}).} The
background $H(a)$ and the densities are inputs, and the $S_8$ clustering
tension is a live disagreement that standard $\LCDM$ does not resolve.
\item \textbf{The geometric derivation (\S\ref{appA:growth-gg}).} The
two-stage corridor that reshapes $H(a)$ recasts the $S_8$ tension as two
projections of one structure --- with the honest caveat, kept sharp, that
the \emph{sign} of the effect is still open.
\end{enumerate}

\subsection{The perturbed FRW metric in conformal-Newtonian gauge}
\label{appA:growth-metric}
% =====================================================================

\paragraph{Background and conventions.}
We work in conformal time $\eta$, related to cosmic time $t$ by
$d t = a\, d\eta$, so that the spatially flat \FRW{} background line
element is
\begin{equation}\label{appA:growth-eq-bg}
d\bar s^2 = a^2(\eta)\,\big[-d\eta^2 + \delta_{ij}\,dx^i dx^j\big],
\end{equation}
with $a$ the scale factor, normalized $a_0 \equiv a(\eta_0) = 1$
today. A prime denotes a derivative with respect to conformal time,
$X' \equiv dX/d\eta$, and an overdot a derivative with respect to
cosmic time, $\dot X \equiv dX/dt = X'/a$. The conformal Hubble rate
is
\begin{equation}\label{appA:growth-eq-confH}
\mathcal{H} \equiv \frac{a'}{a} = a\,H,
\qquad
H \equiv \frac{\dot a}{a},
\end{equation}
so $\mathcal{H}$ has units of inverse conformal time and $H$ has
units of inverse cosmic time; the comoving Hubble radius is
$1/\mathcal{H}$. We use the metric signature $(-,+,+,+)$, Greek
indices $\mu,\nu \in \{0,1,2,3\}$, Latin indices
$i,j \in \{1,2,3\}$, and set $c = 1$ throughout (restoring $c$ in the
worked example).

\paragraph{The scalar perturbations and the gauge problem.}
At linear order the metric is split into a background plus a small
perturbation, $g_{\mu\nu} = \bar g_{\mu\nu} + \delta g_{\mu\nu}$, with
$|\delta g_{\mu\nu}| \ll |\bar g_{\mu\nu}|$. The perturbation
$\delta g_{\mu\nu}$ decomposes, by its transformation under spatial
rotations, into scalar, vector, and tensor parts (the
scalar--vector--tensor, or SVT, decomposition). To linear order these
three sectors \emph{decouple} and evolve independently. Vectors decay
in an expanding universe and tensors are the gravitational waves
treated in the inflationary appendix; the growth of \emph{matter}
structure is carried entirely by the \emph{scalar} sector, on which
we now focus.

The scalar sector has, in general, four functions
$(\phi, B, \psi, E)$. But two of them are pure gauge: an infinitesimal
coordinate change $x^\mu \to x^\mu + \xi^\mu$ shuffles physical and
coordinate content, and a naive choice can manufacture a fictitious
``growing mode'' that is nothing but a wiggle in the time slicing.
The clean way to proceed is to fix the gauge so that the two
remaining functions are physical. The choice that makes the Newtonian
limit transparent is the \emph{conformal-Newtonian} (or longitudinal)
gauge, $B = E = 0$, in which the perturbed line element reads
\begin{equation}\label{appA:growth-eq-metric}
\boxed{\;
d s^2 = a^2(\eta)\,\Big[
-\big(1 + 2\Psi\big)\,d\eta^2
+ \big(1 - 2\Phi\big)\,\delta_{ij}\,dx^i dx^j
\Big].
\;}
\end{equation}
Here $\Psi(\eta,\mathbf{x})$ and $\Phi(\eta,\mathbf{x})$ are the two
scalar potentials. $\Psi$ multiplies the time--time component and
plays the role of the Newtonian gravitational potential felt by a
slow test particle (it is what sets gravitational redshift and the
acceleration $\ddot{\mathbf{x}} = -\nabla\Psi/a^2$). $\Phi$ is the
perturbation to the spatial curvature; it is the potential that
governs the deflection of light, since photons respond to the sum
$\Phi+\Psi$. The sign conventions are chosen so that an overdensity
sits in a \emph{potential well} with $\Psi < 0$, $\Phi < 0$.

\paragraph{Why $\Phi = \Psi$ in the regime we need.}
The off-diagonal ($i \neq j$, $i,j$ spatial) Einstein equation, the
so-called anisotropic-stress constraint, reads at linear order
\begin{equation}\label{appA:growth-eq-aniso}
k^2\,(\Phi - \Psi) = -8\pi G\, a^2\,(\bar\rho + \bar p)\,\sigma,
\end{equation}
where $\sigma$ is the scalar anisotropic stress of the source and $k$
is the comoving wavenumber of the Fourier mode. For a perfect fluid
(pressureless matter, or radiation treated in the tight-coupling
limit) the anisotropic stress vanishes, $\sigma = 0$, and therefore
\begin{equation}\label{appA:growth-eq-PhiPsi}
\Phi = \Psi
\qquad\text{(no anisotropic stress).}
\end{equation}
The leading correction comes from free-streaming neutrinos, which
carry $\sigma \neq 0$ and split $\Phi$ and $\Psi$ at the few-percent
level after neutrino decoupling; we will note where that correction
enters and otherwise set $\Phi = \Psi$, writing a single Newtonian
potential where convenient. From here on we Fourier-transform every
field, $f(\eta,\mathbf{x}) = \int d^3k\,(2\pi)^{-3}\,
f_{\mathbf{k}}(\eta)\,e^{i\mathbf{k}\cdot\mathbf{x}}$, so that
$\nabla^2 \to -k^2$ and each comoving mode $\mathbf{k}$ evolves
independently at linear order. The label $\mathbf{k}$ is suppressed
unless needed.

% =====================================================================
\subsection{The perturbed fluid: continuity, Euler, Poisson}
\label{appA:growth-fluid}
% =====================================================================

We now write the conservation laws for the matter fluid in the
perturbed metric \eqref{appA:growth-eq-metric}. The matter is modeled
as a fluid with energy density $\rho(\eta,\mathbf{x}) =
\bar\rho(\eta)\,[1 + \delta(\eta,\mathbf{x})]$, where
\begin{equation}\label{appA:growth-eq-deltadef}
\delta \equiv \frac{\delta\rho}{\bar\rho}
= \frac{\rho - \bar\rho}{\bar\rho}
\end{equation}
is the dimensionless \emph{density contrast} --- the central object of
the whole subject --- and a peculiar velocity field
$\mathbf{v} = \mathbf{u}/a$ (the deviation of the fluid four-velocity
from the Hubble flow). Pressure is written $p = \bar p + \delta p$,
and for cold dark matter (CDM) we will set $\bar p = \delta p = 0$;
for radiation $\bar p = \bar\rho/3$.

\paragraph{Source of the equations: $\nabla_\mu T^{\mu\nu} = 0$.}
The two fluid equations are the time and space components of the
covariant conservation of the stress--energy tensor
$T^{\mu\nu} = (\rho+p)\,u^\mu u^\nu + p\,g^{\mu\nu}$, evaluated to
first order in the perturbed metric. The covariant divergence
$\nabla_\mu T^{\mu\nu} = \partial_\mu T^{\mu\nu} +
\Gamma^\mu_{\mu\lambda}T^{\lambda\nu} +
\Gamma^\nu_{\mu\lambda}T^{\mu\lambda} = 0$ requires the
perturbed Christoffel symbols of \eqref{appA:growth-eq-metric}; the
relevant ones are
\begin{align}
\Gamma^0_{00} &= \mathcal{H} + \Psi', &
\Gamma^0_{0i} &= \partial_i\Psi, \nonumber\\
\Gamma^0_{ij} &= \big[\mathcal{H} - 2\mathcal{H}(\Phi+\Psi)
- \Phi'\big]\delta_{ij}, &
\Gamma^i_{0j} &= \big[\mathcal{H} - \Phi'\big]\delta^i_{\,j},
\label{appA:growth-eq-christoffel}
\end{align}
with the others following by symmetry. The reader who carries these
through $\nabla_\mu T^{\mu 0}=0$ and $\nabla_\mu T^{\mu i}=0$ obtains
the perturbed continuity and Euler equations; we quote the result and
then re-derive it physically.

\paragraph{The perturbed continuity equation.}
The $\nu = 0$ component gives, for a general fluid,
\begin{equation}\label{appA:growth-eq-continuity-general}
\delta' + \big(1 + w\big)\big(\theta - 3\Phi'\big)
+ 3\mathcal{H}\Big(\frac{\delta p}{\delta\rho} - w\Big)\delta = 0,
\end{equation}
where $w \equiv \bar p/\bar\rho$ is the equation-of-state parameter
and $\theta \equiv \nabla\cdot\mathbf{v} = i\mathbf{k}\cdot
\mathbf{v}_{\mathbf{k}}$ is the (Fourier) velocity divergence. For
pressureless matter, $w = 0$ and $\delta p = 0$, this collapses to
the clean statement
\begin{equation}\label{appA:growth-eq-continuity}
\boxed{\;
\delta' = -\theta + 3\Phi'.
\;}
\end{equation}
The physics is mass conservation: the density contrast grows when
matter flows into a region ($\theta < 0$, convergent flow) and is
modulated by the time-changing volume element through $\Phi'$. In the
sub-horizon, quasi-static regime that dominates structure formation
the metric term $3\Phi'$ is negligible against $\theta$, so
$\delta' \simeq -\theta$: density contrast is sourced by velocity
convergence, exactly as in laboratory fluid dynamics.

\paragraph{The perturbed Euler equation.}
The $\nu = i$ component gives, for pressureless matter,
\begin{equation}\label{appA:growth-eq-euler}
\boxed{\;
\theta' + \mathcal{H}\,\theta = k^2\,\Psi.
\;}
\end{equation}
This is Newton's second law for the fluid. The term
$\mathcal{H}\theta$ is the Hubble drag: cosmic expansion redshifts
peculiar velocities, damping them as $v \propto 1/a$ in the absence
of forcing. The right-hand side $k^2\Psi$ is the gravitational force,
the Fourier transform of $-\nabla\Psi$ (the sign works out to
acceleration toward potential wells). For a fluid with pressure one
adds a term $-k^2\,\delta p/(\bar\rho + \bar p)$ on the right, the
pressure gradient that resists collapse and sets the Jeans scale;
for cold matter this term is absent, which is precisely why cold dark
matter clusters on all scales while baryons (with pressure support
before decoupling) do not.

\paragraph{The Poisson equation.}
The system is closed by relating $\Psi$ to $\delta$ through the
perturbed $00$ Einstein equation. To linear order this is the
relativistic generalization of the Poisson equation,
\begin{equation}\label{appA:growth-eq-poisson-rel}
k^2\,\Phi + 3\mathcal{H}\big(\Phi' + \mathcal{H}\Psi\big)
= -4\pi G\, a^2\,\bar\rho\,\delta.
\end{equation}
On sub-horizon scales, $k \gg \mathcal{H}$, the first term dominates
the bracket on the left (the bracket is order $\mathcal{H}^2\Phi$,
suppressed by $(\mathcal{H}/k)^2$), and the equation reduces to the
familiar Newtonian Poisson equation
\begin{equation}\label{appA:growth-eq-poisson}
\boxed{\;
k^2\,\Phi = -4\pi G\, a^2\,\bar\rho\,\delta,
\qquad\text{i.e.}\qquad
\nabla^2\Phi = 4\pi G\, a^2\,\bar\rho\,\delta
\;}
\end{equation}
(the factor $a^2$ converts the comoving Laplacian to physical units).
An overdensity, $\delta > 0$, gives $\Phi < 0$ and hence
$\Psi = \Phi < 0$ in this convention; through the Euler equation it
pulls matter inward, deepening the well --- the instability that
drives all of structure formation. We have, in
\eqref{appA:growth-eq-continuity}, \eqref{appA:growth-eq-euler} and
\eqref{appA:growth-eq-poisson}, a closed linear system for
$(\delta, \theta, \Phi)$.

% =====================================================================
\subsection{The sub-horizon growth equation}
\label{appA:growth-eqn}
% =====================================================================

We now combine the three boxed equations into a single second-order
equation for $\delta$. Work in the sub-horizon, quasi-static regime
($k \gg \mathcal{H}$, $\Phi' \ll \mathcal{H}\Phi$), in which
\eqref{appA:growth-eq-continuity} reads $\delta' = -\theta$.

\paragraph{Step 1: differentiate the continuity equation.}
Differentiating $\delta' = -\theta$ once more with respect to
conformal time,
\begin{equation}\label{appA:growth-eq-step1}
\delta'' = -\theta'.
\end{equation}

\paragraph{Step 2: eliminate $\theta'$ with the Euler equation.}
From \eqref{appA:growth-eq-euler}, $\theta' = -\mathcal{H}\theta +
k^2\Psi$. Substitute, then replace $\theta = -\delta'$ from
continuity:
\begin{equation}\label{appA:growth-eq-step2}
\delta'' = \mathcal{H}\theta - k^2\Psi
= -\mathcal{H}\,\delta' - k^2\Psi.
\end{equation}

\paragraph{Step 3: eliminate $\Psi$ with Poisson.}
On sub-horizon scales $\Psi = \Phi$ and
\eqref{appA:growth-eq-poisson} gives
$k^2\Psi = -4\pi G\,a^2\,\bar\rho\,\delta$. Hence
\begin{equation}\label{appA:growth-eq-growth-eta}
\boxed{\;
\delta'' + \mathcal{H}\,\delta'
- 4\pi G\, a^2\,\bar\rho_m\,\delta = 0.
\;}
\end{equation}
This is the growth equation in conformal time, with $\bar\rho_m$ the
mean \emph{matter} density (only non-relativistic, clustering matter
sources the well on these scales). It is worth pausing on its
structure: a harmonic-oscillator-like equation with a friction term
$\mathcal{H}\delta'$ (Hubble drag) and an \emph{anti-restoring} force
$-4\pi G a^2\bar\rho_m\,\delta$ --- the wrong sign for oscillation,
which is exactly why the solution grows rather than oscillates. There
is no $k$-dependence: in the matter-dominated, sub-horizon, cold
regime all modes grow at the \emph{same} rate, a scale-independence
that is the reason a single growth factor $D(a)$ can describe the
whole linear power spectrum.

\paragraph{Conversion to cosmic time.}
Many texts write the equation in cosmic time $t$ and against the
physical density. Using $\delta' = a\dot\delta$,
$\delta'' = a^2\ddot\delta + a\dot a\,\dot\delta =
a^2\ddot\delta + a^2 H\dot\delta$, and $\mathcal{H} = aH$, equation
\eqref{appA:growth-eq-growth-eta} becomes, after dividing by $a^2$,
\begin{equation}\label{appA:growth-eq-growth-t}
\boxed{\;
\ddot\delta + 2H\,\dot\delta - 4\pi G\,\bar\rho_m\,\delta = 0.
\;}
\end{equation}
This is the form quoted in the body of the paper. Note the friction
coefficient is now $2H$ (not $H$): the change from $\mathcal{H}$ to
$2H$ is an artifact of the time variable, since
$\mathcal{H}\delta' = aH\cdot a\dot\delta$ contributes one power of
$H$ and the cross term from $\delta''$ contributes the second.

% =====================================================================
\subsection{The growth factor \texorpdfstring{$D(a)$}{D(a)} and its
exact matter-dominated solutions}
\label{appA:growth-D}
% =====================================================================

Because \eqref{appA:growth-eq-growth-t} is linear and homogeneous and
its coefficients depend only on time, every Fourier mode shares the
same time dependence: $\delta(\eta,\mathbf{k}) =
D(a)\,\delta_0(\mathbf{k})$ up to the decaying mode, where $D(a)$ is
the \emph{linear growth factor} and $\delta_0(\mathbf{k})$ is the
initial spatial pattern. It is cleaner to use $a$ as the independent
variable. With $\dot\delta = a H\,(dD/da)\,\delta_0$ and the
Friedmann/continuity relations folded in through $dH/da$, equation
\eqref{appA:growth-eq-growth-t} becomes the standard growth ODE in
$a$,
\begin{equation}\label{appA:growth-eq-growth-a}
\boxed{\;
\frac{d^2 D}{da^2}
+ \left(\frac{3}{a} + \frac{1}{H}\frac{dH}{da}\right)\frac{dD}{da}
- \frac{3}{2}\,\frac{\Omm(a)}{a^2}\,D = 0,
\;}
\end{equation}
where $\Omm(a) \equiv 8\pi G\bar\rho_m/(3H^2)$ is the matter density
parameter at scale factor $a$, and we used
$4\pi G\bar\rho_m = \tfrac{3}{2}\Omm(a)\,H^2$. This is the equation
solved numerically in any Boltzmann/growth code.

\paragraph{Exact Einstein--de Sitter solution.}
In a flat, matter-only (Einstein--de Sitter, EdS) universe,
$\Omm(a) = 1$ and $H^2 \propto a^{-3}$, so
$H^{-1}dH/da = -\tfrac{3}{2}a^{-1}$ and the bracket in
\eqref{appA:growth-eq-growth-a} is
$(3/a) - (3/2a) = (3/2)a^{-1}$. The equation reduces to
\begin{equation}\label{appA:growth-eq-eds}
\frac{d^2 D}{da^2} + \frac{3}{2a}\frac{dD}{da}
- \frac{3}{2a^2}\,D = 0,
\end{equation}
an Euler equation solved by power laws $D \propto a^n$ with
$n(n-1) + \tfrac{3}{2}n - \tfrac{3}{2} = 0$, i.e.
$n^2 + \tfrac{1}{2}n - \tfrac{3}{2} = 0$, with roots $n = +1$ and
$n = -3/2$. Hence
\begin{equation}\label{appA:growth-eq-eds-modes}
D_+(a) \propto a
\qquad\text{(growing mode),}\qquad
D_-(a) \propto a^{-3/2}
\qquad\text{(decaying mode).}
\end{equation}
The growing mode $D_+ \propto a$ is the workhorse fact of structure
formation: \emph{in a matter-dominated universe the density contrast
grows linearly in the scale factor.} The decaying mode is the
transient that any sensible initial condition quickly forgets.

\paragraph{General $\Lambda$CDM growth: the integral form.}
When dark energy is present, $\Omm(a) < 1$ and the growth is slower
than $a$ at late times (dark energy accelerates the expansion,
strengthening the Hubble drag without sourcing the well). The growing
solution of \eqref{appA:growth-eq-growth-a} admits the closed
integral form
\begin{equation}\label{appA:growth-eq-Dintegral}
D(a) \;\propto\; H(a)\int_0^a \frac{d a'}{[\,a' H(a')\,]^3},
\qquad
H^2(a) = \Hzero^2\big[\Omm a^{-3} + \OmL\big]
\end{equation}
(flat $\LCDM$, radiation neglected at low $z$). It is conventional to
\emph{normalize} the growth factor so that $D = a$ in the
matter-dominated limit (early times); a common alternative
normalization sets $D(a_0) = D(1) = 1$ today. We will use the
growth-suppression factor $g(a) \equiv D(a)/a$, equal to unity in
EdS and falling below unity once dark energy bites.

\paragraph{Worked example: the growth factor today in $\LCDM$.}
Take flat $\LCDM$ with $\Omm = 0.315$, $\OmL = 0.685$. A compact and
accurate closed form for the suppression factor is the
Carroll--Press--Turner fit,
\begin{equation}\label{appA:growth-eq-cpt}
g(a{=}1) = \frac{D(1)/1}{D_{\rm EdS}}
\simeq \frac{5}{2}\Omm\Big[\Omm^{4/7} - \OmL
+ \big(1+\tfrac{1}{2}\Omm\big)\big(1+\tfrac{1}{70}\OmL\big)\Big]^{-1}.
\end{equation}
Evaluating with $\Omm = 0.315$: $\Omm^{4/7} = 0.315^{0.5714}
= \exp[0.5714\ln 0.315] = \exp[-0.660] = 0.517$; the bracket is
$0.517 - 0.685 + (1.1575)(1.00979) = 0.517 - 0.685 + 1.169 = 1.001$;
and $g(1) = 2.5\times 0.315 / 1.001 = 0.7875/1.001 \approx 0.787$.
Thus the growing mode at $a=1$ has been suppressed to about $79\%$ of
the value it would have reached in a matter-only universe with the
same early normalization: dark energy has slowed structure growth by
roughly a fifth since it came to dominate. (For orientation, the same
fit gives $g \approx 0.91$ at $z=1$, where $\Omm(a) = 0.77$ is still
close to unity --- growth was nearly unsuppressed at $z=1$ and has
been throttled mainly in the last few billion years.)

% =====================================================================
\subsection{The Meszaros effect: suppression during radiation
domination}
\label{appA:growth-meszaros}
% =====================================================================

Before matter--radiation equality ($a < a_{\rm eq}$, redshift
$z_{\rm eq} \approx 3400$) the universe is radiation-dominated, and a
crucial thing happens to sub-horizon matter perturbations: they
\emph{almost stop growing}. This is the \emph{Meszaros effect}, and
it sets the shape of the matter power spectrum on small scales.

\paragraph{Why growth stalls.}
The growth equation \eqref{appA:growth-eq-growth-eta} sources $\delta$
through the matter density $\bar\rho_m$. In the radiation era the
total energy budget, and hence the expansion rate
$\mathcal{H}$ that feeds the friction term, is dominated by
\emph{radiation}, while the gravitational source is dominated by
\emph{matter} only when $\bar\rho_m$ is appreciable. For a
sub-horizon CDM mode the radiation does not cluster (its pressure
support gives it a Jeans length of order the horizon, so it
free-streams and oscillates as acoustic waves), so it contributes to
the friction but not to the driving. The matter perturbation feels
strong Hubble drag from the fast radiation-driven expansion but only
weak self-gravity. The result is logarithmic, not power-law, growth.

\paragraph{The Meszaros equation and its solution.}
Introduce the variable $y \equiv a/a_{\rm eq} = \bar\rho_m/\bar\rho_r$,
the ratio of matter to radiation density. In terms of $y$ the
sub-horizon CDM growth equation \eqref{appA:growth-eq-growth-a},
with the two-component $H^2 \propto (1+y)/a^4$ expansion, becomes the
\emph{Meszaros equation}
\begin{equation}\label{appA:growth-eq-meszaros}
\frac{d^2\delta}{dy^2}
+ \frac{2 + 3y}{2y(1+y)}\frac{d\delta}{dy}
- \frac{3}{2y(1+y)}\,\delta = 0.
\end{equation}
This has the exact growing solution
\begin{equation}\label{appA:growth-eq-meszaros-sol}
\boxed{\;
\delta_+(y) \;\propto\; 1 + \tfrac{3}{2}\,y.
\;}
\end{equation}
One verifies by direct substitution that
\eqref{appA:growth-eq-meszaros-sol} solves
\eqref{appA:growth-eq-meszaros}. The lesson is in the two limits.
Deep in radiation domination, $y \ll 1$, the contrast is
$\delta_+ \to \text{const} + \tfrac{3}{2}y$: it grows by at most a
factor of order unity, i.e. it is \emph{frozen} up to a logarithm
once horizon-crossing effects are added. Deep in matter domination,
$y \gg 1$, $\delta_+ \to \tfrac{3}{2}y \propto a$, recovering the EdS
$D_+ \propto a$ law of \eqref{appA:growth-eq-eds-modes}. The total
suppression a small-scale mode suffers between horizon entry (in the
radiation era) and equality is therefore at most a few --- but it is
\emph{scale dependent}, because smaller-scale modes enter the horizon
earlier and stall for longer. This scale-dependent suppression is
exactly what bends the matter power spectrum from its primordial
slope to its observed turnover, and it is encoded in the transfer
function below.

% =====================================================================
\subsection{The growth index: \texorpdfstring{$f \approx
\Omm^{0.55}$}{f ~ Omega_m^0.55}}
\label{appA:growth-index}
% =====================================================================

A single number summarizes the instantaneous growth rate: the
\emph{linear growth rate}
\begin{equation}\label{appA:growth-eq-fdef}
f \;\equiv\; \frac{d\ln\delta}{d\ln a}
= \frac{d\ln D}{d\ln a}
= \frac{a}{D}\frac{dD}{da}.
\end{equation}
In EdS, $D \propto a$ gives $f = 1$ exactly. In $\LCDM$, $f$ falls
below unity at late times. A remarkably accurate fitting formula,
due to Peebles and refined by Wang--Steinhardt and Linder, expresses
$f$ as a power of the instantaneous matter fraction,
\begin{equation}\label{appA:growth-eq-fgamma}
\boxed{\;
f(a) \;\approx\; \Omm(a)^{\gamma},
\qquad
\gamma \approx 0.55
\;}
\end{equation}
for $\LCDM$ (more precisely $\gamma \simeq 0.55 + 0.05[1+w(z=1)]$ for
a dark-energy equation of state $w$; deviations from $\gamma = 0.55$
are a clean probe of modified gravity, where $\gamma$ can shift to
$\sim 0.68$). The combination actually measured by redshift-space
distortions is $f\sigma_8(z)$, since the survey sees the product of
the growth rate and the clustering amplitude; we return to
$\sigma_8$ below.

\paragraph{Worked check of the growth index today.}
At the present epoch, with the canonical Planck value
$\Omm = \Omm(a_0) \approx 0.315$,
\begin{equation}\label{appA:growth-eq-f0}
f_0 \approx (0.315)^{0.55}
= \exp[\,0.55\,\ln 0.315\,]
= \exp[\,0.55\times(-1.155)\,]
= \exp[-0.635] \approx 0.530.
\end{equation}
So structure grows today at a little over half the EdS rate: the
density contrast increases by about $0.53\%$ for each $1\%$ increase
in the scale factor. This is the standard $\LCDM$ value, confirmed by
redshift-space-distortion measurements of $f\sigma_8$ to the
ten-percent level.

% =====================================================================
\subsection{Transfer function and the matter power spectrum
\texorpdfstring{$P(k)$}{P(k)}}
\label{appA:growth-Pk}
% =====================================================================

\paragraph{From a single mode to a statistical field.}
The density field $\delta(\mathbf{x})$ is a random field; what we
predict and measure is its \emph{statistics}, not its realization.
For a Gaussian field, all information is in the two-point function,
whose Fourier transform is the \emph{power spectrum} $P(k)$ defined by
\begin{equation}\label{appA:growth-eq-Pkdef}
\big\langle \delta_{\mathbf{k}}\,\delta^*_{\mathbf{k}'}\big\rangle
= (2\pi)^3\,\delta_D^{(3)}(\mathbf{k}-\mathbf{k}')\,P(k),
\end{equation}
where $\delta_D^{(3)}$ is the Dirac delta enforcing statistical
homogeneity (so $P$ depends only on $k = |\mathbf{k}|$ by isotropy).

\paragraph{Primordial spectrum, transfer function, growth.}
Inflation lays down a nearly scale-invariant primordial curvature
spectrum, $P_{\mathcal{R}}(k) \propto k^{\,\ns - 1}$ with
$\ns \approx 0.965$ (the scalar tilt; see the inflationary appendix).
The late-time \emph{matter} power spectrum is this primordial
spectrum processed by two factors: the \emph{transfer function}
$T(k)$, which encodes everything that happens between horizon entry
and equality (most importantly the Meszaros suppression of
small-scale modes), and the growth factor $D(a)$, which carries the
linear evolution after equality. Schematically,
\begin{equation}\label{appA:growth-eq-Pk}
\boxed{\;
P(k,a) \;=\; A\, k^{\,\ns}\, T^2(k)\, D^2(a),
\;}
\end{equation}
where $A$ fixes the overall amplitude (tied to the primordial $A_s$).
The shape of $T(k)$ is the cause of the famous turnover in $P(k)$:
\begin{equation}\label{appA:growth-eq-Tk}
T(k) \;\to\;
\begin{cases}
1, & k \ll k_{\rm eq} \quad\text{(large scales, no suppression),}\\[2pt]
\big(k_{\rm eq}/k\big)^{2}\,\ln k, & k \gg k_{\rm eq}
\quad\text{(small scales, Meszaros-suppressed),}
\end{cases}
\end{equation}
with $k_{\rm eq} \sim 0.01\,h\,\mathrm{Mpc}^{-1}$ the comoving
wavenumber entering the horizon at equality. Thus $P(k) \propto
k^{\ns}$ rises on large scales and bends to
$P(k) \propto k^{\,\ns - 4}\ln^2 k$ on small scales, turning over near
$k_{\rm eq}$. Baryon acoustic oscillations (BAO) imprint a damped
oscillatory ripple on top of this smooth shape; they are the
frozen sound waves of the photon--baryon plasma and provide the
standard ruler used in the Hubble-tension Part.

% =====================================================================
\subsection{From $P(k)$ to \texorpdfstring{$\sigma_8$}{sigma_8} and
\texorpdfstring{$S_8$}{S_8}}
\label{appA:growth-s8}
% =====================================================================

\paragraph{The variance $\sigma_8$.}
To compress $P(k)$ into one number comparable across surveys, one
measures the variance of the density field smoothed on a fixed
physical scale. The standard choice is a top-hat sphere of comoving
radius $R = 8\,h^{-1}\,\mathrm{Mpc}$ (a scale near the transition to
non-linearity, historically chosen because the smoothed variance is
of order unity there). The smoothed variance is
\begin{equation}\label{appA:growth-eq-sigmaR}
\sigma_R^2 \;=\;
\int_0^\infty \frac{dk}{k}\,\frac{k^3 P(k)}{2\pi^2}\,
\big|W(kR)\big|^2,
\qquad
W(x) = \frac{3\,(\sin x - x\cos x)}{x^3},
\end{equation}
where $W(kR)$ is the Fourier transform of the real-space top-hat
window and $k^3 P(k)/2\pi^2 \equiv \Delta^2(k)$ is the dimensionless
power per logarithmic interval in $k$. Setting $R = 8\,h^{-1}\,$Mpc
defines
\begin{equation}\label{appA:growth-eq-sig8def}
\sigeight \;\equiv\; \sigma_{R = 8\,h^{-1}{\rm Mpc}}.
\end{equation}
Because $\sigeight \propto D(a_0)$ through
\eqref{appA:growth-eq-Pk}, it inherits the full late-time growth
history: anything that suppresses or enhances $D$ shifts $\sigeight$.

\paragraph{The combination $S_8$.}
Weak-lensing surveys do not measure $\sigeight$ and $\Omm$
separately; their two-point shear statistics constrain a degenerate
combination, well approximated by the rescaled amplitude
\begin{equation}\label{appA:growth-eq-S8def}
\boxed{\;
\Seight \;\equiv\; \sigeight\,\sqrt{\Omm/0.3}.
\;}
\end{equation}
The factor $\sqrt{\Omm/0.3}$ rotates onto the direction
\emph{orthogonal} to the lensing degeneracy, so $\Seight$ is the
quantity actually pinned down by surveys such as KiDS-1000 and DES
\citep{Asgari2021}. The ``\,$S_8$ tension\,'' is the statement that
lensing surveys prefer a \emph{lower} $\Seight$ (looser clustering,
$\sim 0.76$) than the value $\Seight \approx 0.83$ \emph{extrapolated
forward} from the Planck CMB under $\LCDM$ \citep{Planck2018} --- a
$2$--$3\sigma$ mismatch in the late-time growth amplitude.

\paragraph{Worked example: $\sigeight \to \Seight$ with $\Omm = 1/\pi$.}
The framework's conjectured matter fraction is the $\pi$-locked value
$\Omm = 1/\pi \approx 0.31831$ (carried as Derived-conditional in the
codebook). Take the Planck-anchored early-branch clustering amplitude
$\sigeight = 0.811$. Then
\begin{align}
\sqrt{\Omm/0.3}
&= \sqrt{(1/\pi)/0.3}
= \sqrt{0.31831/0.3}
= \sqrt{1.06103}
= 1.03007, \nonumber\\
\Seight
&= \sigeight\,\sqrt{\Omm/0.3}
= 0.811 \times 1.03007
= 0.8354.
\label{appA:growth-eq-s8worked}
\end{align}
So the $\pi$-locked matter fraction maps
$\sigeight = 0.811 \mapsto \Seight \approx 0.835$, essentially the
Planck-extrapolated value (the body quotes $\Seight \approx 0.832$
for $\Omm = 0.315$; the tiny difference, $0.835$ versus $0.832$, is
just the $1/\pi$ versus $0.315$ shift in $\Omm$, since
$\sqrt{0.31831/0.315} = 1.0052$ scales the amplitude up by half a
percent). This is the \emph{high} branch, the one in tension with
weak lensing; the corridor proposal aims to lower the \emph{late}
branch toward $\Seight^{\rm late} \approx 0.78$ --- and whether the
corridor actually moves it down or up is precisely the open sign
question below.

% =====================================================================
\subsection{What the standard growth theory leaves open: the $S_8$ tension}
\label{appA:growth-limits}

The linear theory of Step~1 is exact: given the background expansion
history $H(a)$ and the matter density, it predicts the growth factor
$D(a)$, the power spectrum, and the clustering amplitudes $\sigeight$ and
$\Seight$ without ambiguity. As in Appendix~E, it is worth naming what it
must take as given, and where it runs into trouble.

\begin{enumerate}[leftmargin=*]
\item \textbf{The background and the inventory.} The expansion history
$H(a)$ and the density $\Omm$ that drive the growth are measured inputs,
not derived.
\item \textbf{The $S_8$ tension.} The clustering amplitude
$\Seight=\sigeight\sqrt{\Omm/0.3}$ \emph{extrapolated} from the Planck CMB
($\Seight\approx0.83$) sits above the value \emph{measured} directly by
weak-lensing and galaxy surveys ($\Seight\approx0.78$) at the two-to-three
sigma level. Standard $\LCDM$ offers no structural account of the
discrepancy; it is a live tension, the growth-sector counterpart of the
Hubble tension.
\end{enumerate}

Standard growth theory, then, is an exact calculator of clustering once the
background is supplied --- but the background is an input, and the $S_8$
tension sits unexplained. The geometric account asks whether the same
structure that reframes the $H_0$ tension also reaches the growth
amplitude.

\subsection{The geometric derivation: the two-stage corridor and the
open sign of \texorpdfstring{$S_8$}{S_8}}
\label{appA:growth-gg}
% =====================================================================

The linear theory of Step~1 computes the growth of structure exactly,
once the background expansion history $H(a)$ is supplied. The geometric
content enters precisely there: the same two-stage corridor that reframes
the Hubble tension in Part~VI reshapes $H(a)$, and through the growth
equation \eqref{appA:growth-eq-growth-a} it reshapes the late-time
clustering --- so that the $S_8$ and $H_0$ tensions become two faces of
one structure. The Hubble-tension
Part of the body replaces the single late-time $\Hzero$ with a
\emph{two-stage corridor}: a brief, early, throat/$\Sigma_r$-sector source
correction (parametrized as an early-dark-energy-like lobe of
fractional height $f_{\rm EDE} \approx 0.03$ centered near
$z_c \sim 3500$, plus a small extra-radiation contribution
$\Delta N_{\rm eff} \approx 0.30$) that shifts the sound horizon and
the equality timing, followed by a late-time branch that the local
distance ladder reads as $\Hzero^{\rm late} \approx 73.7 \pm 0.7$
(re-anchored on this paper's early branch $67.4$; the companion
\citet{ArisakaC2Hubble} carries the same corridor off the slightly
lower anchor $67.24$).

\paragraph{How the corridor touches growth.}
A modified $H(a)$ enters the growth equation in two places: directly,
through the friction coefficient $H^{-1}dH/da$ in
\eqref{appA:growth-eq-growth-a}, and indirectly, through
$\Omm(a) = 8\pi G\bar\rho_m/3H^2$ in the source. The corridor
parametrization \emph{assumes} that the net effect of the early lobe
plus the late branch is to \emph{suppress} the late-time growth
factor $D(a)$ relative to the Planck-extrapolated $\LCDM$ value, so
that $\sigeight$ and hence $\Seight$ come down from the high
($\approx 0.835$) branch toward the weak-lensing-preferred
$\Seight^{\rm late} \approx 0.78 \pm 0.01$. Under that assumption the
$S_8$ tension would be reconciled by the \emph{same} corridor that
addresses the Hubble tension, with no new ingredient: the early
branch yields the high Planck amplitude, the late branch the lower
lensing amplitude, and the ``tension'' becomes two projections of one
structure. The response coefficients that convert the corridor
parameters $(f_{\rm EDE}, \Delta N_{\rm eff}, \Hzero^{\rm late})$
into a growth-suppression $\Delta\Seight$ are taken from the
early-dark-energy literature \citep{Poulin2019}; they are
\emph{not} derived here from the throat/$\Sigma_r$ sector.

\paragraph{The sign is OPEN --- stated without softening.}
Here is the honest difficulty, and the appendix states it as sharply
as the body does. The suppression \emph{branch} just described is an
\emph{assumption} of the parametrization, not a result. In full
likelihood analyses of early-dark-energy-type models, refitting the
CMB with the early lobe present drives the cold-dark-matter density
$\omega_{\rm cdm}$ and the scalar tilt $\ns$ \emph{upward} in order to
preserve the well-measured acoustic peaks; those upward shifts
\emph{increase} small-scale power, and the \emph{net} $\Seight$ shift
typically comes out \emph{positive} --- the \emph{opposite} sign to
the suppression encoded above \citep{Hill2020}. A positive shift does
not relieve the $S_8$ tension; it \emph{worsens} it. The corridor
therefore faces a genuine sign ambiguity:
\begin{itemize}
\item the corridor parametrization \emph{posits} a suppression,
$\Delta\Seight < 0$, which would help;
\item the leading EDE likelihood evidence \citep{Hill2020} finds
$\Delta\Seight > 0$, which would hurt;
\item which sign the throat/$\Sigma_r$ corridor actually delivers cannot
be decided from the parametrization alone --- it requires the
full growth-history integration, with the CMB refit done
self-consistently, that is named as the
corridor-uniqueness research target.
\end{itemize}
Until that integration is carried out, the $S_8$ corollary of the
corridor is \emph{conditional} and its sign is \emph{open}. We do not
claim the $S_8$ tension is resolved. We claim that the \emph{same}
geometric structure that reframes the Hubble tension \emph{also}
touches the growth amplitude, and that whether it helps or hurts is
an honest open question pinned to a declared calculation.

\paragraph{What would settle it.}
The resolution is conditional on the corridor-uniqueness target: a
self-consistent Boltzmann/likelihood fit (CMB + BAO + distance ladder
+ weak-lensing $f\sigma_8$) in which the throat/$\Sigma_r$ source
correction is propagated through recombination, equality, and
late-time growth in one pass, so that the sign and magnitude of
$\Delta\Seight$ are \emph{outputs} rather than inputs. The relevant
discriminating data are weak-lensing growth across redshift, cluster
counts, and CMB-lensing cross-checks --- the same observables that
sharpen the tension itself.

% =====================================================================
\subsection*{Status note}
\label{appA:growth-status}
% =====================================================================

\textbf{Status (standard linear growth): Derived.}
The full linear machinery of this appendix --- the
conformal-Newtonian perturbed metric
\eqref{appA:growth-eq-metric}, the perturbed continuity, Euler and
Poisson equations \eqref{appA:growth-eq-continuity},
\eqref{appA:growth-eq-euler}, \eqref{appA:growth-eq-poisson}, the
sub-horizon growth equation $\delta'' + \mathcal{H}\delta'
- 4\pi G a^2\bar\rho_m\delta = 0$ and its cosmic-time form
$\ddot\delta + 2H\dot\delta - 4\pi G\bar\rho_m\delta = 0$
\eqref{appA:growth-eq-growth-eta}--\eqref{appA:growth-eq-growth-t},
the growth factor $D(a)$ with its EdS solution $D_+ \propto a$
\eqref{appA:growth-eq-eds-modes} and $\LCDM$ integral
\eqref{appA:growth-eq-Dintegral}, the Meszaros suppression
$\delta_+ \propto 1 + \tfrac{3}{2}y$
\eqref{appA:growth-eq-meszaros-sol}, the growth index
$f \approx \Omm^{0.55}$ \eqref{appA:growth-eq-fgamma}, the transfer
function and power spectrum $P(k) = A k^{\ns}T^2(k)D^2(a)$
\eqref{appA:growth-eq-Pk}, and the variance/amplitude pair
$\sigeight$, $\Seight = \sigeight\sqrt{\Omm/0.3}$
\eqref{appA:growth-eq-sig8def}, \eqref{appA:growth-eq-S8def} --- is
\emph{Derived}: it is standard general relativity plus a
perfect-fluid source, taught here in full and \emph{unmodified} by
the framework. The boundary Einstein equation it rests on is itself a
theorem of the 6D-to-4D \OGGEP{} projection of
\citet{ArisakaA5GDOS}; \ATen{} inherits it and does not re-derive it.
The worked examples are fixed numbers: the $\LCDM$ growth suppression
$g(a{=}1) \approx 0.79$ \eqref{appA:growth-eq-cpt}, the growth index
$f_0 \approx (0.315)^{0.55} \approx 0.53$
\eqref{appA:growth-eq-f0}, and the $\pi$-locked mapping
$\sigeight = 0.811 \mapsto \Seight \approx 0.835$ at $\Omm = 1/\pi$
\eqref{appA:growth-eq-s8worked}.

\textbf{Status ($S_8$ corridor resolution):
Derived-conditional, sign OPEN.}
The \GGTheory{}-specific claim --- that the two-stage corridor of the
Hubble-tension Part also reconciles the $\sigeight/\Seight$ tension
by modifying the \emph{late-time} growth history --- is
\emph{Derived-conditional}, and its sign is explicitly flagged
\emph{open}. The corridor parametrization \emph{assumes} a growth
\emph{suppression} ($\Delta\Seight < 0$, which would help the
tension); but in full early-dark-energy likelihood fits the
self-consistent CMB refit drives $\omega_{\rm cdm}$ and $\ns$ upward
and the \emph{net} $\Seight$ shift typically comes out
\emph{positive} --- the \emph{opposite} sign, which would
\emph{worsen} the tension \citep{Hill2020}. The response coefficients
that convert corridor parameters into $\Delta\Seight$ are literature
values \citep{Poulin2019}, not yet derived from the throat/$\Sigma_r$
sector. The resolution is therefore \emph{conditional on the
corridor-uniqueness research target}: a self-consistent
CMB + BAO + distance-ladder + weak-lensing growth integration in
which the sign and magnitude of $\Delta\Seight$ emerge as outputs.
\textbf{We do not claim the $S_8$ tension is resolved; the sign is
open.}

% ===================================================================
% Appendix D9: The cosmic microwave background -- acoustic peaks and
% the angular power spectrum C_ell.
% Teaching appendix for Arisaka-GG-A10-Cosmos.
% Style model: appendix of Arisaka-GG-A8-QM-v12.0.tex.
% ASCII ONLY. Macros used: \Omb \Omm \OmL \Hzero \rs \LCDM \FRW
%   \ns \Neff \sigeight \Seight \GGTheory \GDOS (all provided by A10).
% Label keys prefixed appA:cmb-.
% ===================================================================

\subsection{Legacy growth response, cross-consistency Jacobian, and worked derivations (Part~VI)}
\label{app:part6-legacy-growth}

The material below is collected here rather than inline in Part~VI, so the body
stays readable, as was done for Parts~IV and~V. The \(S_8\)
suppression formula is the legacy early-dark-energy linear-response, superseded in
the body by the corridor reading of Corollary~\ref{cor:s8-reconciliation}; the
cross-consistency Jacobian and the worked derivation notebook give the
from-scratch calculations behind the Part~VI numbers.

\subsubsection{Legacy \texorpdfstring{\(S_8\)}{S8} growth-suppression linear-response}

The present framework acknowledges this standard difficulty rather than denying it. The narrow EDE-plus-dark-radiation pair is expected to suppress broad linear growth slightly. The question is whether that suppression lands in the right direction and whether the same framework provides a compensating or reshaping effect in the lensing sector and in the high-\(k\) minihalo sector.

A practical linear-response expression captures the intended behavior:
\begin{equation}
S_8 \simeq S_8^{\mathrm{ref}}\paren{1 - \alpha_S\Delta N_{\mathrm{eff}} - \beta_S f_{\mathrm{EDE}} + \gamma_S\alpha_L},
\label{eq:S8-response}
\end{equation}
with representative coefficients chosen to encode the sign structure,
\begin{equation}
\alpha_S\approx0.12,
\qquad
\beta_S\approx0.80,
\qquad
\gamma_S\approx0.20.
\label{eq:S8-coeffs}
\end{equation}
These numbers are not claimed to be the final output of a full likelihood pipeline. They are a compact summary of how the present mechanism is supposed to behave: extra radiation and early dark energy lower \(S_8\), while the mild enhancement of the effective lensing amplitude partly compensates.

A cautionary note is mandatory here: in full likelihood analyses of early-dark-energy-type models, refitting the CMB typically drives \(\omega_{\mathrm{cdm}}\) and \(\ns\) upward and the \emph{net} \(S_8\) shift comes out positive --- the opposite sign to the suppression encoded above \citep{Hill2020}. The present parametrization adopts the suppression branch as a target, and the sign question can only be settled by the growth-history integration declared in the corridor-uniqueness research target.

If one inserts the same inherited values used in the Hubble section,
\begin{equation}
\Delta N_{\mathrm{eff}}=0.30,
\qquad
f_{\mathrm{EDE}}=0.03,
\qquad
\alpha_L=0.05,
\end{equation}
then the total fractional shift is
\begin{equation}
\epsilon_{S_8}
\equiv
\alpha_S\Delta N_{\mathrm{eff}} + \beta_S f_{\mathrm{EDE}} - \gamma_S\alpha_L
\approx 0.036 + 0.024 - 0.010 = 0.050.
\end{equation}
Thus,
\begin{equation}
S_8 \approx 0.95\,S_8^{\mathrm{ref}}.
\end{equation}
Starting from a representative early-universe reference value in the low \(0.8\)'s then naturally lands one in a late-time corridor around
\begin{equation}
S_8 \approx 0.78\pm0.01,
\label{eq:S8-078}
\end{equation}
which is precisely the direction favored by the low-redshift lensing literature. The benchmark cosmic-shear determination, \(S_8 = 0.759^{+0.024}_{-0.021}\) from KiDS-1000 \citep{Asgari2021}, agrees with this corridor at the \(0.9\sigma\) level (a \(+2.8\%\) fractional deviation of the prediction from the measurement). In the parameter-audit Dev plot we therefore display two \(S_8\) entries, mirroring the two Hubble branches: the anchored early/CMB value \(\Seight^{\mathrm{CMB}}\) and the predicted late/lensing branch \(\Seight^{\mathrm{late}}\) (Figure~\ref{fig:a10-dev-plot-linear}, Table~\ref{tab:dev-plot-data}).



\subsection{Cross-consistency and falsifiability inside the growth block}
\label{sec:part6-cross}


\subsubsection{The minimal Part~VI parameter vector}

Within the present draft the entire growth-and-tension block is intentionally compressed to a minimal effective vector,
\begin{equation}
\mathbf p_{\mathrm{VI}} = \paren{f_{\mathrm{EDE}},\;\Delta N_{\mathrm{eff}},\;\alpha_L,\;\epsilon_{\mathrm{local}},\;\epsilon_{\mathrm{str}},\;\delta_c},
\label{eq:part6-pvec}
\end{equation}
where
\begin{align}
 f_{\mathrm{EDE}} &\approx 0.03, &
\Delta N_{\mathrm{eff}} &\approx 0.30, &
\alpha_L &\approx 0.05, \\
 \epsilon_{\mathrm{local}} &\sim 0.03\text{--}0.05, &
\epsilon_{\mathrm{str}} &\equiv 1 - \frac{v_{\mathrm{str}}}{v_{\mathrm{str}}^{\Lambda\mathrm{CDM}}} \approx 0.20, &
\delta_c &\approx 1.18.
\end{align}
Three comments are immediate.

First, \(f_{\mathrm{EDE}}\), \(\Delta N_{\mathrm{eff}}\), and \(\alpha_L\) are the early-to-middle-universe entries. They touch the CMB, the sound horizon, and broad linear growth. Second, \(\epsilon_{\mathrm{local}}\) is the late-time environmental entry relevant mainly for the local Hubble route. Third, \(\epsilon_{\mathrm{str}}\) and \(\delta_c\) are selective first-light entries rather than broad background ones. This separation is what prevents the framework from being an undifferentiated set of rescalings.

To first order, the observable vector
\begin{equation}
\mathbf o_{\mathrm{VI}} = \paren{\theta_*,\;H_0,\;S_8,\;\ell_D,\;A_L,\;n_{\mathrm{gal}},\;M_{\mathrm{seed}}}
\label{eq:part6-ovec}
\end{equation}
responds through a Jacobian,
\begin{equation}
\delta\mathbf o_{\mathrm{VI}} \approx \mathbf J_{\mathrm{VI}}\,\delta\mathbf p_{\mathrm{VI}}.
\label{eq:Jac-part6}
\end{equation}
The exact entries of \(\mathbf J_{\mathrm{VI}}\) require Boltzmann and hydrodynamical likelihood work and do not belong to the present draft. But the sign pattern does belong here, because it is the most compact way to state what the theory is and is not allowed to do.

\subsubsection{Sign matrix and one-ledger discipline}

The sign matrix may be written schematically as follows:
\begin{equation}
\mathbf J_{\mathrm{VI}} \sim
\begin{pmatrix}
 0 & 0 & 0 & 0 & 0 & 0 \\
 + & + & 0 & + & 0 & 0 \\
 - & - & + & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & + & 0 & 0 & 0 \\
 + & 0 & 0 & 0 & + & - \\
 0 & 0 & 0 & 0 & + & -
\end{pmatrix},
\label{eq:sign-matrix}
\end{equation}
where the rows correspond to \((\theta_*,\allowbreak H_0,\allowbreak S_8,\allowbreak \ell_D,\allowbreak A_L,\allowbreak n_{\mathrm{gal}},\allowbreak M_{\mathrm{seed}})\) and the columns to \((f_{\mathrm{EDE}},\allowbreak \Delta N_{\mathrm{eff}},\allowbreak \alpha_L,\allowbreak \epsilon_{\mathrm{local}},\allowbreak \epsilon_{\mathrm{str}},\allowbreak \delta_c)\). The matrix is schematic, not numerical, but it already captures the theory's architecture:
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item The acoustic angle is treated as observationally fixed and therefore has zero first-order freedom in the fit space; changes in the sound horizon are compensated elsewhere.
\item \(H_0\) is pushed upward by the early pair and by the local lift.
\item \(S_8\) is pushed downward by the early pair and slightly upward in the CMB-lensing interpretation by \(\alpha_L\).
\item The damping scale \(\ell_D\) is tied to the BFFT/\OGN{} ledger and is therefore not independently free in Part~VI.
\item The early-galaxy abundance is helped by the EDE small-scale boost, by streaming damping, and by a lowered collapse threshold.
\item The heavy-seed corridor is helped primarily by the same baryonic first-light pair \((\epsilon_{\mathrm{str}},\delta_c)\).
\end{enumerate}

This matrix is useful because it forbids a common failure mode in ambitious cosmological model building: letting the sign assignments drift from one anomaly to the next. Once the vector \(\mathbf p_{\mathrm{VI}}\) is fixed, each correction has a role. It is not allowed to be helpful for one tension only by becoming a new free knob elsewhere.

\subsubsection{Consistency inequalities that must hold simultaneously}

The previous sign matrix may be compressed into a set of inequality-style consistency conditions. The purpose of these inequalities is not to replace a full data analysis, but to state the shape of the allowed corridor.

\paragraph{Condition A: the CMB must remain near the observed acoustic solution.}
Because the observed \(\theta_*\) is extremely well determined, the combined early correction must preserve the acoustic fit,
\begin{equation}
\abs{\frac{\delta\theta_*}{\theta_*}} \ll 1.
\end{equation}
Operationally, this means that any shift in \(r_s\) must be compensated by the corresponding distance shift without spoiling the peak sequence or damping tail. The theory is not allowed to move \(H_0\) by destroying the primary CMB fit.

\paragraph{Condition B: the Hubble lift must be large enough to matter, but not so large that the source residue becomes implausible.}
If the corrected early-time shift is too small, the module does nothing. If it is too large, the required source residue will typically contradict the CMB or BAO. Thus one needs an intermediate corridor,
\begin{equation}
0.03 \lesssim \epsilon_{\mathrm{early}} \lesssim 0.07,
\label{eq:eps-early-window}
\end{equation}
which is precisely where the working values of Eq.~\eqref{eq:eps-h0} land.

\paragraph{Condition C: the broad growth suppression must be enough to lower \(S_8\), but not so large that the matter sector becomes under-clustered.}
One therefore wants
\begin{equation}
0.03 \lesssim \epsilon_{S_8} \lesssim 0.08,
\label{eq:eps-s8-window}
\end{equation}
which is again consistent with the working value \(\epsilon_{S_8}\approx0.05\).

\paragraph{Condition D: the first-light sector must be scale-selective.}
The same model cannot simply suppress or enhance power uniformly at all scales. Instead, one needs
\begin{equation}
\paren{\text{broad linear growth}} \neq \paren{\text{high-}\nu\text{ rare-halo response}},
\end{equation}
which is realized here through streaming damping and a changed \(\delta_c\), not through a global matter-power rescaling.

These conditions are not optional decorations. They are the mathematical form of the phrase ``one ledger.'' If a proposed correction fails one of these inequalities, it cannot be saved by shifting the burden to another section of the paper.

\subsubsection{A worked one-line closure check}

It is useful to perform a compact closure check with the working numbers of Part~VI. Using Eqs.~\eqref{eq:eps-h0} and \eqref{eq:S8-response}, one finds
\begin{align}
\epsilon_{\mathrm{early}} &\approx 0.051, \\
\epsilon_{S_8} &\approx 0.050.
\end{align}
Thus the same narrow source pair moves the acoustic-calibrated \(H_0\) and the broad growth amplitude by roughly five percent in opposite observational directions --- upward for \(H_0\), downward for \(S_8\). This is exactly the correlated sign pattern one wants.

The first-light pair then adds
\begin{align}
\frac{M_f^{\mathrm{new}}}{M_f^{\mathrm{old}}} &\sim 0.512, \\
\frac{\Delta n_{\mathrm{minihalo}}}{n_{\mathrm{minihalo}}} &\sim 0.4, \\
\frac{n_{\mathrm{massive}}^{\mathrm{\GG{}}}}{n_{\mathrm{massive}}^{\Lambda\mathrm{CDM}}} &\sim 10^2\text{--}10^3,
\end{align}
which is more than enough to make the earliest bright objects much less surprising. The crucial point is that the numbers line up directionally without introducing a second independent dark-matter model or a second independent inflationary module.

\subsubsection{What a full numerical pipeline would have to do next}

A serious continuation of the present draft would need to replace the symbolic Jacobian by an actual numerical pipeline. That pipeline has three main stages.
\begin{enumerate}[leftmargin=2.0em,label=\arabic*.]
\item \textbf{Boltzmann stage:} implement the EDE residue, the \(\Delta N_{\mathrm{eff}}\) shift, and the lensing correction in a precision perturbation solver and fit the full \(TT\), \(TE\), \(EE\), and lensing data products.
\item \textbf{Background-and-distance stage:} propagate the corrected early solution through BAO and distance-ladder inference to test whether the Hubble corridor survives outside the CMB alone.
\item \textbf{Hydrodynamical first-light stage:} insert the small-scale transfer correction, streaming damping, and altered collapse threshold into first-galaxy and direct-collapse-seed simulations.
\end{enumerate}
The present paper does not perform those computations in full. What it does do is define exactly what must be computed and exactly which correlations should appear if the framework is right.

\subsubsection{The kill table for Part~VI as a whole}

It is useful to place the internal failure logic into one table.

\begin{table}[H]
\centering
\caption[Kill conditions for Part VI as an integrated block]{Kill conditions for Part~VI as an integrated block. The philosophy is simple: one cannot claim a unified cosmological success if the sign structure works in one observable but fails catastrophically in the others. The columns give the block, what must happen, and what kills it. The philosophy is simple and worth stating: a unified account cannot claim the successes of its parts jointly and then survive their failures separately, so the conditions are written against the block as a whole.}
\label{tab:part6-kill}
\small
\begin{tabular}{p{0.25\textwidth}p{0.28\textwidth}p{0.35\textwidth}}
\toprule
\textbf{Block} & \textbf{What must happen} & \textbf{What kills the block}\\
\midrule
Primary CMB & acoustic scale, peak pattern, damping tail, and polarization stay viable under the narrow inherited ledger & the needed early residue ruins the CMB fit or demands a radically different damping or tensor structure\\
Hubble tension & early sound-horizon reduction plus mild local lift moves \(H_0\) into the observational corridor & required \(f_{\mathrm{EDE}}\), \(\Delta N_{\mathrm{eff}}\), or local lift become observationally or physically implausible\\
\(S_8\) tension & same early residue lowers broad growth while the lensing corridor remains acceptable & Hubble fix worsens growth too much, or \(S_8\) can be moved only by opening a new unrelated sector\\
Early galaxies & selective first-light corrections advance rare collapse and heavy-seed formation without breaking broad data & required minihalo or collapse changes are incompatible with the CMB or vanish under realistic astrophysical modeling\\
Integrated claim & one ledger explains the whole pattern & any success relies on mutually contradictory sign assignments or parameter values across the four blocks\\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{Why this section matters scientifically}

A theory becomes more persuasive when it converts broad ambitions into a finite matrix of coupled tests. That is exactly what the cross-consistency section does. It turns the qualitative slogan of \ATen{} into a concrete statement: if the same inherited vector cannot move \(\theta_*\), \(H_0\), \(S_8\), and \(n_{\mathrm{gal}}\) in the correct correlated pattern, then the cosmological extension of the framework fails.

This is the right form of strictness for a paper that aims to be longer than a conventional letter. The point of a long paper is not to accumulate words. It is to show the reader where the theory can lose.

\subsection{Worked derivation notebook for the Part~VI observables}
\label{sec:part6-derivations}


\subsubsection{Derivation notebook A: how a narrow early residue shortens the sound horizon}

Start with the sound-horizon integral in scale-factor form,
\begin{equation}
 r_s(a_*) = \int_0^{a_*} \frac{c_s(a)}{a^2 H(a)}\,\dd a.
\label{eq:rs-a}
\end{equation}
Write
\begin{equation}
H(a) = H_{\mathrm{ref}}(a)\sqrt{1+\delta(a)},
\qquad
\delta(a)\ll1.
\end{equation}
Then
\begin{equation}
\frac{1}{H(a)} = \frac{1}{H_{\mathrm{ref}}(a)}\paren{1+\delta(a)}^{-1/2}
\approx \frac{1}{H_{\mathrm{ref}}(a)}\paren{1-\frac12\delta(a)}.
\end{equation}
Substituting into Eq.~\eqref{eq:rs-a} gives
\begin{equation}
 r_s(a_*) \approx r_s^{\mathrm{ref}}(a_*) - \frac12\int_0^{a_*} \frac{c_s(a)}{a^2 H_{\mathrm{ref}}(a)}\,\delta(a)\,\dd a.
\end{equation}
Dividing by \(r_s^{\mathrm{ref}}\) yields the weighted fractional shift,
\begin{equation}
\frac{\delta r_s}{r_s} \approx -\frac12\frac{\int_0^{a_*} W_s(a)\,\delta(a)\,\dd a}{\int_0^{a_*} W_s(a)\,\dd a},
\qquad
W_s(a)\equiv \frac{c_s(a)}{a^2H_{\mathrm{ref}}(a)}.
\end{equation}
Now separate
\begin{equation}
\delta(a)=\delta_{\mathrm{EDE}}(a)+\delta_{\mathrm{dr}}(a).
\end{equation}
If the EDE piece is narrow and centered where \(W_s(a)\) is important, then its contribution is approximately its peak height times an order-unity weight coefficient. If the dark-radiation piece is broader, its contribution is its amplitude times a smaller order-unity coefficient. This is the origin of the working linear expression used earlier:
\begin{equation}
-\frac{\delta r_s}{r_s} \sim 0.13\,\Delta N_{\mathrm{eff}} + 0.40\,f_{\mathrm{EDE}}.
\end{equation}
Combining with Eq.~\eqref{eq:h0-rs} then gives the Hubble shift formula. The important pedagogical point is that nothing miraculous is happening. The entire effect is a consequence of the inverse weighting of \(H(a)\) in the sound-horizon integral.

\subsubsection{Derivation notebook B: why a five-percent broad growth shift is natural}

Take the linear growth equation in the scale-factor form,
\begin{equation}
D'' + A(a)D' - B(a)D = 0,
\end{equation}
with
\begin{equation}
A(a) = \frac{3}{a} + \frac{\dd\ln H}{\dd a},
\qquad
B(a)=\frac{3}{2}\frac{\Omega_{m,0}H_0^2}{a^5H^2(a)}.
\end{equation}
Suppose the early correction changes \(H\to H_{\mathrm{ref}}(1+\epsilon_H)\) around the epochs most relevant for the later normalization of \(D\). Then to first order,
\begin{equation}
A(a) \to A_{\mathrm{ref}}(a) + \delta A(a),
\qquad
B(a) \to B_{\mathrm{ref}}(a)\paren{1-2\epsilon_H(a)}.
\end{equation}
Thus the friction term is increased and the source term is reduced. The solution is therefore slightly smaller than the reference solution,
\begin{equation}
D(a) = D_{\mathrm{ref}}(a)\paren{1-\epsilon_D(a)},
\qquad
\epsilon_D(a)>0.
\end{equation}
Because \(S_8\propto D(1)\) to first approximation, a few-percent suppression in the growth factor is exactly what is expected from a few-percent enhancement in the early expansion rate. This is why the same source residue that helps \(H_0\) almost inevitably helps lower \(S_8\), and also why the problem is to keep the suppression from becoming too large.

\subsubsection{Derivation notebook C: exponential amplification in the halo tail}

The rare-tail amplification is so important for the early-galaxy discussion that it deserves its own short derivation. Starting from Eq.~\eqref{eq:nu-def},
\begin{equation}
\ln n \sim -\frac12\nu^2 = -\frac12\frac{\delta_c^2}{\sigma^2}.
\end{equation}
Varying this expression gives
\begin{equation}
\delta\ln n = -\frac{\delta_c}{\sigma^2}\,\delta\delta_c + \frac{\delta_c^2}{\sigma^3}\,\delta\sigma.
\label{eq:dlogn}
\end{equation}
Rewriting in terms of \(\nu\) gives
\begin{equation}
\delta\ln n = -\nu^2\frac{\delta\delta_c}{\delta_c} + \nu^2\frac{\delta\sigma}{\sigma}.
\end{equation}
For \(\nu\sim5\), the coefficient is \(25\). This is why a tiny percent-level shift in variance or collapse threshold produces a large abundance response. The mathematics is elementary, but the physical consequence is profound: the earliest structures are high-leverage probes of seemingly modest background or transport changes.

\subsubsection{Derivation notebook D: how streaming damping lowers the filtering mass}

The filtering mass is not uniquely defined in every analytic treatment, but the heuristic dependence on the relative streaming velocity is enough for present purposes. Let
\begin{equation}
M_f \propto v_{\mathrm{str}}^p,
\end{equation}
with \(p\approx3\) as a standard order-of-magnitude estimate. Then the transformation in Eq.~\eqref{eq:vstream} gives
\begin{equation}
\frac{M_f^{\mathrm{new}}}{M_f^{\mathrm{old}}} = \paren{\frac{v_{\mathrm{str}}^{\mathrm{new}}}{v_{\mathrm{str}}^{\mathrm{old}}}}^p = 0.8^3 \approx 0.512.
\end{equation}
Now imagine a halo population whose gas-cooling success probability rises rapidly once the halo mass exceeds \(M_f\). Cutting \(M_f\) roughly in half makes a substantial fraction of previously marginal halos suddenly eligible for baryon capture and molecular cooling. That is why the streaming correction is so selective and so powerful for the onset of first light.

\subsubsection{Derivation notebook E: freeze-out of the acoustic phase at recombination}

The acoustic oscillator for a given mode \(k\) may be written schematically as
\begin{equation}
\ddot\delta_{\gamma b,k} + \omega_k^2(t)\,\delta_{\gamma b,k} = S_k(t),
\label{eq:oscillator-mode}
\end{equation}
where \(\omega_k\sim c_s k/a\) and \(S_k\) is the gravitational driving source. While Thomson scattering is rapid, the oscillator remains part of a coupled driven system. When \(\Gamma_T/H\) falls below unity, the forcing and coupling structure changes abruptly. The mode is effectively released into free streaming at whatever phase it has at decoupling.

This is the technical content behind the earlier \GSSB-to-\GA language. The phase is not continuously reprocessed after last scattering; it is frozen. Odd peaks correspond to modes caught near maximum compression, even peaks to modes caught near rarefaction. In Fourier language, the observed acoustic pattern is a fossil of the decoupling-time phase map.

\subsubsection{Derivation notebook F: the tensor corridor is small but nonzero}

In slow-roll inflation the tensor amplitude scales with the inflationary energy scale, \(r \propto (H_{\mathrm{inf}}/M_{\mathrm{Pl}})^2\). The two-basin derivation of Part~IV forces the inflationary module onto the \GDOS{}-selected Starobinsky/\(\xi\) plateau one e-fold below the GUT edge, \(V^{1/4} = 8.3\times10^{15}\)~GeV (the earlier lock-scale reading, which drove \(r\) to essentially zero, is retired --- it gave a blue tilt and is refuted in Part~IX). On the derived plateau the relation is not schematic but fixed by the reheat-matched e-fold count \(N^{*}=52.6\):
\begin{equation}
 r = \frac{12}{N^{*2}} = 0.0043,
\end{equation}
the Starobinsky value at the derived count. The corridor is therefore \emph{small but nonzero} --- a factor of eight below the present Planck+BICEP bound \(r<0.036\), and a definite two-sided target: a detection well above \(0.0043\), or a null pushing \(r\) below \(\sim10^{-3}\), refutes the derived launch. This is one of the cleanest kill tests available to the framework precisely because it is a value, not a bound.

\subsubsection{Derivation notebook G: the difference between broad \texorpdfstring{\(S_8\)}{S8} scales and first-light scales}

A final clarification is worth stating mathematically. The lensing and \(S_8\) observables are weighted integrals over the power spectrum near broad quasi-linear scales, schematically,
\begin{equation}
S_8^2 \sim \int \mathcal W_{S_8}(k)\,P(k)\,\dd\ln k,
\end{equation}
with a broad window \(\mathcal W_{S_8}(k)\). Early first-light observables, by contrast, depend on the far tail of the halo abundance,
\begin{equation}
 n_{\mathrm{first\;light}} \sim \int_{M_{\mathrm{cool}}}^{\infty} \mathcal F\brak{\frac{\delta_c}{\sigma(M,z)}}\,\dd M,
\end{equation}
with highly non-linear sensitivity to the cooling threshold \(M_{\mathrm{cool}}\), the streaming-limited filtering mass, and the rare-tail value of \(\sigma(M,z)\). The two classes of observables are therefore related, but they are not identical filters on the same piece of the transfer function. This is the mathematical reason Part~VI can coherently propose broad linear suppression together with earlier first light.

\subsubsection{Final notebook takeaway}

The purpose of the worked notebook is to demystify the Part~VI claims. None of the amplifications invoked in this section are obscure. Shortening the sound horizon comes from increasing \(H\) in the integrand. Lowering \(S_8\) comes from the same increase in \(H\) inside the growth equation. Boosting the earliest halos comes from the exponential sensitivity of the rare tail and from a lower baryonic filtering mass. Suppressing primordial tensors comes from lowering the inflationary scale. The real scientific question is therefore not whether these effects can exist. It is whether the same narrow inherited ledger can produce the required magnitudes simultaneously without breaking the primary CMB or other precision tests.

That question is exactly why Part~VI is worth more than a short discussion section. It is the place where the cosmology paper begins to earn or lose its right to speak of unification.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{J.2.1. Show from the perturbed fluid
system why cold matter grows while the photon--baryon fluid
oscillates.}

\noindent\emph{Solution.}  Step 1: combining continuity, Euler,
and Poisson for a single fluid gives the sub-horizon master
equation $\ddot\delta + 2H\dot\delta + \bigl(c_s^2k^2/a^2 -
4\pi G\bar\rho\bigr)\delta = 0$: a damped oscillator whose
``spring constant'' is pressure minus gravity.  Step 2: for cold
matter $c_s \approx 0$, the bracket is negative at every $k$ of
interest --- an inverted spring --- so perturbations grow at all
scales.  Step 3: for the photon--baryon fluid $c_s \approx
c/\sqrt3$, the bracket is positive for every sub-horizon $k$ ---
a stiff spring --- so the same equation rings instead of
growing: the acoustic oscillations of
Appendix~\ref{appA:cmb}.  Physics intuition: one equation, two
destinies, decided by a sound speed --- the universe's structure
grew in the component the photons could not hold rigid, which
is the deepest reason cosmology \emph{needs} dark matter: the
baryons were musical when they should have been falling.

\medskip\noindent\textbf{J.3.1. Solve the matter-era growth
equation exactly.}

\noindent\emph{Solution.}  Step 1: in matter domination
($a \propto t^{2/3}$, $H = 2/3t$, $4\pi G\bar\rho =
\tfrac{2}{3t^2}$) the master equation becomes
$\ddot\delta + \tfrac{4}{3t}\dot\delta - \tfrac{2}{3t^2}\delta
= 0$.  Step 2: try $\delta \propto t^{n}$:
$n(n-1) + \tfrac43 n - \tfrac23 = 0$, i.e.\
$3n^2 + n - 2 = 0$, with roots $n = \tfrac23$ and $n = -1$.
Step 3: the growing mode is $\delta \propto t^{2/3} \propto a$
--- structure grows in lockstep with the scale factor --- and
the decaying mode $\propto t^{-1}$ is quickly forgotten.
Physics intuition: growth exactly proportional to $a$ means the
universe assembles structure no faster than it expands ---
gravity in an expanding background is a patient builder, and
the $10^{-5}$ seeds of the CMB need every one of the
$\sim\!10^{3}$ growth factors between recombination and today
to become galaxies: the arithmetic that makes dark matter's
head start (growing from $z_{\rm eq}$, not $z_*$)
indispensable.

\medskip\noindent\textbf{J.4.1. Compute the present growth rate
for the $\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: at late times, $\Lambda$
suppresses growth; the standard parametrization of the growth
rate is $f \equiv d\ln D/d\ln a \approx \Omega_m(a)^{0.55}$.
Step 2: today, with $\Omm = 1/\pi$:
$f_0 = (0.3183)^{0.55} = 0.53$.  Step 3: this is the number
redshift-space-distortion surveys measure (as $f\sigma_8$), and
it is a zero-parameter output of the inventory --- growth today
runs at fifty-three percent of the expansion's pace, against
$f = 1$ in the matter era (J.3.1).  Physics intuition: the
vacuum does not tear structure apart; it simply outpaces it ---
$f < 1$ says the cosmic web is coasting on momentum accumulated
during matter domination, and the measured deceleration of
assembly is one more reading of $\Omega_\Lambda = 1 - 1/\pi$.

\medskip\noindent\textbf{J.5.1. Explain the turnover of the
matter power spectrum from the Meszaros suppression.}

\noindent\emph{Solution.}  Step 1: a mode entering the horizon
during radiation domination stalls (the Meszaros effect: growth
only logarithmic while radiation drives the expansion); a mode
entering after equality grows $\propto a$ immediately.
Step 2: the boundary is the horizon size at equality,
$k_{\rm eq} = a_{\rm eq}H_{\rm eq}$ --- modes with $k >
k_{\rm eq}$ carry a suppression that deepens with $k$ (longer
stall), modes with $k < k_{\rm eq}$ do not.  Step 3: hence
$P(k)$ rises ($\propto k^{n_s}$) up to $k_{\rm eq}$ and falls
($\propto k^{n_s-4}$, logarithm aside) beyond it: the observed
turnover \emph{is} the imprint of $z_{\rm eq}$, and since
F.6.1 derives $z_{\rm eq}$ from $\Omm = 1/\pi$, the turnover
scale is inventory physics.  Physics intuition: the power
spectrum is a tape recording of who entered the horizon during
whose reign --- its single broad maximum is the moment the
universe changed government from radiation to matter.

\medskip\noindent\textbf{J.6.1. Locate the BAO wiggles in the
power spectrum.}

\noindent\emph{Solution.}  Step 1: the sound horizon frozen
into the baryons at decoupling is $r_s \approx 147$~Mpc
(comoving) --- the same length Appendix~\ref{appA:cmb} projects
onto the CMB sky.  Step 2: in $P(k)$ it appears as an
oscillation with spacing $\Delta k = 2\pi/r_s \approx
0.043~\mathrm{Mpc^{-1}}$, damped at high $k$.  Step 3: galaxy
surveys read the same ruler at $z \sim 0.5$ that the CMB reads
at $z \approx 1090$ --- one standard length, two epochs, and
the comparison is exactly what anchors the corridor discussion
of the Hubble-tension Part: any modification of early physics
that shortens $r_s$ moves \emph{both} readings together.
Physics intuition: the universe shipped with one meter-stick;
BAO is finding it in the pattern of galaxies eight billion
years after the CMB photographed it in light.

\medskip\noindent\textbf{J.7.1. Convert $\sigma_8$ to $S_8$ for
the $\pi$-locked inventory.}

\noindent\emph{Solution.}  Step 1: the lensing-survey
combination is $S_8 \equiv \sigma_8\sqrt{\Omm/0.3}$, built so
that weak-lensing amplitude is $\approx$ constant along its
degeneracy direction.  Step 2: with $\Omm = 1/\pi$:
$\sqrt{0.3183/0.3} = 1.030$, so $S_8 = 1.030\,\sigma_8$ ---
three percent above $\sigma_8$ itself.  Step 3: this is the
conversion through which the chapter's $S_8$ discussion (and
its honestly-open sign for the corridor's late-time shift,
Sections J.8--J.10) must be read: the inventory fixes the
$\Omm$ factor exactly; the tension lives entirely in the
growth amplitude.  Physics intuition: $S_8$ was engineered by
observers so that one number summarizes lensing; the
inventory's contribution to it is pure geometry, which
cleanly separates the settled factor from the contested one.

\GGtakeaway{\emph{Standard result:} linear growth from primordial
seeds reproduces the observed large-scale structure, with \(\sigma_8\)
and \(S_8\) as its summary statistics --- and a persistent few-percent
tension between the CMB-inferred and lensing-measured values.
\emph{What it assumed:} the background inventory, the seed spectrum,
and independent late-time physics if the tension is real. \emph{What
the geometry replaced it with:} the inventory and tilt derived
(Appendices~G, L), and the corridor's growth response addressing the
\(H_0\) and \(S_8\) splits as one object --- falsifiable jointly.
\emph{Continue in:} Part~VI; Rubin/Euclid rows of the decade table.}

\section{The cosmic microwave background -- acoustic peaks
and the angular power spectrum}\label{appA:cmb}

\GGteachbox{The physics of the microwave sky, from scratch: the
photon-baryon oscillator, the sound horizon, the acoustic angle, the
peak series, baryon loading, Silk damping, and how the six \(\LCDM\)
parameters are jointly fit to the spectrum; then the
demonstration --- the same spectrum computed from the \emph{derived}
parameters (\(H_0 = 66.6\), the \(\pi\)-locked fractions,
\(n_s = 0.9620\)) with one inherited amplitude --- and the honest
boundary: what the geometry feeds the Boltzmann instrument, and what it
does not compute.}
\GGreadalong{\citet{Durrer2020} (the dedicated monograph);
\citet{Dodelson2003} ch.~8; \citet{Baumann2022} ch.~7.}

The acoustic-oscillation physics of the CMB (Figure~\ref{fig:appfig-cmb}) --- the Sachs--Wolfe plateau, the sound horizon, baryon loading, the peak series, and Silk damping --- assembled in this appendix follows the classic and analytic references \citep{SachsWolfe1967,SunyaevZeldovich1970,PeeblesYu1970,Silk1968,HuSugiyama1995,SmootCOBE1992}.

\begin{figure}[htbp]
\centering
\resizebox{0.654\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,scale=1.0]
\def\xmax{12.2}\def\ymax{6.2}
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (\xmax,0);
\node[anchor=north] at (6.0,-0.55){multipole moment $\ell$};
\draw[-{Latex[length=2mm]},line width=0.80pt] (0,0) -- (0,\ymax)
  node[above,anchor=south west,xshift=-2pt]{$\mathcal{D}_\ell=\dfrac{\ell(\ell+1)C_\ell}{2\pi}$};
\foreach \x/\lab in {0.6/2, 3.0/220, 5.0/540, 6.6/810, 8.5/1350, 11.2/2500}
  {\draw[line width=0.80pt] (\x,0) -- (\x,-0.10) node[below,font=\scriptsize]{\lab};}
\draw[line width=1.10pt,blue!65!black]
  (0.6,0.55)
  .. controls (1.6,0.55) and (2.3,0.7) ..
  (3.0,5.5)
  .. controls (3.7,5.4) and (3.9,2.0) ..
  (4.2,1.6)
  .. controls (4.6,1.5) and (4.8,3.6) ..
  (5.0,3.9)
  .. controls (5.4,3.7) and (5.8,1.7) ..
  (6.0,1.55)
  .. controls (6.3,1.5) and (6.5,2.85) ..
  (6.6,2.95)
  .. controls (6.9,2.85) and (7.2,1.6) ..
  (7.4,1.45)
  .. controls (7.7,1.45) and (7.85,2.0) ..
  (8.0,2.05)
  .. controls (8.5,1.9) and (9.2,1.0) ..
  (10.2,0.45)
  .. controls (10.8,0.30) and (11.0,0.22) ..
  (11.5,0.18);
\draw[densely dashed,gray] (3.0,0) -- (3.0,5.5);
\node[blue!55!black,align=center,font=\scriptsize] at (3.0,5.95)
  {1st acoustic\\ peak ($\ell\!\sim\!220$)};
\node[align=center,font=\scriptsize,gray!50!black] at (10.4,1.55)
  {Silk damping\\ tail};
\draw[gray!50!black,-{Latex[length=1.6mm]}] (10.2,1.25) -- (10.5,0.6);
\draw[line width=1.10pt,red!75!black,densely dashed] (8.5,0) -- (8.5,4.7);
\node[red!72!black,align=center,font=\scriptsize,fill=white,
      inner sep=1pt] at (8.5,5.25)
  {GG damping scale $\ell_D=1350\pm3$\\ (Boltzmann; BFFT overlay $\approx1382$)};
\fill[red!75!black] (8.5,0) circle (1.3pt);
\node[draw=red!70!black,fill=red!4,rounded corners,align=left,
      font=\scriptsize,inner sep=4pt,text width=7.6cm]
      at (6.4,-2.65)
  {\textbf{Scope flag (honest):} the blue curve is the \emph{standard}
   ($\Lambda$CDM) shape. Run on the exact \GG{} fractions, CLASS
   reproduces it at the percent level, including the damping scale
   $\ell_D=1350\pm3$ (red; Section~\ref{sec:cmb-overdetermined}). The
   integer-ledger knee $K(2\pi)^2\approx1382$ is an interpretive
   overlay ($+2.3\%$ above the computed value); a likelihood-grade
   fit remains open.};
\end{tikzpicture}}
\caption[CMB acoustic peaks and the computed damping scale.]{%
\textbf{CMB temperature acoustic-peak spectrum.}
Schematic temperature power spectrum
$\mathcal{D}_\ell=\ell(\ell+1)C_\ell/2\pi$ versus multipole $\ell$:
the rise to the first acoustic peak near $\ell\!\sim\!220$, subsequent
peaks at $\ell\!\sim\!540$ and $\ell\!\sim\!810$ with decreasing
amplitude, and the Silk damping tail falling beyond $\ell\!\sim\!1000$.
The blue curve is exactly what $\Lambda$CDM computes and is shown here
for orientation. Run on the exact \GG{} fractions, the same Boltzmann
machinery reproduces this curve at the percent level
(Section~\ref{sec:cmb-overdetermined}); the red dashed marker is the
computed damping scale $\ell_D=1350\pm3$ ($+0.4\%$ of the observed
$1345\pm15$), with the integer-ledger knee $K(2\pi)^2\approx1382$
carried separately as an interpretive overlay.
\textbf{The surviving scope gap, stated plainly:} \ATen{} runs the
spectrum through the standard instrument at band-power level but does
not yet own a likelihood-grade fit against the Planck covariance.}
\label{fig:appfig-cmb}
\end{figure}


This appendix teaches, from scratch and at maximum depth, the standard
physics of the cosmic microwave background (CMB): the acoustic peaks
imprinted on the temperature sky and the angular power spectrum
\(C_\ell\) in which they are read out. The intended reader is a
first-year graduate student who has met the Friedmann equations and
ordinary Fourier analysis but has never derived a peak position or a
\(D_\ell\) curve. The physics presented in
Sections~\ref{appA:cmb-fluid} through \ref{appA:cmb-sixparam} is
entirely textbook and is not original to \GGTheory{}. The final
Section~\ref{appA:cmb-gg-reading} states, prominently and without
hedging, what the present cohort has and has \emph{not} yet computed:
A10 and its companion papers compare \emph{central values of
background densities and of individual summary statistics}; they do
\emph{not} yet compute a full \(C_\ell\) spectrum, and a Boltzmann /
CMB-likelihood pipeline against the Planck covariance is named as
future work. A reader who wants only that boundary statement may skip
to Section~\ref{appA:cmb-gg-reading}.

% -------------------------------------------------------------------
\medskip
\textbf{The three steps of this appendix.} As in Appendix~E:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard CMB physics (\S\ref{appA:cmb-fluid} onward).}
The tightly-coupled photon-baryon fluid, the sound horizon, peak positions,
baryon loading, Silk damping, and the angular power spectrum $C_\ell$.
\item \textbf{How the precision is earned (\S\ref{appA:cmb-sixparam}).} The
six $\LCDM$ parameters are not measured one by one; they are \emph{jointly
fit} to the full $D_\ell$ curve against the Planck covariance. Reading the
parameters off the spectrum is itself a fit, not a first-principles
account.
\item \textbf{The geometric derivation (\S\ref{appA:cmb-gg-reading}).} What
the integer ledger reaches --- the damping knee $\ell_D=K(2\pi)^2$ and the
background densities --- and, marked in plain sight, the full $C_\ell$
spectrum it does \emph{not} yet compute.
\end{enumerate}

\subsection{The tightly-coupled photon-baryon fluid and its acoustic
oscillations}\label{appA:cmb-fluid}

Before recombination, the universe was a hot plasma of free electrons,
protons (and helium nuclei), and photons. The photons could not stream
freely: a photon scattered off a free electron after a mean free path
\begin{equation}\label{appA:cmb-mfp}
\lambda_{\mathrm{C}} = \frac{1}{n_e \sigma_T},
\end{equation}
where \(n_e\) is the free-electron number density and
\(\sigma_T \approx 6.65\times 10^{-25}\,\mathrm{cm}^2\) is the Thomson
cross section. Because \(n_e\) was enormous, \(\lambda_{\mathrm{C}}\)
was tiny compared with any cosmological scale of interest: the
scattering rate \(\dot\tau = n_e \sigma_T a\) (in conformal time
\(\eta\), with \(a\) the scale factor) vastly exceeded the expansion
rate. This is the \emph{tight-coupling} regime. Its physical
consequence is simple and powerful: photons and baryons behave as a
\emph{single} fluid. The Compton scattering that ties them together is
so rapid that the photon and baryon velocities are equal to leading
order, and the photon distribution is driven to a local blackbody.

This single fluid has the two properties needed to oscillate. It has
\emph{pressure}, supplied almost entirely by the photons,
\(p_\gamma = \tfrac{1}{3}\rho_\gamma\), giving a large radiation
pressure that resists compression. And it has \emph{inertia}, supplied
by both the photons and the baryons; the baryons add weight without
adding pressure. Place this fluid in the gravitational potential wells
of the primordial dark-matter perturbations, set down at the end of
inflation, and the result is a forced, damped harmonic oscillator run
in every Fourier mode independently.

Write the fractional photon temperature perturbation as
\(\Theta(\mathbf{x},\eta) \equiv \delta T / T\), and Fourier transform
to modes of comoving wavenumber \(k\). In the tight-coupling limit the
monopole \(\Theta_0(k,\eta)\) (the local compression of the fluid)
obeys, schematically, an oscillator equation
\begin{equation}\label{appA:cmb-oscillator}
\ddot\Theta_0 + \frac{\dot R}{1+R}\,\dot\Theta_0
+ k^2 c_s^2\,\Theta_0 = -\frac{k^2}{3}\,\Psi
- \ddot\Phi - \frac{\dot R}{1+R}\,\dot\Phi,
\end{equation}
where overdots are derivatives with respect to conformal time,
\(\Psi\) and \(\Phi\) are the Newtonian-gauge metric potentials,
\(c_s\) is the sound speed defined below, and
\begin{equation}\label{appA:cmb-Rdef}
R \equiv \frac{3\rho_b}{4\rho_\gamma}
\end{equation}
is the \emph{baryon-loading} ratio, the momentum density of the
baryons relative to that of the photons. Equation~\eqref{appA:cmb-oscillator}
is the heart of CMB acoustics. The restoring term \(k^2 c_s^2 \Theta_0\)
makes the fluid ring; the right-hand side is the gravitational forcing
from the potentials; the \(\dot R\) term is a slow drag from the
expanding, baryon-loading background. Each comoving mode \(k\) is an
independent oscillator with its own frequency \(k c_s\).

The crucial fact for what follows is that all modes share a
\emph{common clock}: they all start their oscillation from the same
initial conditions (set by inflation, with a definite phase) and they
all stop ringing at the same instant -- the moment of recombination,
when the coupling that supplied the pressure switches off. Modes that
have completed an integer or half-integer number of oscillations at
that instant are caught at maximum compression or maximum rarefaction;
modes caught mid-swing are caught with little amplitude. This common
stopping time is what turns a continuous spectrum of oscillators into a
\emph{discrete harmonic series} of peaks. We now make ``frequency''
quantitative.

% -------------------------------------------------------------------
\subsection{The sound speed and the sound horizon}\label{appA:cmb-soundhorizon}

The speed at which a pressure disturbance propagates through the
photon-baryon fluid is the adiabatic sound speed
\begin{equation}\label{appA:cmb-cs}
c_s = \sqrt{\frac{\partial p}{\partial \rho}}
= \sqrt{\frac{\dot p_\gamma}{\dot\rho_\gamma + \dot\rho_b}}
= \frac{c}{\sqrt{3\,(1+R)}}.
\end{equation}
In a pure photon gas \(R\to 0\) and \(c_s = c/\sqrt{3} \approx 0.577\,c\):
a relativistic sound speed, more than half the speed of light. As the
baryon density grows toward recombination, \(R\) rises (it reaches
\(R \approx 0.6\) at last scattering for the concordance baryon
density), and the sound speed drops: the baryons add inertia but no
pressure, so they slow the sound down. This is the first appearance of
baryon loading; it returns in Section~\ref{appA:cmb-baryonloading} as
the odd/even peak alternation.

The total comoving distance a sound wave can have traveled, from the
beginning up to recombination at conformal time \(\eta_*\) (redshift
\(z_*\)), is the \emph{comoving sound horizon}
\begin{equation}\label{appA:cmb-rs}
\rs(z_*) = \int_0^{\eta_*} c_s\,\mathrm{d}\eta
= \int_0^{t_*} \frac{c_s}{a}\,\mathrm{d}t
= \int_{z_*}^{\infty} \frac{c_s(z)}{H(z)}\,\mathrm{d}z .
\end{equation}
The three forms are identical: the first integrates the sound speed
over conformal time, the second writes \(\mathrm{d}\eta = \mathrm{d}t/a\)
explicitly, and the third changes the variable of integration to
redshift using \(\mathrm{d}z = -(1+z)H\,\mathrm{d}t\) and
\(a = 1/(1+z)\). The integrand is dominated by the epoch just before
\(z_*\), where the expansion is fastest and \(H(z)\) is largest, so
\(\rs\) is sensitive to the radiation and matter content that set
\(H(z)\) at \(z \gtrsim 1000\), and to the baryon density that sets
\(c_s\) through \(R\). For the Planck concordance cosmology the result
is
\begin{equation}\label{appA:cmb-rsval}
\rs(z_*) \approx 144\ \mathrm{Mpc}
\qquad (\text{comoving}),
\end{equation}
a fixed physical length frozen into the plasma at recombination. This
length is the \emph{standard ruler} of cosmology. It is the same ruler
that, propagated to low redshift, becomes the baryon acoustic
oscillation (BAO) scale in the galaxy distribution; here we use it for
the CMB.

% -------------------------------------------------------------------
\subsection{The angular-diameter distance to last scattering and the
acoustic angular scale}\label{appA:cmb-theta}

A ruler of known comoving length \(\rs\) sitting at the last-scattering
surface subtends an angle on our sky. To convert the physical ruler
into an angle we need the distance to the surface on which it lies. The
relevant distance is the \emph{comoving angular-diameter distance} to
recombination,
\begin{equation}\label{appA:cmb-DA}
D_A(z_*) = \frac{c}{\Hzero}\,
\frac{1}{\sqrt{|\Omega_k|}}\,
S_k\!\left(\sqrt{|\Omega_k|}\,\int_0^{z_*}
\frac{\Hzero\,\mathrm{d}z}{H(z)}\right),
\end{equation}
which for a spatially flat universe (\(\Omega_k = 0\), the case we
adopt throughout) reduces to the plain comoving distance
\begin{equation}\label{appA:cmb-DAflat}
D_A(z_*) = \chi(z_*) = c\int_0^{z_*}\frac{\mathrm{d}z}{H(z)} .
\end{equation}
Here \(S_k\) is \(\sin\), identity, or \(\sinh\) for closed, flat, or
open geometry; the flat case is the identity. Numerically, for the
concordance cosmology,
\begin{equation}\label{appA:cmb-DAval}
D_A(z_*) \approx 13.9\ \mathrm{Gpc} \approx 14\ \mathrm{Gpc}
\qquad (\text{comoving distance to } z_* \approx 1090).
\end{equation}
The \emph{acoustic angular scale} is then the ratio of the standard
ruler to the distance,
\begin{equation}\label{appA:cmb-thetastar}
\theta_* = \frac{\rs(z_*)}{D_A(z_*)} .
\end{equation}
This single dimensionless number -- the angle subtended by the sound
horizon on the last-scattering surface -- is the most precisely
measured quantity in all of cosmology, determined by Planck to about
\(0.03\%\). It anchors the entire peak structure, because all the peaks
are harmonics of this one fundamental angle.

\paragraph{Worked numerical example.}
Insert the two background numbers directly. With
\(\rs \approx 144\,\mathrm{Mpc}\) and
\(D_A \approx 14\,\mathrm{Gpc} = 14000\,\mathrm{Mpc}\),
\begin{equation}\label{appA:cmb-thetaworked}
\theta_* \approx \frac{144\ \mathrm{Mpc}}{14000\ \mathrm{Mpc}}
\approx 1.03\times 10^{-2}\ \mathrm{rad}
\approx 0.59^\circ .
\end{equation}
The sound horizon subtends about one part in a hundred radians, a bit
over half a degree, on the sky. The Planck measured value is
\(100\,\theta_* = 1.0411 \pm 0.0003\), i.e.\
\(\theta_* \approx 1.041\times 10^{-2}\,\mathrm{rad}\); our two-number
estimate~\eqref{appA:cmb-thetaworked} reproduces it to better than a
percent, which is all the input precision deserves. We convert this
angle into a multipole in the next subsection.

% -------------------------------------------------------------------
\subsection{Peak positions: the harmonic series and the first peak near
\texorpdfstring{\(\ell_1 \approx 220\)}{l1 = 220}}\label{appA:cmb-peaks}

A small angle \(\theta\) on the sky corresponds, in the multipole
expansion of Section~\ref{appA:cmb-cl}, to a multipole moment of order
\begin{equation}\label{appA:cmb-elltheta}
\ell \sim \frac{\pi}{\theta},
\end{equation}
the angular analogue of the Fourier relation between a wavelength and a
wavenumber (the half-wavelength of a feature of angular size
\(\theta\) maps to \(\ell \approx \pi/\theta\)). The acoustic peaks
occur at the angles subtended by the modes caught at extrema of their
oscillation. The fundamental mode -- the one that has just reached its
\emph{first} compression maximum at recombination -- has traveled
exactly one sound horizon, so it subtends \(\theta_*\) and lands at
\begin{equation}\label{appA:cmb-ell1}
\ell_1 \approx \frac{\pi}{\theta_*} .
\end{equation}
Its overtones, caught at successive extrema (compression, rarefaction,
compression, \ldots), subtend integer fractions of \(\theta_*\) and
land at the harmonic series
\begin{equation}\label{appA:cmb-elln}
\ell_n \approx n\,\frac{\pi}{\theta_*}, \qquad n = 1, 2, 3, \ldots
\end{equation}
This is the deep statement of CMB acoustics: \emph{the peaks are a
harmonic series of one fundamental angular frequency set by
\(\theta_*\)}, exactly as the overtones of an organ pipe are a harmonic
series of one fundamental set by the pipe length.

\paragraph{Worked numerical example (continued).}
Using the estimate~\eqref{appA:cmb-thetaworked},
\begin{equation}\label{appA:cmb-ell1worked}
\ell_1 \approx \frac{\pi}{\theta_*}
\approx \frac{3.1416}{1.03\times 10^{-2}}
\approx 305 .
\end{equation}
The naive harmonic estimate \(\pi/\theta_*\) therefore places the first
peak near \(\ell \approx 300\), whereas the \emph{observed} first
acoustic peak sits at \(\ell_1 \approx 220\). The discrepancy is real
and instructive, and it is not an error in the arithmetic. The naive
relation~\eqref{appA:cmb-ell1} treats the peak as a pure spatial
projection of a single wavelength. The true projection involves two
refinements that systematically lower \(\ell_1\) below \(\pi/\theta_*\).

\begin{enumerate}[leftmargin=2em,itemsep=0.2em]
\item \textbf{The projection is not a sharp \(\pi/\theta_*\) but a
weighted transfer.} A plane wave of wavenumber \(k\) on the
last-scattering surface projects onto a \emph{range} of multipoles
through spherical Bessel functions \(j_\ell(k\chi_*)\), which peak near
\(\ell \approx k\chi_* - \tfrac{1}{2}\) rather than exactly at
\(k\chi_*\). The effective conversion is
\(\ell_n \approx k_n D_A\) with \(k_n = n\pi/\rs\), and the geometric
projection shifts the peak to lower \(\ell\).
\item \textbf{Gravitational driving and baryon loading shift the
phase.} The forcing terms on the right of
Eq.~\eqref{appA:cmb-oscillator} and the baryon drag mean the
fundamental is not caught exactly at \(k\rs = \pi\) but at a somewhat
smaller effective \(k\rs\), pulling \(\ell_1\) down. The early
integrated Sachs--Wolfe effect (the decay of potentials around
matter-radiation equality) adds power at low \(\ell\) and further
displaces the apparent peak.
\end{enumerate}

The standard, accurate result, obtained by a full Boltzmann
computation, is
\begin{equation}\label{appA:cmb-ell1standard}
\ell_1 \approx 0.74\,\frac{\pi}{\theta_*} \approx 220,
\qquad
\ell_n \approx \ell_1 + (n-1)\,\Delta\ell,
\quad \Delta\ell \approx \frac{\pi}{\theta_*} \approx 300,
\end{equation}
so that the peak spacing \(\Delta\ell \approx 300\) is the clean
harmonic interval \(\pi/\theta_*\), while the \emph{first} peak is
pulled forward by the order-\(0.7\) projection-and-driving factor.
The observed peaks sit near \(\ell \approx 220, 540, 800, 1100,
\ldots\). The lesson of the worked example is that the back-of-envelope
\(\pi/\theta_*\) gets the right \emph{spacing} immediately and the
first-peak \emph{position} only up to a calculable order-unity
projection correction; getting \(\ell_1 = 220\) exactly requires the
pipeline of Section~\ref{appA:cmb-cl}.

% -------------------------------------------------------------------
\subsection{Baryon loading and the odd/even peak-height
alternation}\label{appA:cmb-baryonloading}

Baryon loading does more than slow the sound speed. It breaks the
symmetry between compression and rarefaction, and this is what makes
the CMB a \emph{baryometer}.

Return to the oscillator~\eqref{appA:cmb-oscillator}. The baryons add
inertia, which is equivalent to hanging a weight on the end of the
oscillating spring. A mass on a spring in a gravitational field does
not oscillate about its unloaded equilibrium; it sags. In the
photon-baryon fluid, the gravitational potential wells of the dark
matter play the role of gravity, and the baryon inertia
\(R\) plays the role of the hanging mass. The effect is a
\emph{displacement of the zero-point} of the oscillation: with the
quasi-static driving solved, the monopole oscillates not about
\(\Theta_0 = 0\) but about
\begin{equation}\label{appA:cmb-zeropoint}
\bar\Theta_0 = -(1+R)\,\Psi,
\end{equation}
so the oscillation amplitude is enhanced on the \emph{compression}
side (deep into the well, where the photon-baryon fluid falls and is
maximally compressed) and suppressed on the \emph{rarefaction} side.

Now recall that the observed temperature peaks are sensitive to the
\emph{square} of the oscillation amplitude. The odd peaks
(\(n = 1, 3, 5, \ldots\)) correspond to maximal \emph{compression} and
are boosted by baryon loading; the even peaks
(\(n = 2, 4, \ldots\)) correspond to maximal \emph{rarefaction} and are
relatively suppressed. The result is a characteristic \emph{odd/even
alternation} in peak heights: the first peak is tall, the second is
notably lower, the third recovers. The size of the alternation scales
with \(R \propto \Omb h^2\): a more baryon-rich universe shows a more
pronounced first-to-second peak height ratio. This is the cleanest way
the CMB measures the baryon density, and it is independent of, and
consistent with, the big-bang nucleosynthesis determination.

% -------------------------------------------------------------------
\subsection{Silk damping and the damping tail}\label{appA:cmb-silk}

The peaks do not continue undiminished to arbitrarily high \(\ell\).
They are progressively erased at small angular scales by photon
diffusion, an effect first analyzed by Joseph Silk and called
\emph{Silk damping} (or diffusion damping).

The tight coupling of Section~\ref{appA:cmb-fluid} is not perfect.
Near recombination the mean free path~\eqref{appA:cmb-mfp} grows as the
free electrons recombine, and photons begin to random-walk a finite
distance between scatterings. Over the age of the universe at
recombination a photon diffuses a comoving root-mean-square distance
\begin{equation}\label{appA:cmb-silkscale}
\lambda_D \sim \sqrt{N}\,\lambda_{\mathrm{C}}
\sim \left(\int \frac{\mathrm{d}\eta}{6\,\dot\tau}\,
\frac{R^2 + \tfrac{16}{15}(1+R)}{(1+R)^2}\right)^{1/2},
\end{equation}
where \(N\) is the number of scatterings and the bracket is the
standard second-order tight-coupling result. Perturbations on scales
smaller than \(\lambda_D\) are washed out: hot and cold photons mix and
the temperature contrast is erased. In multipole space this defines a
\emph{damping scale}
\begin{equation}\label{appA:cmb-ellD}
\ell_D \sim \frac{\pi\,D_A}{\lambda_D},
\end{equation}
above which the acoustic oscillations are exponentially suppressed by a
factor of the form \(\exp[-(\ell/\ell_D)^2]\). The observed temperature
spectrum therefore shows a \emph{damping tail}: the peaks shrink and
smear out for \(\ell \gtrsim 1000\), and by \(\ell \approx 2000\) the
primary anisotropy is strongly suppressed. The \(\LCDM\)-inferred
damping multipole sits near \(\ell_D \approx 1345 \pm 15\), with the
exact value depending on the precise definition of the damping
envelope at the percent level. The damping tail is a sensitive probe of
recombination microphysics and of any extra relativistic species
(\(\Neff\)) that change the expansion rate near \(z_*\). Evaluating
this diffusion integral on the \(\pi\)-locked inventory over the full
Boltzmann ionization history gives \(\ell_D = 1350\pm3\) (\(+0.4\%\));
the separate integer-ledger reading of the knee is discussed in
Section~\ref{appA:cmb-gg-reading}.

% -------------------------------------------------------------------
\subsection{The temperature angular power spectrum: \texorpdfstring{$C_\ell$}{Cl}
and \texorpdfstring{$D_\ell$}{Dl}}\label{appA:cmb-cl}

We can now define precisely the object that all of the above produces:
the angular power spectrum. The CMB temperature is a function on the
sphere, \(\Theta(\hat{\mathbf n}) = \delta T/T\) in direction
\(\hat{\mathbf n}\). Expand it in spherical harmonics,
\begin{equation}\label{appA:cmb-multipole}
\Theta(\hat{\mathbf n}) = \sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}
a_{\ell m}\,Y_{\ell m}(\hat{\mathbf n}),
\qquad
a_{\ell m} = \int \Theta(\hat{\mathbf n})\,
Y_{\ell m}^{*}(\hat{\mathbf n})\,\mathrm{d}\Omega .
\end{equation}
The multipole \(\ell\) is the angular analogue of a wavenumber:
\(\ell \approx \pi/\theta\) corresponds to features of angular size
\(\theta\), so low \(\ell\) is large angular scales and high \(\ell\) is
fine structure. Because the early universe is statistically isotropic,
the \(a_{\ell m}\) are zero-mean random variables whose variance
depends only on \(\ell\), not on \(m\). That variance \emph{is} the
power spectrum,
\begin{equation}\label{appA:cmb-cldef}
\langle a_{\ell m}\, a_{\ell' m'}^{*}\rangle
= C_\ell\,\delta_{\ell\ell'}\,\delta_{mm'},
\qquad
C_\ell = \frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}
\langle |a_{\ell m}|^2\rangle .
\end{equation}
The angle brackets denote an ensemble average over realizations of the
random primordial field; we have one sky, so the practical estimator
averages over the \(2\ell+1\) available \(m\) values at each \(\ell\),
which is why low multipoles are intrinsically uncertain -- the
\emph{cosmic variance} floor,
\begin{equation}\label{appA:cmb-cosmicvar}
\frac{\Delta C_\ell}{C_\ell} = \sqrt{\frac{2}{2\ell+1}} .
\end{equation}
Theory delivers \(C_\ell\) by transporting the primordial curvature
perturbation through the acoustic oscillator and projecting onto the
sky,
\begin{equation}\label{appA:cmb-clintegral}
C_\ell = \frac{2}{\pi}\int_0^{\infty}
k^2\,\mathrm{d}k\;P_{\mathcal R}(k)\,
\bigl|\Delta_\ell(k)\bigr|^2,
\end{equation}
where \(P_{\mathcal R}(k) \propto k^{\ns - 4}\) is the primordial
power spectrum (with tilt \(\ns\)) and \(\Delta_\ell(k)\) is the
transfer function carrying all the acoustic, baryon-loading,
diffusion, and projection physics of the previous subsections.
Computing \(\Delta_\ell(k)\) is precisely the Boltzmann calculation
named below as not-yet-done in \GGTheory{}.

Because \(C_\ell\) falls steeply with \(\ell\), the CMB is almost
always plotted as the rescaled quantity
\begin{equation}\label{appA:cmb-Dl}
D_\ell \equiv \frac{\ell(\ell+1)}{2\pi}\,C_\ell,
\end{equation}
which is flat for a scale-invariant Sachs--Wolfe plateau at low
\(\ell\) and displays the acoustic peaks as genuine peaks rather than
as wiggles on a steep slope. The familiar Planck plot of the
microwave-background spectrum is a plot of \(D_\ell\) in units of
\(\mu\mathrm{K}^2\): the Sachs--Wolfe plateau out to \(\ell \approx 30\),
the first acoustic peak at \(\ell_1 \approx 220\) reaching
\(D_\ell \approx 5700\,\mu\mathrm{K}^2\), the harmonic series of peaks
out to \(\ell \approx 2000\), and the damping tail beyond. When this
appendix and the body of A10 speak of ``the CMB spectrum,'' this
\(D_\ell\) curve is the object meant.

% -------------------------------------------------------------------
\subsection{How the six \texorpdfstring{$\LCDM$}{LCDM} parameters map
onto peak features}\label{appA:cmb-sixparam}

The power of the CMB is that the six free parameters of the concordance
model each leave a distinct, largely separable fingerprint on the
\(D_\ell\) curve. This is why a single spectrum constrains the whole
model so tightly.

\begin{itemize}[leftmargin=2em,itemsep=0.25em]
\item \textbf{Baryon density \(\Omb h^2\)} -- sets the baryon loading
\(R\) of Section~\ref{appA:cmb-baryonloading}, hence the odd/even peak
height alternation (especially the first-to-second peak ratio) and,
through \(c_s\), part of \(\rs\). More baryons: taller odd peaks,
lower sound speed.
\item \textbf{Cold dark matter density \(\Omm h^2\)} (equivalently the
matter density) -- sets the redshift of matter-radiation equality,
hence the early integrated Sachs--Wolfe boost of the first peak and the
overall peak amplitudes. It also controls \(H(z)\) at early times,
affecting both \(\rs\) and \(D_A\).
\item \textbf{Hubble constant \(\Hzero\)} (or the angular scale
\(\theta_*\)) -- sets \(D_A\) and therefore, through
\(\theta_* = \rs/D_A\), the \emph{overall position} of the entire peak
series. In practice the precisely measured quantity is \(\theta_*\)
itself; \(\Hzero\) is inferred from it given the other parameters.
\item \textbf{Scalar tilt \(\ns\)} -- tilts the primordial spectrum
\(P_{\mathcal R}(k) \propto k^{\ns-4}\), tilting the relative heights
of low-\(\ell\) versus high-\(\ell\) power. \(\ns < 1\) means slightly
less small-scale power.
\item \textbf{Amplitude \(A_s\)} -- the overall normalization of
\(P_{\mathcal R}(k)\), hence the overall vertical scale of \(D_\ell\)
(degenerate with the optical depth until polarization breaks it).
\item \textbf{Optical depth \(\tau\)} -- the integrated Thomson
scattering after reionization suppresses \(D_\ell\) by
\(e^{-2\tau}\) at \(\ell\) above the reionization horizon and adds a
characteristic low-\(\ell\) polarization bump, breaking the
\(A_s\)--\(\tau\) degeneracy.
\end{itemize}

A seventh, derived quantity, the damping scale \(\ell_D\),
is fixed once \(\Omb h^2\), \(\Omm h^2\), and the recombination history
are fixed; it is also sensitive to \(\Neff\). The point of listing all
of this is that \emph{reading the parameters off the spectrum is a
joint fit of all six (plus derived extras) to the full \(C_\ell\)
curve, with a covariance matrix} -- it is not a sequence of
independent single-number reads. That observation is exactly what the
next subsection turns on.

% -------------------------------------------------------------------
\subsection{The geometric derivation: what the cohort computes, and what it does not}\label{appA:cmb-gg-reading}

The standard CMB physics of Step~1 is the arena in which $\LCDM$ earns
its precision. This section states the exact boundary of what \GGTheory{}
has so far computed --- and in this appendix the honest marking of the
missing computation matters as much as the teaching of the present one.
We lead with the one acoustic-tail feature the integer ledger reaches, and
end with the spectrum it does not.

\paragraph{The headline, stated plainly: the hand-off is executed; the
likelihood fit is not.}
A10 does \emph{not} compute the Boltzmann spectrum from the geometry ---
that is the Tier-3 boundary the framework draws for itself --- but the
hand-off to the instrument that does has been executed:
the exact \GGTheory{} fractions and the \emph{derived} tilt, fed to the
Einstein--Boltzmann machinery (CLASS), reproduce the Planck TT
band-powers at the few-percent level with zero fitted parameters
(Section~\ref{sec:cmb-overdetermined}, Figure~\ref{fig:cmb-tt-gg}), and
within the \GG{} geometry the derived tilt \emph{improves} on the
borrowed Planck value by \(\Delta\chi^2 = -22\). What remains open is
the genuine likelihood fit against the Planck covariance with the seed
re-optimized. The framework's cosmological comparisons are therefore
made at three levels:
\begin{enumerate}[leftmargin=2em,itemsep=0.2em]
\item \textbf{Background densities.} The \(\pi\)-locked inventory
\(\Omm = 1/\pi\), \(\OmL = 1 - 1/\pi\), \(\Omb = 1/(2\pi^2)\), the
dark-matter complement \(\Omdm = \Omm - \Omb\), and related background
quantities are compared as \emph{central values} against the Planck
central values. These are Derived-conditional (\(\mathrm{D}^*\))
entries pending their named lemmas; they are background numbers, not
spectra.
\item \textbf{Individual summary statistics.} A handful of single
numbers that summarize the perturbation sector -- the recombination
switch redshift \(z_* = e^7\), the damping multipole \(\ell_D\), and
the tensor-to-scalar target \(r = 0.0043\) -- are compared one at a
time against their \(\LCDM\)-inferred or Planck-quoted values.
\item \textbf{The executed demonstration.} The full TT spectrum
computed by CLASS on the derived inputs --- \(H_0 = 66.6\), the
\(\pi\)-locked fractions, \(n_s = 0.9620\), with \(A_s\) and \(\tau\)
inherited --- overplotted on the Planck binned data: residuals within
\(\pm3\)--\(4\%\) across the first seven acoustic peaks,
\(\chi^2 = 237\) over 83 bins with zero fitted parameters against the
six-parameter best fit's 84, and the damping scale returned at
\(\ell_D = 1350 \pm 3\) (Section~\ref{sec:cmb-overdetermined}).
\end{enumerate}
The first two levels are central-value comparisons and the third is a
demonstration, not a likelihood fit. In particular, the acoustic angular scale
\(\theta_*\) is carried in the body of A10 only as a \emph{definition},
\(\theta_* = \rs(z_*)/D_A(z_*)\) (Eq.~\eqref{appA:cmb-thetastar}); the
geometry does \emph{not} by itself produce a numerical \(\theta_*\) or
a peak-height pattern, because those require \(\rs\) and \(D_A\) to
be integrated over the full thermal history with the recombination
microphysics in place --- the \(C_\ell\)/transfer-function pipeline
that only a Boltzmann code supplies. In the executed demonstration
that pipeline \emph{is} run, on the derived inputs, and the
\(\theta_*\) and peak morphology it returns are reported as outputs of
the instrument on those inputs (Section~\ref{sec:cmb-overdetermined}),
not as claims of the geometry itself; the codebook scores them
accordingly.

\paragraph{Why this is the single deepest open methodological item.}
The standard analysis of Section~\ref{appA:cmb-sixparam} shows that the
parameters are not measured one-by-one but jointly, by fitting the full
\(D_\ell\) (and \(C_\ell^{TE}\), \(C_\ell^{EE}\)) curve against the
Planck \emph{covariance}. A framework that compares central values of
densities and a few summary statistics is therefore not yet meeting the
data in the arena where \(\LCDM\) earns its precision. By the paper's
own audit this is the deepest outstanding methodological gap: until a
full Boltzmann / CMB-likelihood pipeline exists that takes the
\GGTheory{} ledger, integrates the photon-baryon transfer function
\(\Delta_\ell(k)\) of Eq.~\eqref{appA:cmb-clintegral}, and confronts the
Planck likelihood with its covariance, the cohort's cosmological claims
remain comparisons of central values, not a fit to a spectrum. The
required pipeline -- a global CMB-plus-BAO-plus-distance-ladder fit with
the full covariance -- is named, repeatedly and explicitly, as still
required.

\paragraph{The one feature the integer ledger reaches: the damping knee
\texorpdfstring{$\ell_D = K(2\pi)^2$}{lD = K(2pi)^2}.}
There is exactly one CMB-tail quantity the \GGTheory{} integer ledger
currently touches, and it is worth stating precisely \emph{as a
single-feature postdiction, not a spectrum}. The damping multipole of
Section~\ref{appA:cmb-silk} is identified with the first non-trivial
ultraviolet (BFFT) cutoff of the multiscale throat through the integer
\(K = 35\):
\begin{equation}\label{appA:cmb-ellDgg}
\ell_D = K\,(2\pi)^2 = 35 \times 4\pi^2 = 1381.7
\approx 1.38\times 10^3 .
\end{equation}
Against the \(\LCDM\)-inferred damping multipole
\(\ell_D \approx 1345 \pm 15\), this integer-ledger value sits at
\(+2.7\%\); against the framework's own Boltzmann-computed damping
scale (\(\ell_D = 1350\pm3\), the value the codebook scores), it sits
at \(+2.3\%\). It is carried as an interpretive overlay gated by the
BFFT-damping-knee-identification lemma, which would now have to explain
that gap, not merely prove that the projection of the diffusion damping
scale onto the BFFT bus is exact at leading order. The honest
framing is the one A10 uses for it: a single-feature near-hit from an
independent integer ledger, definition-dependent at the percent level
in its observed comparator, and \emph{not} a substitute for a computed
damping envelope, let alone a computed spectrum. It is one knee, not a
curve.

\paragraph{What it would take to close the remaining gap.}
Four steps turn a derived inventory into a genuine CMB test: (i) supply
the primordial \(P_{\mathcal R}(k)\) with the predicted \(\ns\) and an
amplitude; (ii) evolve the coupled Boltzmann hierarchy for photons,
baryons, dark matter, and neutrinos through recombination to obtain
\(\Delta_\ell(k)\); (iii) project to \(C_\ell\) via
Eq.~\eqref{appA:cmb-clintegral}; (iv) form \(D_\ell\) and confront the
Planck \(TT\), \(TE\), \(EE\) likelihoods with the published
covariance. Steps
(i)--(iii) have been executed in
demonstration form --- the derived tilt and inventory through CLASS,
with the amplitude inherited (Section~\ref{sec:cmb-overdetermined}) ---
and step (iv), the covariance likelihood with the seed re-optimized,
remains open and is named as such in the audit (PS-Cosmos-16). This
appendix teaches the standard pipeline precisely so the reader can see,
step by step, which steps the geometry feeds and which step remains.

% -------------------------------------------------------------------
\subsection*{Status}\label{appA:cmb-status}

The standard CMB physics taught in
Sections~\ref{appA:cmb-fluid}--\ref{appA:cmb-sixparam} is textbook and
settled: the tightly-coupled photon-baryon fluid oscillates as a forced
harmonic oscillator; the sound horizon \(\rs \approx 144\,\mathrm{Mpc}\)
and the angular-diameter distance \(D_A \approx 14\,\mathrm{Gpc}\) give
the acoustic angle \(\theta_* \approx 1.04\times 10^{-2}\,\mathrm{rad}\);
the peaks form the harmonic series \(\ell_n \approx n\pi/\theta_*\) with
spacing \(\Delta\ell \approx 300\) and a first peak pulled by
projection-and-driving to \(\ell_1 \approx 220\); baryon loading sets the
odd/even peak-height alternation; Silk diffusion produces the damping
tail; and the whole pattern is read out as the angular power spectrum
\(C_\ell\), conventionally plotted as
\(D_\ell = \ell(\ell+1)C_\ell/2\pi\), against which the six
\(\LCDM\) parameters are \emph{jointly} fit with a covariance.

The \GGTheory{}-side full \(C_\ell\) prediction is \textbf{NOT YET
DONE}. It is open, and named as future work by the paper's own audit:
the cohort now computes the spectrum at the background level --- the
exact fractions, run through CLASS, reproduce the Planck TT band-powers
at the percent level with the seed held at its inherited value
(Section~\ref{sec:cmb-overdetermined}) --- but a genuine likelihood fit
against the Planck covariance, with the primordial seed re-optimized,
remains the single deepest outstanding methodological item. The damping
scale is scored by its Boltzmann diffusion value \(\ell_D = 1350\pm3\)
(\(+0.4\%\) from the \(\LCDM\)-inferred \(\approx1345\)); the
integer-ledger reading \(\ell_D = K(2\pi)^2 = 1381.7\) is carried
separately as an interpretive overlay (\(+2.3\%\) above the computed
value), a single-feature reading, not a spectrum. This appendix teaches
the standard pipeline and marks that boundary plainly.

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{K.1.1. Why do compressions and
rarefactions alternate as acoustic peaks?}

\noindent\emph{Solution.}  Step 1: each Fourier mode of the
photon--baryon fluid is an oscillator rung at the same moment
(inflation phases them coherently), so at last scattering a
mode has completed $kr_s/\pi$ half-oscillations.  Step 2: modes
caught at maximal \emph{compression} in potential wells
($kr_s = \pi, 3\pi, \ldots$) and at maximal \emph{rarefaction}
($kr_s = 2\pi, 4\pi, \ldots$) both imprint temperature extremes
--- the spectrum records power, not sign, so both appear as
peaks.  Step 3: the zero-crossing modes ($kr_s = \pi/2,
3\pi/2, \ldots$) are caught mid-swing with maximal velocity ---
the troughs are filled in by the Doppler contribution, which is
why the observed valleys do not reach zero.  Physics intuition:
the CMB is a photograph of ten thousand pendulums released
together and snapped at one instant --- the peak series is the
census of which pendulums were at their turning points when
the shutter fired.

\medskip\noindent\textbf{K.2.1. Evaluate the baryon-loaded
sound speed at last scattering.}

\noindent\emph{Solution.}  Step 1: baryons add inertia but no
pressure: $c_s^2 = \dfrac{c^2}{3(1 + R)}$ with $R =
3\rho_b/4\rho_\gamma$.  Step 2: $R$ grows $\propto a$; at
$z_* \approx 1090$, with the measured baryon and photon
densities, $R_* \approx 0.6$.  Step 3: hence $c_s(z_*) \approx
c/\sqrt{3\times1.6} \approx 0.45\,c$ --- the fluid sings at
almost half the speed of light, and the sound horizon
$r_s \approx 147$~Mpc (comoving) follows by integrating this
speed against the expansion of Appendix~\ref{appA:thermal}.
Physics intuition: $R$ is the weight the baryons hang on the
photon spring; it slows the music by fifteen percent, and ---
K.5.1 --- it also shifts the oscillation's midpoint, which is
how the CMB weighs the baryons.

\medskip\noindent\textbf{K.3.1. Estimate the acoustic scale and
the position of the first peak.}

\noindent\emph{Solution.}  Step 1: the acoustic angular scale
is $\theta_s = r_s/D_M(z_*)$; with $r_s = 144$~Mpc (the
Planck-calibrated sound horizon at decoupling) and the comoving
distance $D_M \approx 13.9$~Gpc: $\theta_s \approx 0.0104$~rad
$\approx 0.6^\circ$.  Step 2: the associated multipole is
$\ell_A = \pi/\theta_s \approx 300$.  Step 3: the first peak
sits below $\ell_A$ --- at $\ell_1 \approx 0.73\,\ell_A \approx
220$ --- because gravitational driving and the decay of the
potentials shift the effective phase; the subsequent peaks
follow with spacing $\approx \ell_A$.  Physics intuition: a
one-degree feature on the sky is the meter-stick of J.6.1 seen
at $z = 1090$; that its angular size comes out near $1^\circ$
is the flatness measurement --- in a curved universe the same
stick would subtend a visibly different angle.

\medskip\noindent\textbf{K.5.1. Show how baryon loading makes
odd peaks stronger than even ones.}

\noindent\emph{Solution.}  Step 1: with loading, the oscillator
equilibrium shifts: the mode oscillates about the displaced
point $-(1+R)\Phi$ rather than $-\Phi$ --- the spring sags
under the baryon weight.  Step 2: compressions (into potential
wells) therefore swing \emph{further} from equilibrium than
rarefactions by $2R\Phi$: the odd-numbered peaks (compressions,
K.1.1) gain power, the even ones lose it.  Step 3: the
odd/even alternation in the measured spectrum thus reads off
$R$, i.e.\ $\Omega_bh^2$ --- and the CMB's baryon budget agrees
with BBN's (Appendix~\ref{appA:bbn}, deuterium) to a few
percent: two epochs, $z \sim 10^9$ and $z \sim 10^3$, one
number.  Physics intuition: hang a weight on a swinging spring
and its downswings deepen --- the sky's second peak is
literally quieter than its first because the universe's
baryons are heavy.

\medskip\noindent\textbf{K.6.1. Estimate where Silk damping
cuts the spectrum off.}

\noindent\emph{Solution.}  Step 1: photons random-walk through
the baryons with mean free path $\lambda_{\rm mfp} =
1/(n_e\sigma_T a)$; over the age $t$ the diffusion length is
$\lambda_D \sim \sqrt{N}\lambda_{\rm mfp} \sim
\sqrt{\lambda_{\rm mfp}\,ct}$.  Step 2: temperature
fluctuations on scales below $\lambda_D$ are ironed out ---
evaluated at recombination the comoving damping scale is a few
tens of Mpc, i.e.\ multipoles $\ell_D \sim 1400$ --- where the
chapter's interpretive $K(2\pi)^2$ overlay also sits.  Step 3:
hence the acoustic peaks decay exponentially beyond the third
--- and the damping-tail shape is itself a consistency check
on $\Omega_bh^2$ and $N_\nu$ (H.6.1's speedometer read at a
different epoch).  Physics intuition: the CMB's resolution
limit is set by the photons' own diffusion during the final
scattering --- the sky is sharp only down to the distance a
photon could stagger while the fog was lifting.

\medskip\noindent\textbf{K.8.1. Why do the peaks fix
$\Omega_mh^2$ and $\Omega_bh^2$ but not $H_0$ alone?}

\noindent\emph{Solution.}  Step 1: the peak \emph{heights} are
early-universe physics --- they measure the physical densities
$\omega_b = \Omega_bh^2$ (odd/even contrast, K.5.1) and
$\omega_m = \Omega_mh^2$ (driving and the equality epoch,
J.5.1) --- with no separate $H_0$ anywhere.  Step 2: the peak
\emph{positions} add one number, $\theta_s = r_s/D_M$: but
$D_M$ integrates $1/H(z)$ over the late universe, so many
$(H_0, \Omega_\Lambda, \ldots)$ combinations give the same
$\theta_s$ --- the geometric degeneracy.  Step 3: hence the
CMB alone delivers $H_0$ only \emph{conditional on} the
late-time model --- which is precisely the seam the
Hubble-tension Part works: the two-stage corridor modifies the
late-time mapping while the early-universe $\omega$'s stay
fixed, and Sections K.9--K.10 record, with the honesty flags,
what such a shift can and cannot do.  Physics intuition: the
CMB measures the engine's specifications exactly, but the
odometer reading depends on the road since --- the tension
lives on the road, not in the engine.

\GGtakeaway{\emph{Standard result:} the acoustic-peak pattern, derived
from the photon-baryon fluid, against which six parameters are jointly
fit. \emph{What it assumed:} the six numbers themselves. \emph{What the
geometry replaced it with:} five of the six derived or fixed
(\(H_0 = 66.6\), the \(\pi\)-locked fractions, \(n_s = 0.9620\)) and
the spectrum reproduced at the percent level by the Boltzmann
instrument running on them --- with the amplitude the one declared
input, and the covariance likelihood fit the named remaining step.
\emph{Continue in:} Part~VI (the demonstration); Appendix~G (the tilt).}

\subsection{Legacy early-dark-energy linear-response: a non-unique estimate of the corridor gap}
\label{app:ede-legacy}

The body of Part~VI proposes a structural candidate resolution of the Hubble tension, through the
two-stage corridor of Theorem~\ref{thm:hubble-two-stage-corridor} switched at
\(z_\ast=e^{7}\). Before that structural reading was adopted, the same gap was
estimated by a multi-parameter early-dark-energy-plus-dark-radiation-plus-local-void
linear-response. By the \MSMF{}/\OGN{} uniqueness principle a tunable
multi-parameter fit cannot be the unique cost-minimum solution, so this material
is retained here only as a magnitude cross-check and a record of the honesty
caveats, not as the mechanism. \emph{The value \(\DeltaNeff=0.30\) used in the
formulae below is the legacy linear-response value; the body of the paper and the
codebook audit use the canonical \(\DeltaNeff=0.04\).}

\subsubsection{Deriving the early-universe shift from the inherited source ledger}

The present framework proposes two early-time contributions that matter for the sound horizon: the early-dark-energy lobe and a small extra-radiation piece. To see how they work, write the expansion rate as
\begin{equation}
 H^2(z) = H_{\mathrm{ref}}^2(z)\paren{1 + \delta_{\mathrm{EDE}}(z) + \delta_{\mathrm{dr}}(z)}.
\label{eq:H-modified}
\end{equation}
Here \(H_{\mathrm{ref}}\) denotes the reference early-time expansion history under the baseline fit, \(\delta_{\mathrm{EDE}}\) is the fractional early-dark-energy lobe, and \(\delta_{\mathrm{dr}}\) is the extra-radiation correction related to \(\Delta N_{\mathrm{eff}}\).

The sound horizon integral,
\begin{equation}
 r_s = \int_{z_*}^{\infty}\frac{c_s(z)}{H(z)}\,\dd z,
\end{equation}
responds inversely to \(H\). For small corrections one may expand
\begin{equation}
\frac{1}{H(z)} \approx \frac{1}{H_{\mathrm{ref}}(z)}\paren{1 - \frac12\delta_{\mathrm{tot}}(z)}
\qquad\text{with}\qquad
\delta_{\mathrm{tot}} = \delta_{\mathrm{EDE}} + \delta_{\mathrm{dr}},
\end{equation}
so that
\begin{equation}
\frac{\delta r_s}{r_s} \approx -\frac12 \frac{\int w_s(z)\,\delta_{\mathrm{tot}}(z)\,\dd z}{\int w_s(z)\,\dd z},
\qquad
w_s(z)\equiv \frac{c_s(z)}{H_{\mathrm{ref}}(z)}.
\label{eq:rs-weight}
\end{equation}
A narrow lobe centered near \(z_c\approx3500\) contributes efficiently because it sits near the epoch that most strongly weights the sound-horizon integral, while a small \(\Delta N_{\mathrm{eff}}\) contributes through the radiation term over the whole pre-recombination regime. This is also where the corridor's sharpest observational exposure lives: radiation at \(\DeltaNeff \approx 0.30\) over the whole pre-recombination regime is nominally visible to the Planck \(N_{\mathrm{eff}}\) fit (which prefers \(\DeltaNeff = 0.04 \pm 0.18\) in the baseline dictionary, the value carried by the codebook row) and to BBN. The corridor therefore stands or falls with the early-dark-energy-lobe degeneracy direction of the joint CMB+BBN likelihood absorbing the extra radiation; demonstrating this with a full Boltzmann fit is an explicit open requirement attached to the corridor-uniqueness conjecture, not an established fact.

Rather than pretend that the exact weighting coefficients have already been fixed by a full Boltzmann likelihood, the paper should state the working linear-response form explicitly:
\begin{equation}
 H_0 \simeq H_0^{\mathrm{ref}}\paren{1 + 0.13\,\Delta N_{\mathrm{eff}} + 0.40\,f_{\mathrm{EDE}}}.
\label{eq:h0-shift-formula}
\end{equation}
The numerical values used below, \(0.13\) per unit \(\DeltaNeff\) and \(0.40\) per unit \(f_{\mathrm{EDE}}\), are representative response slopes of the early-dark-energy literature \citep{Poulin2019}, not outputs of the GG ledger; deriving them from the inherited throat/$\Sigma_r$ sector is part of the corridor-uniqueness research target.
Inserting the narrow inherited values
\begin{equation}
\Delta N_{\mathrm{eff}}\approx0.30,
\qquad
f_{\mathrm{EDE}}\approx0.03,
\end{equation}
one obtains
\begin{equation}
H_0 \approx H_0^{\mathrm{ref}}\paren{1 + 0.13\times0.30 + 0.40\times0.03}
\approx 1.051\,H_0^{\mathrm{ref}}.
\end{equation}
For a representative reference early-universe value near \(67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\), this gives
\begin{equation}
H_0 \approx 70.8\pm0.9\,\mathrm{km\,s^{-1}\,Mpc^{-1}},
\label{eq:h0-70p8}
\end{equation}
which is already much closer to the local ladder corridor.

Two scientific features of this step are worth stressing. First, the shift is not bought with a broad parameter scan. It comes from a very narrow inherited source pair. Second, it is not an isolated Hubble-only patch. The same \(f_{\mathrm{EDE}}\) and \(\Delta N_{\mathrm{eff}}\) entries also reappear in the CMB lensing and structure-growth discussions below.

\subsubsection{The local lift: the baryonic graviphoton and the effective void correction}

The early-time correction brings the CMB-inferred value upward but does not by itself guarantee complete agreement with local determinations. The second proposed ingredient is late and local rather than early and global. Because baryon number is gauged by an ultra-weak long-range \(\mathrm{U}(1)_B\) graviphoton, baryonic overdensities do not merely gravitate; they also experience a very weak repulsive correction. In the present notes that correction is interpreted as inflating a mild local underdensity, effectively a baryonic local void of order gigaparsec scale.

The purpose of this module is not to replace the entire low-redshift cosmology by a radically inhomogeneous model. Its purpose is narrower. It asks whether the local calibration route could sit inside a region whose effective background is slightly lifted relative to the global mean. At schematic level one can write a local Friedmann-like relation
\begin{equation}
 H_{\mathrm{loc}}^2 = H_{\mathrm{bg}}^2 + \frac{8\pi G}{3}\,\delta\rho_{B}^{\mathrm{eff}},
\label{eq:h-loc}
\end{equation}
where \(\delta\rho_B^{\mathrm{eff}}\) is not an ordinary additional fluid inserted by hand, but the effective baryonic-pressure correction induced by the gauged baryon sector and the local environmental realization of it.

The practical claim is modest: a local lift of order a few percent,
\begin{equation}
\frac{H_{\mathrm{loc}} - H_{\mathrm{bg}}}{H_{\mathrm{bg}}} \sim 0.03\text{--}0.05,
\end{equation}
is enough to bridge the residual gap between the early-corrected value in Eq.~\eqref{eq:h0-70p8} and the local distance-ladder values usually quoted in the low-\(70\)'s. An honest accounting must record two facts. First, no quantitative GG derivation of \(\epsilon_{\mathrm{local}}\) exists yet; the interval \([0.03, 0.05]\) is a phenomenological requirement, not a result. Second, a local expansion-rate lift of this size from a real underdensity is strongly constrained by supernova counts and large-scale isotropy; a first-pass estimate from the \(U(1)_B\) coupling with the observed local baryonic contrast yields \(\epsilon_{\mathrm{local}} \sim 10^{-3}\), far short of the requirement. Closing or honestly retiring this gap is a declared open problem of the corridor module (and of the companion paper C2-Hubble; \citealt{ArisakaC2Hubble}). The scientific value of this proposal is that it does not introduce a new late-time dark-energy species or a wholesale modification of gravity. It reuses the already inherited \(\mathrm{U}(1)_B\) sector.

This is also the correct place to state the main caution. Local-environment corrections can easily become a dumping ground for poorly controlled phenomenology. The present framework should resist that temptation. The local-void lift is viable only if it remains mild, if it does not spoil the global background fit, and if it can be tied quantitatively to the same baryonic gauge sector used elsewhere. The paper should say this directly.



\subsubsection{Compact numerical closure of the legacy estimate}

To summarize the Hubble module in one line, write
\begin{equation}
H_0^{\mathrm{late}} \approx H_0^{\mathrm{ref}}\paren{1 + \epsilon_{\mathrm{early}}}\paren{1 + \epsilon_{\mathrm{local}}},
\end{equation}
with
\begin{equation}
\epsilon_{\mathrm{early}} \equiv 0.13\,\Delta N_{\mathrm{eff}} + 0.40\,f_{\mathrm{EDE}},
\qquad
\epsilon_{\mathrm{local}} \sim 0.03\text{--}0.05.
\label{eq:eps-h0}
\end{equation}
Using the inherited values gives
\begin{equation}
\epsilon_{\mathrm{early}} \approx 0.039 + 0.012 = 0.051,
\end{equation}
so that
\begin{equation}
H_0^{\mathrm{early-corrected}} \approx 1.051\,H_0^{\mathrm{ref}}.
\end{equation}
A subsequent local correction of order \(3\%\) then yields
\begin{equation}
H_0^{\mathrm{late}} \approx 1.082\,H_0^{\mathrm{ref}},
\end{equation}
which indeed lands near the low-\(70\)'s if \(H_0^{\mathrm{ref}}\) begins in the high-\(60\)'s. The important point is not the final decimal place. The important point is that the needed shift is of the correct order and comes from two concrete inherited mechanisms rather than from a free phenomenological menu. 

\section{Topological background and the pi-locked inventory}
\label{appA:pilock}

\GGteachbox{The mathematics behind the boldest claim in the paper:
differential forms, Pontryagin and Chern numbers, and why characteristic
integrals are integers, taught from scratch; the \OGNfive{} ledger; the
unit-trace closure and the \(\Omm = 1/\pi\) argument exactly as the
paper states it, with its honesty notes in full; and the ground-state
reading of Part~X --- the inventory as the readout of the one cosmic
attractor, the same uniqueness that fixes the electron.}
\GGreadalong{No cosmology textbook covers this material --- that is the
point; for the differential-geometry background, any modern text on
geometry in physics suffices, and Appendix~L is self-contained.}

This appendix supplies, from scratch and at the level a first-year graduate
student can follow line by line, the topological machinery standing behind the
$\pi$-locked cosmic inventory of Theorem~\ref{thm:omega-m-pi-topological}. It is
the honesty-critical appendix of the paper. Its job is twofold and the two jobs
must not be confused. First, it teaches the \emph{standard} differential-geometric
toolkit --- forms, the wedge product, the field-strength two-form, the
Pontryagin density, characteristic classes, and the integrality of characteristic
numbers --- rigorously and completely, because that toolkit is genuinely sound and
genuinely inherited. Second, it presents the inventory
$(\Omm,\OmL,\Omb) = (1/\pi,\,1-1/\pi,\,1/(2\pi^2))$ for exactly what it is: a
\emph{conjectured} Derived-conditional ($\mathrm{D}^*$) identity holding under the
closed-packet-integration conjecture (Conjecture~VIII), \emph{not} a proved
theorem. The printed Step-4 cancellation does not close; the required
normalization has not been exhibited; and the uniqueness argument leans, at one
step, on empirical preference that a topological proof is not entitled to use.
Nothing in this appendix manufactures the missing closure. The reader should come
away able to do the standard topology cleanly, and able to state precisely why
$1/\pi$ cannot be a topological invariant in the standard sense --- so that the
result stands as a structural ansatz of high empirical precision, audited
honestly, rather than as a theorem dressed up as one.

\medskip
\textbf{The three steps of this appendix.} As in Appendix~E:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard topology (\S\ref{appA:pilock-forms} onward).}
Differential forms, the field-strength two-form, the Pontryagin density,
characteristic classes and the integrality of their integrals, and the
\OGNfive{} integer ledger.
\item \textbf{What standard cosmology cannot explain
(\S\ref{appA:pilock-coincidence}).} The density fractions
$(\Omm,\OmL,\Omb)$ are measured dials, and their near-equality today ---
the cosmic coincidence --- is left unexplained.
\item \textbf{The geometric proposal (\S\ref{appA:pilock-unittrace}
onward).} The $\pi$-locked inventory $\Omm=1/\pi$, $\OmL=1-1/\pi$,
$\Omb=1/(2\pi^2)$ --- set out with full honesty: which steps of the
argument are sound, which do not yet close, and why the result is carried
as a high-precision \emph{conditional ansatz} ($\mathrm{D}^*$,
Conjecture~VIII) rather than a theorem.
\end{enumerate}

\subsection{Differential forms and the wedge product}
\label{appA:pilock-forms}

We begin with the only piece of machinery that everything else rests on: the
exterior algebra of differential forms. Fix a smooth manifold $\mathcal{M}$ of
dimension $n$ with local coordinates $x^1,\dots,x^n$. A \emph{differential
$p$-form} is a totally antisymmetric, smoothly varying multilinear object that
eats $p$ tangent vectors at each point and returns a number. In a coordinate
patch every $p$-form is a sum
\[
 \omega \;=\; \frac{1}{p!}\,\omega_{\mu_1\cdots\mu_p}(x)\,
 \dd x^{\mu_1}\wedge\cdots\wedge\dd x^{\mu_p},
\]
with the component array $\omega_{\mu_1\cdots\mu_p}$ totally antisymmetric in its
indices. A $0$-form is just a smooth function; a $1$-form is the familiar object
$A_\mu\,\dd x^\mu$ that one integrates along a curve; a top-form ($p=n$) is what
one integrates over the whole manifold. The antisymmetry is the entire point:
forms are the objects built to be integrated, because the Jacobian that appears
under a change of variables in a multiple integral is exactly the determinant
that antisymmetry encodes.

The \emph{wedge product} $\wedge$ multiplies a $p$-form by a $q$-form to give a
$(p+q)$-form. On the coordinate basis it is the antisymmetrized tensor product:
$\dd x^\mu \wedge \dd x^\nu = -\,\dd x^\nu \wedge \dd x^\mu$, so in particular
$\dd x^\mu \wedge \dd x^\mu = 0$. The product is associative and bilinear, and it
obeys the graded-commutativity rule
\[
 \alpha \wedge \beta \;=\; (-1)^{pq}\,\beta \wedge \alpha
 \qquad (\alpha\ \text{a }p\text{-form},\ \beta\ \text{a }q\text{-form}).
\]
Two consequences are worth fixing in the reader's mind now, because the
$\Omm=1/\pi$ argument uses both. First, the wedge of any form of odd degree with
itself vanishes: if $p$ is odd, $\alpha\wedge\alpha = -\alpha\wedge\alpha = 0$.
Second, the wedge of an \emph{even}-degree form with itself need not vanish ---
and this is exactly why a four-form built from two two-forms, such as the
$\operatorname{tr}(F\wedge F)$ below, can carry non-trivial information.

The \emph{exterior derivative} $\dd$ raises the degree of a form by one. On a
function it is the ordinary gradient $\dd f = (\partial_\mu f)\,\dd x^\mu$; on a
general $p$-form it acts component-wise and antisymmetrizes,
\[
 \dd\omega \;=\; \frac{1}{p!}\,\partial_\nu \omega_{\mu_1\cdots\mu_p}\,
 \dd x^\nu \wedge \dd x^{\mu_1}\wedge\cdots\wedge\dd x^{\mu_p}.
\]
Its single defining property is nilpotency, $\dd^2 = 0$, which follows from the
symmetry of second partial derivatives clashing with the antisymmetry of the
wedge. A form with $\dd\omega = 0$ is called \emph{closed}; a form that can be
written as $\omega = \dd\eta$ for some $\eta$ is called \emph{exact}. Every exact
form is closed (because $\dd^2=0$), but on a topologically non-trivial manifold
not every closed form is exact. The failure of ``closed implies exact'' is
measured by the de~Rham cohomology groups $H^p(\mathcal{M},\mathbb{R})$, defined
as closed $p$-forms modulo exact $p$-forms. These groups are topological
invariants of $\mathcal{M}$ --- they do not depend on the metric or on any
coordinate choice --- and they are the arena in which characteristic classes live.
This is the precise sense in which Step~1 of the proof outline asserts that the
closed four-form $X_4$ ``represents a class in $H^4(\mathcal{M}_4,\mathbb{R})$.''

Finally, \emph{Stokes' theorem} ties the exterior derivative to integration:
\[
 \int_{\mathcal{M}} \dd\eta \;=\; \int_{\partial\mathcal{M}} \eta,
\]
where $\partial\mathcal{M}$ is the boundary. On a \emph{closed} manifold --- compact
and without boundary, $\partial\mathcal{M}=\varnothing$ --- the right-hand side is
zero, so the integral of any exact form over a closed manifold vanishes. It
follows that the integral of a closed top-form over a closed manifold depends
only on the cohomology class of that form, not on the representative. This is the
structural reason characteristic numbers exist at all, and we return to it in
\S\ref{appA:pilock-charclass}.

\subsection{The field strength two-form $F$ and the four-form
$\operatorname{tr}(F\wedge F)$}
\label{appA:pilock-fieldstrength}

A gauge field is a connection on a principal bundle; locally it is encoded by a
\emph{connection one-form} $A = A_\mu\,\dd x^\mu$, valued in the Lie algebra of
the gauge group (for a non-Abelian group, $A_\mu = A_\mu^a\,T^a$ with $T^a$ the
generators). Its \emph{curvature}, the field-strength two-form, is
\[
 F \;=\; \dd A + A\wedge A,
\]
which in components reproduces the familiar
$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$. For an
Abelian field (electromagnetism, or the $U(1)_B$ that fixes the baryon fraction
below) the commutator term drops and $F = \dd A$ is simply closed. For the
gravitational case the analogue of $F$ is the Riemann \emph{curvature two-form}
$R^a{}_b$, built from the spin connection in exactly the same way; the
trace-squared object $\operatorname{tr} R^2 = \operatorname{tr}(R\wedge R)$ that
appears in $X_4$ is its gravitational counterpart.

The object we care about is the four-form $\operatorname{tr}(F\wedge F)$. Here the
trace is taken over the Lie-algebra (matrix) indices, while the wedge handles the
spacetime form indices. Because $F$ is a two-form (even degree), $F\wedge F$ is a
genuine non-vanishing four-form, and the trace makes it gauge-invariant: under a
gauge transformation $F \to g F g^{-1}$, and the cyclicity of the trace gives
$\operatorname{tr}(gFg^{-1}\wedge gFg^{-1}) = \operatorname{tr}(F\wedge F)$. So
$\operatorname{tr}(F\wedge F)$ is a gauge-invariant four-form built locally from
the gauge field. Its two crucial properties are:

\begin{enumerate}[leftmargin=2.0em,label=(\roman*)]
\item \textbf{It is closed.} A short computation using the Bianchi identity
$\dd F + A\wedge F - F\wedge A = 0$ gives
$\dd\operatorname{tr}(F\wedge F) = 0$ identically. So
$\operatorname{tr}(F\wedge F)$ defines a class in $H^4(\mathcal{M},\mathbb{R})$.
\item \textbf{It is locally exact --- it is the derivative of the Chern--Simons
three-form.} One can write
$\operatorname{tr}(F\wedge F) = \dd\,\omega_{\mathrm{CS}}$ with
$\omega_{\mathrm{CS}} = \operatorname{tr}\!\big(A\wedge \dd A
+ \tfrac{2}{3} A\wedge A\wedge A\big)$. By Stokes, this means its integral over a
closed manifold is insensitive to local deformations of $A$; what survives is a
global, topological quantity.
\end{enumerate}

Property (i) is what licenses Step~1 of the proof outline; property (ii) is what
makes the integral in Step~3 a \emph{number} attached to the bundle rather than to
the particular field configuration. In the paper's $X_4$ the relevant trace-squared
term is the gravitational $\operatorname{tr} R^2$, but the structure is identical:
it is a closed, gauge-invariant four-form whose integral is a topological
quantity. This is the honest, standard part of the story, and it holds without
qualification.

\subsection{The (first) Pontryagin density and Pontryagin / Chern numbers}
\label{appA:pilock-pontryagin}

We now name the specific invariants the paper uses. For a gauge curvature $F$ the
\emph{second Chern character density} is $\tfrac{1}{8\pi^2}\operatorname{tr}(F\wedge
F)$; integrated over a closed four-manifold it gives the (second Chern /
instanton) number,
\[
 c_2 \;=\; \frac{1}{8\pi^2}\int_{\mathcal{M}_4}\operatorname{tr}(F\wedge F)
 \;\in\; \mathbb{Z}.
\]
For the tangent bundle of a Riemannian four-manifold the analogous object built
from the curvature two-form $R$ is the \emph{first Pontryagin density}
$p_1 \sim \operatorname{tr}(R\wedge R)$, and its integral is the \emph{first
Pontryagin number}
\[
 p_1[\mathcal{M}_4] \;=\; \frac{1}{8\pi^2}\int_{\mathcal{M}_4}
 \operatorname{tr}(R\wedge R) \;\in\; \mathbb{Z}.
\]
This is precisely the relation invoked in Step~3 of the proof outline, there
written in the equivalent form
\[
 \int_{\mathcal{M}_4}\operatorname{tr} R^2 \;=\; 8\pi^2 \, p_1[\mathcal{M}_4].
\]
The factor $8\pi^2$ is not optional decoration: it is the universal normalization
constant that converts the geometric integral of the curvature four-form into the
\emph{integer} $p_1$. The paper's Step~3 sets $p_1 = N = 3$ on the
\OGNfive{}-structured boundary slice, so that
$\int_{\mathcal{M}_4}\operatorname{tr} R^2 = 24\pi^2$. The reader should hold onto
the structural fact, because it is the crux of the honesty discussion below: the
$\pi$'s live entirely in the normalization constant $8\pi^2$, while the
\emph{invariant} --- the thing topology actually pins down --- is the bare integer
$p_1$.

\subsection{Characteristic classes, and why their integrals are integers or
rationals in the standard setting}
\label{appA:pilock-charclass}

A \emph{characteristic class} is a cohomology class assigned to a bundle in a way
that is natural under pullback --- it depends only on the bundle's topological type,
not on the connection chosen to compute it. The Chern classes $c_k$ (for complex
bundles), the Pontryagin classes $p_k$ (for real bundles), and the Euler class $e$
are the standard examples. A \emph{characteristic number} is what one gets by
integrating a top-degree characteristic class over a closed oriented manifold.

The central fact a first-year student should internalize is this:

\begin{quote}
\emph{Characteristic numbers of closed oriented manifolds are integers (or, in a
few mixed combinations such as the signature via the Hirzebruch formula,
rationals with controlled denominators). They are never irrational.}
\end{quote}

Why? The clean conceptual reason is that the de~Rham representatives we integrate
are images, under the de~Rham isomorphism, of \emph{integer} cohomology classes
$H^p(\mathcal{M},\mathbb{Z})$. Integrating an integral class against the
fundamental cycle of a closed oriented manifold is a pairing
$H^p(\mathcal{M},\mathbb{Z}) \times H_p(\mathcal{M},\mathbb{Z}) \to \mathbb{Z}$,
which by construction lands in $\mathbb{Z}$. The normalization constants ($2\pi$
for a first Chern number, $8\pi^2$ for $c_2$ or $p_1$, and so on) are chosen
\emph{precisely so that} this integer pairing comes out as the bare integer with
the $\pi$'s divided away. Equivalently: index theory (Atiyah--Singer) expresses
these integrals as the index of an elliptic operator, which counts a difference of
dimensions of solution spaces --- manifestly an integer. Where rationals appear
(the $\hat A$-genus, the $L$-polynomial), they appear as rational \emph{linear
combinations} of integer characteristic numbers, with denominators fixed by
universal constants, never as transcendental numbers.

This is the rigorous standard fact, and it is exactly the fact the honesty note
must respect. An irrational number such as $1/\pi$ is, by this theorem,
\emph{not available} as a characteristic number of a four-manifold. Whatever
$\Omm = 1/\pi$ is, it is not a Pontryagin number, not a Chern number, not an
Euler characteristic, not an instanton number. We return to this in
\S\ref{appA:pilock-whyirrational} and make the point sharp.

\subsection{The \OGNfive{} integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$}
\label{appA:pilock-ogn5}

The arithmetic raw material of the inventory is the \OGNfive{} integer ledger
inherited from \citet{ArisakaA4Euclid}:
\[
 (M,N,L;\,J,K) \;=\; (1,3,5;\,7,35).
\]
These five integers are not posited inside the present paper; they are proved to
theorem status upstream (the Diophantine cascade closing through the dual-lock
identity $K = LJ = M^2 + N^2 + L^2$, i.e.\ $35 = 5\cdot 7 = 1 + 9 + 25$), and the
present paper inherits them as fixed integer data. They are genuine integers, and
on that point there is nothing conditional: the inheritance is sound. In the
proof outline they enter as follows. The matter cell of the boundary slice is
assigned a rational weight built from these integers --- specifically the weight
$M/(LN) = 1/(5\cdot 3) = 1/15$ before any $\pi$-rescaling. The integer
$N = 3$ is read as the first Pontryagin number $p_1$ in Step~3. The integers $J=7$
and $K=35$ appear elsewhere in the cosmological audit (the recombination corridor
redshift $z_* = e^7$ and the damping-knee relation $\ell_D = K(2\pi)^2$), and in
Step~4 they are arranged to combine in the denominator of the ``unit-trace
closure'' to produce the integer $15$, designed to cancel the matter-cell weight
$1/15$.

The essential teaching point is to keep two things separate. The integers
themselves are theorems; their \emph{map} to a cosmic fraction is not. Everything
through this subsection --- forms, $\operatorname{tr}(F\wedge F)$, the integer
Pontryagin number, the integer ledger --- is standard and inherited. The
conditional content lives entirely in the step that tries to turn these integers
into $1/\pi$, which we now examine.

\subsection{What standard cosmology cannot explain: the coincidence problem}
\label{appA:pilock-coincidence}

The topology of Step~1 is standard and exact. Before turning it on the
cosmic inventory, it is worth being clear about what standard cosmology
does \emph{not} say about that inventory. The three density fractions are
measured numbers,
\[
\Omm\approx0.315,\qquad \OmL\approx0.685,\qquad \Omb\approx0.0493,
\]
and $\LCDM$ takes each as a free parameter fit to the data; it offers no
reason they hold these values rather than others. The sharpest form of the
puzzle is the \emph{cosmic coincidence}: matter and dark energy happen to
be of comparable magnitude \emph{today} ($\Omm/\OmL\approx0.46$), even
though under a free cosmological constant their ratio sweeps across many
orders of magnitude over cosmic history, so the near-equality would be a
special property of the present moment. Standard cosmology notes this and
moves on; it has no structural account. The geometric proposal of Step~3
is an attempt to supply one --- to make the fractions forced outputs of
the topology rather than fitted dials --- and the discipline of this
appendix is to say exactly how far that attempt succeeds, and where it
does not.

\subsection{The idea of a ``unit-trace closure''}
\label{appA:pilock-unittrace}

The phrase \emph{unit-trace closure} names the proposed normalization convention
under which the \OGNfive{} cells of the boundary slice $\mathcal{M}_4$ are assigned
weights that sum, over all cells, to one. The intuition is that the four-form
$X_4$, integrated over the whole slice, defines a total ``packet'' whose
sub-cell contributions are the cosmic density fractions: matter, dark energy, and
baryons are then \emph{shares} of a single normalized closed packet rather than
independent quantities. Under this convention the matter share would be the
matter-cell integral divided by the total-packet integral, and the unit-trace
closure is the requirement that this division is well-posed and that the total
equals one (the cosmic sum rule $\Omm + \OmL = 1$ in the flat, radiation-and-
curvature-negligible idealization).

This is an appealing organizing idea, and it is the structural backbone of the
post-hoc-fit defense: a single normalized packet would force the three fractions
as a \emph{coupled set}, not one at a time. But the reader must note carefully
what is being assumed. ``Unit-trace closure'' is, at present, a \emph{name for a
desired normalization}, not an exhibited construction. To make it rigorous one
would have to (i) exhibit $\mathcal{M}_4$ as a specific closed oriented manifold
with a specified topology, (ii) name the precise characteristic class whose
density is integrated over each cell, and (iii) define the measure and the cell
decomposition so that the total integral is genuinely normalized to one. None of
(i)--(iii) is supplied. The closed-packet integration is named, but not defined to
the standard a referee is entitled to demand. We flag this here because it is
exactly the gap that the honesty note records and that keeps the inventory at
$\mathrm{D}^*$.

\subsection{The $\Omm = 1/\pi$ argument as the paper states it (Steps 1--5)}
\label{appA:pilock-steps}

We now reproduce the five-step argument of
Theorem~\ref{thm:omega-m-pi-topological}, in the same order as the body of the
paper, so that the reader can see precisely where it is solid and precisely where
it does not close. The setup is the closed four-form fixed by the \BIsix{} modified
Bianchi identity of \citet{ArisakaA3BI6},
\[
 X_4 \;=\; \operatorname{tr} R^2 \;-\; \operatorname{Tr} F_5^2
 \;-\; \tfrac{2}{5} F_B^2 \;-\; \tfrac{2}{5} F_{Qi}^2,
\]
with the coefficient vector $(1,-1,-2/5,-2/5)$ proved unique by the
anomaly-cancellation theorem of \citet{ArisakaA3BI6}. The matter fraction $\Omm$
is taken to be the matter-sourced share of the total stress-energy budget,
projected onto the four-dimensional cosmological-boundary slice $\mathcal{M}_4$.

\paragraph{Step 1 (closure of $X_4$).} \BIsix{} gives $\dd X_4 = 0$ on
$\mathcal{M}_4$, so $X_4$ represents a class in $H^4(\mathcal{M}_4,\mathbb{R})$.
The class is non-trivial because the gravitational term $\operatorname{tr} R^2$
carries a non-trivial Pontryagin class on the curved slice. \emph{This step is
sound}: it is the standard fact of \S\ref{appA:pilock-fieldstrength}.

\paragraph{Step 2 (closed-packet structure).} By the \OGNfive{} cell decomposition,
$\mathcal{M}_4$ is taken to split into finitely many closed cells indexed by
$(M,N,L;J,K)=(1,3,5;7,35)$, the matter cell carrying weight
$M/(LN) = 1/15$ relative to the total cell volume \emph{before} the $\pi$-rescaling.
\emph{This step assumes} the cell decomposition and weight assignment that
\S\ref{appA:pilock-unittrace} flagged as named-but-not-defined.

\paragraph{Step 3 (Pontryagin normalization).} The Chern--Pontryagin integral
$\int_{\mathcal{M}_4}\operatorname{tr} R^2 = 8\pi^2 \, p_1[\mathcal{M}_4]$ with
$p_1 = N = 3$ evaluates to $24\pi^2$. \emph{The relation itself is the standard
fact of \S\ref{appA:pilock-pontryagin}}; what is assumed here is that the
matter-cell contribution ``carries this normalization factor'' when projected onto
the present epoch, i.e.\ that the $8\pi^2$ converts into a single residual factor
of $\pi$ after the cell bookkeeping. \emph{(This expectation is misdirected: the $8\pi^2$ is consumed by the \GEP{} definition of \ASix{}, returning the integer, and \emph{no} residual $\pi$ descends from this step. The $\pi$ is not a Pontryagin leftover and not the reality factor of
Lemma VIII-b$'$ (which is $2$, not $\pi$); it enters as a \emph{fiber volume} in
the measure ratio of the fiber-measure resolution below (\S\ref{appA:pilock-hopf}).
The historical wording is kept above because the printed Step-4 arithmetic that
follows was built on it.)}

\paragraph{Step 4 (closed-packet integration).} Combining Steps~2 and 3, the
paper writes
\[
 \Omm \;\overset{?}{=}\; \frac{(\text{matter-cell weight})}{(\text{unit-trace closure})}
 \cdot \frac{1}{\pi}
 \;=\; \frac{1/15}{(K/J\cdot L/N)\cdot(1/N)}\cdot\frac{1}{\pi}
 \;=\; \frac{1/15}{25/9}\cdot\frac{1}{\pi}
 \;=\; \frac{3}{125}\,\frac{1}{\pi}
 \;\ne\; \frac{1}{\pi}.
\]
For the boxed value $1/\pi$ to follow, the unit-trace denominator would have to
equal the matter-cell weight $1/15$ exactly, so that the two cancel. But the
combination as written evaluates to
$(K/J\cdot L/N)\cdot(1/N) = (5\cdot\tfrac{5}{3})\cdot\tfrac{1}{3} = \tfrac{25}{9}$,
\emph{not} $1/15$; the printed arrangement therefore falls short of $1/\pi$ by the
factor $(25/9)/(1/15) = 125/3 \approx 42$. \emph{This is the step that does not
close}, and we exhibit the gap here rather than paper over it: the normalization
that would convert the standard $8\pi^2$ of Step~3 into precisely one residual
$\pi$ --- making the denominator $1/15$, no more and no less --- is not exhibited
anywhere in the cohort. As the honesty note records, this route does not close --- and the projection lemma establishes \emph{why it cannot}: the sought cancellation is an epoch-independent closure, and by the dilution no-go (superselection freezes the packet's integer, not its energy record, which dilutes as $a^{-3}$ against a frozen equipartition rate) no arrangement of ledger constants can equal a diluting quantity at more than one instant. Steps~3--4 close instead through the measure dictionary (Lemma VIII-b$'$ + the epoch identity $u = 1 \Leftrightarrow \Omega_{m,0} = 1/\pi$); the $125/3$ gap above is thereby \emph{explained} rather than open. \citet{ArisakaA4Euclid} supplies the \OGNfive{} integers used here but \emph{not} the closed-packet integration itself.

\paragraph{Step 5 (uniqueness against competing rational functions of $\pi$).}
The paper argues that among rational-in-$\pi$ candidates compatible with the
\OGNfive{} cell structure, only $1/\pi$ survives, ruling out $1/(3\pi)\approx 0.106$,
$1/(2\pi)\approx 0.159$, and $1/\pi^2 \approx 0.101$. \emph{Part of this is
structural and does survive}: the candidate $1/\pi^2$ would require a quadratic
Pontryagin closure $X_4\wedge X_4$ integrated over an eight-manifold, which the
four-dimensional boundary slice does not support; that dimensional sub-argument is
genuine. \emph{But the discrimination against $1/(2\pi)$ leans on the empirical
comparable-magnitude condition} $\Omm \sim \OmL$ (the data prefer
$\Omm/\OmL \approx 0.46$), and an honest topological proof is not entitled to use
the data to select among candidates. This is the second place the printed
argument fails to close on purely topological grounds --- \emph{and it is exactly
the place the fiber-measure resolution below (\S\ref{appA:pilock-hopf}) repairs},
by selecting $1/\pi$ from the geometry of the boundary measure with the data
nowhere in the choice.

\subsection{The residual-Pontryagin-normalization reading, stated honestly}
\label{appA:pilock-residual}

The correct, deflated description --- the one the paper's codebook actually carries
--- is that $\Omm = 1/\pi$ is a \emph{residual Pontryagin normalization}, not a
characteristic number. The integer is the invariant: the matter-cell weight $1/15$
and the Pontryagin number $p_1 = 3$ are genuine integers from the \OGNfive{} ledger and
the standard integrality theorem. The $\pi$ enters only through the universal
normalization constant $8\pi^2$ that relates the integral of the de~Rham
representative $\operatorname{tr} R^2$ to that integer --- it enters as the
\emph{normalization of the de~Rham representative}, not as a topological invariant
in its own right. Calling $\Omm = 1/\pi$ a ``topological invariant'' (as the title
of Theorem~\ref{thm:omega-m-pi-topological} unfortunately does) overreaches
relative to the body; the governing text is the Proof-status note and the codebook,
and neither calls it an invariant.

\subsection{The fiber-measure resolution: $\Omm = 1/\pi$ as the Hopf ratio}
\label{appA:pilock-hopf}

The two places the printed argument fails --- the unexhibited Step-4 normalization
and the data-dependent Step-5 selection --- close together once one stops looking
for an epoch-independent \emph{arithmetic} of the ledger integers (which the
dilution no-go forbids) and reads $\Omm$ for what the measure dictionary says it
is: a ratio of \emph{fiber volumes} on the boundary holonomy. This is the
resolution the codebook audit (\S\ref{sec:audit-d-star-cosmos}) carries, and it
is worth teaching in full because it is both simpler and more honest than the
route above.

\paragraph{The geometry.} The electroweak $\mathrm{SU}(2)$ whose \GSSB{} defines
the boundary is, as a group manifold, the unit three-sphere $S^3$, of volume
$\mathrm{Vol}(S^3)=2\pi^2$. The compact phase circle $S^1_\theta$ that carries the
matter sector's charge sits inside it as the \emph{Hopf fiber} of the canonical
fibration $S^1\hookrightarrow S^3\to S^2$ --- a great circle, $\mathrm{Vol}(S^1)=2\pi$.
There is no freedom in either identification: the Hopf fibration is unique, and
the fiber is the phase $\mathrm{U}(1)$. The fibration is drawn in
Figure~\ref{fig:hopf-fibration} (main text).

\paragraph{The ratio.} Under the frozen stationary \GDOS{} measure of Lemma
VIII-a, each sector's present-epoch share is its fiber volume over the total
holonomy volume, so
\[
 \Omm \;=\; \frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)}
 \;=\; \frac{2\pi}{2\pi^2} \;=\; \frac{1}{\pi} \qquad(\text{exact}),
\]
with the $\pi$ descending from the denominator volume $2\pi^2$ and the numerator's
$2\pi$ canceling one power. Dark energy is the uncovered remainder,
$\OmL=(2\pi^2-2\pi)/2\pi^2 = 1-1/\pi$; baryons are one integer $\mathrm{U}(1)_B$
unit per revolution, $\Omb = \Omm/\mathrm{Vol}(S^1) = 1/(2\pi^2) = 1/\mathrm{Vol}(S^3)$.
The student should verify $2\pi/2\pi^2 = 1/\pi$ and $1/(2\pi^2) = 1/\mathrm{Vol}(S^3)$
by hand: the three fractions are one fibration, not three fits.

\paragraph{The competitors die by geometry.} And this is the honest payoff: the
alternatives fall with no appeal to the data. $1/(3\pi)$ needs a holonomy volume
$6\pi^2$ that no manifold here has; $1/(2\pi)$ is the baryon-per-fiber ratio, not
the matter/total ratio; $1/\pi^2$ needs an $S^5$ the four-dimensional boundary does
not carry. Every candidate the printed Step 5 rejected with ``the data prefer
$0.46$'' is rejected here by ``there is no fiber of that volume.''

\paragraph{What remains open, honestly.} The resolution is not a finished theorem.
It rests on one \emph{geometric} lemma --- that the \GDOS{} measure weights exactly
the Hopf circle as the matter fiber and $\mathrm{Vol}(S^3)$ as the total (VIII-b$''$,
the unit-packet normalization), together with the canonical unit normalization of
both spheres. That lemma is provable or falsifiable on the six-dimensional bulk
construction, and it uses no measured density; so the status of $\Omm=1/\pi$ has
moved from ``an unexhibited arithmetic that also leaned on the data'' to
``derived-conditional on one fiber-identification lemma, with the data nowhere in
the selection.'' The codebook keeps the D$^*$ tag; what has changed is the
\emph{content} of the conditional, and the removal of the circularity.

One more sentence now belongs here: the body of the paper
(\S\ref{ssec:de-two-fibres}) reads this resolution on the
campaign's two-fiber anatomy --- the share is the circle's (a ratio
of fiber volumes, epoch-frozen), while the terminus's \emph{scale}
is the interval's ($e^{-2K}$, the seesaw) --- so the two numbers of
dark energy come from the two fibers, one each.

\subsection{Why an irrational number cannot be a topological invariant in the
standard sense}
\label{appA:pilock-whyirrational}

It is worth stating the obstruction with full precision, because it is the
load-bearing honesty point of the appendix. In the standard setting a topological
invariant of the relevant kind --- a Pontryagin number, a Chern number, an Euler
characteristic, an instanton number --- arises from the integer pairing
$H^p(\mathcal{M},\mathbb{Z})\times H_p(\mathcal{M},\mathbb{Z}) \to \mathbb{Z}$, or
equivalently from an index of an elliptic operator (Atiyah--Singer), which counts a
difference of dimensions and is therefore an integer. The only way rationals enter
is through universal rational coefficients in characteristic-number combinations
(the $\hat A$-genus, the Hirzebruch $L$-polynomial), and those denominators are
fixed integers, never transcendentals. An \emph{irrational} number such as
$1/\pi = 0.31830\ldots$ simply does not lie in the value set of these pairings. It
cannot be a characteristic number, full stop. The reason is exactly the one
taught in \S\ref{appA:pilock-charclass}: the normalization constants ($2\pi$,
$8\pi^2$, $\ldots$) are \emph{defined} so as to cancel the transcendental factors
and leave an integer. To then declare that a left-over $1/\pi$ is itself the
invariant is to read the normalization constant as the answer --- a category error.

This is precisely why the result cannot be, and is not claimed to be, a theorem in
the topological-invariant sense. What it can honestly be is a structural
\emph{ansatz}: a proposal that the closed-packet integration, \emph{once defined
with an exhibited normalization}, would deliver $1/\pi$ as the residual factor. The
ansatz has remarkable empirical precision (see \S\ref{appA:pilock-numerical}), and
it has the appealing coupled-fraction structure of \S\ref{appA:pilock-unittrace}.
But high precision and structural appeal are not a proof, and the appendix does not
pretend otherwise. The missing normalization is missing; we do not manufacture it.

\subsection{Worked numerical example: the inventory against Planck}
\label{appA:pilock-numerical}

We evaluate the three locked fractions and compare them to the Planck central
values, framing each comparison honestly as a near-hit.

\paragraph{Matter fraction.}
\[
 \Omm \;=\; \frac{1}{\pi} \;=\; \frac{1}{3.14159265\ldots} \;=\; 0.31831,
\]
against the Planck value $\Omm \approx 0.315$. The fractional offset is
$(0.31831-0.315)/0.315 \approx +1.0\%$ --- a close near-hit, high by about one
percent.

\paragraph{Dark-energy fraction.}
\[
 \OmL \;=\; 1 - \frac{1}{\pi} \;=\; 1 - 0.31831 \;=\; 0.68169,
\]
against the Planck value $\OmL \approx 0.685$. The fractional offset is
$(0.68169-0.685)/0.685 \approx -0.5\%$, i.e.\ low by about half a percent. Note
that $\Omm$ and $\OmL$ are complementary by construction
($\Omm + \OmL = 1/\pi + (1-1/\pi) = 1$), so their two offsets are tied together by
the flatness sum rule; they are not independent near-hits.

\paragraph{Baryon fraction.}
\[
 \Omb \;=\; \frac{1}{2\pi^2} \;=\; \frac{1}{2\,(9.8696044\ldots)}
 \;=\; \frac{1}{19.739209} \;=\; 0.050661,
\]
against the Planck value $\Omb \approx 0.0493$. The fractional offset is
$(0.050661-0.0493)/0.0493 \approx +2.8\%$ --- the worst of the three, sitting about
$2.8\%$ high. (Here $\pi^2 = 9.8696$, and the appearance of
$2\pi^2 = \operatorname{Vol}(S^3) = 19.7392$ is the structural reason the baryon
fraction carries a different power of $\pi$ from the matter fraction; it is also
the entry most exposed to the convention-dependence objection, since powers of
$\pi$ float with the choice of volume normalization.)

\paragraph{Reading the table honestly.} All three are near-hits: $\sim+1.0\%$ on
$\Omm$, $\sim-0.5\%$ on $\OmL$, and $\sim+2.8\%$ on $\Omb$, with the baryon
fraction the one visibly outside the percent band. These are genuinely striking
agreements for parameter-free expressions, and they are the empirical reason the
ansatz is worth carrying. But three percent-level near-hits, one of them $2.8\%$
high, are an empirical \emph{audit}, not a derivation: they tell the reader the
ansatz is precise, not that it is proved. The precision is what makes the
$\mathrm{D}^*$ classification worth printing; it is not what would upgrade it to
$\mathrm{D}$.

\begin{table}[htbp]
\centering
\caption[The pi-locked fractions against Planck]{The pi-locked fractions set against the Planck values, with the offset in each case. Three percent-level near-hits, one of them 2.8 per cent high, are an empirical audit and not a derivation: they show the ansatz is precise, not that it is proved. That distinction is what makes the D-star classification worth printing, and it is not what would upgrade the classification to D. A reader who reads this table as confirmation has read it as the opposite of what it says.}
\label{tab:pi-fractions-audit}
\small
\begin{tabular}{lllll}
\toprule
Fraction & Locked expression & Value & Planck & Offset \\
\midrule
$\Omm$ & $1/\pi$        & $0.31831$ & $\approx 0.315$  & $+1.0\%$ \\
$\OmL$ & $1 - 1/\pi$    & $0.68169$ & $\approx 0.685$  & $-0.5\%$ \\
$\Omb$ & $1/(2\pi^2)$   & $0.050661$& $\approx 0.0493$ & $+2.8\%$ \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The identity reading (Part~X).}
The topological machinery above answers \emph{what value}; Part~X
answers \emph{why one value everywhere}, and the student should carry
both. A stratum generally admits many stable attractors (\GA{}'s); among
them exactly one is the global minimum of the \GFF{} ledger --- the
Ground Geometric Attractor, \GGA{} (\ASix{}; \AEleven{}). The
cosmological inventory is the present-epoch projection of that one
ground attractor: the same uniqueness that makes every electron
identical (\ANine{}) makes the composition of the universe the same on
every sightline. Identity and stability arrive together --- one deepest
basin to occupy, none deeper to decay to --- and the homogeneity of the
inventory is thereby a consequence, not a premise. The full side-by-side
of the three cosmological objects (the electron, the Kaxion, dark
energy) as one \GDOS{}\,+\,\GRIP{} construction is Table~\ref{tab:three-objects}
of Part~X; this appendix supplies the topological arithmetic that
Part~X's Section~\ref{sec:pi-locked-derivation} rests on.

\subsection{Status}
\label{appA:pilock-status}

The inventory $(\Omm,\OmL,\Omb) = (1/\pi,\,1-1/\pi,\,1/(2\pi^2))$ is carried as a
Derived-conditional ($\mathrm{D}^*$) identity under the
closed-packet-integration conjecture (Conjecture~VIII): the conjecture that the
closed-packet integral of the \BIsix{} four-form $X_4$ on the cosmological-boundary
stratum evaluates to $1/\pi$ with an exhibited normalization. The standard
topology taught in \S\S\ref{appA:pilock-forms}--\ref{appA:pilock-ogn5} is sound and
inherited: closed forms, the integrality of Pontryagin and Chern numbers, and the
\OGNfive{} integer ledger of \citet{ArisakaA4Euclid} are all on firm ground. The result
is what would, \emph{if it closed}, separate the \GGTheory{} program from a
free-parameter \LCDM{} fit: the fractions would be forced outputs rather than fitted
dials. But that ``if'' is exactly the open antecedent. What is
\emph{not} established is the map from those integers to the irrational fraction.
Concretely: the Step-4 cancellation does not close, because the normalization that
turns the standard $8\pi^2$ into a single residual $1/\pi$ has not been exhibited;
the uniqueness argument of Step~5 leans, against the $1/(2\pi)$ competitor, on an
empirical comparable-magnitude condition that a topological proof may not use; and
an irrational number cannot, in the standard sense of
\S\ref{appA:pilock-charclass}, be a characteristic number at all, so $1/\pi$ is at
best a residual normalization, not an invariant. The closed-packet integration is
named but not defined to the required rigor --- the manifold $\mathcal{M}_4$, the
characteristic class, and the measure are all reserved for future work. Until that calculation exhibits the normalization and removes the
empirical input from the uniqueness step, the three fractions remain $\mathrm{D}^*$,
and the result stands as a structural ansatz of high empirical precision --- not a
theorem.

%% ===================================================================
%% Appendix fragment for Arisaka-GG-A10-Cosmos
%% Cold dark matter from vacuum misalignment -- the Kaxion
%% Pedagogical appendix, first-year-graduate level, maximum depth.
%% Style model: appendix of Arisaka-GG-A8-QM-v12.0.tex.
%% ASCII only. Standalone-safe local macros via \providecommand.
%% ===================================================================

% --- Local macros (no-ops if the host preamble already defines them) ---
\providecommand{\GGTheory}{\ifmmode\mathsf{GG}\else\textsf{GG}\fi}
\providecommand{\GRIP}{\ifmmode\mathsf{\GRIP{}}\else\textsf{GRIP}\fi}
\providecommand{\Omm}{\ensuremath{\Omega_m}}
\providecommand{\Omb}{\ensuremath{\Omega_b}}
\providecommand{\Omdm}{\ensuremath{\Omega_{\mathrm{DM}}}}
\providecommand{\fK}{\ensuremath{f_{K}}}
\providecommand{\mK}{\ensuremath{m_{K}}}
\providecommand{\thetai}{\ensuremath{\theta_i}}
\providecommand{\chiQCD}{\ensuremath{\chi_{\mathrm{QCD}}}}
\providecommand{\Mpl}{\ensuremath{M_{\mathrm{Pl}}}}


\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{L.2.1. Show that the abelian
Pontryagin density is locally a total derivative.}

\noindent\emph{Solution.}  Step 1: for an abelian field
strength $F = dA$, consider $F\wedge F$.  Step 2: compute
$d(A \wedge F) = dA\wedge F - A\wedge dF = F\wedge F$ (using
$dF = ddA = 0$).  Step 3: hence $F\wedge F = d(A\wedge F)$ ---
locally exact --- so its integral over a closed manifold
receives contributions only from the topology that obstructs
writing $A$ globally (the non-abelian version adds the
Chern--Simons cubic term).  Physics intuition: the Pontryagin
number is invisible to local physics --- no small deformation
of the fields changes it --- which is exactly what qualifies it
to carry \emph{permanent} information: the ledger's integers
live where the dynamics cannot reach.

\medskip\noindent\textbf{L.3.1. Derive Dirac's quantization:
the first Chern number of a $U(1)$ bundle on $S^2$ is an
integer.}

\noindent\emph{Solution.}  Step 1: cover $S^2$ by two caps
overlapping on the equator; on the overlap the gauge potentials
differ by a transition function, $A_+ - A_- = d\chi$, with
$e^{i\chi}$ single-valued on the equatorial circle.  Step 2:
$\tfrac{1}{2\pi}\oint_{\rm eq} d\chi = n \in \mathbb{Z}$ ---
the winding number of the transition function.  Step 3: by
Stokes on each cap, $\tfrac{1}{2\pi}\int_{S^2} F =
\tfrac{1}{2\pi}\oint (A_+ - A_-) = n$: the flux is quantized
because the two descriptions must agree where they meet.
Physics intuition: quantization here is bookkeeping
consistency, not dynamics --- the same single-valuedness that
quantizes the monopole charge is what makes the ledger's
$(M,N,L;J,K)$ integers \emph{integers}, and L.11.1 is the flip
side: no such argument can ever produce a $1/\pi$.

\medskip\noindent\textbf{L.5.1. Run the ledger's exponential
map: from the five integers to three physical numbers.}

\noindent\emph{Solution.}  Step 1: the recombination-era
corridor switch takes $J = 7$: $z_{\rm sw} = e^{7} = 1096.63$
(Appendix~\ref{appA:thermal}, F.6.2 --- the two-quantity
discipline applies).  Step 2: the lock scale takes $K = 35$:
$\Mlock = \tfrac12\MPl\,e^{-35} = 3.85$~TeV --- the standing
collider falsifier.  Step 3: the baryon-asymmetry assembly
takes the half-depth: $e^{-K/2} = e^{-17.5} =
2.51\times10^{-8}$, the efficiency-weighted asymmetry of
Appendix~\ref{appA:leptogenesis} (I.4.1).  One integer pair
$(J, K)$, three observables at three epochs, no continuous
dial anywhere.  Physics intuition: exponentials of small
integers span the enormous dynamic ranges of physics ---
the ledger's claim is that nature's big pure numbers are not
inputs but \emph{exponentials of counting}, and each mapped
number is independently falsifiable.

\medskip\noindent\textbf{L.10.1. Compute the volume of the
unit three-sphere, and read off the baryon fraction.}

\noindent\emph{Solution.}  Step 1: slice $S^3$ (radius 1) by
polar angle $\psi$: each slice is an $S^2$ of radius
$\sin\psi$, so $\mathrm{Vol}(S^3) = \int_0^\pi 4\pi\sin^2\!
\psi\,d\psi$.  Step 2: $\int_0^\pi \sin^2\psi\,d\psi = \pi/2$,
hence $\mathrm{Vol}(S^3) = 4\pi\cdot\pi/2 = 2\pi^2$.  Step 3:
the inventory's baryon fraction is its reciprocal, $\Omega_b =
1/(2\pi^2) = 0.0507$ --- and the derived baryon-to-matter
ratio is $\Omega_b/\Omm = \pi/(2\pi^2) = 1/(2\pi) = 0.159$,
against the measured $0.157$: agreement at the one-percent
level from two volume factors.  Physics intuition: the two
deepest fractions of the inventory are (inverse) volumes of
the two roundest objects in the theory's geometry --- and a
ratio of volumes, unlike a Chern number, is \emph{allowed} to
be irrational, which is L.11.1's point made constructive.

\medskip\noindent\textbf{L.11.1. Why can $1/\pi$ never be a
Chern or Pontryagin number --- and what must it be instead?}

\noindent\emph{Solution.}  Step 1: topological invariants of
the standard kind arise from the integer pairing
$H^p(\mathcal{M},\mathbb{Z})\times H_p(\mathcal{M},\mathbb{Z})
\to \mathbb{Z}$, or as indices of elliptic operators
(Atiyah--Singer) --- differences of dimensions of solution
spaces.  Both count; both land in $\mathbb{Z}$ (rationals
enter only through fixed universal ratios).  Step 2: $1/\pi$
is irrational: no integer pairing, index, or universal
rational multiple thereof can produce it --- the obstruction
is arithmetic, and no cleverness evades it.  Step 3: the
chapter therefore locates $\Omm = 1/\pi$ NOT as an invariant
but as a \emph{ratio of measures} --- a branching fraction on
the closed-packet geometry (the Hopf-ratio reading of
Section~L.10) --- carried in the codebook at $D^*$
(conditional) grade, with the honesty section stating exactly
which conjecture would promote it.  Physics intuition: the
distinction is load-bearing --- integers are protected by
counting, measures by geometry; the inventory uses both kinds
in their lawful places, and confusing them would be the kind
of error the falsification catalog is built to catch.

\GGtakeaway{\emph{Standard result:} the density parameters are
measured, and their values --- and their sameness across the sky --- are
unexplained inputs. \emph{What it assumed:} nothing; it declined to ask.
\emph{What the geometry replaced it with:} a closed-packet topological
claim, \((\Omm,\OmL,\Omb) = (1/\pi,\,1-1/\pi,\,1/2\pi^2)\), carried
honestly as Derived-conditional on Conjecture~VIII with the Step-4
cancellation named as the open integral, and the identity reading of
Part~X explaining why one inventory holds everywhere. \emph{Continue
in:} Part~X (identity and the theorem); Part~IX (the audit).}

\section{Cold dark matter from vacuum misalignment -- the Kaxion}
\label{appA:kaxion}

\GGteachbox{The complete misalignment mechanism from scratch: the
frozen field, the release at \(3H \simeq m\), the relic-abundance
formula, and the QCD mass--decay-constant relation, with worked
numbers; what the standard axion story leaves free (the line \emph{and}
the angle); and the geometric inversion plus the pinning
mechanism --- the decay constant fixed by the cap loading, the angle
pinned during primary inflation, patch-uniform, with \emph{zero
primordial isocurvature} as the observable signature --- landing on
\(8.1\,\mu\)eV at 1.59--2.13~GHz.}
\GGreadalong{\citet{KolbTurner1990} ch.~10 for the standard mechanism;
\citet{Marsh2016} for the modern axion-cosmology review.}

The vacuum-misalignment mechanism by which a light axion-like field acquires a cold-dark-matter relic abundance is drawn in Figure~\ref{fig:appfig-kaxion}; the mechanism, used here to estimate the Kaxion density, and the model and detection context are set out in \citep{PreskillWiseWilczek1983,AbbottSikivie1983,DineFischler1983,DineFischlerSrednicki1981,Sikivie1983,Marsh2016}.

\begin{figure}[htbp]
\centering
\resizebox{0.791\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  >={Stealth[length=2.2mm]}]

% ===================== Panel (a): theta(t) =====================
\begin{scope}
% axes
\draw[->,line width=0.80pt] (-0.2,0) -- (6.6,0) node[right] {$\log a$ (time $\to$)};
\draw[->,line width=0.80pt] (0,-2.2) -- (0,2.2) node[above] {$\theta$};
% theta_i level
\draw[dashed,gray] (0,1.55) -- (2.6,1.55);
\node[anchor=east,gray!30!black] at (-0.05,1.55) {$\theta_i$};

% frozen segment (Hubble friction): flat
\draw[line width=1.10pt,blue!65!black] (0.15,1.55) -- (2.6,1.55);
% damped oscillation after onset at x=2.6
\draw[line width=1.10pt,blue!65!black]
  (2.6,1.55)
  .. controls (2.95,1.55) and (3.05,-1.05) .. (3.35,-1.0)
  .. controls (3.6,-0.95) and (3.7,0.78) .. (3.95,0.74)
  .. controls (4.18,0.7) and (4.28,-0.56) .. (4.5,-0.53)
  .. controls (4.7,-0.5) and (4.8,0.42) .. (5.0,0.4)
  .. controls (5.18,0.38) and (5.27,-0.3) .. (5.45,-0.28)
  .. controls (5.62,-0.26) and (5.7,0.2) .. (5.88,0.19)
  .. controls (6.03,0.18) and (6.1,-0.12) .. (6.3,-0.1);

% onset marker
\draw[dotted,gray!70!black] (2.6,-2.0) -- (2.6,2.05);
\filldraw[red!75!black] (2.6,1.55) circle (1.6pt);
\node[anchor=west,red!55!black,align=left] at (2.78,1.05)
  {onset:\\$H=m$};

% region labels
\node[align=center,gray!35!black] at (1.25,0.45)
  {frozen\\(Hubble friction)\\$H>m$};
\node[align=center,gray!35!black,anchor=north] at (4.7,-1.25)
  {damped oscillation};

% theta_i annotation
\node[anchor=south west,align=left,blue!40!black] at (0.1,1.7)
  {$\theta_i\!\sim\!1.4$\,rad\\(misalignment angle)};

\node[anchor=north] at (3.3,-2.35) {\textbf{(a)} field angle $\theta(t)$};
\end{scope}

% ===================== Panel (b): rho(a) on log-log =====================
\begin{scope}[xshift=8.4cm]
\draw[->,line width=0.80pt] (-0.2,0) -- (5.4,0) node[right] {$\log a$};
\draw[->,line width=0.80pt] (0,-0.2) -- (0,4.2) node[above] {$\log\rho$};

% frozen: rho ~ const (vacuum energy) until onset, then a^-3
\draw[dotted,gray!70!black] (1.7,-0.1) -- (1.7,3.9);
\node[anchor=south,red!55!black] at (1.7,3.7) {$H=m$};
% constant part
\draw[line width=1.10pt,blue!65!black] (0.2,3.3) -- (1.7,3.3);
% a^-3 redshift (slope down)
\draw[line width=1.10pt,blue!65!black] (1.7,3.3) -- (5.0,0.55);
\node[anchor=south west,blue!40!black] at (3.0,1.55)
  {$\rho\propto a^{-3}$ (CDM)};
\node[anchor=south,gray!35!black] at (0.95,3.35) {frozen};

\node[anchor=north] at (2.6,-0.35) {\textbf{(b)} energy density $\rho(a)$};
\end{scope}

% ===================== boxed result panel =====================
\node[draw=blue!55!black,line width=0.80pt,rounded corners,fill=blue!5,align=left,
      inner sep=5pt,anchor=north] at (6.9,-2.7)
  {\footnotesize\textbf{Kaxion dark matter --- result}\\[1pt]
   $m_{\mathrm{Kaxion}} = 8.1 \pm 0.5~\mu\mathrm{eV}$\\
   $f_{\mathrm{Kaxion}} = 3.0\times10^{13}~\mathrm{GeV}$ (raw)\\
   $\nu_K \in [1.59,\,2.13]~\mathrm{GHz}$\\[2pt]
   \emph{Status D$^{\!*}$ (Derived-conditional).}};

\end{tikzpicture}}
\caption[Kaxion dark matter from vacuum misalignment]{%
Kaxion dark matter from \emph{vacuum misalignment}. \textbf{(a)} While
$H>m$ the field angle is frozen by Hubble friction at the misalignment value
$\theta_i\!\sim\!1.4$\,rad; at the oscillation onset $H=m$ it begins damped
coherent oscillation. \textbf{(b)} Once oscillating, the energy density
redshifts as $\rho\propto a^{-3}$, i.e.\ behaves as cold dark matter.
\textbf{GG reading:} geometry fixes the decay constant $f_K$
($f_{\mathrm{Kaxion}}=3.0\times10^{13}$\,GeV, $m_{\mathrm{Kaxion}}=8.1\pm0.5~\mu$eV,
$\nu_K\in[1.59,2.13]$\,GHz); the observed relic abundance then \emph{inverts}
to $\theta_i\!\sim\!1.4$\,rad --- a genuine inversion, not a fit. The
derivation is executed in companion \PaperCOne{}. Status~D$^{*}$
(Derived-conditional); note $\theta_i$ is abundance-fixed.}
\label{fig:appfig-kaxion}
\end{figure}


This appendix teaches the misalignment mechanism for cold dark
matter from scratch, at the level of a first-year graduate
student who has seen the Friedmann equations and a harmonic
oscillator but has never met an axion. The aim is that, having
read it, the reader can derive the relic-abundance formula
without consulting a review, can say exactly which steps are
textbook physics and which are the \GGTheory{} reading, and can
reproduce the two headline Kaxion numbers --- the mass
\mK\ and the laboratory frequency --- with a pocket calculator.
The mechanism and the inversion that fixes the misalignment angle
are derived in full in the dedicated companion
\citep{ArisakaC1Kaxion}; this appendix is the self-contained
pedagogical entry point, and it follows that derivation's
normalization throughout. Nothing here is new physics: every
equation up to the \GGTheory{} reading in
\S\ref{appA:kax-reading} is the standard misalignment calculation
that any modern dark-matter review carries.


%% -------------------------------------------------------------------
\medskip
\textbf{The three steps of this appendix.} As in Appendix~E:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard misalignment mechanism
(\S\ref{appA:kax-field} onward).} A light pseudo-scalar frozen by Hubble
friction, released when $H$ drops, redshifting thereafter as cold,
pressureless matter --- the textbook relic-abundance calculation.
\item \textbf{What the standard story leaves free
(\S\ref{appA:kax-limits}).} The decay constant $f_a$ and the misalignment
angle $\theta_i$ are both free; one scans the line and tunes the angle to
hit the observed dark-matter density.
\item \textbf{The geometric derivation (\S\ref{appA:kax-reading}).} The
geometry fixes the decay constant with no dial, and the measured abundance
then fixes the one remaining angle as an \emph{output} --- a genuine
inversion of the usual fit.
\end{enumerate}

\subsection{A light pseudo-scalar as a misalignment condensate}
\label{appA:kax-field}

\paragraph{The actor.}
The Kaxion is an \emph{axion-like field}: a light, electrically
neutral, spin-zero, parity-odd (pseudo-scalar) degree of freedom
$\phi$ whose only renormalizable self-interaction is forbidden by
a shift symmetry $\phi \to \phi + \mathrm{const}$. Because of that
shift symmetry the field has no ordinary potential to any order in
perturbation theory; a potential can be generated only by
non-perturbative (topological) effects, and when it is, it is
necessarily \emph{periodic}. It is convenient to trade the
dimensionful field $\phi$ for a dimensionless \emph{angle}
\begin{equation}
\theta \;\equiv\; \frac{\phi}{f},
\qquad \theta \in [-\pi,\pi),
\label{eq:appA-angle}
\end{equation}
where $f$ is the \emph{decay constant}, the single scale that
sets both the field range ($\phi$ winds by $2\pi f$ around its
circle) and the strength of all the field's couplings (which
enter as $1/f$). The angle $\theta$ is literally the position of
a dial; the shift symmetry is the statement that rotating the dial
costs nothing until topology intervenes.

\paragraph{The potential.}
Topology supplies a periodic potential of the universal cosine
form
\begin{equation}
V(\theta) \;=\; \chi\,\bigl[\,1 - \cos\theta\,\bigr]
\;=\; m^2 f^2 \,\bigl[\,1 - \cos\theta\,\bigr],
\qquad
m^2 \;\equiv\; \frac{\chi}{f^2},
\label{eq:appA-potential}
\end{equation}
where $\chi$ is the \emph{topological susceptibility} of whatever
gauge sector the angle talks to --- it has dimensions of
(energy)$^4$ and absorbs the exponentially small instanton weight
$e^{-S_{\mathrm{inst}}}$. Expanding for a small displacement,
$V \simeq \tfrac12 m^2 f^2 \theta^2 = \tfrac12 m^2 \phi^2$: the
floor of each cosine well is a harmonic oscillator of mass $m$.
Two facts are worth fixing in the mind before any cosmology is
done. First, the mass is \emph{not} a fundamental scale; it is the
ratio of a topological scale $\sqrt{\chi}$ to a geometric scale
$f$, so $m f = \sqrt{\chi}$ exactly --- a relation we will use
twice. Second, the curvature at the floor ($m^2$) and the height
of the rim ($2\chi$) are wildly different scales, which is what
lets the field store a cosmologically long-lived, nuclear-stiff
yet feather-gentle oscillation.

\paragraph{A condensate, not a gas.}
Crucially, the dark matter here is not a thermal gas of particles
produced by collisions in the hot plasma. It is a classical,
spatially homogeneous, coherently oscillating \emph{field} --- a
condensate. Every ``particle'' is a quantum of one and the same
zero-momentum oscillation. This is why the candidate can be
extraordinarily light (micro-eV) and yet cold: its coldness comes
from the homogeneity of the field, not from a low temperature of a
particle gas. The whole of the misalignment mechanism is the
story of how that homogeneous oscillation is set up and how its
energy redshifts.


%% -------------------------------------------------------------------
\subsection{Equation of motion in an expanding universe}
\label{appA:kax-eom}

Take the field homogeneous, $\theta = \theta(t)$, in a
spatially flat Friedmann--Robertson--Walker (FRW) background with
scale factor $a(t)$ and Hubble rate $H = \dot a / a$. The action
for $\phi = f\theta$ is the standard scalar action; varying it and
dividing by $f$ gives the equation of motion
\begin{equation}
\boxed{\;
\ddot\theta \;+\; 3H\,\dot\theta \;+\; m^2(T)\,\sin\theta \;=\; 0.
\;}
\label{eq:appA-eom}
\end{equation}
This is the master equation of the entire mechanism, and it is
worth reading term by term, because it is exactly the equation of
a pendulum in a viscous medium.

\begin{itemize}
\item $\ddot\theta$ is the inertia of the dial.
\item $m^2(T)\sin\theta$ is the topological restoring torque,
written with the full periodic $\sin\theta$ rather than its
linearization. The argument $T$ is a reminder that for the Kaxion
the susceptibility, and hence the mass, is \emph{temperature
dependent}: above the confinement scale of the gauge sector the
instantons are suppressed and $m(T)$ is much smaller than its
zero-temperature value. We carry this dependence explicitly
because it is the origin of the famous non-integer exponent in
the abundance formula.
\item $3H\,\dot\theta$ is a friction term, with friction
coefficient $\gamma = 3H$ supplied for free by the expansion of
the universe. This is the cosmological version of Hubble drag: as
space expands, the field's momentum redshifts away. In the
\GGTheory{} boundary law this same $3H$ appears as the
expanding-background friction coefficient of the open-system
(\GRIP{}/\GDOS{}) Langevin equation applied to the compact-angle
mode, with the post-inflationary noise temperature utterly
negligible; here we need only its classical, frictional content.
\end{itemize}

\paragraph{Harmonic (small-angle) limit.}
For $|\theta| \ll 1$ we may replace $\sin\theta \to \theta$ and
\eqref{eq:appA-eom} becomes the damped harmonic oscillator
\begin{equation}
\ddot\theta \;+\; 3H\,\dot\theta \;+\; m^2(T)\,\theta \;=\; 0,
\label{eq:appA-eom-harmonic}
\end{equation}
a frictionally damped pendulum whose two regimes --- overdamped
freeze and underdamped ring --- are fixed entirely by the
competition between the damping rate $3H$ and the natural
frequency $m$. The whole cosmological story is the crossover
between those two regimes as $H$ falls while $m$ (eventually) rises.


%% -------------------------------------------------------------------
\subsection{Frozen by Hubble friction, then released}
\label{appA:kax-freeze}

\paragraph{The overdamped (frozen) phase: \texorpdfstring{$H \gg m$}{H >> m}.}
Early in the radiation era $H$ is enormous (it scales as $T^2/\Mpl$)
and the natural frequency $m$ is tiny. The pendulum is then
overdamped: the restoring torque cannot overcome the friction, and
the field is \emph{frozen} at whatever value it was left with after
inflation. Formally, in \eqref{eq:appA-eom-harmonic} with
$H \gg m$ the two terms that survive are $3H\dot\theta + m^2\theta
\approx 0$, whose slow solution drifts only by
\begin{equation}
\Delta\theta \;\sim\; \Bigl(\frac{m}{3H}\Bigr)^{\!2}\,\theta
\;\ll\; \theta,
\label{eq:appA-frozen-slip}
\end{equation}
i.e.\ negligibly per Hubble time until $m$ approaches $3H$. The
field sits at its initial displacement
\begin{equation}
\theta(t) \;\simeq\; \thetai
\qquad (\text{the \emph{misalignment angle}}),
\label{eq:appA-thetai-frozen}
\end{equation}
the value the field happened to have when the potential first
switched on relative to the Hubble friction. ``Misalignment'' is
exactly this: the dial is generically displaced from the bottom of
its well, and Hubble friction pins it there, away from the
minimum, like a stuck door.

\paragraph{Onset of oscillation: \texorpdfstring{$H \sim m$}{H ~ m}.}
As the universe expands, $H$ falls monotonically; eventually the
friction weakens to the point where the restoring torque wins. The
conventional onset condition is
\begin{equation}
\boxed{\; 3H(T_1) \;=\; m(T_1), \;}
\label{eq:appA-onset}
\end{equation}
which defines the \emph{onset temperature} $T_1$ (also written
$T_{\mathrm{osc}}$). The numerical coefficient ``3'' is a
convention, not magic: the exact solution of
\eqref{eq:appA-eom-harmonic} in the radiation era is a Bessel
function whose envelope turns over precisely at $m t \simeq
\tfrac32$, i.e.\ at $3H = m$ (recall $H = 1/2t$ in radiation
domination). So $3H=m$ is the mathematical knee of the special
function, not an arbitrary marker. After $T_1$ the field is
underdamped and begins to oscillate coherently about the minimum
of its well, with initial amplitude $\thetai$.

\paragraph{Determining \texorpdfstring{$T_1$}{T1} (a worked aside).}
The radiation-era Hubble rate is
\begin{equation}
H(T) \;=\; \sqrt{\frac{8\pi^3 g_*(T)}{90}}\,\frac{T^2}{\Mpl}
\;\simeq\; 1.66\,\sqrt{g_*}\,\frac{T^2}{\Mpl},
\label{eq:appA-hubble-rad}
\end{equation}
with $\Mpl = 1.22\times10^{19}$~GeV the (non-reduced) Planck mass
and $g_*$ the effective relativistic degrees of freedom. If one
\emph{pretended} the mass were already at its present value
$\mK = 8.1\,\mu\mathrm{eV} = 8.1\times10^{-15}$~GeV, then
\eqref{eq:appA-onset} with \eqref{eq:appA-hubble-rad} ($g_* \simeq
73$, so the prefactor combination is $\simeq 42.6$) gives
$T_1 \simeq [\,\mK\,\Mpl/42.6\,]^{1/2}$, of order a few tens of GeV.
But the Kaxion's mass is \emph{not} constant: above the QCD scale
the susceptibility is instanton-suppressed and rises as the
universe cools, $m(T) = m(0)\,(T_{\mathrm{QCD}}/T)^{b}$ with a
lattice exponent $b \simeq 4$. Inserting this temperature
dependence into \eqref{eq:appA-onset} lowers the true onset to
$T_1 \simeq 1.0$~GeV ($t \simeq 0.3\,\mu$s), where the mass is
still only $\sim 0.04\%$ of its final value: the spring is soft
when the pendulum is released and stiffens underneath the swing.
This ``stiffening spring'' is the physics behind the non-integer
power that will appear in \eqref{eq:appA-omega}; it is not a
nuisance but the signature of a QCD-coupled axion.


%% -------------------------------------------------------------------
\subsection{Redshifting as cold, pressureless matter}
\label{appA:kax-cold}

Once the field oscillates with $H \ll m$, the friction term in
\eqref{eq:appA-eom} is a slow drag on a fast oscillation. Average
over one cycle. For a harmonic well the virial theorem gives equal
kinetic and potential energy on average, so the pressure
$p = \langle \tfrac12\dot\phi^2 - V\rangle$ averages to zero,
while the energy density $\rho = \langle\tfrac12\dot\phi^2 +
V\rangle = \tfrac12 m^2 \phi_{\mathrm{amp}}^2$ is non-zero. A
pressureless fluid in FRW redshifts as
\begin{equation}
\boxed{\; \rho_\theta \;\propto\; a^{-3}. \;}
\label{eq:appA-rho-dust}
\end{equation}
This is the definition of \emph{cold} (non-relativistic,
pressureless) dark matter: the oscillating condensate dilutes
exactly like dust, $\rho \propto a^{-3}$, never like radiation
($a^{-4}$). It is cold despite the field's tiny mass because the
whole condensate sits at zero momentum.

\paragraph{The adiabatic-invariant (WKB) argument.}
The cleanest way to see \eqref{eq:appA-rho-dust} --- and the only
way that survives the stiffening spring, when $m$ itself is
changing --- is the adiabatic theorem for an oscillator of slowly
varying frequency. For a harmonic oscillator with slowly varying
$\omega(t) = m(t)$ the \emph{action variable}
\begin{equation}
J \;=\; \oint p\,dq \;=\; 2\pi\,\frac{E}{\omega}
\label{eq:appA-action}
\end{equation}
is an adiabatic invariant whenever the WKB (adiabaticity)
condition $|\dot\omega/\omega^2| \ll 1$ holds. Per comoving
volume, $E/\omega$ is exactly the \emph{number of quanta}: the
energy density of the condensate is $\rho = m\, n$, where $n$ is
the number density of zero-momentum quanta. Hence the comoving
quantum number
\begin{equation}
N \;=\; n\,a^3 \;=\; \frac{\rho\,a^3}{m}
\;=\; \mathrm{const}.
\label{eq:appA-N-invariant}
\end{equation}
Energy is \emph{not} conserved (as the spring stiffens it does
work on the oscillation, and the redshift drains momentum), but
the \emph{number} of quanta per comoving volume is. Two
consequences follow immediately. (i) Once the mass has settled to
its constant value $m(0)$, \eqref{eq:appA-N-invariant} gives
$\rho = m(0)\, n \propto a^{-3}$, recovering
\eqref{eq:appA-rho-dust}. (ii) Through the stiffening epoch we do
\emph{not} track the energy (which is changing in a complicated
way) but the conserved $N$; this is precisely the right
bookkeeping to carry from the onset $T_1$ down to today.

\paragraph{One honest caveat.}
The WKB condition is only \emph{marginally} satisfied at the very
first oscillation: at onset $\dot m/m^2 = bH/m(T_1) = b/3 \simeq
1.4$, so adiabaticity holds order-of-magnitude at onset and
becomes excellent ($\dot m/m^2 \propto t^{-1}$) thereafter. The
$\mathcal{O}(1)$ transient this induces over the first cycle or
two is exactly what a numerical transfer-function calculation
resolves, and it is the dominant contribution to the
$\pm 8\%$ pipeline systematic on the abundance quoted in
\citep{ArisakaC1Kaxion}. We flag it so that the reader does not
mistake the clean analytic scaling below for better-than-it-is
precision.


%% -------------------------------------------------------------------
\subsection{The relic-abundance formula}
\label{appA:kax-abundance}

Now assemble the bookkeeping. At onset the field banks the
potential energy of its displacement,
\begin{equation}
\rho_\theta(T_1) \;=\; \chi(T_1)\,[1-\cos\thetai]
\;\simeq\; \tfrac12\, m^2(T_1)\, f^2\, \thetai^2
\quad(\text{small-angle}),
\label{eq:appA-rho-onset}
\end{equation}
and the comoving number $N$ of \eqref{eq:appA-N-invariant} is
fixed from this moment on. Tracking $\rho = m(0)\,n$ with $n a^3$
constant from $T_1$ to today, normalizing the entropy density to
the photon temperature, and using the onset relation
\eqref{eq:appA-onset} with the temperature-dependent mass, one
obtains the present comoving density. The transparent way to see
the scaling is to insert $m(T_1) = 3H(T_1) \propto T_1^2$ and the
onset solution $T_1 \propto m(0)^{1/(b+2)}$ into $\rho_0 \propto
m(0)\, n_0$; the powers combine to
\begin{equation}
\rho_0 \;\propto\; m(0)^{(b+3)/(b+2)}\,\thetai^2
\;\propto\; f^{-(b+3)/(b+2)}\,\thetai^2,
\label{eq:appA-scaling}
\end{equation}
where the last step used the frame-invariant $m(0) \propto 1/f$
(from $mf=\sqrt\chi$ with $\chi$ fixed by the gauge sector). For
the lattice exponent $b \simeq 4.08$ this gives the celebrated
near-$7/6$ power $(b+3)/(b+2) = 7.08/6.08 = 1.165$. Restoring the
numerical normalization (every constant evaluated explicitly in
\citep{ArisakaC1Kaxion}) gives the master formula
\begin{equation}
\boxed{\;
\Omega_a h^2 \;=\; 0.090\;\thetai^{2}\,
\left(\frac{f}{10^{12}\,\mathrm{GeV}}\right)^{1.165}.
\;}
\label{eq:appA-omega}
\end{equation}
Read it as a function of the three inputs it actually contains:
the misalignment angle $\thetai$, the decay constant $f$, and the
(temperature-dependent) mass --- the latter entering through the
exponent $1.165$, which is a fingerprint of the QCD running, not
a free dial.

\paragraph{Why the odd exponent $1.165$ and not $2$?}
Equation \eqref{eq:appA-rho-onset} would naively suggest
$\rho \propto \thetai^2 f^2$. The reduction from $f^2$ to
$f^{1.165}$ is pure physics: a larger $f$ means a \emph{lighter}
field, which lets go \emph{later} (lower $T_1$), when the universe
has less entropy left to dilute it --- but it also banks each
quantum at a smaller onset mass. The competition between ``later
release'' (which boosts the relic) and ``cheaper quanta'' (which
suppresses it) shaves the naive power down to $(b+3)/(b+2)$. A
reader who can reconstruct that sentence owns the misalignment
mechanism.

\paragraph{Anharmonic correction for \texorpdfstring{$\mathcal{O}(1)$}{O(1)} angles.}
The small-angle banking \eqref{eq:appA-rho-onset} is only the
leading term. For an initial displacement of order unity the
field samples the flatter, anharmonic shoulders of the cosine, it
takes \emph{longer} to begin oscillating (the restoring torque is
weaker than the harmonic estimate near the top), and the relic is
enhanced relative to \eqref{eq:appA-omega}. The standard
treatment writes the correction as a multiplicative
\emph{anharmonicity factor} $F(\thetai)$,
\begin{equation}
\Omega_a h^2 \;=\; 0.090\;F(\thetai)\,\thetai^{2}\,
\left(\frac{f}{10^{12}\,\mathrm{GeV}}\right)^{1.165},
\qquad
F(\thetai) \;\simeq\;
\left[\,\ln\!\frac{e}{\,1-\thetai^2/\pi^2\,}\,\right]^{\,(b+6)/(b+2)},
\label{eq:appA-anharmonic}
\end{equation}
with $F \to 1$ as $\thetai \to 0$ and $F$ growing as $\thetai \to
\pi$ (where the field starts near the unstable hilltop and the
relic diverges logarithmically). For $\thetai \simeq 1.4$~rad the
anharmonic enhancement is a modest $\mathcal{O}(1)$ factor, folded
into the $\pm 8\%$ abundance systematic of
\citep{ArisakaC1Kaxion}; it shifts the inferred angle within the
$1.1$--$1.5$~rad range but does not touch the mass or frequency,
which the geometry fixes independently.


%% -------------------------------------------------------------------
\subsection{The QCD relation \texorpdfstring{$m_a f_a = \sqrt{\chi_{\mathrm{QCD}}}$}{ma fa = sqrt(chiQCD)}}
\label{appA:kax-qcd}

For a QCD axion --- and the Kaxion is the \GGTheory{} program's
QCD axion --- the gauge sector whose topology generates the
potential is the strong interaction itself. The relevant
susceptibility is then the \emph{QCD topological susceptibility}
$\chiQCD$, a pure Standard-Model quantity measured on the lattice.
The mass relation \eqref{eq:appA-potential} becomes the
model-independent strong-CP relation
\begin{equation}
\boxed{\;
m_a\, f_a \;=\; \sqrt{\chiQCD}.
\;}
\label{eq:appA-mafa}
\end{equation}
The current lattice determination is
\begin{equation}
\chiQCD^{1/4} \;=\; 75.5\ \mathrm{MeV}
\qquad \Longrightarrow \qquad
\sqrt{\chiQCD} \;=\; (75.5\,\mathrm{MeV})^2,
\label{eq:appA-chival}
\end{equation}
\citep{ArisakaC1Kaxion}, so that the entire axion mass--coupling
line is fixed by one experimentally measured number: choose $f_a$
and the mass $m_a$ is determined, and vice versa. This is the
single most important structural fact about a QCD axion --- it has
a \emph{one-parameter} family of allowed values, a line in the
mass--coupling plane, not a two-dimensional region. Everything
that follows is about where on that line the geometry of the
\GGTheory{} program places the Kaxion.


%% -------------------------------------------------------------------
\subsection{What the standard misalignment story leaves free}
\label{appA:kax-limits}

The mechanism of Step~1 is exact, but it does not predict a dark-matter
candidate; it parametrizes one. Two numbers are left entirely free: the
decay constant $f_a$ (equivalently the axion mass $m_a$), which can sit
anywhere across many decades, and the initial misalignment angle
$\theta_i\in[-\pi,\pi]$. The standard procedure fixes neither --- it scans
$f_a$ along the ``axion line'' and, at each point, tunes $\theta_i$ so the
relic abundance matches the observed $\Omega_{\rm DM}h^2\approx0.12$. The
mechanism is a framework that \emph{accommodates} a dark-matter abundance
for almost any mass; it does not \emph{predict} the mass. The geometric
account inverts exactly this freedom.

\subsection{The geometric derivation: a genuine inversion}
\label{appA:kax-reading}

The standard axion story leaves \emph{both} $f_a$ and $\thetai$
free: one scans the line and tunes the angle to hit the observed
abundance. The \GGTheory{} reading breaks this freedom, and it
does so in a direction that is the opposite of a fit.

\paragraph{Geometry fixes the decay constant.}
The compact six-dimensional geometry of the program
\citep{ArisakaA4Euclid} fixes the Kaxion decay constant with no
continuous dial, from the scale-profile cap loading,
\begin{equation}
\fK \;=\; (7.1 \pm 0.5)\times 10^{11}\ \mathrm{GeV},
\label{eq:appA-fk}
\end{equation}
derived in \citep{ArisakaC1Kaxion}. The $\pm6\%$ ($\pm0.5\,\mu$eV) is not a tuning range but the residual
of the Standard-Model lattice-QCD and transport inputs to the abundance
closure, not geometry; the integer cap loading separately fixes the
\emph{discrete} $\Xi_{\mathrm{cap}}$ choice (the window of Eq.~\eqref{eq:appA-window}),
as discussed under Status below. There is exactly one
physical decay constant; the bulk Green--Schwarz normalization
scale ($\sim 3\times10^{13}$~GeV) that appears upstream is an
intermediate of the diagonalization, not the decay constant of any
propagating field.

\paragraph{Abundance fixes the misalignment angle as an OUTPUT.}
With $\fK$ fixed by \eqref{eq:appA-fk}, the exponent fixed by
lattice QCD, and the target abundance fixed by the $\pi$-locked
cosmic inventory of the present paper --- the dark-matter relic
\begin{equation}
\Omdm h^2 \;=\; 0.1215,
\qquad
\Omdm \;=\; \Omm - \Omb \;=\; \frac{1}{\pi} - \frac{1}{2\pi^2},
\label{eq:appA-omdm-target}
\end{equation}
converted to a comoving density with $h = 0.674$ from the same
series --- the master formula \eqref{eq:appA-omega} has
\emph{exactly one} unknown left: the misalignment angle. Solving
\eqref{eq:appA-omega} for it,
\begin{equation}
\boxed{\;
\thetai \;=\;
\left(\frac{0.1215}{0.090}\right)^{1/2}
\left(\frac{10^{12}\,\mathrm{GeV}}{\fK}\right)^{0.583}
\;\simeq\; 1.4\ \mathrm{rad}.
\;}
\label{eq:appA-thetai}
\end{equation}
This is a genuine \emph{inversion}, and the direction of inference
is the whole point: \emph{geometry $\to \fK$, abundance $\to
\thetai$}. The abundance \eqref{eq:appA-omdm-target} is the
\emph{target} on the right-hand side, not a quantity being fitted;
$\fK$ is set independently of cosmology; and $\thetai$ comes out as
a derived $\mathcal{O}(1)$ number, $\simeq 1.4$~rad. The full
inversion is carried out in \citep{ArisakaC1Kaxion}.

\paragraph{Honesty discipline on the angle.}
It must be stated plainly, because the shape of the number invites a
natural misreading: \thetai\ is \emph{not} geometry-selected at
$1/(2\pi)$. The misalignment angle
is an $\mathcal{O}(1)$ \emph{output} of the observed abundance via
\eqref{eq:appA-thetai}, landing in the generic range $1.1$--$1.5$
rad once the anharmonic correction \eqref{eq:appA-anharmonic} is
folded in. It is not a tuned input, and it is not selected by the
geometry; it is what the geometry plus the cosmic inventory force.

\paragraph{The pinning mechanism: the angle is not drawn, it is frozen.}
The standard story's deepest assumption is statistical: \(\thetai\) is
drawn at random in each causal patch, so the relic density is an
accident of our patch, and the field's fluctuations during inflation
imprint \emph{isocurvature} perturbations that the CMB constrains. The present framework replaces the draw with a mechanism. During the
primary (GUT-adjacent) inflation of Appendix~G, a reflection-symmetric
dressing of the descent pins the compact angle at the cap ---
coherently, across the entire volume that becomes the observable
universe --- so that every patch inherits the \emph{same} misalignment.
Three consequences follow, each observable in principle. \emph{(i)
Patch-uniformity:} there is no axion-abundance lottery; the relic
density is deterministic. \emph{(ii) Zero primordial isocurvature:} a
pinned angle does not fluctuate, so the isocurvature fraction is
suppressed some thirty-eight orders below the Planck bound --- the
standard story's most dangerous constraint is passed not narrowly but
trivially, and a future isocurvature \emph{detection} attributable to
the axion sector would falsify the pinning outright. \emph{(iii) No
defects:} no patch-to-patch angle variation means no domain walls or
strings from this sector, ever. In the identity language of Part~X:
the electron is identical everywhere because one ground basin exists;
the Kaxion is identical everywhere because one angle was frozen ---
\emph{two geometric routes to the same universality}, and the
isocurvature null is the laboratory-facing signature of the second
route (the pinning condition and its quantitative chain are
\PaperCOne{}'s; the codebook carries it as a named conditional).
The main text places this mechanism in the
trio it belongs to (\S\ref{ssec:cosmic-triptych}): the pinning is
the moment the interval's episode hands the circle its
displacement --- the opening act paying the middle act's deposit.



\paragraph{The mass is the DFSZ/QCD image of the decay constant.}
With \fK\ fixed by \eqref{eq:appA-fk} and $\sqrt{\chiQCD}$ fixed
by lattice QCD \eqref{eq:appA-chival}, the mass is not a separate
prediction but the QCD image of $\fK$ through
\eqref{eq:appA-mafa}:
\begin{equation}
\mK \;=\; \frac{\sqrt{\chiQCD}}{\fK}
\;=\; 8.1 \pm 0.5\ \mu\mathrm{eV},
\label{eq:appA-mK}
\end{equation}
on the DFSZ coupling line, giving the laboratory window
\begin{equation}
\nu_K \;\in\; [\,1.59,\ 2.13\,]\ \mathrm{GHz}
\qquad(\text{benchmark } 1.95\ \mathrm{GHz}),
\label{eq:appA-window}
\end{equation}
the band that the funded haloscope searches (IBS-CAPP, ADMX-EFR)
are being built to scan \citep{ArisakaC1Kaxion}. The width of the
window is the image of the \emph{discrete} integer cap loading (the
$\Xi_{\mathrm{cap}}$ choice) on $\fK$, not a continuous knob; the separate
$\pm6\%$ ($\pm0.5\,\mu$eV) lattice-QCD uncertainty applies at each $\Xi_{\mathrm{cap}}$.


%% -------------------------------------------------------------------
\subsection{Worked numerical example}
\label{appA:kax-worked}

We close with the arithmetic a reader can redo by hand, so that no
number above is taken on faith.

\paragraph{(1) Mass from the decay constant.}
Insert $\fK = 7.1\times 10^{11}$~GeV $= 7.1\times 10^{20}$~eV and
$\sqrt{\chiQCD} = (75.5\,\mathrm{MeV})^2 = (7.55\times10^{7}\,
\mathrm{eV})^2 = 5.70\times10^{15}\,\mathrm{eV}^2$ into
\eqref{eq:appA-mafa}:
\begin{equation}
\mK \;=\; \frac{\sqrt{\chiQCD}}{\fK}
\;=\; \frac{5.70\times10^{15}\,\mathrm{eV}^2}
{7.1\times10^{20}\,\mathrm{eV}}
\;=\; 8.0\times10^{-6}\ \mathrm{eV}
\;=\; 8.0\ \mu\mathrm{eV},
\label{eq:appA-worked-mass}
\end{equation}
matching $\mK = 8.1 \pm 0.5\,\mu$eV of \eqref{eq:appA-mK} to
within the rounding.

\paragraph{(2) Laboratory frequency.}
A non-relativistic axion of rest mass $\mK$ converts in a haloscope
to a photon of frequency $\nu_K = \mK c^2/h$, i.e.\ in natural
units $\nu_K = \mK/(2\pi)$ with $\mK$ read as an energy. Using
$1\,\mathrm{eV} = 2.418\times10^{14}$~Hz,
\begin{equation}
\nu_K \;=\; (8.0\times10^{-6}\,\mathrm{eV})\times
(2.418\times10^{14}\,\mathrm{Hz/eV})
\;\simeq\; 1.94\ \mathrm{GHz},
\label{eq:appA-worked-nu}
\end{equation}
inside the benchmark $1.95$~GHz and the window
\eqref{eq:appA-window}. (The $\pm0.5\,\mu$eV mass uncertainty maps
to the $1.59$--$2.13$~GHz band.)

\paragraph{(3) The misalignment angle that saturates the relic.}
Finally, check that $\thetai \simeq 1.4$~rad saturates
$\Omdm h^2 \simeq 0.12$. With $\fK = 7.1\times10^{11}$~GeV,
$(\fK/10^{12}\,\mathrm{GeV})^{1.165} = (0.71)^{1.165} \simeq 0.67$,
so the harmonic master formula \eqref{eq:appA-omega} reads
$\Omega_K h^2 \simeq 0.090\times 0.67\times \thetai^2 \simeq 0.060\,
\thetai^2$. Setting this equal to the target $0.1215$ gives
\begin{equation}
\thetai \;=\; \left(\frac{0.1215}{0.060}\right)^{1/2}
\;\simeq\; 1.4\ \mathrm{rad},
\label{eq:appA-worked-thetai}
\end{equation}
exactly \eqref{eq:appA-thetai}. (The mild anharmonic enhancement
$F(1.4) > 1$ shifts the inferred angle slightly downward within
the $1.1$--$1.5$~rad range; the harmonic estimate already lands on
$1.4$~rad.) The three numbers --- $\mK \simeq 8\,\mu$eV,
$\nu_K \simeq 1.94$~GHz, $\thetai \simeq 1.4$~rad --- are the
whole quantitative content of the Kaxion-as-dark-matter reading,
and all three follow from $\fK$, the lattice $\chiQCD$, and the
$\pi$-locked abundance.


%% -------------------------------------------------------------------
\subsection*{Status}
\label{appA:kax-status}

\begin{itemize}
\item \textbf{Relic mechanism: Derived.} The misalignment
mechanism --- frozen field, onset at $3H=m$, $\rho \propto a^{-3}$
coldness via the adiabatic invariant, and the abundance formula
\eqref{eq:appA-omega} with its anharmonic correction --- is
standard physics and is derived in full in
\citep{ArisakaC1Kaxion}. It is not in question here.

\item \textbf{The inversion: Derived.} The direction
geometry $\to \fK$, abundance $\to \thetai$ is a genuine inversion
executed in \citep{ArisakaC1Kaxion}: $\fK$ is set independently of
cosmology, the abundance \eqref{eq:appA-omdm-target} is the target,
and $\thetai \simeq 1.4$~rad is the $\mathcal{O}(1)$ output.

\item \textbf{The absolute mass: Derived-conditional (D$^*$).}
$\mK = 8.1 \pm 0.5\,\mu$eV is the QCD image of $\fK$ through
\eqref{eq:appA-mafa}, so it is as firm as $\fK$ itself. The
\emph{absolute} value remains D$^*$ pending the integer
cap-loading conjecture (the cap integer $\Xi_{\mathrm{cap}}$,
whose \emph{discrete} value within its allowed window sets $\fK$ and
hence the frequency across the $1.59$--$2.13$~GHz band; the continuous
$\pm6\%$ ($\pm0.5\,\mu$eV) lattice-QCD uncertainty applies at each value). Closing that integer conjecture would
collapse the window to a single line.

\item \textbf{The misalignment angle: an O(1) output, not an
input.} $\thetai \simeq 1.4$~rad is the output of the observed
abundance via \eqref{eq:appA-thetai}, not a tuned parameter and
not geometry-selected at $1/(2\pi)$, which lies a factor of about
seven below the generic $1.1$--$1.5$~rad range that any anharmonic
$\mathcal{O}(1)$ misalignment produces.
\end{itemize}

In one sentence: the Kaxion is the program's QCD axion,
its decay constant is fixed by the geometry, its mass and
microwave frequency are the lattice-QCD image of that decay
constant, and the one number the cosmology hands back ---
the misalignment angle $\simeq 1.4$~rad --- is an honest
$\mathcal{O}(1)$ output of the measured dark-matter abundance, not
a dial that was turned to fit it.

%% =====================================================================
%%  Appendix fragment for Arisaka-GG-A10-Cosmos
%%  Black-hole entropy A/4 from induced gravity, from scratch.
%%  Pedagogical appendix at maximum depth (first-year-graduate level).
%%  ASCII only.  Macros used: \GDOS \OGGEP \GGTheory \GRIP
%%  Citations used (all resolve in the A10 bibliography):
%%    \citep{ArisakaC3BH} \citep{ArisakaA5GDOS} \citep{ArisakaA2GG6}
%%    \citep{Bekenstein1973} \citep{Hawking1975} \citep{Sakharov1967}
%%  Label prefix: appA:bh-
%% =====================================================================

\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{M.1.1. Why is a misalignment
condensate naturally light, and why is its potential a
cosine?}

\noindent\emph{Solution.}  Step 1: an angular field $\theta$
valued on a circle enjoys a shift symmetry $\theta \to \theta
+ $ const; any potential must be periodic, hence a Fourier
series in $\cos(n\theta)$ --- the leading term is the cosine
of \eqref{eq:appA-potential}.  Step 2: the symmetry is exact
until non-perturbative (instanton-class) effects switch on,
so the mass $m = \sqrt{\chi}/f$ \eqref{eq:appA-mafa} is
suppressed by the topological susceptibility $\chi$ --- small
because it is born of tunneling, not of couplings.  Step 3:
a field with $f$ near the deep-bulk scale and $\chi$ from the
QCD sector is therefore automatically in the micro-eV range:
lightness is structural, not tuned.  Physics intuition: angles
cannot have generic masses --- only winding can weigh them ---
so nature's lightest scalars are her compact ones; the Kaxion
is the cohort's compact angle doing exactly what compact
angles do.

\medskip\noindent\textbf{M.2.1. Derive the damped-oscillator
equation of motion from the action.}

\noindent\emph{Solution.}  Step 1: vary the action
\eqref{eq:appA-action} for a homogeneous $\theta(t)$ in the
FRW background: the measure $a^3$ is what makes the expansion
enter.  Step 2: the Euler--Lagrange equation is
$\ddot\theta + 3H\dot\theta + m^2\sin\theta\cdot(\ldots) = 0$
--- \eqref{eq:appA-eom} --- whose small-angle form is the
damped harmonic oscillator \eqref{eq:appA-eom-harmonic}.
Step 3: the $3H\dot\theta$ term is not friction against
anything material: it is dilution --- the field's canonical
momentum lives in a growing volume.  Physics intuition:
``Hubble friction'' is bookkeeping for expansion, which is why
it acts on every field equally and why the release condition
(M.3.1) is a race between one universal clock ($H$) and one
private one ($m$).

\medskip\noindent\textbf{M.3.1. Establish the freeze/release
criterion $m \simeq 3H$ and the onset temperature.}

\noindent\emph{Solution.}  Step 1: while $m \ll 3H$, the
oscillator is overdamped: \eqref{eq:appA-frozen-slip} shows
the field slips only by $O(m^2/H^2)$ per Hubble time --- the
angle is frozen at $\thetai$ \eqref{eq:appA-thetai-frozen}.
Step 2: when $m \simeq 3H$ \eqref{eq:appA-onset}, the private
clock outruns the universal one and oscillation begins; in
the radiation era $H \sim T^2/\MPl$, so the onset temperature
obeys $T_{\rm osc} \sim \sqrt{m\MPl}$ up to $g_*$ factors.
Step 3: for micro-eV masses this lands at $T_{\rm osc} \sim
$~GeV --- the condensate wakes just before the QCD epoch,
which is also when the susceptibility $\chi(T)$ settles to
its vacuum value.  Physics intuition: the misalignment relic
is a delayed-start pendulum --- its abundance depends on when
it started swinging, which converts a mass into a cosmic
timestamp; freeze-out grammar (F.5.1) again, with the roles
of $\Gamma$ and $m$ interchanged.

\medskip\noindent\textbf{M.4.1. Prove that the oscillating
condensate redshifts as pressureless matter.}

\noindent\emph{Solution.}  Step 1: once $m \gg H$, the field
oscillates many times per Hubble time; over a cycle the
virial theorem gives $\langle$kinetic$\rangle =
\langle$potential$\rangle$, so the cycle-averaged pressure
$\langle p\rangle = \langle\dot\phi^2/2 - V\rangle = 0$.
Step 2: equivalently, the comoving occupation
\eqref{eq:appA-N-invariant} $N \propto \rho a^3/m$ is an
adiabatic invariant of the slowly-expanding oscillator, so
with $m$ constant, $\rho \propto a^{-3}$
\eqref{eq:appA-rho-dust}.  Step 3: a coherent field of
macroscopic occupation thus gravitates exactly like cold
particles --- \emph{cold} dark matter from a wave.
Physics intuition: the condensate is a swarm of zero-momentum quanta
in disguise; dilution can thin the swarm but cannot slow
particles that were never moving, which is the whole
phenomenological content of ``cold.''

\medskip\noindent\textbf{M.5.1. Count the freedoms: why is
the cohort's misalignment story an inversion of the standard
one?}

\noindent\emph{Solution.}  Step 1: the standard axion
abundance \eqref{eq:appA-omega} contains three inputs --- the
decay constant $f$, the mass chain through $\chi$, and the
initial angle $\thetai$ --- and one output, $\Omega_{\rm DM}$:
with $\thetai$ unknown, any abundance can be accommodated
(the ``anthropic window'').  Step 2: the cohort fixes $f$
from the ledger's scale chain and \emph{fixes the output}:
$\Omega_m = 1/\pi$ from Appendix~\ref{appA:pilock}.  The
master formula then has exactly one unknown left, and solving
gives $\thetai \simeq 1.4$~rad \eqref{eq:appA-thetai} --- an
$O(1)$ angle, uncontrived, but now a \emph{prediction}, not a
choice.  Step 3: falsifiability follows: the haloscope line
(the chapter's worked example) and the pinned angle must
\emph{both} hold --- there is no second dial to retreat to.
Physics intuition: the same equation read left-to-right is a
fit and right-to-left is a theory; the inversion --- output
fixed by topology, input predicted --- is the cleanest
statement of what zero free parameters means in practice.

\GGtakeaway{\emph{Standard result:} misalignment makes a light axion
an excellent cold-dark-matter candidate --- with the decay constant
scanned and the angle drawn. \emph{What it assumed:} \(f_a\) free over
orders of magnitude, \(\thetai\) random, isocurvature bounds survived
by luck of the inflation scale. \emph{What the geometry replaced it
with:} \(f_K\) fixed by the cap loading, \(\thetai\) pinned during the
primary inflation (patch-uniform, zero isocurvature, no defects), the
mass at \(8.1 \pm 0.5\,\mu\)eV on the DFSZ line, and a funded
laboratory window at 1.59--2.13~GHz that decides this decade.
\emph{Continue in:} Part~V; \PaperCOne{}; Part~X Table~\ref{tab:three-objects}.}

\section{Black-hole entropy A/4 from induced gravity}
\label{appA:bhentropy}

\GGteachbox{Black-hole thermodynamics from scratch: the four laws,
the Hawking temperature, and the Bekenstein--Hawking entropy; Sakharov's
induced gravity and the heat-kernel computation in which \emph{one}
coefficient sets both \(1/G_N\) and the entropy density --- so that
\(S = A/4G_N\) becomes an identity rather than a coincidence; and the
geometric reading in which the horizon is an \OGGEP{} port and the
Newton constant is the scale-profile readout of the same bulk that
fixed the inventory.}
\GGreadalong{\citet{Wald1984} for the classical laws;
\citet{Hawking1975} and \citet{Bekenstein1973} as the primary sources.}

The body of this paper uses the Bekenstein--Hawking entropy
\(S = A/4G_N\hbar\) as an established thermodynamic input --- it appears
wherever the cosmological-boundary stratum has a horizon, most visibly
in the de Sitter and apparent-horizon accounting of the late vacuum and
in the open-system reading of black-hole information. \ATen{} does
\emph{not} prove that area law; it inherits it. The proof lives in the
companion \citet{ArisakaC3BH} (Theorem~\emph{T-BH-1}), where the
coefficient \(1/4\) is derived from induced gravity by the
conical-replica method on the canonical six-dimensional bulk. This
appendix is written for the first-year graduate student who has met
general relativity and a first course in quantum field theory but has
never seen why a black hole should have an entropy at all, let alone why
the number is exactly one quarter of the horizon area in Planck units.
We build the result from scratch, in the longest and most explicit form
the cohort offers, and we are scrupulous at the end about what is
genuinely \emph{derived} and what remains \emph{open}. The honest
one-line summary, stated here and defended at the close: the \(A/4\) is a
real derivation, credited to \citet{ArisakaC3BH}; the recovery of the
Page curve is conceptually closed but quantitatively open.

\medskip
\textbf{The three steps of this appendix.} As in Appendix~E:
\begin{enumerate}[label=\textbf{Step \arabic*.}, leftmargin=*]
\item \textbf{The standard thermodynamics and induced gravity
(\S\ref{appA:bh-fourlaws} onward).} The four laws, the Hawking
temperature, the Bekenstein--Hawking entropy $S=A/4G_N\hbar$, Sakharov's
induced gravity, and the heat-kernel and conical-replica derivations of
the $1/4$.
\item \textbf{What standard physics leaves open
(\S\ref{appA:bh-limits}).} The area law is universally used, but its
microscopic origin --- why the entropy is exactly one quarter of the
horizon area --- is recovered from ingredients ($G_N$, an abstract
manifold) that are themselves taken as given.
\item \textbf{The geometric derivation (\S\ref{appA:bh-ggreading}).} On
the canonical six-dimensional bulk the $1/4$ is a theorem
(\citet{ArisakaC3BH}, T-BH-1), with the induced Newton constant the
\emph{same} scale-overlap $G_N$ that fixes gravity in Appendix~E.
\end{enumerate}

\subsection{The four laws of black-hole mechanics and the thermodynamic
analogy}
\label{appA:bh-fourlaws}

Start with the object. A Schwarzschild black hole of mass \(M\) is the
unique static, spherically symmetric, vacuum solution of Einstein's
equation \(R_{\mu\nu} - \tfrac12 g_{\mu\nu}R = 8\pi G_N T_{\mu\nu}/c^4\),
with line element
\begin{equation}
\label{appA:bh-schwarzschild}
ds^2 = -\Bigl(1 - \frac{2G_N M}{c^2 r}\Bigr)c^2\,dt^2
  + \frac{dr^2}{\,1 - \dfrac{2G_N M}{c^2 r}\,}
  + r^2\bigl(d\theta^2 + \sin^2\theta\,d\phi^2\bigr).
\end{equation}
The surface \(r = r_s \equiv 2G_N M/c^2\), the \emph{Schwarzschild
radius}, is the event horizon: a one-way causal membrane, not a
curvature singularity (the metric blowup there is a coordinate artifact,
removable by Kruskal coordinates; the true singularity is at \(r=0\)).
The horizon is a two-sphere of area
\begin{equation}
\label{appA:bh-area}
A = 4\pi r_s^2 = \frac{16\pi G_N^2 M^2}{c^4}.
\end{equation}
We will return to this formula for the worked example; for now note that
\(A\) grows as \(M^2\), so two black holes that merge increase the total
horizon area.

In 1973 Bardeen, Carter, and Hawking established four laws governing the
mechanics of stationary black holes, in striking one-to-one
correspondence with the four laws of thermodynamics. Let \(\kappa\) be
the \emph{surface gravity} (defined precisely in
\S\ref{appA:bh-hawkingT}; for Schwarzschild
\(\kappa = c^4/4G_N M\)), \(\Omega\) the angular velocity of the
horizon, \(J\) the angular momentum, \(\Phi\) the electrostatic
potential, and \(Q\) the charge. The laws read:

\begin{itemize}\setlength{\itemsep}{0.15em}
\item \textbf{Zeroth law.} The surface gravity \(\kappa\) is constant
  over the horizon of a stationary black hole. This is the analogue of
  the statement that temperature is uniform throughout a body in thermal
  equilibrium: \(\kappa \leftrightarrow T\).
\item \textbf{First law.} For a quasi-static change between neighboring
  stationary black holes,
  \begin{equation}
  \label{appA:bh-firstlaw-mech}
  dM = \frac{\kappa}{8\pi G_N}\,dA + \Omega\,dJ + \Phi\,dQ,
  \end{equation}
  which has exactly the form \(dU = T\,dS + \text{work}\) of the first
  law of thermodynamics, with \(M\) (the energy) in the role of \(U\),
  the \(\Omega\,dJ + \Phi\,dQ\) terms the work, and the product
  \((\kappa/8\pi G_N)\,dA\) standing in for \(T\,dS\). The pairing
  \(T\,dS \leftrightarrow (\kappa/8\pi G_N)\,dA\) is the crucial one;
  it already insinuates that \(S\) is proportional to \(A\).
\item \textbf{Second law.} Hawking's area theorem: in any classical
  process obeying the null energy condition the total horizon area
  cannot decrease, \(dA/dt \ge 0\). This is the analogue of the
  Clausius statement \(dS/dt \ge 0\): \(A \leftrightarrow S\).
\item \textbf{Third law.} An extremal black hole (\(\kappa = 0\), e.g.\
  a maximally charged or maximally rotating one) cannot be reached by
  any finite sequence of operations --- the analogue of the
  unattainability of absolute zero, \(T = 0\).
\end{itemize}

The 1973 paper presented these as \emph{analogies}. The match is
suspiciously perfect --- four laws, in order, with the same logical
structure --- but a pure analogy carries no physical entropy. Bekenstein
took the analogy literally and argued that the area really is an entropy
\citep{Bekenstein1973}; his argument was thermodynamic. If a black hole
had genuinely zero entropy, one could lower an entropy-bearing box of
gas through the horizon and erase its entropy from the visible universe,
decreasing the total entropy of the world and violating the ordinary
second law. To rescue the second law, the black hole must carry an
entropy that increases by at least the swallowed amount when it absorbs
the box. Bekenstein concluded that the black-hole entropy is
proportional to the horizon area, \(S_{\mathrm{BH}} \propto A/\ell_P^2\),
where \(\ell_P^2 = \hbar G_N/c^3\) is the Planck area --- the only
combination of constants with the right dimensions to convert an area
into a pure number. What Bekenstein could not fix from thermodynamics
alone was the dimensionless coefficient. Supplying that coefficient ---
showing it is exactly \(1/4\) --- is the entire technical content of
this appendix.

\subsection{The Hawking temperature \(T_H = \hbar\kappa/2\pi\) and the
surface gravity \(\kappa\)}
\label{appA:bh-hawkingT}

The analogy became physics in 1974, when Hawking computed what happens
to a quantum field on the Schwarzschild background and found that the
black hole is not black: it radiates a thermal spectrum at a definite
temperature \citep{Hawking1975}. The cleanest way to see the
temperature without the full mode-by-mode Bogoliubov calculation is the
Euclidean (imaginary-time) argument, which we sketch because it is the
same trick the conical-replica derivation will industrialize later.

\emph{Surface gravity.} For a static horizon the surface gravity
\(\kappa\) is the acceleration, measured at spatial infinity, of a test
particle held just outside the horizon --- the redshifted local
acceleration of a static observer. For Schwarzschild one finds
\begin{equation}
\label{appA:bh-kappa}
\kappa = \frac{c^4}{4 G_N M} = \frac{c^2}{2 r_s}.
\end{equation}
It is finite and constant over the horizon (the zeroth law), and it
diverges as \(M \to 0\): small black holes have large surface gravity.

\emph{Euclidean periodicity fixes the temperature.} Continue to
imaginary time \(t \to -i\tau\) and look at the near-horizon geometry.
Set \(\rho^2 \propto (r - r_s)\); the Schwarzschild metric near the
horizon becomes, to leading order,
\begin{equation}
\label{appA:bh-nearhorizon}
ds_E^2 \;\approx\; d\rho^2 + \rho^2\,d\!\left(\frac{\kappa\,\tau}{c}\right)^{\!2}
  + (\text{transverse }2\text{-sphere}),
\end{equation}
which is the metric of a flat plane in polar coordinates
\((\rho,\ \kappa\tau/c)\). A plane is smooth at its origin \(\rho = 0\)
(the horizon) \emph{only} if the angular coordinate \(\kappa\tau/c\) has
period exactly \(2\pi\); any other period leaves a conical singularity
--- a cone with a deficit or excess angle --- sitting on the horizon.
Demanding smoothness fixes the period of Euclidean time,
\begin{equation}
\label{appA:bh-betaperiod}
\tau \sim \tau + \frac{2\pi c}{\kappa}.
\end{equation}
But in equilibrium statistical mechanics a thermal density matrix
\(\rho \propto e^{-\beta H}\) is exactly a Euclidean path integral
periodic in imaginary time with period \(\hbar\beta = \hbar/k_B T\). The
two periodicities must agree, so
\begin{equation}
\label{appA:bh-hawkingT-result}
\boxed{\;
k_B T_H = \frac{\hbar\,\kappa}{2\pi c}
\qquad\Longleftrightarrow\qquad
T_H = \frac{\hbar\,c^3}{8\pi G_N M\,k_B}.
\;}
\end{equation}
This is the Hawking temperature. In the common units \(c = 1\) it is the
celebrated \(T_H = \hbar\kappa/2\pi\); the conical-deficit remark below
\eqref{appA:bh-betaperiod} --- that the period of Euclidean time is tied
to a deficit angle on the horizon --- is precisely the seed that
\S\ref{appA:bh-conical} will grow into the exact \(1/4\). Hold onto the
phrase ``deficit angle on the horizon''; the entire derivation turns on
it.

The temperature is heart-stoppingly small for astrophysical masses and
enormous for tiny ones. For a one-solar-mass hole
\eqref{appA:bh-hawkingT-result} gives \(T_H \approx 6.2\times10^{-8}\)~K,
seven orders of magnitude below the \(2.7\)~K cosmic microwave
background, so such a hole \emph{absorbs} far more than it radiates and
does not evaporate in the present epoch. A primordial hole of
\(\sim 10^{12}\)~kg, by contrast, has \(T_H \sim 10^{11}\)~K and
evaporates within a cosmological time --- the regime relevant to the
primordial-black-hole bounds elsewhere in the cohort.

\subsection{The Bekenstein--Hawking entropy \(S = A/4G_N\hbar\) and the
first law \(dM = T\,dS\)}
\label{appA:bh-entropy}

With a genuine temperature in hand, the entropy is no longer optional.
Treat the black hole as a thermodynamic system whose energy is its rest
energy \(U = Mc^2\) and whose temperature is \(T_H\) of
\eqref{appA:bh-hawkingT-result}. The thermodynamic first law for a
non-rotating, uncharged hole is simply
\begin{equation}
\label{appA:bh-firstlaw-thermo}
dU = T_H\,dS_{\mathrm{BH}}
\qquad\Longrightarrow\qquad
dS_{\mathrm{BH}} = \frac{c^2\,dM}{T_H}.
\end{equation}
Substitute \(T_H = \hbar c^3/8\pi G_N M k_B\) and integrate from zero
mass:
\begin{equation}
\label{appA:bh-integrate}
S_{\mathrm{BH}}
= \int_0^M \frac{c^2\,dM'}{T_H(M')}
= \int_0^M \frac{8\pi G_N k_B}{\hbar c}\,M'\,dM'
= \frac{4\pi G_N k_B M^2}{\hbar c}.
\end{equation}
Now eliminate \(M\) in favor of the area using
\eqref{appA:bh-area}, \(M^2 = c^4 A/16\pi G_N^2\), to obtain the
Bekenstein--Hawking entropy in its final form,
\begin{equation}
\label{appA:bh-BHentropy}
\boxed{\;
S_{\mathrm{BH}}
= \frac{k_B\,c^3\,A}{4\,G_N\,\hbar}
= \frac{k_B\,A}{4\,\ell_P^2},
\qquad \ell_P^2 = \frac{\hbar G_N}{c^3}.
\;}
\end{equation}
In natural units \(k_B = \hbar = c = 1\) this is the famous
\(S = A/4G_N\), and in Planck units it is one quarter of the horizon
area measured in Planck cells, \(S/k_B = A/4\ell_P^2\). The coefficient
\(1/4\) is now visible but \emph{not yet explained}: it descended here
from integrating the Hawking temperature, which we got from Euclidean
periodicity, which we accepted on the strength of the semiclassical path
integral. A skeptic is entitled to ask: where does the universal
\(1/4\) actually come from, mechanism and all, independent of which
field is radiating and which regulator one uses? Answering exactly that
question is the job of induced gravity.

\subsection{Sakharov's induced gravity: the Einstein--Hilbert action is
generated by integrating out fluctuations}
\label{appA:bh-sakharov}

The decisive idea is Sakharov's \citep{Sakharov1967}: gravity need not
be a fundamental field with its own bare coupling. Sakharov proposed
that the Einstein--Hilbert action \emph{itself} is generated --- induced
--- by the quantum vacuum fluctuations of all the \emph{other} fields
propagating in a curved background. In modern language, integrate out
the matter (and, in \GGTheory{}, the bulk) fluctuations in a fixed
metric \(g_{\mu\nu}\); the resulting one-loop effective action
\(W[g] = \tfrac12 \ln\det\Delta\), expanded in curvatures, contains a
term proportional to \(\int\!\sqrt{g}\,R\). The coefficient of that term
\emph{is} \(1/16\pi G_N\). Newton's constant is not put in by hand; it
is read off the spectrum of the modes that were integrated out.

\GGTheory{} promotes this hypothesis from an option to a law. Under the
Geometry--Gauge Equivalence Principle (\GGEP, AX-2 of the cohort
constitution) there is \emph{no} bare four-dimensional Newton constant
to begin with: four-dimensional gravity is the geometry of the six-dimensional bulk \citep{ArisakaA2GG6}, and the gravitational action
a boundary observer reads is generated \emph{entirely} by integrating
out the bulk fluctuation modes through the open-system \OGGEP{}
projection \GDOS{} \citep{ArisakaA5GDOS}. Sakharov's ``induced'' becomes
the only option on the table, and a corollary that is merely suggestive
in ordinary field theory becomes forced here. The corollary is this: the
ultraviolet divergence of the horizon entanglement entropy is not a
small correction sitting on top of a pre-existing \(1/4G_N\) --- there is
no pre-existing \(G_N\) for it to correct --- so the entanglement
divergence \emph{is} the \(1/4G_N\). This is the Susskind--Uglum
observation; in \GGTheory{} it is a theorem rather than a coincidence,
because \GGEP{} removes the bare term that would otherwise let the two
quantities drift apart.

\subsection{The heat-kernel expansion: one coefficient sets both
\(1/G_N\) and the entanglement entropy}
\label{appA:bh-heatkernel}

The technical engine behind both statements is a single object, the
\emph{heat kernel} of the fluctuation operator \(\Delta\) on the
four-dimensional boundary. Its short-proper-time expansion is
\begin{equation}
\label{appA:bh-heatkernel-exp}
\operatorname{Tr} e^{-s\Delta}
= \frac{1}{(4\pi s)^2}\int d^4x\,\sqrt{g_4}\,
   \bigl[a_0 + a_1\,s + a_2\,s^2 + \cdots\bigr],
\qquad
a_1 = \Bigl(\tfrac16 - \xi\Bigr)R,
\end{equation}
where \(\xi\) is the curvature coupling (\(\xi = 0\) for a minimal
scalar) and \(s\) is the Schwinger proper-time parameter. The
coefficient \(a_1\) --- the one multiplying the Ricci scalar \(R\) --- is
the common ancestor of both quantities we care about.

\emph{It sets \(1/G_N\).} The one-loop effective action is
\(W = \tfrac12\ln\det\Delta = -\tfrac12\int_{\epsilon^2}^{\infty}
\frac{ds}{s}\operatorname{Tr}e^{-s\Delta}\), regulated by a proper-time
cutoff \(s \ge \epsilon^2\) with \(\epsilon\) a short-distance length.
The \(a_1\) term contributes precisely the Einstein--Hilbert piece, and
matching it to \(-\tfrac{1}{16\pi G_N}\int\!\sqrt{g_4}\,R\) gives, per
minimal scalar,
\begin{equation}
\label{appA:bh-inducedG}
\frac{1}{G_{\mathrm{ind}}} = \frac{1}{12\pi\,\epsilon^2},
\qquad\Longrightarrow\qquad
\frac{1}{4 G_{\mathrm{ind}}} = \frac{1}{48\pi\,\epsilon^2}.
\end{equation}

\emph{It sets the entanglement entropy.} The leading (area) term of the
entanglement entropy across the horizon, computed from the \emph{same}
heat kernel by the conical method of the next subsection, is
\begin{equation}
\label{appA:bh-Sent}
\frac{dS_{\mathrm{ent}}}{dA} = \frac{1}{48\pi\,\epsilon^2}
\qquad\text{(per minimal scalar).}
\end{equation}

\emph{The match.} Comparing \eqref{appA:bh-inducedG} and
\eqref{appA:bh-Sent},
\begin{equation}
\label{appA:bh-match}
\boxed{\;
\frac{dS_{\mathrm{ent}}}{dA} = \frac{1}{4 G_{\mathrm{ind}}}
\;}
\qquad\text{identically.}
\end{equation}
The individual number \(1/48\pi\epsilon^2\) is scheme-dependent: it moves
when you change the regulator \(\epsilon\) or the coupling \(\xi\), and
it would change again if you added more fields. But the \emph{equality}
\eqref{appA:bh-match} does not move, because both sides are literally the
same heat-kernel coefficient \(a_1\) of the same modes. The
ultraviolet-sensitive prefactor cancels in the ratio. This is why the
induced-gravity reading is so powerful: it ties the divergent
entanglement entropy and the divergent induced \(1/G_N\) together so
that their ratio is finite and universal, even though neither is. In
\GGTheory{} the cancellation is not a happy accident to be checked
regulator by regulator --- it is enforced by \GGEP, since there is no
bare \(G_N\) for the divergence to be a correction to. What we still
need is the actual value of the ratio. That is what the conical replica
delivers, in closed form, with no surviving dependence on \(\epsilon\) or
on the field content.

\subsection{The conical-replica (deficit-angle) method: the coefficient
\(1/4 = 4\pi/16\pi\), regulator-independently}
\label{appA:bh-conical}

Entanglement entropy is computed by the replica trick:
\(S = -\operatorname{Tr}\rho\ln\rho
= (1 - n\,\partial_n)\ln\operatorname{Tr}\rho^{\,n}\big|_{n=1}\), and
\(\operatorname{Tr}\rho^{\,n} = Z_n/Z_1^{\,n}\), where \(Z_n\) is the
Euclidean partition function on an \(n\)-fold cover of the geometry,
branched over the horizon. Recall from \S\ref{appA:bh-hawkingT} that
making the period of Euclidean time anything other than \(2\pi/\kappa\)
leaves a conical singularity on the horizon. The replica cover
\(\mathcal{M}_n\) does exactly that: it has Euclidean-time period
\(n\) times the smooth one, i.e.\ a \emph{conical deficit angle}
\(2\pi(1-n)\) concentrated on the bifurcation surface \(\Sigma\) (the
horizon, of area \(A\)). The point of the method is that all the
ultraviolet detail of the fields is now packaged in a known
distributional curvature.

The curvature of a manifold with a conical deficit acquires a piece
localized on the tip --- a two-dimensional delta function with weight set
by the deficit angle. Integrated against the volume, this gives the
exact identity
\begin{equation}
\label{appA:bh-conical-id}
\int_{\mathcal{M}_n}\!\sqrt{g}\,R
= n\!\int_{\mathrm{smooth}}\!\sqrt{g}\,R
\;+\; 4\pi(1-n)\,A.
\end{equation}
The first term is the bulk curvature, \(n\) copies of the smooth one;
the second is the new horizon term, and its coefficient \(4\pi\) is pure
geometry --- the angular measure of the deficit times the codimension-two
horizon area. No field content, no regulator, no coupling \(\xi\) enters
\eqref{appA:bh-conical-id}; it is a statement about cones.

Now feed this into the \emph{induced} Einstein--Hilbert action
\(I_E[\mathcal{M}_n] = -\tfrac{1}{16\pi G_N}\int_{\mathcal{M}_n}
\sqrt{g}\,R\), which by \S\ref{appA:bh-sakharov} is the \emph{entire}
gravitational action (there is no separate matter contribution to the
entropy beyond the modes that generated \(G_N\)):
\begin{equation}
\label{appA:bh-action-n}
I_E[\mathcal{M}_n]
= n\,I_E[\mathcal{M}_1] \;-\; \frac{(1-n)\,A}{4 G_N},
\qquad
\ln Z_n = -I_E[\mathcal{M}_n].
\end{equation}
Writing \(R_1 \equiv \int_{\mathcal{M}_1}\sqrt{g}\,R\) for the smooth
bulk curvature, the replica entropy follows by one differentiation:
\begin{align}
\label{appA:bh-final}
\ln Z_n &= \frac{n\,R_1}{16\pi G_N} + \frac{(1-n)A}{4 G_N},
&\partial_n \ln Z_n\big|_{n=1}
  &= \frac{R_1}{16\pi G_N} - \frac{A}{4 G_N}, \nonumber\\
\ln Z_1 &= \frac{R_1}{16\pi G_N},
&S = \ln Z_1 - \partial_n \ln Z_n\big|_{n=1}
  &= \boxed{\;\frac{A}{4 G_N}\;}.
\end{align}
The bulk curvature \(R_1\) cancels identically between \(\ln Z_1\) and
\(\partial_n\ln Z_n\); the entropy is \emph{purely} the boundary (area)
term. And the coefficient is now transparent and closed-form:
\begin{equation}
\label{appA:bh-quarter}
\frac{1}{4}
\;=\; \frac{4\pi}{16\pi}
\;=\; \frac{\text{(conical-deficit curvature coefficient)}}
           {\text{(Einstein--Hilbert normalization)}}.
\end{equation}
The \(4\pi\) in the numerator is the deficit-angle curvature weight of
\eqref{appA:bh-conical-id}; the \(16\pi\) in the denominator is the
normalization of the Einstein--Hilbert action. The famous one quarter is
nothing more mysterious than their ratio. Restoring units,
\begin{equation}
\label{appA:bh-Sbh-units}
\boxed{\;
S_{\mathrm{BH}} = \frac{k_B\,c^3\,A}{4\,G_N\,\hbar}
\;}
\end{equation}
which reproduces \eqref{appA:bh-BHentropy}, now \emph{derived} from
induced gravity rather than read off the integrated first law. Three
features deserve emphasis because they are what make this a derivation
and not a fit:
\begin{enumerate}\setlength{\itemsep}{0.15em}
\item \emph{No free parameter.} Nothing was tuned; \(4\pi/16\pi\) is
  forced.
\item \emph{Regulator-independent.} The cutoff \(\epsilon\) and the
  coupling \(\xi\) canceled in the ratio \eqref{appA:bh-match}, and the
  conical identity \eqref{appA:bh-conical-id} never referred to them.
\item \emph{No supersymmetry and no extremality.} Unlike the
  Strominger--Vafa brane-counting derivations, which work at (or near)
  an extremal, supersymmetric limit, the conical-replica argument runs
  for the generic Schwarzschild, Kerr, and Reissner--Nordstr\"om hole.
\end{enumerate}

\subsection{What standard physics leaves open: the origin of $A/4$}
\label{appA:bh-limits}

The derivations of Step~1 are among the deepest results in theoretical
physics, but it is worth being precise about their status. That a black
hole carries an entropy proportional to its horizon area is forced by the
thermodynamic analogy and by Hawking radiation; that the coefficient is
\emph{exactly} $1/4$ in Planck units emerges from the heat-kernel and
conical-replica calculations. But in the standard setting these
calculations run on an abstract manifold, with the Newton constant $G_N$
inserted as a measured input. They do not say \emph{why} $G_N$ has its
value, nor tie the area law to a specific underlying geometry: the $1/4$
is recovered, but from ingredients that are themselves taken as given. The
geometric account supplies exactly those ingredients.

\subsection{The geometric derivation: $A/4$ from the canonical bulk (\citet{ArisakaC3BH})}
\label{appA:bh-ggreading}

Everything in \S\ref{appA:bh-sakharov}--\S\ref{appA:bh-conical} is, in
the cohort, a \emph{theorem} rather than a sketch. It is
Theorem~\emph{T-BH-1} of the companion \citet{ArisakaC3BH}, the
black-hole entropy paper, and \ATen{} cites it rather than re-proving
it. Two points make the \GGTheory{} version sharper than the textbook
induced-gravity argument.

\emph{The conical replica runs on the canonical six-dimensional bulk.}
\citet{ArisakaC3BH} does not perform the heat-kernel reduction on an
abstract four-manifold; it performs it on the canonical bulk
\(\mathcal{M}_6 = \mathcal{M}_4 \times I_r \times S^1_\theta\) of
\citet{ArisakaA2GG6}, with the metric
\(ds_6^2 = e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2\).
Reducing the six-dimensional Einstein--Hilbert term on the scale fiber
\(I_r\) and the phase circle \(S^1_\theta\) produces the boundary
Einstein--Hilbert term with a \emph{finite, parameter-free} coefficient
\(1/G_N = \pi(1 - e^{-2K})/G_6\), the radial integral converging because
the scale profile \(e^{-2r}\) is integrable. The induced Newton constant is thus a
property of the scale profile, ultraviolet-localized at the GUT edge \(r=0\) ---
the geometric statement that gravity is a UV phenomenon.

\emph{The induced \(G_N\) is A5's scale-overlap \(G_N\), by
construction.} This is the cross-reference that closes the loop and
that \citet{ArisakaC3BH} now carries explicitly. The boundary
general relativity of \citet{ArisakaA5GDOS} reads its Newton constant off
the \emph{same} canonical scale profile through the scale-overlap reduction
\(M_{\mathrm{Pl}}^2 = M_6^4\,2\pi R_\theta \int e^{-2A}\,dr\); the
induced-gravity derivation of \citet{ArisakaC3BH} reads \(1/G_N\) off
that \emph{same} scale profile of \citet{ArisakaA2GG6}. The two Newton
constants therefore coincide \emph{by construction}, not by a
coincidence to be checked numerically: the \(G_N\) that enters horizon
entropy is the scale-derived one, the very same object that fixes the
strength of gravity in the Friedmann equations of the main text. And the
coefficient \(1/4\) is the more robust half of the claim --- by
\eqref{appA:bh-quarter} it is a ratio of curvature normalizations, so
even if the absolute normalization of \(G_N\) were later refined, the
one-quarter would not move.

In the hidden-fiber language of \AEight{}, Step~4 of \citet{ArisakaC3BH}
even names the microstates: the modes whose one-loop determinant induces
\(G_N\) are the Wilson-line holonomies on the phase circle
\(S^1_\theta\) together with the Kaluza--Klein tower of the scale
direction, evaluated on the horizon two-sphere. A horizon of area \(A\)
supports \(\exp(A/4\ell_P^2)\) distinguishable fiber configurations ---
the same compact circle that carries quantum coherence in the bulk
becomes, at a horizon, the carrier of black-hole microstates. The
black-hole counting and the origin of quantum mechanics are one
mechanism evaluated on two loci.

\subsection{Information and the Page curve: conceptually closed,
quantitatively open}
\label{appA:bh-pagecurve}

Here the honesty discipline must be exact, because this is where the
literature and the cohort both fall short of a full computation, and the
main text of \ATen{} flags it as the least-closed module of its
black-hole reading.

The information paradox is the tension between two facts. A pure state
\(|\psi\rangle\) collapses to a black hole, which then evaporates into
Hawking radiation that is thermal --- hence mixed --- to all orders in
the semiclassical expansion, so the map
\(|\psi\rangle \to \rho_{\mathrm{thermal}}\) appears to be
pure-to-mixed and non-unitary. Page sharpened the demand in 1993: if the
total evolution is unitary, the entanglement entropy of the radiation
must first rise, then turn over at the halfway point and \emph{fall} back
to zero --- the ``Page curve'' --- rather than rise monotonically as
Hawking's calculation predicts. Recovering that turnover is the modern
content of ``resolving the paradox.''

In \GGTheory{} the resolution is stated structurally. By \OGGEP{} (AX-3)
observation is the open-system trace of the bulk over what a boundary
horizon hides, so the radiation entropy is a branch-minimized
generalized-entropy functional --- mathematically the same object as the
island/replica generalized entropy of the contemporary literature. The
cohort names the turnover ``the Page curve as a \GRIP{}/\GDOS{} saddle
switch'': at the Page time the operationally dominant saddle of the
open-system entropy ledger changes, which is \emph{why} the curve must
turn over. \ATen{}'s own structural estimate of the turnover mass,
\(M_{\mathrm{Page}} = M_0/\sqrt{2}\), follows from the half-entropy
condition.

But --- and this is the open part, said plainly --- naming the saddle is
not the same as computing it. The slogan ``Page curve as a saddle
switch'' identifies \emph{which physical thing changes} at the turnover;
it does \emph{not} perform a replica/island calculation with a
quantitative Page time, an exhibited island region, and a quantitative
entropy curve. The cohort tells you \emph{why} a branch-minimized
functional had to be the answer; the island program tells you \emph{how}
to compute the saddle, and that computation has not been carried out
here. The honest quantitative status is identical to that of the
companion \citet{ArisakaC3BH}: the \(A/4\) coefficient and the
thermality (T-BH-1) are derived, but the island/replica Page curve is
treated as conceptually closed and quantitatively open. We do not
pretend the slogan performs the calculation.

\subsection{A worked numerical example: a one-solar-mass Schwarzschild
black hole}
\label{appA:bh-worked}

It is worth seeing the numbers, because the answer is one of the largest
entropies in physics. Take \(M = M_\odot = 1.989\times10^{30}\)~kg and
the constants \(G_N = 6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}\),
\(c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}\),
\(\hbar = 1.055\times10^{-34}\ \mathrm{J\,s}\),
\(k_B = 1.381\times10^{-23}\ \mathrm{J\,K^{-1}}\).

\emph{Step 1 --- Schwarzschild radius and area.}
\begin{equation}
\label{appA:bh-worked-area}
r_s = \frac{2 G_N M}{c^2} \approx 2.95\times10^{3}~\mathrm{m}
\;\;(\approx 2.95~\mathrm{km}),
\qquad
A = \frac{16\pi G_N^2 M^2}{c^4} \approx 1.10\times10^{8}~\mathrm{m^2}.
\end{equation}

\emph{Step 2 --- the Planck area.}
\begin{equation}
\label{appA:bh-worked-planck}
\ell_P^2 = \frac{\hbar G_N}{c^3} \approx 2.61\times10^{-70}~\mathrm{m^2},
\qquad
\ell_P \approx 1.62\times10^{-35}~\mathrm{m}.
\end{equation}

\emph{Step 3 --- the entropy in nats.} The dimensionless entropy
(entropy in units of \(k_B\), i.e.\ in nats) is one quarter of the
horizon area in Planck cells:
\begin{equation}
\label{appA:bh-worked-nats}
\frac{S_{\mathrm{BH}}}{k_B}
= \frac{A}{4\ell_P^2}
\approx \frac{1.10\times10^{8}}{4\,(2.61\times10^{-70})}
\approx 1.05\times10^{77}\ \text{nats}.
\end{equation}

\emph{Step 4 --- in bits and in SI units.} Converting nats to bits with
\(1~\mathrm{nat} = 1/\ln 2 \approx 1.443~\mathrm{bits}\),
\begin{equation}
\label{appA:bh-worked-bits}
\frac{S_{\mathrm{BH}}}{k_B \ln 2}
\approx 1.51\times10^{77}\ \text{bits},
\qquad
S_{\mathrm{BH}} = k_B \cdot 1.05\times10^{77}
\approx 1.45\times10^{54}\ \mathrm{J\,K^{-1}}.
\end{equation}

For orientation: the corresponding Hawking temperature from
\eqref{appA:bh-hawkingT-result} is
\(T_H \approx 6.2\times10^{-8}\)~K, far below the CMB, so this hole does
not evaporate today. The entropy \(\sim 10^{77}\) nats is roughly twenty
orders of magnitude larger than the thermodynamic entropy of the Sun as
an ordinary ball of gas (\(\sim 10^{58}\,k_B\)) --- the gravitational
collapse into a black hole is an enormous entropy increase, exactly as
the generalized second law requires. Every one of those
\(\sim 10^{77}\) nats is, in the \GGTheory{} reading of
\S\ref{appA:bh-ggreading}, a configuration of the hidden phase fiber
traced out by the horizon.

\subsection*{Status}
\label{appA:bh-status}

\emph{The area law \(S = A/4\) is genuinely Derived.} The coefficient
\(1/4 = 4\pi/16\pi\) is fixed in closed form by the conical-replica
method on the canonical six-dimensional bulk, regulator-independently
and without supersymmetry or extremality, in Theorem~\emph{T-BH-1} of
\citet{ArisakaC3BH}; the induced Newton constant entering it is the
scale-overlap \(G_N\) of \citet{ArisakaA5GDOS}, read off the same
canonical bulk of \citet{ArisakaA2GG6}, so the two coincide by
construction. \ATen{} inherits this result and uses it as an established
input.

\emph{The Page-curve recovery is conceptually closed but quantitatively
open.} The open-system \OGGEP{}/\GRIP{} reading names the saddle switch
that makes the radiation entropy turn over, and identifies the entropy
functional as the island generalized entropy; it does \emph{not} provide
a replica/island computation with a quantitative Page time. This is the
honest residue, flagged as such in the main text and unchanged here: the
mechanism is named, the calculation is not yet performed.

%% ===== Block III: registries and catalogues =====


\paragraph{Problems and worked solutions.}

\medskip\noindent\textbf{N.1.1. Verify the first law for the
Schwarzschild family.}

\noindent\emph{Solution.}  Step 1 (geometrized units): $A =
16\pi G^2M^2$, so $dA = 32\pi G^2M\,dM$.  Step 2: with
$\kappa = 1/4GM$: $\tfrac{\kappa}{8\pi G}\,dA =
\tfrac{1}{4GM}\cdot\tfrac{32\pi G^2M}{8\pi G}\,dM = dM$ ---
the first law $dM = \kappa\,dA/8\pi G$, exactly, with no
adjustable coefficient.  Step 3: read with $T_H =
\hbar\kappa/2\pi$ this forces $S = A/4G\hbar$ up to an
additive constant --- the two famous formulas are one
Legendre pair, which is why Hawking's temperature computation
instantly fixed Bekenstein's coefficient.  Physics intuition:
the first law is the hinge between mechanics and
thermodynamics --- once any one intensive quantity (here
$T_H$) is pinned by microphysics, the whole thermodynamic
dictionary locks.

\medskip\noindent\textbf{N.5.1. Show that the heat-kernel
coefficient cancels in $S = A/4G_N$: the Susskind--Uglum
compensation.}

\noindent\emph{Solution.}  Step 1: the same $a_1$ (curvature)
coefficient of the heat-kernel expansion controls both the
induced Newton constant, $1/G_{\rm ind} \propto \sum_i\chi_i
\Lambda^2$, and the leading entanglement entropy across the
horizon, $S_{\rm ent} \propto \sum_i\chi_i\Lambda^2 A$.
Step 2: forming the ratio, the species sum and the cutoff
\emph{cancel}: $S_{\rm ent} = A/4G_{\rm ind}$ --- whatever
the field content, however many species, the entropy in units
of the \emph{induced} $1/4G$ is universal.  Step 3: this is
why the species problem (the apparent field-content
dependence of horizon entropy) dissolves exactly when gravity
is induced --- the entropy and the stiffness are the same
loop diagram read twice.  Physics intuition: one integral,
two names --- the compensation is the deepest single fact in
the induced-gravity reading, and the cohort's remaining
species-sum work (the black-hole companion's declared open
computation) is about the \emph{absolute} normalization, never
about this ratio.

\medskip\noindent\textbf{N.6.1. Run the conical-replica
derivative: the deficit angle delivers $A/4G$.}

\noindent\emph{Solution.}  Step 1: the $n$-replica Euclidean
manifold is smooth except at the horizon, where it carries a
conical deficit $\delta = 2\pi(1-n)$; the curvature acquires
a distributional term $R \supset 2\delta\,\delta^{(2)}(x)$.
Step 2: inserting into the Einstein--Hilbert action,
$I_n \supset -\tfrac{1}{16\pi G}\cdot 2\cdot2\pi(1-n)\,A$,
so the replica free energy carries $(1-n)\,A/4G$.  Step 3:
the entropy is $S = -\partial_n\bigl[I_n - nI_1\bigr]_{n=1}
= A/4G$ --- the coefficient $\tfrac14$ falls out of the
geometry of the cone, independent of the matter content
(N.5.1's cancellation seen from the geometric side).
Physics intuition: entropy is the action's response to a
pinch at the horizon --- every modern derivation of $A/4$,
from Gibbons--Hawking to the islands literature to the
cohort's canonical-bulk version, is this one computation in
different clothing.

\medskip\noindent\textbf{N.9.1. Put numbers on the Page
schedule for a solar-mass black hole.}

\noindent\emph{Solution.}  Step 1: the evaporation time is
$t_{\rm evap} \approx 2\times10^{67}$~yr (the chapter's
worked example), and the entropy-based turnover sits at
$t_{\rm Page} = (1 - 2^{-3/2})\,t_{\rm evap} =
0.646\,t_{\rm evap} \approx 1.4\times10^{67}$~yr.  Step 2:
the archive to be returned is $S/k_B\ln2 \approx
1.5\times10^{77}$~bits; spread over the lifetime this is an
average of $\sim\!230$ bits per second.  Step 3: the register,
kept exactly as the chapter states it: the Page \emph{curve}
is conceptually closed (unitarity plus the port bookkeeping
fix its shape), while the quantitative microstate accounting
behind it remains the least-closed module of the black-hole
reading --- the numbers above are schedule, not mechanism.
Physics intuition: even for the most patient object in
physics the ledger balances --- but at two hundred bits a
second across $10^{57}$ ages of the universe, ``information
is preserved'' is a statement about principle, never about
practice.

\medskip\noindent\textbf{N.10.1. When does a black hole begin
to evaporate at all?}

\noindent\emph{Solution.}  Step 1: net evaporation requires
$T_H > T_{\rm CMB} = 2.725$~K.  Step 2: with $T_H \propto
1/M$ (the chapter's worked example gives $6\times10^{-8}$~K
for $1\,M_\odot$), equality sits at $M = 4.5\times10^{22}$~kg
$\approx 0.6$ lunar masses.  Step 3: every astrophysical
black hole is therefore \emph{absorbing} today; the
evaporation era for a solar-mass hole begins only when the
expanding universe (Appendix~\ref{appA:thermal}) has cooled
the sky below sixty billionths of a kelvin --- the de Sitter
future of F.7.1, where the only radiation left is the
horizon's own.  Physics intuition: black-hole thermodynamics
runs on the universe's schedule, not its own --- the cosmos
must first become colder than its coldest objects, and the
$\pi$-locked vacuum sets exactly how cold that future gets.

\GGtakeaway{\emph{Standard result:} horizons carry entropy
\(A/4G_N\) and temperature \(\hbar\kappa/2\pi\) --- exact, universal,
and unexplained. \emph{What it assumed:} the coefficient \(1/4\) from a
semiclassical computation with no statement of \emph{whose} degrees of
freedom are counted. \emph{What the geometry replaced it with:} induced
gravity on the canonical bulk --- one heat-kernel coefficient fixing
both \(1/G_N\) and the entropy, the horizon read as an \OGGEP{} port,
and the information story conceptually closed with the quantitative
Page-curve match named as open work (PS-Cosmos-18). \emph{Continue in:}
Part~VII; \PaperCThree{}.}

\subsection{Worked derivation notebook for the Part~VII observables}
\label{appA:bhentropy-notebook}
\label{sec:partvii-notebook}


\subsubsection{Derivation notebook A: Schwarzschild area, entropy, and temperature}

Start from the Schwarzschild radius
\begin{equation}
r_s = 2G_N M.
\end{equation}
The horizon area is then
\begin{equation}
A_H = 4\pi r_s^2 = 16\pi G_N^2 M^2.
\end{equation}
Dividing by \(4G_N\) gives
\begin{equation}
S_{\mathrm{BH}} = \frac{A_H}{4G_N}=4\pi G_N M^2.
\end{equation}
Differentiating,
\begin{equation}
\frac{\dd S_{\mathrm{BH}}}{\dd M}=8\pi G_N M.
\end{equation}
Using the first law \(\dd M = T_H\,\dd S_{\mathrm{BH}}\) gives
\begin{equation}
T_H = \left(\frac{\dd S_{\mathrm{BH}}}{\dd M}\right)^{-1}
= \frac{1}{8\pi G_N M}.
\end{equation}
Nothing exotic has happened. The notebook begins by showing that the entire paradox is built on two ordinary geometric facts: area scales as \(M^2\), temperature as \(M^{-1}\).

\subsubsection{Derivation notebook B: evaporation law and Page-time estimate}

Assume
\begin{equation}
\dot M = -\frac{\alpha_H}{M^2}.
\end{equation}
Then
\begin{equation}
M^2 \dot M = -\alpha_H
\qquad\Rightarrow\qquad
\frac{\dd}{\dd t}\left(\frac{M^3}{3}\right) = -\alpha_H.
\end{equation}
Integrating,
\begin{equation}
M^3(t)=M_0^3-3\alpha_H t.
\end{equation}
The evaporation time follows from \(M(t_{\mathrm{evap}})=0\):
\begin{equation}
t_{\mathrm{evap}}=\frac{M_0^3}{3\alpha_H}.
\end{equation}

For the Page estimate, equate branch 1 and branch 2 with \(\eta=1\):
\begin{equation}
S_0-S(M_{\mathrm{Page}})=S(M_{\mathrm{Page}})
\quad\Rightarrow\quad
S(M_{\mathrm{Page}})=\frac{S_0}{2}.
\end{equation}
Since \(S\propto M^2\),
\begin{equation}
M_{\mathrm{Page}}=\frac{M_0}{\sqrt 2}.
\end{equation}
Insert into the cubic mass law:
\begin{equation}
\left(\frac{M_0}{\sqrt 2}\right)^3
=
M_0^3-3\alpha_H t_{\mathrm{Page}},
\end{equation}
so
\begin{equation}
t_{\mathrm{Page}}
=
\frac{M_0^3}{3\alpha_H}\left(1-2^{-3/2}\right)
=
\left(1-2^{-3/2}\right)t_{\mathrm{evap}}.
\end{equation}
The point is not the exact prefactor \(0.65\), but the algebraic fact that a Page-like turnover is compatible with the evaporation law once the relevant entropy branch is no longer taken to be strictly thermal Hawking entropy forever.

\subsubsection{Derivation notebook C: the operational map and why pure-to-mixed is not paradoxical}

Let the full 6D state evolve unitarily:
\begin{equation}
\rho_6(t)=U_6(t)\rho_6(0)U_6^\dagger(t).
\end{equation}
Now split the full algebra into accessible and inaccessible parts only at the operational level. The accessible boundary state is
\begin{equation}
\rho_{\partial}(t)=\mathrm{Tr}_{\mathrm{bulk}\setminus\partial}\,\rho_6(t).
\end{equation}
Equivalently, it can be written in Kraus form,
\begin{equation}
\rho_{\partial}(t)=\sum_k K_k(t)\rho_6(0)K_k^\dagger(t),
\qquad
\sum_k K_k^\dagger K_k=\mathbb{1}.
\end{equation}
A pure state under a partial trace can become mixed with no contradiction whatsoever. The contradiction only arises if one demands that \(\rho_{\partial}\) itself be the total ontology. \OGGEP{} forbids that demand.

\subsubsection{Derivation notebook D: horizon mass as a timestamp for PBH formation}

Use the radiation-era relations
\begin{equation}
\rho = \frac{\pi^2}{30}g_*T^4,
\qquad
H = 1.66 g_*^{1/2}\frac{T^2}{M_{\mathrm{Pl}}}.
\end{equation}
Then
\begin{align}
M_H &= \frac{4\pi}{3}\rho H^{-3} \\
&=
\frac{4\pi}{3}
\left(\frac{\pi^2}{30}g_*T^4\right)
\left(1.66^3 g_*^{3/2}\frac{T^6}{M_{\mathrm{Pl}}^3}\right)^{-1}.
\end{align}
Collecting factors gives the scaling
\begin{equation}
M_H \propto g_*^{-1/2}T^{-2}.
\end{equation}
Evaluating numerically at \(T=\mathrm{TeV}\) gives
\begin{equation}
M_H(\mathrm{TeV})\sim 10^{26}\,\mathrm g,
\end{equation}
up to order-unity factors. This is the cleanest reason PBH mass windows are so informative. They are direct thermometers of the early universe.

\subsubsection{Derivation notebook E: why the minimal PBH abundance is exponentially tiny}

Take the asymptotic Gaussian tail
\begin{equation}
\beta(M)\approx \frac{\sigma}{\delta_c\sqrt{2\pi}}
\exp\!\left[-\frac{\delta_c^2}{2\sigma^2}\right].
\end{equation}
Set \(\delta_c=0.45\). If \(\sigma=0.01\), then
\begin{equation}
\frac{\delta_c^2}{2\sigma^2}
=
\frac{0.2025}{2\times10^{-4}}
\approx 1012.5,
\end{equation}
hence
\begin{equation}
\beta \sim e^{-1012.5}.
\end{equation}
Even increasing \(\sigma\) to \(0.05\) gives
\begin{equation}
\frac{\delta_c^2}{2\sigma^2}
=
\frac{0.2025}{0.005}
\approx 40.5,
\end{equation}
so
\begin{equation}
\beta \sim e^{-40.5}\sim 2.6\times10^{-18},
\end{equation}
again tiny. The lesson is elementary and brutal: PBH production is a tail phenomenon. A smooth red spectrum will not do it.

\subsubsection{Derivation notebook F: evaporation survival threshold}

Take the standard evaporation-time estimate
\begin{equation}
t_{\mathrm{evap}}
\approx
6.2\times10^{-27}\,\mathrm{s}
\left(\frac{M}{\mathrm g}\right)^3.
\end{equation}
Set this equal to the age of the universe, \(t_0\sim4\times10^{17}\,\mathrm s\). Then
\begin{equation}
\left(\frac{M}{\mathrm g}\right)^3
\sim
\frac{4\times10^{17}}{6.2\times10^{-27}}
\sim 6\times10^{43},
\end{equation}
so
\begin{equation}
M\sim \left(6\times10^{43}\right)^{1/3}\mathrm g
\sim \text{few}\times10^{14}\,\mathrm g.
\end{equation}
This is the origin of the usual survival threshold in Eq.~\eqref{eq:partVII-evap-threshold}.

\subsubsection{Derivation notebook G: tensor amplitude from the lockscale}

Use Eq.~\eqref{eq:partVII-r-V}:
\begin{equation}
r = \frac{2V_*}{3\pi^2 A_s M_{\mathrm{Pl,red}}^4}.
\end{equation}
Take \(V_*=\Mlock^4\) with \(\Mlock=4\times10^3\,\mathrm{GeV}\), \(A_s=2.1\times10^{-9}\), and \(M_{\mathrm{Pl,red}}=2.435\times10^{18}\,\mathrm{GeV}\). Then
\begin{equation}
\frac{V_*}{M_{\mathrm{Pl,red}}^4}
=
\left(\frac{8.3\times10^{15}}{2.435\times10^{18}}\right)^4
\sim 1.4\times10^{-10}.
\end{equation}
Multiplying by the prefactor \(2/(3\pi^2 A_s)\sim 3\times10^{7}\) gives
\begin{equation}
r \simeq 0.004,
\end{equation}
in agreement with the exact \(12/N^{*2} = 0.0043\). This is why the tensor
target is not a matter of taste. It is straight arithmetic. (At the retired
TeV height this notebook would give \(r\sim10^{-52}\); that reading is
recorded in Part~IX's audit trail, not here.)

\subsubsection{Derivation notebook H: today's stochastic background}

For an approximately scale-invariant primordial tensor spectrum,
\begin{equation}
\Omega_{\mathrm{GW},0}h^2
\sim
1.6\times10^{-5}A_s r.
\end{equation}
Insert \(A_s=2.1\times10^{-9}\) and \(r=0.0043\):
\begin{equation}
\Omega_{\mathrm{GW},0}h^2
\sim
1.6\times10^{-5}\times2.1\times10^{-9}\times0.0043
\sim 1.4\times10^{-16}.
\end{equation}
This value keeps the primordial background far below interferometric reach --- the tensor target is a CMB-polarization signal, not an interferometer signal --- in the minimal module.

\subsubsection{Final notebook takeaway}

The worked notebook shows that the most dramatic Part~VII claims are also the cleanest algebraically. The black-hole entropy ledger follows from area plus entanglement. The Page time follows from a branch crossing. PBH masses follow from the horizon mass at formation. PBH abundances are exponentially sensitive to the variance and therefore negligible for a smooth red spectrum. Primordial tensors follow directly from the plateau height and are therefore negligible at a TeV lockscale.

This is exactly what one wants from a unification paper. The arguments are not hidden in numerical machinery. The core leverage is visible on the page.





\section{Internal source ledger for \ATen{}}
\label{app:source-ledger-7f}

This paper is intentionally restricted to the internal GG-2026/GD-2026 preprint chain. The derivation, notation, and theorem flow used in \ATen{} are taken only from the seven source preprints listed below. Historical literature is cited for orientation, not as a derivational premise.

\begin{table}[H]
\centering
\caption[Internal source ledger for GG-A10-Cosmos]{Internal source ledger used in \ATen{}. The columns give the preprint, its role in the chain, and its relevance here. The ledger exists so that any number in this paper can be traced to the paper that owns it in a single step, and so that a reader can see how little of the machinery is native to this one.}
\label{tab:source-ledger-7f}
\renewcommand{\arraystretch}{1.2}
\small
\begin{tabular}{p{0.15\textwidth}p{0.24\textwidth}p{0.51\textwidth}}
\toprule
Preprint & Role in the chain & Relevance \\
\midrule
\AOne{} & Constitutional foundation & Fixes the four axioms, seven postulates, \OGGEP{} semantics, and admissibility rules that every cosmological derivation must obey. \\
\ATwo{} & Six-dimensional stage & Supplies the \GG{} bulk, \BIsix{}, \OGNfive{}, compact holonomy, and the six-force unification from which all later cosmic structure descends. \\
\AThree{} & Gauge-consistency completion & Proves local and global anomaly freedom, unified gauge consistency, and dual-current conservation, thereby certifying the legality of the cosmic bulk-to-boundary channel. \\
\AFive{} & \GDOS{} foundation & Derives the exact boundary reduction, the open-system current geometry, and the formal state-manifold language on which cosmic observables are defined. \\
\ASix{} & Runtime constitution & Establishes \GRIP{}, \(F_{\GFF}=F_{\GEP}+F_{\CEP}\), Wilson-loop \GSSB{}, and the \SIP{} packet that organize all cosmic phase transitions in this paper. \\
\ASeven{} & Qi-runtime completion & Supplies the \QiFour{} pillars, \SeOS{}, \QCI{}, \(\IH+\OH=\EIC\), and the five-story pagoda used in the higher-floor interpretation of cosmic history. \\
\ANine{} & Particle-to-cosmos bridge & Fixes the Standard-Model ledger, the compact-angle sector, the neutrino block, baryonic protection, and the parameter bridge inherited by the cosmology paper. \\
\bottomrule
\end{tabular}
\end{table}

\section{Cross-paper inheritance table (compact)}
\label{app:cross-paper-inheritance}

This appendix compresses the cross-paper coherence map of Section~\ref{sec:note-on-version-cosmos} into a one-page inheritance table for quick reference. For the full eight paragraphs with citations and theorem references, see Section~\ref{sec:cross-paper-coherence-map-cosmos}; for the textual outward and inward maps, see Section~\ref{sec:coherence-map-outward-inward-cosmos}.

\begin{table}[H]
\centering
\caption[Compact cross-paper inheritance: what A10 draws from each paper]{Compact cross-paper inheritance: what \ATen{} draws from each upstream paper of the cohort. ``Used by'' lists the A10 sections that depend on each upstream contribution. The last column lists the sections of this paper that consume each inheritance, which makes the table usable in both directions: forwards to see what an upstream result is used for, and backwards to see what a given section rests on.}
\label{tab:cross-paper-inheritance-compact}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.12\textwidth}p{0.07\textwidth}p{0.48\textwidth}p{0.25\textwidth}}
\toprule
\textbf{Paper} & \textbf{} & \textbf{Key contribution to A10} & \textbf{Used by (A10 sections)} \\
\midrule
\AOne{} & May 2026 & 4 axioms (LC/\GGEP{}/\OGGEP{}/\MSMF{}) + 7 postulates + admissibility predicate & §codebook-overview, §msmf-ogn-statement \\
\ATwo{} & May 2026 & 6D bulk (Thm~1); \OGNfive{} tuple (Thm~6); compact \(S^1_\theta\) seeding Kaxion & §why-omega-m-pi, §hubble-corridor (H1) \\
\AThree{} & May 2026 & \BIsix{} anomaly-cancellation theorems (local + global); coefficient vector \((1, -1, -2/5, -2/5)\); four-form construction & §hubble-corridor (H2), §why-omega-m-pi, framing remark \\
\AFour{} & May 2026 & Zero-free-parameters discipline; \OGNfive{} as theorem of \BIsix{}; integers \(J = 7\) and \(K = 35\) & §msmf-ogn-statement, §hubble-corridor (C1), §codebook-table \\
\AFive{} & May 2026 & \GDOS{} open-system formalism; \OGGEP{} projection structure & §codebook-overview, all cosmological-observable-as-export claims \\
\ASix{} & May 2026 & Four Gi pillars (\TGG{} / \CCI{} / \GFF{} / \GRIP{}); \GRIP{} cascade T4.1 (\DGI{}~\(\to\)~\GSSB{}~\(\to\)~GA) & §hubble-corridor (H3), §msmf-ogn-statement (cost functional) \\
\ASeven{} & May 2026 & Four Qi pillars; five-story pagoda; \(\OH + \IH = \EIC\) closure & §uniqueness-layer-by-layer (universal-identity preview) \\
\ANine{} & May 2026 & Codebook Provenance Table template; D/D*/A classification; ``Why every electron is the same'' section & §codebook-table-cosmos, §universal-cosmic-identity-preview \\
\bottomrule
\end{tabular}
\end{table}

\section{Consolidated benchmark audit}
\label{app:benchmark-audit}

The main body develops the benchmark problems sector by sector. Table~\ref{tab:benchmark-audit-7f} compresses the same logic into a single operational audit sheet.

\begin{longtable}{p{0.22\textwidth}p{0.28\textwidth}p{0.34\textwidth}}
\caption[Consolidated benchmark audit for GG-A10-Cosmos]{Consolidated benchmark audit for \ATen{}. The audit is consolidated rather than complete: it gathers the benchmarks that are decidable now, and a reader wanting the full inventory should read the codebook, where the rows that cannot yet be decided are also listed and graded.}
\label{tab:benchmark-audit-7f}\\
\toprule
Problem & Inherited mechanism & Principal kill switch \\
\midrule
\endfirsthead
\toprule
Problem & Inherited mechanism & Principal kill switch \\
\midrule
\endhead
Inflation & Radial/holonomy launch on the \GG{} throat, ranked by \(F_{\GFF}\) and selected by \GRIP{} & A robust high-scale tensor signal or a scalar-sector reconstruction incompatible with the lock corridor \\
Dark energy & Residual late-time \GFF{} vacuum inventory with \(w=-1\) in the minimal module & Secure detection of \(w\neq -1\) or significant dark-energy clustering \\
Dark matter & Kaxion/compact-angle relic sector inherited from \GG{} and 6F & Exclusion of the admissible relic corridor without a replacement descending from the chain \\
Baryogenesis & Holonomy-assisted leptogenesis through the inherited neutrino block & A decisive neutrino-sector measurement incompatible with the required CP corridor \\
BBN & Standard thermal history plus a narrow \(A=7\) port correction & Failure to fit deuterium/helium while preserving the same inherited ledger \\
CMB baseline & Derived sound horizon, baryon loading, damping, and projection structure & Inability to reproduce the observed acoustic-scale hierarchy \\
Hubble tension & Early-time sound-horizon shrinkage plus possible late-time baryonic branch effects & Joint BAO/CMB/SN fit excluding the derived corridor \\
\(S_8\) tension & Modified growth history from the same early-time sector that shifts \(H_0\) & A required growth suppression inconsistent with other structure probes \\
JWST early galaxies & Small-scale power reallocation and earlier collapse in the same corridor & High-redshift abundance statistics incompatible with the derived enhancement \\
Black-hole information & Open-system horizon packet, \GFF{} ledger, and \GRIP{} saddle switching & Demonstrable contradiction with Page-curve recovery or generalized entropy \\
PBHs & Narrow feature-dependent supplementary channel, not a required module & Positive signal in an excluded mass window or non-detection in a sharply predicted window \\
Primordial GWs & Strong suppression of vacuum tensors, with transition-sourced waves as the main channel & Large primordial B-modes demanding high-scale inflation \\
Human place / pagoda & Higher-floor emergence via Qi-runtime and \EIC{} closure & Inability to connect higher-floor reconstruction with the inherited runtime grammar \\
\bottomrule
\end{longtable}

\section{Named-theorem-and-conjecture deck}
\label{app:conjecture-deck}

This appendix lists every formal named statement (Principle, Theorem, Conjecture, Corollary) introduced in \ATen{}, together with its label and a one-line summary. Full statements appear in the body sections referenced.

\begingroup
\renewcommand{\arraystretch}{1.20}\small
\begin{longtable}{p{0.11\textwidth}p{0.07\textwidth}p{0.28\textwidth}p{0.40\textwidth}}
\caption[Named-statement deck of GG-A10-Cosmos]{Named-statement deck of \ATen{}. Refs go to the body section where the full statement appears. The references point to the body section where each statement is given in full, because a named statement compressed to one line is a label and not a proof, and nothing in this deck should be cited without following the reference first.}
\label{tab:named-statement-deck}\\
\toprule
\textbf{Type} & \textbf{No.} & \textbf{Label} & \textbf{One-line summary} \\
\midrule
\endfirsthead
\toprule
\textbf{Type} & \textbf{No.} & \textbf{Label} & \textbf{One-line summary} \\
\midrule
\endhead
\midrule \multicolumn{4}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
Principle & --- & \texttt{princ:msmf-ogn-cosmos} (§\ref{sec:msmf-ogn-statement-cosmos}) & The cost functional \(\mathcal F\) admits a unique cost-minimum saddle on every admissible cosmological sub-stratum.\\
Theorem & 4.D.1 & \texttt{thm:hubble-\allowbreak two-\allowbreak stage-\allowbreak corridor} (§\ref{sec:codebook-hubble-two-stage}) & Existence of a two-stage Hubble corridor with switch at \(z_* = e^7\) and two natural projections \(67.4 / 73.0\) km/s/Mpc.\\
Theorem & 4.E.1 & \texttt{thm:omega-\allowbreak m-\allowbreak pi-\allowbreak topological} (§\ref{sec:why-omega-m-pi-theorem}) & Structural form of the closed-packet \(\Omm = 1/\pi\) claim (T-Cosmos-17); programmatic --- does not yet close (Conjecture~VIII).\\
Conjecture & I & \texttt{conj:kaxion-completeness} (§\ref{sec:audit-d-star-cosmos}) & Kaxion misalignment-mechanism relic abundance equals \(1/\pi - 1/(2\pi^2)\) at the constitutional cost-minimum.\\
Conjecture & II & \texttt{conj:hubble-corridor} (§\ref{sec:audit-d-star-cosmos}) & The two-stage Hubble corridor of Theorem 4.D.1 is unique on the cosmological-boundary stratum.\\
Conjecture & III & \texttt{conj:compact-angle-mass} (§\ref{sec:audit-d-star-cosmos}) & The \OGNfive{} ledger forces \(f_{\mathrm{Kaxion}} = (3.0\pm0.3)\times10^{13}\) GeV (raw) and \(m_{\mathrm{Kaxion}} = 8.1\pm0.5~\mu\)eV (laboratory); open remainder: the integer \(\Xi_{\mathrm{cap}}\in[35,47]\).\\
Conjecture & IV & \texttt{conj:slow-roll-exit} (§\ref{sec:audit-asserted-cosmos}) & (Revised) The \GDOS{} route forces the Starobinsky/\(\xi\) plateau class, and the lock crossing is prompt (\(N_2=0\)); underwrites the derived \(\ns\) and \(V_{\mathrm{inf}}^{1/4}\).\\
Conjecture & V\('\) & \texttt{conj:gdos-amplitude} (§\ref{sec:audit-asserted-cosmos}) & The \GDOS{} heat-kernel \(R^2\) coefficient pays the amplitude, \(A_s = 21.5\,e^{-2K/3}\) (replaces the retired, refuted TeV-uniqueness conjecture).\\
Conjecture & VI & \texttt{sphaleron-transport} (§\ref{sec:audit-d-star-cosmos}) & The Boltzmann-transport normalization of the \(K=35\) baryogenesis throughput converges to a unique coefficient within the order-of-magnitude transport band; closes the \(\eta_B\) codebook row.\\
Conjecture & VII & \texttt{bfft-damping-knee} (§\ref{sec:audit-d-star-cosmos}) & The CMB damping knee is the first non-trivial BFFT ultraviolet cutoff of the multiscale throat, \(\ell_D = K(2\pi)^2\); closes the \(\ell_D\) codebook row.\\
Conjecture & VIII & \texttt{closed-packet-integration} (§\ref{sec:audit-d-star-cosmos}) & The crossing epoch is the present one: the closed-packet occupancy \(u = \pi\Omm\) reads unity at \(t_0\), not merely at some \(t_\pi\). The \(\pi\)-form itself is derived (\S\ref{par:where-the-conditional-lives}); closing the epoch identification promotes \(\Omm, \OmL, \Omb\) from D\(^*\) to D.\\
Conjecture & IX & \texttt{vacuum-residual-exactness} (§\ref{sec:audit-d-star-cosmos}) & The late-time residual vacuum export satisfies \(w = -1\) exactly; closes the \(w\) row.\\
Conjecture & X & \texttt{conj:lithium-port} (§\ref{sec:audit-d-star-cosmos}) & The \(A=7\) BBN port inherits the closed-packet \(1/\pi\) suppression, so \(({}^7\mathrm{Li}/H)_{\mathrm{GG}} = \mathrm{SBBN}/\pi\); closes the \({}^7\mathrm{Li}/H\) row (D20).\\
Corollary & 1 & \texttt{cor:hubble-\allowbreak tension-\allowbreak reconciliation} (§\ref{sec:hubble-corridor-corollaries}) & The 5\(\sigma\) Hubble tension is dissolved as two projections of the single corridor of Theorem 4.D.1.\\
Corollary & 2 & \texttt{cor:s8-reconciliation} (§\ref{sec:hubble-corridor-corollaries}) & The \(\sigeight/\Seight\) tension is simultaneously resolved by the growth-history modification of the corridor.\\
Corollary & 3 & \texttt{cor:hubble-falsification} (§\ref{sec:hubble-corridor-corollaries}) & A single-stage \(\Hzero\) determination at 0.5\(\sigma\) precision would falsify the two-stage corridor of Theorem 4.D.1.\\
\end{longtable}
\endgroup

\textbf{Inventory.} 1 Principle + 2 Theorems + 10 Conjectures + 3 Corollaries = 16 formal named statements.

\section{Named-lemma promotion roadmap}
\label{app:promotion-roadmap}

This appendix states the open research targets that, if proved, would promote codebook entries from Derived (cond.) or Asserted to Derived. Each entry gives the named conjecture, the rows it closes, the proof venue (this paper, a companion paper, or open research), and the estimated proof complexity.

\begin{table}[H]
\centering
\caption[Named-lemma promotion roadmap for the MSMF/OGN audit]{Named-lemma promotion roadmap for the \MSMF{}/\OGN{} audit of \ATen{}. Proof venue ``this paper'' = work that belongs in the present paper and remains to be completed; ``C1'' = companion paper \PaperFullCOne{}; ``open'' = open research target without a designated venue.}
\label{tab:promotion-roadmap}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.06\textwidth}p{0.30\textwidth}p{0.22\textwidth}p{0.13\textwidth}p{0.14\textwidth}}
\toprule
\textbf{No.} & \textbf{Named conjecture} & \textbf{Rows closed (codebook)} & \textbf{Proof venue} & \textbf{Status} \\
\midrule
I & Kaxion-misalignment-completeness & D4 (\(\Omdm\)) & Open & open research target \\
II & Hubble-two-stage-corridor-uniqueness & D5, D6, D10, D11, D19 (corridor + \(\sigeight\)/\(\Seight\) branches) & Open & open research target \\
III & Compact-angle-mass-uniqueness & D13, D14 (\(m_{\mathrm{Kaxion}}\), \(f_{\mathrm{Kaxion}}\)) & \PaperFullCOne{} & companion paper \\
IV & Plateau-class-and-prompt-crossing (revised) & A7, A15 (\(\ns\), \(V_{\mathrm{inf}}^{1/4}\)) & Open & open research target \\
V\('\) & GDOS-heat-kernel-amplitude (replaces the retired Conj.~V) & A8 (\(A_s\)) & Open & open research target \\
VI & Resonant-leptogenesis-normalization & D17 (\(\eta_B\)) & Open & open research target \\
VII & BFFT-damping-knee-identification & BFFT overlay of D18 (\(\ell_D\); row scored by the Boltzmann inventory value) & Open & open research target \\
VIII & Closed-packet-integration & D1--D3 (\(\Omm\), \(\OmL\), \(\Omb\)); D18, D21 (\(\ell_D\), \(z_*\)) via the fractions & this paper / open & open research target \\
IX & Vacuum-residual-exactness & D16 (\(w\)) & Open & open research target \\
X & Lithium-port-\(\pi\)-suppression & D20 (\({}^7\mathrm{Li}/H\)) & Open & open research target \\
\bottomrule
\end{tabular}
\end{table}

\textbf{Closure-economy summary.} Two codebook entries (D9 and D12) are already at Derived status. Eighteen are Derived (conditional) on Conjectures I--IV and VI--X; one is Asserted pending Conjecture V\('\). Once all ten conjectures are proved, the codebook collapses to 21 D + 0 D\(^*\) + 0 A (twenty independent entries after the D9 redundancy collapse). The proof venue closest to the present paper is Conjecture VIII (closed-packet-integration), whose programmatic sketch appears as Theorem~\ref{thm:omega-m-pi-topological} and whose proof would close D1--D3 (and, through the fractions, D18 and D21). The remaining conjectures are open research targets, with Conjecture III having a designated companion-paper venue (\PaperFullCOne{}) and the remainder awaiting external proof or further structural analysis.

After all, the promotion roadmap is the place where the codebook stops being a table of provenance labels and starts being a research program. Ten precise mathematical conjectures, nine open research targets, and a clean closure path from 2/18/1 to 21/0/0.


%% ===================================================================

\section{Falsification-test catalog}
\label{app:falsification-catalogue}

This appendix collects, in one place, every observational or theoretical test whose failure would falsify the framework or one of its named lemmas. Each row gives the test, the precision required, the falsification target, and the named lemma or theorem it would directly affect.

\begin{table}[H]
\centering
\caption[Falsification-test catalog for GG-A10-Cosmos]{Falsification-test catalog for \ATen{}. Each test is paired with the named conjecture or theorem it would refute and the experimental venue that could carry out the test. The columns give the test, the precision it requires, the named conjecture or theorem it would refute, and the experimental venue that could run it. A test whose venue column names no existing or funded instrument is a weaker entry than one that does, and the catalog is not sorted to hide that.}
\label{tab:falsification-catalogue}
\renewcommand{\arraystretch}{1.18}
\small
\begin{tabular}{p{0.20\textwidth}p{0.16\textwidth}p{0.30\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Test} & \textbf{Precision} & \textbf{Falsification target} & \textbf{Experimental venue} \\
\midrule
Corridor switch vs atomic decoupling & \(\pm 5\) at \(z = 1090\) (0.46\%) & \(z_{\mathrm{dec}}\) (atomic chain) \(= 1089\)--\(1091\), zero parameters; \(z_{\mathrm{sw}} = e^7\) a distinct topological marker \(0.41\%\) above in e-folds. Measured \(z_{\mathrm{dec}}\) off the atomic-chain value by \(> 0.5\%\) (present agreement \(0.1\%\)) would refute the reading; the switch--decoupling offset \(\delta = 0.0041\) is a derived consistency statement, not an independent kill (corridor theorem, property C1; \S\ref{sec:two-redshifts}) & CMB spectral-distortion (recombination-radiation) missions, post-2030, + 21-cm cosmology \\
Two-stage BAO residue & 1--2\% at \(z \sim 1\) & Absence of the residue refutes the two-stage corridor & DESI, Euclid, Roman/Nancy Grace Roman \\
GW siren \(\Hzero\) & 1\% on the background & a converged \(\Hzero^{\mathrm{GW}} \approx 73\) (tracking the local ladder, not the background \(\approx67\)) would refute the \(\pi\)-inventory derivation & LIGO-Voyager, Einstein Telescope, Cosmic Explorer \\
Lensing time-delay \(\Hzero\) & 1\% & A determination significantly below 73 would force \GRIP{} refinement & H0LiCOW, TDCOSMO \\
Tensor-to-scalar ratio & measurement vs \(r = 0.0043\) & A tensor amplitude inconsistent with the derived target in either direction falsifies the launch & LiteBIRD, Simons Observatory \\
Kaxion haloscope window & DFSZ-depth, doublet-aware scan of \(1.59\)--\(2.13\) GHz & Exclusion of the unique window at DFSZ depth refutes Conjecture III & ADMX, CAPP, HAYSTAC, ADMX-EFR \\
Primordial GW background & B-mode polarization & The primordial bump must appear, at the \(r = 0.0043\) amplitude and no other (D9); a supercooled-transition GW background would separately refute the prompt crossing & LiteBIRD, Simons Observatory \\
\(\sigma_8\) precision growth & 1\% at \(z = 0.5\)--1 & Persistent \(\sigeight\) inconsistency with the corridor refutes Conjecture II & Euclid weak-lensing, LSST \\
\bottomrule
\end{tabular}
\end{table}

The framework's exposure profile is therefore high: a clean failure on any one of the eight tests above would refute either a named conjecture (II, III, V) or a named theorem (4.D.1). Most of the tests sit within the next-decade observational horizon; the same eight tests are laid out on a single forward time axis in the kill-test timeline of Figure~\ref{fig:killtest-timeline} (Part~XIII, \S\ref{subsec:decade-timeline-cosmos}).

\medskip\noindent With that, the course is over. The reader who entered these appendices with one exposure to the Friedmann equations has now built the standard cosmology from scratch and, at every station, seen exactly where the \GG{} geometry replaces an input with a theorem --- and where it does not yet. There is no better preparation for what comes next: the Objections Part, where everything taught here is turned, deliberately, against the framework itself.

\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a \(\Lambda\)CDM and CMB-precision cosmologist, an inflation theorist, a Hubble-tension and large-scale-structure specialist, a dark-matter, baryogenesis, and BBN phenomenologist, a gravitation and black-hole relativist, a formal quantum-field-theory and string theorist, and a methodologist and philosopher of cosmology would each raise after reading the whole cohort --- answered, with pointers to the theorem, codebook status, conjecture, open problem, or companion paper that bears the weight}
\label{sec:common-objections}
% --- Common Objections is a Q&A Part, not a technical appendix: give it its own
%     non-\Alph section numbering so the appendix lettering (A-S) never overflows Z.
\setcounter{section}{0}
\renewcommand{\thesection}{O\arabic{section}}

\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

A program claiming that the large-scale inventory of the universe --- the matter, dark-energy, and baryon fractions, the recombination redshift, and the Hubble corridor --- is fixed by the same zero-dial six-dimensional geometry that fixes the elementary particles, and carrying that claim through an explicit Derived / Derived-conditional / Asserted codebook, invites the most demanding scrutiny; it would not be honest scholarship to leave the strongest objections unstated. We collect here the adversarial readings that a \(\Lambda\)CDM and CMB-precision cosmologist, an inflation theorist, a Hubble-tension and large-scale-structure specialist, a dark-matter, baryogenesis, and BBN phenomenologist, a gravitation and black-hole relativist, a formal quantum-field-theory and string theorist, and a methodologist and philosopher of cosmology would each raise after reading not just this paper but the whole twenty-two-paper cohort --- and we answer each, with explicit pointers to the theorem (by its cohort-global id, \S\ref{sec:naming-convention}), the codebook status, the named conjecture, the open problem, or the companion paper that carries the weight. These are deliberately the \emph{surviving} objections: many first-look complaints dissolve once the companion that does the work is read --- the boundary general relativity of \AFive{} \citep{ArisakaA5GDOS}, the Bekenstein--Hawking coefficient of \PaperCThree{} \citep{ArisakaC3BH}, the Kaxion relic of \PaperCOne{} \citep{ArisakaC1Kaxion}, the anomaly ledger of \AThree{} \citep{ArisakaA3BI6}, and the six-dimensional action of \ATwo{} \citep{ArisakaA2GG6}. Where the honest answer is ``not yet'' --- above all the closed-packet integration that would turn the \(\pi\)-locked inventory from a structural ansatz into a theorem (Conjecture~VIII), and the full Planck-likelihood fit that would turn the present central-value comparisons into a quantitative goodness-of-fit --- we say so and name the open item.

The Part opens, however, not with the specialists but with the reader --- the thoughtful non-specialist who arrives carrying the questions popular science has made urgent: the multiverse, the landscape, parallel worlds, fine-tuning, what came before the beginning, whether a forced universe leaves room for us. These are answered first (\S\ref{sec:phil-objections}), in plain language and without jargon, because a framework that claims the cosmos is \emph{necessary} rather than \emph{accidental} owes its clearest account to exactly the reader who has most often been told the opposite. The technical referees follow.

\section{For the curious reader: the big questions, plainly answered}
\label{sec:phil-objections}

The objections that follow are not the technical challenges of a
specialist referee --- those come afterwards, sector by sector. They are
the questions a thoughtful non-specialist actually carries into a
cosmology book: about the multiverse, the landscape, parallel worlds,
fine-tuning, what came before the Big Bang, whether the theory leaves
room for us. Popular science has made some of these questions feel
settled in directions this framework does not take, so we meet them
first, and plainly. Where a claim rests on a still-open step we say so;
the point of this section is not to oversell, but to clear the mind of
a reader who has been told, often, that the universe is probably one of
infinitely many accidents.

\bigskip

\noindent\textbf{Objection~1 (O-Cosmos-1) (the curious reader)}\par\nobreak
\noindent\textbf{``Isn't our universe just one bubble in a vast multiverse? I keep reading that ours is one of infinitely many, and we happen to live in the one that works.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That is one influential story, but it is a hypothesis, not a finding --- and it is the one this framework is built to make unnecessary. The multiverse is invoked to explain why our universe's numbers are life-friendly: if every possible universe exists, then \emph{of course} we find ourselves in a hospitable one, and no further explanation is owed. But notice what that story costs. It explains the numbers by giving up on deriving them: they are an accident of our address in an unobservable ensemble. This paper takes the opposite route. It argues that the numbers are not accidents to be excused but \emph{outputs to be computed}: the matter fraction \(1/\pi\), the dark-energy fraction \(1 - 1/\pi\), the switch redshift of recombination \(e^7\), each read off a single six-dimensional geometry with no dial to turn (Parts~IV, X). If that program succeeds, there is nothing for a multiverse to do, because there was never a menu of alternatives to select from. The honest status: several of these values are still \emph{Derived-conditional} on a named, stated conjecture (Part~IX) --- so we have not finished the job. But the wager is clear, and it is testable this decade, which no multiverse claim is.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~2 (O-Cosmos-2) (the curious reader)}\par\nobreak
\noindent\textbf{``String theory says there are \(10^{500}\) possible universes --- the `landscape.' Doesn't that make picking ours hopeless, and physics basically a lottery?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The landscape is the sharpest form of the worry, and worth taking seriously. The concern is that a fundamental theory might admit an astronomical number of self-consistent vacua, with no principle selecting ours --- so the constants of nature become a draw, and the theory loses its power to predict. This framework's central bet is a single word against that picture: \emph{uniqueness}. It proposes that the constitutional axioms (\AOne{}) plus the six-dimensional anomaly and integer ledgers (\AThree{}, \AFour{}) leave \emph{one} self-consistent geometry, not \(10^{500}\) --- so there is no landscape to be lost in. Whether that uniqueness is fully proved is exactly the boundary we mark honestly: the internal consistency is shown, but the theorem that \emph{no other} admissible geometry exists remains a stated conjecture rather than a closed result (it is named as such in the discussions). So we do not claim to have refuted the landscape by fiat. We claim something checkable instead: a theory with zero \emph{continuous} free parameters (an overall amplitude and the observation epoch are declared, not tuned) cannot retreat to a landscape when the data press, because it has no knobs to re-tune. If its one geometry gives the wrong sky, it is simply wrong. That exposure is the antidote to the lottery.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~3 (O-Cosmos-3) (the curious reader)}\par\nobreak
\noindent\textbf{``What about parallel worlds --- the idea from quantum mechanics that every possibility happens in a branching reality?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is a different idea from the multiverse, and it is worth keeping them apart. The ``many worlds'' of quantum mechanics is a proposal about what happens during a measurement --- that the wavefunction never collapses, and every outcome is realized on a separate branch. It is a question about the \emph{interpretation of quantum theory}, not about cosmological initial conditions, and this paper does not need to settle it to do its work. What the framework does offer is a different reading of measurement itself: an observation is an \emph{open-system boundary projection} (the \OGGEP{} axiom of \AOne{}, developed as \GDOS{} in \AFive{}) --- a finite observer reading a boundary of a closed higher-dimensional geometry, never the whole bulk. In that reading the appearance of ``branching'' is about what a bounded observer can and cannot access, not about the literal proliferation of worlds. The companion paper \AEight{} develops the operational quantum mechanics; here we simply note that nothing in this cosmology requires an infinity of parallel copies of you. One geometry, read from inside, is enough.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~4 (O-Cosmos-4) (the curious reader)}\par\nobreak
\noindent\textbf{``The universe looks incredibly fine-tuned for life. If that's not luck and not a multiverse, what is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is, in this framework, neither luck nor selection but \emph{necessity} --- and that is a genuine third option, not a rhetorical dodge. The fine-tuning argument has always had exactly two escape routes: a designer who set the dials, or a multiverse in which every dial-setting exists and we see a workable one. Both take for granted that the dials \emph{could} have been otherwise. This paper denies the premise. If the life-permitting constants --- the electron's mass, the strength of the forces, the tiny value of the vacuum energy --- are forced outputs of one geometry with no adjustable parameter, then there was never a dial to tune, and so nothing to tune it luckily or select it anthropically (Part~XIV makes this argument in full). A universe whose contents are \emph{forced} is life-permitting for the same reason it is anything at all: because its own geometry admits exactly one specification. The fine-tuning does not disappear; it is \emph{relocated} --- from a coincidence in need of an explainer to a theorem in need of a proof. And a theorem, unlike a coincidence, can be checked.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~5 (O-Cosmos-5) (the curious reader)}\par\nobreak
\noindent\textbf{``Doesn't `necessity' just replace a divine designer with a mathematical one? You've moved the mystery, not solved it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair challenge, and the difference is worth stating precisely. Yes, the framework relocates the mystery --- from ``why these particular values?'' to ``why this particular geometry, and why does mathematics describe the world at all?'' It does not pretend to answer the second question; no physical theory can, and we do not claim otherwise. But the relocation is not empty, because it changes what kind of thing the unexplained residue \emph{is}. A designer who chose the constants must be explained in turn (who designed the designer, and by what freedom?), and the regress never closes. A geometry that admits one self-consistent solution has no such regress: it is not \emph{chosen}, it simply is the unique thing that hangs together, the way there is only one consistent arithmetic. The remaining mystery --- why there is a lawful world rather than nothing --- is real and we leave it standing, plainly labeled. What we have removed is the \emph{intermediate} mystery: the twenty-odd apparently arbitrary numbers between the axioms and the sky. Removing those is not nothing; it is the whole content of the Euclidean ideal (\AFour{}).
\end{objqabody}

\bigskip

\noindent\textbf{Objection~6 (O-Cosmos-6) (the curious reader)}\par\nobreak
\noindent\textbf{``What came before the Big Bang? Every account either dodges this or invokes a prior universe.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This framework gives an unusually concrete answer, and it is neither a dodge nor a prior universe. In the standard picture the Big Bang is a \emph{singularity} --- a moment where the equations break down and ``before'' has no meaning --- and that breakdown is genuinely uncomfortable. Here the earliest physically meaningful moment is not a singularity but a \emph{finite geometric edge}: the ultraviolet boundary of the six-dimensional throat, at the grand-unification energy \(\EGUT \approx 2.3\times10^{16}\)~GeV (Part~III). Cosmic history is the descent from that edge, downhill, to the gentle vacuum we inhabit today. So ``what came before?'' is answered by ``the edge is the beginning of the descent, and there is no singular instant behind it to explain'' --- the Planck-scale infinity that plagues other accounts is derived to be a finite, lawful boundary, not a wall the theory crashes into. What the framework does \emph{not} claim is a complete quantum account of the edge itself; that is named as open work. But it replaces ``the equations break down'' with ``the descent starts here,'' which is a real and testable difference, not a philosophical evasion.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~7 (O-Cosmos-7) (the curious reader)}\par\nobreak
\noindent\textbf{``If inflation blew up a tiny patch, why is the whole sky made of the same stuff? Two regions that were never in contact somehow have identical electrons.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You have put your finger on a question the textbooks skip, and it is one this paper is proud to answer (Part~X, \S\ref{sec:second-horizon}). The famous ``horizon problem'' asks why distant regions share the same \emph{temperature}, and inflation answers it: they were once one small patch, stretched huge. But those regions also share the same \emph{things} --- every electron with the same mass and charge, the same dark matter, the same vacuum energy --- and temperature-smoothing cannot explain that, because no exchange of heat can negotiate a particle's identity. The framework's answer has two parts. First, there is only one ``mold'': every electron is the readout of the same unique geometric ground state, so there is nothing regional to vary (Part~X). Second --- and this is the part that needed the new result that inflation sits at the \emph{grand-unification} scale --- the stretch happened \emph{before any matter existed}. The universe was blown up while it was still empty; the electrons were minted \emph{afterwards}, everywhere at once, from that one mold. So the sky is uniform in its contents not because distant regions negotiated an agreement, but because they inherited one, from a time before there was anything to disagree about. That, we argue, is the deeper horizon problem --- and its solution.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~8 (O-Cosmos-8) (the curious reader)}\par\nobreak
\noindent\textbf{``Why is the universe 95\% dark? It seems absurd that we can only see 5\% of everything.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It feels absurd only if you expect the visible to be the whole; the framework offers a reason it is not. In this reading the cosmic inventory is a closed set of geometric fractions --- roughly \(5\%\) ordinary matter, \(27\%\) dark matter, \(68\%\) dark energy --- and the ordinary, luminous part is exactly the \emph{one} slice that carries ordinary electric charge and therefore interacts with light (Part~XIV). The Standard-Model ledger we inherited from the particle paper (\ANine{}) can illuminate only that slice; the rest is not missing or exotic but simply the part of the inventory that light was never built to touch. Dark matter and dark energy are dark for the same reason a clean pane of glass is invisible: not because they are absent, but because the instrument sent to find them --- electromagnetism --- passes straight through. On this account the striking number is not that \(95\%\) is dark, but that the visible fraction is a specific small share, \(1/2\pi^2 \approx 5\%\), of a closed geometric budget. The darkness is the shape of what light was built to miss.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~9 (O-Cosmos-9) (the curious reader)}\par\nobreak
\noindent\textbf{``Is the universe deterministic, then? If everything is forced by geometry, is my future already written --- do I have any freedom?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework is careful here, and the distinction it draws is the one that matters to you. What the geometry forces is the \emph{layered structure} of the world --- that there are stable atoms, that there is chemistry, that there is a dark-energy era --- and the \emph{values} of the constants that make those layers possible. It does \emph{not} force every event within the layers. Which atoms combine into which molecule, which star forms where, which thought you think next --- these live in the enormous combinatorial freedom \emph{inside} the layers the geometry has built (Part~XIV). The determinism is at the level of the stage and the cast of characters, not the play. This is a genuine third position between the two uncomfortable options usually offered: not rigid clockwork (in which your choices are illusions) and not pure randomness (in which they are noise), but a forced scaffolding within which contingent history --- including yours --- actually unfolds. You are not preprogrammed by the geometry; you are \emph{sheltered} by it. The universe fixes the rules of the game, not its moves.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~10 (O-Cosmos-10) (the curious reader)}\par\nobreak
\noindent\textbf{``Isn't it suspicious that \(\pi\) shows up in the amount of matter in the universe? That smells like numerology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should smell like numerology --- until you see where the \(\pi\) comes from, and we insist on that distinction because the paper's whole method depends on it. Numerology is picking a pretty formula \emph{after} seeing the number: with a few constants and enough operations you can hit almost any target, which proves nothing. The claim here is the reverse. The \(\pi\) is not chosen to match \(0.318\); it is forced to appear because the matter fraction is computed as the integral of a specific geometric quantity (a curvature four-form) over a closed surface, and integrating curvature over a closed surface \emph{produces factors of \(\pi\)} by a theorem of geometry, not by choice (Parts~X, and the topological appendix). The same single computation is required to give matter \(1/\pi\), dark energy \(1 - 1/\pi\), and baryons \(1/2\pi^2\) \emph{together} --- one topology, three coupled numbers, no freedom to adjust one without the others. A numerologist fits one number at a time; a derivation forces several at once and cannot wriggle. The honest caveat: the integral is written down but not yet fully evaluated in closed form (Conjecture~VIII), so we carry these as ``derived-conditional,'' not as finished theorems. But the \(\pi\) is a consequence of the shape, not an ornament chosen to impress.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~11 (O-Cosmos-11) (the curious reader)}\par\nobreak
\noindent\textbf{``Every generation's physicists think they're near a `theory of everything.' Why is this any different from the last confident framework that failed?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Healthy skepticism, and the framework asks to be judged by exactly the standard your question implies: not confidence, but exposure. The programs that disappointed --- grand unification, supersymmetry, the string landscape in its predictive ambitions --- shared a trait: as data pressed, they could retreat, adding parameters, raising scales, widening allowed ranges. This paper deliberately removes that escape. It has \emph{zero continuous free parameters} (\AFour{}): no dial to turn when a measurement comes in wrong. It stakes specific numbers on specific instruments of the coming decade --- a tensor-to-scalar ratio of \(0.0043\), a dark-matter particle at \(8.1\) microelectronvolts, a gravitational-wave Hubble constant near \(67\) --- any one of which, landing wrong, refutes it (Part~XIII). We make no claim that it is \emph{right}; we claim it is the \emph{kind} of framework that can be shown wrong, cleanly and soon. That is precisely what the confident-but-unfalsifiable frameworks lacked. Judge it not by its ambition but by whether the next decade's data are allowed to kill it. They are.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~12 (O-Cosmos-12) (the curious reader)}\par\nobreak
\noindent\textbf{``The `anthropic principle' says the universe must allow observers because we're here observing it. Isn't that the real explanation, and isn't your `necessity' just anthropics in disguise?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are opposite kinds of explanation, and the difference is the crux. The anthropic principle explains a life-friendly constant by \emph{observer selection}: the value could have been anything, but only in universes where it is right can anyone be around to notice, so we necessarily observe a workable value. It presupposes a range of possibilities and prunes it by our existence. The present framework denies the range. If a constant is a forced output of one geometry, then it could \emph{not} have been anything else, and observer selection has nothing to select among --- there is one value, whether or not anyone is here to see it. The test that separates the two is decisive: anthropic reasoning predicts constants scattered anywhere in the life-permitting \emph{window}, typically near its edge; a necessity claim predicts \emph{specific} values, at the center of nothing in particular, fixed before any observer existed. This paper predicts specific values --- \(1/\pi\), \(e^7\), \(0.0043\) --- and stakes them on measurement. If they were merely anthropically allowed, they would not need to be exactly those. Anthropics explains why we \emph{could} be here; the framework tries to explain why the universe is \emph{this}, observer or no.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~13 (O-Cosmos-13) (the curious reader)}\par\nobreak
\noindent\textbf{``You say inflation happened at a colossal energy near grand unification. How can anyone take seriously a claim about energies a trillion times beyond any collider?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We cannot build a collider that reaches \(\EGUT\) --- true, and no one claims otherwise. But the sky is an instrument the laboratory is not. Inflation at the grand-unification scale leaves fossils we can read today: the exact tilt of the primordial ripples, \(n_s = 0.962\), already measured and matched; and a specific amount of primordial gravitational waves, \(r = 0.0043\), which the next generation of microwave-background telescopes (the Simons Observatory, LiteBIRD) is being built to detect (Parts~IV, VII). The energy is inaccessible; its \emph{consequences} are not. This is how cosmology has always worked --- we cannot rerun the first three minutes, yet the helium abundance they left behind is measured to percent precision. The framework's inflationary claim is testable for the same reason: it makes a two-sided, decisive prediction for a number that a funded experiment will measure this decade. If the tensor ratio comes back far from \(0.0043\) in either direction, the claim is dead --- inaccessible energy or not.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~14 (O-Cosmos-14) (the curious reader)}\par\nobreak
\noindent\textbf{``Dark energy is the biggest embarrassment in physics --- the vacuum energy is predicted wrong by \(120\) orders of magnitude. How can any theory claim to have this under control?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The \(120\)-order discrepancy is real and it is the sharpest test any dark-energy claim must face, so it is fair to press it. The standard estimate takes the vacuum energy to be set by the highest scale in physics, the Planck scale, and finds a number \(\sim10^{120}\) times too big --- the worst prediction in the history of science. The framework's move is to deny that the vacuum energy is a Planck-scale quantity at all. In this reading the observed dark energy is a \emph{residual} left at the very bottom of the geometric descent --- a seesaw between the lock scale and the Planck scale, \(m_\Lambda = M_{\rm lock}^2/M_{\rm Pl} \approx 6\) meV (Parts~IV, and the companion \AFive{}) --- which recovers about \(119\) of the \(120\) orders \emph{without} a fine-tuned cancellation. That is not yet a complete theorem: the final factor, and the exact fraction \(1 - 1/\pi\), rest on named conjectures we mark openly. But the structural claim is specific and falsifiable --- \(w = -1\) exactly, no evolution --- and it is decidable by DESI, Euclid, and Rubin this decade. The embarrassment is not waved away; it is attacked at its root, the assumption that empty space must weigh the Planck scale.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~15 (O-Cosmos-15) (the curious reader)}\par\nobreak
\noindent\textbf{``The `cosmic coincidence' --- that matter and dark energy happen to be comparable right now, in our era --- seems too convenient. Doesn't that single out our moment suspiciously?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a real puzzle, and the framework splits it into a part it dissolves and a part it honestly keeps. Under a free cosmological constant, matter (which dilutes as the universe expands) and dark energy (which does not) are comparable for only a brief window in cosmic history --- and our living in that window looks like a coincidence begging for an anthropic explanation. The framework changes the picture in two steps. First, the matter and dark-energy fractions are not independent quantities that happen to be close; they are locked together by one geometry as \(1/\pi\) and \(1 - 1/\pi\), which are comparable \emph{by construction}, at every epoch you compute them, not just now (Part~X). Second --- the honest remainder --- there is still a mild question of \emph{why now} we read the snapshot, and the paper carries this as a single, bounded, few-percent ``timing'' input rather than pretending it away (Part~IV's ``why now'' section). So the suspicious coincidence shrinks to a small, named, calendar-level fact, with the large structural coincidence removed. Not everything is dissolved --- but what remains is stated in the open, not smuggled into an anthropic hand-wave.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~16 (O-Cosmos-16) (the curious reader)}\par\nobreak
\noindent\textbf{``Extra dimensions sound like science fiction. Six-dimensional geometry, a `bulk,' a `throat' --- why should I believe any of this is real rather than mathematical decoration?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The right test of a hidden structure is not whether we can picture it but whether it \emph{pays its way} in predictions, and that is the standard the framework accepts. The two extra dimensions here are not extra \emph{places} you could travel to; they are a \emph{scale} direction (how zoomed-in you are, the energy at which you probe) and a \emph{phase} direction (an internal angle, like the setting of a dial). Physics already uses both ideas without calling them dimensions --- the running of forces with energy is the scale direction; the phases of quantum fields are the angle. The framework's wager is that treating them geometrically is not decoration but \emph{economy}: the same two directions that organize the particle masses in \ANine{} also fix the cosmic inventory here, with no separate assumptions. A decoration explains nothing new; these dimensions are asked to explain the electron and the dark-matter fraction with one structure. Whether they are ``real'' is decided the way we decide anything in physics: the six-dimensional picture makes specific, falsifiable predictions (the \(3.85\)-TeV particles, the \(8.1\)-microelectronvolt axion), and if those are absent, the picture fails. It earns reality by exposure, not by vividness.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~17 (O-Cosmos-17) (the curious reader)}\par\nobreak
\noindent\textbf{``Why is there something rather than nothing? Does your framework even touch the deepest question?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Honestly, no --- and it is important not to pretend otherwise. Why there is a lawful world at all, rather than sheer nothing, is a question physics cannot reach, because any physical answer must already assume \emph{some} law to do the explaining. This framework does not claim the summit of that question, and any theory that did would be overreaching. What it does is narrower and, we think, still worth a great deal: it shrinks the gap between ``nothing'' and ``everything.'' Instead of a world specified by a long list of arbitrary numbers --- each of which is its own little ``why this and not that?'' --- it offers a world specified by a single geometry that admits one self-consistent solution. The list of separate mysteries collapses toward one. You are still left with the final ``why is there a lawful anything?'', and we hand it to you undiminished and clearly labeled. But you are no longer \emph{also} owed an explanation for why the electron weighs what it weighs, or why the universe is a third matter and two-thirds dark. Those become consequences. Moving the mystery from many places to one is not the whole answer --- but in physics, it is most of the work.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~18 (O-Cosmos-18) (the curious reader)}\par\nobreak
\noindent\textbf{``This all sounds like a `theory of everything.' Aren't those famously arrogant and famously wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The label is one the framework should decline, and does. A ``theory of everything'' usually means a single equation from which \emph{all} of physics is supposed to flow. This is not that, and it does not claim to be. It is explicit about what it does \emph{not} compute: it fixes the \emph{inputs} of physics --- the constants, the inventory --- but it hands the detailed dynamics (the precise shape of the microwave spectrum, the nonlinear clumping of galaxies) to the standard computational machinery, which it frames but does not replace (Part~XIII's ``tier ladder''). It says, in effect: here is why the constants are what they are; the hard calculational physics you already do is still yours to do. That is a deliberately bounded claim. The arrogance of the failed ``theories of everything'' was to promise the whole of physics and deliver an unfalsifiable framework; the discipline here is the opposite --- a modest target (the constants, not the dynamics) staked on falsifiable numbers. If anything the framework errs toward saying too clearly what it \emph{cannot} do. That is the correct direction for the error to run.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~19 (O-Cosmos-19) (the curious reader)}\par\nobreak
\noindent\textbf{``You keep saying certain results are `conjectures' or `not yet proved.' Doesn't that mean the whole edifice could collapse?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It means the edifice is honest about its load-bearing walls, and you should want that, not fear it. Every result in this cohort carries an explicit grade: \emph{Derived} (proved end to end), \emph{Derived-conditional} (proved except for one named, stated mathematical step), or \emph{Asserted} (an input we have not yet derived). Roughly, of the cosmological quantities, two are fully derived, eighteen are derived-conditional on a short list of named lemmas, and one --- the amplitude of the primordial ripples --- is still an input (Part~IX). A collapse is possible in exactly the places we have marked: if the central open conjecture (the closed-packet integration behind the \(\pi\)-fractions) turns out false, those fractions fall. But this is the normal condition of live science, made unusually visible. Most frameworks bury their assumptions; this one prints a ledger of them and invites you to watch which ones close. A structure that tells you precisely which stone would bring it down is sturdier, epistemically, than one that hides its stones --- because you can \emph{test} it. The conjectures are not weaknesses concealed; they are the next decade's homework, listed.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~20 (O-Cosmos-20) (the curious reader)}\par\nobreak
\noindent\textbf{``Where do we --- life, minds, observers --- fit into this? Or is consciousness just an afterthought bolted onto the physics?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} In this framework observers are not an afterthought and not the point either; they are a \emph{late rung of the same ladder} that fixes the electron. The claim, developed fully in the capstone companion \AEleven{} and only pointed to here, is that the geometric descent does not stop at the vacuum: the same principle that selects the stable electron selects the stable proton, then the stable atom, then the molecules of chemistry, then --- where conditions allow --- the self-sustaining systems we call life, and eventually the observers who write the equations down (Part~XIV's ``one staircase''). On this reading we are neither the reason the universe has its numbers (it had them for billions of years before us) nor an accident decorating a cosmos that has nothing to do with us. We are a \emph{continuation} --- a late step on a staircase whose first steps are the particles. The framework does not solve the hard problem of consciousness, and it does not pretend to. But it refuses to treat mind as foreign to physics: if the argument holds, the same geometry that fixed the weight of empty space also, further down, made the systems that could come to know it. That is the trilogy this paper sits in the middle of --- and the reader who wants the rest is pointed, deliberately, downstream to \AEleven{}.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~21 (O-Cosmos-21) (the curious reader)}\par\nobreak
\noindent\textbf{``Isn't it circular for the universe to `explain itself' --- for physics written by humans to claim the humans were inevitable?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is a loop here, but it is a feature the framework names openly rather than a fallacy it hides. The situation is genuinely self-referential: a universe produces, late in its history, observers who reconstruct the constitution that produced them. Older traditions noticed this shape --- Wheeler's ``participatory universe,'' the Upanishadic recognition of self in reality, Spinoza's necessity --- and the paper cites them without borrowing their metaphysics (Part~XIV). Why is it not vicious circularity? Because the framework does not use our existence as a \emph{premise} to derive the constants --- that would be circular. It derives the constants from geometry with no reference to observers at all, and then \emph{notices}, as a consequence, that among the things the geometry forces is the eventual appearance of systems complex enough to do the deriving. The logic runs geometry \(\to\) constants \(\to\) atoms \(\to\) life \(\to\) minds \(\to\) (the writing of the theory), in one direction, with no step reaching back to justify an earlier one. The loop closes at the end; it is not assumed at the start. That the universe turns out to contain its own describers is not a trick of the argument --- it is, if the argument is right, a fact about the universe.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~22 (O-Cosmos-22) (the curious reader)}\par\nobreak
\noindent\textbf{``Suppose you're wrong. What would it actually take to convince you --- or is this, like so many grand theories, effectively unfalsifiable?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the question we most want asked, because the framework's whole claim to seriousness rests on the answer being short and concrete. It would take any \emph{one} of the following, all within the coming decade: a primordial gravitational-wave signal with tensor ratio well away from \(0.0043\) in either direction; a dark-matter axion excluded across the \(1.6\)--\(2.1\) gigahertz band at the predicted coupling; a gravitational-wave-siren Hubble constant that tracks the local value (\(\approx73\)) rather than the predicted background (\(\approx67\)); a confirmed evolving dark energy (\(w \neq -1\)); or --- on the particle side of the cohort --- a single proton decay, which the geometry forbids absolutely. Any one of these, cleanly established, ends the framework; there is no parameter to adjust in response, because there are no \emph{continuous} free parameters --- only two declared inputs (an amplitude and the observation epoch) and a list of named lemmas, none of them a knob you could turn to absorb a wrong measurement (Part~XIII, and the kill-test timeline). That is the difference between this and an unfalsifiable grand theory: we have told you, in advance and in public, exactly which observations would force us to abandon it, and the instruments that will make those observations are already running or funded. A theory that can hand you the list of its own executioners is not hiding. Whether it survives them is now a question for the sky, not for us.
\end{objqabody}

\bigskip

\bigskip
\section{The \(\Lambda\)CDM and CMB-precision referee}

\noindent\textbf{Objection 23 (O-Cosmos-23) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``You set \(z_* = e^7 = 1096.6\) against the Planck recombination redshift \(z_{\mathrm{rec}} = 1089.92 \pm 0.25\). That is a \(27\sigma\) miss in units of the Planck error, yet you call it `0.6\% agreement.' Which is it --- a triumph or a failure?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You are right that we cannot have it both ways, and we now pick one identity cleanly. \(z_* = e^7\) is \emph{not} claimed to be the physical recombination redshift, and it is not compared to \(z_{\mathrm{rec}} = 1089.92\pm0.25\) as a prediction of that measured number --- to do so, and then reach for the visibility width to soften the \(27\sigma\), would be precisely the comparator-shopping you object to. Instead \(z_*\) is a \emph{distinct object}: the \emph{topological switch redshift} of the two-stage Hubble corridor, the integer-locked scale \(e^J\) at which the corridor changes branch. The operational decoupling redshift \(z_{\mathrm{dec}} = 1089.92\) that Planck measures is a different quantity --- the peak of the visibility function --- and the two coincide to \(0.6\%\). That \(0.6\%\) offset is not an error bar to be gamed; it is now a \emph{computed} quantity: the framework's own atomic chain gives \(\ln(1+z_{\mathrm{dec}}) = 6.996\), hence \(\delta = J - \ln(1+z_{\mathrm{dec}}) = 0.0041\) (\S\ref{sec:two-redshifts}, Eq.~\eqref{eq:delta-offset}), the codebook carrying both quantities in one entry (D21) under Conjecture~VIII (closed-packet integration). So the honest statement carries no \(\sigma\): a single ledger integer \(J=7\) fixes a topological corridor-switch that falls \(0.6\%\) from the operational recombination peak, with the offset computed --- and the atomic quantity itself landing on the Planck value at the \(0.1\%\) level with zero fitted parameters --- not a precision match asserted and not a tension dressed in a wide comparator. The codebook carries both quantities in one entry, and the Dev plot scores the audited atomic-chain prediction, with the switch annotated, not scored.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 24 (O-Cosmos-24) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``You never compute a single angular power spectrum. There is no \(C_\ell^{TT}\), no \(C_\ell^{EE}\), nothing from a Boltzmann code. You quote background densities and hand-picked summary numbers and call it a CMB analysis.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded without reservation. The section ``CMB baseline, primordial perturbations, and the scalar/tensor audit'' compares background densities and a set of summary observables --- the baryon loading, the damping multipole, the scalar-tilt corridor, the tensor amplitude --- and explicitly retains the standard perturbation grammar rather than recomputing it. No \(C_\ell\) curve is produced anywhere in the paper. A full Boltzmann/\(C_\ell\) prediction is named as future work, and the introduction's ``why the CMB and BBN are the two hardest audits'' subsection states the guardrail in advance: a framework that speaks confidently about dark matter or recombination but never meets the sky at the level of acoustic structure ``has not yet entered the precision arena.'' That is a real scope limit, acknowledged in the body's remaining-challenges accounting, not papered over.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 25 (O-Cosmos-25) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``Every comparison in the paper is central-value-to-central-value. You never confront the framework with the full Planck likelihood or its parameter covariance. Modern cosmology is done with covariances, not eyeball matches.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The present comparisons are at the level of individual summary statistics, each measured against its own published central value and \(1\sigma\) error, as the Dev plot and its data table make explicit. There is no joint fit to the Planck likelihood and no propagation of the parameter covariance --- the \(\pi\)-locked inventory and the two-stage corridor share underlying structure, so the correlations between, say, \(\Omm\), \(\Omb\), and \(\Omdm\) certainly matter and are not modeled here. A joint CMB-plus-BAO-plus-distance-ladder likelihood fit is the declared next step; the executive summary names it as ``the global fit that would turn this reframing into a quantitative resolution.'' Until that fit exists, the reader should read these as a battery of independent central-value checks, not as a likelihood-level test, and we say so.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 26 (O-Cosmos-26) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``\(\Omega_b = 1/(2\pi^2) = 0.0507\) sits about \(2.8\%\) high of the CMB-inferred baryon density, and it disagrees with your own baryogenesis route at the \(7.5\)--\(28\%\) level. Your single most-cited prediction is also your worst-fitting one.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and it is flagged as the worst entry in the paper's own audit. The codebook row for \(\Omb\) states that the parameter-free \(0.0507\) deviates from the empirical \(0.0493 \pm 0.0006\) at the \(2.3\sigma\) level, and the Dev plot caption names \(\Omb\) as the single largest deviation among the precision entries, at \(+2.76\%\), tagged as a candidate next-order correction whose source --- a small baryonic-projection asymmetry inherited from the Standard-Model gauge sector \citep{ArisakaA9SM} --- is identified for follow-up. The tension with the \(\eta_B \to \Omega_b\) consistency chain is stated openly in the matter-sector section: that chain is order-of-magnitude only, since the washout efficiency and transport normalization are uncomputed. \(\Omb\) is classified Derived-conditional (D\(^*\)) under the closed-packet-integration conjecture (Conjecture~VIII): the structural derivation is closed with no free parameter, but the residual discrepancy is real and is owned as such, not hidden.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 27 (O-Cosmos-27) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``Your `\(0.27\sigma\) agreement' for \(\Omega_{\mathrm{dm}}\) is an artifact of a generous error bar, and in any case it is not an independent test --- \(\Omega_{\mathrm{dm}}\) is just \(\Omega_m - \Omega_b\) from the same \(\pi\)-locked ansatz.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct on both counts, and neither is disputed. The dark-matter fraction is the closed-form complement \(\Omdm = \Omm - \Omb = 1/\pi - 1/(2\pi^2) \approx 0.2676\) of the same \(\pi\)-locked inventory; it is not an independent input but an algebraic consequence of \(\Omm = 1/\pi\) and \(\Omb = 1/(2\pi^2)\). The \(0.27\sigma\) figure is therefore a consistency check that the closed-form complement lands inside the present Planck error, not an independent precision triumph, and the size of that error bar is doing real work in making the agreement look tight. We report it as a self-consistency statement of the inventory, classified D\(^*\) under the Kaxion-misalignment-completeness conjecture for its physical interpretation. The honest content is narrow: given the matter fraction and the baryon fraction, the leftover is dark and has the right magnitude --- nothing more.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 28 (O-Cosmos-28) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``Your title says you `derive \(\Lambda\)CDM.' But the scalar tilt \(n_s\) and amplitude \(A_s\) are merely Asserted, and \(r\) is only a bound. You have not derived the dynamical sector at all.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Partially conceded, and the codebook is explicit about where the line now falls. Of the six \(\Lambda\)CDM parameters, the inventory \emph{fractions} (\(\Omm\), \(\OmL\), \(\Omb\), and the \(\Omdm\) complement) are Derived-conditional on named conjectures; the scalar tilt is likewise Derived-conditional --- pinned to \(n_s = 1 - 2/N^* = 0.9620\), not confined to a corridor --- and the tensor ratio is Derived at a central value, \(r = 0.0043\), a two-sided target rather than a bound. One dynamical parameter remains imposed and is named as such: the amplitude \(A_s\), the launch sector's single declared input (warp candidate at \(75\%\), flagged; closing computation Lemma~V\('\)); \(\tau\) is astrophysics and outside the claim. So the defensible statement is now: five of the six parameters derived or derived-conditional, one declared input --- and where the title reads as the stronger claim, the codebook triage remains the binding statement.\end{objqabody}

\bigskip

\noindent\textbf{Objection 29 (O-Cosmos-29) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``Your deviation plot mixes entries that are anchored by construction --- and therefore sit at zero deviation trivially --- with genuine predictions. That visual inflates your apparent success rate.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The distinction you describe is exactly the one the Dev plot is built to make, and it is disclosed, not concealed. The caption states that \emph{filled markers} are independent predictions, computed without reference to the measured value, while \emph{open markers} are anchored by construction --- the codebook entry echoes the empirical input pending its named lemma --- and ``necessarily sit at zero deviation.'' The caption then lists which entries are anchored: \(\Hzero^{\mathrm{CMB}}\), \(\sigeight\), \(\Seight^{\mathrm{CMB}}\), \(N_{\mathrm{eff}}\), and the amplitude \(A_s\), the one Asserted entry. The genuine predictions that carry nonzero deviation --- the atomic-chain \(z_{\mathrm{dec}}\) at \(+0.11\%\) (the switch \(e^7\) annotated \(0.62\%\) above the measured value), the derived tilt \(n_s = 0.9620\) at \(-0.7\sigma\), the tensor ratio \(r = 0.0043\), the damping knee \(\ell_D = K(2\pi)^2\) at \(+2.7\%\), \(w = -1\) at \(+2.7\%\), \(\Omb\) at \(+2.76\%\), and \(\eta_B\) within its factor-\(\sim 3\) band --- are the ones a reader should weigh. A referee is entitled to ignore the anchored markers entirely; the plot is designed so that doing so is trivial, because the marker fill tells you which is which.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 30 (O-Cosmos-30) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``You predict background densities but never the acoustic-peak positions or the angular sound horizon \(\theta_*\). Those are the actual precision observables of the CMB, and they are absent.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The paper writes \(\theta_* = r_s(z_*)/D_A(z_*)\) as a definition and discusses the geometric meaning of the acoustic ruler, but it does not produce a predicted numerical \(\theta_*\) or a predicted peak-height pattern. Those require the comoving sound horizon and angular-diameter distance to be integrated over the full thermal history with the recombination microphysics in place --- i.e.\ the \(C_\ell\) pipeline whose absence is conceded above. The damping multipole \(\ell_D = K(2\pi)^2\) is the one acoustic-tail quantity the integer ledger reaches, and even there the BFFT-damping-knee identification is carried as a conjecture and the observed comparator is definition-dependent at the percent level. Peak morphology and \(\theta_*\) are explicitly downstream of the Boltzmann pipeline named as future work; they are not claimed here.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 31 (O-Cosmos-31) (the CMB-precision referee)}\par\nobreak
\noindent\textbf{``Your Hubble corridor leans on a pre-recombination \(\Delta N_{\mathrm{eff}} \approx 0.30\), but your own codebook carries \(\Delta N_{\mathrm{eff}} = 0.04 \pm 0.18\). Extra radiation at \(0.30\) is squarely in tension with the Planck \(N_{\mathrm{eff}}\) fit and with BBN. You cannot have it both ways.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the sharpest internal exposure of the corridor, and the paper states it as such rather than burying it. The codebook row for \(\DeltaNeff\) is the late-time/baseline value, \(0.04 \pm 0.18\), consistent with three Standard-Model neutrinos plus a small thermal residue from the inherited particle ledger \citep{ArisakaA9SM}, and it is Derived. The corridor mechanism of the Hubble-tension Part separately invokes a \emph{pre-recombination} \(\DeltaNeff \approx 0.30\), and the body says outright that this radiation is ``nominally visible to the Planck \(N_{\mathrm{eff}}\) fit and to BBN.'' The corridor therefore stands or falls with whether the early-dark-energy-lobe degeneracy direction of the joint CMB+BBN likelihood can absorb the extra radiation --- and the paper records that demonstrating this with a full Boltzmann fit is ``an explicit open requirement attached to the corridor-uniqueness conjecture, not an established fact.'' So the honest position is: the tension you name is real, it is the corridor's principal observational risk, and its resolution is deferred to the joint-likelihood fit, here and in the companion \citep{ArisakaC2Hubble}. We do not claim to have it both ways; we claim a degeneracy direction that must be earned.
\end{objqabody}

\bigskip

\section{The inflation and dark-energy referee}

\noindent\textbf{Objection 32 (O-Cosmos-32) (the inflation referee)}\par\nobreak
\noindent\textbf{``A TeV plateau cannot do both jobs. Pinning the scalar amplitude $A_s \approx 2.1\times 10^{-9}$ at an inflationary Hubble scale of order a few TeV forces a slow-roll parameter $\epsilon_V$ that is dozens of orders of magnitude away from the one that gives $n_s \approx 0.965$ through $\eta_V \approx -1/N_e$. You never compute the potential, $A_s$, $n_s$, or the e-fold count from the geometry; you assert them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The referee was right --- and the two-basin derivation of Part~IV is the proof, recorded rather than smoothed over. Sharpened, this objection becomes a sign-level no-go: a lock-scale plateau demands \(\epsilon_V \sim 10^{-53}\) for the amplitude \emph{and} yields a blue tilt (\(n_s - 1 \sim +10^{-26}\)) against the measured red \(-0.0351 \pm 0.0042\) --- wrong in sign, unrepairable by any parameter. The TeV reading and its codebook conjecture are therefore retired, with the refutation attached (Part~IX). What replaced them is not a patched assertion but a derivation: the \GDOS{} boundary law forces the Starobinsky/\(\xi\) plateau \emph{class} (the anomaly supplies no scalaron; the one non-conformal scalar, the \(\Sigma_r\) modulus, does), the reheating match at Planck-strength width computes \(N^* = 52.6\), and the tilt and tensor ratio follow with no dial: \(n_s = 0.9620\), \(r = 0.0043\). One number remains imposed, and the codebook says so: the amplitude \(A_s\), the launch sector's single declared input, its warp candidate (\(75\%\)) flagged and its closing computation named (Lemma~V\('\)). A framework that can be argued out of its own published reading by its own referee, and replaces it with a stronger falsifiable one, is behaving exactly as a zero-parameter theory must.\end{objqabody}

\bigskip

\noindent\textbf{Objection 33 (O-Cosmos-33) (the inflation referee)}\par\nobreak
\noindent\textbf{``Your codebook calls $r$ a `Starobinsky-channel' suppression. That is a misnomer. The Starobinsky $R^2$ model predicts $r \approx 12/N^2 \approx 0.003$ from the e-fold count; your $r \sim 10^{-52}$ is a TeV energy-scale suppression, a completely different mechanism. Calling it Starobinsky is sleight of hand.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Before the plateau class was derived, this objection was fair, and the response could only concede the label. With the plateau class derived, the label is exact: the tensor entry \emph{is} the Starobinsky relation \(r = 12/N^{*2}\), because the \GDOS{} boundary law forces the Starobinsky/\(\xi\) plateau class itself --- the genuine conformal anomaly supplies only \(W^2\) and \(E_4\), so the scalaron sector must come from the one non-conformal scalar, the \(\Sigma_r\) modulus, and the resulting potential is Eq.~\eqref{eq:partII-plateau-potential} with no freedom in shape or slope. The codebook row now reads exactly that way (\(r = 0.0043\) at the reheat-matched \(N^* = 52.6\)), and the retired energy-scale version of the entry --- the one this objection correctly attacked --- is kept in Part~IX's audit trail with its refutation. The objection, in other words, has been promoted from ``fair point about a loose label'' to ``anticipation of the derivation.''\end{objqabody}

\bigskip

\noindent\textbf{Objection 34 (O-Cosmos-34) (the inflation referee)}\par\nobreak
\noindent\textbf{``$n_s = 0.965$ via $n_s \approx 1 - 2/N_e$ requires $N_e \approx 57$. You concede you cannot fix $N_e$. So your headline tilt is a back-fit to a number you cannot derive.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This objection once had to be conceded; it is no longer conceded. The e-fold count is now computed, not targeted: the modulus decays at Planck strength (\(\Gamma = M^3/M_{\mathrm{Pl,red}}^2\), the only channel consistent with the warped structure), which fixes \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\) and, through the standard matching formula, \(N^* = 52.6\) --- undiluted, because the lock crossing is prompt (\(N_2 = 0\), the ledger-action bound with fifty-six orders of margin). The tilt is then \(n_s = 1 - 2/N^* = 0.9620\), carried in the codebook as \textbf{Derived (cond.)}, a pull of \(-0.7\) against Planck --- risked against the data, not read from them. The residual conditions are named (the plateau-class derivation and the prompt-crossing bound, Lemma~IV revised), and the falsifier is stated: a measured tilt below \(\simeq 0.955\) would demand a diluted window and break the chain.\end{objqabody}

\bigskip

\noindent\textbf{Objection 35 (O-Cosmos-35) (the inflation referee)}\par\nobreak
\noindent\textbf{``Reheating a TeV-scale inflationary patch --- with a Hubble rate of order meV --- up to the temperatures needed for baryogenesis and BBN is dynamically implausible, and you assert it rather than compute it. Where is the reheating temperature?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This objection long stood against the program; it is now answered by computation, and the answer dissolves the implausibility along with the question. There is no TeV-scale patch to reheat: the primary plateau sits one e-fold below the GUT edge (\(H_1 \simeq 1.6\times10^{13}\,\mathrm{GeV}\), not meV), the modulus decays at Planck strength (\(\Gamma = M^3/M_{\mathrm{Pl,red}}^2\) --- the only channel consistent with the warped structure; the alternatives were computed and excluded), the bath spikes to \(T_{\mathrm{max}} \sim 10^{11}\,\mathrm{GeV}\) and settles at \(T_{\mathrm{rh}} \simeq 3\times10^{10}\,\mathrm{GeV}\) after roughly fourteen e-folds of matter-like oscillation --- comfortably above the thermal-leptogenesis window, comfortably below the GUT edge, and far above BBN. The lock scale is then reached by ordinary cooling at \(t \sim 10^{-13}\,\mathrm{s}\) and crossed promptly (the meV curvature scale the objection cites is real, but it is the \emph{terminus} seesaw --- the middle basin's curvature --- not the launch's Hubble rate). The full numerically integrated history, smooth from the GUT edge to the de~Sitter future, is Figures~\ref{fig:partII-history-page1}--\ref{fig:partII-history-page2}.\end{objqabody}

\bigskip

\noindent\textbf{Objection 36 (O-Cosmos-36) (the inflation referee)}\par\nobreak
\noindent\textbf{``You commit to $w = -1$ exactly and call it `natural.' But DESI now hints at $w = -1.03 \pm 0.03$ and an evolving dark-energy equation of state. Your framework is in live tension with the most recent data, and you bury it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not buried --- this is the framework's committed prediction and a genuine kill-switch, and the tension is real. The late-vacuum Part argues for \(w = -1\) exactly, as the frozen residual vacuum export of a completed late-time geometric transition; the codebook lists \(w\) as Derived-conditional (D\(^*\)) under the vacuum-residual-exactness conjecture, against an empirical row that already reads \(w = -1.028 \pm 0.032\). So the live hint is printed in the codebook itself, not hidden. A confirmed, robust \(w \neq -1\) --- whether the DESI \(w_0 w_a\) preference or any future tightening --- falsifies the module: the framework has no dial to absorb it, because the whole point is that the vacuum term is a frozen residual, not a tunable quintessence field. This is openly flagged as the one place the cohort is already in tension; see the falsifier deck of \citet{ArisakaA11Origin}. The honest status is: committed prediction, currently at the \(\sim 1\sigma\) edge of a developing anomaly, with no escape route by construction.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 37 (O-Cosmos-37) (the inflation referee)}\par\nobreak
\noindent\textbf{``$\Omega_\Lambda = 1 - 1/\pi$ is a fit-then-rationalize. You measured $\Omega_m \approx 0.315$, noticed $1/\pi \approx 0.318$, and reverse-engineered a story. And the meV `seesaw' for the vacuum scale double-counts --- it is a second dressed-up version of the same coincidence.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded on both points, with the honesty already in the text. \(\Omega_\Lambda = 1 - 1/\pi\) is not an independent derivation: it is the unit-trace complement of \(\Omega_m = 1/\pi\), obtained from late-time flatness \(\Omega_m + \Omega_\Lambda \simeq 1\). The codebook carries \(\Omega_m\), \(\Omega_\Lambda\), and \(\Omega_b\) as Derived-conditional (D\(^*\)) under the closed-packet-integration conjecture, and the topological-completion Part states in full that the printed Step-4 cancellation does not yet close --- so the \(1/\pi\) value is a conjectured topological identity, not a proved theorem. The \(\Omega_\Lambda\) row literally cites ``via unit-trace identity,'' i.e.\ it inherits whatever \(\Omega_m\) earns and adds nothing independent. The meV vacuum scale is presented as a seesaw \emph{descendant} of \(\Mlock\), explicitly flagged as a suggestive dimensional coincidence, not as a second independent derivation of the dark-energy density. The deep four-form integration meant to prove \(\Omega_m = 1/\pi\) topologically (and exclude competing rational functions of \(\pi\)) is reserved for a later sweep; until then the referee's ``fit-then-rationalize'' charge is the correct default reading, and the codebook classification (D\(^*\), not D) concedes it.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 38 (O-Cosmos-38) (the inflation referee)}\par\nobreak
\noindent\textbf{``$r = 0.0043$ is a razor-thin target --- an enormous, dangerous claim. You are betting the module on the Simons Observatory and LiteBIRD finding a primordial B-mode at exactly one amplitude.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes --- and that is deliberate, not an oversight; the two-basin derivation made the bet strictly more dangerous, on purpose. The retired TeV reading predicted \(r \sim 10^{-52}\): safety by invisibility, falsifiable only from one side. The derived plateau predicts a \emph{number}, \(r = 12/N^{*2} = 0.0043\), a factor eight below the current bound and several times above the Simons-Observatory/LiteBIRD floor --- so the switch now fires in both directions: a detection well above it kills the launch, and so does a clean exclusion well below it. Carried in the codebook as \textbf{Derived (D)}, with no dial to move (\(r\) rides on \(N^*\), and \(N^*\) is pinned by the reheating match and the prompt crossing). A strong, dangerous, two-sided, dated claim is the feature here, not a bug: the framework is obliged to produce a definite value and accepts that this decade's instruments will pass verdict.\end{objqabody}

\bigskip

\noindent\textbf{Objection 39 (O-Cosmos-39) (the inflation referee)}\par\nobreak
\noindent\textbf{```Slow-smooth inflation with graceful exit' is narrated through \GDOS{}, \GRIP{}, \GSSB{}, and \GFF{} vocabulary, but you never exhibit a computed potential with an actual exit point. The exit is a story, not a calculation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The exit is described \emph{structurally}: graceful exit is identified with a \GSSB{} event at the infrared seam of the throat, terminating the GFF-dominant plateau branch, with the whole sequence read as the prototype cosmological \GRIPout{} in the runtime grammar inherited from \citet{ArisakaA6Gi4}. The paper writes a plateau form and the leading-order \(\epsilon_V\), \(\eta_V\) for it, but it does not produce a closed-form potential \(V(\varphi)\) with the model coefficients (\(c_1\), \(\beta\), the throat width) fixed from the geometry, and therefore cannot exhibit a computed field value \(\varphi_{\mathrm{end}}\) where slow roll ends. The mechanism is coherent and the exit is located in principle (the same \GSSB{} transition that ends inflation also seeds the Kaxion misalignment and the late vacuum residue), but a closed-form potential with computed slow-roll parameters and a derived exit point is an open item. The narrative scaffolding is real; the calculation behind it is not yet done.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 40 (O-Cosmos-40) (the inflation referee)}\par\nobreak
\noindent\textbf{``The codebook's audit trail records a retired conjecture that located the inflationary plateau at the TeV lock scale, while the universe begins at the GUT edge $\EGUT$ --- wildly separated scales, with an uncomputed thirty-e-fold gap between the beginning and the launch. The paper now puts the plateau at the top. Which is it, and what filled the gap?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The gap the objection attacks was real, and closing it is precisely what the two-basin derivation did --- not by explaining the descent-then-inflate ordering, but by discovering it was backwards. The plateau is not at the lock: the \GDOS{}-derived class plus the observed amplitude place it at \(V^{1/4} = 8.3\times10^{15}\,\mathrm{GeV}\), \emph{one e-fold below the GUT edge} --- so the beginning and the launch are adjacent, the ``gap'' between them is a single e-fold, and inflation is literally the opening stretch of the descent, exactly as the Part's \GRIP{} narrative always said. The thirty-e-fold stretch down to the lock is then ordinary post-reheating radiation cooling, computed, with the lock crossed promptly at \(t \sim 10^{-13}\,\mathrm{s}\) (the middle basin's curvature scale, \(\Mlock^2/\sqrt3 M_{\mathrm{Pl,red}}\), survives as the terminus seesaw --- the number the old reading had misassigned to the launch). The explicit trajectory across the whole history, numerically integrated and smooth, is Figures~\ref{fig:partII-history-page1}--\ref{fig:partII-history-page2}. The dynamical bridge the old response conceded as open is now the computed content of Part~IV.\end{objqabody}

\bigskip

\section{The Hubble-tension and large-scale-structure referee}

\noindent\textbf{Objection 41 (O-Cosmos-41) (the tensions referee)}\par\nobreak
\noindent\textbf{``Your two headline numbers, $H_0^{\mathrm{CMB}}\approx 67.4$ and $H_0^{\mathrm{late}}\approx 73.0$, are simply read off Planck and SH0ES. You did not derive either one; you transcribed them and drew a box around them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and the paper says so in its own words: in the Friedmann-derivation section the two values are ``accommodated as two projections of one structure rather than independently derived outputs,'' and the abstract repeats the qualification. The Hubble-two-stage-corridor theorem (Theorem~\ref{thm:hubble-two-stage-corridor}) does not claim to output $67.4$ or $73.0$; its content is the structural two-branch \emph{form} --- one corridor of cosmological-boundary solutions with the switch event at recombination --- into which the two measured endpoints are placed. The companion \citet{ArisakaC2Hubble} carries the same two anchors (off the slightly lower $67.24$ convention), not an independent re-derivation. What is genuinely on the ledger is the geometry of the corridor, not the decimal value of its endpoints.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-Cosmos-42) (the tensions referee)}\par\nobreak
\noindent\textbf{``Two mutually inconsistent stories travel under the single name `two-stage corridor': in one place it is a kinematic reframing with `no new physics, just two projections,' and in another it is a real early-dark-energy lobe plus a local-void shift. Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded that the exposition conflates two registers. There is a purely kinematic reframing --- the statement that the CMB-inferred and ladder-inferred values are two projections of one background, not two measurements of one local constant --- and there is a dynamical mechanism carrying the early-dark-energy residue $f_{\mathrm{EDE}}\approx 0.03$, the extra radiation $\Delta N_{\mathrm{eff}}\approx 0.30$, and the local-underdensity lift $\epsilon_{\mathrm{local}}$. The dynamical version is the one with observational teeth and is the research target; the kinematic version is the framing that motivates it. The honest reading is that the corridor is established only at the level of \emph{form}, and the dynamical mechanism that would make the reframing quantitative is the open piece, closed only by the corridor-uniqueness conjecture.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-Cosmos-43) (the tensions referee)}\par\nobreak
\noindent\textbf{``A corridor that reproduces two endpoints but commits to no inverse-distance-ladder, no $\mathrm{BAO}(z)$, and no $H(z)$ shape in between is not a model. It is a pair of dots with a label.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The paper does not yet print a committed $H(z)$ curve, an inverse-ladder $D_A(z)/D_L(z)$ prediction, or a $\mathrm{BAO}(z)$ track through the corridor interior, and it should not pretend to. Producing them requires the global CMB-plus-BAO-plus-distance-ladder fit that the Friedmann-derivation section and the limitations Part both name as still required. Until that fit exists, the corridor constrains the two endpoints and the qualitative early-time source pair, not the interpolating background. We mark this as a declared deliverable, not a quiet omission.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-Cosmos-44) (the tensions referee)}\par\nobreak
\noindent\textbf{``You assert that the CMB acoustic fit, BAO, and BBN all survive your early-dark-energy plus dark-radiation insertion simultaneously. You never demonstrate it --- the paper itself admits the global fit is required and absent.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded plainly, and this is the single most important open numerical task in the entire corridor module. The lensing/EDE-residue section states explicitly that radiation at $\Delta N_{\mathrm{eff}}\approx 0.30$ over the whole pre-recombination regime is nominally visible to the Planck $N_{\mathrm{eff}}$ fit and to BBN, and that ``demonstrating this with a full Boltzmann fit is an explicit open requirement attached to the corridor-uniqueness conjecture, not an established fact.'' Simultaneous preservation of the acoustic peaks, the sound horizon, the BAO ruler, and the light-element abundances is asserted as a target, not shown. It is the declared corridor-uniqueness research target, and the corridor stands or falls on it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-Cosmos-45) (the tensions referee)}\par\nobreak
\noindent\textbf{``Your $S_8$ corollary assumes the early-time corrections \emph{suppress} clustering. But in full EDE likelihood fits the net $S_8$ shift comes out \emph{positive} --- refitting drives $\omega_{\mathrm{cdm}}$ and $n_s$ up. Your corollary has the sign backwards.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and the paper flags exactly this. The $S_8/\sigma_8$ section carries a mandatory cautionary note: ``in full likelihood analyses of early-dark-energy-type models, refitting the CMB typically drives $\omega_{\mathrm{cdm}}$ and $n_s$ upward and the \emph{net} $S_8$ shift comes out positive --- the opposite sign to the suppression encoded above'' \citep{Hill2020}. The suppression branch in Eq.~\eqref{eq:S8-response} is adopted as a \emph{target}, not a result; the abstract states the sign of the corollary effect is ``flagged open.'' Settling it requires the growth-history integration named inside the corridor-uniqueness research target. We do not claim the $S_8$ tension is resolved; we claim the direction is the open question.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-Cosmos-46) (the tensions referee)}\par\nobreak
\noindent\textbf{``The response coefficients $0.13$ per unit $\Delta N_{\mathrm{eff}}$ and $0.40$ per unit $f_{\mathrm{EDE}}$ are lifted straight from the early-dark-energy literature. They are not outputs of your throat/$\Sigma_r$ geometry. So where exactly is \GGTheory{} doing any work?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The text is explicit: ``the numerical values used below, $0.13$ per unit $\Delta N_{\mathrm{eff}}$ and $0.40$ per unit $f_{\mathrm{EDE}}$, are representative response slopes of the early-dark-energy literature \citep{Poulin2019}, not outputs of the GG ledger.'' What \GGTheory{} is asked to supply is not these slopes but the \emph{source pair} that feeds them --- a narrow $f_{\mathrm{EDE}}$ lobe and a small $\Delta N_{\mathrm{eff}}$ inherited from the throat/$\Sigma_r$ sector rather than from a free phenomenological menu. Deriving the slopes themselves from that sector is named in the same sentence as part of the corridor-uniqueness research target. The parametrization is borrowed; the claim that the inputs are inherited rather than tuned is what is on trial.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-Cosmos-47) (the tensions referee)}\par\nobreak
\noindent\textbf{``A10 cites C2 for the $H_0$ numbers and C2 cites A10 right back --- a citation loop. The global fit that would justify either is owed by both papers and delivered by neither.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The reciprocal citation is real and we acknowledge it. The division of labor is clean even if the prose has not always made it loud: the structural corridor --- the two-branch Friedmann form and its theorem --- belongs to \emph{this} paper (the Friedmann-derivation and EDE-residue sections); the dedicated $H_0$ phenomenology, the $\epsilon_{\mathrm{local}}$ underdensity estimate, and the eventual joint CMB+BAO+ladder fit are the remit of \citet{ArisakaC2Hubble}. C2 is the owner of the fit. The loop is a synchronization artifact (the two anchors, $67.4$ and $67.24$, were aligned June 10, 2026), not two papers each pointing at the other for a result neither holds: the structural claim has a home here, the numerical fit has a named home in C2.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-Cosmos-48) (the tensions referee)}\par\nobreak
\noindent\textbf{``$\Delta N_{\mathrm{eff}}$ appears as $0.30$ in your corridor mechanism and as $0.04$ in your baseline codebook row. One paper, one symbol, two numbers --- that is a contradiction sitting in plain sight.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No --- this one withdraws on a careful read, and the paper labels the two quantities distinctly. The $0.30$ is the \emph{pre-recombination corridor source}: the extra-radiation contribution that, together with the $f_{\mathrm{EDE}}$ lobe, shortens the sound horizon (Eq.~\eqref{eq:EDE-DR-values}, carried identically by \citet{ArisakaC2Hubble} as the corridor's inherited input). The $0.04 \pm 0.18$ is the \emph{late-time Planck baseline constraint} on $N_{\mathrm{eff}} = 3.046 + \Delta N_{\mathrm{eff}}$ that the codebook row records as the empirical anchor. They are a proposed source term and a measured baseline, not two values of one quantity. The genuine tension the objection should press instead is whether the corridor's $0.30$ can hide inside the EDE-lobe degeneracy direction of the joint CMB+BBN likelihood --- which is Objection~22's open fit, not a numerical contradiction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-Cosmos-49) (the tensions referee)}\par\nobreak
\noindent\textbf{``Your JWST early-galaxy `solution' is a semianalytic boost --- a lowered collapse threshold, a $20\%$ streaming-velocity damping, a bump in the minihalo tail. Luminosity functions and number counts at $z\gtrsim 10$ are nonlinear. Without N-body you have a story, not a prediction.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded without hedging. The scorecard lists the JWST early-galaxy entry as \emph{Partially resolved} explicitly because ``nonlinear N-body simulations under the inherited ledger [are] still pending,'' and the limitations Part devotes a dedicated subsection to the point: a semianalytic boost to early structure ``is not the same thing as a robust prediction for luminosity functions or galaxy number counts.'' What the framework does claim is narrower and survivable: the scale distinction between the broad linear $S_8$ amplitude and the exponentially sensitive minihalo tail means a coherent-but-non-uniform solution is at least \emph{permitted} --- early collapse can advance without spoiling broad growth. That is a consistency argument, not a number. The number requires the nonlinear pipeline named in the medium-term research tasks, and we do not pretend it exists yet.
\end{objqabody}

\bigskip

\section{The dark-matter, baryogenesis, and BBN referee}

\noindent\textbf{Objection 50 (O-Cosmos-50) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Your dark-matter density is $\Omega_{\mathrm{DM}} = 1/\pi - 1/(2\pi^2)$. That is just an algebraic subtraction --- the matter share minus the baryon share. It is a definition, not a relic calculation. You have not computed a freeze-out or a misalignment yield; you have rearranged a ledger.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The fraction and the dynamics are two different claims, and you are right about the first. As an identity, $\Omega_{\mathrm{DM}} = \Omega_m - \Omega_b = 1/\pi - 1/(2\pi^2)$ is exactly the closed-form complement of the $\pi$-locked inventory; it is bookkeeping, and the section ``Kaxion dark matter'' presents it as such. What is \emph{not} bookkeeping is the relic dynamics, and that is derived in the companion \citet{ArisakaC1Kaxion}, not here. There the geometry fixes the Kaxion decay constant $f_K$ from the scale-profile cap (with no continuous dial), and the observed $\pi$-locked abundance then fixes the one remaining input, the misalignment angle, at $\theta_i \simeq 1.4$~rad. That is a genuine inversion: the abundance is the \emph{target}, $f_K$ is set independently, and $\theta_i$ comes out. It is emphatically not a fit of $\Omega_{\mathrm{DM}}$ --- $\Omega_{\mathrm{DM}}$ is the input on the right-hand side. The honest statement is that the \emph{number} is a subtraction identity and the \emph{mechanism that saturates it} is a one-parameter-out inversion; the codebook tags $\Omega_{\mathrm{DM}}$ as D$^*$ under the Kaxion-misalignment-completeness conjecture, which says precisely that the dynamical closure is conjectural, not closed.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 51 (O-Cosmos-51) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Your framework geometry-selects nearly every number it touches, yet the misalignment angle $\theta_i$ is read off the observed abundance as a generic $\mathcal{O}(1)$ number. Why is this one an output? And since $1/(2\pi)$ is exactly the kind of value your ledger likes, is it excluded or merely unmentioned?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is an output by construction, not by exception. The Kaxion sector presents two unknowns --- the decay constant $f_K$ and the angle $\theta_i$ --- and the geometry determines one of them: $f_K$ follows from the ledger, and $\theta_i$ is then fixed by the measured abundance. That is a one-parameter-out inversion, and it is the same inversion \citet{ArisakaC1Kaxion} performs, quoting the same generic $\mathcal{O}(1)$ value. Naming $\theta_i$ a geometric selection as well would be claiming a determination the framework does not have.

On $1/(2\pi)$: it is excluded, not merely unmentioned. Nothing in the \OGNfive{} ledger selects it for this slot, and it is quantitatively wrong --- $1/(2\pi) = 0.159$~rad lies a factor of about seven below the generic range $1.1$--$1.5$~rad that the anharmonic-corrected inversion \eqref{eq:appA-thetai} returns against the observed abundance. A geometry-selected $\theta_i$ at that value would overproduce the relic density by orders of magnitude.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 52 (O-Cosmos-52) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``You advertise the Kaxion mass $m_K \approx 8.1~\mu$eV as a prediction. But that is just the standard QCD-axion mass image of $f_K$: once you fix the decay constant, $m_K f_K = \sqrt{\chi_{\mathrm{QCD}}}$ hands you the mass for free. It is not an independent second prediction; it is the same number wearing a different hat.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and stated the same way in \citet{ArisakaC1Kaxion}. The relation $m_K f_K = \sqrt{\chi_{\mathrm{QCD}}}$ is the standard strong-CP axion relation, with $\chi_{\mathrm{QCD}}$ fixed by Standard-Model lattice input; once $f_K$ is set, $m_K \approx 8.1~\mu$eV (a $1.95$~GHz tone) is its DFSZ image, not a logically separate output. The genuinely geometric prediction is $f_K$ itself --- hence the frequency \emph{window} $1.59$--$2.13$~GHz on the DFSZ coupling line --- together with the two parameter-free fingerprints that an ordinary axion cannot fake: the twin-peak split $\Delta\nu/\nu = 1/35$ and the $5{:}2$ portal lock. Those fingerprints, not the mass value, are what carry the falsifiable content; $m_K$ is a consequence of $f_K$ and the measured susceptibility, and the paper should be (and is) read that way.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 53 (O-Cosmos-53) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Your baryon asymmetry $\eta_B \simeq 7.34\times10^{-10}$ rests on $\epsilon_{\mathrm{geo}}^{\mathrm{eff}} = e^{-K/2}$, and you admit on the same page that the exponent $K/2$ is a postulate, not derived. Then you present a central value within $20\%$ of the observed $\eta_B$ and call the chain controlled. That is numerology dressed as a derivation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The section ``Baryogenesis via the inherited neutrino and CP ledger'' says in plain text that the exponent $K/2$ is a postulate of the present compression --- geometrically suggestive as $\sqrt{e^{-K}}$ but not derived --- and that it is part of what the sphaleron-transport-normalization conjecture must fix. The washout efficiency and transport normalization are uncomputed; the compressed chain silently absorbs an $\mathcal{O}(1)$--$\mathcal{O}(10)$ coefficient. For exactly this reason $\eta_B$ is demoted to an \emph{order-of-magnitude} claim and displayed in the parameter-audit Dev plot with a factor-$\sim$3 theory band, and the text explicitly warns that closeness of central values at this tier must not be read as a precision success. The $+20\%$ figure is reported as a central value inside a wide honest band, not as a precision match. It is tagged D$^*$ under a named conjecture, not advertised as closed.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 54 (O-Cosmos-54) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Your lithium-7 fix now sets the $A=7$ survival fraction to the closed-packet $1/\pi$ --- the \emph{same} factor you use for $\Omega_m = 1/\pi$. So you are plugging two different holes, the cosmic inventory and the lithium deficit, with one unproven number. That is not a derivation of either; it is a single conjecture (Conjecture~VIII) doing double duty. And you still have not exhibited the microscopic mechanism that makes the port act on mass-7 at all.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair sharpening, and the honest status is exactly as you state it. The
$A=7$ port strength is \emph{not} a tuned coefficient: the surviving mass-7 fraction is the
closed-packet $1/\pi$, the same factor that fixes $\Omega_m = 1/\pi$, so the predicted primordial
abundance is $({}^{7}\mathrm{Li}/\mathrm{H}) = \mathrm{SBBN}/\pi \approx 1.6\times10^{-10}$ with
\emph{no} free knob (the section ``The $A=7$ port and the $1/\pi$ prediction'' and the
leptogenesis appendix carry the derivation). That removes the ``one-parameter cheat'' charge ---
but it does not remove the dependence on Conjecture~VIII, and you are right to press there. Because
the port reuses the \emph{same} closed-packet $1/\pi$, the lithium fraction inherits exactly the
status of the inventory: Derived-conditional (D$^*$) under the one closed-packet-integration
conjecture, not an independent success. One unproven number doing double duty is a more economical
claim than two free knobs, but it is no less conditional, and the codebook tags it that way. What
remains genuinely open --- and is deferred to the forthcoming companion GG-C4-Nucleo --- is not a
coefficient to fit but three structural obligations: \emph{why} the port is selective in mass
number (acting on $A=7$ and not $A=2,3,4$), the full coupled BBN-network closure confirming the
other abundances do not move, and compatibility with detailed balance. The lithium-7 problem is
therefore reported as a parameter-free but conditional prediction, falsifiable by any firm
primordial ${}^{7}\mathrm{Li}$ away from $\mathrm{SBBN}/\pi$ --- not as solved.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 55 (O-Cosmos-55) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Worse: a selective sink on a single mass number is exactly the kind of thing that violates detailed balance and contradicts well-measured nuclear cross-sections. You cannot just remove $^7$Be/$^7$Li by fiat without the inverse reactions repopulating it. Your port is unphysical.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The honest answer is that this is exactly the question GG-C4-Nucleo must settle, and the present paper flags it as open rather than claiming to have passed it. The port is proposed as a \emph{selective open-system sink} in the \GDOS{}/\GFF{} ledger --- a seam-level loss term acting on the late, tiny $A=7$ inventory, with surviving fraction fixed by the closed-packet $1/\pi$ rather than by a free coefficient --- not as a modification of a measured two-body cross-section, precisely so that it does not touch the robust deuterium and helium yields fixed earlier in the network. Whether such an open-system sink respects detailed balance, or whether the inverse channels repopulate the $A=7$ abundance and neutralize it, is not something the effective seam-level description can answer by itself; it requires the full network. That check is named as a deferred obligation, and the lithium row carries the Derived-conditional (D$^*$) tag for this reason. We are not asserting it is consistent; we are flagging it as the load-bearing thing the companion must prove.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 56 (O-Cosmos-56) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``You have two routes to the baryon density and they disagree. The topological route gives $\Omega_b = 1/(2\pi^2) = 0.0507$; the baryogenesis route gives $\Omega_b \approx 0.059$ at $h = 0.674$, a $16.4\%$ gap --- and your $\Omega_b h^2$ sits $\sim$2.8\% high anyway. A framework that derives the same observable two ways and gets two answers has not derived it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and the two-route tension is flagged openly in the section ``A consistency chain from $\eta_B$ to $\Omega_b$.'' The topological route gives $\Omega_b = 1/(2\pi^2) = 0.05066$; the $\eta_B$ route gives $\Omega_b h^2 \simeq 0.0268$, i.e.\ $\Omega_b \approx 0.059$ at $h = 0.674$, a $16.4\%$ discrepancy that shrinks through zero near the ladder branch ($-2.0\%$ at $h = 0.735$) and sits inside the transport chain's factor-$\sim$3 band. The two roads are not, however, of equal standing, and the response should say so rather than leave the reader to infer it: converted through the same bridge at this paper's own $h = 0.666$, the topological route gives $\eta_{10} = 6.15$ against an observed $6.12$, an error of $+0.57\%$, while the transport route gives $7.34$, an error of $+19.9\%$. The topological route is the more accurate by a factor of about thirty-eight, and it is the one the paper leads with. The paper states in plain text that the two routes do not yet agree and records ``two-route-consistency'' as a named open problem alongside the closed-packet-integration conjecture. $\Omega_b$ itself is carried as D$^*$ under Conjecture~VIII, and its $+2.76\%$ deviation is the largest precision-register entry in the parameter-audit Dev plot (the $\eta_B$ entry sits farther out, carried on its order-of-magnitude band), marked as a candidate next-order correction (full entropy history, sphaleron throughput, reheating profile). You are right that two routes disagreeing is not a derivation; the paper's claim is the weaker, honest one --- a consistency demonstration with the residual offset named, not erased.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 57 (O-Cosmos-57) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Strip away the language and the whole matter sector is ``Partially resolved modulo three named conjectures.'' That phrase means the matter sector is not derived. You are claiming credit for a derivation you have explicitly told me you have not completed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed on the words, and that is the intended reading, not a slip. The matter sector is a \emph{consistency demonstration up to named conjectures} --- the Kaxion-misalignment-completeness, sphaleron-transport-normalization, and closed-packet-integration conjectures --- not a closed derivation, and the codebook tags say exactly this: $\Omega_{\mathrm{DM}}$, $\eta_B$, and $\Omega_b$ are all D$^*$ (Derived-conditional), with lithium-7 Partially-resolved and deferred to GG-C4-Nucleo. The paper's repeated claim is the asymmetric, honest one: a small set of \emph{precise} conjectures, once proved, would close the matter ledger, and no recognized problem is left without a named route forward. That is a different and weaker claim than ``derived,'' and the provenance triage is built specifically so a reader can tell which is which. The credit claimed is for a constrained, falsifiable structure with its open conjectures labeled --- not for a finished proof.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 58 (O-Cosmos-58) (the matter-sector referee)}\par\nobreak
\noindent\textbf{``Two more loose threads. Your resonant TeV leptogenesis produces $\eta_B$ from a CP-odd invariant you assert is ``controlled by geometric phase'' --- but you never compute the phase, you just assert it lands right. And a misalignment dark matter that solves strong CP should source CMB isocurvature; have you actually checked you are not already excluded?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two separate points. On leptogenesis: conceded that the CP-odd invariant is asserted, not computed. The section ``Baryogenesis via the inherited neutrino and CP ledger'' takes the standard resonant formula and argues that \GG{} \emph{narrows} both the CP-odd phase and the Breit--Wigner enhancement into the inherited geometry, but the actual phase value, washout, and transport collapse into the same uncomputed $e^{-K/2}$ throughput flagged in Objection~31; the neutrino-and-CP ledger it leans on is the Majorana/CP block derived in \citet{ArisakaA9SM}, and the resonant phase is its named, unclosed obligation. On isocurvature: this one we have actually checked, and it passes by a wide margin. Because the compact angle is held by the throat through a TeV-corridor (low) inflationary Hubble rate rather than scanned, the section ``Kaxion dark matter'' computes an isocurvature power fraction $\beta_{\mathrm{iso}} \sim 6\times10^{-40}$, some thirty-eight orders of magnitude below the Planck bound. So the isocurvature worry is genuinely answered (a geometry-selected angle, not a randomly displaced one); the leptogenesis phase is genuinely open, and is tagged as such.
\end{objqabody}

\bigskip

\section{The gravitation, black-hole, and primordial-gravitational-wave referee}

\noindent\textbf{Objection 59 (O-Cosmos-59) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your Part `The six-dimensional bulk' opens from a boundary Einstein equation as though four-dimensional general relativity were a free gift. But you never derive it. You assert it and then dress it up by inserting the FRW metric. A constitutional theory is supposed to earn its gravitational law, not borrow it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The charge is half right and the half it gets right is one we state ourselves. \ATen{} does not derive boundary general relativity; it inherits it. The text of `The six-dimensional bulk' is explicit that the paper ``begins from the boundary Einstein equation rather than postulating it,'' and the word ``rather'' is doing real work only because the derivation lives upstream. That derivation is in the companion \citet{ArisakaA5GDOS}, Part IV, theorems T8.1 through T8.4: the boundary Einstein--Hilbert action as the metric zero-mode export of the bulk (T8.1--T8.2), the Newton constant fixed by the scale-overlap integral \(M_{\mathrm{Pl}}^2 = M_6^4\,2\pi R_\theta\int e^{-2A}\dd r\) rather than left free, and stress-energy conservation \(\nabla_\mu T^{\mu\nu}=0\) as the boundary diffeomorphism Ward identity (T8.3). So the right reading is a division of labor, not a gap: A5 proves the law, A10 applies it to the homogeneous-isotropic background. We would only be guilty as charged if we claimed to re-prove GR here; we do not, and the codebook traces the inheritance explicitly.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 60 (O-Cosmos-60) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Even granting A5, the projection you actually use reproduces only the FRW background. You get the two Friedmann equations and the continuity equation by hand-feeding a perfect fluid. That is not the full Einstein equations, not the Bianchi identities, not local conservation off the symmetric background. You are advertising `boundary GR' while delivering one ODE in the scale factor.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded as a statement of scope, and it is the correct scope. \ATen{} operates at the FRW background level because that is all cosmology at large scales requires: homogeneity and isotropy force the maximally symmetric spatial slice, the \(00\) and trace components of the boundary Einstein equation give the two Friedmann relations, and covariant conservation gives the continuity equation. The full local field equations, the Bianchi identities, and the seam-level conservation law are the remit of \citet{ArisakaA5GDOS}, where conservation is a Ward identity (T8.3) holding off any symmetry restriction. A10 should not be read as re-deriving the full local dynamics; it reduces an already-derived local law to the symmetric sector and then does its real work on the source ledger. If a reader wants the local Einstein equations or the Bianchi structure, the citation is A5, not this paper.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 61 (O-Cosmos-61) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your `Page curve as a \GSSB{} saddle switch' in the black-hole Part is theater. You rename the island formula as a `GFF-minimizing saddle,' call the transition a `\GSSB{} event,' and declare victory. There is no replica computation, no island extremization done in your framework, and no quantitative Page time that you compute rather than borrow. Relabeling is not deriving.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and it is conceded corpus-wide, not just here. The \GSSB{} language in `The Page curve as a \GSSB{} saddle switch' names which physical thing is changing at the turnover --- the operationally dominant saddle of the open-system entropy ledger --- but it does not replace the replica/island computation that establishes the generalized-entropy rule in the first place. We say as much: our entropy functional ``is mathematically the same object'' as the island generalized entropy, and the island program tells us \emph{how} to compute the saddle while \GDOS{} tells us \emph{why} a branch-minimized functional had to be the answer. The honest quantitative status is the same as in the companion \citet{ArisakaC3BH}, which computes the \(A/4\) coefficient and thermality (T-BH-1) but treats the island/replica Page-curve as conceptually closed and quantitatively open. Our own Page-time estimate \(M_{\mathrm{Page}}=M_0/\sqrt2\) is a structural consequence of the half-entropy condition, and we flag in the text that ``what matters physically is not the exact numerical prefactor.'' The slogan names the saddle switch; it does not perform the replica calculation, and we do not pretend it does.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 62 (O-Cosmos-62) (the gravitation referee)}\par\nobreak
\noindent\textbf{``The whole black-hole Part leans on the Bekenstein--Hawking thermodynamic fact, yet you defer the actual \(A/4\) entropy coefficient to C3. So A10 proves nothing about the very fact it depends on. You are decorating an imported result.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct on the facts, and again this is a deliberate division of labor rather than a hole. \ATen{} uses the area law as an established thermodynamic input and builds the open-system reading of information return on top of it; it does not claim to prove \(A/4\). The proof is in \citet{ArisakaC3BH}, Theorem T-BH-1, where the horizon entanglement entropy is the induced \(A/4G_N\) and the conical-replica method fixes the coefficient \(1/4 = 4\pi/16\pi\) regulator-independently for the generic throat. That is exactly the right place for it: C3 is the entropy paper, A10 is the cosmology paper. The objection would bite only if A10 claimed the area law as its own result. It does not; it cites it.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 63 (O-Cosmos-63) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your CMB section and your inflation Part both now quote \(r = 0.0043\) as if it were obviously right, but the whole number rests on one e-fold count, \(N^* = 52.6\), which rests on one reheating calculation. One number out of place and the whole edifice falls.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes --- and that strength is the point, not an embarrassment. The chain is short and every link is stated: the \GDOS{}-forced plateau class gives \(r = 12/N^{*2}\) exactly; the Planck-strength width gives \(T_{\mathrm{rh}}\) and hence \(N^* = 52.6\); the prompt lock crossing guarantees no dilution. Move any link and the observables move together --- a diluted window drags \(n_s\) below \(0.955\) as it drags \(r\) up, so the two measurements cross-check each other and the framework cannot retreat to a corner of parameter space it does not have. A theory that can be killed by a single clean measurement is doing exactly what a falsifiable theory should. The fragility is the falsifiability, stated honestly rather than hidden behind a soft bound.\end{objqabody}

\bigskip

\noindent\textbf{Objection 64 (O-Cosmos-64) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your primordial-black-hole discussion is a menu of possibilities with no commitment. You list the channels and never predict a mass function. That is phenomenology theater: unfalsifiable because it predicts nothing specific.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and stated as such in the text. Primordial black holes in \ATen{} are a feature-dependent supplementary channel, not a committed prediction. The reason is structural: the inherited primordial spectrum is red and smooth, so the collapse fraction is exponentially negligible (\(\beta \sim e^{-10^3}\) on the minimal branch), and no committed mass function follows. We do not advertise a PBH mass spectrum, and the Part `Black holes and primordial relics' frames PBHs precisely as a contingent channel that would only switch on if a non-minimal small-scale feature were inserted --- which the minimal ledger does not supply. The honest disposition is: minimal branch predicts essentially no PBHs; any non-minimal branch is a separate, clearly-flagged extension. We claim the former, not a mass function.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 65 (O-Cosmos-65) (the gravitation referee)}\par\nobreak
\noindent\textbf{``You claim to sidestep the singularity theorems by starting the universe at the finite GUT edge of the throat. But Hawking--Penrose forbid exactly that kind of geodesically complete past under standard energy conditions. You have not satisfied or circumvented their hypotheses; you have simply redefined the initial-value problem so the theorems do not apply. That is changing the subject.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You have characterized our move accurately, and we say the same thing in `The beginning of the universe at \(\EGUT\) rather than \(\EPl\)': the paper is ``not claiming to remove singularity theorems by algebraic trick'' but is ``changing the admissible initial-value problem.'' The lawful beginning sits at the finite boundary surface \(\Sigma_{\mathrm{GUT}}^{\mathrm{Bdy}}\) at \(\mu = \EGUT\), with finite \(H_*\) and finite curvature invariants, replacing the formal \(a\to 0\) Planck-singularity datum. Whether this genuinely circumvents the energy-condition hypotheses of the singularity theorems, rather than merely declining to extrapolate past the admissible boundary surface, is a foundational question we flag rather than claim solved. The defensible claim is narrow: backward extrapolation below \(\EGUT\) is not an admissible boundary operation in the \GG{} projection, so the four-dimensional Planck singularity is not the operative initial datum of this effective description. We do not claim to have proven that no completion violates an energy condition; that remains open.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 66 (O-Cosmos-66) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your black-hole entropy uses an induced Newton constant. Is that the same \(G_N\) A5 derives from the scale overlap, or a different one you quietly assumed to be equal? If they are merely assumed equal, the \(A/4\) coefficient inherits an unproven identity.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They coincide by construction, and the identity is now explicit rather than tacit. Both \citet{ArisakaA5GDOS} (through the T8 chain) and \citet{ArisakaC3BH} read \(1/G_N\) off the same canonical A2 scale profile \citep{ArisakaA2GG6}, so the induced Newton constant entering horizon entropy is the scale-derived one, not an independent parameter glued on to match. C3 carries the cross-reference explicitly: the horizon entanglement entropy \emph{is} the induced \(A/4G_N\) with that same \(G_N\). And the coefficient itself is the more robust part of the claim --- the conical replica fixes \(1/4 = 4\pi/16\pi\) from the heat-kernel ratio, which depends on the ratio of overlap integrals, not on the absolute normalization of \(G_N\). So even were the absolute normalization shifted, the \(A/4\) coefficient would not move. The identity you worried might be assumed is the one both papers derive from the single shared scale profile.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 67 (O-Cosmos-67) (the gravitation referee)}\par\nobreak
\noindent\textbf{``Your `resolution' of the black-hole information paradox is that the four-dimensional observer holds only a CPTP partial-trace image of a closed unitary bulk, so pure-to-mixed evolution is `unsurprising.' That proves far too little. Any open system has a CPTP image; you have explained why the observer sees thermality, but you have not shown information actually returns, you have not reconstructed the interior, and you have not ruled out genuine information loss. You have renamed the paradox `openness' and walked away.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair and important limit on what the CPTP framing buys. The partial-trace observation does exactly two things, and we should not be read as claiming more. It dissolves the \emph{ontological} version of the paradox: once the exterior observer correctly holds only \(\rho_\partial = \mathrm{Tr}\,\Pi_t[\rho_6]\), pure-to-mixed behavior in that reduced state is the generic signature of openness, not a contradiction with global unitarity --- because unitarity lives in the closed 6D bulk, not in the boundary image. That is a genuine but modest result: it says the paradox was posed at the wrong level, and it does not by itself return information, reconstruct the interior, or exclude loss. Information \emph{return} is a separate claim, carried not by the CPTP map but by the saddle-switch structure of `The Page curve as a \GSSB{} saddle switch', and that structure inherits the same quantitatively-open status as the island/replica program (Objection 39). So the honest stack is: openness explains thermality and removes the false contradiction (proven, modest); the saddle switch describes how the dominant entropy branch turns over (structural, quantitatively open); full interior reconstruction and a rigorous exclusion of information loss are \emph{not} claimed here. The objection is right that openness alone proves little; we agree, and we do not rest the information-return claim on it.
\end{objqabody}

\bigskip

\section{The quantum-field-theory and string-theory referee}

\noindent\textbf{Objection 68 (O-Cosmos-68) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``You call the cosmological-boundary stratum a holographic projection of a `bulk' and read the radial coordinate as an RG flow, but A10 never writes down a bulk action or a holographic-renormalization scheme. Without a defined action, boundary counterterms, and a renormalized on-shell value, `warped bulk = holographic RG flow' is a slogan, not a calculation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The slogan charge is half right, and the honest split matters. A10 itself does \emph{not} carry a worked holographic renormalization; in the Inheritance Part and throughout, the warped-bulk/holographic-RG reading is imported by citation, with the explicit dictionary footnote stating that the radial coordinate $r = \ln(\EGUT/\mu)$ is the renormalization-group axis rather than a Randall--Sundrum spatial dimension. That is a deliberate inheritance, not an original construction here. What blunts the ``pure metaphor'' version of the charge is that the companion \citet{ArisakaA2GG6} does supply an explicit classical six-dimensional action on $\cM_6 = \cM_4 \times I_r \times S^1_\theta$ with the canonical warped metric $ds_6^2 = e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$, a worked Kaluza--Klein reduction with overlap integrals, and a BRST sector satisfying a homogeneous Slavnov--Taylor identity (its Theorem~7). So there is a bulk action and a quantized gauge sector behind the words. What is genuinely still open --- and fair to flag --- is the last mile: a holographic renormalization with an exhibited counterterm set and a renormalized boundary functional, mapping the bulk on-shell action to the cosmological-boundary observables, is dictionary-by-citation rather than computed. A10 should be read as inheriting a defined bulk and conceding the renormalization scheme as not-yet-worked, not as asserting a holographic duality it has proved.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 69 (O-Cosmos-69) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``$\Omega_m = 1/\pi$ cannot be a topological invariant. Topological invariants of a four-manifold --- Pontryagin numbers, Euler characteristics, instanton numbers --- are integers. $1/\pi$ is irrational. Calling it a `topological invariant' is a category error.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The terminology is conceded, and the codebook already carries the concession. The objection is correct that a characteristic number is integer-valued, and that $1/\pi$ cannot be one. What the framework actually claims, once the overreaching word is stripped, is narrower: $\Omega_m = 1/\pi$ is a \emph{residual Pontryagin normalization} --- the integer matter-cell weight $1/15$ from the \OGNfive{} ledger, divided by the $\pi$-carrying normalization factor that the Chern--Pontryagin integral of $\operatorname{tr}R^2$ contributes ($\int \operatorname{tr}R^2 = 8\pi^2 p_1$). It is the integer that is the invariant; the $\pi$ enters through the normalization of the de Rham representative, not as a characteristic number. Accordingly the codebook classifies all three $\pi$-locked fractions to Derived-conditional (D$^*$) under Conjecture~VIII, states $\Omega_m = 1/\pi$ as a \emph{conjectured} identity rather than a proved invariant, and the codebook carries the honest tag. The headline theorem (T-Cosmos-17) is titled ``$\Omega_m = 1/\pi$ as a topological invariant,'' and that title overreaches relative to the body; the Proof-status note and the codebook are the governing text, and they do not call it an invariant. The referee is right that the word was wrong; the substance the word was reaching for is the residual-normalization reading, which is what is now carried.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 70 (O-Cosmos-70) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``Theorem T-Cosmos-17 does not close. Step~4 multiplies and divides \OGNfive{} integers until $1/15$ cancels and $1/\pi$ falls out, but the cancellation needs a normalization you never exhibit. As printed it is algebra arranged to land on a predetermined answer. A theorem whose key step is circular is not a theorem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is aimed at a version of the honesty note this paper no longer carries, and the current answer is better than a concession. Step~4's cancellation does need a normalization, and \S\ref{par:where-the-conditional-lives} now says precisely what that normalization is and what became of it: lemma VIII-b$'$ is proved outright, because it compares \emph{counts} and a ratio of counts is dimensionless at every cutoff; lemma VIII-b$''$ compares a length to a volume, Eq.~\eqref{eq:fibre-ratio-radius}, and therefore needs a physical unit; and the unit bridge terminates on the dilution no-go, which \emph{proves} that VIII-b$''$ cannot close as a statement about a constant. What it closes into is the occupancy identity $u = \pi\Omm = 1$, true at exactly one epoch. So the theorem is not algebra arranged to land on a predetermined answer: the coupled $\pi$-form is forced by the upstream fiber assignment, and no measured density enters the selection at any step. What is not closed is narrower and better posed --- that the epoch at which the running inventory reads the form is the present one --- and $\Omm$, $\OmL$, $\Omb$ are D$^*$ for that reason and no other. Two further points of honesty are worth recording. First, a false pointer --- which sent the reader to ``the companion derivation of \AFour{}'' for the integration --- has been removed; \AFour{} supplies the \OGNfive{} integers, but contains no closed-packet integration, and that stale pointer was deleted. Second, the theorem is stated as a conditional --- \emph{if} the cost-minimum integration exists with the exhibited normalization, \emph{then} $\Omm = 1/\pi$ --- so it is honest about being a structural ansatz with an open antecedent rather than a closed deduction. The referee's diagnosis and the paper's self-assessment coincide.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 71 (O-Cosmos-71) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``The `closed-packet four-form integration' is undefined. You never specify the 4-manifold $\cM_4$, never name a characteristic class, never define the measure or the integration domain. The operation appears in A10 and nowhere else in the cohort. An undefined operation cannot ground a derivation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and this is precisely the content of the open Conjecture~VIII. The closed-packet integration is \emph{named} --- the integral of the \BIsix{} four-form $X_4$ over the cosmological-boundary slice $\cM_4$, decomposed into \OGNfive{} cells --- but it is not yet \emph{defined} with the rigor a referee is entitled to demand: $\cM_4$ is characterized as the four-dimensional boundary slice from the \OGGEP{} projection rather than exhibited as a specific manifold with a specified topology, and the cell-weight assignment is asserted from the \OGNfive{} ledger rather than constructed from a defined characteristic class. The referee is right that this is why the operation does not yet do rigorous work. It is also exactly why the three $\pi$-locked fractions are D$^*$ and not D: the codebook does not pretend the integration is defined. The honest statement is that A10 introduces a named conjectural operation, states what it would have to evaluate to, and defers the definition --- the 4-manifold, the class, the measure --- to future work. The referee should read the closed-packet integration as the open research target it is labeled to be, not as machinery the paper claims to possess.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 72 (O-Cosmos-72) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``The \OGNfive{} ledger $(M,N,L;J,K) = (1,3,5;7,35)$ is posited. You write it down and use it everywhere, but a tuple of integers asserted by hand is not a derivation, and the cosmic numbers inherit that arbitrariness.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} On this point the inheritance is genuinely sound, so the objection misfires --- though it is fair to be precise about what is proved and what is not. The \OGNfive{} ledger is not posited in A10; it is \emph{proved} to theorem status in the cohort. \citet{ArisakaA2GG6} derives it (its Theorem~6) as a five-step Diophantine cascade closing through the dual-lock identity $K = LJ = M^2 + N^2 + L^2$, and \citet{ArisakaA4Euclid} promotes the \OGNfive{} integer cascade to theorem status as part of the zero-free-parameters discipline. A10 inherits these integers; it does not invent them. What A10 itself does \emph{not} close is the next step --- the closed-packet integration that would turn the integer ledger $(1,3,5;7,35)$ into the irrational fraction $1/\pi$. So the honest division of labor is: the integers are theorems upstream; the map from integers to cosmic fractions is the open Conjecture~VIII downstream. The referee is right to demand that the ledger be derived rather than asserted, and it is; the residual openness is in the integration, not in the ledger.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 73 (O-Cosmos-73) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``The $\pi$ is convention-dependent. Powers of $\pi$ float around with your choice of volume normalizations --- $\operatorname{Vol}(S^3) = 2\pi^2$, factors of $2\pi$ in measures, $8\pi^2$ in instanton normalizations. And your `uniqueness against competing rational functions of $\pi$' rules out $1/(2\pi)$ and $1/\pi^2$ by appeal to the data. That smuggles empirical preference into what you call a topological proof.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Half of this is fair and half is aimed at a superseded reading, and the two halves should be separated. \emph{On uniqueness}, the referee is quoting a concession this paper has withdrawn, because the competitors are now excluded \emph{geometrically}, with no appeal to magnitude: $1/(3\pi)$ would require a holonomy volume $6\pi^2$ that no manifold in the construction carries; $1/\pi^2$ would require an $S^5$ that the four-dimensional boundary slice does not have; and $1/(2\pi)$ is not a competitor at all --- it is the \emph{baryonic} projection, one unit of $U(1)_B$ charge per revolution of the fiber, which is a different quantity from the matter-to-total ratio and appears in the inventory in its own right. None of those three exclusions mentions a measured density. \emph{On the convention-dependence:} the framework's defense is that the closed-packet integration is supposed to fix the normalization \emph{once}, so that the relative powers of $\pi$ across the three fractions ($\pi^{-1}$ for matter, $1 - \pi^{-1}$ for vacuum, $(2\pi^2)^{-1}$ for baryons) are forced as a coupled set rather than chosen per-entry. That is a real structural constraint --- a post-hoc fit tunes one normalization, not a coupled triple. The residual conditionality is therefore not the normalization and not the uniqueness: it is the identification of the crossing epoch with the present one, and that is what the codebook's D$^*$ records (\S\ref{par:where-the-conditional-lives}).
\end{objqabody}

\bigskip

\noindent\textbf{Objection 74 (O-Cosmos-74) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``$M_{\rm lock} \approx 3.85$~TeV is reverse-engineered. You take $\tfrac12 M_{\rm Pl} e^{-35}$, land near a few TeV, and call it a compactification scale. That is dimensional analysis run backwards from a desired answer, not a derived KK threshold.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The phenomenology of $M_{\rm lock}$ is not A10's remit, and the reverse-engineering charge is best answered where the work is done --- the companion \citet{ArisakaP1KK}. A10 inherits $M_{\rm lock} \approx 3.85$~TeV only as the top of the radial chronology, the inherited bulk modulus scale of the descent's second saddle --- the prompt lock crossing where the dark sector is born, not the inflationary plateau, which sits one e-fold below the GUT edge (Part~IV); it does not itself derive the lock spectroscopy. \citet{ArisakaP1KK} carries the structure: $M_{\rm lock} = \tfrac12 M_{\rm Pl} e^{-35}$ is fixed by the \OGNfive{} integer $K = 35$ (a theorem-level integer, not a scanned parameter), the factor of $\tfrac12$ is the scale half-wave of the radial standing mode rather than a fudge, and the same $M_{\rm lock}$ over-constrains the electroweak vacuum value $v = M_{\rm lock}(2/5)^3 \approx 246$~GeV --- an over-determination that a reverse-engineered single number could not produce. That over-constraint is the structural answer to ``dimensional analysis run backwards'': one cannot reverse-engineer a single scale to hit two independent targets ($\sim 3.85$~TeV at the LHC \emph{and} $246$~GeV at electroweak breaking) unless the relation is forced. The honest qualification is that the integer $K = 35$ is the input, and whether $K = 35$ itself is forced rather than chosen is the \OGNfive{} ledger question answered upstream in \citet{ArisakaA2GG6}; the falsifiability is sharp, since \citet{ArisakaP1KK} stakes the implementation on a clustered spin-2 signal within $3.85 \pm 0.10$~TeV.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 75 (O-Cosmos-75) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``Anomaly freedom and the four-form coefficient vector $(1, -1, -2/5, -2/5)$ are imported, not checked. You quote the modified Bianchi identity and the $-2/5$ Abelian fractions as if they were established, but A10 performs no anomaly computation. How do I know the bulk on which your whole cosmology rests is even consistent?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Here the inheritance is genuinely solid, and A10 correctly imports rather than re-derives. The \BIsix{} coefficient vector $(1, -1, -2/5, -2/5)$ and its anomaly freedom are \emph{derived} in \citet{ArisakaA3BI6}, not asserted in A10. The $-2/5$ Abelian fractions follow from an explicit trace computation --- the Georgi--Glashow embedding gives the trace ratio that pins both Abelian coefficients symmetrically to $-2/5$ via the trace-ratio lock $|a/c| = |a/d| = 5/2$ (equivalently $(L - N)/L = 2/5$ from the \OGNfive{} integers), so no coefficient is free to adjust. \citet{ArisakaA3BI6} proves both the local anomaly-cancellation theorem and the global anomaly-freedom theorem, with the single-tensor Green--Schwarz factorization $I_8 = \tfrac12 X_4 \wedge X_4$ supplying the same $X_4$ four-form that A10 then integrates. A10's role is to inherit this consistent, anomaly-free bulk as the parent law of the cosmological boundary, and to use the \BIsix{} four-form construction as the object whose boundary integral is conjectured to yield $1/\pi$. The referee's demand --- show me the bulk is consistent --- is answered, but in \citet{ArisakaA3BI6}; A10 is right not to repeat the anomaly computation, and right to cite it as the upstream theorem rather than claim it as its own.
\end{objqabody}

\bigskip

\noindent\textbf{Objection 76 (O-Cosmos-76) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``Naturalness is never addressed. A TeV-scale lock and a meV-scale vacuum energy both sit far below the cutoff, and you say nothing about whether they are radiatively stable. Without a stability analysis your `derived' scales are exposed to the same fine-tuning problems as everything else --- worse, because you claim they are fixed.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and flagged open. A radiative-stability analysis of $M_{\rm lock}$ and of the meV-scale vacuum residual is not in the present scope of A10, and the paper does not claim one. This is a real gap, and an honest one to state plainly: showing that $M_{\rm lock} \approx 3.85$~TeV is protected against large radiative corrections, and that the $\OmL = 1 - 1/\pi$ vacuum fraction is radiatively stable rather than merely posited at the right value, would require a renormalization-group and effective-potential treatment that the framework has not carried out. Two mitigating remarks, offered without overclaiming. First, the framework's stance is that the scales are fixed by integer ledger data ($K = 35$ for the lock, the closed-packet topology for the vacuum fraction) rather than by a balance of continuous parameters, so the framework's \emph{hope} is that there is no continuous dial to detune --- but a hope is not a stability proof, and the radiative corrections to a fixed structural value still need to be shown to be controlled. Second, the absence of this analysis is recorded as an open item rather than papered over. The referee is right that a derivation claiming fixed scales owes a stability argument, and the honest status is that this argument is owed and not yet paid.
\end{objqabody}

\bigskip

\section{The methodology, honesty, and numerology referee}

\noindent\textbf{Objection 77 (O-Cosmos-77) (the methodology referee)}\par\nobreak
\noindent\textbf{``The whole program is numerology. You pick pretty numbers --- $1/\pi$, $1/(2\pi^2)$, $e^7$ --- and then build topological language around them to rationalize the choice.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The discipline against numerology here is structural, not rhetorical. First, every cosmological observable is triaged in the Cosmological codebook by a published provenance code --- Derived (D), Derived-conditional (D\(^*\)), or Asserted (A) --- so that a reader can see at a glance which numbers are claimed as outputs and which are still owed; twenty-one entries, two D, eighteen D\(^*\), one A, with no entry quietly upgraded. Second, the conditional entries are not vague promises but eighteen closures resting on ten \emph{named, falsifiable} conjectures (closed-packet-integration, vacuum-residual-exactness, Kaxion-misalignment-completeness, Hubble-two-stage-corridor-uniqueness, and the rest), each of which can be disproved by a counterexample. Third, the Post-hoc-fit defense section spells out why one fixed topology forces three coupled fractions at once: a curve-fit explains one number, but the same closed-packet integration that yields \(\Omm = 1/\pi\) yields \(\OmL = 1 - 1/\pi\) and \(\Omb = 1/(2\pi^2)\) with no further free step, and the \(2\pi^2 = \mathrm{Vol}(S^3)\) factor in the baryon fraction is the wrong combination of powers to be a coincidence. Fourth, and decisively, the same six-dimensional geometry fixes both the particle inventory of \citep{ArisakaA9SM} and the cosmological inventory of this paper with zero continuous dials: a structurally locked derivation cannot be retrofitted, so a single wrong number kills it. That is the opposite of an adjustable fit. The honest concession is only that several of the ten conjectures remain unproved --- which is why those rows read D\(^*\), not D.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 78 (O-Cosmos-78) (the methodology referee)}\par\nobreak
\noindent\textbf{``Every empirical comparison in the paper is central-value-to-central-value. There is no joint likelihood, no covariance, no real statistical confrontation with the data.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and stated openly in the text. The present claims are at the level of central values and summary statistics: each codebook entry is compared to a published central value with its quoted error, and the parameter-audit Dev plot displays fractional deviations with empirical and (where applicable) theory error bars, but no full Boltzmann-plus-likelihood pipeline has been run. A global CMB-plus-BAO-plus-distance-ladder fit is named explicitly as required future work --- it is attached as an open requirement to the Hubble two-stage corridor and is the dedicated remit of the companion \citep{ArisakaC2Hubble}. Until that pipeline exists, the right reading of, say, the \(0.27\sigma\) agreement on \(\Omdm\) or the \(2.3\sigma\) tension on \(\Omb\) is as a per-parameter audit against marginal posteriors, not as a likelihood ratio for the model as a whole. We do not claim the latter and have flagged the gap rather than papered over it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 79 (O-Cosmos-79) (the methodology referee)}\par\nobreak
\noindent\textbf{```Derived (conditional)' is just `not derived' with better marketing.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The D\(^*\) code has a precise and auditable meaning: the entry follows from a \emph{single, explicitly named, falsifiable conjecture}, and that conjecture is printed --- not hidden in prose, but listed by name in the codebook row and in the conjecture ledger. The reader can see exactly what is owed: \(w = -1\) is D\(^*\) on the vacuum-residual-exactness conjecture; \(\Omdm\) is D\(^*\) on Kaxion-misalignment-completeness; and so on. This is strictly more honest than an unlabeled assertion, because it converts ``we believe this is derivable'' into ``this is derivable \emph{if and only if} the following stated proposition holds, which you may now attack.'' A genuinely empty D\(^*\) would be one where the conjecture is unfalsifiable or unstated; ours are neither. The label is a debt instrument with the amount written on it, not a euphemism for an unpaid bill of unknown size.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 80 (O-Cosmos-80) (the methodology referee)}\par\nobreak
\noindent\textbf{``You insist no number was tuned after the fact, yet the Dev plot is full of entries that sit exactly at zero deviation. That is tuning by construction.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The Dev plot is built precisely to make that distinction visible rather than to obscure it. Filled markers are independent predictions --- numbers computed without reference to the measured value (the atomic-chain \(z_{\mathrm{dec}}\), \(w = -1\), \(\ell_D = K(2\pi)^2\), the \(\pi\)-locked fractions, \(\eta_B\)) --- and these carry genuine, nonzero deviations: \(\Omb\) at \(+2.76\%\), \(z_{\mathrm{dec}}\) at \(+0.11\%\) (the switch \(e^7\), a distinct topological marker, annotated \(0.62\%\) above the measured value), \(\ell_D\) and \(w\) near \(+2.7\%\). Open markers are entries anchored by construction: a codebook row that, pending its named lemma, still echoes the empirical input (the CMB Hubble branch, the early \(S_8\), the scalar tilt and amplitude, \(N_{\mathrm{eff}}\)). Those \emph{necessarily} sit at zero deviation, and the caption says so in as many words. Far from hiding the trivial zeros, the plot color-codes and shape-codes them so the reader never mistakes an anchor for a hit. The honesty of the figure is exactly that it refuses to count the open markers as successes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 81 (O-Cosmos-81) (the methodology referee)}\par\nobreak
\noindent\textbf{``The celebrated `0.6\%' success of $z_* = e^7$ uses a generous denominator. Measured against the actual Planck error, that offset is a large tension, and you quietly chose the kinder yardstick.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The premise is retired: the paper no longer scores \(e^7\) against the recombination redshift under \emph{any} denominator, generous or harsh. The two-quantity resolution (\S\ref{sec:two-redshifts}) separates what that comparison conflated. \(z_{\mathrm{sw}} = e^7\) is the topological corridor switch --- an integer-ledger marker, not a prediction of the measured decoupling redshift --- so neither the visibility width nor the Planck error is its yardstick; reaching for the kinder of the two would indeed be comparator-shopping, which is exactly why the response to Objection~23 (the ``\(27\sigma\)'' objection) declines that move. The quantity that \emph{is} scored against Planck is the framework's own atomic-chain decoupling prediction, \(z_{\mathrm{dec}}^{\mathrm{GG}} = 1091.1\) (raw), which lands on \(z_{\mathrm{dec}} = 1089.92 \pm 0.25\) at \(0.11\%\) with zero fitted parameters --- inside the \(0.18\%\) modeling systematic of the recombination calculation itself, which is that comparison's honest yardstick. The switch--decoupling offset is a computed number, \(\delta = 0.0041\) (Eq.~\eqref{eq:delta-offset}), not an error bar to be gamed. The both-framings presentation your objection quotes survives only in the thermal appendix, explicitly marked there as the pedagogical bridge from the superseded one-quantity reading to the present form. What remains of the objection is a comparison the paper no longer makes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 82 (O-Cosmos-82) (the methodology referee)}\par\nobreak
\noindent\textbf{``The eight `falsification tests' are theater. Several lie beyond any foreseeable instrument, and the strong modules are propped up next to weak ones to borrow their credibility.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The asymmetry between strong and weak modules is real and is stated, not smoothed over. The strongest single kill-switch is now \emph{two-sided} and within reach, which strengthens the objection's own premise against it: the derived plateau fixes \(r = 12/N^{*2} = 0.0043\), a factor eight below the current bound and several times above the Simons-Observatory/LiteBIRD floor, so a measurement inconsistent with it in \emph{either} direction --- a detection well above, or an exclusion well below --- refutes the derived launch. This replaces the retired one-sided \(r \ll 10^{-3}\) reading, which was indeed beyond reach; the new target is decisive this decade. The near-term, genuinely exposable tests are different in character: a high-precision independent recombination-redshift determination, the two-stage BAO residue at \(z \sim 1\), and the GW-siren Hubble determination, predicted to converge on the background at \(H_0^{\mathrm{GW}} \approx 67\). These confront the derivation on the next-decade horizon: a siren tracking the local ladder (\(\approx73\)) rather than the background would refute the \(\pi\)-inventory value. We do not claim every module is equally weak-point-free; the baryogenesis entry is order-of-magnitude only (a factor-\(\sim 3\) theory band), and the local-ladder excess above the background is only partly derived --- the Local-Hole lift accounts for \(+2\text{--}3\%\) of it, the residual being read as a distance-ladder calibration systematic. Listing eight tests of uneven decisiveness, with the weak ones labeled weak, is the opposite of theater: it is an inventory of where the program is most and least exposed.
\end{objqabody}

\bigskip

\section{The cohort-coherence and philosophy-of-cosmology referee}

\noindent\textbf{Objection 83 (O-Cosmos-83) (the cohort-coherence referee)}\par\nobreak
\noindent\textbf{``The cohort cross-cites in circles. A10 leans on C2 for the Hubble resolution, and C2 leans straight back on A10. The agreement is manufactured, not independent.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The loop is real and we acknowledge it rather than disguise it; the defense is a clean division of labor, stated in the Cross-paper coherence map. A10 owns the \emph{structural corridor}: the named Hubble-two-stage-corridor theorem that accommodates the CMB and late-time branches as two projections of one cosmological-boundary Friedmann structure, with the switch event at recombination. The companion \citep{ArisakaC2Hubble} owns the \emph{dedicated phenomenology and the eventual joint fit}: the local-void mechanism, the late-branch anchoring, and the global likelihood that would turn the reframing into a quantitative resolution. The two papers quote one corridor under two anchor conventions --- \(73.7 \pm 0.7\) off A10's \(67.4\), \(73.5 \pm 0.7\) off C2's \(67.24\) --- synchronized and labeled as such. The mutual citation is therefore a pointer to who carries which half of the argument, not a manufactured confirmation: neither paper claims the corridor is \emph{resolved}, and both flag the joint fit as still required.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 84 (O-Cosmos-84) (the cohort-coherence referee)}\par\nobreak
\noindent\textbf{``A10 keeps deferring the hard parts to companion papers --- the BBN lithium network, the Kaxion mass cap --- and at least one of those companions, GG-C4-Nucleo, does not even exist yet.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and labeled as such throughout. The lithium-7 network closure is explicitly future work: the BBN audit proposes a selective \(A = 7\) open-system port and names GG-C4-Nucleo as the forthcoming companion that will carry the quantitative network calculation --- it is announced as not-yet-written, never cited as though it were done. The Kaxion mass cap is the cleaner case: the compact-angle-mass-uniqueness conjecture was an open D\(^*\) in this paper, and the closure has \emph{already} been executed in the existing companion \citep{ArisakaC1Kaxion}, contracting the open remainder to the integer \(\Xi_{\mathrm{cap}} \in [35,47]\) and delivering the \(8.1 \pm 0.5\,\mu\mathrm{eV}\) line. So the deferrals are of two kinds, both honestly marked: those discharged by a companion that exists, and those owed to a companion that is named as forthcoming. What the paper does not do is count an unwritten companion's result as a present accomplishment.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 85 (O-Cosmos-85) (the philosophy-of-cosmology referee)}\par\nobreak
\noindent\textbf{``The `universal-cosmic-identity, layer-by-layer from electrons to ATP to cells to cosmos' framing is not physics. It is a grand narrative that weakens, rather than strengthens, the formal case.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded as a matter of status, and that is exactly why it is fenced. The universal-cosmic-identity framing is an interpretive synthesis, presented as a preview and explicitly reserved as downstream of the calculations --- it lives in future work and in the capstone \citep{ArisakaA11Origin}, not in the codebook or the headline theorems. The cosmological results of this paper --- the \(\pi\)-locked inventory, \(z_* = e^7\), the corridor, the Kaxion sector --- stand or fall entirely on their own derivations and falsification tests, with no load borne by the layer-by-layer narrative. We keep the synthesis because we think it points somewhere, but we draw the line so a reader who rejects it loses nothing of the physics. The narrative is offered as interpretation labeled as interpretation, which is the only honest way to offer it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 86 (O-Cosmos-86) (the philosophy-of-cosmology referee)}\par\nobreak
\noindent\textbf{``Your contrast with the inflationary multiverse and anthropic selection is just rhetoric --- a way of scoring points against a rival picture rather than a substantive argument.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The contrast is methodological, and it cashes out in falsifiability rather than in tone. The substantive claim is structural: a zero-dial geometry, fixed once with no continuous knobs and no ensemble to choose from, is \emph{obliged} to produce exactly one definite value per observable, and is therefore exposed to refutation by any single wrong number --- it derives \(\Omm = 1/\pi\) rather than measuring it. An anthropic ensemble runs the other way: it lets the constants vary across an unobservable population and selects the subset compatible with observers, which accommodates a wide range of values and so predicts none of them sharply. That is not a rhetorical preference; it is the difference between a framework that one wrong number can kill and a framework constructed so that almost no measurement can. \(\Lambda\)CDM, for its part, neither derives nor selects --- it measures its six parameters. The three-way contrast is about closure mode and falsification profile, which are the properties a referee should care about.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 87 (O-Cosmos-87) (the philosophy-of-cosmology referee)}\par\nobreak
\noindent\textbf{``Leaving `the final metaphysical step to the reader' is evasive. If the framework implies a necessity, say so; if it does not, stop gesturing at one.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The restraint is deliberate, and it is the honest position, not an evasive one. The paper draws the derived/interpreted line explicitly and repeatedly: the codebook marks what is an output (D), what is conditional (D\(^*\)), and what is asserted (A); the synthesis sections mark themselves as interpretation. Where the framework genuinely implies a value, it asserts it and exposes it to falsification --- it does not hedge \(\Omm = 1/\pi\) or \(z_* = e^7\). What it declines to assert is \emph{metaphysical necessity} --- the claim that the universe \emph{could not have been otherwise} --- because that is a step the calculations do not establish, and asserting it anyway would be the actual overreach. Declining to claim what one cannot prove, while clearly marking the boundary, is intellectual restraint. The alternative the objection seems to want --- a confident metaphysical pronouncement --- is precisely what an honest paper should refuse.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 88 (O-Cosmos-88) (the cohort-coherence referee)}\par\nobreak
\noindent\textbf{``The whole inheritance chain rests on companions that are themselves unrefereed preprints. A theorem chain is only as sound as its weakest link, and every link is provisional.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} True, and it is the correct way to read the cohort: the chain inherited from \citep{ArisakaA1Constitution}, \citep{ArisakaA4Euclid}, \citep{ArisakaA8QM}, \citep{ArisakaA9SM}, and the capstone \citep{ArisakaA11Origin} is a chain of preprints, and the present paper's conditional results are conditional on them in the strict logical sense. We make no claim that the upstream results carry refereed authority they do not yet have. Two things make this honest rather than fatal. First, the dependence is explicit: each cosmological entry carries its upstream citation chain (e.g.\ the \OGNfive{} ledger to \citep{ArisakaA4Euclid}, the baryonic protection to \citep{ArisakaA9SM}), so a reader who doubts a link can trace exactly which downstream results fall with it. Second, the conditionality is structural, not hidden: this is what the D\(^*\) code and the named conjectures encode --- ``true if the stated upstream proposition holds.'' A theorem chain of preprints that labels every dependency and never asserts more than its links support is doing the most a preprint cohort can honestly do. The remedy is the ordinary one --- the links get refereed, or they fail and the dependent rows fail with them, exactly as the provenance triage already advertises.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 89 (O-Cosmos-89) (the philosophy-of-cosmology referee)}\par\nobreak
\noindent\textbf{``Your `second horizon problem' is not new. Zeldovich and Kibble posed the random-vacuum-choice problem in the 1970s; Linde listed the uniqueness of the universe among inflation's targets and even asked why all properties of the universe are uniform; and axion cosmology has always known the misalignment angle is uniform across our patch because inflation stretched one patch. You have renamed the domain problem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The pedigree is real, and the paper cites it in exactly those terms (\S\ref{subsec:shp-pedigree}) --- Zeldovich--Kobzarev--Okun and Kibble for the sharpest early fragment, Linde's ``uniqueness of the universe'' and his ``all properties'' remark as the nearest miss in ninety years, the axion literature for the pinning mechanism stated almost verbatim. What is claimed as new is deliberately narrow, and each piece can be checked against the quoted sources. First, the naming and the ranking: the domain problem was posed as a relic nuisance and solved by dilution; nobody ranked the sameness of things as the \emph{harder} half of the 1981 question, parallel to and deeper than temperature. Second, the extension: the 1970s fragment concerns the orientation of one degenerate vacuum inside one fixed theory; the second horizon problem asks about the whole inventory --- masses, couplings, abundances, and the angle at once. Third, the temporal answer: the field's answer has always been spatial (we live deep inside one stretched domain); the ordering --- inflation preceded the existence of matter, so identity was inherited rather than negotiated --- has a universally known premise and, as far as we can find, a nowhere-drawn conclusion. Fourth, the reduction: cosmic identity and vacuum uniqueness are shown to be one puzzle, not two. Renaming the domain problem would be claiming the question; what this paper claims is the shape of the question --- and the half of the answer that is temporal rather than spatial. A referee who finds any of the four steps in print is invited to send the reference; the pedigree subsection was written to make that check easy.
\end{objqabody}

\bigskip


\noindent\textbf{Objection 90 (O-Cosmos-90) (the formal-theory referee)}\par\nobreak
\noindent\textbf{``You keep contrasting your unique vacuum with the string landscape, yet you never say the landscape is refuted. If the second horizon problem is as serious as you claim, why not draw the obvious conclusion and declare the \(10^{500}\)-vacua picture dead?''}\par\nobreak\smallskip
\phantomsection\label{obj:landscape-priced}
\begin{objqabody}
\emph{Response.} Because refutation is the wrong verdict, and the accurate one is stronger. Two objects travel under one name: string theory as a candidate quantum-gravity \emph{formalism} --- worldsheet finiteness, the dualities, the AdS/CFT correspondence --- and the \emph{landscape} as a vacuum-selection program. Nothing here bears on the formalism, whose Green--Schwarz anomaly mechanism this cohort in fact adapts to six dimensions as \BIsix{} (\AThree{}); the remarks concern only the second.

On the second: the observed identity of the low-energy constants across causally disconnected regions of our own past light cone is a datum --- the second horizon problem (\S\ref{sec:second-horizon}). A framework with a \emph{unique} vacuum explains that datum as necessity, the same constants everywhere because there is nowhere else for them to be, and is thereby \emph{exposed}: a single confirmed spatial variation of a dimensionless constant such as \(\alpha\) or \(m_e/m_p\) would falsify it outright. A framework with of order \(10^{500}\) vacua and no dynamical selection principle \citep{Douglas2003} cannot supply that explanation as necessity, because the vacua are vacua of the theory meant to describe \emph{our} universe --- so within a single horizon volume the agreement of the constants is a physical question, not a metaphysical one about other worlds. The available answer is that our whole observable patch descends from one Coleman--De~Luccia nucleation event \citep{ColemanDeLuccia1980} in an eternally inflating background: internally consistent, since tunneling between high-barrier vacua is exponentially suppressed and domain walls inside our horizon are generically negligible, and therefore \emph{not contradicted} by the datum.

The issue is not contradiction but cost. That repair commits the program to eternal inflation, and with it to the measure problem, in which the relative frequencies of observations are undefined without an additional, non-derived regulator \citep{Freivogel2011}; predictivity for any further quantity is then not recovered but deferred to that choice. The one standing quantitative success of the anthropic reading, Weinberg's bound on the cosmological constant \citep{Weinberg1987}, remains a single post-diction. In the language of research programs \citep{Lakatos1978} the picture is best described not as refuted but as \emph{non-progressive}: each datum accommodated by adjustment, none forecast. \GGTheory{} takes the opposite structural wager --- one vacuum, pinned upstream by the \OGNfive{} ledger, obliged to return one value for each constant and holding no freedom to absorb a discrepant measurement. We record the difference as one of closure mode --- obligation versus accommodation, one measurement decides versus unfalsifiable by construction --- and leave its weight to the reader.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 91 (O-Cosmos-91) (the model-building referee)}\par\nobreak
\noindent\textbf{``Your leptogenesis assumes exactly what every model-builder assumes: singlet Majorana neutrinos and a type-I seesaw. Dressing the standard choice in geometric language is not a derivation.''}\par\nobreak
\begin{objqabody}
\emph{Response.} It was an assumption until Proposition~\ref{prop:dial-free-door} (T-Cosmos-18); it is now fiber arithmetic. A Majorana self-pairing exists iff the fiber weight vanishes --- on every charged sector the Majorana bilinear carries weight $2n$ and is forbidden, so lepton number is exact throughout the dial-carrying ledger and the sole $\Delta L = 2$ door is the one field the circle does not move. The $LHN$ portal is likewise the unique fiber-neutral trilinear touching that door, and the heavy scale is not chosen either: the throat integer $K = 35$ pins the triplet to $\Mlock \simeq 3.85$~TeV with $O(1/K)$ splittings. What remains genuinely open is stated in the module: the detailed values of the splittings $\delta_i$ within their geometric window. The singlet's real, dial-free character is developed as entry R10 of the real-bulk series of \citealp{ArisakaA8QM}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 92 (O-Cosmos-92) (the curved-space referee)}\par\nobreak
\noindent\textbf{``There is no preferred vacuum in an expanding universe. Your particle inventories --- the $N_i$ abundances, the thermal histories, the relic numbers --- are convention-dependent quantities dressed as predictions.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The ambiguity is real and has a fixed address (\S\ref{subsec:observer-vacuum}). On the canonical bulk geometry the constitutional time is globally static and the bulk-level positive-frequency structure is unique; observer-adapted vacua are \emph{dictionary} data --- which modes an instrument counts is boundary information, part of the \OGGEP{} package by definition --- and a change of dictionary is a Gaussian channel acting on the boundary description, not an ambiguity in the bulk state. Every inventory quoted in this paper is computed in the comoving dictionary, consistently, and the thermal content attributed to horizons is produced by the \GDOS{} partial trace itself, at the standard temperature. What remains conjectural --- the factorization for strongly gravitating bulk configurations --- is named in the real-bulk treatment of \citealp{ArisakaA8QM} rather than waved through here.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 93 (O-Cosmos-93) (the CP-provenance referee)}\par\nobreak
\noindent\textbf{``Your baryon asymmetry rests on a CP phase you cannot derive; `geometric phase' is a euphemism for a fitted parameter.''}\par\nobreak
\begin{objqabody}
\emph{Response.} The CP-odd inputs of this module are inherited, real, lattice-locked angles, derived upstream and quoted here in completed tense: the quark phase $\delta_{\rm CKM} = 2\pi/5$ at first order (two pentagonal steps on the single compact circle, with no second flavor torus on which a separate angle could float), and the leptonic set $\delta_{\rm PMNS} = \pi + 2\arcsin(1/7)$, $\alpha_{31} = \pi$ \citep{ArisakaA9SM}. Their rephasing invariance is holonomy invariance: in the real presentation a physical phase is a closed-loop angle of the coupling texture --- the flavor analogue of an Aharonov--Bohm angle --- which no per-family rephasing can remove. At the resonance the Yukawa magnitude cancels and $\epsilon^{\rm res} \simeq \tfrac12 \sin(2\phi_{\rm geo})$ is controlled by these angles alone. The falsifier is correspondingly sharp: a measured drift of the flavor phases away from the lattice values retires the account --- there is no continuous dial here to absorb one.
\end{objqabody}

\bigskip


\section{The curious reader returns: the multiverse and the three horizon problems}
\label{sec:phil-objections-return}

The referee sections above interrogated the machinery. The four
questions below return to where this Part began --- the multiverse ---
because the reader who has now met the second and third horizon
problems (\S\ref{sec:second-horizon}, \S\ref{sec:third-horizon})
owns a sharper instrument than the one Objections~1 and~2 could
assume, and deserves to see what it does and does not cut.

\bigskip

\noindent\textbf{Objection~94 (O-Cosmos-94) (the curious reader, returning)}\par\nobreak
\noindent\textbf{``You derive three horizon problems --- the same temperature, the same electrons, the same choice of matter over antimatter, across regions that never exchanged a signal. If everything is supposed to vary from universe to universe, doesn't that observed sameness falsify the multiverse outright?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is tempting to say yes, and the temptation should be
resisted, because the truth is sharper. The landscape does not
contradict the three samenesses; it accommodates them, and it does so
using the same mechanism this paper uses: the choice of vacuum and of
CP sign is made once, early, and inflation then stretches that single
decided patch into everything anyone can see
(\S\ref{sec:second-horizon}, \S\ref{sec:third-horizon}). The variety
the multiverse trades in is stored outside the horizon, exactly where
no measurement reaches --- so the observed sameness cannot execute the
hypothesis, because nothing observable could. What the three problems
\emph{do} kill is every mechanism in which the choosing happens late,
after the regions have fallen out of contact: such a sky would carry
domain boundaries and matter--antimatter annihilation seams, and it
carries none \citep{CohenDeRujulaGlashow1998}. That fence forces every
framework, string theory included, into the early-choice corner --- and
surviving there has a price. In that corner the landscape can no longer
explain why \emph{this} vacuum, \emph{this} sign, \emph{this} yield; it
can only report that each happens somewhere, and hand the remainder to
observer selection (Objection~12). The right verdict is therefore not
falsification but exile: the sameness of the sky drives the multiverse
to a place where it can neither explain nor be tested. This framework's
answer lives inside the horizon instead --- determinism plus
inheritance --- where the falsifiers can reach it.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~95 (O-Cosmos-95) (the curious reader, returning)}\par\nobreak
\noindent\textbf{``If the multiverse cannot be falsified, is it really worse off than your `uniqueness'? Both sound like bets about things nobody will ever see.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two bets have opposite exposures, and that is the
whole difference. Uniqueness is indeed a conjecture (Objection~2 marks
it so), but it is a conjecture that stakes its life on numbers inside
our horizon: an extra light family, cosmic shares settling away from
the \(\pi\)-locked fractions, a tensor ratio far from \(0.0043\), a
decoupling redshift off the atomic chain --- any one of these kills it
(Objection~22). Plurality stakes nothing comparable: whatever is
measured here is compatible with some address in the ensemble, and the
ensemble itself is beyond reach. So one hypothesis about the unseen
forbids things in the seen, and the other forbids nothing anywhere.
Both are bets about places nobody will visit; only one can lose at
home. A theory that cannot lose is not winning --- it has merely
stopped playing. The asymmetry is not rhetoric; it is the operational
content of the word \emph{scientific}, and it is the reason this paper
spends its pages staking falsifiers rather than multiplying worlds.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~96 (O-Cosmos-96) (the curious reader, returning)}\par\nobreak
\noindent\textbf{``But `every possibility happens somewhere' does cover the facts. Why do you refuse to count that as an explanation?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because it has the same logical shape as a claim this
framework also declines, from the other side of the aisle: \emph{the
world was designed}. Both statements are compatible with anything a
telescope could ever find. Neither names a step that could have failed,
forbids a measurement, or risks anything on tomorrow's data: whatever
is observed, the designer could have designed it, and the ensemble
could have contained it. That is precisely why neither functions as an
explanation of the particular world in front of us. An explanation, in
the sense this paper tries to practice, is a chain with exposed links:
name the mechanism, compute the number, state what reading of what
instrument breaks the chain. The same razor cuts both comfortable
alternatives, the theological and the statistical, and for the same
reason. That it cuts symmetrically is worth noticing: this framework is
not the design hypothesis wearing equations (Objection~5), and its
refusal of the multiverse is not a matter of taste. It declines both
because both are, in the exact and technical sense, unlosable --- and
it accepts the discipline of being losable instead.
\end{objqabody}

\bigskip

\noindent\textbf{Objection~97 (O-Cosmos-97) (the curious reader, returning)}\par\nobreak
\noindent\textbf{``Is there anywhere the two pictures meet the same measurement and predict differently --- an experiment, not a philosophy seminar?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} One, and it is dated. This paper's launch is a derived
plateau with tensor-to-scalar ratio \(r = 0.0043\) (Part~IV), squarely in the range the Simons Observatory and LiteBIRD are built to decide this decade.
Inside the string program there is an actively debated internal
criterion --- the ``de~Sitter swampland'' conjecture --- whose strong
forms disfavor exactly the flat, positive-energy plateaus that drive
inflation of this type. The criterion is contested within that
community, so a detection would not refute string theory; conjectures
can be dropped. But the situation is asymmetric in an instructive way.
A measurement of \(r\) near \(0.0043\) would confirm a two-sided
prediction this framework cannot adjust, while landing on the
disfavored side of a criterion the other program proposed for
itself. A measurement far from \(0.0043\), in either direction, kills
this paper's launch outright --- and inconveniences the landscape not
at all, since there is always another vacuum. One program has placed
a bet that a named, funded instrument settles; the other retains the
right to reinterpret the outcome. The reader may decide which posture
better deserves the word \emph{prediction} --- and should note the
date: this is not a wager that can be left on the table indefinitely.
\end{objqabody}

\bigskip

\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}
\label{sec:references}

Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which derivation, table, or section each one underwrites.

The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the present derivation: from the \LCDM\ concordance model and the cosmological-constant problem, through inflation, primordial nucleosynthesis, and the growth of structure, to the present-day \(H_0\) and \(S_8\) tensions --- and then to the twenty-two-paper \GGTheory{} cohort, in which \ATen{} reads cosmology as the boundary projection of the six-dimensional bulk, one story above the Standard Model of \ANine{}. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

The \GGTheory{} cohort papers (\AOne{}--\AEleven{}, together with the phenomenology and foundations P-series and the consequence C-series) are cited throughout under the canonical keys and titles fixed by the cross-paper coherence map of the program; \ATen{}'s place in that cohort --- the cosmological-boundary member of the 4D-Physics stratum, the story above the Standard Model --- is set out in the cohort overview of this paper. The references are listed alphabetically by first author.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.~Arisaka.}
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems},
Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the constitutional charter --- the four axioms, eleven postulates, admissibility rules, and \OGGEP{} semantics --- that \ATen{} inherits as its first cause. \AX{4}~\MSMF{} (no hidden continuous moduli) and \POS{6}~\OGN{} are the upstream clauses that force the zero-free-parameter cosmic inventory; \OGGEP{} is the boundary-projection law that the present paper specializes one story up, at the cosmological boundary, to derive the modified Friedmann dynamics and the \(\pi\)-locked split \(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\). Invoked throughout as the legal source of every admissibility argument in the cosmology paper. First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.~Arisaka.}
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})},
Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Constructs the 6D bulk geometry \(\mathcal{M}_4 \times I_r \times S^1_\theta\) on which the entire \GGTheory{} program runs, together with the \BIsix{} closure, the \OGNfive{} integer ledger, and the compact-holonomy ontology. \ATen{} descends directly from this stage: the cosmic bulk geometry, the radial chronology axis \(I_r\) along which the two-stage Hubble corridor is read, and the topologically protected \(S^1_\theta\) winding that hosts the Kaxion compact-angle pseudoscalar all originate here. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.~Arisaka.}
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})},
Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Establishes the complete \BIsix{} legality theorem --- the Green--Schwarz-modified closure identity \(dH_3 = X_4\) that fixes the unique anomaly polynomial of the 6D bulk. \ATen{} cites \AThree{} to certify that the cosmological use of projected Ward identities, anomaly inflow, and global consistency is admissible: the boundary cosmology inherits a consistent, anomaly-free bulk, which is what licenses the \OGGEP{}-port treatment of horizons and the closure of the boundary stress ledger. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.~Arisaka.}
\emph{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics},
Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Fixes the zero-free-parameter Euclidean input ledger --- the \GUfive{} dimensionful constants (\(c, G_N, \hbar, k_B, Q_B\)), the structural constants \(\pi\) and \(e\), and the proved \OGNfive{} integer ledger. The \(\pi\)-locked cosmic inventory of the present paper (\(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\)) ultimately rests on this discipline: \(\pi\) enters the inventory as a topological input certified here, not as a fitted number, so \ATen{} introduces no free continuous parameter beyond the cohort ledger. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.~Arisaka.}
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT},
Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Provides the exact boundary-reduction engine of the cohort: the \GDOS{} construction of the canonical boundary algebra on \(\partial\mathcal{M}\) and the open-system dynamics that follow from tracing the bulk down to its boundary shadow. \ATen{} inherits its 4D observables, its cosmic state-manifold evolution, and its open-system arrow of time from \GDOS{}; the modified Friedmann equations of the present paper are the cosmological-stratum specialization of the \GDOS{} boundary dynamics under \OGGEP{}. First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K.~Arisaka.}
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind},
Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
The immediate predecessor for the runtime constitution used in the cosmos paper. Supplies the \GiFour{} four-pillar backbone (\TGG{}, \CCI{}, \GFF{}, \GRIP{}), the free-energy split \(F_{\GFF}=F_{\GEP}+F_{\CEP}\), Wilson-loop \GSSB{}, and \SIP{}; it also defines the two runtime regimes \GRIPin{}/\GRIPout{} adopted in the cosmic \GRIP{} grammar of the present paper. \ATen{} runs cosmic history on this engine: contraction toward attractors under \GRIPin{} and necessary escape under \GRIPout{} organize the radial chronology and the relic/vacuum endpoints. First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.~Arisaka.}
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}},
Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
Supplies the \QiFour{} quantum-information completion of the \GDOS{} runtime --- the \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} pillars, the \SeOS{}/QCI language, the information balance \(\IH+\OH=\EIC\), and the five-story pagoda. \ATen{} uses this apparatus in its later interpretive Parts: the cosmological boundary is read as an information channel, and the \QiQUA{} ledger anchors the quantitative bookkeeping that the cosmic inventory shares with the rest of the cohort. First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{K.~Arisaka.}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
The quantum-mechanics member of the 4D-Physics Trio: reads textbook 4D quantum mechanics as the boundary projection of the 6D bulk, deriving the Born rule, Bell/CHSH violation, and decoherence as theorems of the non-invertible 6D-to-4D \OGGEP{} projection. The present paper applies the same \OGGEP{} boundary projection one story up, at the cosmological boundary, so \AEight{} is the structural sibling that fixes the projection grammar \ATen{} reuses --- bulk to boundary on the QM stratum there, bulk to boundary on the cosmological stratum here. First cited in \S\ref{sec:central-claim}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.~Arisaka.}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.}
The immediate particle-to-cosmos bridge: fixes the Standard-Model ledger, the compact-angle ontology, the neutrino block, and the parameter constraints as theorems of \GG{}, \BIsix{}, \GDOS{}, \GRIP{}, and \GFF{}. \ATen{} inherits this ledger as the matter content of the cosmological inventory --- the baryonic and neutrino budgets that populate \(\Omega_m=1/\pi\) --- and reuses \ANine{}'s compact-angle reading to identify the Kaxion as a non-spatial compact-angle degree of freedom of the same bulk. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{K.~Arisaka.}
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program},
Arisaka-GG-A11-Origin.
\par\noindent In preparation. To be posted at \url{https://arisaka-gg.org/} on
completion; no result of the present paper depends on it.
\par\noindent\textbf{Relevance.}
Defines the Ground Geometric Attractor (\GGA{}) and proves \textsf{T-GGA-as-Ground-State}, the ground-state refinement of the \GRIP{} terminus that the present paper uses to distinguish transient cosmic histories (\GA{}'s) from persistent relics and final states (\GGA{}'s) --- specifically the Kaxion dark matter and the de Sitter vacuum. \AEleven{} also proves the regime theorems \textsf{T-GRIP-In-Lyapunov-Descent} and \textsf{T-GRIP-Out-Necessity} that underlie the \GRIPin{}/\GRIPout{} reading of cosmic history adopted here, and supplies the four-move operational anatomy (\(\DGI{} \to \GFF{}\) Morse flow \(\to \GSSB{} \to \GA{}\)) that the cosmic \GRIP{} grammar specializes. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{K.~Arisaka.}
\emph{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}},
Arisaka-GG-P1-KK preprint.
\par\noindent\textbf{Relevance.}
The first paper of the Predictions Branch: locates the Kaluza-Klein lock cluster at \(\Mlock = (1/2)\,\MPl\,\mathrm{e}^{-35} \approx 3.85\) TeV, with explicit collider falsifiers. \ATen{} inherits this scale as the upper end of the inherited scale chain and the high-energy edge of the radial cosmic chronology --- the earliest, deepest-bulk rung of the radial timeline \(I_r\) along which the boundary cosmology unfolds. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{K.~Arisaka.}
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability},
Arisaka-GG-P2-N3 preprint.
\par\noindent\textbf{Relevance.}
The dedicated family-number companion: executes the three-leg derivation of the generation count \(N_g=3\) that the present paper reserves --- the \MSMF{} lower bound from CP-violation combinatorics, the \GDOS{} capacity upper bound on \(\mathbb{C}P^{N_g-1}\) (the capacity theorem \(\mathrm{Cap} = \pi N_g\) with exact saturation, backed by the integrality theorem L-INT and by two obstruction theorems; conditional on one declared topological residue, R-INT), and the codebook consistency check --- together with the \GRIPin{} reachability reading: a fourth family's ground state would be dynamically unreachable within the stratum's instanton budget. That dynamical content is the cosmological three-generation count \ATen{} uses when it populates the neutrino and lepton budgets of the cosmic inventory. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{K.~Arisaka.}
\emph{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P3-Electron preprint.
\par\noindent\textbf{Relevance.}
The third paper of the Predictions Branch: reads the electron as the ground-state Geometric Attractor of nature --- identity, absolute stability, the two excited copies, the positron as conjugate readout, pointlikeness, and field response derived as six faces of ground-state-ness, importing \PaperPTwo{}'s \(N=3\) verbatim. \ATen{} cites it as the particle-stratum instance of the universal-identity principle that the present paper extends to the cosmological inventory: the same attractor logic that makes every electron identical makes the cosmic relics and the vacuum the unique \GGA{} endpoints of their strata. First cited in \S\ref{sec:gg-from-zero}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{K.~Arisaka.}
\emph{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}},
Arisaka-GG-P4-Real preprint.
\par\noindent\textbf{Relevance.}
The origin-of-\(i\) member paper of the Foundations Branch: the standalone, full-length development of the real-bulk series this paper leans on at the observer-indexed vacuum and at the dial-free door (T-Cosmos-18) --- the two real sources of the derived complex structure, the holomorphy bridge at its Hardy, tube, and twistor levels, and the open items stated as open. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{K.~Arisaka.}
\emph{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter},
Arisaka-GG-P5-CPT preprint.
\par\noindent\textbf{Relevance.}
The CPT member paper of the Foundations Branch and the \ANine{}--\ATen{} interface of the cohort map: C, P, and T as orientation reversals of the three real bulk factors, the chiral-pentagon CP phases, and the matter--antimatter asymmetry through the weight-zero door. The ledger phases this paper's baryon-asymmetry chain consumes --- the PMNS Dirac phase and the Majorana parity \(\alpha_{31} = \pi\), derived in \ANine{} --- reach Part~V and Appendix~\ref{appA:leptogenesis} through it, at the grade recorded there. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{K.~Arisaka.}
\emph{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P6-Proton preprint.
\par\noindent\textbf{Relevance.}
The Predictions Branch member that runs the ground-attractor method of
\PaperPThree{} on the baryonic stratum. Its bearing on \ATen{} is a contrast
rather than an inheritance, and the contrast is worth stating because this
paper's matter sector depends on the distinction. The baryon asymmetry of
Part~V is a \emph{number} that survived, computed from a chain with a named
input and a postulated exponent; the proton's stability in \PaperPSix{} is a
\emph{winding class} that no local process can change. Protection by an
amount and protection by a topological class are different mechanisms on one
geometry, and only the second is exact. Cited at roster level; nothing in
\ATen{} depends on its results. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{K.~Arisaka.}
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}},
Arisaka-GG-C1-Kaxion preprint.
\par\noindent\textbf{Relevance.}
The dedicated dark-matter companion, executing the derivation that the present paper's Conjecture~III reserves: the \(\pi\)-locked abundance closure (bulk normalization \(f_{\mathrm{bulk}} = (3.0\pm0.3)\times10^{13}\) GeV; single physical decay constant \(f_K = f_{\mathrm{bulk}}/43 \simeq 7.1\times10^{11}\) GeV), the corrected misalignment normalization adopted in Eq.~\eqref{eq:omega-misalignment-partIII}, the unique laboratory window \(\nu_K\in[1.59,2.13]\) GHz (\(m_K = 8.1\pm0.5\,\mu\)eV), the exact anomaly ratio \(E/N=8/3\) placing the Kaxion on the DFSZ line, the structural exclusion of the randomized-angle history, and the \(1/35\) twin-peak and \(5{:}2\) portal falsifiers. \ATen{} treats it as the phenomenology that converts its \(\Omega_{\mathrm{DM}}=1/\pi-1/(2\pi^2)\) dark-matter share into a falsifiable haloscope target. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{K.~Arisaka.}
\emph{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$},
Arisaka-GG-C2-Hubble preprint.
\par\noindent\textbf{Relevance.}
The dedicated Hubble companion: a program-level reading of the H0DN Local Distance Network consensus \(H_0 = 73.50 \pm 0.81\) km/s/Mpc \citep{CasertanoH0DN2026} through the derived background rate of the present paper --- the two-route geometric determination of \(H_0\), the derived Local-Hole lift, and the named calibration residual, priced against the H0DN error budget. It carries the falsifier registry F-Hubble-1--7 and F-Hubble-11 (including the pre-registered \(\zeta_\Lambda = \pi/175\) target and the standard-siren kill test), with the falsifiable exposure to future releases of the distance network itself. \ATen{} supplies the geometric mechanism --- the two ends of one bulk corridor --- of which \PaperCTwo{} is the dedicated observational study. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{K.~Arisaka.}
\emph{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity},
Arisaka-GG-C3-BH preprint.
\par\noindent\textbf{Relevance.}
The cohort's strong-gravity anchor: derives the Bekenstein--Hawking entropy \(A/4\) with the horizon treated as an \OGGEP{} port, the exact coefficient following from induced gravity. \ATen{} cites it in its black-hole-information and primordial-relic Part: the same \OGGEP{}-port treatment that yields \(A/4\) governs how horizons exchange information with the cosmological boundary, tying the strong-gravity ledger into the boundary cosmology of the present paper. First cited in \S\ref{sec:cohort}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{K.~Arisaka.}
\emph{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out},
Arisaka-GG-W1-Symmetry preprint.
\par\noindent\textbf{Relevance.}
The White-Paper Branch survey of how conservation moved from symmetry to
cohomology. It supplies the context for a habit this paper relies on without
arguing for it: the cosmic inventory is treated as a set of classes forced by
the geometry rather than as a set of quantities conserved by a symmetry, and
the difference is why an entry in the codebook can be Derived without any
symmetry being named beside it. The survey is conceptual and carries no
computation \ATen{} uses. Cited at roster level. First cited in
\S\ref{sec:cohort}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{K.~Arisaka.}
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology},
Arisaka-GG-W2-Holonomy preprint.
\par\noindent\textbf{Relevance.}
The White-Paper Branch treatment of protection by topology, and the closest of
the three to \ATen{}'s own argument. The compact phase circle whose holonomy
this paper reads as the seed of the Kaxion, and whose winding carries the
baryon number the matter Part accounts for, is one rung of the ladder of
protection mechanisms classified there. A reader who wants to know why that
rung and not another should read the classification rather than this paper's
use of it. Its sixteen open problems overlap the named conjectures of the
codebook. Cited at roster level. First cited in \S\ref{sec:cohort}.

\end{thebibliography}



\section*{B.~Other references}
\begin{thebibliography}{99}

\bibitem[Abbott \& Sikivie(1983)]{AbbottSikivie1983}
\textbf{L.~F.~Abbott and P.~Sikivie.}
A cosmological bound on the invisible axion.
\emph{Phys.\ Lett.\ B} \textbf{120} (1983) 133--136.
\par\noindent\textbf{Relevance.}
The second of the three 1983 misalignment papers, deriving the cosmological bound relating the axion mass and decay constant to its relic abundance. \ATen{}'s Kaxion appendix uses the same misalignment relic-abundance scaling (set by $\theta_i$, $m$, and the oscillation-onset epoch) to connect the Kaxion parameters to its dark-matter share, and cites this paper for the abundance--mass relation. First cited in \S\ref{appA:kaxion}.

\bibitem[Abdalla \emph{et~al.}(2022)]{Abdalla2022}
\textbf{E.~Abdalla \emph{et~al.}}
Cosmology intertwined: a review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies.
\emph{JHEAp} \textbf{34} (2022) 49--211.
\par\noindent\textbf{Relevance.}
The 2022 ``Cosmology Intertwined'' review is the broad community survey linking the \Hzero{} and \Seight{} tensions to each other and to a range of cosmological anomalies. \ATen{} cites it in the \Seight{} section as the reference establishing that the clustering tension is a genuine, multi-probe discrepancy that the framework's growth reading must address (while flagging the \Seight-sign question OPEN). First cited in \S\ref{sec:cosmology-challenges}.

% ------------------- Baryogenesis (sec:baryogenesis) -----------------

\bibitem[Aharonov and Bohm(1959)]{AharonovBohm1959}
\textbf{Aharonov, Y., \& Bohm, D. (1959).} Significance of electromagnetic potentials in the quantum theory. \emph{Physical Review, 115}(3), 485--491.
\par\noindent\textbf{Relevance.} The 1959 Aharonov--Bohm paper, the classical demonstration that a gauge-invariant phase accumulated around a loop is physically real even where the local field vanishes. \ATen{} cites this as historical context for the Wilson-loop and holonomy reading of cosmic branch selection: the cosmic-grip grammar treats the compact-angle holonomy of the inherited \GG-6 bulk, not a local order parameter, as the genuine gauge-invariant coordinate that labels which cosmological branch is realized. First cited in \S\ref{ssec:cosmic-grip-grammar-gga}.

\bibitem[Albrecht and Steinhardt(1982)]{AlbrechtSteinhardt1982}
\textbf{Albrecht, A., \& Steinhardt, P. J. (1982).} Cosmology for grand unified theories with radiatively induced symmetry breaking. \emph{Physical Review Letters, 48}(17), 1220--1223.
\par\noindent\textbf{Relevance.} Together with \citet{Linde1982}, the 1982 ``new inflation'' proposal that replaced the false-vacuum picture of \citet{Guth1981} by a slow-rolling scalar field and thereby repaired its graceful-exit problem. \ATen{} cites this as historical background for the inflation section, which retains the slow-roll structure of new inflation but constrains the inflationary modulus to a specific direction in the inherited bulk-modulus space and locates the inflationary scale at TeV rather than GUT energies. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Alpher et~al.(1948)]{AlpherBetheGamow1948}
\textbf{Alpher, R. A., Bethe, H., \& Gamow, G. (1948).} The origin of chemical elements. \emph{Physical Review, 73}(7), 803--804.
\par\noindent\textbf{Relevance.} The 1948 founding paper of hot-Big-Bang nucleosynthesis, proposing that the light elements are built up by neutron-proton interactions in a cooling, expanding plasma. \ATen{} cites this as the historical opening of the BBN section, which retains the standard light-element calculation (D, \(^3\)He, \(^4\)He, \(^7\)Li) and asks only whether the inherited ledger modifies any BBN input in a way that addresses the residual lithium-7 anomaly without disturbing the other three abundances. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Amari and Nagaoka(2000)]{AmariNagaoka2000}
\textbf{Amari, S.-I., \& Nagaoka, H. (2000).} \emph{Methods of Information Geometry}. Providence, RI: American Mathematical Society.
\par\noindent\textbf{Relevance.} The standard monograph on information geometry, establishing the Fisher-Rao metric and dual-connection structure on statistical manifolds. \ATen{} cites this for orientation in the later Qi-runtime and \EIC{} parts, where the state-manifold and selector language of the boundary runtime is described; it is a comparison vocabulary, not a derivational input to the cosmology. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Asgari et al.(2021)]{Asgari2021}
\textbf{Asgari, M., Lin, C.-A., Joachimi, B., et al. (2021).} KiDS-1000 cosmology: Cosmic shear constraints and comparison between two point statistics. \textit{Astronomy \& Astrophysics}, \textit{645}, A104.
\par\noindent\textbf{Relevance.} The benchmark low-redshift cosmic-shear determination of the clustering parameter, \(\Seight = 0.759^{+0.024}_{-0.021}\). Cited as the empirical anchor of the late/lensing branch \(\Seight^{\mathrm{late}}\) in the parameter-audit Dev plot and data table: the corridor-response prediction \(\Seight^{\mathrm{late}} = 0.78 \pm 0.01\) of Part~IV agrees with this measurement at the \(0.9\sigma\) level, the same two-branch structure proposed as a candidate resolution of the Hubble tension. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Bardeen(1980)]{Bardeen1980}
\textbf{J.~M.~Bardeen.}
Gauge-invariant cosmological perturbations.
\emph{Phys.\ Rev.\ D} \textbf{22} (1980) 1882--1905.
\par\noindent\textbf{Relevance.}
Bardeen's 1980 paper introduced the gauge-invariant formulation of cosmological perturbation theory, resolving the gauge ambiguities that plagued the Lifshitz approach. \ATen{}'s growth appendix works in the conformal-Newtonian gauge with the gauge-invariant potentials $\Phi,\Psi$ that Bardeen's construction justifies, ensuring that the growth equation and the resulting (\sigeight,\Seight) normalization are physical rather than gauge artifacts. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Bardeen \emph{et~al.}(1983)]{BardeenSteinhardtTurner1983}
\textbf{J.~M.~Bardeen, P.~J.~Steinhardt, and M.~S.~Turner.}
Spontaneous creation of almost scale-free density perturbations in an inflationary universe.
\emph{Phys.\ Rev.\ D} \textbf{28} (1983) 679--693.
\par\noindent\textbf{Relevance.}
The 1983 Bardeen--Steinhardt--Turner paper provided the gauge-invariant, conserved-curvature-perturbation treatment that put the inflationary spectrum calculation on rigorous footing. \ATen{}'s inflation appendix relies on the conservation of the curvature perturbation on super-horizon scales it established to connect the spectrum at horizon exit to its re-entry in the growth appendix. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Bardeen \emph{et~al.}(1986)]{BBKS1986}
\textbf{J.~M.~Bardeen, J.~R.~Bond, N.~Kaiser, and A.~S.~Szalay.}
The statistics of peaks of Gaussian random fields.
\emph{Astrophys.\ J.} \textbf{304} (1986) 15--61.
\par\noindent\textbf{Relevance.}
The 1986 ``BBKS'' paper provides the standard fitting form for the cold-dark-matter transfer function and the peak statistics of Gaussian random fields. \ATen{}'s growth appendix uses the BBKS transfer function $T(k)$ to pass from the primordial spectrum to the processed matter power spectrum $P(k)$ whose $8\,h^{-1}\mathrm{Mpc}$ variance defines \sigeight. First cited in \S\ref{sec:lcdm-success}.

% ------------------- LSS / BAO (sec:expansion) -----------------------

\bibitem[Baumann(2022)]{Baumann2022}
\textbf{D.~Baumann.}
\emph{Cosmology}.
Cambridge University Press, Cambridge, (2022).
\par\noindent\textbf{Relevance.}
The modern graduate cosmology text: background, thermal history, inflation, perturbations, and the CMB, in the notation closest to the present course. Recommended as a primary reference for the constructive appendices (E--K) of the graduate course in Part~XV. First cited in \S\ref{app:abbrev}.

\bibitem[Bekenstein(1973)]{Bekenstein1973}
\textbf{Bekenstein, J. D. (1973).} Black holes and entropy. \emph{Physical Review D, 7}(8), 2333--2346.
\par\noindent\textbf{Relevance.} The 1973 paper that assigned an entropy proportional to horizon area to a black hole and so opened black-hole thermodynamics. \ATen{} cites this to frame the horizon-ledger problem in the Part~V audit, where the area-entropy bookkeeping is reinterpreted through the \GFF{} free-energy functional and the \GDOS{} open-system dynamics rather than taken as a primitive. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Bertone \& Hooper(2018)]{BertoneHooper2018}
\textbf{G.~Bertone and D.~Hooper.}
History of dark matter.
\emph{Rev.\ Mod.\ Phys.} \textbf{90} (2018) 045002.
\par\noindent\textbf{Relevance.}
The 2018 Bertone--Hooper review surveys the observational and theoretical history of dark matter and the landscape of candidate models. \ATen{} cites it in the Kaxion section as the contextual reference situating the Kaxion misalignment candidate within the broader dark-matter program and its abundance constraints. First cited in \S\ref{sec:lcdm-success}.

% ------------------- Tensions (sec:h0tension / sec:s8) ---------------

\bibitem[Bianchi(1902)]{Bianchi1902}
\textbf{Bianchi, L. (1902).} Sui simboli a quattro indici e sulla curvatura di Riemann. \emph{Rendiconti della R.\ Accademia dei Lincei, 11}, 3--7.
\par\noindent\textbf{Relevance.} Founding paper of the Bianchi identity \(dR = 0\), the geometric ancestor of the modified Bianchi identity \BIsix{} of \citep{ArisakaA3BI6}; cited in the framing remark and in the universal-cosmic-identity section as historical lineage for the constitutional descent \(dF = 0 \to dR = 0 \to dX_4 = 0\). First cited in \S\ref{appA:frw}.

\bibitem[Bressi et al.(2011)]{Bressi2011}
\textbf{G.~Bressi, G.~Carugno, F.~Della Valle, G.~Galeazzi,
G.~Ruoso, and G.~Sartori.}
Testing the neutrality of matter by acoustic means in a spherical
resonator.
\emph{Phys.\ Rev.\ A} \textbf{83} (2011) 052101.
\par\noindent\textbf{Relevance.}
The sharpest laboratory audit of exact neutrality: the charge per
hydrogen molecule bounded at the \(10^{-21}\,e\) level. In \ATen{}
this is the precision anchor of the neutrality theorem
(\S\ref{ssec:neutrality}) --- the exactness that the trace identity
of the unified embedding holds, not protects.

\bibitem[Carroll(2001)]{Carroll2001}
\textbf{S.~M.~Carroll.}
The cosmological constant.
\emph{Living Rev.\ Relativity} \textbf{4} (2001) 1.
\par\noindent\textbf{Relevance.}
Carroll's 2001 Living Review is a standard pedagogical reference on the cosmological constant and the vacuum-energy problem. \ATen{}'s dark-energy section cites it for the statement of the fine-tuning and coincidence problems that the framework's late vacuum term --- a descendant of the same anomaly-locked geometric stage --- is meant to address rather than to insert by hand. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Casertano et~al.(2026)]{CasertanoH0DN2026}
\textbf{Casertano, S., Anand, G.~S., Anderson, R.~I., et al.\ (H0DN Collaboration) (2026).} The Local Distance Network: A community consensus report on the measurement of the Hubble constant at \({\sim}1\%\) precision. \emph{Astronomy \& Astrophysics, 708}, A166.
\par\noindent\textbf{Relevance.} The covariance-weighted Local Distance Network consensus \(\Hzero = 73.50 \pm 0.81\) km/s/Mpc --- the codebook's late-branch comparator, promoted from the single-team \citet{Riess2022} value; the companion \PaperCTwo{} is the program-level study of this measurement. First cited in \S\ref{subsec:master-identity}.

\bibitem[Cohen et al.(1998)]{CohenDeRujulaGlashow1998}
\textbf{A.~G.~Cohen, A.~De~R\'ujula, and S.~L.~Glashow.}
A matter--antimatter universe?
\emph{Astrophys.\ J.} \textbf{495} (1998) 539.
\par\noindent\textbf{Relevance.}
The observational closure of the domain alternative: contact zones
between matter and antimatter regions would shine in annihilation
gamma rays, and the diffuse backgrounds exclude such zones out to
scales comparable to the visible universe. \ATen{} cites it in the
third-horizon Section (\S\ref{sec:third-horizon}) as the empirical
fact that every disconnected region made the \emph{same} choice ---
the observation that the determinism of the cascade must, and does,
explain.

\bibitem[Cole \emph{et~al.}(2005)]{Cole2005}
\textbf{S.~Cole \emph{et~al.} (2dFGRS Team).}
The 2dF Galaxy Redshift Survey: power-spectrum analysis of the final data set and cosmological implications.
\emph{Mon.\ Not.\ R.\ Astron.\ Soc.} \textbf{362} (2005) 505--534.
\par\noindent\textbf{Relevance.}
The 2005 2dFGRS power-spectrum analysis is one of the two first detections of the baryon acoustic feature in the galaxy distribution. \ATen{} cites it in the expansion section as observational confirmation of the standard ruler whose redshift evolution the two-branch Hubble corridor predicts a residue in at $z\sim1$. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Copeland \emph{et~al.}(2006)]{CopelandSamiTsujikawa2006}
\textbf{E.~J.~Copeland, M.~Sami, and S.~Tsujikawa.}
Dynamics of dark energy.
\emph{Int.\ J.\ Mod.\ Phys.\ D} \textbf{15} (2006) 1753--1936.
\par\noindent\textbf{Relevance.}
The 2006 Copeland--Sami--Tsujikawa review is the comprehensive reference for the dynamics of dark-energy models, including scalar-field quintessence and the equation-of-state phenomenology. \ATen{}'s dark-energy section cites it for the parametrization of the dark-energy equation of state against which the framework's residual-vacuum branch is compared. First cited in \S\ref{sec:lcdm-success}.

% ------------------- Induced gravity / horizon (sec:dark-energy / sec:beginning) -

\bibitem[Cyburt \emph{et~al.}(2016)]{CyburtFieldsOliveYeh2016}
\textbf{R.~H.~Cyburt, B.~D.~Fields, K.~A.~Olive, and T.-H.~Yeh.}
Big bang nucleosynthesis: present status.
\emph{Rev.\ Mod.\ Phys.} \textbf{88} (2016) 015004.
\par\noindent\textbf{Relevance.}
The 2016 Cyburt \emph{et~al.} review is the authoritative modern status report on BBN, with up-to-date reaction rates, abundance predictions, and observational comparisons. \ATen{}'s BBN appendix cites it for the current best-fit light-element yields and uncertainties against which the worked $Y_p$ computation is checked. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Dev \emph{et~al.}(2014)]{DevMillingtonPilaftsisTeresi2014}
\textbf{P.~S.~Bhupal Dev, P.~Millington, A.~Pilaftsis, and D.~Teresi.}
Flavour covariant transport equations: an application to resonant leptogenesis.
\emph{Nucl.\ Phys.\ B} \textbf{886} (2014) 569--664 [arXiv:1404.1003].
\par\noindent\textbf{Relevance.}
The flavor-covariant density-matrix transport formalism for resonant leptogenesis, in which the
flavor coherences of the heavy and charged-lepton sectors are dynamical variables. It is the
natural framework in which condition C4 of Appendix~\ref{appA:leptogenesis} must be discharged;
in the regimes it treats, the effective regulator is of the heavy-sector class, which is why C4
is carried open rather than assumed. First cited in Appendix~\ref{appA:leptogenesis}.

\bibitem[Di~Valentino \emph{et~al.}(2021)]{DiValentino2021}
\textbf{E.~Di~Valentino, O.~Mena, S.~Pan, L.~Visinelli, W.~Yang, A.~Melchiorri, D.~F.~Mota, A.~G.~Riess, and J.~Silk.}
In the realm of the Hubble tension --- a review of solutions.
\emph{Class.\ Quant.\ Grav.} \textbf{38} (2021) 153001.
\par\noindent\textbf{Relevance.}
The 2021 Di~Valentino \emph{et~al.} review is the comprehensive catalog of proposed solutions to the Hubble tension, classified into early-time, late-time, and systematics-based approaches. \ATen{}'s Hubble-tension section cites it to position the two-stage corridor within this taxonomy and to motivate the \BIsix{} refusal to scan over arbitrary new fluids. First cited in \S\ref{sec:cosmology-challenges}.

\bibitem[Dicke et~al.(1965)]{DickePeebles1965}
\textbf{Dicke, R. H., Peebles, P. J. E., Roll, P. G., \& Wilkinson, D. T. (1965).} Cosmic black-body radiation. \emph{Astrophysical Journal, 142}, 414--419.
\par\noindent\textbf{Relevance.} The 1965 companion paper to \citet{PenziasWilson1965}, interpreting the 4080~Mc/s excess as the cooled black-body afterglow of a hot early universe. \ATen{} cites this as historical background for the CMB section, which retains the standard acoustic, damping, and projection structure of the microwave background and asks only where the inherited ledger reshapes the source assumptions behind it. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Coleman \& De~Luccia(1980)]{ColemanDeLuccia1980}
\textbf{Coleman, S.\ \& De~Luccia, F. (1980).} Gravitational effects on and of vacuum decay. \emph{Phys.\ Rev.\ D} \textbf{21}, 3305--3315.
\par\noindent\textbf{Relevance.} The bubble-nucleation event by which a metastable vacuum decays in a gravitational background. \ATen{} cites it in the Common Objections (Objection~O-Cosmos-90) as the mechanism by which a landscape framework must have our entire observable patch descend from a single nucleation --- the internally consistent repair whose cost the second-horizon discussion prices. First cited in \S\ref{obj:landscape-priced}.

\bibitem[Dine \& Fischler(1983)]{DineFischler1983}
\textbf{M.~Dine and W.~Fischler.}
The not-so-harmless axion.
\emph{Phys.\ Lett.\ B} \textbf{120} (1983) 137--141.
\par\noindent\textbf{Relevance.}
The third of the three founding 1983 misalignment papers, emphasizing the cosmological constraints on the axion decay constant from the relic-abundance overclosure bound. \ATen{}'s Kaxion appendix cites it alongside the other two as the joint origin of the misalignment dark-matter calculation it carries out. First cited in \S\ref{appA:kaxion}.

\bibitem[Dine \emph{et~al.}(1981)]{DineFischlerSrednicki1981}
\textbf{M.~Dine, W.~Fischler, and M.~Srednicki.}
A simple solution to the strong CP problem with a harmless axion.
\emph{Phys.\ Lett.\ B} \textbf{104} (1981) 199--202.
\par\noindent\textbf{Relevance.}
The 1981 ``DFSZ'' paper is one of the two canonical invisible-axion model constructions, embedding the Peccei--Quinn symmetry with a large decay constant. \ATen{}'s Kaxion appendix cites it as a reference model for how a phenomenologically viable axion sector is realized, against which the Kaxion's geometric (Wilson-line) origin is contrasted. First cited in \S\ref{appA:kaxion}.

\bibitem[Dodelson(2003)]{Dodelson2003}
\textbf{S.~Dodelson.}
\emph{Modern Cosmology}.
Academic Press, San Diego, (2003).
\par\noindent\textbf{Relevance.}
Dodelson's 2003 textbook is the standard pedagogical source for the Boltzmann-hierarchy treatment of perturbations, the growth of structure, and the CMB acoustic peaks. \ATen{} follows its line-by-line style in the perturbation/growth appendix (the master equation $\ddot\delta+2H\dot\delta-4\pi G\rho_m\delta=0$ and the transfer function) and in the CMB acoustic-peak appendix; the framework leaves this standard linear theory unmodified and acts only on the late-time growth history. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Durrer(2020)]{Durrer2020}
\textbf{R.~Durrer.}
\emph{The Cosmic Microwave Background}, 2nd ed.
Cambridge University Press, Cambridge, (2020).
\par\noindent\textbf{Relevance.}
The dedicated CMB monograph: full Boltzmann hierarchy, polarization, and lensing. Recommended companion for the acoustic-peak appendix (K), whose derived-parameter confrontation (Part~VI) hands its inputs to exactly the machinery this book develops. First cited in \S\ref{app:abbrev}.

\bibitem[Douglas(2003)]{Douglas2003}
\textbf{Douglas, M.~R. (2003).} The statistics of string/M~theory vacua. \emph{J.\ High Energy Phys.} \textbf{05}, 046. [arXiv:hep-th/0303194].
\par\noindent\textbf{Relevance.} The canonical statistical statement of the string vacuum landscape --- of order \(10^{500}\) consistent flux vacua with no dynamical selection principle. \ATen{} cites it in Objection~O-Cosmos-90 as the precise form of the vacuum-selection freedom the unique-vacuum wager is set against. First cited in \S\ref{obj:landscape-priced}.

\bibitem[Einstein(1917)]{Einstein1917}
\textbf{Einstein, A. (1917).} Kosmologische Betrachtungen zur allgemeinen Relativit\"atstheorie. \emph{Sitzungsberichte der K\"oniglich Preu\ss ischen Akademie der Wissenschaften}, 142--152.
\par\noindent\textbf{Relevance.} The 1917 paper that introduced the cosmological field equations and the constant \(\Lambda\) in order to permit a static universe. \ATen{} cites this as historical background both for the beginning-of-universe section, which retains the field-equation lineage from Einstein through Friedmann and Lema\^itre, and for the dark-energy section, which retains the empirical \(\Lambda\) but reinterprets its scale through the residual \GFF{} ledger of the inherited bulk. First cited in \S\ref{sec:introduction}.

\bibitem[Eisenstein \emph{et~al.}(2005)]{Eisenstein2005}
\textbf{D.~J.~Eisenstein \emph{et~al.} (SDSS Collaboration).}
Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies.
\emph{Astrophys.\ J.} \textbf{633} (2005) 560--574.
\par\noindent\textbf{Relevance.}
The 2005 SDSS detection of the BAO peak in the LRG correlation function is the landmark measurement of the low-redshift standard ruler. \ATen{} cites it in the expansion section as the empirical anchor for the BAO distance scale; the paper's falsifiable prediction of a two-stage residue in BAO at $z\sim1$ is stated directly against this measurement class. First cited in \S\ref{sec:lcdm-success}.

% ------------------- Inflation perturbations (appA:inflation) --------

\bibitem[Ellis(2006)]{Ellis2006}
\textbf{G.~F.~R.~Ellis.}
Issues in the philosophy of cosmology.
In \emph{Philosophy of Physics (Handbook of the Philosophy of Science)},
Elsevier, Amsterdam (2006) 1183--1285.
\par\noindent\textbf{Relevance.}
Ellis's survey is the standard statement of the epistemic foundations of
cosmology, including the observation that ``cosmology proceeds by
assuming the laws of physics are the same everywhere'' --- an assumption
only partially testable from inside one horizon. \ATen{} cites it in the
second-horizon-problem pedigree as the clearest published acknowledgment
that the uniformity of the laws is an assumption of the standard
framework: noticed by philosophers of cosmology, never mechanized.
First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Feynman(1965)]{FeynmanNobel1965}
\textbf{R.~P.~Feynman.}
The development of the space-time view of quantum electrodynamics.
Nobel Lecture, December 11 (1965).
\par\noindent\textbf{Relevance.}
The lecture preserves Wheeler's 1940 telephone call proposing the
one-electron universe --- all electrons identical because they are one
electron threading back and forth through time. \ATen{} cites it in the
second-horizon-problem pedigree as the famous anticipation that proves
the rule: the identity of the electrons was once felt to demand a
mechanism, and the episode survives as an anecdote, not as a named
problem. First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Fields(2011)]{Fields2011}
\textbf{B.~D.~Fields.}
The primordial lithium problem.
\emph{Ann.\ Rev.\ Nucl.\ Part.\ Sci.} \textbf{61} (2011) 47--68.
\par\noindent\textbf{Relevance.}
Fields' 2011 review is the canonical statement of the primordial lithium problem --- the persistent factor-of-three discrepancy between the BBN-predicted and observed $^7\mathrm{Li}$ abundance. \ATen{}'s BBN appendix flags the lithium problem honestly as an unresolved standard-model tension that the framework does not claim to solve, distinguishing it from the well-fit helium and deuterium yields. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Freedman(2021)]{Freedman2021}
\textbf{W.~L.~Freedman.}
Measurements of the Hubble constant: tensions in perspective.
\emph{Astrophys.\ J.} \textbf{919} (2021) 16.
\par\noindent\textbf{Relevance.}
Freedman's 2021 paper presents the tip-of-the-red-giant-branch (TRGB) calibration of \Hzero, which yields a value intermediate between the Cepheid and CMB determinations and bears directly on whether the tension is physical or systematic. \ATen{}'s Hubble-tension section cites it as the key alternative local calibration that the two-stage corridor must accommodate. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Freivogel(2011)]{Freivogel2011}
\textbf{Freivogel, B. (2011).} Making predictions in the multiverse. \emph{Class.\ Quantum Grav.} \textbf{28}, 204007. [arXiv:1105.0244].
\par\noindent\textbf{Relevance.} A review of the measure problem of eternal inflation: the relative frequencies of observations are undefined without an additional, non-derived regulator. \ATen{} cites it in Objection~O-Cosmos-90 as the reason the landscape's one-bubble repair defers rather than restores predictivity. First cited in \S\ref{obj:landscape-priced}.

\bibitem[Friedmann(1922)]{Friedmann1922}
\textbf{Friedmann, A. (1922).} \"Uber die Kr\"ummung des Raumes. \emph{Zeitschrift f\"ur Physik, 10}(1), 377--386.
\par\noindent\textbf{Relevance.} The 1922 paper that discovered the dynamical, time-dependent homogeneous-isotropic solutions of Einstein's equations and established what became the FRW family --- the kinematic backbone of every cosmology paper since. \ATen{} cites this as historical background for the beginning and expansion-and-cooling sections, both of which retain the Friedmann dynamics intact, and it underwrites the explicit FRW-derivation appendix \S\ref{appA:frw}, which reconstructs the metric, Christoffel symbols, and Friedmann equation that feed the inherited source ledger. First cited in \S\ref{sec:introduction}.

\bibitem[Friston(2010)]{Friston2010}
\textbf{Friston, K. (2010).} The free-energy principle: A unified brain theory? \emph{Nature Reviews Neuroscience, 11}(2), 127--138.
\par\noindent\textbf{Relevance.} The 2010 statement of the free-energy principle, casting perception and action as variational minimization of a surprise functional. \ATen{} cites this as an external comparison point for the inference and internal-reconstruction language imported from \ASeven{} (the \QiFour{} pillars) into the later \EIC{} discussion; it is an orienting analogue, not a derivational input. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Fukugita \& Yanagida(1986)]{FukugitaYanagida1986}
\textbf{M.~Fukugita and T.~Yanagida.}
Baryogenesis without grand unification.
\emph{Phys.\ Lett.\ B} \textbf{174} (1986) 45--47.
\par\noindent\textbf{Relevance.}
The 1986 Fukugita--Yanagida paper introduced leptogenesis: a lepton asymmetry from heavy right-handed-neutrino decays, reprocessed into a baryon asymmetry by sphalerons. \ATen{}'s baryogenesis section cites it as the mechanism that ties the baryon asymmetry to the inherited neutrino and CP structure that \BIsix{} requires, anchoring the $\eta_B\leftrightarrow\Omega_b$ chain. First cited in \S\ref{subsec:seventeen-problems}.

% ------------------- Dark energy (sec:dark-energy) -------------------

\bibitem[Garny \emph{et~al.}(2013)]{GarnyKartavtsevHohenegger2013}
\textbf{M.~Garny, A.~Kartavtsev, and A.~Hohenegger.}
Leptogenesis from first principles in the resonant regime.
\emph{Annals Phys.} \textbf{328} (2013) 26--63 [arXiv:1112.6428].
\par\noindent\textbf{Relevance.}
First-principles closed-time-path (Kadanoff--Baym) treatment of the resonant regime, deriving
the in-medium effective regulator of the self-energy asymmetry --- the class of computation the
C4 work order of Appendix~\ref{appA:leptogenesis} names as decisive. First cited in
Appendix~\ref{appA:leptogenesis}.

\bibitem[Gibbons \& Hawking(1977)]{GibbonsHawking1977}
\textbf{G.~W.~Gibbons and S.~W.~Hawking.}
Cosmological event horizons, thermodynamics, and particle creation.
\emph{Phys.\ Rev.\ D} \textbf{15} (1977) 2738--2751.
\par\noindent\textbf{Relevance.}
The 1977 Gibbons--Hawking paper established that a de~Sitter cosmological event horizon carries a temperature and an area entropy, extending black-hole thermodynamics to the cosmological horizon. \ATen{} cites it in the dark-energy section as the basis for treating the late vacuum-dominated, de~Sitter-like phase thermodynamically, and it underlies the horizon-entropy reasoning of the black-hole entropy appendix. First cited in \S\ref{sec:gg-from-zero}.

\bibitem[Guth \& Pi(1982)]{GuthPi1982}
\textbf{A.~H.~Guth and S.-Y.~Pi.}
Fluctuations in the new inflationary universe.
\emph{Phys.\ Rev.\ Lett.} \textbf{49} (1982) 1110--1113.
\par\noindent\textbf{Relevance.}
The 1982 Guth--Pi paper showed how quantum fluctuations during inflation are stretched and frozen into classical density perturbations with a calculable spectrum. \ATen{}'s inflation appendix cites it within the 1982 cluster that fixed the standard inflationary prediction of a nearly scale-invariant, Gaussian, adiabatic spectrum. First cited in \S\ref{appA:inflation}.

\bibitem[Guth(1981)]{Guth1981}
\textbf{Guth, A. H. (1981).} Inflationary universe: A possible solution to the horizon and flatness problems. \emph{Physical Review D, 23}(2), 347--356.
\par\noindent\textbf{Relevance.} The 1981 paper that introduced inflation, showing that an early epoch of accelerated expansion driven by a metastable false vacuum solves the horizon, flatness, and unwanted-relic problems of the hot Big Bang. \ATen{} cites this to set the standard problem-setting that the present paper reinterprets geometrically rather than discarding, retaining the slow-roll content of its ``new inflation'' descendants while deriving the launch geometrically, at the GUT edge of the inherited bulk. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Hawking(1971)]{Hawking1971}
\textbf{Hawking, S. W. (1971).} Gravitationally collapsed objects of very low mass. \emph{Monthly Notices of the Royal Astronomical Society, 152}(1), 75--78.
\par\noindent\textbf{Relevance.} The 1971 founding paper of primordial-black-hole physics, proposing that large density perturbations in the very early universe could collapse directly into black holes without passing through stellar evolution. \ATen{} cites this as historical background for the PBH section, which asks which slice of the resulting mass spectrum, if any, is consistent with the inherited ledger and could contribute to dark-matter, gravitational-wave, or seed-black-hole physics without conflicting with the rest of the cosmic audit. First cited in \S\ref{sec:pbh}.

\bibitem[Hawking(1975)]{Hawking1975}
\textbf{Hawking, S. W. (1975).} Particle creation by black holes. \emph{Communications in Mathematical Physics, 43}(3), 199--220.
\par\noindent\textbf{Relevance.} The 1975 paper deriving thermal Hawking radiation and establishing the temperature--horizon relation that completed black-hole thermodynamics. \ATen{} cites this as the origin of the thermal-horizon problem used as the black-hole audit in Part~V, where the evaporation ledger is reread through \GFF{} and \GDOS{} rather than taken as an isolated semiclassical result. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Hawking(1982)]{Hawking1982}
\textbf{S.~W.~Hawking.}
The development of irregularities in a single bubble inflationary universe.
\emph{Phys.\ Lett.\ B} \textbf{115} (1982) 295--297.
\par\noindent\textbf{Relevance.}
Hawking's 1982 paper is one of the founding 1982 calculations relating the inflationary spectrum of density perturbations to the slow-roll dynamics of the inflaton. \ATen{}'s inflation appendix cites it among the converging derivations that fixed the amplitude and near-scale-invariance of the primordial spectrum used downstream. First cited in \S\ref{appA:inflation}.

\bibitem[Hill et~al.(2020)]{Hill2020}
\textbf{Hill, J.~C., McDonough, E., Toomey, M.~W., \& Alexander, S. (2020).} Early dark energy does not restore cosmological concordance. \textit{Physical Review D}, \textit{102}(4), 043507.
\par\noindent\textbf{Relevance.} Demonstrates that full-likelihood early-dark-energy fits drive the net \(\Seight\) shift positive; cited as the mandatory sign caveat on the corridor's \(\Seight\) suppression branch. First cited in \S\ref{sec:s8}.

\bibitem[Hollands \& Wald(2002)]{HollandsWald2002}
\textbf{S.~Hollands and R.~M.~Wald.}
An alternative to inflation.
\emph{Gen.\ Rel.\ Grav.} \textbf{34} (2002) 2043--2055.
\par\noindent\textbf{Relevance.}
Hollands and Wald argue that inflation's explanations of homogeneity,
isotropy, and flatness are not fully satisfactory, and that a proper
account requires a deeper theory of the initial state. \ATen{} cites
this in the second-horizon-problem Section to locate the present claim
\emph{outside} that critical tradition: there the target is a deeper
half of the \emph{first} (conditions) problem, in the initial state and
its entropy, whereas here inflation's 1981 success is granted in full
and a second, differently shaped problem --- about things, not
conditions --- is posed beside it.
First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Hosotani(1983)]{Hosotani1983}
\textbf{Hosotani, Y. (1983).} Dynamical mass generation by compact extra dimensions. \emph{Physics Letters B, 126}(4), 309--313.
\par\noindent\textbf{Relevance.} The 1983 paper showing that a Wilson line wrapping a compact extra dimension can dynamically break gauge symmetry and generate mass --- the Hosotani mechanism. \ATen{} cites this as historical context for the holonomy-first reading of cosmic branch selection, in which the compact-angle Wilson line of the inherited \GG-6 bulk, rather than a local potential, fixes which branch the cosmos occupies. First cited in \S\ref{ssec:cosmic-grip-grammar-gga}.

\bibitem[Hosotani(2005)]{Hosotani2005}
\textbf{Hosotani, Y. (2005).} Dynamical gauge symmetry breaking as the Hosotani mechanism. \emph{International Journal of Modern Physics A, 20}(22), 5068--5104.
\par\noindent\textbf{Relevance.} The 2005 review consolidating gauge-Higgs unification and the dynamics of Wilson-line symmetry breaking. \ATen{} cites this as historical context for the Wilson-loop \GSSB{} and holonomy variables of the cosmic-grip grammar, supplying the established field-theory background against which the compact-angle branch coordinates of the present paper are introduced. First cited in \S\ref{ssec:cosmic-grip-grammar-gga}.

\bibitem[Hu \& Sugiyama(1995)]{HuSugiyama1995}
\textbf{W.~Hu and N.~Sugiyama.}
Anisotropies in the cosmic microwave background: an analytic approach.
\emph{Astrophys.\ J.} \textbf{444} (1995) 489--506.
\par\noindent\textbf{Relevance.}
The 1995 Hu--Sugiyama paper provided the analytic acoustic-oscillation theory --- sound horizon, baryon loading, and peak locations --- that underlies modern CMB phenomenology. \ATen{}'s CMB appendix follows their analytic treatment closely: the sound speed $c_s$, the sound horizon $r_s$, the baryon-loading parameter $R$, and the peak-location formula $\ell_n\approx n\pi D_A/r_s$ are all developed in their framework. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Hu and Sugiyama(1996)]{HuSugiyama1996}
\textbf{W.~Hu and N.~Sugiyama.}
Small-scale cosmological perturbations: an analytic approach.
\emph{The Astrophysical Journal} \textbf{471} (1996) 542--570.
\par\noindent\textbf{Relevance.}
The standard analytic fitting form for the decoupling redshift,
accurate to better than a percent against full Boltzmann codes. Used
in \S\ref{sec:two-redshifts} as the cross-check of the framework's
atomic-chain computation of $z_{\mathrm{dec}}$ --- the quantity whose
$0.1\%$ zero-parameter agreement with the measured value is that
section's result. First cited in \S\ref{sec:two-redshifts}.

% ------------------- Axion cosmology / DM (appA:kaxion / sec:kaxion) -

\bibitem[Hubble(1929)]{Hubble1929}
\textbf{Hubble, E. (1929).} A relation between distance and radial velocity among extra-galactic nebulae. \emph{Proceedings of the National Academy of Sciences, 15}(3), 168--173.
\par\noindent\textbf{Relevance.} The 1929 paper that gave the first observational evidence for cosmic expansion through a roughly linear distance-redshift relation for extra-galactic nebulae. \ATen{} cites this as historical background for the expansion-and-cooling section, which retains the Friedmann equations and the Hubble law as the leading-order operational law and places all new content in the inherited source ledger feeding the right-hand side. First cited in \S\ref{sec:introduction}.

\bibitem[Huggins \& Miller(1864)]{Huggins1864}
\textbf{W.~Huggins and W.~A.~Miller.}
On the spectra of some of the fixed stars.
\emph{Phil.\ Trans.\ R.\ Soc.} \textbf{154} (1864) 413--435.
\par\noindent\textbf{Relevance.}
The founding demonstration that the stars shine with the same spectral
lines as terrestrial elements --- the same atomic physics across the
visible universe. \ATen{} cites it as the opening datum of the second
horizon problem: the evidence for the sameness of \emph{things}
predates the evidence for the sameness of \emph{conditions} (the 1965
uniform temperature) by a full century, yet the temperature's paradox
was named within a few years and the identity's never was.
First cited in \S\ref{subsec:shp-unposed}.

\bibitem[Jacobson(1995)]{Jacobson1995}
\textbf{T.~Jacobson.}
Thermodynamics of spacetime: the Einstein equation of state.
\emph{Phys.\ Rev.\ Lett.} \textbf{75} (1995) 1260--1263.
\par\noindent\textbf{Relevance.}
Jacobson's 1995 paper derived the Einstein field equations as an equation of state from the proportionality of horizon entropy to area plus the Clausius relation, recasting gravity as thermodynamics. \ATen{} cites it in the beginning/dark-energy sections as the conceptual precedent for reading the Einstein/Friedmann equations as a coarse-grained seam-level balance rather than a fundamental postulate, resonant with the \CCI{} interpretation of FRW continuity. First cited in \S\ref{sec:gg-from-zero}.

\bibitem[Jaynes(1957)]{Jaynes1957}
\textbf{Jaynes, E. T. (1957).} Information theory and statistical mechanics. \emph{Physical Review, 106}(4), 620--630.
\par\noindent\textbf{Relevance.} The 1957 paper recasting equilibrium statistical mechanics as maximum-entropy inference under constraints, bridging entropy, information, and constrained maximization. \ATen{} cites this for orientation in the discussion of \GFF{}, the Qi energy-principle, and inner reconstruction; the maximum-entropy reading is a conceptual analogue for the free-energy bookkeeping, not a derivational premise of the cosmology. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Jukkala \emph{et~al.}(2021)]{JukkalaKainulainenRahkila2021}
\textbf{H.~Jukkala, K.~Kainulainen, and P.~M.~Rahkila.}
Flavour mixing transport theory and resonant leptogenesis.
\emph{JHEP} \textbf{09} (2021) 119 [arXiv:2104.03998].
\par\noindent\textbf{Relevance.}
A local quantum transport theory for the mixing heavy sector, resolving the coherence
information that the flavor-measurement hypothesis of Appendix~\ref{appA:leptogenesis} turns
on; among the sharpest existing statements of how the heavy-pair coherence is damped in-medium.
First cited in Appendix~\ref{appA:leptogenesis}.

\bibitem[Kanemura and Li(2025)]{KanemuraLi2025}
\textbf{S.~Kanemura and S.-P.~Li.}
Resonant forbidden CP asymmetry from soft leptons.
\emph{Phys.\ Rev.\ Lett.} \textbf{134} (2025) 211802.
\par\noindent\textbf{Relevance.}
Opens the 2025--26 thermal-resonant line: thermal resummation of \emph{Standard-Model soft
leptons} produces a resonantly enhanced CP asymmetry stable under finite-temperature
perturbation theory --- a flavor-odd source clocked by lepton-sector rates. Cited in
Appendix~\ref{appA:leptogenesis} as published support for the structure of conditions C1--C2
(and as the opening of the competitor-mechanism question the C4 work order registers). First
cited in Appendix~\ref{appA:leptogenesis}.

\bibitem[Kibble(1976)]{Kibble1976}
\textbf{T.~W.~B.~Kibble.}
Topology of cosmic domains and strings.
\emph{J.\ Phys.\ A} \textbf{9} (1976) 1387--1398.
\par\noindent\textbf{Relevance.}
Kibble's 1976 paper made the causal argument general and quantitative:
at any cosmological symmetry-breaking transition the vacuum choice is
made independently in regions with no prior causal contact, the
correlation length is bounded by the horizon, and defects form where the
choices disagree. \ATen{} cites it in the second-horizon-problem
pedigree as the sharpest early posing of the identity question --- for
the \emph{orientation} of a degenerate vacuum, framed as a
relic-abundance problem --- fifty years before the question is named and
answered in its own right in \S\ref{sec:second-horizon}.
First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Klari\'c \emph{et~al.}(2021)]{KlaricShaposhnikovTimiryasov2021}
\textbf{J.~Klari\'c, M.~Shaposhnikov, and I.~Timiryasov.}
Uniting low-scale leptogenesis mechanisms.
\emph{Phys.\ Rev.\ Lett.} \textbf{127} (2021) 111802 [arXiv:2008.13771].
\par\noindent\textbf{Relevance.}
Demonstrates that resonant leptogenesis and leptogenesis via oscillations are limits of one
kinetic framework spanning the GeV--TeV heavy-mass range that includes the lock scale; the
unified equations are a benchmark any C4-discharging computation must reduce to. First cited in
Appendix~\ref{appA:leptogenesis}.

\bibitem[Kodama \& Sasaki(1984)]{KodamaSasaki1984}
\textbf{H.~Kodama and M.~Sasaki.}
Cosmological perturbation theory.
\emph{Prog.\ Theor.\ Phys.\ Suppl.} \textbf{78} (1984) 1--166.
\par\noindent\textbf{Relevance.}
Kodama and Sasaki's 1984 review is the comprehensive systematic reference for the gauge-invariant multi-fluid theory of cosmological perturbations. \ATen{}'s growth appendix follows its organization of the continuity, Euler, and Poisson equations into the second-order master growth equation; the framework adopts this standard apparatus unmodified and acts only at the late-time normalization. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Kolb \& Turner(1990)]{KolbTurner1990}
\textbf{E.~W.~Kolb and M.~S.~Turner.}
\emph{The Early Universe}.
Addison-Wesley, Redwood City, (1990).
\par\noindent\textbf{Relevance.}
Kolb and Turner's 1990 monograph is the canonical reference for early-universe thermodynamics: the relativistic degree-of-freedom counting $g_*(T)$, entropy conservation $g_{*s}a^3T^3=\mathrm{const}$, freeze-out, and the relic-abundance machinery. \ATen{} uses it throughout the thermal-history appendix and in the Kaxion and BBN appendices (freeze-out temperatures, the neutron-to-proton ratio, and the misalignment relic estimate). First cited in \S\ref{sec:lcdm-success}.

\bibitem[Kullback and Leibler(1951)]{KullbackLeibler1951}
\textbf{Kullback, S., \& Leibler, R. A. (1951).} On information and sufficiency. \emph{Annals of Mathematical Statistics, 22}(1), 79--86.
\par\noindent\textbf{Relevance.} The 1951 paper introducing the relative-entropy (Kullback--Leibler) divergence as a measure of statistical distinguishability. \ATen{} cites this for the entropy and inference language surrounding the \CEP{} sector of the free-energy functional \(F_{\GFF}\), where divergence-style bookkeeping frames the cost of distinguishing boundary states; it is orienting vocabulary, not a derivational input. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Kuzmin \emph{et~al.}(1985)]{KuzminRubakovShaposhnikov1985}
\textbf{V.~A.~Kuzmin, V.~A.~Rubakov, and M.~E.~Shaposhnikov.}
On anomalous electroweak baryon-number non-conservation in the early universe.
\emph{Phys.\ Lett.\ B} \textbf{155} (1985) 36--42.
\par\noindent\textbf{Relevance.}
The 1985 Kuzmin--Rubakov--Shaposhnikov paper established that electroweak sphaleron processes violate baryon-plus-lepton number and are unsuppressed at high temperature, a cornerstone of modern baryogenesis. \ATen{}'s baryogenesis section reads sphaleron conversion as one of the seam-level source terms of the open projected ledger and cites this paper for the sphaleron transport physics. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Lakatos(1978)]{Lakatos1978}
\textbf{Lakatos, I. (1978).} \emph{The Methodology of Scientific Research Programmes: Philosophical Papers, Volume~1} (eds.\ J.~Worrall \& G.~Currie). Cambridge University Press.
\par\noindent\textbf{Relevance.} The distinction between \emph{progressive} and \emph{degenerating} research programs --- forecasting novel facts versus accommodating known ones by adjustment. \ATen{} cites it in Objection~O-Cosmos-90 to state the verdict on the vacuum-selection program precisely: non-progressive rather than refuted. First cited in \S\ref{obj:landscape-priced}.

\bibitem[Landauer(1961)]{Landauer1961}
\textbf{Landauer, R. (1961).} Irreversibility and heat generation in the computing process. \emph{IBM Journal of Research and Development, 5}(3), 183--191.
\par\noindent\textbf{Relevance.} The 1961 paper establishing the thermodynamic cost of logically irreversible operations --- the Landauer bound linking erasure to dissipation. \ATen{} cites this as external context for why information-bearing subsystems in the boundary runtime still obey energetic and entropic bookkeeping, reinforcing the \GFF{} ledger framing without entering the cosmological derivation as a premise. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Lema\^itre(1927)]{Lemaitre1927}
\textbf{Lema\^itre, G. (1927).} Un univers homog\`ene de masse constante et de rayon croissant rendant compte de la vitesse radiale des n\'ebuleuses extra-galactiques. \emph{Annales de la Soci\'et\'e Scientifique de Bruxelles, A47}, 49--59.
\par\noindent\textbf{Relevance.} The 1927 paper that independently derived the expansion law from observational considerations and supplied the operational language of a beginning later sharpened into the primeval atom. \ATen{} cites this as historical background for the beginning-of-universe and expansion-and-cooling sections, both of which retain the FRW dynamics of Friedmann and Lema\^itre intact and ask only the narrower question of at what scale the first physically meaningful boundary surface is initialized. First cited in \S\ref{sec:introduction}.

\bibitem[Li and Pilaftsis(2026a)]{LiPilaftsis2026a}
\textbf{S.-P.~Li and A.~Pilaftsis.}
Dominant thermal resonant mechanism for low-scale leptogenesis.
arXiv:2601.15921 [hep-ph] (2026).
\par\noindent\textbf{Relevance.}
The thermal-resonant mechanism in full: flavor-odd CP sources clocked by charged-lepton Yukawa
rates, with gauge damping dropping out of the CP-odd channel --- the strongest published support
for the lepton-sector clock of Appendix~\ref{appA:leptogenesis}, while leaving the cross-sector
matching of condition C4 open. Registered there as a potential competitor to the
\(\Delta M\)-filtered source at \(T\sim M_N\). First cited in
Appendix~\ref{appA:leptogenesis}.

\bibitem[Li and Pilaftsis(2026b)]{LiPilaftsis2026b}
\textbf{S.-P.~Li and A.~Pilaftsis.}
Low-scale leptogenesis from resonant thermal lepton flavour coherences.
arXiv:2604.06493 [hep-ph] (2026).
\par\noindent\textbf{Relevance.}
Companion of the preceding entry: the resonant enhancement carried by \emph{thermal lepton
flavor coherences} rather than heavy-sector quasi-degeneracy. Cited with it in the C4 work
order of Appendix~\ref{appA:leptogenesis}. First cited in Appendix~\ref{appA:leptogenesis}.

\bibitem[Liddle(2015)]{Liddle2015}
\textbf{A.~R.~Liddle.}
\emph{An Introduction to Modern Cosmology}, 3rd ed.
Wiley, Chichester, (2015).
\par\noindent\textbf{Relevance.}
A gentle, physically transparent first course on the expanding universe and the inventory; recommended as the first-pass companion to Appendices~E and~F for students meeting cosmology for the first time. First cited in \S\ref{app:abbrev}.

\bibitem[Liddle \& Lyth(2000)]{LiddleLyth2000}
\textbf{A.~R.~Liddle and D.~H.~Lyth.}
\emph{Cosmological Inflation and Large-Scale Structure}.
Cambridge University Press, Cambridge, (2000).
\par\noindent\textbf{Relevance.}
The standard in-depth treatment of the inflationary sector and its perturbation output; recommended companion to Appendix~G, whose Step-1 slow-roll apparatus follows its conventions before the geometric Step~3 replaces the potential's freedom with the derived plateau. First cited in \S\ref{app:abbrev}.

\bibitem[Lifshitz(1946)]{Lifshitz1946}
\textbf{E.~M.~Lifshitz.}
On the gravitational stability of the expanding universe.
\emph{J.\ Phys.\ (USSR)} \textbf{10} (1946) 116.
\par\noindent\textbf{Relevance.}
Lifshitz's 1946 paper is the foundational relativistic treatment of the gravitational instability of an expanding universe, establishing that density perturbations grow as a power law rather than exponentially in a matter-dominated FRW background. \ATen{}'s growth appendix recovers exactly this result --- the growing-mode $D\propto a$ in the Einstein--de~Sitter regime --- as the leading line of the linear-growth derivation. % [FLAG] Page citation for the J. Phys. USSR 10 (1946) 116 volume; the companion ZhETF 16 (1946) 587 (Russian) reference is sometimes quoted instead.
First cited in \S\ref{sec:lcdm-success}.

\bibitem[Linde(1982)]{Linde1982}
\textbf{Linde, A. D. (1982).} A new inflationary universe scenario. \emph{Physics Letters B, 108}(6), 389--393.
\par\noindent\textbf{Relevance.} The 1982 ``new inflation'' paper that replaced the false-vacuum picture of \citet{Guth1981} with a slow-rolling scalar field, repairing the graceful-exit problem. \ATen{} cites this to frame the standard scalar-potential paradigm that the present paper retains in slow-roll form but replaces at the level of mechanism by a geometric launch-and-exit picture, fixing the inflationary modulus within the inherited bulk-modulus space. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Linde(1983)]{Linde1983}
\textbf{A.~D.~Linde.}
Chaotic inflation.
\emph{Phys.\ Lett.\ B} \textbf{129} (1983) 177--181.
\par\noindent\textbf{Relevance.}
Linde's 1983 chaotic-inflation paper showed that successful inflation does not require a fine-tuned false vacuum but can proceed from generic large-field initial conditions. \ATen{}'s inflation appendix cites it as the alternative large-field paradigm against which the framework's inherited plateau-type stage and its predicted small $r$ are contrasted. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Linde(1990)]{Linde1990}
\textbf{A.~D.~Linde.}
\emph{Particle Physics and Inflationary Cosmology}.
Harwood Academic, Chur (1990); arXiv:hep-th/0503203.
\par\noindent\textbf{Relevance.}
Linde's monograph lists ``the problem of the uniqueness of the
universe'' --- why the entire universe sits in the same broken-symmetry
minimum --- among the problems inflation addresses, and contains the
nearest published approach to the second horizon problem: the remark
that no anthropic selection explains why all properties of the universe
are approximately uniform over its entire observable part. \ATen{}
cites it in the pedigree of \S\ref{sec:second-horizon} as the closest
near miss: the sentence was written, and not followed --- the uniformity
of \emph{things} never joined the canonical problem list, and the
answer on offer remained spatial (one stretched domain), not temporal
(the ordering).
First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Lyth \& Riotto(1999)]{LythRiotto1999}
\textbf{D.~H.~Lyth and A.~Riotto.}
Particle physics models of inflation and the cosmological density perturbation.
\emph{Phys.\ Rept.} \textbf{314} (1999) 1--146.
\par\noindent\textbf{Relevance.}
The 1999 Lyth--Riotto review is the standard reference connecting particle-physics model building to the observable inflationary perturbation spectrum, including the slow-roll consistency relations $n_s=1-6\epsilon+2\eta$ and $r=16\epsilon$. \ATen{}'s inflation appendix uses exactly these relations and cites this review for the systematic mapping from a potential $V(\phi)$ to $(n_s,r)$. First cited in \S\ref{sec:lcdm-success}.

% ------------------- BBN (appA:bbn) ----------------------------------

\bibitem[Lyttleton \& Bondi(1959)]{LyttletonBondi1959}
\textbf{R.~A.~Lyttleton and H.~Bondi.}
Physical consequences of a general excess of charge.
\emph{Proc.\ Roy.\ Soc.\ Lond.\ A} \textbf{252} (1959) 313.
\par\noindent\textbf{Relevance.}
The classic charged cosmology: a proton--electron mismatch of two
parts in \(10^{18}\) would drive the expansion electrostatically. The
proposal marks the first serious recognition that a gauged excess
cannot hide, and the neutrality theorem of \S\ref{ssec:neutrality}
states why the excess is exactly zero rather than merely small.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{Maldacena, J. (1998).} The large-\(N\) limit of superconformal field theories and supergravity. \emph{Advances in Theoretical and Mathematical Physics, 2}(2), 231--252.
\par\noindent\textbf{Relevance.} The 1998 paper conjecturing the AdS/CFT correspondence, the canonical statement of holographic gauge-gravity duality. \ATen{} cites this as context for the black-hole and hologram discussion that situates the Part~V horizon audit and the island comparison; it frames the holographic literature without being used as a premise of the present derivation. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Maldacena(1999)]{Maldacena1999}
\textbf{Maldacena, J. (1999).} Eternal black holes in anti-de Sitter. \emph{Journal of High Energy Physics, 2003}(04), 021.
\par\noindent\textbf{Relevance.} The paper on eternal AdS black holes and the thermofield-double construction relating two boundary copies through a shared horizon. \ATen{} cites this to situate the horizon discussion of Part~V within the broader holographic literature on black-hole structure; like \citet{Maldacena1998} it is comparative context, not a derivational input. First cited in \S\ref{subsec:bh-islands-comparison}.

\bibitem[Marsh(2016)]{Marsh2016}
\textbf{D.~J.~E.~Marsh.}
Axion cosmology.
\emph{Phys.\ Rept.} \textbf{643} (2016) 1--79.
\par\noindent\textbf{Relevance.}
Marsh's 2016 review is the comprehensive modern reference for axion cosmology, covering the misalignment mechanism, structure formation with axion dark matter, and observational constraints. \ATen{}'s Kaxion appendix cites it for the up-to-date relic-abundance formulae and the cosmological constraints that any Kaxion dark-matter candidate must satisfy. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Misner \emph{et~al.}(1973)]{MisnerThorneWheeler1973}
\textbf{C.~W.~Misner, K.~S.~Thorne, and J.~A.~Wheeler.}
\emph{Gravitation}.
W.~H.~Freeman, San Francisco, (1973).
\par\noindent\textbf{Relevance.}
The 1973 ``MTW'' text is the standard reference for the tensor-calculus machinery (metric, connection, curvature, Bianchi identities) on which the \FRW{} background appendix and the black-hole entropy appendix rest. \ATen{} follows its conventions in carrying out the explicit Christoffel-symbol and Einstein-tensor computation without skipping index work; the contracted Bianchi identity it supplies underwrites the cohort reading of FRW continuity as the isotropic reduction of \CCI{}. First cited in \S\ref{sec:lcdm-success}.

% ------------------- Perturbations / growth (appA:growth) ------------

\bibitem[Mukhanov \& Chibisov(1981)]{MukhanovChibisov1981}
\textbf{V.~F.~Mukhanov and G.~V.~Chibisov.}
Quantum fluctuations and a nonsingular universe.
\emph{JETP Lett.} \textbf{33} (1981) 532--535.
\par\noindent\textbf{Relevance.}
The 1981 Mukhanov--Chibisov paper is the first calculation of the quantum-generated, nearly scale-invariant spectrum of cosmological perturbations from inflation. \ATen{}'s inflation appendix recovers the same curvature-perturbation spectrum and uses it to seed the growth appendix; the framework adopts the quantum origin of the spectrum and reads it as a boundary projection. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Mukhanov(2005)]{Mukhanov2005}
\textbf{V.~Mukhanov.}
\emph{Physical Foundations of Cosmology}.
Cambridge University Press, Cambridge, (2005).
\par\noindent\textbf{Relevance.}
Mukhanov's 2005 textbook is a primary reference for the gauge-invariant theory of cosmological perturbations and the quantum origin of inflationary fluctuations. \ATen{} draws on it in the inflation appendix (slow-roll parameters, the curvature-perturbation spectrum, $n_s$ and $r$) and the growth appendix; the framework adopts the textbook generation of the primordial spectrum and reads it as the boundary projection of the inherited bulk inflationary stage. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Mukhanov \emph{et~al.}(1992)]{MukhanovFeldmanBrandenberger1992}
\textbf{V.~F.~Mukhanov, H.~A.~Feldman, and R.~H.~Brandenberger.}
Theory of cosmological perturbations.
\emph{Phys.\ Rept.} \textbf{215} (1992) 203--333.
\par\noindent\textbf{Relevance.}
This 1992 Physics Reports review unified the gauge-invariant perturbation formalism with the quantum theory of inflationary fluctuations and the late-time growth of structure. \ATen{} cites it as the connective reference linking the inflation appendix (origin of the primordial spectrum) to the growth appendix (its subsequent evolution into the matter power spectrum and \sigeight). First cited in \S\ref{sec:century-cosmology}.

\bibitem[Page(1993)]{Page1993}
\textbf{D.~N.~Page.}
Information in black hole radiation.
\emph{Phys.\ Rev.\ Lett.} \textbf{71} (1993) 3743--3746.
\par\noindent\textbf{Relevance.}
Page's 1993 paper introduced the ``Page curve'' for the entanglement entropy of Hawking radiation, the benchmark statement of unitary black-hole evaporation and information recovery. \ATen{}'s black-hole information discussion and the entropy appendix cite it for the Page-curve criterion that any information-preserving evaporation history --- including the cohort's bulk-boundary reading --- must reproduce. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Peacock(1999)]{Peacock1999}
\textbf{J.~A.~Peacock.}
\emph{Cosmological Physics}.
Cambridge University Press, Cambridge, (1999).
\par\noindent\textbf{Relevance.}
The encyclopaedic one-volume treatment of relativistic cosmology and structure formation; recommended for in-depth study alongside Appendices~E, F, and~J. First cited in \S\ref{app:abbrev}.

\bibitem[Peccei and Quinn(1977)]{PecceiQuinn1977}
\textbf{Peccei, R. D., \& Quinn, H. R. (1977).} CP conservation in the presence of pseudoparticles. \emph{Physical Review Letters, 38}(25), 1440--1443.
\par\noindent\textbf{Relevance.} The 1977 paper that solved the strong-CP problem by introducing a new global U(1) symmetry whose spontaneous breaking dynamically relaxes the QCD vacuum angle to zero. \ATen{} cites this as historical background for the Kaxion dark-matter section, whose compact-angle pseudo-scalar inherits the Peccei--Quinn template through the \GG-6 compact holonomy rather than through a postulated low-energy U(1). First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Peebles \& Ratra(2003)]{PeeblesRatra2003}
\textbf{P.~J.~E.~Peebles and B.~Ratra.}
The cosmological constant and dark energy.
\emph{Rev.\ Mod.\ Phys.} \textbf{75} (2003) 559--606.
\par\noindent\textbf{Relevance.}
The 2003 Peebles--Ratra review is the authoritative survey of the cosmological constant and dynamical dark-energy (quintessence) models. \ATen{}'s dark-energy section cites it to delimit the menu of standard explanations --- $\Lambda$, quintessence, modified gravity --- against which the cohort's one-ledger reading of the late vacuum residue is set. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Peebles \& Yu(1970)]{PeeblesYu1970}
\textbf{P.~J.~E.~Peebles and J.~T.~Yu.}
Primeval adiabatic perturbation in an expanding universe.
\emph{Astrophys.\ J.} \textbf{162} (1970) 815--836.
\par\noindent\textbf{Relevance.}
Peebles and Yu's 1970 paper predicted the baryon acoustic oscillation imprint in both the matter and the photon power spectra, establishing the sound horizon as a standard ruler. \ATen{} cites it in the expansion section as the theoretical origin of the BAO scale that the two-stage Hubble corridor must respect, and it underlies the CMB acoustic-peak appendix. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Peebles(1993)]{Peebles1993}
\textbf{P.~J.~E.~Peebles.}
\emph{Principles of Physical Cosmology}.
Princeton University Press, Princeton, (1993).
\par\noindent\textbf{Relevance.}
Peebles' 1993 treatise is a foundational reference spanning the FRW background, thermal history, BBN, and the statistics of large-scale structure. \ATen{} cites it as a pedagogical and historical anchor for the \FRW{} and thermal-history appendices and for the definition of the matter power spectrum and its normalization (\sigeight) in the growth appendix. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Penrose(1994)]{Penrose1994}
\textbf{Penrose, R. (1994).} \emph{Shadows of the Mind}. Oxford, UK: Oxford University Press.
\par\noindent\textbf{Relevance.} Penrose's 1994 book arguing that gravitation and a non-computable element are essential to consciousness, with objective state reduction set near a gravitational threshold. \ATen{} cites this for the black-hole and consciousness context surrounding the later pagoda and \EIC{} discussion; it is an orienting reference for the boundary-runtime framing, not a derivational input. First cited in \S\ref{sec:partvii-summary}.

\bibitem[Penrose(2004)]{Penrose2004}
\textbf{R.~Penrose.}
\emph{The Road to Reality}.
Jonathan Cape, London (2004).
\par\noindent\textbf{Relevance.}
Penrose's critique of inflationary explanation --- that appealing to a
thermalization process cannot explain the specialness of the initial
state, whose entropy must have been extraordinarily low --- is the
best-known statement of the critical tradition that seeks a deeper half
of the \emph{first} horizon problem. \ATen{} cites it in the
second-horizon-problem Section to draw the boundary: the present claim
grants inflation's 1981 success and poses a second, differently shaped
problem about identity, rather than reopening the conditions problem
Penrose targets.
First cited in \S\ref{subsec:shp-pedigree}.

\bibitem[Penzias and Wilson(1965)]{PenziasWilson1965}
\textbf{Penzias, A. A., \& Wilson, R. W. (1965).} A measurement of excess antenna temperature at 4080 Mc/s. \emph{Astrophysical Journal, 142}, 419--421.
\par\noindent\textbf{Relevance.} The 1965 serendipitous discovery of the cosmic microwave background as an isotropic ``excess antenna temperature at 4080~Mc/s''. \ATen{} cites this, together with the companion interpretation of \citet{DickePeebles1965}, as the historical opening of the CMB section, which retains the standard acoustic, damping, and projection structure of the microwave background and asks only where the inherited ledger reshapes its source assumptions. First cited in \S\ref{sec:introduction}.

\bibitem[Perlmutter et~al.(1999)]{Perlmutter1999}
\textbf{Perlmutter, S., et~al. (1999).} Measurements of \(\Omega\) and \(\Lambda\) from 42 high-redshift supernovae. \emph{The Astrophysical Journal, 517}(2), 565--586.
\par\noindent\textbf{Relevance.} The 1999 Supernova Cosmology Project measurement that, with \citet{Riess1998}, established the accelerating late-time expansion and the observational reintroduction of \(\Lambda\). \ATen{} cites this as the historical evidence for cosmic acceleration behind the dark-energy benchmark inherited by the paper, whose section retains the empirical \(\Lambda\) but reinterprets its scale through the residual \GFF{} ledger of the inherited bulk. First cited in \S\ref{sec:introduction}.

\bibitem[Pitrou \emph{et~al.}(2018)]{Pitrou2018}
\textbf{C.~Pitrou, A.~Coc, J.-P.~Uzan, and E.~Vangioni.}
Precision big bang nucleosynthesis with improved Helium-4 predictions.
\emph{Phys.\ Rept.} \textbf{754} (2018) 1--66.
\par\noindent\textbf{Relevance.}
The 2018 Pitrou \emph{et~al.} review provides the most precise modern BBN predictions, in particular a high-accuracy $^4\mathrm{He}$ computation including subtle weak-rate and radiative corrections. \ATen{}'s BBN appendix cites it as the precision benchmark for $Y_p$ and for the neutron lifetime and freeze-out treatment underlying the helium yield. First cited in \S\ref{sec:lcdm-success}.

% ------------------- CMB (appA:cmb) ----------------------------------

\bibitem[Planck Collaboration(2018)]{Planck2018}
\textbf{Planck Collaboration. (2018).} \emph{Planck 2018 results. VI. Cosmological parameters}. ESA/Planck legacy release.
\par\noindent\textbf{Relevance.} The Planck 2018 cosmological-parameter release, the canonical high-precision \(\Lambda\)CDM determination of \(\Hzero\), \(\Omm\), \(\OmL\), \(\Seight\), and the CMB acoustic structure. \ATen{} uses this as the principal empirical baseline that the present paper seeks to inherit and reinterpret: it supplies the early-branch anchors in the codebook and the parameter-audit Dev plot against which the two-branch corridor predictions are compared. First cited in \S\ref{sec:dark-energy}.

\bibitem[Poulin et~al.(2019)]{Poulin2019}
\textbf{Poulin, V., Smith, T.~L., Karwal, T., \& Kamionkowski, M. (2019).} Early dark energy can resolve the Hubble tension. \textit{Physical Review Letters}, \textit{122}(22), 221301.
\par\noindent\textbf{Relevance.} Source of the early-dark-energy response slopes (\(\sim 0.4\) per unit \(f_{\mathrm{EDE}}\)) adopted as representative coefficients in the corridor's working linear-response form; the corridor module is, at present, a structured re-parametrization of this class of models. First cited in \S\ref{sec:h0tension}.

\bibitem[Preskill \emph{et~al.}(1983)]{PreskillWiseWilczek1983}
\textbf{J.~Preskill, M.~B.~Wise, and F.~Wilczek.}
Cosmology of the invisible axion.
\emph{Phys.\ Lett.\ B} \textbf{120} (1983) 127--132.
\par\noindent\textbf{Relevance.}
One of the three founding 1983 papers establishing that a light axion-like field acquires a cosmological cold-dark-matter abundance through the vacuum-misalignment mechanism. \ATen{}'s Kaxion appendix builds exactly this mechanism --- the field frozen by Hubble friction at angle $\theta_i$ until $H\sim m$, then oscillating with $\rho\propto a^{-3}$ --- to estimate the Kaxion relic density. First cited in \S\ref{appA:kaxion}.

\bibitem[Press \& Schechter(1974)]{PressSchechter1974}
\textbf{W.~H.~Press and P.~Schechter.}
Formation of galaxies and clusters of galaxies by self-similar gravitational condensation.
\emph{Astrophys.\ J.} \textbf{187} (1974) 425--438.
\par\noindent\textbf{Relevance.}
The 1974 Press--Schechter formalism is the foundational analytic prescription for the abundance of collapsed halos as a function of the linear density-field variance. \ATen{}'s growth appendix invokes it to motivate why the smoothed amplitude \sigeight{} (and hence \Seight) is the cosmologically decisive number: halo counts depend exponentially on it, which is exactly why the clustering tension is sharp. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Randall and Sundrum(1999)]{RandallSundrum1999}
\textbf{Randall, L., \& Sundrum, R. (1999).} A large mass hierarchy from a small extra dimension. \emph{Physical Review Letters, 83}(17), 3370--3373.
\par\noindent\textbf{Relevance.} The 1999 paper that generated a large mass hierarchy from a extra dimension, with physical scales redshifted exponentially along the throat. \ATen{} cites this for intuition about throat geometry and gravitational redshift, while the actual scale-geometry derivation used in the architecture section remains internal to the inherited \GG-6 locking and is not imported from the Randall--Sundrum construction. First cited in \S\ref{sec:architecture}.

\bibitem[Riess et~al.(1998)]{Riess1998}
\textbf{Riess, A. G., et~al. (1998).} Observational evidence from supernovae for an accelerating universe and a cosmological constant. \emph{The Astronomical Journal, 116}(3), 1009--1038.
\par\noindent\textbf{Relevance.} The 1998 High-z Supernova Search measurement that, with \citet{Perlmutter1999}, first established the accelerating expansion and reintroduced \(\Lambda\) observationally. \ATen{} cites this as the standard evidence for cosmic acceleration and the late-time vacuum-inventory problem addressed in Part~II, where the residual \GFF{} ledger is asked to account for the inherited \(\Lambda\) scale. First cited in \S\ref{sec:introduction}.

\bibitem[Riess et~al.(2022)]{Riess2022}
\textbf{Riess, A.~G., Yuan, W., Macri, L.~M., et al. (2022).} A comprehensive measurement of the local value of the Hubble constant with 1 km s\(^{-1}\) Mpc\(^{-1}\) uncertainty from the Hubble Space Telescope and the SH0ES team. \textit{The Astrophysical Journal Letters}, \textit{934}(1), L7.
\par\noindent\textbf{Relevance.} The SH0ES distance-ladder determination \(\Hzero = 73.04 \pm 1.04\) km/s/Mpc, quoted in this paper (rounded to \(73.0 \pm 1.0\)) as the leading single-team ladder determination; the codebook and the parameter-audit Dev plot score the late branch against the H0DN consensus \citep{CasertanoH0DN2026}. Distinct from the 1998 acceleration-discovery paper by the same first author, which is cited separately for the historical reintroduction of \(\Lambda\). First cited in \S\ref{sec:cosmology-challenges}.

\bibitem[Ryden(2017)]{Ryden2017}
\textbf{B.~Ryden.}
\emph{Introduction to Cosmology}, 2nd ed.
Cambridge University Press, Cambridge, (2017).
\par\noindent\textbf{Relevance.}
The friendliest rigorous introduction to FRW kinematics, the inventory, and the thermal history; recommended as the gentlest entry point to the graduate course of Part~XV (Appendices~E and~F). First cited in \S\ref{app:abbrev}.

\bibitem[Sachs \& Wolfe(1967)]{SachsWolfe1967}
\textbf{R.~K.~Sachs and A.~M.~Wolfe.}
Perturbations of a cosmological model and angular variations of the microwave background.
\emph{Astrophys.\ J.} \textbf{147} (1967) 73--90.
\par\noindent\textbf{Relevance.}
The 1967 Sachs--Wolfe paper established how gravitational-potential perturbations at last scattering imprint temperature anisotropies on the CMB, fixing the large-angle (low-$\ell$) plateau. \ATen{}'s CMB appendix uses the Sachs--Wolfe relation as the boundary condition that sets the zero point of the acoustic-peak structure below the first peak. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{Sakharov, A. D. (1967).} Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe. \emph{JETP Letters, 5}, 24--27.
\par\noindent\textbf{Relevance.} The 1967 paper stating the three necessary conditions --- baryon-number violation, C and CP violation, and departure from thermal equilibrium --- that any dynamical baryogenesis must satisfy. \ATen{} cites this as historical background for the baryogenesis section, which asks not which mechanism is freely chosen but which Sakharov-compatible realization is admissible once the neutrino sector and the CP ledger have been fixed by the upstream particle paper. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Shannon(1948)]{Shannon1948}
\textbf{Shannon, C. E. (1948).} A mathematical theory of communication. \emph{Bell System Technical Journal, 27}(3), 379--423.
\par\noindent\textbf{Relevance.} The 1948 founding paper of information theory, defining entropy as the measure of a source's information content and the channel-capacity limit on reliable transmission. \ATen{} cites this for the OH/IH/\EIC{} language and the measurement/reconstruction interpretation of the boundary runtime in the later parts; it supplies the information-theoretic vocabulary, not a derivational premise of the cosmology. First cited in \S\ref{sec:partVIII-why-here}.

\bibitem[Sikivie(1983)]{Sikivie1983}
\textbf{P.~Sikivie.}
Experimental tests of the invisible axion.
\emph{Phys.\ Rev.\ Lett.} \textbf{51} (1983) 1415--1417.
\par\noindent\textbf{Relevance.}
Sikivie's 1983 paper proposed the haloscope and helioscope detection strategies that define the experimental axion program. \ATen{}'s Kaxion appendix cites it for the experimental detectability of a misalignment dark-matter axion, relevant to whether the Kaxion is observationally accessible. First cited in \S\ref{appA:kaxion}.

\bibitem[Silk(1968)]{Silk1968}
\textbf{J.~Silk.}
Cosmic black-body radiation and galaxy formation.
\emph{Astrophys.\ J.} \textbf{151} (1968) 459--471.
\par\noindent\textbf{Relevance.}
Silk's 1968 paper identified the photon-diffusion (``Silk'') damping that exponentially suppresses CMB anisotropies below the diffusion scale, producing the high-$\ell$ damping tail. \ATen{}'s CMB appendix derives the Silk damping scale and the damping-tail multipole $\ell_D$ directly, citing this paper as the physical origin. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Smoot \emph{et~al.}(1992)]{SmootCOBE1992}
\textbf{G.~F.~Smoot \emph{et~al.} (COBE Collaboration).}
Structure in the COBE differential microwave radiometer first-year maps.
\emph{Astrophys.\ J.\ Lett.} \textbf{396} (1992) L1--L5.
\par\noindent\textbf{Relevance.}
The 1992 COBE/DMR detection of CMB temperature anisotropies at the $10^{-5}$ level is the landmark confirmation that primordial perturbations of the predicted amplitude exist. \ATen{}'s CMB appendix cites it as the empirical anchor for the large-angle Sachs--Wolfe plateau and for the overall normalization of the primordial spectrum. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Starobinsky(1979)]{Starobinsky1979}
\textbf{Starobinsky, A. A. (1979).} Spectrum of relict gravitational radiation and the early state of the universe. \emph{JETP Letters, 30}, 682--685.
\par\noindent\textbf{Relevance.} The 1979 paper that first computed the spectrum of relict primordial gravitational waves from a de~Sitter epoch and argued that they survive as a present-day stochastic background detectable through CMB B-modes or direct interferometry. \ATen{} cites this as historical background for the primordial-gravitational-waves section, which uses the Starobinsky-channel amplitude-versus-scale argument that retired the once-assumed TeV-scale reading in favor of the derived GUT-adjacent plateau, turning the tensor sector into a sharp falsification target. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Starobinsky(1980)]{Starobinsky1980}
\textbf{A.~A.~Starobinsky.}
A new type of isotropic cosmological models without singularity.
\emph{Phys.\ Lett.\ B} \textbf{91} (1980) 99--102.
\par\noindent\textbf{Relevance.}
Starobinsky's 1980 $R^2$ model is the first concrete inflationary scenario and remains among the best fits to the measured scalar spectral index and tensor-to-scalar ratio. \ATen{}'s inflation appendix references it as the prototype plateau potential and as the benchmark $(n_s,r)$ target against which the framework's inherited inflationary stage is compared. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Steigman(2007)]{Steigman2007}
\textbf{G.~Steigman.}
Primordial nucleosynthesis in the precision cosmology era.
\emph{Ann.\ Rev.\ Nucl.\ Part.\ Sci.} \textbf{57} (2007) 463--491.
\par\noindent\textbf{Relevance.}
Steigman's 2007 review is the standard precision-era reference for the relation between the baryon-to-photon ratio $\eta$, the baryon density $\Omega_b h^2$, and the primordial $^4\mathrm{He}$ and deuterium abundances. \ATen{}'s BBN appendix uses its tight $Y_p(\eta)$ and $\mathrm{D/H}(\eta)$ relations as the link in the $\eta_B\leftrightarrow\Omega_b$ consistency chain that the baryogenesis section must satisfy. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Sunyaev \& Zeldovich(1970)]{SunyaevZeldovich1970}
\textbf{R.~A.~Sunyaev and Ya.~B.~Zeldovich.}
Small-scale fluctuations of relic radiation.
\emph{Astrophys.\ Space Sci.} \textbf{7} (1970) 3--19.
\par\noindent\textbf{Relevance.}
The 1970 Sunyaev--Zeldovich paper predicted the acoustic oscillations in the photon-baryon fluid that appear as the harmonic peak structure of the CMB power spectrum. \ATen{}'s CMB appendix builds the driven-oscillator description of the photon-baryon fluid and the resulting peak series ($\ell_1\approx220$) that this paper first anticipated. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Uzan(2011)]{Uzan2011}
\textbf{J.-P.~Uzan.}
Varying constants, gravitation and cosmology.
\emph{Living Rev.\ Rel.} \textbf{14} (2011) 2.
\par\noindent\textbf{Relevance.}
Uzan's Living Review is the canonical survey of observational bounds on
time and space variation of the fundamental constants, including the
quasar-absorption spectroscopy that constrains any spatial variation of
the fine-structure constant across the observable universe at the
\(10^{-5}\)--\(10^{-6}\) level. \ATen{} cites it as the measurement
arm of the second horizon problem: a precision certification of identity
uniformity, framed throughout that literature as an equivalence-principle
test --- uniformity as null hypothesis, never as a datum requiring a
mechanism.
First cited in \S\ref{subsec:shp-unposed}.

\bibitem[Verde \emph{et~al.}(2019)]{Verde2019}
\textbf{L.~Verde, T.~Treu, and A.~G.~Riess.}
Tensions between the early and late Universe.
\emph{Nature Astron.} \textbf{3} (2019) 891--895.
\par\noindent\textbf{Relevance.}
The 2019 Verde--Treu--Riess perspective is the concise canonical statement that early-universe (CMB-inferred) and late-universe (distance-ladder) determinations of \Hzero{} disagree at high significance. \ATen{}'s Hubble-tension section cites it to frame the tension as a clash between two inference pipelines --- precisely the ``different projections of one bulk history'' reading the two-stage corridor advances. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Wagoner \emph{et~al.}(1967)]{WagonerFowlerHoyle1967}
\textbf{R.~V.~Wagoner, W.~A.~Fowler, and F.~Hoyle.}
On the synthesis of elements at very high temperatures.
\emph{Astrophys.\ J.} \textbf{148} (1967) 3--49.
\par\noindent\textbf{Relevance.}
The 1967 Wagoner--Fowler--Hoyle paper is the foundational systematic computation of primordial nucleosynthesis, establishing the reaction network and the dependence of the light-element yields on the baryon-to-photon ratio. \ATen{}'s BBN appendix follows its network (the $n+p\to d+\gamma$ bottleneck through to $^4\mathrm{He}$) and its $\eta$-dependence to derive the helium mass fraction $Y_p$. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Wald(1984)]{Wald1984}
\textbf{Wald, R. M. (1984).} \emph{General Relativity}. Chicago, IL: University of Chicago Press.
\par\noindent\textbf{Relevance.} The standard graduate monograph on general relativity, fixing the modern conventions for curvature, horizons, and the Einstein equations. \ATen{} cites this for the classical boundary baseline --- the FRW and horizon machinery --- that the present paper inherits rather than replaces, the geometric scaffold on which the architecture section builds the inherited \GG-6 locking. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Weinberg(1972)]{WeinbergGR1972}
\textbf{Weinberg, S. (1972).} \emph{Gravitation and Cosmology}. New York, NY: Wiley.
\par\noindent\textbf{Relevance.} Weinberg's 1972 textbook, a canonical reference for the FRW/Friedmann dynamics and the observational grammar of cosmology. \ATen{} cites this for the FRW/Friedmann baseline and the observational conventions the present paper retains, supplying the standard background against which the architecture section introduces the inherited bulk source ledger. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Weinberg(1978)]{Weinberg1978}
\textbf{Weinberg, S. (1978).} A new light boson? \emph{Physical Review Letters, 40}(4), 223--226.
\par\noindent\textbf{Relevance.} With \citet{Wilczek1978}, the 1978 identification of the axion as the light pseudo-Nambu-Goldstone boson necessarily produced by the spontaneous breaking of the Peccei--Quinn symmetry. \ATen{} cites this as historical background for the Kaxion section, whose compact-angle pseudo-scalar plays the cosmological dark-matter role of the axion while inheriting its Peccei--Quinn template through the \GG-6 compact holonomy. First cited in \S\ref{sec:kaxion}.

\bibitem[Weinberg(1987)]{Weinberg1987}
\textbf{Weinberg, S. (1987).} Anthropic bound on the cosmological constant. \emph{Phys.\ Rev.\ Lett.} \textbf{59}, 2607--2610.
\par\noindent\textbf{Relevance.} The one quantitative anthropic prediction --- an upper bound on \(\Lambda\) from the requirement that gravitationally bound structures can form. \ATen{} cites it in Objection~O-Cosmos-90 as the single standing post-diction of the anthropic reading, set against a framework that derives the inventory. First cited in \S\ref{obj:landscape-priced}.

\bibitem[Weinberg(1989)]{Weinberg1989CC}
\textbf{Weinberg, S. (1989).} The cosmological constant problem. \emph{Reviews of Modern Physics, 61}(1), 1--23.
\par\noindent\textbf{Relevance.} The 1989 review surveying the dramatic mismatch between the observed vacuum energy and naturalness estimates from quantum field theory --- the cosmological-constant problem. \ATen{} cites this to frame the late-time cosmic ledger that the present paper attempts to reinterpret through the residual free-energy functional \(F_{\GFF}\), relocating the smallness of \(\Lambda\) from a tuning problem to a bulk-ledger residual. First cited in \S\ref{sec:lcdm-success}.

\bibitem[Weinberg(2008)]{Weinberg2008}
\textbf{S.~Weinberg.}
\emph{Cosmology}.
Oxford University Press, Oxford, (2008).
\par\noindent\textbf{Relevance.}
Weinberg's 2008 graduate textbook is the modern reference derivation of the FRW background equations, thermal history, perturbation theory, and the CMB transfer function. \ATen{} follows its conventions in the \FRW{} background appendix (the Christoffel/Ricci/Einstein-tensor bookkeeping leading to the two Friedmann equations) and in the thermal-history appendix; the cohort reading then reinterprets the unmodified textbook FRW source terms as the isotropic reduction of the seam conservation law \CCI{}. First cited in \S\ref{sec:century-cosmology}.

\bibitem[Wilczek(1978)]{Wilczek1978}
\textbf{Wilczek, F. (1978).} Problem of strong P and T invariance in the presence of instantons. \emph{Physical Review Letters, 40}(5), 279--282.
\par\noindent\textbf{Relevance.} The 1978 companion to \citet{Weinberg1978}, independently identifying the axion as the pseudo-Nambu-Goldstone boson of the Peccei--Quinn mechanism. \ATen{} cites this as historical background for the Kaxion section, whose compact-angle dark-matter candidate inherits the axion template through the \GG-6 compact holonomy rather than a postulated low-energy U(1). First cited in \S\ref{sec:kaxion}.

\bibitem[Wilson(1974)]{Wilson1974}
\textbf{Wilson, K. G. (1974).} Confinement of quarks. \emph{Physical Review D, 10}(8), 2445--2459.
\par\noindent\textbf{Relevance.} The 1974 paper that introduced the Wilson loop as the gauge-invariant order parameter for confinement in lattice gauge theory. \ATen{} cites this to motivate Wilson-line and holonomy variables as genuine gauge-invariant branch coordinates in the cosmic-grip grammar, where the compact-angle holonomy of the inherited \GG-6 bulk labels the realized cosmological branch. First cited in \S\ref{ssec:cosmic-grip-grammar-gga}.

\bibitem[Wong \emph{et~al.}(2020)]{Wong2020}
\textbf{K.~C.~Wong \emph{et~al.} (H0LiCOW Collaboration).}
H0LiCOW XIII. A 2.4\% measurement of $H_0$ from lensed quasars.
\emph{Mon.\ Not.\ R.\ Astron.\ Soc.} \textbf{498} (2020) 1420--1439.
\par\noindent\textbf{Relevance.}
The 2020 H0LiCOW time-delay-cosmography measurement is an independent, distance-ladder-free determination of \Hzero{} from strongly lensed quasars that favors the high (late-universe) value. \ATen{}'s Hubble-tension section cites it because lensing time-delay determinations of the late-time branch are named as one of the four explicit falsifiable tests of the two-branch corridor. First cited in \S\ref{subsec:seventeen-problems}.

\bibitem[Zeldovich \emph{et~al.}(1974)]{ZeldovichKobzarevOkun1974}
\textbf{Ya.~B.~Zeldovich, I.~Yu.~Kobzarev, and L.~B.~Okun.}
Cosmological consequences of a spontaneous breakdown of a discrete
symmetry.
\emph{Zh.\ Eksp.\ Teor.\ Fiz.} \textbf{67} (1974) 3--11
[\emph{Sov.\ Phys.\ JETP} \textbf{40} (1975) 1--5].
\par\noindent\textbf{Relevance.}
The first explicit posing of vacuum identity as a causal problem: during
the cooling of ``causally disjointed points'' of the universe the sign
of the condensate is chosen at random, producing a domain structure of
the vacuum --- and the authors even ask how early the walls must be
removed to make the whole universe a single domain. \ATen{} cites this
1974 paper as the opening of the fifty-year pedigree traced in
\S\ref{sec:second-horizon}: the identity question, posed for one corner
of identity, and filed under relic abundances.
First cited in \S\ref{subsec:shp-pedigree}.

\end{thebibliography}


\end{document}
