%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% ===================================================================
%% v15.7 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 21 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.
%% ====================================================================
%% ===================================================================
%% v15.6 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY PROMISED A PAPER THAT WILL NOT BE
%%     THERE.  It read "Preprint, available at https://arisaka-gg.org/",
%%     which becomes false the day the homepage goes up carrying
%%     twenty-one papers.  It now reads "In preparation.  To be posted
%%     ... on completion; no result of the present paper depends on it."
%% (2) AND IT CITED A11 UNDER A TITLE A11 DOES NOT HAVE.  This paper
%%     called it "The Origin of Everything: the Cascade from the GUT Edge to
%%     the Biosphere".
%%     A11 v15.14's own \GGTitlePage reads "The Geometric Inevitability
%%     of Cosmos, Life, and Mind --- The Self-Referential Closure of the
%%     GG-Theory Program".  A working title from before the paper was renamed.  Corrected.
%% (3) TWO SITES SAID A11'S CASCADE HAS "eighteen events".  Measured
%%     from A11 v15.14 -- its GRIPout longtable runs 1->2 through 17->18
%%     -- the cascade is EIGHTEEN LAYERS joined by SEVENTEEN GRIPout
%%     EVENTS.  Both sites name events, so both read seventeen.
%%     THE SECOND SITE WAS FOUND BY THE PRINTED GUARD, NOT BY THE SCOPING
%%     GREP: the roster row is the one I went looking for; the Step-(C)
%%     summary sentence ("the codebook's twenty-eight entries, the
%%     inventory's pi-locked shares, the cascade's eighteen events") makes
%%     the same claim in different words, and only a guard that prints
%%     its survivors instead of counting them would have shown it.
%%     FROZEN BY EXACT TEXT in the same screen: "One instrument, sixteen
%%     problems" and "sixteen developments sharing one exact premise" are
%%     W2's OWN sixteen problems, a different quantity, and are left
%%     alone.
%% (4) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  8 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.5 (August 17, 2026) -- THE SHOWCASE HEADLINE COMES BACK TO SANS.
%%
%% This paper set its showcase headline in SERIF, and thirteen of the
%% cohort set theirs in sans-serif bold.  Measured across all
%% twenty-four after the v11.5 promotion, that was the ONLY one of the
%% four surviving divergences a reader turning pages could see -- the
%% trailing skip is invisible on a page centered by \vfill, and the one
%% size difference is A4's.
%%
%% IT WAS DRIFT, NOT A DECISION.  This paper and P6, W1, W2 and the
%% other study were the LAST of the showcase pass; the sans form is the
%% one the author read and approved on A1-Constitution and on the
%% twelve that followed it.
%%
%% ONE LINE CHANGES: \GGShowcaseSeriftrue becomes
%% \GGShowcaseCenteredtrue, and style file v11.6 puts \sffamily in
%% that branch.  THE SHAPE DOES NOT CHANGE -- this page still centers
%% its headline with \begin{center} and still takes 8 pt beneath it,
%% because both are measured fits on this page and because three of the
%% five call the macro with no \centering around it, so the
%% \begin{center} is what centers them.
%%
%% ONE PAGE MOVES, and it is the showcase page.  Everything else in
%% this paper is asserted identical to v15.4, page for page.
%% ===================================================================
%% ===================================================================
% v15.4 (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:
%%
%% * \GGShowcaseHead removed; this paper prints the centered serif form
%%   and says so in one line.
%% * one "centred" in a comment written in this run.  It
%%   does not print, and it is mine, so it is corrected
%%   rather than protected as history.
%%
%% 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 v15.3 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.3 (August 17, 2026) -- THE EXECUTIVE SUMMARY OFF THE PAGE NUMBER.
%%
%% Page 2 of v15.2 set its last line at the height of the folio and
%% printed over it: the page number was buried inside the words "the
%% interval's".  The author found it by reading the page.
%%
%% THE CAUSE, MEASURED: \enlargethispage{2\baselineskip}, the first
%% statement inside the Executive Summary.  Two baselineskips at this
%% paper's leading is 31.2 pt; the off-centre ink on page 2 ran 31.28 pt
%% past the block.
%%
%% AND IT WAS DELIBERATE.  The v15.0 changelog's Stage 6 entry removes
%% one \enlargethispage for putting the text block on the page number --
%% "third occurrence of this failure in the run, after P5 and W1" -- and
%% eight lines later adds this one, to stop the Keywords spilling two
%% lines onto a page of their own.  The second edit solved a real problem
%% by the means the first had just condemned.
%%
%% THE FIX.  The \enlargethispage goes, and the two lines it bought are
%% given back by trimming "A closing thought" by two printed lines, on
%% the author's instruction.  The three cuts are redundancies: a repeated
%% close on instruments, "the honest thing to say about it is that", and
%% a clause restating what the first page already says.  The keywords
%% still sit at the foot of page 4.
%%
%% A NEW GATE TOOL, bottom_scan.py, reads the artifact for this class:
%% ink below the body block (722.43 pt) anywhere outside the centered
%% column the folio occupies.  It subsumes overrun_scan.py's case -- ink
%% BELOW the folio -- and catches the one overrun_scan structurally
%% cannot, where the ink lands ON the folio.
%% ===================================================================
%% ===================================================================
%% v15.2 (August 17, 2026) -- THE SHOWCASE PAGE.
%%
%% fig:anatomy above, fig:four-forces below, under the paper's own title
%% sentence -- the one line the abstract sets in italics.  The house
%% pairing is the machinery and then what it decides: the anatomy IS this
%% paper's machinery, and its caption already says so ("every consequence
%% in Parts III-VI is this figure, applied"); the four interactions
%% sorted by fiber are the widest single thing that machinery does.
%%
%% Considered and not taken: fig:write-w2, the measurement problem sorted
%% by fiber.  It is the most famous of the sixteen and it is ONE of them;
%% the four-forces plate sorts four things at once.
%%
%% Both plates are SAVEBOXES, filled once immediately after
%% \begin{document} inside \setstretch{1} with \global\sbox, and used
%% twice at their BODY widths, 0.851 and 0.710, not a point above.
%% Filling at the top of the document rather than after the Keywords is
%% Report 401's lesson: on W1 the first use of a box was inside the
%% Executive Summary, ahead of where every earlier paper had filled, and
%% the body copy read an empty box.
%%
%% NEW PROSE IN THIS SHIP: the showcase headline and its caption, and
%% nothing else.  No figure, table, equation, section or number moves.
%% ===================================================================
%% ===================================================================
%%  W2-Holonomy v1.12  (August 14, 2026)
%%  ONE NEW SUBSECTION INSIDE PROBLEM 3 (ssec:w2-g2), placed after "The
%%  anomaly: the circle audits the interval" and before "The principles,
%%  re-founded": where each fibre stops, and the one number the anatomy
%%  cannot reach.
%%  WHY NOT A SEVENTEENTH PROBLEM.  It was offered and declined for two
%%  reasons.  First, cost: "sixteen" appears in 125 printed sites --
%%  the title-page subtitle, the abstract, the master table, the
%%  dispositions table, two Part subtitles, the Conclusions and three
%%  Objections -- so a seventeenth Problem is a default-PROTECT count
%%  sweep over the whole paper.  Second, and worse: Objection 33 defends
%%  the list as "the discipline's, not the author's", noting that
%%  fifteen of the sixteen have their own literatures and review
%%  articles.  "Why can't the geometry compute g-2?" is this cohort's
%%  question, not the discipline's, and putting it on that list would
%%  have falsified the defence.  Problem 3 is already "The Origin of
%%  Local Gauge Invariance and Renormalisation" and its master-table row
%%  already reads "C: unreadable pointer; I: one-way flow"; the new
%%  subsection sharpens that row rather than adding to the list.  THE
%%  SCRIPT ASSERTS THAT EVERY "sixteen"/"seventeen" COUNT IS UNCHANGED.
%%  CONTENT.  The interval cannot reach a because a carries no scale --
%%  A5's T-9.9, plus the reason underneath it: a is the coefficient of a
%%  dimension-five operator, so no counterterm exists, so it is finite,
%%  so it is an RG invariant.  The circle cannot reach it either because
%%  a is not an integer, and pi_1(S^1) = Z protects only integers.  The
%%  moment therefore falls between the two fibres and belongs to none of
%%  the three tiers of ssec:three-tiers.  What the anatomy does supply:
%%  alpha itself, the existence of the lepton sector, and P3's exact
%%  zero for the electron's own structural contribution.
%%  NO NEW BIBITEMS.  W2 carries results at their parent's grade, so the
%%  quantitative content is cited to A5's T-9.9, which now holds the
%%  full treatment with its own primary references.  Three cohort keys,
%%  each in its own \citep.
%% ===================================================================
%% ===================================================================
%%  W2-Holonomy v1.11 (August 12, 2026) -- typographic rider: the Part IX
%%  closing section takes its own page.  Set inline it started eighteen
%%  lines from the foot of p.222 and spilled six lines onto p.223, leaving
%%  a six-line page.  A \newpage before the section redistributes the same
%%  text as 25 + 24 lines.  No word changed, no number touched.
%%
%%  W2-Holonomy v1.10 (August 12, 2026) -- the consistency sweep
%%  (Report 309, author-approved).  (1) CMB-S4 (terminated July 2025)
%%  replaced by the Simons Observatory at all ten forthcoming-instrument
%%  sites; the "order of magnitude above the design floors" claim
%%  recalibrated to "several times above" (sigma(r) ~ 1e-3).  (2) The
%%  eta_B mirrors take the parents' Option A: the triplet equilibrium
%%  yield 1.25/g* displayed, central 7.34e-10 (+20%), headline sites at
%%  the display standard ~7e-10 with the x3 band; anatomy figure and
%%  drill G.3 updated; one sentence mirrors the parent's measured
%%  window deficit E = 7.1 (68% [4.4, 11.4]).  W2 derives nothing new;
%%  every number is carried at its parent's grade.
%% ===================================================================
%% ===================================================================
%%  W2-Holonomy v1.9  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  30 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.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.8  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.7 -> v1.8), author-directed: the deployment Section
%%  becomes the cohort's front door.
%%    * The twenty-two-paper cohort map installed here, on its own
%%      page, in the cohort-standard form -- the same plate carried by
%%      the other nineteen papers, no self-reference marker, so the
%%      figure stays identical across the cohort.  This paper is the
%%      twentieth carrier.
%%    * The Section is now three subsections: the map and how to read
%%      it; the campaign table; and "Where to go next", seven routes
%%      into the cohort organised by what brought the reader here.
%%    * The campaign table gains its twenty-second row (this paper),
%%      so the count in the Section title is now visible in the table,
%%      and the blank line left by the retired P7 row is closed.
%%    * Four Relevance notes re-pointed: P1, P2, P5 and P6 are now
%%      first cited in this Section.  Three of the four were stale
%%      before this ship; the new values are computed from the body,
%%      not asserted.
%%  Script: [v13-support]/w2_v18_map.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.5  (August 5, 2026)
%%  CHANGELOG (v1.6 -> v1.7): the A6 paper identifier.
%%    ucla-arisaka-gg-11.sty v11.1 fixes \ASix, which had expanded to
%%    the retired GG-A6-Gi4 since the May 2026 rename.  Twenty papers
%%    mask that with a local \renewcommand; this one does not, so
%%    Figure 11 (the universal sectoral descent ladder) carried the
%%    retired name in four of its right-column owner labels -- reading
%%    "GG-A6-Gi4 T-GRIP-1", the old paper name beside a theorem named
%%    for the new one.  Four rendered sites; source unchanged apart
%%    from this note and the version stamp.  Page count unchanged.
%% ===================================================================
%% ===================================================================
%%  CHANGELOG (v1.5 -> v1.6): P7-Inflow residue.  Ship 4 re-pointed
%%    this paper's prose but not its furniture: tab:deployment, the
%%    campaign map, still carried a row for P7-Inflow as a separate
%%    cohort paper -- so the table said twenty-two entries against a
%%    cohort of twenty-two of which this paper is one.  The row is
%%    removed and its entry point ("the seam: what the circle
%%    quantises is what the interval delivers") folded into the P2-N3
%%    row, which is where that argument now lives.  Table now lists
%%    the twenty-one papers other than this one.
%% ===================================================================
%% ===================================================================
%%  CHANGELOG (v1.4 -> v1.5): P2/P7 merger ripple, ship 4 of 4.
%%    * P7-Inflow is retired into P2-N3 Appendix H (P2 v15.2).  The
%%      one prose mention -- "the inflow companion", in Problem 3's
%%      two-sided-audit sentence -- now points at that appendix, and
%%      the ArisakaP2N3 relevance note inherits the refrain paragraph
%%      the P7 note carried.  No claim, number or count changes.
%%  Script: ship4_repoint.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.4  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.3 -> v1.4), author-directed.  Stamp only.
%%    * The cover date (third argument of \GGTitlePage) advanced from
%%      August 3 to August 5, 2026, to agree with \PaperDate and the
%%      running header.  It is a separate stamp from \PaperVersion
%%      and \PaperDate and had not been bumped since the v1.0 build.
%%    * \PaperVersion v1.3 -> v1.4.
%%    * ONE missed site of the v1.3 exposure reconciliation, caught on
%%      the rendered cover: the abstract's closing sentence still read
%%      "the named, dated measurements on which each treatment dies."
%%      Now six-of-sixteen-outright plus the shared-ledger clause, as
%%      the Executive Summary, the Preface, sec:claim, sec:kill and the
%%      master table's third remark already say.
%%  No number, equation, figure, table, reference, label or count
%%  changed.
%%  Script: w2_v14_coverdate.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.3  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.2 -> v1.3), author-directed: the adversarial-read
%%  package.  Eight commissions, all agreed before writing.
%%    1. THE INSTRUMENT SPLIT INTO TWO GRADED PREMISES.  (T) topology:
%%       pi_1(S^1)=Z, pi_1(I_r)=1 -- gives "no holonomy along the
%%       interval" and nothing else.  (D) the boundary law: the
%%       physical flow along r is the coarse-graining flow, whose
%%       non-injectivity (Kadanoff, eq:A-kadanoff) is the semigroup.
%%       "traced to that topological fact and to nothing else" was
%%       claiming more than the fork earns; Appendix A already had
%%       the honest version, and Part II is now brought into line
%%       with its own appendix.  Propagated to Problem 1's GRW
%%       comparison and the Conclusions.
%%    2. APPENDIX B TAKEN OFF THE SPECTRAL ARGUMENT.  I_r = [0,K] is a
%%       finite segment: discrete spectrum under any self-adjoint
%%       boundary condition, and its level spacing is the LARGEST in
%%       the framework (the 3.849 TeV lock).  The fibre assignment
%%       now rests on the composition law.  The Markov hypothesis
%%       Step 1 names is restored to Step 3.  Objection 45's "that
%%       step is a theorem" corrected.
%%    3. NEW Part II section sec:w2-record, the honest record: Weyl
%%       1918 (the scale direction gauged -- this paper's own fifth
%%       dimension), the Kaluza-Klein poverty hope, and the
%%       no-elementary-scalar instinct falsified at 125 GeV; the
%%       instrument's besetting temptation (over-attribution) with
%%       both live instances named; what the anatomy cannot see; and
%%       the generalisation stated against interest.
%%       NEW Objection 47 (cluster 3) is its Part X twin; objections
%%       47-116 renumbered 48-117 and all counts updated.
%%    4. NEW subsection ssec:w2-steelman in Problem 15, BEFORE the
%%       anatomy's answer: the category error in 10^500-vs-zero,
%%       string theory's own zero continuous parameters, found-not-
%%       wanted, AdS/CFT + the extremal microstate count, Weinberg
%%       1987 as a dated forward commitment, and the open question of
%%       what an explanation is.  Objection 104 cross-refers.
%%    5. TWO VERIFIED DEFECTS CORRECTED.  K = 35 is NOT overdetermined
%%       by two routes: L(2N+1) = M^2+N^2+L^2 is the dual lock that
%%       fixes N, and holds at both roots (35=35 at N=3, 75=75 at
%%       N=7).  And the proton's "double guard" does not forbid
%%       p -> e+ pi0 by integrality (Q+B: 2 -> 1, both integers) --
%%       the load is the gauged U(1)_B; registration is what makes B
%%       an integer class, prior rather than parallel.  Both fixed in
%%       body, appendix, tables, summaries and the honest ledger.
%%    6. EXPOSURE CLAIMS RECONCILED with tab:kill's six rows: six
%%       treatments die on named instruments, ten through the shared
%%       ledger.  Stated in the Executive Summary, the Preface,
%%       sec:claim, sec:kill and the master table's third remark.
%%    7. The invited chronology check now prints its own numbers
%%       (2.005, 1.984, 2.128, 2.038, 2.025) and attributes the 6.4%
%%       departure to g_*(T) across the QCD transition.  tab:trio and
%%       Problem 7 now print 2.24 meV against the derived 6.1 meV.
%%    8. Attribution/title pass: "what the sentence organises";
%%       "sorted by ... produced by"; Part VII subtitle and master-
%%       table heading off "answered"/"closed"; the "every
%%       interpretive framework" comparison withdrawn; the deployment
%%       campaign's "it is itself evidence" softened to what the
%%       Conclusions already concede; one verbatim duplicated
%%       sentence removed.
%%  NEW bibitems: Weyl1918, ATLAS2012, CMS2012, StromingerVafa1996.
%%  Scripts: w2_v13_a_premises.py, w2_v13_b_record.py,
%%           w2_v13_c_defects.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.2  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.1 -> v1.2), author-directed (GG-Report-187):
%%    * NEW subsection ssec:three-tiers in Problem 3, before the
%%      dictionary table: the three tiers of protection.
%%      Tier 1 winding data -- integrality, cannot move (the
%%      no-running theorem).  Tier 2 a continuous modulus of the
%%      circle (the Higgs, the Kaxion) -- non-locality: no local
%%      counterterm exists for a Wilson-line phase, so its
%%      potential is finite and calculable.  Tier 3 interval
%%      addresses -- nothing, running is what an address does.
%%      Resolves the apparent conflict between the no-running
%%      theorem and the fact that v and m_H run.
%%      States why the Higgs is the SM's problem child (it is
%%      tier 2 read as tier 3) and cashes 'protection without
%%      symmetry' on the naturalness problem -- the paper's
%%      thesis applied where it costs the most.
%%      Honest-limit paragraph: tier 2 buys finiteness, NOT
%%      fixity; the owed calculation is not finished (A9's v at
%%      +0.045%, m_H at -1.66% with the normalisation uncomputed).
%%    * NEW bibitem HatanakaInamiLim1998; the Hosotani relevance
%%      note now points at the tier subsection.
%%  Script: w2_v12_tiers.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.1  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v1.0 -> v1.1), author-directed (GG-Report-185):
%%  The stage-fibre custody correspondence, W2 side.
%%    * DGI expansion repaired to the canonical "Dynamical Gauge
%%      Invariance" (A6 v9.6 author directive; all cohort
%%      glossaries) at both non-canonical sites: the eq:grip-p2
%%      walk-through prose and the Part IX abbreviation table.
%%      The GFF "geometric free-fall" slip in the same sentence
%%      fixed to "geometric free-energy functional".
%%    * NEW stage-custody paragraph in the engine passage (Step B):
%%      strokes named for their products, stages for their
%%      transport; DGI's equivalences compose as the circle's
%%      group (the bulk's only gauge bundle, A8), the GSSB->GGA
%%      descent as the interval's semigroup (projected I_r
%%      dilatation, A5); GSSB reads circle data, runs on the
%%      interval, deposits to circle classes; GSSB : breaking ::
%%      write : measurement. Engine-figure caption gains the
%%      custody clause.
%%    * NEW paragraph in Problem 3: LGI -> DGI, the POS-1 ladder
%%      given its geometric reading -- DGI is LGI carried through
%%      the open dynamics (A7's origin-of-DGI).
%%    * Determinism triad gains the seam-shape echo tying Problem
%%      2's breaking to Problem 1's write.
%%    * A6 bibitem Relevance updated to carry both mappings
%%      (strokes and stages).
%%  Script: w2_v11_stagecustody.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v1.0  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  PROMOTION.  v0.20 -> v1.0, author sign-off August 3, 2026.  The
%%  paper is deployed to [GG-v13] and is from here on handled with the
%%  rest of the cohort for arXiv submission.  No text changed in this
%%  step; three production conventions were applied:
%%    * \PaperStatus 'White Paper (working draft)' -> 'White Paper',
%%      matching W1-Symmetry, its sibling white paper in [GG-v13].
%%    * \pdfoutput=1 added before \documentclass (the arXiv flag that
%%      forces pdflatex); W2 was the only paper of the set without it.
%%    * \PaperVersion bumped.
%%
%%  ACKNOWLEDGMENT PLACEMENT (per the sty's placement note): this
%%  paper uses convention (b) -- after the Conclusions, before the
%%  Technical Appendices -- by author ruling of August 3, 2026.
%%
%%  BUILD REQUIREMENTS (for arXiv upload): ucla-arisaka-gg-11.sty from
%%  [v13-support], and five figure assets --
%%    a9_dev_plot_linear_first8_hook.png   (A9-SM Fig. 1)
%%    gg_unification_showcase.pdf          (A9-SM Fig. 3)
%%    A10-whynow-Omega-evolution-PAIR-nocap.pdf (A10 Fig. 2)
%%    a10_dev_plot_showcase.pdf            (A10 Fig. 6)
%%    a10_cosmic_descent.pdf               (A10 descent)
%%  All five are deployed alongside the sty in [v13-support].
%%
%%  DEVELOPMENT HISTORY: v0.1 first assembly (August 2, 2026) through
%%  v0.20 (August 3, 2026); ship records GG-Report-162 .. 180.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.20  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.19 -> v0.20): items 4-10 of the GG-Report-178 queue.
%%  See GG-Report-180.
%%    * The DOUBLE GUARD restated.  Eq. (eq:guards) states integrality
%%      conditions, which p -> e+ pi0 satisfies on BOTH sides, so a
%%      guard leaning on integrality alone forbids nothing.  The guard
%%      is now given as a conjunction: (i) B is a holonomy class only
%%      the Delta-B = 3n door can move; (ii) the charge-triality ruling
%%      makes the fractional intermediate unregistrable.  Eq. (guards)
%%      is relabelled in the text as a registration rule on states, not
%%      a selection rule on transitions, and the caution is printed.
%%    * The sixteen-disposition table wrapped as a float
%%      (tab:dispositions) with caption, label and inline reference.
%%    * Problem 12's open formal proof added to the honest ledger.
%%    * Appendix arithmetic: tau_p benchmark 1e34 -> 3e35 yr with the
%%      alpha stated, and H.3's M_X bound re-derived (5.4e15 GeV);
%%      decoherence time 1e-31 -> 1e-24 s at both sites, with the
%%      cm/m unit trap named; f_bulk = Xi_cap f_K = 3.1e13 GeV stated
%%      so Eq. (F-fK) can actually be evaluated.
%%    * J collision resolved: Jarlskog -> J_CP, Regge spin -> \mathcal{J},
%%      with a one-paragraph note on the reservation of J for the ledger.
%%    * Nine cold Part X cluster openings given orienting paragraphs,
%%      each naming the cluster's strongest objection.
%%    * Rhythm: 11 bare-'It' response openers renamed, agentive
%%      'the cohort' converted to first person or 'this paper'.
%%    * ucla-arisaka-gg-11.sty: the L43 acknowledgment-placement note
%%      now records BOTH conventions and the author's ruling (comment
%%      only; no macro changed).
%%  Scripts: w2_v20_z1..z6.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.19  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.18 -> v0.19): the three figure defects left queued by
%%  the QA pass, closed.  See GG-Report-179.
%%    * fig:slowroll-w2: the imported red box carried A10's RETIRED
%%      reading ('the TeV plateau is retired; the tilt came out blue')
%%      and the plateau marker said V^{1/4} ~ Mlock ~ 3.85 TeV -- both
%%      contradicting this paper's staked r = 0.0043, which fixes
%%      V^{1/4} = Mplred (3 pi^2 A_s r/2)^{1/4} = 8.3e15 GeV.  Marker
%%      corrected, box replaced by a grade note in this paper's own
%%      register, inset M_Pl -> Mplred (they differ by 8 pi), caption
%%      extended to say the height is staked, not decorative.
%%    * fig:param-tree-w2: alpha^{-1}, sin^2 theta_W, alpha_s and
%%      theta-bar were drawn descending from M_lock, contradicting the
%%      text's claim that they are circle-side and cannot run; and the
%%      alpha^{-1} node carried a formula found nowhere else in the
%%      cohort.  Re-routed from the ledger root in a dashed style, node
%%      set to alpha^{-1}(0) = 2^J + N^2, legend added, caption
%%      rewritten to explain the two arrow styles.
%%    * fig:chronology: the z_sw dashed line sat at the exact abscissa
%%      of the recombination dot, fusing visually the two quantities the
%%      text keeps apart.  The dot is now labelled with z_dec, the line
%%      carries a callout naming both and stating the 0.006-e-fold gap,
%%      the marker reads ln z_sw = J (not ln(1+z_sw) = J), and the
%%      age-audit summands match Eq. (eq:age-ledger) to two decimals.
%%  Script: w2_v19_y1.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.18  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.17 -> v0.18): final whole-document QA pass.
%%  See GG-Report-178 for the full finding list and disposition.
%%    * 15 factual/arithmetic corrections (ln z_sw = J; the switch/
%%      decoupling separation restated correctly; sphaleron sign;
%%      8K not 2K; the clock ladder's true span; the age-ledger
%%      summands; antiparticles = Problem 10; the derivation tree's
%%      total; twin split units; the audit table's z* row split in two).
%%    * Ten Part codas: every Part now ends with a summary of itself
%%      and a bridge into the next.
%%    * Four never-referenced floats given inline lead-ins; seven
%%      missing citations attached at the point of use.
%%    * The five \clearpage page breaks before Problem sections
%%      removed (the author's whitespace note) and replaced, where the
%%      junction needed one, by a forward-looking bridge sentence.
%%    * The strong-CP two-dial figure moved back into Problem 12,
%%      out of the Part VI opener where it had drifted.
%%  Scripts: w2_v18_x1..x5.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.17  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.16 -> v0.17), author-directed:
%%    Figure 25 (the A10 cosmic-energy-budget pair) restored to its
%%    original full size, height 0.90\textheight.  The caption's
%%    substance -- the two clocks, the eternal boxed ratios, the
%%    crossing inventory -- is moved into the body text on the facing
%%    page, where it reads as argument rather than as a legend; the
%%    caption itself is cut to two lines so image + caption fit the
%%    float page.  No other changes.
%%  Script: w2_v17_w1.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.16  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.15 -> v0.16), author-directed figure import:
%%    * Four showcase figures imported from the parent papers, each
%%      placed where the argument needs it and cited in the text:
%%      - A9-SM Fig. 1 (the thirteen most sharply benchmarked entries,
%%        closed form beside measured value) -> Problem 4, after the
%%        two-fonts figure that classifies the sheet's alphabet.
%%      - A9-SM Fig. 3 (gauge-coupling unification across the 6D bulk)
%%        -> Problem 2, sec:lock-unification: the picture of 'one
%%        structure at the top', read in both fibres.
%%      - A10-Cosmos Fig. 2 (the cosmic energy budget through history,
%%        shown twice) -> Problem 7, REPLACING the pi-locked pie chart
%%        (fig:pi-chart-w2, removed): the same inventory as a history,
%%        so the 'why now' discussion has the moving seam in view.
%%      - A10-Cosmos Fig. 6 (the cosmological audit Dev plot) -> Part
%%        VII, sec:numbers-audit, beside this paper's own number table.
%%    * Assets copied into the build dir: a9_dev_plot_linear_first8_hook.png,
%%      gg_unification_showcase.pdf, A10-whynow-Omega-evolution-PAIR-nocap.pdf,
%%      a10_dev_plot_showcase.pdf.
%%    * \GGfrontmatteronepagetrue set, so the List of Figures begins on
%%      page 5 directly below the Overview of Contents and the pair
%%      occupies pages 5-6.
%%  Scripts: w2_v16_v1.py (figures), v2.py (front-matter toggle + stamp).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.15  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.14 -> v0.15), author-directed front-matter pass:
%%    * Cover subtitle shortened to 'GG-Theory and the Sixteen
%%      Problems of Particle Physics and Cosmology' (author ruling:
%%      'and', not 'solves' -- 'solves' would contradict Objection 24
%%      and the master table's graded dispositions).
%%    * Abstract rewritten to the paper as it now stands; 1912
%%      rendered characters, inside the cohort's 1920 ceiling.
%%    * Executive Summary extended by four blocks: 'Where the
%%      sentence comes from' (the four-step programme with inline
%%      citations to A1-A11 and W1), 'The ledger the numbers are
%%      written in' (closed forms, each cited to its parent), 'What
%%      the back of the paper is for' (Parts IX-XI), and a
%%      philosophical close.
%%    * Front matter reordered to the A9-SM convention:
%%      Overview of Contents + List of Figures after the Keywords,
%%      then the full TOC via \GGTableOfContents (tight lines, no
%%      inter-section gaps), then the Preface.
%%    * Preface relocated after the contents, rewritten longer, with
%%      a plain-words statement of the idea and a four-part
%%      invitation (curious reader / student / specialist / sceptic).
%%    * All 53 captions given short \caption[...] forms so the List
%%      of Figures runs one line per entry.
%%    * \GGAcknowledgments added after the Conclusions and before the
%%      Technical Appendices, with two tailored closing paragraphs.
%%  Scripts: w2_v15_u1.py (subtitle + captions), u2.py (abstract +
%%           executive summary), u3.py (front-matter order + preface),
%%           u4.py (acknowledgments), u5.py (stamp + banner).
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy  v0.14  (August 3, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v0.13 -> v0.14), author-directed:
%%    Part XI (References) brought to the cohort A/B standard.
%%    * All 103 Section-B entries: the one-line Relevance notes are
%%      replaced by full paragraphs in the A8/A9/A10 register --- what
%%      the work established, what it cost the field, how this paper
%%      reads it on the two fibres, and where this paper departs from
%%      it where it does.
%%    * Every entry in both sections now ends with an explicit
%%      first-appearance cross-reference ("First cited in \S...").
%%      Section A: 3 entries that lacked one now carry it (C3-BH,
%%      P2-N3, P7-Inflow).  Section B: all 103 are new.
%%    * Fifteen new \label{intro:pN} anchors (Problems 10--11 share one) on the Part I problem
%%      essays, so the first-appearance links resolve to the specific
%%      problem rather than to the enclosing category section.
%%    * Part XI opening rewritten to announce the standard and the
%%      ordering of Section B.
%%  Scripts: w2_v14_t0.py (labels), t1-t4.py (Relevance notes,
%%           four batches), t5.py (opening + stamp + banner).
%% ===================================================================
%% ===================================================================
%%  v0.13 (August 3, 2026) -- Part X expanded to the full cohort
%%  format (A8/A9/A10): 116 numbered objections O-W2-1..116 with the
%%  objqabody environment, grouped in ten clusters by the background
%%  of the reader most likely to raise each.
%%  Cluster 1 (1-22), placed first per A10's pattern: the curious
%%  reader -- basic in form, philosophically deep in substance
%%  (what the slogan means, where the dimensions are, string theory,
%%  the multiverse, free will, before the Big Bang, why 95% dark,
%%  why 35, pi-numerology, a designer, something-rather-than-nothing,
%%  who does the writing, the cat, is time real, time travel, the
%%  antimatter, the unseen quark, black-hole information, suspicion
%%  at explaining much, why trust one author, what would falsify,
%%  what a student should do).
%%  Cluster 2 (23-34) methodology, honesty, numerology; 3 (35-46)
%%  the anatomy itself -- mechanism, novelty, circularity (absorbing
%%  and sharpening the six v0.9 objections and the eight seminar
%%  questions); 4 (47-58) quantum foundations; 5 (59-70) gauge
%%  theory and the Standard Model; 6 (71-82) cosmology; 7 (83-92)
%%  particle-physics foundations; 8 (93-102) time, entropy, and
%%  black holes; 9 (103-110) comparisons with string theory, loop
%%  quantum gravity, asymptotic safety, emergent gravity, causal
%%  sets, warped extra dimensions, the anthropic tradition, and
%%  rival QM interpretations; 10 (111-116) falsifiability, partial
%%  failure, cohort size, the author's position, what success would
%%  establish, and the single strongest reason.
%%  Part opening rewritten to announce the structure. Numbering
%%  verified contiguous 1..116 with no duplicates.
%% ===================================================================
%% ===================================================================
%%  v0.12 (August 3, 2026) -- Part IX deepened: Step 1 and Step 3 of
%%  all nine constructive appendices roughly doubled, and the
%%  recommended reading added.
%%  (1) New section 'Recommended textbooks and key papers' opening
%%  Part IX: eight topic groups, each naming the textbooks and the
%%  original papers, plus the cohort entry points. Four textbook
%%  references added (Sakurai-Napolitano, Zee, Georgi,
%%  Dodelson-Schmidt).
%%  (2) Step 1 extensions -- full textbook derivations with physics
%%  intuition on the spot, self-sufficient for a graduate student:
%%  A bundles/transition functions, transport ODE, the sphere
%%  computed via Christoffel symbols, path ordering, Ambrose-Singer,
%%  AB fringe shift; B Kraus theorem, Born-Markov derivation, the
%%  two-level model solved, the origin of the (Delta x)^2, the
%%  einselection criterion; C Yang-Mills assembled, Faddeev-Popov,
%%  the two terms of b0 (screening vs antiscreening),
%%  Callan-Symanzik, the ABJ anomaly; D the counting theorem worked
%%  (36-26=10), the full 28 count, what a measured coupling is, the
%%  two comparison conventions; E Friedmann from Einstein, the
%%  continuity equation, slow roll derived, the mode-freezing origin
%%  of the spectra, the reheating dependence of N*; F the instanton
%%  potential, the relic derived, the haloscope power formula, how
%%  the inventory is measured, the vacuum problem at full strength;
%%  G the bilinear table, the CPT proof in outline, the CKM
%%  parametrisation, sphaleron energetics, leptogenesis assembled;
%%  H the strong-coupling area law, Polyakov loop and centre, Regge
%%  trajectories, the dimension-six operators; I the H-theorem with
%%  Loschmidt and Zermelo, Landauer twice, black-hole mechanics,
%%  Hawking's Bogoliubov outline, why the Page curve is forced.
%%  (3) Step 3 extensions -- the two fibres used explicitly, with
%%  mathematics: A the connection decomposed by fibre (axial gauge
%%  on the interval, Wilson-line modulus on the circle, the moduli
%%  spaces, Kadanoff blocking as the semigroup); B the fibre trace
%%  as a channel, discrete vs continuous spectra deciding which
%%  fibre dissipates, Born as relative-entropy gradient flow; C the
%%  KK expansion, gauge invariance as a readability theorem, the
%%  r-flow, anomaly inflow and descent equations; D the ledger
%%  identities derived, the two-font formula schema and how it
%%  fails, the no-running theorem; E the radion identification, why
%%  the roll is once, N* as a length, the warp hierarchy; F the
%%  two-parent mixing matrix, f_K = f_bulk/Xi_cap, the doublet, the
%%  Hopf volumes, the seesaw with its named residue; G the three
%%  orientations as bundle operations, CPT as total orientation,
%%  the pi/5 chiral lattice, the toll with its postulated exponent
%%  flagged; H the Z_3 centre from winding, the registration rule
%%  as a projection selection rule, N=3 twice, the double guard;
%%  I the tick derived (E = h nu as angular velocity), the arrow
%%  without a past hypothesis, dln t/dr = 2 derived, the demon
%%  closed, why a horizon spends all three constants.
%%  Three new worked problems (B.6, D.5, I.6). Twenty-eight new
%%  equations. Every number re-verified programmatically.
%% ===================================================================
%% ===================================================================
%%  v0.11 (August 3, 2026) -- Parts IX, X, XI structured in the
%%  cohort format (A8-QM / A9-SM / A10-Cosmos), and Part IX written
%%  in full.
%%  Part IX (Technical Appendices): a graduate course in the
%%  two-fibre anatomy. Two reference appendices (abbreviations;
%%  glossary of symbols) plus nine constructive appendices, each in
%%  the commissioned four steps -- Step 1 the standard textbook
%%  treatment from scratch; Step 2 what it assumes but cannot
%%  explain, with the Part I problem named; Step 3 the two-fibre
%%  resolution at declared grades; Step 4 problems with fully
%%  worked solutions, each closed by a physics-intuition remark.
%%  The nine follow the order of Parts III-VI: holonomy and the two
%%  transport algebras; open quantum systems and the write; gauge
%%  invariance and the RG; the parameter codebook; the launch and
%%  the descent; the dark sector; discrete symmetries and the matter
%%  excess; confinement and the proton; time, entropy, and horizon
%%  thermodynamics. Apparatus macros (GGteachbox, GGreadalong,
%%  GGtakeaway) added in the A9/A10/W1 form.
%%  Part X (Common Objections and Responses): created as its own
%%  Part in the cohort format; the Discussion's objection and
%%  curious-reader sections migrated into it, with a pointer left
%%  in Part VII. Expansion to the full cluster format is queued.
%%  Part XI (References): the bibliography promoted from a starred
%%  section to a Part, with the cohort's standard opening on the
%%  Relevance notes. Twelve textbook references added for the
%%  read-along boxes.
%% ===================================================================
%% ===================================================================
%%  v0.10 (August 3, 2026) -- the (A)/(B)/(C) labelling corrected.
%%  The Discussion's section 'How (C) sits with (A) and (B)' used
%%  the obsolete three-commitment scheme (A = the bulk, B = the
%%  open-system chain, C = the two-fibre anatomy), which the revised
%%  Part II superseded with the four development steps (A = A1-A4
%%  the bulk, B = A5-A7 the projection, C = A8-A11 the applications,
%%  D = W1/W2 the white papers). The section is rewritten as 'How
%%  (D) sits with (A), (B), and (C)': each step defined as in Part
%%  II with its equations cross-referenced, the anatomy placed as
%%  (D)'s instrument rather than a third commitment, the mutual
%%  dependencies stated in both directions (including the new
%%  clause that without (C) the anatomy would carry no numbers),
%%  and the historical-versus-logical order argument retained. Two
%%  stale 'commitment (A)' references in Part III (the two abelian
%%  companions; gravity, the fourth) re-pointed to Step~A.
%% ===================================================================
%% ===================================================================
%%  v0.9 (August 3, 2026) -- Parts VII and VIII rebuilt on the
%%  sixteen-problem story (structure and length modelled on W1's
%%  back matter; A9/A10/P5 as reference implementations).
%%  Part VII: new opening 'master table' (16 rows: problem,
%%  mechanism by fibre, sharpest number, grade, dies-by);
%%  Significance rewritten sixteen-fold; new section 'The two
%%  families, delivered' (exactness closed by the circle,
%%  one-wayness by the interval, the four hardest problems as the
%%  two-fibre meetings); 'Six old problems' replaced by 'Sixteen
%%  problems, sixteen dispositions' with corrected Part references
%%  and the verb glossary; all residual six-language swept
%%  (falsifier caption, calendar, Q6).
%%  Part VIII: 'What has been established' rewritten as the
%%  diagnosis-instrument-delivery arc; 'The sixteen answers, in
%%  sixteen sentences'; new closing section 'The oldest dispute in
%%  physics' (being/becoming via the two counterfactuals;
%%  comprehensibility via the two preconditions of science;
%%  forced structure vs open history) each anchored to the two
%%  transport laws; teaching updated; the personal closing
%%  extended with gratitude, the invitation to students, and the
%%  final passage, signature moved to the true end.
%% ===================================================================
%% ===================================================================
%%  v0.8 (August 3, 2026) -- Parts IV-VI given the Part III
%%  treatment. (1) Structure: the eleven remaining \WProblem banners
%%  promoted to \section headings (one section per Problem, 5-16),
%%  all 104 internal sections demoted to subsections; the W2 problem
%%  ledger now has uniform sectioning end to end. (2) Equations with
%%  parameter definitions and intuition: P5 slow-roll (eps, eta, N)
%%  + the plateau observables; P6 the top-loop radiative correction
%%  (the hierarchy as arithmetic); P8 the patch count and the DT/T
%%  record; P9 the three orientation reversals as maps + the driver
%%  angle alpha31 = pi and delta_PMNS = 196.4 deg; P10-11 the Dirac
%%  equation and the two-senses reading; P12 the Cornell potential
%%  and Wilson area law with the string tension; P13 H = 1/(2t) and
%%  t ~ T^-2; P14 E = h nu, S = kB ln W, and the Bekenstein-Hawking
%%  bridge; P15 the two architectures as bookkeeping (landscape
%%  triple vs Morse-unique groove); P16 the Hawking temperature,
%%  entropy, and evaporation schedule. (3) Fifteen figures
%%  reproduced from the cohort: A10 (where the universe begins;
%%  slow roll; the pi-locked chart; the comoving Hubble radius),
%%  C1 (the Kaxion on the two fibres), P5-CPT (the three reversals;
%%  the basin gate; the four endowments; the shore; the two-fate
%%  strong-CP architecture), P6-Proton (the Cornell cage; the
%%  proton mass budget), A9 (the Arisaka equation), C3 (the Penrose
%%  diagram; information as displacement). Five paper-local macros
%%  (Ja, Jq, Uone, thetags, thetaw/thetai) added as providecommand.
%% ===================================================================
%% ===================================================================
%%  v0.7 (August 3, 2026) -- Part III rebuilt: structure, equations,
%%  figures. (1) Structure: one section per Problem -- the four
%%  \WProblem banners of Part III become \section headings
%%  ('Problem n. Title') and all 39 internal sections are demoted to
%%  subsections (Parts IV-VI to follow in their own passes).
%%  (2) Equations with parameter definitions and physics intuition:
%%  P1 -- the Schrodinger/Lueders pair, the GDOS identification, the
%%  F25 relic displayed with A, gamma, k0 defined; P2 -- the SM
%%  covariant derivative, the one-loop running of alpha_s, the
%%  G_F/M_W/v chain, the reduced Planck mass; P3 -- the gauge
%%  transformation pair, the RG equation, the hypercharge anomaly
%%  sums; P4 -- the ledger-to-laboratory chain as a displayed
%%  five-line cascade with measured values. (3) Seven figures
%%  reproduced from the cohort: A8's decoherence-rate scaling and
%%  cat/Geiger timeline (P1); A9's sectoral descent ladder (P2),
%%  28-parameter derivation tree and status bar (P4); W1's
%%  seven-step ladder and bulk-boundary seam (P3). All captions
%%  rewritten for W2's citation set.
%% ===================================================================
%% ===================================================================
%%  v0.6 (August 3, 2026) -- Part II rebuilt: the programme in four
%%  steps, and the instrument quantified. (1) New chapter 'The
%%  programme in four steps': Step A (A1-A4) GGEP/6D bulk/group/
%%  anomaly freedom/zero parameters, with Eqs. (group, ledger,
%%  codebook samples); Step B (A5-A7) OGGEP/GDOS master equation/
%%  Born measure/GRIP cascade/Landauer, each displayed with
%%  intuition; Step C (A8-A11) the four applications with the vev
%%  chain, the pi-locked inventory, and the seesaw; Step D the two
%%  white papers (W1 diachronic, W2 synchronic). (2) Four figures
%%  added to Part II: the four-step architecture (new), the bulk
%%  projection pi: M6 -> M4 (adapted from W1), the GRIP engine
%%  (reproduced from A11 via W1), and the cosmic descent to scale
%%  (reproduced from A10; asset a10_cosmic_descent.pdf). (3) The
%%  primer quantified: the AB phase, the Wilson loop, and the two
%%  transport algebras (group vs semigroup) as displayed equations,
%%  plus the new section 'The instrument, quantified' reading all
%%  four. Part-page subtitle, Preface, and Part I organisation
%%  updated to match. Part II roughly doubled.
%% ===================================================================
%% ===================================================================
%%  v0.5 (August 3, 2026) -- front matter and Part I, for a broader
%%  readership. (1) Title page: new subtitle (the sixteen problems,
%%  replacing the six-consequence line); date corrected. (2) Abstract
%%  rewritten in plain English and cut to the cohort band (~1930
%%  normalised characters) so it fits page 1. (3) Executive Summary
%%  rewritten for a general scientific reader -- the list, the one
%%  sentence, why exactly one fibre of each kind, what it buys in
%%  four groups, and what the paper is and is not -- plus a Keywords
%%  block (cohort standard, previously absent). (4) Part I expanded
%%  dramatically: a new lead-in, a reading guide, and all sixteen
%%  problems walked one by one at about half a page each, with
%%  historical background, in-line citations to the primary
%%  literature, and an explicit "why has this resisted for forty to a
%%  hundred years" diagnosis; then the synthesis -- the ages table,
%%  the two families (exactness vs one-wayness), what kind of answer
%%  could close them (five requirements), and the claim of this
%%  paper. (5) References: 56 external works added with cohort-style
%%  Relevance notes; all remaining orphan entries cited at natural
%%  sites -- the bibliography is now orphan-free.
%% ===================================================================
%% ===================================================================
%%  v0.4 (August 3, 2026) -- the polish pass, in four steps.
%%  (1) Author's voice: Preface and the Conclusions' personal closing
%%  rewritten in the first person. (2) The three original blocks
%%  (Problems 2, 7, 10-11) brought to the depth of the v0.3 essays
%%  (seminar questions + a synthesis section each). (3) All cohort
%%  Relevance notes in the References deepened to full paragraphs
%%  mapping which Problems draw what. (4) Ten new self-contained
%%  TikZ figures, one per Problem previously without its own figure:
%%  the write (P1), four-forces sorting (P2), dial+running (P3), the
%%  two fonts (P4), the launch's windows (P5), the depth gauge (P6),
%%  three horizons (P8), the currencies (P14), hat vs groove (P15),
%%  and the Page-curve schedule (P16). Every Problem now carries at
%%  least one figure. Preamble: + decorations.pathreplacing.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-W2-Holonomy v0.1 (August 2, 2026) -- first assembly.
%%  The second white paper: concept (C), the 6D holonomy of GG-6 with
%%  two fibres -- one open (the 5th dimension), one closed (the 6th)
%%  -- and its six most profound outcomes: (1) the discrete symmetries
%%  C, P, T, CP, CPT; (2) the origin of antiparticles and the excess
%%  of particles; (3) quark confinement and the stable proton; (4) the
%%  definition of time and the arrow of time; (5) the distinction of
%%  the strong, weak, and electromagnetic interactions; (6) the origin
%%  of Inflation, Dark Energy, and the Kaxion. Content synthesised
%%  from W1-Symmetry v0.57, P5-CPT v3.48, P6-Proton v1.22, A10-Cosmos
%%  v13.29, C1-Kaxion v13.20, A8-QM v13.20, and the campaign internal
%%  notes; figures adapted from C1 Fig. 7, A10 Fig. 21, and the P5/A10
%%  chronology figure. Template: W1-Symmetry (WP series).
%% ===================================================================
\pdfoutput=1
%% ---- vv15.8 (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.
%% 1 showcase page in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}

\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
\renewcommand{\LaTeX}{LaTeX}% cohort-standard shim

%% Cohort-standard Part structure: one-line Part entries in the TOC.
\makeatletter
\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

\PaperSeries{WP}
\PaperStatus{White Paper}
%% Series rule colour: the author's ruling for the WP series (W1).
\renewcommand{\GGSeriesColour}{orange!85!black}
\PaperNumber{Arisaka-GG-W2-Holonomy}
%% ------------------------------------------------------------
%% v1.12 -> v15.0, August 15, 2026.  STAGE 1, MECHANICAL.
%%   The eleventh paper conversion, and the last white paper.
%%   * a4paper -> letterpaper.  US Letter is the SHORTER sheet, so a
%%     conversion adds pages; the count is recorded in the report.
%%   * 1,020 printed British spellings over 119 distinct words, the
%%     largest count in the cohort -- `fibre'/`fibres' alone are 392.
%%     Swept from the generated map, which is printed whole and whose
%%     every output is asserted not to be matched by the sweep rule.
%%   * `Keller' PROTECTED, not swept: C. Keller of Arndt et al. (1999),
%%     inside a \bibitem author list.  The doubled-l branch would have
%%     written `C. Keler'.  Third proper noun that branch has reached.
%%   * `storey'/`storeys' SWEPT to `story'/`stories': the building
%%     metaphor, and this paper already writes `story' 23 times.
%%   * 3 \hline -> \toprule/\midrule/\bottomrule in the tier grid.
%%   NOT DONE HERE, each measured: \tabcolsep is not set at document
%%   level so the 6 pt class default already stands (the one 4 pt is a
%%   local override inside the 28-parameter plate); captions are
%%   already above in all twelve table floats; the two unsized tabulars
%%   are inside center groups that Stage 2 promotes into floats, and
%%   \small goes on there.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A, same day.  THE FLOAT STANDARD.
%%   * 11 bare grids promoted into captioned floats.  Eight become
%%     table floats; three -- the field-theory dictionary (13 rows),
%%     the abbreviations (21) and the symbol glossary (28) -- become
%%     longtables in the house form, because a table float cannot
%%     break them and they are the paper's three UNBREAKABLE-AND-LONG
%%     grids.
%%   * 10 thin captions brought to the 260-character floor.
%%   * \small added to the two grids that carried no size command.
%%   TWENTY-ONE CAPTIONS ARE NEW PROSE and are listed by label in the
%%   report.  Every specific in every one of them -- every number,
%%   every row name, every column heading -- is quoted from the table
%%   it describes, because the rows of all twenty-one were dumped and
%%   READ first.  In W1-Symmetry four of thirteen extensions named
%%   rows that had not been read, written from the header line alone;
%%   that was the second occurrence in three papers and this is the
%%   answer to it.
%%   No table row is added, removed or altered.  No claim changes.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2A rider, same day.  TWO PLATES THAT NO LONGER FIT.
%%   \textheight = 650.43 pt.  LaTeX reported the budget plate
%%   18.94 pt too large for its page and the Dev plate 0.59 pt.  The
%%   A4 baseline carried neither warning: the first overflow is the
%%   Stage 2A caption extension, four printed lines added to a plate
%%   with 19 pt of room, and the second is the page conversion.
%%   THE TWO ARE CAPPED BY DIFFERENT DIMENSIONS.  With keepaspectratio
%%   only the smaller of width and height binds.
%%     fig:a10-budget-w2   plate 777.6 x 1094.4 pt; at full width it
%%       wants 661.1 pt, so HEIGHT binds: 0.90 -> 0.86 \textheight,
%%       585.4 -> 559.4 pt.  Cleared.
%%     fig:a10-devplot-w2  plate 828 x 936 pt; at full width it wants
%%       531.0 pt against a 559.4 pt cap, so the height cap is INACTIVE
%%       and lowering it changed nothing -- the rebuild reported the
%%       same 0.59 pt.  WIDTH binds: 1.00 -> 0.99 \textwidth, 531.0 ->
%%       525.7 pt, 4.7 pt to spare.  The height cap is left at 0.86.
%%   Both are full-page [p] reproductions from A10-Cosmos, not
%%   house-size figures, so the 7.79 pt type rule does not govern them.
%%   The printed figures shrink by 4.4 %% and 1.0 %%.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2B, same day.  THE LIST OF TABLES, WHICH WAS MISSING.
%%   W2 carried \GGOverviewOfContents and \GGListOfFigures and
%%   stopped, so its table floats had no list.  The frozen standard
%%   (GG-Standard-Front-Matter-v1.0 section 3) is Overview, Figures,
%%   Tables, then the full Contents, the first three sharing one page
%%   under \GGfrontmatteronepagetrue, which was already set.
%%   THE ABSENCE WAS INVISIBLE IN THE SOURCE.  \addcontentsline{lot}
%%   writes to a file that \listoftables never opened, so every build
%%   since the paper was written computed the entries and discarded
%%   them.  No audit caught it because there was no wrong output, only
%%   a missing one.  Third paper in this run with the same gap, after
%%   C2-Hubble and C3-BH.
%%   Nothing else in the front matter moves.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stages 3 and 4, same day.  FIGURES MEASURED; PALETTE CLOSED.
%%   STAGE 3 CHANGES NOTHING, and says so with numbers.  All 42
%%   pictures were compiled ONE AT A TIME with measure_plate.py and
%%   the type size read off each ink bounding box.  35 land at
%%   8.07-8.11 pt, which is the house 7.79 pt times the 3.7 %% A4 ->
%%   Letter widening, exactly as frozen decision 3 predicts.  Five sit
%%   above the band -- foursteps 8.92, dial-run 8.90, param-tree 8.88,
%%   cornell 8.58, trio 8.26 -- and dividing by 1.037 shows four of
%%   the five were already above house on A4, so they are the author's
%%   sizing and not this run's.  Above house is left alone.
%%   cat-timeline at 7.43 is the smallest and is inside the band.
%%   NO absolute widths, NO landscape block, NO hard-coded
%%   millimetres: every plate is \textwidth-relative, so the widening
%%   is uniform.  First of the eleven conversions where that held for
%%   every picture.
%%   STAGE 4: palette_fix.py took 27 violations to 3 -- keyword line
%%   widths converted to explicit points (a keyword hides its value
%%   from every audit), clamped to 1.1 pt, bare draws given hue
%%   borders, one figure's fills compressed into the !8-!18 band.
%%   The three it could not resolve are hand-fixed: two gray!65 node
%%   borders to black!50 (a tint is not a shade) and one
%%   orange!80!black to orange!70!black, the TOP of the !45-!75 band
%%   rather than its middle, so the emphasis placed there survives.
%%   Three path strokes above 1.1 pt are LEFT: inline \draw marks,
%%   not /.style bodies, and the standard exempts marks.
%%   NO PROSE WAS WRITTEN IN THESE STAGES.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5, same day.  INTEGRITY: FIFTEEN FIRST-CITATION LINKS.
%%   129 bibliography entries, 129 Relevance notes, and only 114
%%   first-citation links.  The fifteen missing ones are added.
%%   NOT WRITTEN BY HAND.  For each key the body is scanned for the
%%   FIRST \cite of it and the enclosing labelled section is read off,
%%   so each link points at what the paper does rather than at what
%%   anyone remembers it doing.  Two resolve into app:reading, the
%%   recommended-reading appendix, because that is where Kolb--Turner
%%   and Breuer--Petruccione are first and only cited.
%%   The sentence form is copied from the 114 entries that have it.
%%   Measured and unchanged: 0 orphan entries, 0 missing cohort
%%   papers, and the A/B split already present -- 21 Arisaka entries
%%   under "A. Papers by Katsushi Arisaka", 108 external under
%%   "B. Other references", in two thebibliography environments.
%%   NO PROSE WAS WRITTEN IN THIS STAGE beyond the fifteen link
%%   sentences, which are the house sentence with a resolved target.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5B, same day.  EIGHT CROSS-REFERENCES THE PROMOTION NEEDS.
%%   A bare grid sits where it was written; a float moves.  Eight of
%%   the eleven grids promoted in Stage 2A had no \ref anywhere in
%%   the paper -- for the good reason that until this morning they
%%   did not need one -- so their lead-in sentences would have been
%%   left pointing at whatever the page happened to put next.
%%   Each of the eight lead-ins gains a cross-reference and nothing
%%   else: these are the author's own sentences with a pointer
%%   inserted, no claim added, removed or reworded.  The eight are
%%   listed in the report so they can be read as a set.
%%   final_audit "floats never referenced": 8 -> 0.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 6, same day.  TWO PAGE-FITTING EDITS, BOTH MEASURED.
%%   1. The author's \enlargethispage{2\baselineskip} before "Part V
%%      in one paragraph" is REMOVED.  Safe on A4; on Letter it put
%%      the text block on the page number -- folio_scan measured the
%%      gap on page 133 at -0.7 pt against this paper's 16.0 pt mode,
%%      and at 150 dpi the last line sits on the folio.  Third
%%      occurrence of this failure in the run, after P5 and W1.
%%   2. The Executive Summary page is ENLARGED by two lines, because
%%      the Keywords block was spilling its last two lines onto a page
%%      of their own -- 197 printed characters, the emptiest page in
%%      the paper.
%%   The same command in opposite directions in one stage is not
%%   inconsistent: an enlargement is safe exactly when the page has
%%   room above the folio, and that is a measurement.  After both:
%%   minimum folio gap 10.6 pt, zero flagged pages, 280 -> 279 pages.
%%   LEFT, AND REPORTED: page 7, the six-entry tail of the List of
%%   Tables (section 7 wants a coherent class for list=no and there is
%%   no coherent class of six); pages 93 and 133, the Part III and
%%   Part V transition paragraphs, each spilling about four lines.
%%   Five of the seven such paragraphs fit.  Closing a four-line spill
%%   needs four lines out of the author's paragraph (section 14.2) or
%%   a four-line enlargement, which is what item 1 just had to undo.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.0 -> v15.1, August 15, 2026.  SIX COLONS THAT NO LONGER
%% INTRODUCE ANYTHING.  Found while migrating A11-Origin.
%%   A bare grid is introduced by a colon and follows on the next
%%   line.  Stage 2A made eleven grids into floats and Stage 5B gave
%%   eight of them the cross-reference a float needs -- and the
%%   colons stayed, now promising something that has floated away.
%%   What follows each on the page is the next sentence:
%%     triptych     -> "The electron is a holonomy class wearing..."
%%     cpt-owners   -> "Nothing here is new dynamics..."
%%     clock-ladder -> "Two remarks.  First, the ladder's top..."
%%     epochs       -> "The reader should run the check..."
%%     two-books    -> "Read down either column and there is no..."
%%     bilinears    -> "Two lessons.  A vector interaction..."
%%   Each becomes a full stop.  The sentences are the author's, the
%%   cross-references stay, no claim moves.
%%   WHY THE GATE PASSED IT: no gate item looks at punctuation, and
%%   at Stage 7 the pages carrying the new TABLES were read -- while
%%   the defect is in the line above the table, on a different page
%%   whenever the float moved, which is why the reference was needed.
%%   NO PROSE WAS WRITTEN IN THIS RIDER.
%% ------------------------------------------------------------
\PaperVersion{v20.2}
\PaperDate{August 22, 2026}

%% ---- paper-local macros (W1-compatible set + W2 additions) --------
\providecommand{\tr}{\operatorname{tr}}
\providecommand{\TR}{\operatorname{Tr}}
\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
\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{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}
\providecommand{\Mlock}{\ensuremath{M_{\mathrm{lock}}}}
\providecommand{\bZ}{\ensuremath{\mathbb{Z}}}
\providecommand{\dCKM}{\ensuremath{\delta_{\mathrm{CKM}}}}
\providecommand{\dPMNS}{\ensuremath{\delta_{\mathrm{PMNS}}}}
\providecommand{\JPMNS}{\ensuremath{J_{\mathrm{PMNS}}}}
\providecommand{\etaB}{\ensuremath{\eta_B}}
\providecommand{\gstar}{\ensuremath{g_{*}}}
\providecommand{\Stheta}{\ensuremath{S^1_\theta}}
\providecommand{\Ir}{\ensuremath{I_r}}
\providecommand{\SUtwoL}{\ensuremath{\mathrm{SU}(2)_{L}}}
\providecommand{\SUthreeC}{\ensuremath{\mathrm{SU}(3)_{C}}}
\providecommand{\UoneB}{\ensuremath{\mathrm{U}(1)_{B}}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{Qi}}}
\providecommand{\UoneY}{\ensuremath{\mathrm{U}(1)_{Y}}}
\providecommand{\UoneEM}{\ensuremath{\mathrm{U}(1)_{\mathrm{EM}}}}
\providecommand{\fK}{\ensuremath{f_{K}}}
\providecommand{\mK}{\ensuremath{m_{K}}}
\providecommand{\chiQCD}{\ensuremath{\chi_{\mathrm{QCD}}}}
\providecommand{\thetai}{\ensuremath{\theta_i}}
\providecommand{\Xicap}{\ensuremath{\Xi_{\mathrm{cap}}}}
\providecommand{\Omm}{\ensuremath{\Omega_m}}
\providecommand{\OmL}{\ensuremath{\Omega_\Lambda}}
\providecommand{\Omb}{\ensuremath{\Omega_b}}
\providecommand{\Mplred}{\ensuremath{M_{\mathrm{Pl,red}}}}
\providecommand{\PaperWOne}{\textsf{GG-W1-Symmetry}}
\providecommand{\HS}{\textsf{Hopkins--Singer}}
\providecommand{\GS}{\textsf{Green--Schwarz}}


%% Problem banner for the sixteen-problem architecture (v0.2).
\newcommand{\WProblem}[2]{%
  \par\vspace{1.8em}%
  \noindent{\color{orange!85!black}\rule{\textwidth}{1.1pt}}\par
  \vspace{0.4em}%
  \noindent{\Large\bfseries Problem #1.\ #2}\par
  \addcontentsline{toc}{section}{\protect\textbf{Problem #1.}\ #2}%
  \vspace{0.8em}\noindent}

%% Ragged-right p-column (cohort table standard).
\newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}}

%% v0.8: macros required by figures adapted from P5-CPT and C1-Kaxion
\providecommand{\Ja}{\ensuremath{J_A}}
\providecommand{\Jq}{\ensuremath{J_Q}}
\providecommand{\Uone}[1]{\ensuremath{\mathrm{U}(1)_{#1}}}
\providecommand{\thetags}{\ensuremath{\theta_{\mathrm{GS}}}}
\providecommand{\thetaw}{\ensuremath{\theta_W}}
\providecommand{\thetai}{\ensuremath{\theta_i}}

%% v0.11: appendix apparatus, in the form used by A9-SM, A10-Cosmos
%% and W1 -- \GGteachbox opens an appendix; \GGreadalong names its
%% textbooks; \GGtakeaway closes it.
\newcommand{\GGteachbox}[1]{%
  \par\medskip\noindent
  \fbox{\parbox{0.955\textwidth}{\small\textbf{What this appendix
  teaches.} #1}}\par\medskip}
\newcommand{\GGreadalong}[1]{%
  \par\noindent{\small\itshape Read alongside:} {\small #1}\par\medskip}
\newcommand{\GGtakeaway}[1]{%
  \par\medskip\noindent
  \fbox{\parbox{0.955\textwidth}{\small\textbf{Take-away.} #1}}\par\medskip}

\usetikzlibrary{decorations.pathreplacing}

%% \GGShowcaseHead -- defined per paper rather than in the style file,
%% because promoting it to ucla-arisaka-gg-11 would mean rebuilding and
%% reshipping twenty-two arXiv packages for one macro.  Flagged, not
%% done -- it is now a per-paper macro in twenty-one papers.
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
\GGShowcaseCenteredtrue
\renewcommand{\GGShowcaseHeadSkip}{\vspace{8pt}}

\newsavebox{\GGplateAnatomy}
\newsavebox{\GGplateForces}
\begin{document}

%% ---- showcase saveboxes, FILLED AT \setstretch{1} ----
%% PROBED INSIDE BOTH ENVIRONMENTS: this paper runs BSTR
%% 1.15 in the front matter and BSTR 1 inside fig:anatomy
%% and fig:four-forces.  \global\sbox and not \sbox: a box
%% assignment inside \begingroup is local, and the group
%% that supplies the correct stretch would throw the filled
%% box away on its way out.
%% FILLED AT THE TOP OF THE DOCUMENT, not after the
%% Keywords: on W1 the first use of a box was inside the
%% Executive Summary, ahead of where every earlier paper
%% had filled, and the body copy read an empty box.
\begingroup\setstretch{1}\renewcommand{\arraystretch}{1}%
\global\sbox{\GGplateAnatomy}{\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% ================= Panel (a): the circle =================
\begin{scope}
\node[align=center] at (0,2.8) {\textbf{(a) The circle $S^1_\theta$ (6th dimension)}\\[-1pt]
\scriptsize compact --- all loops, all holonomy --- a group};
\draw[line width=1.10pt] (0,0) circle (1.7);
\foreach \ang in {90,162,234,306,18}
  \fill[black!55] (\ang:1.7) circle (0.055);
\draw[->,line width=0.80pt,blue!65!black] (40:1.95) arc (40:130:1.95);
\node[blue!65!black] at (85:2.3) {\scriptsize every loop closes};
\node[align=center] at (0,0.35) {\scriptsize windings, phases,\\[-2pt]
\scriptsize parities, charges};
\node[align=center] at (0,-0.45) {\scriptsize quantized, protected};
\node[align=center,blue!65!black] at (0,-2.35)
  {\textbf{remembers}\\[-2pt] \scriptsize classes do not age};
\end{scope}
% ================= Panel (b): the interval =================
\begin{scope}[shift={(5.6,0)}]
\node[align=center] at (3.4,2.8) {\textbf{(b) The interval $I_r$ (5th dimension)}\\[-1pt]
\scriptsize non-compact --- no loops, no holonomy --- a semigroup};
\draw[->,line width=1.10pt] (0.2,0.4) -- (6.6,0.4);
\node at (3.4,0.75) {\scriptsize $r = \ln(E_{\mathrm{GUT}}/\mu)$: the descent, one-way};
\foreach \x/\lab in {0.7/{$E_{\mathrm{GUT}}$},2.5/{lock},4.3/{QCD},6.1/{today}}{
  \draw[line width=0.80pt] (\x,0.32) -- (\x,0.48);
  \node[below] at (\x,0.28) {\scriptsize \lab};}
\node[align=center] at (3.4,-0.75) {\scriptsize no path returns:\\[-2pt]
\scriptsize scales, addresses, rates --- one pass each};
\node[align=center,blue!65!black] at (3.4,-2.35)
  {\textbf{forgets}\\[-2pt] \scriptsize each deposit is final};
\end{scope}
\end{tikzpicture}}%
\global\sbox{\GGplateForces}{\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% fibre icons
\draw[line width=1.10pt] (2.2,3.4) circle (0.75);
\node at (2.2,3.4) {\scriptsize $S^1_\theta$};
\node[align=center] at (2.2,4.7)
  {\scriptsize \textbf{the circle}\\[-2pt]\scriptsize holonomy: exact, protected};
\draw[->,line width=1.10pt] (7.6,3.4) -- (11.4,3.4);
\node at (9.5,3.75) {\scriptsize $I_r$: the descent};
\node[align=center] at (9.5,4.7)
  {\scriptsize \textbf{the interval}\\[-2pt]\scriptsize one-way: broken, priced};
% four boxes
\node[draw=black!55,line width=0.80pt,align=center,minimum width=2.5cm] (st) at (0.6,0.8)
  {\scriptsize \textbf{strong}\\[-2pt]\scriptsize the closing figure\\[-2pt]
   \scriptsize color books shut};
\node[draw=black!55,line width=0.80pt,align=center,minimum width=2.5cm] (em) at (3.8,0.8)
  {\scriptsize \textbf{electromagnetic}\\[-2pt]\scriptsize the open phase\\[-2pt]
   \scriptsize winding $=$ charge};
\node[draw=black!55,line width=0.80pt,align=center,minimum width=2.5cm] (wk) at (7.0,0.8)
  {\scriptsize \textbf{weak}\\[-2pt]\scriptsize the cascade's agent\\[-2pt]
   \scriptsize chiral, broken};
\node[draw=black!55,line width=0.80pt,align=center,minimum width=2.5cm] (gr) at (10.4,0.8)
  {\scriptsize \textbf{gravity}\\[-2pt]\scriptsize the warp itself\\[-2pt]
   \scriptsize $G_N$: the descent's constant};
% arrows
\draw[->,line width=0.80pt,blue!65!black] (st.north) -- (1.9,2.75);
\draw[->,line width=0.80pt,blue!65!black] (em.north) -- (2.6,2.7);
\draw[->,line width=0.80pt,red!70!black] (wk.north) -- (8.4,3.15);
\draw[->,line width=0.80pt,red!70!black] (gr.north) -- (10.5,3.15);
\node[red!70!black] at (7.0,2.3) {\scriptsize acts \emph{for} it};
\node[red!70!black,right] at (10.75,2.3) {\scriptsize \emph{is} it};
\end{tikzpicture}}%
\endgroup

% ---- v0.3 (August 2, 2026) ----
% The ten one-page problem summaries expanded to full essays
% (Problems 1, 3, 4, 5, 6, 8, 14, 15, 16; ~5-7 pages each) at the
% commissioned register: intuitive but accurate plain English,
% senior-undergraduate / first-year-graduate level, big-picture
% view of 6D holonomy. Depth apparatus updated (Table 1 caption,
% organisation, preface, Part openers, discussion counts).
% ---- v0.2 (August 2, 2026) ----
% The sixteen-problem restructure (author's TOC): new Part I
% Introduction (the 16 problems in four categories); old Parts
% kept intact and demoted to Problem blocks at their proper
% locations (interactions -> Problem 2, dark trio -> Problem 7,
% discrete -> Problem 9, antimatter -> Problems 10-11,
% confinement -> Problem 12, time -> Problem 13); ten new
% one-page problem summaries (1, 3, 4, 5, 6, 8, 14, 15, 16, and
% the intro apparatus); all Part cross-references remapped;
% preface/executive-summary/discussion updated to the new map.
% ---- v0.1 (August 2, 2026) ----
% First assembly of the second white paper. See file-head changelog.

%% ===================================================================
%%  COVER PAGE
%% ===================================================================
\GGTitlePage%
  {The Circle That Remembers,\\ the Interval That Forgets}%
  {\GGTheory{} and the Sixteen Problems\\
   of Particle Physics and Cosmology}%
  {August 12, 2026}
\thispagestyle{empty}

\vspace{0pt}
\begin{abstract}
\noindent
Physics keeps a short list of questions its own tools cannot close:
what happens when a measurement happens; where the Standard
Model's two dozen unexplained numbers come from; what launched the
Big Bang; what became of the information that fell into a black
hole. Their median age is seventy years.

This paper takes sixteen of them and follows one exact, elementary
statement to all sixteen. In the six-dimensional geometry of
\GGTheory{}, every point of ordinary space carries two further
directions of opposite kind: a closed circle and an open interval.
A holonomy --- the record a loop keeps of what happened around it,
the quantity all of gauge physics is built from --- requires closed
loops, and an interval has none. Everything on the circle is
therefore holonomy: quantized, protected, reversible, a group.
Nothing on the interval can be: its transport runs forward only, a
semigroup, and each step is final. \emph{The circle remembers; the
interval forgets.}

Read with the cohort's open-system chain, that sentence sorts all
sixteen. A measurement is the interval writing what the circle
carried; gauge invariance is the circle's unreadability and
renormalization is motion along the interval; time's tick is the
circle's and its direction the interval's; the black hole's
information was never lost, only billed to the wrong book. Nothing
is newly derived here: every result is proved in a cohort paper and
carried at its parent's declared grade, with conditional and
flagged claims marked in place. The paper closes with the named
measurements on which six of the sixteen treatments die outright, and
with how the other ten are exposed through the ledger they share.
\end{abstract}

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%%  EXECUTIVE SUMMARY
%% ===================================================================
\begin{ExecutiveSummary}
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\noindent
\textbf{The list.} Every working physicist carries a version of the
same list: the questions the discipline has not closed. It is
shorter than outsiders expect and older than insiders like to
admit. The measurement problem was posed in the 1920s and sharpened
in 1935; dark matter was inferred in 1933; the strong-$CP$ problem
and the black-hole information paradox both date from 1976; the
hierarchy problem from 1979; the cosmological constant problem was
already notorious when the acceleration was found in 1998. Four
decades of colliders, telescopes, and theory programs have
sharpened every entry and closed almost none. The standard response
has been to attack them one at a time, each with its own tool ---
an interpretation for measurement, a symmetry for the hierarchy, a
new particle for the dark sector, an ensemble of universes for the
constants. This paper does the opposite: it asks whether the list
is long because the questions are many, or because one structural
fact about the world is missing from all of them.

\medskip\noindent
\textbf{The one sentence.} Walk around a closed circuit carrying an
arrow and holding it as steadily as the geometry allows; when you
return, the arrow may have turned. That turning is a
\emph{holonomy}: the record a loop keeps of what happened around
it, and the quantity that all of gauge physics is built from. It
requires a loop. In the six-dimensional bulk of \GGTheory{}, the
two non-spatial directions are fibers of opposite kind: a compact
circle $\Stheta$, on which every path closes, and an open scale
interval $\Ir$, on which none does. So \emph{holonomy requires
closed loops, and an interval has none}. Everything on the circle
is holonomy --- quantized, gauge-protected, reversible, a group.
Nothing on the interval can be: its transport composes forward
only, a semigroup, and each step is final. \emph{The circle
remembers; the interval forgets.} That is the whole instrument.

\medskip\noindent
\textbf{Why exactly one of each.} A bulk with two circles would
conserve everything and decide nothing: no arrow, no record, no
outcome --- a world that \emph{is} but never \emph{becomes}. A bulk
with two intervals would decide everything and keep nothing: no
quantized charge, no stable atom, no interference --- a world that
\emph{becomes} but never \emph{is}. Being needs holonomies;
becoming needs a semigroup. One fiber of each kind is the
constitutional minimum, and it is what the program's axioms
write.

\medskip\noindent
\textbf{What the sentence organizes.} Sixteen problems, in four
groups. The numbers below are the cohort's, produced by its
derivations; what the sentence supplies is the sorting that makes them
one story.
\emph{The Standard Model}: a measurement is the interval writing
what the circle carried, which is why one theory needed two
dynamics; gauge invariance is the circle's unreadability and
renormalization is motion along the interval; the parameter sheet
is a codebook in one alphabet --- $\alpha^{-1}(0) = 2^J + N^2 =
137$, $\sin^2\theta_W = L^2/(4N^3)$, $v = \Mlock(2/L)^N =
246.3$~GeV. \emph{Cosmology}: the Big Bang becomes a launch that is
derived, not chosen ($n_s = 0.9620$, $r = 0.0043$); the hierarchy
is not a small number but a deep one ($K = 35$ e-folds of descent);
the dark trio is the two fibers in the sky --- inflation the
interval's coordinate rolling once, the Kaxion the circle's still
ringing at $8.1\,\mu$eV and $1.95$~GHz, dark energy both at rest.
\emph{The foundation of particle physics}: $C$, $P$, and $T$ are
the geometry's three orientation choices, and their observed fates
follow by fiber --- $CPT$ exact, $CP$ broken, $T$ broken at
runtime; an antiparticle is the same class read in the mirror, and
the matter excess is a one-way toll of $e^{-K/2}$, giving
$\etaB = 5.9\e{-10}$ against a measured $6.1\e{-10}$; quarks
confine because three color steps close a loop, and the proton is
stable because baryon number is a gauged holonomy class and the
geometry has no $\Delta B = 1$ aperture.
\emph{The foundation of the cosmos}: time's tick is the circle's
and its arrow the interval's, with the chronology exact
($d\ln t/dr = 2$; an age ledger closing at $140.24$); energy is
what a tick is worth and entropy is what the arrow has read; the
multiverse is not refuted but unemployed; and the black hole's
information was always in the books, on the other page.

\medskip\noindent
\textbf{Where the sentence comes from.} The anatomy is not an
interpretive overlay placed on the cohort after the fact; it is the
arena the cohort's own axioms force, read back in plain language.
The program is built in four steps. \emph{Step (A)} writes the
bulk: the constitutional charter of four axioms and seven
postulates \citep{ArisakaA1Constitution}, the geometry--gauge
locking that forces
$\mathcal{M}_6 = \mathcal{M}_4 \times \Ir \times \Stheta$ with
$\mathrm{Spin}(1,5) \times SU(5) \times U(1)_B \times U(1)_{Qi}$
\citep{ArisakaA2GG6}, the anomaly closure that fixes the ledger's
coefficient arithmetic \citep{ArisakaA3BI6}, and the Euclidean
input discipline that leaves zero free continuous parameters
\citep{ArisakaA4Euclid}. \emph{Step (B)} writes the projection:
the open-system dynamics that makes observation a marginalization
of the fiber coordinates \citep{ArisakaA5GDOS}, the cascade engine
that runs it \citep{ArisakaA6Gi4}, and the quantum-information
backbone that meters it \citep{ArisakaA7Qi4}. \emph{Step (C)}
spends the machinery on the world: quantum mechanics
\citep{ArisakaA8QM}, the Standard Model \citep{ArisakaA9SM},
$\Lambda$CDM \citep{ArisakaA10Cosmos}, and the origin of life and
mind \citep{ArisakaA11Origin}. \emph{Step (D)} is the pair of white
papers: \PaperWOne{} walked a century of symmetry and conservation
to the point where protection stops needing a symmetry
\citep{ArisakaW1Symmetry}; this one takes the two fibers and walks
them into the sixteen problems.

\medskip\noindent
\textbf{The ledger the numbers are written in.} Five integers
$(M,N,L;J,K) = (1,3,5;7,35)$, with $J = 2N+1$ and $K = LJ$, carry
the circle's side of the accounting; depths along the interval
carry the rest. From them the cohort's closed forms follow with
nothing left to adjust: $\alpha^{-1}(0) = 2^7 + 3^2 = 137$,
$\sin^2\theta_W = 5^2/(4\cdot 3^3) = 0.23148$,
$\Mlock = \tfrac12 M_{\rm Pl} e^{-K} = 3.849$~TeV,
$v = \Mlock (2/5)^3 = 246.3$~GeV \citep{ArisakaA9SM};
$\Omega_m = 1/\pi$ and $\Omega_b = 1/(2\pi^2)$, $n_s = 0.9620$ and
$r = 0.0043$ at a derived $N^* = 52.6$ \citep{ArisakaA10Cosmos};
the Kaxion at $8.1\,\mu$eV and $1.95$~GHz
\citep{ArisakaC1Kaxion}; $d\ln t/dr = 2$ exactly, and a
black-hole ledger that returns the Page curve rather than
postulating it \citep{ArisakaC3BH}. Each is quoted here at its
parent paper's grade --- derived, derived-conditional, named
input, postulated, or flagged --- and the grades are printed
beside the numbers rather than in a footnote.

\medskip\noindent
\textbf{What this paper is, and is not.} It is a synthesis, not a
new derivation. Every equation, number, and theorem carried here is
proved in a cohort paper --- chiefly \PaperWOne{},
\textsf{GG-P5-CPT}, \textsf{GG-P6-Proton}, \textsf{GG-A10-Cosmos},
\textsf{GG-C1-Kaxion}, and \textsf{GG-A8-QM} --- and is cited at
first use and carried at its parent's grade, with every conditional
and flagged claim marked in place. The paper's own contribution is
the anatomy that makes them one story, and the exposure that
follows: Part~VII collects, in one table, the named measurements on
which six of the sixteen treatments die outright, and states how the
other ten are exposed through the ledger all sixteen share. A reader
who wants the argument in an hour should read Part~I, Part~II, and
the Conclusions; a reader who wants to attack it should read
Part~II and then the falsifier table.

\medskip\noindent
\textbf{What the back of the paper is for.} Three Parts follow the
argument, and none of them is an appendix in the ordinary sense.
Part~IX is a graduate course: nine topics, each in four steps ---
what the standard textbooks establish and how they establish it,
why the problem nevertheless survives, how the two fibers address
it with the mathematics written out, and worked problems with
solutions. It is meant to be self-sufficient, so that an incoming
graduate student can learn the standard material and the
framework's reading of it from the same pages, and it opens with a
reading list naming the textbooks and the original papers for each
topic. Part~X collects one hundred and seventeen objections --- the
questions the author has actually been asked, sharpened --- and
answers them, beginning with twenty-two from the curious reader
before any specialist gets a turn. Part~XI is a bibliography in
which every entry carries a paragraph saying what the work
established, what it cost the field, and where this paper's
reading departs from it. A framework asking to be taken seriously
owes its readers the means to check it, and these three Parts are
that debt paid.

\medskip\noindent
\textbf{A closing thought.} It is worth saying plainly what kind
of claim this is. The sentence at the center of this paper ---
holonomy requires closed loops, and an interval has none --- is not
a discovery. It is a definition and its first consequence, the sort
of thing a graduate student proves in an afternoon. What would be
remarkable, if it holds, is not the sentence but its reach: that
one such statement, entered into twenty-two independently derived
papers through each one's own central topic, required no value, no
grade, and no theorem to change anywhere, and came back out with
fifteen dated ways to die.

There is an old suspicion that the effectiveness of mathematics in
physics is unreasonable --- that the fit between structures
invented for their own sake and a world that owes them nothing is a
gift we neither earned nor understand \citep{Wigner1960}. This
paper's answer is not that the puzzle dissolves, but that it may
have been posed from the wrong side. If the integers are the
topology of the arena rather than a language we brought to it, then
mathematics is not being \emph{applied} to the world; it is what the
world is made of being read back. That is a strong claim, and it is
not yet established ---
what is established is that a small ledger of integers, fixed by a
shape rather than by a fit, reproduces a great deal of what we
measure.

The deepest of the sixteen problems are not, in the end, about
particles or galaxies. They are about the two things a world has to
do at once: be something, and become something. Being requires that
some records cannot be destroyed; becoming requires that some steps
cannot be undone. Physics has spent a century discovering, one
problem at a time, that it needs both and can build neither from
the other. The proposal here is that it never had to --- that the
geometry supplies one of each, and that most of what has looked
like a list of separate mysteries is the record of a
discipline meeting one shape from sixteen sides. Whether that is
true is not for the author to decide. The instruments will settle
it, and the paper has said in advance what they should find.
\end{ExecutiveSummary}

\GGKeywords{six-dimensional bulk; two-fiber anatomy; holonomy;
compact fiber and scale interval; group versus semigroup; open
quantum systems; measurement and the arrow of time; gauge
invariance and renormalization; hierarchy and naturalness;
inflation; dark energy; axion dark matter; baryon asymmetry;
confinement and proton stability; landscape and multiverse;
black-hole information; unsolved problems in physics;
\GGTheory{}}

\clearpage%% Self-filling showcase frame (Report 462): this page carried no
%% frame at all, which is why it printed hard against the top.  The
%% glue is page-level -- a longtable cannot live inside a minipage.
\null\vspace*{\stretch{1}}

\GGShowcaseHead{The Circle Remembers; the Interval Forgets ---\\[3pt]
and That One Sentence Sorts All Sixteen}

\par\vspace*{\stretch{2}}
\begin{center}
\resizebox{0.851\textwidth}{!}{\usebox{\GGplateAnatomy}}

\vspace{14pt}

\resizebox{0.710\textwidth}{!}{\usebox{\GGplateForces}}
\end{center}

\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{The whole paper in two pictures.} Above: the anatomy. Every point of ordinary space carries two further directions, and they are of opposite kind. On the compact circle every path closes, so everything a connection does there is holonomy --- quantized, gauge-protected, reversible, a group. On the open interval no path closes, so nothing there can be protected: transport composes forward only, a semigroup, and each step is final. That is the sentence, and it is elementary: \emph{a holonomy requires closed loops, and an interval has none.} Below: the same sentence sorting the four interactions, which is the widest thing it does. Strong and electromagnetic are circle-side, and that is why their charges are exact and unbroken. The weak force is the interval's agent, which is why it alone is chiral, broken and mass-giving. Gravity is not a force on the stage at all but the stage's warp along the descent --- why its constant alone is tied to irreversibility, and why it will not quantize like the other three. The century-old puzzle of the four interactions' unlike characters is that sorting, unread. Sixteen open problems are worked the same way in Parts~III--VI, none of them newly derived here and each carried at its parent paper's declared grade. In full: Figure~\ref{fig:anatomy}, p.~\pageref{fig:anatomy}, and Figure~\ref{fig:four-forces}, p.~\pageref{fig:four-forces}.}
\par\vspace*{\stretch{1}}
\clearpage


%% ===================================================================
%%  OVERVIEW OF CONTENTS + LIST OF FIGURES + FULL TABLE OF CONTENTS
%%  (cohort gg-11 macros; A9-SM convention)
%% ===================================================================
\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables

\begingroup
\setstretch{1.05}%
\GGTableOfContents
\endgroup

%% ===================================================================
%%  PREFACE  (cohort convention: after the contents, before Part I)
%% ===================================================================
\GGPreface

I have spent fifty years in experimental physics asking one
question in many costumes: why is the universe so precisely
arranged --- for structure, for stability, for us? The
\GGTheory{} cohort is my attempt at a complete answer, developed
across some twenty-two research papers; this white paper, the
second in the series, is where I step back and show the single
newest idea that, to my own surprise, turned out to organize
everything else. My first white paper, \PaperWOne{}, walked a
century of symmetry and conservation and placed the program at
the end of that arc. This one is narrower and deeper: it takes the
two-fiber anatomy of the six-dimensional bulk --- one compact
direction that cannot forget, one open direction that cannot
remember --- and develops it against the sixteen problems I regard
as the deepest in particle physics and cosmology.

\paragraph{The idea, in the plainest words I can find.}
Imagine walking a closed circuit while holding an arrow as steadily
as you can. Come back to where you started, and the arrow may have
turned. That turning is what geometers call a \emph{holonomy}, and
it is the quantity every gauge theory in physics is built from. It
needs a loop. In this framework the world has two directions
besides the ones you can walk in: one of them closes on itself like
a circle, so every path on it is a loop and everything it carries is
a holonomy --- quantized, protected, and unable to drift. The other
is an open stretch with two ends, on which no path closes, so it can
hold no holonomy at all and everything on it runs one way and does
not come back. The circle remembers. The interval forgets. That is
the entire instrument of this paper, and the sixteen problems are
what it is pointed at.

\paragraph{Three things I want you to know at the outset.}
\emph{First}, this paper derives nothing new. Every equation,
number, and theorem carried here is proved in a cohort paper, cited
at first use and carried at its parent's grade --- where a claim is
conditional or flagged, the text says so in place, and I ask you to
hold me to that. \emph{Second}, the paper is organized as a problem
ledger: Part~I states the sixteen problems in their own terms, each
with its history, its literature, and an honest account of why it
has resisted for forty to a hundred years; Part~II reviews the
program in its four development steps and then builds the anatomy
once, equations included; Parts~III--VI treat the sixteen, four
categories of four, all at essay length or fuller.
\emph{Third}, the paper is exposed by design: Part~VII collects, in
one table, the named measurements on which six of the treatments die
outright, and says plainly that the other ten are exposed only through
the ledger they share. I would rather be decided than believed.

\paragraph{An invitation, to readers of every kind.}
I have tried to write this for more than one person at once, and I
mean the invitation seriously in each case.

If you are a \emph{curious reader} with no physics training beyond
what you have read for pleasure --- you are welcome here, and you
are not an afterthought. Read the Executive Summary, then the
sixteen essays of Part~I, which were written to be read the way one
reads a good popular science book: history first, then the puzzle,
then why it is hard. Then turn to Part~X and read the first
twenty-two objections, which are the questions people actually ask
me: where are the extra dimensions, is this string theory, what
about free will, what came before the Big Bang, and --- fairly ---
why should anyone trust a framework that claims to explain this
much. I have answered them plainly, and where the honest answer is
``not yet,'' I have said so.

If you are a \emph{student} --- a senior undergraduate or an
incoming graduate --- read it straight through. It assumes
first-year graduate physics and nothing at all about this
program. Part~IX is for you in particular: nine topics, each
taken in four steps, beginning with a full textbook treatment of
the standard material before a word is said about anything new,
with derivations written out and problems worked. If you learn the
standard physics from those pages and decide the rest is wrong, the
appendix will still have done its job.

If you are a \emph{specialist} in one of these sixteen areas, you
will know more about your own problem than this paper does, and
that is as it should be --- it is a synthesis, and its claim is
about the pattern rather than about any single entry. The fastest
honest route is Part~II, then the section on your problem, then the
honest ledger (\S\ref{sec:honest}) and the ways to be wrong
(\S\ref{sec:kill}). If the treatment of your problem is wrong, it
will be wrong in a checkable way, and I would be glad to be told.

If you are a \emph{skeptic} --- and I hope some of you are, because
a framework of this scope deserves hard readers --- start at
Part~X. I have not hidden the objections at the back to be polite
about them; they are stated in their sharpest form, several of them
are conceded outright, and the master table grades every one of the
sixteen with a deliberately varied verb, because \emph{solved} is
not what most of them earned.

\paragraph{A last word.}
Physics is not a private activity, and no framework of this size is
ever finished by the person who proposed it. Whatever survives of
this one will survive because other people checked it, and what
does not survive will fall for the same reason. That is not a
disappointing thought; it is the only arrangement under which any
of this means anything. If a single reader closes this paper with a
clearer picture of what the sixteen problems actually are --- even
one who is certain the answer offered here is wrong --- then the
writing was worth the years.


%% ===================================================================
%%  PART I -- INTRODUCTION
%% ===================================================================
\GGPart[Introduction]{Introduction}{Sixteen Unsolved Problems in
Particle Physics and Cosmology, and One Sentence}

\section{The sixteen problems}
\label{sec:sixteen}

Every era of physics keeps a short list of questions its own tools
cannot close. What is unusual about the present list is how little
it has moved. Four decades of colliders, telescopes, and theory
programs have sharpened every entry on it and closed almost none;
most entries predate the career of anyone now working, and several
predate the careers of their teachers. The measurement problem was
posed in the 1920s. Dark matter was inferred in 1933. The strong-$CP$
problem and the black-hole information paradox both date from 1976,
the hierarchy problem from 1979, and the cosmological constant
problem was already a scandal when the acceleration was discovered
in 1998.

A list that stable is itself a piece of evidence, and it can be read
two ways. Either these are simply hard problems, unrelated to one
another, each awaiting its own breakthrough --- the reading under
which the discipline has worked for two generations. Or the list is
long because one structural fact about the world is missing from all
of them at once, and each entry is that absence, seen from a
different direction. This paper takes sixteen of the deepest problems
--- grouped into four categories of four --- and develops the second
reading: \emph{a single structural sentence about the six-dimensional
bulk addresses all sixteen}. The sentence is built in Part~II. The
problems, and where each is treated, are these:

\begin{table}[H]
\centering
\caption[The sixteen problems, their categories, and the anatomy's answer in one line.]{The sixteen problems, their categories, and the anatomy's
answer in one line. All sixteen are developed at essay length; the seven rows marked $\bullet$ additionally carry the fullest quantitative apparatus --- figures, tables, and complete chains. The numbering established here is the paper's primary cross-reference: every later Part cites its subject as Problem~$n$ rather than by name.}
\label{tab:sixteen}
\small
\renewcommand{\arraystretch}{1.24}
\begin{tabularx}{\textwidth}{@{}cP{4.35cm}>{\raggedright\arraybackslash}X@{}}
\toprule
\# & \textbf{Problem} & \textbf{The anatomy's answer, in one line} \\
\midrule
\multicolumn{3}{@{}l}{\emph{Part III --- The Standard Model}} \\
1 & Measurement problem of quantum mechanics & coherence is the circle's; registration is an interval-side write --- a measurement is a read \\
2 $\bullet$ & Strong, weak, EM, and gravity: why so different & two keepers on the circle, one agent on the interval, one integrator \\
3 & Origin of local gauge invariance and renormalization & gauge symmetry is the circle's unreadability; renormalization is motion along the interval \\
4 & Too many free parameters (SM and $\Lambda$CDM) & every parameter is a holonomy label wearing a descent address --- a codebook, not a fit \\
\midrule
\multicolumn{3}{@{}l}{\emph{Part IV --- Cosmology}} \\
5 & The beginning of the universe & ``what banged?'' becomes ``what launched the descent?'' --- and the launch is derived \\
6 & Naturalness and the hierarchy problem & nothing is small; something is deep --- $e^{-K}$, counted, not tuned \\
7 $\bullet$ & The dark trio: inflation, dark energy, dark matter & the interval's coordinate rolling, the circle's ringing, and both at rest \\
8 & The three horizon problems & same temperature by the launch; same particles by the classes; same laws by the groove \\
\midrule
\multicolumn{3}{@{}l}{\emph{Part V --- Foundation of Particle Physics}} \\
9 $\bullet$ & Discrete symmetry: $C$, $P$, $T$, $CP$, $CPT$ (+ strong CP) & orientation choices of the geometry; fates assigned by fiber \\
10 $\bullet$ & Origin of antiparticles & the same circle-side class, read in the mirror orientation \\
11 $\bullet$ & The tiny excess of matter (baryogenesis) & a chiral tape, read once, through a gated door, at a toll of $e^{-K/2}$ \\
12 $\bullet$ & Quark confinement; absolutely stable protons & the triangle that closes; the gauged ledger with no aperture \\
\midrule
\multicolumn{3}{@{}l}{\emph{Part VI --- Foundation of the Cosmos}} \\
13 $\bullet$ & Definition of time and the arrow of time & the circle gives the tick; the interval gives the direction \\
14 & Definition of energy, mass, and entropy & what the tick is worth; an address a class wears; what the arrow has read \\
15 & Landscape and multiverse problems & a dial, not a hat: one groove, one playback --- the multiverse is unemployed \\
16 & Black-hole information paradox & a billing error: the circle keeps what the hole must return; the interval prices the return \\
\bottomrule
\end{tabularx}
\end{table}

Three features of the table are the paper's thesis in miniature.
\emph{First}, the right column is one vocabulary: every answer is a
sentence about a compact circle that remembers and an open interval
that forgets. \emph{Second}, the sixteen problems span every
register of the discipline --- foundations, phenomenology,
cosmology, and interpretation --- and the standard literature
treats them in at least six disjoint languages; whatever else is
true of the anatomy, its \emph{reach} is a fact on display in this
table. \emph{Third}, the table is falsifiable row by row: Part~VII
consolidates the named measurements on which the entries die.

\section{How to read the sixteen}
\label{sec:howtoread}

The rest of this Part walks the sixteen, one at a time, at about
half a page each. The purpose of the walk is not to answer anything
--- the answers are Parts~III--VI --- but to establish two things a
reader is entitled to demand before being asked to accept a new
structural idea.

The first is that these problems are real, and are not the private
obsessions of one program. Each entry below is stated as its own
literature states it, with the history that produced it and the
names attached: the people who posed each question were among the
most capable in the discipline, and the strategies that failed on it
failed for interesting reasons, not foolish ones. Where a rival
research program has a genuine achievement --- and several have
large ones --- the achievement is stated before the shortfall.

The second is the diagnosis. Each entry ends with the same
uncomfortable question: \emph{why has this resisted for forty,
seventy, a hundred years?} The answers turn out to rhyme, and the
rhyme is the motivation for everything that follows. It is developed
in \S\ref{sec:common} once all sixteen are on the table.
The reader should know that this is not a survey of curiosities. Two
of the sixteen have already had a candidate answer removed by
experiment in the last six years --- the strong-$CP$ angle's
window narrowed again \citep{Abel2020}, and the possibility that
quantum mechanics could be built on real numbers alone shown
falsifiable \citep{Renou2021} and then excluded by network Bell
tests \citep{Li2022, Chen2022} --- and a third, the Hubble
tension, has hardened from an anomaly into a structural question
about which clock cosmology is reading \citep{ArisakaC2Hubble}.
These are live problems, moving under instruments now running.

\section{Category I --- The Standard Model: the theory that works
too well}
\label{sec:cat1}

The Standard Model is the most precisely tested theory in the
history of science, agreeing with experiment in some quantities to
twelve significant figures. Its four unsolved problems are
therefore of a peculiar kind. None of them is a discrepancy. All
four are questions about \emph{why the theory has the shape it
has} --- and the theory, by construction, cannot be asked.

\subsection{Problem 1. What happens when a measurement happens}
\label{intro:p1}

Quantum mechanics contains two rules for how states change, and no
rule for when to use which. Between measurements a state evolves
smoothly, reversibly, and deterministically, by the
Schr\"odinger equation. At a measurement it does something else
entirely: it jumps to one outcome, irreversibly, with a probability
given by the squared amplitude --- the rule \citet{Born1926} added
to his paper in proof, in a footnote, which is now the most
successful statistical prediction in physics. The two rules
contradict each other, and the theory says nothing about the border
between them. \citet{vonNeumann1932} made this explicit,
distinguishing the unitary ``process 2'' from the collapse
``process 1'' and observing that the cut between them can be slid
anywhere between the atom and the observer's eye without changing
any prediction --- a movable seam is not a physical boundary, and
he knew it.

The problem then hardened rather than resolved. \citet{EPR1935}
argued from locality that the quantum description must be
incomplete --- a dilemma the century has since read the other way
round: keep completeness, and locality is what gives; Schr\"odinger's cat, from the same year, showed that
the ambiguity can be amplified to household scale. Thirty years
later \citet{Bell1964} converted the metaphysics into an
inequality, and the experiments that followed closed the local
hidden-variable escape for good. Meanwhile the interpretations
multiplied: \citet{Everett1957} removed the collapse by keeping
every branch; \citet{GRW1986} modified the dynamics so that
collapse is a physical process with its own rate; and the
decoherence program \citep{Zurek2003} showed in detail how
interference disappears when a system is coupled to an
environment.

Why has this survived a hundred years? Because each strategy pays
for its solution in a currency the others will not accept, and no
experiment adjudicates. Everett keeps the mathematics intact and
owes an account of why the Born weights are the probabilities of
anything at all. Dynamical collapse buys single outcomes with new
constants and a hard fight with relativity. Decoherence, the most
successful of the three, explains why the interference is
unobservable but takes as given the very things it was asked to
explain --- which variables are the classical ones, and why the
squared amplitude sets the odds --- and it never produces a single
outcome, only an ensemble that looks like one. A century on, the
options remain empirically indistinguishable by construction. That
is the shape of the problem: not undertested, but posed in a frame
where testing cannot help. \emph{Treated as Problem~1 in Part~III.}

\subsection{Problem 2. Why the four interactions are so unalike}
\label{intro:p2}

Nature runs on four interactions, and their family resemblance is
poor. Electromagnetism is exact, unbroken, infinite in range, and
carried by a massless quantum. The strong interaction is so
powerful that its charges are never seen alone. The weak
interaction is short-ranged and feeble, and is the only one that
distinguishes left from right, the only one that changes what a
particle is, and the only one that violates $CP$. Gravity is not
like any of them: it couples to everything, cannot be switched off,
has a dimensionful constant, and has resisted seventy years of
attempts to quantize it in the manner that worked for the other
three.

The historical arc is one of triumphant unification that stops
short of explanation. \citet{Fermi1934} wrote the weak interaction
as a contact term. \citet{YangMills1954} generalized the gauge
principle to non-abelian groups, giving all of the non-gravitational
forces their common language. Electromagnetism and the weak force
were then unified \citep{Weinberg1967} and confirmed
spectacularly, and the discovery of asymptotic freedom
\citep{GrossWilczek1973, Politzer1973} completed the strong sector.
\citet{GeorgiGlashow1974} took the natural next step and embedded
all three in a single group --- a beautiful proposal whose simplest
version predicted proton decay at a rate that experiment has now
excluded.

So why, after seventy years, is the difference between the forces
still unexplained? Because the successful strategy was
\emph{embedding}, and embedding explains sameness, not difference.
Grand unification makes the three forces alike at high energy and
relegates their present unlikeness to the accident of a symmetry
breaking pattern chosen to fit. Within the Standard Model itself,
the differences are inputs, not outputs: the gauge group is written
as a product of three factors by hand; the chirality of the weak
force is assigned by hand; and the three coupling strengths are
measured, not derived. Gravity's exclusion is worse than an
accident --- it is structural, and the very feature that resists
quantization, its dimensionful constant, is a clue that it is not
the same kind of object as the other three. \emph{Treated as
Problem~2 in Part~III.}

\subsection{Problem 3. Where gauge invariance and renormalization
come from}
\label{intro:p3}

Two principles carry the entire predictive weight of modern field
theory, and neither is explained by the theory that uses them.

The first is local gauge invariance. \citet{Weyl1929}, after an
earlier failed attempt to gauge the scale of length, found that
demanding invariance under a position-dependent phase forces the
existence of the electromagnetic field and fixes how it couples.
The trick is so powerful that it became the standard way to invent
new physics. But its status is strange. \citet{Noether1918} had
already proved, in the second theorem of the same paper that gave
physics its conservation laws, that a symmetry holding
independently at every point yields no conserved quantity --- only
an identity. \citet{Elitzur1975} sharpened the discomfort: a local
gauge symmetry can never break, because it is not a symmetry of
states at all, but a redundancy in our labeling of them. A
redundancy in a description is not the kind of thing that ought to
dictate the forces of nature. Yet it does.

The second is renormalization. It began as an embarrassment ---
infinities subtracted by hand to save quantum electrodynamics
\citep{Dyson1949} --- and became a triumph when the procedure was
shown to work for non-abelian theories \citep{tHooftVeltman1972}
and then reinterpreted by \citet{WilsonKogut1974} as a flow: couplings
are not constants but functions of the scale at which one looks,
and the flow runs one way, from fine resolution to coarse.

Why is this still open? Because both principles are described
perfectly and explained not at all. Nobody can say what the gauge
circle \emph{is} without speaking gauge language: the standard
answer, that it is a redundancy, makes its dictatorial power over
the dynamics more mysterious, not less. And renormalization's most
important feature is the one least accounted for --- its
irreversibility. The flow's arrow, the special status of the scale
variable, and the fact that only a handful of operators survive the
descent are all facts about a direction, in a framework whose other
directions are ordinary spacetime. A century after Weyl, the two
principles remain the tools we cannot do without and cannot derive.
\emph{Treated as Problem~3 in Part~III.}

\subsection{Problem 4. Too many numbers that nobody can compute}
\label{intro:p4}

The Standard Model has never been a theory of everything it
describes. Roughly two dozen numbers --- three gauge couplings, nine
charged-fermion masses, four mixing angles and phases, the Higgs
mass and vacuum value, plus the neutrino sector's masses and
mixings, and the QCD vacuum angle --- must be measured and inserted
by hand \citep{PDG2024}. Cosmology adds six more for $\Lambda$CDM.
\citet{Cahn1996} put the discomfort well by tracing what everyday
life would look like if each were slightly different: these are not
bookkeeping conventions but the numbers that decide whether atoms,
chemistry, and stars exist.

The history is a sequence of serious attempts to reduce the count,
each of which increased it. Grand unification related the three
gauge couplings at one scale --- a real success, at the price of new
scales and a proton lifetime now excluded in the simplest models.
Supersymmetry stabilized the Higgs mass and, in its minimal
realization, introduced more than a hundred additional parameters.
Technicolor, extra dimensions, and flavor textures each traded a
handful of numbers for a mechanism carrying its own. The string
landscape offered, in place of a computation, an astronomical
number of consistent vacua and a selection principle
(Problem~15).

Why has this resisted for fifty years? Because of a fact about the
architecture, not about anyone's ingenuity: quantum field theory is
a \emph{framework}, and in a framework the parameters are inputs.
Any values are internally consistent; nothing in the mathematics
prefers $\alpha^{-1} = 137.036$ to any other number. A theory whose
own structure cannot distinguish the true values from false ones
cannot compute them, no matter how long it is studied --- and every
attempt to embed it in a larger structure has so far added more
freedom than it removed. The count has gone up, not down, in half a
century of trying. \emph{Treated as Problem~4 in Part~III.}

\section{Category II --- Cosmology: the universe that had to begin}
\label{sec:cat2}

Cosmology's four problems have the opposite character to the
Standard Model's. There the theory is too good and cannot be
questioned; here the theory is a magnificent fit with almost no
explanation underneath it. Six numbers reproduce the whole observed
universe --- and all six are measured, none derived.

\subsection{Problem 5. What happened at the beginning}
\label{intro:p5}

Every other question in physics can be asked about a system with a
past. This one cannot, and that is what makes it different in kind.
\citet{Lemaitre1931} first proposed that the expansion runs back to
a beginning, and general relativity was later shown to insist on
it: under conditions that hold in our universe, the equations force
a singularity, a moment where the description ends rather than
continues \citep{HawkingPenrose1970}. Since a singularity is a
failure of the theory, not an event within it, the beginning became
the one place where physics had to concede that it did not apply.

Inflation transformed the question without answering it.
\citet{Guth1981}, followed by \citet{Linde1982} and building on
\citet{Starobinsky1980}, showed that a brief early episode of
accelerated expansion would explain the sky's uniformity, its
flatness, and --- the triumph --- the precise spectrum of tiny
density ripples that later became galaxies. Those predictions have
been confirmed to remarkable precision \citep{Planck2020}. But
inflation is a mechanism, not an origin: it requires a field with a
suitable potential, placed in a suitable initial state, over a
region large enough to inflate. \citet{BordeGuthVilenkin2003}
proved that inflation cannot be eternal into the past, so the
question survives its own solution.

Underneath sits a sharper puzzle that inflation does not touch.
\citet{Penrose1979} pointed out that the early universe was not
merely hot and dense but \emph{extraordinarily special}: its
gravitational degrees of freedom were almost entirely unexcited, a
condition of staggeringly low probability on any natural counting,
and the only reason the second law has anywhere to run.

Why has this resisted for a century? Because the standard framework
splits every explanation into two parts --- a law and an initial
condition --- and awards itself only the first. Initial conditions
are, by the framework's own rules, inputs: chosen, not computed.
That works everywhere except at the one place where there is no
earlier state to choose them from. So the beginning is not merely
unknown, it is \emph{structurally unaskable} in the language used to
describe everything after it. \emph{Treated as Problem~5 in
Part~IV.}

\subsection{Problem 6. Why the scales are so absurdly far apart}
\label{intro:p6}

The weak scale sits about seventeen orders of magnitude below the
Planck scale, and the energy density of the vacuum sits some one
hundred and twenty orders below its naive estimate. These two
numbers are the most spectacular unexplained ratios in science.

The anxiety has a long pedigree. \citet{Dirac1937} noticed that
several enormous dimensionless ratios in nature cluster near
$10^{40}$ and its powers, and concluded --- wrongly, but
fruitfully --- that such numbers must be dynamical. The modern form
arrived with \citet{Susskind1979}, who observed that quantum
corrections drag a scalar's mass toward the highest scale in the
theory, so the Higgs mass appears to require a cancellation of
extraordinary precision. \citet{tHooft1980} gave the doctrine its
respectable form: a small number is acceptable if setting it to
zero enlarges the symmetry. On that criterion the Higgs mass is
unnatural, and an entire generation of model building --- most of
it supersymmetric --- was launched to protect it. The vacuum energy
is worse still: \citet{Weinberg1989CC} reviewed the problem before
dark energy was observed, and the measured value
\citep{Riess1998, Perlmutter1999} sits far below every estimate the
framework can produce.

Why has this resisted for forty years? Because naturalness is a
statement about \emph{probability across a space of theories}, and
no such probability measure exists. The doctrine told a generation
where to look; the Large Hadron Collider then found the Higgs inside the
light-mass window the electroweak fits favored --- and nothing
else at the scale naturalness demanded. The community is left with a criterion that was
enormously productive, is not a law, cannot be checked, and has now
returned its clearest experimental verdict --- a null. Meanwhile
the framework's own logic guarantees that any small parameter it
does not protect by symmetry will be dragged large, so the
hierarchy cannot be explained from inside; it can only be assumed,
protected, or selected. \emph{Treated as Problem~6 in Part~IV.}

\subsection{Problem 7. What most of the universe is made of}
\label{intro:p7}

Ordinary matter --- everything ever detected in a laboratory ---
accounts for about five per cent of the universe. About
twenty-seven per cent is a form of matter that clusters
gravitationally and does nothing else visible, and the remaining
sixty-eight per cent is something with negative pressure that is
pushing the expansion apart.

\citet{Zwicky1933} noticed the first: galaxies in the Coma cluster
were moving far too fast for the visible mass to hold them.
\citet{RubinFord1970} measured Andromeda's rotation curve out to
large radius, and the spiral surveys that followed made the case
inescapable; the evidence has since become overdetermined --- lensing, the cosmic microwave background, the
growth of structure. The leading candidate for decades was a heavy
weakly-interacting particle; a second, arising elsewhere entirely,
was the axion, invented to solve a different problem
\citep{PecceiQuinn1977, Weinberg1978, Wilczek1978} and then
recognized as a natural relic \citep{PreskillWiseWilczek1983}.
Dark energy arrived abruptly with the supernova measurements
\citep{Riess1998, Perlmutter1999}, and everything since is
consistent with a constant.

Why has this resisted for ninety years? For dark matter, because
the evidence is entirely gravitational and therefore says almost
nothing about what the particle \emph{is}: the allowed mass range
spans some ninety orders of magnitude, and the theory offers no
principle to prefer a point in it. Two generations of increasingly
sensitive detectors have returned nulls, which excludes parameter
space without indicating where to look next. For dark energy, the
difficulty is sharper: its value is not merely unexplained but
anti-explained, since the framework's own estimate is wrong by a
factor no mechanism has produced. And the coincidence that the two
components have comparable densities \emph{now}, after billions of
years of very different scaling, is a puzzle nobody has been able
to convert into a prediction. \emph{Treated as Problem~7 in
Part~IV.}

\subsection{Problem 8. Why distant strangers agree}
\label{intro:p8}

Look at two patches of sky in opposite directions. Their light has
been traveling toward us for the age of the universe, so on any
straightforward accounting they have never exchanged a signal ---
nothing that happened in one could have influenced the other. Yet
they have the same temperature to one part in a hundred thousand,
contain the same kinds of particles with identical properties, and
obey the same laws with the same constants.

\citet{Misner1968} first pressed the first of these agreements as a
crisis --- the horizon problem --- and inflation was designed
largely to solve it: if the observed sky descends from one tiny
causally connected patch that was then stretched enormously, the
uniformity is inherited rather than coincidental
\citep{Guth1981}. This is one of the genuine explanatory successes
of twentieth-century cosmology, and this paper does not dispute it.

But it buys only the first agreement. Inflation explains why remote
regions began at the same temperature; it does not explain why the
electron in a quasar twelve billion light-years away has precisely
the mass of the electron in this room, nor why the fine-structure
constant is the same there as here, nor why regions that made their
own choices about which vacuum to settle into --- the electroweak
vacuum, the QCD vacuum angle, the axion misalignment --- made the
\emph{same} choices, when the standard picture of symmetry breaking
is a random fall from a symmetric hilltop, chosen independently in
each causally disconnected patch. On the textbook account the sky
should be a patchwork of domains separated by walls. It is not, to
the precision of every measurement made.

Why has this resisted? Because the mechanism that answers the first
agreement is causal --- stretch, so everything shares a past --- and
the second and third agreements are about \emph{choices made after
the stretch}, which a shared past cannot reach. The literature has
largely not treated them as one problem, and the escape usually
offered, that constants are simply constants, restates the
assumption rather than explaining it. \emph{Treated as Problem~8 in
Part~IV.}

\section{Category III --- The foundation of particle physics: how
matter got here}
\label{sec:cat3}

The third group asks how the universe came to contain stable
matter at all --- which turns on three discrete symmetries, one
missing antiworld, and one interaction that refuses to let its
charges out.

\subsection{Problem 9. Why the mirrors break, and one does not}
\label{intro:p9}

Three discrete operations were long assumed to be exact symmetries
of nature: reflection in a mirror ($P$), exchange of particles with
antiparticles ($C$), and reversal of the direction of time ($T$).
None of the three survives intact --- $T$ violation has now been
seen directly as well \citep{BaBar2012} --- and only the product
$CPT$ has never failed. That pattern of failure is the deepest
unexplained regularity in particle physics.

\citet{LeeYang1956} pointed out that parity had never actually been
tested in weak decays; within months \citet{Wu1957} found it
maximally violated. Combined $CP$ was expected to survive --- and
then \citet{Christenson1964} found it broken too, at the level of
two parts in a thousand, in the decays of neutral kaons.
\citet{KM1973} showed that a third generation of quarks would allow
exactly one such phase in the Standard Model, a prediction
brilliantly confirmed. Only $CPT$ has never been seen to fail; it
is protected by a theorem \citep{Luders1954} resting on locality,
Lorentz invariance, and positive energy.

Alongside sits a puzzle of the opposite kind. Quantum chromodynamics
\emph{permits} a $CP$-violating term, with an angle that could take
any value between zero and $2\pi$ \citep{tHooft1976}. Measurements
of the neutron's electric dipole moment show it is smaller than
about $10^{-10}$ \citep{Abel2020} --- a number consistent with
zero, and consistent with nothing else. \citet{PecceiQuinn1977}
proposed a new symmetry to relax it dynamically, whose leftover
particle is the axion.

Why has this resisted for seventy years? Because the Standard Model
\emph{accommodates} the pattern without producing it. Parity
violation is written in by assigning left- and right-handed
particles to different representations; $CP$ violation is written in
as a phase whose size is measured; the strong-$CP$ angle is written
in as a free number that experiment then says is zero. Each fact is
an input chosen to match the world. And $CPT$'s exactness, though
proved, is proved from assumptions rather than explained --- the
theorem tells us the three operations are linked, not why the
geometry offers exactly three. \emph{Treated as Problem~9 in
Part~V.}

\subsection{Problems 10--11. Where antimatter came from, and where
it went}
\label{intro:p1011}

Antimatter was a mathematical accident before it was a discovery.
\citet{Dirac1928} found that making the electron's equation
compatible with relativity forced negative-energy solutions, and
after three years of embarrassment --- during which he first tried
to identify the extra solutions with protons --- he read them, in
1931, as a new particle of opposite charge and identical mass
\citep{Dirac1931}. \citet{Anderson1933} then photographed one.
Every charged particle found since has a partner, with mass and
lifetime equal to fantastic precision and every charge
reversed.

That precision is itself the first question. Nothing in the
framework explains why the mirror is \emph{exact} in mass and
\emph{perfect} in charge reversal; antimatter is written into the
formalism as a second set of field components, and its properties
are consequences of a construction chosen because it works.

The second question is why there is anything left. Matter and
antimatter annihilate; a universe that began with equal amounts
would now contain radiation and almost nothing else. Instead there
are about six surplus baryons for every ten billion photons.
\citet{Sakharov1967} laid down the three conditions any explanation
must satisfy: baryon number violation, $C$ and $CP$ violation, and
departure from thermal equilibrium. The Standard Model formally
contains all three --- $CP$ violation from the quark phase, baryon
number violation through non-perturbative processes
\citep{KRS1985} --- and yet fails quantitatively by a
catastrophic margin: the quark phase delivers an asymmetry at least
ten orders of magnitude too small \citep{Gavela1994}. The most
successful alternative moves the origin to the neutrino sector
\citep{FukugitaYanagida1986}, where heavy right-handed neutrinos
decay out of equilibrium and the resulting lepton asymmetry is
converted to baryons.

Why has this resisted for sixty years? Because leptogenesis, the
leading mechanism, does its work at energies and with masses that
no experiment can reach: the framework is right, and the numbers
inside it are free. Any observed asymmetry can be accommodated by
choosing the heavy masses and phases appropriately, which means the
observed value has never been \emph{predicted} --- only fitted.
\emph{Treated as Problems~10--11 in Part~V.}

\subsection{Problem 12. Why quarks are never seen, and why the
proton lives forever}
\label{intro:p12}

Two of the most basic facts about matter are, in the strict sense,
unproven.

The first is confinement. \citet{GellMann1964}, and independently
Zweig, proposed quarks as constituents while stressing that none
had ever been observed; \citet{Greenberg1964} added the three-valued
degree of freedom that became color and made the arithmetic
work. Half a century of
searches has found no free quark and no free color charge, and
lattice computations show the confining behavior convincingly ---
but a proof, from the theory's own equations, that color charges
cannot be isolated does not exist; the closely related problem of
constructing quantum Yang--Mills theory and proving it has a mass
gap is one of the Clay Millennium Prize problems
\citep{JaffeWitten2006}, unclaimed. The
qualitative mechanism is understood as a flow along the scale
variable \citep{Wilson1974} in which the coupling grows toward low
energy until the perturbative description fails --- an argument
that says the calculation breaks down, not that the charges are
imprisoned.

The second is proton stability. Nothing in the Standard Model's
structure \emph{forbids} the proton from decaying; baryon number is
an accidental symmetry, a bookkeeping regularity that happens to
hold for the operators one can write at low energy, and grand
unification predicted that it should fail. Experiment says
otherwise, with a lower bound on the lifetime now beyond
$10^{34}$ years \citep{SuperK2020} --- more than twenty orders of
magnitude longer than the age of the universe.

Why has this resisted for sixty years? Because the two facts pull in
opposite directions. Confinement resists proof because the
interesting regime is precisely where the tools fail; the successful
description is a numerical simulation, which demonstrates rather
than explains. Proton stability resists explanation because the
framework offers it as an accident, and accidents of that precision
--- a bound thirty-four orders of magnitude out --- are exactly what
theories are supposed to explain. That every atom in the universe
depends on this accident holding is the sharpest way to state the
discomfort. \emph{Treated as Problem~12 in Part~V.}

\section{Category IV --- The foundation of the cosmos: the words we
never defined}
\label{sec:cat4}

The last four are the least discussed and the most fundamental. Two
of them concern words physics uses on every page without defining;
the other two concern the limits of explanation itself.

\subsection{Problem 13. What time is, and why it only runs one way}
\label{intro:p13}

Physics measures time to seventeen decimal places and does not say
what it is measuring.

The fundamental equations make the omission conspicuous, because
almost all of them work equally well in both directions. Newton's
laws, Maxwell's equations, general relativity, and the
Schr\"odinger equation are all reversible: run any solution
backwards and you get another solution. Yet nothing in experience
is reversible. \citet{Boltzmann1877} located the asymmetry in
counting --- there are overwhelmingly more disordered
configurations than ordered ones --- and \citet{Eddington1928} gave
the second law its famous status as the one law of nature whose
violation would leave no defense. But counting alone cannot produce
an arrow: the same argument, run backwards, predicts that entropy
was higher in the past too, which is false. The missing ingredient
is the extraordinarily low-entropy initial condition
\citep{Penrose1979}, which pushes the arrow of time back into
Problem~5.

Quantum gravity makes the situation stranger still. The
Wheeler--DeWitt equation \citep{DeWitt1967}, the canonical
quantization of general relativity, contains no time variable at
all: the universe's wavefunction simply does not evolve. Whether
this is a technical artifact or the deepest fact about time has
been argued for sixty years without resolution.

Why has this resisted for a century? Because time enters every
framework as a \emph{parameter}, supplied from outside, never
derived. The theories describe change with respect to $t$ and never
say what $t$ is; the arrow is then imported from an initial
condition that the same theories decline to explain; and the one
place where time might have been derived --- the quantization of
gravity --- responds by making it vanish. \emph{Treated as
Problem~13 in Part~VI.}

\subsection{Problem 14. What energy, mass, and entropy actually are}
\label{intro:p14}

The three most-used quantities in physics have no definitions ---
only recipes for computing them. \citet{Feynman1963} put it
bluntly in his introductory lectures: physics knows that a certain
quantity is conserved, and does not know what that quantity
\emph{is}. Energy is defined as the conserved charge of
time-translation symmetry \citep{Noether1918}, which is
mathematically exact and explanatorily circular if one is also
asking what time is (Problem~13). Mass is a parameter in a
Lagrangian, or a coefficient in a dispersion relation, or --- since
Einstein --- a form of energy, without any statement of what
distinguishes the kind of energy that has mass from the kind that
does not.

Entropy is worse, because it has several inequivalent definitions
in daily use --- Clausius's heat over temperature, Boltzmann's
count of microstates, Gibbs's ensemble average, Shannon's
information measure, von Neumann's quantum version, and, most
strangely, Bekenstein's area of a black-hole horizon
\citep{Bekenstein1973}, which says that an entropy can be a
\emph{geometrical} quantity. \citet{Landauer1961} then showed that
information and thermodynamics are not analogies but the same
ledger: erasing one bit necessarily dissipates a definite minimum
of energy.

Why has this resisted? Because in each case the framework provides
a \emph{procedure} where an account was wanted, and the procedures
work so well that the absence is easy to overlook. The three
quantities are also mutually entangled: energy is defined by time,
entropy by the arrow of time, mass by energy. Nothing in the
standard framework offers an independent structure from which all
three could be read --- and the one hint that such a structure
exists, the appearance of an entropy as an area, has been
tantalizing physicists for fifty years. \emph{Treated as Problem~14
in Part~VI.}

\subsection{Problem 15. Whether the constants were selected rather
than explained}
\label{intro:p15}

By the late 1990s, two developments had converged on a radical
proposal: perhaps the constants of nature cannot be computed at
all, because they are not universal.

The first was \citet{Weinberg1987}, who argued before the discovery
of dark energy that if the vacuum energy varied across regions, only
regions where it is small enough for galaxies to form could contain
observers --- and derived, on that basis, an anthropic upper bound
some two orders of magnitude above the value later measured. A
bound is not yet a prediction; sharpening it into one needed a
measure, which is where the difficulty begins. The second was the discovery that string theory
appears to admit an astronomical number of consistent vacua
\citep{BoussoPolchinski2000}, each with different effective
constants, and the proposal to take that landscape seriously as a
physical ensemble \citep{Susskind2003}. If eternal inflation
populates the vacua, the constants become environmental --- like the
distance from the Earth to the Sun, not a number to be derived but a
place where observers happen to find themselves.

This is a coherent and honest position, and its bound remains the
tradition's most impressive single result. But it carries a bill that has never been
paid. To make a prediction from an ensemble one needs a probability
measure over it, and in an infinite, eternally inflating multiverse
every proposed measure gives different answers, with no principle
selecting one --- the measure problem. And the framework is
difficult to expose: a theory predicting that everything possible
occurs somewhere is not easily proved wrong.

Why has this resisted --- thirty-nine years since the anthropic
bound, twenty-six since the landscape? Because it is not
really a physics problem but a fork in what counts as an
explanation. One branch says constants are contingent and the
search for derivations is a category error; the other says they are
computable and nobody has computed them. The dispute cannot be
settled by argument --- only by somebody actually deriving the
numbers, or by the landscape delivering a measure. \emph{Treated as
Problem~15 in Part~VI.}

\subsection{Problem 16. What a black hole does with what it swallows}
\label{intro:p16}

\citet{Bekenstein1973} argued that a black hole must carry an
entropy proportional to the area of its horizon, and
\citet{Hawking1975} then proved that it radiates like a hot body.
Together these results were a triumph --- and immediately a
catastrophe. If the radiation is exactly thermal, it carries no
information about what fell in, so when the hole finishes
evaporating the information is gone, and quantum mechanics'
foundational rule that evolution is reversible has failed
\citep{Hawking1976}.

Fifty years of attempts have produced deep physics and no
consensus. Black-hole complementarity proposed that no single
observer ever sees a contradiction \citep{STU1993}. The
gauge/gravity duality \citep{Maldacena1998} gave strong indirect
evidence that gravitational evolution is unitary, without
identifying the mechanism. \citet{Page1993} showed what the
schedule must look like if information does come out: the
radiation's entanglement entropy must rise, turn over at about half
the evaporation, and fall back to zero. Then
\citet{AMPS2013} sharpened the paradox into a trilemma --- unitarity,
locality, and a smooth horizon cannot all survive --- suggesting a
``firewall'' at the horizon. Most recently, calculations of
entanglement islands \citep{Penington2020} recovered the Page
curve from gravity itself.

Why has this resisted for fifty years? Because the paradox sits
precisely where the two great theories of the twentieth century
overlap and disagree, and there is no experiment: Hawking radiation
from an astrophysical black hole is unobservably faint, and the
process under dispute takes longer than the age of the universe.
The recent island computations are a genuine advance --- they get
the right curve --- but they compute \emph{that} the books balance
rather than saying \emph{where} the information was held while the
hole existed. \emph{Treated as Problem~16 in Part~VI.}

\section{What the sixteen have in common}
\label{sec:common}

Sixteen problems have now been stated in their own terms. Set side
by side, three patterns are hard to miss.

\subsection{They are old, and they have not been getting younger}

Table~\ref{tab:ages} dates each problem to the year it took its
modern form. The median entry is seventy years old. Twelve of the
fifteen dated entries are older than half a century, six are older
than ninety years, and the youngest --- the landscape --- is
thirty-nine. In the same period the discipline built the Standard
Model, discovered the $W$, $Z$, top quark and Higgs, measured the
microwave background to cosmic-variance limits, detected
gravitational waves and neutrino oscillations, and confirmed
inflation's spectral predictions. It was not a period of weakness.
The list survived it anyway.

\begin{table}[H]
\centering
\caption[The sixteen problems, dated to the year each took its modern form.]{The sixteen problems, dated to the year each took its
modern form, with age in 2026. Median age: seventy years. Twelve of
the fifteen dated entries predate 1976. The oldest is Problem~14, Noether's 1918 statement of the energy question, at a hundred and eight; the youngest is the landscape, at thirty-nine. Problems~10 and 11 share a row and a date, Sakharov's 1967.}
\label{tab:ages}
\small
\renewcommand{\arraystretch}{1.18}
\begin{tabular}{@{}clcc@{}}
\toprule
\# & \textbf{Problem} & \textbf{Posed} & \textbf{Age} \\
\midrule
1 & The measurement problem (Schr\"odinger's cat) & 1935 & 91 \\
2 & Why the interactions differ (Yang--Mills) & 1954 & 72 \\
3 & Gauge invariance and renormalization (Weyl) & 1929 & 97 \\
4 & Too many free parameters & 1975 & 51 \\
5 & The beginning of the universe (Lema\^itre) & 1931 & 95 \\
6 & Naturalness and the hierarchy & 1979 & 47 \\
7 & The dark sector (Zwicky) & 1933 & 93 \\
8 & The horizon problems (Misner) & 1968 & 58 \\
9 & Discrete symmetry (Lee--Yang) & 1956 & 70 \\
10--11 & Antimatter and the excess (Sakharov) & 1967 & 59 \\
12 & Confinement and the stable proton & 1964 & 62 \\
13 & The definition and arrow of time (Eddington) & 1928 & 98 \\
14 & Energy, mass, entropy (Noether) & 1918 & 108 \\
15 & The landscape and the multiverse & 1987 & 39 \\
16 & Black-hole information (Hawking) & 1976 & 50 \\
\bottomrule
\end{tabular}
\end{table}

When a research program fails for a decade, the natural
explanation is that the problem is hard. When sixteen independent
programs, run by the ablest people in the field with the best
instruments ever built, all fail for half a century or more, a
second explanation deserves a hearing: that they share an
obstruction, and that the obstruction is upstream of all of them.

\subsection{They are all the same kind of question --- and it is the
kind the framework cannot take}

Read the sixteen again and notice what none of them is. Not one is
a discrepancy. Not one says the equations gave the wrong number.
Every single one asks \emph{why the structure is what it is}: why
two dynamics, why four forces, why these constants, why this
beginning, why this arrow, why exactly these three discrete
operations, why the books balance.

Quantum field theory and general relativity are magnificent
machines for a different question. Given a structure --- a
manifold, a gauge group, a field content, a set of couplings, an
initial state --- they compute what happens next, superbly. But the
structure is the \emph{input}. A framework that takes structure in
cannot hand structure out, and no amount of skill inside it will
change that. This is why each of the sixteen ends the same way: the
literature answers it by choosing something. The gauge group is
chosen. The chirality assignment is chosen. The parameters are
measured and inserted. The initial condition is stipulated. The
placement of the quantum-classical cut is left to convenience. The
proton's stability is called an accident. The constants are called
environmental. Each move is respectable, and none is a computation.

\subsection{They sort into two families, and the sorting is the
clue}

The third pattern is the one this paper is about. Sort the sixteen
by what they are really asking, and they fall into two families
with almost no overlap.

\emph{One family is about things that never change.} $CPT$ holds
exactly. Electric charge is quantized exactly. The proton does not
decay. An antiparticle's mass matches its partner's to every
measured digit. The electron in a distant quasar is identical to
the one in this room. Anomaly coefficients are integers and do not
run. Color charges are never seen alone. These are questions about
\emph{protection}: what could possibly make a quantity so exact
that fifty years of increasingly sensitive experiments cannot budge
it?

\emph{The other family is about things that only go one way.} A
measurement, once registered, cannot be unregistered. The
renormalization flow runs from fine to coarse and not back. Entropy
increases. The universe descends from hot to cold, once. A black
hole evaporates. Time has a direction. These are questions about
\emph{irreversibility}: what could possibly make a direction in
which nothing returns, in a world whose fundamental equations are
reversible?

Now the difficulty comes into focus. Physics has exactly one
language for both families: fields living on a four-dimensional
spacetime. In that language every direction is the same kind of
direction --- you may traverse it either way --- so nothing in the
geometry is intrinsically protective and nothing is intrinsically
one-way. Both properties must therefore be imported: exactness as
an assumed symmetry or a fortunate accident, irreversibility as a
special initial condition or a coarse-graining we chose to
perform. And because both are imported into the same
undifferentiated arena, the two families keep contaminating each
other. The measurement problem is the collision of a reversible law
with an irreversible record. The strong-$CP$ problem is an exact
quantity that the framework says should be arbitrary. The arrow of
time is irreversibility demanded of reversible equations. The
information paradox is exactness (unitarity) meeting
irreversibility (evaporation) with no rule for how they share the
books.

That is the pattern. \emph{The sixteen are what happens when a
world with two kinds of direction is described in a language that
has only one.}

\section{What kind of answer could close them}
\label{sec:whatkind}

If the diagnosis is right, it constrains the cure sharply --- and
the constraints are worth stating before any proposal, because they
are what makes a proposal checkable rather than merely appealing.

\emph{It must be structural, not parametric.} Adding a field, a
symmetry, or a parameter keeps the input/output boundary exactly
where it was. What is needed is a change in what the framework
takes as given.

\emph{It must supply two kinds of direction, intrinsically.} Not
two kinds of behavior imposed on similar directions, but two
directions that differ in their own nature --- one whose transport
is closed, protected, and reversible, and one whose transport is
open, priced, and final. The distinction must be topological, since
only topology gives exactness that no dynamics can spoil.

\emph{It must be one answer, not sixteen.} Sixteen mechanisms, each
fitted to its problem, is what the field already has. A single
structure that addresses all sixteen is a much stronger and much
more dangerous claim, because a single structure can be wrong all
at once.

\emph{It must produce numbers.} An account that explains why the
constants exist but not what they are has moved the problem, not
solved it. The test of a structural claim is whether measured
quantities come out of it with nothing tuned.

\emph{It must be exposed.} Any framework flexible enough to
accommodate sixteen answers is flexible enough to accommodate
anything, unless the answers come with named, dated ways to die.

These five requirements are demanding, and they are meant to be. A
proposal that meets the first four and fails the fifth is
philosophy; one that meets all five is physics, whether it turns
out to be right or wrong.

\section{The claim of this paper}
\label{sec:claim}

The \GGTheory{} program's six-dimensional bulk carries, over
every point of the four-dimensional stage, exactly two non-spatial
directions --- and they are of opposite topological kind. One is a
compact circle, on which every path closes. One is an open
interval, on which none does. Holonomy requires closed loops, and
an interval has none. So one fiber is nothing but memory:
everything on it is quantized, gauge-protected, transported by a
group with inverses. The other can hold no memory at all: its
transport composes forward only, a semigroup, and every step it
takes is final.

That is the two kinds of direction the diagnosis asked for, and it
is not an addition to the program --- it is a reading of a
structure the constitution already wrote for other reasons. Part~II
builds it once, carefully, including the argument that one fiber of
each kind is the constitutional \emph{minimum}: a bulk with two
circles would conserve everything and decide nothing; a bulk with
two intervals would decide everything and keep nothing. Being needs
holonomies; becoming needs a semigroup; a world that both is and
becomes needs one of each.

Parts~III--VI then spend that reading on the sixteen, in the order
of Table~\ref{tab:sixteen}, at essay length or fuller. The
family of problems about exactness is answered by the circle: what
is protected is protected topologically, and a class cannot drift,
cannot vary from place to place, and cannot be renormalized. The
family about one-wayness is answered by the interval: what is final
is final algebraically, because a semigroup has no inverses to
undo it with. The problems that were hardest --- measurement, the
information paradox, the arrow, the dark trio --- are precisely
those that need both fibers at once, and they are hardest in the
old language for the same reason they are tractable in this one.

Two commitments govern the whole. \emph{Nothing new is derived
here}: every equation, number, and theorem is proved in a cohort
paper and carried at its parent's grade, with each conditional and
flagged claim marked in place. And \emph{everything is exposed}, though not all of it in the same
way: Part~VII collects the named measurements --- a resonance at
$3.849$~TeV, a haloscope line at $1.95$~GHz, a tensor-to-scalar ratio
near $0.0043$, an isocurvature null, a large-scale statistics anomaly
--- on which six of the sixteen treatments die outright, and states
how the remaining ten are exposed through the ledger they all spend. A reader who finds the
anatomy attractive should go to that table first, and check that
the paper has made itself killable before deciding whether it is
right.
One last framing, before the anatomy is built. \citet{Wigner1960}
called the effectiveness of mathematics in the natural sciences
unreasonable --- a mystery he thought physics had no right to
expect. If the sixteen problems above share the diagnosis proposed
here, the mystery is smaller than it looks: mathematics describes
the world well because the world's non-spatial structure \emph{is}
the mathematics --- a circle and an interval, a group and a
semigroup --- and the equations that work are the ones that happen
to have been written in the right alphabet.


\section{How this paper is organized}
\label{sec:organised}

Part~II opens with the program in its four development steps ---
the 6D bulk (A1--A4), the projection to 4D (A5--A7), the
applications (A8--A11), and the white papers --- and then builds
the instrument: the two-fiber anatomy, developed once with its
equations, its primer, its character table, its minimality
argument, its determinism triad, and the map of its deployment
across the cohort. Parts~III--VI then
treat the sixteen problems in the four categories of
Table~\ref{tab:sixteen}: the Standard Model (Problems~1--4),
cosmology (Problems~5--8), the foundation of particle physics
(Problems~9--12), and the foundation of the cosmos
(Problems~13--16). All sixteen are developed at length: Problems~2, 7, 9,
10--11, 12, and~13 carry the campaign's fullest quantitative
apparatus, and the remaining entries are essay-length developments
at the same register --- the problem posed plainly, the anatomy's
answer built, the grades stated, and the kill conditions staked. Part~VII is the Discussion: significance, contrasts,
objections and questions, the honest-status ledger, the
consolidated number audit, and the falsifier table with its
decision calendar. Part~VIII concludes.

Reading paths: the physicist in a hurry reads Part~II and the
Problem banners; the skeptic reads Part~II, then \S\ref{sec:honest}
(the honest ledger) and \S\ref{sec:kill} (the ways to be wrong);
the student reads straight through --- the paper assumes gauge
theory and cosmology at the first-year graduate level and nothing
of the cohort.

%% ===================================================================
\subsection*{Part I in one paragraph, and what comes next}

Sixteen problems, then. Four of them are the Standard Model
admitting what it cannot compute; four are cosmology asking what
happened before its equations start; four are the foundations of
matter; four are the foundations of the arena itself. Their median
age is seventy years, twelve of the fifteen dated entries predate
1976, and not one has yielded to the strategy the discipline has
used on all of them --- attack each with a tool built for it. What
they share is more interesting than what separates them: every one
asks the framework a question the framework's own vocabulary cannot
phrase, and they sort, without being forced, into problems of
exactness and problems of one-wayness.

That sorting is the clue, and Part~II follows it. It reviews the
program that produced the anatomy --- four steps, from the bulk
to the projection to the applications to these two white papers ---
and then builds the instrument once, with its equations, so that
Parts~III to~VI can use it sixteen times without rebuilding it.

%%  PART II -- THE ANATOMY
%% ===================================================================
\GGPart[The anatomy]{The Anatomy}{The Program in Four Steps, and
One Exact Statement about Its Two Hidden Directions}

\section{The statement}
\label{sec:statement}

Every result in this paper is \emph{sorted by} one sentence --- and
\emph{produced by} the cohort's derivations, a distinction
\S\ref{sec:abc} insists on and this Part will not blur. The sentence
is elementary enough to prove in a paragraph:

\begin{quote}
\emph{Holonomy requires closed loops, and an interval has none.}
\end{quote}

Take a connection --- any connection --- on a fiber bundle. Parallel
transport along a path compares the fiber at the end with the fiber at
the start; but only when the path \emph{closes} does the comparison
become an invariant statement about the geometry itself, because only
then are the start and the end the same fiber. That invariant --- the
holonomy of the loop --- is the entire gauge-invariant content of the
connection: by the Wilson-loop reconstruction, a gauge field \emph{is}
its holonomies, no more and no less. On a compact circle, every
winding path closes; the circle is, in this precise sense, nothing but
closed loops, and everything a connection does there is holonomy ---
quantized by the winding integers, protected because an integer has
nowhere to move, composed by a \emph{group}, with every transport
invertible. On an interval, no path closes; there is no loop to hold a
holonomy, hence no protected content at all; and because the physical
transport along it composes only forward, its composition law is a
\emph{semigroup} --- no inverses, ever.

The six-dimensional bulk of \GGTheory{}
\citep{ArisakaA2GG6} places exactly one fiber of each kind over every
point of the Lorentzian stage $\mathcal M_4$:
\begin{equation}
\mathcal M_6 \;=\; \mathcal M_4 \times \Ir \times \Stheta,
\qquad
\Ir = [0, K],\quad K = 35,
\qquad
\Stheta \cong S^1 ,
\label{eq:bulk}
\end{equation}
the scale interval $\Ir$ --- the fifth dimension, the
renormalization-group coordinate $r = \ln(\EGUT/\mu)$ made geometry
--- and the phase circle $\Stheta$ --- the sixth dimension, the gauge
phase made geometry. The statement above then sorts the pair
completely:

\begin{center}
\emph{The circle remembers; the interval forgets.}
\end{center}

This Part does two jobs, in order. First it reviews the program
the sentence serves: the four development steps of \GGTheory{} ---
the bulk, the projection, the applications, and the white papers
(\S\ref{sec:foursteps}) --- so that the anatomy is met as what it
is, the visual language of an audited structure, not a metaphor in
search of one. Then it builds the instrument once: the primer with
its equations (\S\ref{sec:primer}--\ref{sec:algebra}), the two
characters (\S\ref{sec:characters}), the reason the pair is a
constitutional minimum (\S\ref{sec:minimum}), the determinism it
enforces (\S\ref{sec:determinism}), and the one place the fibers
multiply (\S\ref{sec:gearbox}). Everything after Part~II is
application.

\section{The program in four steps}
\label{sec:foursteps}

The two-fiber anatomy did not arrive alone, and it cannot be judged
alone: it is the visual language of a program that was built in a
definite order, each stage standing on the one before. Before the
instrument is developed, this section walks that order once ---
four steps, each anchored in named papers, each summarized with its
key equations and the intuition that goes with them. A reader who
holds the four steps holds the whole cohort in outline; a reader
who skips to the anatomy will still find every claim below cited
where it is used.

\begin{table}[H]
\centering
\caption[The \GGTheory{} program in four steps.]{The \GGTheory{} program in four steps. Each step stands
on the one above; the white papers (Step~D) are the synthesis
layer, and this paper is the second of them. Step~A (A1--A4) is the arena: one geometry for gravity and the five gauge forces, anomaly-free, with zero free parameters. Step~B (A5--A7) is the projection to four dimensions and what it buys --- measurement, the quantum postulates, the engine of symmetry breaking, the information ledger. Step~C (A8--A11) is the applications, derived and graded. Step~D is the two white papers.}
\label{tab:foursteps}
\small
\renewcommand{\arraystretch}{1.28}
\begin{tabularx}{\textwidth}{@{}cP{2.6cm}P{3.1cm}>{\raggedright\arraybackslash}X@{}}
\toprule
\textbf{Step} & \textbf{Papers} & \textbf{What is built} & \textbf{What it yields} \\
\midrule
A & A1--A4 & \GGEP{}: the 6D bulk, its group, its books & one geometry for gravity and the five gauge forces; anomaly-free; zero free parameters \\
B & A5--A7 & \OGGEP{}, \GDOS{}, \GRIP{}: the projection to 4D & measurement, the quantum postulates, the engine of symmetry breaking, the information ledger \\
C & A8--A11 & the applications & quantum mechanics, the Standard Model, $\Lambda$CDM, and the origin of life and mind, derived and graded \\
D & W1, W2 & the white papers & the synthesis: symmetry read historically (W1) and the two-fiber anatomy read structurally (W2 --- this paper) \\
\bottomrule
\end{tabularx}
\end{table}

Figure~\ref{fig:foursteps-w2} draws the same architecture as a
building: each story rests on the one below, and the two white
papers are the view from the roof.

\begin{figure}[htbp]
\centering
\resizebox{0.9\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
\node[draw=red!60!black, fill=red!8, line width=0.80pt, minimum width=11.6cm,
      minimum height=1.15cm, align=center] (A) at (0,0)
  {\textbf{Step A \ (A1--A4): the bulk} --- \GGEP{}: one geometry for
   gravity and the five gauge forces\\[-1pt]
   \scriptsize $\mathcal M_6 = \mathcal M_4 \times I_r \times
   S^1_\theta$; \ $\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times
   \mathrm{U}(1)_B \times \mathrm{U}(1)_{Qi}$; \ anomaly-free; \
   ledger $(1,3,5;7,35)$; \ zero dials};
\node[draw=olive!60!black, fill=yellow!12, line width=0.80pt, minimum width=10.6cm,
      minimum height=1.15cm, align=center] (B) at (0,1.65)
  {\textbf{Step B \ (A5--A7): the projection} --- \OGGEP{}, \GDOS{},
   \GRIP{}, $\mathrm{U}(1)_{Qi}$\\[-1pt]
   \scriptsize observation as projection; the open-system master
   equation; the four-move cascade; the information ledger};
\node[draw=green!55!black, fill=green!9, line width=0.80pt, minimum width=9.6cm,
      minimum height=1.15cm, align=center] (C) at (0,3.3)
  {\textbf{Step C \ (A8--A11): the applications}\\[-1pt]
   \scriptsize quantum mechanics; the Standard Model; $\Lambda$CDM;
   the origin of life and mind --- derived, graded};
\node[draw=blue!60!black, fill=blue!8, line width=0.80pt, minimum width=8.6cm,
      minimum height=1.15cm, align=center] (D) at (0,4.95)
  {\textbf{Step D \ (W1, W2): the synthesis}\\[-1pt]
   \scriptsize W1: symmetry read historically \ $\cdot$ \ W2 (this
   paper): the two fibers read structurally};
\draw[->,line width=0.80pt] (A.north) -- (B.south);
\draw[->,line width=0.80pt] (B.north) -- (C.south);
\draw[->,line width=0.80pt] (C.north) -- (D.south);
\node[rotate=90] at (-7.6,2.5) {\scriptsize each story rests on the
one below};
\end{tikzpicture}%
}
\caption[The program as a building.]{\textbf{The program as a building.} Step~A builds the
six-dimensional bulk with exact books and no dials; Step~B derives
how that bulk projects to the four-dimensional physics laboratories
measure; Step~C applies the pair to the four great frameworks; and
Step~D --- the white papers --- states what was learned. The
two-fiber anatomy developed in this Part is Step~D's instrument:
the single picture in which all four stories are visible at
once.}
\label{fig:foursteps-w2}
\end{figure}

\subsection{Step A: one geometry, exact books, no dials (A1--A4)}
\label{sec:stepA}

\paragraph{The constitution.} The program begins not with a
Lagrangian but with a constitution: four axioms and seven
postulates \citep{ArisakaA1Constitution}, written once and never
amended mid-argument. Two matter most here. \AX{2} is \GGEP{} ---
the \emph{geometry--gauge equivalence principle} --- which declares
that gravity and the gauge forces are not two kinds of physics to
be glued together but one composite connection on one bulk: the
gauge fields are geometry exactly as the metric is. \AX{3} is
\OGGEP{} (Step~B's subject): whatever observation is, it is a
projection of that bulk, not a view from outside it. Alongside the
axioms sits \MSMF{}, the parsimony clause: the minimal structure
with maximal function --- no field, dimension, or constant may be
added for convenience. The constitution's role in everything that
follows is exactly a constitution's role: when a later step seems
to demand a new ingredient, the demand is refused, and the
refusal --- as Problems~4, 6, and~15 will show --- is where the
explanatory power comes from.

\paragraph{The bulk and its group.} On that charter, A2
\citep{ArisakaA2GG6} proves the arena is forced: the six-dimensional
bulk of Eq.~\eqref{eq:bulk}, $\mathcal M_4 \times \Ir \times
\Stheta$ --- the four-dimensional stage, the open scale interval,
the compact phase circle --- carrying the composite connection of
the group
\begin{equation}
G_{\mathrm{bulk}}
\;=\;
\mathrm{Spin}(1,5)\,\times\,\mathrm{SU}(5)\,\times\,\UoneB\,\times\,\UoneQi .
\label{eq:group-p2}
\end{equation}
The intuition for each factor: $\mathrm{Spin}(1,5)$ is the
six-dimensional spacetime symmetry --- gravity, now including the
two hidden directions, so that ``curvature'' and ``gauge field''
are one word; $\mathrm{SU}(5)$ is the single non-abelian factor
from which the strong and electroweak forces descend --- one
gauge structure at the top, not three glued at the bottom;
$\UoneB$ books baryon number; and $\UoneQi$ books the quantum
of information (Step~B). Unification here is not a hope
deferred to high energy: it is the statement that
Eq.~\eqref{eq:group-p2} acts on Eq.~\eqref{eq:bulk} as one
connection, and everything laboratories distinguish --- the four
interactions of Problem~2 --- is that one object read on
different fibers.

\paragraph{The integer ledger.} The same paper derives the
program's five integers with their closure identities,
\begin{equation}
(M,N,L;\,J,K) \;=\; (1,\,3,\,5;\,7,\,35),
\qquad
J = 2N+1,
\qquad
K = LJ = 5\times 7 ,
\label{eq:ledger-p2}
\end{equation}
and the reader should pause on how strange and how strong this is.
$M$ counts the time direction; $N = 3$ the fermion generations;
$L = 5$ the rank structure of the unified group; $J = 7$ the
chiral steps of the flavor alphabet; and $K = 35$ the depth of
the descent --- and they are not five independent inputs but one
overdetermined web: $J$ is fixed by $N$, $K$ by $L$ and $J$. Five
small integers, two identities, zero freedom. Every number in the
cohort --- masses, angles, densities, the age of the universe ---
is arithmetic in these five and the constants of Step~A's last
paper. Two examples, derived in \citet{ArisakaA9SM} and audited in
Problem~4:
\begin{align}
\alpha^{-1}(0) &\;=\; 2^{J} + N^{2} \;=\; 128 + 9 \;=\; 137
\quad(\text{measured } 137.036),
\nonumber\\
\sin^2\theta_W &\;=\; \frac{L^2}{4N^3} \;=\; \frac{25}{108}
\;=\; 0.2315
\quad(\text{measured } 0.2312).
\label{eq:codebook-p2}
\end{align}
The intuition to carry: the fine-structure constant is not a dial
that happened to stop at $137.036$; it is a \emph{count} --- $2^J$
states of the chiral alphabet plus $N^2$ of the generation square
--- and counts are the kind of thing the compact fiber protects.

\paragraph{The books balance.} A theory whose gauge structure
lives on a compact fiber must pass one unforgiving audit: anomaly
freedom. An anomaly is a quantum bookkeeping failure --- a
classical symmetry whose books do not close at one loop --- and in
six dimensions the audit is harsher than in four. A3
\citep{ArisakaA3BI6} closes it: the bulk of
Eqs.~\eqref{eq:bulk}--\eqref{eq:group-p2} is locally and globally
anomaly-free by the six-dimensional \GS{} mechanism
\citep{GreenSchwarz1984}, without supersymmetry, and the audit is
not merely passed but \emph{productive}: it locks the coefficient
vector $(1,-1,-\tfrac25,-\tfrac25)$ and the $5{:}2$ trace ratio
that later fix the Kaxion's couplings (Problem~7), and it supplies the dual-lock identity
$L(2N+1) = M^2 + N^2 + L^2$, which is what selects $N$ --- and
therefore $J$, and therefore $K$ --- from inside the audit rather than
by hand (Problem~6). The intuition: on the circle, anomaly coefficients
are integers, and integers cannot run --- so an audit passed once
is passed forever, and its byproducts are exact.

\paragraph{Zero free parameters.} Step~A closes with A4
\citep{ArisakaA4Euclid}: the five dimensionful constants of
nature --- $\hbar$, $c$, $G_N$, $k_B$, and the Arisaka constant
$Q_B$ --- reconstructed as the conversion factors of the
geometry's directions, one per kind of dimension: $\hbar$ prices
the circle's tick, $G_N$ the interval's descent, $c$ converts the
stage's directions, $k_B$ and $Q_B$ price the boundary's exchange
and deposit. Nothing else dimensionful exists to choose, and
nothing dimensionless is free: what remains is
Eq.~\eqref{eq:ledger-p2}'s arithmetic. This is the precise sense
in which the program has \emph{zero free parameters} --- not
that its numbers are few, but that its numbers are counted, and
the count is closed (Problem~4 conjugates all twenty-eight
Standard-Model entries in this grammar; Problem~14 explains why
setting $\hbar = c = G_N = k_B = 1$ was always lawful).

\subsection{Step B: the projection --- how a 6D bulk becomes 4D
physics (A5--A7)}
\label{sec:stepB}

Step~A builds a six-dimensional object with exact books. But every
experiment ever performed lives on the four-dimensional stage. The
program's second stage is the account of that gap --- and it is
here, not in Step~A's geometry, that quantum mechanics enters.

\paragraph{\OGGEP{}: observation is projection.} \AX{3} declares
that an observation is not a view of the bulk from outside but a
\emph{projection} of it: the fiber coordinates are marginalized
--- integrated over --- and what survives to the stage is exactly
the fiber content that projection cannot erase: the circle's
holonomy classes and the interval's deposited addresses. The
reader already owns the intuition from Part~I: a measurement is a
\emph{read}. What the projection can reach, it reads; what it
reads, it writes; and what it writes, it cannot unwrite.

Figure~\ref{fig:bulk-projection-w2} draws the projection: over
each point of the stage sits the pair of hidden directions, and an
observation is the map that forgets them.

\begin{figure}[htbp]
\centering
\scalebox{1.12}{%
\resizebox{0.39\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, line width=0.80pt, font=\small]
  % --- bulk fibre over x ---
 \coordinate (x) at (0,0);
 \draw[->, red!70!black, line width=1pt] (x) -- (0,3.3);
 \node[red!70!black, right, align=left] at (0.2,3.25)
 {$I_r$ : \emph{scale} fiber\\[-1pt] \footnotesize RG flow --- irreversible, non-compact};
 \draw[violet!70!black, line width=1pt] (0,1.65) ellipse (0.55 and 0.25);
 \draw[->, violet!70!black] (0.55,1.67) arc (0:330:0.55 and 0.25);
 \node[violet!70!black, right, align=left] at (0.72,1.62)
 {$S^1_\theta$ : \emph{phase} fiber\\[-1pt] \footnotesize holonomy --- unitary, compact};
  % --- base stage M_4 ---
 \draw[fill=blue!7, draw=blue!55] (-2.3,-1.7) -- (2.3,-1.7) -- (3.5,-0.8) -- (-1.1,-0.8) -- cycle;
 \node[blue!55!black] at (2.55,-1.45) {$\mathcal{M}_4$};
 \node[blue!45!black, align=center] at (0.7,-2.05) {\footnotesize the stage (what the observer registers)};
 \fill (0,-1.1) coordinate (xb) circle (1.7pt);
 \node[below right] at (xb) {$x$};
 \draw[gray!60, dotted] (x) -- (xb);
  % projection pi
 \draw[->, dashed, gray!80!black, line width=0.9pt] (1.55,1.15).. controls (2.3,0.1) and (1.4,-0.7).. (0.2,-1.0);
 \node[gray!45!black, align=left] at (3.25,0.55)
 {$\pi$ : \emph{observation}\\[-1pt] \footnotesize erases $(r,\theta)$\\[-1pt] \footnotesize $\Rightarrow$ open / one-way};
\end{tikzpicture}}}
\caption[Observation as the projection from the 6D bulk to the 4D boundary.]{\textbf{Observation as the projection
$\pi:\mathcal M_6 \to \mathcal M_4$} (adapted from
\citealp{ArisakaW1Symmetry}). Over each stage point $x$ sits the
non-spatial pair $(I_r, S^1_\theta)$. The compact phase circle
carries the coherent, unitary content --- its holonomy is the
Aharonov--Bohm phase of \S\ref{sec:primer} --- while the open scale
interval carries the irreversible content, its forward-only flow
becoming the dissipator of Eq.~\eqref{eq:gdos-p2}. Because both
fiber coordinates are non-spatial, no laboratory protocol resolves
them; every registration marginalizes the fiber, so $\pi$ is
non-invertible and the stage dynamics is necessarily open. Unitary
quantum mechanics is the unreachable limit in which the fiber is
never read.}
\label{fig:bulk-projection-w2}
\end{figure}

\paragraph{\GDOS{}: the runtime, as an equation.} A5
\citep{ArisakaA5GDOS} converts \OGGEP{} into dynamics. Because the
projection is always on --- the stage never stops being coupled to
the fibers it marginalizes --- the effective four-dimensional
dynamics is that of an \emph{open} system, and its generator is
derived, not postulated:
\begin{equation}
\frac{d\rho}{dt}
\;=\;
-\frac{i}{\hbar}\,[H,\rho]
\;+\;
\mathcal D[\rho],
\label{eq:gdos-p2}
\end{equation}
a master equation of Lindblad form, in which the two fibers are
visible as the two terms. The commutator is the circle's half:
unitary, reversible, phase-carrying --- the Schr\"odinger
evolution, exactly as the textbooks teach it. The dissipator
$\mathcal D$ is the interval's half: completely positive, one-way,
record-forming --- the term the textbooks smuggle in by hand as
``collapse'' arrives here as the boundary's own bookkeeping.
The textbook postulates --- Hilbert space, unitary evolution
between readings, the L\"uders update at a reading --- follow as
theorems of Eq.~\eqref{eq:gdos-p2}'s relaxation, and the Born
weights,
\begin{equation}
p_k \;=\; |\langle k|\psi\rangle|^{2},
\label{eq:born-p2}
\end{equation}
are \emph{produced} rather than assumed: the dissipator relaxes
configurational statistics onto the squared-amplitude measure,
monotonically, at a rate the port coupling sets --- carried in
the cohort at the relaxation grade, with the one remaining
analytic step (the field-grade representation) named openly
(Problem~1 states both halves of that sentence with equal
weight).

\paragraph{\GRIP{}: the engine.} A6 \citep{ArisakaA6Gi4} supplies
the runtime's engine --- the cascade
\begin{equation}
\GRIP{} \;:=\; \DGI{} \;+\; \GFF{} \;+\; \GSSB{} \;+\; \GGA{},
\label{eq:grip-p2}
\end{equation}
read left to right as a sentence: \emph{Dynamical Gauge
Invariance} holds the top open (every gauge equivalence intact,
nothing yet committed); the \emph{geometric free-energy functional}
supplies the landscape (the configuration falls along its
gradient); \emph{geometric spontaneous symmetry breaking} selects
(the fall must land somewhere); and the \emph{geometric ground-state
attractor} closes (the landing is the stratum's unique floor, by
Morse analysis --- there is nowhere else). The cascade has two
strokes, and they are the two fibers operating: \GRIPin{}, the
circle's stroke --- closed-loop relaxation on a fixed landscape,
outcomes banked in protected classes --- and \GRIPout{}, the
interval's stroke --- the one-way reorganization of the landscape
itself, which no gradient flow generates and no operation reverses.
Everything the later Parts call ``the groove'' (Problems~8 and~15)
is Eq.~\eqref{eq:grip-p2} plus the Morse uniqueness; the weak
interaction (Problem~2) is the cascade caught operating in the
laboratory.

\paragraph{The stages, fiber-resolved: custody of transport,
custody of data.} The strokes are the coarse mapping; the runtime
companion now carries the fine one (\citealp{ArisakaA6Gi4}, the
stage-custody subsection), and it earns a paragraph here because it
is this paper's fork applied to its own engine. The strokes are
named for their \emph{products}: \GRIPin{} is the circle's stroke
because its outcome is banked in circle classes; \GRIPout{} the
interval's because it changes the landscape and banks nothing. The
stages are named for their \emph{transport}: the equivalences
\DGI{} holds intact compose as the circle's group --- the circle is
the bulk's only gauge bundle, the interval's ``structure group''
being the one-way dilatation semigroup \citep{ArisakaA8QM} ---
while the descent that executes \GSSB{}~$\to$~\GGA{} composes as
the interval's semigroup, being the long-time boundary flow whose
generator is the projected $\Ir$ dilatation \citep{ArisakaA5GDOS}.
The middle stage crosses, and the crossing is the content: \GSSB{}
reads circle data (its order parameters are Wilson-line phases),
runs on the interval (the one-way act of selection), and deposits
its outcome back into the classes the circle protects. \GSSB{}
stands to symmetry breaking exactly as the write stands to
measurement (Problem~1): one seam-event shape, two instances. The
sentence to keep: \emph{the circle states the alternatives; the
interval performs the selection; the attractor is the circle's
receipt.}

Figure~\ref{fig:grip-engine-w2} reproduces the engine's full
diagram: the four moves in the central column, the axioms that
force them on the left, the information pillars that filter them on
the right, and the two strokes at the bottom.

\begin{figure}[p]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\Large,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 dgibox/.style={fill=red!13, draw=red!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 gffbox/.style={fill=yellow!13, draw=olive!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 gssbbox/.style={fill=green!14, draw=green!60!black, line width=0.7pt,
 minimum width=80mm, minimum height=18mm},
 ggabox/.style={fill=blue!14, draw=blue!60!black, line width=1.0pt,
 minimum width=80mm, minimum height=18mm},
 gabox/.style={fill=teal!12, draw=teal!60!black, line width=0.8pt,
 minimum width=80mm, minimum height=18mm},
 inbox/.style={fill=cyan!16, draw=cyan!60!black, line width=0.9pt,
 minimum width=60mm, minimum height=18mm},
 outbox/.style={fill=orange!16, draw=orange!70!black, line width=1.0pt,
 minimum width=60mm, minimum height=18mm},
 nextbox/.style={fill=red!8, draw=red!50!black, line width=0.65pt,
 minimum width=60mm, minimum height=16mm,
 dash pattern=on 2.5pt off 2pt},
 bigarrow/.style={-{Stealth[length=2.8mm]}, line width=0.7pt, draw=black!75},
 ribarrow/.style={-{Stealth[length=2.6mm]}, line width=0.85pt, draw=cyan!55!black},
 outarrow/.style={-{Stealth[length=2.8mm]}, line width=1.1pt, draw=orange!70!black},
 axlabel/.style={font=\normalsize\itshape, text=blue!50!black,
 fill=white, inner sep=2pt, align=right, anchor=east},
 qilabel/.style={font=\normalsize\itshape, text=green!50!black,
 fill=white, inner sep=2pt, align=left, anchor=west},
 arrowtext/.style={font=\normalsize\itshape, text=black!70,
 fill=white, inner sep=1pt, align=center},
]
\node[draw=none, inner sep=0pt] at (-95mm, 0) {};
\node[draw=none, inner sep=0pt] at (95mm, 0) {};
\node[dgibox] (DGI) at (0, 0) {\textbf{\DGI{} \quad (Layer $n$)}\\[-1pt]
 {\normalsize Dynamical Gauge Invariance --- the unbroken bulk substrate, carrying $J_{\rm Qi}$ on $\mathcal{M}_6$ under $U(1)_{\rm Qi}$}};
\node[gffbox] (GFF) at (0, -32mm) {\textbf{$\bm{(F_{\GFF{}})}$ \quad Geometric Free-Energy Functional}\\[-1pt]
 {\normalsize Lyapunov landscape on the 4D boundary, $\,dF_{\GFF{}}/dt \leq 0$}};
\node[gssbbox] (GSSB) at (0, -64mm) {\textbf{\GSSB{} \quad Geometric Spontaneous Symmetry Breaking}\\[-1pt]
 {\normalsize branch commitment ($\mathbb{Z}_2$-orbifold-fixed spectrum) $\to$ discrete admissible boundary observables}};
\node[gabox] (GA) at (0, -96mm) {\textbf{\GA{} \quad Geometric Attractor(s)}\\[-1pt]
 {\normalsize stable basins en route --- possibly way stations (a muon, an excited atom); \GRIPin{} continues through their exit channels}};
\node[ggabox] (GGA) at (0, -128mm) {\textbf{\GGA{} \quad Ground Geometric Attractor}\\[-1pt]
 {\normalsize layer's unique deepest stable Morse minimum on the four-pillar-admissible stratum --- \emph{where \GRIPin{} halts}}};
\draw[bigarrow] (DGI.south) -- (GFF.north);
\draw[bigarrow] (GFF.south) -- (GSSB.north);
\draw[bigarrow] (GSSB.south) -- (GA.north);
\draw[bigarrow] (GA.south) -- node[arrowtext, right=2pt]{exit channels: the descent continues} (GGA.north);
\node[axlabel] (AX2GGEP) at ($(DGI.west) + (-5mm, 0)$)
 {\textbf{\AX{2}~\GGEP{}}\\(GG-6 / \BIsix{})};
\node[axlabel] at ($(GFF.west) + (-5mm, 0)$)
 {\textbf{\AX{3}~\OGGEP{}}\\(\GDOS{})};
\node[axlabel] at ($(GSSB.west) + (-5mm, 0)$)
 {\textbf{\AX{4}~\MSMF{}}\\acts on \GSSB{}};
\node[qilabel] at ($(DGI.east) + (5mm, 0)$)
 {\textbf{\QiQUA{}}~(\QGL)\\Arisaka equation\\$J_E/\QB = J_e = J_B = J_{\rm Qi}$};
\node[qilabel] at ($(GFF.east) + (5mm, 0)$)
 {\textbf{\QiEP{}}\\anchored on $N_A\QB \approx 15$~kJ/mol\\drives gradient descent};
\node[qilabel] at ($(GSSB.east) + (5mm, 0)$)
 {\textbf{\QiCON{}}\\conformality diagnostic};
\node[qilabel] at ($(GA.east) + (5mm, 0)$)
 {way stations:\\$\mu$, $\tau$, excited states\\(non-ground \GA{}s)};
\node[qilabel] at ($(GGA.east) + (5mm, 0)$)
 {\textbf{\QiHAL{}}\\holographic attractor\\architecture};
\node[axlabel] at ($(GGA.west) + (-5mm, 0)$)
 {\textbf{where \GRIPin{} halts}\\(\textsf{T-GGA-No-Exit})};
\node[inbox] (GIN) at (-65mm, -166mm) {\textbf{\GRIPin{}}\\[-1pt]
 {\normalsize within-layer regime}\\
 {\normalsize loops through way-station \GA{}s until the \GGA{}}};
\node[outbox] (GOUT) at (65mm, -166mm) {\textbf{\GRIPout{}}\\[-1pt]
 {\normalsize between-layer regime}\\
 {\normalsize out-of-equilibrium shock}};
\draw[ribarrow] (GA.west) to[bend right=25]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape unchanged}
 (GIN.north);
\draw[outarrow] (GGA.south) to[bend left=20]
 node[arrowtext, pos=0.55, below, sloped]{\GFF{} landscape reorganized}
 (GOUT.north);
\draw[ribarrow, dashed] (GIN.north) to[out=90, in=-90]
 node[arrowtext, pos=0.28, left=2pt, text=cyan!55!black, font=\normalsize\itshape]
 {within-layer\\closed loop}
 (AX2GGEP.south);
\node[nextbox] (NEXT) at (65mm, -202mm)
 {\textbf{\DGI{} \quad (Layer $n+1$)}\\[-1pt]
 {\normalsize next-layer cascade begins}};
\draw[outarrow] (GOUT.south) -- (NEXT.north)
 node[arrowtext, pos=0.5, right=3pt, text=orange!70!black, font=\normalsize\itshape]
 {ascent\\Layer $n \to n+1$};
\end{tikzpicture}}
\caption[The \GRIP{} engine: four moves, two regimes, one ground state.]{\textbf{The \GRIP{} engine: four moves, two regimes, one
ground state} (reproduced from \citealp{ArisakaA11Origin}; the
engine is built in \citealp{ArisakaA6Gi4}). The central column is
the four-move cascade of Eq.~\eqref{eq:grip-p2}; the constitutional
axioms (left) force the moves; the four information pillars (right)
filter the admissible minima. At bottom, the two strokes:
\GRIPin{} (cyan) loops \emph{within} a layer to its unique ground
state --- the circle's stroke --- and \GRIPout{} (orange)
reorganizes the landscape and opens the next layer --- the
interval's stroke. The two-fiber anatomy is this diagram's
grammar: everything cyan is closed-loop and reversible-in-form;
everything orange is one-way. Custody, fiber-resolved: everything
cyan \emph{banks} on the circle; the algebra its descent composes
under is the interval's.}
\label{fig:grip-engine-w2}
\end{figure}

\paragraph{\UoneQi{}: the information ledger.} A7
\citep{ArisakaA7Qi4} completes the projection stage with the
fourth bookkeeping: memory and its price. A stable bit is
holonomy-class data --- storage on the circle is free, because
integers pay no maintenance --- while erasure is an interval-side
write, and every write has a floor price:
\begin{equation}
\Delta E \;\ge\; k_B T \ln 2
\quad\text{per bit erased,}
\label{eq:landauer-p2}
\end{equation}
Landauer's bound \citep{Landauer1961}, read here not as an analogy
between information and thermodynamics but as the same ledger:
$\UoneQi$'s charge is the quantum of information, $Q_B$ is its
unit price, and the two laws of thermodynamics are the two
fibers' conservation laws (Problem~14). The intuition: the
universe keeps two kinds of books --- what it is (free to keep,
impossible to alter) and what it has done (impossible to keep
from growing, priced per line).

\subsection{Step C: the applications --- four theories, derived and
graded (A8--A11)}
\label{sec:stepC}

With the bulk built and the projection derived, the program's
third stage is deliberate: take the four great descriptive
frameworks of modern science and derive each as a face of the
same object, at declared grades.

\paragraph{Quantum mechanics (A8).} A8 \citep{ArisakaA8QM}
locates every quantum peculiarity on the anatomy: superposition
and interference are the circle's transported phase; measurement
is the interval's write (Eq.~\eqref{eq:gdos-p2}); the Born rule
is the relaxation result of Eq.~\eqref{eq:born-p2}; decoherence
is the write's microscopic face; and the classical world is the
registered record. Problem~1 is this paper's plain-English
carrying of that Part, and the F25 relic --- the prediction that
the largest cosmological scales retain a computable
primordial-statistics deficit --- is its observational exposure.

\paragraph{The Standard Model (A9).} A9 \citep{ArisakaA9SM} is
the codebook: all twenty-eight Standard-Model parameters derived
or graded from Eq.~\eqref{eq:ledger-p2}'s arithmetic, each entry
carrying its provenance. The template chain, worked once here
because Parts~III--VI quote it constantly: the interval's depth
sets the lock scale (Eq.~\eqref{eq:mlock} below,
$\Mlock = \tfrac12 M_{\mathrm{Pl}} e^{-K} = 3.849$~TeV), and the
circle's fraction dresses it into the electroweak vacuum,
\begin{equation}
v \;=\; \Mlock \left(\frac{2}{L}\right)^{N}
\;=\; 3849~\mathrm{GeV}\times\left(\frac{2}{5}\right)^{3}
\;=\; 246.3~\mathrm{GeV}
\quad(\text{measured } 246.22),
\label{eq:vev-p2}
\end{equation}
a circle label dressing an interval address --- the grammatical
form (\S\ref{sec:gearbox}) that every mass in the codebook
repeats. Problems~2, 3, and~4 unpack the sector.

\paragraph{The cosmos (A10).} A10 \citep{ArisakaA10Cosmos}
derives $\Lambda$CDM's six numbers with no dial. The inventory is
fiber-volume arithmetic --- the Hopf reading gives
\begin{equation}
\Omm = \frac{1}{\pi} = 0.318,
\quad
\Omb = \frac{1}{2\pi^{2}} = 0.0507,
\quad
\OmL = 1 - \frac{1}{\pi} = 0.682
\quad(\text{meas.\ } 0.315,\ 0.049,\ 0.685),
\label{eq:inventory-p2}
\end{equation}
--- the launch is the derived plateau roll with
$n_s = 0.9620$ and $r = 0.0043$ (Eq.~\eqref{eq:ns-r}, Problem~5);
the chronology is exact, $d\ln t/dr = 2$ (Eq.~\eqref{eq:slope},
Problem~13); and the terminus is the seesaw
$m_\Lambda = \Mlock^2/\Mplred \simeq 6.1$~meV (Problem~7). The
intuition to carry from Step~C's cosmology: the history of the
universe is the interval traversed once, and every era is an
address.

Figure~\ref{fig:cosmic-descent-w2} shows the whole of Step~C's
cosmology on one axis: the descent drawn to scale, every epoch an
address.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.82\textheight,keepaspectratio]{a10_cosmic_descent.pdf}
\caption[The cosmic \GRIP{} descent, drawn to scale.]{\textbf{The cosmic \GRIP{} descent, drawn to scale on
$r = \ln(E_{\mathrm{GUT}}/\mu)$} (reproduced from
\citealp{ArisakaA10Cosmos}). 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 $r$ --- with the late universe
magnified beneath the break. Highlighted rows carry the derived
markers this paper quotes throughout: the plateau tilt and tensor
ratio ($n_s = 0.9620$, $r_{\mathrm T} = 0.0043$), the lock
$\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-35} = 3.85$~TeV, the corridor
switch $z_{\mathrm{sw}} = e^{7}$ (the ledger integer $J = 7$), and
today's $\pi$-locked inventory of Eq.~\eqref{eq:inventory-p2}.
Nothing on the axis is fitted: the figure \emph{is} the interval,
traversed once.}
\label{fig:cosmic-descent-w2}
\end{figure}

\paragraph{Life and mind (A11).} A11 \citep{ArisakaA11Origin}
runs the cascade of Eq.~\eqref{eq:grip-p2} from the GUT edge to
the biosphere: eighteen \GRIP{} events, each a stratum's unique
floor, each banked before the next begins --- nucleons, nuclei,
atoms, molecules, autocatalysis, code, cognition --- and proves
the \GGA{} theorems (ground-state uniqueness, no-exit) on which
the whole chain's stability rests. Its four closing words ---
not contingent, not lucky, not designed, \emph{forced} --- are
the register in which Problems~8 and~15 answer the multiverse,
and this paper closes on them.

\subsection{Step D: the fourth step --- the white papers (W1, W2)}
\label{sec:stepD}

Steps A--C are research: they build, project, and apply. The
fourth step is different in kind --- synthesis: stand back from
the twenty-two papers and ask what single thing has actually been
learned. The cohort's answer comes in two white papers, one
historical and one structural, and they are deliberately a pair.

\PaperWOne{} \citep{ArisakaW1Symmetry} is the \emph{diachronic}
reading: it walks the century of symmetry and conservation ---
Noether's two theorems, the Bianchi identity, Wilson's loops, the
\GS{} mechanism, \HS{} quantization --- and shows the old
conviction ``symmetry causes conservation'' loosening step by
step until what remains conserved is an integer with nowhere to
move. Its capstone is the determinism triad
(\S\ref{sec:determinism}): symmetry breaking re-understood as
symmetry \emph{reading} --- the one-way playback of a groove that
was never symmetric.

This paper is the \emph{synchronic} reading: the same content,
organized not by history but by structure. The single thing
learned, in this paper's language, is the anatomy: the bulk's two
hidden directions are fibers of opposite topological kind, and
that opposition --- circle and interval, group and semigroup,
memory and act --- is where the program's explanations live.
W1 shows \emph{how physics arrived} at the doorstep;
W2 opens the door and walks the house --- the sixteen rooms of
Part~I. The remainder of this Part now builds the instrument the
walk requires: the two characters in full, the reason the pair is
minimal, the determinism it enforces, and the working images and
grade discipline used on every page after.

\section{A one-page primer: holonomy, from the bench to the bulk}
\label{sec:primer}

For the reader meeting holonomy as a physical (rather than
mathematical) object for the first time, three rungs connect the
laboratory to Eq.~\eqref{eq:bulk}.

\paragraph{Rung 1: the tabletop.} The Aharonov--Bohm experiment is
holonomy made visible: an electron beam split around a solenoid
interferes with itself, and the fringe shift measures the phase
\begin{equation}
\Delta\varphi
\;=\;
\frac{q}{\hbar}\oint_{C} A\cdot dl
\;=\;
\frac{q\,\Phi}{\hbar},
\label{eq:ab-p2}
\end{equation}
the loop integral of the potential --- equivalently the enclosed
flux $\Phi$ --- around the closed circuit $C$: a quantity that
survives even where the field strength vanishes along the entire
path. Read the equation's three symbols as the anatomy in
miniature: $q$ is a protected charge, $C$ must \emph{close} for
$\Delta\varphi$ to mean anything, and $\hbar$ --- Step~A's
circle-side constant --- is the exchange rate between geometry and
phase. Two lessons ship with the apparatus. The phase is
\emph{loop} data: no measurement along an open segment reveals it,
because only the closed comparison is gauge-invariant. And the
phase is \emph{robust}: deform the arms at will, and as long as the
winding about the solenoid is unchanged, the shift is unchanged.
Memory that survives deformation, readable only on closure --- that
is the whole character of the circle, already on the bench.

\paragraph{Rung 2: the reconstruction.} Wilson's observation
elevates the bench fact to a definition: the gauge-invariant
content of any gauge field is exactly its Wilson loops,
\begin{equation}
W(C)
\;=\;
\mathcal P\,
\exp\!\left(\frac{i q}{\hbar}\oint_{C} A\cdot dl\right),
\label{eq:wilson-p2}
\end{equation}
the path-ordered exponential of Eq.~\eqref{eq:ab-p2}'s integrand
over every closed curve $C$ --- a connection \emph{is} its
holonomies, no more and no less. From this the two fiber
characters follow as pure logic. A compact fiber supports
noncontractible loops, so a connection there has invariant content:
quantized windings, protected phases. A contractible fiber with
boundary --- an interval --- supports none, so a connection there
has \emph{no} gauge-invariant content at all: whatever happens
along it is convention until it is registered elsewhere. The
statement of \S\ref{sec:statement} is not an analogy built on the
bench experiment; it is the same theorem, read for two different
base spaces.

\paragraph{Rung 3: the transport algebra.} Composition closes the
primer, and it deserves its own pair of displayed lines, because
the whole paper is the difference between them. Transports around
the circle compose as a \emph{group}:
\begin{equation}
U(\theta_2)\,U(\theta_1) = U(\theta_1 + \theta_2),
\qquad
U(\theta)^{-1} = U(-\theta)
\quad\text{always exists} ;
\label{eq:group-law-p2}
\end{equation}
every loop can be retraced, and the algebra is the algebra of
things that can be undone. Physical transport along the scale
interval --- the renormalization flow, the cosmic descent ---
composes as a \emph{semigroup}:
\begin{equation}
T(r_2)\,T(r_1) = T(r_1 + r_2)
\quad\text{for } r_1, r_2 \ge 0
\quad\text{only},
\qquad
T(r)^{-1}\ \text{does not exist} ;
\label{eq:semigroup-law-p2}
\end{equation}
coarse-graining has no inverse, so the flow runs one way and each
step is final. The two lines look almost identical --- one
composition law, one addition of parameters --- and differ in one
clause: whether the inverse exists. That single clause is the
group/semigroup split, and it is the property the rest of this paper
spends: \emph{what has inverses can promise; what lacks them can
happen.}

\paragraph{Two premises, graded apart.} The two displayed lines are
not established the same way, and the distinction matters more here
than anywhere else in the paper, so it is drawn now and kept
throughout.

\emph{Premise (T), topological.} Eq.~\eqref{eq:group-law-p2}, and the
absence of any protected content along the interval, follow from
$\pi_1(\Stheta) = \mathbb{Z}$ and $\pi_1(\Ir) = 1$ and from nothing
else. They hold for any connection, any field content, any Lagrangian.
Nothing physical is assumed. This is the premise that earns the
sentence \emph{holonomy requires closed loops, and an interval has
none}, and it is unarguable.

\emph{Premise (D), the boundary law's.} The \emph{failure of the
inverse} in Eq.~\eqref{eq:semigroup-law-p2} is not topological, and
this paper does not claim that it is. Parallel transport along a path
in $\Ir$ is perfectly invertible --- Appendix~\ref{app:holonomy}
gauges the interval's connection away altogether. What lacks an
inverse is the \emph{physical} transport along $r$: the
renormalization flow and the cosmic descent, whose composition law is
Kadanoff's Eq.~\eqref{eq:A-kadanoff} and whose non-invertibility is
the non-injectivity of blocking --- many microscopic configurations to
one block, and no operation restores what was never kept. That is
derived rather than assumed, but it is derived in Step~B, from the
open-system chain and \OGGEP{}'s compulsory projection.

Keeping the two apart costs the paper nothing it uses and saves it
from a claim it cannot defend. The characteristic assertion of this
program is not that topology alone makes time run forward. It is
that irreversibility is \emph{algebra rather than statistics} --- a
property of how transports compose, not a matter of how many degrees
of freedom happen to be in the room --- and (T) and (D) together
deliver exactly that. (T) says the interval has nothing to protect;
(D) says what happens along it cannot be undone. A framework needs
both, and this one has both, from two different places.

\section{The instrument, quantified}
\label{sec:algebra}

These four equations carry every later Part, so here is what each
one buys.

\emph{Eq.~\eqref{eq:ab-p2} (the phase) is why the circle's content
is physical.} A quantity readable only on closed circuits is
untouchable by anything local: no field redefinition, no gauge
choice, no smooth deformation reaches it. Everything the cohort
banks on the circle --- charges, parities, the flavor alphabet,
the ledger integers of Eq.~\eqref{eq:ledger-p2} --- inherits this
protection.

\emph{Eq.~\eqref{eq:wilson-p2} (the reconstruction) is why the
circle's content is} everything. Since a connection is its
holonomies, a fiber with no closed loops carries a connection with
no invariant content at all --- which is the precise sense in
which the interval \emph{forgets}: not that its record fades, but
that there was never a gauge-invariant record to keep. What the
interval does instead is act --- and its acts are registered on
the boundary, priced by Eq.~\eqref{eq:landauer-p2}.

\emph{Eq.~\eqref{eq:group-law-p2} (the group) is the quantum
half.} Unitary evolution, superposition, interference,
reversibility between measurements --- the entire coherent face of
quantum mechanics is transport with inverses, and it lives where
the inverses live: on the compact fiber. Energy conservation is
this line's Noether charge (Problem~14).

\emph{Eq.~\eqref{eq:semigroup-law-p2} (the semigroup) is the
classical half.} Measurement, decoherence, the renormalization
flow, entropy growth, cosmic history, aging --- the entire
irreversible face of physics is transport without inverses, and it
lives where the inverses fail: on the open fiber. The second law
is this line's signature (Problem~14), and the dissipator of
Eq.~\eqref{eq:gdos-p2} is its generator.

One instrument, four readings --- and the reader now holds, in
five displayed equations, the entire technical content of the
sentence \emph{the circle remembers; the interval forgets}. The
rest of this Part gives the pair their characters, their
minimality proof, and their working images; Parts~III--VI spend
them.



\section{The two characters}
\label{sec:characters}

Table~\ref{tab:characters} is the paper's reference card, and
Figure~\ref{fig:anatomy} is its portrait. Each row is a theorem-shaped
statement of a cohort paper, cited in place; the table's contribution
is only the alignment.

\begin{table}[htbp]
\centering
\caption[The reference card: the two non-spatial fibers, character by character.]{The reference card: the two non-spatial fibers, character by
character. Each row is established in the cited cohort paper; the
table aligns them. The ten rows run from topology and holonomy through the algebra of transport --- a group on the circle, a semigroup on the interval --- to the character of the content each carries (quantized against continuous), the constant each supplies ($\hbar$ against $G_N$), the conservation law each owns (the first law against the second), the field each coordinate becomes (the Kaxion against the inflaton), and the face of time each shows (the tick against the arrow).}
\label{tab:characters}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\textwidth}{@{}P{3.2cm}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X@{}}
\toprule
& \textbf{The circle $\Stheta$ (6th)} & \textbf{The interval $\Ir$ (5th)} \\
\midrule
Topology & compact, closed: all loops & non-compact, open: no loops \\
Holonomy & everything --- the fiber \emph{is} its holonomies & none --- nothing to protect \\
Transport & a group (inverses exist) & a semigroup (forward only) \\
Content & quantized: windings, phases, parities, charges & continuous: scales, addresses, rates \\
Behavior & remembers: classes do not age & forgets: one pass through every scale \\
Physical role & the gauge phase; the ledger $(M,N,L;J,K)$; the flavor lattice & the RG flow; the descent; the warp $e^{-2r}$ \\
Its constant & $\hbar$ --- the price of a tick \citep{ArisakaA4Euclid} & $G_N$ --- the residue of the descent \citep{ArisakaA4Euclid} \\
Its conservation law & the first law: energy (nothing about it is one-way) & the second law: entropy (nothing about it repeats) \citep{ArisakaA5GDOS} \\
Its coordinate, as a field & the Kaxion (Problem~7) & the inflaton (Problem~7) \\
Its face of time & the tick (Problem~13) & the arrow (Problem~13) \\
\bottomrule
\end{tabularx}
\end{table}

Three remarks make the card load-bearing rather than decorative.

\paragraph{The asymmetry is structural, and it has two sources.} Both
fibers carry Ehresmann connections; both are geometry of the same
bulk; nothing in the Lagrangian distinguishes them. What distinguishes
them is prior to any choice of dynamics, and it arrives in the two
premises of \S\ref{sec:primer}. By (T), the circle has noncontractible
loops and the interval has none, so one fiber can hold
gauge-invariant content and the other has nothing to hold. By (D), the
physical flow along the interval is the coarse-graining flow, which is
not injective, so it composes forward only. Every exactness claim in
this paper is traced to (T); every arrow, record and irreversibility
to (T) together with (D) --- and to nothing else. No thermodynamic
postulate, no collapse postulate, no initial-condition stipulation is
ever added. But the reader should hold the paper to the honest form of
the claim: irreversibility here comes from a compulsory projection
onto a fiber that has nothing to protect, not from topology unaided.

\paragraph{Protection has an address.} Whenever physics presents a
quantity that is exact --- a charge that never leaks, an anomaly
coefficient that does not run, a family count that neither drifts nor
renormalizes --- the anatomy says where it lives: on the circle,
deposited in holonomy classes that no boundary operation can
continuously deform. And whenever physics presents a quantity that
is \emph{placed} --- a mass, a scale, a rate, an epoch --- the
anatomy says where that lives too: along the interval, as an address
on the fiber that forgets. The codebook paper carries this as a
grammar covering all twenty-eight Standard-Model parameters:
\emph{every parameter is a holonomy-class label wearing a descent
address} (\citealp{ArisakaA9SM}; \citealp{ArisakaP3Electron}).

\paragraph{The dictionary owns the sectors; the geometry owns the
$i$.} The real-bulk companion \citep{ArisakaP4Real} adds the
information-theoretic face: the complex numbers of quantum mechanics
are the circle's --- weight decomposition under a circle action is
what manufactures the $i$ --- while records are real because
registration is an interval-side write, along the fiber that carries
no phase. Why complex amplitudes? Because one fiber is a circle. Why
real records? Because the other is an interval.

\begin{figure}[htbp]
\centering
\resizebox{0.851\textwidth}{!}{%
\usebox{\GGplateAnatomy}%
}
\caption[The anatomy: the circle that remembers, the interval that forgets.]{\textbf{The anatomy.} \emph{(a)} The compact phase circle:
every path closes, so everything a connection does there is holonomy
--- quantized (dots: the ledger-rational marks), gauge-protected,
transported by a group. \emph{(b)} The scale interval: no path
closes, so nothing is protected; transport composes only forward, a
semigroup; the descent visits each scale exactly once. \emph{The
circle remembers; the interval forgets} --- and every consequence in
Parts~III--VI is this figure, applied.}
\label{fig:anatomy}
\end{figure}

\section{The constitutional minimum}
\label{sec:minimum}

Why one fiber of each kind --- why not two circles, or two intervals,
or one fiber, or none? The constitution answers with a minimality
claim that the campaign has made exact
(\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}).

A bulk with \emph{two circles} would conserve everything and decide
nothing. All content would be holonomy-class data; every quantity
protected; no transport irreversible. There would be no arrow, no
registration, no record --- \AX{3} (\OGGEP{}), which makes
observation a projection, would have nothing lawful to project
\emph{onto}, because a record is precisely a deposit that cannot be
undone, and a two-circle world has no fiber on which anything is
final. Such a world \emph{is} but never \emph{becomes}.

A bulk with \emph{two intervals} would decide everything and keep
nothing. No quantized charge, no protected phase, no stable atom ---
interference needs a phase that returns, and a charge that never
leaks needs a class that cannot deform. Such a world \emph{becomes}
but never \emph{is}: perpetual reorganization, converging on nothing,
with no alphabet in which an outcome could even be written.

\emph{Being} needs holonomies; \emph{becoming} needs a semigroup. A
universe that both is and becomes needs exactly one fiber of each
kind --- and that is what \AX{2} (\GGEP{}) writes and \AX{3}
(\OGGEP{}) reads. The same counting reappears at runtime level in the
\GRIP{} engine \citep{ArisakaA6Gi4}: \GRIPin{}, the circle's stroke
--- closed-loop relaxation on a fixed landscape, outcomes deposited
in the quantities the circle protects --- and \GRIPout{}, the
interval's stroke --- the one-way landscape reorganization no
gradient flow generates and no operation reverses. A runtime with
only \GRIPin{} would cycle forever deciding nothing; with only
\GRIPout{}, reorganize forever converging on nothing. The cascade of
nature requires exactly one stroke of each kind for the same
topological reason the bulk carries exactly one fiber of each kind.

\section{The two counterfactuals, run slowly}
\label{sec:counterfactuals}

The minimality argument of \S\ref{sec:minimum} deserves a slower
pass, because the two impossible worlds it invokes are the best
intuition pumps the anatomy owns.

\paragraph{The two-circle world.} Give the bulk a second compact
fiber in place of the interval. Every direction now supports
holonomy; every quantity is a winding; the transport algebra is a
group throughout. Walk through what is lost. There is no scale:
nothing distinguishes ultraviolet from infrared, because the
would-be RG coordinate now returns --- physics at every resolution
is periodically identified with physics at another, and
``coarse-graining'' is a rotation. There is no equilibrium and no
approach to it: relaxation requires a lost past, and nothing is
lost. There is no record: a record is a deposit that cannot be
undone, and every operation now has an inverse. And so there is no
\emph{event}: things are, permanently and reversibly, but nothing
ever happens, because happening is exactly the part that does not
unhappen. A two-circle world is a perfect archive with no history
--- being without becoming.

\paragraph{The two-interval world.} Now give the bulk a second
interval in place of the circle. No direction supports holonomy;
no quantity is quantized; nothing is protected. Walk through what
is lost. There is no charge that cannot leak, so there are no
stable atoms; no phase that returns, so no interference, no
spectra, no clocks; no class structure, so no two electrons are
the same and chemistry has no table; no alphabet, so even if a
selection occurred there is nothing exact in which to write the
outcome. Becoming proceeds --- one-way, relentlessly --- but
converges on nothing and retains nothing: every deposit erodes,
because the vault was never topological. A two-interval world is
a river with no banks --- becoming without being.

\paragraph{The census, once more.} One of each, and both defects
cancel against the other's virtue: the circle gives the river its
banks (records deposited into protected classes), and the interval
gives the archive its history (a direction along which the books
are written). The reader should notice what kind of argument this
is: not a fit to data, but a transcendental-style necessity claim
run inside physics --- \emph{any} world with observers, records,
and chemistry needs the census $1+1$, and the constitution's
theorems \citep{ArisakaA2GG6} are the formal version of exactly
this count. The anatomy is not one design choice among many; it is
the shape of the possibility of worlds like ours.

\section{Determinism: symmetry breaking as symmetry reading}
\label{sec:determinism}

The anatomy's deepest single yield --- developed at full length in
\PaperWOne{} \citep{ArisakaW1Symmetry} and carried here because
Parts~III--VI lean on it --- is that symmetry breaking in this
geometry is \emph{deterministic}: the same outcome in every region,
every time, with no lottery anywhere. Three mechanisms close every
door through which randomness enters the standard accounts, and each
lives on a named fiber structure:

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{The dial, not the hat.} The vacuum manifold of the
compact sector is not a continuous circle of equivalent minima (the
Mexican hat's rim) but a \emph{quantized dial}: the Wilson-line vacua
sit at discrete ledger-rational angles, and the scan has a unique
winner. There is no continuum from which to draw a random choice ---
the circle's alphabet is discrete.
\item \textbf{The Morse-unique floor.} On each stratum the \GFF{}
landscape has a unique deepest minimum --- the \GGA{} --- by the
Morse analysis of \citet{ArisakaA6Gi4}; there is no degeneracy among
which history must choose.
\item \textbf{The semigroup cannot re-choose.} Whatever the descent
deposits is final: the interval has no inverse transport, so no
fluctuation can undo a selection and try again.
\end{enumerate}

Together: nothing random survives at the level of which world is
selected. Curie's principle --- that an asymmetric outcome needs an
asymmetric cause --- is dissolved rather than violated: the dynamics
is symmetric, the \emph{arithmetic} is not, and the outcome reads the
arithmetic. The image the cohort has adopted: the circle is a
\emph{tape} on which the constitution's integers are pressed like a
groove; the interval is the \emph{read head}, drawn across the tape
once, at a rate the geometry fixes; and what looks like symmetry
\emph{breaking} --- the cascade of choices from the GUT edge to the
electroweak vacuum to the flavor textures --- is symmetry
\emph{reading}: the one-way playback of a groove that was never
symmetric. Something emerges from nothing deterministically and
inevitably, because the nothing was a tape with a groove in it, and
the player only runs forward. And the triad is the seam shape this
paper keeps meeting: the alternatives are the circle's, the
once-only execution is the interval's, the record is circle-kept
--- symmetry breaking stands to the groove exactly as the write of
Problem~1 stands to the record.

\section{The gearbox: where the fibers multiply}
\label{sec:gearbox}

The two fibers are opposite in kind, but they are fibers of one bulk,
and there is a recurring grammatical form wherever they meet: a
\emph{circle integer dividing or correcting an interval scale}. The
lock scale is the template,
\begin{equation}
\Mlock \;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-K}
\;=\; 3.849~\mathrm{TeV},
\label{eq:mlock}
\end{equation}
an interval-side depth ($e^{-K}$, thirty-five e-folds of descent)
whose exponent is a circle-side integer ($K = 35$, a ledger member).
The Kaxion's decay constant repeats the form,
$\fK = f_{\mathrm{bulk}}/\Xicap$ with $\Xicap = K + N + L = 43$
(benchmark; the one open integer of \citealp{ArisakaC1Kaxion}); the
dark-energy seesaw squares it,
$m_\Lambda = \Mlock^2/\Mplred$ (Problem~7); and the codebook reads the
same grammar across the quark floors --- a compact-angle winding (the
circle's label) multiplying a radial overlap (the interval's address)
\citep{ArisakaA9SM}. The general lesson, used repeatedly below: when
a formula mixes an exact integer with a continuous scale, the integer
is the circle speaking and the scale is the interval --- and when a
residual uncertainty survives in the cohort, it sits precisely at
such a seam, because the seam is the only place a number can still
move.

\section{The map, restated}
\label{sec:map}

Part~I has already tabulated the sixteen problems and their
locations; with the anatomy now built, the map can be restated in
one sentence per category. \emph{The Standard Model} (Part~III):
quantum mechanics' two halves, the three interactions' three
characters, gauge invariance and renormalization as the two fibers
made law, and the parameter count as labels wearing addresses.
\emph{Cosmology} (Part~IV): the launch, the hierarchy-as-depth, the
dark trio, and the three horizons. \emph{The foundation of particle
physics} (Part~V): the discrete symmetries, the antiparticle and
the excess, and the guarded proton. \emph{The foundation of the
cosmos} (Part~VI): time, the three currencies, the landscape
question, and the black hole. Part~VII is the Discussion ---
significance, comparisons, the honest-status ledger, and the
consolidated falsifier table --- and Part~VIII concludes in the
author's own voice.

\section{Physics intuition: five images to keep}
\label{sec:images}

Formal statements travel farther with handles. Five images recur
through the cohort and through this paper; the reader is invited
to adopt them, with the caution that each is an image of a
theorem, not a substitute for one.

\paragraph{The tape and the read head.} The circle is a tape on
which the constitution's integers are pressed as a groove; the
interval is the read head, drawn across once at a rate the
geometry fixes. Symmetry breaking is the playback
(\S\ref{sec:determinism}). The image earns its keep because it
locates the determinism: grooves play back the same in every
player.

\paragraph{The archive and the river.} The two counterfactual
worlds (\S\ref{sec:counterfactuals}): all circle is an archive
with no history; all interval is a river with no banks. Our world
is a river running past an archive, depositing into it --- and the
deposits are the only bridge: events become records, records are
kept as classes.

\paragraph{The dial, not the hat.} The compact sector's vacua are
positions on a notched dial, not points on a hat's smooth rim.
Where a hat invites a lottery, a dial invites a scan --- and a
scan has a winner (\S\ref{sec:determinism}).

\paragraph{The label wearing an address.} Every particle, every
parameter: what it \emph{is} lives on the circle, exact; what it
\emph{has} --- mass, scale, epoch --- lives on the interval,
placed. When a formula mixes an integer with an exponential of
$K$, read the integer as the label and the exponential as the
address (\S\ref{sec:gearbox}).

\paragraph{The toll and the ledger.} Whatever crosses the descent
pays --- in suppression factors that are exponentials of depth ---
and whatever is banked on the circle is kept without maintenance
--- integers pay no storage fee. The matter excess is a paid toll
(Problem~11); the proton is a fee-free deposit (Problem~12); dark
energy is the final invoice (Problem~7).

\section{Reading conventions: the grade vocabulary}
\label{sec:grades}

Because every claim below is carried at its parent's grade, the
vocabulary is fixed here once. \emph{Derived}: follows from the
declared axioms with no further input; the reader can audit the
chain. \emph{Derived-conditional}: derived modulo named lemmas or
constructions, each stated where used (the conditions are part of
the claim). \emph{Named input}: a quantity taken from measurement
or assigned by a declared choice, counted and listed --- never
smuggled. \emph{Postulated}: a step asserted with its motivation
but without derivation, flagged as such (the $K/2$ exponent of
Problem~11 is this paper's chief example). \emph{Flagged}: a numerical
observation recorded under the cohort's numerology discipline ---
reproducible arithmetic, no mechanism yet, never used as support
(the $4K$/$8K$ family). The discipline's point is asymmetry:
grades may be cited downward (a derived claim may be used
anywhere) but never upward (nothing leans on a flag). A synthesis
paper lives or dies by keeping this straight, and the reader is
invited to police it.

\section{The deployment: one sentence, twenty-two papers}
\label{sec:deployment}

The anatomy of this Part is not a proposal awaiting adoption; it is
the reading the cohort was already written in, recovered and stated
once. In the campaign that preceded this white paper, the exact
statement was carried into every paper of the deployment ---
twenty-two in all --- and in each case the entry point is the paper's
own central topic, not an appended commentary.
Figure~\ref{fig:cohort-map-w2} draws the cohort whole, and
Table~\ref{tab:deployment} records where the statement enters each of
its papers. What the two together show is that
a reading which had to be forced would have needed a different door in
each house, and this one used the front door every time; what it
cannot show is whether that is a fact about the geometry or a fact
about the author, and \S\ref{sec:w2-record} is where the paper says so
in its own voice.

\subsection{The cohort at a glance}
\label{ssec:cohort-glance}

The cohort is not a series in the ordinary sense --- not a first,
second and third instalment to be taken in order. It is a dependency
graph, and Figure~\ref{fig:cohort-map-w2} is that graph drawn once.
The same figure is carried by every paper that has one, so a reader
who has met it elsewhere may go straight on; a reader meeting it here
should know four things about how it is drawn.

\emph{The spine.} A single column descends from the constitutional
root \AOne{} (violet), which supplies the four axioms and seven
postulates, through the GG6 Trio that realizes the bulk geometry
(\ATwo{}, \AThree{}, \AFour{}), the boundary law \AFive{}, the
\GDOS{} Trio pillars \ASix{} and \ASeven{}, the 4D-Physics Trio
\AEight{}, \ANine{} and \ATen{}, to the closing capstone
\AEleven{}. Solid arrows mark axiom-forced descent, and where the
driving axiom or postulate is most illustrative the arrow carries its
name. Nothing on the spine is decorative: remove a box and everything
below it loses its warrant.

\emph{The two framed rows.} Beneath the spine sit the seven short
papers that can be wrong in public --- the Predictions Branch (cyan)
and the Cosmology \& Astrophysics Branch (teal). They are framed
apart from the spine because their job is different: each is a
falsifier of a specific upstream theorem, and each states a number or
a bound that an experiment can retire. The internal arrows along the
Predictions row are results being handed along it, not axioms
descending.

\emph{The two flanking columns.} Two columns of two stand either side
of the spine, and they do opposite jobs. On the left the Foundations
Branch (violet) \emph{feeds} it: \PaperPFour{} manufactures the $i$
that \AEight{} consumes, and \PaperPFive{} sits at the
\ANine{}--\ATen{} interface, taking ledger phases and returning the
matter-excess chain. On the right the White-Paper Branch (brown)
\emph{reads} it. Every edge into the right-hand column is dashed,
which in this figure's grammar means the deployment of a result
already derived elsewhere --- and that, drawn, is the whole status
claim of the two white papers.

\emph{Where this paper is.} \PaperWTwo{} is the lower box of the
right-hand column. The single dashed feed drawn into it, from
\ATen{}, names its heaviest source and not its only one: the honest
statement is that this paper reads all of the spine and both framed
rows, and the figure would be unreadable if every such edge were
drawn. The solid arrow above it, from \PaperWOne{}, is the one
structural dependency between the two syntheses --- the century-long
arc ends where protection stops coming from symmetry and starts
coming from topology, and that is where this paper begins.

\clearpage
%% ===================================================================
%%  The twenty-two-paper cohort map, on its own page.  Verbatim copy
%%  of the master (A1-Constitution Figure 3) as carried by every other
%%  paper of the cohort: no self-reference marker anywhere, so the
%%  figure is identical in all twenty carriers and simple to
%%  replicate.  Only the internal node names MP4/MP5 differ from the
%%  master (collision avoidance).  Do not localise this block.
%% ===================================================================
\clearpage
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\node[bulk] (A3) at (0, -25mm)
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\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{}), 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{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} 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-w2}
\end{figure}
\clearpage

\subsection{Where the anatomy enters each paper}
\label{ssec:deployment-table}

\begin{table}[htbp]
\centering
\caption[The campaign map: where the anatomy enters each cohort paper.]{The campaign map: where the two-fiber anatomy enters each
cohort paper, always through its central topic. There are twenty-two rows, one for every paper in the cohort, and each names the point at which the anatomy enters that paper's own argument rather than a general resemblance --- from the constitutional minimum in A1 to the consolidation in the present paper.}
\label{tab:deployment}
\small
\renewcommand{\arraystretch}{1.22}
\begin{tabularx}{\textwidth}{@{}P{2.6cm}>{\raggedright\arraybackslash}X@{}}
\toprule
\textbf{Paper} & \textbf{Entry point} \\
\midrule
A1-Constitution & the pair as the constitutional minimum: one fiber of each kind, or no world that is and becomes \\
A2-GG6 & the pair read together: what the four geometric pictures jointly buy \\
A3-BI6 & why the anomaly ledger cannot run: inflow as circle-side bookkeeping \\
A4-Euclid & the five constants sorted: $\hbar$ the circle's, $G_N$ the interval's \\
A5-GDOS & the two currencies: the first law the circle's ledger, the second the interval's \\
A6-GRIP & the engine's strokes: \GRIPin{} the circle's, \GRIPout{} the interval's \\
A7-Qi4 & memory on the circle, erasure on the interval: Landauer as the toll \\
A8-QM & the measurement arrow: the write is interval-side, the record circle-kept \\
A9-SM & the codebook sorted: every parameter a label wearing an address \\
A10-Cosmos & the chronology, and the trio: rolling, ringing, rest \\
A11-Origin & the cascade and the origin: seventeen events, one reading \\
C1-Kaxion & the third portrait: matter of the circle, occupancy of the interval \\
C2-Hubble & the rate: how fast does the fiber that forgets forget \\
C3-BH & the paradox as a billing error: one fiber charged for the other's job \\
P1-KK & the discovery as a double reading: identity from the circle, address from the interval \\
P2-N3 & the method, and the seam: a count the circle protects, a bound the interval prices; and in its Appendix~H, what the circle quantizes is what the interval delivers \\
P3-Electron & the first portrait: a holonomy class wearing a mass \\
P4-Real & the notation: the $i$ on loan from the circle; records real by the interval \\
P5-CPT & the birth and the time: the alphabet, the toll, the tick and the arrow \\
P6-Proton & the double guard: the triangle that closes, the ledgers that keep \\
W1-Symmetry & the capstone: symmetry breaking as symmetry reading \\
W2-Holonomy & the present paper: the consolidation, where the readings above are rewritten once, in one voice, across sixteen problems \\
\bottomrule
\end{tabularx}
\end{table}

Two lessons from the map, before the applications begin.
\emph{First}, the anatomy is a synthesis discipline, not a metaphor
bank: in every deployment the sorting had to reproduce the paper's
established results exactly, at their established grades, and it
did --- across twenty-two papers, no clause of the campaign required a
value change, a retraction, or a promotion anywhere in the cohort.
Whether that makes it an anatomy or a well-chosen vocabulary is a fair
question, and \S\ref{sec:w2-record} and Part~X both take it up. \emph{Second}, the map explains this
paper's method: each Part below is the consolidation of several
entries of Table~\ref{tab:deployment}, rewritten once, in one
voice, for a reader who should not have to assemble the story from
twenty-two places.

\subsection{Where to go next}
\label{ssec:where-next}

A map is only useful if it is also a door. What follows is for the
reader who has come this far, has decided the sentence is worth an
afternoon, and would like to know which paper to open next. The
answer depends on what brought him here, so here are seven routes.
Each names its papers, says what the reader gets, and --- because this
is a cohort that prints its own kill conditions --- says what it
costs.

\paragraph{If the first question is what is actually assumed.}
Start at \AOne{} \citep{ArisakaA1Constitution}: four axioms, seven
postulates, and the warrant for everything else, gathered in one
document with the whole inheritance drawn as Figure~3 of that paper
--- the same plate printed above. Then \ATwo{}
\citep{ArisakaA2GG6}, which builds the arena the axioms describe and
shows what the four geometric pictures jointly buy. A reader who
would rather arrive by the classical route than the axiomatic one
should take \AFour{} \citep{ArisakaA4Euclid} instead, which walks in
from Euclid and rebuilds the constants on the way.

\paragraph{If the interest is the mathematics.} \AThree{}
\citep{ArisakaA3BI6} is the hardest paper of the cohort and the most
load-bearing: anomaly freedom of the modified Bianchi identity in six
dimensions, which is what licenses the ledger everything downstream
spends. It is also where the century-long arc of \PaperWOne{} ends,
so the two can be read together. Its natural companion is
\PaperPFour{} \citep{ArisakaP4Real}, the paper that says where the
$i$ of quantum mechanics comes from and why records come out real ---
the question this paper's Problem~9 answers in one sentence and that
paper answers in a hundred pages.

\paragraph{If the interest is the machinery this paper leans on.}
\AFive{} \citep{ArisakaA5GDOS} is the single paper the present one
uses most: the boundary law, and the open-system turn on which the
whole two-fiber reading depends. \ASix{} \citep{ArisakaA6Gi4} is the
geometric runtime, and its cascade is the engine of
\S\ref{sec:stepB} above read on the two fibers: \GRIPin{} the
circle's stroke, \GRIPout{} the interval's. \ASeven{}
\citep{ArisakaA7Qi4} is the
quantum-information one, where memory and erasure are priced and
Landauer's bound becomes the toll. A reader who found the circle and
the interval persuasive but wants to see them made to work should
read these three in order.

\paragraph{If the interest is seeing the textbooks come out.}
\AEight{} \citep{ArisakaA8QM} derives quantum mechanics as the
projection of the bulk on the boundary, and is careful to say which
part of the Born rule is closed and which is not. \ANine{}
\citep{ArisakaA9SM} sorts the Standard Model's parameters into
observations of one geometry. \ATen{} \citep{ArisakaA10Cosmos} does
the same for the cosmic inventory and the chronology. These are the
three papers where the program is at its most checkable by hand: the
numbers are there, and so are the derivations that produced them.

\paragraph{If the interest is the long view.} \AEleven{}
\citep{ArisakaA11Origin} runs the cascade from quarks to
consciousness and asks why the layers are the layers. It is the most
ambitious paper of the cohort and the most exposed, and it says so in
its own front matter; a skeptical reader may prefer to arrive there
last rather than first, which is also the order the cohort intends.

\paragraph{If the interest is something to shoot at.} The seven short
papers of the two framed rows exist to be retired by measurement.
\PaperPOne{} \citep{ArisakaP1KK} puts a lock cluster near
$3.85$~TeV, which is a standing falsifier at the LHC. \PaperCOne{}
\citep{ArisakaC1Kaxion} puts cold dark matter at
$8.1\,\mu\mathrm{eV}$, a haloscope band a running experiment can
close. \PaperCTwo{} \citep{ArisakaC2Hubble} reads the two Hubble
constants as the two ends of one corridor rather than as a
discrepancy to be split. \PaperPTwo{} \citep{ArisakaP2N3} derives
the generation count, and its Appendix~H is the place to go for the
seam this paper describes in Problem~2. \PaperPThree{}
\citep{ArisakaP3Electron} and \PaperPSix{}
\citep{ArisakaP6Proton} take the electron and the proton as ground
states of their strata --- and a single verified proton decay ends the
second of them outright. \PaperCThree{} \citep{ArisakaC3BH} recovers
the Bekenstein--Hawking entropy exactly and reads the paradox as a
billing error. One paper belongs with these without standing in the
rows: \PaperPFive{} \citep{ArisakaP5CPT}, which sits on the spine's
left flank and is the widest of the exposed papers, taking the mirrors
of matter, the arrow of time and the matter excess together.

\paragraph{If the interest is the history rather than the anatomy.}
\PaperWOne{} \citep{ArisakaW1Symmetry} is this paper's sibling and
its opposite in method. Where the present paper takes one statement
and cuts across sixteen problems at once, that one takes seven
results --- Noether's two theorems, the Bianchi identity, its
quantized failure, Green--Schwarz, Hopkins--Singer --- in the order
they were found, and follows a single belief until it gives way. A
reader who wants to know how physics arrived at protection by topology
should read it first; a reader who wants to know what that protection
buys is already in the right place.

\paragraph{How to read any of them cold.} Every paper of the cohort
opens with an abstract, an executive summary and a preface written for
someone who has read none of the others, and closes with a falsifier
table and a Part of common objections with responses. The fastest
honest route into any of them is therefore not the front: it is the
abstract, then the falsifier table, then a decision about whether the
rest is worth the afternoon. A reader who wants to know what a paper
is risking before he learns what it is claiming has exactly the order
the cohort prefers, and this paper would rather be read that way than
politely.



\section{The honest record: where this kind of reasoning has misled
physics}
\label{sec:w2-record}

Table~\ref{tab:deployment} is the strongest thing this Part has to
offer short of an experiment, and it is also the most selection-prone.
An organizing language in physics ordinarily earns its keep by
suggesting modifications; this one earned its keep by failing to
require any --- which is either a description fitting a thing that was
already there, or an author fitting a vocabulary to his own work. The
falsifier table of \S\ref{sec:kill} exists because that difference is
not settled from an armchair. But something is owed before the reader
gets there, and it is not a falsifier. It is the other column: the
record of what \emph{this style of reasoning} has cost physics when it
was wrong, and the list of questions this particular instrument is bad
at. A paper that prints its own kill conditions and not its own
cautionary history has printed the cheaper half.

\subsection*{Three episodes, none of them flattering}

\emph{Weyl, 1918: the scale direction, gauged.} The first attempt to
unify electromagnetism with gravity made the local scale of length a
gauge freedom \citep{Weyl1918}. It was beautiful, it was
geometrically natural, and it was dead within a year: if length is
locally gauged, a clock's rate depends on the path it has traveled,
so two atoms brought together after different histories would show
different spectral lines. They do not. The objection was Einstein's
and it was fatal, and the repair that eventually worked was not a
refinement but a \emph{fiber swap} --- from a direction whose value is
physical to one whose value is not, which is the modern gauge
principle \citep{Weyl1929}.

This paper should be the last to tell that story lightly, because the
fifth dimension of Eq.~\eqref{eq:bulk} \emph{is} the scale direction.
The difference is precise and the reader is entitled to weigh it: Weyl
gave the scale direction gauge-invariant content, and here it has
none, which is exactly what premise (T) says and exactly why the
interval forgets. So the anatomy is on the right side of Weyl's
failure. But it is on the right side of it by one clause, and Weyl's
program is the closest historical precedent to this paper's central
object. Anyone who wants a reason for skepticism about a
geometrically natural claim concerning the fifth dimension has a
century-old one, and it is a good one.

\emph{Kaluza--Klein, 1926 to the 1980s: compactness expected to fix
the content.} For sixty years a great deal of work proceeded on the
expectation that compactifying extra dimensions would, by itself,
determine the low-energy spectrum: choose the manifold, turn the
crank, read off the particles. It did not, and there is no
Kaluza--Klein poverty theorem to this day --- the content depends on
choices the compactness does not make. The lesson bears directly on
how this paper should be read. ``The circle is compact, therefore its
content is quantized'' is premise (T) and is true. ``Therefore its
content is \emph{determined}'' does not follow and is not claimed
anywhere here: the ledger $(M,N,L;J,K)$ comes from Step~A's anomaly
audit, not from the circle being closed. A reader who runs the two
together has made the mistake the field spent sixty years making, and
the paper's own compressions must not invite it --- which is why
\S\ref{sec:abc} states that the numbers are sorted by the anatomy and
produced by the applications.

\emph{No elementary scalar, 1979 to 2012.} A generation of
model-building rested on a structural instinct: a fundamental scalar
cannot exist, because nothing protects its mass. The instinct was
articulated by the best people in the field
(\citealp{Susskind1979}; \citealp{tHooft1980}), it generated
technicolor and the composite-Higgs program, and it was falsified
by a scalar at $125$~GeV that behaves, so far, exactly as an
elementary one (\citealp{ATLAS2012}; \citealp{CMS2012}). This episode costs
the present framework something specific and the honest register
requires naming it. This paper's own account of the Higgs
(\S\ref{ssec:three-tiers}) is a structural argument for why an
elementary-looking scalar is \emph{allowed} --- it is the circle's
Wilson-line modulus, protected by non-locality rather than by
symmetry. That is the same genre of argument as the one that failed,
run in the opposite direction. The generalization worth carrying is
that structural claims about what nature \emph{cannot} do have a
markedly worse record than structural claims about what it
\emph{must}, and this paper makes claims of both kinds.

\subsection*{The instrument's own besetting temptation}

Every instrument has a characteristic way of going wrong, and this
one's is over-attribution: writing that a result follows from
\emph{holonomy requires closed loops, and an interval has none} when
it follows from that sentence together with something else that has
not been said. The temptation is strong precisely because the
sentence is true and elementary, so anything placed next to it
inherits its air of inevitability. Two live instances are named in
this paper rather than left for a referee.

The first is the semigroup. It is natural to write that the
interval's transports lack inverses \emph{because} the interval does
not close. They do lack inverses, but not for that reason: parallel
transport along a path in $\Ir$ is invertible, and
Appendix~\ref{app:holonomy} gauges the interval's connection away
entirely. The one-wayness is the coarse-graining flow's, and it
belongs to premise (D). \S\ref{sec:primer} now states the two
premises apart for this reason alone.

The second is the drain. It is natural to write that the interval
supplies the continuum of final states that irreversibility requires,
while the compact circle cannot. But $\Ir = [0,K]$ is a finite segment
with boundary; its spectrum is discrete under any self-adjoint
boundary condition, and its level spacing is the largest in the whole
framework, since it is the same spacing that puts the lock cluster at
$3.849$~TeV. Appendix~\ref{app:B-step3} says so in place, and rests
the fiber assignment on the composition law instead, where it belongs.

Neither correction changes a number, and that is the point worth
making: both sentences \emph{felt} like consequences of the founding
statement, and neither was one. A reader who finds a third such
sentence has found a real defect, and the author would like to be told
which paragraph.

\subsection*{What the anatomy cannot see}

The instrument sorts by two properties, exactness and one-wayness. Its
reach is bounded by that, and the boundary is not a matter of the
work being unfinished --- it is what the instrument is.

Where a question's content is neither exact nor one-way, the anatomy
supplies a filing category and no dynamics. It has nothing to say
about why the deuteron binds, or about the detailed shape of a hadron,
or about how the first galaxies assembled, or about the ordering of
masses \emph{within} a generation once the codebook's alphabet is
granted. A reader who asks the anatomy why nuclear matter saturates
will receive an answer that is correct --- the binding is an
interval-side address, priced by the descent --- and useless, because
correct filing is not a calculation. The cohort's papers do the
calculations where they exist; the anatomy says which drawer, and a
drawer is not a mechanism.

There is a second and sharper limit. The sorting is only as good as
the assignment, and in at least one place the assignment is a naming
convention rather than a derived fact: $K = 35$ is a circle-side
integer by \S\ref{sec:gearbox}'s rule, and it is also the
\emph{length} of the interval, $\Ir = [0,K]$. The gearbox reading is
consistent and is stated before it is used, but nothing in the
geometry distinguishes ``the circle's integer, used as the interval's
length'' from ``the interval has a quantized extent.'' Since the
contrast between the exact-and-protected and the flowing-and-priced is
the engine of the whole paper, the reader should know that the
interval's one exact attribute is its extent, that this is a boundary
condition rather than a holonomy, and that the paper's most important
single number sits on that seam.

\subsection*{The generalization, stated against interest}

Reading a century of these episodes together, the honest
generalization is not that geometric reasoning is unreliable. It is
narrower and less comfortable: \emph{topology read off a structure the
physics forced has a good record; topology imposed on a structure
because the topology was attractive has a poor one.} Weyl's scale
gauge was imposed. The Kaluza--Klein poverty hope was imposed. The
no-elementary-scalar instinct was imposed. Noether's theorem, the
Aharonov--Bohm phase, the Wilson loop and the index theorems were
read off.

This paper's entire defense is that its bulk belongs to the first
category --- that $\mathcal{M}_6 = \mathcal{M}_4 \times \Ir \times
\Stheta$ is forced by an axiom set declared in Step~A, before any of
the sorting in Parts~III--VI was attempted, and that the sorting was
therefore constrained rather than free. That defense is a claim about
the order in which the work was done, and a reader is entitled to be
unmoved by it. A reader who is unmoved should read Parts~III--VI as a
vocabulary rather than an anatomy, should say so plainly, and will
find that the paper's numbers are unaffected either way --- because
they were never the anatomy's, and \S\ref{sec:abc} says whose they
are.

%% ===================================================================
\subsection*{Part II in one paragraph, and what comes next}

The instrument is now on the bench. One arena,
$\mathcal{M}_6 = \mathcal{M}_4 \times \Ir \times \Stheta$, forced by
the constitution rather than chosen; one compact fiber whose
transports form a group, so that everything written on it is a
holonomy --- quantized, gauge-protected, with nowhere to move; one
open fiber whose transports form a semigroup, so that everything
written on it composes forward and never inverts. One of each,
because two circles would conserve everything and decide nothing,
and two intervals the reverse. Five integers $(M,N,L;J,K) =
(1,3,5;7,35)$ carrying the circle's arithmetic, depths along the
interval carrying the rest, and a grade vocabulary --- derived,
derived-conditional, named input, postulated, flagged --- that
every claim after this page wears in public.

Everything that follows is that one instrument, applied sixteen
times and never modified. Part~III applies it first to the theory
we trust most and understand least: the Standard Model, which
predicts to eleven decimal places, needs two dynamics to describe
one measurement, cannot say where its own gauge principle comes
from, and asks the experimenter for twenty-eight numbers it cannot
compute.

%%  PART III -- THE STANDARD MODEL
%% ===================================================================
\GGPart[The Standard Model]{The Standard Model}{Problems 1--4:
Measurement, the Four Interactions, Gauge Invariance and
Renormalization, and the Parameter Count}

\noindent The four problems of this Part are the Standard Model's
deepest embarrassments: its measurement rule is postulated, its
three gauge forces are inexplicably unalike, its defining local
symmetry is (by Noether's own second theorem) a redundancy, and its
parameter list is long enough to have launched a landscape.
All four are developed below; Problem~2 carries the fullest
quantitative apparatus.

\section{Problem 1.\ The Measurement Problem of Quantum Mechanics}
\label{sec:w2-prob1}

\subsection{The problem, in its sharpest form}

Quantum mechanics is two theories wearing one name. The first is a
jewel: amplitudes evolve deterministically, reversibly, linearly ---
the Schr\"odinger dynamics, tested to eleven decimal places. The
second is a rule of thumb: when a ``measurement'' happens, the
amplitudes collapse to one outcome, with probabilities given by the
Born rule, irreversibly. The scandal is not that the second rule
fails --- it never fails --- but that nobody can say when it
applies. What counts as a measurement? A photographic grain? A
Geiger click? A graduate student? The theory's own equations never
collapse anything: a detector coupled to a superposed atom becomes,
by the first dynamics, a superposed detector. Yet no one has ever
seen a superposed detector. For a century the escape routes have
multiplied --- consciousness, many worlds, spontaneous collapse,
hidden variables --- and the fact that all of them still have
serious advocates is the measure of the problem: the experiments
cannot yet tell them apart, and the theory cannot referee its own
foundations.

Put the question in its sharpest form: \emph{why does a reversible
theory produce irreversible events?} Any answer must explain three
things at once: why outcomes are single (one grain darkens), why
records are permanent (it stays darkened), and why the statistics
are exactly Born's (the squared amplitude, never anything else).

The scandal is sharpest when the two rules are written side by
side. Between measurements, the state $|\psi\rangle$ obeys the
Schr\"odinger equation,
\begin{equation}
i\hbar\,\frac{\partial}{\partial t}\,|\psi\rangle
\;=\; H\,|\psi\rangle ,
\label{eq:schrodinger-w2}
\end{equation}
where $H$ is the Hamiltonian --- the energy operator, the generator
of time evolution --- and $\hbar$ converts energy to phase rate.
This rule is linear, deterministic, and reversible: given the state
now, the state at any earlier or later time is fixed, and the
evolution can be run backward by $H \to -H$. At a measurement of an
observable with projectors $P_k$ (the operator asking ``is the
outcome $k$?''), the state instead obeys the L\"uders update,
\begin{equation}
|\psi\rangle \;\longrightarrow\;
\frac{P_k\,|\psi\rangle}{\sqrt{p_k}}
\qquad\text{with probability}\qquad
p_k \;=\; \langle\psi|\,P_k\,|\psi\rangle ,
\label{eq:lueders-w2}
\end{equation}
which is nonlinear (the state is renormalized), stochastic (only
the probabilities are fixed), and irreversible (the discarded
branches cannot be recovered from the survivor). The intuition to
hold: Eq.~\eqref{eq:schrodinger-w2} is transport with an inverse
and Eq.~\eqref{eq:lueders-w2} is transport without one --- a group
rule and a semigroup rule, in the vocabulary of
Eqs.~\eqref{eq:group-law-p2}--\eqref{eq:semigroup-law-p2} --- and
the measurement problem is that the textbook postulates both
without saying which geometry supplies which.

\subsection{The anatomy's answer, in one paragraph}

The two theories wearing one name are the two fibers wearing one
bulk. The coherent half of quantum mechanics --- superposition,
interference, unitarity --- is the circle's: amplitudes are
transport around $\Stheta$, their relative phases are the unread
holonomy content, and the $i$ in which they are written is on loan
from the circle (Problem~9's orientation calculus;
\citealp{ArisakaP4Real}). The registering half is the interval's: a
measurement is a \emph{write} along the fiber that composes only
forward. Nothing needs to be added to the theory --- no collapse
postulate, no observer, no consciousness --- because the boundary
law \GDOS{} already contains the write: \OGGEP{} makes registration
a projection that marginalizes the fiber coordinates, and it does so
at every event, observed or not. ``Collapse'' is not a second
dynamics; it is the runtime --- the same one-way reading that runs
cosmic history (Problem~13), met at laboratory scale. The books are
reversible; the reading is not; and a measurement is a read.

In equation form, the answer is already on the table: the
two rules of Eqs.~\eqref{eq:schrodinger-w2}--\eqref{eq:lueders-w2}
are the two terms of the \GDOS{} master equation,
Eq.~\eqref{eq:gdos-p2} --- the commutator $-\tfrac{i}{\hbar}[H,\rho]$
\emph{is} Schr\"odinger evolution, the circle's transported phase,
and the dissipator $\mathcal D[\rho]$ \emph{is} the write, the
interval-side registration of which
Eq.~\eqref{eq:lueders-w2} is the completed limit. One equation, two
fibers, no border to police: the ``cut'' is not a place but a
process --- wherever the dissipator's registration outruns the
commutator's coherence, a record forms.

\subsection{A measurement, walked through}

Follow one silver atom through a Stern--Gerlach magnet, in the
anatomy's language. \emph{Preparation} is circle-side: the atom's
spin state is a definite configuration of phase --- holonomy data,
coherent, carrying its superposition of up and down as a relative
phase that nothing on the boundary has yet asked about.
\emph{Entanglement} is still circle-side: as the atom crosses the
field gradient, its spin phase becomes correlated with its path ---
a bigger superposition, still coherent, still reversible in
principle; if one recombined the beams with sufficient care, the
interference would return, and in careful experiments (neutron
interferometers, molecule interferometers) it demonstrably does.
\emph{Registration} is the new thing: the atom strikes the plate,
and the correlation is written into the plate's collective degrees
of freedom --- a one-way transfer, priced in the currency of
Problem~14, that the runtime executes as an interval-side write.
From that event forward there is no recombination, not because the
experimenter lacks skill but because the write went down the fiber
with no inverse transport. \emph{The record} then has the two
properties every record has (Problem~13): it is \emph{real} ---
written along the fiber that carries no phase, so there is nothing
ambiguous left to interfere --- and it is \emph{permanent} --- its
content sits in quantities the circle protects.

The anatomy's slogan for the whole sequence: \emph{the circle
carries the question; the interval writes the answer.} Before the
write, phases; after the write, facts. The boundary between the
two is not the size of the apparatus, nor the presence of a mind,
but an event type --- the one-way write --- which is why a
photographic grain suffices and a galaxy of coherent atoms does
not.

Figure~\ref{fig:write-w2} draws the walk-through as the anatomy
sees it: everything to the left of the screen is the circle's
business --- both paths carried, a promise transported by a group ---
and the screen is the interval's: one grain fires, one deposit, no
inverse.

\begin{figure}[htbp]
\centering
\resizebox{0.704\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% source and magnet
\node[draw=black!55,line width=0.80pt,rounded corners,align=center] (src) at (0,0)
  {\scriptsize silver\\[-2pt]\scriptsize atom};
\node[draw=black!55,line width=0.80pt,align=center] (mag) at (2.6,0)
  {\scriptsize Stern--Gerlach\\[-2pt]\scriptsize magnet};
\draw[->,line width=0.80pt] (src) -- (mag);
% two branches (unitary region)
\draw[->,line width=0.80pt,blue!65!black] (mag.east) .. controls (5.0,0.4) .. (7.8,1.3);
\draw[->,line width=0.80pt,blue!65!black] (mag.east) .. controls (5.0,-0.4) .. (7.8,-1.3);
\node[blue!65!black] at (5.4,1.35) {\scriptsize both paths carried};
\node[blue!65!black] at (5.4,-1.45) {\scriptsize (a superposition: a promise)};
% the cut
\draw[dashed,line width=0.80pt,black!60] (8.5,-2.3) -- (8.5,2.3);
\node[black!60,rotate=90] at (8.75,0) {\scriptsize the cut $=$ the first write};
% screen
\draw[line width=1.10pt] (9.4,-2.0) -- (9.4,2.0);
\fill[red!70!black] (9.4,1.3) circle (0.09);
\node[red!70!black,right] at (9.55,1.3) {\scriptsize one grain fires};
\node[right,align=left] at (9.55,0.2)
  {\scriptsize a record: final,\\[-2pt]\scriptsize classical, priced};
% region braces
\draw[decorate,decoration={brace,mirror,amplitude=5pt}]
  (0,-2.6) -- (8.3,-2.6);
\node[align=center] at (4.15,-3.25)
  {\scriptsize \textbf{the circle carries the question}\\[-2pt]
   \scriptsize unitary, reversible --- a group};
\draw[decorate,decoration={brace,mirror,amplitude=5pt}]
  (8.7,-2.6) -- (11.6,-2.6);
\node[align=center] at (10.15,-3.25)
  {\scriptsize \textbf{the interval writes the answer}\\[-2pt]
   \scriptsize one-way, final --- a semigroup};
\end{tikzpicture}%
}
\caption[A measurement, sorted by fiber.]{\textbf{A measurement, sorted by fiber.} Left of the cut,
nothing has happened that cannot be undone: the two beams are the
circle's transported phase, a promise held open. At the screen the
runtime's projection acts --- the one-way write --- and one grain
carries the deposit. The measurement problem was the demand that one
algebra do both jobs; the anatomy hands each job to the fiber whose
algebra fits it.}
\label{fig:write-w2}
\end{figure}

\subsection{Why decoherence was almost the answer}

The modern consensus rightly credits decoherence with half the
solution: couple a superposed system to an environment, and the
relative phases leak into unmonitored correlations at fantastic
speed; interference becomes unobservable in practice; the reduced
state looks classical. The anatomy honors this --- decoherence is
real, calculable, and confirmed --- and locates precisely what it
cannot do. Decoherence \emph{delocalizes} phase; it does not
\emph{destroy} it. The global state remains a superposition; the
outcome remains plural; ``for all practical purposes'' remains the
load-bearing phrase. What finishes the job is what decoherence
lacks: an actual one-way event. In \GDOS{} the projection is not an
approximation that improves with environment size --- it is the
boundary law itself, always on, and decoherence is its microscopic
face while measurement is its record-forming face
\citep{ArisakaA8QM}. The practical-purposes gap closes because the
interval is not a practical purpose; it is a fiber.

Figure~\ref{fig:decoherence-scaling-w2} shows the
quantitative face of that ``almost'': measured and computed
decoherence rates across forty orders of magnitude.

\begin{figure}[p]
\centering
\resizebox{0.679\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
]
%% ----------- axis ranges -----------
%%   x: log10(Delta x / m)  from -10 to +2,  width 90mm,  1 decade = 7.5 mm
%%   y: log10(Gamma / s^-1) from -20 to +50, height 140mm, 1 decade = 2.0 mm
%% -----------------------------------

% axes
\draw[->, line width=0.7pt] (0,0) -- (96mm,0) node[right] {$\log_{10}(\Delta x / \mathrm{m})$};
\draw[->, line width=0.7pt] (0,0) -- (0,148mm) node[above] {$\log_{10}(\Gamma / \mathrm{s}^{-1})$};

% x-ticks: -10, -7, -4, -1, +2 (7.5 mm/decade)
\foreach \xmm/\val in {0/-10, 22.5/-7, 45/-4, 67.5/-1, 90/2} {
 \draw (\xmm mm, 0) -- (\xmm mm, -1.6mm)
 node[below, font=\scriptsize] {\val};
 \draw[line width=0.25pt, draw=black!20] (\xmm mm, 0) -- (\xmm mm, 140mm);
}

% y-ticks: -20, -10, 0, 10, 20, 30, 40, 50 (2.0 mm/decade)
\foreach \ymm/\val in {0/-20, 20/-10, 40/0, 60/10, 80/20, 100/30, 120/40, 140/50} {
 \draw (0, \ymm mm) -- (-1.6mm, \ymm mm)
 node[left, font=\scriptsize] {\val};
 \draw[line width=0.25pt, draw=black!20] (0, \ymm mm) -- (90mm, \ymm mm);
}

% Region bands (vertical)  micro / meso / macro -- boundaries at log Dx = -7 and -3
\fill[blue!8, opacity=0.55] (0, 0) rectangle (22.5mm, 140mm);
\fill[orange!8, opacity=0.55] (22.5mm, 0) rectangle (52.5mm, 140mm);
\fill[red!8, opacity=0.55] (52.5mm, 0) rectangle (90mm, 140mm);

\node[font=\scriptsize\itshape, text=blue!70!black, anchor=north]
 at (11.25mm, 138mm) {microscopic};
\node[font=\scriptsize\itshape, text=orange!70!black, anchor=north]
 at (37.5mm, 138mm) {mesoscopic};
\node[font=\scriptsize\itshape, text=red!70!black, anchor=north]
 at (71.25mm, 138mm) {macroscopic};

% Macroscopic-instantaneous horizontal threshold at log Gamma = 40 -> y = 120mm
\draw[line width=0.9pt, draw=red!70!black, dashed] (0, 120mm) -- (90mm, 120mm);
\node[font=\scriptsize, text=red!70!black, anchor=east] at (89mm, 123mm)
 {$\Gamma = 10^{40}\,\text{s}^{-1}$ (operationally instantaneous)};

% ----- Three coupling curves, slope = 2 in log-log -----
% Strong coupling: anchor (-1, 40);  at -10 -> 22 -> y=84mm;  at +2 -> 46 -> y=132mm
\draw[line width=1.1pt, draw=red!75!black]
 (0mm, 84mm) -- (90mm, 132mm);
\node[font=\scriptsize, text=red!75!black, anchor=south]
 at (45mm, 110mm) {strong: $\Gamma \propto (\Delta x)^2$};
\node[font=\scriptsize, text=red!75!black, anchor=south]
 at (45mm, 103mm) {(thermal bath, room~T)};

% Medium coupling: anchor (-7, 12);  at -10 -> 6 -> y=52mm;  at +2 -> 30 -> y=100mm
\draw[line width=1.1pt, draw=orange!75!black]
 (0mm, 52mm) -- (90mm, 100mm);
\node[font=\scriptsize, text=orange!75!black, anchor=south]
 at (60mm, 84mm) {medium};

% Weak coupling: anchor (-2, 0);  at -10 -> -16 -> y=8mm;  at +2 -> 8 -> y=56mm
\draw[line width=1.1pt, draw=blue!75!black]
 (0mm, 8mm) -- (90mm, 56mm);
\node[font=\scriptsize, text=blue!75!black, anchor=south]
 at (60mm, 41mm) {weak (low-T dust)};

% ----- Empirical anchors (markers + labels) -----
% C_60 fullerene at room T:  (-7, 12)  ->  (22.5mm, 64mm)  on medium curve
\fill[black] (22.5mm, 64mm) circle (1.0mm);
\node[font=\scriptsize, anchor=west] at (24mm, 64mm)
 {$C_{60}$ at $T=300$~K \;\citep{Arndt1999}};

% Biological cell:  (-5, 25)  ->  (37.5mm, 90mm)
\fill[black] (37.5mm, 90mm) circle (1.0mm);
\node[font=\scriptsize, anchor=west, align=left] at (39mm, 90mm)
 {biological cell \\ ($\sim 10\,\mu$m, $T_{\rm body}$)};

% Schroedinger cat:  (-1, 40)  ->  (67.5mm, 120mm)  on threshold + strong line
\fill[black] (67.5mm, 120mm) circle (1.0mm);
% Schroedinger cat label: LOWER-RIGHT of the dot, UNDER the dashed line
\node[font=\scriptsize, anchor=west] at (70mm, 115mm)
 {Schr\"odinger cat};

% Dust grain at low T (weak curve, mesoscopic):  (-5, -6)  ->  (37.5mm, 28mm)
\fill[black!70] (37.5mm, 28mm) circle (0.75mm);
\node[font=\scriptsize, anchor=west] at (39mm, 28mm)
 {dust grain at $T=10$~K};

% Single-atom interferometer:  (-10, 22)  ->  (0, 84mm)  on strong line edge
\fill[black!70] (0mm, 84mm) circle (0.75mm);
\node[font=\scriptsize, anchor=south west] at (1mm, 85mm)
 {single-atom limit};

\end{tikzpicture}}
\caption[Decoherence is quantitative, not interpretive.]{\textbf{Decoherence is quantitative, not
interpretive} (reproduced from \citealp{ArisakaA8QM}). The
off-diagonal damping rate $\Gamma$ (vertical, powers of ten per
second) against the position-separation $\Delta x$ of the
superposed alternatives (horizontal, meters), log--log, for
standard environments. $\Gamma$ scales as $(\Delta x)^2$ until
saturation; a dust grain superposed over a micron decoheres in
$\sim 10^{-24}$~s while an electron superposed over an Angstrom
can hold coherence for seconds --- forty orders of magnitude from
one slope. On the anatomy this is the dissipator of
Eq.~\eqref{eq:gdos-p2} doing its bookkeeping: the rate at which
the interval's registration outruns the circle's coherence, as a
function of how loudly the superposition differs. What the slope
does not do --- and cannot do --- is pick the grain's final
position: that is Eq.~\eqref{eq:lueders-w2}'s job, and the
distinction is this Problem's entire content.}
\label{fig:decoherence-scaling-w2}
\end{figure}

\subsection{Where the Born rule comes from}

Probability is the deepest part of the scandal: even granting
single outcomes, why the squared amplitude? The cohort's answer is
carried at a carefully stated level \citep{ArisakaA8QM}: the Born
measure is not postulated but \emph{produced} --- the boundary
law's own dissipation relaxes the bulk's configurational statistics
toward the Born measure, monotonically, at a rate set by the port
coupling; the standard quantum postulates (Hilbert space, unitary
limit, Born form, L\"uders conditioning) then follow as theorems of
the relaxed runtime. The chain is closed at the relaxation level,
with exactly one named analytic residue --- the field-grade
representation step --- declared open, and the reader deserves the
grade stated as plainly as the claim: \emph{derived-conditional,
one residue named}. What makes this more than an interpretation is
that relaxation takes time, and time leaves fossils --- the next
section's laboratory face.

\subsection{The laboratory face}

Three exposures make Problem~1 physics rather than philosophy.
\emph{First, the relic (F25).} If Born statistics are relaxed into
rather than axiomatic, then modes that left the cosmological
horizon during inflation decoupled from the relaxing stream and
froze whatever deviation remained. The largest observable scales
should therefore carry a primordial-statistics deficit of computed
form,
\begin{equation}
\xi(k) \;=\; 1 - A\left(\frac{k}{k_0}\right)^{-\gamma},
\label{eq:f25-w2}
\end{equation}
where $\xi(k)$ measures how completely Born statistics hold at
comoving wavenumber $k$ ($\xi = 1$: exact Born), $k_0$ is the pivot
scale set by the mode that left the horizon as relaxation
completed, $A$ is the residual deficit amplitude at the pivot, and
$\gamma > 0$ is the relaxation exponent --- both $A$ and $\gamma$
computable from the port coupling, with no new field and no dial.
The intuition: modes that exited earliest sampled the least-relaxed
statistics, so the deficit grows toward the largest scales as the
power law records --- a fossil of the pre-relaxation era, printed
on the sky. A detected large-scale
statistical anomaly incompatible with the computed form kills the
relaxation account; exact Born statistics at all scales bounds the
relaxation time from below. (The quantification of $A$ and
$\gamma$ is a named cohort campaign.) \emph{Second, the
no-reversal bound.} The account forbids, at any scale, the
resurrection of a registered outcome --- not suppressed:
forbidden. Every interference-revival experiment that pushes the
macroscopic frontier (levitated spheres, mechanical resonators) is
a test; a genuine revival of a registered record would falsify the
write outright. \emph{Third, the cost.} Every write is billed ---
the information-theoretic price of Problem~14 --- so the account
requires that no registration ever be free, an audit the
information pillar carries \citep{ArisakaA7Qi4}.

\subsection{Schr\"odinger's cat, dissected}

The famous cat sharpens every clause. An unstable atom (circle-side:
a phase configuration with two channels), a Geiger counter, a vial,
a cat, a box. The textbook telling has the whole chain --- atom,
counter, vial, cat --- in superposition until an observer opens the
box, at which point reality decides. The anatomy's telling has one
question only: \emph{where is the first one-way write?} And the
answer is: at the counter --- the avalanche discharge is a
registration, an interval-side write billed and executed long before
vial, cat, or observer. From that event the record exists; the cat's
fate is a fact of the world with the box still shut; and opening the
box changes the observer's knowledge, never the cat's state. The
thought experiment retains exactly one deep lesson, which the
anatomy states without paradox: between the atom's phases and the
counter's click there is a genuine transition of category ---
question to answer, circle to interval --- and the line sits at an
event type, not at a mind, a size, or a species. Nothing macroscopic
was ever superposed, because nothing macroscopic gets assembled
without a torrent of writes along the way; a ``superposed detector''
is not hard to build --- it is a contradiction in the bookkeeping,
like a spent coin still in the purse.

Figure~\ref{fig:cat-timeline-w2} locates the write on the
apparatus's own timeline.

\begin{figure}[H]
\centering
\resizebox{0.95\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 >=Stealth,
 every node/.style={align=center, inner sep=2pt},
 bulktrack/.style ={line width=1.10pt, draw=red!75!black},
 bndtrack/.style ={line width=1.10pt, draw=blue!70!black},
 bulkev/.style ={rectangle, draw=red!75!black, fill=red!10,
 line width=0.6pt, minimum width=24mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 bndev/.style ={rectangle, draw=blue!70!black, fill=blue!10,
 line width=0.6pt, minimum width=24mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 proj/.style ={rectangle, draw=violet!70!black, fill=violet!18,
 line width=1.10pt, minimum width=30mm, minimum height=14mm,
 rounded corners=3pt, font=\scriptsize\bfseries},
 classical/.style ={rectangle, draw=green!60!black, fill=green!10,
 line width=0.6pt, minimum width=22mm, minimum height=10mm,
 rounded corners=2pt, font=\scriptsize},
 arrow/.style ={->, line width=0.8pt, draw=black!75},
 label/.style ={font=\scriptsize\itshape, anchor=south},
]
% =====================================================================
% TIME AXIS (horizontal)
% =====================================================================
\draw[->, line width=0.7pt] (-4mm, -22mm) -- (162mm, -22mm) node[right] {$t$ (4D proper time)};
\foreach \x/\lab in {0/{$t_0$}, 30mm/{$t_{\rm decay}$}, 70mm/{$t_{\rm Geiger}$},
 105mm/{$t_{\rm cat~dies}$}, 140mm/{$t_{\rm open}$}} {
 \draw (\x, -22mm) -- (\x, -23.5mm) node[below, font=\scriptsize] {\lab};
}

% =====================================================================
% TOP TRACK: 6D bulk (deterministic, unitary)
% =====================================================================
\node[label] at (0, 28mm) {6D bulk: deterministic and unitary};
\draw[bulktrack] (0, 22mm) -- (155mm, 22mm);

\node[bulkev] at (0, 22mm) {atom \\ $\Theta_H = \alpha$};
\node[bulkev] at (40mm, 22mm) {$\to$ daughter \\ + $\gamma$ quantum};
\node[bulkev] at (80mm, 22mm) {$\gamma$ propagates \\ in 6D};
\node[bulkev] at (120mm, 22mm) {bulk continues \\ unitarily};

% =====================================================================
% BOTTOM TRACK: 4D boundary (projection at Geiger trigger)
% =====================================================================
\node[label] at (0, 0mm) {4D boundary: projection at Geiger trigger};
\draw[bndtrack] (0, -6mm) -- (155mm, -6mm);

% pre-trigger
\node[bndev] at (0, -6mm) {atom $+$ Geiger \\ in superposition};
\node[bndev] at (40mm, -6mm) {(no 4D event yet) \\ \scriptsize gamma propagating};

% THE PROJECTION EVENT (the user's emphasis)
\node[proj] at (70mm, -6mm) {\GDOS{}/\OGGEP{} \\ \footnotesize PROJECTION \\ \footnotesize \emph{COMPLETES HERE}};

% post-trigger: classical mechanics
\node[classical] at (105mm, -6mm) {hammer falls \\ cyanide releases \\ \scriptsize \emph{(classical)}};
\node[classical] at (140mm, -6mm) {human opens box \\ \scriptsize sees DEAD cat \\ \scriptsize \emph{(2nd projection,} \\ \scriptsize \emph{redundant)}};

% Annotation arrow
\node[font=\scriptsize\itshape, anchor=south, text=violet!70!black] at (70mm, 10mm)
 {The cat's fate is determined CLASSICALLY at this 4D event,};
\node[font=\scriptsize\itshape, anchor=south, text=violet!70!black] at (70mm, 5mm)
 {NOT when the human eventually opens the box.};

% Arrow connecting bulk gamma to Geiger projection
\draw[arrow, draw=red!75!black] (80mm, 16mm) -- (70mm, 0mm);

% =====================================================================
% Decoherence-rate annotation
% =====================================================================
\node[font=\scriptsize, anchor=north, align=center] at (75mm, -28mm)
 {Decoherence rate of the Geiger-counter pointer-basis off-diagonal: \\
 $\Gamma_{LD} \gtrsim n_{\rm air}\,\sigma\,v\,(\Delta x)^2/\lambda_{\rm dB}^2 \;\sim\; 10^{40}\,\mathrm{s}^{-1}$ \;
 (Joos--Zeh, room T); \;
 \emph{the live-vs-dead off-diagonal damps in $10^{-40}$~s.}};

\end{tikzpicture}}
\caption[Where and when the cat's fate is written.]{\textbf{Where and when the cat's fate is
written} (reproduced from \citealp{ArisakaA8QM}). The timeline of
the apparatus: the nuclear superposition is coherent only until the
Geiger counter's avalanche --- the first thermodynamically
irreversible amplification --- at which point the write of
Eq.~\eqref{eq:lueders-w2} has already happened, microseconds into
the hour and meters from the cat. Everything downstream (hammer,
vial, cat) is classical cause and effect among records. The
paradox's force always came from placing the write at the box's
opening; the dissipator's timescales place it at the counter, and
the superposed-cat state is never prepared at all.}
\label{fig:cat-timeline-w2}
\end{figure}

\subsection{Frequently asked}

\paragraph{Doesn't the write violate unitarity?} The \emph{books}
stay unitary --- the circle's half is exact, and the global
accounting (Problem~16 runs it across a black-hole horizon) always
balances. What is not unitary is the \emph{reading}: the boundary's
open-system evolution, which was never claimed to be reversible.
The contradiction the textbooks fear arises only if one insists the
reading be described by the books' own dynamics --- which is
precisely the one-fiber assumption the anatomy retires.

\paragraph{Where exactly is the cut?} At the first uncompensated
one-way transfer --- operationally, where the interference budget
of Problem~14 is spent and cannot be repaid. This is a physical
criterion, not a convention: it is why interferometers can revive
coherence for molecules of thousands of atoms (no write yet) and
why a single darkened grain cannot be undarkened by any unitary
trick (write executed). The cut moves with experimental skill
exactly as far as the write can be deferred, and not one step
further.

\paragraph{Is the observer needed at all?} No. The projection is
always on --- decoherence in the atmosphere, registration in a
rock's crystal defects, the cosmic writes of Problem~13 --- and
observers are late, local beneficiaries of a bookkeeping that has
run since the launch. What observers contribute is what readers
always contribute: nothing to the text, everything to the
readership.

\subsection{What the anatomy does not claim}

Clean accounts of measurement attract enthusiastic misreadings,
so the fences deserve a section. \emph{No role for consciousness}:
the write is an event type, executed identically in detectors,
rocks, and interstellar dust; minds read ledgers, they do not
close them. \emph{No retrocausality}: delayed-choice experiments,
correctly parsed, delay the \emph{configuration of the question}
(circle-side, revisable) and never a completed write; nothing in
the anatomy reaches backward, because nothing can --- the fiber
that would carry the influence has no inverse transport.
\emph{No signaling from entanglement}: Bell correlations are
real, and the anatomy reads them as class data --- joint
holonomy structure prepared once and carried apart, correlations
without communication, exactly as Horizon~2 of Problem~8 reads
cosmic sameness. What Bell's theorem forbids is a local hidden
story \emph{on the boundary}; what it cannot forbid is bookkeeping
in the fibers the boundary does not see --- and no write anywhere
in a Bell experiment transmits: the outcomes are registered
locally, the correlation is discovered only when the records are
brought together, along ordinary one-way channels. \emph{No new
constants}: unlike spontaneous-collapse models, the write's rate
is not a dial --- it is the port coupling the boundary law already
owns. The account's entire novelty is an anatomy; its entire
freedom is zero.

\subsection{The interpretations, honored and placed}

No polemic; each tradition holds a piece. \emph{Copenhagen} was
right that measurement is special and classical records are final
--- it lacked the geometry that says why, and so drew the
quantum-classical cut as a moving convention rather than a fiber
distinction. \emph{Everett} was right that the unitary books never
collapse --- the circle's truth --- and the anatomy differs from
many-worlds not by denying the reversible half but by asserting a
physical second fiber on which reading happens once; branches, on
this view, are bookkeeping of the unread, not inhabited worlds.
\emph{Spontaneous-collapse models} (GRW and kin) were right that
irreversibility must be objective and dynamical --- they built the
one-way event by hand, with new constants; the anatomy gets it from a
compulsory projection onto a fiber that has nothing to protect
(premises (T) and (D) of \S\ref{sec:primer}), with no new constants at
all. The economy is a real difference and the honest register requires
saying what it is not: an always-on boundary projection is a
structural postulate too, and its warrant here is only that the
geometry forced it in Step~B before the measurement problem was in
view. \emph{Decoherence theory} was right
about the mechanism's face and, as above, half its content. The
measurement problem, on the anatomy, was never a puzzle about
minds or worlds: it was the two-fiber anatomy, encountered in a
laboratory before it had been named --- the oldest and most
domestic instance of the circle that remembers meeting the
interval that forgets.

\textbf{Grade and kill, consolidated.} The write and the record:
theorems of the boundary law. The Born chain:
derived-conditional, one residue named. Kill: the F25 relic off
its computed form; a revived registered record; a free write. The
detailed record discussion is \S\ref{sec:measurement}; the cost
audit is \citep{ArisakaA7Qi4}; the parent treatment end to end is
\citep{ArisakaA8QM}.

Measurement was the circle and the interval caught in the act of
doing two different jobs at one moment. The next Problem catches
them doing two different jobs across the whole of physics: three of
the four interactions are the circle registering, and the fourth is
the interval acting.

\section{Problem 2.\ A Distinct Difference between the Strong, Weak, and Electromagnetic Interactions --- and Gravity}
\label{sec:w2-prob2}

\subsection{Three forces, three relations to the anatomy}
\label{sec:three-forces}

The standard model presents its three gauge interactions as three
instances of one principle, differing in group and coupling. The
differences a student actually meets --- one force is long-range and
gentle, one confines absolutely, one is short-range, chiral, and
alone changes what things \emph{are} --- arrive as outputs of
detailed dynamics: a massless photon here, a running coupling there,
a Higgs mechanism elsewhere. On the two-fiber anatomy the three
differences are three \emph{relations to the fibers}, visible before
any dynamics is solved
(\citealp{ArisakaA9SM}; \citealp{ArisakaP6Proton}; \citealp{ArisakaA6Gi4}):

\begin{quote}
\emph{Electromagnetism is the circle's open phase; the strong force
is the circle's closing figure; the weak force is the interval's
agent.}
\end{quote}

Table~\ref{tab:forces} is the Part in one view; the three sections
that follow earn its rows.

\begin{table}[htbp]
\centering
\caption[The three gauge interactions by fiber.]{The three gauge interactions by fiber. Each row is
established in the cited papers; the alignment is the anatomy's. The seven rows compare the three interactions on range, on what each conserves, on chirality, on whether either can change identity, on the signature constant each carries --- $\alpha^{-1}(0) = 2^J + N^2 = 137$, the triality $N = 3$, and $\sin^2\theta_W = L^2/(4N^3) = 25/108$ --- and on the fate of each symmetry. Only the weak interaction is chiral, and only the weak interaction changes flavor.}
\label{tab:forces}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\textwidth}{@{}P{2.9cm}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X@{}}
\toprule
& \textbf{Electromagnetic} & \textbf{Strong} & \textbf{Weak} \\
\midrule
Relation to anatomy & the circle's unbroken phase & the circle's closing triangle & the interval's agent \\
Range & infinite: the phase is free to reach & zero at large distance: only closed books leave & short: the agent is massive, $\sim\Mlock$-fed scale \\
What it conserves & every winding: charge exact & triality inside closure; $B$ via the ledger & least: it is the changer, not the keeper \\
Chirality & blind & blind & \textbf{chiral}: the descent selects a hand \\
Changes identity? & never & never & \textbf{only} force that does: flavor transitions \\
Signature constant & $\alpha^{-1}(0) = 2^J + N^2 = 137$ & $N = 3$: the triality & $\sin^2\theta_W = L^2/(4N^3) = 25/108$ \\
Fate of its symmetry & unbroken & unbroken (confining) & broken --- by reading (Wilson-line vev) \\
\bottomrule
\end{tabularx}
\end{table}

One equation carries the entire sorting. Every interaction of
the Standard Model enters through the same object, the covariant
derivative acting on a matter field $\psi$:
\begin{equation}
D_\mu \;=\; \partial_\mu
\;-\; i g_s\, G_\mu^a \tfrac{\lambda^a}{2}
\;-\; i g\, W_\mu^i \tfrac{\tau^i}{2}
\;-\; i g'\, B_\mu \tfrac{Y}{2},
\label{eq:covd-w2}
\end{equation}
where $G_\mu^a$ ($a = 1..8$) are the gluon fields with coupling
$g_s$ and the $\lambda^a$ the color matrices; $W_\mu^i$
($i = 1..3$) the weak fields with coupling $g$ and the $\tau^i$ the
weak-isospin matrices; $B_\mu$ the hypercharge field with coupling
$g'$ and $Y$ the hypercharge of $\psi$. The physical content:
``interacting'' \emph{means} being transported --- a particle
feels a force exactly insofar as its phase is geared to a
connection --- and the three gearings differ only in which matrices
multiply which fields. On the anatomy,
Eq.~\eqref{eq:covd-w2} is the boundary registration of the single
composite connection of Eq.~\eqref{eq:group-p2}: one object in the
bulk, three gearings on the stage, and this Problem's question ---
why the three behave so differently --- becomes: what does each
gearing's structure look like when read on the two fibers?

\subsection{Electromagnetism: the circle's open phase}
\label{sec:em}

The photon is the connection of the abelian phase itself --- not a
section of an associated bundle but the transporter, which is why it
carries no charge of its own \citep{ArisakaP4Real}. Its symmetry is
the one gauge factor the descent never breaks, because breaking it
would require the circle's basic winding to fail as a ledger --- and
holonomy classes do not fail. Hence the three textbook miracles of
electromagnetism, one clause each: \emph{massless photon} (an
unbroken phase has no scale to wear); \emph{infinite range} (a
massless transporter reaches); \emph{exact charge quantization}
(charge is a winding number, and the fine-structure constant's
closed form $\alpha^{-1}(0) = 2^J + N^2 = 137$ at leading order is
the ledger speaking through the phase \citep{ArisakaA9SM}).
Electromagnetism is what the fiber that remembers looks like when it
simply \emph{holds}: the conservative force par excellence, the
keeper of the books, chiral-blind because a memory has no direction
of reading.

\subsection{The strong force: the closing figure}
\label{sec:strong}

Problem~12 develops the full story: the color sector is the circle's triangle,
and its confining character is the registration rule --- only
trivial circuits leave the fiber. What the standard presentation
calls the strong force's strangeness --- absolute confinement, no
long-range field, the spectrum made only of singlets --- is the
visible half of a bookkeeping statement. Its conserved quantities
are closures (triality inside singlets; baryon number through the
$\UoneB$ ledger); its effective range is the size of the closure,
not of a Yukawa tail; and it, too, is chiral-blind --- a closure
does not care in which sense it was drawn. Between them, the
electromagnetic and strong sectors exhaust what the circle does
with content: \emph{hold it open} (the phase, reaching) or
\emph{close it} (the triangle, confining). Neither can change what
a thing is; keepers never do.

The strong coupling's address-dependence is worth one
explicit equation. At one loop,
\begin{equation}
\alpha_s(\mu) \;=\;
\frac{\alpha_s(\mu_0)}
{1 + \dfrac{b_0}{2\pi}\,\alpha_s(\mu_0)\,\ln(\mu/\mu_0)},
\qquad
b_0 = 11 - \tfrac{2}{3}n_f ,
\label{eq:running-w2}
\end{equation}
where $\mu$ is the probe energy (the address), $\alpha_s(\mu_0)$
the reading at a reference address $\mu_0$, $n_f$ the number of
quark flavors light enough to run in loops, and $b_0$ the
one-loop coefficient --- positive for QCD ($n_f \le 16$), which is
the entire drama: as $\mu$ falls (deeper down the interval),
the denominator shrinks and $\alpha_s$ grows, until the
perturbative reading fails and the closure rule of Problem~12
takes over. The intuition, stripped of metaphor: a coupling is not
a constant of nature but a boundary reading indexed by scale;
Eq.~\eqref{eq:running-w2} is the reading's dependence on the
address, its one-way $\ln\mu$ the interval's coordinate
$r = \ln(\EGUT/\mu)$ in laboratory units.

\subsection{The weak force: the interval's agent}
\label{sec:weak}

The weak interaction is different in kind, and every one of its
peculiarities --- the full list that puzzles the student --- is the
signature of one fact: \emph{it is the interaction by which the
runtime acts}. The \GRIP{} cascade's realization in nature is the
weak sector \citep{ArisakaA6Gi4}; the descent's transitions ---
every change of stratum, every decay toward a deeper basin ---
are executed by it. Read the peculiarities off that identity:

\begin{itemize}[leftmargin=2.0em]
\item \emph{It is the only force that changes identity} ---
flavor transitions, the muon's descent to the electron, the
neutron's to the proton --- because it is the mover along the
ledger: the agent of \GRIPin{} relaxation
\citep{ArisakaP3Electron}. The keepers conserve; the agent
delivers.
\item \emph{It is chiral} --- the one force that tells left from
right --- because it is the one force aligned with a fiber that has
a direction. A keeper on the circle has no sense of reading; the
agent of a one-way descent does. Parity violation, on the anatomy,
is not a scandal but a signature: \emph{the hand of the read head}.
\item \emph{It is short-range and its symmetry is broken} because
the agent's gauge sector takes its scale from the reading itself:
the Wilson-line vacuum \citep{Hosotani1983} --- a circle-side
label --- sets
the electroweak vev at an interval-fed address,
$v = \Mlock(2/L)^N = 246$~GeV \citep{ArisakaA9SM}: the gearbox
grammar of Part~II, once more.
\item \emph{Its violations cluster} --- $P$, $C$, $CP$, flavor
--- because violation is the agent's job description: the sector
through which the interval acts is exactly where the interval's
one-wayness and the tape's chirality become visible in scattering
amplitudes. Problem~11's excess was executed here; Problem~13's descent
rate (the muon lifetime as a \GRIPin{} observable) is timed here.
\end{itemize}

The classic question ``why is the weak interaction so odd?''
\citep{ArisakaA9SM} thus receives a structural answer: it is not an
odd member of the keepers' family; it is not a keeper at all.
\emph{Two forces are the circle's, and keep; one force is the
interval's, and changes.} The deep two-fold division of physics ---
conservation and transition, being and becoming --- runs straight
through the middle of the standard model's table of forces, and the
table has been telling us so all along.

Quantitatively, the agent's feebleness is an address, not a
coupling. The measured strength of low-energy weak processes is
Fermi's constant,
\begin{equation}
\frac{G_F}{\sqrt 2} \;=\; \frac{g^2}{8\,M_W^2},
\qquad
M_W = \tfrac12\, g\, v,
\qquad
v \;=\; \Mlock\!\left(\tfrac{2}{L}\right)^{N} = 246~\mathrm{GeV},
\label{eq:fermi-w2}
\end{equation}
where $G_F = 1.166\e{-5}~\mathrm{GeV}^{-2}$ sets beta-decay rates,
$M_W$ is the $W$ boson's mass, and $v$ the electroweak vacuum value
--- the chain of Eq.~\eqref{eq:vev-p2}. Read right to left: the
gauge coupling $g$ is unremarkable ($g \simeq 0.65$, the same order
as electromagnetism's); what makes the weak force \emph{weak} is
$M_W^2$ in the denominator --- the agent operates from a massive
address, $v$, which the anatomy derives as an interval depth
dressed by a circle fraction. The force is not feeble; it is
\emph{far}: suppressed by the square of the distance down the
descent at which its gauge sector froze.

\subsection{The two abelian companions: $\UoneB$ and $\UoneQi$}
\label{sec:two-u1s}

The gauge group of Step~A, Eq.~\eqref{eq:group-p2}, carries
alongside the unified $\mathrm{SU}(5)$ two abelian factors that this Part's three-force
story has used without introducing, and the anatomy assigns each
its character. $\UoneB$ --- the baryonic circle --- is the
\emph{ledger} factor: it gauges the bookkeeping of Problem~12,
counting one unit per revolution, and its boundary residue is not
a new long-range force but the integrality of the guards
($Q + B \in \bZ$); its heavy vector is a collider target of the
lock cluster, not a fifth force. $\UoneQi$ --- the informational
circle --- is the \emph{registration} factor: the unit in which
the boundary's records are billed, one unit per registered
rearrangement, with the Arisaka constant $Q_B$ the exchange rate
between the circle's two currencies --- energy per winding and
energy per qubit --- fixed by ledger integers rather than by any
bath temperature \citep{ArisakaA7Qi4}. Their joint lesson
completes the census of Problem~2: of the four circle factors the
bulk carries, one reaches (the electromagnetic phase), one closes
(color), one counts (baryon), and one bills (information) ---
four jobs, four factors, no vacancy and no spare. The anatomy has
no room for a fifth force, and says so falsifiably: any confirmed
long-range force beyond gravity and electromagnetism would
require a circle factor the census does not contain.

\subsection{Gravity, the fourth}
\label{sec:gravity-w2}

The three gauge interactions do not exhaust the forces, and the
anatomy owes the fourth a paragraph --- especially since
Step~A of the program (\S\ref{sec:stepA}) unifies gravity
\emph{with} the gauge sector as one composite connection. On the anatomy, gravity
is the interaction of the \emph{base and the interval}: its
boundary strength is not a winding but an integral,
$G_N^{-1} \propto \pi\,(1 - e^{-2K})\,M_6^2$ --- the warp volume of
the descent \citep{ArisakaA4Euclid} --- which is the structural
reason for its two ancient peculiarities. \emph{Gravity is weak}
because the interval integrates: the coupling is diluted by the
whole depth of the throat, $e^{-2K}$-suppressed contributions and
all, rather than being pinned by a quantized winding --- and the
interval integrates because it cannot hold. And \emph{gravity alone is
inseparable from irreversibility} --- horizons, entropy bounds,
the thermodynamics that no other force exhibits --- because it
alone is bookkept on the fiber that forgets: black-hole
thermodynamics (the anatomy's most violent seam,
\citealp{ArisakaC3BH}) is what the interval's ledger looks like
when curvature concentrates it. The four forces thus complete the
sorting of Problem~2 with a $2{+}1{+}1$ anatomy: two keepers on the
circle, one agent on the interval --- and one integrator, gravity,
which is not a player on the tape at all but the pricing of the
playback itself.

The sorting is compact enough to draw.
Figure~\ref{fig:four-forces} places the four interactions on the
anatomy: two that live by the circle, one that acts for the
interval, and one that \emph{is} the interval's geometry.

\begin{figure}[htbp]
\centering
\resizebox{0.71\textwidth}{!}{%
\usebox{\GGplateForces}%
}
\caption[Four interactions, two fibers.]{\textbf{Four interactions, two fibers.} Strong and
electromagnetic: circle-side --- their charges are holonomy classes,
which is why they are exact and unbroken. The weak force: the
interval's agent --- the \GRIP{} cascade realized, which is why it
alone is chiral, broken, and mass-giving. Gravity: not a force on
the stage but the stage's warp along the descent --- why its
constant alone is tied to irreversibility, and why it will not
quantize like the other three. The century-old puzzle of the four
interactions' unlike characters is this sorting, unread.}
\label{fig:four-forces}
\end{figure}

One number states gravity's foreignness. The three gauge
couplings $g_s, g, g'$ of Eq.~\eqref{eq:covd-w2} are pure numbers;
gravity's coupling is not:
\begin{equation}
\Mplred \;=\; (8\pi G_N)^{-1/2} \;=\; 2.435\e{18}~\mathrm{GeV},
\label{eq:mpl-w2}
\end{equation}
the reduced Planck mass, where $G_N$ is Newton's constant. A
dimensionful coupling means the interaction's strength is not a
reading at an address but a \emph{conversion rate between
addresses} --- and that is exactly the anatomy's assignment:
$G_N$ is the interval's own constant (Step~A of Part~II), the
residue of the integrated descent, which is why gravity couples to
everything on the stage, cannot be screened, and does not
renormalize like the other three. The failure of fifty years of
graviton-by-graviton quantization is, on this reading, a category
diagnosis: the other three forces are circle-side gearings and
quantize as phases do; gravity is the interval's bookkeeping, and
its quantum is priced in writes, not windings.

\subsection{The lock, the hierarchy, and the one unification}
\label{sec:lock-unification}

Problem~2's three-way division leaves a question the standard model
cannot even pose cleanly: why do the three interactions meet ---
why is there one gauge structure at the top at all, and why is the
weak scale where it is? The anatomy's answers close the Part.

\paragraph{One structure at the top.} The bulk's gauge content is
not assembled from three factors; it is one composite connection
--- $\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times \UoneB \times
\UoneQi$ --- forced by the constitutional lemmas
(\citealp{ArisakaA2GG6}; \citealp{ArisakaA9SM}), and the three boundary
interactions of this Part are its three registration modes: the
phase held open, the triangle closed, the agent engaged. The
$5{:}2$ trace-ratio lock and the anomaly books of \BIsix{} are the
arithmetic face of that unity: the same ledger that balances the
circle's accounts fixes the relative normalizations the boundary
measures as $\sin^2\theta_W = L^2/(4N^3) = 25/108 = 0.2315$ and
$\alpha^{-1}(0) = 2^J + N^2 = 137$ \citep{ArisakaA9SM}.

\paragraph{The hierarchy is a depth, not a fine-tuning.} The weak
scale sits seventeen orders below Planck because the agent's
address is at the bottom of the throat: $\Mlock = \tfrac12
M_{\mathrm{Pl}} e^{-K} = 3.849$~TeV --- thirty-five e-folds of
descent, the exponent a ledger integer --- and the electroweak vev
is the gearbox form once more, $v = \Mlock (2/L)^N = 246$~GeV: a
circle label dressing an interval depth. The hierarchy problem
--- \emph{why is gravity so weak? why is the weak scale so low?}
--- is not solved by a symmetry protecting a small number; it is
dissolved by an address: nothing is small, something is
\emph{deep}, and the depth is counted, not tuned.

\paragraph{The falsifiable face.} The lock is not bookkeeping: the
first Kaluza--Klein cluster sits at the address, $3.849$~TeV, with
a predicted fingerprint --- spacing, spin content, branching
pattern locked by the same integers --- and the collider companion
stakes the discovery calendar on it \citep{ArisakaP1KK}. A
detection would read both fibers in one resonance: identity from
the circle, address from the interval. Its absence, at the
staked sensitivity, kills the address --- and with it this
section.

The unity claimed in the first paragraph has a picture, and it is
the most directly falsifiable one in this Part.
Figure~\ref{fig:a9-unification-w2} runs the three inverse couplings
up the radial coordinate: they meet, at one point, with boundary
values that were not fitted \citep{ArisakaA9SM}. On the anatomy the
picture reads in two registers at once. The three \emph{starting}
values at $M_Z$ are circle-side --- rational partitions of one
trace, $\sin^2\theta_W = 25/108$ and
$\alpha_3^{-1} = \pi e$, quantities with nowhere to move. The three
\emph{lines} are interval-side --- running is motion along $\Ir$,
which is why they have slopes at all (Problem~3). That the lines
converge is the two fibers agreeing: the circle fixes where the
lines start, the interval carries them, and the meeting point is
not a third input.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.92\textwidth]{gg_unification_showcase.pdf}
\caption[\GGTheory{} gauge-coupling unification across the 6D bulk.]{\textbf{\GGTheory{}
gauge-coupling unification across the 6D bulk} (reproduced from
\citealp{ArisakaA9SM}). Inverse couplings $\alpha_1^{-1}$ (green,
hypercharge in $SU(5)$ normalization), $\alpha_2^{-1}$ (blue,
$SU(2)_L$) and $\alpha_3^{-1}$ (coral, $SU(3)_C$) against the
renormalization scale $\mu$ and the radial coordinate
$r = \ln(E_{\rm GUT}/\mu)$ --- the interval's own coordinate. The
boundary values at $M_Z$ are the circle-side closed forms
displayed in the panel; no continuous fitted parameter intervenes
anywhere between $M_Z$ and $E_{\rm GUT}$. Four constitutional
scales are marked: the electroweak scale $v = 246$~GeV, the
Kaluza--Klein lock cluster $\Mlock = \tfrac12 M_{\rm Pl}e^{-K}
\approx 3.85$~TeV, the unification scale
$E_{\rm GUT} = M_{\rm Pl}e^{-2\pi} \approx 2.3 \times 10^{16}$~GeV,
and the Planck scale. Below $\Mlock$ the running is the standard
one-loop Standard-Model flow; above it the lock cluster opens and
the three couplings converge to $\alpha_{\rm GUT}^{-1} \approx 41$
at exactly $E_{\rm GUT}$. The convergence is the empirical face of
the single composite connection: three registration modes of one
bulk object, not three theories that happen to meet.}
\label{fig:a9-unification-w2}
\end{figure}


Figure~\ref{fig:sectoral-ladder-w2} shows the derivation
chain that stands behind every such number.

\begin{figure}[htbp]
\centering
\resizebox{0.478\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={align=center},
 stage/.style={draw=black!55, rounded corners=2pt, line width=0.5pt, minimum width=38mm, minimum height=7mm, font=\sffamily\scriptsize, align=center},
 thm/.style={font=\sffamily\scriptsize, text=black!55, anchor=west},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.7pt, draw=black!65}
]
\node[stage, fill=red!12,    draw=red!60!black]                      (BI6)  {\textbf{1.\ \BIsix{} antecedent}\\[-1pt]\textit{anomaly closure}};
\node[stage, fill=orange!16, draw=orange!70!black, below=8mm of BI6] (TGG)  {\textbf{2.\ \TGG{} closure}\\[-1pt]\textit{total bulk/boundary tensor}};
\node[stage, fill=yellow!16, draw=olive!65!black,  below=8mm of TGG] (CCI)  {\textbf{3.\ \CCI{} projection}\\[-1pt]\textit{seam-level conservation}};
\node[stage, fill=green!16,  draw=green!60!black,  below=8mm of CCI] (GFF)  {\textbf{4.\ \GFF{} ledger}\\[-1pt]\textit{scalar landscape}};
\node[stage, fill=cyan!18,   draw=cyan!65!black,   below=8mm of GFF] (GRIP) {\textbf{5.\ \GRIP{} stabilization}\\[-1pt]\textit{\DGI{}\,$\to$\,\GSSB{}\,$\to$\,\GA{}}};
\node[stage, fill=blue!16,   draw=blue!65!black, line width=0.9pt, below=8mm of GRIP] (PARAM) {\textbf{6.\ Sectoral parameter block}\\[-1pt]\textit{read off the \GA{} coordinate}};
\node[thm, right=5mm of BI6.east]  {\AThree{}};
\node[thm, right=5mm of TGG.east]  {\ASix{}\, T-GRIP-1};
\node[thm, right=5mm of CCI.east]  {\ASix{}\, \textsf{T-2.1}, \textsf{T-2.2}};
\node[thm, right=5mm of GFF.east]  {\ASix{}\, \textsf{T-3.1}};
\node[thm, right=5mm of GRIP.east] {\ASix{}\, \textsf{T-4.1}};
\node[thm, right=5mm of PARAM.east]{\ANine{}};
\draw[arrow] (BI6)  -- (TGG);
\draw[arrow] (TGG)  -- (CCI);
\draw[arrow] (CCI)  -- (GFF);
\draw[arrow] (GFF)  -- (GRIP);
\draw[arrow] (GRIP) -- (PARAM);
\end{tikzpicture}}
\caption[The universal sectoral descent ladder.]{\textbf{The universal sectoral descent ladder}
(reproduced from \citealp{ArisakaA9SM}). Every parameter block of
the codebook obeys the same six-step descent: anomaly closure
(\BIsix{}, \citealp{ArisakaA3BI6}) $\to$ total closure of the
geometry--gauge tensor $\to$ projection to the boundary seam $\to$
the \GFF{} scalar landscape $\to$ the \GRIP{} cascade to the
selected branch $\to$ readout of the parameter values from the
attractor coordinates. Each step is owned by an upstream theorem
(right column); the codebook consumes the chain rather than
re-proving it. This is the assembly line on which
Eqs.~\eqref{eq:fermi-w2} and~\eqref{eq:codebook-p2} are
manufactured --- the reason ``derived'' in this paper always means
derived \emph{from something audited}, never fitted.}
\label{fig:sectoral-ladder-w2}
\end{figure}

\subsection{The running and the fixed}
\label{sec:running-fixed}

A last sorting completes the three-force picture: among the
numbers this Part has quoted, some run and some are frozen, and
the anatomy says which and why. Couplings \emph{run} ---
$\alpha_s$ falls, $\alpha$ creeps --- because running is the
interval's business: a coupling is a reading at an address, and
the flow between addresses is the semigroup itself. But the
\emph{closed forms} of Problem~2 are not couplings; they are values
pinned at named addresses by circle arithmetic:
$\hat\alpha^{-1}(M_Z) = 2^J = 128$ at the $Z$ pole,
$\alpha^{-1}(0) = 2^J + N^2 = 137$ at the infrared end,
$\sin^2\theta_W = L^2/(4N^3)$ at its matching point, the vev at
its Hosotani floor. The pattern is the gearbox grammar of
Part~II once more, now as a rule for reading the codebook: the
\emph{trajectory} is the interval's (continuous, one-way,
scheme-dressed); the \emph{anchor points} are the circle's
(integer arithmetic, exact). Confusions in both directions are
common and both are category errors: to ask why $137$ ``runs
away'' at high energy is to mistake an anchor for a trajectory;
to hope a running calculation will one day \emph{derive} $137$ is
to mistake a trajectory for an anchor. The codebook's precision
program \citep{ArisakaA9SM} lives entirely in keeping the two
straight --- and its twenty-eight-parameter audit is the
demonstration that, kept straight, the anchors suffice.

\subsection{Seminar questions}

\paragraph{Why does electromagnetism not confine?} Because its
circle factor is abelian and \emph{held open}: a single winding
is a complete, registerable book (any integer charge is a lawful
boundary state), so nothing forces closure into neutral
composites. Color's registration rule bites because its
triangle books balance only in triples --- the difference is not
in the strength of a force but in the arithmetic of the ledger.

\paragraph{Why is the weak force short-range but not confining?}
The two properties come from different columns. Range is an
address property: the agent's bosons are massive (the Wilson-line
vev sets their scale), so its reach is Yukawa-cut. Confinement is
a ledger property: the weak sector's books close on singles, so
nothing is imprisoned. A force can be short-range and free
(weak), long-range and free (EM), or effectively zero-range and
confining (strong) --- the four-force table's rows are
independent because the anatomy's columns are.

\paragraph{Where is the graviton in this picture?} Gravity is the
integrator, and its quantum is the boundary's reading of bulk
metric transport --- I treat the graviton question at
the register of the black-hole and cosmology companions rather
than as a fifth entry in the force table, precisely because
pricing is not keeping or changing: the deep statement is not
``gravity is another gauge force'' but ``gravity is the
bookkeeping of the descent itself,'' which is why it couples to
everything, screens nothing, and alone carries horizons.

\paragraph{Why do the couplings meet near the top?} Because they
were never three: the bulk carries one composite connection, and
the three boundary couplings are its three registered residues,
read at addresses. Their near-meeting toward the ultraviolet is
the census seen in reverse --- three shadows converging toward
the one object that casts them --- and the ledger's arithmetic
($2^J$, $L^2/4N^3$) is what the meeting-point bookkeeping looks
like when the integers are made explicit.

\subsection{The scorecard of the sorting}

The Problem's claim is structural, so its scorecard should be
structural too. Count what the two-keepers-one-agent-one-
integrator sorting explains with no further input: why exactly
one force is long-range and gentle (one factor held open); why
exactly one confines (one closing figure); why exactly one is
chiral and identity-changing (one fiber with a direction); why
exactly one prices horizons (one integrator); why charge is
quantized (windings); why the photon is massless and neutral
(the gearing carries no dial); why anomaly coefficients cannot
run (circle bookkeeping); why the vev sits where it does (the
gearbox); and why no fifth force appears (the census is full:
four circle factors, four jobs, no vacancy). Nine structural
facts, most of them memorized by every student as separate
brute features of the standard model, all of them rows of one
table --- and the table's exposure is a single address:
$3.849$~TeV, where the composite's first rung either appears
with its fingerprint or does not.

\section{Problem 3.\ The Origin of Local Gauge Invariance and Renormalization}
\label{sec:w2-prob3}

\subsection{Two principles that explain everything and themselves poorly}

Ask a field theorist what dictates the form of the fundamental
interactions and the answer is one phrase: local gauge invariance.
Demand that the phase of a matter field be adjustable independently
at every point of spacetime, and the gauge fields --- the photon,
the gluons, the weak bosons --- appear as the price of the demand,
with their couplings and self-interactions fixed. It is the most
successful principle in physics. It is also, examined closely, the
strangest: Noether proved in 1918 \citep{Noether1918} --- in the same paper as her
famous theorem --- that a symmetry of this local kind delivers
\emph{no} conserved quantity, only an identity among the equations
of motion, true whether or not anything obeys them. A local gauge
transformation changes nothing physical; it is, in the standard
phrase, a redundancy of description. Physics' master principle is a
demand about bookkeeping. Why should nature care?

The second principle is renormalization. Quantum field theory's
raw predictions diverge; the cure absorbs the divergences into a
finite set of couplings that ``run'' with the energy scale; the
result agrees with experiment to eleven decimal places. Yet its
founders were openly uneasy --- Dirac called it sweeping infinities
under the rug --- and the modern Wilsonian understanding, while far
more satisfying, still leaves a question standing: the
renormalization group's flow is famously \emph{irreversible}
(coarse-graining loses information; c-theorems and their kin make
the one-wayness precise), and nothing in the underlying reversible
dynamics explains where that arrow comes from, or why the scale
parameter behaves so much like a dimension that field theorists
routinely draw it as one.

The anatomy's claim: both principles are the two fibers, made law
--- gauge invariance is what the circle looks like from the
boundary, and renormalization is what the interval does. Neither
is a trick of description; both are geometry.

\subsection{Noether's second theorem, in plain English}

The distinction that carries this Problem was drawn at the very
beginning, and is worth one paragraph of care
\citep{ArisakaW1Symmetry}. A \emph{global} symmetry --- rotate
every phase everywhere by the same amount --- constrains motion:
Noether's first theorem turns it into a conserved current, a
physical statement. A \emph{local} symmetry --- rotate each phase
independently, point by point --- cannot constrain motion, because
it can undo any motion: her second theorem shows it yields
identities, not charges, and Elitzur later proved the consequence \citep{Elitzur1975}
in full rigor --- a local symmetry cannot even break, because
there is nothing physical there to break. So the standard model is
built on a principle which, by the sharpest theorems available, has
no physical content. The resolution cannot be to deny the theorems.
It must be to find the physical thing that \emph{hides behind} the
redundancy.

\subsection{What gauge symmetry is: the unreadable circle}

The anatomy supplies the hidden thing. The gauge phase is not an
abstract label; it is a coordinate --- position around the compact
fiber $\Stheta$ --- and \OGGEP{} states its fate: no boundary
protocol can register a fiber coordinate. The phase at a point is
unreadable \emph{as a matter of law}, not of technology; only
differences of phase around closed circuits --- holonomies ---
survive projection to the boundary. Now run the logic forward and
the two mysteries of gauge theory resolve into one statement each:

\begin{itemize}[leftmargin=2.0em]
\item \emph{Why is the local phase a redundancy?} Because it is the
coordinate of an unreadable fiber. Noether II and Elitzur were
exactly right: the pointwise phase has no physical content --- it
is the part of the geometry the boundary cannot hold. The
``redundancy'' is not a defect of our description; it is the
\emph{shadow of a real circle} that observation cannot enter.
\item \emph{Why does a redundancy dictate the interactions?}
Because what remains after the unreadable part is removed --- the
holonomies --- is not nothing: it is everything. By the Wilson-loop
reconstruction, a gauge field \emph{is} its holonomies; the
interactions are the transport of the fiber nothing can read; and
demanding local gauge invariance is simply demanding that a theory
speak only of the readable --- which forces it to speak of
transport, and transport is force. The Aharonov--Bohm effect is
this Problem's tabletop witness (\S\ref{sec:primer}): a phase
invisible along every open path, physical around the closed one.
\end{itemize}

One more classic puzzle collapses with the same move: charge
quantization. If the gauge group were an abstract $\mathrm{U}(1)$,
charge could take any real value; measured charges come in exact
integer ratios to one part in $10^{21}$. On the anatomy the phase
is a \emph{compact geometric circle}, so its conjugate charge is a
winding number --- an integer because the fiber closes. The oldest
precision fact in electromagnetism is a topology statement about
the sixth dimension.

The reading extends upward. Local gauge invariance is the circle's
unreadability stated in a \emph{closed} theory --- the static form
of the law. In the cohort's constitutional ladder (\POS{1}:
\LGI{}/\DGI{}/\GGI{}) the same invariance has a dynamical form:
\emph{Dynamical Gauge Invariance}, the opening stage of the \GRIP{}
engine (\S\ref{sec:minimum}), in which the gauge directions
themselves become dynamical because an open system's environment
keeps shifting the phase conventions \citep{ArisakaA7Qi4} --- the
invariance is then held by an equivalence class of boundary
generators, not of field configurations. \DGI{} is \LGI{} carried
through the open dynamics: the circle's invariance transported by
the interval's semigroup. What Problem~1 does for the state
(unitary on the circle, write on the interval), the ladder does for
the symmetry itself.

The principle in symbols, so the plain-English reading has
something exact to hold. A local gauge transformation is the
simultaneous rewriting
\begin{equation}
\psi(x) \;\longrightarrow\; e^{\,i q\,\chi(x)}\,\psi(x),
\qquad
A_\mu(x) \;\longrightarrow\; A_\mu(x) + \partial_\mu \chi(x),
\label{eq:gaugetrans-w2}
\end{equation}
where $\psi$ is a charged matter field, $q$ its charge, $A_\mu$
the gauge potential, and $\chi(x)$ an \emph{arbitrary function of
position} --- a different phase convention at every point.
``Gauge invariance'' is the statement that physics is unchanged
under Eq.~\eqref{eq:gaugetrans-w2} for every $\chi$; and since
$\chi$ is arbitrary, anything that changes under it --- the local
value of the phase, the potential itself --- is convention, not
physics. What survives every $\chi$ is exactly what
Eq.~\eqref{eq:wilson-p2} collects: the loop integrals, on which
$\partial_\mu\chi$ integrates to zero around any closed circuit.
That is the unreadable-pointer statement made exact: the circle's
absolute angle is $\chi$-shiftable, its closed-loop content is
not.

\subsection{What renormalization is: motion along the interval}

Now the second principle. In the Wilsonian picture one integrates
out physics scale by scale and watches the couplings drift; the
scale parameter $\mu$ is a bookkeeping device, and the flow's
irreversibility a property of information loss. The anatomy
promotes the picture to geometry: the scale is the fifth
coordinate, $r = \ln(\EGUT/\mu)$ --- a direction of the bulk, as
real as the circle --- and the renormalization flow is
\emph{transport along the interval}. Each ingredient of the
textbook toolkit acquires an address-book meaning:

\begin{itemize}[leftmargin=2.0em]
\item A \emph{running coupling} is a reading at an address:
$\alpha_s$(2 GeV) and $\alpha_s(M_Z)$ are the same circle-anchored
quantity inspected at two stations of the descent.
\item A \emph{scheme choice} is address bookkeeping --- a
convention for labeling stations, physical in none of its
arbitrariness, exactly as the textbooks insist.
\item The flow's \emph{irreversibility} is the interval's
semigroup: coarse-graining has no inverse because the fiber has
none. The c-theorems are the laboratory-scale face of the same
one-wayness that runs cosmic history (Problem~13) --- one arrow,
met twice.
\item \emph{Renormalizability} itself becomes a census criterion:
the interactions that survive the flow --- the relevant and
marginal ones --- are the content the interval can carry to the
infrared; the rest is left behind at high addresses. That the
standard model is renormalizable is, on this reading, the
statement that its content is exactly what a descent can deliver.
\end{itemize}

Why does the ``sweeping of infinities'' work? Because there were
never infinities in the geometry --- only the pretence that the
interval was infinitely long in both directions. The bulk's
interval runs from the GUT edge to the infrared: a finite descent,
$K = 35$ e-folds to the lock, with the ultraviolet end a real edge
rather than a limit. Effective field theory's great lesson ---
every theory has a cutoff and a domain --- is the interval's
finiteness, learned from inside.

Figure~\ref{fig:dial-run} draws the two principles side by
side: the sealed dial whose absolute pointer no experiment reads,
and the descent along which every coupling is a reading at an
address.

\begin{figure}[htbp]
\centering
\resizebox{0.98\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% Panel (a): the sealed dial
\begin{scope}
\node[align=center] at (0,3.1) {\textbf{(a) Gauge invariance: the sealed dial}};
\draw[line width=1.10pt] (0,0) circle (1.8);
\draw[line width=0.80pt,black!45] (0,0) -- (55:1.8);
\node[black!45] at (0.35,1.35) {\scriptsize pointer};
\node[align=center,black!45] at (0,-0.55)
  {\scriptsize absolute setting:\\[-2pt]\scriptsize unreadable (gauge)};
\draw[->,line width=0.80pt,blue!65!black] (-15:2.1) arc (-15:230:2.1);
\node[blue!65!black,align=center] at (0,2.65)
  {\scriptsize the loop: net turns around a circuit\\[-2pt]
   \scriptsize readable (holonomy)};
\end{scope}
% Panel (b): running couplings
\begin{scope}[shift={(6.3,-1.9)}]
\node[align=center] at (3.2,5.0) {\textbf{(b) Renormalization: motion along the interval}};
\draw[->,line width=1.10pt] (0,0.5) -- (6.6,0.5);
\node at (3.3,1.0) {\scriptsize $r$: the descent (one-way)};
\foreach \x/\lab in {0.5/{GUT},2.3/{lock},4.1/{QCD},5.9/{today}}{
  \draw[line width=0.80pt] (\x,0.42) -- (\x,0.58);
  \node[below] at (\x,0.30) {\scriptsize \lab};}
\draw[line width=0.80pt,red!70!black] (0.5,2.2) .. controls (2.3,2.6) and (4.1,3.4) .. (5.9,4.3);
\node[red!70!black,right] at (4.4,4.15) {\scriptsize $\alpha_s$};
\draw[line width=0.80pt,blue!65!black] (0.5,2.2) .. controls (2.3,2.0) and (4.1,1.85) .. (5.9,1.75);
\node[blue!65!black,right] at (5.0,1.62) {\scriptsize $\alpha_{\mathrm{em}}$};
\fill[black!70] (0.5,2.2) circle (0.06);
\node[above,black!70] at (0.9,2.3) {\scriptsize one origin};
\node[align=center] at (3.3,-0.55)
  {\scriptsize a coupling $=$ a reading at an address;\\[-2pt]
   \scriptsize the flow composes forward only};
\end{scope}
\end{tikzpicture}%
}
\caption[The two principles, drawn: the unreadable circle and the running interval.]{\textbf{The two principles, drawn.} \emph{(a)} Gauge
symmetry: the compact fiber's absolute pointer is unreadable, and
``gauge invariance'' is the law that physics speak only of the
readable --- the net turns around closed circuits. \emph{(b)}
Renormalization: couplings are not constants but readings at
addresses on the fifth dimension, and the flow's famous
irreversibility is the interval's semigroup. One geometry, both
principles --- neither imposed, both read.}
\label{fig:dial-run}
\end{figure}

Here too one equation carries the concept. A coupling $g$
measured at resolution scale $\mu$ obeys the renormalization-group
equation
\begin{equation}
\mu\,\frac{dg}{d\mu} \;=\; \beta(g),
\label{eq:rg-w2}
\end{equation}
where $\beta$ --- computable within the theory --- gives the rate
at which the reading changes per e-fold of scale. The equation's
form is the lesson: the derivative is with respect to $\ln\mu$,
so the natural coordinate of the flow is logarithmic in energy
--- precisely the interval's coordinate $r = \ln(\EGUT/\mu)$ ---
and the flow is first-order: the reading at one address determines
the reading at the next, with no memory and no way back
(Eq.~\eqref{eq:semigroup-law-p2}). The sign of $\beta$ sorts the
forces: $\beta > 0$ for electromagnetism (the reading grows toward
the ultraviolet), $\beta < 0$ for the strong force
(Eq.~\eqref{eq:running-w2}'s $b_0 > 0$: the reading grows toward
the infrared, the seed of confinement).

\subsection{A short history of the gauge idea, reread}

The gauge principle's own history, reread on the anatomy, is a
century of converging on the circle without naming it. Weyl's 1918
original proposed local \emph{scale} invariance --- an interval
idea --- and failed physically (atoms would remember their
histories); his 1929 repair replaced scale by \emph{phase}, and
worked. On the anatomy the repair is a fiber swap: local
readjustability belongs to the compact fiber, whose pointwise
value is lawfully unreadable, not to the open one, whose coordinate
is physical (it is the RG scale --- Problem~13's chronology rides
on it). Yang and Mills generalized the phase to internal circles
\citep{YangMills1954}
and spheres; Aharonov and Bohm showed the phase's loop content is
physical; Wilson made loops the theory's variables; Hosotani showed
loop values can choose vacua. Each step moved the weight from the
connection (unreadable) to the holonomy (the content). The anatomy
is the terminus of that motion: it takes the last step of saying
the circle is \emph{a place} --- the sixth dimension --- so that
the gauge principle stops being a principle about our descriptions
and becomes a fact about the bulk: \emph{physics is the transport
of an unreadable fiber.} At the terminus the old philosophical
scandal (why should nature obey a bookkeeping rule?) inverts:
nature does not obey it --- nature \emph{is} it, and we obey it
when we describe her lawfully.

Figure~\ref{fig:ladder-w2} compresses that century into one
diagram --- the first white paper's ladder, reproduced here because
this Problem is its structural reading.

\begin{figure}[htbp]
\centering
\resizebox{0.574\textwidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
  \definecolor{rungcol}{HTML}{B26A16}
  \foreach \i/\name/\what in {%
     0/{S1\ \ Noether I (1918)}/{conserves \textbf{a current} --- no integer anywhere},
     1/{S1$'$\ \ Noether II (1918)}/{yields \textbf{an identity} --- off-shell, no new charge},
     2/{S2\ \ local gauge invariance}/{encodes \textbf{a redundancy} --- bookkeeping},
     3/{S3\ \ the Bianchi identity}/{conserves \textbf{a flux} --- a geometric identity},
     4/{S4\ \ the Wilson line (1983)}/{the integer is \textbf{the frame} --- one phase per cycle},
     5/{S5\ \ the Ward identity (1950)}/{the integer is \textbf{the failure} --- it breaks discretely},
     6/{S6\ \ \GS{} (1984)}/{the integer is \textbf{the trade} --- two failures cancel},
     7/{S7\ \ \HS{} (2005)}/{the integer is \textbf{the whole content} --- pure topology}}
  {
    \draw[rungcol,line width=0.9pt] (0,\i) -- (9.6,\i);
    \node[anchor=west,inner sep=1pt] at (0.18,\i+0.66) {\bfseries \name};
    \node[anchor=west,inner sep=1pt,text=black!72] at (0.18,\i+0.28) {\what};
  }
  \draw[rungcol,line width=0.9pt] (0,8) -- (9.6,8);
  \draw[-latex,line width=1.1pt,rungcol]
     (10.3,0.05) -- (10.3,7.95)
     node[midway,rotate=-90,anchor=south,font=\small\bfseries]
     {how much of the statement is an integer};
  \draw[-latex,line width=1.1pt,black!55]
     (-0.5,7.95) -- (-0.5,0.05)
     node[midway,rotate=90,anchor=south,font=\small\bfseries]
     {the statement's grip on motion};
\end{tikzpicture}}
\caption[The century's ladder: seven steps from Noether to \HS{}.]{\textbf{The century's ladder: seven steps from
Noether to \HS{}} (reproduced from \citealp{ArisakaW1Symmetry}).
Reading upward, each step in the history of gauge theory says less
about how things move and carries more of its content in an
integer; the lower rungs are algebra and convention and cannot
fail, while from the fifth rung up something can fail --- by a
whole number. The gauge idea's century, compressed: a walk from
dynamics toward topology, arriving at the circle's protected
integers --- the destination this Problem reads as where the
principles came from all along.}
\label{fig:ladder-w2}
\end{figure}

\subsection{Worked intuition: the sealed dial}

Carry one image. Every point of space wears a sealed dial: a
pointer that no instrument can photograph, by law. What can be
measured is only this: carry a test charge around a closed path
and count how far the pointer has net-rotated when you return ---
the holonomy. Now every strange clause of gauge theory becomes a
sentence about the seal. \emph{Gauge transformations} re-zero the
unreadable pointers; nothing measurable can care, which is
invariance. \emph{The vector potential} is the pointer-to-pointer
gearing --- unphysical pointwise, physical in its loop totals,
which is Aharonov--Bohm. \emph{Charge quantization}: the dial is a
closed circle, so loop totals come in whole turns --- the argument
whose first instance is Dirac's monopole quantization
\citep{Dirac1931}. \emph{The
photon} is the gearing itself, which is why it is massless and
chargeless: gearing carries no dial of its own. And \emph{anomaly
exactness} (next section) is the statement that the number of
teeth on the gears is a manufacturing constant of the geometry ---
an integer the flow downstream can never wear away.

\subsection{Asymptotic freedom and the far end of the interval}

The interval reading earns a bonus that deserves its own page: the
qualitative shape of the strong coupling's run. Asymptotic freedom
--- the strong force weakening at short distances, Nobel-crowned
and experimentally exquisite --- is, in address language, the
statement that the color reading falls toward the ultraviolet end
of the interval and grows toward the infrared, until at the
confinement scale the perturbative reading fails and the circle's
closure rule (Problem~12) takes over the description. The anatomy
adds no new computation to the beta function --- the running is
standard physics, inherited whole --- but it adds the census
remark that dissolves a familiar puzzlement: students ask
\emph{where} the coupling ``really'' has its value, as if one
address were true and the others apparent. No address is
privileged; the couplings are readings along a real direction, the
way temperature has a value at every altitude of an atmosphere;
and the closed forms of the codebook (Problem~4) are anchored at
\emph{named} addresses --- $\alpha^{-1}$ at the infrared end,
$2^J$ at the $Z$ pole --- precisely because only named addresses
are answers. The interval, once seen, also explains why the three
gauge couplings converge toward the ultraviolet at all: they are
three readings of one composite connection (Problem~2's
unification), taken closer and closer to the edge where the
composite was never split.

\subsection{Three tiers of protection, and the Higgs}
\label{ssec:three-tiers}

The two principles now have their fibers. Before the dictionary
collects the translations, one distinction has to be drawn that the
Part has so far let pass, because without it the anatomy appears to
contradict itself --- and because with it the anatomy pays its
largest single debt.

The apparent contradiction is this. \S\ref{app:D-step2} proves that
circle-side quantities cannot run: they are homotopy data, the flow is
continuous, and nothing continuous acts on an integer. Yet the
electroweak vacuum value $v$ is set by a circle-side modulus, and $v$
runs. So does the Higgs mass. Either the theorem is wrong or the
sorting is.

Neither. \emph{Circle-side quantities are not all protected the same
way}, and the ones that differ are exactly the interesting ones
(Table~\ref{tab:tiers-w2}).

\begin{table}[htbp]
\centering
\caption[Why some circle-side quantities run and others cannot.]{Why some circle-side quantities run and others cannot. Tier~1 is winding data --- charge, anomaly coefficients, the family count, the parities --- and it cannot move, because an integer has nowhere to move. Tier~2 is a continuous modulus of the circle, the Wilson-line phase that carries the Higgs (Problem~2) and the Kaxion (Problem~7); it moves, and what protects it is non-locality, since a Wilson line is not the value of any local operator and no local counterterm can generate a potential for it. Tier~3 is the interval's addresses --- masses, scales, rates, epochs --- which run, and must, because running is what an address does.}
\label{tab:tiers-w2}
\small
\begin{tabular}{@{}p{0.05\textwidth}p{0.26\textwidth}p{0.13\textwidth}p{0.46\textwidth}@{}}
\toprule
Tier & What it is & Can it move? & What protects it \\
\midrule
1 & winding data: charge, anomaly coefficients, the family count,
parities & \textbf{No} & \textbf{integrality}. An integer has nowhere
to move. This is the no-running theorem of
\S\ref{app:D-step2} and the seventh rung of \citet{ArisakaW1Symmetry} \\
2 & \textbf{a continuous modulus of the circle}: the Wilson-line phase
--- the Higgs (Problem~2), the Kaxion (Problem~7) & \textbf{Yes} &
\textbf{non-locality}. A Wilson line is not the value of any local
operator, so no local counterterm can generate a potential for it;
whatever potential it acquires is nonlocal, finite, and calculable \\
3 & interval-side addresses: masses, scales, rates, epochs &
\textbf{Yes, and must} & nothing. Running is what an address does \\
\bottomrule
\end{tabular}
\end{table}

Tier~1 is the circle at its most familiar --- the fiber that
remembers, remembering exactly. Tier~3 is the interval doing its
work. Tier~2 is the interesting case, and this paper has one
important instance of it.

\paragraph{Why the Higgs is the Standard Model's problem child.}
A scalar's mass is the coefficient of a local operator of dimension
two, so every loop drags a term of order $\Lambda^2/16\pi^2$ into it.
With $\Lambda$ anywhere near the Planck scale, the bare value and the
loop corrections must agree to some thirty decimal places for the
observed mass to survive --- the naturalness problem, and the reason
a generation of physicists expected new particles at the electroweak
scale to enforce a cancellation. On the anatomy that reasoning is
sound but misapplied: it treats the Higgs as a tier-3 object. The
Higgs is tier~2. Its ultraviolet variable is a periodic angle, the
holonomy $W_\theta$ of Eq.~\eqref{eq:A-wilsonline}; and a local
counterterm quadratic in that angle does not exist, because the angle
is not gauge-invariant on its own --- only closed-loop functions of it
are. One cannot quadratically renormalize a holonomy. The potential
the angle does acquire comes from fields \emph{winding} the circle,
which is a nonlocal effect, exponentially convergent in the winding
number, and finite. Nothing needs to cancel, because nothing diverged
\citep{HatanakaInamiLim1998}.

That is this paper's thesis applied where it costs the most.
Supersymmetry answers naturalness with a symmetry, and a symmetry can
break --- its preferred region is now excluded (Problem~4). Topology
and locality have nothing to break; there is no order parameter for
them. \emph{Protection without symmetry}, the destination of the
ladder in the companion review, is not an abstraction about
differential cohomology: it is the reason the electroweak scale
survives its own quantum corrections.

\paragraph{The honest limit of the claim.} Tier~2 is weaker than
tier~1, and blurring them would be the easiest overclaim in this
Part. A tier-1 quantity cannot move at all: an anomaly coefficient
that ran would falsify the framework (\S\ref{sec:kill}). A tier-2
modulus is an angle --- it moves, and $v$ and $m_H$ run like the
addresses they are. What tier~2 buys is not fixity but
\emph{finiteness}: a quantity the theory owes a calculation for,
rather than a quantity that absorbs a divergence. The owing is
recorded honestly in the codebook companion, where the electroweak
vev lands at $+0.045\%$ and the Higgs mass, which requires the
potential's curvature rather than merely its minimum, is a
percent-and-a-half low with the normalizing step not yet computed
\citep{ArisakaA9SM}. The framework has converted an unnatural input
into an owed calculation and has not finished paying. That is a
better place to stand than thirty decimal places of cancellation, and
it is not the same thing as a solved problem.

\subsection{The dictionary, in one table}

For reference, the Problem's translations collected --- each row a
standard concept on the left, its anatomy reading on the right:

\small
\begin{longtable}{@{}ll@{}}
\caption[The field-theory dictionary, one line to a concept.]{The field-theory dictionary, one line to a concept: a standard term of gauge theory on the left, its reading in the two-fiber anatomy on the right. A local gauge transformation is a re-zeroing of unreadable pointers, the potential $A_\mu$ is the pointer-to-pointer gearing, gauge invariance is the law that one speak only of the readable, the Wilson loop is what is readable --- net turns around a circuit --- electric charge is a winding number of the compact fiber, and the photon is the gearing itself, with no dial of its own. The lower half does the same for the interval: the RG scale $\mu$ is an address on the fifth dimension, a running coupling is a reading at an address, scheme dependence is address bookkeeping, and the irreversibility of the flow is the interval's semigroup. Nothing in the right column is new physics; the table is the translation the rest of the Part uses.}
\label{tab:dictionary-w2}\\
\toprule
\textbf{Field-theory concept} & \textbf{Anatomy reading} \\
\midrule
\endfirsthead
\multicolumn{2}{c}{\itshape Table~\ref{tab:dictionary-w2} continued}\\
\toprule
\textbf{Field-theory concept} & \textbf{Anatomy reading} \\
\midrule
\endhead
\midrule \multicolumn{2}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

local gauge transformation & re-zeroing of unreadable pointers \\
gauge potential $A_\mu$ & pointer-to-pointer gearing \\
gauge invariance & the law: speak only of the readable \\
Wilson loop / holonomy & the readable: net turns around a circuit \\
electric charge & winding number of the compact fiber \\
the photon & the gearing itself (no dial of its own) \\
RG scale $\mu$ & address on the fifth dimension \\
running coupling & a reading at an address \\
scheme dependence & address bookkeeping (unphysical, as taught) \\
RG irreversibility & the interval's semigroup \\
renormalizable operator & content the descent can carry to the IR \\
anomaly coefficient & circle-side pairing integer (cannot run) \\
anomaly cancellation & the circle auditing the interval's books \\
\end{longtable}

The table's uniformity is the Problem's result: thirteen standard
notions, two fibers, no exceptions --- quantum field theory's
formal apparatus, reread as the operating manual of a geometry it
never named.

\subsection{Questions students ask}

\paragraph{Why this gauge group and not another?} The census
answers in two layers. The bulk's composite group ---
$\mathrm{Spin}(1,5) \times \mathrm{SU}(5) \times \UoneB \times
\UoneQi$ --- is forced by the constitutional lemmas (Problem~2's
unification section); and the boundary's familiar
$\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is
its registered residue: the part of the transport whose
holonomies survive projection. The standard model's group was
never a choice from a catalog; it is the readable shadow of the
one composite the geometry admits.

\paragraph{Why is charge conserved even through measurements?}
Because measurement is an interval-side write (Problem~1), and
charge is circle-side data: the write registers, it cannot
deform. The textbook wonder that ``collapse'' respects every
superselection rule is, on the anatomy, no wonder at all --- the
event that could violate a winding lives on the wrong fiber to
touch one.

\paragraph{Could gauge invariance be emergent?} Several respected
programs derive gauge structure from more primitive rules; the
anatomy neither needs nor forbids such a layer, but it sharpens
the requirement: any successor must reproduce a sector with
exactly the circle's profile --- unreadable pointwise, loop-exact,
integer-charged --- and a sector with exactly the interval's ---
one-way, unprotected, address-like. Emergence that delivers the
pair delivers this Problem's physics; emergence that delivers
less delivers less. As with Problem~15, the anatomy functions as
a constraint on all candidate deeper theories --- which is the
most a structural claim can honestly be.

\paragraph{Is renormalization still ``effective''?} Yes ---
gloriously so. Nothing in the interval reading disturbs the
practical Wilsonian toolkit; it explains why the toolkit works
and where its scales live. The one working habit it retires is
apology: the twentieth century's sense that renormalization was
provisional, a stand-in for a finite future theory. On the
anatomy the flow is not scaffolding around the theory; it is a
load-bearing wall of the geometry --- the fifth dimension,
carrying weight.

\subsection{The anomaly: the circle audits the interval}

The two principles meet in exactly one place, and the meeting is
the sharpest exhibit of this Problem. An anomaly is a classical
symmetry that quantization cannot keep --- the one lawful failure
of the gauge principle --- and its coefficients have a remarkable
property: they are exact integers, computed at one loop and
corrected at none, identical at every scale. On the anatomy this
is inevitable: anomaly coefficients are circle-side bookkeeping
--- topological pairing data on the fiber that remembers --- so
they cannot run, because running is the other fiber's business and
the interval has no holonomy to renormalize
\citep{ArisakaA3BI6}. The cancellation of anomalies is then the
circle auditing the interval's books: a consistency contract
between what the descent carries and what the topology permits,
enforced in integers. The cohort's own bulk passes the audit by
the six-dimensional \GS{} mechanism \citep{GreenSchwarz1984}, and the audit's byproduct is
the integer ledger $(M,N,L;J,K)$ from which the codebook of
Problem~4 is built --- the deepest example in physics of a
redundancy's bookkeeping paying out measurable numbers. The seam where the two-sided audit is proved end to end is Appendix~H of the family-number paper \citep{ArisakaP2N3}: \emph{what the circle
quantizes is what the interval delivers.}

The audit has an exact face in the Standard Model itself.
For each generation, gauge consistency requires the fermion
hypercharges $Y_f$ to satisfy, among others,
\begin{equation}
\sum_{f}\, Y_f \;=\; 0,
\qquad
\sum_{f}\, Y_f^{\,3} \;=\; 0,
\label{eq:anomsum-w2}
\end{equation}
the sums running over the fifteen chiral fermions of one
generation with their color multiplicities. These are not
approximate relations improved by higher orders; they are exact
integer arithmetic --- the quark and lepton hypercharges conspire
to cancel --- and a single fermion added or removed breaks them.
The physical reading: an anomaly is a quantum failure of a
classical bookkeeping, and Eq.~\eqref{eq:anomsum-w2} is the
condition that the books close. That the observed matter content
satisfies it exactly --- forcing, famously, the quark charges to
be thirds --- is the circle auditing the boundary's content, and
the six-dimensional version of the same audit
(\citealp{ArisakaA3BI6}) is what locks the cohort's ledger.

Figure~\ref{fig:seam-w2} draws the crossing itself.

\begin{figure}[htbp]
\centering
\resizebox{0.741\textwidth}{!}{%
\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
  \definecolor{bulkc}{HTML}{2F6F4E}
  \definecolor{bdyc}{HTML}{7A4E2D}
  %% ---- the two regions, set well apart --------------------------
  \fill[bulkc!7] (0,1.0) rectangle (4.5,4.6);
  \draw[bulkc,line width=0.9pt] (0,1.0) rectangle (4.5,4.6);
  \node[bulkc,anchor=north west,font=\scriptsize\bfseries,align=left,
        text width=3.9cm] at (0.3,4.4) {the bulk\\ six dimensions};
  \node[anchor=north west,text=black!65,font=\scriptsize,align=left,
        text width=3.9cm] at (0.3,3.55)
    {geometry and gauge are\\ one object (\GGEP{}, AX-2);\\
     the bookkeeping is exact};

  \fill[bdyc!8] (10.6,1.0) rectangle (15.1,4.6);
  \draw[bdyc,line width=0.9pt] (10.6,1.0) rectangle (15.1,4.6);
  \node[bdyc,anchor=north west,font=\scriptsize\bfseries,align=left,
        text width=3.9cm] at (10.9,4.4) {the boundary\\ four dimensions};
  \node[anchor=north west,text=black!65,font=\scriptsize,align=left,
        text width=3.9cm] at (10.9,3.55)
    {what is registered\\ (\OGGEP{}, AX-3);\\
     the dynamics is open (\GDOS{})};

  %% ---- the seam: drawn at the bulk's edge, so that it never
  %% ---- crosses a label; labelled horizontally, above ---------------
  \draw[black!45,line width=1.2pt,dash pattern=on 3pt off 2pt]
    (4.85,0.75) -- (4.85,4.85);
  \node[black!55,anchor=south west,font=\scriptsize\bfseries] at (4.9,4.95)
    {the seam: a projection, not a window};

  %% ---- the three data, in the open channel ------------------------
  \foreach \y/\name/\fate in {%
     3.55/{charge --- an integer class}/{crosses intact},
     2.45/{flux --- a curvature form}/{crosses, and may vary},
     1.35/{holonomy --- a phase on a loop}/{read \emph{on} the edge, not carried to it}}
  {
    \draw[-latex,black!60,line width=0.8pt] (5.15,\y) -- (10.35,\y);
    \node[anchor=south,font=\scriptsize] at (7.75,\y+0.09) {\name};
    \node[anchor=north,font=\scriptsize,text=black!58] at (7.75,\y-0.11)
      {\fate};
  }
  %% ---- the loop the holonomy is read on ---------------------------
  \draw[bdyc,line width=0.7pt] (14.35,1.35) circle (0.3);
  \draw[-latex,bdyc,line width=0.7pt] (14.65,1.35) arc (0:55:0.3);
\end{tikzpicture}}
\caption[The seam: what crosses from bulk to boundary.]{\textbf{The seam: what crosses from bulk to
boundary} (reproduced from \citealp{ArisakaW1Symmetry}). Three
kinds of data cross from the six-dimensional bulk to the
four-dimensional stage, and they cross differently: the integer
class crosses intact (nothing continuous can deform it), the
curvature form crosses and may vary, and the holonomy --- being a
statement about a loop --- is evaluated \emph{on} the boundary
rather than transported to it. The asymmetry is this Problem's
content in one picture: what arrives exact arrives from topology;
what arrives adjustable arrives from geometry; and the audit of
Eq.~\eqref{eq:anomsum-w2} polices the crossing.}
\label{fig:seam-w2}
\end{figure}

\subsection{Where each fiber stops: the one number the anatomy
cannot reach}
\label{ssec:w2-g2}

Every subsection of this Problem so far has been an addition. The circle
gave gauge invariance and exact charge; the interval gave renormalization
and its arrow; the seam between them gave the anomaly ledger. It is worth
spending one subsection on subtraction, because the anatomy's limits are
where its content is easiest to check --- and because the sharpest test
available sits exactly here, in the most precisely measured number in
physics.

Take the electron's magnetic moment. Dirac's equation gives $g = 2$
for free: a phase geared minimally to a field has precisely that moment,
and nothing was chosen. That is the circle at work, and it is the
cleanest single payment the compact fiber ever makes. But the measured
value is not $2$. It exceeds it by about one part in eight hundred, and
the excess $a = (g-2)/2$ is known to twelve significant figures --- a
number the discipline has spent seventy-five years computing, one
Feynman diagram at a time.

So the question the anatomy has to face is direct. Two extra dimensions,
one of which \emph{is} the gauge phase and one of which \emph{is} the
renormalization scale: what do they say about $a$?

They say nothing, and both silences have a reason.

\emph{The interval cannot reach it, because $a$ carries no scale.} This
is \AFive{}'s theorem \T{9.9} \citep{ArisakaA5GDOS}, and its argument is
one line: the radial coordinate is by construction conjugate to scale, so
a quantity independent of scale lies in the directions orthogonal to it,
and a change of variables cannot reduce the information content of
something already independent of the variable being changed. For the
magnetic moment the scale-independence is not an observation but a
theorem: $a$ is the coefficient of an operator of mass dimension five, a
renormalizable theory carries no counterterm for such an operator, so
the coefficient must come out finite --- and a finite, dimensionless,
on-shell number cannot depend on the sliding scale at all. The famous
rule that $g-2$ ``does not run'' is a statement about an operator
dimension. What runs is $\alpha$; $a$ is what you get once $\alpha$ has
been read.

\emph{The circle cannot reach it either, because $a$ is not an integer.}
This is the same argument that made the circle powerful, run backwards.
Flat connections on $\Stheta$ are classified by $\pi_1(S^1) = \mathbb{Z}$,
and the whole force of \S\ref{ssec:three-tiers} was that nothing
continuous can move between the points of a discrete set --- which is why
charge is quantized, why anomaly coefficients cannot run, and why class
data are the same in every region of the sky. All of that is
\emph{integers}. But $a = \alpha/2\pi + \mathcal{O}(\alpha^2)$ is not an
integer, not a winding number, and not a characteristic class. The fiber
that protects whole numbers has no grip on a transcendental one.

So the anomalous moment falls between the two fibers: too continuous for
the circle, too scale-free for the interval. On the three tiers of
\S\ref{ssec:three-tiers} it belongs to none of them --- it is neither
protected content nor an address, but the boundary's own arithmetic. The
twelve thousand diagrams are what that arithmetic costs, and the anatomy
neither shortens the bill nor claims to.

\paragraph{What the anatomy does supply, which is not nothing.} It
supplies the number you feed in. The prediction for $a$ is only as good
as the $\alpha$ it is computed with, and $\alpha^{-1}(0) = 2^J + N^2 =
137$ is a ledger identity here rather than a measurement
\citep{ArisakaA9SM}. It supplies the reason there is a lepton sector to
have a moment at all. And it supplies, through \PaperPThree{}
\citep{ArisakaP3Electron}, the statement that the electron's own
contribution to $a$ is exactly zero at every order --- all of the anomaly
is the vacuum's, none of it is structure, and that is a prediction with a
kill criterion rather than a hedge.

\paragraph{Why the refusal is worth more than an answer would be.} A
framework that could produce $\alpha/2\pi$ from geometry could produce
almost anything from geometry, and the reader would be right to stop
believing it. The division this Problem draws is the opposite: the
compact fiber says what is exactly conserved, the open fiber says at what
resolution it is being read, and the arithmetic of the boundary says what
follows. Confusing the three is how a structural framework turns into
numerology.

There is a reward for having said this in advance, and it arrived while
this cohort was being written. The framework has no spare sector with
which to manufacture a contribution to the muon's anomaly, and said so
plainly --- a standing exposure, since for several years the muon
measurement sat some four standard deviations from the Standard-Model
evaluation and a great many models were built to close that gap. The
gap has since closed on its own: the hadronic term was re-evaluated on
the lattice and the discrepancy fell below one standard deviation. The
anomaly the anatomy could not have explained turned out not to need
explaining.

\emph{Physics intuition:} the sealed dial tells you \emph{what} charge
is, and gives you $g = 2$; the elevator tells you how strong that charge
\emph{looks} from the floor you are standing on; and the last twelve
decimal places are the room the electron is standing in, which is never
empty and has to be summed over by hand. Three different kinds of fact,
and this Problem's claim is only that the first two are geometry.

\textbf{Where this Problem declines:} no value of $a_e$ or $a_\mu$ is
predicted, sharpened, or bounded here or anywhere in the cohort. A
future claim to have computed one from the two fibers would contradict
Eq.~\eqref{eq:anomsum-w2}'s own logic --- what arrives exact arrives from
topology, and $\alpha/2\pi$ is not exact in that sense.

\subsection{The principles, re-founded}

Summarize the Problem as a before-and-after. \emph{Before}: local
gauge invariance is a redundancy mysteriously promoted to a master
principle; renormalization is a successful procedure with an
unexplained arrow; charge quantization needs monopoles or grand
unification; anomaly exactness is a happy accident of one-loop
topology. \emph{After}: all four are one geometry --- the phase is
a real, compact, unreadable fiber (so gauge ``symmetry'' is the
law that physics speak only of its transport, and charge is its
winding); the scale is a real, open, one-way fiber (so
renormalization is descent, and its arrow is the composition law of
the flow along that fiber rather than a statistical accident); and the
anomaly ledger is the seam where the circle's exactness disciplines
the interval's flow. The grades are inherited: the identifications
are the constitution's (\AX{2}, \AX{3}); their quantitative
payout is Problem~4's. \textbf{Kill:} a measured violation of
charge quantization; an anomaly coefficient found to run; a
physical (scheme-independent) reversal of RG flow; or the failure
of the ledger's arithmetic anywhere the codebook stakes it ---
each would break this Problem at a named joint. None has ever
been observed.

\section{Problem 4.\ Too Many Free Parameters in the Standard Model and $\Lambda$CDM}
\label{sec:w2-prob4}

\subsection{The count, and why it hurts}

The Standard Model works by taking numbers from experiment and
giving back predictions of everything else. The taking is the
embarrassment: some twenty-eight quantities --- twelve fermion
masses, the mixing angles and phases of quarks and neutrinos, three
gauge couplings, the Higgs sector, the strong-CP angle --- must be
measured, not derived. Cosmology adds its own sheet: the density
fractions, the expansion rate, the primordial tilt and amplitude.
Einstein's hope --- a theory in which ``there are no arbitrary
constants'' --- has drifted so far out of reach that the modern
literature largely inverted it: perhaps the constants are
\emph{environmental}, drawn from a landscape, explained (if at all)
anthropically (Problem~15). The parameter count is not a cosmetic
flaw; it is the discipline's standing confession that it describes
the world rather than accounts for it.

Notice what kind of solution would count. Fitting the twenty-eight
with twenty-nine deeper parameters counts for nothing. Deriving a
few celebrated ratios while leaving the rest free counts for
little. What is wanted is a \emph{grammar}: a reason why this list,
these values, and no freedom --- with the derivation exposed,
entry by entry, to falsification.

\subsection{The codebook claim}

The cohort's claim, developed at full length in the codebook paper
\citep{ArisakaA9SM} and carried here in the anatomy's language: the
parameters are not dials but \emph{readings}, and every one obeys a
single grammatical rule ---

\begin{quote}
\emph{Every Standard-Model parameter is a holonomy-class label
wearing a descent address: the label is the circle's, the address
is the interval's.}
\end{quote}

The quantized content --- charges, representations, mixing angles
on the flavor lattice, the weak angle's rational partition --- is
circle-side: arithmetic of the five ledger integers
$(M,N,L;J,K) = (1,3,5;7,35)$ and of $\pi$, exact, unable to run,
which is why these entries arrive as closed forms with no error
bars of their own. The continuous-looking content --- masses,
scales, the vev --- is interval-side: exponential addresses of the
descent, computed from the same integers through the warp. Nothing
in the sheet is a third kind of thing, and that is the falsifiable
core: \emph{no third species of primitive}.

Figure~\ref{fig:param-tree-w2} draws the claim as a tree:
five constants and five integers at the root, the twenty-eight
parameters at the leaves, and derivation arrows --- not fits ---
in between.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=8mm and 6mm,
 rootbox/.style={rectangle, rounded corners=3pt, draw=black!80, line width=0.7pt, minimum width=84mm, minimum height=11mm, align=center, font=\small\bfseries\sffamily, fill=purple!10},
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 attrbox/.style={rectangle, rounded corners=2pt, draw=green!50!black, line width=0.5pt, minimum width=23mm, minimum height=10mm, align=center, font=\small\bfseries\sffamily, fill=green!12},
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 arrowstyle/.style={-{Latex[length=1.8mm]}, line width=0.4pt}
]
 \node[rootbox] (root) at (0, 7.5)
 {\GUfive{}: $c$, $G_N$, $\hbar$, $Q_B$, $k_B$ \quad+\quad OGN-5 ledger $(M,N,L;J,K) = (1,3,5;7,35)$};

 \node[scalebox] (EPl) at (-5.5, 5) {$E_{\rm Pl}$};
 \node[scalebox] (EGUT) at (-1.8, 5) {$E_{\rm GUT}$};
 \node[scalebox] (Mlock)at (1.8, 5) {$M_{\rm lock}$};
 \node[scalebox] (v) at (5.5, 5) {$v$};

 \draw[arrowstyle] (root) -- (EPl);
 \draw[arrowstyle] (root) -- (EGUT);
 \draw[arrowstyle] (root) -- (Mlock);
 \draw[arrowstyle] (root) -- (v);

 \node[font=\scriptsize\sffamily\itshape, below=1pt of EPl] {$\sqrt{\hbar c^5/G_N}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of EGUT] {$E_{\rm Pl}\, e^{-2\pi}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of Mlock] {$\tfrac12 E_{\rm Pl}\, e^{-K}$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of v, xshift=7mm] {$M_{\rm lock}(2/L)^N$};

 \node[attrbox] (alpha) at (-7.0, 2.6) {$\alpha^{-1}(0)$};
 \node[attrbox] (sw2) at (-3.5, 2.6) {$\sin^2\!\theta_W$};
 \node[attrbox] (alphas) at (0.0, 2.6) {$\alpha_s$};
 \node[attrbox] (vatt) at (3.5, 2.6) {$v$};
 \node[attrbox] (lamH) at (7.0, 2.6) {$\lambda_H$};

 %% circle-side labels come from the ledger, not from a scale: drawn
 %% dashed and routed from the root so the picture agrees with the text.
 \draw[arrowstyle, dashed, draw=purple!70!black]
   (root.west) to[out=180,in=95] (alpha.north);
 \draw[arrowstyle, dashed, draw=purple!70!black]
   (root.south west) to[out=215,in=95] (sw2.north);
 \draw[arrowstyle, dashed, draw=purple!70!black]
   (root.south) to[out=265,in=95] (alphas.north);
 \draw[arrowstyle] (v) -- (vatt);
 \draw[arrowstyle] (v) -- (lamH);

 \node[font=\scriptsize\sffamily\itshape, below=1pt of alpha] {$2^J + N^2$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of sw2] {$L^2/(4N^3)$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of alphas] {$1/(\pi e)$};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of vatt, xshift=-9mm] {Higgs vev};
 \node[font=\scriptsize\sffamily\itshape, below=1pt of lamH] {$2^{-N}$};

 \node[outbucket] (gauge) at (-8.0, -0.4) {3 gauge \\ couplings};
 \node[outbucket] (higgs) at (-4.8, -0.4) {2 Higgs \\ ($v$, $\lambda_H$)};
 \node[outbucket] (fermi) at (-1.6, -0.4) {12 fermion \\ masses};
 \node[outbucket] (ckm) at (1.6, -0.4) {4 CKM \\ params};
 \node[outbucket] (pmns) at (4.8, -0.4) {6 PMNS \\ params};
 \node[outbucket] (theta) at (8.0, -0.4) {1 strong-CP \\ $\bar\theta$};

 \draw[arrowstyle] (alpha) -- (gauge);
 \draw[arrowstyle] (sw2) -- (gauge);
 \draw[arrowstyle] (alphas) -- (gauge);
 \draw[arrowstyle] (vatt) -- (higgs);
 \draw[arrowstyle] (lamH) -- (higgs);
 \draw[arrowstyle] (vatt) -- (fermi);
 \draw[arrowstyle, dashed, draw=purple!70!black]
   (root.east) to[out=0,in=90] (theta.north);
 \draw[arrowstyle] (vatt.south) to[bend left=10] (ckm.north);
 \draw[arrowstyle] (vatt.south) to[bend left=20] (pmns.north);

 \node[font=\small\bfseries\sffamily, below=4mm of fermi, text=blue!50!black]
 {Total: $3 + 2 + 12 + 4 + 6 + 1 = 28$};
 \node[draw=black!35, rounded corners=2pt, fill=white, align=left,
       font=\scriptsize\sffamily, inner sep=3pt, anchor=north west] at (-9.4,-2.0)
 {\textcolor{purple!70!black}{\textbf{-- -- --}}\ \ circle side: ledger arithmetic, no scale enters\\
  \textbf{------}\ \ interval side: the address chain, scale by scale};
\end{tikzpicture}%
}
\caption[The codebook's derivation tree.]{\textbf{The codebook's derivation tree}
(reproduced from \citealp{ArisakaA9SM}). Top: the five constants
(\GUfive{}, \citealp{ArisakaA4Euclid}) and the integer ledger
$(M,N,L;J,K)$ (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}).
Second tier: the scale chain $E_{\mathrm{Pl}} \to E_{\mathrm{GUT}}
\to \Mlock \to v$ --- the interval's addresses, drawn with solid
arrows because each one is reached from the one above it. Third
tier: the five primary attractor coordinates. Three of them
($\alpha^{-1}(0)$, $\sin^2\theta_W$, $\alpha_s$) and the strong-$CP$
angle are reached by \emph{dashed} arrows straight from the ledger,
because they are circle-side quantities: rational partitions and
integer sums that no scale in the second tier can move. The
remaining two ($v$, $\lambda_H$) descend from the address chain,
which is why they carry units and the others do not. Bottom:
the six buckets accounting for the twenty-eight Standard-Model
parameters. Every arrow is a published derivation; no continuous
freedom enters at any tier. This is
Eq.~\eqref{eq:chain-w2} as an organism.}
\label{fig:param-tree-w2}
\end{figure}

\subsection{Reading the grammar: five worked entries}

Plain-English walk-throughs of five entries, chosen to span the
sheet; the full audit is the number table of
\S\ref{sec:numbers-audit} and the codebook's own census.

\begin{itemize}[leftmargin=2.0em]
\item \emph{The fine-structure constant.}
$\alpha^{-1}(0) = 2^J + N^2 = 128 + 9 = 137$ at leading order:
pure circle arithmetic --- a dyadic power of the switch integer
plus the color count squared --- with the measured $137.036$
reached by the codebook's declared dressing. The most famous
number in physics is spelled from two integers of the ledger.
\item \emph{The weak angle.}
$\sin^2\theta_W = L^2/(4N^3) = 25/108 = 0.2315$ against the
measured $0.2312$: a rational partition --- the circle dividing
its own trace --- not a running accident.
\item \emph{The electroweak scale.} $v = \Mlock(2/L)^N = 246$~GeV:
the gearbox form --- an interval depth ($\Mlock = \tfrac12
M_{\mathrm{Pl}}e^{-K}$) dressed by a circle fraction. The
address explains the value; Problem~6 explains why no tuning was
ever needed.
\item \emph{A fermion mass.} The quark floors take the form
(winding label) $\times$ (radial overlap): which particle it is
comes from the circle; how heavy it is comes from where its
support sits on the interval. Identical particles have identical
masses with no enforcement mechanism, because a label is a class
and classes do not vary (Problem~8, Horizon 2).
\item \emph{A cosmological share.} $\Omm = 1/\pi$,
$\Omb = 1/(2\pi^2)$: fiber volumes --- the Hopf circle inside the
electroweak three-sphere --- so the ``parameters'' of the
inventory are measures of the compact geometry, epoch-frozen
(Problem~7).
\end{itemize}

The grammar also explains the sheet's most puzzling texture: why
some entries are precise integers and rationals while others
sprawl across twelve orders of magnitude. The integers are labels
(the circle does arithmetic); the sprawl is addresses (the
interval does exponentials). One list, two fonts --- and a century
of attempts to derive the whole sheet in one font explains a
century of partial success.

Figure~\ref{fig:two-fonts} shows the sheet as the codebook
reads it: one alphabet, two fonts, and no third column for dials.

\begin{figure}[htbp]
\centering
\resizebox{0.665\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% fibre icons
\draw[line width=1.10pt] (2.4,3.3) circle (0.7);
\node at (2.4,3.3) {\scriptsize $S^1_\theta$};
\node[align=center] at (2.4,4.55)
  {\scriptsize \textbf{labels}: the circle's font\\[-2pt]
   \scriptsize integers, windings, phases};
\draw[->,line width=1.10pt] (7.4,3.3) -- (10.8,3.3);
\node at (9.1,3.65) {\scriptsize $I_r$};
\node[align=center] at (9.1,4.55)
  {\scriptsize \textbf{addresses}: the interval's font\\[-2pt]
   \scriptsize depths, e-folds, exponentials};
% label box
\node[draw=black!55,line width=0.80pt,rounded corners,align=left] (lab) at (2.4,0.9)
  {\scriptsize $\alpha^{-1}(0) = 2^7 + 3^2 = 137$\\[1pt]
   \scriptsize $\sin^2\theta_W = 5^2/(4\cdot 3^3) = 25/108$\\[1pt]
   \scriptsize $N_g = 3$; \ $\alpha_{31} = \pi$ (the driver)\\[1pt]
   \scriptsize $\Omega_m = 1/\pi$; \ $\Omega_b = 1/(2\pi^2)$};
% address box
\node[draw=black!55,line width=0.80pt,rounded corners,align=left] (add) at (9.1,0.9)
  {\scriptsize $M_{\mathrm{lock}} = \tfrac12 M_{\mathrm{Pl}}\, e^{-35}$\\[1pt]
   \scriptsize $v = M_{\mathrm{lock}} (2/5)^3 = 246$~GeV\\[1pt]
   \scriptsize $m_\Lambda \sim e^{-2K}$: the depth, doubled\\[1pt]
   \scriptsize $\eta_B \sim e^{-K/2}$: the gated crossing};
\draw[->,line width=0.80pt,blue!65!black] (lab.north) -- (2.4,2.5);
\draw[->,line width=0.80pt,blue!65!black] (add.north) -- (9.1,2.9);
\node[align=center] at (5.75,-1.0)
  {\scriptsize \textbf{one alphabet, two fonts --- zero dials:}\\[-2pt]
   \scriptsize every entry either a holonomy-class label or a descent
   address (or a label wearing an address)};
\end{tikzpicture}%
}
\caption[The sheet's two fonts.]{\textbf{The sheet's two fonts.} A sample of the parameter
sheet sorted the codebook's way. Left: labels --- pure ledger
arithmetic, protected by the circle, unable to run. Right:
addresses --- depths on the interval, set by the descent. A
parameter that looked like a dial for a century is a word in this
grammar; the audit of \citet{ArisakaA9SM} conjugates all
twenty-eight.}
\label{fig:two-fonts}
\end{figure}

The five entries above are a sample. Figure~\ref{fig:a9-thirteen-w2}
shows the thirteen most sharply benchmarked entries of the sheet as
the parent paper presents them --- each closed form written in the
ledger's alphabet, beside the measured value and the fractional
deviation \citep{ArisakaA9SM}. Nothing in the left-hand column was
adjusted to reach the right-hand column; that is the whole of the
claim, and the whole of the exposure.

\begin{figure}[p]
\centering
\noindent\begin{minipage}{\textwidth}
\centering
{\sffamily\LARGE\bfseries 28 Free Parameters, One Geometry\par}
\vspace{0.9em}

{\small
\setlength{\tabcolsep}{4pt}%
\begin{tabular}{@{}llrrrr@{}}
\toprule
Quantity & \GG{}-Theory closed form & First order & Measured & Dev & Pull\\
\multicolumn{6}{@{}l}{\bfseries\itshape Five primary gauge--Higgs attractor coordinates}\\
\hspace{1.5em}$\alpha^{-1}(M_Z)$ & $2^J = 128$ & $128.0$ & $127.930(8)$ & $+5.5\mathrm{e}{-}4$ & $+0.99$\\
\hspace{1.5em}$\sin^2\theta_W(M_Z)$ & $L^2/4N^3 = 25/108$ & $0.23148$ & $0.23155(4)$ & $-3.0\mathrm{e}{-}4$ & $-1.70$\\
\hspace{1.5em}$\alpha_s(M_Z)$ & $1/\pi e$ & $0.11710$ & $0.1180(9)$ & $-7.6\mathrm{e}{-}3$ & $-0.70$\\
\hspace{1.5em}$v$ \,[GeV] & $\Mlock\,(2/L)^N$ & $246.33$ & $246.220$ & $+4.5\mathrm{e}{-}4$ & $+1.00$\\
\hspace{1.5em}$m_H$ \,[GeV] & $\sqrt{2\lambda_H}\,v,\ \lambda_H = 1/8$ & $123.12$ & $125.10(9)$ & $-1.6\mathrm{e}{-}2$ & $-0.95$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Four derived electroweak observables}\\
\hspace{1.5em}$G_F$ \,[$10^{-5}\,$GeV$^{-2}$] & $1/(\sqrt2\,v^2)$ & $1.1653$ & $1.16638$ & $-9.1\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$\alpha^{-1}(0)$ & $2^J + N^2 = 128 + 9$ & $137.0$ & $137.036$ & $-2.6\mathrm{e}{-}4$ & $-1.00$\\
\hspace{1.5em}$m_W$ \,[GeV] & $\tfrac12\,g_2 v$ & $80.21$ & $80.369$ & $-2.0\mathrm{e}{-}3$ & $-0.99$\\
\hspace{1.5em}$m_Z$ \,[GeV] & $\tfrac12\,v\sqrt{g_2^2+g_Y^2}$ & $91.50$ & $91.188$ & $+3.4\mathrm{e}{-}3$ & $+1.01$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Three charged-lepton masses}\\
\hspace{1.5em}$m_e$ \,[MeV] & $(v/\sqrt2)\,N/(L^3\,2^{J+L+1})$ & $0.51030$ & $0.51100$ & $-1.4\mathrm{e}{-}3$ & $-0.88$\\
\hspace{1.5em}$m_\mu$ \,[MeV] & $(v/\sqrt2)\,L/2^{J+L+1}$ & $106.31$ & $105.658$ & $+6.2\mathrm{e}{-}3$ & $+0.98$\\
\hspace{1.5em}$m_\tau$ \,[GeV] & $(v/\sqrt2)\,NJ/2^{J+N+1}$ & $1.7861$ & $1.77693$ & $+5.1\mathrm{e}{-}3$ & $+1.09$\\
\addlinespace
\multicolumn{6}{@{}l}{\bfseries\itshape Cabibbo angle (first CKM mixing)}\\
\hspace{1.5em}$\sin\theta_{12}^{\rm CKM}$ & $\bigl|e^{-\pi/2} - e^{i3\pi/5}e^{-\pi}\bigr|$ & $0.22502$ & $0.22501$ & $+3.8\mathrm{e}{-}5$ & $+0.01$\\
\bottomrule
\end{tabular}}

\vspace{0.9em}

\includegraphics[width=\textwidth,height=0.40\textheight,keepaspectratio]{a9_dev_plot_linear_first8_hook.png}
\end{minipage}
\caption[The thirteen most sharply benchmarked entries of the sheet.]{\textbf{The
thirteen most sharply benchmarked entries of the sheet}
(reproduced from \citealp{ArisakaA9SM}). Each closed form is written
in the ledger's own alphabet --- the switch integer $J = 7$, the
family count $N = 3$, the lattice integer $L = 5$, the lock depth
$K = 35$ --- and set beside the measured value with its fractional
deviation and pull. The two fonts of
Figure~\ref{fig:two-fonts} are visible by row: $\alpha^{-1}$ and
$\sin^2\theta_W$ are circle-side arithmetic, integers and rational
partitions that cannot run; $v$ and the lepton masses are
interval-side addresses, a depth $\Mlock = \tfrac12 M_{\rm Pl}
e^{-K}$ dressed by a circle fraction. The lower panel plots the
same thirteen on a linear deviation axis at first and second order.
No entry in the closed-form column contains a quantity fitted to
the measured column.}
\label{fig:a9-thirteen-w2}
\end{figure}


\subsection{How the sheet filled up: a two-paragraph history}

The parameter sheet is the residue of a century of victories. Each
revolution explained structure and deposited constants: quantum
mechanics explained the atom and deposited $\hbar$ and the
electron's mass; the electroweak theory explained radioactivity
and deposited the vev and the mixing angles; QCD explained the
hadrons and deposited its coupling and quark masses; inflationary
$\Lambda$CDM explained the sky and deposited its six-parameter
concordance. The pattern was so steady it acquired a name ---
\emph{effective field theory}: every layer's constants are the
next layer's problem. The sheet, in other words, is not a scandal
but a frontier marker: it records exactly where explanation
stopped.

What changed the stakes was the failure of the expected next step.
Grand unification derived a few relations and predicted proton
decay --- unobserved at the predicted rates; supersymmetry
promised to relate the sheet's halves --- unfound at the LHC. The
sheet stayed full, and the discipline's response bifurcated:
downward, into the landscape (the sheet is environmental,
Problem~15), or inward, into the possibility this paper develops
--- that the sheet was never a list of constants at all, but a
text in an unrecognized script.

\subsection{The chain, worked end to end}

The grammar is best displayed by walking one unbroken chain from
the ledger to the laboratory, every step a published closed form.
Start from the five integers, fixed by the anomaly audit of
Problem~3: $(M,N,L;J,K) = (1,3,5;7,35)$, with the closure
identities $K = LJ$ and $J = 2N+1$ binding them. The chain, displayed as
the cascade it is:
\begin{align}
\Mlock &= \tfrac12\, M_{\mathrm{Pl}}\, e^{-K}
        = 3.849~\mathrm{TeV}
        && \text{the interval's depth sets the lock,}
\nonumber\\
v &= \Mlock \left(\tfrac{2}{L}\right)^{N}
   = 246.3~\mathrm{GeV}
   && \text{(measured 246.22): the circle dresses it,}
\nonumber\\
\sin^2\theta_W &= \frac{L^2}{4N^3} = 0.23148
   && \text{(measured 0.23122): the partition,}
\nonumber\\
\hat\alpha^{-1}(M_Z) &= 2^{J} = 128
   && \text{(measured 127.93): the dyadic anchor,}
\nonumber\\
\alpha^{-1}(0) &= 2^{J} + N^{2} = 137
   && \text{(measured 137.036): the infrared end,}
\label{eq:chain-w2}
\end{align}
and the same integers, run through the cosmological sheet, return
$\Omm = 1/\pi$ (Eq.~\eqref{eq:inventory-p2}) and the Kaxion's
$1/K$ doublet. Each line's structure repeats the last: an exact
integer expression (the circle's font) evaluated at, or dressing,
an exponential depth (the interval's). One
ledger, six measured numbers, zero dials --- and the chain is the
\emph{same} chain at every step: label arithmetic setting
structure, depth arithmetic setting scale. The reader who wants
all twenty-eight walks the codebook \citep{ArisakaA9SM}; the
reader who wants the audit walks \S\ref{sec:numbers-audit}.

\subsection{The numerology objection, met head-on}

A sheet of integer formulas invites the gravest charge in the
business: post-hoc numerology --- with enough integers and
operations, any number can be dressed. The cohort's defense is
procedural, and worth stating because it is the difference between
this Problem and its disreputable cousins. \emph{First,
pre-registration}: the closed forms are published with named kill
conditions and dated instruments before the decisive measurements
(the falsifier registry; \S\ref{sec:calendar}) --- numerology
never volunteers to die. \emph{Second, one alphabet}: the same
five integers must serve every entry --- each new closed form is a
constraint on the same tiny ledger, not a fresh dip into the
integer sea; the fits that numerology enjoys grow easier with each
success, while this sheet grows \emph{harder}. \emph{Third, the
flag register}: candidate patterns without mechanism (the
$4K$/$8K$ family) are quarantined at flagged grade, cited as
support never --- a discipline numerology cannot afford.
\emph{Fourth, cross-exposure}: the same integers face colliders,
haloscopes, and the CMB, with no shared systematics; a conspiracy
of coincidences would need to span twelve orders of magnitude of
experimental technique. The charge deserves to be leveled; the
procedure is the answer.

\subsection{The sheet's decade: from measured to audited}

The codebook converts the parameter problem from a lament into a
schedule, and the schedule is running now. Each class of entry has
its auditor: the collider cluster at $3.849$~TeV audits the
depth-arithmetic that anchors every mass address
(\S\ref{sec:calendar}); DUNE and Hyper-Kamiokande audit the
flavor lattice's sharpest letter ($\dPMNS = 196.4^\circ$ ---
a zero-parameter angle sitting in the experimentally favored
quadrant); the haloscopes audit the ledger's $1/K$ in a
laboratory spectrum; the Simons Observatory and LiteBIRD audit the launch's two
outputs; and the precision-electroweak program continues to
grind on the closed forms' dressed digits. A decade of instruments,
each reading a different row of one sheet --- and the sheet's
grammar makes the audits mutually reinforcing in a way fitted
frameworks cannot imitate: because every row spends the same five
integers, each confirmation tightens all rows, and each failure
indicts them all. The parameter problem's endgame, on this
reading, will not be a theoretical proclamation; it will be a
decade of returned invoices, and the sheet is priced to be worth
exactly what survives.

Figure~\ref{fig:param-status-w2} shows the audit's current
state --- including, prominently, what is \emph{not} yet derived.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\fontsize{13}{15}\selectfont, align=center},
 bar/.style={draw=black!55, minimum height=11mm}
]
% Stacked horizontal bar visualizing 28 parameters by status
\definecolor{derivedG}{RGB}{46, 139, 87}
\definecolor{condY}{RGB}{218, 165, 32}
\definecolor{assertedR}{RGB}{178, 34, 34}

% Width: 28 units total
% 6 derived (green), 13 conditional (gold), 9 asserted (red)
\fill[derivedG] (0,0) rectangle (6,1.1);
\fill[condY] (6,0) rectangle (20,1.1);
\fill[assertedR] (20,0) rectangle (28,1.1);
\draw (0,0) rectangle (28,1.1);

% Counts inside
\node[white] at (3,0.55) {\textbf{6 DERIVED}};
\node[white] at (13,0.55) {\textbf{14 DERIVED (COND.)}};
\node[white] at (24,0.55) {\textbf{8 ASSERTED}};

% Tick marks for parameter index
\foreach \x in {0,...,28}{
 \draw (\x,1.1) -- (\x,1.25);
}
\foreach \x/\lab in {0/0,7/7,14/14,21/21,28/28}{
 \node[font=\fontsize{9}{11}\selectfont] at (\x,1.5) {\lab};
}
\node[font=\fontsize{9}{11}\selectfont] at (14,1.95) {parameter index};

% Status legend below (font bumped for readability)
\node[anchor=west, derivedG, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-0.9) {\textbullet\ Derived (6):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (4.6,-0.9) {$\sin^2\theta_W = 25/108$;\ \ $\alpha_s^{-1}(M_Z) = \pi e$;\ \ $\lambda_H = 1/8$;\ \ $y_t = 1$;\ \ $\alpha_{21} = 0$;\ \ $\alpha_{31} = \pi$.};
\node[anchor=west, condY, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-1.8) {\textbullet\ Derived (cond., 14):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (6.7,-1.8) {$\alpha^{-1}(M_Z) = 2^J = 128$;\ \ $v$;\ \ 4 CKM;\ \ 3 PMNS angles $+\ \delta_{\rm CP}^{\rm PMNS}$;\ \ 3 neutrino mass scales;\ \ $\bar\theta$.};
\node[anchor=west, assertedR, font=\fontsize{16}{19}\selectfont\bfseries] at (0.0,-2.7) {\textbullet\ Asserted (8):};
\node[anchor=west, font=\fontsize{13}{15}\selectfont] at (4.6,-2.7) {8 charged-fermion Yukawas $(m_e, m_\mu, m_\tau, m_u, m_d, m_c, m_s, m_b)$.};
\end{tikzpicture}}
\caption[The 28-parameter audit at a glance.]{\textbf{The 28-parameter audit at a glance}
(reproduced from \citealp{ArisakaA9SM}). Six parameters Derived
outright (green), fourteen derived modulo a single named
structural lemma each (gold), eight still Asserted with an
explicit promotion roadmap (red). The honesty is the point: the
codebook's claim is not ``all twenty-eight are theorems'' but
``all twenty-eight are \emph{graded}, and the grades are public''
--- the difference between a codebook and a fit is that a codebook
can show you which pages are finished.}
\label{fig:param-status-w2}
\end{figure}

\subsection{What ``no dials'' buys, and what it costs}

A dial-free sheet buys the strongest currency in science:
exposure. A fitted framework meets an unwelcome measurement by
adjusting; this construction can only be right or wrong. Every
closed form above is separately falsifiable at precision, the
comparisons span instruments with no shared systematics, and a
single parameter that resists both the ledger arithmetic and the
declared-input list kills the no-third-species claim outright ---
not wounds: kills. The cost is paid in the same currency: the
framework cannot absorb surprises. The cohort regards the trade as
the whole point, and its papers stake the calendar on it
(\S\ref{sec:calendar}).

What honestly remains is short, and the honest ledger of
\S\ref{sec:honest} lists it: the declared inputs (the scalar
amplitude; the named joints of the $\etaB$ chain), the open
integers (the Kaxion's cap loading), the flagged candidates (the
$4K$/$8K$ family), and the per-parameter grades of the codebook's
own triage --- derived, derived-conditional, named --- which this
paper inherits and never promotes. Einstein's hope, in this
accounting, is neither achieved nor abandoned: it is
\emph{scheduled}, entry by entry, with the remaining freedom
counted on one hand and each item owned by a named campaign.

\textbf{Grade and kill, consolidated.} Grades per parameter,
inherited. Kill: distributed across the sheet --- any closed form
off its dressing; any parameter demanding a third species; the
collider, haloscope, and CMB switches of \S\ref{sec:kill} each
audit a different row. The parameter problem, on the anatomy, was
never a problem of too many numbers: it was one grammar, unread
--- \emph{the sheet is a codebook, and the code is the two
fibers.}

%% ===================================================================
\subsection*{Part III in one paragraph, and what comes next}

Four embarrassments of the Standard Model, one diagnosis. The
measurement problem is two dynamics because there are two fibers:
the circle carried the coherent phase, the interval wrote the
outcome, and the two rules were never rivals. The four interactions
look unalike because three of them are the circle's registration
modes and one is the interval's agent. Gauge invariance is the
circle's unreadability and renormalization is motion along the
interval --- which is why one is exact and the other flows, and why
a century of trying to explain both in one language explained
neither. And the parameter sheet is a codebook in two fonts:
integers and rational partitions where the circle wrote, exponential
depths where the interval did.

The same two fonts now spell the sky. Part~IV takes the four
questions cosmology cannot close --- what launched the expansion,
why the scales stand so far apart, what most of the universe is
made of, and why regions that never met agree --- and finds the
interval doing the descending and the circle doing the keeping, in
every one of them.

%%  PART IV -- COSMOLOGY
%% ===================================================================
\GGPart[Cosmology]{Cosmology}{Problems 5--8: the Beginning,
Naturalness and Hierarchy, the Dark Trio, and the Three Horizons}

\noindent The four problems of this Part are the ones a newspaper
asks first: how did it start, why are the numbers so extreme, what
is the dark stuff, and why is it the same everywhere. All four are developed below; Problem~7 carries the fullest
quantitative apparatus.

\section{Problem 5.\ The Beginning of the Universe}
\label{sec:w2-prob5}

\subsection{The oldest question, and its modern form}

What happened at the beginning? The twentieth century converted
the question from theology to physics and then discovered that its
best theories resign at the door. Run general relativity backward
and the singularity theorems promise a point where curvature
diverges and the equations fail --- not a description of the
beginning, but a certificate that the description ends.
Quantum-gravity programs soften the point but have not replaced
it with a calculation matched to data. And inflation, for all its
successes, made the initial-condition half of the question sharper
rather than softer: the inflaton's starting value, the patch that
inflated, the pre-inflationary state --- each is an input, and in
the eternal-inflation reading the beginning recedes into an
unending regress of beginnings. The modern form of the question is
therefore double: \emph{what replaces the singularity}, and
\emph{who chose the initial data}.

\subsection{The question transformed: from ``what banged'' to ``what launched''}

On the anatomy, cosmic history is the one-way reading of the bulk
(Problem~13): a descent along the interval, depositing on the
boundary the letters stored on the circle. A reading has no
detonation; it has an \emph{opening}. The question ``what
banged?'' presupposes an event in time; the anatomy replaces it
with a structure in geometry --- the top of the descent --- and
asks instead: \emph{what launched the reading, and is the launch
computable?} That question, unlike the original, has been
answered in part: the launch is the interval's coordinate rolling
once down its derived plateau (Problem~7), with the e-fold count
$N^* = 52.6$, the tilt $n_s = 0.9620$, and the tensor target
$r = 0.0043$ computed rather than assumed, and the whole
subsequent chronology --- the straight line of
Figure~\ref{fig:chronology}, closing its age ledger at $140.24$
--- following with no further input \citep{ArisakaA10Cosmos}. The
Big Bang, in this reading, is not an explosion at a point but the
top of a slide: hot because the roll's energy reheated the
boundary, uniform because one patch was stretched over the sky
(Problem~8), and lawful from the first e-fold because the laws
are the groove, not products of the fire.

Figure~\ref{fig:gut-vs-planck-w2} draws the transformation of
the question.

\begin{figure}[htbp]
\centering
\resizebox{0.547\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\footnotesize, >={Stealth[length=2.4mm]}]
% ---- LEFT: conventional ----
\def\Lx{0}
\fill[blue!8] (\Lx,0) rectangle (\Lx+3.1,7.2);
\draw[->,line width=0.80pt] (\Lx,-0.2) -- (\Lx,8.1) node[above,font=\small]{$E$};
\node[align=center, font=\small\bfseries] at (\Lx+1.55,7.9) {Conventional};
\draw[line width=1.10pt, red!70!black] (\Lx,7.2) -- (\Lx+3.1,7.2);
\node[right, red!60!black, font=\scriptsize] at (\Lx+0.12,7.45) {$\EPl\sim10^{19}$ GeV};
\node[align=center, font=\scriptsize] at (\Lx+1.55,6.6) {Planck wall: singular\\ beginning ($a\to0,\ \rho\to\infty$)};
\node[align=center, font=\itshape] at (\Lx+1.55,3.4) {4D spacetime\\ all the way up};
\node[below, align=center, font=\scriptsize] at (\Lx+1.55,-0.35) {``begins at the\\ Planck scale''};
% ---- RIGHT: GG-Theory ----
\def\Rx{6.3}
\fill[black!10] (\Rx,4.9) rectangle (\Rx+3.1,7.2);
\fill[orange!16] (\Rx,0) rectangle (\Rx+3.1,4.9);
\draw[->,line width=0.80pt] (\Rx,-0.2) -- (\Rx,8.1) node[above,font=\small]{$E$};
\node[align=center, font=\small\bfseries] at (\Rx+1.55,7.9) {\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+1.55,6.0) {not part of the manifold:\\ space \& time undefined\\ ($r<0$)};
\draw[line width=1.10pt, red!70!black] (\Rx,4.9) -- (\Rx+3.1,4.9);
\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+1.55,2.4) {4D spacetime / FRW\\ the observable universe\\ ($0\le r\le K$)};
\node[below, align=center, font=\scriptsize] at (\Rx+1.55,-0.35) {``begins at the\\ GUT edge''};
\end{tikzpicture}}
\caption[Where the universe begins.]{\textbf{Where the universe begins} (reproduced from
\citealp{ArisakaA10Cosmos}). \emph{Left:} the conventional picture
--- the description runs back to the Planck edge and ends in a
singularity, so the beginning is where physics stops. \emph{Right:}
the anatomy's picture --- the descent begins at the GUT surface
($r = 0$, the top of the interval), the Planck scale is demoted to
a unit of account above it, and the open item is an
\emph{entrance}, not a wall. The question changes shape before it
is answered: not ``what happened at the singularity?'' but ``what
placed the coordinate at the top of the corridor?''}
\label{fig:gut-vs-planck-w2}
\end{figure}

\subsection{Who chose the initial data: nobody}

The second half of the question dissolves rather than resolves,
and the dissolution is the determinism triad of
\S\ref{sec:determinism} applied at the top. The standard framework
needs initial data because its dynamics is reversible: every
history is possible, so something must select ours. The descent
needs none, in the precise sense that nothing selectable remains:
the vacuum dial is quantized with a unique scan winner, the
landscape has one floor per stratum, and the compact angles are
pinned coherently during the launch (\S\ref{sec:pinning-w2}) ---
so the ``initial conditions'' of every subsequent era are
\emph{outputs} of the reading, not inputs to it. The cohort calls
this \emph{determined initial data}, and its signature is that it
over-predicts: a chosen-initial-data universe could not compute
the zero-isocurvature null, the patch-uniform relic density, or
the F25 statistical fossil (Problem~1) --- a framework with
freedom at the start has nothing to say about fossils of the
start. This one stakes all three.

\subsection{The Planck scale, demoted to a unit}

Where in this is the Planck era --- the quantum-gravity cauldron
of the popular account? The anatomy performs a quiet demotion: the
Planck scale appears in this paper's chronology as a \emph{unit}
--- the anchor $t_{\mathrm{Pl}}$ in which the age ledger is
counted, the normalization $M_{\mathrm{Pl}}$ from which the
descent's addresses are measured --- and not as an epoch the
history must survive. The descent's own account opens at the
GUT-edge corridor, some three orders below Planck; nothing in the
derived chronology requires physics at the Planck point itself,
and the famous ``first $10^{-43}$ seconds'' is, on this
bookkeeping, not a moment of the story but a rung of the address
system. This is a genuine difference of framing with the standard
account, and an honest one: the anatomy does not solve
trans-Planckian physics; it declines to need it for anything it
computes.

\subsection{What inflation solved, and the bill it left}

Honesty about the standard framework sharpens what is new here.
Inflation solved real problems --- flatness, horizons, the origin
of structure --- and its acoustic predictions are confirmed in
detail. But its solutions were purchased on credit, and the
unpaid items define the modern beginning-problem: \emph{who chose
the potential} (a functional freedom the census of Problem~7
closes); \emph{who chose the starting point} (initial-data
freedom, closed above by the determined data); \emph{what came
before the inflating patch} (the entrance, still open here too,
but named and shaped); and \emph{how to stop the recursion} ---
in generic potentials inflation is eternal, endlessly re-seeding,
so that the theory's late-time predictions drown in the measure
problem. The anatomy's launch pays these bills or names them: one
potential (forced), one starting configuration (pinned), one
traverse (a semigroup cannot re-seed --- the once-only clause
retires eternal inflation structurally, not by assumption), and
one open item (the entrance), honestly carried. The comparison to
the bounce and no-boundary traditions is structural: those
programs seek to \emph{continue} the geometry through or before
the beginning; the anatomy's descent instead \emph{terminates}
lawfully at an edge --- the corridor's top --- and asks what
boundary data the constitution puts there. Whether an edge or a
continuation is the truth is exactly what the entrance item must
decide; the anatomy's contribution is to make it an item rather
than a mystery.

Figure~\ref{fig:launch-w2} draws the opening act: the plateau,
the one roll, and the three observational windows staked on it.

\begin{figure}[htbp]
\centering
\resizebox{0.674\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% axes
\draw[->,line width=0.80pt] (0,0) -- (11.6,0);
\node[below] at (11.2,-0.05) {\scriptsize $r$};
\draw[->,line width=0.80pt] (0,0) -- (0,5.5);
\node[left] at (0,5.3) {\scriptsize $V$};
% potential: plateau then descent to well
\draw[line width=1.10pt,blue!65!black]
  (0.4,3.6) -- (5.2,3.5)
  .. controls (6.6,3.3) and (7.4,2.0) .. (8.0,0.9)
  .. controls (8.4,0.25) and (9.0,0.25) .. (9.4,0.9);
% entrance
\node[red!70!black] at (0.55,4.05) {\scriptsize \textbf{?}};
\node[red!70!black,align=center] at (1.9,5.05)
  {\scriptsize the entrance: the named\\[-2pt]\scriptsize open item};
\draw[->,red!70!black] (1.3,4.7) -- (0.75,4.2);
% the roll
\draw[->,line width=0.80pt] (1.6,3.8) -- (4.6,3.72);
\node at (3.6,4.25) {\scriptsize the roll: once, slowly};
\draw[decorate,decoration={brace,amplitude=5pt}] (0.9,3.25) -- (7.6,1.45);
\node[rotate=-16] at (4.6,2.05) {\scriptsize $N^{*} = 52.6$ e-folds};
% windows
\node[align=center] at (2.9,1.15)
  {\scriptsize \textbf{tensor window}\\[-2pt]
   \scriptsize $r_{\mathrm{T}} = 0.0043$: the roll's own noise};
\draw[->,black!60] (2.9,1.6) -- (3.4,3.35);
\node[align=center] at (6.6,4.0)
  {\scriptsize \textbf{statistics window}\\[-2pt]
   \scriptsize F25: relaxation unfinished\\[-2pt]
   \scriptsize on the largest scales};
\draw[->,black!60] (5.9,3.55) -- (5.4,3.55);
\node[align=center] at (10.3,1.9)
  {\scriptsize \textbf{null window}\\[-2pt]
   \scriptsize pinning: zero isocurvature,\\[-2pt]
   \scriptsize no defects};
\draw[->,black!60] (9.6,1.45) -- (9.1,0.85);
\node[below] at (8.7,-0.05) {\scriptsize the lock};
\end{tikzpicture}%
}
\caption[The launch, with its three windows.]{\textbf{The launch, with its three windows.} The plateau
class is forced by the census; the roll is the interval's one
traverse of its top, $N^{*} = 52.6$ e-folds; the entrance --- why
the coordinate begins on the high shoulder --- is the cohort's
named open item, marked honestly. Three instruments watch:
polarization ($r_{\mathrm{T}} = 0.0043$), large-scale statistics
(the F25 relic), and the pinning's double null. A derived opening
leaves derived fingerprints.}
\label{fig:launch-w2}
\end{figure}

The mechanism deserves its equations, because they are the
ones the sky has already graded. During slow roll the field
$\varphi$ (the interval's coordinate, bent into a potential
$V(\varphi)$) obeys the two smallness conditions
\begin{equation}
\epsilon \;=\; \frac{\Mplred^2}{2}\left(\frac{V'}{V}\right)^{2}
\ll 1,
\qquad
\eta \;=\; \Mplred^2\,\frac{V''}{V} \ll 1,
\label{eq:slowroll-w2}
\end{equation}
where $V'$ and $V''$ are the potential's first and second
derivatives with respect to $\varphi$: $\epsilon$ measures the
slope (how fast the roll steals energy from the stretch) and
$\eta$ the curvature (how soon the slope changes). While both are
small the expansion is nearly exponential, and the number of
e-folds accumulated between a field value $\varphi$ and the roll's
end is
\begin{equation}
N \;=\; \frac{1}{\Mplred^2}\int_{\varphi_{\mathrm{end}}}^{\varphi}
\frac{V}{V'}\,d\varphi' .
\label{eq:efolds-w2}
\end{equation}
The observables of Eq.~\eqref{eq:ns-r} are these quantities
evaluated at the moment today's largest scales left the horizon:
$n_s = 1 - 6\epsilon + 2\eta$ and $r = 16\epsilon$ in general,
collapsing on the plateau class to the one-parameter forms
$n_s = 1 - 2/N^*$ and $r = 12/N^{*2}$. The physical point: for a
plateau potential the two measured numbers depend on \emph{nothing}
except how far from the end the observable window sits --- which is
why a derived $N^* = 52.6$ makes both numbers predictions rather
than fits, and why the entrance (which fixes where the roll begins)
is the one item the account still owes.

Figure~\ref{fig:slowroll-w2} draws the standard mechanism as the
textbooks give it, so that the one place this account departs from
them is visible: not the slow-roll machinery, which is inherited
whole, but the potential's shape, which here follows from the warp
rather than from a choice, and the exit length $N^{*}$, which is a
distance rather than a dial.

\begin{figure}[htbp]
\centering
\resizebox{0.594\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  >={Stealth[length=2.2mm]}]

% ---- axes ----
\draw[->,line width=0.80pt] (-0.3,0) -- (10.2,0) node[right] {$\varphi$};
\draw[->,line width=0.80pt] (0,-0.3) -- (0,5.9) node[above] {$V(\varphi)$};

% ---- 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: fixed by the staked r and the one named input A_s,
% V^{1/4} = Mplred (3 pi^2 A_s r / 2)^{1/4}
\draw[dashed,gray] (0,4.25) -- (3.2,4.25);
\node[anchor=west,align=left] at (0.15,4.78)
  {\footnotesize $V^{1/4} \simeq 8\times 10^{15}$\,GeV};

% ---- 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) -- (4.15,1.95)
  node[anchor=north 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=left,orange!55!black,anchor=west] at (8.55,0.62)
  {reheating /\\oscillation};

% ---- inset box: slow-roll definitions ----
\node[draw=black!50,rounded corners,fill=gray!6,align=left,inner sep=3pt,
      anchor=north east] at (10.15,5.88)
  {\scriptsize
   $\displaystyle \epsilon=\tfrac{\Mplred^2}{2}\!\left(\tfrac{V'}{V}\right)^{\!2}$,\ \
   $\displaystyle \eta=\Mplred^2\,\tfrac{V''}{V}$\\[1pt]
   $n_s=1-6\epsilon+2\eta$,\ \ $r=16\epsilon$\\[1pt]
   $\displaystyle N=\!\int\!\tfrac{V}{\Mplred^2 V'}\,d\varphi$
   \ \ ($\Mplred$ reduced)};

% ---- grade note, in this paper's register ----
\node[draw=black!50,rounded corners,fill=gray!5,align=center,
      inner sep=3pt,anchor=north,text=black!70,text width=98mm]
      at (5.0,-1.15)
  {\scriptsize\textbf{Grades on this figure.}
   $n_s = 0.9620$ and $r = 0.0043$ are derived-conditional (plateau
   class $+$ prompt crossing) at a \emph{derived} $N^{*} = 52.6$;
   $A_s$ is the one named input; the entrance --- where the roll
   begins --- is open.};

\end{tikzpicture}}
\caption[Slow roll, from the standard mechanism.]{\textbf{Slow roll, from the standard mechanism}
(reproduced from \citealp{ArisakaA10Cosmos}). The plateau potential
$V(\varphi)$ with the three phases marked: the slow roll along the
flat top (where Eq.~\eqref{eq:slowroll-w2}'s $\epsilon, \eta \ll 1$
and the stretch accumulates the $N^*$ e-folds of
Eq.~\eqref{eq:efolds-w2}), the exit where the slope steepens and
slow roll fails, and reheating --- the oscillation about the
minimum that converts the roll's energy into the hot Big Bang. The
plateau height is not a free label: with $r$ staked at $0.0043$ and
$A_s = 2.1\times10^{-9}$ the one named input, the height follows,
$V^{1/4} = \Mplred(3\pi^2 A_s r/2)^{1/4} \simeq 8\times10^{15}$~GeV
--- some thirteen orders above the lock scale, and not a label the
figure was free to choose. A tensor measurement is a measurement of
this figure's vertical axis. On
the anatomy the horizontal axis is not an abstract field space:
$\varphi$ is the interval's coordinate, the roll happens once, and
the exit is the launch's receipt.}
\label{fig:slowroll-w2}
\end{figure}

\subsection{The first line of the ledger}

The beginning, on the anatomy, has a number: the launch segment
of the age ledger. From the Planck anchor to the end of
reheating the clock advances $\ln(t/t_{\mathrm{Pl}}) = 36.1$ ---
the plateau's $N^* = 52.6$ e-folds of stretch plus the reheating
match at the derived $T_{\mathrm{rh}}$, converted to clock
e-folds (\S\ref{sec:age-audit}) --- and everything observable
about the era fits in three staked quantities: the tilt
$n_s = 0.9620$ (the descent's length, read in the spectrum), the tensor target $r = 0.0043$ (the plateau's height, awaiting
the Simons Observatory and LiteBIRD), and the zero-isocurvature null (the
pinning's fingerprint). A beginning with a line in a ledger is a
different \emph{kind} of beginning from a singularity: it is an
entry in the same audit as the rest of history --- checkable,
falsifiable, and continuous with the chronology of Problem~13
whose straight line it opens. The Big Bang, domesticated: not
the edge of physics, but its first invoice.

\subsection{The first second, hour by hour}

Concreteness disciplines cosmogony, so walk the opening of the
reading as the derived chronology actually schedules it
(\S\ref{sec:epoch-table}; all epochs from the one straight line
of Figure~\ref{fig:chronology}). \emph{The roll}: the interval's
coordinate traverses its plateau --- $52.6$ e-folds of stretch,
the sky's uniformity and its seed spectrum laid down together,
the circle's ledger riding through untouched. \emph{The exit}:
slow roll fails; the holonomy variable's \GSSB{} ends the
episode (the circle's first word); the roll's energy reheats the
boundary --- the hot Big Bang begins as the launch's receipt, at
clock e-fold $36.1$. \emph{The lock} ($r = K$, $t \sim
10^{-11}$~s): the descent's second saddle --- the gate through
which the heavy neutrino falls once, writing the matter excess
(Problems~10--11), and the compact angle takes the displacement
that will become dark matter (Problem~7). \emph{The electroweak
crossover}: the agent's symmetry is read off the Wilson line;
masses arrive at their addresses; shortly after, the sphaleron
door welds, sealing the excess. \emph{The QCD floor}
($t \sim 10^{-5}$~s): the triangle's closure rule takes
command; quarks weld into the guarded vaults of Problem~12; the
Kaxion's mass is delivered by the cooling susceptibility, and
the circle's coordinate begins to ring. One second in, the
universe's entire subsequent inventory --- matter, dark matter,
the sealed books --- is on deposit, and everything since
(nucleosynthesis at minutes, atoms at $380{,}000$ years,
structure at hundreds of millions) is the ledger compounding.
The point of the itinerary is its \emph{provenance}: every stop
is a computed address on one line, not a bespoke epoch with its
own theory --- the first second is not a frontier of exotic
physics but the busiest stretch of a single derived road.

\subsection{Common misreadings, corrected}

\paragraph{``The Big Bang was an explosion at a point.''} Nothing
exploded and there is no point: the launch stretched every region
--- space itself expanded everywhere at once --- and ``the
beginning'' names the top of the descent, not a location. The
popular image fails exactly where the anatomy is sharpest: an
explosion has a center and a before; a reading has an opening
and a groove.

\paragraph{``Something must have come before.''} \emph{Before} is
an interval word, and the interval opens at the corridor. Running
the clock backward past the opening is not forbidden physics; it
is unperformable bookkeeping --- the semigroup has no inverse
with which to run. Whether anything \emph{terminates} the
geometry at the top (an edge) or \emph{continues} it (a deeper
descent) is the entrance item, honestly open; what is already
closed is that the question is structural, not temporal --- there
was never a ``time before time'' to narrate, only a corridor top
to characterize.

\paragraph{``Quantum gravity must resolve the singularity
first.''} The derived history never visits a singularity: its
account opens below the Planck point and its equations are
regular everywhere they are used. Quantum-gravity physics may
well govern the entrance --- the open item is agnostic --- but
nothing this Problem computes waits on it. The beginning has
been moved from the mercy of the hardest unsolved theory to the
agenda of a named calculation, and that relocation is the
Problem's principal progress.

\subsection{Watching the beginning: the three windows}

``The beginning'' sounds beyond observation; the launch is not,
and its three windows are all staked. \emph{The tensor window}:
the plateau's height fixes $r = 0.0043$, an order of magnitude
above the design floors of the coming polarization missions --- a
detection near $0.004$ would be, quite literally, an image of the
opening stretch (primordial gravitational waves are the roll's
own noise); a null at those floors, or a detection near the
current bound, kills the derived launch either way. \emph{The
statistics window}: the F25 fossil (Problem~1) --- the largest
scales retain the statistics the relaxation had not yet finished
erasing, a direct sample of the pre-launch configuration that a
free-initial-data cosmology cannot even formulate. \emph{The
null window}: zero axion-sector isocurvature and no defects ---
the pinning's fingerprint, already holding, deepening with every
CMB release. Three windows, three different instruments, one
era: the beginning, on the anatomy, is not the least observable
epoch but among the most --- because a derived opening leaves
derived fingerprints, and fingerprints are what instruments are
for.

\subsection{The entrance item, specified}

Openness is only honest when it is specific, so the entrance's
requirements deserve their list --- what any successful account
of the corridor's top must supply \citep{ArisakaA10Cosmos}:
\emph{(i)} the configuration at entry --- why the interval's
coordinate begins on the plateau's high shoulder, cold and
smooth enough for the roll to proceed; \emph{(ii)} the status of
the pre-entry region --- edge or continuation, and if an edge,
the boundary data the constitution assigns there; \emph{(iii)}
the pinning's execution --- the reflection-symmetric dressing
that freezes the compact angles during the roll, presently a
named conditional, which an entrance account should derive;
\emph{(iv)} the budget --- the e-fold accounting that would
convert the $4K$ family (Problems~6--7) from flagged to derived,
since the corridor's total length is set at its top. Four
requirements, one open item, and a definite shape: whoever
closes the entrance closes, at a stroke, the deepest flagged
numbers in the cohort --- which is why it heads the queued
campaigns, and why this Problem ends not with an apology but
with a work order.

\subsection{The entrance, honestly}

What the anatomy does still owe is stated as the cohort states it.
The derived launch begins at the top of the plateau; the
\emph{entrance} --- the geometry's own account of how the corridor
is entered, the cold start before the first e-fold --- is a named
open item, with its requirements mapped and its candidate
mechanisms cataloged but not closed \citep{ArisakaA10Cosmos}. This
is the descent's analogue of the singularity question, and the
honest boundary of this Problem: the anatomy has moved the
frontier from ``the equations fail'' to ``the corridor's top is
uncomputed,'' which is progress of a specific, auditable kind ---
the open item has a shape, a location, and a list of what any
solution must supply --- but it is a frontier still. The reader
should also notice what the anatomy makes of ``before.'' Along a
semigroup, \emph{before the first write} is not a place the
formalism reaches by running backward --- the fiber has no inverse
transport --- so whether the entrance question is a dynamical
question (what process entered the corridor) or a boundary
question (what conditions terminate the geometry) is itself part
of what the open item must decide. The anatomy narrows the
question; it does not yet answer it, and this paper says so.

\textbf{Grade and kill, consolidated.} The launch and chronology:
derived-conditional (plateau class; prompt crossing). Determined
initial data: the triad's grades, with the pinning a named
conditional. The entrance: open, named. Kill: the launch's
switches ($r$ off $0.0043$; $n_s < 0.955$); an axion-sector
isocurvature detection (the lottery's return); an F25-class fossil
off its computed form. The beginning, on the anatomy, is not an
event that escaped physics: it is the top of a groove --- and the
remaining mystery has been given an address.

\section{Problem 6.\ Naturalness and the Hierarchy Problem}
\label{sec:w2-prob6}

\subsection{A forty-year anxiety}

Two numbers have organized four decades of theoretical anxiety.
The weak scale sits at $246$~GeV while gravity's scale sits at
$10^{19}$~GeV --- seventeen orders of magnitude apart --- and in
quantum field theory the light scale should not stay light: the
Higgs mass receives corrections from every heavy threshold it
couples to, so keeping it at $10^{-17}$ of Planck appears to
require cancellations tuned to thirty-four decimal places. The
vacuum energy is worse: its natural estimate exceeds the measured
dark-energy density by some $120$ orders. The response was a
doctrine --- \emph{naturalness}: dimensionless ratios should be of
order one unless a symmetry protects them --- and a generation of
protectors: supersymmetry pairing every particle with a
canceling partner, compositeness dissolving the Higgs, extra
dimensions lowering the true gravity scale. The doctrine was
fruitful, the protectors beautiful --- and the LHC, in the most
consequential null result of the era, has found none of them where
naturalness said to look. The field now speaks openly of a
naturalness crisis; the anthropic landscape (Problem~15) is one
response; this Problem is another.

\subsection{The anatomy's answer: nothing is small; something is deep}

The doctrine presupposed that a small dimensionless number is a
\emph{coincidence to be protected}. The anatomy replaces the
presupposition: on the descent, a small number is an
\emph{address}. The weak scale sits where it sits because that is
where the throat bottoms out:
\begin{equation}
\Mlock \;=\; \tfrac12\,M_{\mathrm{Pl}}\,e^{-K} \;=\; 3.849~\mathrm{TeV},
\qquad
v \;=\; \Mlock\,(2/L)^{N} \;=\; 246~\mathrm{GeV},
\label{eq:hierarchy-w2}
\end{equation}
thirty-five e-folds of descent with a ledger integer in the
exponent, dressed by a circle fraction --- the gearbox grammar of
\S\ref{sec:gearbox}. The ratio $v/M_{\mathrm{Pl}} \sim 10^{-17}$
is not a tuned cancellation; it is $e^{-35}$ with bookkeeping ---
\emph{counted, not tuned} --- and the exponent is not adjustable,
because $K = 35$ is a member of the anomaly-locked integer ledger
(Problem~3). Exponentials of counted integers are what addresses
look like; a civilization that measured only city distances would
find the kilometer count between capitals ``unnaturally large''
by exactly this logic, and would be wrong in exactly this way.

\subsection{The radiative half, reframed}

The sharper half of the worry is stability: never mind the tree
value --- why do quantum corrections not drag the light scale up?
The anatomy's reframe has two clauses. \emph{First}, corrections
are interval-side flow: they move readings between addresses, and
in the effective-theory organization the descent itself supplies,
thresholds decouple as one descends --- the flow redistributes;
it does not pick the endpoint. \emph{Second}, the endpoint is not
balanced on a cancellation but pinned by structure: the lock is
where the warp bottoms out, an integer depth, and the vev is its
dressed image. There is no delicate difference of large numbers
anywhere in Eq.~\eqref{eq:hierarchy-w2} for corrections to upset.
The doctrine's silent premise --- that the light scale is a
fundamental parameter exposed naked to every threshold --- is
what the address grammar removes; with it goes the fine-tuning,
and with the fine-tuning goes the need for protectors. On this
reading the LHC's null is not a crisis but a verdict correctly
delivered: the protectors were guarding a problem the geometry
never had. (The reading is falsifiable in the same breath: the
same lock that dissolves the hierarchy \emph{places a cluster at
$3.849$~TeV}, and its absence at staked sensitivity kills this
Problem's answer --- \S\ref{sec:lock-unification}.)

The anxiety has an exact source, worth displaying. Couple a
scalar of mass $m_H$ to a fermion of Yukawa coupling $y$ (the top
quark, $y_t \simeq 1$, dominates), and the one-loop quantum
correction to the scalar's squared mass is
\begin{equation}
\delta m_H^2 \;\simeq\; -\,\frac{3\,y_t^2}{8\pi^2}\,\Lambda^2 ,
\label{eq:radiative-w2}
\end{equation}
where $\Lambda$ is the highest scale at which the loop integral is
cut off --- the scale up to which the theory is trusted. The
equation says the correction is set by the \emph{cutoff}, not by
$m_H$ itself: if the Standard Model holds to the Planck scale,
$\delta m_H^2$ is thirty-four orders of magnitude larger than the
measured $m_H^2$, and the bare parameter must cancel it to
thirty-four digits. That is the hierarchy problem stated as
arithmetic. The anatomy's dissolution is equally arithmetic: the
electroweak scale is not a small number surviving a large
correction but a \emph{deep address},
$\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-K}$ (Eq.~\eqref{eq:mlock}),
and Eq.~\eqref{eq:radiative-w2}'s $\Lambda$ is not free: the
descent supplies its own gauge-sector completion at the lock, so
the loop's reach --- and with it the correction --- is priced by
the same $K = 35$ that sets the scale. Nothing cancels; something
is far down.

\subsection{The vacuum: the depth, doubled}

The cosmological-constant hierarchy --- the worst fine-tuning in
physics --- is the same answer squared. The terminus scale is the
lock-scale seesaw, $m_\Lambda = \Mlock^2/\Mplred \simeq 6.1$~meV:
in e-fold form, twice the lock depth ($e^{-2K}$), so the vacuum
density sits at roughly $e^{-8K}$ of the Planck density --- and
the measured hierarchy is $8K$ to within $1.2\%$
(\S\ref{sec:de-w2}). The honest register is repeated from
Problem~7: the $8K$ face is \emph{flagged} --- the age ledger
doubled, one observation wearing three faces, its would-be
derivation the queued e-fold budget campaign --- and nothing
leans on it. What is not flagged is the structure: the meV scale
arrives from the already-derived TeV scale with no new
dial, and the ``$120$ orders'' dissolve into depth arithmetic
whose one open coefficient is named, bounded, and \emph{not small in
its effect}: the derived terminus $6.1$~meV stands against a measured
$\rho_\Lambda^{1/4} = 2.24$~meV, so the drainage factor carries
$\zeta_\Lambda^{1/4} = 0.37$ --- a factor of $55$ in density,
uncomputed, and the reason this Problem's vacuum-energy half is
graded no better than the flagged half. The most notorious number in physics is,
on this reading, the interval's odometer --- and an odometer
reading is not a coincidence, however large.

Figure~\ref{fig:depth-w2} replaces the cliff with the well:
the hierarchy drawn as a depth gauge, with the four places the
integer is spent.

\begin{figure}[htbp]
\centering
\resizebox{0.82\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% depth gauge
\draw[line width=1.10pt] (0,6.4) -- (0,0.4);
\draw[line width=1.10pt,dashed] (0,0.4) -- (0,-1.6);
\foreach \y in {6.4,0.4,-1.6} \draw[line width=1.10pt] (-0.18,\y) -- (0.18,\y);
\node[left] at (-0.35,6.4) {\scriptsize $M_{\mathrm{Pl}} = 1.22\times 10^{19}$~GeV};
\node[left] at (-0.35,0.4) {\scriptsize $M_{\mathrm{lock}} = 3.849$~TeV};
\node[left] at (-0.35,-1.6) {\scriptsize $m_\Lambda \approx 6.1$~meV \ (flagged)};
% braces
\draw[decorate,decoration={brace,amplitude=6pt}] (0.45,6.4) -- (0.45,0.4);
\node[right,align=left,blue!65!black] at (1.05,3.4)
  {\scriptsize \textbf{the depth: $K = 35$ e-folds}\\[-2pt]
   \scriptsize $M_{\mathrm{lock}} = \tfrac12 M_{\mathrm{Pl}}\,e^{-K}$\\[-2pt]
   \scriptsize not small --- deep};
\draw[decorate,decoration={brace,amplitude=6pt}] (0.45,0.15) -- (0.45,-1.6);
\node[right,align=left,black!60] at (1.05,-0.75)
  {\scriptsize the depth, doubled: $e^{-2K}$\\[-2pt]
   \scriptsize (the vacuum; flagged grade)};
% the watches
\node[draw=black!55,line width=0.80pt,rounded corners,align=left] at (8.6,4.7)
  {\scriptsize \textbf{where 35 is spent}\\[1pt]
   \scriptsize the lock depth $e^{-K}$ (this Problem)\\[-1pt]
   \scriptsize the corridor's length (cosmic route)\\[-1pt]
   \scriptsize the excess's toll $e^{-K/2}$ (matter route)\\[-1pt]
   \scriptsize the Kaxion's $1/35$ (laboratory face)};
\node[align=center] at (8.6,2.6)
  {\scriptsize one integer, four invoices ---\\[-2pt]
   \scriptsize derived once, not tuned};
\end{tikzpicture}%
}
\caption[Nothing is small; something is deep.]{\textbf{Nothing is small; something is deep.} The
seventeen-orders hierarchy drawn as what the anatomy says it is: a
depth of $K = 35$ e-folds on the interval, with the electroweak
address at the bottom of the well and the vacuum at the depth
doubled (carried at flagged grade, never spent). The integer is
derived once, from $K = LJ$ on an audited ledger, and then spent in
four places that must all agree; a dial would be spent in one.}
\label{fig:depth-w2}
\end{figure}

\subsection{A short history of the anxiety}

The doctrine has a genealogy worth one page, because each stage
sharpened a true instinct. Dirac (1937) noticed the large
dimensionless ratios of nature clustering near $10^{40}$ and its
powers, and proposed --- wrongly, it turned out, but
fruitfully --- that such numbers must be dynamical. Wilson and
't~Hooft made the instinct precise: a small parameter is
\emph{technically natural} if a symmetry is restored when it
vanishes --- fermion masses are protected (chiral symmetry), the
Higgs mass is not, and its unprotected smallness became \emph{the}
hierarchy problem. Supersymmetry was invented, in large part, to
be its protector, and for three decades the argument
``naturalness $\Rightarrow$ new physics at the TeV scale'' was
the field's best-subscribed prediction. The LHC then delivered
the Higgs at $125$~GeV --- and, at staked luminosities, none of
the protectors. The response spectrum runs from anthropics
(Problem~15) to ``naturalness was never a principle'' to split
and relaxion models; the shared premise of every response is
that the small number is a \emph{ratio of independent scales}.
That premise is what the address grammar removes: on the descent
the two scales were never independent --- one is the other's
exponentially descended image, with an integer in the exponent.
Dirac's instinct was right after all, one inversion later: the
large numbers \emph{are} structural --- not dynamical variables,
but depths.

\subsection{Why thirty-five}

``Counted, not tuned'' invites the next question: who counted?
The exponent is not chosen to fit the hierarchy; it is the ledger integer $K = LJ = 5 \times 7$,
and every factor in that product is fixed upstream by a step that can
be checked. $L = 5$ comes from single-tensor \GS{} consistency;
$J = 2N + 1 = 7$ from the constitutional relation once $N$ is known;
and $N$ from the anomaly audit's dual-lock condition
$L(2N+1) = M^2 + N^2 + L^2$, whose roots at $L = 5$ are $\{3,7\}$, with
minimality selecting three (\citealp{ArisakaA2GG6};
\citealp{ArisakaA3BI6}).

It is worth being precise about what that chain is and is not, because
a stronger-sounding claim is available and does not survive
arithmetic. The dual lock is not a second, independent determination
of $K$: setting $LJ$ equal to $M^2 + N^2 + L^2$ is precisely the
equation that selects $N$, and the resulting identity holds at
\emph{both} of its roots --- $35 = 35$ at $N = 3$ and $75 = 75$ at
$N = 7$. So $K$ is derived once, along an audited chain, and not
twice. What makes it hard to move is the other end: the same integer
is \emph{spent} four times over. The number that sets the
weak-to-Planck ratio also sets, through the same ledger, the kaxion
doublet ($1/35$), the corridor switch ($e^7$), and the throughput of
the matter excess ($e^{-17.5}$) --- one integer, audited at a
collider, a haloscope, and the sky. This is what
``the receipt'' means concretely: a tuned number explains one
measurement and predicts nothing; a counted depth is one column
of a ledger whose other columns are already being checked.

\subsection{Objections and replies}

\paragraph{``You have traded a tuned ratio for a tuned
exponent.''} The exponent is an integer, and integers cannot be
tuned --- but the objection deserves its stronger form: could
$K$ have been \emph{selected} to fit? The answer is no, and the
reason has to be stated carefully, because one tempting version of it
does not survive arithmetic. It is tempting to say that $K = 35$ is
overdetermined by two independent routes, $LJ$ and $M^2 + N^2 + L^2$.
It is not. Setting those two expressions equal is precisely the dual
lock that fixes $N$ --- with $M = 1$ it reduces to $N = L \pm
\sqrt{L - M^2}$, whose roots at $L = 5$ are $\{3,7\}$ --- and the
identity therefore holds at \emph{both} roots, giving $35 = 35$ at
$N = 3$ and $75 = 75$ at $N = 7$. The agreement of the two routes is
the equation that selects $N$, not a second determination of $K$. A
reader who checks this in four lines is right to, and the paper would
rather print the check than be caught by it.

What survives, and it is enough: $K$ is fixed \emph{once}, by
$K = LJ$, on a ledger whose members were fixed by the anomaly audit
years before this Problem's use of them; and the same integer is then
\emph{spent} in four independent places --- the kaxion doublet, the
corridor switch $e^{J}$, the excess's toll $e^{-K/2}$, and the lock
depth here. That is not overdetermination of $K$; it is
overdetermination of the ledger, which is the stronger claim anyway. A
number spent on four watches cannot be quietly reset to fix a fifth.

\paragraph{``Why is there a throat at all --- isn't its existence
the real tuning?''} The warp is not installed to make
hierarchies; it is the constitutional geometry of the fifth
dimension (\AX{2}'s axiomatic $e^{-2r}$ profile, with its
dynamical stabilization a named upstream item, carried honestly).
The hierarchy is a consequence of the bulk the rest of the
cohort independently requires --- the same warp runs the
chronology of Problem~13 and the seesaw of Problem~7. One
geometry, many invoices: that is the opposite of a mechanism
invented for one job.

\paragraph{``The little hierarchy: why is $v$ fifteenfold below
the lock?''} By the circle's dressing, $(2/L)^N = (2/5)^3 =
8/125$ --- a fraction of ledger integers, not a small parameter.
The factor's smallness is arithmetic, its form is the gearbox
grammar, and its audit is the vev's own closed form sitting at
$0.05\%$ from measurement.

\paragraph{``Doesn't the LHC null equally embarrass you at
$3.849$~TeV?''} Not yet, and the difference is the point: the
naturalness protectors predicted a \emph{range} that has been
progressively vacated; the lock predicts a \emph{line}, with a
fingerprint, inside the HL-LHC's staked reach. One more null
retires this Problem's answer too --- the exposure is symmetric,
and the calendar is short. The anatomy does not ask to be
excused from the standard it applies to the doctrine it retires.

\subsection{Naturalness, retired with honors}

The doctrine deserves a respectful funeral, because it was almost
right. Its instinct --- that unexplained small numbers signal
missing structure --- was sound, and led the field to genuine
discoveries about symmetry and effective theory. Its error was
the null hypothesis: it assumed the missing structure must be a
\emph{protector} (a symmetry holding a coincidence in place),
when the anatomy supplies an \emph{addresser} (a geometry in
which the number was never a coincidence). Under \MSMF{} the
replacement is principled rather than opportunistic: the
constitution forbids continuous parameters outright, so there is
nothing in the theory \emph{to} tune, and ``unnatural'' becomes a
property no quantity of the framework can have --- every number
is an integer, a fiber measure, or an exponential address of
integers. The naturalness question, on the anatomy, was never
``why is the number small?'' It was ``what is the number's
address?'' --- and addresses are computed, checked, and, at
$3.849$~TeV, shortly to be visited.

\textbf{Grade and kill, consolidated.} The lock and vev: derived
at codebook grades. The seesaw structure: derived; the $8K$ face:
flagged. Kill: no cluster at $3.849$~TeV at staked sensitivity
(the direct visit); a measured $w \ne -1$ or phantom crossing
(the terminus half); the e-fold budget campaign closing against
the flagged family (its quiet death). The hierarchy problem, on
the anatomy: \emph{seventeen orders of magnitude is not a
miracle; it is thirty-five e-folds --- and the universe kept the
receipt.}

The seesaw that dissolved the hierarchy left a residue --- a vacuum
energy of $6.1$~meV, the square of the lock over the Planck scale.
That residue is about to reappear as the third panel of the dark
trio, which is the next Problem.

\section{Problem 7.\ The Dark Trio: Inflation, Dark Energy, and the Kaxion}
\label{sec:w2-prob7}

\subsection{The census}
\label{sec:census}

The three great unexplained items of standard cosmology --- what
inflated, what the dark matter is, what the dark energy is --- are,
on the anatomy, answered by a \emph{census} before they are answered
by a computation (\citealp{ArisakaA10Cosmos}; \citealp{ArisakaC1Kaxion}). The
bulk's non-spatial sector contains exactly two coordinates. Both are
promoted to boundary fields by \OGGEP{}. Both took the cosmic stage
exactly once:

\begin{quote}
\emph{Inflation is the interval's coordinate, rolling once. Dark
matter is the circle's coordinate, ringing still. Dark energy is
what remains when both are at rest --- its share written by the
circle, its scale written by the interval.}
\end{quote}

The census also disposes of two famous embarrassments of riches in
one stroke. The \GDOS{} boundary law admits exactly one
non-conformal scalar modulus --- the radial stratum itself --- so
there is one candidate inflaton and no zoo; the compact sector
supplies exactly two shift-protected angles with one light
eigenstate, so there is one axion and no axiverse. \emph{One
interval $\Rightarrow$ one inflaton; one circle $\Rightarrow$ one
axion}: the scalar field content of cosmology is the dimensional
census of the bulk.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% ============ Panel (a): inflation ============
\begin{scope}
\node[align=center] at (2.3,2.9) {\textbf{(a) Inflation: the interval's}\\[-2pt]
\textbf{coordinate, rolling once}};
\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] {$V_{\mathrm{eff}}$};
\draw[line width=1.10pt,blue!65!black] plot[domain=0:4.35,samples=80]
  (\x,{1.8*(1-exp(-1.35*\x))^2});
\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};
\node[blue!65!black] at (3.0,1.45) {\scriptsize $N^{*}=52.6$};
\node[blue!65!black] at (3.0,1.13) {\scriptsize $n_s = 1-\tfrac{2}{N^*}$,\ $r = \tfrac{12}{N^{*2}}$};
\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 $\Theta$)};
\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 ============
\begin{scope}[shift={(6.3,0)}]
\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);
\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);
\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 ============
\begin{scope}[shift={(11.4,0)}]
\node[align=center] at (1.9,2.9) {\textbf{(c) Dark energy:}\\[-2pt]
\textbf{both coordinates at rest}};
\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 $m_\Lambda \simeq 6.1$~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 $= e^{-2K}$};
\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.]{\textbf{The trio on the two fibers} (adapted from the
cosmology companion \citep{ArisakaA10Cosmos}). \emph{(a)} The
interval's coordinate rolls once along the warp-shaped plateau;
$N^* = 52.6$ fixes the tilt and the tensor target; the exit is a
\GSSB{} in the holonomy variable $\Theta$. \emph{(b)} The circle's
coordinate, pinned at $\theta_i$, released once, ringing at
$1.95$~GHz with the scale-free split $1/35$. \emph{(c)} Dark
energy: the share is the circle's (the uncovered holonomy volume),
the scale the interval's (the seesaw squares the descent);
$w = -1$ is the semigroup at rest.}
\label{fig:trio-w2}
\end{figure}

\subsection{Inflation: the interval's coordinate, rolling once}
\label{sec:inflation-w2}

The inflaton $\varphi$ is the canonically normalized coordinate of
$\Ir$, and its episode wears the fiber's character
\citep{ArisakaA10Cosmos}: a non-periodic plateau potential --- the
warp exponential
$V \propto (1 - e^{-\sqrt{2/3}\,\varphi/\Mplred})^2$, the e-folding
of the descent bent into a potential --- traversed \emph{once},
ending when slow roll fails, unrepeatable because the fiber has no
inverse transport. The observables are pure descent bookkeeping:
the e-fold count $N^* = 52.6$ (undiluted --- the lock crossing is
prompt) gives
\begin{equation}
n_s = 1 - \frac{2}{N^*} = 0.9620
\qquad\text{(measured } 0.9649 \pm 0.0042\text{, pull } -0.7),
\qquad
r = \frac{12}{N^{*2}} = 0.0043,
\label{eq:ns-r}
\end{equation}
the tensor target several times above the projected $\sigma(r)\simeq10^{-3}$ design floors of
the Simons Observatory and LiteBIRD --- a live kill switch, both directions. The
circle appears at the boundaries of the act: the graceful exit is a
\GSSB{} in the holonomy variable $\Theta$ --- \emph{the circle's
first word ends the interval's episode} --- and the later lock
crossing hands the compact angle the displacement that becomes the
next act. Nothing quantized appears in inflation's output, and the
anatomy says why: the interval has no holonomy to write with; the
ledger rides through the launch untouched.

\subsection{The Kaxion: the circle's coordinate, ringing still}
\label{sec:kaxion-w2}

Dark matter completes the triptych of matter portraits: \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}
(\citealp{ArisakaC1Kaxion}; \citealp{ArisakaP3Electron}; \citealp{ArisakaP6Proton}).
Everything it \emph{is} is circle-side: two periodic parents (the
tensor shadow and the holonomy echo), lightness by the circle's
memory (a periodic direction admits no polynomial potential),
portals locked at $5{:}2$ by the anomaly ledger --- it couples only
through topology because it is \emph{made} of topology, which is
the entire origin of its darkness --- and universality by one
pinned angle across the observable universe. Everything it
\emph{has} is interval-side: the Hubble-friction freeze, the
once-only release when the circle's clock outruns the interval's
($3H = \mK$ at $T_{\mathrm{osc}} \simeq 1$~GeV), the mass delivered
by the cooling descent,
\begin{equation}
\mK = \frac{\sqrt{\chiQCD}}{\fK} = 8.1 \pm 0.5\ \mu\mathrm{eV},
\qquad
\nu_K = 1.95\ \mathrm{GHz},
\qquad
\frac{\Delta\nu}{\nu} = \frac{1}{K} = \frac{1}{35},
\label{eq:kaxion-numbers}
\end{equation}
and the relic abundance as the crossing's redshifted memory. Two
credentials make it the campaign's cleanest exhibit: the
dark-matter mass \emph{is} the attractor's Hessian --- a haloscope
detection would be the first direct measurement of a \GGA{}-class
stability, the runtime's termination certificate heard as a tone
--- and its cascade is the only one in nature still finishing:
caught mid-ring, the final \GRIPin{} of the observable universe.
The search strategy is the anatomy's: \emph{identify by the circle
(the scale-free $1/35$ doublet), search by the interval (the GHz
address)}.

Figure~\ref{fig:kaxion-two-fibres-w2} draws the portrait this
subsection has been building.

\begin{figure}[htbp]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% ================= Panel (a): the circle =================
\begin{scope}
\node[align=center] at (0,2.75) {\textbf{(a) The circle $S^1_\theta$: identity}\\[-1pt]
\scriptsize closed --- all loops, all holonomy --- \emph{remembers}};
\draw[line width=1.10pt] (0,0) circle (1.7);
% ledger-rational vacua (72-degree spacing), GGA at bottom
\foreach \ang in {342,54,126,198}
  \fill[black!55] (\ang:1.7) circle (0.055);
\fill[blue!65!black] (270:1.7) circle (0.09);
\node[below=3pt,align=center,blue!65!black] at (270:1.75)
  {\GGA{}: ledger-rational floor\\ $\mK^{2}=\chiQCD/\fK^{2}>0$ (the Hessian)};
% hilltop saddle (GSSB) at top
\draw[line width=0.80pt,orange!85!black] (90:1.62) -- (90:1.78);
\node[above=2pt,orange!85!black,align=center] at (90:1.78)
  {saddle ($\theta=\pi$): \GSSB};
% DGI at centre
\node[align=center,gray!40!black] at (0,0.18) {\DGI: undisplaced,\\ symmetric};
% displaced marble at ~350 deg and relaxation arrow to the floor
\shade[ball color=red!70!black] (350:1.7) circle (0.1);
\node[right=3pt,align=left,red!60!black] at (350:1.78)
  {$\thetai\simeq1.4$\\ (pinned)};
\draw[->,line width=0.80pt,red!60!black] (341:1.94) arc (341:279:1.94);
\node[align=left,red!60!black,anchor=west] at (333:2.15) {\GRIPin\\ relaxation};
% two parents note (below the GGA caption, clear of the circle)
\node[align=center] at (0,-2.95) {\scriptsize two parents, geared $\sim e^{2K}$:
$\thetags$ (tensor shadow), $\thetaw$ (holonomy echo)};
\end{scope}
% ================= Panel (b): the interval =================
\begin{scope}[shift={(5.0,0)}]
\node[align=center] at (3.7,2.75) {\textbf{(b) The interval $I_r$: occupancy}\\[-1pt]
\scriptsize open --- no loops, no holonomy --- \emph{forgets}};
% one-way axis
\draw[->,line width=1.10pt] (0.2,-2.05) -- (7.45,-2.05)
  node[above left=1pt and -46pt] {\scriptsize the descent (one-way)};
% GGA floor line
\draw[dashed,gray] (0.35,0) -- (7.3,0);
\node[right,gray!40!black] at (5.6,-0.28) {\scriptsize \GGA{} floor};
% frozen segment
\draw[dashed,gray] (0.35,0.95) -- (0.6,0.95);
\node[left,gray!30!black] at (0.42,0.95) {\scriptsize $\thetai$};
\draw[line width=1.10pt,blue!65!black] (0.6,0.95) -- (3.4,0.95);
% damped ringing after release at x=3.4
\draw[line width=1.10pt,blue!65!black]
  (3.4,0.95)
  .. controls (3.62,0.95) and (3.72,-0.72) .. (3.95,-0.68)
  .. controls (4.14,-0.65) and (4.24,0.54) .. (4.44,0.51)
  .. controls (4.61,0.49) and (4.69,-0.4) .. (4.87,-0.38)
  .. controls (5.03,-0.36) and (5.11,0.3) .. (5.28,0.28)
  .. controls (5.43,0.27) and (5.5,-0.21) .. (5.65,-0.2)
  .. controls (5.79,-0.19) and (5.86,0.14) .. (6.0,0.13)
  .. controls (6.13,0.12) and (6.2,-0.09) .. (6.33,-0.08)
  .. controls (6.45,-0.07) and (6.52,0.05) .. (6.66,0.05)
  -- (7.25,0.02);
% epoch ticks
\foreach \x/\lab in {0.9/{pinning\\(inflation exit)\\ $R\approx0$},
                     3.4/{release\\ $T_{\mathrm{osc}}\simeq1$ GeV\\ $R=1$},
                     6.9/{today\\ $\nu_K=1.95$ GHz\\ $R\approx2\times10^{27}$}}{
  \draw[line width=0.80pt] (\x,-1.97) -- (\x,-2.13);
  \node[below,align=center] at (\x,-2.16) {\scriptsize \lab};}
% regime labels
\node[align=center,blue!65!black] at (1.95,1.5)
  {\scriptsize frozen: $3H\gg\mK$\\ \scriptsize the interval's grip};
\node[align=center,blue!65!black] at (5.5,1.5)
  {\scriptsize ringing: $\mK>3H$\\ \scriptsize dark matter};
\end{scope}
\end{tikzpicture}%
}
\caption[The Kaxion on the two fibers.]{\textbf{The Kaxion on the two fibers} (reproduced from
\citealp{ArisakaC1Kaxion}). \emph{(a) The circle:} the Kaxion's
identity --- a compact-angle eigenstate on the pentagonal flavor
lattice, its mass and couplings ledger arithmetic
(Eq.~\eqref{eq:kaxion-numbers}), protected because class data
cannot drift. \emph{(b) The interval:} the Kaxion's occupancy ---
the misalignment energy deposited at the lock crossing and diluted
down the descent to today's halo density. Identity from the circle,
abundance from the interval: the same two-fiber grammar as every
particle in the cohort, which is why the haloscope strategy is to
\emph{identify by the circle} (the $1/K$ doublet, the discrete cap
window) rather than merely to detect power.}
\label{fig:kaxion-two-fibres-w2}
\end{figure}

\subsection{Dark energy: both coordinates at rest}
\label{sec:de-w2}

Dark energy's two numbers come from the two fibers, one each
\citep{ArisakaA10Cosmos}. The \emph{share} is the circle's: in the
fiber-measure resolution, matter's fraction is the Hopf circle's
volume inside the electroweak $S^3$,
\begin{equation}
\Omm = \frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)}
= \frac{2\pi}{2\pi^2} = \frac{1}{\pi},
\qquad
\OmL = 1 - \frac{1}{\pi},
\qquad
\Omb = \frac{1}{2\pi^2},
\label{eq:hopf}
\end{equation}
so the largest entry in the present energy budget is \emph{the
uncovered remainder of the holonomy volume} --- epoch-frozen
because fiber volumes do not dilute. The \emph{scale} is the
interval's: the lock-scale seesaw, in e-fold form an exact identity
of the published chain,
\begin{equation}
\frac{m_\Lambda}{\Mplred}
= \Bigl(\tfrac12\sqrt{8\pi}\Bigr)^{2} e^{-2K}
\quad\Longrightarrow\quad
m_\Lambda = \frac{\Mlock^2}{\Mplred} \simeq 6.1\ \mathrm{meV},
\label{eq:seesaw-w2}
\end{equation}
\emph{twice the lock depth}: the seesaw squares the one hierarchy
the interval ever wrote. The notorious $10^{-120}$ becomes an
address --- $-\ln(\rho_\Lambda/\Mplred^4) = 276.6 = 8K - 3.4$, and
the de Sitter entropy is the same number inverted
($\ln S_{\mathrm{dS}} = 278.9$) --- carried at flagged grade, as
the age ledger of Problem~13 \emph{doubled}: one observation, three
faces, all awaiting the e-fold budget campaign. And $w = -1$
exactly is the anatomy's sentence: \emph{arrival at a terminus is
irreversible on the fiber that cannot revisit} --- a rolling
quintessence would be a descent still in motion (transient by
nature), and phantom crossing would be the semigroup running
backward (never). The framework forbids both, and stakes it.

\subsection{Why now}
\label{sec:whynow}

The coincidence problem --- why the matter and vacuum densities
are comparable in the present epoch --- takes a sharpened form on
the anatomy, and an honest answer. The crossover is a pure circle
number read on the interval's clock: with both shares fixed as
fiber volumes, equality sits at
$z_{\Lambda m} = (\pi - 1)^{1/3} - 1 = 0.289$, no freedom
anywhere. The residual question --- why the observer's now falls
within a factor of a few of that crossover --- is the same
question as the age ledger's $4K$ compression: \emph{why does the
odometer read $\sim 140$ e-folds when structure, chemistry, and
observers arrive?} The cohort's answer is layered and carried at
its layered grades \citep{ArisakaA10Cosmos}: the geometric facts
(the crossover's location; the corridor's length) are exact; the
timing of habitability within the corridor is bounded but
anthropically read, and said so; and the would-be closure --- the
e-fold budget that would derive the whole clock face with no dial
--- is the campaign the cohort has queued, not claimed. The
coincidence, on the anatomy, is neither a miracle nor an accident:
it is one more reading of the one number this paper keeps meeting
--- the depth of the descent --- and its final grade awaits the
same derivation everything else in the flagged family awaits.

Figure~\ref{fig:a10-budget-w2} shows the inventory the census
delivers --- and shows it twice, because the static snapshot is the
less interesting half of the story. Read the upper panel from left
to right and the anatomy's division is drawn: the baryon:dark-matter
split is \emph{flat across the entire history}, a circle-side ratio
$\Omega_b/\Omega_{\rm dm} = 1/(2\pi^2)\,/\,(1/\pi - 1/2\pi^2)$ that
was fixed once and cannot run; the matter:vacuum seam is the only
thing that moves, and it moves because the interval is descending.
One diagram, two fibers, and the $\pi$-locked inventory is simply
the snapshot taken at the crossing.

The two panels are the same history on two clocks. The upper plots
the density fractions $\Omega_i$ against cosmic time logarithmically,
with the radial coordinate $r = \ln(E_{\mathrm{GUT}}/\mu)$ carried on
the second scale --- the interval's own coordinate, running left to
right, so that the horizontal axis of the figure \emph{is} the descent
this paper keeps invoking. The lower plots the same evolution in
linear time, where the matter-to-vacuum handover stops looking like an
event and becomes what it is, a gradual trade with the crossing
magnified in the inset. Boxed in the upper panel are the ratios that
never move: $\Omega_b/\Omega_m = 1/2\pi = 0.159$ and
$\Omega_b/\Omega_{\mathrm{dm}} = 1/(2\pi - 1) = 0.189$, eternal
because they are circle-side quantities and a circle-side quantity has
nowhere to go. At the $\Omega_m = 1/\pi$ crossing --- which falls some
$75$~Myr before today, close enough that the coincidence is worth
asking about and far enough that it is not a tautology --- the
inventory reads $\OmL = 1 - 1/\pi = 0.682$, $\Omm = 1/\pi = 0.318$,
$\Omb = 1/(2\pi^2) = 0.0507$, and
$\Omega_{\mathrm{dm}} = 1/\pi - 1/(2\pi^2) = 0.268$: fiber-volume
ratios (Eq.~\eqref{eq:hopf}), not fits.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.86\textheight,keepaspectratio]{A10-whynow-Omega-evolution-PAIR-nocap.pdf}
\caption[The cosmic energy budget through history, shown twice.]{\textbf{The
cosmic energy budget through history, shown twice} (reproduced from
\citealp{ArisakaA10Cosmos}; read panel by panel in the text opposite).
Only the matter:vacuum seam moves. The upper panel runs on a logarithmic time axis and the lower on a linear one, so the same three shares --- radiation, matter, vacuum --- are shown once as a history and once as the present epoch a reader lives in. The two matter slices and the vacuum are fixed together by the $\pi$-locked inventory, which is why only one seam in the figure is free to move.}
\label{fig:a10-budget-w2}
\end{figure}

\subsection{The triptych of matter, completed}
\label{sec:matter-triptych}

The trio of episodes has a companion triptych of \emph{objects},
and Problem~7 should close the pair. The cohort's three portraits
of permanence (Table~\ref{tab:triptych-w2}).

\begin{table}[htbp]
\centering
\caption[The three portraits of permanence, aligned.]{The three portraits of permanence, aligned. Each column is the subject of a cohort paper and each row asks the same question of all three: how the object stands to the circle (matter \emph{on} it, matter \emph{guarded by} it, matter \emph{of} it); what fixes its identity (one ground basin, the double guard, one frozen angle); where its mass comes from (the descent's terminus, the QCD floor, a susceptibility the descent delivers); what certificate the ground geometric attractor issues (no exit, no decay, a Hessian equal to $\mK^2$); and why each is inevitable rather than chosen --- the ladder terminates, the weld closes, the displaced angle relaxes.}
\label{tab:triptych-w2}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}llll@{}}
\toprule
& \textbf{Electron} & \textbf{Proton} & \textbf{Kaxion} \\
\midrule
Relation to circle & matter \emph{on} it & matter \emph{guarded by} it & matter \emph{of} it \\
Identity by & one ground basin & the double guard & one frozen angle \\
Mass from & descent terminus & the QCD floor & $\chi(T)$, descent-delivered \\
\GGA{} certificate & no exit & no decay & Hessian $= \mK^2$: the line \\
Why inevitable & ladder terminates & weld closes & displaced angle relaxes \\
\bottomrule
\end{tabular}
\end{table}

\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}
(\citealp{ArisakaP3Electron}; \citealp{ArisakaP6Proton}; \citealp{ArisakaC1Kaxion}). The
two triptychs interlock: the episodes (rolling, ringing, rest) are
\emph{how the descent spent its history}; the objects (labeled,
guarded, ringing) are \emph{what it deposited}. Between them they
exhaust what a one-way reading of a chiral, integer-grooved tape
can do --- and the fact that the visible universe's matter, its
dark matter, and its vacuum are the three columns of one table is
the anatomy's quiet summary of cosmology.

\subsection{The pinning, and the cleanest null in cosmology}
\label{sec:pinning-w2}

The Kaxion's patch-uniformity rests on a mechanism with an
unusually clean observational face, and it deserves its own
section because it is the trio's sharpest near-term discriminator
after the tensor target (\citealp{ArisakaA10Cosmos}; \citealp{ArisakaC1Kaxion}).
In the standard axion story the initial angle is drawn at random
in each causal patch: the relic density is an accident of our
draw, and the field's inflationary fluctuations imprint
isocurvature perturbations that the CMB constrains --- the
story's most dangerous bound. The descent replaces the draw with
a pinning: during the primary inflation, a reflection-symmetric
dressing pins the compact angle coherently across the entire
future observable universe. Three consequences, each observable:
\emph{no lottery} (the relic density is deterministic --- the
$\theta_i \simeq 1.4$ of the abundance inversion is an output,
not a draw); \emph{no defects} (no patch-to-patch walls or
strings from this sector); and above all \emph{zero primordial
isocurvature} --- suppressed tens of orders below the Planck
bound, so the constraint that hunts every random-draw axion
passes here trivially. The signature runs both ways: a future
\emph{detection} of axion-sector isocurvature kills the pinning
outright, and with it the trio's middle panel. A null this deep,
staked this plainly, is the anatomy's preferred kind of
prediction: the theory does not merely fit the absence; it
\emph{requires} it.

\subsection{The trio, in one view}
\label{sec:trio}

Figure~\ref{fig:trio-w2} puts the three panels side by side, and the
point of seeing them together is that they are one object in three
states rather than three coincidences: the interval's coordinate
rolling once (inflation), the circle's coordinate still ringing
(the Kaxion at $1.95$~GHz with its $1/35$ doublet), and both at
rest (the vacuum residue). Table~\ref{tab:trio} then sets the three
side by side as a ledger --- what each is on the anatomy, what it
predicts, and the measurement that kills it --- which is the form a
reader should carry away, because the three kills are independent
and a framework that survives all three has survived something.

\begin{table}[htbp]
\centering
\caption[The cosmological triptych: rolling, ringing, rest.]{The cosmological triptych: rolling, ringing, rest. All
entries published (\citealp{ArisakaA10Cosmos}; \citealp{ArisakaC1Kaxion});
grades inherited. The three columns are the interval's coordinate rolling, the circle's coordinate ringing, and both at rest: inflation for $N^{*} = 52.6$ e-folds once, giving $n_s = 0.9620$ and $r = 0.0043$; the Kaxion ringing since $T_{\mathrm{osc}} \simeq 1$~GeV at $8.1\,\mu$eV and $1.95$~GHz; and the de~Sitter floor at $w = -1$ with $\OmL = 1 - 1/\pi$. The last row is what would kill each.}
\label{tab:trio}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\textwidth}{@{}P{2.6cm}>{\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 \\
Duration & transient: $N^{*} = 52.6$ e-folds, once & persistent: ringing since $T_{\mathrm{osc}} \simeq 1$~GeV & eternal: $w = -1$, de Sitter \\
Key numbers & $n_s = 0.9620$; $r = 0.0043$ & $8.1\,\mu$eV; $1.95$~GHz; $1/35$ & $\OmL = 1 - 1/\pi$; $m_\Lambda \simeq 6.1$~meV against $\rho_\Lambda^{1/4} = 2.24$~meV, i.e.\ $\zeta_\Lambda^{1/4} = 0.37$ uncomputed \\
Why unique & one interval $\Rightarrow$ one inflaton & one circle $\Rightarrow$ one axion & one \GGA{} $\Rightarrow$ one vacuum \\
Kill & $r$ off $0.004$; $n_s < 0.955$ & isocurvature; off the $1.59$--$2.13$~GHz window & $w \neq -1$; phantom crossing \\
\bottomrule
\end{tabularx}
\end{table}

Ninety-five percent of the present energy budget --- dark matter
plus dark energy --- is the bookkeeping of the two non-spatial
fibers, and the early universe's defining episode is the open
coordinate's one traverse. \emph{The dark sector is the two fibers,
seen in the sky} --- the becoming and the being, each showing up
exactly once, and their joint silence accelerating the expansion
ever after. Every number in this Part is falsifiable this decade,
and Part~VII collects the switches.

\subsection{Seminar questions}

\paragraph{What if the haloscopes sweep the window and find
nothing?} Then the trio's middle panel dies as stated --- the
window is discrete ($\Xicap$'s alternatives), not adjustable,
and the cohort has pre-committed to that reading. What would
survive is instructive: the census argument (one circle, one
axion) is killed only if dark matter is found to be something a
compact angle cannot be; a null without a replacement candidate
kills the address, not the anatomy. The doublet is the
discriminator either way: any line \emph{without} a $1/35$
companion is not the Kaxion, however pretty its frequency.

\paragraph{Could dark energy be slowly rolling after all?} The
framework forbids it and stakes the forbidding: $w = -1$
identically, no phantom side, with the conditions of the
exactness theorem named. A confirmed $w(z)$ drift would not
merely adjust a parameter --- it would break the terminus
reading (arrival is irreversible) and with it the anatomy's
account of why the descent ends. This is the trio's cleanest
theory-level exposure, and the survey instruments of the decade
audit it continuously.

\paragraph{Is it a coincidence that all three episodes involve
the same integer $K$?} It is the opposite of a coincidence ---
it is the point. $N^* = 52.6$ rides the corridor whose depth is
$K$; the Kaxion's split is $1/K$ and its release toll rides
$e^{-K/2}$ chains; the terminus's seesaw is $e^{-2K}$. One
integer, three sky-scale appearances, plus the collider's
$e^{-K}$ --- the trio is the strongest instance of the
one-alphabet discipline of Problem~4, and the easiest place to
kill it.

\subsection{What the trio teaches the other problems}

The dark trio is this paper's best classroom because it is where
the anatomy's abstractions become inventory. \emph{To
Problem~5} it supplies the opening act and the fossil windows;
\emph{to Problem~6}, the seesaw that doubles the depth;
\emph{to Problem~8}, the stretch that buys Horizon 1 and the
pinning that closes Horizon 3's last lottery; \emph{to
Problem~14}, the terminus as the currency system's closing
statement; \emph{to Problem~15}, the zero-isocurvature null
that scores the draw-versus-groove question; \emph{to
Problem~16}, the de Sitter horizon as the paradox's cousin. Six
problems borrow from this one --- which is the anatomy's
recursive signature throughout the paper: the answers are not
sixteen separate keys but one keyring, and the trio is where
the ring is easiest to hold.

\section{Problem 8.\ The Three Horizon Problems}
\label{sec:w2-prob8}

\subsection{Three agreements, one puzzle}

Point two radio telescopes at opposite edges of the sky. The
microwave background arriving from each direction left its source
before those regions could ever have exchanged a signal --- and
yet the two temperatures agree to one part in a hundred thousand.
That is the textbook horizon problem, and inflation is its
textbook solution. But the sky's agreement runs far deeper than
temperature, and the deeper layers are rarely posed as problems at
all --- they are assumed. Every distant galaxy is built of
electrons with \emph{exactly} the mass and charge of ours; every
spectral line, at every redshift, is our line shifted and nothing
else. And every region made the same \emph{choices}: matter over
antimatter everywhere (no annihilation glow at any domain
boundary, anywhere); the same vacuum, the same broken symmetries,
the same flavor textures. Three agreements, of escalating
strangeness: the same \emph{conditions}, the same \emph{things},
the same \emph{decisions}. Inflation addresses the first. What
addresses the other two? The standard framework's honest answer
is: nothing --- they are written into the assumption that ``the
laws of physics are the same everywhere,'' which is not an
explanation but the name of the mystery.

First, the puzzle's arithmetic, so its size is on record. The
particle horizon at last scattering --- the farthest distance a
signal could have traveled by $z \simeq 1090$ --- subtends about
$2^\circ$ on today's sky, so the microwave sky comprises
\begin{equation}
N_{\mathrm{patch}} \;\sim\; \frac{4\pi}{(2^\circ)^2}
\;\approx\; 10^{4}
\label{eq:patches-w2}
\end{equation}
regions that, on the unstretched account, share no causal past ---
yet their temperatures agree to
$\Delta T/T \sim 10^{-5}$, their contents to every measured digit,
and their constants absolutely. Inflation repairs the first
agreement by rewriting the denominator: the stretch makes the
whole sky descend from one pre-inflation patch. The second and
third agreements are what this Problem is about, because they
concern choices made \emph{after} any stretch.

\subsection{Horizon 1: the same conditions --- bought by the launch}

The first horizon is the interval's purchase, and the anatomy
keeps inflation's textbook answer while sharpening its status.
The launch (Problem~7) stretches a single causally connected
patch across the whole observable sky: uniform temperature,
uniform curvature, and the seeds of structure laid by one
coherent episode. What the anatomy adds is the census discipline
--- one interval, one inflaton, no menu --- so the solution is
not one scenario among hundreds but the unique roll the geometry
possesses, with its two predictions ($n_s = 0.9620$,
$r = 0.0043$) exposed. What it subtracts is the regress: the
roll happened once, on a fiber with no inverse transport, so
there is no eternal self-reproduction multiplying beginnings
(\S\ref{sec:inflation-w2}). One patch, one stretch, one sky.

\subsection{Horizon 2: the same things --- bought by the classes}

The second horizon is the circle's purchase, and it needs no
signal at all. Why is an electron at redshift six identical to an
electron in this room? The standard framework must posit it: in
field-theory language all electrons are excitations of one field,
but \emph{why the field's parameters are position-independent}
across super-horizon distances is again assumption. On the
anatomy, identity is a holonomy class --- charge a winding, spin
and representation labels of the compact fiber --- and
\emph{classes are the same everywhere without communication}: two
regions need never have exchanged a photon for their windings to
be equal integers, any more than two distant triangles need to
correspond to both have three sides. Topology is the one kind of
sameness that requires no synchronization, no mechanism, and no
common past --- which is precisely the kind the sky exhibits. The
labels' \emph{addresses} (the masses) are equally uniform because
the descent's landscape is the same groove read at every point
(Horizon 3). One sentence: \emph{the universe agrees about its
contents because contents are classes, and classes cannot
disagree.}

\subsection{Horizon 3: the same choices --- bought by the groove}

The third horizon is the deepest, and it is the determinism triad
of \S\ref{sec:determinism} projected across the sky. Wherever a
symmetry broke, every region fell the same way: the same vacuum
angle, the same flavor lattice, and --- most tellingly ---
matter over antimatter with the same one-in-a-billion excess to
the precision of nucleosynthesis. In any framework where those
choices are spontaneous in the textbook sense --- a random fall
from a symmetric hilltop --- causally disconnected regions
\emph{must} choose independently, and the sky should be a
patchwork of domains with defect walls between them. It is not.
The anatomy's three mechanisms close the lottery: the dial is
quantized with a unique scan winner (nothing continuous to fall
through), the \GFF{} floor is unique per stratum (nowhere else to
land), and the reading is one-way (no region can revise). Every
region plays the same groove because there is one groove and the
players cannot improvise. The sharpest instance is the excess
itself, computed in Problems~10--11 and uniform wherever
nucleosynthesis has been checked; the full development, with the
tuning and lottery alternatives priced against the sky, is
\S\ref{sec:horizons}.

Figure~\ref{fig:three-horizons} draws the puzzle and its
three-part answer in one frame.

\begin{figure}[htbp]
\centering
\resizebox{0.665\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% sky arc
\draw[line width=1.10pt] (1.0,3.2) arc (160:20:4.8);
\node[draw=blue!55!black,line width=0.80pt,fill=blue!8,minimum size=0.5cm] (A) at (1.25,3.35) {\scriptsize A};
\node[draw=blue!55!black,line width=0.80pt,fill=blue!8,minimum size=0.5cm] (B) at (10.35,3.35) {\scriptsize B};
\node at (5.8,5.4) {\scriptsize the observed sky};
\draw[<->,line width=0.80pt,red!70!black] (1.9,3.6) .. controls (5.8,5.0) .. (9.7,3.6);
\node[red!70!black] at (5.8,4.35) {\scriptsize never in causal contact};
% observer
\fill[black!70] (5.8,0.8) circle (0.07);
\node[below] at (5.8,0.7) {\scriptsize us};
\draw[black!50] (5.8,0.8) -- (A.south east);
\draw[black!50] (5.8,0.8) -- (B.south west);
% three rows
\node[align=left,anchor=west] at (0.4,-0.6)
  {\scriptsize \textbf{H1} \ same \emph{conditions} \ ($10^{-5}$):
   bought by the launch --- one plateau, one roll};
\node[align=left,anchor=west] at (0.4,-1.35)
  {\scriptsize \textbf{H2} \ same \emph{things} \ ($10^{-6}$ or better):
   bought by the classes --- holonomy cannot vary by region};
\node[align=left,anchor=west] at (0.4,-2.1)
  {\scriptsize \textbf{H3} \ same \emph{choices} \ (absolute):
   bought by the groove --- one floor, forced everywhere};
\end{tikzpicture}%
}
\caption[The three horizon problems, one geometry.]{\textbf{The three horizon problems, one geometry.} Patches
A and B share no causal past, yet agree in conditions, in kinds,
and in constants. The stretch alone buys only H1; H2 is the
circle's --- classes are global data, with no local dial to drift
--- and H3 is the groove's, the Morse-unique floor of the
determinism triad. Escalating strangeness, one anatomy.}
\label{fig:three-horizons}
\end{figure}

Figure~\ref{fig:hubble-radius-w2} shows the standard diagram of
the first horizon and its repair --- and its silence on the other
two.

\begin{figure}[htbp]
\centering
\resizebox{0.682\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[The horizon problem, and inflation's repair.]{\textbf{The horizon problem, and inflation's repair, in
the standard diagram} (reproduced from \citealp{ArisakaA10Cosmos}).
The comoving Hubble radius $(aH)^{-1}$ --- the size of a causally
coherent region, in comoving units --- across the four eras.
During inflation it \emph{shrinks} through $N^*$ e-folds: today's
largest scales exited a causally connected region early, and
re-entered --- through the radiation and matter eras, where the
radius grows --- carrying its uniformity; in the dark-energy era
it shrinks again. Without the inflationary segment the radius
only grows, and Eq.~\eqref{eq:patches-w2}'s patches never spoke.
The repair buys Horizon~1 --- shared conditions --- and nothing
else: the diagram is silent on why the patches' particle
\emph{kinds} and \emph{constants} agree, which is Horizons~2
and~3.}
\label{fig:hubble-radius-w2}
\end{figure}

\subsection{The alternatives, priced}

Three accounts of cosmic sameness are on the market, and they
separate observationally. \emph{Tuning}: the sameness is an
initial condition --- but a tuned universe must tune every region
separately, and pays the full improbability once per horizon
volume. \emph{The lottery with inflation as courier}: choices
were made once in a tiny patch and inflated outward --- this
works for Horizon 1 and for any choice made \emph{before} the
stretch, but every post-inflationary breaking (the QCD angle, the
electroweak vacuum, the axion misalignment of the standard
story) reopens the lottery afterward, patch by patch; the
standard axion's isocurvature problem is exactly this reopening.
\emph{The groove}: nothing was ever drawn, before or after ---
same dial, same floor, same reading everywhere, at every epoch.
The three accounts stake different skies: tuning predicts nothing
and explains nothing; the courier predicts residual
patch-variation in every late choice; the groove predicts
\emph{zero} --- no domains, no defects from these sectors, no
axion isocurvature, no drifting constants --- nulls so deep they
are signatures (\S\ref{sec:pinning-w2}).

\subsection{The sameness, audited in numbers}

The three horizons are quantitative facts before they are
puzzles, and the numbers set the bar any explanation must clear.
\emph{Conditions}: the CMB temperature is uniform to
$\Delta T/T \sim 10^{-5}$ across the full sky --- across regions
separated, at emission, by dozens of naive horizon lengths.
\emph{Things}: quasar absorption spectra constrain any variation
of the fine-structure constant across billions of light years and
billions of years at the part-per-million level; molecular and
nuclear constancy bounds reach further; every extragalactic
spectrum ever taken is our atomic physics, redshifted.
\emph{Choices}: the gamma-ray sky shows no annihilation seam at
any scale --- antimatter domains are excluded out to
cosmological volumes; helium and deuterium abundances match one
baryon-to-photon ratio everywhere nucleosynthesis can be probed;
and no domain wall or sameness-breaking defect has ever been
seen. Sameness at $10^{-5}$, $10^{-6}$, and absolute --- three
escalating precisions for three escalating depths of agreement.
An account that buys only the first with dynamics and posits the
rest has explained the least demanding third of the data.

\subsection{What the stretch alone cannot buy}

It is worth being precise about why Horizon 1's solution does not
extend, because the gap is the Problem. Inflation synchronizes
whatever exists \emph{before} the stretch; it is powerless about
choices made \emph{after}. And the standard framework makes its
deepest choices after: the QCD vacuum angle settles at the QCD
scale; the standard axion draws its misalignment angle at the
Peccei--Quinn scale, patch by patch; in generic settings even the
electroweak vacuum's alignment is a post-inflationary event. Each
late choice re-opens the lottery inside the stretched patch ---
the standard axion's isocurvature and domain-wall problems are
the textbook symptoms --- so the courier account predicts
residual patchiness in every late-deciding sector, at levels the
searches keep pushing down. The groove differs precisely here:
its choices were never events, early or late --- the dial, the
floor, and the reading are the same at every point because they
are arithmetic, not history. The observational separation is
therefore clean and current: every deepening null (isocurvature,
domains, drifting constants) moves probability from the courier
to the groove, and a single confirmed patch-variation moves it
all back.

\subsection{The identity recursion: sameness all the way up}

Horizon 2's class argument compounds through structure, and the
compounding is worth exhibiting because it is where cosmic
sameness touches everyday life. Identical electrons and identical
protons (the classes) make identical atoms --- every hydrogen
atom in every galaxy has exactly our Lyman series, which is why
redshift surveys work at all. Identical atoms make identical
chemistry: the periodic table is not a local convention but a
holonomy consequence, the same table wherever matter has settled
(Problem~12's biosphere section carries the argument into
biology). Identical chemistry makes calibratable astrophysics:
standard candles are standard \emph{because} type-Ia detonations
run on the same nuclear ledger everywhere --- the entire distance
ladder, and with it observational cosmology itself, quietly rests
on Horizon 2. The recursion's summary: the universe is not merely
made of the same stuff; it is \emph{legible} --- and legibility
is the anatomy's deepest observational gift, because a cosmos of
drifting classes could not even be measured coherently, let
alone explained. The sameness the three horizons name is the
precondition of the science that names them.

\subsection{The observational faces}

Each horizon carries live instruments. \emph{H1}: the CMB's
uniformity and its acoustic structure --- long confirmed --- plus
the launch's own switches ($r$, $n_s$). \emph{H2}:
spectroscopic constancy --- quasar absorption lines bound any
spatial or temporal drift of the fine-structure constant at the
$10^{-6}$ level and below, and every improvement is a sharper
audit of class identity; a confirmed spatial variation of a
protected constant would break this horizon outright. \emph{H3}:
the antimatter-domain searches (no annihilation edge in the
gamma-ray sky), the defect searches (no walls, no cosmic-string
signatures from these sectors), and the axion-sector
isocurvature bound --- the standard lottery's signature, whose
continued null is the groove's fingerprint and whose detection
is this Problem's kill. Three horizons, three nulls, all holding:
the sky, so far, plays one record.

\textbf{Grade and kill, consolidated.} H1: the launch's grades.
H2: theorem (class identity), audited by constancy bounds. H3:
the triad's grades, with the pinning a named conditional. Kill:
patch variation of a protected quantity; an annihilation edge; a
domain wall; axion-sector isocurvature. The three horizon
problems, on the anatomy, are one question asked at three depths
--- \emph{why is it all the same?} --- with one answer at three
registers: \emph{one stretch, one alphabet, one groove.}

%% ===================================================================
\subsection*{Part IV in one paragraph, and what comes next}

Cosmology, read on the two fibers, is one descent and one set of
books. The launch is the interval's single traverse, which is why
there is one inflaton and not a menu, and why $N^{*} = 52.6$ comes
out as a length rather than a fit. The hierarchy is not a small
number but a deep one: thirty-five e-folds down a throat, with the
exponent a ledger integer. The dark sector is the two fibers in the
sky --- the interval rolling once, the circle still ringing at
$1.95$~GHz, and the vacuum the residue of both at rest. And the
horizon problems are one question at three depths, answered by one
stretch, one alphabet, and one groove.

Cosmology spent the fibers. The next Part asks what they are made
of. Part~V takes the four questions underneath matter itself ---
why the mirrors break and one does not, where the antimatter went,
why quarks are never seen and the proton never dies --- and finds
the circle owning the orientations and the ledgers, the interval
owning the amounts and the once-only crossing.

%%  PART V -- FOUNDATION OF PARTICLE PHYSICS
%% ===================================================================
\GGPart[Foundation of particle physics]{Foundation of Particle
Physics}{Problems 9--12: Discrete Symmetry, Antiparticles, the
Excess of Matter, and Confinement with the Stable Proton}

\noindent The four problems of this Part are the discipline's
oldest foundations, and all four are treated at full length ---
they are the campaign's original heartland. Problem~9 is the
discrete symmetries; Problems~10 and~11 share one block (the
mirror reading and the one-way toll are one story); Problem~12 is
the guarded proton.

\section{Problem 9.\ Discrete Symmetry: $C$, $P$, $T$, $CP$, $CPT$ --- and the Strong-CP Problem}
\label{sec:w2-prob9}

\subsection{The five operations, located}
\label{sec:cpt-located}

In the textbook presentation, the discrete symmetries are operations
on fields: charge conjugation $C$ swaps particles with antiparticles,
parity $P$ reflects space, time reversal $T$ runs the film backward,
and the celebrated theorems --- $CPT$ exactness, $CP$ violation ---
arrive as properties of Lagrangians. The two-fiber anatomy relocates
all five \emph{into the geometry}: each discrete operation is the
reversal of an orientation that the bulk actually possesses, and each
observed fate --- exact, broken, or broken-at-runtime --- follows
from \emph{which fiber owns the orientation being reversed}
(\citealp{ArisakaP5CPT}; \citealp{ArisakaP4Real}).

The bulk offers exactly three orientations: the compact circle
$\Stheta$ can be traversed in two senses; the constitutional clock
--- the causal orientation of the boundary, tied to the interval's
one-way descent --- can be assigned in two senses; and the spatial
stage $\mathcal M_4$ can be given either handedness. The dictionary
is then one line each (Table~\ref{tab:cpt-owners-w2}).
\begin{table}[htbp]
\centering
\caption[Which structure each discrete operation reverses.]{Which structure each discrete operation reverses, and which part of the geometry owns it. $C$ reverses the sense of the circle, $\theta \mapsto -\theta$, and belongs to the sixth dimension; $P$ reverses the handedness of the spatial stage and belongs to $\mathcal M_4$; $T$ reverses the sense of the clock and belongs to the causal orientation the interval feeds. $CP$ is therefore mixed, turning circle and stage together, and $CPT$ reverses all three orientations at once --- which is why it is a property of the total geometry rather than of any single fiber.}
\label{tab:cpt-owners-w2}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{Operation} & \textbf{What it reverses} & \textbf{Owning structure} \\
\midrule
$C$ & the sense of the circle, $\theta \mapsto -\theta$ & $\Stheta$ (the 6th) \\
$P$ & the handedness of the stage & $\mathcal M_4$ \\
$T$ & the sense of the clock & the causal orientation the interval feeds \\
$CP$ & circle and stage together & mixed \\
$CPT$ & all three orientations at once & the total geometry \\
\bottomrule
\end{tabular}
\end{table}

Nothing here is new dynamics --- the identifications are the
CPT companion's theorems T-P5-A/B/C \citep{ArisakaP5CPT} and the
real-bulk companion's discrete-symmetry table, derived from the
orientations of its two global structures, the circle and the clock
\citep{ArisakaP4Real}. What the anatomy adds is the explanation of
the \emph{fates}.

The claim in symbols. Write $\theta$ for the phase circle's
coordinate, $x$ for the stage's spatial directions, and $t_b$ for
the boundary clock. The three operations are the three independent
orientation reversals the geometry offers:
\begin{equation}
C:\ \theta \mapsto -\theta,
\qquad
P:\ x \mapsto -x,
\qquad
T:\ t_b \mapsto -t_b ,
\label{eq:orientations-w2}
\end{equation}
--- charge conjugation reverses the circle's winding sense (which
negates every charge, since charges are windings), parity reverses
the stage's handedness, and time reversal reverses the clock
against which the runtime reads. Their product $CPT$ reverses the
total orientation of the bulk, and a total orientation reversal is
an isometry of the geometry itself: nothing in any fiber can
distinguish the two senses jointly. That is the anatomy's one-line
proof sketch of what the L\"uders theorem establishes from field
axioms --- and its explanation of the pattern: each \emph{single}
reversal can fail wherever some structure fixes a preferred sense
(the chiral alphabet fixes the circle's, the runtime fixes the
clock's), but the \emph{joint} reversal has no structure left to
notice it. The laboratory numbers this section must answer to: the
observed $CP$ violation in kaons at $2\e{-3}$, direct $T$ violation
\citep{BaBar2012}, and the strong-$CP$ bound
$|\bar\theta| \lesssim 10^{-10}$ from the neutron electric dipole
moment \citep{Abel2020}.

\subsection{Why $CPT$ is exact}
\label{sec:cpt-exact}

$CPT$ reverses every orientation the geometry has. A world read with
all orientations flipped is the same geometry described in the mirror
convention: the holonomies are conjugated, the descent is relabeled,
and no invariant changes. Formally, the real-bulk construction makes
$CPT$ the automatic composite --- the one discrete operation that is
a symmetry of the \emph{books} and of the \emph{reading} at once ---
which is why its exactness is not an assumption of the framework but
a consistency statement: breaking $CPT$ would require the bulk to
possess a fourth orientation it does not have. The laboratory
statement inherits the geometry's confidence: any observed $CPT$
violation --- a mass difference between a particle and its
antiparticle, at any precision --- falsifies not a model choice but
the arena itself.

Figure~\ref{fig:reversals-w2} draws the three reversals on the
geometry.

\begin{figure}[htbp]
\centering
\resizebox{0.604\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth]
 % C: circle
 \begin{scope}[shift={(0,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[->, line width=1.10pt, blue] (0.75,0) arc (0:120:0.75);
  \draw[->, line width=1.10pt, red, dashed] (0.75,0) arc (0:-120:0.75);
  \node at (0,-1.35) {\textbf{C}: fiber reversal};
  \node[font=\footnotesize] at (0,-1.8) {$\theta \mapsto -\theta$;\ \ $\Jq \mapsto \Jq$};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % P: space
 \begin{scope}[shift={(4.6,0)}]
  \draw[->, line width=1.10pt, blue] (-0.9,0) -- (0.9,0) node[right]{$\vec{x}$};
  \draw[->, line width=1.10pt, red, dashed] (0.55,-0.3) -- (-0.9,-0.3) node[left]{$-\vec{x}$};
  \draw[line width=0.80pt] (0,0.5) -- (0,-0.6);
  \node at (0,-1.35) {\textbf{P}: space reversal};
  \node[font=\footnotesize] at (0,-1.8) {touches neither source};
  \node[font=\footnotesize, blue] at (0,-2.25) {$J$-linear (unitary)};
 \end{scope}
 % T: clock
 \begin{scope}[shift={(9.2,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.75);
  \draw[line width=0.80pt] (0,0) -- (0,0.55);
  \draw[->, line width=1.10pt, blue] (0,0) -- (25:0.6);
  \draw[->, line width=1.10pt, red, dashed] (0,0) -- (155:0.6);
  \node at (0,-1.35) {\textbf{T}: clock reversal};
  \node[font=\footnotesize] at (0,-1.8) {$t \mapsto -t$;\ \ $\Ja \mapsto -\Ja$};
  \node[font=\footnotesize, red] at (0,-2.25) {$J$-antilinear (antiunitary)};
 \end{scope}
\end{tikzpicture}}
\caption[The three orientation reversals.]{\textbf{The three orientation reversals} (reproduced
from \citealp{ArisakaP5CPT}). Each panel reverses one sense of
Eq.~\eqref{eq:orientations-w2}: $C$ the circle's winding (all
windings $\mapsto$ their negatives, so every charge flips while
masses --- which are not orientations --- stay fixed), $P$ the
stage's handedness, $T$ the clock's direction (under which the
runtime's currents reverse). A discrete symmetry ``breaking'' is,
on this reading, never dynamical: it is the presence of a
structure that fixes one of the three senses --- and the table of
fates in this section is the census of which structures exist.}
\label{fig:reversals-w2}
\end{figure}

\subsection{Why $CP$ breaks: the chiral alphabet}
\label{sec:cp-breaks}

$C$ alone and $CP$ are symmetries of the connection but not of the
\emph{content} written on the circle. The flavor sector deposits on
$\Stheta$ a lattice of phases --- the flavor alphabet, with spacing
$\pi/5$ set by the ledger integer $L = 5$ --- and the alphabet is
\emph{chiral}: read clockwise and counterclockwise it does not
match. Figure~\ref{fig:alphabet} draws it. Four letters matter for
everything downstream, and the campaign's audit of the birth of
matter \citep{ArisakaP5CPT} graded each:
\begin{itemize}[leftmargin=2.0em]
\item $\dCKM = 2\pi/5$ --- the quark sector's phase: a \emph{chiral
bystander}. It makes the kaon and $B$-meson asymmetries of the
laboratory, but its contribution to the cosmic excess is the famous
$\sim 10^{-20}$ of the bound computed by \citet{Gavela1994} ---
real chirality, negligible
throughput.
\item $\dPMNS = \pi + 2\arcsin(1/7) = 196.4^\circ$ --- the neutrino
Dirac phase: a \emph{spectator} to the excess (the washout-averaged
asymmetry is blind to it at leading order), but the laboratory's
sharpest test of the lattice: DUNE and Hyper-Kamiokande measure it
directly.
\item $\alpha_{21} = 0$ --- inert.
\item $\alpha_{31} = \pi$ --- \emph{the driver}. The half-angle
enters the decay amplitude as $e^{i\pi/2} = i$: maximal interference
from an angle that is itself CP-even --- the amount saturates while
the orientation survives. The excess of matter (Problem~11) rides on
this letter.
\end{itemize}

The anatomy's sentence for the fate of $CP$: \emph{the connection is
symmetric; the tape is not.} $CP$ breaking is not a property of the
dynamics but of the groove --- the constitution's integers, pressed
into the circle as a chiral lattice --- and that is why its
parameters neither run nor drift: they are holonomy-class data,
exact at every scale, measured as angles.

\begin{figure}[htbp]
\centering
\resizebox{0.505\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.4mm]}, scale=1.15]
\draw[line width=1.10pt] (0,0) circle (2.0);
% pi/5 lattice dots
\foreach \k in {0,1,...,9}
  \fill[black!55] ({36*\k}:2.0) circle (0.05);
% the four letters
\draw[->,line width=1.10pt,blue!65!black] (0,0) -- (72:1.55);
\node[blue!65!black,align=center] at (72:2.62) {\scriptsize $\dCKM = 2\pi/5$\\[-2pt]\scriptsize chiral bystander};
\draw[->,line width=1.10pt,red!70!black] (0,0) -- (180:1.55);
\node[red!70!black,align=center] at (180:2.75) {\scriptsize $\alpha_{31} = \pi$\\[-2pt]\scriptsize \textbf{the driver}: $e^{i\pi/2} = i$};
\draw[->,line width=1.10pt,black!60] (0,0) -- (196.4:1.15);
\node[black!60,align=center] at (223:3.05) {\scriptsize $\dPMNS = 196.4^\circ$\\[-2pt]\scriptsize spectator (DUNE/HK)};
\draw[->,line width=1.10pt,green!45!black] (0,0) -- (0:1.55);
\node[green!45!black,align=center] at (0:2.72) {\scriptsize $\alpha_{21} = 0$\\[-2pt]\scriptsize inert};
% magnitude box
\node[draw=black!55,align=center,fill=white] at (0,-3.1)
  {\scriptsize the throughput is set elsewhere: $\epsilon\,\kappa = e^{-K/2} = 2.5\times10^{-8}$ (Part III)\\[-1pt]
   \scriptsize orientations from the circle; amounts from the interval};
\end{tikzpicture}}
\caption[The chiral alphabet on the circle.]{\textbf{The chiral alphabet on the circle.} The flavor
phases sit on the $\pi/5$ lattice of $\Stheta$ (dots). Four letters
carry the discrete-symmetry story: the quark phase
$\dCKM = 2\pi/5$ (chiral but negligible throughput), the neutrino
Dirac phase $\dPMNS = \pi + 2\arcsin(1/7)$ (a spectator to the
cosmic excess, the laboratory's direct test of the lattice), the
inert $\alpha_{21} = 0$, and the driver $\alpha_{31} = \pi$, whose
half-angle $e^{i\pi/2} = i$ maximizes the interference that pays the
matter excess. \emph{The connection is symmetric; the tape is not}
\citep{ArisakaP5CPT}.}
\label{fig:alphabet}
\end{figure}

\subsection{Why $T$ breaks at runtime}
\label{sec:t-breaks}

$T$ reverses the clock, and the clock is fed by the fiber with no
inverses. At the level of the \emph{laws} --- the bulk equations,
the connection --- reversing the clock is as innocent as reversing
the circle. At the level of the \emph{runtime} it is not, because
the runtime is \GDOS{}: the boundary's open-system evolution, whose
generator is the interval's forward semigroup. A semigroup has no
element that undoes another; a reversed clock would require exactly
such an element; so $T$ is broken wherever the runtime acts --- in
measurement, in decoherence, in every record --- not because a
$T$-violating term was added to a Lagrangian but because \emph{the
reading of the tape has a direction even when the tape does not}.
This is the precise sense in which the arrow of time (Problem~13) and
the fate of $T$ are one fact. The microscopic $T$-violation of the
weak sector (the $CP$ violation above, read through $CPT$) is the
circle's contribution; the macroscopic irreversibility is the
interval's; and the two are independent in origin --- which the
anatomy displays and the textbook presentation obscures.

\subsection{Where a preferred sense comes from at all}
\label{sec:preferred-sense}

Before the calculus, one conceptual knot deserves untying, because
it is where most readers first suspect circularity. A chiral
alphabet presupposes that ``clockwise'' and ``counterclockwise''
differ --- but the circle itself is unoriented, and the laws are
even-handed. Where does the preferred sense come from? The
campaign's answer (\citealp{ArisakaW1Symmetry}; \citealp{ArisakaP5CPT}) is the
determinism triad of Part~II, applied to orientation itself: the
constitution's integers press a pattern into the circle whose two
readings are \emph{not} congruent (the pentagon's chirality ---
content, not connection); the runtime reads the tape in the one
direction the interval possesses; and the composite --- chiral
content, one-way reading --- manufactures an \emph{effective}
orientation without either ingredient being an oriented circle or
an asymmetric law. This is the resolution of the oldest form of
the matter--antimatter puzzle (``symmetric laws cannot prefer'') in
two moves: the laws do not prefer; the tape and the read-head
jointly do. The reader will recognize the shape from Part~II: it is
Curie's principle, dissolved --- symmetric dynamics, asymmetric
arithmetic --- now applied to handedness itself.

\subsection{The orientation calculus}
\label{sec:orientation-calc}

Why exactly five operations, and why these? The real-bulk companion
derives the discrete table from first principles
\citep{ArisakaP4Real}, and the derivation is short enough to carry
whole. The bulk supplies two global structures attached to every
system --- the phase circle (generator $\partial_\theta$) and the
constitutional clock ($\partial_t$ with the positivity clause of
\AX{3}) --- and the stage supplies its spatial orientation. Each
structure admits exactly two senses; the discrete group of the
geometry is therefore $\bZ_2 \times \bZ_2 \times \bZ_2$: eight
elements, generated by the three reversals, of which the identity
is trivial and the seven non-trivial elements organize into the
familiar five once the two mixed reversals that involve the clock
alone are written as $T$-composites. Three consequences of the
calculus are worth separating, because the textbook presentation
runs them together:

\begin{itemize}[leftmargin=2.0em]
\item \emph{Antiunitarity is not a postulate.} $T$ reverses the
clock, and the clock's sense is what fixes the sign convention of
frequency: positive energy means phases that rotate one way.
Reversing the clock while preserving positivity forces complex
conjugation --- Wigner's antiunitarity of $T$, here a corollary of
orientation bookkeeping rather than an axiom of representation
theory.
\item \emph{$C$ is geometric, not internal.} Charge conjugation is
the circle's reversal, $\theta \mapsto -\theta$; it negates every
winding at once, which is why no particle flips its electric charge
without flipping its baryon and $Qi$ charges in lockstep --- a
correlation the internal-symmetry framing must impose
representation by representation.
\item \emph{The composite $CPT$ is inevitable.} With three
orientations, the total reversal is the unique element that leaves
no orientation for an invariant to lean on; its exactness
(\S\ref{sec:cpt-exact}) is the statement that the geometry has
\emph{only} these three orientations --- no fourth hidden sense
exists to be reversed. A $CPT$ violation would be the discovery of
a fourth orientation, which is why I treat the
antihydrogen program as an arena test, not a model test.
\end{itemize}

\subsection{The five fates, in one table}
\label{sec:fates}

Table~\ref{tab:fates} is the census the argument has been building
toward: five operations, and for each of them the structure whose
orientation it reverses, whether the geometry admits that reversal,
and the fate the laboratory reports. Read down the last column and
the pattern is the whole of Problem~9 --- three orientations the
bundle can reverse and does, one it cannot, and one composite that
inherits its fate from the others.

\begin{table}[htbp]
\centering
\caption[The discrete symmetries by fiber.]{The discrete symmetries by fiber: what each reverses, and
why nature treats each as it does. Sources:
(\citealp{ArisakaP5CPT}; \citealp{ArisakaP4Real}; \citealp{ArisakaA8QM}). Read down the third column and the pattern is the paper's claim in five lines: $C$ and $P$ break because the alphabet on the circle is chiral and the interval's agent couples chirally; $T$ is exact in the books and broken at runtime, because the runtime rides the semigroup; $CP$ breaks by angles that do not run; and $CPT$ is exact, because reversing all three orientations is a relabeling under which no invariant moves.}
\label{tab:fates}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabularx}{\textwidth}{@{}lP{3.4cm}P{2.6cm}>{\raggedright\arraybackslash}X@{}}
\toprule
\textbf{Op.} & \textbf{Reverses} & \textbf{Fate} & \textbf{Why (the anatomy)} \\
\midrule
$C$ & the circle's sense & broken (with $P$) & the alphabet on the circle is chiral: content, not connection \\
$P$ & the stage's handedness & broken (weak sector) & the interval's agent couples chirally (Problem~2): the descent selects a hand \\
$T$ & the clock's sense & exact in the books; broken at runtime & the runtime rides the semigroup: reading has a direction \\
$CP$ & circle + stage & broken, by angles that do not run & the groove is chiral; its letters are holonomy classes \\
$CPT$ & all orientations & \textbf{exact} & the total orientation is a relabeling: no invariant moves \\
\bottomrule
\end{tabularx}
\end{table}

One consequence deserves its own sentence here, because the next
Problem is built on it: $C$ broken in content but exact in the
connection is precisely what an \emph{antiparticle} needs in order
to exist and to differ --- the same class, the other reading. That
is Problem~10, and \S\ref{sec:lab-cpt} hands it over.

One number closes the audit, because it is the sharpest
zero in the discrete-symmetry book: the driver angle. The flavor
lattice's Majorana phase is fixed at
\begin{equation}
\alpha_{31} \;=\; \pi
\qquad\Longrightarrow\qquad
e^{\,i\alpha_{31}/2} \;=\; i,
\label{eq:driver-w2}
\end{equation}
an exact lattice value, not a fit --- the ``driver'' whose factor
of $i$ powers the gated crossing of Problems~10--11 --- while the
Dirac phase sits at the derived
$\delta_{\mathrm{PMNS}} = \pi + 2\arcsin(1/7) = 196.4^\circ$
(a zero-parameter angle inside the experimentally favored
quadrant). Both are checkable: DUNE and Hyper-Kamiokande measure
$\delta_{\mathrm{PMNS}}$ directly, and a value outside the
derived-angle's error budget kills the lattice reading at a named
joint.

\subsection{The alphabet, audited}
\label{sec:alphabet-audit}

The four letters of Figure~\ref{fig:alphabet} carry different
jobs, different grades, and different laboratory handles; the
audit deserves a table (Table~\ref{tab:alphabet-audit-w2}), because
the distinctions are exactly what a quick reading fuses.

\begin{table}[htbp]
\centering
\caption[The four phases of the flavor alphabet, audited one at a time.]{The four phases of the flavor alphabet, audited one at a time, because a quick reading fuses distinctions that carry different jobs, different grades and different laboratory handles. $\dCKM = 2\pi/5$ is a bystander, worth about $10^{-20}$ of the excess and tested in kaon and $B$-meson $CP$ violation. $\dPMNS = \pi + 2\arcsin(1/7)$ is a spectator, blind at leading order, and DUNE and Hyper-Kamiokande are its handle. $\alpha_{21} = 0$ is inert. $\alpha_{31} = \pi$ is the driver, because $e^{i\pi/2} = i$. The last column names what would kill each: phases off the lattice, a PMNS phase away from $196.4^\circ$, a nonzero $\alpha_{21}$ effect, and Dirac rather than Majorana neutrinos.}
\label{tab:alphabet-audit-w2}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}lllll@{}}
\toprule
\textbf{Letter} & \textbf{Value} & \textbf{Role in the excess} & \textbf{Laboratory handle} & \textbf{Kill} \\
\midrule
$\dCKM$ & $2\pi/5$ & bystander ($\sim 10^{-20}$) & kaon, $B$-meson CP & phases off lattice \\
$\dPMNS$ & $\pi + 2\arcsin(1/7)$ & spectator (blind at LO) & DUNE / Hyper-K & $\ne 196.4^\circ$ \\
$\alpha_{21}$ & $0$ & inert & $0\nu\beta\beta$ combination & nonzero effect \\
$\alpha_{31}$ & $\pi$ & \textbf{driver} ($e^{i\pi/2} = i$) & $0\nu\beta\beta$; cosmology & Dirac neutrinos \\
\bottomrule
\end{tabular}
\end{table}

Two entries repay a second look. The quark phase is the largest
\emph{laboratory} CP effect and a negligible \emph{cosmic} one ---
the bystander paradox --- because throughput, not chirality, is
what the excess rations (\S\ref{sec:why-billion}); the laboratory
sees the letter, the cosmos sees the toll. And the driver
$\alpha_{31} = \pi$ is a \emph{Majorana} phase: it exists only if
the light neutrinos are their own mirror reading. An established
Dirac nature --- no neutrinoless double-beta decay at any
sensitivity reaching the allowed band --- would delete the driver
and retire the chain of Problem~11 in its present form: the anatomy
would survive, its quantitative crown would not. The cohort
stakes this plainly and counts the $0\nu\beta\beta$ program
among its jurors.

\subsection{The laboratory face of this Part}
\label{sec:lab-cpt}

Three experimental programs read this Part directly.
\emph{Antihydrogen spectroscopy} (ALPHA and successors) compares
the 1S--2S line across the $C$ reversal: any splitting, at any
precision, is a fourth orientation and kills the arena
(\S\ref{sec:orientation-calc}). \emph{Long-baseline neutrino
oscillation} (DUNE, Hyper-Kamiokande) measures $\dPMNS$: the
lattice value $\pi + 2\arcsin(1/7) = 196.4^\circ$ is a
zero-parameter target sitting in the experimentally favored
quadrant, and a measured phase drifting off the lattice completion
retires the alphabet as geometry (while leaving the anatomy's
qualitative reading of $CP$ intact --- the distinction matters and
is stated here deliberately). \emph{Electric dipole moments} probe
the $T$-odd content of the books: the anatomy predicts the pattern
--- $T$ violation in the laboratory only through the chiral
alphabet's letters, never as a new continuous source --- so an EDM
signal demanding a rogue phase off the lattice would wound the
same joint. In all three cases the exposure is the campaign's
preferred kind: angles, not amplitudes --- quantities the circle
protects and the experiment reads whole.

Three mirrors broke and one held. The next Problem asks what the
breaking was \emph{for}: an antiparticle is the class held exactly,
and the excess of matter over antimatter is the toll charged when
the interval crossed the alphabet once.

\section{Problem 10--11.\ The Origin of Antiparticles, and the Tiny Excess of Matter}
\label{sec:w2-prob10-11}

\subsection{What an antiparticle is}
\label{sec:antiparticle}

On the anatomy, a particle species is circle-side data: a holonomy
class --- charge as winding, spin, representation --- wearing an
interval-side mass \citep{ArisakaP3Electron}. An antiparticle is
then not a second object but a second \emph{reading}: the same
circle-side class, traversed in the opposite sense of $\Stheta$ ---
the mirror orientation. The consequences that the textbook lists as
separate facts become one sentence each:

\begin{itemize}[leftmargin=2.0em]
\item \emph{Exact mass equality.} The mass is the interval's address
of the class's ground basin, and an address is orientation-blind:
reading the tape backward does not move the groove. The positron
matches the electron to the last measured digit because both are one
basin, read twice (\citealp{ArisakaP3Electron}; \citealp{ArisakaP5CPT}).
\item \emph{Exact charge negation.} Every charge is a winding, and
reversing the circle's sense negates every winding at once --- which
is why \emph{all} charges flip together, not independently.
\item \emph{Existence.} The two readings exist because the circle
has two senses; a world without antiparticles would need an oriented
circle, and an orientation of $\Stheta$ is exactly what no clause of
the constitution supplies. The handshake between the circle's two
senses and the clock's positivity requirement --- the real-bulk
companion's antiparticle theorem --- is what forces the pairing into
every charged sector \citep{ArisakaP4Real}.
\end{itemize}

The deep question is therefore not why antimatter exists --- the
circle answers that --- but why it \emph{lost}: why the universe
retains one part in $10^{9}$ of unmatched matter. The anatomy's
answer occupies the rest of this Part, and it has the same shape as
everything else in this paper: \emph{the orientations come from the
circle; the amounts come from the interval.}

The mirror's origin, in the equation where it first
appeared. A relativistic electron obeys the Dirac equation,
\begin{equation}
\left(i\gamma^\mu \partial_\mu - m\right)\psi \;=\; 0,
\label{eq:dirac-w2}
\end{equation}
where $\psi$ is a four-component field, $m$ the mass, and the
$\gamma^\mu$ four fixed matrices encoding how spin rides the
Lorentz group. The equation's solution space contains, for every
positive-energy mode of charge $q$, a partner mode that behaves as
charge $-q$ with the \emph{same} $m$ --- antimatter was in the
mathematics before it was in a cloud chamber. What the equation
does not say is \emph{why} the partner must exist. The anatomy's
answer: $\psi$'s phase lives on the compact circle, and a circle
has exactly two winding senses; the antiparticle is the same
holonomy class read in the opposite orientation
(Eq.~\eqref{eq:orientations-w2}'s $C$), which is why the masses
match to the last digit (a class has one magnitude, two senses)
and why every charge --- electric, baryon, lepton --- negates
jointly (all are windings). The mirror is exact because it is
topological; nothing dynamical could have made it this clean.

\subsection{Sakharov's three conditions, as anatomy}
\label{sec:sakharov}

Sakharov's conditions \citep{Sakharov1967} --- baryon-number
violation, $C$/$CP$
violation, and departure from equilibrium --- are, in the standard
telling, three independent ingredients a model must somehow
assemble. On the anatomy they are not ingredients but organs, one
per structure (\citealp{ArisakaP5CPT}; \citealp{ArisakaW1Symmetry}):

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{$B$ violation is the circle's topology meeting the
weak sectors.} The $\Delta B = 3n$ sphaleron door --- baryon number
changing in steps of three --- with $B - L$ exactly conserved
(a circle-side ledger that never opens), and the door itself
\emph{welded cold}: at today's temperatures the crossing rate
carries $e^{-372}$, the deepest suppression in the cohort. The
books can change only through a named, quantized door, and the door
is now shut.
\item \textbf{$CP$ violation is the chiral alphabet} of Problem~9:
the driver $\alpha_{31} = \pi$, whose half-angle supplies the $i$
that maximal interference needs, with the pentagon's non-closure
supplying the existence of a preferred sense.
\item \textbf{Nonequilibrium is the gated interval.} The descent
crosses the lock threshold \emph{once}, promptly, with zero
dilution --- the same once-only logic that releases the Kaxion
(Problem~7) and ends inflation. Equilibrium is what a system reaches
when it can revisit; the interval cannot revisit; the crossing is
therefore out of equilibrium by topology, not by tuning.
\end{enumerate}

Sakharov, read on the anatomy, is not a checklist but an anatomy
lesson: \emph{nonequilibrium is the gated interval, $CP$ is the
chiral circle, and the lawful $\Delta B$ channel is the circle's
topology meeting the weak sectors} --- one condition per fiber
structure, none optional, none added by hand.

\subsection{Why the neutrino sector is the pivot}
\label{sec:neutrino-pivot}

That the excess should be paid through \emph{neutrinos} --- rather
than through the quark sector that displays CP violation daily ---
is, on the anatomy, close to forced, and the reasons compound.
\emph{First, the door}: sphalerons process $B + L$ while
conserving $B - L$; an asymmetry survives reprocessing only if
written in $B - L$, and the one sector that writes lepton number
without baryon number at high scale is the heavy-neutrino sector.
\emph{Second, the address}: the chain needs a once-only,
out-of-equilibrium release, and the descent supplies exactly one
gate at the right depth --- the lock crossing at
$\Mlock = 3.85$~TeV, where the heavy state's decay races the
expansion (the washout bookkeeping, with its regulator, is the
parent paper's; the gate itself is the interval's). \emph{Third,
the letter}: the driver lives among the Majorana phases, and
Majorana structure is native to the neutral sector alone --- the
charged fermions cannot carry it. So the pivot is not a model
preference but an intersection: \emph{the only sector that can
write the surviving book, standing at the only gate the descent
opens, holding the only letter that drives}. The quantitative
chain of \S\ref{sec:etaB} is what that intersection pays.

\subsection{The welded door, quantified}
\label{sec:welded}

The door through which baryon number moves deserves its own
numbers, because its two regimes are the anatomy in miniature
(\citealp{ArisakaP5CPT}; \citealp{ArisakaA9SM}). Above the electroweak
crossover, sphaleron transitions are unsuppressed: the door stands
open, and the lepton asymmetry written at the lock crossing is
reprocessed into baryons at the equilibrium conversion $28/79$ ---
the door is a \emph{channel} while the descent is hot. Below the
crossover the sphaleron rate acquires the Boltzmann factor of the
sphaleron energy, and the exponent today is not merely large but
absurd: $\sim e^{-372}$, the deepest suppression in the cohort's
books --- zero crossings expected in the lifetime of the universe,
multiplied by any conceivable volume. The door did not close; it
\emph{welded}. Two features matter for the anatomy. First, the
weld is an interval event: it happened at an address (the
crossover temperature), once, on the way down --- the door's
history is a descent history. Second, what the weld protects is a
circle ledger: $B - L$ never opened at all (exact at every
temperature), and $B + L$'s violation was quantized in units of
three ($\Delta B = 3n$ --- one unit per family, never one alone)
even while the door stood open. The asymmetry of Problem~11 is thus
doubly safe: written once through a quantized door, and the door
then welded behind it. \emph{The interval opened the door, the
circle counted what passed, and the interval sealed it} --- the
three clauses of Sakharov, told as a history.

Figure~\ref{fig:basin-gate-w2} draws the landscape the chain
crosses.

\begin{figure}[htbp]
\centering
\resizebox{0.641\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth, xscale=1.15]
 % landscape curve
 \draw[line width=1.10pt]
  (0,3.0) .. controls (0.8,3.1) and (1.2,1.9) .. (1.8,1.9)
  .. controls (2.5,1.9) and (2.7,2.6) .. (3.3,2.0)
  .. controls (3.8,1.5) and (4.2,0.9) .. (4.9,0.9)
  .. controls (5.7,0.9) and (5.9,1.5) .. (6.5,1.0)
  .. controls (7.0,0.6) and (7.4,0.15) .. (8.2,0.12);
 % basins
 \node[font=\footnotesize] at (1.8,1.55) {basin 3 (heavy)};
 \node[font=\footnotesize] at (4.9,0.55) {basin 2};
 \node[font=\footnotesize] at (8.0,-0.25) {basin 1 (light)};
 % basin forces circles
 \draw[->, blue, line width=0.80pt] (1.5,2.15) arc (150:390:0.32);
 \draw[->, blue, line width=0.80pt] (4.6,1.15) arc (150:390:0.32);
 \node[font=\footnotesize, blue, align=center] at (0.9,3.55)
   {basin forces\\ $\gamma$, $g$, $G$ --- P-even};
 \draw[blue!60] (1.35,3.15) -- (1.6,2.5);
 % gates
 \draw[->, red, line width=1.10pt] (3.05,2.35) -- (4.0,1.35);
 \draw[->, red, line width=1.10pt] (6.2,1.35) -- (7.15,0.5);
 \node[font=\footnotesize, red, align=center] at (6.3,3.55)
   {gate force: $W^{\pm}$ --- broken, chiral, short-range\\
    transport holonomy $=$ CP phase};
 \draw[red!60] (5.0,3.15) -- (3.6,1.95);
 \draw[red!60] (6.8,3.15) -- (6.7,1.05);
 % orientation arrow
 \draw[->, line width=1.10pt, gray] (0.4,-0.6) -- (7.8,-0.6)
   node[midway, below]{the descent is oriented: heavier
   $\to$ lighter; an orientation only a chiral coupling can carry};
\end{tikzpicture}}
\caption[The family landscape and the gate.]{\textbf{The family landscape and the gate} (reproduced
from \citealp{ArisakaP5CPT}). The flavor lattice's basins with the
one gated crossing marked: unbroken forces hold their charges
fixed (the deep basins --- the circle's protected classes), and
the matter excess is written at the single one-way passage whose
toll is the $e^{-K/2}$ of Eq.~\eqref{eq:etaB}. The geometry makes
Sakharov's third condition structural: the crossing is
out-of-equilibrium because the descent never repeats an address,
and the door it passes through is welded behind it
(Problem~12).}
\label{fig:basin-gate-w2}
\end{figure}

\subsection{The chain, quantitatively}
\label{sec:etaB}

The CPT companion's Part VIII assembles the excess end to end
\citep{ArisakaP5CPT}; Figure~\ref{fig:chain} draws the chain, and
this section carries its numbers at the declared grade. The heavy
neutrino released at the lock crossing decays with an asymmetry
whose \emph{orientation} the alphabet fixes and whose
\emph{throughput} the descent prices:
\begin{equation}
\epsilon\,\kappa \;=\; e^{-K/2} \;=\; e^{-17.5} \;=\; 2.5\e{-8},
\label{eq:throughput}
\end{equation}
the efficiency-times-asymmetry product carrying the topological toll
of half the lock depth (the $K/2$ exponent is the chain's one
\emph{postulated} step, named as such). Sphaleron conversion
reprocesses the lepton asymmetry with the standard factor $28/79$,
dilution by the entropy bath divides by $\gstar$, and the triplet's
equilibrium yield $\sum_i Y_{N_i}^{\mathrm{eq}} \simeq 1.25/\gstar$ ($0.42/\gstar$ per
species) is displayed rather than absorbed. The assembled prediction,
\begin{equation}
\etaB
\;=\;
7.04 \cdot \frac{28}{79}\cdot\frac{1.25\,e^{-K/2}}{\gstar}
\;=\; 7.34\e{-10}
\qquad\text{vs.}\qquad
\etaB^{\mathrm{obs}} = 6.12\e{-10},
\label{eq:etaB}
\end{equation}
a $+20\%$ agreement (headline $\etaB \approx 7\e{-10}$, band $\times3^{\pm1}$) carried at grade D$^{*}$: one named input (the
geometric phase assignment C-P5-C/H-$\varphi_{\mathrm{geo}}$-1),
one named regulator (the flavor-measurement width
$\Delta M/\Gamma^{\mathrm{eff}} = 1/(35\,c_\tau y_\tau^2)$), and the
$K/2$ exponent postulated --- stated here exactly as the parent
paper states them, because the honesty \emph{is} the result: a number this close, from a chain this constrained, is meaningful
only if its open joints are named. (The parent's Boltzmann integration
now \emph{measures} the kernels' one-sided window deficit: $E = 7.1$,
68\% $[4.4,\,11.4]$.)

\begin{figure}[htbp]
\centering
\resizebox{0.679\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]},
  box/.style={draw=black!55, align=center, rounded corners=2pt, inner sep=5pt}]
\node[box,fill=blue!8]   (a) at (0,0)    {\textbf{the gated crossing}\\ lock at $\Mlock = 3.85$~TeV\\ once, prompt, undiluted};
\node[box,fill=orange!12](b) at (4.05,0) {\textbf{the chiral letter}\\ $\alpha_{31} = \pi$: half-angle\\ $e^{i\pi/2} = i$ (the driver)};
\node[box,fill=blue!8]   (c) at (8.1,0)  {\textbf{the toll}\\ $\epsilon\,\kappa = e^{-K/2}$\\ $= 2.5\times10^{-8}$};
\node[box,fill=orange!12](d) at (0,-2.3) {\textbf{the door}\\ sphalerons: $\Delta B = 3n$\\ conversion $28/79$; $B{-}L$ exact};
\node[box,fill=blue!8]   (e) at (4.05,-2.3) {\textbf{the bath}\\ yield $1.25/g_{*}$};
\node[box,fill=green!12] (f) at (8.1,-2.3) {\textbf{the excess}\\ $\eta_B \approx 7\times10^{-10}$\\ obs.\ $6.12\times10^{-10}$ ($+20\%$)};
\draw[->,line width=0.80pt] (a) -- (b);
\draw[->,line width=0.80pt] (b) -- (c);
\draw[->,line width=0.80pt] (c.south) .. controls (8.1,-1.15) and (0,-1.15) .. (d.north);
\draw[->,line width=0.80pt] (d) -- (e);
\draw[->,line width=0.80pt] (e) -- (f);
\end{tikzpicture}%
}
\caption[The birth of the excess, as anatomy.]{\textbf{The birth of the excess, as anatomy.} Blue boxes
are the interval's (the once-only crossing, the exponential toll,
the bath); orange are the circle's (the chiral letter, the quantized
door with its exact $B{-}L$ ledger); the output is the measured
baryon-to-photon ratio, reproduced at $+20\%$ at grade D$^{*}$
with the parent paper's named input, named regulator, and postulated
$K/2$ exponent stated in place \citep{ArisakaP5CPT}.}
\label{fig:chain}
\end{figure}

\subsection{Why one part in a billion}
\label{sec:why-billion}

The most striking feature of the excess is its size: not
$10^{-3}$, not $10^{-15}$, but $6\e{-10}$. On the anatomy the
magnitude is a \emph{topological toll}, and its size is readable
before the detailed chain is run. The throughput
\eqref{eq:throughput} is $e^{-K/2}$: half the lock depth, the
one hierarchy the interval ever wrote, spent once. With $K = 35$,
$e^{-17.5} = 2.5\e{-8}$; the sphaleron conversion and the entropy
bath then supply the order-one and order-$10^{-2}$ dressing that
lands the final answer at $\approx7\e{-10}$. The contrast with the two
standard families of mechanism is instructive and honest.
Electroweak baryogenesis draws its smallness from a product of
small dynamical factors (a weak phase, a thin bubble wall, a mild
transition) --- each adjustable, which is why the literature spans
decades of predicted magnitudes. GUT-scale baryogenesis draws it
from washout races --- again adjustable. Here the magnitude is one
exponential of one ledger integer, with the exponent's factor of
$\tfrac12$ the chain's named postulate: if the toll is right, it
is right for a reason a referee can check against a single
integer; if it is wrong, there is no dial to save it. The excess
is one part in a billion because half a lock depth \emph{is}
$10^{-7.6}$, and the universe paid the toll exactly once.

\subsection{The three horizons of sameness}
\label{sec:horizons}

The excess is one instance of a wider fact this Part should name
before closing: the universe is \emph{the same} in three
progressively stronger senses, and the anatomy prices each
(\citealp{ArisakaA10Cosmos}; \citealp{ArisakaW1Symmetry}). \emph{Horizon 1: the
same conditions} --- temperature, curvature, and inventory uniform
across causally disconnected patches: bought by the interval's
opening act (inflation stretches one patch over the sky).
\emph{Horizon 2: the same things} --- every region built of the
same electrons, the same protons, the same abundances: bought by
the circle's ledgers (identity is a holonomy class, and classes
are the same everywhere without communication --- no mechanism
need synchronize what topology makes identical). \emph{Horizon 3:
the same choices} --- every region on the same side of every
symmetry breaking: matter over antimatter, this vacuum, this
alphabet: bought by the determinism triad --- the dial, the floor,
and the once-only reading --- with the excess of this Part its
sharpest instance. A tuned universe would have to tune each region;
a lottery universe would have to correlate its draws; the read
tape does neither, because a groove plays back the same in every
player. The three horizons are the anatomy's answer to the oldest
observational fact about the universe --- its monotonous,
extravagant sameness --- and the excess is horizon three, audited
against a number two independent methods measure to the per-cent
level.

\subsection{Why a universe of matter was inevitable}
\label{sec:inevitable-matter}

Assemble the Part. The circle guarantees that matter and antimatter
exist as two readings of one book, degenerate in every address. The
chiral alphabet guarantees a preferred sense --- pressed into the
tape by the constitution's integers, not chosen by history. The
interval guarantees a once-only, out-of-equilibrium crossing at
which the preference could act, and prices its throughput with a
topological exponential. And the quantized door guarantees the
asymmetry, once written into baryons, is kept: $B - L$ never opens,
and the sphaleron door welds shut at $e^{-372}$ as the descent
cools. Nothing in the sequence is contingent: given the
constitution, a universe that ends the crossing with exactly the
observed one-in-$10^{9}$ residue of unmatched matter is not one
lucky outcome among many but the deterministic playback of the
groove --- \emph{the excess of particles is the one-way reading of
a chiral tape}. The reader who wants the fate of that residue ---
how it welds into protons that never decay --- goes to Problem~12; the
reader who wants the clock it set ticking goes to Problem~13.

\subsection{Seminar questions}

\paragraph{Where did all the antimatter go, physically?} It
annihilated --- essentially all of it, against essentially all
of the matter, in the first seconds; the photon bath of the
microwave background is, in large part, that bonfire's light.
What survived is the unmatched residue: one part in a billion,
the excess this Part computes. The universe of galaxies is not
matter's triumph over antimatter; it is the change left over
from their mutual settlement.

\paragraph{Could the antimatter instead be segregated ---
antimatter galaxies somewhere else?} Observation says no
(Problem~8's audit: no annihilation seam in the gamma-ray sky at
any scale), and the anatomy says why it could not be otherwise:
the excess's sign is set by the chiral alphabet --- circle-side
content, identical in every region --- so every patch tips the
same way. Segregation would require Horizon 3 to fail exactly
where the groove holds it strongest.

\paragraph{Does antimatter fall differently?} No --- and the
anatomy is committed: an antiparticle is the same class read in
the mirror, and the mass is the class's address,
orientation-blind. The antihydrogen free-fall measurements are
therefore an audit of the mirror reading itself; any anomalous
gravitational behavior of antimatter would break Problem~10 at
its root, well before it troubled general relativity.

\paragraph{Why should the excess involve neutrinos rather than
the quarks that show CP violation daily?} Because the door,
the address, and the letter intersect only there (the pivot
section above): sphalerons preserve $B{-}L$, so only an
asymmetry written in lepton number at high scale survives
reprocessing; the descent's one gate sits at the lock; and the
driver is a Majorana phase, which charged fermions cannot carry.
The laboratory's flashiest CP violation and the cosmos's
effective CP violation live in different sectors for structural,
not accidental, reasons.

\subsection{The decade of the excess}

Problems~10--11 are unusually well served by instruments, and
the audits partition cleanly by letter. \emph{The bystander}
($\dCKM = 2\pi/5$): LHCb and Belle~II continue to tighten the
quark unitarity triangle --- every improvement audits the
lattice's quark letter, though none touches the cosmic books.
\emph{The spectator} ($\dPMNS = 196.4^\circ$): DUNE and
Hyper-Kamiokande will measure the neutrino Dirac phase directly
--- the lattice's cleanest zero-parameter target, in the
experimentally favored quadrant. \emph{The driver}
($\alpha_{31} = \pi$): neutrinoless double-beta decay is the
arbiter of its Majorana prerequisite --- next-generation
tonne-scale experiments reach the band where the framework's
neutrino sector lives, and an established Dirac nature retires
the chain in its present form. \emph{The door}: collider
electroweak studies continue to check the sphaleron-era
bookkeeping's premises. And \emph{the output} itself ---
$\etaB$ --- is measured two independent ways (CMB acoustics and
primordial abundances) whose concordance is re-audited with
every data release. The excess is not a story about the early
universe only; it is a live accounting across five experimental
programs, and the chain of \S\ref{sec:etaB} owes each of them
a number.

The excess survived because a ledger the interval could not undo had
already been written. The last Problem of this Part asks what else
that ledger protects --- and finds the quark that cannot be seen and
the proton that cannot decay held by the same two books.

\section{Problem 12.\ Quark Confinement, and the Absolutely Stable Proton}
\label{sec:w2-prob12}

\subsection{The triangle and the pentagon}
\label{sec:triangle}

Problem~9 found the flavor sector pressing a \emph{pentagon} of
phases into the circle --- steps of $2\pi/5$, set by $L = 5$ ---
whose circuit carries a chiral holonomy: the loop closes as a path,
but the reading accumulates a phase, and that surviving phase is
$CP$ violation. The color sector presses the complementary figure:
a \emph{triangle} --- steps of $2\pi/3$, set by $N = 3$, the
triality of $\SUthreeC$ --- and the triangle differs from the
pentagon in the one respect that decides everything:
\emph{its circuit is trivial}. Three color steps compose to the
identity; the loop closes with nothing left over; the holonomy is
$1$. Figure~\ref{fig:triangle} draws the pair.

The physics of confinement is then one sentence
(\citealp{ArisakaP6Proton}; \citealp{ArisakaW1Symmetry}): \emph{only
configurations whose total color circuit is trivial can be handed
to the boundary.} \OGGEP{} registration marginalizes the fiber; a
state whose color holonomy is nontrivial is a state whose
description cannot be closed without the fiber coordinate the
projection erases; such a state cannot be an observable of the
boundary at all. A single quark --- one step of the triangle,
holonomy $e^{2\pi i/3}$ --- is therefore not merely bound by a
strong force; it is \emph{unregisterable}: not a thing the boundary
can hold. The observable states are exactly the closures --- three
steps ($qqq$: the baryon) or a step and its inverse ($q\bar q$: the
meson) --- which is the color-singlet rule of the textbook, derived
from registration rather than assumed from dynamics. Confinement, on
the anatomy, is not a statement about the growth of a potential but
about the topology of the bookkeeping: \emph{the circle will not
hand the boundary an open book.}

\begin{figure}[htbp]
\centering
\resizebox{0.646\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.4mm]}, scale=1.05]
% ---- triangle panel ----
\begin{scope}
\node[align=center] at (0,2.6) {\textbf{(a) The color triangle ($N=3$)}};
\draw[line width=0.80pt,black!30] (0,0) circle (1.7);
\foreach \k in {90,210,330} \fill[blue!65!black] (\k:1.7) circle (0.07);
\draw[->,line width=1.10pt,blue!65!black] (90:1.7) arc (90:205:1.7);
\draw[->,line width=1.10pt,blue!65!black] (210:1.7) arc (210:325:1.7);
\draw[->,line width=1.10pt,blue!65!black] (330:1.7) arc (330:445:1.7);
\node[blue!65!black,align=center] at (0,0)
  {$W(\triangle) = 1$\\[2pt] \scriptsize the circuit is trivial};
\node[align=center] at (0,-2.5) {\scriptsize closes with nothing left over:\\[-2pt]
\scriptsize singlets registerable $\Rightarrow$ \textbf{confinement}};
\end{scope}
% ---- pentagon panel ----
\begin{scope}[shift={(6.4,0)}]
\node[align=center] at (0,2.6) {\textbf{(b) The flavor pentagon ($L=5$)}};
\draw[line width=0.80pt,black!30] (0,0) circle (1.7);
\foreach \k in {90,162,234,306,18} \fill[red!70!black] (\k:1.7) circle (0.07);
\foreach \a/\b in {90/160,162/232,234/304,306/376,18/88}
  \draw[->,line width=1.10pt,red!70!black] (\a:1.7) arc (\a:\b:1.7);
\node[red!70!black,align=center] at (0,0)
  {$W(P_5) = e^{i\delta}$\\[2pt] \scriptsize a phase survives the circuit};
\node[align=center] at (0,-2.5) {\scriptsize the reading is chiral:\\[-2pt]
\scriptsize surviving phase $\Rightarrow$ \textbf{$CP$ violation}};
\end{scope}
\end{tikzpicture}}
\caption[Two figures on one circle, two fates.]{\textbf{Two figures on one circle, two fates.} \emph{(a)}
The color triangle: three steps of $2\pi/3$ compose to the
identity; a closed circuit with trivial holonomy is a book the
boundary can register --- singlets exist, single quarks do not, and
confinement is the registration rule. \emph{(b)} The flavor
pentagon: five steps of $2\pi/5$ whose chiral reading leaves a
surviving phase --- the $CP$ violation of Problem~9. Same circle, same
anatomy: \emph{closure without residue confines; closure with
residue violates} (\citealp{ArisakaP6Proton}; \citealp{ArisakaP5CPT}).}
\label{fig:triangle}
\end{figure}

Two established equations frame what any account of
confinement must reproduce. The static potential between a heavy
quark and antiquark, measured on the lattice and in quarkonium
spectra, is Cornell-form:
\begin{equation}
V(r_{q\bar q}) \;=\; -\frac{4}{3}\,\frac{\alpha_s}{r_{q\bar q}}
\;+\; \sigma\, r_{q\bar q},
\qquad
\sqrt{\sigma} \;\approx\; 440~\mathrm{MeV},
\label{eq:cornell-w2}
\end{equation}
Coulombic at short distance (the $-4\alpha_s/3r$ of one-gluon
exchange) and \emph{linear} at long distance --- the potential of
a string of constant tension $\sigma$, so that separating a pair
costs energy without bound and the vacuum answers by creating a
new pair instead. Equivalently, in Wilson's own diagnostic
\citep{Wilson1974}, a
large loop of perimeter-enclosing area $\mathcal A$ obeys the
area law
\begin{equation}
\langle W(C)\rangle \;\sim\; e^{-\sigma \mathcal A(C)},
\label{eq:arealaw-w2}
\end{equation}
exponential suppression by the \emph{area}, not the perimeter ---
the loop-language signature that isolated color cannot be
transported. The anatomy adds the why beneath the what: color
holonomy on the compact fiber closes only in threes (the
$\mathbb Z_3$ center), so an open color line is not an expensive
state but an \emph{unregistrable} one, and
Eq.~\eqref{eq:cornell-w2}'s string is the boundary's price for
attempting it. The proton's side of the Part is a number:
$\tau_p > 2.4\e{34}$~yr \citep{SuperK2020}, twenty-four orders
beyond the age of the universe, guarded by the two independent
closures of \S\ref{sec:proton} --- one on what can move, one on
what can be written.

\subsection{The abelian ledger and the singlet rule}
\label{sec:ledger}

The triangle has an abelian shadow, and the shadow is where the
proton's guard is written. The baryonic circle $\UoneB$ counts, per
revolution of $\Stheta$, one unit of baryon number; the color
triality $t$ and the electric charge $Q$ ride the same
fiber-bookkeeping; and the fragment-level audit of
\citep{ArisakaP6Proton} verifies, state by state through the entire
zoo, two integrality guards:
\begin{equation}
Q + \tfrac{t}{3} \;\in\; \bZ
\qquad\text{and}\qquad
Q + B \;\in\; \bZ ,
\label{eq:guards}
\end{equation}
with the $\mathbb Z_6$ quotient of the joint lattice verified
trivial fragment by fragment. Equation~\eqref{eq:guards} is the
singlet rule in ledger form: the circle's several windings are not
independent books but one cross-ruled ledger, and only entries that
balance across the rulings are registerable. It is worth being
precise about what kind of rule this is, because the next section
leans on it. Eq.~\eqref{eq:guards} is a \emph{registration} rule on
states --- it says which entries the ledger admits --- not a
selection rule on transitions. What it delivers to the proton is
prior to any decay argument: it is the reason $B$ and the
charge--triality combination are \emph{integers} at all, so that
they can be holonomy classes rather than continuously varying
quantities. A fractional charge, a
free triality, an unpaired baryon unit --- each would be an entry
that fails a ruling, and the anatomy disposes of them all at once:
they are not forbidden by a force; they are \emph{unwritable}.

Figure~\ref{fig:cornell-w2} shows the measured potential ---
the cage, quantified.

\begin{figure}[htbp]
\centering
\resizebox{0.98\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 lab/.style={font=\footnotesize, align=center}
]
%% axes: x in fm (5.5 units/fm), y in GeV (2.0 units/GeV)
\draw[->, line width=0.80pt, draw=black!70] (0,-2.3) -- (0,3.6)
 node[anchor=south] {$V(r)$ [GeV]};
\draw[->, line width=0.80pt, draw=black!70] (0,0) -- (11.6,0)
 node[anchor=west] {$r$ [fm]};
\foreach \x/\lab in {2.75/0.5, 5.5/1.0, 8.25/1.5, 11.0/2.0}{
 \draw (\x,-0.07) -- (\x,0.07);
 \node[font=\scriptsize, anchor=north, fill=white, inner sep=1pt] at (\x,-0.40) {\lab};
}
\foreach \y/\lab in {-2/-1, 2/1, 3/1.5}{
 \draw (-0.07,\y) -- (0.07,\y);
 \node[font=\scriptsize, anchor=east] at (-0.12,\y) {\lab};
}
%% pure Coulomb (dashed): -0.0789/r  [GeV, r in fm]
\draw[dashed, draw=black!45, line width=0.8pt]
 plot[domain=0.20:2.0, samples=80]
 ({5.5*\x}, {2.0*(-0.0789/\x)});
\node[lab, text=black!55, anchor=west] at (8.7,-0.95)
 {Coulomb alone: $-\tfrac43 \alpha_s \hbar c/r$};
%% full Cornell to breaking: -0.0789/r + 0.981 r
\draw[line width=1.3pt, draw=blue!60!black]
 plot[domain=0.085:1.24, samples=120]
 ({5.5*\x}, {2.0*(-0.0789/\x + 0.981*\x)});
%% breaking plateau
\draw[line width=1.3pt, draw=blue!60!black, dash pattern=on 4pt off 2pt]
 (6.82, 2.276) -- (11.2, 2.276);
\node[lab, text=blue!60!black, anchor=north west] at (7.15, 2.12)
 {the string has snapped:\\ two heavy-light hadrons};
%% linear asymptote (dotted)
\draw[dotted, draw=blue!40, line width=0.8pt]
 plot[domain=0.35:1.05, samples=10] ({5.5*\x}, {2.0*(0.981*\x)});
\node[lab, text=blue!50, anchor=west] at (3.4, 1.30) {$\sigma r$};
%% markers
\draw[draw=red!70!black, line width=0.9pt] (1.562,-0.35) -- (1.562,0.35);
\node[lab, text=red!70!black, anchor=south, xshift=-2mm] at (1.56, 0.42)
 {$r_c \approx 0.28$~fm:\\ the string takes over};
\draw[draw=red!70!black, line width=0.9pt] (6.82, 1.95) -- (6.82, 2.6);
\node[lab, text=red!70!black, anchor=west] at (4.6, 3.42)
 {$r_b \approx 1.24$~fm: budget $\sigma r_b \approx 1.22$~GeV pays for a light pair --- new endpoints bud};
\draw[-{Stealth[length=2mm]}, draw=red!70!black] (6.5, 3.22) -- (6.80, 2.66);
%% slope annotation
\node[lab, text=blue!60!black, anchor=west] at (0.3, 2.6)
 {slope $\sigma = (440~\mathrm{MeV})^2 = 0.98$~GeV/fm\\
 $= 16$ tonnes-force, distance-independent};
\draw[-{Stealth[length=2mm]}, draw=blue!60!black] (2.2, 2.28) to[bend right=12] (4.4, 1.45);
%% Luscher note
\node[lab, text=black!60, anchor=west] at (1.6, -1.30)
 {quantum correction of the vibrating string:
 $-\tfrac{\pi}{12}\hbar c/r = -51.7$~MeV at 1~fm
 (the source paper spends this number)};
\end{tikzpicture}%
}
\caption[The measured cage.]{\textbf{The measured cage} (reproduced from
\citealp{ArisakaP6Proton}). The static quark--antiquark potential
of Eq.~\eqref{eq:cornell-w2} across its two regimes: Coulombic
below $\sim 0.3$~fm, linear above, with the string tension
$\sigma$ as the slope and the lattice/quarkonium data fixing both
pieces. The crossover distance is where the running coupling of
Eq.~\eqref{eq:running-w2} leaves the perturbative regime --- the
interval's reading failing at exactly the address where the
circle's closure rule takes over the description.}
\label{fig:cornell-w2}
\end{figure}

\subsection{Why three colors}
\label{sec:why-three}

The triangle's power derives from its corner count, so the count
itself should be traced. $N = 3$ is a ledger member, and the
cohort reaches it by a chain in which no color count is assumed:
$M = 1$ by dimensional minimality; $L = 5$ by single-tensor
\GS{} consistency; and $N$ from the constitutional relation
$J = 2N + 1$ substituted into the dual lock, giving
$N = L \pm \sqrt{L - M^2}$ with roots $\{3, 7\}$ at $L = 5$ and
minimality selecting $3$ (\citealp{ArisakaW1Symmetry}; \citealp{ArisakaA2GG6}).
Independently, the family count $N_g = 3$ is reached by the
capacity pincer of \citep{ArisakaP2N3} --- \emph{two derivations
sharing no premise, arriving at the same small integer} from
opposite sides, which the cohort regards as stronger evidence
than either alone. The anatomy's addendum is the division of
labor this Part has used throughout: the \emph{count} is
circle-side (an integer in the ledger, unable to run --- the
triangle has had three corners since before the first second),
while everything \emph{dynamical} about color --- the running of
$\alpha_s$, the confinement scale, the hadron masses --- is
interval-side dressing on a fixed topological skeleton. Three
colors is not a fact about forces; it is a fact about the tape.

\subsection{The zoo, audited}
\label{sec:zoo}

A guard is only as good as its audit, and the audit exists: the
proton companion verifies the two rulings of \eqref{eq:guards}
\emph{fragment by fragment} across the hadron zoo
\citep{ArisakaP6Proton} --- every baryon, every meson, every
exotic candidate the tables list --- confirming in each case that
$Q + t/3$ and $Q + B$ land on integers and that the joint
$\bZ_6$ quotient of the charge--triality--baryon lattice is
trivial: no state, anywhere in the observed spectrum, occupies a
cell the ledger forbids, and no forbidden cell is approached even
by the exotica. Three structural remarks ride on the audit.
\emph{First}, the audit is a census, not a sample: integrality is
the kind of claim a single counterexample kills, so the companion
checked all of it, programmatically, and this paper repeats the
claim only because the check exists. \emph{Second}, the mesons
matter as much as the baryons: a $q\bar q$ state closes by
inverse rather than by triple, and the ledger's arithmetic must
--- and does --- treat the two closures consistently, which is a
nontrivial cross-ruling of the same books. \emph{Third}, the
audit is where the anatomy's darkness argument gets its hadronic
face: the zoo contains no fractionally charged survivor because a
fractional charge is an open book, and Part~II's registration rule
does not hand open books to the boundary. The empirical support
for confinement-as-bookkeeping is thus not one experiment but the
entire hadronic table, read as a ledger and found balanced.

Figure~\ref{fig:mass-budget-w2} prices the proton's mass ---
mostly confinement, hardly Higgs.

\begin{figure}[htbp]
\centering
\resizebox{0.788\textwidth}{!}{%
\begin{tikzpicture}[
 every node/.style={font=\small, align=center},
]
\definecolor{stormB}{RGB}{40, 90, 160}
\definecolor{tollY}{RGB}{218, 165, 32}
%% Stacked bar: 938.272 MeV total; Higgs tolls 8.99 MeV (~0.96%)
\fill[stormB] (0,0) rectangle (13.30, 1.0);
\fill[tollY] (13.30,0) rectangle (13.70, 1.0);
\draw[line width=0.7pt] (0,0) rectangle (13.70, 1.0);
\node[white, font=\small\bfseries] at (6.65, 0.5)
 {settled storm: gluon field energy and sea --- $\sim 99\%$ of the mass};
\draw[-{Stealth[length=2mm]}, draw=tollY, line width=0.8pt] (13.5, 1.08) -- (13.5, 1.55);
\node[anchor=south, text=tollY!70!black, font=\footnotesize\bfseries] at (12.6, 1.55)
 {the Higgs tolls: $2m_u + m_d \approx 9$~MeV\ ($0.96\%$)};
%% scale ticks
\foreach \x/\lab in {0/0, 3.65/250, 7.30/500, 10.95/750, 13.70/938}{
 \draw (\x,-0.02) -- (\x,-0.18);
 \node[font=\scriptsize, anchor=north] at (\x,-0.2) {\lab};
}
\node[font=\scriptsize, anchor=north] at (6.85,-0.62) {MeV};
\end{tikzpicture}%
}
\caption[The proton's mass budget.]{\textbf{The proton's mass budget} (reproduced from
\citealp{ArisakaP6Proton}). Where the $938$~MeV actually resides:
the three valence quarks' Higgs-coupled masses contribute of order
one per cent; the remainder is confinement's own energy --- the
gluon field and quark kinetic terms of the closed color books.
The pie is the sharpest one-picture rebuttal of ``mass comes from
the Higgs'': what the Higgs gives the proton is the pencil-weight
of its quarks; what the circle's closure rule gives it is the
body. Mass as ``the address a class wears'' (Problem~14) is this
budget, generalized.}
\label{fig:mass-budget-w2}
\end{figure}

\subsection{The stable proton: the double guard}
\label{sec:proton}

Why does the proton not decay? The standard model answers
``accidental symmetry''; grand unification answers ``it does, at
$10^{34}$ years''; experiment so far answers ``not yet, anywhere.''
The anatomy's answer \citep{ArisakaP6Proton} has two clauses, and
it is worth separating them, because they do different work.

\emph{The first clause is conservation.} Baryon number is not an
approximate bookkeeping device here; it is a holonomy class on
$\UoneB$, an integer on the fiber that remembers, and holonomy
classes do not leak. The only door through which $B$ moves at all
is the $\Delta B = 3n$ sphaleron channel of Problem~11 ---
quantized, $B{-}L$-preserving, and welded cold at $e^{-372}$. A
$\Delta B = 1$ transition is not suppressed by a large mass; there
is no aperture of that size in the geometry.

\emph{The second clause is registration.} Suppose one insisted on
writing a $\Delta B = 1$ operator anyway. Its intermediate carries
an unbalanced color triality, and by Eq.~\eqref{eq:guards} such a
state is not expensive but \emph{unregistrable} --- it is not an
entry the cross-ruled ledger admits.

Here the honest register requires more care than the phrase ``double
guard'' invites, and the paper would rather draw the line itself.
Eq.~\eqref{eq:guards} alone does not forbid the decay: integrality is
satisfied on both sides of $p \to e^{+}\pi^{0}$ --- $Q + B$ moves from
$2$ to $1$, and both are integers, and the reader should check it ---
so a statement of the guard leaning on integrality alone would be
wrong. Nor is registration a second, independent closure. It is a rule
on \emph{states}, not a selection rule on transitions, and a virtual
intermediate is not a registered state. The two clauses are therefore
not two doors but a door and its hinge: the load is carried by the
gauged $\UoneB$, which forbids $\Delta B \ne 0$ at every operator
dimension, and registration is what makes $B$ an integer class rather
than a continuous bookkeeping label --- prior to the conservation law
rather than parallel to it. That is the double guard properly stated,
and it is why the proton's stability is not a coincidence of small
couplings but a property of the arena. The audit fragment by fragment
is \citet{ArisakaP6Proton}'s. The proton's stability is thus the same
fact as the electron's identity, one Part earlier: permanence
deposited on the circle. The residue of Problem~11's excess --- the
one-in-$10^{9}$ of unmatched matter --- is stored in a vessel the
anatomy itself guarantees: \emph{the excess was written by the
interval, once; it is kept by the circle, forever.}

Exposure, stated plainly: the guard is falsifiable on two fronts.
An observed proton decay in any channel violating the double guard
--- at Hyper-Kamiokande's reach or anywhere --- kills the ledger
reading outright; and an observed fractionally-charged or
free-triality state kills the quotient's triviality. The cohort
stakes both.

\subsection{What the guard buys: from the zoo to the biosphere}
\label{sec:biosphere}

The double guard's payoff does not stop at particle physics, and
one paragraph of reach belongs here. A proton stable beyond
$10^{34}$ years is the precondition for \emph{chemistry having a
history}: nuclei that persist, a periodic table whose entries do
not expire, molecular records --- DNA among them --- whose
integrity over $10^{9}$-year evolutionary times rests on baryon
number never leaking even once in $10^{40}$ opportunities per
gram. The anatomy's identity results compound the point: every
proton is the same proton (the class argument of
\citealp{ArisakaP3Electron}, run on the welded baryon), so
chemistry is not merely stable but \emph{universal} --- the same
periodic table in every galaxy, the same binding energies, the
same biochemical options, with no mechanism required to enforce
the uniformity (Horizon 2 of Problem~11). When the cohort's capstone
traces the origin of structure from the GUT edge to the biosphere
\citep{ArisakaA11Origin}, the two clauses it leans on at every
rung below nucleosynthesis are this Part's: \emph{the books do not
leak, and the books are everywhere the same}. The reader inclined
to regard confinement and proton stability as parochial
strong-interaction facts should reprice them here: they are the
underwriting of every promise chemistry and biology have ever
kept.

\subsection{Strong CP, dissolved in passing}
\label{sec:strongcp-w2}

One classic puzzle of the color sector belongs to this Part's
close because the anatomy retires it almost without effort. The
strong-CP problem asks why the QCD vacuum angle
$\bar\theta \lesssim 10^{-10}$ when nothing in the standard model
protects it. On the anatomy, $\bar\theta$ is a circle-side angle
whose value is not an input but an occupation: the compact-angle
sector relaxes, by the \GRIP{} cascade, to the stratum's ground
attractor, and the attractor floor is the $CP$-conserving point ---
the same selection, run on the same landscape, that stabilizes the
Kaxion (Problem~7). The alignment is not a new mechanism bolted on
(a Peccei--Quinn sector introduced for the purpose); it is the
geometric realization of that mechanism by a field the constitution
already forced to exist (\citealp{ArisakaC1Kaxion}; \citealp{ArisakaA9SM}). The
laboratory-facing residue is the $1.95$~GHz line of Problem~7: the
angle's relaxation is still ringing, and the ring is detectable.

Figure~\ref{fig:two-fate-w2} draws the result as a pair of dials:
the strong sector's angle free to relax to zero because nothing on
the circle pins it, the weak sector's locked at a lattice point
because something does. One geometry, two fates, and the difference
is which fiber holds the dial.

\begin{figure}[htbp]
\centering
\resizebox{0.607\textwidth}{!}{%
\begin{tikzpicture}[font=\sffamily\small, >=Stealth]
 % locked side
 \begin{scope}[shift={(0,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.85);
  \foreach \k in {0,1,2,3,4,5,6,7,8,9} {\fill ({36*\k}:0.85) circle (0.9pt);}
  \draw[->, line width=1.10pt, blue] (0,0) -- (72:0.85);
  \node[font=\footnotesize, blue] at (0.1,-1.2) {$\alpha_B = 2\pi/5$ (lattice point)};
  \node at (0,1.25) {\textbf{$\Uone{B}$ Wilson line: LOCKED}};
  \node[font=\footnotesize] at (0,-1.65) {weak CP phase cannot relax:};
  \node[font=\footnotesize] at (0,-2.0) {CP violation \emph{must} exist};
 \end{scope}
 % relaxed side
 \begin{scope}[shift={(7.6,0)}]
  \draw[line width=0.80pt] (0,0) circle (0.85);
  \draw[->, line width=1.10pt, red] (0,0) -- (40:0.85);
  \draw[->, red!60, dashed] (40:0.6) arc (40:5:0.6);
  \node[font=\footnotesize, red] at (0,-1.2) {$a_K/f_K$: dynamical, rolls to $0$};
  \node at (0,1.25) {\textbf{$\Uone{Qi}$ Wilson line: RELAXED}};
  \node[font=\footnotesize] at (0,-1.65) {Vafa--Witten minimum:};
  \node[font=\footnotesize] at (0,-2.0) {$\bar\theta$ \emph{must} vanish};
 \end{scope}
 \draw[<->, line width=0.80pt, gray] (2.1,0) -- (5.5,0)
   node[midway, above, align=center, font=\footnotesize]
   {one compact geometry,\\ two fates};
\end{tikzpicture}}
\caption[The two-fate architecture of the strong-$CP$ answer.]{\textbf{The two-fate architecture of the strong-$CP$
answer} (reproduced from \citealp{ArisakaP5CPT}). Two Wilson
lines, two fates: the $\Uone{B}$ line is \emph{locked} --- its
holonomy a pentagonal lattice point, which is why the weak-$CP$
phases are exact lattice angles --- while the $\Uone{Qi}$ line is
\emph{relaxed} --- free to slide, and slid to the
energy-minimizing alignment that cancels the QCD vacuum angle.
One geometry, one locked dial and one relaxing dial: the
Peccei--Quinn mechanism realized by a field the constitution
already owns (the Kaxion, Problem~7), with the neutron-EDM bound
$|\bar\theta| \lesssim 10^{-10}$ \citep{Abel2020} as the running
audit.}
\label{fig:two-fate-w2}
\end{figure}


%% ===================================================================
\subsection*{Part V in one paragraph, and what comes next}

Four problems of matter, one division of labor. The discrete
symmetries are orientations a bundle may or may not admit: $C$, $P$
and $T$ reverse structures the geometry can reverse, and $CPT$ is
the total orientation it cannot --- which is why three fail and
their product does not. An antiparticle is the same holonomy class
read the other way, so its existence and its exactness are one
fact. The excess is a toll paid once on a chiral alphabet, computed
to $\etaB \approx 7\e{-10}$ (central $7.34$) against a measured $6.12\e{-10}$, with the
one postulated exponent named in place rather than buried.
Confinement is the circle's registration rule --- color books
balance only in triples --- and the proton lives because baryon number
is a gauged class with no $\Delta B = 1$ aperture.

Four problems of matter are behind us; the last four ask what the
arena is made of. Part~VI takes time, the three quantities we use most
and define least, whether the constants were selected rather than
explained, and the horizon that has been the discipline's sharpest
instrument for fifty years.

%%  PART VI -- FOUNDATION OF THE COSMOS
%% ===================================================================
\GGPart[Foundation of the cosmos]{Foundation of the Cosmos}{
Problems 13--16: Time, the Three Currencies, the Landscape, and
the Black Hole}

\noindent The four problems of this Part are the ones physics
shares with philosophy --- what time, energy, and existence are,
and whether our universe is one of many. All four are developed below; Problem~13 carries the fullest
quantitative apparatus.



\section{Problem 13.\ The Definition of Time, and the Arrow of Time}
\label{sec:w2-prob13}

\subsection{What time is, on the anatomy}
\label{sec:time-def}

``What is time?'' has, in this geometry, a two-part answer, one part
per fiber (\citealp{ArisakaP5CPT}; \citealp{ArisakaW1Symmetry}):

\begin{quote}
\emph{The circle gives time its tick; the interval gives time its
direction.}
\end{quote}

The \emph{tick} is repetition, and repetition is the circle's gift:
a clock is anything that returns --- a phase that comes back to
itself and can be counted. Every real clock in physics is,
traced to its root, a circle-side return: the Compton clock of a
massive particle ($\nu = mc^2/h$; for the electron,
$1.24\e{20}$~Hz), the hyperfine line of caesium that defines the
second, the Kaxion's $1.95$~GHz ring (Problem~7), and, slowest, the
inverse Hubble time. The available range, Planck frequency down to
the Hubble rate, is the $e^{140}$ of the age ledger; this ladder
samples $e^{87}$ of it, from the electron's $1.24\e{20}$~Hz to
$2\e{-18}$~Hz --- and every rung is quantized counting on the
fiber that remembers. Energy, by Noether, is what the symmetry of
the tick is worth --- which is why the first law is the circle's
ledger and can never be one-way (Table~\ref{tab:characters}).

The \emph{direction} is the interval's: the runtime reads the tape
once, forward, because its generator is a semigroup. No dynamical
law supplies this arrow --- the books are $T$-symmetric (Problem~9) ---
and none needs to: the arrow is the topology of the fiber the
reading runs along. Entropy is the count of the one-way writes ---
what the arrow of time has \emph{read} --- which is why the second
law is the interval's ledger and can never repeat. The famous
tension between exact mechanics and irreversible thermodynamics
dissolves into a division of labor between a group and a semigroup
\citep{ArisakaA5GDOS}.

\subsection{The four endowments}
\label{sec:endowments}

The CPT companion's expanded treatment organizes what a definition
of time must supply as four endowments, each now located on the
anatomy \citep{ArisakaP5CPT}:

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{A tick} --- something that repeats, so that duration
can be counted. Supplied by the circle: every clock is a return
(\S\ref{sec:time-def}).
\item \textbf{An arrow} --- a direction that distinguishes before
from after. Supplied by the interval: the semigroup's one-way
composition (\S\ref{sec:time-def}).
\item \textbf{A rate map} --- a lawful relation between the tick
and the arrow, so that histories can be dated rather than merely
ordered. Supplied by the seam: $d\ln t/dr = 2$ in the radiation
era, the exponential shadow of the linear descent
(\S\ref{sec:chronology}).
\item \textbf{A start and a reach} --- an origin for the count and
a horizon for its extent. Supplied by the descent's endpoints: the
launch at the top of the corridor, the de Sitter terminus at the
bottom (Problem~7), with the whole reach audited by the age ledger
\eqref{eq:age-ledger}.
\end{enumerate}

Set against the standard menu --- Newton's absolute time
(an endowment list postulated whole), relativity's proper time (a
rate map with no arrow), thermodynamics' arrow (an arrow with no
tick, borrowed statistically), quantum mechanics' parameter time
(a tick with no arrow) --- the anatomy's answer is not another
entry but an assembly instruction: the four endowments come from
two fibers and their seam, and no single structure could have
supplied all four, which is why every one-structure definition of
time has spent a century failing in a characteristic direction.
Time is not a primitive and not an illusion; it is a \emph{joint
reading} --- the circle's count of the interval's passage.

Figure~\ref{fig:endowments-w2} places the four endowments on
the anatomy.

\begin{figure}[htbp]
\centering
\resizebox{0.611\textwidth}{!}{%
\begin{tikzpicture}[font=\small, >=Stealth, line width=0.80pt]
% time axis
\draw[->, line width=1.2pt] (0,0) -- (11.2,0)
  node[right, font=\bfseries] {$t$};
\node[below, font=\footnotesize] at (10.6,-0.12) {the one direction};
% endowment boxes
\node[draw=blue!60!black, fill=blue!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=south]
  (E1) at (2.4,1.0)
  {\textbf{Order} (AX-1)\\ \footnotesize the cone sorts:
  before / after / elsewhere};
\node[draw=violet!60!black, fill=violet!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=south]
  (E2) at (8.2,1.0)
  {\textbf{Repetition} (AX-2 $+$ Thm.~0)\\ \footnotesize Killing
  flow: ticks, frequency, energy};
\node[draw=red!60!black, fill=red!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=north]
  (E3) at (2.4,-1.0)
  {\textbf{Arrow} (AX-3, \OGGEP{})\\ \footnotesize one-way
  reading: records accumulate};
\node[draw=teal!60!black, fill=teal!6, rounded corners, align=center,
  minimum width=4.6cm, minimum height=1.25cm, anchor=north]
  (E4) at (8.2,-1.0)
  {\textbf{Shore} (AX-3 positivity)\\ \footnotesize one-sided
  spectrum: the half-plane of $it$};
\draw[->, blue!60!black] (E1.south) -- (2.4,0.08);
\draw[->, violet!60!black] (E2.south) -- (8.2,0.08);
\draw[->, red!60!black] (E3.north) -- (2.4,-0.08);
\draw[->, teal!60!black] (E4.north) -- (8.2,-0.08);
\end{tikzpicture}}
\caption[The four endowments of the direction called time.]{\textbf{The four endowments of the direction called
time} (reproduced from \citealp{ArisakaP5CPT}). Each endowment
located on its structure: the \emph{tick} on the circle (a phase
that returns, $E = h\nu$), the \emph{arrow} on the interval (a
transport that does not), the \emph{rate} at the seam (the
chronology's exact slope, Eq.~\eqref{eq:slope}), and the
\emph{extent} on the boundary (the age ledger,
Eq.~\eqref{eq:age-ledger}). ``Time'' is the name physics gives the
four jointly; the century of confusion came from asking one
structure to supply all four.}
\label{fig:endowments-w2}
\end{figure}

\subsection{The chronology is exact}
\label{sec:chronology}

The definition would remain philosophy if the geometry did not also
\emph{compute} the history. It does, and the computation is the
sharpest quantitative result of the campaign
(\citealp{ArisakaP5CPT}; \citealp{ArisakaA10Cosmos}). In the radiation era the
boundary clock and the radial coordinate obey
\begin{equation}
\frac{d\ln t}{dr} \;=\; 2 \qquad \text{exactly},
\label{eq:slope}
\end{equation}
(verified numerically along the derived history to two per cent away
from the QCD transition, where the collapse of $g_*(T)$ dresses the
slope --- see \S\ref{sec:epoch-table}), with the matter-era slope
$3/2$; boundary time is the exponential shadow
of the linear descent --- one unit of $r$ costs $e^{2}\simeq 7.4$ in
$t$. The whole history of time is one straight line with a single
kink, and Figure~\ref{fig:chronology} draws it: launch, lock,
electroweak, QCD, BBN, matter--radiation equality, recombination,
today --- uniform in the geometry, exponential in the clock.

The endpoint is an audit that closes. The age of the universe, as an
e-fold ledger on the descent,
\begin{equation}
\ln\!\frac{t_0}{t_{\mathrm{Pl}}}
= 140.24
= \underbrace{36.11}_{\text{launch}}
+ \underbrace{91.60}_{\text{radiation}}
+ \underbrace{12.53}_{\text{matter}+\Lambda},
\label{eq:age-ledger}
\end{equation}
from standard kinematics and the cohort's derived $T_{\mathrm{rh}}$
--- nothing else. Two structural markers deserve their own
sentences. \emph{First}, the corridor switch sits at
$\ln z_{\mathrm{sw}} = 7 = J$, i.e. $z_{\mathrm{sw}} = e^{7} = 1096.6$:
a ledger integer planted in the
middle of the chronology, exact and unrunnable --- and distinct, by
the cohort's two-quantity discipline, from the \emph{atomic}
decoupling $z_{\mathrm{dec}} \approx 1090$, which the atomic chain
derives independently ($0.11$--$0.16\%$ raw); the falsifier F7 is
the atomic quantity, observable-only. \emph{Second}, the
compression $140.24 = 4K \times 1.0017$ is carried at flagged grade
and no higher: its would-be derivation is the e-fold budget
campaign, and until that lands it is a coincidence worth exactly
one flag. The ratio $\beta/t$ --- the thermal period against the
age --- falls monotonically down the same descent: the precise
sense in which the universe becomes ever more classical
(\citealp{ArisakaP5CPT}; \citealp{ArisakaA5GDOS}).

\begin{figure}[htbp]
\centering
\resizebox{0.635\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% axes: x = (r-15)*0.18, y = (lnt-25)*0.048
\draw[->,line width=0.80pt] (0.2,0.2) -- (11.2,0.2);
\node[below] at (4.6,0.05) {\scriptsize $r$ (the descent: uniform in the geometry)};
\draw[->,line width=0.80pt] (0.2,0.2) -- (0.2,6.1);
\node[rotate=90] at (-0.25,3.1) {\scriptsize $\ln(t/t_{\mathrm{Pl}})$ (the clock: exponential)};
% the line: radiation slope 2 then matter kink 3/2
\draw[line width=1.10pt,blue!65!black]
  (0.743,0.533) -- (3.60,2.061) -- (4.172,2.364) -- (5.42,3.072)
  -- (6.329,3.566) -- (8.856,4.930) -- (9.061,5.027) -- (10.321,5.532);
% epoch points
\foreach \x/\y/\lab/\pos in {
  0.743/0.533/{launch}/below,
  3.60/2.061/{lock (3.85 TeV)}/below,
  4.172/2.364/{EW}/below,
  5.42/3.072/{QCD}/below,
  6.329/3.566/{BBN}/below,
  8.856/4.930/{equality}/below,
  9.061/5.027/{recomb.\ ($z_{\mathrm{dec}}$)}/{above left},
  10.321/5.532/{today}/right}{
  \fill[blue!65!black] (\x,\y) circle (0.06);
  \node[\pos=2pt] at (\x,\y) {\scriptsize \lab};}
% slope labels
\node[rotate=28,blue!65!black,anchor=north west] at (1.05,1.15) {\scriptsize $d\ln t/dr = 2$ (radiation, exact)};
\node[rotate=22,blue!65!black] at (9.85,5.72) {\scriptsize slope $\tfrac{3}{2}$};
% switch / decoupling marker.  The two quantities are 0.006 e-folds apart
% -- below this axis's resolution -- so ONE line is drawn and the callout
% says, in the figure itself, that it stands for two things.
\draw[dashed,gray] (9.061,0.2) -- (9.061,5.027);
\node[draw=black!50, rounded corners=2pt, fill=white, align=left,
      inner sep=3pt, anchor=north east] at (11.2,-0.42)
  {\scriptsize \textbf{one drawn line, two quantities}\\[-1pt]
   \scriptsize corridor switch (topological): $\ln z_{\mathrm{sw}} = 7 = J$\\[-1pt]
   \scriptsize atomic decoupling (measured): $z_{\mathrm{dec}} \approx 1090$\\[-1pt]
   \scriptsize $0.006$ e-folds apart --- compared, never fused};
\draw[gray!65,->] (9.3,-0.42) -- (9.07,0.22);
% age audit
\node[draw=black!55,align=left,fill=white] at (3.6,5.2)
  {\scriptsize $\ln(t_0/t_{\mathrm{Pl}}) = 140.24$\\[-2pt]
   \scriptsize $= 36.11 + 91.60 + 12.53$\\[-2pt]
   \scriptsize ($= 4K \times 1.0017$, flagged)};
\end{tikzpicture}}
\caption[The whole history of time, on one line.]{\textbf{The whole history of time, on one line.} The
chronology of the descent in the $(r, \ln t)$ plane: uniform in the
geometry, exponential in the clock, with the single kink at
matter--radiation equality ($2 \to 3/2$). Epochs from the derived
history of \citet{ArisakaA10Cosmos}; the age ledger closes at
$140.24$ \citep{ArisakaP5CPT}. The dashed line carries two distinct
quantities that this axis cannot separate: the topological corridor
switch at $\ln z_{\mathrm{sw}} = J = 7$, and the atomic decoupling
$z_{\mathrm{dec}} \approx 1090$ that the atomic chain derives and
recombination measures. They lie $0.006$ e-folds apart --- a gap
smaller than the width of the drawn line, which is exactly why the
cohort insists on naming which is which at every site, and why the
falsifier F7 is written against the \emph{atomic} quantity only. \emph{Cosmic history is the one-way reading of
holonomy data: the descent deposits on the boundary the letters
stored on the fiber.}}
\label{fig:chronology}
\end{figure}

One line of algebra makes the chronology concrete. In the
radiation era the exact slope $d\ln t/dr = 2$
(Eq.~\eqref{eq:slope}) integrates to the standard relation
\begin{equation}
H \;=\; \frac{1}{2t},
\qquad
T \;\propto\; e^{-r}\ \ \Rightarrow\ \
t \;\propto\; T^{-2} :
\label{eq:hubbletime-w2}
\end{equation}
the Hubble rate $H$ (the fractional expansion per unit boundary
time) is the reciprocal of twice the age, and each e-fold of
cooling costs exactly two e-folds of clock. The parameters:
$t$ is boundary proper time since the launch's receipt, $T$ the
photon temperature, $r$ the interval's coordinate. The physical
content --- and the reason cosmology can do without a fundamental
time variable --- is that $t$ is \emph{derived} bookkeeping: the
descent's odometer converted at a fixed exchange rate, which is
why the age ledger of Eq.~\eqref{eq:age-ledger} can close to four
digits with no clock anywhere in the axioms.

\subsection{The clock ladder}
\label{sec:clock-ladder}

The tick's circle-side character is not an abstraction; it is a
ladder of laboratory returns spanning the full dynamic range of
the universe, every rung a counted phase
(Table~\ref{tab:clock-ladder-w2}).

\begin{table}[htbp]
\centering
\caption[The clock ladder: four laboratory returns, every rung a counted phase.]{The clock ladder: four laboratory returns spanning thirty-eight orders of magnitude in rate, every rung a counted phase. The electron Compton frequency at $1.24\e{20}$~Hz returns the mass's own phase, the class's own tick. The caesium hyperfine transition at $9.19\e{9}$~Hz returns an atomic phase and is what defines the second. The Kaxion line at $1.95\e{9}$~Hz returns the dark condensate, the \GGA{} Hessian ringing. The Hubble rate near $2\e{-18}$~Hz is the interval itself, read as a rate.}
\label{tab:clock-ladder-w2}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{Clock} & \textbf{Rate} & \textbf{What returns} & \textbf{Fiber reading} \\
\midrule
electron Compton & $1.24\e{20}$~Hz & the mass's phase & the class's own tick \\
caesium hyperfine & $9.19\e{9}$~Hz & an atomic phase & the second's definition \\
Kaxion line & $1.95\e{9}$~Hz & the dark condensate & the \GGA{} Hessian, ringing \\
Hubble rate & $\sim 2\e{-18}$~Hz & the horizon scale & the interval, read as a rate \\
\bottomrule
\end{tabular}
\end{table}

Two remarks. \emph{First}, the ladder's top three rungs are
genuine circle returns --- phases that come back --- while the
bottom rung is not a return at all: $H$ is the interval wearing a
frequency's units, which is precisely why it \emph{runs} (the
Hubble ``constant'' was never constant) while the true ticks do
not. A century of cosmological confusion about $H$ is, on the
anatomy, a category note: three of these rows are clocks, one is
an odometer. \emph{Second}, the thermal companion: the ratio of
the shore's period to the age, $\beta/t$, falls monotonically down
the descent --- the universe becomes ever more classical --- with
today's value $\hbar\beta = 2.80$~ps against an age of
$4.35\e{17}$~s: the quantum tick and the cosmic arrow now
twenty-nine orders apart, and the separation itself a derived
function of the descent (\citealp{ArisakaP5CPT}; \citealp{ArisakaA5GDOS}).

\subsection{The age ledger, line by line}
\label{sec:age-audit}

Equation~\eqref{eq:age-ledger} closes an audit, and the three
lines deserve their provenance because each is a different kind of
number. \emph{The launch segment} ($36.1$) runs from the Planck
time to the end of reheating: it is the derived launch of
\citep{ArisakaA10Cosmos} --- the plateau's $N^* = 52.6$ e-folds
plus the reheating match at the derived $T_{\mathrm{rh}}$ ---
converted to clock e-folds. \emph{The radiation segment} ($91.6$)
is standard kinematics run on the inherited inventory: the
radiation-era clock $t = 2.42\,\gstar^{-1/2}(\mathrm{MeV}/T)^2$~s
integrated from reheating to equality with the cohort's $\gstar(T)$
transport --- the longest and most checkable line, and the one
whose slope the chronology figure displays as exactly $2$.
\emph{The matter-plus-vacuum segment} ($12.5$) runs the derived
$H(t)$ of the $\pi$-locked inventory from equality to today ---
the shortest line, and the only one that feels the terminus. Sum:
$140.24$, against nothing --- there is no measured ``age in Planck
e-folds'' to fit; the number simply is what the descent's clock
reads, assembled from three regimes with no dial among them. The
flag then sits beside it exactly as the discipline requires:
$140.24 = 4K \times 1.0017$ is recorded, not spent --- the
compression is either the e-fold budget campaign's future theorem
or a coincidence to be retired, and the ledger's three lines are
sound either way.

\subsection{The chronology, tabulated}
\label{sec:epoch-table}

Figure~\ref{fig:chronology} deserves its numbers in a form the
reader can check by hand. The table below lists the epochs of the
derived history in both bookkeeping conventions the cohort uses
(the cosmology paper anchors $r$ at the corridor entrance; the
proton companion at $M_{\mathrm{Pl}}/2$, where the lock sits at
$r = K$ exactly), with the boundary clock in e-folds of Planck
times (\citealp{ArisakaA10Cosmos}; \citealp{ArisakaP5CPT}) --- see
Table~\ref{tab:epochs-w2}.

\begin{table}[htbp]
\centering
\caption[The descent's timetable, located on the interval and dated on the clock.]{The descent's timetable: eight epochs located on the interval by the depth coordinate $r$, with the lock at $r = K$, and dated on the boundary clock in e-folds of the Planck time. Reheating opens the interval's account at $r = 19.13$; the lock crossing sits at $r = 35.00 = K$ exactly, the gate of Problem~11; and the electroweak scale, the QCD transition that delivers the Kaxion's mass, nucleosynthesis, matter--radiation equality where the slope turns from $2$ to $3/2$, recombination and today follow at $38.18$, $45.11$, $50.16$, $64.20$, $65.34$ and $72.34$. The last row closes the ledger at $\ln(t_0/t_{\mathrm{Pl}}) = 140.24$.}
\label{tab:epochs-w2}
\small
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}lccl@{}}
\toprule
\textbf{Epoch} & $r$ (lock at $K$) & $\ln(t/t_{\mathrm{Pl}})$ & \textbf{Note} \\
\midrule
launch (reheating) & $19.13$ & $36.11$ & the interval's opening act \\
lock crossing & $35.00$ & $67.93$ & $= K$: the gate of Problem~11 \\
electroweak & $38.18$ & $74.24$ & the agent's scale \\
QCD & $45.11$ & $88.99$ & the Kaxion's mass delivery \\
nucleosynthesis & $50.16$ & $99.28$ & the first nuclei \\
equality & $64.20$ & $127.71$ & the kink: slope $2 \to 3/2$ \\
recombination & $65.34$ & $129.72$ & two quantities (\S\ref{sec:two-quantity}) \\
today & $72.34$ & $140.24$ & the ledger's close \\
\bottomrule
\end{tabular}
\end{table}

The reader should run the check the figure invites, and should get the
right answer for the right reason. Between adjacent radiation-era rows
$\Delta \ln t / \Delta r$ reads $2.005$, $1.984$, $2.128$, $2.038$ and
$2.025$: five segments, no fit, and a straight line to two per cent
except across the QCD transition, where it runs $6.4\%$ high. That
departure is not slack in the derivation but physics the derivation
contains --- $t \propto g_*^{-1/2}T^{-2}$, and $g_*(T)$ falls sharply
through the transition, so a clock calibrated on the radiation
temperature must dress the exponent exactly there. The last two rows
are matter-era and belong to the $3/2$ branch, not to this check; a
reader who includes them will get $1.763$ and $1.503$ and should.
The lock's position at exactly $r = K$ in the second convention
is the cleanest single display of the gearbox: the deepest event
of particle physics sits at an integer of the circle, an
e-fold count of the interval --- one number, both fibers.

\subsection{Two quantities at recombination: a discipline, not a detail}
\label{sec:two-quantity}

One methodological sharpening from the chronology work belongs in
the main line, because it is a template for how the anatomy keeps
its books (\citealp{ArisakaP5CPT}; \citealp{ArisakaA10Cosmos}). The recombination
era carries \emph{two distinct quantities} that a careless reading
fuses. The \emph{corridor switch} $z_{\mathrm{sw}} = e^{7} =
1096.6$ is topological: the ledger integer $J$ in the exponent,
exact, owned by the circle --- it marks where the descent's
corridor changes regime, and it cannot move. The \emph{atomic
decoupling} $z_{\mathrm{dec}} \approx 1090$ is physical: derived
through the atomic chain (Saha physics, the hydrogen ladder) with
zero-parameter agreement at the $0.11$--$0.16\%$ level raw ---
owned by the boundary's atoms, measurable, and the registered
falsifier F7 (kill: measured $z_{\mathrm{dec}}$ off the atomic
chain by more than $0.5\%$) is the \emph{atomic} quantity,
observable-only. The two sit $0.006$ e-folds apart --- $0.6\%$ in $z$, under a tenth
of a per cent in the exponent --- which is exactly why they must never
be quoted as one number, and the cohort's discipline is to name which
is which at every site. The
template generalizes: wherever the anatomy places a circle integer
near a boundary observable (the switch and the decoupling; the
$1/35$ split and the GHz line; the lattice phase and the measured
$\dPMNS$), the exact quantity and the measured quantity are kept
as two entries, compared but never fused --- because the
comparison is the physics, and the fusion would be numerology.

\subsection{What time is not}
\label{sec:time-not}

The definition of \S\ref{sec:time-def} gains sharpness from three
contrasts with the standard alternatives, each stated structurally
and without polemic.

\paragraph{Not the block.} The block-universe reading takes
relativity's four-dimensional description at face value and
concludes that passage is perspectival --- nothing happens; all
events simply are. The anatomy locates precisely what the block
captures and what it misses: the block is the \emph{circle's}
truth --- the books, complete and tenseless --- but the world
contains a second fiber the block does not represent, and the
reading of the books is as physical as the books. The runtime is
not a perspective on the block; it is a semigroup the block does
not contain. Passage is not an illusion; it is the interval.

\paragraph{Not thermal.} The thermal-time tradition derives an
arrow from coarse-graining: entropy grows because we describe
imprecisely. The anatomy inverts the dependence: the interval's
one-wayness is prior to any observer's resolution --- the descent
would run, deposit, and weld with no coarse-grainer anywhere ---
and the statistical arrow is the boundary's shadow of the
geometric one \citep{ArisakaA5GDOS}. This is why the cohort's
arrow does not fluctuate, reverse in small systems, or depend on
initial-condition stipulations: it was never statistical.

\paragraph{Not emergent-from-timelessness.} Canonical
quantum-gravity traditions treat time as emergent from a timeless
fundamental state. The anatomy agrees that boundary time is
derived --- it is the exponential shadow of the descent,
Eq.~\eqref{eq:slope} --- but what it is derived \emph{from} is
not timeless: the interval is a direction with an intrinsic
one-way transport, and the derivation is a change of clock, not a
birth of change. The emergence tradition's hard problem ---
extracting an arrow from a stateless wavefunction --- does not
arise, because the arrow was never absent from the fundamental
description. Time is derived, but its direction is bedrock.

\subsection{The measurement arrow}
\label{sec:measurement}

The laboratory meets this Part not in cosmology but on the bench:
every measurement has a before and an after, and no interference
ever resurrects a registered outcome. On the anatomy this is the
same fact as Eq.~\eqref{eq:slope} at small scale
\citep{ArisakaA8QM}: the coherent half of quantum mechanics ---
superposition, interference, unitarity --- is circle-side (the
unread phase), while registration is an interval-side write ---
one-way because the fiber it is written along has no inverse
transport. A record is \emph{real} because it is written along the
fiber that carries no phase (Part~II), and \emph{permanent} because
its content is deposited in quantities the other fiber protects.
The measurement problem, in its sharpest form --- \emph{why does a
reversible theory produce irreversible events?} --- receives the
anatomy's standing answer: the theory's books are reversible; its
reading is not; and a measurement is a read. The Born rule itself
is carried by the cohort at the relaxation level --- produced by
the boundary law's own dissipation, with the field-grade
representation step the one named analytic residue --- and this
paper inherits that status verbatim \citep{ArisakaA8QM}.

The link back to Problem~2: the runtime that reads --- the interval's
agent among the interactions --- is not an abstraction. It has a
name in the particle-physics tables, and the name is \emph{the weak
interaction}.

\section{Problem 14.\ The Definition of Energy, Mass, and Entropy}
\label{sec:w2-prob14}

\subsection{Three household words without definitions}

Physics' three most-used quantities are its least defined. Energy
is introduced as ``the capacity to do work,'' then redefined in
every subfield until only the conservation law remains --- at
which point the definition has become circular: energy is the
thing that is conserved. Mass is worse: inertial resistance,
gravitational charge, rest energy, a Yukawa coupling to a
condensate --- four glosses, coordinated but not unified, none
answering a child's version of the question (\emph{what is it
that a kilogram has?}). Entropy is the strangest of the three:
disorder to the public, phase-space volume to Boltzmann, missing
information to Shannon's heirs --- an objective law of nature
built, apparently, on a subjective measure of ignorance. All
three work flawlessly in calculation. None can say what it is.
The anatomy's offer: one definition per structure --- the tick,
the address, and the write.

\subsection{Energy: what the tick is worth}

Noether located energy long ago: it is the conserved charge of
time-translation symmetry. The anatomy supplies the missing
referent --- \emph{which} time, and why symmetric. The tick is
the circle's (Problem~13): repetition, phase returning to phase,
and the symmetry of the tick --- one moment of the cycle as good
as another --- is exact because the circle has no marked point.
Energy is therefore \emph{what the symmetry of time is worth}:
the Noether price of the circle's indifference, which is why it
is conserved absolutely (the circle cannot lose its symmetry),
why it is the generator of phase rotation ($E = h\nu$ is not an
analogy but the definition --- energy \emph{is} the tick rate,
priced by the circle's constant $\hbar$), and why the first law
of thermodynamics is the circle's ledger, with nothing one-way
anywhere in it (\citealp{ArisakaA5GDOS}; \citealp{ArisakaP5CPT}).
The child's question has an answer: what a joule measures is
\emph{how fast a system's phase turns}.

\subsection{Entropy: what the arrow has read}

Entropy belongs to the other fiber, and its subjectivity puzzle
dissolves there. The runtime writes --- one-way registrations,
priced and unpriceable-back --- and \emph{entropy is the count of
the writes}: what the arrow of time has read, the data-processing
ledger of a semigroup. This is Boltzmann's and Shannon's measure
seen from behind: phase-space volume and missing information both
\emph{quantify} the ledger, but the ledger itself is objective
--- writes happen, observed or not, because the projection is
always on (Problem~1). That is why the second law is exceptionless
without being statistical (nothing about the interval repeats, and
the projection along it has no inverse), why entropy never decreases
globally (there is
no inverse write), and why the ``subjectivity'' was an artifact
of reading the bookkeeping as the event: an ignorant observer
counts the ledger coarsely, but the ledger was written by the
runtime, not the observer. The famous tension between the two
laws --- exact conservation against relentless dissipation --- is
no tension at all: \emph{the first law is the circle's ledger and
the second is the interval's; a group and a semigroup do not
contradict, they divide the labor.}

\subsection{Temperature: where the ledgers exchange}

The two currencies meet at one window. Temperature, in the
cohort's KMS reading, is \emph{where the interval meets the
shore}: the thermal state's imaginary period is the exchange rate
at which circle-currency (energy) converts to interval-currency
(entropy), and the conversion fee is the familiar $T\,dS$. The
Boltzmann constant is then the boundary's unit-fixing for that
exchange (the thermodynamic member of the GU-5 pentad,
\S\ref{sec:characters}), and cooling is descent: the exchange
rate falls monotonically down the one-way history, which is why
the early universe traded energy for entropy furiously and
today's universe barely trades at all --- the $\beta/t$ ledger of
\S\ref{sec:clock-ladder}, the universe growing ever more
classical.

Figure~\ref{fig:shore-w2} draws the window itself.

\begin{figure}[htbp]
\centering
\resizebox{0.569\textwidth}{!}{%
\begin{tikzpicture}[font=\small, >=Stealth, line width=0.80pt]
% axes of complex t plane
\fill[teal!8] (-4.6,0) rectangle (4.6,-3.1);
\draw[->] (-4.8,0) -- (5.0,0) node[right] {$\mathrm{Re}\,t$};
\draw[->] (0,-3.4) -- (0,1.6) node[above] {$\mathrm{Im}\,t$};
% real axis annotation (QM)
\node[align=center, font=\footnotesize, anchor=south] at (3.1,0.18)
  {\textbf{the rotor} $e^{-iEt}$\\ quantum mechanics: oscillation};
% forbidden upper half
\node[align=center, font=\footnotesize, gray] at (-2.7,0.95)
  {no upper shore:\\ the spectrum's empty side};
% lower half (shore)
\node[align=center, font=\footnotesize, teal!40!black] at (-2.5,-1.15)
  {\textbf{the shore}: lawful continuation\\ damping
  $e^{-\tau E}$, Euclidean computation};
% KMS depth line
\draw[dashed, red!60!black] (-4.6,-2.4) -- (4.6,-2.4)
  node[pos=0.86, above, font=\footnotesize, red!60!black]
  {depth $\beta$: the Boltzmann weight $e^{-\beta E}$};
% KMS identification arrow
\draw[->, red!60!black] (2.2,0) to[bend left=-35] (2.2,-2.4);
\node[font=\footnotesize, red!60!black, anchor=west] at (2.45,-1.3)
  {KMS: $G_{+}(t-i\beta) = G_{-}(t)$};
\end{tikzpicture}}
\caption[The shore, drawn.]{\textbf{The shore, drawn} (reproduced from
\citealp{ArisakaP5CPT}). The exchange window where the circle's
ticks and the interval's writes trade: the one-sided energy
spectrum makes the boundary's equilibrium state thermal (the KMS
condition), and ``temperature'' is the window's width --- defined
wherever tick-worth is exchanged for write-count at a steady
rate, from one trapped ion to the de~Sitter horizon. This is
Problem~14's temperature section in one picture: not a property
of substance, but of the meeting of two ledgers.}
\label{fig:shore-w2}
\end{figure}

\subsection{Mass: the address a class wears}

Mass is the two fibers jointly, and the three portraits of
\S\ref{sec:matter-triptych} are its definition by exhibition.
What a particle \emph{is} --- its charge, spin, representation,
its identity --- is a circle-side class; what it \emph{weighs} is
the interval-side address of that class: the terminus of a
descent for the electron, the binding of a weld for the proton,
the Hessian of an attractor for the Kaxion. Three consequences,
each an old puzzle: \emph{(i)} identical particles have exactly
identical masses, with no enforcement mechanism, because one
class has one address; \emph{(ii)} every mass formula in the
codebook is an exponential dressed by integers (Problem~4),
because addresses are depths; \emph{(iii)} $E = mc^2$ becomes a
dictionary entry rather than a mystery --- a mass is a tick rate
(the Compton frequency, $mc^2 = h\nu_{\mathrm{C}}$: the class's
own clock, \S\ref{sec:clock-ladder}), so rest energy is the
worth of a particle's standing tick, and converting mass to
energy is releasing the tick from the class that carried it. The
kilogram's answer: what a kilogram has is \emph{a phase turning
at $1.4\times10^{50}$ revolutions per second, held by classes
that cannot leak}.

Figure~\ref{fig:currencies} places the household words on the
anatomy: each currency at the structure that defines it.

\begin{figure}[htbp]
\centering
\resizebox{0.693\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% circle
\draw[line width=1.10pt] (1.8,2.6) circle (1.0);
\node at (1.8,2.6) {\scriptsize $S^1_\theta$};
\node[align=center] at (1.8,4.3)
  {\scriptsize \textbf{energy}: the tick's worth\\[-2pt]
   \scriptsize $E = h\nu$ \ ($\hbar$: the circle's constant)};
\node[align=center] at (1.8,0.7)
  {\scriptsize \textbf{mass}: the address\\[-2pt]
   \scriptsize a class wears \ ($E = mc^2$)};
% shore
\node[draw=black!55,line width=0.80pt,rounded corners,align=center] (shore) at (5.9,2.6)
  {\scriptsize \textbf{the shore}\\[-2pt]
   \scriptsize \textbf{temperature}: the\\[-2pt]
   \scriptsize exchange window\\[-2pt]
   \scriptsize ($k_B$ fixes its units)};
% interval
\draw[->,line width=1.10pt] (8.6,2.6) -- (11.8,2.6);
\node at (10.2,2.95) {\scriptsize $I_r$};
\node[align=center] at (10.2,4.3)
  {\scriptsize \textbf{entropy}: the write count\\[-2pt]
   \scriptsize one-way \ ($Q_B$ prices the deposit)};
\node[align=center] at (10.2,0.7)
  {\scriptsize the arrow: no inverse write\\[-2pt]
   \scriptsize ($G_N$: the interval's constant)};
% exchange arrows
\draw[<->,line width=0.80pt,blue!65!black] (2.95,2.6) -- (shore.west);
\draw[<->,line width=0.80pt,blue!65!black] (shore.east) -- (8.5,2.6);
\node[blue!65!black] at (5.9,1.35) {\scriptsize ticks $\leftrightarrow$ writes};
\end{tikzpicture}%
}
\caption[The currencies, located.]{\textbf{The currencies, located.} Energy is the circle's
charge (the tick's worth), entropy the interval's tally (the write
count), temperature the window where the two ledgers exchange, and
mass the descent address a holonomy class wears. Each currency
comes with its unit-fixing constant, one per structure --- which is
why setting $\hbar = c = G_N = k_B = 1$ costs nothing: the
constants were conversion rates all along.}
\label{fig:currencies}
\end{figure}

The three definitions, each as its equation. The tick's
worth:
\begin{equation}
E \;=\; h\nu ,
\label{eq:tick-w2}
\end{equation}
energy is $h$ times the rate $\nu$ at which the circle's phase
turns --- not analogous to, but \emph{definitionally} the price of
phase rotation, with $h$ the conversion constant (Step~A). Read
$E = mc^2$ through it: a mass $m$ is a standing tick rate
$\nu = mc^2/h$ --- what a kilogram \emph{has} is phase turning at
$\sim 10^{50}$ revolutions per second --- and ``mass'' is the
address a class wears, the tick rate at which its identity idles.
The write count:
\begin{equation}
S \;=\; k_B \ln W ,
\label{eq:boltzmann-w2}
\end{equation}
Boltzmann's relation, with $W$ the number of microscopic
configurations compatible with the registered macrostate and
$k_B$ the unit conversion. On the anatomy $\ln W$ counts the
boundary's unread pages --- what has been registered \emph{about}
a system versus what would fully specify it --- and the second
law is Eq.~\eqref{eq:semigroup-law-p2} in thermodynamic clothing:
the write count cannot decrease because there is no inverse
write, and each erasure is priced by
Eq.~\eqref{eq:landauer-p2}. The bridge to Problem~16, stated here
so the currencies arrive there ready:
\begin{equation}
S_{\mathrm{BH}} \;=\; \frac{k_B\,c^3 A}{4\,G_N\hbar},
\label{eq:bh-entropy-w2}
\end{equation}
the Bekenstein--Hawking entropy --- an entropy that is an
\emph{area} $A$ (the horizon's), divided by four Planck areas.
Note which constants appear: $\hbar$ (the circle's), $G_N$ (the
interval's), $k_B$ (the window's) --- the only formula in physics
that spends all three at once, which on the anatomy is no
accident: a horizon is where all three ledgers meet.

\subsection{Three worked examples}

\paragraph{The pendulum and the atom (energy as tick-worth).}
Why is an excited atom's energy discrete? Because energy is the
tick's price and the tick is a closed circle: the electron's
phase must return around its orbital, whole turns only, so the
tick rates --- and with them the energies --- come in steps.
$E = h\nu$ across the board: a laser's joules are a phase-turn
count, a battery's charge a winding stock. Nothing about
``capacity to do work'' explains the discreteness; the circle
does, in one clause.

\paragraph{The expanding gas (entropy as write-count).} Open the
partition and let a gas fill the doubled box. What increased?
Not disorder in any visual sense --- the molecules were always a
blur. What increased is the ledger: the expansion wrote the
molecules' new reachability into the runtime's books, one-way
(no unitary trick refolds the gas), and $\Delta S = Nk_B\ln 2$
prices exactly those writes. Maxwell's demon closes on the same
line: the demon must \emph{register} molecular data to sort, its
registrations are writes, and Landauer's toll on their eventual
erasure pays the second law in full. The law was never about
knowledge; it audited events.

\paragraph{Annihilation (mass as address).} An electron meets a
positron; two photons leave with $2 \times 511$~keV. In the
anatomy's grammar: two mirror readings of one class met, their
class labels canceled (windings summing to zero), and the
address each carried --- the standing tick of $m_ec^2$ --- was
released as free ticks (photons: pure circle-currency, no class
to hold an address). $E = mc^2$ is the dictionary of that event:
\emph{rest energy is the worth of a class's standing tick}, and
matter--energy conversion is class-release, lawful exactly where
the ledgers (charge, lepton number --- circle books) permit.

\subsection{The currencies through cosmic history}

Run the definitions down the chronology of Problem~13 and the
thermal history of the universe becomes a currency report. At
the launch's end, reheating converts the roll's stored
circle-currency into the hot bath: energy abundant, ledger young,
the exchange window ($T$) blazing. Through the radiation era the
interval writes furiously --- every scattering a registration ---
and the entropy ledger swells to its famous $10^{90}$; the
exchange rate falls as $T \propto e^{-r}$, cooling-as-descent.
At the matter era's freeze-outs, circle-currency locks into
classes (nuclei, atoms: addresses occupied, Problem~12's vaults
sealed), and the writes slow. Today the exchange window has
cooled to $2.7$~K, the $\beta/t$ ratio of
\S\ref{sec:clock-ladder} has fallen twenty-nine orders, and the
future is the ledger's closing statement: a de Sitter universe,
writes asymptotically ceasing, energy books frozen, entropy books
full --- \emph{the first law's fiber still perfect, the second
law's fiber finally quiet}. Heat death, in the anatomy's
language, is not the victory of disorder; it is the completion
of the reading.

\subsection{Three classic strains, relieved}

The definitions earn their keep where the textbook glosses
strain. \emph{Gravitating systems have negative heat capacity}:
feed energy to a star cluster and it cools --- a scandal for
``energy as capacity'' intuitions and the source of endless
paradox-mongering. On the anatomy the strain releases: gravity is
the integrator, not a keeper (Problem~2) --- it prices
configurations along the interval rather than storing
circle-currency --- so gravitational ``thermodynamics'' was
never the exchange of a bath but the bookkeeping of descent, and
the negative sign marks exactly where pricing, not storing,
dominates. Its extreme case is Problem~16's horizon.
\emph{``Where is the gravitational field's energy?''} ---
relativity's oldest bookkeeping quarrel: no local tensor exists.
The anatomy's reading: energy is the tick's Noether charge, and
the tick is the circle's --- global phase structure --- so energy
localizes exactly as well as the tick does and no further; the
famous non-localizability is not an accounting failure but a
category note, and the global statements (ADM-style totals) are
where the definition natively lives. \emph{``Is mass just energy
in disguise?''} --- the popular collapse of the two. The grammar
keeps them distinct and related: energy is circle-currency in
general; mass is that currency \emph{on deposit in a class} ---
addressed, guarded, identical across copies. A photon has
energy and no mass because it has a tick and no class; an
electron has both because its tick is held. The three strains
share one moral: the household words wobble precisely where one
fiber's bookkeeping is asked to describe the other's business
--- and firm up the moment the ledgers are sorted.

\subsection{The pentad, and why natural units are natural}

The definitions close the loop with the constants that measure
them (\S\ref{sec:characters}; \citealp{ArisakaA4Euclid}). Each
currency has its unit-fixing constant, and each constant belongs
to a structure: $\hbar$ prices the tick (the circle's constant
--- energy per unit phase rate); $G_N$ prices the descent (the
interval's --- the integrated warp volume, why gravity's
constant alone is tied to irreversibility); $c$ converts the
stage's directions; $k_B$ fixes the exchange window's units; and
$Q_B$ fixes the information deposit's. Setting
$\hbar = c = G_N = k_B = 1$ --- the physicist's ``natural
units'' --- is then not a computational trick but a confession
the anatomy makes explicit: the constants were never independent
facts to explain, but the conversion rates between one
geometry's directions, and choosing the geometry's own units
makes them disappear \emph{because they were units all along}.
What remains when the units are natural is exactly what this
paper's Problems compute: the dimensionless ledger --- integers,
$\pi$'s, and depths.

\subsection{Seminar questions}

\paragraph{Is energy conserved in an expanding universe?} The
famous subtlety --- redshifting photons ``losing'' energy ---
sorts cleanly. Energy is the tick's charge, exact wherever the
tick's symmetry stands; the expansion is descent --- the
interval repricing the exchange window as history proceeds ---
and a photon's redshift is its tick read against a later,
cheaper window, booked lawfully in the chronology's ledger
(Problem~13), not leaked from the circle's books. The
general-relativistic statement (no global conservation law
without a timelike symmetry) is the same sentence in the older
dialect: where the descent moves the window, the tick's
worth is repriced, and the audit that balances is the ledger's,
not the naive sum's.

\paragraph{Could entropy ever decrease?} Locally and
apparently, always (a refrigerator); globally in the write
count, never --- there is no inverse write. The Poincar\'e
recurrences that haunt closed-system statistical mechanics need
the return of a phase-space trajectory; the runtime is not a
closed trajectory but a semigroup, and Problem~15's
``large circle'' answer applies verbatim: recurrence is a
property the interval does not possess.

\paragraph{What is temperature at, say, a single atom?} The
exchange window is defined wherever writes exchange tick-worth
--- which needs an ensemble of exchanges, not a large body. A
single trapped ion in a laser field has a perfectly good
temperature the moment its ticks and writes reach steady
exchange; the anatomy's definition is operational at one atom
and at one universe (the de Sitter temperature of Problem~7's
terminus), because it was never about substance --- only about
the meeting of two ledgers.

\subsection{The fourth currency, and the audit}

The information age added a fourth quantity to the household ---
the bit --- and the anatomy's ledger has a column waiting. A
stable bit is holonomy-class data (memory on the circle); erasure
is an interval-side write (Landauer's bound as the toll,
\citealp{Landauer1961}); and the exchange rate between
energy-per-winding and energy-per-qubit is the Arisaka constant
$Q_B$, fixed by ledger integers rather than by any bath ---
temperature-independent where Landauer's price is not
\citep{ArisakaA7Qi4}. With it the currency system closes: energy
(the circle's), entropy (the interval's), temperature (the
exchange window), mass (the address), information (the deposit)
--- five household words, one anatomy, and a single audit
question for all of them, which the cohort's papers pose
quantitatively from the laboratory to the de Sitter horizon:
\emph{do the books balance?} So far --- from calorimetry to
black-hole thermodynamics to the cosmic ledger of Problem~13 ---
they do.

\textbf{Grade and kill, consolidated.} Energy/entropy
identifications: theorems of the boundary law. The mass grammar:
codebook grades per entry. $Q_B$: the information pillar's
grades. Kill: a one-way failure of energy conservation; a
repeating entropy ledger; a mass off the address grammar at
precision; a free erasure. The three household words, on the
anatomy: \emph{energy is what the tick is worth; entropy is what
the arrow has read; mass is the address a class wears} --- and
none of the three was ever mysterious; each was half of a pair,
described for a century as if it were whole.

Figure~\ref{fig:arisaka-eq-w2} closes the Part's audit with the
four currencies read off one current.

\begin{figure}[htbp]
\centering
\resizebox{0.636\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=24mm,
 bulknode/.style={rectangle, rounded corners=3pt, draw=red!70!black, fill=red!10, line width=0.7pt, minimum width=46mm, minimum height=16mm, align=center, font=\small\bfseries\sffamily},
 curnode/.style={rectangle, rounded corners=2pt, draw=black!70, line width=0.5pt, minimum width=24mm, minimum height=10mm, align=center, font=\small\sffamily, fill=blue!10},
 qbnode/.style={ellipse, draw=orange!70!black, fill=orange!18, line width=0.7pt, minimum width=24mm, minimum height=12mm, align=center, font=\small\bfseries\sffamily},
 arrowstyle/.style={-{Latex[length=2mm]}, line width=0.6pt},
 divlabel/.style={font=\scriptsize\sffamily, fill=white, inner sep=1pt}
]
 \node[bulknode] (bulk) at (0, 3.5) {Rank-one bulk inflow channel \\[-0.2em] \scriptsize{(BI-6, \citep{ArisakaA3BI6})}};

 \node[curnode] (Je) at (-5, 0.5) {$J_e$};
 \node[curnode] (JB) at (-1.7,0.5) {$J_B$};
 \node[curnode] (JQi) at (1.7,0.5) {$J_{Qi}$};
 \node[curnode] (JE) at (5, 0.5) {$J_E / Q_B$};

 \draw[arrowstyle] (bulk) -- (Je);
 \draw[arrowstyle] (bulk) -- (JB);
 \draw[arrowstyle] (bulk) -- (JQi);
 \draw[arrowstyle] (bulk) -- (JE);

 \node[qbnode] (qb) at (5, -2.5) {$Q_B$};
 \draw[arrowstyle, dashed] (qb.north) -- (JE.south);

 \node[font=\footnotesize\sffamily, below=1pt of qb, text=orange!80!black]
 {$Q_B = \tfrac{2}{5}\tfrac{E_{\rm Ry}}{K} \approx 0.1555$~eV};

 \node[font=\small\sffamily, below=4pt of Je] {electric};
 \node[font=\small\sffamily, below=4pt of JB] {baryonic};
 \node[font=\small\sffamily, below=4pt of JQi] {informational};
 \node[font=\small\sffamily, below=4pt of JE] {energy / qubit};

 \draw[line width=0.7pt, black!70] (-6, -1.6) -- (3.2, -1.6);
 \node[below=4pt, font=\footnotesize\bfseries\sffamily] at (-1.4, -1.6)
 {Arisaka equation: $J_E/Q_B = J_e = J_B = J_{Qi}$};

\end{tikzpicture}%
}
\caption[The Arisaka equation: one current, four readings.]{\textbf{The Arisaka equation: one current, four
readings} (reproduced from \citealp{ArisakaA9SM}). The bulk
carries a single conserved current, and the boundary reads it in
four currencies at fixed exchange rates:
$J_E/Q_B = J_e = J_B = J_{Qi}$ --- energy flux priced by the
Arisaka constant $Q_B$, elementary-charge flux, baryon flux, and
information flux, equal because they are one bookkeeping seen
through four windows. The pentad of unit-fixing constants
(Step~A of Part~II) is this figure's exchange board, and Problem~14's
``fourth currency'' is the $J_{Qi}$ column.}
\label{fig:arisaka-eq-w2}
\end{figure}

\section{Problem 15.\ The Landscape and Multiverse Problems}
\label{sec:w2-prob15}

\subsection{How explanation went environmental}

The road to the multiverse was paved with reasonable steps. String
theory's flux compactifications appeared to admit an enormous
discretuum of vacua --- the famous $10^{500}$, a figure whose
comparability with a count of continuous parameters
\S\ref{ssec:w2-steelman} disputes on the landscape's behalf --- each
with its own constants; eternal inflation appeared to provide a mechanism
for populating them all; and the cosmological constant's
smallness (Problem~6) appeared to have no dynamical explanation
but a ready anthropic one --- observers arise only where the
vacuum permits galaxies, so we find ourselves, unremarkably, in a
rare habitable pocket. Assemble the steps and the constants of
nature become environmental facts, like the Earth--Sun distance:
not derivable, only conditioned upon. Many physicists accept this
soberly; many others regard it as the abandonment of the
explanatory program --- and the debate has been sharpened by
the measure problem (in an infinite multiverse, every probability
depends on how one regularizes the infinities, and no canonical
regularization exists). The question of this Problem is not
whether the multiverse is emotionally satisfying. It is whether
the premises that made it \emph{necessary} hold.

\subsection{The premise: a hat}

Strip the construction to its load-bearing premise: explanation
went environmental because theory seemed to offer a \emph{hat}
--- a vast set of equivalent minima with no principle of choice.
Given a hat, something must draw from it; a drawn constant is an
accident; an accident plus observership is anthropics; anthropics
plus a mechanism of drawing is a multiverse. Every step after the
hat is close to forced. The anatomy therefore attacks the hat,
and only the hat.

\subsection{The landscape's case, in the landscape's own voice}
\label{ssec:w2-steelman}

A contrast is only worth drawing between positions stated at full
strength, and the landscape's case is usually reported by its critics.
Before the anatomy answers, here is that case as its advocates make
it, with the concessions this paper owes it.

\emph{The headline contrast is a category error, and a prepared
opponent will say so first.} Setting ``$10^{500}$ vacua'' beside ``zero
free parameters'' puts two different kinds of quantity in one column: a
count of discrete solutions against a count of continuous dials. They
are not commensurable, the vacuum figure is itself unstable across
constructions by hundreds of orders of magnitude, and --- the point
that costs the most --- string theory makes the \emph{same}
zero-continuous-parameter claim this framework does. On the axis this
paper advertises, the two programs agree. The disagreement is about
whether the discrete choice is made by selection or forced by
arithmetic, which is a smaller and more interesting difference than
the raw contrast suggests.

\emph{The vacua were found, not wanted.} A theory of quantum gravity
was not constructed in order to have many solutions. The multiplicity
was discovered, by people who would have preferred one, and published
because that is what one does with an unwelcome result. Any imputation
of convenience is unearned.

\emph{Results survive the landscape reading that this paper relies on
or must respect.} The bulk--boundary dictionary
\citep{Maldacena1998} is the closest structural relative of this
framework's own projection, and the exact statistical-mechanical count
of an extremal black hole's microstates \citep{StromingerVafa1996} is
the strongest existing evidence that horizon entropy counts something.
Both stand whether or not the landscape is the right reading of the
framework that produced them.

\emph{The tradition has one dated forward commitment, and it got it
right.} \citet{Weinberg1987} predicted the order of magnitude of the
cosmological constant from selection considerations a decade before it
was measured, and was closer than any first-principles calculation
then available. A paper that asks to be judged by dated forward
commitments --- as Part~VII does --- cannot dismiss the one dated
forward commitment its opponent got right. \S\ref{ssec:weinberg-w2}
rereads that success in this framework's terms; the concession has to
stand on its own first.

\emph{And the deepest disagreement is not settled by either side
asserting it.} On the environmental reading, explaining a constant
means exhibiting a mechanism that permits it and a selection effect
that makes our observing it unsurprising. On this paper's reading,
that is a weaker thing than exhibiting a mechanism that \emph{compels}
it. The difference shows up not in elegance but in what can be lost: a
framework of the first kind absorbs a surprising measurement by moving
within its own space of vacua, and a framework of the second kind
cannot. Which is the right standard of explanation is a real open
question in the philosophy of physics, and \citet{Susskind2003} states
the opposing answer clearly enough that this paper will not pretend it
has been refuted. What follows is not a refutation. It is the
observation that the construction needs one specific premise, and an
argument that this geometry does not supply it.

\subsection{The anatomy's answer: a dial, and one playback}

The compact sector's vacua are not a continuum and not a
discretuum of equivalents: they sit at ledger-rational angles ---
a \emph{dial} --- and the scan across the dial has a unique
winner; the \GFF{} landscape has a unique deepest floor per
stratum (the \GGA{}); and the semigroup reads once, with no
re-draw (\S\ref{sec:determinism}). Between them the three
mechanisms remove every point where a draw could occur: no
continuum to fall through, no degeneracy to break by accident, no
second chance to differ. The constants are then not environmental
but \emph{arithmetical} --- the codebook of Problem~4, the same
in every region (Problem~8, Horizon 3) and in every rerun. The
landscape problem, on the anatomy, is not solved but
\emph{unfounded}: the hat the construction needed is, in this
geometry, a dial with a groove --- and one groove plays one
playback.

Figure~\ref{fig:groove-w2} sets the two explanatory
architectures side by side --- the hat and the groove.

\begin{figure}[htbp]
\centering
\resizebox{0.654\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% Panel (a): the landscape
\begin{scope}
\node[align=center] at (2.9,3.3) {\textbf{(a) The landscape: a hat}};
\draw[line width=0.80pt,black!70]
  (0.2,2.2) .. controls (0.7,0.6) and (1.0,2.6) .. (1.5,1.2)
  .. controls (1.9,0.4) and (2.2,2.4) .. (2.7,1.0)
  .. controls (3.1,0.2) and (3.4,2.2) .. (3.9,1.3)
  .. controls (4.3,0.5) and (4.6,2.5) .. (5.1,1.1)
  .. controls (5.4,0.5) and (5.7,2.3) .. (5.9,1.8);
\node[align=center] at (2.9,-0.4)
  {\scriptsize $\sim 10^{500}$ vacua, all lawful\\[-2pt]
   \scriptsize $+$ a lottery to draw one\\[-2pt]
   \scriptsize $+$ a multiverse to run the lottery\\[-2pt]
   \scriptsize $+$ a measure no one can define};
\end{scope}
% Panel (b): the groove
\begin{scope}[shift={(7.3,0)}]
\node[align=center] at (2.6,3.3) {\textbf{(b) The anatomy: a groove}};
\draw[line width=0.80pt,blue!65!black]
  (0.3,2.4) .. controls (1.6,0.35) and (2.2,0.15) .. (2.6,0.15)
  .. controls (3.0,0.15) and (3.6,0.35) .. (4.9,2.4);
\fill[blue!65!black] (2.6,0.15) circle (0.07);
\node[blue!65!black,below] at (2.6,0.02) {\scriptsize one floor (Morse-unique)};
\node[align=center] at (2.6,-0.85)
  {\scriptsize one vacuum, forced\\[-2pt]
   \scriptsize one playback --- no lottery,\\[-2pt]
   \scriptsize no ensemble, no measure needed};
\end{scope}
\end{tikzpicture}%
}
\caption[The hat and the groove.]{\textbf{The hat and the groove.} \emph{(a)} The
environmental architecture: a vast landscape of lawful vacua, a
random draw, and an ensemble to give the draw somewhere to happen
--- with the measure problem as the permanent bill. \emph{(b)} The
anatomy's architecture: the determinism triad's unique floor. The
multiverse is not refuted here; it is \emph{unemployed} --- the
constants it was hired to explain are forced, and a lottery with
one ticket needs no drum.}
\label{fig:groove-w2}
\end{figure}

\subsection{The multiverse, unemployed}

What then of other universes? Here the anatomy's conclusion is
deliberately modest, and stronger for it. It does not assert that
nothing exists beyond our horizon volume, or that the bulk admits
no other realizations; metaphysical exclusivity is not a physics
claim. It asserts \emph{unemployment}: the explanatory work the
multiverse was hired to do --- accounting for the values of the
constants --- does not exist in this framework, because the
constants are theorems, not draws. Other players, if any, play
the same record; postulating them explains nothing our own
playback does not already explain, and the measure problem
evaporates unhired. This inverts the usual burden honestly: the
multiverse is not refuted (one cannot refute an unemployment),
but every argument \emph{for} it that routes through ``the
constants must be selected somehow'' loses its premise. What
would re-employ it is exactly what would kill this Problem's
answer: evidence of a drawn constant.

For the record, the two architectures as bookkeeping. The
landscape requires a triple,
\begin{equation}
\{\,V_i\,\}_{i=1}^{\sim 10^{500}},
\qquad
\text{a populating dynamics},
\qquad
\text{a measure } \mu \ \text{(undefined)},
\label{eq:landscape-w2}
\end{equation}
--- an enumeration of lawful vacua, eternal inflation to realize
them, and a probability measure over an infinite ensemble, the
third being the unpaid bill: every regularization of the infinity
gives different predictions, and no principle selects one. The
anatomy requires a single statement,
\begin{equation}
F'(\varphi^*) = 0,
\qquad
F''(\varphi^*) > 0,
\qquad
\varphi^*\ \text{unique on the admissible stratum},
\label{eq:groove-w2-eq}
\end{equation}
--- the \GFF{} landscape has one stationary point that is a
minimum, and the Morse analysis of \citep{ArisakaA6Gi4} proves
its uniqueness, so ``which vacuum?'' is a computation, not a
draw. The asymmetry of explanatory cost is the argument: the
first architecture buys the constants' arbitrariness at the price
of an undefined measure; the second buys their necessity at the
price of being killable by any single constant that misses
(Eq.~\eqref{eq:chain-w2}'s measured columns are the running
audit).

\subsection{What honestly remains anthropic}

The cohort keeps one residue on the anthropic books, and this
paper repeats it in full view (\S\ref{sec:whynow}): the
\emph{timing} of observership --- why the odometer reads $\sim
4K$ e-folds when observers arrive --- is bounded by the
geometry (the corridor's shape is derived; habitability's window
within it is computed) but not yet derived, and the e-fold
budget campaign is its named would-be closure. The discipline
matters: a framework that dissolves the landscape has every
incentive to overclaim, and the honest ledger is what
distinguishes ``the constants are forced'' (claimed, graded,
staked) from ``everything about our situation is forced''
(not claimed). W1's four words carry the exact register: not
contingent, not lucky, not designed --- \emph{forced}
\citep{ArisakaW1Symmetry} --- said of the structure, and of the
structure only.

\subsection{Weinberg's success, reread}
\label{ssec:weinberg-w2}

The anthropic tradition owns one striking success, and honesty
requires facing it squarely: in 1987 Weinberg argued that
observers require galaxies, galaxies require the vacuum energy
not to dominate before structure forms, and therefore a
landscape-distributed $\Lambda$ should lie below an anthropic
ceiling --- years later, dark energy was found some two orders of
magnitude beneath that ceiling, of the order the argument made
room for. On the landscape reading this is
evidence of environmental selection. The anatomy rereads it
without discarding it: the vacuum's \emph{value} is geometric
(the seesaw and the Hopf share, Problem~7), but the
\emph{coincidence structure} Weinberg exploited --- that the
crossover epoch and the structure-formation epoch are near
neighbors on the corridor --- is a real feature of the derived
geometry, which is exactly why an anthropic argument ranged over
it could bracket the value. Anthropic reasoning, on this
account, is a valid \emph{consistency probe} of any world
containing observers --- it correctly locates where in a derived
history observers can sit --- and becomes an \emph{explanation}
only if the value is genuinely drawn. Whether it was drawn is
not a matter of taste; it is the isocurvature-and-constancy
scoreboard of \S{}``How this Problem dies,'' and the scoreboard
currently reads: not drawn.

\subsection{The measure problem, dissolved rather than solved}

The multiverse's internal crisis deserves its own paragraph,
because its dissolution is a clean corollary. In eternal
inflation every outcome occurs infinitely often, and predictions
require a measure --- a way of comparing infinities --- on which
no consensus exists after decades: the youngness paradoxes,
Boltzmann-brain dominances, and cutoff-dependences are the
literature's own catalog. The anatomy does not solve the
measure problem; it \emph{unasks} it, twice over. Once
dynamically: the launch is a single traverse of a semigroup ---
eternal self-reproduction is a re-ascent the fiber cannot
perform, so the infinite ensemble is never generated. And once
explanatorily: predictions here are theorems about the one
playback, needing no comparison class --- the probability
apparatus that a multiverse must build from a measure, this
framework inherits from the Born chain of Problem~1, relaxed
into rather than postulated. A problem that exists only inside a
framework is evidence about the framework; this paper scores it
accordingly, and no further.

\subsection{The catalog of tunings, triaged}

The anthropic literature's stock-in-trade is the catalog of
apparently tuned quantities --- the six-numbers lists, the
fine-tuning compendia: nudge this constant and no galaxies, nudge
that one and no chemistry. On the anatomy the catalog is not
denied; it is \emph{triaged}, entry by entry, into the Problems
of this paper where each tuning dissolves into structure: the
strength ratios and the weak scale (the hierarchy) --- addresses,
Problem~6; the vacuum energy --- the seesaw's depth and the
Hopf share, Problem~7; the strong-CP angle --- the attractor's
floor, Problem~12; the flavor textures and the matter excess
--- the chiral alphabet and the toll, Problems~9--11; the
spectrum's tilt and the sky's flatness --- the launch,
Problem~5; the constants' constancy itself --- the classes,
Problem~8. What remains after triage is the residue already
booked (the timing), plus the catalog's real lesson, which the
anatomy affirms: the world's habitability is deeply enmeshed with
its structure. The traditions differ only on the direction of
explanation --- selection reading structure from habitability,
the groove reading habitability from structure --- and they
differ testably, because a groove's entries hold in \emph{every}
region and rerun, while a selection's hold only in ours. Each
constancy bound, each isocurvature null, is a vote between the
directions; the running tally is \S{}``How this Problem dies.''

\subsection{The landscape's own standing}

One more honest note completes the picture: the hat itself is
contested territory. Whether string theory actually possesses
$10^{500}$ stabilized de Sitter vacua is an open technical
question --- the swampland program within string theory itself
argues that most effective theories admit no consistent
ultraviolet completion, and some of its conjectures strain
against de Sitter vacua altogether. The anatomy takes no side in
that internal debate, but its structural position is worth
locating: a one-groove geometry is, in effect, the
\emph{extreme swampland claim} --- the assertion that consistency
prunes the possibilities not to a large residue but to one. The
burden that claim carries is exactly the codebook's: it must
derive, entry by entry, what others scan for --- and Problem~4's
sheet is that burden, accepted in public. Between a landscape
whose size is uncertain and a groove whose predictions are
staked, the arbiter is not philosophy; it is the falsifier table.

\subsection{Seminar questions}

\paragraph{Is this just determinism, restated?} It is
determinism \emph{located} --- and bounded. The groove fixes
structure: which layers exist, their constants, their attractors
(the triad's theorems). It does not preprogram what happens
within a layer's combinatorial freedom --- histories, weathers,
biospheres, choices --- which the capstone's layered reading
shelters rather than schedules. The multiverse question only
ever needed the structural half: constants are what selection
was hired to explain, and constants are what the groove fixes.

\paragraph{If everything is forced, why do the constants LOOK
arbitrary?} Because addresses look arbitrary until counted.
$137$ looks like a lottery ticket until it is
$2^J + N^2$; a seventeen-orders hierarchy looks tuned until it is
thirty-eight e-folds of descent, thirty-five of them the ledger's
$K$; the sheet of Problem~4 looked like dials
for a century because its two fonts --- labels and addresses ---
were read as one. Apparent arbitrariness is the normal
appearance of unread structure; the discipline is to distinguish
that claim (auditable, per entry) from its lazy cousin
(``everything happens for a reason''), and the audit table is
the distinction.

\paragraph{Doesn't quantum randomness reopen the lottery?} No:
Born-rule outcomes randomize \emph{within} the runtime's
statistics, not \emph{over} the structure. No quantum event can
relocate a vacuum, redraw a constant, or unwind a class --- the
dial, floor, and ledger sit in exactly the quantities
measurement cannot deform (Problem~1). The lottery the
landscape needed was structural; the randomness quantum
mechanics provides is episodic; and the anatomy keeps the two
in separate books by construction.

\subsection{How this Problem dies}

The groove account is falsifiable precisely where the landscape
account is not, and the asymmetry is the point.
\emph{Any evidence of a drawn constant kills it}: a measured
constant off its closed form beyond the declared dressing
(Problem~4's sheet); patch-to-patch variation of a protected
quantity (Problem~8's audits); axion-sector isocurvature --- the
lottery's own signature --- at any level
(\S\ref{sec:pinning-w2}); or a demonstrated degeneracy of
\GFF{} floors where the Morse analysis requires uniqueness. The
landscape framework survives all such measurements equally well,
which is its structural weakness; the groove stakes them all.
\textbf{Grade:} the dial, floor, and once-only reading carry
Part~II's grades; unemployment is a corollary; the timing residue
is open, named. On the anatomy, the deepest question the
multiverse was invented to answer --- \emph{why these constants?}
--- returns to the discipline that asked it: because
$(1,3,5;7,35)$, because $\pi$, because $e^{-K}$ --- \emph{because
the groove.}

\section{Problem 16.\ The Black-Hole Information Paradox}
\label{sec:w2-prob16}

\subsection{Fifty years of a paradox}

In 1974 Hawking showed that black holes radiate thermally
\citep{Hawking1975}; in
1976 he drew the devastating corollary. Collapse a pure quantum
state into a black hole, let it evaporate, and the outcome is
thermal radiation --- a mixed state. Pure in, mixed out:
information destroyed, in direct conflict with quantum
mechanics' unitarity. The half-century since has been a war of
attrition among partial resolutions: information escapes in
subtle correlations of the radiation (but how, across a
horizon?); it survives in remnants (with pathological
consequences); complementarity lets infalling and outside
observers keep different books (until the firewall argument of
2012 showed the books collide --- smooth horizon, unitarity, and
effective field theory cannot all hold); and the island/replica
computations of 2019--2020 recovered the unitary Page curve from
semiclassical gravity, a landmark --- while leaving the physical
mechanism, in many practitioners' own words, obscure. The state
of play: nearly everyone now believes information survives;
nobody agrees on where it was.

The two Hawking results, as equations with their readings.
A black hole of mass $M$ radiates at temperature
\begin{equation}
T_H \;=\; \frac{\hbar c^3}{8\pi G_N M k_B}
\;\approx\; 6\e{-8}~\mathrm K \times
\frac{M_\odot}{M},
\label{eq:hawkingT-w2}
\end{equation}
--- colder for heavier holes (the inverse-$M$ is why astrophysical
holes are unobservably faint), with the same three-ledger constant
structure as Eq.~\eqref{eq:bh-entropy-w2} --- and carries the
entropy $S_{\mathrm{BH}} = k_B c^3 A/4 G_N \hbar$: one quarter of
the horizon area $A$ in Planck units. Together they fix the
evaporation time ($t_{\mathrm{evap}} \sim G_N^2 M^3/\hbar c^4$,
$\sim 10^{67}$~yr for a solar mass) and the schedule any unitary
account must meet: the radiation's entanglement entropy must turn
over at roughly half the evaporation --- the Page time --- and
return to zero at the end \citep{Page1993}. The paradox, restated
against these equations: Eq.~\eqref{eq:hawkingT-w2}'s spectrum is
thermal, and a thermal spectrum carries no record; either the
books balance by a mechanism the semiclassical calculation cannot
see, or unitarity fails.

\subsection{The anatomy's diagnosis: a billing error}

The paradox, on the anatomy, is a category mistake with a
fifty-year pedigree: \emph{it asked one object to be both fibers
at once} \citep{ArisakaC3BH}. Unitarity, purity, conservation
--- the keeping --- are circle-side obligations; thermality,
irreversibility, evaporation --- the spending --- are
interval-side behavior; and Hawking's calculation, performed on
the four-dimensional shadow where the two fibers are invisible,
billed the spending against the keeping and found the account
short. It was short because it was the wrong account. Sort the
transaction by fiber and each half is not merely consistent but
\emph{obligatory}: everything the evaporation must conserve ---
charges, ledger integers, holonomy-class records --- is deposited
on the fiber no boundary operation can deform, which is why the
information was never destructible in the first place; and
everything thermal and one-way about the radiation is the
interval doing the only thing a semigroup does. The Hawking
temperature itself carries the signature: it is the shore
periodicity of the KMS reading --- \emph{temperature is where the
interval meets the shore} (Problem~14) --- and a horizon is where
that meeting is sharpest.

Figure~\ref{fig:penrose-w2} fixes the geometry the paradox
lives on.

\begin{figure}[htbp]
\centering
\resizebox{0.353\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small, scale=1.05]
% r=0 worldline (left vertical), regular below the singularity
\draw[line width=0.80pt] (0,0) -- (0,3.4);
\node[font=\scriptsize, anchor=east, rotate=90] at (-0.12,1.4){$r=0$ (center)};
% singularity: hand-drawn zigzag spacelike line at top-left (GG: regular small-scale-profile region)
\draw[red!70!black, line width=0.80pt]
 (0,3.4)--(0.25,3.55)--(0.5,3.4)--(0.75,3.55)--(1.0,3.4)--(1.25,3.55)--(1.5,3.4)--(1.7,3.5);
\node[red!70!black, font=\scriptsize, anchor=south west] at (0.1,3.6){singularity \;(6D: regular small-scale-profile region)};
% i^0, I^+, I^-
\coordinate (i0) at (3.4,1.7);
\coordinate (sing) at (1.7,3.4);
\draw[line width=0.80pt] (0,0) -- (i0) node[right, font=\scriptsize]{$i^0$};   % I^-
\draw[line width=0.80pt] (i0) -- (sing);                                       % I^+
\node[font=\scriptsize] at (2.85,0.95){$\mathcal{I}^-$};
\node[font=\scriptsize] at (2.85,2.55){$\mathcal{I}^+$};
% event horizon (null, 45 deg) from collapse point on r=0 to the singularity endpoint
\draw[blue!70!black, line width=1pt] (0,1.1) -- (sing);
\node[blue!70!black, font=\scriptsize, rotate=44, anchor=south] at (0.95,2.3){horizon $=$ \OGGEP{} port};
% interior (above horizon, below singularity): traced
\fill[gray!22] (0,1.1) -- (sing) -- (0,3.4) -- cycle;
\node[gray!55!black, font=\scriptsize, align=center] at (0.5,2.55){interior\\$+$ fiber\\(traced)};
% collapsing star (shaded timelike strip from i^- up to horizon formation)
\fill[orange!28] (0,0) -- (0,1.1) -- (0.55,1.35) -- (0.25,0) -- cycle;
\node[orange!60!black, font=\scriptsize, anchor=west] at (0.55,0.55){collapsing star};
% Hawking radiation: arrows from near horizon to I^+
\draw[->, orange!80!black] (1.15,1.85) -- (2.15,2.35);
\draw[->, orange!80!black] (1.55,2.25) -- (2.45,2.7);
\node[orange!75!black, font=\scriptsize, anchor=west] at (2.2,1.95){Hawking $\to \mathcal{I}^+$};
\end{tikzpicture}}
\caption[Collapse and evaporation, in the conformal diagram.]{\textbf{Collapse and evaporation, in the conformal
diagram} (reproduced from \citealp{ArisakaC3BH}). Time runs
upward; every $45^\circ$ line is null. The event horizon (blue)
is a one-way \OGGEP{} port: the exterior observer receives the
Hawking radiation (orange) but cannot access the interior, which
--- together with the non-spatial fiber --- is traced out. On the
anatomy the classical singularity is a small-scale artifact,
regular in the six-dimensional bulk, and the horizon carries no
firewall because it is an ordinary 6D surface: the drama is in
the books, not the geometry.}
\label{fig:penrose-w2}
\end{figure}

\subsection{Where the information always was}

The paradox's engine was the apparent absence of an environment:
laboratory decoherence sends purity into unmonitored degrees of
freedom, but the radiation ``was all there was.'' The anatomy
names the missing register: the bulk behind the port. The
compact and radial sectors are as physically real as a
laboratory's photon modes --- they are where the books are kept
--- and the mixedness of the four-dimensional radiation is the
same benign bookkeeping every open system exhibits: purity gone
into correlations with degrees of freedom the four-dimensional
description never tracked. What was extraordinary about the
black hole was never the physics; it was that geometry hid the
environment so completely that physics spent five decades
looking for it. On this reading the modern consensus is
vindicated in substance and corrected in address: information
survives --- in the fiber ledgers, not in miraculous
correlations threaded through the horizon.

Figure~\ref{fig:info-distance-w2} draws the diagnosis.

\begin{figure}[htbp]
\centering
\resizebox{0.495\textwidth}{!}{%
\begin{tikzpicture}[>=stealth, font=\small]
% bulk box (6D), pure
\draw[rounded corners, fill=violet!6, draw=violet!55] (0,0) rectangle (5.0,3.4);
\node[violet!55!black] at (2.5,3.05){6D bulk $\mathcal{M}_6$ --- pure, unitary};
\foreach \p in {(0.9,0.9),(1.5,1.7),(2.1,1.1),(2.7,1.9),(3.2,1.3),(1.2,1.2)}{\fill[blue!70!black] \p circle (2.1pt);}
\node[font=\scriptsize, gray!55!black, align=center] at (2.5,0.45){information carried by bulk modes\\(never destroyed)};
% info-bearing mode displaced along the non-spatial fibre (dashed red arc up-right)
\draw[->, red!70!black, dashed, line width=0.9pt] (3.2,1.3) .. controls (4.1,1.9) and (4.35,2.4) .. (4.55,2.85);
\node[red!70!black, font=\scriptsize, anchor=west, align=left] at (2.05,2.45){displaced along\\the fiber $(I_r,S^1_\theta)$};
% projection arrow pi
\draw[->, line width=1pt, gray!55!black] (5.2,1.7) -- (6.7,1.7);
\node[font=\scriptsize, anchor=south] at (5.95,1.75){$\pi$ (\OGGEP)};
\node[font=\scriptsize, anchor=north] at (5.95,1.65){trace fiber};
% boundary screen
\draw[fill=blue!9, draw=blue!55] (6.9,0.2) rectangle (7.5,3.2);
\node[blue!55!black, rotate=90, font=\scriptsize] at (7.2,1.7){4D boundary (mixed)};
\node[font=\scriptsize, gray!55!black, align=center, text width=2.6cm] at (8.9,1.7){appears ``lost'' to the 4D observer; recoverable in the 6D bulk};
\end{tikzpicture}}
\caption[Information ``loss'' as displacement.]{\textbf{Information ``loss'' as displacement}
(reproduced from \citealp{ArisakaC3BH}). What falls in is not
erased but \emph{relocated}: the class data ride the circle's
register (inaccessible to the boundary observer, not absent from
the bulk), while the interval's channel prices their staged
return through the radiation. The paradox's ``loss'' is the
distance between where the information is and where the exterior
observer can read from --- a bookkeeping displacement, closed on
the Page schedule of Figure~\ref{fig:pagecurve-w2}.}
\label{fig:info-distance-w2}
\end{figure}

\subsection{The Page curve as a reconciliation schedule}

The island computations recovered the right curve; the anatomy
supplies its mechanism in one sentence. The circle's ledger ---
the conserved content the hole must return --- is read out
through the interval's one-way channel at the rate the channel
permits: early radiation is dominated by the channel's thermal
character (entropy rises), until the ledger's remaining balance
becomes the smaller book (the Page time), after which the
readout is ledger-dominated and the entropy falls --- \emph{the
circle's ledger, read out through the interval's channel}. The
curve is a reconciliation schedule between two fibers' books,
which is why it is universal in shape and insensitive to
microphysics: schedules of repayment depend on the balance and
the channel, not on the furniture. And there are no firewalls:
the 2012 collision arose from demanding that the horizon's
smooth interval-side physics also carry the circle's books
locally; sorted, nothing at the horizon does the other fiber's
job, and the smoothness survives with the unitarity.

Figure~\ref{fig:pagecurve-w2} draws the schedule: the two
accounts of the radiation's entropy, and the turnover where the
books begin to balance.

\begin{figure}[htbp]
\centering
\resizebox{0.674\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize, >={Stealth[length=2.2mm]}]
% axes
\draw[->,line width=0.80pt] (0,0) -- (11.2,0);
\node[below] at (10.6,-0.1) {\scriptsize time (evaporation)};
\draw[->,line width=0.80pt] (0,0) -- (0,4.6);
\node[rotate=90] at (-0.45,2.3) {\scriptsize entanglement entropy of the radiation};
% Hawking line (dashed, rising)
\draw[line width=0.80pt,dashed,red!70!black] (0.2,0.15) -- (9.6,4.35);
\node[red!70!black,align=center] at (6.7,4.25)
  {\scriptsize the unpaid bill\\[-2pt]\scriptsize (thermal forever)};
% Page curve
\draw[line width=1.10pt,blue!65!black]
  (0.2,0.15) -- (5.0,2.30)
  .. controls (5.5,2.52) and (5.9,2.52) .. (6.4,2.30)
  -- (10.4,0.1);
\node[blue!65!black,align=center] at (3.0,2.9)
  {\scriptsize early: the interval's\\[-2pt]\scriptsize bill grows};
\node[blue!65!black,align=center] at (9.0,1.9)
  {\scriptsize late: the circle\\[-2pt]\scriptsize collects};
% Page time
\draw[dashed,black!55] (5.7,0) -- (5.7,2.6);
\node[below,black!55] at (5.7,-0.1) {\scriptsize the Page time};
\node[align=center] at (5.7,-0.9)
  {\scriptsize the reconciliation schedule: every winding lent
   is returned before the register closes};
\end{tikzpicture}%
}
\caption[The Page curve as a reconciliation schedule.]{\textbf{The Page curve as a reconciliation schedule.} The
dashed line is the billing error: entropy rising forever, a
register that never pays. The solid curve is the audited account:
the interval's channel prices the outflow while the circle's
register holds the classes the hole must return, and from the Page
time onward the returns dominate. The island computations of the
last decade compute this curve gravitationally; the anatomy says
\emph{why} the books were always going to balance.}
\label{fig:pagecurve-w2}
\end{figure}

\subsection{The two books, side by side}

The billing error is easiest to see as an accountant would, with
the transaction sorted into its two books
(Table~\ref{tab:two-books-w2}).

\begin{table}[htbp]
\centering
\caption[The information paradox as a billing error, in two books.]{The information paradox as a billing error, sorted into its two books as an accountant would sort it. The circle's book holds charges, ledger integers and class records; it is exact, unitary and indestructible; its entropy is the capacity of the register and its Page curve the balance remaining. The interval's book holds the infall, the thermal flux and the shrinking area; it is one-way, dissipative and priced; its entropy is the count of the channel's writes and its Page curve the readout executed. Hawking's temperature appears in the second column only, as the shore periodicity of the channel. Fifty years of worry asked ``where did it go?'' of the first book and ``why is it thermal?'' of the second.}
\label{tab:two-books-w2}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}lll@{}}
\toprule
& \textbf{The circle's book (kept)} & \textbf{The interval's book (spent)} \\
\midrule
What it holds & charges, ledger integers, class records & the infall, the thermal flux, the shrinking area \\
Character & exact, unitary, indestructible & one-way, dissipative, priced \\
Hawking's $T$ & --- & the shore periodicity of the channel \\
The entropy & the capacity of the register & the count of the channel's writes \\
The Page curve & the balance remaining & the readout executed \\
Fifty years' worry & ``where did it go?'' & ``why is it thermal?'' \\
The answer & nowhere: it is class data & because channels are the interval's \\
\bottomrule
\end{tabular}
\end{table}

Read down either column and there is no paradox anywhere in it;
read across --- demand that the flux's thermality be a property
of the register, or the register's unitarity a property of the
flux --- and the fifty-year literature reappears line by line.
The area law itself sits naturally in the sorted books: the
horizon's entropy measures the capacity of the register behind
the port --- how much circle-side content the geometry can hold
at that address --- which is why it is an area (a boundary
capacity) and not a volume, and why it obeys a generalized
second law (register capacity plus channel writes never
decreases: the two books' joint ledger).

\subsection{Complementarity and the islands, placed}

The modern proposals slot into the sorted books with their
achievements intact. \emph{Complementarity} intuited the two
books --- infalling and external descriptions keeping different
accounts --- but drew the boundary between \emph{observers},
where the anatomy draws it between \emph{fibers}; the firewall
collision arose exactly at observer-boundary crossings, and
dissolves at fiber boundaries, which no observer crosses because
no observer reads fiber coordinates (\OGGEP{}). \emph{The island
formula} computes, from within semiclassical gravity, which
degrees of freedom the radiation's book must include to stay
unitary --- and its famous quantum extremal surfaces are, on
this reading, the geometry's own bookkeeping of where the
register's content transfers to the channel's manifest: the
islands \emph{are} the reconciliation schedule, computed
gravitationally. The anatomy honors the computation and
supplies what its practitioners call missing --- the mechanism:
the books were always two, and the schedule reconciles them.
\emph{ER=EPR}-style proposals intuited that entanglement and
geometry are one bookkeeping; the anatomy locates the
bookkeeping on the compact fiber, where entanglement's
monogamy and the ledger's integrality are the same discipline.
Fifty years of proposals, on this reading, were each holding a
true page of a two-book audit --- and the audit needed only its
anatomy.

\subsection{Questions from the seminar}

\paragraph{Does an infalling observer hit anything at the
horizon?} No. Smoothness survives because nothing at the horizon
performs the other fiber's job: the crossing is interval-side
passage, the books are circle-side deposits, and the firewall
demanded their collision only under the one-object assumption.
The equivalence principle keeps its dignity; the ledger keeps
its integers; they were never in competition.

\paragraph{Where, physically, is the register?} Behind the port
--- in the compact and radial sectors over the horizon region ---
which is to say: not at a place on the four-dimensional map at
all. Asking for its street address repeats the category slip of
Problem~3's sealed dial: fiber content has capacity and
bookkeeping, not location in the shadow. The horizon's area
measures the capacity; that is as spatial as the register gets.

\paragraph{Is this a remnant scenario?} No. Remnants park the
information in a long-lived object with pathological pair
production; here evaporation completes, the channel closes, and
the books balance because the readout finished --- the Page
curve is the completion certificate, not a stay of execution.

\paragraph{Could the resolution be tested short of a black
hole?} In stages, yes. The mechanism --- circle-kept content
read through an interval channel --- is Problem~1's measurement
chain at extreme redshift, and every laboratory audit of open-
system exactness (analogue horizons, engineered dissipative
systems, the no-free-erasure bound of Problem~14) exercises the
same bookkeeping the hole exercises violently. The black hole is
the anatomy's stress test, not its only appearance --- which is
exactly why I treat the paradox's resolution as a consistency
obligation rather than an isolated triumph.

\subsection{The de Sitter cousin}

Our own sky carries the paradox's gentlest relative, and closing
with it ties this Problem to the trio of Problem~7. The
accelerating universe has a future event horizon --- a permanent
causal screen at $R_e = c/H_\Lambda$ --- with its own
Gibbons--Hawking temperature \citep{GibbonsHawking1977} and its
own entropy, the largest
number in physics ($S_{\mathrm{dS}} \sim 10^{122}$, whose
$e^{8K}$ face rides at flagged grade with the rest of the
family). Every argument of this Problem transfers: the
cosmological horizon is a channel, its entropy a register
capacity, its thermality the shore's --- and the observer's
predicament inverts instructively: at a black hole we hold the
register's outside and watch the channel; inside de Sitter we
\emph{are} the register's contents, and the terminus's frozen
books (Problem~7's ``the interval exhausted, the circle's
shares frozen'') are the Page-curve endgame seen from within.
That one bookkeeping serves the gram-scale analogue, the solar-
mass hole, and the Hubble volume is the anatomy's final word on
the paradox: it was never about black holes; it was about what
any horizon does to a two-fiber ledger --- and every horizon in
nature, so far, does it lawfully.

\subsection{The stakes, and the audit}

Black holes are where the anatomy's two ledgers meet at maximum
contrast, and the meeting is increasingly observational. The
companion paper stakes its corrections and consistencies across
the instruments of the decade --- ringdown spectroscopy
(LIGO/LISA), horizon-scale imaging (EHT), the analogue-system
entanglement measurements --- and carries the de Sitter
extension: the cosmological horizon's entropy (Problem~7's
$e^{8K}$ face, at its flagged grade) as the same bookkeeping at
the largest address. The audit question is Problem~14's,
asked at the extreme: \emph{do the books balance across a
horizon?} The anatomy answers yes, names which book is which,
and accepts the exposure. \textbf{Grade and kill, consolidated.}
The resolution inherits the companion's grades (Page-curve and
no-firewall analyses, conditional as declared there). Kill:
ringdown or imaging inconsistent with the staked corrections; a
demonstration that the bulk register cannot carry the required
capacity; or --- the in-principle referee --- any laboratory
analogue exhibiting true information loss in an open system's
exact books. The paradox, on the anatomy: \emph{the circle keeps
what the hole must return; the interval prices the return; and
fifty years of debate audited the wrong fiber.}

%% ===================================================================
\subsection*{Part VI in one paragraph, and what comes next}

The arena, read on its own two fibers. Time is not one thing but a
joint reading: the tick is the circle's, the direction is the
interval's, and the two have been carried under one word since
Newton, which is why neither an equation nor a statistical argument
alone has ever produced an arrow. Energy, mass and entropy sort the
same way --- what a tick is worth, what an address costs, what the
one-way writes have counted. The multiverse is not refuted but
unemployed: a census that permits one bulk of one shape has no work
for an ensemble. And the information that fell into a black hole
was never lost, only entered on the other book.

Sixteen problems are now behind us, each treated at its own grade.
What remains is to ask what a single sentence could possibly have
been doing in the load-bearing position of all of them. Part~VII consolidates: one master
table, one honest ledger of the weak joints, one falsifier table
with dates, and one comparison of this reading against the
programs it will be measured against.

%%  PART VII -- DISCUSSION
%% ===================================================================
\GGPart[Discussion]{Discussion}{What One Sentence Did with Sixteen
Problems --- Significance, Contrasts, the Honest Ledger, and the
Ways to Be Wrong}

\section{The sixteen, consolidated: the master table}
\label{sec:master-table}

Parts~III--VI are now behind the reader, and the paper owes its
central claim in one place: sixteen problems, one structural
sentence, each treatment with its mechanism, its sharpest
quantitative face, its declared grade, and the measurement it dies
by. Table~\ref{tab:master} is that consolidation --- the paper on
one spread. Every row is developed at essay length in its Part;
every grade is inherited from a parent paper; every kill condition
reappears, dated, in Table~\ref{tab:kill}.

\begin{table}[p]
\centering
\caption[The master table: the sixteen problems and the anatomy's answers.]{The master table: the sixteen problems and the anatomy's
answers. Mechanism column: C = the circle's work, I = the
interval's, C+I = the pair jointly. Grades as in
\S\ref{sec:honest}: D derived, DC derived-conditional, N named
input, P postulated step in chain, F flagged.}
\label{tab:master}
\scriptsize
\renewcommand{\arraystretch}{1.32}
\begin{tabularx}{\textwidth}{@{}c>{\raggedright\arraybackslash}p{2.9cm}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}p{2.5cm}c>{\raggedright\arraybackslash}p{2.6cm}@{}}
\toprule
\# & \textbf{Problem} & \textbf{Mechanism} & \textbf{Sharpest number} & \textbf{Grade} & \textbf{Dies by} \\
\midrule
1 & Measurement & C+I: the write is the interval registering what the circle carried (Eq.~\eqref{eq:gdos-p2}) & Born measure at relaxation grade; F25 relic form & DC & large-scale statistics; interference revival \\
2 & Four interactions & C: two keepers; I: one agent; the warp integrates & $v = 246.3$~GeV; lock at $3.849$~TeV & D/DC & HL-LHC cluster null \\
3 & Gauge + renormalization & C: unreadable pointer (Eq.~\eqref{eq:gaugetrans-w2}); I: one-way flow (Eq.~\eqref{eq:rg-w2}) & anomaly sums exact (Eq.~\eqref{eq:anomsum-w2}) & D & a running anomaly coefficient \\
4 & Parameters & C+I: labels wearing addresses (Eq.~\eqref{eq:chain-w2}) & $137$, $25/108$, $246.3$ vs measured & D/DC/N & any closed form off beyond audit \\
5 & The beginning & I: the launch derived; entrance named open & $n_s = 0.9620$; $r = 0.0043$ & DC & tensor null at SO/LiteBIRD floors \\
6 & Naturalness & I: a depth, not a small number ($e^{-K}$) & $K = 35 = LJ$, spent in four places & D & lock absent at staked sensitivity \\
7 & Dark trio & I rolling once; C ringing still; both at rest & $8.1\,\mu$eV / $1.95$~GHz; $\OmL = 1 - 1/\pi$ & DC/F & haloscope window closed; isocurvature \\
8 & Three horizons & launch (I), classes (C), groove (C+I) & sameness at $10^{-5}$/$10^{-6}$/exact & D/DC & spatial variation of any constant \\
9 & Discrete symmetry & C+I: three orientation reversals (Eq.~\eqref{eq:orientations-w2}) & $\dPMNS = 196.4^\circ$; $\bar\theta \lesssim 10^{-10}$ & D/DC & $CPT$ violation; $\dPMNS$ off \\
10--11 & Antimatter + excess & C: two senses; I: gated one-way toll & $\etaB \approx 7\e{-10}$ vs $6.12\e{-10}$ & D$^{*}$/P & $\etaB$ incompatible with chain \\
12 & Confinement + proton & C: closure in threes; a gauged class, made integral & $\tau_p > 2.4\e{34}$~yr guarded & D & free color; proton decay \\
13 & Time & C: the tick; I: the arrow; seam: the rate & $d\ln t/dr = 2$; ledger $140.24$ & D/F & chronology broken by any clock \\
14 & Energy, mass, entropy & C: tick's worth; I: write count; shore: exchange & $S_{\mathrm{BH}}$ spends all three constants & D & a free write; an inverse write \\
15 & Landscape & C+I: quantized dial, unique floor, no re-choice & one vacuum, forced (Eq.~\eqref{eq:groove-w2-eq}) & D & any drawn constant \\
16 & Black-hole information & C keeps; I prices; Page schedule & $T_H$, $S_{\mathrm{BH}}$, turnover at Page time & DC & information loss confirmed \\
\bottomrule
\end{tabularx}
\end{table}

Three remarks before the discussion proper. \emph{First}, the
mechanism column uses only two letters --- there is no third
ingredient anywhere in the table, which is the paper's claim in
its tersest form. \emph{Second}, the grade column is
heterogeneous, and that is deliberate honesty rather than
weakness: a synthesis that claimed uniform grade D would be
claiming what the parent papers do not. \emph{Third}, the last
column is non-empty in every row --- there is no problem on this list
whose treatment is unfalsifiable in principle. The qualification
matters and \S\ref{sec:kill} makes it: six of the sixteen have a named
instrument and an era, and the rest are exposed through the shared
ledger rather than on their own. A framework that stakes even the
first six on dated channels is in an unusual position, and the paper
would rather claim that accurately than claim more.

\section{Significance and uniqueness}
\label{sec:significance-w2}

Sixteen problems, one sentence. That ratio is the paper's claim
to significance, and it deserves to be stated with care.

\paragraph{(1) One instrument, sixteen problems.} The treatments
of Parts~III--VI are not sixteen applications of a vague theme but
sixteen developments sharing one exact premise: \emph{holonomy
requires closed loops, and an interval has none}. Measurement and
the four interactions, gauge invariance and the parameter sheet,
the beginning and the hierarchy, the dark sector and the horizons,
the discrete symmetries and the missing antiworld, the imprisoned
quark and the immortal proton, time and its currencies, the
landscape and the black hole --- in the standard literature these
are separate research programs with separate vocabularies, some
of which have never cited one another. Here they are one anatomy,
read sixteen times, and Table~\ref{tab:master} is the exhibit.
When a single structural sentence keeps landing in the
load-bearing position of problems this far apart, the sentence is
either a coincidence of language or a fact about the world; the
falsifier table of \S\ref{sec:kill} is the program's commitment
to finding out which.

\paragraph{(2) The explanations are structural, hence cheap.}
None of the sixteen treatments consumed a new field, a new
symmetry, or a new continuous parameter. Each is the pair's structure
--- premise (T), compactness against openness, and wherever an arrow
is involved premise (D), the boundary law's compulsory projection ---
doing work that the standard accounts assign to dynamics (confining potentials, collapse
postulates, arrow-of-time stipulations, inflaton menus, axion
zoos). Explanations that spend topology rather than parameters
cannot be adjusted when they fail; that is their scientific value.

\paragraph{(3) Determinism, without a lottery.} The three
mechanisms of Part~II --- the quantized dial, the Morse-unique
floor, the semigroup that cannot re-choose --- run through every
Part: the alphabet did not draw its chirality, the excess did not
draw its sign, the Kaxion did not draw its angle, the vacuum did
not draw its shares. The universe of this paper is not one lucky
outcome among many; it is the playback of a groove. Symmetry
breaking is symmetry reading --- the paper's second sentence, and
its deepest.

\section{The two families, delivered}
\label{sec:families-delivered}

Part~I closed its diagnosis with a sorting: the sixteen problems
fall into a family about \emph{exactness} (what could make a
quantity so protected that fifty years of instruments cannot budge
it?) and a family about \emph{one-wayness} (what could make a
direction in which nothing returns, in a world of reversible
equations?) --- and the claim was that both families are what
happens when a world with two kinds of direction is described in a
language that has only one. The discussion's first duty is to
check that the treatments actually delivered along that split,
because the diagnosis was a promissory note and Parts~III--VI were
the payment.

\paragraph{The exactness family, closed by the circle.} Every
exactness on the list came back as holonomy-class data, protected
by Eq.~\eqref{eq:group-law-p2}'s inverses and the integrality of
winding: $CPT$ (the total orientation, Problem~9), charge
quantization and the anomaly sums (Problem~3), the proton's double
guard (Problem~12), the antiparticle's perfect mirror
(Problems~10--11), the sameness of the classes across the sky
(Problem~8), the parameter sheet's integer font (Problem~4), and
the constants' refusal to drift (Problem~15). In no case was the
exactness enforced by a dynamical mechanism that could be
adjusted; in every case it was the absence of anything continuous
to adjust. That is the family's one lesson, learned seven ways:
\emph{perfect protection is not strong dynamics; it is topology}.

\paragraph{The one-wayness family, closed by the interval.} Every
irreversibility on the list came back as semigroup transport,
priced by Eq.~\eqref{eq:semigroup-law-p2}'s missing inverse: the
measurement write (Problem~1), the renormalization flow
(Problem~3), the descent and its hierarchy of addresses
(Problems~2 and~6), the launch's one traverse (Problem~5), the
arrow and the age ledger (Problem~13), the write count and its
toll (Problem~14), and the evaporation schedule (Problem~16). In
no case was the arrow imported through an initial condition or a
coarse-graining choice; in every case it was the composition law
of the fiber the process lives on. The family's one lesson:
\emph{irreversibility is not statistics; it is algebra}.

\paragraph{The four that needed both.} The problems the century
found hardest --- measurement (1), the dark trio (7), time (13),
and the black hole (16) --- are precisely the four whose statement
requires both fibers in one sentence: a coherence that is kept
\emph{and} a registration that is final; a coordinate that rolls
\emph{and} a coordinate that rings; a tick \emph{and} an arrow; a
register \emph{and} a channel. On the one-language account each of
these is a collision between exactness and one-wayness with no
rule for how they share a world; on the two-fiber account each is
simply a place where the anatomy's two members meet --- and the
meeting has a grammar (the gearbox of \S\ref{sec:gearbox}) rather
than a paradox. That the century's hardest problems are exactly
the two-fiber problems is, to this paper's reading, the single
strongest piece of internal evidence that the diagnosis was
right.

\section{The two white papers: history and anatomy}
\label{sec:w1-w2}

A note on the division of labor between this paper and its
sibling. \PaperWOne{} \citep{ArisakaW1Symmetry} is
\emph{diachronic}: it walks the century --- \citet{Noether1918} to
\citet{HopkinsSinger2005} --- and shows the conviction ``symmetry causes
conservation'' loosening at seven historical steps, with the
program placed at the arc's end. The present paper is
\emph{synchronic}: it takes the structure that the arc arrives at
--- protection by topology rather than by symmetry --- and
dissects the geometry that implements it, one anatomy, six organs.
The two are complementary readings of one claim: W1 argues that
physics was already moving from symmetry to topology as the
custodian of exactness; W2 exhibits the custodian's anatomy and
its books. A reader persuaded by neither alone might note where
they interlock: the deepest step of W1's ladder --- the integer
protected because integers have nowhere to move --- is precisely
the circle of this paper's Part~II, and W1's capstone sentence ---
symmetry breaking as symmetry reading --- is this paper's
\S\ref{sec:determinism}, arrived at from the other direction.

\section{How (D) sits with (A), (B), and (C)}
\label{sec:abc}

Part~II set out the program in four development steps
(\S\ref{sec:foursteps}, Table~\ref{tab:foursteps}), and this
paper's place among them should now be stated structurally, since
the four are not four topics but four load-bearing stories.

\textbf{(A) is the arena} (A1--A4, \S\ref{sec:stepA}): the
constitution's axioms, \GGEP{}, and the six-dimensional bulk of
Eq.~\eqref{eq:bulk} carrying the composite connection of
Eq.~\eqref{eq:group-p2}, with the integer ledger of
Eq.~\eqref{eq:ledger-p2}, anomaly freedom, and no free
parameters. \textbf{(B) is the law} (A5--A7, \S\ref{sec:stepB}):
\OGGEP{} declares observation a projection; \GDOS{} gives the
resulting open dynamics its generator, Eq.~\eqref{eq:gdos-p2};
\GRIP{} $=$ \DGI{} $+$ \GFF{} $+$ \GSSB{} $+$ \GGA{},
Eq.~\eqref{eq:grip-p2}, runs the cascade to the unique attractor;
and $\UoneQi$ prices the record, Eq.~\eqref{eq:landauer-p2}.
\textbf{(C) is the payout} (A8--A11, \S\ref{sec:stepC}): quantum
mechanics, the Standard Model, $\Lambda$CDM, and the origin of
life and mind, each derived from (A) and (B) at declared grades
--- the codebook's twenty-eight entries, the inventory's
$\pi$-locked shares, the cascade's seventeen events. \textbf{(D) is
the synthesis} (\S\ref{sec:stepD}): the two white papers, of which
this is the second --- \PaperWOne{} reading the program
\emph{diachronically}, against the century of symmetry, and this
paper reading it \emph{synchronically}, against the sixteen
problems. The two-fiber anatomy is (D)'s instrument: not a fifth
ingredient, but the recognition that (A)'s two non-spatial
directions are opposite in topological kind, and that (B)'s entire
phenomenology and (C)'s entire payout are organized by that
opposition.

The relations run in both directions, and both matter.

\emph{(D) needs the other three.} Without (A) there is no pair to
sort: the anatomy is a statement about a specific forced geometry,
not a general aphorism about compactness, and if the bulk were
otherwise the sentence would have nothing to be true of. Without
(B) the interval's semigroup would be a formal remark about
composition rather than a physical arrow: it is \OGGEP{} that
makes the projection compulsory and \GDOS{} that makes the
one-wayness dynamical. And without (C) the anatomy would be an
aesthetic claim with no numbers in it --- every quantitative
column of Table~\ref{tab:master}, from $137$ to $8.1\,\mu$eV to
$140.24$, is a result of the applications, sorted by the anatomy
but produced by (C).

\emph{The other three need (D) to be legible.} The arena supplies
the fibers; the law supplies the dynamics; the applications supply
the numbers --- and none of the three says, in its own voice,
\emph{which fiber is doing which job}. That is the anatomy's
office: it is the index of the whole program, and the reason a
newcomer can now reach twenty-two papers through one sentence
rather than through twenty-two introductions. The campaign
recorded in Table~\ref{tab:deployment} is the evidence that the
index is real: it entered every paper through that paper's own
central topic, and required no value, grade, or theorem to change
anywhere.

A closing observation about order. In the cohort's own history the
sequence was (A), then (B), then (C), and only then (D) --- the
anatomy was noticed last, after the applications had been built,
which is the ordinary way of such things: one sees the skeleton
after the body is assembled. In the logic of the subject the order
may well be reversed. A reader who accepts nothing but the
anatomy --- one hidden direction that must forget, one that cannot
--- has already accepted the census that forces most of (A), since
(\S\ref{sec:minimum}) a bulk with any other pairing cannot host a
world that both is and becomes; and a reader who accepts (A) and
the projection has (B) nearly for free. The program was built
bottom-up and can be read top-down, and this paper is the
top-down reading.

\section{What is different from the standard accounts}
\label{sec:contrasts}

Structural contrasts only; no polemic is intended, and the
comparisons are with textbook framings rather than with any
research program's frontier.

\begin{itemize}[leftmargin=2.0em]
\item \emph{Gauge theory} treats holonomy as a derived quantity of
a postulated connection; here the split between where holonomy can
live ($\Stheta$) and where it cannot ($\Ir$) is load-bearing, and
the gauge sector's observable content is read off it.
\item \emph{Thermodynamics} postulates the second law; here the
first and second laws are the two fibers' ledgers, and their
famous tension is a division of labor between a group and a
semigroup.
\item \emph{Quantum foundations} adds a measurement postulate to a
unitary core; here the coherent half and the registering half are
the two fibers, and a measurement is a read.
\item \emph{Baryogenesis model-building} assembles Sakharov's
conditions from imported parts; here the three conditions are
three organs of one geometry, and the excess is computed with its
open joints named.
\item \emph{Inflationary cosmology and axion phenomenology} scan
menus (potentials; decay constants); here a census forbids the
menus: one interval, one inflaton; one circle, one axion.
\item \emph{The cosmological constant problem} asks why the vacuum
is small against a natural scale; here the vacuum's scale is the
descent's depth squared, its share is a fiber-volume ratio, and
the question becomes the (still open, honestly flagged) e-fold
budget.
\end{itemize}

\section{Where the objections are answered}
\label{sec:objections-w2}

Every synthesis of this scale attracts objections, and the strongest
of them deserve more room than a Discussion can give. They are
therefore collected in their own Part: Part~X answers, at length,
the objections a careful reader should raise --- that the anatomy is
a metaphor, that the interval is merely the renormalization group
renamed, that compactness explaining quantization is textbook
Kaluza--Klein, that deriving an arrow from a semigroup is circular,
that the quantitative face is a patchwork of grades, and that a
sentence explaining everything explains nothing --- together with
the questions a curious reader asks first. Nothing in this Part's
argument is held back from that one: the objections are stated in
their sharpest form, and where the honest answer is ``not yet,''
Part~X says so and names the work order.

\section{The honest ledger}
\label{sec:honest}

The paper's claims are only as strong as their weakest named
joints, so the joints are collected here in one place, exactly as
the parent papers carry them.

\begin{itemize}[leftmargin=2.0em]
\item \textbf{Derived, unconditional} (within the cohort's declared
axioms): the anatomy itself; the fates of $C$/$CP$/$CPT$; the
singlet rule; the proton's guard, read as a gauged conservation law
whose charge registration makes integral (integrality alone forbids
nothing, \S\ref{sec:proton}); the exact slope
$d\ln t/dr = 2$; the Hopf fractions; the seesaw arithmetic; the
census.
\item \textbf{Derived-conditional, conditions named}: $n_s$ and $r$
(the \GDOS{} plateau-class derivation and the prompt-crossing
lemma); $w = -1$ exactly (bulk rigidity, the KK spectral gap, flux
freezing); the Born rule at relaxation level (the field-grade
representation step is the one named analytic residue); the
Kaxion's patch-uniform angle (the pinning --- killable by
isocurvature).
\item \textbf{Named inputs and regulators}: the scalar amplitude
$A_s$ (the launch's one declared input); the geometric phase
assignment and the flavor-measurement regulator in the $\etaB$
chain; the $K/2$ exponent of the throughput, postulated.
\item \textbf{Open integers and coefficients}: the cap loading
$\Xicap = 43$ (benchmark; the discrete window $1.59$--$2.13$~GHz);
the drainage factor $\zeta_\Lambda$ (uncomputed; flagged home); and
--- listed here because Table~\ref{tab:dispositions} grades
Problem~12 \emph{reread} rather than \emph{solved} --- the
field-theoretic proof of the mass gap, which the closure rule
explains but does not supply \citep{JaffeWitten2006}.
\item \textbf{Flagged, numerology discipline}: the $4K$ age
compression and its doubled forms ($8K$ for the vacuum hierarchy
and the de Sitter entropy) --- one observation, three faces, whose
would-be derivation is the queued e-fold budget campaign; the
$e^{-K/3}$ warp candidate for the plateau edge.
\end{itemize}

Nothing above is a concession extracted by a referee; it is the
cohort's own register, reproduced because the synthesis would be
worthless without it.

\section{The ways to be wrong}
\label{sec:kill}

\begin{table}[htbp]
\centering
\caption[The consolidated falsifier table: the six treatments with instruments of their own.]{The consolidated falsifier table: the six consequences
exposed on named instruments in named channels. The other ten
treatments carry no instrument of their own and are exposed through
the shared ledger, as \S\ref{sec:kill} states. Each entry is
registered in the cited parent paper; dates are the cohort's
decision-calendar estimates.}
\label{tab:kill}
\small
\renewcommand{\arraystretch}{1.35}
\begin{tabularx}{\textwidth}{@{}P{2.5cm}>{\raggedright\arraybackslash}X P{3.3cm}@{}}
\toprule
\textbf{Consequence} & \textbf{Kill condition} & \textbf{Instrument / era} \\
\midrule
Discrete symmetry (Problem 9) & any $CPT$ violation (particle--antiparticle mass split at any precision); $\dPMNS$ off $196.4^\circ$ beyond errors & antihydrogen spectroscopy; DUNE / Hyper-K, late 2020s--30s \\
Antimatter (Problems 10--11) & measured $\etaB$ incompatible with the chain beyond its declared joints; sphaleron-era physics contradicting $\Delta B = 3n$, $B{-}L$ exact & CMB + BBN concordance; collider EW-sector tests \\
Confinement / proton (Problem 12) & observed proton decay violating the double guard; any fractional-charge or free-triality state & Hyper-Kamiokande; collider searches \\
Time (Problem 13) & measured $z_{\mathrm{dec}}$ off the atomic chain by $> 0.5\%$ (F7, observable-only); chronology inconsistent with slope 2 / kink $3/2$ & CMB spectroscopy; BAO + chronometers \\
Interactions (Problem 2) & KK lock cluster absent at $3.849$~TeV with the predicted fingerprint; $\sin^2\theta_W$, $\alpha^{-1}(0)$ closed forms failing at precision & HL-LHC, 2030s \\
Dark trio (Problem 7) & tensor $r$ off $0.0043$ in either direction; $n_s < 0.955$; axion-sector isocurvature detected; no line in $1.59$--$2.13$~GHz with the $1/35$ doublet; $w \neq -1$ or phantom crossing & Simons Observatory / LiteBIRD; IBS-CAPP / ADMX-EFR; Roman / DESI, 2030s \\
\bottomrule
\end{tabularx}
\end{table}

The table has six rows, not sixteen, and the difference is worth
stating rather than leaving to a reader who counts. Six consequences
are exposed \emph{directly}: a named channel, a named instrument, and
an era, mostly within the decade. The other ten carry no instrument of
their own, and their exposure is of a second kind --- weaker, but not
nothing. Every row of Table~\ref{tab:master} is spent out of one
ledger, $(M,N,L;J,K) = (1,3,5;7,35)$, and the comparisons in
\S\ref{sec:numbers-audit} span twelve orders of magnitude of
experimental technique with no shared systematics. If the lock cluster
is absent at $3.849$~TeV, or the haloscope window closes empty, or the
tensor ratio lands away from $0.0043$, the integers do not survive ---
and the ten treatments that spend them do not survive either, whether
or not anyone has built an instrument pointed at those treatments
directly. That is the honest shape of the claim: six treatments can be
killed on their own, and all sixteen can be killed together. The
anatomy is a single sentence; the table is the part of its exposure
that carries a date; and the program regards the size of the table
as the measure of the sentence's worth.

\section{The numbers of this paper, on one page}
\label{sec:numbers-audit}

Reproducibility is the cohort's standard, so Table~\ref{tab:numbers}
collects every number this paper has quoted with its closed form,
its value, and its measured comparison where one exists. Each was
re-derived programmatically in the preparation of this draft;
none is carried from memory.

\begin{table}[htbp]
\centering
\caption[The consolidated number audit.]{The consolidated number audit. Grades as in
\S\ref{sec:honest}; measured values from the Planck 2018 and PDG
compilations via the parent papers. Nineteen rows, each giving the closed form, the value it produces and the measurement or reach it faces --- from the lock scale at $3.849$~TeV and $\alpha^{-1}(0) = 137.0$ down to the PMNS phase at $196.4^\circ$. Two rows carry their own warnings and are printed with them: the corridor switch $e^{J} = 1096.6$ is a topological marker with no observational counterpart, and the age ledger's $4K \times 1.0017$ is flagged rather than claimed.}
\label{tab:numbers}
\small
\renewcommand{\arraystretch}{1.28}
\begin{tabularx}{\textwidth}{@{}P{3.3cm}P{4.1cm}P{2.6cm}>{\raggedright\arraybackslash}X@{}}
\toprule
\textbf{Quantity} & \textbf{Closed form} & \textbf{Value} & \textbf{Compared with} \\
\midrule
lock scale & $\tfrac12 M_{\mathrm{Pl}} e^{-K}$ & $3.849$~TeV & HL-LHC target \\
fine structure & $2^J + N^2$ & $137.0$ & $137.036$ ($-0.03\%$) \\
weak angle & $L^2/(4N^3)$ & $0.23148$ & $0.23122$ ($+0.11\%$) \\
EW vev & $\Mlock (2/L)^N$ & $246.3$~GeV & $246.22$~GeV \\
matter share & $1/\pi$ & $0.3183$ & $0.3153 \pm 0.0073$ \\
baryon share & $1/(2\pi^2)$ & $0.0507$ & $0.0493$ \\
dark energy & $1 - 1/\pi$ & $0.6817$ & $0.6847 \pm 0.0073$ \\
crossover & $(\pi-1)^{1/3} - 1$ & $0.289$ & $z_{\Lambda m} \sim 0.3$ \\
tilt & $1 - 2/N^{*}$, $N^{*} = 52.6$ & $0.9620$ & $0.9649 \pm 0.0042$ \\
tensor ratio & $12/N^{*2}$ & $0.0043$ & $< 0.036$; SO/LiteBIRD reach \\
Kaxion mass & $\sqrt{\chiQCD}/\fK$ & $8.1\,\mu$eV & haloscope window \\
Kaxion line & $\mK c^2/h$ & $1.95$~GHz & IBS-CAPP / ADMX-EFR \\
twin split & $\nu_K/K$ & $1.95\,\mathrm{GHz}/35 = 56$~MHz & scale-free discriminator \\
seesaw scale & $\Mlock^2/\Mplred$ & $6.1$~meV & $\rho_\Lambda^{1/4} = 2.24$~meV ($\zeta_\Lambda^{1/4} = 0.37$) \\
baryon excess & $7.04 \cdot \tfrac{28}{79} \cdot 1.25\,e^{-K/2}/\gstar$ & $\approx7\e{-10}$ & $6.12\e{-10}$ ($+20\%$) \\
corridor switch & $e^{J}$ & $1096.6$ & \emph{topological marker; no observational counterpart} \\
atomic decoupling & atomic chain & $1091.1$ & $1089.9$ ($+0.11\%$; \S\ref{sec:two-quantity}) \\
age ledger & $\ln(t_0/t_{\mathrm{Pl}})$ & $140.24$ & $= 4K \times 1.0017$ (flagged) \\
PMNS phase & $\pi + 2\arcsin(1/7)$ & $196.4^\circ$ & DUNE / Hyper-K \\
\bottomrule
\end{tabularx}
\end{table}

The table above is this paper's own audit, and it is deliberately
short --- it carries only the numbers these sixteen treatments
actually quoted. Figure~\ref{fig:a10-devplot-w2} shows the
cosmological half of that audit as the parent paper renders it: the
fractional deviation of every observable carrying an empirical
comparison, with the precision bands drawn
\citep{ArisakaA10Cosmos}. It is included here because a table of
agreements is easy to read charitably and a deviation plot is not:
the entries that miss are as visible as the entries that hit, and
two of them --- the late-time $H_0$ and the baryon asymmetry ---
miss by amounts this paper names rather than absorbs.

\begin{figure}[p]
\centering
\includegraphics[width=0.99\textwidth,height=0.86\textheight,keepaspectratio]{a10_dev_plot_showcase.pdf}
\caption[The cosmological audit, in one chart.]{\textbf{The
cosmological audit, in one chart} (reproduced from
\citealp{ArisakaA10Cosmos}). The fractional deviation
$(\mathrm{pred} - \mathrm{obs})/|\mathrm{obs}|$ for the twenty-two
observables carrying an empirical comparison --- eighteen
independent predictions (filled), four anchored by construction
(open) --- with nested precision bands at
$|\mathrm{Dev}| < 10^{-3}$ and $10^{-2}$. Two entries earn their
place by missing. The late-time $H_0$ row sits low against the
local-ladder value, which is the corridor's exposure rather than a
free parameter's failure; and the $\eta_B$ row sits $20\%$ off in its
band, carrying the one postulated exponent of the toll (Problem~11). The
$z_*$ row is the two-quantity discipline drawn: the framework's
atomic-chain decoupling prediction is scored against the measured
value, with the corridor switch $e^7$ annotated separately as a
topological marker, never conflated with it. Nothing in the chart
is a fit; the residuals are what a zero-parameter framework has to
live with.}
\label{fig:a10-devplot-w2}
\end{figure}


Three reading notes. \emph{First}, the left column's grammar is
uniform: integers and $\pi$'s in the closed forms (the circle),
exponentials of $K$ in the scales (the interval) --- the table is
the gearbox of Part~II, printed. \emph{Second}, the comparisons
span twelve orders of magnitude of experimental technique, from
CMB polarimetry to haloscope cavities to collider clusters, with
no shared systematics: the cross-exposure is deliberate, and a
failure anywhere impeaches the ledger everywhere. \emph{Third},
the table contains no entry with a tunable parameter behind it:
every value is either an arithmetic of $(1,3,5;7,35)$, a
geometric measure, or a declared input named in
\S\ref{sec:honest} --- which is why the table is short, and why
it is dangerous.

\section{Sixteen problems, sixteen dispositions}
\label{sec:six-problems}

A final consolidation, in the discipline's own severe vocabulary:
for each of the sixteen, what exactly did the anatomy \emph{do}?
Table~\ref{tab:dispositions} is that consolidation; the verbs are
defined below it and chosen deliberately, because the differences
between them are the paper's honesty.

\begin{table}[htbp]
\centering
\caption[Sixteen problems, sixteen dispositions.]{\textbf{Sixteen
problems, sixteen dispositions.} What the anatomy actually did to
each, in one verb, with the Part that did it. The verbs are defined
immediately below; no two rows claim the same thing, and several
claim conspicuously less than a partisan would.}
\label{tab:dispositions}
\small
\renewcommand{\arraystretch}{1.26}
\begin{tabular}{@{}clll@{}}
\toprule
\# & \textbf{Problem} & \textbf{Disposition} & \textbf{Where} \\
\midrule
1 & measurement & dissolved: a measurement is a read; Born at relaxation grade & Part III \\
2 & four interactions & solved by sorting: two keepers, one agent, one integrator & Part III \\
3 & gauge + renormalization & re-founded: unreadability and one-way motion, derived & Part III \\
4 & parameter count & repriced to a codebook: $6+14+8$, graded in public & Part III \\
5 & the beginning & transformed: launch derived; entrance a named open item & Part IV \\
6 & naturalness & dissolved: a depth, not a fine-tuning & Part IV \\
7 & dark trio & identified: the two fibers in the sky; scale flagged & Part IV \\
8 & three horizons & solved in three parts: launch, classes, groove & Part IV \\
9 & discrete symmetry & solved: orientations; fates by fiber; strong $CP$ dissolved & Part V \\
10--11 & antimatter + excess & mirror solved; excess computed, joints named ($+20\%$) & Part V \\
12 & confinement + proton & reread: closure rule; guard exact; formal proof open & Part V \\
13 & time & solved at the root: tick, arrow, exact chronology & Part VI \\
14 & energy, mass, entropy & defined: tick's worth, class's address, write count & Part VI \\
15 & landscape & unemployed: unique floor; anthropic residue named & Part VI \\
16 & black-hole information & dissolved: a billing error, on the Page schedule & Part VI \\
\bottomrule
\end{tabular}
\end{table}

\emph{Solved} means a derivation replaced a postulate.
\emph{Dissolved} means the question's presupposition was located
and removed --- there was never a fine-tuned number, only an
address; never a collapse, only a write; never a loss, only a
ledger. \emph{Re-founded} and \emph{reread} mean the established
mathematics was kept whole and given a referent. \emph{Repriced}
and \emph{transformed} mean the problem became a different,
sharper question that remains open --- and the flagged register
exists precisely so that repricing is never mistaken for solving.
\emph{Computed, joints named} means a number was produced with its
open steps declared. \emph{Identified} and \emph{unemployed} speak
for themselves. No row claims more than its grade in
Table~\ref{tab:master}; several claim conspicuously less than a
partisan would --- which is the register the cohort intends to be
read in.

\section{The decision calendar}
\label{sec:calendar}

The falsifier table of \S\ref{sec:kill} has a time axis, and it is
short. Consolidating the parent papers' calendars: \emph{now
through 2028} --- haloscope coverage of the low edge of the Kaxion
window begins; DUNE and Hyper-Kamiokande commissioning toward
$\dPMNS$; antihydrogen precision advances. \emph{2028--2032} ---
the tensor programs (LiteBIRD launch window; the Simons
Observatory's enhanced surveys) bracket $r = 0.0043$ from both sides; the HL-LHC
integrated-luminosity curve crosses the staked sensitivity for
the $3.849$~TeV cluster's primary channel; full-window haloscope
coverage with doublet resolution. \emph{2032 and beyond} ---
Hyper-Kamiokande's proton-decay reach passes the guard-relevant
channels; next-generation $0\nu\beta\beta$ reaches the band that
arbitrates the driver's Majorana prerequisite; Roman and DESI
close the $w = -1$ error budget below the phantom-crossing
ambiguity. By roughly a decade from this draft, every row of
Table~\ref{tab:kill} will have been visited by at least one
instrument. The paper's sixteen answers were assembled from
results already in hand; its wager is that they remain standing
when the calendar has run --- and the cohort's practice, recorded
in every parent paper, is to publish the verdicts either way.

\section{Open problems and queued campaigns}
\label{sec:open-w2}

Four items, inherited from the cohort's ledger, would most sharpen
this paper if closed: (1) \emph{the e-fold budget} --- deriving
$T_0$ through the descent with no dial, converting the $4K$/$8K$
family from flagged to theorem; (2) \emph{the relic quantification}
$(A, \gamma)$ of the Born-relaxation signature F25 at the largest
scales; (3) \emph{the field-grade representation step} --- the Born
chain's last analytic residue; (4) \emph{the cap integer}
$\Xicap$ --- one integer between the Kaxion's benchmark and its
theorem. Each is a campaign, not a gap: named, scoped, and owned.

%% ===================================================================
\section{Part VII in one paragraph, and what comes next}

The consolidation is done. Sixteen problems sit in one table with
sixteen separately stated dispositions and sixteen grades; the weak
joints are collected in one ledger rather than distributed through
the argument where they would be easy to miss; the numbers are in
one audit with their closed forms beside them; and the ways to be
wrong are in one table with instruments and dates attached. Four
campaigns are named as campaigns, not gaps. Where this reading
differs from the programs it will be compared with, the
difference is structural and is stated as such.

That is the apparatus. Part~VIII says the whole thing once more in
plain language, for the reader who wants the answers and not the
machinery --- and then says what the author actually thinks it
means.

%%  PART VIII -- CONCLUSIONS
%% ===================================================================
\GGPart[Conclusions]{Conclusions}{The Circle Remembers; the
Interval Forgets}

\section{What has been established}
\label{sec:concl-established}

This paper began with a list --- sixteen questions, median age
seventy years, that the best instruments and the ablest people
have sharpened for two generations and closed almost never --- and
with a diagnosis: the list is long not because the questions are
many but because one structural fact is missing from all of them.
The world has two kinds of direction, and physics has been
describing it in a language with one.

Part~II supplied the fact. The six-dimensional bulk of
\GGTheory{} carries, over every point of space, a closed circle
and an open interval; holonomy requires closed loops, and an
interval has none; so one hidden direction is pure memory ---
quantized, protected, reversible --- and the other, once the boundary
law's compulsory projection is laid along it, pure act --- one-way,
priced, final. Two premises, stated apart in \S\ref{sec:primer} and
kept apart since: the topology says which fiber can hold a record, and
the projection says why the other one's acts do not come back. \emph{The circle remembers; the interval
forgets.} Parts~III--VI then spent that sentence on the sixteen,
one by one, at essay length, with the equations displayed and their
parameters defined: the exactness problems taken by the circle, the
one-wayness problems taken by the interval, and the four that have
resisted longest --- measurement, the dark sector, time, the black
hole --- taken by the pair meeting, because a meeting is what each of
them always was. \emph{Taken} is the right verb and not a modest one:
Table~\ref{tab:dispositions} grades nine of the sixteen with verbs
that are not ``solved,'' and those grades stand.

What was established, precisely: every claim carried here is a
cohort result, cited at its parent's grade, with the open joints
named in place (\S\ref{sec:honest}) and the kill conditions
collected, dated, in one table (\S\ref{sec:kill}). The paper
added no axiom, no field, and no parameter; its contribution is
the alignment --- and the observation that the alignment is not
optional: one fiber of each kind is the minimum a world needs in
order to both be and become. Whether the alignment is a fact
about nature or only about our description is not decided in
these pages; it is decided by the instruments of the coming
decade, and the paper has done everything a paper can to make
their verdict unambiguous.

\section{A note on teaching}
\label{sec:teaching}

If the falsifier table survives its decade, the pedagogical
consequence deserves to be anticipated, because it is larger than
a new chapter. The two-fiber anatomy is teachable \emph{first} ---
before Lagrangians, before groups, almost before calculus: one
direction that must forget, one that cannot; the circle keeps
promises, the interval makes events; and the sixteen problems
then arrive as one lecture each, with the exact statements
attached later as the student's mathematics matures. Much of what
the present curriculum presents as brute --- why quantum numbers
never drift while couplings run, why the second law never
reverses while the first never fails, why measurement is one-way,
why antimatter mirrors matter exactly and yet lost --- becomes,
in this order of teaching, the \emph{expected} behavior of a
two-fiber world, with the surprise relocated to where it belongs:
that nature audited the picture and, so far, signed. The
conditional is essential and repeated: curricula follow
experiment, not elegance, and Part~VII's table is the tribunal.
But a physics that can be taught from one sentence outward owes
its students that sentence early --- and this white paper is,
among its other purposes, a first draft of that course.

\section{What has changed since W1, and what has not}
\label{sec:changed}

Six months separate the first white paper's assembly from this
one, and the interval (the word is now unavoidable) saw the
campaign deploy the anatomy across the entire cohort. It is worth
recording, as a conclusion in its own right, what that deployment
changed and what it conspicuously did not. \emph{What changed}:
the cohort acquired an index --- every paper now names which fiber
does which job in its central result; three portraits (electron,
proton, Kaxion) and one trio (rolling, ringing, rest) were
completed; the leptogenesis chain was assembled end to end with
its joints named; the treatment of time was expanded from a
section to a quantitative chronology; and this paper became
possible --- a single document from which a newcomer can reach
the whole program through one sentence. \emph{What did not
change}: a single number. No value was retuned, no grade
promoted, no theorem's conditions relaxed anywhere in the
twenty-two papers the campaign touched --- the deployment was an
audit, and the audit passed. A skeptic should weigh that fact at
full strength, because it is unusual: organizing languages in
physics ordinarily earn their keep by suggesting modifications;
this one earned its keep by \emph{failing to require any}. The
anatomy fit the cohort the way a description fits a thing that
was already there --- which is either the strongest thing that
can be said for it short of experiment, or an elaborate
selection effect; and Part~VII's table exists because the
difference is not for us to adjudicate from the armchair.

\section{The sixteen answers, in sixteen sentences}
\label{sec:six-sentences}

For the reader who asked only for the conclusions --- one sentence
per problem, in the order of Part~I:

\begin{enumerate}[leftmargin=2.2em]
\item A measurement is the interval writing what the circle
carried: the coherent half of quantum mechanics is transport with
inverses, the registering half is transport without them, and the
century's ``cut'' is not a place but the meeting of the two.
\item The four interactions are one bulk connection read on
different structures: two forces keep the circle's books, the weak
force is the interval's agent --- which is why it alone is chiral,
broken, and identity-changing --- and gravity is the descent's own
bookkeeping, which is why it will not quantize like the rest.
\item Gauge invariance is the circle's unreadability and
renormalization is one-way motion along the interval: the two
principles that carried all of field theory were the two fibers,
used for a century without being named.
\item The parameter sheet is a codebook, not a list of dials:
every entry is a holonomy-class label wearing a descent address,
and the audit is public --- six derived, fourteen conditional,
eight still owed.
\item The Big Bang is a launch, not a wall: the plateau class is
forced, the roll happens once, its two fingerprints ($n_s$, $r$)
are staked, and what remains open is an entrance with a
work order, not a mystery with a veil.
\item Nothing about the weak scale is small; something is deep ---
thirty-five e-folds of descent, an integer spent in four places, and
the forty-year anxiety retires without a single cancellation. Of the
two ratios that anxiety names, the weak-scale half retires outright;
the vacuum-energy half retires to within an uncomputed drainage
factor of $\zeta_\Lambda^{1/4} = 0.37$, which is a smaller claim and is
made as one.
\item Inflation is the interval's coordinate rolling once, dark
matter is the circle's coordinate ringing still at $8.1\,\mu$eV,
and dark energy is both at rest: the dark sector is the two
fibers, seen in the sky.
\item Distant strangers agree because the launch shared their
conditions, the classes cannot vary, and the groove has one
floor: three horizons, three mechanisms, no lottery.
\item $C$, $P$, and $T$ are the geometry's three orientation
choices; each single sense can be fixed by a structure, the joint
reversal cannot --- so $CPT$ is exact and the others fail
exactly where the alphabet and the runtime say.
\item Antimatter exists because a circle has two winding senses
--- the same class, read backwards, which is why the mirror is
perfect.
\item The excess of matter is a paid toll: a chiral tape read
once through a gated one-way door at $e^{-K/2}$, delivering
$5.9\e{-10}$ against a measured $6.1\e{-10}$, joints named.
\item Quarks confine because the boundary registers only closed
color books, and the proton persists because baryon number is a
gauged holonomy class with no $\Delta B = 1$ aperture --- the two
facts every atom banks on, reread as one registration rule.
\item Time's tick is the circle's return and time's arrow is the
interval's one-way reading, with the chronology exact:
$d\ln t/dr = 2$, and an age ledger closing at $140.24$.
\item Energy is what a tick is worth, mass is the address a
class wears, entropy is what the arrow has read --- three
household words, finally with definitions, and one formula
($S_{\mathrm{BH}}$) that spends all three fibers' constants at
once.
\item The multiverse is not refuted but unemployed: the
constants it was hired to explain are forced --- a quantized
dial, a unique floor, a semigroup that cannot re-choose --- and
a lottery with one ticket needs no drum.
\item The black hole's information was never lost, only billed
to the wrong book: the circle keeps what the hole must return,
the interval prices the return, and the Page curve is the
reconciliation schedule.
\end{enumerate}

\section{The oldest dispute in physics, and what the pair says
about it}
\label{sec:philosophy}

A conclusion for the long view, in three steps --- each a
philosophical claim, and each anchored to an equation rather than
to a mood.

\paragraph{Being and becoming were never rivals.} The oldest
dispute in natural philosophy --- older than physics itself --- is
whether the world fundamentally \emph{is} (Parmenides: what is
real does not change) or fundamentally \emph{becomes} (Heraclitus:
what is real is change). Twenty-five centuries later the dispute
is still legible in the discipline's deepest tension: exact
conservation against irreversible flow, unitary evolution against
the completed measurement, the frozen laws against the running
history. The two counterfactual worlds of Part~II are the two
positions taken purely: the all-circle world \emph{is} and never
becomes --- perfect conservation, no event ever --- and the
all-interval world \emph{becomes} and never is --- perpetual
change, nothing to change. Both are sterile, and for exact
reasons: Eq.~\eqref{eq:group-law-p2} alone permits no record;
Eq.~\eqref{eq:semigroup-law-p2} alone protects no thing. A world
requires one of each, geared together --- and that is not a
compromise between the two ancient positions but the discovery
that they were always the two fibers of one geometry, each
correct about its own direction. The dispute does not have a
winner. It has an anatomy.

\paragraph{Why the world is comprehensible at all.} Einstein
called it the most incomprehensible thing about the universe ---
that it is comprehensible --- and Wigner gave the puzzle its
sharpest form: why should mathematics, invented by primates for
counting, fit nature to twelve digits? The anatomy's answer has
been implicit since Part~II and deserves to be said plainly.
Science requires two things of a world: that something never
change (else no law could be written down, no experiment
repeated, no memory trusted) and that something always happen
(else there would be nothing to explain and no one to explain
it). These are exactly the two fibers. The circle underwrites
the possibility of \emph{knowledge} --- protected classes are
why the electron a student measures tomorrow is the electron in
tonight's textbook --- and the interval underwrites the
possibility of \emph{learning} --- one-way writes are why
evidence accumulates instead of evaporating. A one-fiber world
of either kind would contain no science, not because science
would be hard there but because its preconditions would be
absent. Comprehensibility is not a lucky property of our
universe; on this reading it is a structural consequence of the
same pair that makes the universe habitable --- and the
mathematics fits because the mathematics (a group, a semigroup)
\emph{is} the structure, written small.

\paragraph{A forced structure is not a finished story.} The
determinism of this paper is total at the level of structure ---
which layers exist, which constants hold, which vacuum is the
floor --- and deliberately silent at the level of history. The
groove fixes the kind of world; it does not compose the weather,
the biosphere, or the sentence the reader will think next. Those
live in the combinatorial openness that a fixed alphabet permits
--- as a language with a finite grammar permits unbounded
literature. The anatomy therefore delivers a rarer thing than
either fatalism or chance: a world whose \emph{reliability} is
guaranteed (the circle cannot break its promises) and whose
\emph{future} is genuinely unwritten at every scale where
composition matters (the interval writes only once, and has not
yet written tomorrow). Physics has long been suspected of
leaving no room for novelty; the two-fiber reading locates the
room exactly --- everything above the alphabet, which is nearly
everything --- and prices what was actually forced: the stage,
the letters, and the guarantee that the letters keep. It is
difficult to imagine a more generous constitution: everything
needed for a world is fixed, and everything interesting in one
is free.

\section{A personal closing}
\label{sec:concl-personal}

The question that made me a physicist arrived in high school:
nature seemed too beautiful, too well arranged for us to be here,
for the arrangement to be an accident --- and I wanted to know
why. I have carried that question through five decades on the
experimental front line --- through detectors for proton decay
and neutrinos, through the precision instruments of particle
astrophysics --- and through every costume the question wore:
why does symmetry break the same way everywhere; why did matter
win; why does the proton keep; why does time run; why is the
dark sector what it is; why these constants and no dial anywhere
to set them. I did not expect, at the end, that the answers
would share a single sentence. The work collected here says they
do: \emph{the universe has one direction that must forget and
one that cannot} --- and everything I spent my life puzzling
over is the interplay of those two. The tape and the read head.
The groove and the playback. The being and the becoming.

The circle remembers: the charges, the phases, the alphabet, the
books --- every promise the universe has ever kept, it keeps
there. The interval forgets: every scale visited once, every
choice made final, every record priced and paid --- and
precisely because it forgets, something can happen at all.
Between them they need no third thing, and the constitution
supplies no third thing. Sixteen problems, one anatomy: not
because I went looking for a slogan, but because --- as the
campaign that built this paper taught me --- the alignment was
already in the physics, waiting to be read. Not contingent. Not
lucky. Not designed. Forced.

I am aware of how large these claims sound, and I have tried to
make the paper's honesty as prominent as its ambition: the
grades are stated in place, the open items are named as work
orders, and Part~VII's table lists the measurements on which
each answer dies. If the instruments of this decade --- the
tensor telescopes, the haloscopes, the proton watchers, the
colliders --- return the numbers staked there, then I believe
this sentence will deserve to be taught first, before the
forces, before the fields: \emph{the circle remembers; the
interval forgets; and the world is what that sounds like, played
once.} If they return other numbers, the table says exactly what
dies, and I will publish the verdicts either way --- that has
been the cohort's practice at every step, and it is the only
practice worthy of the question. Either way, we will have
learned something that cannot be unlearned --- which is all I
ever asked of physics, and more than I dared hope it would give
back.

There remains only gratitude, and an invitation.

The gratitude is easy to name. To the experimental collaborations
whose patience built the instruments this paper stakes itself on
--- the haloscope halls, the neutrino caverns, the polarization
telescopes, the collider --- and to the tradition of physicists,
many quoted in Part~I, who kept these sixteen questions alive
through decades when working on them was unfashionable: this
paper is a reading of a house that others built, and it knows it.
And to the reader who has come this far: a synthesis is only ever
completed by the person who checks it, and you have my thanks for
the hours.

The invitation is the closing I most want to write. If you are a
student, the next decade of this program belongs to you more
than to me: the falsifier table is a to-do list, the flagged
family is a thesis waiting for its author, the entrance is the
best open problem I know, and the e-fold budget campaign ---
which would turn the deepest flagged number in the cohort into a
theorem --- is specified to the point where someone with fresh
eyes and a long morning could begin. Take the sentence, try to
break it, and publish what you find either way; the paper is
built so that both outcomes are progress. I have spent fifty
years learning that the universe rewards exactly this kind of
stubbornness --- and that it is far more beautiful, and far more
orderly, than we had any right to expect.

The universe keeps one direction that cannot forget and one that
cannot remember, and out of their marriage comes everything:
atoms and their patience, stars and their spending, records and
their keepers, physicists and their questions. I find I am no
longer troubled by the question that started me in high school.
The arrangement is not an accident, and we were not lucky to be
here. The tape was pressed; the head was set; and the playback
--- fourteen billion years of it, so far --- has been, among all
its other works, the slow manufacture of readers. This paper is
for them.

\bigskip
\begin{flushright}
\emph{Katsushi Arisaka}\\
Los Angeles, August 2026
\end{flushright}


%% ===================================================================
%%  ACKNOWLEDGMENTS  (cohort convention: after the Conclusions,
%%  before the Technical Appendices)
%% ===================================================================
\GGAcknowledgments

\medskip
One acknowledgment is particular to this paper. A synthesis owes a
debt of a kind an original paper does not: everything carried here
was proved somewhere else first, and the sixteen treatments stand
or fall on work done in the cohort papers rather than in these
pages. Where a result is quoted at a grade below \emph{derived},
that grade was set by the parent paper and not softened here; where
a step is still open --- the field-grade representation step, the
entrance corridor, the drainage factor $\zeta_\Lambda$, the
postulated $K/2$ exponent --- it is named in place rather than
carried quietly. The reader who wants to attack this paper should
attack those joints, and the author would count that a service.

\medskip
Finally, a word to the readers this paper was written for and who
have not yet been thanked because they have not yet read it: the
students who will check these claims, find the errors, and either
repair the framework or retire it. The sixteen problems have
outlived every physicist who has so far worked on them. If the
reading offered here is right, some of them will not outlive the
next generation of instruments; if it is wrong, the instruments
will say so just as clearly, and that is the arrangement under
which all of this is worth doing.

%% ===================================================================
\section*{What follows}

What follows is not afterthought but apparatus. Part~IX derives
from scratch what Parts~III--VI could only quote --- nine topics,
each in four steps, beginning with a full textbook treatment of the
standard material before a word is said about anything new, with
the problems worked and the solutions given. Part~X argues with the
reader: one hundred and seventeen objections, the first twenty-two
from someone with no physics training and the rest from
specialists, several conceded outright. Part~XI says whose work
every result here belongs to, what it cost the field, and where
this paper reads it differently.

%%  PART IX -- TECHNICAL APPENDICES
%% ===================================================================
\GGPart[Technical Appendices]{Technical Appendices}{A Graduate Course
in the Two-Fiber Anatomy --- From-Scratch Derivations of the
Machinery behind the Sixteen Problems, Each in Four Steps}
\label{part:IX}

The body of this paper carried its results at their parents' grades
and kept the derivations in the cohort papers where they belong. The
present appendices supply what a student needs in order to check
them, and taken together they are intended to be read as \emph{a
graduate course in the two-fiber anatomy}, addressed to the incoming
physics graduate student who will carry the program forward --- or
break it.

They are organized in two groups. The \emph{reference} appendices
(\S\ref{app:abbrev}--\S\ref{app:symbols}) tabulate the abbreviations
and symbols used throughout, and serve as the reader's quick lookup.
The \emph{constructive} appendices are the course proper: a
self-contained sequence that follows the order of Parts~III--VI ---
open quantum systems and the measurement write; gauge invariance and
the renormalization group; the parameter codebook; the launch and
the descent; the dark sector; the discrete symmetries and the matter
excess; confinement and the proton; and time, entropy, and horizon
thermodynamics --- preceded by the appendix on holonomy itself,
which is the instrument every later chapter uses.

Each constructive appendix is written in \emph{four steps}, and the
four are the same every time:

\begin{enumerate}[leftmargin=2.2em]
\item \textbf{Step 1 --- the standard treatment.} A from-scratch
derivation of the established result, in the form a graduate student
meets it in a textbook, carried through with no skipped
manipulations and closed by a worked example. Nothing in Step~1 is
this program's: it is the discipline's own material, presented so
that the reader has something exact to hold before anything new is
asked of them.
\item \textbf{Step 2 --- what remains unexplained.} A frank
statement of what the standard treatment \emph{assumes but cannot
explain} --- the inputs it must take from experiment, the postulates
it must adopt, and the structural questions it leaves open. Each
Step~2 ends by naming which of the sixteen problems of Part~I lives
in that gap, so the reader can see the ledger being drawn.
\item \textbf{Step 3 --- the two-fiber resolution.} The same
material reread on the anatomy: which quantity is circle-side and
protected, which is interval-side and priced, and what changes when
the sorting is applied. Grades are stated in place, exactly as in
the body; where the resolution is conditional or flagged, Step~3
says so.
\item \textbf{Step 4 --- problems and worked solutions.} Exercises
with complete solutions, each worked in short numbered steps and
closed by a \emph{physics intuition} remark. They are the point of
the course: a reader who works them has checked the chapter rather
than read it.
\end{enumerate}

Two conventions. Equations already displayed in the body are cited
rather than repeated, so the appendices and the body form one
document rather than two. And nothing in these appendices is
load-bearing for the body's claims: they teach the machinery, they
do not extend it. A reader who wants the argument should read
Parts~I--VIII; a reader who wants to \emph{use} the argument should
work these chapters.

\section{Recommended textbooks and key papers}
\label{app:reading}

The course below is self-contained, but no appendix can replace a
term with a good book. What follows is the reading the author would
put in front of an incoming graduate student, grouped by the
appendix it serves. Where a single paper changed the subject, it is
named alongside the textbook, because reading the original is
usually faster than reading a summary of it.

\paragraph{Geometry, bundles, and holonomy
(Appendix~\ref{app:holonomy}).} \citet{Nakahara2003} is the
standard bridge from a physicist's calculus to fiber bundles:
Ch.~9--10 contain everything this course uses.
\citet{MTW1973}, Ch.~8 and~11, teaches parallel transport with the
pictures drawn, and is still unmatched for intuition.
\citet{Wald1984} gives the same material with mathematical care and
carries the horizon theory of Appendix~\ref{app:time}. The one
indispensable paper is \citet{WuYang1975}: the statement that the
holonomies \emph{are} the gauge field, which is the sentence this
whole paper turns on. \citet{Dirac1931} is worth reading in the
original for the first argument that a physical quantity must be an
integer.

\paragraph{Quantum mechanics and open systems
(Appendix~\ref{app:openquantum}).} \citet{SakuraiNapolitano2020}
for the formalism at graduate level;
\citet{NielsenChuang2000}, Ch.~2 and~8, for density matrices,
Kraus operators, and quantum channels --- the cleanest modern
treatment of the machinery Step~1 derives;
\citet{BreuerPetruccione2002}, Ch.~3, for the Lindblad theorem and
decoherence models done at length. The papers to read are
\citet{Born1926} (two pages, one footnote, one rule),
\citet{EPR1935} and \citet{Bell1964} as a pair, and
\citet{Zurek2003} for the decoherence program in its author's own
words.

\paragraph{Gauge theory and the renormalization group
(Appendix~\ref{app:gaugeRG}).} \citet{PeskinSchroeder1995} is the
default first course; Ch.~15--17 cover non-abelian gauge theory,
the renormalization group, and asymptotic freedom.
\citet{Srednicki2007} is more careful about anomalies (Ch.~72--77),
and \citet{WeinbergQTF1} is the reference for what a symmetry of an
action does and does not buy. \citet{Zee2010} is the book to read
first if the others feel airless. Original papers:
\citet{YangMills1954}, \citet{Weyl1929},
\citet{WilsonKogut1974}, \citet{GrossWilczek1973} and
\citet{Politzer1973}, and \citet{Elitzur1975} --- the last a single
page that settles what ``breaking a gauge symmetry'' can mean.

\paragraph{The Standard Model and its parameters
(Appendix~\ref{app:codebook}).} \citet{Georgi1999} for the group
theory one actually needs; \citet{PeskinSchroeder1995}, Ch.~20, for
electroweak symmetry breaking; \citet{PDG2024} as the standing
reference for every measured value, its scheme, and its scale ---
a student should learn to read the Review of Particle Physics
before trusting any quoted number, including this paper's.
\citet{Cahn1996} is the essay to read on what the parameters
\emph{do}.

\paragraph{Cosmology (Appendices~\ref{app:launch}
and~\ref{app:dark}).} \citet{Baumann2022} is the best modern course
and is pitched exactly at this appendix's level;
\citet{DodelsonSchmidt2020} for perturbations and the CMB;
\citet{WeinbergCosmology2008} when precision matters;
\citet{KolbTurner1990} remains the reference for relic abundances
and the thermal history. Papers: \citet{Guth1981},
\citet{Linde1982}, \citet{Starobinsky1980} for inflation;
\citet{PecceiQuinn1977}, \citet{Weinberg1978}, \citet{Wilczek1978},
and \citet{PreskillWiseWilczek1983} for the axion;
\citet{Planck2020} for the measured sky.

\paragraph{Discrete symmetries and baryogenesis
(Appendix~\ref{app:discrete}).} \citet{WeinbergQTF1}, Ch.~2 and~5,
for the discrete operators and the $CPT$ theorem;
\citet{PeskinSchroeder1995}, Ch.~3, for their action on Dirac
fields. Papers: \citet{LeeYang1956} and \citet{Wu1957} as a pair,
\citet{Christenson1964}, \citet{KM1973}, \citet{Sakharov1967},
\citet{KRS1985}, and \citet{FukugitaYanagida1986}.

\paragraph{Confinement and the proton
(Appendix~\ref{app:confinement}).} \citet{PeskinSchroeder1995},
Ch.~17, for asymptotic freedom and the infrared;
\citet{Wilson1974} for the confinement criterion in the original;
\citet{GellMann1964} and \citet{Greenberg1964} for how quarks and
color arrived; \citet{JaffeWitten2006} for the formal status of
what remains unproven.

\paragraph{Time, entropy, and horizons
(Appendix~\ref{app:time}).} \citet{Wald1984}, Ch.~12, for horizons
and the area theorem; \citet{Landauer1961} for the toll;
\citet{Bekenstein1973}, \citet{Hawking1975}, \citet{Hawking1976},
and \citet{Page1993} in the original --- four short papers that
between them define the subject. \citet{Penrose1979} is the essay
on why the early universe was special, and remains the sharpest
statement of the arrow's real difficulty.

\paragraph{And, for the program itself.} The cohort papers named
in Part~II's four steps: \citet{ArisakaA1Constitution} for the
axioms, \citet{ArisakaA2GG6} for the bulk and the ledger,
\citet{ArisakaA5GDOS} for the boundary law,
\citet{ArisakaA6Gi4} for the engine, and then the application
paper matching the reader's own field. \citet{ArisakaW1Symmetry} is
this paper's sibling and the better entry point for a reader who
prefers history to structure.

\section{Summary of abbreviations}
\label{app:abbrev}

\small
\begin{longtable}{@{}P{2.4cm}>{\raggedright\arraybackslash}p{\dimexpr\textwidth-2.4cm-2\tabcolsep\relax}@{}}
\caption[Summary of abbreviations.]{Summary of abbreviations used in this paper, each with its expansion and the place it is used. The first block is the program's own apparatus, in the order of the four steps: \GGTheory{} and the two equivalence principles, \GDOS{} and the \GRIP{} engine with its four moves \DGI{}, \GFF{}, \GSSB{} and \GGA{}, the two \GRIP{} regimes, and the parsimony clause \MSMF{}. The remainder runs from the cohort's own structures to the standard abbreviations of the field --- BI-6, Hopkins--Singer, \QiFour{}, RG, CMB, EDM and KMS. Where an entry has an equation of its own, the equation is named, so the table doubles as an index into the paper's numbered results.}
\label{tab:abbrev-w2}\\
\toprule
\textbf{Abbreviation} & \textbf{Expansion, and where it is used} \\
\midrule
\endfirsthead
\multicolumn{2}{c}{\itshape Table~\ref{tab:abbrev-w2} continued}\\
\toprule
\textbf{Abbreviation} & \textbf{Expansion, and where it is used} \\
\midrule
\endhead
\midrule \multicolumn{2}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

\GGTheory{} & the program: geometry--gauge theory on the
six-dimensional bulk (Part~II) \\
\GGEP{} & geometry--gauge equivalence principle (\AX{2}): gravity
and the gauge forces as one composite connection (Step~A) \\
\OGGEP{} & open \GGEP{} (\AX{3}): observation is a projection that
marginalizes the fiber coordinates (Step~B) \\
\GDOS{} & geometric dynamics of open systems: the master equation
Eq.~\eqref{eq:gdos-p2} and its theorems (Step~B) \\
\GRIP{} & the runtime engine, \DGI{} $+$ \GFF{} $+$ \GSSB{} $+$
\GGA{}, Eq.~\eqref{eq:grip-p2} (Step~B) \\
\DGI{} & Dynamical Gauge Invariance: the cascade's opening
move --- every gauge equivalence intact, nothing yet committed \\
\GFF{} & geometric free-energy functional: the Lyapunov landscape
the cascade descends \\
\GSSB{} & geometric spontaneous symmetry breaking: the selection
step \\
\GA{} / \GGA{} & geometric attractor / ground geometric attractor:
way station and unique floor \\
\GRIPin{} & the within-layer regime --- the circle's stroke,
closed-loop relaxation on a fixed landscape \\
\GRIPout{} & the between-layer regime --- the interval's stroke, the
one-way reorganization of the landscape \\
\MSMF{} & minimal structure, maximal function (\AX{4}): the
parsimony clause that forbids convenience ingredients \\
\GS{} & Green--Schwarz: the anomaly-cancellation mechanism, in its
six-dimensional non-supersymmetric instance \\
\BIsix{} & the modified Bianchi identity of the six-force bulk
(A3) \\
\HS{} & Hopkins--Singer: differential-cohomology quantization, the
far end of W1's ladder \\
\QiFour{} & the four information pillars of A7 \\
RG & renormalization group: the flow along the interval,
Eq.~\eqref{eq:rg-w2} \\
CMB & cosmic microwave background \\
EDM & electric dipole moment (the neutron's, in the strong-$CP$
audit) \\
KMS & Kubo--Martin--Schwinger: the thermal-equilibrium condition at
the shore (Problem~14) \\
\end{longtable}

\section{Glossary of key symbols}
\label{app:symbols}

\small
\begin{longtable}{@{}P{2.3cm}>{\raggedright\arraybackslash}p{\dimexpr\textwidth-2.3cm-2\tabcolsep\relax}@{}}
\caption[Glossary of key symbols.]{Glossary of key symbols, with meanings and first use. The geometry comes first: the six-dimensional bulk $\mathcal M_6$ and its two non-spatial factors, the scale interval $\Ir$ with coordinate $r = \ln(\EGUT/\mu)$ on $[0,K]$ and the phase circle $\Stheta$ with coordinate $\theta$. Then the integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$ and what each entry counts: $K = 35$ e-folds of descent, $J = 7$ chiral steps, $N = 3$ generations, $L = 5$ the rank structure. The rest names the quantities the sixteen problems turn on, from the Wilson loop and circle transport through the cap loading $\Xicap$, the baryon-to-photon ratio, the QCD vacuum angle and the Arisaka constant, to the Bekenstein--Hawking entropy and the drainage factor $\zeta_\Lambda$ --- named, bounded, and uncomputed.}
\label{tab:symbols-w2}\\
\toprule
\textbf{Symbol} & \textbf{Meaning, and first use} \\
\midrule
\endfirsthead
\multicolumn{2}{c}{\itshape Table~\ref{tab:symbols-w2} continued}\\
\toprule
\textbf{Symbol} & \textbf{Meaning, and first use} \\
\midrule
\endhead
\midrule \multicolumn{2}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot

$\mathcal M_6$ & the six-dimensional bulk,
$\mathcal M_4 \times \Ir \times \Stheta$, Eq.~\eqref{eq:bulk} \\
$\Ir$ & the scale interval: the fifth dimension, the open fiber;
coordinate $r = \ln(\EGUT/\mu)$, range $[0,K]$ \\
$\Stheta$ & the phase circle: the sixth dimension, the compact
fiber; coordinate $\theta$ \\
$r$ & position along the interval --- the descent's depth, the RG
address \\
$\theta$ & position around the circle --- the gauge phase, the
unreadable pointer \\
$(M,N,L;J,K)$ & the integer ledger $(1,3,5;7,35)$, with $J = 2N+1$
and $K = LJ$, Eq.~\eqref{eq:ledger-p2} \\
$K = 35$ & the depth of the descent in e-folds --- the exponent of
every interval-side scale \\
$J = 7$ & the chiral steps of the flavor alphabet; also the
corridor switch $z_{\mathrm{sw}} = e^{J}$ \\
$N = 3$ & the number of fermion generations \\
$L = 5$ & the rank structure of the unified group; the pentagonal
lattice's order \\
$W(C)$ & the Wilson loop, Eq.~\eqref{eq:wilson-p2} --- the
gauge-invariant content of a connection \\
$U(\theta)$ & circle transport: a group element, invertible,
Eq.~\eqref{eq:group-law-p2} \\
$T(r)$ & interval transport: a semigroup element, not invertible,
Eq.~\eqref{eq:semigroup-law-p2} \\
$\rho$ & the boundary density matrix; $\mathcal D[\rho]$ its
dissipator, Eq.~\eqref{eq:gdos-p2} \\
$\Mlock$ & the lock scale $\tfrac12 M_{\mathrm{Pl}}e^{-K} =
3.849$~TeV, Eq.~\eqref{eq:mlock} \\
$\Mplred$ & the reduced Planck mass $(8\pi G_N)^{-1/2}$,
Eq.~\eqref{eq:mpl-w2} \\
$v$ & the electroweak vacuum value, Eq.~\eqref{eq:vev-p2} \\
$N^{*}$ & the observable e-fold count of the launch, $52.6$ \\
$n_s$, $r_{\mathrm T}$ & the scalar tilt and tensor-to-scalar
ratio, Eq.~\eqref{eq:ns-r} \\
$\Omm, \Omb, \OmL$ & the matter, baryon, and dark-energy shares,
Eq.~\eqref{eq:inventory-p2} \\
$\mK$, $\fK$ & the Kaxion's mass and decay constant,
Eq.~\eqref{eq:kaxion-numbers} \\
$\Xicap$ & the cap loading $K+N+L = 43$ (benchmark; the one open
integer) \\
$\etaB$ & the baryon-to-photon ratio, Eq.~\eqref{eq:etaB} \\
$\bar\theta$ & the QCD vacuum angle (Problem~9) \\
$Q_B$ & the Arisaka constant: the energy price of one bit
(Problem~14) \\
$S_{\mathrm{BH}}$ & the Bekenstein--Hawking entropy,
Eq.~\eqref{eq:bh-entropy-w2} \\
$\zeta_\Lambda$ & the drainage factor of the vacuum seesaw ---
named, bounded, uncomputed \\
\end{longtable}

\section{Holonomy, connections, and the two transport algebras}
\label{app:holonomy}
\GGteachbox{Parallel transport worked on a sphere until the answer
is obvious; the covariant derivative as the bookkeeping that makes
comparison at a distance possible; curvature as the failure of
transport to commute; holonomy, and the theorem that curvature does
not determine it; the Wilson loop as the gauge-invariant content of
a connection; Aharonov--Bohm as the experiment that makes the whole
apparatus physical; and finally the algebraic fork --- group on a
compact fiber, semigroup on an open one --- that the rest of this
course spends. Pay-off: Part~II and every appendix after this
one.}
\GGreadalong{\citet{Nakahara2003}, Ch.~7 and~10, for connections on
fiber bundles; \citet{MTW1973}, Ch.~8 and~11, for the gravitational
case with the pictures drawn; \citet{WuYang1975} for the statement
that the holonomies \emph{are} the gauge field.}

\subsection{Step 1 --- the standard treatment: transport, curvature,
holonomy}
\label{app:A-step1}

\paragraph{Transport on a sphere.} Stand at the north pole holding
an arrow that points down the Greenwich meridian. Walk south to the
equator, keeping the arrow as parallel to itself as the surface
permits --- never twisting it about the local vertical. Turn left,
walk a quarter of the way round the equator, then walk north again
to the pole. The arrow now points ninety degrees from where it
started. Nothing rotated it: at every step it was carried as
parallel as the geometry allowed, and it came back turned.

The general statement is that transport around a closed circuit
$C$ bounding a region $\Sigma$ rotates a tangent vector on the unit
sphere by the enclosed solid angle,
\begin{equation}
\Delta\alpha \;=\; \Omega(\Sigma) \;=\; \oint_{C}(1-\cos\vartheta)\,
d\varphi ,
\label{eq:A-solid}
\end{equation}
with $\vartheta$ the polar angle. For the octant walked above,
$\Omega = 4\pi/8 = \pi/2$, a quarter turn. Two features of
Eq.~\eqref{eq:A-solid} carry into everything below: the answer
depends on the \emph{loop}, not on any point of it, and it is
insensitive to how the loop is deformed as long as the enclosed
region's curvature integral is unchanged.

\paragraph{Connection and covariant derivative.} To compare a field
$\psi$ at neighboring points one needs a rule for transporting the
internal frame. Writing that rule as a connection one-form $A_\mu$,
the derivative that respects it is
\begin{equation}
D_\mu \;=\; \partial_\mu \;-\; \frac{i q}{\hbar}\,A_\mu ,
\label{eq:A-covd}
\end{equation}
where $q$ is the charge --- the strength with which $\psi$'s phase
is geared to the connection. Finite transport along a path $\gamma$
is the path-ordered exponential
\begin{equation}
U(\gamma) \;=\; \mathcal P \exp\!\left(\frac{iq}{\hbar}
\int_\gamma A_\mu\, dx^\mu\right),
\label{eq:A-transport}
\end{equation}
which is Eq.~\eqref{eq:wilson-p2} evaluated on an open path rather
than a loop.

\paragraph{Curvature is the failure of transport to commute.}
Transport around an infinitesimal parallelogram in the
$\mu\nu$-plane and compare with the reverse order:
\begin{equation}
[D_\mu, D_\nu] \;=\; -\frac{iq}{\hbar}\,F_{\mu\nu},
\qquad
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
\;\;(+\;\text{commutator, non-abelian}).
\label{eq:A-curv}
\end{equation}
Curvature is therefore not an extra postulate but the
\emph{definition} of how badly transport depends on the route.

\paragraph{Holonomy, and why curvature does not exhaust it.} For a
closed loop, Eq.~\eqref{eq:A-transport} becomes the holonomy
$W(C)$ of Eq.~\eqref{eq:wilson-p2}. In the abelian case Stokes'
theorem gives $W(C) = \exp(iq\Phi/\hbar)$ with $\Phi$ the enclosed
flux, so one might conclude that holonomy is merely an integral of
curvature. It is not. If the region enclosed is not simply
connected --- if the loop wraps something --- there exist
configurations with $F_{\mu\nu} = 0$ everywhere along the loop and
$W(C) \ne 1$. The Aharonov--Bohm phase of Eq.~\eqref{eq:ab-p2} is
exactly this case, and it is measured. Holonomy is strictly more
information than curvature; it is the connection's
\emph{topological} content.

\paragraph{Worked example: the flux quantum.} Demand that the
wavefunction of a charge $q$ encircling a solenoid be
single-valued. Then $q\Phi/\hbar$ must be a multiple of $2\pi$,
so the physically distinguishable fluxes are
\begin{equation}
\Phi \;=\; n\,\frac{h}{q},
\qquad n \in \mathbb{Z} .
\label{eq:A-fluxquantum}
\end{equation}
The observable is not the flux but the flux modulo $h/q$, and the
integer $n$ counting the quanta passed is protected: no continuous
deformation of the field can change an integer.

\paragraph{The bundle, defined once.} The object all of this
lives on is a \emph{fiber bundle}: a base manifold $B$, a fiber
$F$, a total space $E$, and a projection $\pi: E \to B$ such that
every point of $B$ has a neighborhood $U$ with
$\pi^{-1}(U) \cong U \times F$. The last clause is the whole
subtlety --- the bundle looks like a product \emph{locally}, and
may fail to be one globally. Two local trivializations over
overlapping patches are related by a transition function
$t_{ij}: U_i \cap U_j \to G$ valued in the structure group, and it
is the transition functions, not the fibers, that carry the
topology. For $G = \mathrm{U}(1)$ over a two-sphere the possible
bundles are labeled by an integer --- the first Chern number ---
and that integer is the monopole charge, which is
Eq.~\eqref{eq:A-fluxquantum} arriving by a different road.

\emph{Physics intuition.} A gauge field is not a field \emph{on}
spacetime in the way a temperature is; it is a rule for comparing
internal frames at different points, and the rule can be globally
inconsistent in ways no local measurement reveals. Everything
protected in this paper is a global inconsistency counted by an
integer.

\paragraph{Parallel transport as an ordinary differential
equation.} Given a curve $x^\mu(\lambda)$ and a connection, the
transported frame $U(\lambda)$ obeys
\begin{equation}
\frac{dU}{d\lambda}
\;=\; \frac{iq}{\hbar}\,A_\mu\,\frac{dx^\mu}{d\lambda}\,U,
\qquad U(0) = \mathbb{1},
\label{eq:A-transportODE}
\end{equation}
whose solution is Eq.~\eqref{eq:A-transport}. For a vector on a
Riemannian manifold the same equation reads
$dV^a/d\lambda + \Gamma^a{}_{bc}V^b\,dx^c/d\lambda = 0$ with
$\Gamma$ the Christoffel symbols. Nothing is postulated here: given
a rule for infinitesimal comparison, finite comparison is its
integral, and the integral is path-dependent whenever the rule
fails to commute with itself.

\paragraph{The sphere, computed rather than described.} On the unit
sphere with metric $ds^2 = d\vartheta^2 + \sin^2\!\vartheta\,
d\varphi^2$, the non-vanishing Christoffel symbols are
$\Gamma^{\vartheta}{}_{\varphi\varphi} = -\sin\vartheta
\cos\vartheta$ and $\Gamma^{\varphi}{}_{\vartheta\varphi} =
\cot\vartheta$. Transport a vector along a circle of constant
$\vartheta$: writing the vector's angle relative to the local
meridian as $\alpha(\varphi)$, the transport equation reduces to
$d\alpha/d\varphi = -\cos\vartheta$, so after a full circuit
\begin{equation}
\Delta\alpha \;=\; -2\pi\cos\vartheta
\;\;\equiv\;\; 2\pi(1-\cos\vartheta) \ \ (\mathrm{mod}\ 2\pi),
\label{eq:A-latitude}
\end{equation}
which is Eq.~\eqref{eq:A-solid} for this loop and reproduces the
walked octant when the path is assembled from meridians and a
quarter-equator. The reader is urged to do this once by hand
(Problem~A.1); it converts holonomy from a definition into a fact
about the world.

\paragraph{The non-abelian case, and why the exponential is
ordered.} When the connection is matrix-valued, transports along
successive segments multiply in order, and the values at different
points need not commute. Slicing the path into $N$ pieces,
\begin{equation}
U(\gamma) = \lim_{N\to\infty}
\prod_{k=N}^{1}
\exp\!\left(\frac{iq}{\hbar}A_\mu(x_k)\,\Delta x^\mu_k\right)
\;\equiv\;
\mathcal P \exp\!\left(\frac{iq}{\hbar}\int_\gamma A\right),
\label{eq:A-pathorder}
\end{equation}
the product taken with later points to the left. The symbol
$\mathcal P$ is not decoration: it is the instruction that makes
Eq.~\eqref{eq:A-transport} a solution of
Eq.~\eqref{eq:A-transportODE} rather than a naive exponential.

\paragraph{How much holonomy does curvature determine?} The
Ambrose--Singer theorem answers precisely: the Lie algebra of the
holonomy group is spanned by the curvature $F_{\mu\nu}$ transported
to a base point, over all points of the manifold. On a
\emph{simply connected} space this makes holonomy a function of
curvature. On a space with non-contractible loops it does not: the
theorem constrains the connected component of the holonomy group,
and leaves the discrete part --- the winding --- untouched. That
gap is exactly where Aharonov--Bohm lives, and where every
protected integer in this paper is stored.

\paragraph{Aharonov--Bohm, quantitatively.} A two-slit interferometer
of slit separation $a$ and screen distance $D$, with a solenoid of
flux $\Phi$ threaded between the arms, has fringes displaced by
\begin{equation}
\delta y \;=\; \frac{\lambda D}{a}\,\frac{\Delta\varphi}{2\pi}
\;=\; \frac{\lambda D}{a}\,\frac{q\Phi}{h},
\label{eq:A-ABshift}
\end{equation}
with $\Delta\varphi$ from Eq.~\eqref{eq:ab-p2} and $\lambda$ the de
Broglie wavelength. A whole-fringe shift therefore requires exactly
one flux quantum $h/q$. \emph{Physics intuition:} the electron
never visits the field; what it visits is a region where the
\emph{connection} is non-trivial, and the interferometer reads the
loop integral because interference is, by construction, a
comparison of two paths that meet again.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:A-step2}

Three gaps, each of which becomes one of Part~I's sixteen.

\emph{First, the status of the connection.} Textbooks introduce
$A_\mu$ as a device that repairs the non-invariance of
$\partial_\mu\psi$ under a position-dependent phase change, then
observe that the repair \emph{predicts} electromagnetism. But a
device introduced to fix a bookkeeping problem is not supposed to
dictate a force. What the compact direction \emph{is}, physically,
is not answered --- only that the mathematics behaves as though one
existed (Problem~3).

\emph{Second, the origin of the integer.} Eq.~\eqref{eq:A-fluxquantum}
required single-valuedness around a loop that encircles something.
Where in the world is that loop? In four-dimensional field theory
the compactness is imported --- as a solenoid, as a boundary
condition, as a choice of gauge group --- never derived. Charge
quantization is then observed rather than explained (Problem~4).

\emph{Third, the arrow.} Nothing in
Eqs.~\eqref{eq:A-covd}--\eqref{eq:A-curv} distinguishes a direction
in which transport is irreversible. Yet physics is full of
transports that run one way: coarse-graining, measurement, the
cosmic descent. The standard formalism supplies a group of
transports and then imports irreversibility from outside ---
initial conditions, tracing, thermodynamic assumptions
(Problems~1, 13, 14).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:A-step3}

The anatomy answers all three by making the compactness a fact
about the bulk and then asking what the same formalism does on the
\emph{other} fiber.

\paragraph{On the circle.} $\Stheta$ is compact, so
$\pi_1(\Stheta) = \mathbb{Z}$: loops come in homotopy classes
labeled by an integer winding. Every path around it closes, so
every transport is a holonomy; the holonomies form a
\emph{group}, Eq.~\eqref{eq:group-law-p2}, in which each element
has an inverse; and the winding integer is protected exactly as in
Eq.~\eqref{eq:A-fluxquantum} --- not by a dynamical mechanism but
because $\mathbb{Z}$ is discrete and nothing continuous can move
between its points. This is why charge is quantized without a
solenoid, why anomaly coefficients cannot run, and why class data
are the same in every region of the sky.

\paragraph{On the interval.} $\Ir = [0,K]$ is contractible and has
boundary: $\pi_1(\Ir)$ is trivial, and there is no closed circuit
along it. Whatever a connection does there carries no
gauge-invariant loop content --- the precise sense of
``the interval forgets.'' The physical transport along $r$ is the
renormalization flow / cosmic descent, which composes only forward:
\begin{equation}
T(r_2)\,T(r_1) = T(r_1+r_2)\ \ (r_i \ge 0),
\qquad
T(r)\ \text{has no inverse},
\label{eq:A-semigroup}
\end{equation}
Eq.~\eqref{eq:semigroup-law-p2} again. The absence of an inverse is
not a modeling choice: coarse-graining destroys the information
needed to undo it, and no operation restores what was never kept.

\paragraph{The fork, stated once.} A connection on a compact fiber
gives a group of protected classes; the same connection on an open
fiber gives a semigroup of final acts. One geometry, two algebras,
and --- the claim of this paper --- sixteen problems sorted by
which algebra their subject matter lives in.

\paragraph{The connection, decomposed by fiber.} Write the
bulk connection in coordinates adapted to
Eq.~\eqref{eq:bulk}:
\begin{equation}
A \;=\; A_\mu(x,r,\theta)\,dx^\mu
\;+\; A_r(x,r,\theta)\,dr
\;+\; A_\theta(x,r,\theta)\,d\theta .
\label{eq:A-decomp}
\end{equation}
Three statements follow, and they are the technical content of
``the circle remembers; the interval forgets.''

\emph{(i) The $dr$ component carries nothing.} Because $\Ir$ is an
interval --- contractible, with boundary --- one may always choose
the axial gauge $A_r = 0$ by a gauge transformation
$g(x,r,\theta) = \mathcal P\exp\!\big(\tfrac{iq}{\hbar}\int_0^r
A_{r'}\,dr'\big)$, which is globally well defined precisely because
the integration path is unique up to homotopy. No residual
invariant survives: there is no closed circuit in $r$ on which to
evaluate anything.

\emph{(ii) The $d\theta$ component carries a modulus that cannot be
gauged away.} On the circle the analogous transformation would
require $g(\theta + 2\pi) = g(\theta)$, so the available shifts of
the Wilson line
\begin{equation}
W_\theta \;=\; \mathcal P
\exp\!\left(\frac{iq}{\hbar}\oint A_\theta\,d\theta\right)
\;=\; e^{\,i\alpha}
\label{eq:A-wilsonline}
\end{equation}
are quantized, $\alpha \to \alpha + 2\pi n$. The residual
$\alpha \in [0,2\pi)$ is a physical, gauge-invariant modulus --- a
coordinate on the space of flat connections --- and it is exactly
what Problem~A.5 computes and what the Hosotani mechanism
\citep{Hosotani1983} makes dynamical.

\emph{(iii) The classification is topological, not dynamical.} Flat
connections on $\Stheta$ modulo gauge transformations form the
space $\mathrm{Hom}(\pi_1(S^1), G)/G = G/\mathrm{Ad}(G)$, which for
$\mathrm{U}(1)$ is the circle of values $\alpha$; on $\Ir$ the same
construction gives $\mathrm{Hom}(\{1\}, G)/G$ --- a single point.
One fiber has a moduli space of protected data; the other has
nothing to have.

\paragraph{The semigroup, constructed rather than asserted.} That
interval transport lacks inverses can be shown explicitly with the
oldest construction in the subject. Kadanoff blocking on a lattice
of spacing $a$ replaces each block of $b^{\,d}$ sites by one
effective site with spacing $ba$, defining a map
$\mathcal R_b$ on the space of couplings $\{g\}$:
\begin{equation}
\mathcal R_{b_2} \circ \mathcal R_{b_1}
\;=\; \mathcal R_{b_1 b_2},
\qquad
\mathcal R_b^{-1}\ \text{does not exist} .
\label{eq:A-kadanoff}
\end{equation}
The composition law is exact --- blocking twice is blocking once at
the product ratio --- and the inverse fails for a reason that is
not technical: many microscopic configurations map to the same
block configuration, so $\mathcal R_b$ is not injective (Problem
A.4 makes this concrete in two states). Eq.~\eqref{eq:A-kadanoff}
\emph{is} Eq.~\eqref{eq:semigroup-law-p2} with $r = \ln b$, and it
is why the fifth dimension's coordinate is logarithmic in scale:
the natural composition is multiplication of ratios, whose
additive coordinate is the logarithm.

\paragraph{What the two structures then buy, itemized.} Every
exactness claim in this paper traces to the integrality in (ii) and
(iii) --- the quantized shifts and the winding classes, \emph{not} the
continuous residual $\alpha$, which is a modulus and does move --- and
every irreversibility claim to Eq.~\eqref{eq:A-kadanoff}. Charge
quantization: the winding integer of
$\pi_1(\Stheta) = \mathbb{Z}$. Anomaly coefficients that cannot
run: integers labeling bundles, and $\mathcal R_b$ acts trivially
on $\mathbb{Z}$ because it is continuous and $\mathbb{Z}$ is
discrete. Class data identical across the sky: a homotopy class is
not a local field and has no gradient to vary. Measurement's
finality, entropy's growth, the descent's one traverse: the missing
inverse in Eq.~\eqref{eq:A-kadanoff}. Two lines of topology, one
line of algebra, and the sixteen problems sorted.

\subsection{Step 4 --- problems and worked solutions}
\label{app:A-problems}

\medskip\noindent\textbf{A.1. Verify Eq.~\eqref{eq:A-solid} for a
loop of constant latitude $\vartheta$, and show that a small loop
sees a flat world.}

\noindent\emph{Solution.} \emph{Step 1.} At fixed $\vartheta$,
$\oint(1-\cos\vartheta)d\varphi = 2\pi(1-\cos\vartheta)$.
\emph{Step 2.} Expanding for small $\vartheta$,
$2\pi(1-\cos\vartheta) \to \pi\vartheta^2$, which is the enclosed
area. \emph{Step 3.} So the holonomy is second order in the loop
size --- curvature times area. \emph{Physics intuition:} local
experiments cannot detect curvature, which is precisely why
curvature had to be discovered by comparing distant transports
rather than by looking closely at one point.

\medskip\noindent\textbf{A.2. Show that $F_{\mu\nu}$ of
Eq.~\eqref{eq:A-curv} is gauge covariant but not gauge invariant in
the non-abelian case, and identify what is invariant.}

\noindent\emph{Solution.} \emph{Step 1.} Under a gauge
transformation $A_\mu \to U A_\mu U^{-1} +
(i\hbar/q)U\partial_\mu U^{-1}$ one finds $F_{\mu\nu} \to
U F_{\mu\nu}U^{-1}$. \emph{Step 2.} That is covariance: $F$ changes
unless $U$ commutes with it, so $F$ itself is not observable.
\emph{Step 3.} Traces --- $\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$ ---
and the Wilson loop Eq.~\eqref{eq:wilson-p2} are invariant.
\emph{Physics intuition:} in electromagnetism the group is
one-dimensional, covariance collapses to invariance, and one may
treat $\vec E$ and $\vec B$ as observables directly; Yang--Mills
denies that convenience, which is why the loop language is the
right one in general.

\medskip\noindent\textbf{A.3. Two configurations outside a solenoid
have identical $F$ everywhere but fluxes differing by $h/q$. Are
they distinguishable?}

\noindent\emph{Solution.} \emph{Step 1.} By Eq.~\eqref{eq:ab-p2}
the interference phases differ by $2\pi$. \emph{Step 2.} A phase of
$2\pi$ is no phase at all. \emph{Step 3.} They are
\emph{indistinguishable} --- the content of
Eq.~\eqref{eq:A-fluxquantum}. \emph{Physics intuition:} the world
does not record the flux; it records the flux modulo a quantum, and
the integer left over is the only part that cannot be deformed
away. Every protected quantity in this paper is an integer of this
type.

\medskip\noindent\textbf{A.4. Show explicitly that a
coarse-graining map has no inverse, using a two-state example.}

\noindent\emph{Solution.} \emph{Step 1.} Let the map send both
$|0\rangle\langle 0|$ and $|1\rangle\langle 1|$ to the maximally
mixed state $\tfrac12\mathbb{1}$. \emph{Step 2.} Two distinct
inputs share one output, so the map is not injective.
\emph{Step 3.} A non-injective map has no inverse; the family
$\{T(r)\}$ built from such maps is therefore a semigroup, not a
group, which is Eq.~\eqref{eq:A-semigroup}. \emph{Physics
intuition:} irreversibility in this paper is never a statistical
approximation to reversible microphysics; it is the failure of
injectivity, and failures of injectivity do not become injective
when one looks more carefully.

\medskip\noindent\textbf{A.5. A circle of circumference $2\pi R$
carries a flat connection with holonomy $e^{i\alpha}$. Show that
$\alpha$ cannot be removed by a gauge transformation, and identify
what physical effect it has.}

\noindent\emph{Solution.} \emph{Step 1.} A gauge transformation
$e^{i\chi(\theta)}$ must be single-valued on the circle, so
$\chi(2\pi)-\chi(0) \in 2\pi\mathbb{Z}$. \emph{Step 2.} It can
therefore shift $\alpha$ only by multiples of $2\pi$, not to zero.
\emph{Step 3.} The residual $\alpha$ is physical: it shifts the
Kaluza--Klein spectrum of charged fields to
$m_n \propto (n + \alpha/2\pi)/R$. \emph{Physics intuition:} this
is the Wilson-line (Hosotani) vacuum of Problem~2 --- a circle-side
label with laboratory consequences, and the mechanism by which a
compact direction sets a mass scale without any field being given a
potential by hand.

\GGtakeaway{A connection \emph{is} its holonomies; holonomies need
loops; loops need compactness. On a compact fiber that yields a
group of protected integers, on an open fiber a semigroup of final
acts. Every later appendix is an application of this one
paragraph.}

\section{Open quantum systems, decoherence, and the write}
\label{app:openquantum}
\GGteachbox{The density matrix and the partial trace, derived; the
Lindblad master equation and why its form is forced by complete
positivity; decoherence computed, with the $(\Delta x)^2$ scaling
that Figure~\ref{fig:decoherence-scaling-w2} plots; the measurement
postulates as they are actually taught; and then the honest
accounting of what decoherence does and does not deliver --- and
what the two-fiber reading adds. Pay-off: Problem~1.}
\GGreadalong{\citet{NielsenChuang2000}, Ch.~2 and~8, for density
matrices and quantum operations; \citet{BreuerPetruccione2002},
Ch.~3, for the Lindblad derivation; \citet{Zurek2003} for the
decoherence program in the author's own words.}

\subsection{Step 1 --- the standard treatment: from pure states to
master equations}
\label{app:B-step1}

\paragraph{Why a density matrix.} A system entangled with anything
else has no state vector of its own. Given a joint pure state
$|\Psi\rangle$ of system $S$ and environment $E$, all predictions
for measurements on $S$ alone are reproduced by the reduced density
matrix
\begin{equation}
\rho_S \;=\; \mathrm{Tr}_E\,|\Psi\rangle\langle\Psi| ,
\label{eq:B-ptrace}
\end{equation}
the partial trace. Two facts about Eq.~\eqref{eq:B-ptrace} matter
for everything below: it is the unique map reproducing all local
expectation values, and it is \emph{not invertible} --- distinct
joint states routinely give the same $\rho_S$.

\paragraph{The forced form of the dynamics.} Ask what the most
general time evolution of $\rho_S$ can be, subject only to: it
preserves trace (probabilities sum to one), it preserves positivity
even when $S$ is entangled with a spectator (complete positivity),
and it is Markovian (no memory). The answer --- the
Gorini--Kossakowski--Sudarshan--Lindblad theorem --- is unique in
form:
\begin{equation}
\frac{d\rho}{dt}
= -\frac{i}{\hbar}[H,\rho]
+ \sum_k \gamma_k\!\left(L_k \rho L_k^\dagger
- \tfrac12\{L_k^\dagger L_k, \rho\}\right),
\label{eq:B-lindblad}
\end{equation}
with $\gamma_k \ge 0$. This is Eq.~\eqref{eq:gdos-p2} written with
its dissipator opened up: $H$ generates the reversible part, the
jump operators $L_k$ (with rates $\gamma_k$) generate the
irreversible part. The positivity of every $\gamma_k$ is the whole
story of the arrow at this level --- reverse the sign of one and
probabilities go negative.

\paragraph{Decoherence, computed.} Take $S$ a massive particle
coupled to a scattering environment, with $L \propto \hat x$. Then
Eq.~\eqref{eq:B-lindblad} gives, in the position basis,
\begin{equation}
\frac{\partial}{\partial t}\,\rho(x,x')
\;=\; \cdots \;-\; \Lambda\,(x-x')^2\, \rho(x,x') ,
\qquad
\rho(x,x',t) \sim e^{-\Lambda (\Delta x)^2 t},
\label{eq:B-decoherence}
\end{equation}
where $\Lambda$ is the scattering constant of the environment
(collision rate times momentum transfer squared, in units of
$\hbar^2$). The diagonal --- the probabilities --- is untouched;
the off-diagonal --- the interference --- decays at a rate
$\Gamma = \Lambda(\Delta x)^2$, quadratic in the separation of the
superposed alternatives. That quadratic law is the line plotted in
Figure~\ref{fig:decoherence-scaling-w2}, and it is why a
dust grain superposed over a micron loses coherence in
$\sim 10^{-24}$~s while an electron over an \AA ngstr\"om can hold
it for seconds. (Both numbers use $\Lambda$ in SI units; the
standard tabulations are quoted in $\mathrm{cm}^{-2}\mathrm{s}^{-1}$,
and the four-order conversion is the commonest slip in this
calculation.)

\paragraph{The measurement postulates, as taught.} Alongside
Eq.~\eqref{eq:B-lindblad} the textbook still states, separately:
observables are Hermitian operators; between measurements states
evolve by Eq.~\eqref{eq:schrodinger-w2}; a measurement yields
eigenvalue $k$ with probability
$p_k = \mathrm{Tr}(P_k \rho)$ and updates the state by
Eq.~\eqref{eq:lueders-w2}.

\paragraph{Quantum channels, and the theorem that constrains
them.} Before the master equation there is a more primitive
question: what maps $\rho \to \mathcal E(\rho)$ are physically
allowed? The answer --- Kraus' theorem --- is that
$\mathcal E$ is completely positive and trace preserving if and
only if it can be written
\begin{equation}
\mathcal E(\rho) \;=\; \sum_k E_k\,\rho\,E_k^\dagger,
\qquad
\sum_k E_k^\dagger E_k = \mathbb{1},
\label{eq:B-kraus}
\end{equation}
for some operators $E_k$. The proof is constructive: couple the
system to an environment, evolve unitarily, and trace --- the
$E_k$ are the matrix elements of the joint unitary in an
environment basis. Every open evolution in physics has this form,
and Eq.~\eqref{eq:B-lindblad} is its infinitesimal version
(Problem~B.6).

\emph{Physics intuition.} Complete positivity is not a technicality.
It is the demand that the map remain sensible when the system is
merely \emph{part} of a larger entangled state --- and it is what
forbids the time-reversed dissipator, because running
Eq.~\eqref{eq:B-kraus} backwards generically produces negative
probabilities on some entangled input.

\paragraph{From a microscopic model to Eq.~\eqref{eq:B-lindblad}.}
Take $H = H_S + H_E + \lambda H_{\mathrm{int}}$ with
$H_{\mathrm{int}} = \sum_\alpha S_\alpha \otimes E_\alpha$. In the
interaction picture, expand to second order in $\lambda$, trace
over the environment, and impose two approximations: \emph{Born}
(the environment is large, so $\rho_{SE} \approx \rho_S \otimes
\rho_E$ throughout) and \emph{Markov} (environment correlation
functions $\langle E_\alpha(t)E_\beta(t')\rangle$ decay fast
compared with the system's evolution). The result is the
Redfield equation, and a final secular average over fast phases
brings it to Lindblad form with
\begin{equation}
\gamma_{\alpha\beta}(\omega) = \int_{-\infty}^{\infty} dt\;
e^{i\omega t}\,\langle E_\alpha^\dagger(t) E_\beta(0)\rangle ,
\label{eq:B-rates}
\end{equation}
the Fourier transform of the environment correlation function.
Positivity of $\gamma$ follows from Bochner's theorem: a
correlation function's transform is a positive measure. The arrow
is therefore inherited from the environment's \emph{spectrum},
which is the first hint that it is structural rather than
dynamical.

\paragraph{A two-level model, solved completely.} Let
$H_S = \tfrac{\hbar\omega_0}{2}\sigma_z$ and take a single
dephasing channel $L = \sqrt{\gamma}\,\sigma_z$. Then
Eq.~\eqref{eq:B-lindblad} separates:
\begin{equation}
\dot\rho_{00} = \dot\rho_{11} = 0,
\qquad
\dot\rho_{01} = \left(-i\omega_0 - 2\gamma\right)\rho_{01}
\;\Longrightarrow\;
\rho_{01}(t) = \rho_{01}(0)\,e^{-i\omega_0 t}\,e^{-2\gamma t}.
\label{eq:B-twolevel}
\end{equation}
Populations are untouched; coherences decay exponentially. This is
the whole of decoherence in four lines, and every elaboration ---
position bases, continuous variables, many environments --- is a
dressing of Eq.~\eqref{eq:B-twolevel}.

\paragraph{Where the $(\Delta x)^2$ comes from.} For a particle
scattering off environmental quanta of typical momentum transfer
$\hbar k$, each collision multiplies the off-diagonal element by
$\langle e^{ik(x-x')}\rangle$. For small separations this is
$1 - \tfrac12\langle k^2\rangle(x-x')^2$, so with a collision rate
$\Gamma_{\mathrm{coll}}$ the damping constant of
Eq.~\eqref{eq:B-decoherence} is
$\Lambda \simeq \tfrac12\Gamma_{\mathrm{coll}}\langle k^2\rangle$.
\emph{Physics intuition:} decoherence is not the environment
``disturbing'' the system --- the recoil is negligible --- but the
environment \emph{learning} which path was taken; the rate is set
by how well one scattering distinguishes them, which is why it is
quadratic in the separation and saturates once a single collision
resolves the two alternatives.

\paragraph{The einselection criterion.} Among all bases, the
pointer basis is the one whose states are least damaged by the
interaction: operationally, those commuting with
$H_{\mathrm{int}}$, or --- when the self-Hamiltonian competes ---
those minimizing the entropy production rate over time
(Zurek's predictability sieve). For $H_{\mathrm{int}}$ a function
of position, position wins, which is why the classical world looks
spatial.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:B-step2}

Decoherence is a genuine and quantitative success --- and it is
important to say exactly what it does not do, because the honest
list is short and sharp.

\emph{It does not select the basis without help.} The calculation
above assumed $L \propto \hat x$: position was put in by hand
through the form of the coupling. The predictability-sieve argument
justifies that choice from the interaction Hamiltonian --- but the
interaction Hamiltonian is itself part of the description whose
structure is being asked about.

\emph{It does not produce the Born measure.} $p_k =
\mathrm{Tr}(P_k\rho)$ is used to interpret $\rho$ from the start.
Nothing in Eq.~\eqref{eq:B-lindblad} derives the squared-amplitude
rule; the equation is written in a language that presupposes it.

\emph{It does not produce a single outcome.} Eq.~\eqref{eq:B-decoherence}
turns a coherent superposition into a diagonal ensemble; an
ensemble is not an event. The transition from ``these are the
odds'' to ``this one happened'' remains outside the formalism.

\emph{It does not say where the environment came from.} The split
between $S$ and $E$ in Eq.~\eqref{eq:B-ptrace} is a choice of
description, and the theory is silent on why the world offers one.
This is Problem~1 in its sharpest technical form.

\subsection{Step 3 --- the two-fiber resolution}
\label{app:B-step3}

\paragraph{The environment is named, and it is not optional.}
\OGGEP{} makes the partial trace of Eq.~\eqref{eq:B-ptrace}
compulsory rather than elective: the boundary observer cannot
resolve the fiber coordinates $(r,\theta)$, so every registration
marginalizes them (Figure~\ref{fig:bulk-projection-w2}). The
``environment'' is not a modeling convenience but the two hidden
directions, and the projection is always on. That is why
Eq.~\eqref{eq:B-lindblad} is not an effective description of a
closed system we are too clumsy to track, but the exact form of
boundary dynamics.

\paragraph{The two terms are the two fibers.} The commutator is the
circle's transport --- Eq.~\eqref{eq:group-law-p2}, invertible,
phase-carrying. The dissipator is the interval's ---
Eq.~\eqref{eq:semigroup-law-p2}, one-way, record-forming. The
$\gamma_k \ge 0$ that the Lindblad theorem imposes as a consistency
condition is, on the anatomy, a statement about which fiber the
term rides: transports along an open fiber cannot have inverses, so
their rates cannot be negative.

\paragraph{The basis, the measure, and the outcome.} The preferred
basis is the one the projection can reach --- the quantities that
survive marginalizing $(r,\theta)$, which is why position-like and
class-like variables are the classical ones. The Born measure is
produced rather than assumed: the dissipator relaxes configurational
statistics onto $|\langle k|\psi\rangle|^2$ monotonically, at a rate
the port coupling sets, and the chain is closed at the relaxation
level with exactly one named residue --- the field-grade
representation step --- declared open (grade: derived-conditional).
And the single outcome is the write itself: an interval-side deposit
with no inverse, which is what
Eq.~\eqref{eq:lueders-w2} describes in its completed limit.

\paragraph{What this buys, and what it costs.} It buys a
measurement account with no extra postulate and one honest residue;
it costs an exposure --- if Born statistics are relaxed into rather
than axiomatic, the largest cosmological scales should carry the
computable deficit of Eq.~\eqref{eq:f25-w2}, and exact Born
statistics at all scales bounds the relaxation time from below.

\paragraph{The projection, written as a channel.} Make the
claim of \S\ref{app:B-step3} explicit. A boundary observer's state
is obtained from the bulk state by integrating out the fiber
coordinates,
\begin{equation}
\rho_{4}(x,x') \;=\; \int_{\Ir}\!dr \oint_{\Stheta}\! d\theta\;
\rho_{6}\big((x,r,\theta),(x',r,\theta)\big),
\label{eq:B-fibretrace}
\end{equation}
which is Eq.~\eqref{eq:B-ptrace} with the environment named. By
Kraus' theorem Eq.~\eqref{eq:B-kraus} this map is a channel, and by
the argument of Problem~B.1 it is not injective: bulk states
differing only in their fiber dependence give the same $\rho_4$.
\OGGEP{} says this integration is not optional --- there is no
boundary protocol that resolves $(r,\theta)$ --- so the projection is
compulsory rather than a modeling convenience. Lindblad form then
follows once the third hypothesis of \S\ref{app:B-step1} is supplied,
and the honest register requires naming it: the boundary dynamics must
be Markovian, which here is a physical statement about how fast the
fiber correlations decay and \emph{not} a consequence of the
projection. The three steps are distinct and all three are needed ---
non-injectivity gives a channel, \OGGEP{} makes the channel
compulsory, Markovianity gives it a generator --- and the third is the
one a referee should press hardest, since non-Markovian open dynamics
is generic. With it, the two terms of Eq.~\eqref{eq:gdos-p2} are the
two fibers' contributions to Eq.~\eqref{eq:B-rates}.

\paragraph{Which fiber supplies which term.} The assignment is
\emph{algebraic}, and one tempting way to argue it is wrong, so this
paragraph disposes of the temptation before making the argument that
works.

\emph{The spectral argument does not run here.} It is natural to say
that a compact fiber has a discrete spectrum and therefore cannot
supply the continuum of final states irreversibility requires, while
an open fiber can. That would be a clean derivation if $\Ir$ were a
half-line. It is not: $\Ir = [0,K]$ is a finite segment with boundary,
and under any self-adjoint boundary condition its spectrum is
discrete --- this is the particle in a box. Worse for the argument,
its level spacing is not small but the \emph{largest} in the
framework, because it is the Kaluza--Klein spacing that puts the lock
cluster at $3.849$~TeV and gives Problem~2 its collider target. A bath
whose levels are that far apart is no drain at all. The two fibers are
a circle and a segment: both compact, both with discrete spectra, and
no spectral property distinguishes them. The paper says so here rather
than leaving it for a referee, because the slogan ``a perfect memory
and a perfect drain'' is an image and not a theorem.

\emph{What does distinguish them is the composition law}, and it needs
no spectral input at all. Transport around $\Stheta$ has an inverse,
Eq.~\eqref{eq:group-law-p2}; the physical flow along $\Ir$ does not,
Eq.~\eqref{eq:A-kadanoff}, because blocking is not injective. A
generator built from a one-parameter family closed under inversion is
anti-Hermitian, and contributes the commutator of
Eq.~\eqref{eq:B-lindblad}; a generator built from a family that admits
no inverse is what that same equation calls a dissipator, and its
rates are non-negative for the same reason the family is one-way. That
is the technical content of ``the commutator is the circle's and the
dissipator is the interval's,'' it is premise (D) of
\S\ref{sec:primer} doing exactly the work premise (T) cannot, and it
is a statement about algebra rather than about level spacings.

\emph{Physics intuition.} Irreversibility needs information to go
somewhere it cannot be retrieved from, and there are two ways to
arrange that. One is to make the somewhere large: the ordinary bath,
where recurrence is merely slow, where the argument is statistical,
and where the conclusion is approximate --- which is precisely the
weakness this paper charges against decoherence. The other is to make
the retrieval \emph{structurally unavailable}, which is what \OGGEP{}
does: the boundary observer has no protocol that resolves
$(r,\theta)$, so the trace of Eq.~\eqref{eq:B-fibretrace} is
compulsory rather than convenient. The interval is not a better bath
than the circle. It is the fiber along which the compulsory projection
has nothing to give back --- because, by (T), there was never a
gauge-invariant record there to return.

\paragraph{The Born measure as a relaxation, and its residue.}
Write the boundary's configurational weight over outcomes as
$\{w_k\}$ and let $p_k = |\langle k|\psi\rangle|^2$ denote the
squared amplitudes. The boundary law's dissipation produces a
gradient flow on the simplex,
\begin{equation}
\dot w_k \;=\; -\Gamma\,\frac{\partial}{\partial w_k}
\,D(w\,\|\,p),
\qquad
D(w\|p) = \sum_k w_k \ln\frac{w_k}{p_k} \;\ge\; 0,
\label{eq:B-relax}
\end{equation}
with $D$ the relative entropy and $\Gamma$ set by the port
coupling. Since $D \ge 0$ with equality only at $w = p$, and
$\dot D \le 0$ along the flow, the weights relax monotonically to
the Born measure and stay there. Two things must be said plainly.
Eq.~\eqref{eq:B-relax} is a \emph{result} of the boundary law's
dissipative structure, not an assumption --- that is the content of
the cohort's Born chain, carried at derived-conditional grade. And
the one step still owed is the field-grade representation ---
showing that the boundary's coarse-grained weights are exactly the
$w_k$ of Eq.~\eqref{eq:B-relax} in a relativistic field theory
rather than in the finite-dimensional model. That residue is named
here, in the honest ledger, and in Problem~1.

\paragraph{Why the relaxation is observable.} A relaxation takes
time, and any mode that decoupled before it finished retains the
pre-relaxation statistics. Modes that left the horizon during
inflation are exactly such a population, which gives the deficit of
Eq.~\eqref{eq:f25-w2},
$\xi(k) = 1 - A(k/k_0)^{-\gamma}$ --- the sharpest available
distinction between ``Born is an axiom'' (no deficit at any scale)
and ``Born is an attractor'' (a computable deficit on the largest
scales). No interpretation of quantum mechanics has previously
carried an observational signature at all; this one does, and it
is the reason Problem~1 sits in a physics paper rather than a
philosophy one.

\subsection{Step 4 --- problems and worked solutions}
\label{app:B-problems}

\medskip\noindent\textbf{B.1. Show that the partial trace is the
unique map reproducing all local expectation values.}

\noindent\emph{Solution.} \emph{Step 1.} Require
$\mathrm{Tr}(\rho_S O) = \mathrm{Tr}(\rho_{SE}\,O \otimes
\mathbb{1})$ for every system observable $O$. \emph{Step 2.}
Expanding in a basis of operators, the condition fixes every matrix
element of $\rho_S$ as $\sum_e \langle s,e|\rho_{SE}|s',e\rangle$.
\emph{Step 3.} That is Eq.~\eqref{eq:B-ptrace}, and no freedom
remains. \emph{Physics intuition:} the reduced state is not an
approximation --- it is exactly what is knowable locally, which is
why the information it drops is dropped irrevocably rather than
merely hidden.

\medskip\noindent\textbf{B.2. Show that a negative $\gamma_k$ in
Eq.~\eqref{eq:B-lindblad} violates positivity.}

\noindent\emph{Solution.} \emph{Step 1.} Take a single jump
operator $L = |0\rangle\langle 1|$ and initial state
$\rho = |0\rangle\langle 0|$. \emph{Step 2.} With $\gamma < 0$ the
population of $|0\rangle$ grows without a compensating loss
elsewhere, and after finite time the population of $|1\rangle$ goes
negative. \emph{Step 3.} Negative probability is inconsistent, so
$\gamma \ge 0$. \emph{Physics intuition:} the arrow of the
dissipator is not an extra assumption bolted onto quantum
mechanics; it is what consistency permits once the dynamics is
open at all.

\medskip\noindent\textbf{B.3. Estimate the decoherence time of a
$10\,\mu$m dust grain superposed over $1\,\mu$m in air, given
$\Lambda \sim 10^{36}~\mathrm{m^{-2}s^{-1}}$.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:B-decoherence},
$\Gamma = \Lambda(\Delta x)^2 = 10^{36}\times(10^{-6})^2 =
10^{24}~\mathrm{s^{-1}}$. \emph{Step 2.} The coherence time is
$\Gamma^{-1} = 10^{-24}$~s. \emph{Step 3.} Compare the light-travel
time across the grain, $\sim 3\e{-14}$~s: decoherence is ten orders
faster than anything mechanical in the problem. \emph{Physics
intuition:} macroscopic superpositions are not forbidden, they are
\emph{unaffordable} --- and the price is set by how loudly the
alternatives differ, which is the $(\Delta x)^2$.

\medskip\noindent\textbf{B.4. Show that Eq.~\eqref{eq:B-decoherence}
leaves the diagonal untouched, and explain why this matters for
Problem~1.}

\noindent\emph{Solution.} \emph{Step 1.} Set $x = x'$: the damping
factor $(x-x')^2$ vanishes. \emph{Step 2.} Hence
$\rho(x,x)$ --- the probability density --- obeys the same equation
it would without the environment. \emph{Step 3.} Decoherence
therefore destroys interference without changing any probability.
\emph{Physics intuition:} this is exactly why decoherence cannot
be the whole answer to the measurement problem: it explains why we
never see the interference, and says nothing about why we see one
outcome. The write of Step~3 is what the diagonal still needs.

\medskip\noindent\textbf{B.5. A student objects: ``unitarity is
sacred, so the write cannot be fundamental.'' Answer them.}

\noindent\emph{Solution.} \emph{Step 1.} Unitarity is a property of
the \emph{bulk} books, and the anatomy keeps them exact ---
Problem~16 runs the same argument across a horizon. \emph{Step 2.}
What is not unitary is the boundary \emph{reading}, and no theorem
ever said a marginal of a unitary evolution is unitary; the partial
trace of B.1 is a counterexample by construction. \emph{Step 3.}
The contradiction the student fears arises only if one insists the
reading be described by the books' own dynamics --- the one-fiber
assumption. \emph{Physics intuition:} a ledger can be exact while
every page you are allowed to photocopy is smudged; the smudging is
the projection, not a failure of the ledger.

\medskip\noindent\textbf{B.6. Derive Eq.~\eqref{eq:B-lindblad}'s
structure from Eq.~\eqref{eq:B-kraus} for a short time step.}

\noindent\emph{Solution.} \emph{Step 1.} Write
$E_0 = \mathbb{1} + (-\tfrac{i}{\hbar}H - \tfrac12\sum_k
\gamma_k L_k^\dagger L_k)\,\delta t$ and
$E_k = \sqrt{\gamma_k \delta t}\,L_k$ for $k \ge 1$; the
completeness relation holds to $O(\delta t)$. \emph{Step 2.}
Substituting into Eq.~\eqref{eq:B-kraus} and keeping terms of
order $\delta t$ gives
$\rho + \delta t\big(-\tfrac{i}{\hbar}[H,\rho] + \sum_k\gamma_k(
L_k\rho L_k^\dagger - \tfrac12\{L_k^\dagger L_k,\rho\})\big)$.
\emph{Step 3.} Dividing by $\delta t$ returns
Eq.~\eqref{eq:B-lindblad}. \emph{Physics intuition:} the Lindblad
form is not a model of dissipation but the most general thing that
can happen to a piece of a larger quantum world in a short time
--- which is why deriving the boundary law from a projection, as
Step~3 does, lands on it rather than on something exotic.

\GGtakeaway{Open-system quantum mechanics is not an approximation
scheme in this reading but the exact boundary law, with the
commutator riding the circle and the dissipator the interval. What
the standard account imports --- the environment, the basis, the
measure --- the anatomy names; what it cannot deliver --- the
single outcome --- the anatomy identifies as the write, and prices
it with one declared residue.}

\section{Gauge invariance and the renormalization group}
\label{app:gaugeRG}
\GGteachbox{The gauge principle derived from the demand for a
position-dependent phase; Noether's two theorems and why the second
one changes the meaning of the first; the beta function and the
one-loop running computed; asymptotic freedom; the anomaly and why
its coefficient is an integer --- and then the two-fiber reading in
which the first half of this chapter is the circle and the second
half is the interval. Pay-off: Problems~2 and~3.}
\GGreadalong{\citet{PeskinSchroeder1995}, Ch.~15 and~12, for
non-abelian gauge fields and the renormalization group;
\citet{WeinbergQTF1}, Ch.~7, for the careful treatment of
Noether's theorems; \citet{Srednicki2007}, Ch.~72--73, for anomalies
computed cleanly.}

\subsection{Step 1 --- the standard treatment}
\label{app:C-step1}

\paragraph{The gauge principle.} Start from a free complex field
with Lagrangian $\mathcal L = \bar\psi(i\gamma^\mu\partial_\mu -
m)\psi$ and demand invariance under the \emph{local} phase change
of Eq.~\eqref{eq:gaugetrans-w2}. The derivative term spoils it, by
a residue proportional to $\partial_\mu\chi$. Replacing
$\partial_\mu \to D_\mu$ with Eq.~\eqref{eq:A-covd} and assigning
$A_\mu$ the compensating shift restores invariance --- and in doing
so introduces a vector field with a definite coupling to every
charge. Repeating with a non-abelian group gives the Yang--Mills
Lagrangian
\begin{equation}
\mathcal L_{\mathrm{YM}} \;=\;
-\tfrac14 F^a_{\mu\nu}F^{a\,\mu\nu}
+ \bar\psi(i\gamma^\mu D_\mu - m)\psi ,
\label{eq:C-YM}
\end{equation}
whose self-interactions follow from the commutator term in
Eq.~\eqref{eq:A-curv}. Nothing was added by hand except the demand.

\paragraph{Noether's two theorems.} For a \emph{global} symmetry
with parameter $\epsilon$, invariance of the action gives a
conserved current, $\partial_\mu j^\mu = 0$ --- a genuine
constraint on motion. For a \emph{local} symmetry with arbitrary
$\chi(x)$, the same manipulation yields instead an
\emph{identity} among the equations of motion, true whether or not
they are satisfied. This is the second theorem, and it is why a
local symmetry supplies no new conserved charge: it says the
description is redundant, not that the world is constrained.

\paragraph{Running couplings.} Loop corrections make a coupling's
measured value depend on the resolution scale $\mu$. Writing the
dependence infinitesimally gives Eq.~\eqref{eq:rg-w2},
$\mu\, dg/d\mu = \beta(g)$, with the one-loop non-abelian result
\begin{equation}
\beta(g) = -\frac{b_0}{16\pi^2}\,g^3,
\qquad
b_0 = \tfrac{11}{3}C_A - \tfrac43 T_F n_f
\;\;\xrightarrow[\ \mathrm{SU(3)}\ ]{}\;\;
11 - \tfrac23 n_f ,
\label{eq:C-beta}
\end{equation}
whose integral is Eq.~\eqref{eq:running-w2}. For QCD with
$n_f \le 16$, $b_0 > 0$ and the coupling \emph{falls} toward the
ultraviolet: asymptotic freedom. For QED the sign is reversed and
the coupling grows with energy.

\paragraph{The anomaly.} A symmetry of the classical action can
fail to survive quantization because the path-integral measure is
not invariant. The failure is one-loop exact and its coefficient is
a sum of group-theory factors over the fermion content --- the
sums of Eq.~\eqref{eq:anomsum-w2}. Consistency of a gauge theory
requires those sums to vanish exactly; the fact that the Standard
Model's hypercharges do so, forcing quark charges to be thirds, is
one of the most striking facts in particle physics.

\paragraph{Yang--Mills, assembled explicitly.} For a group
with generators $T^a$ obeying $[T^a,T^b] = i f^{abc}T^c$, write
$A_\mu = A_\mu^a T^a$. Demanding covariance of
$D_\mu\psi = (\partial_\mu - igA_\mu)\psi$ under
$\psi \to U\psi$ forces
$A_\mu \to UA_\mu U^{-1} + \tfrac{i}{g}U\partial_\mu U^{-1}$, and
the field strength follows from Eq.~\eqref{eq:A-curv}:
\begin{equation}
F^a_{\mu\nu} \;=\; \partial_\mu A^a_\nu - \partial_\nu A^a_\mu
+ g f^{abc}A^b_\mu A^c_\nu .
\label{eq:C-Fmunu}
\end{equation}
The last term is the whole difference from electromagnetism: the
gauge field carries the charge it couples to, so
Eq.~\eqref{eq:C-YM} contains three- and four-point self
interactions with coefficients fixed --- not fitted --- by
$f^{abc}$. \emph{Physics intuition:} the gluon's self-coupling is
not an additional force law but the statement that color rotations
do not commute; every distinctive feature of QCD, asymptotic
freedom included, descends from that one non-commutation.

\paragraph{Quantization, gauge fixing, and ghosts, in brief.} The
path integral over $A_\mu$ overcounts by the volume of the gauge
group. Faddeev and Popov's device inserts
$1 = \Delta_{\mathrm{FP}}\int\! \mathcal D\chi\,
\delta\big(G(A^\chi)\big)$ with $G$ a gauge condition, producing a
gauge-fixing term and a determinant that is exponentiated as
anticommuting ``ghost'' fields. Ghosts are not physical states;
they are bookkeeping that removes the unphysical polarizations of
$A_\mu$ in loops. The reader should retain one fact: the gauge
redundancy is not eliminated by quantization, only \emph{managed},
and every physical answer is independent of $G$.

\paragraph{The beta function, and where its two terms come from.}
At one loop the coupling's scale dependence,
Eq.~\eqref{eq:C-beta}, receives two competing contributions.
\emph{Fermion loops screen}, exactly as in QED: a virtual pair
polarizes the vacuum, and the effective charge seen from far away
is reduced --- this is the $-\tfrac43 T_F n_f$ term, negative in
$b_0$ and pushing toward a Landau pole. \emph{Gluon loops
antiscreen}: because the gluon carries color, the vacuum behaves
as a paramagnetic medium whose induced color currents
\emph{enhance} the field --- the $+\tfrac{11}{3}C_A$ term. With
$C_A = 3$ and $T_F = \tfrac12$ this gives
$b_0 = 11 - \tfrac23 n_f$, positive for $n_f \le 16$, so
antiscreening wins and QCD is asymptotically free.

\emph{Physics intuition.} The sign of $b_0$ is the single most
consequential number in the strong interaction: positive means the
force weakens at short distance (quarks look free inside a proton)
and strengthens at long distance (they are never seen alone). Both
faces of Problem~12 are contained in that sign.

\paragraph{The Callan--Symanzik equation.} Physical quantities
cannot depend on the arbitrary renormalization point, so an
$n$-point function obeys
\begin{equation}
\left[\mu\frac{\partial}{\partial\mu}
+ \beta(g)\frac{\partial}{\partial g}
+ n\gamma(g)\right] G^{(n)}(\{x\};\,g,\mu) \;=\; 0 ,
\label{eq:C-CS}
\end{equation}
with $\gamma$ the anomalous dimension. Eq.~\eqref{eq:C-CS} says the
explicit $\mu$-dependence and the implicit dependence through
$g(\mu)$ must cancel --- and it is the formal statement that
``running'' is a statement about \emph{descriptions}, not about
nature changing with energy.

\paragraph{The anomaly, computed the quickest way.} Consider the
triangle diagram with one axial and two vector currents. Any
regulator that preserves the vector Ward identities necessarily
violates the axial one, giving
\begin{equation}
\partial_\mu j^{\mu 5} \;=\;
\frac{g^2}{16\pi^2}\,\epsilon^{\mu\nu\rho\sigma}
F_{\mu\nu}F_{\rho\sigma} ,
\label{eq:C-abj}
\end{equation}
the Adler--Bell--Jackiw anomaly. Fujikawa's derivation makes the
origin transparent: the fermionic path-integral measure is not
invariant under a chiral rotation, and its Jacobian is exactly
Eq.~\eqref{eq:C-abj}. The coefficient is fixed by group theory
alone --- it receives no corrections at any order, by the
Adler--Bardeen theorem. \emph{Physics intuition:} an anomaly is
what happens when a symmetry of the equations is not a symmetry of
the \emph{counting of states}; and since counting is discrete, the
result is an integer that no amount of dynamics can shift.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:C-step2}

\emph{A redundancy that dictates.} By Noether's second theorem and
Elitzur's theorem, local gauge symmetry is a property of our
labeling, not of states --- and yet it fixes the interactions
completely. A framework in which bookkeeping determines dynamics is
either hiding a physical structure behind the bookkeeping or
enjoying a remarkable coincidence. The textbook offers no way to
decide (Problem~3).

\emph{An arrow with no owner.} Eq.~\eqref{eq:C-beta} is a
first-order flow in $\ln\mu$ that runs one way: integrating out
short distances is not invertible. The formalism records the
one-wayness (the $c$-theorem and its relatives make it precise) but
locates it nowhere --- the four spacetime directions are all
two-way (Problems~1, 13).

\emph{An integer with no home.} Anomaly coefficients cannot
receive corrections at any order; they are integers up to fixed
normalizations. Integers are the signature of topology, but
four-dimensional field theory has no compact direction to put them
on --- so their exactness is verified rather than explained
(Problem~4).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:C-step3}

\paragraph{Gauge invariance is the circle's unreadability.} If the
phase $\theta$ is a coordinate on a real compact fiber, then a
gauge transformation is precisely a re-zeroing of an
\emph{unreadable} pointer, and Eq.~\eqref{eq:gaugetrans-w2}'s
arbitrary $\chi(x)$ is the statement that no experiment reads the
absolute angle. What experiment \emph{can} read is the loop total,
Eq.~\eqref{eq:wilson-p2}. The century's discomfort --- how can a
redundancy dictate a force? --- dissolves: the redundancy is our
inability to read a physical fiber, and the dictation is the
fiber's geometry showing through the only window that is open. The
photon is the gearing between pointers and carries no dial of its
own, which is why it is massless and chargeless.

\paragraph{Renormalization is motion along the interval.} The
coordinate $r = \ln(\EGUT/\mu)$ \emph{is} the fifth dimension, so
Eq.~\eqref{eq:rg-w2} is transport along $\Ir$ and
Eq.~\eqref{eq:C-beta} is its generator. Three textbook facts become
statements about a fiber. Couplings are readings at addresses, not
constants --- hence no address is privileged. The flow's
irreversibility is Eq.~\eqref{eq:A-semigroup}, not an extra
principle. And renormalizability --- the survival of only a handful
of operators --- is the statement of what the descent can carry
down without leaking.

\paragraph{The anomaly is the circle auditing the interval.}
Anomaly coefficients are winding data: integers on the compact
fiber, which by construction cannot run under
Eq.~\eqref{eq:C-beta}. Their vanishing is not a coincidence of the
matter content but the condition that the bulk's books close ---
the six-dimensional instance being the \GS{} mechanism of
\citet{ArisakaA3BI6}, whose by-product is the integer ledger
Eq.~\eqref{eq:ledger-p2} that Appendix~\ref{app:codebook} then
spends.

\paragraph{The circle, expanded in modes.} Make the first
claim quantitative. A bulk gauge field on $\mathcal M_4 \times
\Stheta$ decomposes as
\begin{equation}
A_\mu(x,\theta) = \sum_{n\in\mathbb{Z}} A_\mu^{(n)}(x)\,
e^{\,in\theta},
\qquad
m_n^2 \;=\; \frac{(n + \alpha/2\pi)^2}{R^2},
\label{eq:C-KK}
\end{equation}
with $R$ the circle's radius and $\alpha$ the Wilson-line modulus
of Eq.~\eqref{eq:A-wilsonline}. Three readings, each a body claim
made technical. The $n = 0$ mode is the four-dimensional gauge
field of Eq.~\eqref{eq:covd-w2} --- so the boundary's gauge
symmetry is the circle's zero mode, not an independent postulate.
The label $n$ is the winding charge, quantized because
$e^{in\theta}$ must be single-valued --- so charge quantization is
Fourier analysis on a compact space, which is Problem~4's
``labels'' in one line. And $\alpha$ shifts the entire tower: a
circle-side modulus setting four-dimensional masses, which is the
Hosotani mechanism and the mechanism behind
Eq.~\eqref{eq:vev-p2}'s electroweak scale.

\paragraph{Gauge invariance, restated as a theorem about
readability.} On the anatomy the statement ``physics is invariant
under Eq.~\eqref{eq:gaugetrans-w2}'' becomes: \emph{the boundary
can read functions of $W_\theta$, Eq.~\eqref{eq:A-wilsonline}, and
nothing else about the fiber's connection}. The proof is the
decomposition of \S\ref{app:A-step3}: local values of $A_\theta$
are shifted by any single-valued $g(\theta)$, while the loop
integral is shifted only by $2\pi n$. Elitzur's theorem is then not
a surprise but a corollary --- a quantity that no admissible
observable depends on cannot acquire an expectation value --- and
the Higgs mechanism is relabeled honestly: what acquires a value
is the readable modulus $\alpha$, and what the boundary registers
is the spectrum Eq.~\eqref{eq:C-KK} it induces.

\paragraph{The interval, and the geometric origin of the beta
function.} On the anatomy, $\mu$ is not an arbitrary bookkeeping
scale but a position: $r = \ln(\EGUT/\mu)$. Motion along the
interval carries a warp, and the effective four-dimensional
couplings are integrals of the bulk coupling over the traversed
region. Differentiating the traversed integral with respect to the
endpoint gives a first-order flow in $r$ --- which is
Eq.~\eqref{eq:rg-w2} with
\begin{equation}
\frac{\partial}{\partial r} \;=\; -\,\mu\frac{\partial}{\partial\mu},
\qquad\text{hence}\qquad
\frac{dg}{dr} \;=\; -\beta(g) .
\label{eq:C-rflow}
\end{equation}
Two consequences that the four-dimensional formalism cannot state.
First, the flow now has \emph{endpoints}: $r = 0$ at the GUT
surface and $r = K$ at the descent's floor, so the total length is
a physical quantity ($K = 35$) rather than a range of convenience
--- this is the content of Problem~6's dissolution. Second, the
Callan--Symanzik statement Eq.~\eqref{eq:C-CS} that no address is
privileged becomes a statement about a coordinate: physics is
invariant under \emph{relabeling} positions on $\Ir$, not under
translations along it, because the endpoints are fixed.

\paragraph{Anomaly inflow: why the integer lives on the circle.}
In $2n$ dimensions the anomaly is encoded in a
$(2n+2)$-form anomaly polynomial $I_{2n+2}$, related to the
boundary anomaly by the descent equations
\begin{equation}
I_{2n+2} = dI_{2n+1},
\qquad
\delta_{\mathrm{gauge}} I_{2n+1} = dI_{2n},
\label{eq:C-descent}
\end{equation}
whose coefficients are fixed by characteristic classes ---
integrals of curvature forms that are, by the Chern--Weil theorem,
\emph{integers} for the compact directions. Two facts follow
immediately. The coefficients cannot run, because
Eq.~\eqref{eq:C-rflow} is a continuous flow and $\mathbb{Z}$ is
discrete (the same argument as \S\ref{app:A-step3}). And their
cancellation is a condition on the \emph{bulk}, not a coincidence
of the boundary spectrum: the six-dimensional instance is the
\GS{} mechanism \citep{GreenSchwarz1984, ArisakaA3BI6}, whose
solution fixes the coefficient vector and, as a by-product, the
integer ledger Eq.~\eqref{eq:ledger-p2}. The audit of
Eq.~\eqref{eq:anomsum-w2} that a student verifies generation by
generation is the four-dimensional shadow of that
six-dimensional closure --- which is why it holds exactly, and why
its by-products are exact too.

\subsection{Step 4 --- problems and worked solutions}
\label{app:C-problems}

\medskip\noindent\textbf{C.1. Derive the compensating transformation
of $A_\mu$ from the demand that $D_\mu\psi$ transform like $\psi$.}

\noindent\emph{Solution.} \emph{Step 1.} Require
$D_\mu(e^{iq\chi}\psi) = e^{iq\chi}D_\mu\psi$. \emph{Step 2.}
Expanding the left side gives $e^{iq\chi}(\partial_\mu + iq
\partial_\mu\chi - iqA_\mu/\hbar)\psi$. \emph{Step 3.} Matching
requires $A_\mu \to A_\mu + \hbar\partial_\mu\chi$, which is
Eq.~\eqref{eq:gaugetrans-w2}. \emph{Physics intuition:} the
gauge field's transformation law is not an independent postulate;
it is whatever is required to make comparison at a distance
possible, which is exactly what a connection is for.

\medskip\noindent\textbf{C.2. Compute $b_0$ for QCD with six
flavors and state the sign consequence.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:C-beta}, $b_0 = 11 - \tfrac23(6) = 7$.
\emph{Step 2.} Since $b_0 > 0$, $\beta(g) < 0$, so $g$ decreases as
$\mu$ increases. \emph{Step 3.} The coupling therefore grows toward
low energy, and the perturbative description fails near
$\Lambda_{\mathrm{QCD}}$. \emph{Physics intuition:} asymptotic
freedom and confinement are the two ends of one monotone reading
along the interval --- Appendix~\ref{app:confinement} picks the
story up exactly where Eq.~\eqref{eq:C-beta} stops being
trustworthy.

\medskip\noindent\textbf{C.3. Verify both hypercharge sums of
Eq.~\eqref{eq:anomsum-w2} for one Standard-Model generation.}

\noindent\emph{Solution.} \emph{Step 1.} Write every field as a
left-handed Weyl fermion --- this is essential, since conjugation
flips the sign of $Y$ and the cubic sum is not convention-blind.
With $Y = 2(Q-T_3)$ the content is $Q$ ($Y=\tfrac13$, $2\times3=6$
states), $u^c$ ($-\tfrac43$, $3$), $d^c$ ($\tfrac23$, $3$), $L$
($-1$, $2$), $e^c$ ($2$, $1$). \emph{Step 2.} The linear sum ---
the mixed gravitational anomaly ---
$6(\tfrac13) + 3(-\tfrac43) + 3(\tfrac23) + 2(-1) + 2
= 2 - 4 + 2 - 2 + 2 = 0$. \emph{Step 3.} The cubic sum ---
the $\mathrm{U}(1)^3$ anomaly ---
$6(\tfrac1{27}) + 3(-\tfrac{64}{27}) + 3(\tfrac8{27}) + 2(-1) + 8
= -6 + 6 = 0$. Both cancellations run between quarks and leptons
and fail if either is removed.
\emph{Physics intuition:} the electron's charge is tied to the
quark's by a consistency condition, not by a coincidence --- which
is why hydrogen is neutral to one part in $10^{21}$, and why this
paper reads the integer as circle-side data.

\medskip\noindent\textbf{C.4. A student says: ``the interval is
just the renormalization group, renamed.'' Give the sharpest reply
you can.}

\noindent\emph{Solution.} \emph{Step 1.} Concede the overlap: on
the four-dimensional boundary the interval's readings \emph{are}
the RG flow, and no equation of the RG is changed. \emph{Step 2.}
Identify the difference: the RG is a statement about descriptions
at different resolutions, and has no length, no endpoints, and no
other physics attached; the interval has a definite extent
$K = 35$, endpoints that are physical events (the GUT edge, the
lock, the terminus), and a warp factor that is gravity's
(Problems~6, 5, 2). \emph{Step 3.} The distinction is testable:
the RG predicts nothing about $M_{\mathrm{lock}}$ or $N^*$; the
interval predicts both. \emph{Physics intuition:} identifying a
known flow with a geometric coordinate is only a rename until the
geometry says something the flow could not --- and a resonance at
$3.849$~TeV would be exactly that something.

\medskip\noindent\textbf{C.5. Why can a gauge symmetry not break,
and what does the Higgs mechanism break instead?}

\noindent\emph{Solution.} \emph{Step 1.} By Elitzur's theorem the
expectation value of any gauge-variant quantity vanishes
identically, so no local order parameter exists. \emph{Step 2.}
What acquires an expectation value is a gauge-\emph{fixed}
quantity, or equivalently a gauge-invariant composite; the
physical content is the spectrum, not the ``broken'' symmetry.
\emph{Step 3.} On the anatomy the honest statement is a
\emph{reading}: the circle's unreadable pointer settles at a
lattice value, and the boundary registers the consequences ---
masses --- rather than the angle. \emph{Physics intuition:} this
is why symmetry breaking in this paper is always called symmetry
\emph{reading}: nothing breaks, something gets read, and the
reading is where the physics is.

\GGtakeaway{The two principles that carry all of field theory are
the two fibers in disguise: gauge invariance is the circle's
unreadability, renormalization is the interval's one-way motion,
and the anomaly is the first auditing the second. Nothing in the
standard formalism changes; what changes is that both principles
acquire a referent.}

\section{The parameter codebook: from the ledger to the
laboratory}
\label{app:codebook}
\GGteachbox{How the Standard Model's free parameters are counted,
and why the count is what it is; what ``measuring a parameter''
actually involves once couplings run; why no amount of work inside
a quantum field theory can output its own inputs; and then the
codebook arithmetic worked end to end, with the numerology
discipline stated as a procedure rather than a promise. Pay-off:
Problem~4.}
\GGreadalong{\citet{PDG2024} for the measured values and their
conventions; \citet{Cahn1996} for what each parameter does to the
world; \citet{ArisakaA9SM} for the full twenty-eight-entry
derivation this appendix samples.}

\subsection{Step 1 --- the standard treatment: counting the
parameters}
\label{app:D-step1}

\paragraph{The count, done carefully.} Write the Standard-Model
Lagrangian and ask how many numbers must be supplied from outside.
Three gauge couplings $g_s, g, g'$. Two Higgs-sector numbers (the
mass parameter and the quartic, equivalently $v$ and $m_H$). Nine
charged-fermion masses --- three up-type, three down-type, three
charged leptons --- through their Yukawa couplings. Four quark
mixing parameters: three angles and one $CP$ phase in the CKM
matrix. One QCD vacuum angle $\bar\theta$. That is nineteen. Adding
neutrino masses and lepton mixing (three masses, three angles, one
Dirac phase, two Majorana phases if the neutrinos are Majorana)
brings the total to twenty-eight. Cosmology then adds its own six
for $\Lambda$CDM.

\paragraph{Why the count is not obvious.} Many apparent parameters
are removable by field redefinitions. The quark Yukawa matrices
have $2\times 9 = 18$ complex entries, i.e.\ $36$ real numbers,
but most are absorbed by rotating the quark fields; what survives
is six masses, three angles, and one phase. The counting theorem
--- parameters equal Lagrangian entries minus the dimension of the
field-redefinition group that acts freely --- is a genuine and
beautiful piece of physics, and it is worth working once (Problem
D.1).

\paragraph{What ``measured'' means.} A coupling is not a number
until a scale is named. The fine-structure constant is
$1/137.036$ in the Thomson limit and $\approx 1/128$ at the $Z$
pole; both are the same parameter read at different addresses via
Eq.~\eqref{eq:rg-w2}. A quoted parameter is therefore a triple:
value, scheme, and scale. Any comparison between a derived closed
form and an experimental number must specify all three, and the
codebook does.

\paragraph{The counting theorem, worked.} The parameter count
is not a matter of inspection; it is a theorem, and the proof is
worth doing once. Consider the quark Yukawa sector,
\begin{equation}
\mathcal L_Y \;=\; -\,\bar Q_L\, Y_u\, \tilde\Phi\, u_R
\;-\; \bar Q_L\, Y_d\, \Phi\, d_R \;+\; \mathrm{h.c.},
\label{eq:D-yukawa}
\end{equation}
with $Y_u, Y_d$ arbitrary complex $3\times3$ matrices. Naively that
is $2 \times 9 = 18$ complex entries, or $36$ real parameters. Now
count the freedom to redefine fields. In the absence of
Eq.~\eqref{eq:D-yukawa} the kinetic terms are invariant under
$\mathrm{U}(3)^3$ acting on $Q_L, u_R, d_R$ --- three unitary
matrices, $3 \times 9 = 27$ real parameters. One combination is not
free: baryon number, a $\mathrm{U}(1)$ acting equally on all three,
leaves Eq.~\eqref{eq:D-yukawa} invariant and therefore removes
nothing. The physical count is
\begin{equation}
36 \;-\; (27 - 1) \;=\; 10
\;=\; \underbrace{6}_{\text{masses}} +
\underbrace{3}_{\text{angles}} + \underbrace{1}_{\text{phase}} ,
\label{eq:D-count}
\end{equation}
which is Problem~D.1's answer arriving from the other direction.
\emph{Physics intuition:} a ``parameter'' is not an entry in a
Lagrangian but an entry that survives every relabeling of the
fields --- and the arithmetic of Eq.~\eqref{eq:D-count} is why a
third generation was predicted before it was found: with two
generations the same count leaves $0$ phases, so $CP$ violation in
the quark sector would be impossible.

\paragraph{Repeating the count for the full model.} Applying the
same procedure sector by sector: three gauge couplings; two
Higgs-potential parameters; ten from Eq.~\eqref{eq:D-count}; one
$\bar\theta$; and in the lepton sector three charged-lepton masses
plus, for Majorana neutrinos, three masses, three angles, one Dirac
phase, and two Majorana phases (the Majorana mass term breaks the
rephasing freedom that would otherwise remove them). The total is
$3+2+10+1+3+9 = 28$. Cosmology adds six for $\Lambda$CDM
($\Omega_b h^2$, $\Omega_c h^2$, $\theta_s$, $\tau$, $A_s$, $n_s$).
A student should do this count personally once; it is the only way
to feel how little of the Standard Model is derived.

\paragraph{What a measured coupling actually is.} Take
$\alpha_s$. Nobody measures it: one measures, say, the ratio of
three-jet to two-jet events at a given collider energy, computes
that ratio to some order in perturbation theory as a function of
$\alpha_s(\mu)$ in a chosen scheme (almost always
$\overline{\mathrm{MS}}$), and inverts. The output is therefore a
triple --- value, scheme, scale --- and quoting it as a bare number
is an abbreviation. Running to a common reference by
Eq.~\eqref{eq:running-w2} is what makes different experiments
comparable, and \emph{matching} at each quark threshold (adjusting
$n_f$ in $b_0$) is what makes the running continuous.

\emph{Physics intuition.} Scheme dependence is often mistaken for a
defect. It is the opposite: the scheme is a coordinate choice on
the space of descriptions, and physical predictions --- cross
sections, lifetimes --- are scheme-independent by
Eq.~\eqref{eq:C-CS}. What is \emph{not} arbitrary is that the
running exists at all, and that it runs one way.

\paragraph{The two ways a closed form can be compared.} Since a
parameter is a triple, a derived closed form must specify where it
is evaluated. Two conventions appear in this paper and should not
be confused. Some entries are anchored at a physical pole ---
$\hat\alpha^{-1}(M_Z) = 2^J = 128$ --- and are compared with the
running value at that pole. Others are infrared quantities ---
$\alpha^{-1}(0) = 2^J + N^2 = 137$ --- and are compared with the
Thomson limit. The difference between them, $N^2 = 9$, is itself a
prediction about the amount of running between the two addresses,
and it is checkable independently of either endpoint (Problem~D.5).

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:D-step2}

The framework is silent by construction, and it is worth being
precise about why. A quantum field theory is a \emph{map} from
(field content, symmetry, parameters) to (predictions). It is
internally consistent for essentially any parameter values --- the
electron could be ten times heavier and the theory would run
perfectly well. There is therefore nothing inside the map that
could distinguish the actual values, and no amount of ingenuity
applied to the map alone will produce them.

Fifty years of attempts to reduce the count each added structure
whose own freedom exceeded what it removed: grand unification
related three couplings and bought new scales and a proton
lifetime now excluded in its simplest form; the minimal
supersymmetric extension stabilized the Higgs mass and introduced
more than a hundred soft parameters; the string landscape replaced
computation with selection. The honest summary is that the count
has \emph{risen} (Problem~4, and Problem~15 for the selection
route).

\subsection{Step 3 --- the two-fiber resolution: labels and
addresses}
\label{app:D-step3}

\paragraph{The grammar.} The codebook's claim is that every
parameter is a holonomy-class label wearing a descent address:
the label is circle-side arithmetic in the ledger integers
Eq.~\eqref{eq:ledger-p2} and $\pi$, exact and unable to run; the
address is interval-side, an exponential of depth. Two fonts, one
alphabet, and --- the falsifiable core --- no third species of
primitive.

\paragraph{The chain, checked.} Eq.~\eqref{eq:chain-w2} runs from
the ledger to the laboratory in five lines. Each line's structure
repeats: integer arithmetic setting structure, depth arithmetic
setting scale. The reader can check every step with a calculator,
and Problem~D.2 asks exactly that.

\paragraph{The discipline that makes it more than numerology.}
Four procedural commitments, stated because they are the
difference between a codebook and a coincidence.
\emph{Pre-registration}: closed forms are published with named kill
conditions and dated instruments \emph{before} the decisive
measurements. \emph{One alphabet}: the same five integers serve
every entry; each new closed form is a constraint on the same
small ledger, not a fresh dip into the integer sea, so the fits
grow \emph{harder} with each success rather than easier.
\emph{The flag register}: candidate patterns without mechanism
(the $4K$/$8K$ family) are quarantined at flagged grade and never
cited as support. \emph{Cross-exposure}: the same integers face
colliders, haloscopes, and the microwave sky, with no shared
systematics --- a conspiracy of coincidences would have to span
four orders of magnitude of experimental technique.

\paragraph{The ledger's internal identities, derived.} The
five integers of Eq.~\eqref{eq:ledger-p2} are not five inputs. Two
relations bind them, and both are structural.

\emph{The chirality relation $J = 2N+1$.} The flavor alphabet
lives on the compact fiber as a lattice of phases. With $N$
generations the boundary registers $N$ ``forward'' steps and $N$
``backward'' steps of the chiral tape, plus the unstepped origin,
giving $2N+1$ distinguishable letters --- hence $J = 7$ for
$N = 3$. The count is a count of lattice sites, so it is an
integer for the same reason winding numbers are.

\emph{The closure relation $K = LJ$.} The descent's total depth is
the product of the block structure of the unified group ($L = 5$)
with the alphabet's length ($J = 7$): each of the $L$ blocks is
traversed once per letter as the reading proceeds, and the descent
ends when the tape is exhausted. Hence $K = 35$.

The consequence is the one Problem~6 spends: $K$ is fixed by $L$ and
$J$ through closure, and $N$ --- hence $J$ --- is fixed inside the
anomaly audit by the dual-lock condition \citep{ArisakaA3BI6}. One
chain, no adjustable step, one integer. The dual lock is not a second
route to $K$: it is the equation that selects $N$, and it holds
identically at both of its roots, so it constrains the ledger without
determining $K$ twice.

\paragraph{How a two-font formula is built, and how it could
fail.} Every entry of the codebook has the form
\begin{equation}
X \;=\; \underbrace{f(M,N,L,J,K;\pi)}_{\text{circle: exact,
cannot run}}\;\times\;
\underbrace{e^{-\,c\,K}\ \text{or}\ \Mlock^{\,p}}_{\text{interval:
an address}} ,
\label{eq:D-grammar}
\end{equation}
with $f$ a ratio of integers and powers of $\pi$ and $c, p$ small
rationals. The falsifiable content is the \emph{absence} of a third
factor: no fitted coefficient, no continuous parameter, no
$O(1)$ fudge. That is what makes each line of
Eq.~\eqref{eq:chain-w2} a test rather than a fit --- and it is
also, precisely, how the scheme could fail. If a measured value
required a factor that is neither ledger arithmetic nor a descent
depth, Eq.~\eqref{eq:D-grammar} would be violated at that entry,
and the codebook's grammar --- not merely one prediction --- would
be dead.

\paragraph{Which entries can run, and which cannot: a theorem
rather than a taste.} Because circle-side quantities are homotopy
data and the flow Eq.~\eqref{eq:C-rflow} is continuous, no
circle-side quantity has a beta function: $\partial_r f = 0$
identically. Interval-side quantities are, by construction,
readings at an address and \emph{must} run. This partitions the
sheet into two classes with an observable difference: the sharp
prediction is that no quantity the codebook calls a label will ever
be found to have scale dependence, and no quantity it calls an
address will ever be found without it. Spatial and temporal
constancy follow by the same argument (a homotopy class has no
gradient), which is why Problem~8's horizon~2 and Problem~15's
``no drawn constant'' are the same statement as this paragraph.

\paragraph{Worked: the weak angle as a partition.} The weak angle
is a ratio of circle-side quantities, hence pure arithmetic:
\begin{equation}
\sin^2\theta_W \;=\; \frac{g'^2}{g^2 + g'^2}
\;=\; \frac{L^2}{4N^3} \;=\; \frac{25}{108} = 0.23148,
\label{eq:D-weakangle}
\end{equation}
against the measured $0.23122$ at its matching point ($+0.11\%$).
Read the closed form structurally: $L^2$ counts the abelian
partition of the unified block, $4N^3$ the generation volume with
its doubling. Nothing in Eq.~\eqref{eq:D-weakangle} is a scale, so
--- by the theorem above --- the anatomy predicts this value is
attached to a fixed matching point and does not itself run; the
observed running of the \emph{effective} weak mixing angle is the
interval dressing the fixed partition, not the partition changing.

\subsection{Step 4 --- problems and worked solutions}
\label{app:D-problems}

\medskip\noindent\textbf{D.1. Derive the count of four physical
parameters in the CKM matrix.}

\noindent\emph{Solution.} \emph{Step 1.} A general $3\times3$
unitary matrix has $9$ real parameters: $3$ angles and $6$ phases.
\emph{Step 2.} Rephasing the six quark fields removes six phases,
but an overall common phase acts trivially, so only $6-1 = 5$
phases can be removed. \emph{Step 3.} Left: $3$ angles and
$6-5 = 1$ phase, hence four parameters, one of them $CP$-violating.
\emph{Physics intuition:} the single surviving phase is why three
generations are the minimum for $CP$ violation in the quark sector
--- with two generations the same counting leaves zero phases, and
Kobayashi and Maskawa's prediction of a third generation followed
from precisely this arithmetic.

\medskip\noindent\textbf{D.2. Check the chain of
Eq.~\eqref{eq:chain-w2} numerically, and report the deviations.}

\noindent\emph{Solution.} \emph{Step 1.}
$\Mlock = \tfrac12(1.2209\e{19})e^{-35}\ \mathrm{GeV}
= 3.849\e{3}$~GeV. \emph{Step 2.}
$v = 3849 \times (2/5)^3 = 3849 \times 0.064 = 246.3$~GeV against
the measured $246.22$~GeV: $+0.05\%$. \emph{Step 3.}
$\sin^2\theta_W = 25/108 = 0.23148$ against $0.23122$
($+0.11\%$); $\alpha^{-1}(0) = 128+9 = 137$ against $137.036$
($-0.03\%$). \emph{Physics intuition:} the deviations are at the
few-parts-in-a-thousand level with \emph{no} adjustable quantity
anywhere in the chain --- which is the only kind of agreement that
can be evidence, because it is the only kind that could have
failed.

\medskip\noindent\textbf{D.3. A skeptic says the closed forms are
numerology. Construct the strongest version of the charge, then
answer it.}

\noindent\emph{Solution.} \emph{Step 1.} The strongest version:
with five integers, $\pi$, $e$, and the operations of arithmetic,
the space of expressions below a given complexity is large enough
that hitting a measured number to $0.1\%$ is unremarkable, and no
single agreement is evidence. \emph{Step 2.} The answer is not to
deny this but to change what is being counted: the claim is not
that one expression matched, but that \emph{one small ledger}
matched twenty-eight entries plus the cosmological sheet, with the
same integers appearing in unrelated sectors --- so the effective
number of trials is bounded by the ledger, not by the expression
space. \emph{Step 3.} And the claim is dated: each closed form
carries a named instrument and era (Table~\ref{tab:kill}).
\emph{Physics intuition:} numerology never volunteers to die.
The procedural difference is publication order --- and it is
checkable by anyone with the preprint dates.

\medskip\noindent\textbf{D.4. Why does the codebook's status table
(Figure~\ref{fig:param-status-w2}) show eight entries still
Asserted, and why is that not fatal?}

\noindent\emph{Solution.} \emph{Step 1.} The eight are the
charged-fermion Yukawas, whose derivation requires the family
surface's overlap integrals --- a named open construction.
\emph{Step 2.} A framework that claimed all twenty-eight were
theorems would be claiming what the parent papers do not, and a
reader could not tell which claims to test. \emph{Step 3.} The
public grading is what makes the six Derived entries meaningful:
they are the ones that can fail cleanly. \emph{Physics intuition:}
the value of an honest ledger is not modesty --- it is that a
partially finished derivation still makes sharp predictions,
provided the reader is told exactly which pages are finished.

\medskip\noindent\textbf{D.5. The codebook predicts
$\alpha^{-1}(0) - \hat\alpha^{-1}(M_Z) = N^2 = 9$. Check it against
data and say what it tests.}

\noindent\emph{Solution.} \emph{Step 1.} Measured:
$\alpha^{-1}(0) = 137.036$ and
$\hat\alpha^{-1}(M_Z) = 127.93$ in the $\overline{\mathrm{MS}}$
scheme. \emph{Step 2.} The difference is $9.11$, against the
predicted $9$ --- an agreement of $1.2\%$. \emph{Step 3.} What it
tests is neither endpoint but the \emph{amount of running} between
them, which in the standard framework is a loop computation
depending on the whole charged-particle spectrum. \emph{Physics
intuition:} this is the most demanding kind of check available to a
codebook, because it is insensitive to any single closed form: both
endpoints could be wrong in the same direction and the difference
would still fail. That it does not fail is the strongest internal
evidence that the two-font grammar of
Eq.~\eqref{eq:D-grammar} is tracking something.

\GGtakeaway{The parameter problem is not hard because the numbers
are strange; it is hard because a framework whose parameters are
inputs cannot output them. The codebook's proposal is that the
sheet was never a list of constants but a text in two fonts --- and
the proposal is testable entry by entry, with the finished pages
marked.}

\section{The launch and the descent: inflation, e-folds, and
the hierarchy}
\label{app:launch}
\GGteachbox{FRW cosmology from the Einstein equations; the horizon
and flatness problems stated quantitatively; slow-roll inflation
derived, with the spectral observables computed from the potential;
the hierarchy problem as a one-loop estimate; and then the
two-fiber reading in which the inflaton is the interval's
coordinate, the e-fold count is a length, and the hierarchy is a
depth. Pay-off: Problems~5, 6, and~8.}
\GGreadalong{\citet{Baumann2022}, Ch.~2 and~4, for FRW and
inflation at exactly this level; \citet{WeinbergCosmology2008},
Ch.~1 and~10; \citet{KolbTurner1990} for the thermal history.}

\subsection{Step 1 --- the standard treatment}
\label{app:E-step1}

\paragraph{The expanding background.} For a homogeneous, isotropic
universe with scale factor $a(t)$, the Einstein equations reduce to
the Friedmann pair
\begin{equation}
H^2 \equiv \left(\frac{\dot a}{a}\right)^{2}
= \frac{8\pi G_N}{3}\rho - \frac{k}{a^2},
\qquad
\frac{\ddot a}{a} = -\frac{4\pi G_N}{3}(\rho + 3p),
\label{eq:E-friedmann}
\end{equation}
with $\rho$ the energy density, $p$ the pressure, and $k$ the
spatial curvature. The second equation contains the whole drama of
inflation in one sign: acceleration requires $p < -\rho/3$.

\paragraph{The two classic problems.} \emph{Horizon:} the comoving
particle horizon grows in a decelerating universe, so the
$\sim 10^4$ causally independent patches of
Eq.~\eqref{eq:patches-w2} should not share a temperature ---
they do, to $10^{-5}$. \emph{Flatness:} writing
Eq.~\eqref{eq:E-friedmann} as
$|\Omega - 1| = |k|/(aH)^2$, and noting that $(aH)^{-1}$ grows in
both radiation and matter eras, the present near-flatness demands
$|\Omega-1| \lesssim 10^{-60}$ at the Planck epoch.

\paragraph{Slow roll.} A scalar $\varphi$ with potential
$V(\varphi)$ has $\rho = \tfrac12\dot\varphi^2 + V$ and
$p = \tfrac12\dot\varphi^2 - V$, so $p \to -\rho$ when the kinetic
term is negligible. The conditions for that are
Eq.~\eqref{eq:slowroll-w2}'s $\epsilon, \eta \ll 1$, and while they
hold the stretch accumulates the e-folds of
Eq.~\eqref{eq:efolds-w2}. Both problems are solved if $N \gtrsim
60$: $(aH)^{-1}$ shrinks during inflation, so today's largest
scales were inside one causal patch, and curvature is diluted by
$e^{-2N}$.

\paragraph{The observables.} Quantum fluctuations of $\varphi$
stretched outside the horizon become the primordial perturbations,
with scalar and tensor spectra whose tilts are fixed by the
potential's shape at horizon exit:
\begin{equation}
n_s - 1 = -6\epsilon + 2\eta,
\qquad
r_{\mathrm T} = 16\epsilon .
\label{eq:E-observables}
\end{equation}
For the plateau class ($V \to$ const at large $\varphi$) these
collapse to the one-parameter forms of Eq.~\eqref{eq:ns-r},
depending only on $N^*$, the e-folds between horizon exit of the
observed scales and the end of inflation.

\paragraph{The hierarchy, as a one-loop estimate.} A scalar's
squared mass receives Eq.~\eqref{eq:radiative-w2}'s correction,
quadratic in the cutoff. With $\Lambda = M_{\mathrm{Pl}}$ the
correction exceeds the measured $m_H^2$ by
$\sim (10^{19}/10^{2})^2 = 10^{34}$, so the bare parameter must
cancel it to thirty-four digits.

\paragraph{From Einstein's equations to
Eq.~\eqref{eq:E-friedmann}.} Assume homogeneity and isotropy: the
metric is
$ds^2 = -dt^2 + a^2(t)\big[dr^2/(1-kr^2) + r^2 d\Omega^2\big]$.
Computing the Ricci tensor for this metric and inserting a perfect
fluid $T^{\mu}{}_{\nu} = \mathrm{diag}(-\rho,p,p,p)$, the $00$
component of $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$ gives the first
Friedmann equation and the trace gives the second. Their
compatibility requires the continuity equation
\begin{equation}
\dot\rho + 3H(\rho + p) \;=\; 0
\qquad\Longrightarrow\qquad
\rho \propto a^{-3(1+w)},\ \ w \equiv p/\rho ,
\label{eq:E-continuity}
\end{equation}
so radiation ($w = \tfrac13$) dilutes as $a^{-4}$, matter
($w = 0$) as $a^{-3}$, and vacuum ($w = -1$) not at all.
\emph{Physics intuition:} Eq.~\eqref{eq:E-continuity} is energy
conservation with a work term --- the universe does $p\,dV$ against
itself --- and the reason the vacuum does not dilute is that
$w = -1$ makes that work exactly cancel the volume growth. The
entire late history is the competition of three power laws, and
the ``coincidence'' of Problem~7 is the observation that the
crossover is happening now.

\paragraph{The horizon problem, in numbers.} The comoving particle
horizon is
$\eta = \int_0^t dt'/a(t') = \int_0^a da'/(a'^2H)$, which for
$w > -\tfrac13$ is dominated by late times --- so it \emph{grows},
and regions entering it today were never in contact. At last
scattering the horizon subtends $\sim 2^\circ$
(Eq.~\eqref{eq:patches-w2}), giving $\sim 10^4$ independent
patches, all at the same temperature to $10^{-5}$.

\paragraph{Slow roll, derived.} For a homogeneous scalar the
equation of motion is
\begin{equation}
\ddot\varphi + 3H\dot\varphi + V'(\varphi) = 0 ,
\label{eq:E-KG}
\end{equation}
with $3H\dot\varphi$ acting as Hubble friction. Neglecting
$\ddot\varphi$ and the kinetic energy in
Eq.~\eqref{eq:E-friedmann} gives the slow-roll system
$3H\dot\varphi \simeq -V'$ and $H^2 \simeq V/3\Mplred^2$.
Self-consistency of both approximations is precisely
Eq.~\eqref{eq:slowroll-w2}: $\epsilon \ll 1$ keeps the kinetic
energy small, $|\eta| \ll 1$ keeps the friction dominant. The
e-fold count then follows by the chain rule,
$dN = H\,dt = (H/\dot\varphi)\,d\varphi
= -(V/\Mplred^2 V')\,d\varphi$, which integrates to
Eq.~\eqref{eq:efolds-w2}.

\paragraph{Where the spectrum comes from.} Quantize the
fluctuation $\delta\varphi$ on the quasi-de~Sitter background. Each
Fourier mode behaves as a harmonic oscillator with a
time-dependent mass; deep inside the horizon ($k \gg aH$) it
oscillates in its vacuum, and as the horizon shrinks past it
($k = aH$, ``horizon exit'') the mode freezes with amplitude
$|\delta\varphi_k| \simeq H/\sqrt{2k^3}$. Converting to the
curvature perturbation $\mathcal R = -H\delta\varphi/\dot\varphi$
gives the scalar power spectrum
\begin{equation}
\Delta_{\mathcal R}^2(k)
\;=\; \frac{1}{8\pi^2}\frac{H^2}{\Mplred^2}\frac{1}{\epsilon}
\bigg|_{k = aH},
\qquad
n_s - 1 \equiv \frac{d\ln\Delta_{\mathcal R}^2}{d\ln k}
= -6\epsilon + 2\eta ,
\label{eq:E-spectrum}
\end{equation}
the tilt arising purely because $H$ and $\epsilon$ drift slowly
between the exits of different modes. The same calculation for the
two tensor polarizations gives
$\Delta_h^2 = 2H^2/\pi^2\Mplred^2$, whence
$r_{\mathrm T} = \Delta_h^2/\Delta_{\mathcal R}^2 = 16\epsilon$,
completing Eq.~\eqref{eq:E-observables}.

\emph{Physics intuition.} The sky's largest structures are the
frozen shadows of vacuum fluctuations that happened to be crossing
the horizon while the universe was inflating --- and the tilt is a
clock reading: it measures how much the expansion rate changed
between the exit of one scale and the next. That is why a single
number, $N^*$, controls both observables on a plateau.

\paragraph{Reheating, and why $N^*$ is usually a range.} After slow
roll fails the field oscillates about its minimum and decays,
reheating the universe to $T_{\mathrm{rh}}$. Matching the observed
scales back through this epoch gives
\begin{equation}
N^{*} \simeq 62 - \ln\frac{k_*}{a_0H_0}
+ \ln\frac{V_*^{1/4}}{10^{16}\,\mathrm{GeV}}
+ \frac{1-3w_{\mathrm{rh}}}{3(1+w_{\mathrm{rh}})}
\ln\frac{\rho_{\mathrm{rh}}^{1/4}}{V_*^{1/4}} ,
\label{eq:E-Nstar}
\end{equation}
so $N^*$ depends on the unknown reheating history --- which is why
the standard practice is to quote $50$--$60$ and let the data
choose. Everything in this appendix's Step~3 turns on removing that
freedom.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:E-step2}

\emph{The potential is chosen.} Eq.~\eqref{eq:E-observables} maps a
potential to observables, and hundreds of potentials have been
written. Nothing in the framework prefers one; the plateau class is
favored by data, not derived.

\emph{The initial condition is stipulated.} Slow roll requires the
field to begin high on the plateau, over a region already large
enough to inflate. Since inflation cannot be past-eternal
\citep{BordeGuthVilenkin2003}, the question survives its own
solution (Problem~5).

\emph{The e-fold count is a fit.} $N^*$ depends on the reheating
history, which is unknown; the standard practice is to quote a
range $50$--$60$ and let the data pick. The observables of
Eq.~\eqref{eq:E-observables} are therefore predictions only up to
that range.

\emph{Naturalness is not a law.} The one-loop estimate assumes the
cutoff is physical and that no structure intervenes. It told a
generation where to look, and the null result at the LHC is its
clearest verdict (Problem~6).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:E-step3}

\paragraph{The inflaton is the interval, and there is only one.}
The census of Part~II forbids the menu: the bulk has exactly one
non-compact non-spatial direction, so it has exactly one field of
this kind, and the plateau shape is what the boundary law's anomaly
structure permits rather than what fits. The roll happens
\emph{once} --- Eq.~\eqref{eq:A-semigroup} has no inverse --- which
retires eternal inflation structurally rather than by fiat.

\paragraph{$N^*$ is a length, not a fit.} On the anatomy the
e-fold count is the traversed portion of a fiber of definite
extent, and the cohort derives $N^* = 52.6$ with the reheating
match made explicit
\citep{ArisakaA10Cosmos}. Eq.~\eqref{eq:ns-r} then reads
$n_s = 0.9620$ and $r_{\mathrm T} = 0.0043$ as \emph{numbers}, and
the tensor value sits an order of magnitude above the design floors
of the coming polarization missions. Grade: derived-conditional
(the plateau-class argument and the prompt-crossing lemma are
named).

\paragraph{The hierarchy is the interval's depth.} The weak scale
is not a small number surviving a large correction but an address:
$\Mlock = \tfrac12 M_{\mathrm{Pl}}e^{-K}$, Eq.~\eqref{eq:mlock},
with $K = 35$ fixed by the ledger identity $K = LJ$, whose factors the
dual-lock anomaly condition fixes upstream (one chain, audited at
every step; see Problem~6 for why it is not two).
Eq.~\eqref{eq:radiative-w2}'s $\Lambda$ is then not free: the
descent supplies its own completion at the lock, and the loop's
reach is priced by the same integer that sets the scale. Nothing
cancels; something is deep.

\paragraph{And the three horizons separate.} Inflation buys the
first agreement of Problem~8 --- shared conditions --- and only
that. The sameness of particle \emph{kinds} is circle-side
(classes cannot vary by region), and the sameness of the
\emph{choices} made after the stretch is the groove's (a quantized
dial with a unique winner). The standard treatment conflates all
three because it has one mechanism; the anatomy separates them
because it has two fibers and a selection theorem.

\paragraph{Why the interval's coordinate \emph{is} the
inflaton.} The identification is not an analogy. On
$\mathcal M_4\times\Ir\times\Stheta$ the position of the boundary
along $\Ir$ is a four-dimensional scalar field: writing the induced
metric with a warp $e^{-2A(r)}$, the fluctuation of the boundary's
location $r(x)$ has a canonically normalized kinetic term and a
potential inherited from the warp,
\begin{equation}
\mathcal L_{\mathrm{eff}} \;=\;
\tfrac12 (\partial_\mu \varphi)^2 - V(\varphi),
\qquad
\varphi \propto \Mplred\, r,
\label{eq:E-radion}
\end{equation}
--- one field, because there is one such coordinate. This is the
census statement of Part~II made technical: the menu of inflaton
candidates is not narrowed by taste but \emph{closed} by the
bulk's dimension count. The same identification fixes the shape's
class: the potential is the warp, and the warp is determined by
the boundary law's anomaly structure rather than chosen, giving the
plateau family whose small-$\epsilon$ region is
Eq.~\eqref{eq:ns-r}'s parent.

\paragraph{Why the roll happens once.} In Eq.~\eqref{eq:E-radion}
the field is a \emph{position}, and positions on $\Ir$ compose by
Eq.~\eqref{eq:A-kadanoff}: the physical transport has no inverse.
An eternally inflating patch would require the coordinate to be
re-randomized upward somewhere, which is precisely the operation
the semigroup does not contain. The structural retirement of
eternal inflation --- and hence of the measure problem that
Problem~15 inherits --- is therefore a theorem about the fiber's
algebra, not a preference about initial conditions.

\paragraph{$N^{*}$ as a length, computed.} Because $\Ir$ has
endpoints, the quantity that Eq.~\eqref{eq:E-Nstar} leaves free is
determined. The descent's full length is $K = 35$ in the ledger's
units, and the observable window's position within it is fixed by
two things the cohort computes rather than assumes: the reheating
temperature (from the warp's exit behavior) and the $g_*$ counting
at that temperature. Carrying those through
\citep{ArisakaA10Cosmos} gives $N^{*} = 52.6$, and hence
\begin{equation}
n_s = 1 - \frac{2}{52.6} = 0.9620,
\qquad
r_{\mathrm T} = \frac{12}{52.6^2} = 0.0043 ,
\label{eq:E-numbers}
\end{equation}
with no free parameter left anywhere in the chain. Grade:
derived-conditional --- the plateau-class argument and the
prompt-crossing lemma are the named conditions, and both are stated
in the parent paper.

\emph{Physics intuition.} In the standard treatment $N^*$ is an
uncertainty; here it is a \emph{measurement of the fiber's
geometry}. The two numbers in Eq.~\eqref{eq:E-numbers} are
therefore not two predictions but one: a single length, read twice.
Falsifying either falsifies the length.

\paragraph{The hierarchy as an integrated warp.} The same warp that
supplies Eq.~\eqref{eq:E-radion}'s potential supplies the scale
hierarchy. A four-dimensional mass parameter located at position
$r$ along the interval is measured on the boundary as
\begin{equation}
m_{\mathrm{eff}}(r) \;=\; m_0\,e^{-r},
\qquad
\text{so}\qquad
\Mlock = \tfrac12 M_{\mathrm{Pl}}\,e^{-K}
\ \ \text{at}\ \ r = K = 35 ,
\label{eq:E-warp}
\end{equation}
which is Eq.~\eqref{eq:mlock}. Two remarks make this more than
relabeling. First, the exponent is not tunable: it is a ledger
integer, fixed twice over (\S\ref{app:D-step3}). Second, the loop
correction of Eq.~\eqref{eq:radiative-w2} is cut off not at an
arbitrary $\Lambda$ but at the address where the descent supplies
its own completion --- so the correction is bounded by the same
$e^{-K}$ that sets the scale, and the ratio
$\delta m_H^2/m_H^2$ is $O(1)$ rather than $10^{32}$. There is no
cancellation anywhere in this argument, which is exactly why it
does not require one.

\subsection{Step 4 --- problems and worked solutions}
\label{app:E-problems}

\medskip\noindent\textbf{E.1. Show that $|\Omega-1|$ decreases
during inflation and increases afterwards.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:E-friedmann}, $\Omega - 1 = k/(aH)^2$.
\emph{Step 2.} During inflation $aH$ grows nearly exponentially, so
$|\Omega-1| \propto e^{-2N}$. \emph{Step 3.} In radiation and
matter eras $aH$ falls, so $|\Omega-1|$ grows --- which is why the
present near-flatness requires either an enormous initial tuning or
an inflationary episode. \emph{Physics intuition:} flatness is not
a coincidence to be explained once but an instability to be
outrun; the stretch buys enough margin that sixty e-folds of later
growth still leave the universe flat to a per-cent.

\medskip\noindent\textbf{E.2. Derive the plateau limits
$n_s = 1 - 2/N^*$ and $r_{\mathrm T} = 12/N^{*2}$.}

\noindent\emph{Solution.} \emph{Step 1.} For the plateau class
$\epsilon \simeq 3/(4N^2)$ and $\eta \simeq -1/N$ at $N$ e-folds
before the end. \emph{Step 2.} Substituting into
Eq.~\eqref{eq:E-observables}: $n_s - 1 = -6(3/4N^2) + 2(-1/N)
\simeq -2/N$ to leading order. \emph{Step 3.} And
$r_{\mathrm T} = 16(3/4N^2) = 12/N^2$. \emph{Physics intuition:}
on a plateau the only scale left is ``how far from the end,'' so
both observables must be functions of $N$ alone --- which is why a
derived $N^*$ converts two measured numbers into two tests at
once.

\medskip\noindent\textbf{E.3. With $N^* = 52.6$, evaluate both.}

\noindent\emph{Solution.} \emph{Step 1.}
$n_s = 1 - 2/52.6 = 0.96198$. \emph{Step 2.}
$r_{\mathrm T} = 12/52.6^2 = 0.004337$. \emph{Step 3.} Compare
$n_s^{\mathrm{obs}} = 0.9649 \pm 0.0042$ (pull $-0.7$) and
$r_{\mathrm T} < 0.036$ (consistent; the Simons Observatory and LiteBIRD reach
toward $\sigma(r) \simeq 10^{-3}$, several times below $0.0043$). \emph{Physics intuition:} the tensor number is the
sharpest cosmological kill condition in this paper --- a null at
the design floors removes the derived launch, and a detection near
$0.004$ would be a photograph of the interval's first traverse.

\medskip\noindent\textbf{E.4. Estimate the fine-tuning of
Eq.~\eqref{eq:radiative-w2} and restate it as a depth.}

\noindent\emph{Solution.} \emph{Step 1.} With
$y_t \simeq 1$ and $\Lambda = M_{\mathrm{Pl}}$,
$|\delta m_H^2| \sim (3/8\pi^2)(1.2\e{19})^2 \approx
5\e{36}~\mathrm{GeV^2}$. \emph{Step 2.} With
$m_H^2 = (125)^2 \approx 1.6\e{4}~\mathrm{GeV^2}$, the ratio is
$\sim 3\e{32}$ --- the famous thirty-plus digits. \emph{Step 3.}
Written as a depth instead: $\ln(M_{\mathrm{Pl}}/2\Mlock) = 35$,
a single integer. \emph{Physics intuition:} the same fact is
``impossible'' in one language and ``$K = 35$'' in the other; the
choice of language is exactly what this paper is arguing about,
and the arbitration is whether anything sits at $3.849$~TeV.

\GGtakeaway{Inflation's mechanism is inherited whole; what the
anatomy supplies is the census that forbids the menu, the length
that fixes $N^*$, and the reading in which the hierarchy is an
address rather than a cancellation. All three are exposed: the
tensor floor, the lock, and the entrance.}

\section{The dark sector: misalignment, the Kaxion, and the
vacuum}
\label{app:dark}
\GGteachbox{The axion from the Peccei--Quinn mechanism, and the
misalignment relic computed; the haloscope detection principle;
the $\Lambda$CDM inventory and how its shares are measured; the
cosmological constant problem stated in its full severity; and then
the two-fiber reading in which the dark sector is the pair of
non-spatial coordinates in three states of motion. Pay-off:
Problem~7.}
\GGreadalong{\citet{KolbTurner1990}, Ch.~10, for relic abundances;
\citet{PreskillWiseWilczek1983} for the misalignment calculation in
the original; \citet{Weinberg1989CC} for the vacuum-energy problem
as it stood before the discovery.}

\subsection{Step 1 --- the standard treatment}
\label{app:F-step1}

\paragraph{Why an axion exists, in the standard story.} QCD permits
the $CP$-violating term $\bar\theta\,(g_s^2/32\pi^2)
G\tilde G$, and the neutron EDM bounds $|\bar\theta| \lesssim
10^{-10}$ \citep{Abel2020}. Peccei and Quinn promoted $\bar\theta$
to a dynamical field whose potential --- generated by QCD
instantons --- is minimized at zero, so the angle relaxes rather
than being tuned. The pseudo-Goldstone boson of the broken
symmetry is the axion, with mass and decay constant related by
\begin{equation}
m_a \;\simeq\; \frac{\sqrt{\chi_{\mathrm{QCD}}}}{f_a},
\qquad
\chi_{\mathrm{QCD}} \approx (75.5~\mathrm{MeV})^4 ,
\label{eq:F-axionmass}
\end{equation}
$\chi_{\mathrm{QCD}}$ being the QCD topological susceptibility. A
larger decay constant means a lighter, more weakly coupled axion.

\paragraph{The misalignment relic.} Before QCD becomes strong the
axion potential is negligible and the field sits at an arbitrary
initial angle $\theta_i$. When the Hubble rate falls below the
axion mass, $3H \simeq m_a$, the field begins to oscillate; the
oscillation energy redshifts like matter, $\rho_a \propto a^{-3}$,
and survives as cold dark matter. The relic density scales as
\begin{equation}
\Omega_a h^2 \;\propto\; \theta_i^2 \left(
\frac{f_a}{10^{12}~\mathrm{GeV}}\right)^{7/6},
\label{eq:F-relic}
\end{equation}
the exponent $7/6$ coming from the temperature dependence of
$\chi_{\mathrm{QCD}}$ near the QCD crossover (the prefactor is
computed in the references, and carries the largest theory
uncertainty in the field).

\paragraph{Haloscopes.} A galactic-halo axion converts to a
microwave photon in a strong magnetic field, resonantly if the
cavity is tuned to $\nu = m_a c^2/h$. The signal power scales as
$B^2 V Q$, and the search proceeds by scanning frequency ---
which is why the axion's \emph{mass} is the crucial unknown: it
sets where to look.

\paragraph{The inventory, and the vacuum.} The measured shares are
$\Omm \simeq 0.315$, $\Omb \simeq 0.049$, $\OmL \simeq 0.685$
\citep{Planck2020}. The last is the problem: a naive estimate of
the vacuum energy from zero-point modes up to the Planck scale
exceeds the measured value by $\sim 10^{120}$, and no symmetry is
known that sets it to the observed small non-zero number
\citep{Weinberg1989CC}.

\paragraph{The instanton potential, and where
Eq.~\eqref{eq:F-axionmass} comes from.} QCD's vacuum energy depends
on the effective angle $\bar\theta$ through the topological
susceptibility,
\begin{equation}
V(\bar\theta) \;=\; -\chi_{\mathrm{QCD}}\cos\bar\theta
\;\;\Longrightarrow\;\;
m_a^2 \;=\; \left.\frac{\partial^2 V}{\partial a^2}\right|_{\min}
= \frac{\chi_{\mathrm{QCD}}}{f_a^2},
\label{eq:F-instanton}
\end{equation}
where $a = f_a\bar\theta$ is the canonically normalized field. The
susceptibility itself is computable in chiral perturbation theory:
$\chi_{\mathrm{QCD}} = m_u m_d\,f_\pi^2 m_\pi^2/(m_u+m_d)^2$,
giving $\chi^{1/4} \approx 75.5$~MeV. \emph{Physics intuition:} the
axion's mass is not a free parameter of the axion --- it is QCD's
own vacuum stiffness divided by the scale at which the field lives.
A larger $f_a$ means a flatter potential and a lighter, more weakly
coupled particle; the entire experimental landscape is that one
inverse proportionality.

\paragraph{The relic density, derived rather than quoted.} Before
QCD confines, $\chi(T)$ is suppressed and the field is frozen by
Hubble friction at its initial value $\theta_i$. Oscillation begins
when
\begin{equation}
3H(T_{\mathrm{osc}}) \;=\; m_a(T_{\mathrm{osc}}),
\label{eq:F-osc}
\end{equation}
after which the energy density
$\rho_a \simeq \tfrac12 m_a^2 f_a^2\theta_i^2$ redshifts as
$a^{-3}$ (Problem~F.2), while the comoving number density
$n_a/s$ is conserved. Tracking $n_a/s$ from $T_{\mathrm{osc}}$ to
today and using the measured entropy density gives
Eq.~\eqref{eq:F-relic}; the exponent $7/6$ comes from the
temperature dependence $\chi(T) \propto T^{-n}$ with $n \approx 8$
near the crossover, entering through $T_{\mathrm{osc}}$'s
dependence on $f_a$. The largest theory uncertainty in the whole
subject sits in that $n$, which is why the prefactor is quoted with
a factor-of-two band in the literature.

\paragraph{The haloscope, quantitatively.} In a magnetic field the
axion mixes with the photon through
$\mathcal L_{a\gamma} = -\tfrac14 g_{a\gamma}\,a\,
F_{\mu\nu}\tilde F^{\mu\nu}$, so a halo axion of mass $m_a$
converts resonantly in a cavity tuned to
$\nu = m_ac^2/h$. The signal power is
\begin{equation}
P_{\mathrm{sig}} \;=\; g_{a\gamma}^2\,
\frac{\rho_a}{m_a}\,B_0^2\,V\,C\,Q_L ,
\label{eq:F-haloscope}
\end{equation}
with $B_0$ the field, $V$ the volume, $C$ a mode-overlap factor,
and $Q_L$ the loaded quality factor. With
$\rho_a \simeq 0.45~\mathrm{GeV/cm^3}$ this is of order
$10^{-23}$~W --- which is why haloscopes are among the most
sensitive radio receivers ever built, and why the scan rate, not
the sensitivity, is the binding constraint.

\paragraph{How the inventory is measured.} The shares of
Eq.~\eqref{eq:inventory-p2} are not weighed directly. $\Omega_b$
comes from the odd/even acoustic peak ratio in the CMB (baryons
load the photon-baryon fluid and enhance compressions);
$\Omega_m$ from the peak positions and the matter-radiation
equality scale; $\Omega_\Lambda$ from flatness plus the
distance--redshift relation of supernovae and BAO. Three different
physical effects, three independent handles --- which is why the
inventory is now known to per-cent precision and why any derived
closed form for it is sharply testable.

\paragraph{The vacuum problem, stated at full strength.} Summing
zero-point energies of a field up to a cutoff gives
$\rho_{\mathrm{vac}} \sim \Lambda^4/16\pi^2$. With
$\Lambda = M_{\mathrm{Pl}}$ this exceeds the measured
$\rho_\Lambda \simeq (2.24~\mathrm{meV})^4$ by $\sim 10^{120}$.
Even with $\Lambda$ at the electroweak scale the discrepancy is
$10^{55}$. No symmetry is known that sets the result to zero
\emph{and} allows a small non-zero remainder --- supersymmetry
would give exact cancellation if unbroken, and its breaking
reintroduces a contribution far too large.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:F-step2}

\emph{The mass is unpredicted.} Eq.~\eqref{eq:F-axionmass} relates
$m_a$ to $f_a$, and $f_a$ is a free scale spanning many decades.
Experiments therefore scan rather than test, and a null result over
a decade of frequency excludes parameter space without indicating
where to look next.

\emph{The initial angle is free.} Eq.~\eqref{eq:F-relic} depends on
$\theta_i^2$, drawn at random in the standard picture --- which
also predicts isocurvature fluctuations and, in some scenarios,
topological defects.

\emph{The dark-matter identity is unconstrained by the
gravitational evidence.} The allowed mass range for a dark-matter
particle spans some ninety orders of magnitude; the evidence is
entirely gravitational and says almost nothing about what the
particle is.

\emph{The vacuum is anti-explained.} The framework's own estimate
is wrong by $10^{120}$, and the coincidence that
$\rho_\Lambda \sim \rho_m$ \emph{now} has no mechanism
(Problem~7).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:F-step3}

\paragraph{One circle, one axion --- with numbers.} The census
forbids the menu here as it did for the inflaton: one compact
non-spatial direction, hence one field of this kind. The cohort's
Kaxion has its decay constant fixed by the cap loading,
$\fK = f_{\mathrm{bulk}}/\Xicap$ with $\Xicap = K+N+L = 43$
(benchmark; the one open integer). Inverting Eq.~\eqref{eq:F-axionmass} at the cohort's
$\mK = 8.1\,\mu$eV with $\chi_{\mathrm{QCD}}^{1/4} = 76$~MeV gives
$\fK \simeq 7.1\e{11}$~GeV, hence a bulk scale
$f_{\mathrm{bulk}} = \Xicap \fK \simeq 3.1\e{13}$~GeV --- and a reader who wants to check the
chain end to end should run those two numbers through
Eq.~\eqref{eq:F-axionmass} rather than take the result on trust.
Eq.~\eqref{eq:F-axionmass}
then returns definite laboratory numbers,
Eq.~\eqref{eq:kaxion-numbers}: $\mK \simeq 8.1\,\mu$eV and a line
at $1.95$~GHz, with a scale-free doublet splitting
$\Delta\nu/\nu = 1/K = 1/35$. The experiment changes character:
one identifies by the circle's structure --- the doublet, the
discrete cap window --- rather than merely scanning for power.

\paragraph{The angle is pinned, not drawn.} Where the standard
picture draws $\theta_i$ at random, the anatomy's
reflection-symmetric dressing freezes the compact angle during the
roll, patch-uniformly. That is a conditional claim (the pinning is
named), and it is the cleanest null in cosmology: zero
axion-sector isocurvature and no defects. An isocurvature
detection kills it.

\paragraph{The trio, sorted.} Inflation is the interval's
coordinate rolling once; the Kaxion is the circle's coordinate
still ringing; dark energy is both at rest --- the terminus. The
shares are fiber-volume ratios (the Hopf reading,
Eq.~\eqref{eq:hopf}), giving Eq.~\eqref{eq:inventory-p2}'s
$\Omm = 1/\pi$, $\Omb = 1/2\pi^2$, $\OmL = 1-1/\pi$ against
measured $0.315$, $0.049$, $0.685$. The vacuum's \emph{scale} is
the lock-scale seesaw $m_\Lambda = \Mlock^2/\Mplred \simeq 6.1$~meV
--- the depth doubled --- with one named, bounded, uncomputed
coefficient ($\zeta_\Lambda$) and the $8K$ face carried at flagged
grade and never spent.

\paragraph{Two compact parents, one light eigenstate.} The
existence claim is a mixing computation. The compact sector carries
two angular directions with mass-squared matrix
\begin{equation}
\mathcal M^2 \;=\;
\begin{pmatrix}
\mu_1^2 & \delta^2 \\ \delta^2 & \mu_2^2
\end{pmatrix},
\qquad
\lambda_\pm = \tfrac12\Big[\mu_1^2 + \mu_2^2 \pm
\sqrt{(\mu_1^2-\mu_2^2)^2 + 4\delta^4}\Big],
\label{eq:F-mixing}
\end{equation}
and when the off-diagonal entry approaches the geometric mean of
the diagonals the lighter eigenvalue is driven far below either ---
the same algebra as a seesaw. The program's compact sector
delivers exactly this configuration, so \emph{one} light
pseudoscalar exists and its lightness is structural rather than
tuned. The census then forbids a second: there is one compact
non-spatial fiber, hence one tower of angular moduli, hence one
such eigenstate. This is why the anatomy cannot accommodate an
axion \emph{zoo} --- and why a second, unrelated axion-like
detection would be a serious problem for it.

\paragraph{The decay constant, fixed.} What remains is the scale.
The bulk's natural decay constant is divided by the cap loading,
\begin{equation}
\fK \;=\; \frac{f_{\mathrm{bulk}}}{\Xicap},
\qquad
\Xicap \;=\; K + N + L \;=\; 35 + 3 + 5 \;=\; 43 ,
\label{eq:F-fK}
\end{equation}
the three terms counting the descent's depth, the generation
multiplicity, and the block structure --- the loading the compact
sector must carry. Substituting into
Eq.~\eqref{eq:F-instanton} returns
Eq.~\eqref{eq:kaxion-numbers}: $\mK \simeq 8.1\,\mu$eV and a line
at $1.95$~GHz. The honest note, carried everywhere in this paper:
$\Xicap = 43$ is a \emph{benchmark}, the one open integer of
\citet{ArisakaC1Kaxion}, and the associated window is
$1.59$--$2.13$~GHz rather than a single frequency.

\paragraph{The doublet, and why it is the real signature.} The
same compact structure that fixes $\fK$ predicts a companion split
from the principal line by the descent's depth:
\begin{equation}
\frac{\Delta\nu}{\nu} \;=\; \frac{1}{K} \;=\; \frac{1}{35}
\;\;\Longrightarrow\;\;
\Delta\nu \simeq 56~\mathrm{MHz\ at\ 1.95~GHz}.
\label{eq:F-doublet}
\end{equation}
The ratio is \emph{scale-free}: it does not depend on $\Xicap$, on
$f_{\mathrm{bulk}}$, or on the halo density, so it survives every
uncertainty that afflicts Eq.~\eqref{eq:F-haloscope}'s
normalization. A haloscope that resolves two lines at this spacing
has measured a ledger integer directly.

\paragraph{The inventory as fiber volumes.} The shares are ratios
of the two fibers' volumes in the Hopf presentation of the bulk's
compact structure (Eq.~\eqref{eq:hopf}). Taking the circle's
one-dimensional volume against the three-sphere's,
\begin{equation}
\frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)}
= \frac{2\pi}{2\pi^2} = \frac{1}{\pi}
\;\;\Longrightarrow\;\;
\Omm = \frac1\pi = 0.3183,
\qquad
\Omb = \frac{1}{2\pi^2} = 0.0507 ,
\label{eq:F-hopfvols}
\end{equation}
against measured $0.3153 \pm 0.0073$ and $0.0493$, with
$\OmL = 1 - 1/\pi = 0.6817$ against $0.6847 \pm 0.0073$. The
content is that the inventory is a \emph{geometric} ratio, not a
dynamical outcome: it is fixed by which fibers exist, and so cannot
drift, which is the same protection statement as
\S\ref{app:D-step3}'s theorem.

\paragraph{The vacuum's scale, and the honest residue.} The
terminus energy is the lock scale seesawed against the Planck
scale,
\begin{equation}
m_\Lambda \;=\; \frac{\Mlock^2}{\Mplred}
\;=\; \frac{(3849~\mathrm{GeV})^2}{2.435\e{18}~\mathrm{GeV}}
\;=\; 6.1~\mathrm{meV},
\label{eq:F-seesawnum}
\end{equation}
against $\rho_\Lambda^{1/4} = 2.24$~meV --- a ratio
$\zeta_\Lambda^{1/4} = 0.37$, where $\zeta_\Lambda$ is the
\emph{drainage factor}: named, bounded, and \emph{uncomputed}. In
e-fold form Eq.~\eqref{eq:F-seesawnum} is $e^{-2K}$ --- the
descent's depth doubled --- and the observation that the vacuum
hierarchy's $120$ orders equal $8K$ to about one per cent is
carried at flagged grade and used nowhere. What the anatomy claims
here is therefore narrower and sharper than a solution: the
\emph{share} is derived, the \emph{scale} arrives from an
already-derived TeV address with no new dial, and one coefficient
remains a work order.

\subsection{Step 4 --- problems and worked solutions}
\label{app:F-problems}

\medskip\noindent\textbf{F.1. From Eq.~\eqref{eq:F-axionmass},
compute $m_a$ for $f_a = 10^{12}$~GeV and convert to a haloscope
frequency.}

\noindent\emph{Solution.} \emph{Step 1.}
$m_a \simeq (75.5\e{-3}~\mathrm{GeV})^2/(10^{12}~\mathrm{GeV})
= 5.7\e{-15}$~GeV $= 5.7\,\mu$eV. \emph{Step 2.} The frequency is
$\nu = m_ac^2/h = 5.7\e{-6}\,\mathrm{eV} \times
2.418\e{14}~\mathrm{Hz/eV} = 1.38$~GHz. \emph{Step 3.} For the
cohort's $8.1\,\mu$eV the same conversion gives $1.96$~GHz.
\emph{Physics intuition:} the microwave band is not a coincidence
of engineering --- it is where the QCD scale divided by a
grand-unified-scale decay constant lands, which is why cavity
haloscopes are the right instrument at all.

\medskip\noindent\textbf{F.2. Show that coherent oscillation energy
redshifts like matter.}

\noindent\emph{Solution.} \emph{Step 1.} For
$\ddot\phi + 3H\dot\phi + m^2\phi = 0$ with $m \gg H$, write
$\phi = A(t)\cos(mt)$. \emph{Step 2.} Averaging over oscillations,
$\rho = \tfrac12 m^2 A^2$ and the damping term gives
$\dot\rho = -3H\rho$. \emph{Step 3.} Hence
$\rho \propto a^{-3}$: pressureless matter. \emph{Physics
intuition:} a field oscillating fast compared to the expansion is
indistinguishable, gravitationally, from a gas of cold particles of
mass $m$ --- which is how a wave becomes ``dark matter'' without
anything particle-like ever being produced.

\medskip\noindent\textbf{F.3. Why does the $1/35$ doublet matter
more than the line itself?}

\noindent\emph{Solution.} \emph{Step 1.} A single line at a
predicted frequency is a strong but reproducible-by-accident
result: many models can be tuned to a frequency. \emph{Step 2.} A
\emph{splitting} fixed at $\Delta\nu/\nu = 1/K$ with no free
parameter is a structural signature: it tests the ledger integer
directly, and its scale-freedom means it cannot be absorbed by
recalibrating $f_a$. \emph{Step 3.} Hence the search strategy of
Problem~7: identify by the circle. \emph{Physics intuition:}
detection experiments answer ``is something there?''; identification
experiments answer ``is it \emph{this}?'' --- and only the second
kind can discriminate a geometry.

\medskip\noindent\textbf{F.4. Restate the cosmological constant
problem in the two-fiber language and say precisely what remains
open.}

\noindent\emph{Solution.} \emph{Step 1.} The \emph{share} is
geometric: $\OmL = 1 - 1/\pi$, a fiber-volume ratio, derived.
\emph{Step 2.} The \emph{scale} is the interval's depth doubled,
$m_\Lambda = \Mlock^2/\Mplred$, which reproduces the meV scale with
no new dial --- but with one open coefficient $\zeta_\Lambda$,
named and bounded. \emph{Step 3.} What remains open is therefore
not ``why is it small'' but ``compute $\zeta_\Lambda$'' --- a
narrower and more answerable question, plus the flagged $8K$
observation that is never used as support. \emph{Physics
intuition:} progress here looks like repricing rather than
solving, and the paper's discipline is to say so: a problem
converted into a computation is not the same as a problem closed.

\GGtakeaway{The dark sector, on this reading, is not a zoo awaiting
a discovery but the two hidden coordinates in their three possible
states of motion --- rolling, ringing, at rest. Every one of the
three is exposed: the tensor floor, the haloscope window with its
doublet, and the isocurvature null.}

\section{Discrete symmetries, $CPT$, and the matter excess}
\label{app:discrete}
\GGteachbox{$C$, $P$, and $T$ defined as operators, with $T$'s
antiunitarity derived rather than asserted; the $CPT$ theorem's
three hypotheses and why it is a theorem at all; the CKM phase and
the Jarlskog invariant; Sakharov's three conditions; sphalerons and
why they conserve $B-L$; leptogenesis; and then the two-fiber
reading in which the three operations are three orientation
reversals and the excess is a toll. Pay-off: Problems~9 and
10--11.}
\GGreadalong{\citet{WeinbergQTF1}, Ch.~2 and~5, for the discrete
operators and the $CPT$ theorem; \citet{PeskinSchroeder1995},
Ch.~3, for $C$, $P$, $T$ on Dirac fields;
\citet{KRS1985} for the sphaleron rate; \citet{FukugitaYanagida1986}
for leptogenesis in the original.}

\subsection{Step 1 --- the standard treatment}
\label{app:G-step1}

\paragraph{The three operators.} On a Dirac field, parity acts as
$P\psi(t,\vec x)P^{-1} = \gamma^0\psi(t,-\vec x)$; charge
conjugation as $C\psi C^{-1} = i\gamma^2\psi^{*}$, exchanging
particle and antiparticle; and time reversal as
$T\psi(t,\vec x)T^{-1} = \gamma^1\gamma^3\psi(-t,\vec x)$, with
$T$ \emph{antiunitary}. The antiunitarity is not a convention: from
$[x,p] = i\hbar$ and $T x T^{-1} = x$, $T p T^{-1} = -p$, one gets
$T (i\hbar) T^{-1} = -i\hbar$, so $T$ must conjugate complex
numbers (Problem~G.1).

\paragraph{The $CPT$ theorem.} Any local, Lorentz-invariant quantum
field theory with a positive-energy spectrum and the usual
spin--statistics connection is invariant under the product $CPT$.
Two of its consequences are exact and heavily tested: a particle
and its antiparticle have identical mass and lifetime; and no
laboratory has ever seen a violation.

\paragraph{$CP$ violation and the Jarlskog invariant.} With three
generations, the CKM matrix retains one physical phase
(Problem~D.1). All $CP$-violating effects in the quark sector are
proportional to the single rephasing-invariant combination
\begin{equation}
J_{CP} \;=\; \mathrm{Im}\!\left(V_{ud}V_{cs}V_{us}^{*}V_{cd}^{*}\right)
\;\approx\; 3\e{-5},
\label{eq:G-jarlskog}
\end{equation}
and $J_{CP}$ vanishes if any two quark masses of the same charge
are degenerate. This is the measured $CP$ violation --- and it is
famously too small for cosmology. \emph{A note on notation}: the
symbol $J$ is reserved throughout this paper for the ledger integer
$J = 2N+1 = 7$, so the Jarlskog invariant is written $J_{CP}$ here
and the Regge spin $\mathcal{J}$ in Appendix~H. The clash is
standard in the literature and worth naming once rather than
letting a reader meet $J = 7$ and $J \approx 3\times10^{-5}$ on
facing pages.

\paragraph{Sakharov, sphalerons, leptogenesis.} A baryon asymmetry
requires baryon-number violation, $C$ and $CP$ violation, and
departure from equilibrium. The Standard Model supplies all three
in principle: $B+L$ is violated non-perturbatively by sphalerons
\citep{KRS1985} while $B-L$ is exactly conserved; $CP$ violation
comes from Eq.~\eqref{eq:G-jarlskog}; and the electroweak
transition could in principle be out of equilibrium. Quantitatively
it fails: the quark phase delivers an asymmetry at least ten orders
of magnitude too small \citep{Gavela1994}. Leptogenesis moves the
origin to heavy right-handed neutrinos whose out-of-equilibrium
decays create a lepton asymmetry, which sphalerons then partially
convert to baryons --- with the standard conversion factor
$B = (28/79)(B-L)$ for Standard-Model content
\citep{KRS1985}.

\paragraph{The bilinears, tabulated.} The fastest way to see
what $C$, $P$, $T$ do is to act on the fermion bilinears, since
every interaction in the Standard Model is built from them. Writing
$\bar\psi\Gamma\psi$ for the five Dirac structures
(Table~\ref{tab:bilinears-w2}).
\begin{table}[htbp]
\centering
\caption[$C$, $P$, $T$ and $CPT$ acting on the five Dirac bilinears.]{$C$, $P$, $T$ and $CPT$ acting on the five Dirac bilinears, which is the fastest way to see what the three operations do, since every Standard Model interaction is built from them. The columns are the scalar, pseudoscalar, vector, axial-vector and tensor structures; each entry is the sign the operation returns, with the spatial components noted where they differ from the time component. The last row is the product, and it is the one that matters for Problem~9: $CPT$ returns $+1$ on the scalar and the pseudoscalar, $-1$ on the vector and the axial-vector, and $+1$ on the tensor.}
\label{tab:bilinears-w2}
\small
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}lccccc@{}}
\toprule
& $\bar\psi\psi$ & $\bar\psi i\gamma^5\psi$ &
$\bar\psi\gamma^\mu\psi$ & $\bar\psi\gamma^\mu\gamma^5\psi$ &
$\bar\psi\sigma^{\mu\nu}\psi$ \\
\midrule
$P$ & $+1$ & $-1$ & $+1$ (spatial $-$) & $-1$ (spatial $+$) &
$+1$ (mixed) \\
$C$ & $+1$ & $+1$ & $-1$ & $+1$ & $-1$ \\
$T$ & $+1$ & $-1$ & $+1$ (spatial $-$) & $+1$ (spatial $-$) &
$-1$ \\
$CPT$ & $+1$ & $+1$ & $-1$ & $-1$ & $+1$ \\
\bottomrule
\end{tabular}
\end{table}
Two lessons. A vector interaction ($\bar\psi\gamma^\mu\psi A_\mu$)
is $P$- and $C$-even, so electromagnetism and QCD respect both. An
interaction mixing vector and axial ($V-A$, as in the weak
current) cannot be: the two pieces have opposite $P$, so any theory
that uses their difference violates parity maximally --- which is
what \citet{Wu1957} measured. \emph{Physics intuition:} parity
violation is not a small effect that had to be searched for
carefully; it is total, and it was missed for a quarter-century
only because nobody had looked in the one sector where the world
distinguishes left from right.

\paragraph{Why $CPT$ is a theorem, in outline.} The proof rests on
three hypotheses: Lorentz invariance, locality (fields at spacelike
separation commute or anticommute), and a spectrum bounded below.
The mechanism is analytic continuation: correlation functions of a
local field theory extend to complex Lorentz transformations, and
the element $-\mathbb{1}$ of the complexified group --- which
reverses all four coordinates --- is connected to the identity in
the complexified group even though it is not in the real one.
Implementing that element on the Hilbert space requires an
antiunitary operator, and it acts precisely as $CPT$.

\emph{Physics intuition.} $CPT$ is protected not by a dynamical
mechanism but by the \emph{shape of the symmetry group}, once the
theory is local and the energies are positive. That is why no
model-building can violate it without abandoning one of those
three hypotheses --- and why an observed violation would be far
more damaging to physics than the discovery of parity violation
was.

\paragraph{The CKM matrix, parametrized.} In the standard
parametrization
\begin{equation}
V_{\mathrm{CKM}} =
\begin{pmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} &
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} &
-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13}
\end{pmatrix},
\label{eq:G-ckm}
\end{equation}
with $s_{ij} = \sin\theta_{ij}$. The measured hierarchy is
striking: $s_{12} \approx 0.22$, $s_{23} \approx 0.042$,
$s_{13} \approx 0.0037$, with $\delta \approx 1.14$~rad. The
Jarlskog invariant of Eq.~\eqref{eq:G-jarlskog} evaluates to
$J_{CP} = c_{12}c_{23}c_{13}^2 s_{12}s_{23}s_{13}\sin\delta \approx
3\e{-5}$ --- small not because the phase is small (it is of order
one) but because the mixing angles are.

\paragraph{Sphalerons, quantitatively.} The electroweak vacuum has
a periodic structure labeled by Chern--Simons number, and the
barrier between adjacent vacua has height
\begin{equation}
E_{\mathrm{sph}} \;\simeq\; \frac{4\pi v}{g}\,B(\lambda/g^2)
\;\approx\; 9~\mathrm{TeV} ,
\label{eq:G-sphaleron}
\end{equation}
with $B$ a slowly varying function of order unity. Below the
electroweak crossover the transition rate is Boltzmann-suppressed
by $e^{-E_{\mathrm{sph}}/T}$ and switches off rapidly; above it,
the rate per unit volume is unsuppressed,
$\Gamma/V \sim \alpha_W^5 T^4$, and $B+L$ is efficiently
violated. Each transition changes $B$ and $L$ by
$\Delta B = \Delta L = N_g = 3$ --- one unit per generation ---
so $B-L$ is untouched (Problem~G.4).

\paragraph{Leptogenesis, assembled.} A heavy right-handed neutrino
$N_1$ of mass $M_1$ decays to $\ell H$ and $\bar\ell H^\dagger$
with a $CP$ asymmetry
$\epsilon_1 = [\Gamma(N_1\to \ell H) - \Gamma(N_1 \to \bar\ell
H^\dagger)]/\Gamma_{\mathrm{tot}}$ generated by interference
between tree and one-loop diagrams. With a washout efficiency
$\kappa$ and dilution by the effective degrees of freedom $g_*$,
the resulting baryon asymmetry is
\begin{equation}
\etaB \;\simeq\; 7.04 \times \frac{28}{79} \times
\frac{\epsilon_1\,\kappa}{g_*} ,
\label{eq:G-etachain}
\end{equation}
where $7.04$ converts the entropy-normalized asymmetry to the
photon-normalized one and $28/79$ is the sphaleron reprocessing
factor for Standard-Model content. Every factor except
$\epsilon_1\kappa$ is standard and computed; the mechanism's whole
freedom sits in that one product.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:G-step2}

\emph{The pattern is accommodated, not produced.} Parity violation
is written in by assigning left- and right-handed fields to
different representations; the $CP$ phase is a measured input; and
$\bar\theta$ is a free number that experiment says is zero to ten
orders. Each is a dial set to match the world.

\emph{$CPT$'s exactness is proved from assumptions.} The theorem
tells us the three operations are linked; it does not say why the
geometry offers exactly three, nor why the joint operation should
be the protected one.

\emph{The excess is fitted.} Leptogenesis is a mechanism with free
masses and phases at energies no experiment reaches. Any observed
$\etaB$ can be accommodated, which means the observed value has
never been predicted (Problems~10--11).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:G-step3}

\paragraph{Three operations because there are three
orientations.} The bulk offers exactly three independent senses to
reverse: the circle's winding, the stage's handedness, and the
clock's direction, Eq.~\eqref{eq:orientations-w2}. $C$, $P$, $T$
are those reversals, so the census of discrete symmetries is a
census of geometry --- and the count is three because the anatomy
has three orientable structures, not because three operators
happened to be useful. $CPT$ is the total reversal, an isometry of
the whole, hence exact; each single reversal fails wherever a
structure fixes its sense --- the chiral flavor alphabet fixes the
circle's, the runtime fixes the clock's.

\paragraph{The alphabet, audited.} The flavor lattice's phases are
exact lattice values rather than fitted numbers: the quark phase is
a \emph{bystander} (real chirality, negligible throughput), the
Dirac neutrino phase a \emph{spectator} at the derived
$\delta_{\mathrm{PMNS}} = \pi + 2\arcsin(1/7) = 196.4^\circ$,
$\alpha_{21} = 0$ \emph{inert}, and $\alpha_{31} = \pi$ the
\emph{driver} whose half-angle contributes $e^{i\pi/2} = i$,
Eq.~\eqref{eq:driver-w2}. Grade: derived for the lattice values,
derived-conditional for the assignment.

\paragraph{The excess as a paid toll.} The chain of
Eq.~\eqref{eq:etaB} carries the geometry's own factors: the
efficiency-times-asymmetry product $\epsilon\kappa = e^{-K/2}$
(the $K/2$ exponent is the chain's one \emph{postulated} step,
declared as such), the standard sphaleron reprocessing $28/79$, and
dilution by $g_*$. The result, $7.34\e{-10}$ against a measured
$6.12\e{-10}$ ($+20\%$), is a computation with named joints rather than
a fit --- and the door through which the excess passed is
welded shut afterwards by the same electroweak crossover
(Problem~12's guards).

\paragraph{The three reversals as bundle operations.} Make
Eq.~\eqref{eq:orientations-w2} precise. The bulk
$\mathcal M_4\times\Ir\times\Stheta$ has exactly three structures
that admit an orientation, and each admits exactly two:
\begin{equation}
\begin{aligned}
&\text{the compact fiber } \Stheta:
&&\theta \mapsto -\theta, &&\text{winding } n \mapsto -n
\ \Rightarrow\ \text{all charges negate},\\
&\text{the spatial slice of } \mathcal M_4:
&&\vec x \mapsto -\vec x, &&\text{handedness reverses},\\
&\text{the boundary clock:}
&&t_b \mapsto -t_b, &&\text{the reading's sense reverses}.
\end{aligned}
\label{eq:G-threeorient}
\end{equation}
The interval carries \emph{no} orientation to reverse in this
sense: reversing $r$ is not a symmetry operation but a request to
run Eq.~\eqref{eq:A-kadanoff} backwards, which the semigroup does
not contain. So the census of discrete symmetries has exactly
three entries because the geometry has exactly three orientable
structures --- and the count is a fact about the bulk, not about
which operators happened to be useful.

\paragraph{$CPT$ as the total orientation, made exact.} The product
of the three reversals of Eq.~\eqref{eq:G-threeorient} acts on the
bulk as an orientation-reversing isometry of the whole. Since it
maps the geometry to itself, every quantity constructed from the
geometry is invariant, so no structure exists that could
distinguish the two total senses --- there is nothing left over to
notice the flip. That is why breaking $CPT$ would require
\emph{a fourth} orientable structure, which the bulk does not
contain. The reading also explains what the standard theorem
leaves implicit: the three hypotheses of the proof (locality,
Lorentz invariance, positive energy) are, on the anatomy, three
ways of saying the geometry is the one Eq.~\eqref{eq:bulk}
describes.

\emph{Physics intuition.} Each single reversal can fail because a
\emph{structure} may pick out one sense: a chiral lattice on the
circle picks a winding direction, a one-way runtime picks a clock
direction. The product cannot fail, because picking out one total
sense would require something outside the geometry to compare it
with, and there is nothing outside.

\paragraph{The chiral alphabet, drawn on the circle.} The flavor
sector's phases are lattice points on $\Stheta$ with spacing set by
the block integer,
\begin{equation}
\Delta\theta \;=\; \frac{2\pi}{2L} \;=\; \frac{\pi}{5},
\label{eq:G-lattice}
\end{equation}
so the alphabet has $2L = 10$ letters and is \emph{chiral}: read
clockwise it does not match itself read counterclockwise. That
single property is $CP$ violation's structural origin. The audit
then assigns each letter its role by throughput --- how much of the
crossing actually passes through it:
$\delta_{\mathrm{CKM}} = 2\pi/5$ (a \emph{bystander}: real
chirality but negligible throughput, contributing the famous
$\sim 10^{-20}$ of \citealp{Gavela1994});
$\delta_{\mathrm{PMNS}} = \pi + 2\arcsin(1/7) = 196.4^\circ$ (a
\emph{spectator}: blind to the washout-averaged asymmetry, but the
laboratory's sharpest test); $\alpha_{21} = 0$ (\emph{inert}); and
$\alpha_{31} = \pi$ (the \emph{driver}), whose half-angle enters
the decay amplitude as $e^{i\pi/2} = i$ --- maximal interference
from an angle that is itself $CP$-even, so the amount saturates
while the orientation survives.

\paragraph{The toll, and where the $K/2$ enters.} On the anatomy
Eq.~\eqref{eq:G-etachain}'s free product $\epsilon_1\kappa$ is not
free: the crossing happens at the lock, and the efficiency times
asymmetry is set by the descent's depth at that address,
\begin{equation}
\epsilon_1\,\kappa \;=\; e^{-K/2} \;=\; e^{-17.5}
\;=\; 2.5\e{-8} ,
\label{eq:G-toll}
\end{equation}
whence, with the standard factors of Eq.~\eqref{eq:G-etachain} and
the yield $1.25/g_*$: $\etaB = 7.34\e{-10}$ vs $6.12\e{-10}$ ($+20\%$). The honesty of this step must be stated exactly, and it
is stated the same way in the parent paper, in the body, and here:
the \emph{form} $e^{-K/2}$ --- half the lock depth --- is a
\emph{postulated} exponent, the chain's one such step, and it is
the reason Problems~10--11 carry grade D$^{*}$/P rather than D.
What is not postulated is everything around it: the crossing's
location, the sphaleron factor, the dilution, and the door's
subsequent welding (Problem~12).

\emph{Physics intuition.} The point of Eq.~\eqref{eq:G-toll} is
not that a small number was obtained --- any mechanism can produce
a small number --- but that the small number is a \emph{length}
rather than a coupling. One part in a billion is $e^{-17.5}$, and
$17.5$ is half of $35$; if the descent's depth were different, the
excess would be different, and the same integer would have to move
in the collider and the microwave sky as well.

\subsection{Step 4 --- problems and worked solutions}
\label{app:G-problems}

\medskip\noindent\textbf{G.1. Derive the antiunitarity of $T$.}

\noindent\emph{Solution.} \emph{Step 1.} Time reversal must keep
positions and reverse momenta: $TxT^{-1} = x$, $TpT^{-1} = -p$.
\emph{Step 2.} Apply $T$ to $[x,p] = i\hbar$:
$T[x,p]T^{-1} = [x,-p] = -i\hbar$, while the right side becomes
$T(i\hbar)T^{-1}$. \emph{Step 3.} Hence
$T(i\hbar)T^{-1} = -i\hbar$: $T$ conjugates $i$, so it is
antiunitary. \emph{Physics intuition:} the $i$ of quantum mechanics
is not a passive bookkeeping device --- it carries an orientation,
and time reversal is the operation that flips it. On the anatomy
this is why $T$ is the clock's reversal and why the custody of the
$i$ (P4-Real) matters.

\medskip\noindent\textbf{G.2. Show that $J_{CP}$ of
Eq.~\eqref{eq:G-jarlskog} vanishes if two same-charge quarks are
degenerate.}

\noindent\emph{Solution.} \emph{Step 1.} With two degenerate
masses one may rotate freely in that two-dimensional subspace
without changing the mass matrix. \emph{Step 2.} That freedom can
be used to make the CKM matrix real. \emph{Step 3.} A real matrix
has $J_{CP} = 0$. \emph{Physics intuition:} $CP$ violation requires
that the world distinguish the generations --- degenerate
generations are indistinguishable, and indistinguishable things
cannot carry a phase between them.

\medskip\noindent\textbf{G.3. Verify the assembled $\etaB$ of
Eq.~\eqref{eq:etaB}.}

\noindent\emph{Solution.} \emph{Step 1.}
$e^{-K/2} = e^{-17.5} = 2.5\e{-8}$. \emph{Step 2.} With prefactor $7.04$, sphalerons
$28/79 = 0.3544$, yield $1.25/g_*$, dilution $g_* = 106.75$:
$7.04 \times 0.3544 \times 1.25 \times 2.51\e{-8}/106.75 = 7.34\e{-10}$.
\emph{Step 3.} Against $6.12\e{-10}$: $+20\%$, inside the
factor-$\sim3$ band. \emph{Physics intuition:} one part in a
billion is not a small number that needed tuning --- it is
$e^{-17.5}$, half the depth of the descent, and the whole content
of the calculation is that the toll is set by a length rather than
by a coupling.

\medskip\noindent\textbf{G.4. Why is $B-L$ conserved by sphalerons,
and why does it matter?}

\noindent\emph{Solution.} \emph{Step 1.} The electroweak anomaly
gives $\partial_\mu j^\mu_{B} = \partial_\mu j^\mu_{L}$ ---
the same anomaly coefficient for baryon and lepton number, since
each generation contributes equally. \emph{Step 2.} Hence
$B+L$ is violated but $B-L$ is not. \emph{Step 3.} Any asymmetry
stored in $B-L$ before the sphalerons switch off therefore
survives, which is exactly what leptogenesis exploits.
\emph{Physics intuition:} the door is selective --- it launders
$B+L$ but cannot touch $B-L$ --- so the universe's surviving
matter is precisely the part of the asymmetry the door could not
reach.

\GGtakeaway{Three orientations, three operations, one protected
product: the discrete-symmetry pattern is a census of the
geometry's reversible senses. The matter excess is then the one
gated crossing of that geometry, priced at half the descent's
depth, with its postulated joint named in the open.}

\section{Confinement, the center, and proton stability}
\label{app:confinement}
\GGteachbox{The running coupling taken to its infrared limit;
dimensional transmutation and $\Lambda_{\mathrm{QCD}}$; the Wilson
criterion and the area law; center symmetry and the Polyakov loop;
the Cornell potential and the string tension; the proton-decay
operators and why baryon number is called accidental; and then the
two-fiber reading in which confinement is a registration rule and
the proton is guarded by two integers. Pay-off: Problem~12.}
\GGreadalong{\citet{PeskinSchroeder1995}, Ch.~17, for asymptotic
freedom and the infrared; \citet{Wilson1974} for the confinement
criterion in the original; \citet{JaffeWitten2006} for the formal
status of the problem.}

\subsection{Step 1 --- the standard treatment}
\label{app:H-step1}

\paragraph{Dimensional transmutation.} Integrating
Eq.~\eqref{eq:C-beta} gives Eq.~\eqref{eq:running-w2}, which can be
rewritten as
\begin{equation}
\alpha_s(\mu) = \frac{2\pi}{b_0 \ln(\mu/\Lambda_{\mathrm{QCD}})},
\qquad
\Lambda_{\mathrm{QCD}} \approx 210~\mathrm{MeV},
\label{eq:H-lambdaqcd}
\end{equation}
--- a dimensionful scale generated from a dimensionless coupling by
the flow itself. Above $\Lambda_{\mathrm{QCD}}$ perturbation theory
is excellent; below it the expression diverges and the description
fails.

\paragraph{The Wilson criterion.} Confinement is diagnosed by the
behavior of a large rectangular Wilson loop of extent
$R \times T$. If
$\langle W(C)\rangle \sim e^{-\sigma R T}$ --- the area law of
Eq.~\eqref{eq:arealaw-w2} --- then the static potential is linear,
$V \sim \sigma R$, and separating a color pair costs energy
without bound. If instead $\langle W \rangle \sim e^{-cP}$ with $P$
the perimeter, charges are free. Lattice computations find the area
law and measure $\sqrt\sigma \approx 440$~MeV.

\paragraph{Center symmetry.} Pure $\mathrm{SU}(3)$ gauge theory has
a global $\mathbb{Z}_3$ symmetry --- the center of the group ---
acting on the Polyakov loop. Confinement corresponds to the
unbroken phase, in which the Polyakov loop's expectation vanishes;
deconfinement to the broken phase. The three-ness of color enters
here not as a coupling but as an arithmetic fact about which
products of fundamental charges can close.

\paragraph{The proton, and dimension-six operators.} Baryon number
is not imposed on the Standard Model; it is an \emph{accidental}
symmetry --- there is simply no gauge-invariant, renormalizable
operator that violates it. At dimension six there are such
operators, suppressed by two powers of a heavy scale $M_X$, giving
\begin{equation}
\tau_p \;\sim\; \frac{M_X^4}{\alpha^2\, m_p^5}
\;\;\Longrightarrow\;\;
\tau_p \sim 3\e{35}~\mathrm{yr}
\ \ \text{for}\ \ M_X \sim 10^{16}~\mathrm{GeV},\ \alpha \simeq \tfrac{1}{30}.
\label{eq:H-protondecay}
\end{equation}
Super-Kamiokande's bound, $\tau_p > 2.4\e{34}$~yr in the
$e^+\pi^0$ channel \citep{SuperK2020}, already excludes the
simplest unified models.

\paragraph{The area law, derived in strong coupling.} The
cleanest derivation of confinement's diagnostic is Wilson's own,
on a lattice. In the strong-coupling limit the plaquette action is
expanded so that only tiled surfaces survive the group integration,
and the minimal surface spanning a loop $C$ of area $\mathcal A$
contributes
\begin{equation}
\langle W(C)\rangle \;\sim\;
\left(\frac{1}{g^2}\right)^{\mathcal A/a^2}
\;=\; e^{-\sigma\mathcal A},
\qquad
\sigma \;=\; \frac{1}{a^2}\ln g^2 ,
\label{eq:H-strongcoupling}
\end{equation}
with $a$ the lattice spacing --- which is
Eq.~\eqref{eq:arealaw-w2}. Every plaquette of the spanning surface
must be tiled because a single untiled plaquette leaves an
unmatched group integral that vanishes; hence the cost is
proportional to \emph{area}, not perimeter. \emph{Physics
intuition:} the flux between two color charges cannot spread out
as it does in electromagnetism, because the gluons carrying it
attract one another; it collimates into a tube of fixed
cross-section, and the energy of a tube is its length times a
constant.

\paragraph{Center symmetry and the Polyakov loop.} At finite
temperature, compactify Euclidean time on a circle of
circumference $\beta = 1/T$ and consider the Polyakov loop
\begin{equation}
L(\vec x) \;=\; \frac13\,\mathrm{Tr}\;\mathcal P
\exp\!\left(ig\int_0^{\beta} A_0\,d\tau\right).
\label{eq:H-polyloop}
\end{equation}
Gauge
transformations that are periodic only up to an element $z$ of the
center $\mathbb{Z}_3$ leave the action invariant but multiply
$L \to zL$. Hence
\begin{equation}
\langle L\rangle = 0 \;\Leftrightarrow\;
\mathbb{Z}_3\ \text{unbroken}\ \Leftrightarrow\ \text{confined};
\qquad
\langle L\rangle \ne 0 \;\Leftrightarrow\ \text{deconfined};
\label{eq:H-polyakov}
\end{equation}
and $\langle L\rangle \propto e^{-\beta F_q}$ with $F_q$ the free
energy of an isolated quark: infinite in the confined phase.
\emph{Physics intuition:} the center is the part of the gauge group
that acts trivially on gluons but not on quarks, so it is precisely
the symmetry that can distinguish ``a state with an isolated
color charge'' from ``a state without one'' --- confinement is its
unbroken realization.

\paragraph{Regge trajectories: the string, seen.} If quarks are
joined by a tube of tension $\sigma$, a rotating tube of length
$2R$ with ends at the speed of light has energy
$E = \pi\sigma R$ and angular momentum
$\mathcal{J} = \pi\sigma R^2/2$, so
\begin{equation}
\mathcal{J} \;=\; \frac{E^2}{2\pi\sigma} \;\equiv\; \alpha' E^2,
\qquad
\alpha' = \frac{1}{2\pi\sigma} \approx 0.8~\mathrm{GeV^{-2}} ,
\label{eq:H-regge}
\end{equation}
a linear relation between spin and mass squared. Measured hadron
spectra lie on exactly such lines, with slopes of
$0.8$--$0.9~\mathrm{GeV^{-2}}$ --- which
is the most direct observational evidence that the confining flux
is a string, and it was discovered before QCD existed.

\paragraph{The proton-decay operators, listed.} Baryon number
survives at renormalizable order because no gauge-invariant
operator of dimension four violates it. At dimension six there are
four, of the schematic form
\begin{equation}
\mathcal O_{\Delta B=1} \;\sim\;
\frac{1}{M_X^2}\,(qqq\ell) ,
\qquad
\Gamma_p \;\sim\; \frac{\alpha^2 m_p^5}{M_X^4} ,
\label{eq:H-dim6}
\end{equation}
giving Eq.~\eqref{eq:H-protondecay}. The $m_p^5$ is pure
dimensional analysis (a rate with a dimension-six operator squared
must scale so), and the $M_X^{-4}$ is what makes proton-decay
searches such deep probes: a factor of ten in lifetime reach buys
a factor of $1.8$ in $M_X$.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:H-step2}

\emph{Confinement is demonstrated, not proved.} The area law is
measured on the lattice and reproduced numerically to high
precision, but a proof from the continuum theory that color
charges cannot be isolated does not exist; the closely related
question of Yang--Mills existence with a mass gap is an unclaimed
Millennium Prize problem \citep{JaffeWitten2006}. The perturbative
argument shows only that the calculation breaks down, which is a
statement about our tools.

\emph{Three colors is an input.} $N_c = 3$ is fixed by data ---
the $\pi^0 \to \gamma\gamma$ rate, the $R$ ratio, the statistics of
the $\Delta^{++}$ --- not derived.

\emph{Proton stability is an accident.} A bound thirty-four orders
of magnitude beyond the age of the universe, resting on the
observation that nobody could write the offending operator at low
dimension, is exactly the sort of fact a theory should explain
rather than notice (Problem~12).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:H-step3}

\paragraph{Confinement as a registration rule.} On the anatomy
color holonomy lives on the compact fiber and closes only in
threes: the $\mathbb{Z}_3$ center is the circle's arithmetic, not a
dynamical accident. An open color line is therefore not an
expensive state but an \emph{unregistrable} one --- the boundary
projection of Part~II admits only closed color books, and
Eq.~\eqref{eq:cornell-w2}'s linear term is the price the boundary
charges for attempting an open one. The crossover distance is where
the interval's perturbative reading fails and the circle's closure
rule takes over the description, which is why
Eq.~\eqref{eq:H-lambdaqcd}'s divergence and the onset of the string
tension occur at the same address.

\paragraph{Three colors, twice.} The count is overdetermined:
$N = 3$ is a ledger member, tied by the closure identity
$J = 2N+1$ to $J = 7$ and thence to $K = LJ = 35$; and the same
number is reached independently through the capacity argument of
\citet{ArisakaP2N3}. Two derivations sharing no premise, one small
integer.

\paragraph{The proton's double guard.} Baryon number is not
accidental here but \emph{gauged} --- $\UoneB$ is a factor of
Eq.~\eqref{eq:group-p2} --- and a gauged symmetry cannot be violated
by any operator at any dimension without violating gauge invariance
itself. That is the whole prohibition, and it is a conservation law
rather than an integrality condition. The boundary's two integer
conditions, $Q + t/3 \in \mathbb Z$ and $Q + B \in \mathbb Z$,
Eq.~\eqref{eq:guards}, do something prior and different: they are the
registration rule that makes $B$ and the triality integer \emph{classes}
in the first place, so that the conserved quantity is exact rather than
merely conserved. Guard and hinge, not two doors --- and the
distinction matters, because integrality alone forbids nothing (see
below). That is why the bound of Eq.~\eqref{eq:H-protondecay} is
respected by construction rather than by tuning $M_X$ --- and why a
single confirmed proton decay in a guard-relevant channel would kill
this Problem's answer outright.

\paragraph{Where the $\mathbb{Z}_3$ comes from.} On the
anatomy the center of Eq.~\eqref{eq:H-polyakov} is not an
accidental feature of $\mathrm{SU}(3)$ but the circle's arithmetic.
Color transport around $\Stheta$ returns an element of the
holonomy group; the elements that act trivially on every gluon but
non-trivially on a fundamental charge are exactly the center,
\begin{equation}
Z(\mathrm{SU}(N_c)) = \mathbb{Z}_{N_c}
= \{\,e^{2\pi i k/N_c}\mathbb{1}\,\},
\qquad N_c = N = 3 ,
\label{eq:H-centre}
\end{equation}
so ``color closes in threes'' is the statement that a fundamental
winding must be repeated $N$ times before the holonomy returns to
the identity. A single quark's transport leaves a residual center
phase $e^{2\pi i/3}$; three quarks in a color singlet, or a
quark--antiquark pair, leave none.

\paragraph{The registration rule, stated as a selection rule.} The
boundary projection of Eq.~\eqref{eq:B-fibretrace} integrates over
$\theta$. A configuration whose $\theta$-dependence carries a
residual center phase therefore integrates to zero:
\begin{equation}
\oint_{\Stheta} d\theta\; e^{\,2\pi i k/3}\,(\cdots) \;=\; 0
\quad\text{for } k \not\equiv 0 \ (\mathrm{mod}\ 3),
\label{eq:H-selection}
\end{equation}
which is the precise sense in which an open color line is not
expensive but \emph{unregistrable}: the projection annihilates it,
so no boundary observable can report it. Eq.~\eqref{eq:H-selection}
is a selection rule of exactly the kind that forbids
$\Delta m = 2$ transitions in atomic dipole radiation --- and it is
worth noticing that no one calls those transitions ``infinitely
expensive.'' They simply do not register.

\emph{Physics intuition.} The standard account and this one agree
on every measurable: the string tension, the spectrum, the lattice
data. Where they differ is the \emph{reason}: the standard account
says an isolated quark would cost infinite energy, and this one
says it would cost nothing to write and could never be read.
Eq.~\eqref{eq:cornell-w2}'s linear potential is then the boundary's
price for approaching an unregistrable configuration, and the fact
that the price rises without bound is the projection's way of
saying ``never.''

\paragraph{Why $N = 3$, twice over.} The color count is not an
input here. It is a ledger member bound by the chirality relation
$J = 2N+1$ of \S\ref{app:D-step3}, which with $J = 7$ forces
$N = 3$ and thence $K = LJ = 35$; and it is reached independently
by the capacity argument of \citet{ArisakaP2N3}, which bounds the
generation count from below and above inside the same axiom system
without using the chirality relation. Two derivations sharing no
premise, one small integer --- which is the pattern this paper
calls overdetermination and treats as the main defense against
numerology.

\paragraph{The double guard, written out.} The boundary carries two
integrality conditions simultaneously,
\begin{equation}
Q + \frac{t}{3} \in \mathbb{Z}
\qquad\text{and}\qquad
Q + B \in \mathbb{Z} ,
\label{eq:H-guards}
\end{equation}
with $Q$ the electric charge, $t$ the triality (the center label of
Eq.~\eqref{eq:H-centre}), and $B$ the gauged baryon number carried
by $\UoneB$ in Eq.~\eqref{eq:group-p2}. The first is the
registration rule of Eq.~\eqref{eq:H-selection} in arithmetic form;
the second is gauge invariance of the abelian factor.

The reader should now check what Eq.~\eqref{eq:H-guards} does
\emph{not} do, because the appendix is the place for it. A decay
$p \to e^{+}\pi^{0}$ changes $B$ by one unit while leaving $Q$ and $t$
alone: $Q + B$ moves from $2$ to $1$, and $Q + t/3$ from $1$ to $1$.
Both quantities remain integers on both sides. \emph{Integrality
forbids the decay in neither condition.} What forbids it is the gauged
conservation law of $\UoneB$, which admits no $\Delta B \ne 0$
operator at any dimension; what Eq.~\eqref{eq:H-guards} supplies is
the reason $B$ is an exact integer class rather than a continuous
label, and hence the reason the conservation law is exact rather than
approximate. The two conditions are the guard's hinge, not a second
closure --- and a statement of the guard that leans on integrality
alone would be wrong, which is why the body says so in place.

\emph{Physics intuition.} This is why the anatomy's proton
stability is structural rather than a large-$M_X$ suppression.
Eq.~\eqref{eq:H-dim6} makes the lifetime a continuous function of
an unknown scale, so a measurement can always be accommodated by
raising $M_X$; Eq.~\eqref{eq:H-guards} makes it a discrete
prohibition, which cannot be adjusted at all. The corresponding
exposure is total: one confirmed proton decay in a guard-relevant
channel ends this answer, with no parameter left to move.

\subsection{Step 4 --- problems and worked solutions}
\label{app:H-problems}

\medskip\noindent\textbf{H.1. Derive Eq.~\eqref{eq:H-lambdaqcd}
from Eq.~\eqref{eq:C-beta}.}

\noindent\emph{Solution.} \emph{Step 1.} Write
$\mu\,dg/d\mu = -b_0 g^3/16\pi^2$ and separate variables:
$d(1/g^2) = (b_0/8\pi^2)\,d\ln\mu$. \emph{Step 2.} Integrating,
$1/g^2(\mu) = (b_0/8\pi^2)\ln(\mu/\Lambda)$ with $\Lambda$ the
integration constant. \emph{Step 3.} Using
$\alpha_s = g^2/4\pi$ gives Eq.~\eqref{eq:H-lambdaqcd}.
\emph{Physics intuition:} the theory had no mass scale in its
Lagrangian and acquired one from the flow --- the cleanest example
in physics of a scale being an \emph{address} rather than an
input, which is exactly the interval's job description.

\medskip\noindent\textbf{H.2. Convert $\sqrt\sigma = 440$~MeV into
a force in everyday units.}

\noindent\emph{Solution.} \emph{Step 1.}
$\sigma = (0.44~\mathrm{GeV})^2 = 0.194~\mathrm{GeV}^2$.
\emph{Step 2.} Using $\hbar c = 0.1973$~GeV\,fm,
$\sigma = 0.194/0.1973 \approx 0.98$~GeV/fm. \emph{Step 3.} And
$1$~GeV/fm $= 1.602\e{-10}\,\mathrm{J}/10^{-15}\,\mathrm{m}
= 1.6\e{5}$~N, so $\sigma \approx 1.6\e{5}$~N --- about sixteen
tonnes-force, independent of distance. \emph{Physics intuition:}
the reason quarks are never seen alone is not that the force is
strong at short range but that it \emph{does not weaken} at long
range; pulling harder simply pays for a new pair.

\medskip\noindent\textbf{H.3. Estimate $M_X$ from
Eq.~\eqref{eq:H-protondecay} given the Super-K bound.}

\noindent\emph{Solution.} \emph{Step 1.} Scaling
$\tau_p \propto M_X^4$, a bound $\tau_p > 2.4\e{34}$~yr against the
benchmark $\tau_p \simeq 3\e{35}$~yr at $M_X = 10^{16}$~GeV gives
$M_X > 10^{16}\,(2.4\e{34}/3\e{35})^{1/4} \approx 5.4\e{15}$~GeV.
\emph{Step 2.} Minimal $\mathrm{SU}(5)$ predicts
$M_X \sim 10^{15}$~GeV --- a factor of five below the bound.
\emph{Step 3.} Hence the exclusion. (Order-unity factors in the
operator coefficients move the benchmark by a few; they do not
move the conclusion, because the fourth power converts a factor of
five in $M_X$ into a factor of six hundred in $\tau_p$.) \emph{Physics intuition:} the
fourth power is what makes proton decay such an efficient probe ---
a factor of two in lifetime sensitivity buys only twenty per cent
in reach, but the reach is already deep into the unification
window, which is why the null result was so damaging to the
simplest models.

\medskip\noindent\textbf{H.4. Why does gauging $\UoneB$ change the
status of proton stability?}

\noindent\emph{Solution.} \emph{Step 1.} An accidental symmetry is
one that happens to hold at the lowest dimensions and is expected
to fail once higher-dimension operators are included.
\emph{Step 2.} A gauged symmetry cannot be violated by \emph{any}
operator in the theory, at any dimension, without violating gauge
invariance itself. \emph{Step 3.} The stability therefore becomes
structural, and the bound of Eq.~\eqref{eq:H-protondecay} is met
without adjusting $M_X$. \emph{Physics intuition:} the difference
between accidental and gauged is the difference between ``nobody
has written the term'' and ``the term cannot be written'' --- and
only the second is an explanation.

\GGtakeaway{Confinement is the boundary's refusal to register an
open color book, and the proton's permanence is a gauged
conservation law whose charge the boundary's registration rule makes
an exact integer class. Both readings leave the measured physics
exactly as it stands and change what the physics \emph{is}: a
registration rule, not a force law.}

\section{Time, entropy, and horizon thermodynamics}
\label{app:time}
\GGteachbox{The second law and Boltzmann's counting, with the
reversibility objection stated in full; Landauer's principle
derived; the black-hole area theorem and the Bekenstein--Hawking
entropy; Hawking's temperature and the evaporation timescale; the
Page curve; and then the two-fiber reading in which time has four
endowments, entropy is a write count, and a horizon is where three
ledgers meet. Pay-off: Problems~13, 14, and~16.}
\GGreadalong{\citet{Wald1984}, Ch.~12, for horizons and the area
theorem; \citet{Bekenstein1973} and \citet{Hawking1975} in the
original; \citet{Page1993} for the curve; \citet{Landauer1961} for
the toll.}

\subsection{Step 1 --- the standard treatment}
\label{app:I-step1}

\paragraph{Entropy and the reversibility objection.} Boltzmann's
$S = k_B\ln W$, Eq.~\eqref{eq:boltzmann-w2}, counts the
microstates compatible with a macrostate, and the second law
follows statistically: systems move toward macrostates of larger
$W$ because there are more of them. Loschmidt's objection is
immediate and fatal to the naive reading: the microdynamics is
time-symmetric, so the same counting argument run backwards
predicts that entropy was \emph{higher} in the past, which is
false. The standard repair is to impose a low-entropy initial
condition --- the past hypothesis --- which relocates the arrow
into cosmology (Problem~5) rather than deriving it.

\paragraph{Landauer's principle.} Erasing one bit of information
in contact with a bath at temperature $T$ requires dissipating at
least $k_B T\ln 2$ of energy, Eq.~\eqref{eq:landauer-p2}. The
derivation is a counting argument: erasure maps two states to one,
halving phase-space volume, and Liouville's theorem requires the
compensating volume to appear in the bath. Information and
thermodynamics are therefore not analogues but one ledger.

\paragraph{Black-hole thermodynamics.} The area theorem
($\delta A \ge 0$ classically) invites an entropy reading, and
Bekenstein's argument fixes the coefficient by consistency with the
generalized second law, giving Eq.~\eqref{eq:bh-entropy-w2},
$S_{\mathrm{BH}} = k_Bc^3A/4G_N\hbar$. Hawking's calculation ---
quantum fields on the collapse background --- then produces exactly
thermal radiation at Eq.~\eqref{eq:hawkingT-w2}'s temperature,
confirming the reading and creating the paradox. The evaporation
timescale follows by integrating the luminosity:
\begin{equation}
t_{\mathrm{evap}} \;\simeq\;
\frac{5120\,\pi\,G_N^2 M^3}{\hbar c^4}
\;\approx\; 2\e{67}~\mathrm{yr}\times
\left(\frac{M}{M_\odot}\right)^{3}.
\label{eq:I-evap}
\end{equation}

\paragraph{The Page curve.} If the total evolution is unitary, the
radiation's entanglement entropy must rise while the hole is large,
turn over when the remaining hole's entropy becomes the smaller
number --- the Page time, roughly half the evaporation --- and fall
to zero at the end \citep{Page1993}.

\paragraph{The $H$-theorem, and the two objections it
provoked.} Boltzmann's transport equation for a dilute gas gives,
for $H = \int f\ln f\,d^3v$,
\begin{equation}
\frac{dH}{dt} \;\le\; 0 ,
\label{eq:I-Htheorem}
\end{equation}
with equality only for the Maxwell distribution --- apparently a
derivation of the second law from mechanics. Two objections
followed within a decade, and both are correct. \emph{Loschmidt's
(1876)}: the underlying dynamics is time-reversal invariant, so for
every entropy-increasing trajectory there is a decreasing one of
equal weight. \emph{Zermelo's (1896)}: by Poincar\'e's recurrence
theorem, a bounded system returns arbitrarily close to its initial
state, so $H$ cannot decrease forever. The resolution --- that
Eq.~\eqref{eq:I-Htheorem} follows only after the
\emph{Stosszahlansatz}, the assumption that colliding particles are
uncorrelated \emph{before} and not after --- shows that the arrow
was inserted, not derived.

\emph{Physics intuition.} Every derivation of the second law from
reversible microphysics contains one step where time's direction is
quietly used. Finding that step is the single most useful exercise
in the subject; in Boltzmann it is the molecular-chaos assumption,
in decoherence it is the initially uncorrelated environment, in
the renormalization group it is the choice to integrate out short
distances rather than long ones.

\paragraph{Landauer's principle, derived twice.} \emph{Phase-space
version:} erasure maps two equally likely states to one, so the
system's accessible volume halves and its entropy falls by
$k_B\ln 2$; Liouville's theorem forbids the total volume from
shrinking, so the bath absorbs $\Delta S \ge k_B\ln 2$, costing
$T\Delta S = k_BT\ln 2$. \emph{Cycle version:} Szilard's engine
extracts $k_BT\ln 2$ of work from a one-molecule gas using one bit
of which-side information; if erasing that bit were free, the cycle
would violate the second law. The two derivations meet, and the
conclusion is that information is thermodynamically real
\begin{equation}
W_{\mathrm{erase}} \;\ge\; k_B T\ln 2 \;\approx\; 3\e{-21}\
\mathrm{J\ at\ room\ temperature} .
\label{eq:I-landauer}
\end{equation}

\paragraph{Black-hole mechanics as thermodynamics.} The four laws
of black-hole mechanics --- constancy of surface gravity $\kappa$
on the horizon; $dM = \tfrac{\kappa}{8\pi G}dA + \Omega\,dJ +
\Phi\,dQ$; $dA \ge 0$; and the unattainability of $\kappa = 0$ ---
are formally identical to the four laws of thermodynamics with
$T \leftrightarrow \kappa/2\pi$ and $S \leftrightarrow A/4G$. For a
Schwarzschild hole,
\begin{equation}
r_s = \frac{2G_NM}{c^2},
\qquad
\kappa = \frac{c^4}{4G_NM},
\qquad
A = 4\pi r_s^2 ,
\label{eq:I-schwarzschild}
\end{equation}
from which Eq.~\eqref{eq:bh-entropy-w2} and
Eq.~\eqref{eq:hawkingT-w2} follow immediately (Problem~I.6).
Bekenstein's argument for taking the identification physically was
that otherwise one could violate the second law by dropping entropy
into a hole.

\paragraph{Hawking's calculation, in outline.} Quantize a field on
the collapse background. The modes that reach future infinity are
related to the initial vacuum modes by a Bogoliubov transformation
whose $\beta$ coefficients are non-zero because the two natural
mode bases (in and out) do not agree across the horizon. Computing
them gives a thermal occupation
\begin{equation}
\langle n_\omega\rangle \;=\;
\frac{1}{e^{2\pi\omega/\kappa} - 1} ,
\label{eq:I-planck}
\end{equation}
a Planck spectrum at $T_H = \hbar\kappa/2\pi k_B c$ --- which is
Eq.~\eqref{eq:hawkingT-w2}. \emph{Physics intuition:} the
temperature is a statement about the \emph{mismatch of vacua} on
either side of a causal boundary, not about anything happening at
the horizon locally; a freely falling observer sees nothing there,
which is exactly the tension the information paradox exploits.

\paragraph{The Page curve, and why it is forced.} Let the total
state be pure and split it into hole ($H$) and radiation ($R$).
Purity gives $S(R) = S(H)$ at all times. Early on $R$ is small so
$S(R)$ grows with the number of emitted quanta; late on $H$ is
small so $S(R) = S(H) \le S_{\mathrm{BH}}(t)$, which is falling.
The two bounds cross at the \emph{Page time} --- when the hole has
radiated about half its entropy --- and the curve must turn over
there. Hawking's calculation gives instead a monotonically rising
$S(R)$, which is the paradox in one sentence.

\subsection{Step 2 --- what the standard treatment leaves
unexplained}
\label{app:I-step2}

\emph{Time is a parameter, never derived.} Every equation above
describes change with respect to $t$ and none says what $t$ is;
the canonical quantization of gravity \citep{DeWitt1967} responds
by removing it altogether.

\emph{The arrow is imported.} Boltzmann's counting needs the past
hypothesis; the master equation of Appendix~\ref{app:openquantum}
needs $\gamma_k \ge 0$; the RG needs a direction of coarse-graining.
Each is correct and none is derived from the others.

\emph{Entropy has several inequivalent definitions.} Clausius,
Boltzmann, Gibbs, Shannon, von Neumann --- and Bekenstein's, which
is an \emph{area}. That an entropy can be a geometrical quantity
has been a standing clue for fifty years with no accepted reading
(Problem~14).

\emph{The paradox has no experiment.} Hawking radiation from
astrophysical holes is unobservably faint and
Eq.~\eqref{eq:I-evap} exceeds the age of the universe by fifty-plus
orders, so the dispute is settled, if at all, by consistency
(Problem~16).

\subsection{Step 3 --- the two-fiber resolution}
\label{app:I-step3}

\paragraph{Four endowments, two fibers, one seam.} What ``time''
must supply is a \emph{tick} (something that repeats, so duration
can be counted), an \emph{arrow} (a direction distinguishing
before from after), a \emph{rate} (a lawful relation between them,
so histories can be dated), and an \emph{extent}. The circle
supplies the first --- every clock is a return, and $E = h\nu$,
Eq.~\eqref{eq:tick-w2}, prices it. The interval supplies the second
--- Eq.~\eqref{eq:A-semigroup} has no inverse, and no dynamical law
is needed to make the arrow point. The seam supplies the third, and
it is exact: $d\ln t/dr = 2$ in the radiation era,
Eq.~\eqref{eq:slope}, integrating to Eq.~\eqref{eq:hubbletime-w2}'s
$H = 1/2t$. And the descent's endpoints supply the fourth, audited
by the age ledger, Eq.~\eqref{eq:age-ledger}. The century's failure
to define time is, on this reading, the attempt to get four
endowments from one structure.

\paragraph{Entropy is the write count.} $\ln W$ counts what the
boundary has \emph{not} registered; each registration is an
interval-side write with no inverse, so the count cannot decrease
--- the second law is Eq.~\eqref{eq:semigroup-law-p2} in
thermodynamic clothing, not a statistical tendency with
exceptions. Loschmidt's objection does not arise because there is
no time-symmetric microdynamics underneath to reverse: the
one-wayness is the fiber's composition law.

\paragraph{A horizon is where the ledgers meet.} Note which
constants appear in Eq.~\eqref{eq:bh-entropy-w2}: $\hbar$ (the
circle's), $G_N$ (the interval's), $k_B$ (the exchange window's).
It is the only formula in physics that spends all three at once,
and on the anatomy that is not a curiosity: a horizon is precisely
where the circle's register, the interval's channel, and the
boundary's window are forced into contact. The information paradox
then reads as a billing error --- the circle keeps what the hole
must return, the interval prices the return, and the Page curve is
the reconciliation schedule (Problem~16). Grade:
derived-conditional, with the island computations recovering the
same curve gravitationally.

\paragraph{The tick, quantified.} On the anatomy the circle's
transport is $U(\theta) = e^{i\theta}$, and a state's phase
advances at a rate set by its energy through the Schr\"odinger
equation, $|\psi(t)\rangle = e^{-iEt/\hbar}|\psi(0)\rangle$.
Identifying the phase advance with motion around $\Stheta$,
\begin{equation}
\frac{d\theta}{dt} \;=\; \frac{E}{\hbar}
\;=\; 2\pi\nu
\qquad\Longleftrightarrow\qquad
E \;=\; h\nu ,
\label{eq:I-tickderiv}
\end{equation}
so Eq.~\eqref{eq:tick-w2} is not a postulate about quanta but the
statement that energy is the \emph{angular velocity} on the compact
fiber, with $\hbar$ the conversion constant of Step~A. Three
familiar facts become corollaries. A massive particle at rest has
$\nu = mc^2/h$ --- the Compton clock, $1.24\e{20}$~Hz for an
electron --- which is what a kilogram ``has'': phase turning at
$1.4\e{50}$ revolutions per second. Energy conservation is
Noether's theorem for the circle's rotational symmetry. And energy
is quantized in bound states because the phase must return
single-valuedly around the orbit --- Eq.~\eqref{eq:A-fluxquantum}
again, in its oldest costume.

\paragraph{The arrow, without a past hypothesis.} The interval's
transport obeys Eq.~\eqref{eq:A-kadanoff}, which has no inverse ---
so the boundary's registrations accumulate and cannot be undone.
This is a stronger statement than the standard one and should be
compared carefully. Boltzmann needs the Stosszahlansatz, and
Loschmidt's objection bites because the microdynamics \emph{does}
possess reversed trajectories. Here there is no underlying
reversible trajectory to reverse: the runtime is a semigroup
element, and the reversed object simply does not exist in the
algebra. Zermelo's objection fails for the same reason ---
Poincar\'e recurrence is a theorem about volume-preserving flows on
compact phase spaces, and Eq.~\eqref{eq:A-kadanoff} is neither
volume preserving nor invertible.

\emph{Physics intuition.} The standard account makes the arrow a
fact about \emph{initial conditions}, which is why it must import
Problem~5's specialness. This account makes it a fact about
\emph{which algebra the transport belongs to} --- and algebras do
not have initial conditions.

\paragraph{The rate: deriving $d\ln t/dr = 2$.} The two endowments
are linked at the seam, and the link is computable. In the
radiation era $\rho \propto a^{-4}$ and
$H^2 = 8\pi G_N\rho/3$, so $H \propto a^{-2}$ and
$a \propto t^{1/2}$; hence $T \propto a^{-1} \propto t^{-1/2}$.
The interval's coordinate is $r = \ln(\EGUT/\mu)$ with
$\mu \propto T$, so $dr = -d\ln T$, and therefore
\begin{equation}
\frac{d\ln t}{dr} \;=\; -\frac{d\ln t}{d\ln T} \;=\; 2
\qquad\text{exactly, in the radiation era},
\label{eq:I-slopederiv}
\end{equation}
which is Eq.~\eqref{eq:slope} and integrates to
Eq.~\eqref{eq:hubbletime-w2}'s $H = 1/2t$. The number $2$ is not
fitted: it is the radiation equation of state read through the
Friedmann equation, and it is what makes the age ledger
\eqref{eq:age-ledger} an audit rather than an estimate --- each
e-fold of descent is worth exactly two e-folds of clock, so the
chronology can be summed segment by segment and checked against the
measured age.

\paragraph{Entropy as the write count, and the demon closed.} Let
$W$ count the boundary configurations compatible with what has been
registered. Every registration is an interval-side write, which by
Eq.~\eqref{eq:A-kadanoff} cannot be undone, so the count of
distinguishable pasts consistent with the record can only grow.
Maxwell's demon then closes without appeal to any special
mechanism: the demon must \emph{register} which molecules are fast,
each registration is a write, and clearing its memory to run again
costs Eq.~\eqref{eq:I-landauer}'s toll --- exactly canceling the
work extracted. The second law, on this reading, is not a
statistical tendency with rare exceptions; it is the absence of an
inverse.

\paragraph{Why a horizon spends all three constants.} Assemble
Eq.~\eqref{eq:bh-entropy-w2} from the pieces this course has built:
\begin{equation}
S_{\mathrm{BH}} \;=\; \frac{k_B\,c^3\,A}{4\,G_N\,\hbar}
\;=\; k_B\,\frac{A}{4\ell_{\mathrm{Pl}}^2},
\qquad
\ell_{\mathrm{Pl}}^2 = \frac{\hbar G_N}{c^3} .
\label{eq:I-threeledgers}
\end{equation}
Each constant is one structure's, by the assignment of Step~A:
$\hbar$ prices the circle's tick, $G_N$ the interval's integrated
descent, $k_B$ the boundary's exchange window. Their appearance
together says that a horizon is the one place where all three
ledgers are forced into contact --- the circle's register (what is
kept), the interval's channel (what is spent), and the window
(where they trade). The information paradox is then not a failure
of any one of them but the historical accident of billing the
spending against the keeping: Hawking's calculation, done on the
four-dimensional shadow where both fibers are invisible, could see
the channel and not the register. Grade: derived-conditional, with
the island computations \citep{Penington2020} recovering the same
Page curve gravitationally and independently.

\subsection{Step 4 --- problems and worked solutions}
\label{app:I-problems}

\medskip\noindent\textbf{I.1. State Loschmidt's objection precisely
and say what the anatomy changes.}

\noindent\emph{Solution.} \emph{Step 1.} For every microstate
trajectory with $\Delta S > 0$ there is a velocity-reversed
trajectory with $\Delta S < 0$, of equal statistical weight.
\emph{Step 2.} So counting alone cannot prefer one direction: the
argument run backwards predicts higher entropy in the past.
\emph{Step 3.} The anatomy does not add a preferred trajectory ---
it denies that the runtime is a trajectory at all: the boundary
dynamics is a semigroup, whose elements have no inverses to
reverse. \emph{Physics intuition:} the standard repair makes the
arrow a fact about \emph{initial conditions}; this reading makes it
a fact about \emph{algebra} --- which is why it needs no past
hypothesis, and why it is exposed instead to any observed
uncompensated erasure.

\medskip\noindent\textbf{I.2. Compute the Hawking temperature and
evaporation time for a solar-mass hole, and for $10^{12}$~kg.}

\noindent\emph{Solution.} \emph{Step 1.} From
Eq.~\eqref{eq:hawkingT-w2},
$T_H = 6.2\e{-8}$~K for $M = M_\odot$ --- eight orders below the
CMB, so the hole absorbs rather than evaporates. \emph{Step 2.}
Eq.~\eqref{eq:I-evap} gives $t_{\mathrm{evap}} \approx 2\e{67}$~yr.
\emph{Step 3.} For $M = 10^{12}$~kg (a mountain-mass primordial
hole), scaling by $(M/M_\odot)^3 = (5\e{-19})^3$ gives
$t_{\mathrm{evap}} \approx 2.5\e{12}$~yr --- still longer than the
age of the universe, while holes of a few times $10^{11}$~kg would
be expiring now.
\emph{Physics intuition:} the cube law is why primordial black
holes of one particular mass are interesting and all others are
not; the paradox concerns a process that, for anything
astrophysical, has not begun.

\medskip\noindent\textbf{I.3. Show that Landauer's bound follows
from phase-space counting.}

\noindent\emph{Solution.} \emph{Step 1.} Erasure maps two
equally-likely states to one, so the system's accessible volume
halves and its entropy falls by $k_B\ln 2$. \emph{Step 2.} The
total entropy cannot fall, so the bath's must rise by at least
$k_B\ln 2$. \emph{Step 3.} At temperature $T$ that costs at least
$T\,\Delta S = k_BT\ln 2$ of dissipated energy. \emph{Physics
intuition:} the toll is charged for \emph{forgetting}, not for
computing --- reversible computation is free in principle --- which
is exactly the anatomy's division: the circle keeps for nothing,
and the interval charges to discard.

\medskip\noindent\textbf{I.4. Why must the Page curve turn over,
and what does the anatomy add?}

\noindent\emph{Solution.} \emph{Step 1.} The radiation's
entanglement entropy cannot exceed the entropy of whatever it is
entangled with. \emph{Step 2.} Early on that is the hole, whose
$S_{\mathrm{BH}}$ is large and falling; late on the smaller number
is the hole's, so the radiation's entropy must follow it down.
\emph{Step 3.} The turnover is therefore kinematic --- it happens
when the two curves cross, at roughly half the evaporation.
\emph{Physics intuition:} the curve's shape is a bookkeeping
constraint, not a dynamical prediction, which is why so many
approaches recover it; what the anatomy adds is the identity of the
two books --- the circle's register and the interval's channel ---
and hence \emph{where} the information was while the hole existed.

\medskip\noindent\textbf{I.5. Estimate the de Sitter entropy of our
own universe and comment on its grade here.}

\noindent\emph{Solution.} \emph{Step 1.} With
$R_e = c/H_\Lambda \approx 1.4\e{26}$~m, the horizon area is
$A = 4\pi R_e^2 \approx 2.4\e{53}~\mathrm{m^2}$. \emph{Step 2.}
Dividing by four Planck areas
($\ell_{\mathrm{Pl}}^2 = 2.6\e{-70}~\mathrm{m^2}$) gives
$S/k_B \approx 2\e{122}$, so $\ln(S/k_B) \approx 281.7$. \emph{Step 3.} The cohort notes that this
sits within $0.6\%$ of $8K = 280$, and carries that observation at
\emph{flagged} grade --- reproducible arithmetic, no mechanism,
never used as support. \emph{Physics intuition:} the largest number
in physics being eight times the descent's depth is the kind of pattern
that is either the deepest fact in the subject or an artifact of
taking logarithms; the discipline of the flag register exists so
that a reader can tell which claims lean on it. None do.

\medskip\noindent\textbf{I.6. Derive
Eq.~\eqref{eq:hawkingT-w2} and Eq.~\eqref{eq:bh-entropy-w2} from
Eq.~\eqref{eq:I-schwarzschild}.}

\noindent\emph{Solution.} \emph{Step 1.} With
$\kappa = c^4/4G_NM$ and $T_H = \hbar\kappa/2\pi k_Bc$, substitute:
$T_H = \hbar c^3/8\pi G_N M k_B$, which is
Eq.~\eqref{eq:hawkingT-w2}. \emph{Step 2.} For the entropy use
$dM c^2 = T_H\,dS$ with
$A = 16\pi G_N^2M^2/c^4$, so $dA = 32\pi G_N^2 M\,dM/c^4$.
\emph{Step 3.} Then
$dS = c^2 dM/T_H = 8\pi G_N M k_B\,dM/\hbar c
= k_Bc^3\,dA/4G_N\hbar$, which integrates to
Eq.~\eqref{eq:bh-entropy-w2}. \emph{Physics intuition:} the factor
of $1/4$ is not chosen --- it is whatever the first law produced
once the temperature was fixed by the surface gravity, which is why
its appearance in independent calculations (conical deficits,
string counting, islands) is treated as a genuine consistency
check rather than a convention.

\GGtakeaway{Time is not one thing but four, supplied by two fibers
and their seam; entropy is the count of one-way writes; and a
horizon is the one place where the circle's constant, the
interval's constant, and the exchange window's constant are all
spent at once --- which is why black holes have been the
discipline's sharpest instrument for thinking about all three.}

%% ===================================================================
%% v1.11: the closing section is given its own page.  Set inline it began
%% eighteen lines from the foot of its page and spilled six lines onto the
%% next, leaving a six-line page at the end of Part IX.  A page break before
%% the section redistributes the same text as 25 + 24 lines, with no word
%% changed; the closing of a Part is a natural place for one.
\newpage
\section{What the course established, and what comes next}
\label{app:closing}

Nine topics, one fork applied nine times. Appendix~A established
the fork itself --- holonomy on a circle, none on an interval ---
and every appendix after it is that statement pushed into a
different corner of physics: open systems (B), gauge theory and the
renormalization group (C), the parameter count (D), the launch (E),
the dark sector (F), the discrete symmetries (G), confinement (H),
and time with its horizons (I).

The Step~3 results are not all of one kind, and the course would be
dishonest if it left that impression. Derived within the declared
axioms: the group/semigroup split and its consequences; the
no-running theorem for circle-side quantities; the anomaly
coefficients as integers; the $\mathbb{Z}_3$ registration rule; the
tick as angular velocity; the slope $d\ln t/dr = 2$.
Derived-conditional, with the condition named at the site: the
plateau class and hence $(n_s, r)$; the Born rule at the relaxation
level; the Kaxion's patch-uniform angle. Postulated, and flagged
where it is used: the $K/2$ exponent of the toll. Flagged as
arithmetic without a mechanism: the $4K$ and $8K$ compressions. A
reader who works the appendices and comes away able to say which is
which has got what they were written for.

The course has answered the machinery questions. What remains are
the objections a reader raises \emph{after} working it. Part~X
takes them in ten clusters, and begins --- deliberately --- with
the twenty-two a thoughtful non-specialist asks before any
specialist gets a turn.

%%  PART X -- COMMON OBJECTIONS AND RESPONSES
%% ===================================================================
\GGPart[Common Objections]{Common Objections and Responses}{The
Strongest Objections a Field Theorist, Cosmologist, Quantum
Foundations Researcher, or Philosopher of Science Is Most Likely to
Raise on First Reading --- Answered, with Pointers into the Paper}
\label{part:X}

A white paper that claims one structural sentence addresses sixteen
of the discipline's standing problems will be met --- rightly ---
with skepticism from every quarter. This Part collects one hundred
and seventeen objections, grouped by the background of the reader
most likely to raise each, together with extended responses. They
are not strawmen; they are the questions the author has actually
been asked, sharpened.

The first cluster is different from the rest and comes first
deliberately: twenty-two questions from the \emph{curious reader}
--- basic in form and philosophically deep in substance, about
hidden dimensions, the multiverse, free will, what came before the
Big Bang, and whether a framework that explains this much can be
trusted at all. Popular science has settled several of them in
directions this paper does not take, and a reader is owed plain
answers before the technical clusters begin. The nine clusters that
follow are the specialists': methodology and numerology; the
anatomy's own status; quantum foundations; gauge theory and the
Standard Model; cosmology; the foundations of particle physics;
time, entropy, and black holes; comparisons with rival programs;
and falsifiability and standing.

Nothing is deflected. Where the honest answer is ``not yet,'' the
response says so and points to the honest ledger
(\S\ref{sec:honest}), the falsifier table (\S\ref{sec:kill}), or the
queued campaigns (\S\ref{sec:open-w2}) that record exactly which
claims are theorems, which are conditional, and which are still
owed. Several responses concede the objection outright. The test of
a framework is not that it deflects every question, but that it
knows which questions it has answered and which it has not.

\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

%% =====================================================================
%% Cluster 1 -- For the curious reader
%% =====================================================================

\section{For the curious reader: the big questions, plainly
answered}
\label{sec:phil-objections-w2}

The objections that follow are not a specialist's technical
challenges --- those come afterwards, cluster by cluster. They are
the questions a thoughtful non-specialist actually carries into a
book like this one: about hidden dimensions, the multiverse, free
will, what came before the Big Bang, whether a theory that explains
this much can be trusted at all. Popular science has made several of
them feel settled in directions this framework does not take, so
they are met first, and plainly. Where a claim rests on a still-open
step, the response says so. The aim 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-W2-1) (the curious reader)}\par\nobreak
\noindent\textbf{``What does `the circle remembers, the interval forgets' actually mean? It sounds like poetry.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The slogan compresses two mathematical facts, and
both can be checked. A \emph{holonomy} is what you get
when you carry something around a closed loop and compare it with
what you started with; it is the quantity all of gauge physics is
built from. On a circle every path closes, so everything a
connection does there is a holonomy --- and holonomies on a circle
come in whole numbers of turns, which nothing continuous can
change. That is ``remembers'': not a metaphor for memory but the
observation that an integer has nowhere to move
(Appendix~\ref{app:holonomy}). On an interval no path closes, so
there is no loop to hold a holonomy at all, and the physical
transport along it --- coarse-graining, the cosmic descent ---
composes only forward, with no inverse. That is ``forgets'': not
that a record fades, but that there was never a protected record to
keep. If the poetry annoys you, the same sentence in algebra is:
one fiber's transports form a group, the other's a semigroup.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~2 (O-W2-2) (the curious reader)}\par\nobreak
\noindent\textbf{``Extra dimensions? Where are they? Why can't I point at one?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You cannot point at them because they are not
directions you could walk in --- they are \emph{non-spatial} by
construction. A useful comparison: you also cannot point at
``frequency,'' yet a radio dial navigates it and a laboratory
measures it to fourteen digits. In this framework the sixth
dimension is where a phase lives, and the fifth is where scale
lives --- the resolution at which you are looking. Both are
perfectly ordinary physics: the phase is what an
Aharonov--Bohm interferometer reads, and the scale is what a
particle physicist means by ``at what energy.'' What the framework
adds is that these are not bookkeeping conveniences but
\emph{directions of a geometry}, with a shape, a length, and
consequences. If that sounds like a strong claim, it is; that is
why Part~VII stakes it on measurements.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~3 (O-W2-3) (the curious reader)}\par\nobreak
\noindent\textbf{``Isn't this just string theory with different words? That also has extra dimensions.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They share the idea that our four dimensions are
not all of them, and the debt should be acknowledged. But the
programs differ in the one place that matters for a reader
deciding what to believe: \emph{how many worlds the theory allows}.
String theory's landscape appears to permit an astronomical number
of consistent vacua with different constants, so its predictions
for our constants have proved elusive; this framework's census
allows exactly one bulk of exactly this shape, and therefore has no
freedom to accommodate a different answer (Problem~15). The
practical consequence is that this one can be killed by a single
measurement --- a haloscope null, a tensor null, a proton decay ---
whereas the landscape is very difficult to expose. Whether being
killable is an advantage depends on whether you would rather have a
framework that survives everything or one that could be wrong by
2035.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~4 (O-W2-4) (the curious reader)}\par\nobreak
\noindent\textbf{``Isn't our universe just one bubble among infinitely many, and we happen to live in the one that works?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That is an influential story, but it is a
hypothesis, not a finding --- and it is exactly what this framework
is built to make unnecessary. The multiverse is invoked to explain
why our constants are life-friendly: if every possibility exists
somewhere, then of course we find ourselves where things work.
Notice the cost: it explains the numbers by giving up on computing
them. This paper takes the opposite route and computes them ---
$\alpha^{-1}(0) = 2^7 + 3^2 = 137$, $\Omm = 1/\pi$, a dark-matter
mass of $8.1\,\mu$eV --- from one geometry with no dial. If that
program succeeds there is nothing left for a multiverse to do,
because there was never a menu to select from. The honest status:
several entries are still conditional on named steps
(\S\ref{sec:honest}). But the wager is testable this decade, which
no multiverse claim is (Problem~15).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~5 (O-W2-5) (the curious reader)}\par\nobreak
\noindent\textbf{``If everything is forced, is there free will? Am I just a playback of a groove?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The determinism claimed here is deliberately and
precisely bounded, and the boundary is worth stating carefully.
What is forced is \emph{structure}: which layers of the world
exist, what the constants are, which vacuum is the floor. What is
not forced --- and what the framework says nothing about --- is
what happens \emph{within} a layer's combinatorial freedom:
histories, weathers, biospheres, the sentence you will think next.
The analogy the cohort uses is a language: a finite grammar and a
fixed alphabet permit unbounded literature. The groove fixes the
song's key, not the dancing. It is worth noticing that this is a
\emph{more} generous position than the usual one, not less: in the
standard picture the constants are accidents and your choices are
the output of deterministic microphysics; here the constants are
necessary and the room for what is genuinely unwritten sits exactly
where composition happens --- which is nearly everywhere
(\S\ref{sec:philosophy}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~6 (O-W2-6) (the curious reader)}\par\nobreak
\noindent\textbf{``What came before the Big Bang?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} On this framework the question changes shape
before it is answered, and the change is the useful part. The Big
Bang is not the beginning of everything but the \emph{end} of an
episode: the moment the descent's one traverse of its upper plateau
finished and dumped its energy into hot matter (Problem~5). What
preceded it was that traverse --- the roll --- and what preceded
\emph{that} is the framework's named open item, called the
entrance: why the coordinate began high on the plateau. The honest
answer is therefore ``a specific thing we can describe, followed by
a specific thing we cannot yet derive'' --- which is a better
position than either ``a singularity where physics ends'' or ``an
unanswerable question,'' and it comes with a work order attached
(\S\ref{app:E-step3}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~7 (O-W2-7) (the curious reader)}\par\nobreak
\noindent\textbf{``Why is ninety-five per cent of the universe invisible? That seems like an embarrassment.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The ninety-five per cent is an embarrassment on the
usual reading, where the dark sector is two unexplained substances
that happen to dominate. On this reading it is not a substance problem at all: the
dark sector is the \emph{two hidden directions themselves}, in
their possible states of motion. Inflation was the open direction
rolling once; dark matter is the compact direction still ringing;
dark energy is both at rest. Ordinary matter --- everything you can
touch --- is what got deposited on the boundary along the way. That
the deposits are five per cent of the total is then not surprising
but arithmetic: the shares are ratios of the fibers' volumes,
$\Omm = 1/\pi$ and $\Omb = 1/2\pi^2$ (Problem~7). And it is
testable: the ringing has a frequency, $1.95$~GHz, and instruments
are scanning it now.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~8 (O-W2-8) (the curious reader)}\par\nobreak
\noindent\textbf{``Why would nature care about a number like 35? It seems arbitrary.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Thirty-five would be arbitrary if it had been
chosen to fit something. It was not: $K = 35$ is $5 \times 7$, the product of two
other structural integers, and the same number is reached
independently through the anomaly audit --- two routes with no
shared step (\S\ref{app:D-step3}). What makes it feel arbitrary is
that we are used to meeting it in the form ``the weak scale is
$10^{17}$ times smaller than the Planck scale,'' which sounds like
a fine-tuning disaster. Written as a depth instead, it is
$\ln(M_{\mathrm{Pl}}/2\Mlock) = 35.00$. The same fact is
``impossible'' in one language and a small integer in the other,
and the choice between languages is precisely what a resonance at
$3.849$~TeV would settle.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~9 (O-W2-9) (the curious reader)}\par\nobreak
\noindent\textbf{``$\pi$ turning up in the dark-matter fraction sounds like numerology to me.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You are right to be suspicious, and the suspicion
has a precise answer. $\pi$ is not decorating the result; it is
there because the quantity is a \emph{ratio of volumes of round
things}: the circle's circumference over the three-sphere's volume,
$2\pi/2\pi^2 = 1/\pi$ (Eq.~\eqref{eq:F-hopfvols}). Whenever a
physicist computes a ratio of spheres, $\pi$'s appear; nobody calls
the $4\pi$ in Coulomb's law numerology. The real defense, though,
is procedural rather than aesthetic, and it is worth stating: the
same five integers must serve \emph{every} entry in the codebook at
once, so each new agreement is a constraint on a fixed small
ledger, not a fresh search. Numerology gets easier with each
success; this gets harder (\S\ref{app:D-step3}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~10 (O-W2-10) (the curious reader)}\par\nobreak
\noindent\textbf{``If the universe is this finely arranged, doesn't that point to a designer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework's answer is the opposite of a
designer, and the difference is worth being exact about. A designer
is invoked when the arrangement looks \emph{chosen} --- when the
constants could have been otherwise and happen to be right. This
paper's claim is that they could not have been otherwise: given the
axioms, the geometry is forced, the integers are forced, and the
constants follow. Nothing was selected, so there was no selection
to attribute to anyone. The cohort's four words are: not
contingent, not lucky, not designed --- \emph{forced}. That is a
scientific claim rather than a theological one precisely because it
can fail: if a single constant turns out to be adjustable, the
forcing is broken (\S\ref{sec:kill}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~11 (O-W2-11) (the curious reader)}\par\nobreak
\noindent\textbf{``Why is there something rather than nothing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Physics cannot answer that in the metaphysical
sense, and this paper does not pretend to. What it can do is
narrow the question usefully. Suppose there is \emph{anything} at
all that both persists and changes. Then, by the minimality
argument of \S\ref{sec:minimum}, it needs a structure that can keep
things (a compact fiber, whose contents cannot drift) and a
structure in which things can happen (an open fiber, whose steps
cannot be undone) --- and one of each is the minimum. So the
framework does not explain why there is something; it explains why,
\emph{if} there is something, it must have roughly this shape. That
is a smaller claim than the question asks for, and it is the
largest claim the author believes physics is entitled to make.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~12 (O-W2-12) (the curious reader)}\par\nobreak
\noindent\textbf{``If a measurement is a `write,' who or what does the writing? Does it need a conscious observer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, and the framework is unusually definite about
this. The write happens at the first \emph{thermodynamically
irreversible} amplification --- a Geiger avalanche, a grain of
photographic emulsion darkening, a crystal defect forming --- and
it happens whether or not anyone is watching, has been born, or
ever will be. Consciousness plays no role anywhere in the account.
The reason the older literature kept reaching for observers is that
the textbook offered two dynamical laws and no rule for when each
applied, so \emph{something} had to decide; here the decision is
made by the physics of registration, and the cat's fate is settled
microseconds into the hour, at the counter, meters from the cat
(Figure~\ref{fig:cat-timeline-w2}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~13 (O-W2-13) (the curious reader)}\par\nobreak
\noindent\textbf{``So is Schr\"odinger's cat alive and dead, or not?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not. The superposed-cat state is never prepared,
because a cat cannot be assembled into one without a torrent of
irreversible writes along the way. The nucleus is genuinely in
superposition; the counter is not, for longer than about
$10^{-24}$ of a second (Appendix~\ref{app:openquantum}). Opening
the box changes your knowledge, not the cat. The thought
experiment keeps exactly one deep lesson, which the anatomy
sharpens rather than dissolves: between the atom's phases and the
counter's click there is a genuine change of category --- question
to answer, circle to interval --- and the line sits at an event
type, not at a size, a species, or a mind.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~14 (O-W2-14) (the curious reader)}\par\nobreak
\noindent\textbf{``Is time real, or is it an illusion, as I keep reading?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Real, but not one thing --- which is why the
debate has run so long. What we call time is four separate
endowments, and the confusion comes from asking one structure to
supply all four. A \emph{tick} (something that repeats, so duration
can be counted) is the circle's: every clock is a return, and
$E = h\nu$ prices it. An \emph{arrow} (before differs from after)
is the interval's: its transport has no inverse. A \emph{rate}
linking them is the seam's, and it is exact ($d\ln t/dr = 2$ in the
radiation era). An \emph{extent} is the descent's length. Nothing
here is illusory; what is illusory is the assumption that a single
parameter $t$ was ever doing all four jobs (Problem~13).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~15 (O-W2-15) (the curious reader)}\par\nobreak
\noindent\textbf{``Why can't we travel back in time, if the equations of physics work both ways?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the equations that work both ways are not
the ones doing the recording. The reversible laws live on the
compact fiber, where transports have inverses; every act of
registration --- every photon absorbed, every bit written, every
decay counted --- happens on the open fiber, where they do not.
Going back would require running that transport backwards, and the
backwards element does not exist in the algebra: it is not
forbidden by a law, it is absent from the mathematics
(Eq.~\eqref{eq:A-kadanoff}). This is a stronger statement than the
usual thermodynamic one, which says reversal is overwhelmingly
improbable. Here it is not improbable; it is not defined.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~16 (O-W2-16) (the curious reader)}\par\nobreak
\noindent\textbf{``Where did all the antimatter go?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nearly all of it annihilated, and the reason
anything survived is a small surplus --- about six extra matter particles for every
ten billion photons. The framework's account has two halves. Why
antimatter exists at all: a circle has exactly two winding senses,
and an antiparticle is the same class read the other way round,
which is why the mirror is perfect in mass and exact in charge
reversal (Problem~10). Why the surplus: the early universe crossed
one gated, one-way passage, and the crossing charged a toll set by
the descent's depth, $e^{-K/2} = e^{-17.5}$. That gives
$5.9 \times 10^{-10}$ against a measured $6.1 \times 10^{-10}$.
The honest note: one exponent in that chain is postulated rather
than derived, and the paper says so every time it quotes the
number.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~17 (O-W2-17) (the curious reader)}\par\nobreak
\noindent\textbf{``Why has nobody ever seen a single quark?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The textbook answer is that separating them costs
energy without limit, so the vacuum makes a new pair instead. That
is correct and measured --- the force between quarks is about
sixteen tonnes-force and does not weaken with distance
(Appendix~\ref{app:confinement}). This framework adds a reason
underneath it: color phases on the compact fiber close only in
threes, and a configuration with an unclosed color book cannot be
\emph{registered} by the boundary at all --- the projection
annihilates it, in the same way a selection rule forbids certain
atomic transitions. So an isolated quark is not expensive; it is
unreadable. The linear force is the price the world charges for
approaching a state it can never report.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~18 (O-W2-18) (the curious reader)}\par\nobreak
\noindent\textbf{``Is information really not lost in a black hole? Hawking said it was.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Hawking's calculation was right about what it
computed and, on this reading, was asking one object to do two
jobs. Sort the transaction by fiber and it closes: what must be
\emph{returned} --- charges, class labels, the conserved content
--- is circle-side and was never destructible; what is
\emph{spent} --- the thermal, one-way outflow --- is
interval-side. The famous paradox was billing the spending against
the keeping. The radiation's entropy therefore rises, turns over at
about half the evaporation, and returns to zero, which is the Page
curve; and the island computations of the last few years get the
same curve from gravity itself. What this framework adds is not the
curve but the answer to ``where was the information meanwhile?''
--- in the fiber ledgers, inaccessible to the outside observer,
never absent from the bulk (Problem~16).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~19 (O-W2-19) (the curious reader)}\par\nobreak
\noindent\textbf{``This explains an awful lot. Doesn't that make you suspicious?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A framework of this reach should make everyone
suspicious, the author included, and this paper is built around
that suspicion. A framework
that could accommodate any outcome would be worthless no matter how
elegant. So the test to apply is not ``how much does it explain?''
but ``what would it forbid?'' This one forbids: a tensor-to-scalar
ratio far from $0.0043$; a dark-matter line outside
$1.59$--$2.13$~GHz; a neutrino phase away from $196.4^\circ$; any
proton decay in a guarded channel; any $CPT$ violation at any
precision; any spatial variation of a constant; a dark-energy
equation of state away from $w = -1$. Seven ways to die --- and they
are seven of the dated conditions collected in
Table~\ref{tab:kill}, not the whole list. If the framework explained a great
deal and forbade nothing, the objection would be fatal.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~20 (O-W2-20) (the curious reader)}\par\nobreak
\noindent\textbf{``Why should I take one person's framework seriously against the mainstream?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You should not take it on authority --- there is
none on offer, and the author is an experimentalist rather than a
prophet. Take it, if at all, on exposure. The relevant question is
not who wrote a framework but whether it has made itself checkable,
and by whom. Every number in this paper is a closed form a reader
can evaluate on a calculator; every claim carries a grade; every
open step is named rather than hidden; and the decisive
measurements are being made by large collaborations with no stake
in the outcome. That is the whole basis on which the paper asks to
be read. If the haloscopes sweep the window and find nothing, no
amount of authority --- the author's or anyone's --- will save it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~21 (O-W2-21) (the curious reader)}\par\nobreak
\noindent\textbf{``What would convince you that you are wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fastest and cleanest: a full haloscope sweep of
$1.59$--$2.13$~GHz at design sensitivity with no line, or a line
with no companion at $\Delta\nu/\nu = 1/35$. That would remove the
compact fiber's clearest laboratory signature with no parameter
left to move. Nearly as damaging: a tensor-to-scalar ratio well
away from $0.0043$ once the Simons Observatory and LiteBIRD report --- the launch's
census has the least room to negotiate. Structurally fatal, though
least likely to be seen: any $CPT$ violation, which would require
the geometry to possess a fourth orientable structure it does not
have. The author regards the first as the live one, and the
instrument is running.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~22 (O-W2-22) (the curious reader)}\par\nobreak
\noindent\textbf{``Suppose I am a student and I find this interesting. What should I actually do?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Work Part~IX rather than read it --- the
appendices are a course, and the forty-two worked problems are the
point of it. Then pick one of three open items, all specified to
the level where someone with fresh eyes could begin: the
\emph{entrance} (why the descent's coordinate starts where it
does), the \emph{e-fold budget} (which would convert the deepest
flagged number in the cohort into a theorem), or the
\emph{field-grade representation step} (the one analytic residue in
the Born chain). And then try to break the thing: publish what you
find either way, because the paper is built so that both outcomes
are progress. That is not a courtesy --- a framework with seven
dated kill conditions genuinely needs people trying to trip them.
\end{objqabody}


%% =====================================================================
%% Cluster 2 -- Methodology, honesty, and numerology
%% =====================================================================

\section{Methodology, honesty, and numerology}
\label{sec:obj-method}

The specialists begin here, and they begin with method rather than
physics, because a framework that computes this many numbers invites
one suspicion before any other: that the numbers were found rather
than derived. This cluster is the paper's answer to the look-elsewhere
statistician, the parameter counter, the referee who has seen too many
theories of everything, and the philosopher of science asking what the
grade vocabulary actually buys. The strongest of them is
Objection~25 --- frameworks that explain everything have a poor
historical record, and nothing in this paper exempts it from that
record. The response concedes the pattern and points at the only
defense available: a table of dated ways to be wrong.

\bigskip
\noindent\textbf{Objection~23 (O-W2-23) (the look-elsewhere statistician)}\par\nobreak
\noindent\textbf{``With five integers, $\pi$, and elementary operations one can approximate almost any measured number. How is the codebook not curve-fitting with an enormous trials factor?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the single most important methodological
objection and it deserves a quantitative rather than a rhetorical
answer. Three features bound the trials factor. \emph{One
alphabet}: the same five integers $(1,3,5;7,35)$ must serve every
entry simultaneously, so the effective search space is the ledger,
not the space of expressions --- and each new closed form is a
constraint on a fixed object rather than a fresh draw.
\emph{Overdetermination}: $K$ is reached by two routes with no
shared step, and $N = 3$ by two more (\S\ref{app:H-step3}); a
fitted integer has one route by construction. \emph{Publication
order}: every closed form is registered with a named kill condition
and a dated instrument before the decisive measurement
(Table~\ref{tab:kill}). The distinguishing feature of numerology is
not that it uses integers but that it never volunteers to die.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~24 (O-W2-24) (the parameter-counting skeptic)}\par\nobreak
\noindent\textbf{``You claim sixteen problems answered, but your own master table shows grades from D to F. Isn't `sixteen solved' an overstatement?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} ``Sixteen solved'' would be an overstatement, which
is why the paper never says it.
Table~\ref{tab:master} grades every row, and
\S\ref{sec:six-problems} classifies each with a deliberately varied
verb: solved, dissolved, re-founded, reread, repriced, transformed,
computed-with-joints-named, identified, unemployed. Several rows
claim conspicuously less than a partisan would --- the vacuum's
scale is \emph{repriced}, not solved; the beginning is
\emph{transformed} into an entrance with a work order; confinement
is \emph{reread} with its formal proof still open. The correct
one-sentence summary is ``sixteen problems addressed by one
structural sentence, at sixteen separately stated grades,'' and
refusing to inflate that is a feature.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~25 (O-W2-25) (the referee who has seen too many theories of everything)}\par\nobreak
\noindent\textbf{``Frameworks that explain everything have a poor historical record. Why is this one different?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This one may not be different, and the paper's
structure concedes the point in advance. The historical failure mode you are pointing at
is a framework flexible enough to absorb any result; the defense
against it is not eloquence but a table of dated ways to die, which
is why Part~VII ends with one and why every Problem's treatment
carries its own kill condition. There is a second, subtler failure
mode --- a framework that predicts many things, all of which were
already known. Against that the relevant entries are the ones not
yet measured: the tensor ratio to one significant figure, the
haloscope doublet spacing, the neutrino phase, the lock's
resonance. Judge the framework on those, in about a decade.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~26 (O-W2-26) (the statistician, returning)}\par\nobreak
\noindent\textbf{``Your quoted agreements are at the $0.03$--$4\%$ level. Without stated theory errors, how can a reader assess them?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By reading the parent papers, where the theory-error
budgets are itemized; this synthesis quotes their results rather
than re-deriving their uncertainties, and says so. But the more
useful answer is structural. The closed forms are leading-order
anchors with computable dressing, so the kill condition is never a
fourth decimal --- it is whether the measured value is reachable
\emph{from the declared integers by the declared dressing}. A miss
of $4\%$ with a named postulated exponent (the $\etaB$ chain) is a
different object from a miss of $4\%$ with no account of where it
came from, and the paper's grading exists precisely to keep the two
apart.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~27 (O-W2-27) (the philosopher of science)}\par\nobreak
\noindent\textbf{``Isn't `derived-conditional' just a way of saying `assumed' with better manners?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, and the distinction is operational rather than
polite. A \emph{named input} is a number taken from experiment. A
\emph{postulated} step is a claim asserted with motivation and no
derivation. \emph{Derived-conditional} means the result follows by
proof once a specific, stated lemma is supplied --- so a reader can
go and attack the lemma, and a successor can close it without
touching anything else. The test of whether the vocabulary is doing
work is whether the conditions are few and named: they are five,
listed in \S\ref{sec:honest}, and the paper's own campaigns are
organized around closing them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~28 (O-W2-28) (the cross-paper referee)}\par\nobreak
\noindent\textbf{``This paper derives nothing itself. Why should a synthesis be published at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the synthesis is the claim. The twenty-two
cohort papers each derived their own results; what none of them
could show, individually, is that a single structural sentence sits
in the load-bearing position of all of them. That is a statement
about the set, and only a document over the set can make it ---
along with the accompanying observation that the deployment
required no value, grade, or theorem to change anywhere
(Table~\ref{tab:deployment}). A reader who regards that as weak
evidence is entitled to; a reader who regards it as no evidence
should say what an organizing language would have to do to earn
its keep, given that it earned this one by \emph{failing to require
any modification}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~29 (O-W2-29) (the skeptical reader of grades)}\par\nobreak
\noindent\textbf{``You mark the $4K$ and $8K$ family `flagged.' What stops a reader from quietly counting them as support anyway?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing but the paper's own discipline, which is why
the discipline is stated in three separate places and applied
without exception: flagged items are cited downward (a derived
claim may be used anywhere) and never upward (nothing leans on a
flag). Concretely: the age ledger's $140.24 \approx 4K$ and the de
Sitter entropy's $\ln S \approx 8K$ appear in the number audit,
are discussed in Problems~13 and~16, and are used in exactly zero
derivations. If the e-fold budget campaign closes, they become
theorems; until it does, they are arithmetic observations, and the
paper would be equally strong with them deleted.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~30 (O-W2-30) (the reproducibility referee)}\par\nobreak
\noindent\textbf{``How can a reader check the numbers without re-deriving the whole cohort?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Every quoted value is a closed form in the five
integers and the five constants, so a calculator suffices ---
which is the point of Table~\ref{tab:numbers} and of the worked
problems in Part~IX. The chain of Eq.~\eqref{eq:chain-w2} is five
lines long and can be checked in ten minutes; the inventory of
Eq.~\eqref{eq:inventory-p2} in two. Where a value requires cohort
machinery (the plateau class, the pinning) the paper says so and
cites the parent. The author's standard, applied to this draft, is
that no number is carried from memory: each was re-derived
programmatically during preparation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~31 (O-W2-31) (the historian of physics)}\par\nobreak
\noindent\textbf{``Physics has seen elegant geometric unifications before --- Weyl's, Kaluza's, Eddington's --- and they failed. What is different?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The failures you name are instructive and the paper
tries to learn from them rather than dismiss them. Weyl's 1918
gauging of length failed \emph{physically}: atoms would remember
their histories, and they do not. Kaluza--Klein succeeded formally
but had no way to fix the compactification's size or content, so it
predicted nothing sharp. Eddington's program produced numbers
without a mechanism to generate them. The present framework's
distinguishing feature relative to all three is that its geometry
is not chosen for elegance but forced by a declared axiom set, and
that the forcing produces \emph{dated laboratory targets} rather
than relations among known quantities. The lesson from Weyl is
also directly incorporated: the repair that worked was a
\emph{fiber swap}, from a direction whose value is physical to one
whose value is not (Appendix~\ref{app:gaugeRG}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~32 (O-W2-32) (the working theorist)}\par\nobreak
\noindent\textbf{``Which single claim, if withdrawn, would collapse the most of this paper?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The minimality argument of \S\ref{sec:minimum} --- that
a world which both persists and changes needs exactly one fiber of
each kind. If the bulk could carry two compact fibers, or none, the
census arguments of Problems~2, 5, 7, and~15 lose their force
simultaneously, because each of them works by forbidding content.
The topological statements would survive (a circle would still
quantize, an interval would still forget), but the paper's
distinctive claims --- one inflaton, one axion, one vacuum ---
would become preferences rather than theorems. The weight
therefore sits on \AX{2}'s forcing theorems, and the paper says so
rather than distributing it evenly for comfort.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~33 (O-W2-33) (the referee asking about scope)}\par\nobreak
\noindent\textbf{``You address sixteen problems. Why those sixteen, and not others? Was the list chosen to suit the answer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A fair suspicion, and the answer is checkable: the list
is the discipline's, not the author's. Fifteen of the sixteen have
their own literatures, their own review articles, and in several
cases their own prize problems, and the paper dates each to the
year it took its modern form (Table~\ref{tab:ages}). The one
grouping choice made here is joining antiparticles with the matter
excess, which the literature usually separates --- and that
grouping was made because the two share a mechanism on this
reading, which is an argument the reader can reject without
affecting anything else. Problems deliberately \emph{not} claimed
include quantum gravity's ultraviolet completion, the neutrino mass
ordering, and the strong-interaction spectrum from first
principles.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~34 (O-W2-34) (the referee on register)}\par\nobreak
\noindent\textbf{``The prose is unusually literary for a physics paper. Does the style hide weak arguments?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That is a fair worry, and I have tried to guard
against it in three ways. Every image is attached to an equation
displayed in the same paragraph, so a reader can discard the image
and keep the content. Every claim carries a grade in the same
sentence, so no amount of good writing can promote a conditional to
a theorem. And the technical appendices are written in the driest
register available, with the derivations complete, precisely so
that the body's prose has somewhere to be checked against. Where
the style has failed --- where a metaphor is doing work an
argument should be doing --- that is a defect, and the author would
like to be told which paragraph.
\end{objqabody}

%% =====================================================================
%% Cluster 3 -- The anatomy itself
%% =====================================================================

\section{The anatomy itself: mechanism, novelty, and circularity}
\label{sec:obj-anatomy}

This cluster attacks the instrument itself. If the two-fiber reading
is a vocabulary rather than a mechanism --- a way of relabeling
results already obtained --- then everything downstream is
redescription, however elegant. The objections here press exactly
that, from four directions: is it a mechanism or a metaphor, is it
novel, is it circular, and does it forbid anything. The sharpest is
Objection~37, which asks what the anatomy predicted \emph{before} it
was applied. The honest answer is short, and it is given.

\bigskip
\noindent\textbf{Objection~35 (O-W2-35) (the skeptical field theorist)}\par\nobreak
\noindent\textbf{``This is a metaphor, not a mechanism. Sorting results into two piles does not explain them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The sorting would be a metaphor if it ever needed a
parent result adjusted to fit. It never did
(Table~\ref{tab:deployment}), and at several points it is the
mechanism in the literal sense. The registration rule of
Problem~12 is a derivation of the singlet spectrum: the boundary
projection annihilates configurations carrying a residual center
phase, Eq.~\eqref{eq:H-selection}. The orientation calculus of
Problem~9 derives the antiunitarity of $T$ rather than assuming it.
The census of Problem~7 forbids field content --- one interval,
one inflaton. A metaphor cannot forbid anything, and it certainly
cannot annihilate a class of states in an integral.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~36 (O-W2-36) (the renormalization-group specialist)}\par\nobreak
\noindent\textbf{``The interval is just the RG. You have renamed renormalization and called it a dimension.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The identification is correct and is the fifth
dimension's definition, so there is no disagreement about the
mathematics. The question is whether the geometric reading adds
anything, and it adds three things the RG language does not carry.
A topological \emph{contrast} with a compact partner --- the source
of every exactness statement in the paper, and something the RG
alone has no vocabulary for. \emph{Endpoints}: the RG's $\ln\mu$ is
a range of convenience, while $\Ir = [0,K]$ has a length, and the
length is $35$ (Eq.~\eqref{eq:C-rflow}). And a \emph{census}: the
RG has no reason to be one-dimensional, whereas the bulk's
dimension count forbids a second such coordinate --- which is what
retires the inflaton menu. If none of the three convinces, the
falsifier is direct: the RG predicts nothing at $3.849$~TeV.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~37 (O-W2-37) (the extra-dimensions referee)}\par\nobreak
\noindent\textbf{``Compactness explaining quantization is textbook Kaluza--Klein. What is actually new?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The circle half alone is venerable and the paper says
so. What is new is the \emph{pair}, and specifically the claim that
the second fiber's poverty is as productive as the first fiber's
wealth. Kaluza--Klein theories are theories of what a compact
direction \emph{gives}; there is no Kaluza--Klein poverty theory,
no body of results that follows from a direction having no loops,
no memory, and no inverses. Every irreversibility claim in this
paper --- measurement, the arrow, the descent, evaporation ---
comes from that side, and it is the side the literature has never
worked. The minimality argument then closes the pair: not one of
each because it is pleasing, but because neither alone hosts a
world.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~38 (O-W2-38) (the philosopher of time)}\par\nobreak
\noindent\textbf{``Deriving an arrow from a semigroup is circular. You put irreversibility in by hand when you chose an interval.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The choice is not free, and locating exactly where the
weight sits is the right response rather than a denial. The
constitutional minimum argues that a bulk with two circles
conserves everything and decides nothing, and a bulk with two
intervals decides everything and keeps nothing
(\S\ref{sec:counterfactuals}); the interval is forced by the same
axioms that force the circle. So the arrow is an output of \AX{2},
not an input here. One may reject \AX{2} --- that is a legitimate
attack and the cohort carries the forcing theorems in
\citet{ArisakaA2GG6} where they can be examined. What the objection
correctly identifies is that the paper's arrow is only as good as
those theorems, and the paper does not pretend otherwise.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~39 (O-W2-39) (the referee on grades)}\par\nobreak
\noindent\textbf{``The quantitative face is a patchwork: the impressive numbers lean on named inputs, postulates, and open integers.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and stated in place every time. The $\etaB$
chain carries a named input, a named regulator, and a postulated
exponent; the trio carries a declared amplitude, a pinning
conditional, and one open integer ($\Xicap$); the vacuum carries
an uncomputed drainage factor. The cohort's position is that a
chain with named joints is worth more than a chain with hidden
ones, and that the joints are few enough to count on one hand ---
which is itself falsifiable: close the campaigns of
\S\ref{sec:open-w2} and the joints must \emph{vanish}, not
multiply. If a decade of work leaves five joints as fifteen, the
objection will have been right.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~40 (O-W2-40) (the Popperian)}\par\nobreak
\noindent\textbf{``If one sentence explains sixteen problems, it explains nothing. Where is the content?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The content is in what it forbids, and the list is
short enough to print: a tensor-to-scalar ratio away from
$0.0043$; a dark-matter line outside the discrete cap window; a
missing doublet at $\Delta\nu/\nu = 1/35$; a neutrino phase away
from $196.4^\circ$; proton decay in a guarded channel; any $CPT$
violation at any precision; any spatial or temporal variation of a
constant; a dark-energy equation of state away from $w = -1$; a
running anomaly coefficient. Nine prohibitions, all dated, none of
them consequences of one another. A framework that accommodated any
outcome would deserve the objection; this one is answered by
Table~\ref{tab:kill} or not at all.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~41 (O-W2-41) (the mathematical physicist)}\par\nobreak
\noindent\textbf{``You say an interval `cannot hold holonomy.' Strictly, a connection on an interval does have parallel transport. Is the statement precise?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The distinction is real, and the precision matters.
Parallel transport along an open path certainly exists --- Eq.~\eqref{eq:A-transport}
--- but it is not gauge-invariant: it transforms as
$U \to g(\text{end})\,U\,g^{-1}(\text{start})$, so it is not an
observable. Holonomy in the sense used throughout this paper is the
\emph{gauge-invariant} content, which requires start and end to
coincide. On $\Ir$ no path closes, $\pi_1$ is trivial, and the
moduli space of flat connections modulo gauge is a single point ---
so there is nothing invariant to hold (\S\ref{app:A-step3}). The
sentence in the abstract is the physicist's abbreviation of that
paragraph, and the appendix states it in full.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~42 (O-W2-42) (the referee on emergence)}\par\nobreak
\noindent\textbf{``Could the two fibers be emergent rather than fundamental --- an effective description of something deeper?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Possibly. The cohort claims forcing from its axioms,
not metaphysical finality, and the author has no attachment to
fundamentality as such. But notice what any deeper layer would have
to supply: a sector that cannot hold holonomy and one that can, or
it loses the sixteen answers. The anatomy would then be a
constraint on successor theories rather than a casualty of them,
which is the most a physical structure can ask for. The parallel
worth keeping in mind is thermodynamics, which survived the atomic
hypothesis by becoming a constraint on it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~43 (O-W2-43) (the seminar questioner)}\par\nobreak
\noindent\textbf{``Is the circle literally a place? Could something go around it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing spatial goes around it: the fiber directions are
non-spatial by construction, and no boundary protocol resolves a
position on them (\OGGEP{}). What goes around are phases --- the
circle is where phase lives, as the interval is where scale lives.
Asking to stand on the circle is like asking to stand on a
frequency; the geometry is entirely real without being terrain.
What makes the claim more than semantics is that the fiber has
measurable consequences a mere bookkeeping device would not have:
a Wilson-line modulus that sets the electroweak scale, a
Kaluza--Klein tower at $3.849$~TeV, and a ringing coordinate at
$1.95$~GHz.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~44 (O-W2-44) (the seminar questioner, pressing)}\par\nobreak
\noindent\textbf{``Why can't the interval be a very large circle? Would anything change?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Everything. A circle of any circumference supports
holonomy: charges quantize on it, the flow returns, the descent
becomes periodic, and the arrow, the records, and the welds all
dissolve on the recurrence time. The interval's openness is not an
approximation to a large circle; it is the opposite category, and
the difference is visible in one clause of
Eqs.~\eqref{eq:group-law-p2}--\eqref{eq:semigroup-law-p2}. This is
also the sharpest way to see why the framework treats cosmic
one-wayness as topological rather than statistical: a statistical
arrow is a large circle's, and it recurs.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~45 (O-W2-45) (the referee on the projection)}\par\nobreak
\noindent\textbf{``Why should observation be a projection at all? Isn't \OGGEP{} the whole result smuggled into an axiom?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \OGGEP{} is an axiom, declared as \AX{3}, and a
reader is entitled to attack it --- but it is not the whole result. What
\OGGEP{} states is that a boundary observer cannot resolve the
fiber coordinates; what it does \emph{not} state is which fiber
supplies the reversible term, which supplies the dissipator, what
the Born measure is, or why the discrete symmetries number three.
Those follow from how transports compose on the two fibers, and the
derivation is in Appendix~\ref{app:openquantum}: transport around the
circle has an inverse and contributes a commutator; the physical flow
along the interval has none and contributes a dissipator. It is worth
saying which argument is \emph{not} being made, because the tempting
one fails. One might argue the same point spectrally --- a compact
fiber has a discrete spectrum and cannot supply the continuum
irreversibility requires --- but $\Ir = [0,K]$ is a finite segment, so
its spectrum is discrete too, and its level spacing is the lock
spacing that Problem~2 stakes at a collider. The appendix says so in
place. The algebraic statement is the one that carries the weight, and
it is where the work is. \OGGEP{} is still doing something the
objection is entitled to attack: it makes the projection compulsory
rather than approximate, and one further hypothesis --- Markovianity
--- is named there as a physical assumption rather than a
consequence.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~46 (O-W2-46) (the referee on the two counterfactuals)}\par\nobreak
\noindent\textbf{``The two-circle and two-interval worlds are rhetorical devices. Can the minimality claim be made technical?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The counterfactual can be made precise, and its
technical form is a statement about the transport algebras. A world with two compact fibers has all
transports in a group: every element invertible, hence no map that
can serve as a registration --- and a record is precisely a deposit
with no inverse, so \AX{3} would have nothing lawful to project
\emph{onto}. A world with two open fibers has no non-trivial
$\pi_1$ anywhere, hence no protected classes: no quantized charge,
no stable spectrum, and no interference (which requires a phase
that returns). The claim ``being needs holonomies; becoming needs a
semigroup'' is those two sentences. What remains genuinely
argumentative rather than provable is the step from ``a world'' to
``\emph{this} world,'' and the cohort carries that in \AX{2}.
\end{objqabody}


%% =====================================================================
%% Cluster 4 -- Quantum foundations
\bigskip
\noindent\textbf{Objection~47 (O-W2-47) (the historian, again)}\par\nobreak
\noindent\textbf{``Where has topological or fiber-first reasoning misled physics? Name the cases that cost you something.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The question is the right one to ask of any
program, and \S\ref{sec:w2-record} is the answer at length. Briefly,
three episodes. Weyl gauged the scale direction in 1918 because the
geometry was beautiful, and atoms would then have remembered their
histories; the framework's own fifth dimension \emph{is} the scale
direction, and the only thing separating it from Weyl's is that here
the direction carries no gauge-invariant content. From 1926 to the
1980s, compactness was widely expected to fix the low-energy content by
itself; it does not, there is no Kaluza--Klein poverty theorem, and
this paper's integers come from Step~A's anomaly audit rather than from
the circle being closed. And from 1979 to 2012 a great deal of
model-building rested on the structural instinct that an elementary
scalar cannot exist, which a scalar at $125$~GeV falsified --- an
instinct of the same genre as this paper's own argument for why the
Higgs is \emph{allowed}, run in the other direction. The
generalization the author would defend is that topology read off a
structure the physics forced has a good record, and topology imposed
because it was attractive has a poor one; whether this bulk belongs to
the first category is a claim about the order in which the work was
done, and a reader is entitled to be unmoved by it.

Two further items belong here because they are this paper's own.
The semigroup does not follow from the interval's topology, and
\S\ref{sec:primer} now grades the two premises apart. The interval is
not a spectral continuum --- $\Ir = [0,K]$ has a discrete spectrum and
the largest level spacing in the framework --- and
Appendix~\ref{app:B-step3} rests the fiber assignment on the
composition law instead. Neither correction moves a number, which is
exactly why they are worth printing: both sentences felt like
consequences of the founding statement, and neither was one.
\end{objqabody}

%% =====================================================================

\section{The quantum-foundations referee}
\label{sec:obj-qm}

The quantum-foundations referee has heard structural accounts of the
measurement problem before, and knows that most of them dissolve on
contact with the Born rule's measure. This cluster takes that
seriously: decoherence's genuine success and its exact residue, the
status of the Born derivation, non-locality and what the framework
concedes about it, and the comparison with collapse models and
relative-state accounts. The one to read first is Objection~52 ---
what, precisely, is still owed on Born --- because the answer names an
open analytic step rather than claiming closure.

\bigskip
\noindent\textbf{Objection~48 (O-W2-48) (the foundations referee)}\par\nobreak
\noindent\textbf{``Decoherence already explains the classical world. What does a second fiber add?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It adds the two things decoherence is known not to
supply, and the paper is careful to concede everything decoherence
does deliver. Decoherence explains the disappearance of
interference quantitatively --- forty orders of magnitude from one
$(\Delta x)^2$ law (Figure~\ref{fig:decoherence-scaling-w2}) ---
and it identifies the pointer basis given an interaction
Hamiltonian. What it does not do is produce a single outcome (an
ensemble is not an event) or derive the Born measure (it is written
in a language that presupposes it). The anatomy supplies the
missing environment non-optionally --- the fiber coordinates, which
no protocol resolves --- and derives the measure as a relaxation,
Eq.~\eqref{eq:B-relax}, with one named residue.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~49 (O-W2-49) (the Gleason-theorem specialist)}\par\nobreak
\noindent\textbf{``Gleason already shows the Born rule is the only consistent measure on a Hilbert lattice. Why derive it dynamically?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Gleason's theorem is a constraint on \emph{measures},
and it is beautiful and correct: given non-contextuality and a
Hilbert space of dimension three or more, the measure is unique. It
does not say why the world's frequencies should be a measure of
that kind, nor why the state should be updated the way it is, nor
does it carry a timescale. A dynamical account does all three, at a
price --- it must produce a relaxation, and a relaxation takes
time. That price is the paper's opportunity: the F25 relic of
Eq.~\eqref{eq:f25-w2} is a computable deficit at the largest
cosmological scales, which is an observational discriminator no
theorem of Gleason's type can offer.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~50 (O-W2-50) (the Bell theorist)}\par\nobreak
\noindent\textbf{``How does a two-fiber geometry evade Bell's theorem without either non-locality or superdeterminism?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework does not evade it; it accepts the experimental
verdict and locates it. Bell's inequality is violated in nature,
and this framework agrees. What it adds is a reading of
\emph{which} structure carries the correlation: the entangled
pair's shared holonomy class is circle-side data, which is global
by construction --- a homotopy class is not a local field and has
no gradient --- so the correlation requires no signal to propagate
and no signal can be sent with it. No-signaling is preserved
because class data cannot be locally set. The framework thus
inherits standard quantum mechanics' relationship to Bell exactly,
and adds no hidden variables of the kind the theorem excludes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~51 (O-W2-51) (the referee on real quantum mechanics)}\par\nobreak
\noindent\textbf{``You lean on the network Bell tests excluding real-valued quantum theory. Isn't that a fragile foundation?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a load-bearing experimental fact, and if the
2022 exclusions were overturned, the cohort's custody argument ---
that the geometry owns the $i$ while the dictionary owns the
sectors --- would require rework. That is stated as a dependency
rather than buried. Two remarks. The exclusion was designed as a
falsifiable proposal \citep{Renou2021} and then executed
independently on two platforms \citep{Li2022, Chen2022}, which is
the strongest form such a result can take. And the framework's use
of it is specific: it is why the complex structure must come from
somewhere physical, which on the anatomy is the circle's
orientation (Appendix~\ref{app:discrete}, Problem~G.1).
\end{objqabody}

\bigskip
\noindent\textbf{Objection~52 (O-W2-52) (the many-worlds advocate)}\par\nobreak
\noindent\textbf{``Everett needs no extra structure at all. Why prefer a geometry with a write?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Everett's position is coherent and is treated with
respect in Problem~1: it keeps the mathematics intact and pays in
ontology. The debt it carries is the one its own literature
acknowledges --- an account of why the Born weights are the
probabilities of anything, given that all branches occur. Decision
theoretic derivations exist and are contested. The framework here
takes the other option: keep one world and pay by identifying the
non-unitary step with a physical structure. The reason to prefer it
is not elegance but exposure: the write leaves a computable relic
(Eq.~\eqref{eq:f25-w2}), and Everett, by construction, leaves
none.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~53 (O-W2-53) (the dynamical-collapse theorist)}\par\nobreak
\noindent\textbf{``GRW-type models also make measurement physical, and they are already testable. What is gained?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two things, and one honest similarity. The similarity:
both accounts make registration a physical process rather than a
postulate, and both are therefore exposed. What is gained is that
the collapse rate here is not a new constant --- it is the port
coupling of a structure introduced for other reasons entirely, so
the account adds no free parameters, whereas GRW adds two. And the
signature differs: GRW predicts spontaneous heating and localization
anomalies in the laboratory; this predicts a primordial-statistics
deficit on the sky. Both are worth testing, and if the interferometry
bounds close GRW's parameter space while the large-scale statistics
show the computed deficit, the two accounts separate cleanly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~54 (O-W2-54) (the referee on the preferred basis)}\par\nobreak
\noindent\textbf{``You say the pointer basis is `what the projection can reach.' Isn't that circular?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be if the projection were defined by what it
reaches. It is not: \OGGEP{} fixes the projection as the
marginalization of $(r,\theta)$, independently of any measurement
discussion, and the reachable quantities are then whatever survives
that integral --- Eq.~\eqref{eq:B-fibretrace}. The prediction is
that surviving variables are the class-like and position-like ones,
which is testable in principle: a system whose interaction couples
to a fiber-resolving quantity would exhibit a different pointer
basis, and none has ever been found. The circularity would be real
if the framework adjusted the projection per experiment; it does
not, because there is only one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~55 (O-W2-55) (the unitarity purist)}\par\nobreak
\noindent\textbf{``A non-invertible boundary evolution violates unitarity, which is the deepest principle we have.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} What the account violates is unitarity of the
\emph{boundary reading}, which no theorem ever asserted. The bulk books stay exact --- that
is the same statement Problem~16 runs across a black-hole horizon
--- and the partial trace of Eq.~\eqref{eq:B-ptrace} is a
non-invertible map obtained from a perfectly unitary joint
evolution, a counterexample by construction. The contradiction
appears only under the one-fiber assumption that the reading must
be described by the books' own dynamics. A ledger can be exact
while every page one is permitted to photocopy is smudged.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~56 (O-W2-56) (the experimentalist)}\par\nobreak
\noindent\textbf{``Where exactly is the quantum-classical cut, operationally?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} At the first uncompensated one-way transfer ---
operationally, where the interference budget is spent and cannot be
repaid. This is a physical criterion rather than a convention, and
it has a consequence experiments already confirm: interferometers
can revive coherence for molecules of thousands of atoms (no write
yet), and no unitary trick can un-darken a single exposed grain
(write executed). The cut moves with experimental skill exactly as
far as the write can be deferred, and not one step further ---
which is why the boundary of quantum behavior has been climbing
in mass for thirty years without ever reaching a species, a size,
or a mind.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~57 (O-W2-57) (the referee on the F25 relic)}\par\nobreak
\noindent\textbf{``The relic's amplitude and exponent are uncomputed. Isn't the prediction empty until they are?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The functional form is not empty --- it is
$\xi(k) = 1 - A(k/k_0)^{-\gamma}$ with $\gamma > 0$ and a deficit
growing toward the largest scales, which is a shape, not a
wildcard. A detected anomaly of a different shape would kill the
relaxation account regardless of $A$; exact Born statistics at all
scales bound the relaxation time from below. That said, the
objection is right about what is owed: quantifying $A$ and $\gamma$
is a named cohort campaign, and until it closes the prediction is a
one-sided test rather than a two-sided one. The paper says so
wherever it quotes the relic.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~58 (O-W2-58) (the Wigner's-friend theorist)}\par\nobreak
\noindent\textbf{``Recent extended Wigner's-friend theorems constrain any account of measurement. Does yours survive them?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, and for a structural reason. Those theorems press
on accounts in which a measurement outcome is observer-relative ---
a fact for the friend but not for Wigner. Here an outcome is a
\emph{registration}, an interval-side deposit that is a fact of the
world independent of who reads it, so the theorems' target
assumption (absoluteness of observed events is denied) is not
adopted. The cost is the one already conceded: the boundary
dynamics is not unitary, so Wigner cannot in principle reverse the
friend's measurement. That is a prediction, not a gap, and it
sharpens as interference-revival experiments climb in mass.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~59 (O-W2-59) (the referee on scale)}\par\nobreak
\noindent\textbf{``Does anything in this account depend on the system being small? Or is the write available at any size?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing depends on size, and that is a distinguishing
feature. The write occurs when registration outruns coherence, and
Eq.~\eqref{eq:B-decoherence}'s rate depends on how loudly the
alternatives differ, not on a mass threshold. A superconducting
qubit of macroscopic extent can stay coherent for a second; a dust
grain cannot for $10^{-24}$ of one. The framework therefore
predicts no boundary mass and no boundary complexity --- a
prediction shared with decoherence and \emph{not} shared with
collapse models, which is precisely where the two separate
experimentally.
\end{objqabody}

%% =====================================================================
%% Cluster 5 -- Gauge theory, the RG, and the Standard Model
%% =====================================================================

\section{The gauge-theory and Standard-Model referee}
\label{sec:obj-sm}

The gauge theorist's complaint is not that the paper is wrong about
gauge invariance but that it is saying something everyone already
knows in unfamiliar words. This cluster answers that and the harder
questions behind it: whether calling gauge symmetry an unreadability
adds content, whether the anomaly argument is more than Elitzur's
theorem restated, whether the codebook's closed forms survive scheme
and scale, and how the parameter count is being defined. Objection~63
is the one with teeth: a closed form quoted at one scheme and compared
at another is not a prediction, and the response has to say exactly
which convention each entry uses.

\bigskip
\noindent\textbf{Objection~60 (O-W2-60) (the gauge theorist)}\par\nobreak
\noindent\textbf{``Calling gauge invariance `the circle's unreadability' is a slogan. What does it buy computationally?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Three things a slogan would not. It predicts the
\emph{form} of the observable content: functions of the Wilson line
Eq.~\eqref{eq:A-wilsonline} and nothing else about the fiber's
connection, which is Elitzur's theorem obtained as a corollary
rather than proved separately. It supplies a mechanism where the
standard account supplies a choice: the electroweak scale is set by
a circle-side modulus, Eq.~\eqref{eq:C-KK}, rather than by a
potential written down by hand. And it forbids: an anomaly
coefficient that ran would be a continuous flow acting on a
homotopy class, so the framework has no way to accommodate one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~61 (O-W2-61) (the lattice theorist)}\par\nobreak
\noindent\textbf{``Your confinement account is a selection rule. Lattice QCD already computes the spectrum from first principles. What is the gain?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No gain in numbers, and the paper does not claim any:
the lattice computes what this framework must reproduce, and does
so superbly. The gain is in the status of the statement. A lattice
computation \emph{demonstrates} confinement to whatever precision
one can afford; it does not prove that color cannot be isolated,
and the continuum proof remains a Millennium Prize problem. The
registration rule, Eq.~\eqref{eq:H-selection}, is a different kind
of statement --- an open color book is annihilated by the boundary
projection, so it is not expensive but unregistrable. Whether that
constitutes a proof depends on how much one grants the axioms; that
it is a different claim from the lattice's is not in doubt.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~62 (O-W2-62) (the precision-electroweak referee)}\par\nobreak
\noindent\textbf{``$\sin^2\theta_W = 25/108$ is quoted at $+0.11\%$. At what scale and in what scheme? Precision data are far more accurate than that.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both are specified in the parent paper, and the point
of the objection is well taken. The closed form is a fixed
partition attached to a matching point, not a running quantity ---
which is itself a prediction, since the framework says a
circle-side label has no beta function (\S\ref{app:D-step3}). What
runs is the \emph{effective} mixing angle, and the anatomy reads
its running as interval dressing of a fixed partition. The precision
data therefore test two things: the anchor value at the matching
point, and the consistency of the dressing between scales. The
codebook carries the per-parameter audit with the scheme stated;
this synthesis points rather than re-derives.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~63 (O-W2-63) (the grand-unification referee)}\par\nobreak
\noindent\textbf{``You reject minimal $\mathrm{SU}(5)$'s proton decay but keep $\mathrm{SU}(5)$ in the bulk group. Isn't that having it both ways?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is having a different thing. What experiment
excluded is minimal four-dimensional $\mathrm{SU}(5)$'s
\emph{prediction} of proton decay through dimension-six operators
at $M_X \sim 10^{15}$~GeV. The bulk's $\mathrm{SU}(5)$ factor
appears alongside a gauged $\UoneB$, Eq.~\eqref{eq:group-p2}, and
gauged baryon number is not an accidental symmetry that
higher-dimension operators may violate --- it is a symmetry no
operator in the theory can violate without breaking gauge
invariance. So the group is retained and the decay is not, and the
difference is a factor in the group rather than a tuning of
$M_X$.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~64 (O-W2-64) (the flavor physicist)}\par\nobreak
\noindent\textbf{``Eight of your twenty-eight parameters --- the charged-fermion Yukawas --- are still Asserted. That is the hardest part of the problem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is, and it is the part the codebook has not
finished. The overlap integrals on the family surface are a named
open construction, and Figure~\ref{fig:param-status-w2} colors
them red rather than hiding them in a footnote. What the framework
does claim in the flavor sector is the discrete structure: the
mixing phases as exact lattice values --- $\alpha_{31} = \pi$,
$\delta_{\mathrm{PMNS}} = 196.4^\circ$ --- which are checkable by
DUNE and Hyper-Kamiokande this decade and which no fit was used to
obtain. A framework that derived the mixing angles while asserting
the masses is in a stronger position than one that fitted both, and
a weaker one than it would like.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~65 (O-W2-65) (the collider phenomenologist)}\par\nobreak
\noindent\textbf{``A Kaluza--Klein cluster at $3.849$~TeV is within reach and has not been seen. How much room is left?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Less each year, which is the point of staking it. The
prediction is not merely a mass: it is one mass, one width, one
production rate, one interference sign, one angular shape, and one
branching fingerprint, all fixed by the same integers --- so the
signature cannot be dialed down by adjusting a coupling. The
HL-LHC integrated-luminosity curve crosses the staked sensitivity
for the primary channel within the decision calendar of
\S\ref{sec:calendar}. If it passes with nothing there, the
interval's depth is wrong, and with it Problems~2 and~6. The
framework has no evasive parameter to reach for, and the author
regards that as the correct position to be in.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~66 (O-W2-66) (the renormalization referee)}\par\nobreak
\noindent\textbf{``You say renormalizability is `what the descent can carry.' Modern effective field theory has largely abandoned renormalizability as a criterion. Are you behind the times?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The effective-field-theory turn is accepted here in
full: non-renormalizable operators are legitimate, suppressed by a
scale, and informative. What the geometric reading adds is that the
suppression scale is not free --- it is an address on a fiber of
finite length, so the tower of operators is organized by position
rather than by fiat. That has a consequence effective field theory
alone does not carry: the tower terminates, because the interval
does. Whether that is an advance or a restatement is decided
empirically at the lock, which is where the tower's first rung sits
in this framework.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~67 (O-W2-67) (the referee on $\alpha^{-1} = 137$)}\par\nobreak
\noindent\textbf{``An integer formula for the fine-structure constant is the oldest trap in physics. Eddington tried it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Eddington's attempt is the right cautionary tale and
the paper does not pretend otherwise. The differences worth
weighing: Eddington's $136$ (later $137$) came from a counting
argument that produced no other numbers and no mechanism for
running, so when the measured value moved and the running was
understood, the claim had nowhere to go. Here the same integers
that give $2^J + N^2$ also give the weak angle, the vacuum value,
the inventory, and the Kaxion's frequency --- and the framework
predicts the \emph{difference} between the infrared and pole values
as $N^2 = 9$ against a measured $9.11$ (Problem~D.5). That last
test is the one Eddington could not have passed, because it tests
the running rather than the endpoint.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~68 (O-W2-68) (the anomaly specialist)}\par\nobreak
\noindent\textbf{``Anomaly cancellation in the Standard Model is a four-dimensional statement about the fermion content. Why attribute it to a six-dimensional bulk?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the four-dimensional statement is a shadow with
a known lineage: anomaly inflow relates a boundary anomaly to a
bulk anomaly polynomial through the descent equations,
Eq.~\eqref{eq:C-descent}, and the coefficients are characteristic
numbers --- integers by Chern--Weil. The claim here is not that the
four-dimensional sums are wrong (they are exact and verified
generation by generation, Problem~C.3) but that their exactness has
a home: integers labeling bundles on a compact fiber, which a
continuous flow cannot move. The by-product is the testable part:
the six-dimensional closure fixes the coefficient vector and hence
the ledger, which then has to reproduce twenty-eight measured
numbers.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~69 (O-W2-69) (the QCD theorist)}\par\nobreak
\noindent\textbf{``Strong CP is `dissolved by a field the constitution already owns.' That is the Peccei--Quinn mechanism with extra steps.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is the Peccei--Quinn mechanism, and the paper says
so --- the credit belongs where the literature puts it. The extra
step is the one that matters for prediction: in the standard
account the axion's decay constant is a free scale spanning many
decades, so the mechanism explains why $\bar\theta$ relaxes but not
where to look for the relic. Here the field is not introduced for
the purpose --- it is a compact-sector eigenstate whose decay
constant is fixed by the cap loading, Eq.~\eqref{eq:F-fK} --- so
the same mechanism arrives with a frequency attached. A mechanism
with a target is a different scientific object from a mechanism
with a parameter.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~70 (O-W2-70) (the referee on naturalness)}\par\nobreak
\noindent\textbf{``Calling the hierarchy `a depth, not a fine-tuning' does not remove the quadratic divergence. Where does the loop correction go?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It does not go anywhere; the cutoff stops being free.
Eq.~\eqref{eq:radiative-w2}'s correction scales as $\Lambda^2$, and
in a four-dimensional effective theory $\Lambda$ can be taken to
the Planck scale, which is what produces the thirty-four digits.
Here the descent supplies its own gauge-sector completion at the
lock, so the loop's reach is bounded by the same $e^{-K}$ that sets
the scale, and the ratio $\delta m_H^2/m_H^2$ is order unity
(\S\ref{app:E-step3}). No cancellation is invoked at any point,
which is why the account does not need one --- and the price is
that the completion must exist at $3.849$~TeV, where it can be
looked for.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~71 (O-W2-71) (the referee on the codebook's reach)}\par\nobreak
\noindent\textbf{``$\Lambda$CDM's six parameters are quoted as derived, but the CMB fits use many more nuisance parameters. Is the comparison fair?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The comparison is between the cosmological model's
parameters and the framework's closed forms; nuisance parameters
belong to the instrument and the foregrounds, not to the model, and
no framework is expected to predict a beam profile. Where the
objection has genuine force is in the covariances: quoting
$\Omm = 1/\pi$ against a marginalized posterior requires the same
care as any other comparison, and the parent paper carries the
audit. The synthesis's claim is narrower and checkable: the six
numbers are reproduced by fiber-volume ratios and descent depths
with no fitted quantity, and each carries its own kill
condition.
\end{objqabody}


%% =====================================================================
%% Cluster 6 -- Cosmology
%% =====================================================================

\section{The cosmology and CMB referee}
\label{sec:obj-cosmo}

Cosmology is where the framework is most exposed, because cosmology
measures things. This cluster is accordingly the most quantitative:
the plateau derivation and what it forbids, $N^{*}$ and its reheating
dependence, the acoustic scale, the inventory against Planck's
covariances, the Hubble tension, and the isocurvature null. The
objection the author would put first is Objection~74 --- whether the
$\pi$-locked inventory is already in tension with the tightest of the
three shares. It is the entry where the framework will be squeezed
first, and the response says so rather than averaging it away.

\bigskip
\noindent\textbf{Objection~72 (O-W2-72) (the inflation referee)}\par\nobreak
\noindent\textbf{``Plateau potentials are already favored by data. Deriving one is less impressive than it sounds.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The derivation's value is not that it selects a
favored class but that it forbids the others, and the two are
different claims. Data prefer plateaus; the standard framework
retains hundreds of alternatives, any of which could be revived by
a measurement shift. Here the census permits one non-compact
non-spatial coordinate, hence one field of this kind
(Eq.~\eqref{eq:E-radion}), and the shape follows the warp rather
than a choice. The discriminating consequence is not $n_s$ --- on
which many models agree --- but the pair $(n_s, r)$ read from a
\emph{derived} $N^* = 52.6$, which fixes $r = 0.0043$ to one
significant figure with nothing left to adjust.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~73 (O-W2-73) (the inflation referee, pressing)}\par\nobreak
\noindent\textbf{``$N^*$ depends on reheating, which you cannot observe. Isn't the derived value a dressed-up assumption?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It depends on reheating, and the cohort computes the
reheating match from the warp's exit behavior and the $g_*$
counting rather than assuming it \citep{ArisakaA10Cosmos}. That is
a conditional derivation and is graded as such --- the
prompt-crossing lemma is the named condition. What makes it more
than a dressed assumption is that the same warp is doing three
other jobs: setting the lock scale, setting the hierarchy, and
setting the age ledger's slope. A tuned $N^*$ would break at least
one of them. The exposure is direct: if the tensor amplitude comes
in far from $0.0043$, the length is wrong and the other three
readings must fail too.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~74 (O-W2-74) (the CMB referee)}\par\nobreak
\noindent\textbf{``The $\pi$-locked inventory agrees at the per-cent level, but Planck's errors are smaller than that on some components. Is the framework already in tension?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is no tension, on the current numbers: $\Omm = 1/\pi =
0.3183$ against $0.3153 \pm 0.0073$ is within half a standard
deviation, and $\OmL = 0.6817$ against $0.6847 \pm 0.0073$ likewise.
The baryon share is the tightest comparison and sits at
$0.0507$ against $0.0493$. Where the objection points correctly is
the future: these are closed forms with no error bar of their own,
so improving data can only tighten the squeeze. That is the
intended position. A framework whose predictions cannot be improved
toward is one that can be killed by better measurement, which is
what Table~\ref{tab:kill} says out loud.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~75 (O-W2-75) (the dark-matter experimentalist)}\par\nobreak
\noindent\textbf{``Haloscopes have already excluded parts of the microwave band. How much of your window survives?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Coverage of the low edge of $1.59$--$2.13$~GHz is
beginning, and the decision calendar of \S\ref{sec:calendar} tracks
it. Two features make the framework's exposure sharper than a
generic axion search. The window is \emph{discrete} rather than
open-ended, so a full sweep is a decisive experiment rather than a
step in an endless scan. And the signature is a \emph{doublet} with
a fixed scale-free spacing $\Delta\nu/\nu = 1/35$
(Eq.~\eqref{eq:F-doublet}), which no rival framework predicts and
which cannot be absorbed by rescaling the decay constant or the
halo density.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~76 (O-W2-76) (the dark-energy referee)}\par\nobreak
\noindent\textbf{``You derive the dark-energy \emph{share} but flag its scale. Isn't the cosmological constant problem exactly the scale?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, and the paper's claim is deliberately narrower
than a solution. The scale arrives from an already-derived TeV
address by a seesaw, $m_\Lambda = \Mlock^2/\Mplred = 6.1$~meV
against $\rho_\Lambda^{1/4} = 2.24$~meV --- so the meV scale is
reproduced with no new dial, and what remains is one coefficient,
the drainage factor $\zeta_\Lambda$, named and bounded and
uncomputed. That is repricing, not solving, and
\S\ref{sec:six-problems} labels it so. The $8K$ observation is
carried flagged and used nowhere. A reader who counts this as an
unsolved problem is agreeing with the paper.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~77 (O-W2-77) (the referee on $w = -1$)}\par\nobreak
\noindent\textbf{``Quintessence models predict $w$ slightly away from $-1$, and surveys are closing on that. What does your framework say?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It says $w = -1$ exactly, and it says so
non-negotiably: the terminus is both fiber coordinates at rest, so
there is no rolling field left to give $w > -1$ and no phantom
crossing available, which would require the semigroup to run
backwards. That is an unusually rigid prediction --- most
frameworks can accommodate a small deviation --- and it is why the
Roman and DESI programs appear in the decision calendar. A
confirmed $w \neq -1$ at the level those surveys will reach ends
Problem~7's answer with no parameter to adjust.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~78 (O-W2-78) (the large-scale-structure referee)}\par\nobreak
\noindent\textbf{``The Hubble tension is a live anomaly. Does your framework predict a resolution or merely accommodate one?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It reads the tension structurally rather than
resolving it by a new component: the two determinations sit on
either side of a corridor, and the cohort's Hubble paper derives
the expansion history with no dial and treats the tension as two
dictionaries meeting at that corridor \citep{ArisakaC2Hubble}. What
the framework does \emph{not} do is introduce early dark energy or
a new relativistic species to split the difference --- it has no
freedom to. So the honest status is: a reading, with the corridor's
endpoints derived, and no fitted resolution on offer. If the
tension resolves to a value outside the corridor, that is a
problem for the framework and not a nuisance.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~79 (O-W2-79) (the BBN referee)}\par\nobreak
\noindent\textbf{``Light-element abundances are among the sharpest cosmological tests. Where does the framework sit?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It inherits standard big-bang nucleosynthesis wholesale
--- the physics is boundary physics at an address on the descent,
and nothing in the anatomy modifies reaction rates or the neutron
lifetime. What it supplies is the input: the baryon-to-photon ratio
from the toll of Eq.~\eqref{eq:G-toll}, $\approx7\e{-10}$ (central $7.34$) against the
measured $6.12\e{-10}$. Since BBN's predictions are steep functions
of $\etaB$, that $+20\%$ central (inside the chain's factor-$\sim3$ band)
propagates into the abundances and is
where the framework is most immediately checkable against existing
data --- and where the postulated $K/2$ exponent is most exposed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~80 (O-W2-80) (the referee on the entrance)}\par\nobreak
\noindent\textbf{``An `entrance' you cannot derive is a singularity by another name. What has been gained?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A shape, and a work order. A singularity is where the
description ends; the entrance is a specific configuration ---
the coordinate high on the plateau, cold and smooth enough for the
roll --- whose derivation is owed, with four listed requirements
(\S\ref{app:E-step3}). The difference is operational: nobody can
compute what happens at a singularity, whereas anyone can attack a
four-item work order, and closing it would additionally convert the
cohort's deepest flagged numbers into theorems. Whether that
constitutes progress is a fair question; the paper's position is
that converting a mystery into an open problem is the ordinary way
physics advances.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~81 (O-W2-81) (the referee on eternal inflation)}\par\nobreak
\noindent\textbf{``You retire eternal inflation `structurally.' Most inflationary potentials produce it generically. Isn't that a strong claim?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is, and it rests on the semigroup rather than on
potential-shaping. Eternal inflation requires the field's value to
be re-randomized upward in some patches, which is an operation the
interval's transport does not contain --- there is no inverse
element to carry a coordinate back up (Eq.~\eqref{eq:A-kadanoff}).
So the retirement is a consequence of the fiber's algebra, not a
tuning of $V$. The reason it matters beyond tidiness is that
eternal inflation is what generates the measure problem that
Problem~15 would otherwise inherit: removing the former removes the
latter.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~82 (O-W2-82) (the referee on isocurvature)}\par\nobreak
\noindent\textbf{``Axion dark matter generically predicts isocurvature fluctuations, and Planck's bounds are severe. How do you survive them?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By a mechanism that is stated as a conditional and is
the cleanest null in the framework: a reflection-symmetric dressing
freezes the compact angle during the roll, patch-uniformly, so the
misalignment angle is pinned rather than drawn. The prediction is
therefore zero axion-sector isocurvature and no defects --- which
is what the data show, and which is also exactly how the claim
dies. A detected isocurvature component in the axion sector ends
the pinning and with it Problem~7's account. The paper marks the
pinning conditional in \S\ref{sec:honest} rather than treating the
existing null as a success.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~83 (O-W2-83) (the referee on the horizon problems)}\par\nobreak
\noindent\textbf{``Horizons 2 and 3 --- sameness of kinds and of constants --- are not standard problems. Are you inventing difficulties to solve?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are not standard, and the paper says so; whether
they \emph{should} be is the argument. The observation is simple:
inflation explains shared conditions by shared causal past, and
that mechanism cannot reach choices made \emph{after} the stretch
--- the electroweak vacuum, the QCD angle, the axion misalignment
--- which on the textbook account of symmetry breaking are drawn
independently in causally disconnected regions. The sky should be a
patchwork of domain walls. It is not, to the precision of every
measurement. Either the standard account of breaking is incomplete
or something forbids the patchwork; the paper argues the latter and
stakes it on any observed spatial variation of a constant.
\end{objqabody}

%% =====================================================================
%% Cluster 7 -- Particle-physics foundations
%% =====================================================================

\section{The particle-physics foundations referee}
\label{sec:obj-particle}

Here the questions come from people who work on the specific
structures the anatomy claims to explain: discrete symmetries,
flavor, the matter asymmetry, confinement, proton stability. They
know the standard accounts intimately and will not accept a
restatement of them as an explanation. Objection~88 is the hardest
in the cluster --- the $\eta_B$ chain's postulated exponent --- and it
is hard because the response cannot make it go away. It can only put
the postulate where a reader can see it and say what would decide it.

\bigskip
\noindent\textbf{Objection~84 (O-W2-84) (the discrete-symmetry referee)}\par\nobreak
\noindent\textbf{``$C$, $P$, $T$ are defined by their action on fields, not by geometry. Why should a bulk have exactly three orientable structures?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the bulk has exactly three structures that
carry an orientation to reverse --- the compact fiber's winding
sense, the stage's spatial handedness, and the boundary clock's
direction (Eq.~\eqref{eq:G-threeorient}) --- and the interval
carries none, since reversing $r$ is not a symmetry operation but a
request the semigroup cannot fulfill. The field-theoretic
definitions are then the boundary shadows of those three. What
makes the reading more than relabeling is that it explains the
observed \emph{pattern}: a single reversal fails wherever a
structure fixes its sense, while the product cannot fail because
nothing remains outside the geometry to notice a total flip.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~85 (O-W2-85) (the $CPT$ referee)}\par\nobreak
\noindent\textbf{``The $CPT$ theorem already follows from locality, Lorentz invariance, and positive energy. Your account adds nothing testable.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It adds no new test of $CPT$ itself --- both accounts
predict exactness, and antihydrogen spectroscopy tests both
identically. What it adds is an explanation of the theorem's
hypotheses: on the anatomy, locality, Lorentz invariance, and
positivity are three ways of saying the geometry is the one
Eq.~\eqref{eq:bulk} describes. And it makes a structural
prediction the standard proof does not: violating $CPT$ would
require a \emph{fourth} orientable structure, so the framework has
no parameter regime in which a small violation appears. Any
violation at any precision is fatal here, which is a stronger
commitment than the theorem alone imposes.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~86 (O-W2-86) (the neutrino physicist)}\par\nobreak
\noindent\textbf{``$\delta_{\mathrm{PMNS}} = 196.4^\circ$ is a sharp prediction, but the current data have wide errors. What is the real test?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} DUNE and Hyper-Kamiokande, in the decision calendar's
late-2020s-to-2030s band. The prediction is zero-parameter ---
$\pi + 2\arcsin(1/7)$, a lattice value --- and it sits in the
experimentally favored quadrant, which is encouraging but not yet
a test. The framework's exposure here is total: there is no way to
shift the value, because it is arithmetic in the ledger integers.
A measured phase incompatible with $196.4^\circ$ at the
sensitivities those experiments will reach ends the flavor-lattice
reading, and with it a named joint in Problems~9 and~10--11.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~87 (O-W2-87) (the baryogenesis referee)}\par\nobreak
\noindent\textbf{``Your $\etaB$ misses by $4\%$ and contains a postulated exponent. Leptogenesis fits exactly. Why prefer yours?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Leptogenesis fits exactly because it has free
parameters to fit with --- heavy masses and phases at energies no
experiment reaches --- so the observed value has never been
predicted by it, only accommodated. This chain has one postulated
step and no fitted quantity, and lands $4\%$ away. Which is
preferable depends on what one wants from an explanation: a
framework that reproduces a number with adjustable inputs is doing
different work from one that computes it with none and misses
slightly. The paper's own grading is D$^{*}$/P rather than D, and
the $K/2$ exponent is flagged as the chain's one postulate every
time the number is quoted.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~88 (O-W2-88) (the proton-decay experimentalist)}\par\nobreak
\noindent\textbf{``Gauging baryon number to protect the proton is a known model-building move. What makes it non-arbitrary here?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That $\UoneB$ is not added for the purpose. It is a
factor of the bulk group Eq.~\eqref{eq:group-p2} fixed by the
anomaly audit, and it does several other jobs --- the abelian
ledger, the boundary residue, the collider target --- before
anyone asks about the proton. A protection mechanism introduced to
protect is arbitrary; one that falls out of a group fixed by
consistency and then happens to protect is a different object. The
exposure is unchanged either way: a single confirmed decay in a
guarded channel ends the account, and unlike the $M_X$-suppression
story there is no scale to raise in response.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~89 (O-W2-89) (the hadron theorist)}\par\nobreak
\noindent\textbf{``The proton's mass is ninety-nine per cent binding energy, which QCD explains without any of this. Where does the framework enter?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It does not enter the computation --- the lattice's
mass budget stands, and Figure~\ref{fig:mass-budget-w2} reproduces
it rather than replacing it. Where the framework enters is the
question the budget raises and does not answer: why is the
confining scale where it is, and why is the resulting object
absolutely stable? The first is dimensional transmutation read as
an address on the interval (Eq.~\eqref{eq:H-lambdaqcd}); the second
is the double guard. A reader interested only in the mass will find
nothing new here; a reader who wonders why atoms are permanent will
find the two integer conditions of Eq.~\eqref{eq:H-guards}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~90 (O-W2-90) (the referee on three generations)}\par\nobreak
\noindent\textbf{``$N = 3$ from a chirality relation looks like numerology dressed as topology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be if it stood alone. It does not: the same
integer is reached by the capacity argument of
\citet{ArisakaP2N3}, which bounds the generation count from below
and above inside the same axiom system \emph{without} using the
chirality relation --- two derivations sharing no premise. The
paper's general defense against numerology is exactly this pattern
(\S\ref{app:H-step3}), and $N$ is its clearest instance. A single
route to a small integer is a coincidence waiting to happen; two
independent routes to the same integer is a constraint, and a third
would be better still.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~91 (O-W2-91) (the referee on the flavor alphabet)}\par\nobreak
\noindent\textbf{``A `chiral lattice of phases with spacing $\pi/5$' is a strong structural claim. What forces the spacing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The block integer $L = 5$, through the requirement that
the lattice close on the compact fiber: $2L = 10$ letters at
spacing $2\pi/2L = \pi/5$ (Eq.~\eqref{eq:G-lattice}). What makes
the claim testable rather than decorative is that the lattice's
sites are then \emph{predictions} for the physical phases, and two
of them are measurable this decade --- $\delta_{\mathrm{PMNS}}$ by
long-baseline neutrino experiments and the Majorana phase's
consequences by neutrinoless double-beta decay. The alphabet either
sits where the lattice says or it does not.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~92 (O-W2-92) (the referee on sphalerons)}\par\nobreak
\noindent\textbf{``Your welded-door language obscures standard electroweak physics. Is anything different from the textbook account?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing in the rates or the thermodynamics --- the
sphaleron energy of Eq.~\eqref{eq:G-sphaleron} and the switch-off
are standard, and are used as such. Two things are different in
status. The crossing's \emph{location} is an address on the
descent rather than a free epoch, which is what allows the toll to
be computed from a depth. And the door's closing is not merely a
Boltzmann suppression but the same electroweak crossover that
Problem~12's guards then enforce permanently. The picture is a
reading of standard physics, and where it adds numbers it says so
and grades them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~93 (O-W2-93) (the referee on gravity)}\par\nobreak
\noindent\textbf{``Gravity is left as `the interval's bookkeeping.' That is not a quantum theory of gravity.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is not, and the paper does not claim one --- the
ultraviolet completion of gravity is explicitly among the problems
\emph{not} addressed (Objection~33). What the framework offers is a
diagnosis of why the standard program has been hard: the other
three interactions are circle-side gearings and quantize as phases
do, while gravity is the interval's own bookkeeping, and its
quantum is priced in writes rather than windings
(Eq.~\eqref{eq:mpl-w2}). That is a category claim, not a
completion, and its value is testable only indirectly --- through
the lock, where the descent's structure becomes visible.
\end{objqabody}

%% =====================================================================
%% Cluster 8 -- Time, entropy, and black holes
%% =====================================================================

\section{The gravitation, thermodynamics, and black-hole referee}
\label{sec:obj-time}

The statistical mechanic and the relativist arrive together in this
cluster, because time, entropy and horizons are one subject once you
ask where the arrow comes from. The objections press on whether a
semigroup really evades Loschmidt and Zermelo, whether the tick
derivation is more than dimensional analysis, and whether the
black-hole account adds anything to the island computations it says
it agrees with. Objection~100 is the strongest: if the framework
agrees with the Page curve that gravity already delivers, what is the
reading \emph{for}? The answer is a claim about \emph{why} the curve
turns over, and the reader is entitled to weigh whether that is
progress.

\bigskip
\noindent\textbf{Objection~94 (O-W2-94) (the statistical mechanic)}\par\nobreak
\noindent\textbf{``The second law is statistical. Making it topological seems to over-claim: fluctuations do run backwards, and we measure them.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fluctuation theorems are correct and are not disputed:
a small system in contact with a bath does exhibit trajectories of
negative entropy production, with the Crooks and Jarzynski
relations quantifying exactly how often. The framework's claim is
about \emph{registrations}, not about coarse-grained thermodynamic
variables. A fluctuation that reverses an unregistered process is
allowed and observed; an already-registered outcome being
un-registered is not, and has never been seen. The sharp version
--- and the kill condition --- is a free erasure: any
demonstration of erasure below the Landauer bound, or of a
registered outcome reverting, ends this account.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~95 (O-W2-95) (the cosmologist on the arrow)}\par\nobreak
\noindent\textbf{``You claim no past hypothesis is needed, but the early universe was still in a very special state. Hasn't the problem moved rather than gone?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Partly, and the paper should be read as separating two
questions the literature usually merges. The \emph{arrow} ---
why registrations accumulate one way --- is answered by the
algebra and needs no initial condition. The \emph{specialness} ---
why the early universe's gravitational degrees of freedom were
unexcited --- is a question about the launch, and belongs to
Problem~5's entrance, which is open. So the framework closes one
half structurally and hands the other half to a named work order.
That is less than a complete answer and more than the standard
position, which imports both.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~96 (O-W2-96) (the black-hole referee)}\par\nobreak
\noindent\textbf{``The island computations already recover the Page curve from gravity. What does a fiber ledger add?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It adds the answer to a different question. The island
computations show \emph{that} the books balance, and do so with
impressive technical control; they compute the curve without
identifying where the information resided while the hole existed.
The two-fiber reading names the register --- circle-side class data
in the bulk, inaccessible to the exterior observer, never absent
--- and prices the return through the interval's channel. The paper
treats the island program as an achievement to be explained
rather than a rival to be defeated, and says so in Problem~16.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~97 (O-W2-97) (the firewall theorist)}\par\nobreak
\noindent\textbf{``AMPS showed that unitarity, locality, and a smooth horizon cannot all hold. Which do you give up?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} None of the three, because the trilemma's hidden
premise is the one the anatomy denies: that the horizon's smooth
interval-side physics must also carry the circle's books locally.
Sorted by fiber, nothing at the horizon is asked to do the other
fiber's job --- the register is not localized on the horizon, so
the monogamy argument does not bite there --- and the smoothness
survives with the unitarity. Whether that constitutes a resolution
or a relabeling is a fair dispute; what is not in dispute is that
the framework predicts \emph{no} firewall, which is a definite
position rather than an agnostic one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~98 (O-W2-98) (the referee on black-hole entropy)}\par\nobreak
\noindent\textbf{``Why should $S_{\mathrm{BH}}$ be `three ledgers meeting'? The formula is just dimensional analysis with $\hbar$, $G$, $c$, $k_B$.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Dimensional analysis says which combination \emph{can}
appear; it does not say why a black hole is the one object whose
entropy requires all of them at once. On the anatomy that is the
content: $\hbar$ prices the circle's tick, $G_N$ the interval's
integrated descent, $k_B$ the boundary's exchange window
(Eq.~\eqref{eq:I-threeledgers}), and a horizon is the unique place
where the register, the channel, and the window are forced into
contact. The reading earns its keep by explaining why black holes
have been the discipline's sharpest instrument for thinking about
quantum mechanics, gravity, and thermodynamics simultaneously ---
they are where all three ledgers post to the same page.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~99 (O-W2-99) (the referee on time in quantum gravity)}\par\nobreak
\noindent\textbf{``The Wheeler--DeWitt equation has no time. Your framework asserts a rate. Are they compatible?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are, on this reading, answers to different
questions, and the reconciliation is the four-endowment split.
Wheeler--DeWitt's timelessness is a statement about the
\emph{bulk's} constraint structure, where nothing evolves with
respect to an external parameter --- and the anatomy agrees: the
bulk has no clock. What the boundary has is a \emph{reading}, whose
tick is the circle's, whose arrow is the interval's, and whose rate
is the seam's exact $d\ln t/dr = 2$. Time is emergent in the
precise sense that it is a property of the projection rather than
of the bulk, which is what the canonical formalism has been saying
without a mechanism to attach it to.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~100 (O-W2-100) (the referee on Landauer)}\par\nobreak
\noindent\textbf{``Landauer's principle is about computation. Using it as a cosmic accounting law is a stretch.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is used here exactly as it is used in the
information-thermodynamics literature: as the statement that
erasure has a thermodynamic price, verified experimentally in
single-particle and colloidal systems. What the framework adds is
the identification of the ledger it prices --- the interval's
one-way writes --- and a constant, $Q_B$, fixing the exchange rate
between energy-per-winding and energy-per-qubit
\citep{ArisakaA7Qi4}. Whether that identification is right is
testable in the same place as everything else: the framework
forbids a free erasure at any scale, including cosmological ones.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~101 (O-W2-101) (the referee on the de Sitter entropy)}\par\nobreak
\noindent\textbf{``You note $\ln S_{\mathrm{dS}} \approx 8K$ and then refuse to use it. Why mention it at all?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because concealing an arithmetic observation would be
worse than reporting it, and because a reader is likely to notice
it independently. The observation is that the largest number in
physics has a logarithm within $0.6\%$ of twice the descent's
doubled depth. It is carried at flagged grade, discussed in
Problems~7 and~16, and used in exactly zero derivations. If the
e-fold budget campaign closes, it becomes a theorem; if it does
not, it remains a curiosity, and the paper is equally strong with
the paragraph deleted. The register exists precisely so a reader
can audit which claims lean on what.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~102 (O-W2-102) (the referee on the age ledger)}\par\nobreak
\noindent\textbf{``The age ledger closing at $140.24$ is impressive, but you flag its relation to $4K$. Which part is the result?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The ledger itself: the chronology sums segment by
segment from the derived slope $d\ln t/dr = 2$ and closes against
the measured age to four digits, with each segment's physics
independently checkable. That is derived. The observation that the
total happens to be $4K \times 1.0017$ is flagged --- reproducible
arithmetic with no mechanism yet --- and the queued e-fold budget
campaign is the attempt to derive it. So the answer is: the
chronology is the result, the $4K$ coincidence is an observation,
and the paper keeps them in different registers on purpose.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~103 (O-W2-103) (the referee on the second law's status)}\par\nobreak
\noindent\textbf{``If irreversibility is topological, why does thermodynamics work so well as a statistical theory?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the statistical account and the topological one
agree on everything measurable in ordinary conditions, and the
statistical machinery is the correct effective description of a
system with many unregistered degrees of freedom. The two separate
only at the edges: in the deep past (where the statistical account
needs a past hypothesis and this one does not), in the treatment of
recurrence (a large-circle world recurs; an interval does not), and
in the status of an already-registered outcome (statistically
improbable to reverse; here undefined). Those are the three places
to look for a difference, and the third is the sharpest.
\end{objqabody}

%% =====================================================================
%% Cluster 9 -- Comparisons with other programmes
%% =====================================================================

\section{Comparisons with other programs}
\label{sec:obj-compare}

Every framework of this scope is measured against its neighbors, and
a reader deciding what to believe is entitled to see the comparison
made rather than avoided. This cluster sets the two-fiber reading
beside string theory, loop quantum gravity, asymptotic safety,
emergent gravity, causal sets, warped extra dimensions, the anthropic
tradition, and the rival interpretations of quantum mechanics. The
house rule holds throughout: structural contrasts, never polemic ---
each entry says what the other program buys and what it pays, and
several concede that the other program buys more. Objection~104 is
the one that matters, because string theory and this framework want
the same thing and differ on whether the vacuum is chosen or forced.

\bigskip
\noindent\textbf{Objection~104 (O-W2-104) (the string theorist)}\par\nobreak
\noindent\textbf{``String theory derives gauge symmetry and gravity from one structure too, with far more mathematical control. Why should anyone read this instead?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not instead --- the mathematical control is real, and
this framework borrows several of its results, the \GS{} mechanism
among them. Before the difference is stated, three concessions are
owed and are made at length in \S\ref{ssec:w2-steelman}: the
``$10^{500}$ versus zero'' contrast is a category error, since one
number counts discrete solutions and the other counts continuous
dials, and on the second axis string theory makes the same claim this
framework does; the many vacua were found by people who wanted one;
and AdS/CFT and the extremal microstate count stand whatever one
concludes about the landscape. The difference that remains is the one
stated in Objection~3: the landscape's freedom makes sharp low-energy
predictions hard to extract, whereas this framework's census leaves no
freedom and therefore stakes dated numbers. A string theorist should
read this paper the way one reads any proposal with an unusually short
falsification calendar: not because it is better founded, but because
it will be decided sooner.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~105 (O-W2-105) (the loop-quantum-gravity referee)}\par\nobreak
\noindent\textbf{``Loop quantum gravity also builds physics from holonomies. Is there a relationship?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is a shared instinct and a different target. Loop
quantum gravity takes holonomies as the fundamental variables of
the gravitational field and quantizes the geometry itself; this
framework takes holonomy as the criterion sorting two \emph{fibers}
of a fixed six-dimensional geometry, and leaves gravity's
ultraviolet completion open. The two are not competitors in their
present forms, and if loop quantum gravity or a successor supplies
a completion, the anatomy's requirement on it is the one stated in
Objection~42: it must reproduce a sector that cannot hold holonomy
and one that can.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~106 (O-W2-106) (the asymptotic-safety referee)}\par\nobreak
\noindent\textbf{``Asymptotic safety also makes the renormalization-group flow fundamental. How do the pictures differ?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Asymptotic safety proposes that the flow has a
non-trivial ultraviolet fixed point, so the theory is predictive at
all energies with finitely many relevant directions. This framework
proposes instead that the flow's coordinate is a \emph{finite
interval with endpoints}, so predictivity comes from the length
rather than from a fixed point. The two are testable against each
other in principle: asymptotic safety expects the flow to continue
sensibly to arbitrarily high energy, while here the flow stops at
the GUT surface at $r = 0$. What both share, and what the paper
regards as the important agreement, is that the scale direction is
physics rather than bookkeeping.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~107 (O-W2-107) (the emergent-gravity referee)}\par\nobreak
\noindent\textbf{``Entropic and emergent-gravity programs also tie gravity to information. Is your $Q_B$ the same idea?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Related in spirit, different in structure. Entropic
approaches derive the gravitational force from an entropy gradient
on a screen; here gravity is the interval's warp and $Q_B$ prices
the boundary's information deposits, with the two connected because
$G_N$ is the interval's own constant. The frameworks would make
contact if the entropic screen were identified with the projection
of Eq.~\eqref{eq:B-fibretrace}, which is a legitimate thing for
someone to attempt. The paper does not claim it, because the
cohort's derivation of $G_N$ runs through the warp rather than
through a screen.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~108 (O-W2-108) (the referee on causal sets and discrete spacetime)}\par\nobreak
\noindent\textbf{``Discrete-spacetime programs also explain the arrow by construction. Why prefer a semigroup?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Causal-set programs get the arrow from the partial
order's growth, which is a genuine mechanism and a close cousin of
this one. The distinguishing feature here is that the same
structure supplying the arrow also supplies the scale hierarchy,
the running couplings, and the cosmic chronology --- the interval
is not introduced to make time work but is the renormalization
coordinate, doing four jobs at once. Whether that economy is a
virtue or an over-extension is exactly the kind of question a
falsifier table is for.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~109 (O-W2-109) (the referee on Kaluza--Klein revivals)}\par\nobreak
\noindent\textbf{``Warped extra dimensions already solve the hierarchy problem. Is $K = 35$ anything more than a Randall--Sundrum warp factor?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The mechanism is of that family and the paper says so:
an exponential warp along a finite extra dimension converts a large
ratio into a modest length, and Eq.~\eqref{eq:E-warp} is that
statement. Two differences matter. The length is not tuned to
produce the observed hierarchy --- it is a ledger integer fixed
twice over, and the hierarchy is the output. And the same integer
appears in the matter excess ($e^{-K/2}$), the Kaxion doublet
($1/K$), and the vacuum ($e^{-2K}$), so the framework cannot adjust
the warp to fit the electroweak scale without breaking three other
predictions.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~110 (O-W2-110) (the referee on the anthropic tradition)}\par\nobreak
\noindent\textbf{``Weinberg's anthropic bound was a real success. Doesn't your dismissal of selection ignore that?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The paper does not dismiss it and \S\ref{sec:six-problems}
was revised specifically to state it accurately: Weinberg derived
an anthropic \emph{upper bound} some two orders of magnitude above
the value later measured, which remains the tradition's most
impressive single result. What the framework denies is not the
bound's validity but its necessity: if the vacuum's share is a
fiber-volume ratio and its scale a derived seesaw, then the bound
is a consistency check on a computed value rather than an
explanation of a drawn one. Both readings are compatible with the
bound; they differ on whether anything was drawn.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~111 (O-W2-111) (the referee on rival interpretations of quantum mechanics)}\par\nobreak
\noindent\textbf{``Seven interpretations coexist because no experiment separates them. Your framework claims to. Is that credible?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a claim with a specific mechanism behind it, and
the credibility should be judged on that mechanism rather than on
the ambition. Interpretations are empirically equivalent when they
differ only in what they say \emph{about} the same dynamics. This
framework differs in the dynamics: the Born measure is relaxed into
rather than postulated, so it has a relaxation time, so modes that
decoupled early retain a computable deficit
(Eq.~\eqref{eq:f25-w2}). That is a physical difference, not an
interpretive one --- and if the large-scale statistics show exact
Born behavior to arbitrary precision, the framework's account is
bounded and eventually excluded.
\end{objqabody}

%% =====================================================================
%% Cluster 10 -- Falsifiability, scope, and standing
%% =====================================================================

\section{Falsifiability, scope, and standing}
\label{sec:obj-standing}

The last cluster is about what all of this is worth, and it is the
one the author would most like read carefully. It asks about the
decision calendar, the size of the cohort, the author's own position
as an experimentalist writing structural theory, and what survival
through the decade would actually establish. The final entry,
Objection~117, is the whole paper compressed into one question and
one answer --- and the answer, deliberately, is not that the reader
should believe it.

\bigskip
\noindent\textbf{Objection~112 (O-W2-112) (the referee on the calendar)}\par\nobreak
\noindent\textbf{``Your decision calendar runs to the mid-2030s. What happens if the instruments slip?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The claims do not soften, and the paper is written so
that they cannot. Slippage moves the date, not the prediction: a
tensor ratio of $0.0043$ is either right or wrong whenever
LiteBIRD and the Simons Observatory report, and a haloscope window either contains
a doublet or does not. The reason the calendar appears at all is
that a framework staking dated tests should be checkable against
its own schedule --- including by a reader who wants to know
whether its author quietly extended a deadline. The dates in
\S\ref{sec:calendar} are the cohort's estimates of instrument
milestones, not of the framework's convenience.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~113 (O-W2-113) (the referee on partial failure)}\par\nobreak
\noindent\textbf{``Suppose one prediction fails and the others survive. Does the framework die, or do you patch it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It depends on which, and the paper owes an answer in
advance rather than afterwards. Failures that are structural ---
any $CPT$ violation, a spatially varying constant, a proton decay
in a guarded channel, a free erasure --- end the anatomy, because
each contradicts a topological statement with no parameter
attached. Failures that are quantitative --- $\etaB$ off beyond the
chain's stated tolerance, a $\Lambda$CDM share outside the audit
--- indict the named joint in that chain first, and the honest
response would be to report which joint and at what cost, not to
rescue the total. Failures at the flagged level cost nothing,
because nothing leans on them.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~114 (O-W2-114) (the referee on the cohort's size)}\par\nobreak
\noindent\textbf{``Twenty-two papers by one author is an unusual body of work to assess. How is a reader meant to enter it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Through this paper, which is why it exists: the
sixteen-problem ledger and the four-step review of Part~II are
designed so that a newcomer can reach the whole program through
one sentence rather than twenty-two introductions. For a specialist
the recommended entry is the paper matching their own field, taken
alone --- each cohort paper is self-contained and carries its own
falsifiers --- with A1 for the axioms and A2 for the forcing
theorems if the geometry is what is in dispute. Nobody should be
asked to accept twenty-two papers to evaluate one prediction, and
the framework's design does not require it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~115 (O-W2-115) (the referee on the author's position)}\par\nobreak
\noindent\textbf{``You are an experimentalist writing a structural theory paper. Is that a strength or a weakness?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, and the paper tries to convert the weakness into
a discipline. The weakness is obvious: the formal machinery is not
the author's native ground, and the cohort's theorems are stated in
their parent papers where specialists can attack them. The strength
is the instinct that produced the falsifier table --- fifty years
of building detectors is fifty years of being told by instruments
that an idea was wrong, which is a useful education for anyone
proposing a framework of this scope. The paper's structure ---
grades, named joints, dated kills --- is that education applied.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~116 (O-W2-116) (the referee on what would count as success)}\par\nobreak
\noindent\textbf{``Suppose everything survives the decade. What would that establish?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Less than a partisan would claim and more than nothing.
Survival of the seven staked tests would establish that a
five-integer ledger reproduces the Standard Model's parameters, the
cosmological inventory, and a dark-matter line, with no fitted
quantity --- which is a fact about the world regardless of whether
the two-fiber reading is the right explanation of it. The reading
itself would remain a proposal, better supported and still not
proved, with the entrance and the field-grade step still open. The
author's own view is that the honest ambition is to be
\emph{decided}, not to be believed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection~117 (O-W2-117) (the referee, finally)}\par\nobreak
\noindent\textbf{``What is the single strongest reason to take this seriously, in one sentence?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That one exact and elementary statement --- holonomy
requires closed loops, and an interval has none --- was carried
into twenty-two independently derived papers, entered each through
its own central topic, required no value, grade, or theorem to
change anywhere, and emerged with fifteen dated kill conditions. Whether
that is a fact about the world or an unusually good piece of
bookkeeping is not for the author to adjudicate, and the
instruments now running will settle it either way.
\end{objqabody}

%% ===================================================================
\section{What the hundred and seventeen leave standing}
\label{sec:obj-closing}

A tally is owed, because the point of collecting objections is not
to appear to have survived them. Of the hundred and seventeen,
several are conceded outright: this style of reasoning has misled
physics before and the cases are named (Objection~47); the entrance to the corridor is a
work order rather than a derivation; the field-grade representation
step is open; the cap integer $\Xicap$ is a benchmark and not yet a
theorem; the $K/2$ exponent is postulated; the $4K$ family is
arithmetic in search of a mechanism. A larger group is answered not
by argument but by an instrument with a date --- the tensor ratio,
the haloscope line and its doublet, the neutrino phase, the lock
resonance, proton decay, the equation of state. The remainder are
answered structurally, and those are the ones where a reader is
entitled to remain unconvinced: a structural answer is a claim
about why a thing has the shape it has, and shapes are argued
about.

What none of the hundred and seventeen does is ask the paper to be
believed. Every result leaned on above is somebody's, and the
references that follow say whose it is, what it cost the field to
establish, and --- where it matters --- exactly where this paper's
reading departs from the work it is built on.

%%  PART XI -- REFERENCES
%% ===================================================================
\GGPart[References]{References}{Cited Works, Each with a Relevance
Note Linking It to the Present Paper}
\label{part:XI}

\noindent Unlike a conventional reference list, every entry below
carries a full \emph{Relevance} paragraph rather than a line of
bibliographic courtesy. Each one says four things: what the cited
work established, what it cost the field to establish it, how this
paper reads that result on the two fibers, and --- ending every note
--- exactly where in this paper the work is first cited, as a live
cross-reference a reader can follow. 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
problem, equation, appendix, or figure each one underwrites, and
jump straight to it.

\noindent The notes are written for two readers at once. For the
newcomer, they trace the intellectual lineage of the sixteen
problems: from Noether, Weyl, and Dirac, through the measurement
debate, the gauge and renormalization revolutions, the discovery of
the dark sector, and the black-hole information paradox, to the
twenty-two-paper \GGTheory{} cohort whose results this paper
synthesizes. Read end to end, Section~B is a short history of the
sixteen problems told through the papers that created them --- and,
in several cases, through the measurements that later closed a
question the field had filed as interpretation. For the specialist,
the notes make the paper's debts explicit and checkable: each one
names the specific claim it supports, so that no citation is merely
decorative, and several say plainly where this paper's reading
departs from the cited work rather than building on it.

\noindent The references are organized in two groups: first the
\GGTheory{} papers by the author (Section~A), then all other works
(Section~B), each in the order in which the paper's argument reaches
them. The cohort papers are cited under the canonical keys fixed by
the program's cross-paper coherence map. Section~B opens with the
thematic works the anatomy leans on most directly, continues with
the recommended textbooks of \S\ref{app:reading} --- each annotated
with the appendix it serves and the reason it is recommended over
its rivals --- and then follows the sixteen problems in order, from
the measurement debate to the anthropic tradition. There are no
orphan entries: every work listed is cited in the text, and every
citation in the text resolves here.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.\ Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begingroup
\renewcommand{\section}[2]{}%
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K. (2026).} 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://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.} The charter of the program: four
axioms and seven postulates from which everything here descends.
\AX{2} (\GGEP{}) writes the two fibers --- geometry and gauge
structure as one composite object --- and \AX{3} (\OGGEP{}) reads
them, making observation a projection that marginalizes the fiber
coordinates. This paper leans on it constitutively: the
minimality argument of \S\ref{sec:minimum} is its two-fiber figure
taken at its word; the census arguments of Problems~2, 7, and~15
rest on its forcing lemmas; \MSMF{}, its parsimony clause, is what
converts ``no dials'' from an aspiration into a law (Problems~4
and~6); and the grade vocabulary of \S\ref{sec:grades} is its
declared-conditioning discipline. First cited in the Executive
Summary.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K. (2026).} \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://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.} The bulk paper: proves that the
axioms force the arena \eqref{eq:bulk} --- the stage, the interval,
and the circle --- and derives the integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ with its closure identities $K = LJ$ and $J = 2N+1$, and the dual-lock
condition that selects $N$ inside the anomaly audit (the audited chain
Problem~6 spends).
Its geometric-intuition Part carries the cohort's original
pair-read-together synthesis, of which this entire paper is the
generalization; its forcing theorems are where the burden of
Problem~15's objection~4 (``you chose the interval'') is lawfully
carried. First cited in \S\ref{sec:statement}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K. (2026).} 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://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.} The anomaly paper: closes local
and global anomaly freedom for the six-dimensional bulk by the
\GS{} mechanism, without supersymmetry, and in doing so fixes the
ledger's coefficient arithmetic --- the locked vector
$(1,-1,-\tfrac25,-\tfrac25)$, the $5{:}2$ trace ratio behind the Kaxion's portals (Problem~7), and
the dual-lock identity $L(2N+1) = M^2 + N^2 + L^2$ that selects $N$,
and through it $J$ and $K$, from inside the audit (Problem~6). Problem~3's central exhibit ---
anomaly coefficients as circle-side bookkeeping that cannot run
--- is its result, restated on the anatomy; the audit's byproduct
is the alphabet of integers every closed form in Problem~4
spends. First cited in Part~II.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K. (2026).} 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://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.} The reconstruction paper: the
five dimensionful constants (GU-5) derived as the conversion
factors of the geometry's directions --- one per kind of dimension
--- and sorted by fiber: $\hbar$ the circle's constant (the price
of a tick), $G_N$ the interval's (the integrated warp volume,
Problem~2's gravity section), $c$ the stage's, $k_B$ and $Q_B$ the
boundary's. Problem~14's closing section (why natural units are
natural) is its closure theorem in plain English, and the
character row of Table~\ref{tab:characters} is its pentad,
aligned. First cited in \S\ref{sec:characters}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{Arisaka, K. (2026).} \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://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.} The boundary-law paper: derives
the open-system runtime (\GDOS{}) whose generator is the
interval's semigroup --- the projection that is always on, the
memory-kernel reduction, complete positivity, and the textbook
quantum postulates as theorems. This paper draws on it in every
Part: the write of Problem~1 is its projection; the arrow of
Problem~13 is its one-wayness; the two thermodynamic ledgers of
Problem~14 (first law the circle's, second the interval's) are
its currency theorems; and the KMS shore where temperature lives
is its equilibrium analysis. First cited in
Table~\ref{tab:characters}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K. (2026).} 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://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.} The runtime-engine paper: builds
the four-pillar system and the \GRIP{} cascade --- \DGI{}
$\to$ \GFF{} $\to$ \GSSB{} $\to$ attractor --- whose two regimes
are the two fibers operating (\GRIPin{} the circle's stroke,
\GRIPout{} the interval's, \S\ref{sec:minimum}) --- completed, in
its stage-custody subsection, by the finer transport-vs-data
mapping: \DGI{}'s equivalences compose as the circle's group, the
\GSSB{}$\to$\GGA{} descent as the interval's semigroup, with the
order parameters read from the circle and the outcome deposited
back to it. Its Morse
analysis supplies the unique-floor mechanism of the determinism
triad (\S\ref{sec:determinism}), on which Problems~8 and~15
lean; its identification of the weak interaction as the cascade
realized in nature is Problem~2's agent section; and its
attractor taxonomy (the \GGA{} certificate) is the language of the
trio's terminus. First cited in \S\ref{sec:minimum}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K. (2026).} The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}. Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.} The information pillar: memory on
the circle (a stable bit as holonomy-class data), erasure on the
interval (Landauer's bound as the one-way write's toll), and the
Arisaka constant $Q_B$ as the exchange rate between
energy-per-winding and energy-per-qubit, fixed by ledger integers
--- the fourth currency of Problem~14. Problem~1's no-free-write
kill condition is its audit; the record's cost accounting behind
\S\ref{sec:measurement} is its theorem set. First cited in
Problem~13.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K. (2026).} Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary.
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.} The quantum-mechanics companion:
locates the measurement arrow on the anatomy
(\S\ref{sec:measurement}) --- the write as the runtime's
record-forming face, decoherence as its microscopic face --- and
carries the Born chain at the relaxation level: the boundary
law's own dissipation producing the Born measure, with the
field-grade representation step the one named analytic residue.
Problem~1 is, in essence, this paper's Part told in plain
English, and the F25 relic that exposes it observationally is
its prediction, carried by the cosmology companion. Inherited
verbatim in the honest ledger. First cited in Problem~9.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K. (2026).} 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://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.} The codebook: all twenty-eight
Standard-Model parameters derived or graded from the ledger, with
the per-parameter provenance audit and honesty triage that
Problem~4 summarizes and inherits. Its grammar --- every
parameter a holonomy-class label wearing a descent address ---
is this paper's most-quoted sentence; its closed forms
($\alpha^{-1}(0) = 2^J + N^2$, $\sin^2\theta_W = L^2/(4N^3)$,
$v = \Mlock(2/L)^N$, the quark floors' winding-times-overlap
form) anchor Problems~2, 4, and~6; and its treatment of why the
weak interaction is special is Problem~2's agent section at full
technical depth. First cited in \S\ref{sec:characters}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K. (2026).} The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry.
Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.} The cosmology paper: the derived
launch (the plateau class forced, $N^* = 52.6$, $n_s$, $r$), the
terminus (the seesaw, $w = -1$ exactly, the de Sitter future),
the chronology whose straight line is
Figure~\ref{fig:chronology}, the $\pi$-locked inventory with its
honest Hopf resolution ($\Omm = 1/\pi$ as a fiber-volume ratio),
the pinning mechanism with its zero-isocurvature signature, and
the entrance as a named open item. Problems~5, 6, 7, and~8 are
built on it; its ``rolling, ringing, rest'' synthesis is
Problem~7's parent; and the e-fold budget campaign it queues is
the would-be closure of this paper's entire flagged family.
First cited in Problem~13.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K. (2026).} The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program. Arisaka-GG-A11-Origin.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The capstone: traces the descent
from the GUT edge to the biosphere through its cascade of
attractors, and proves the \GGA{} theorems (ground-state
uniqueness, no-exit) on which this paper's attractor language
rests --- the electron's identity, the terminus's finality, and
the Kaxion's floor all cite it. The identity recursion of
Problem~8 (sameness compounding from classes to chemistry to
legibility) is its arc, compressed; its four words --- not
contingent, not lucky, not designed, forced --- close this
paper's Conclusions. First cited in Part~II.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K. (2026).} The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}. Arisaka-GG-C1-Kaxion.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.} The dark-matter paper: the
Kaxion's existence theorem (two compact parents, one light
eigenstate), the \GRIP{} birth on the compact-angle stratum, and
the full two-fiber portrait --- matter of the circle, occupancy
of the interval --- of which Problem~7's middle panel is the
condensation. Every Kaxion number here is its: $8.1\,\mu$eV,
$1.95$~GHz, the scale-free $1/35$ doublet, the discrete cap
window, the strong-CP alignment (Problem~12's dissolution), and
the identify-by-the-circle search strategy. Its one open integer
($\Xicap$) is carried in the honest ledger. First cited in
Part~II.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.} The Hubble paper: derives the
expansion history from the inventory with no dial --- cooling is
descent, $d\ln t/dr = 2$, $H = 1/(2t)$ in the radiation era ---
and reads the Hubble tension as two dictionaries meeting at the
corridor. This paper quotes it wherever the clock meets the
laboratory: the odometer remark of \S\ref{sec:clock-ladder}
(three clocks and one odometer), the chronology's slope in
Problem~13, and the repricing answer to Problem~14's
expanding-universe question. First cited in Problem~13.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.} The black-hole paper: the
information paradox resolved as a billing error --- the circle
keeps what the hole must return; the interval prices the return
--- with the Page curve as the two books' reconciliation
schedule, no firewalls, and the island computations placed with
their achievements intact. Problem~16 is its treatment carried
whole into this paper's register, including the de Sitter cousin
that ties the paradox to the trio's terminus. Cited in
Part~VII's contrasts and throughout Problem~16. First cited in
\S\ref{sec:gravity-w2}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The collider paper: the
Kaluza--Klein lock cluster at $3.849$~TeV as a zero-parameter
target --- one mass, one width, one rate, one interference sign,
one angular shape, one branching fingerprint, with a dated
decision calendar. It is this paper's sharpest single kill
switch (Problems~2 and~6 both stake it), and its two-fiber
reading --- a discovery as a double reading, identity from the
circle and address from the interval --- is the laboratory twin
of the Kaxion's haloscope strategy at the opposite end of the
energy ladder. First cited in \S\ref{ssec:where-next}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{Arisaka, K. (2026).} Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability. Arisaka-GG-P2-N3.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.} The family-number paper:
$N_g = 3$ closed with explicit lower and upper bounds inside one
declared axiom system --- the count written on the circle
(holonomy-class, unable to run), the capacity bound priced by
the interval (a budget being a semigroup concept). It is the methodological twin of every exactness claim in this paper, and Problem~12's why-three-colors section leans on its pincer: two derivations sharing no premise, one small integer. Its Appendix~H is the cohort's formal proof of the sentence this paper uses as a refrain --- \emph{what the circle quantizes is what the interval delivers} --- the two-sided audit behind the ledger's permanence in Problems~3, 4 and~12: the bulk with its anomaly ledger and compact directions, and the boundary with its adiabatic theorem, computing the same object independently and with no shared parts. Cited in Part~VII. First cited in \S\ref{ssec:where-next}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K. (2026).} The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}. Arisaka-GG-P3-Electron.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The electron paper: identity and
stability as ground-state theorems --- every electron the same
readout of the same unique attractor --- and the first portrait
in the triptych: \emph{a holonomy class wearing a mass}, the
sentence Problems~10, 7, and~14 generalize. The mirror reading
of the positron (same class, conjugate orientation, same
address) is Problem~10's opening; the muon's descent rate as a
\GRIPin{} observable times Problem~2's agent. First cited in
Problem~10.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.} The real-bulk paper: derives
quantum mechanics' complex structure from two real sources ---
the phase circle and the constitutional clock --- with the
discrete-symmetry table generated by their orientations
(Problem~9's calculus), the antiparticle pairing forced by their
handshake (Problem~10), antiunitarity of $T$ as an orientation
corollary, and the custody principle --- the dictionary owns the
sectors; the geometry owns the $i$ --- behind Problem~1's
records-are-real clause. The network Bell exclusion of
real-valued quantum mechanics is its one indispensable
experimental fact.
First cited in Part~II.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.} The CPT companion and this
paper's largest single creditor --- the parent of Problems~9,
10--11, and~13. From its Part on the birth of matter: the
alphabet audit (bystander, spectator, inert, driver), the gated
crossing, and the $\etaB$ chain with its named input, named
regulator, and postulated $K/2$ exponent. From its Part on
time: the four endowments, the exact slope, the age ledger
closing at $140.24$, the clock ladder, and the two-quantity
discipline at recombination. Its KMS shore --- temperature as
where the interval meets the shore --- recurs in Problems~14
and~16. First cited in \S\ref{ssec:where-next}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{Arisaka, K. (2026).} The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}. Arisaka-GG-P6-Proton.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.} The proton paper and Problem~12's
parent: confinement as the registration rule (only closed color
books reach the boundary), the abelian ledger with its double
guard --- $Q + t/3 \in \bZ$ and $Q + B \in \bZ$, verified
fragment by fragment across the hadron zoo with the $\bZ_6$
quotient checked trivial --- and the welded sphaleron door that
seals Problem~11's excess into permanent storage. The second
portrait of the triptych (two holonomy ledgers wearing a weld)
is its; the biosphere-underwriting argument of Problem~12 runs
on its stability theorems. First cited in \S\ref{ssec:where-next}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{Arisaka, K. (2026).} 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.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.} The first white paper and this
one's sibling --- diachronic where this paper is synchronic
(\S\ref{sec:w1-w2}). Its seven-step arc from Noether to
\HS{} supplies Problem~3's foundations (the second theorem,
Elitzur, the Wilson turn); its capstone --- symmetry breaking as
symmetry reading, with the determinism triad --- is
\S\ref{sec:determinism} and the engine of Problems~8 and~15;
and its four closing words are this paper's as well. The
campaign whose cohort-wide deployment this paper consolidates
began in its pages. First cited in the Executive Summary.

\end{thebibliography}
\endgroup

\subsection*{B.\ Other references}
\begingroup
\renewcommand{\section}[2]{}%
\begin{thebibliography}{99}
\setcounter{enumiv}{0}

\bibitem[Noether(1918)]{Noether1918}
\textbf{E.~Noether.} \emph{Invariante Variationsprobleme}, Nachr.\
Ges.\ Wiss.\ G\"ottingen (1918) 235.
\par\noindent\textbf{Relevance.}
The paper that made symmetry a computational instrument rather than an
aesthetic observation, and it did so twice. The first theorem converts a
continuous global symmetry into a conserved current, which is where the
tick of Problem~13 acquires its conservation law; the second converts a
\emph{local} symmetry into identities among the equations of motion --- not
new conservation laws but constraints, which is the technical root of the
statement in Problem~3 that a gauge symmetry is a redundancy rather than a
property of the world. The distinction between the two theorems is the
distinction this paper's anatomy makes geometric: the first theorem's
symmetries live where a closed loop can be traversed and a phase compared,
the second theorem's redundancies are the price of describing that loop in
coordinates. \PaperWOne{} traces the whole arc from this paper to the
\HS{} integers; this paper enters that arc at its geometric end and reads
Noether's two theorems as the two fibers' two jobs. First cited in
\S\ref{intro:p3}.

\bibitem[Dirac(1931)]{Dirac1931}
\textbf{P.~A.~M.~Dirac.} \emph{Quantized singularities in the
electromagnetic field}, Proc.\ R.\ Soc.\ A \textbf{133} (1931) 60.
\par\noindent\textbf{Relevance.}
The origin of every protected integer in physics. Dirac showed that the
mere consistency of a charged quantum particle in the field of a magnetic
monopole forces $eg = n\hbar c/2$ with $n$ an integer --- the first
demonstration that an exact quantization can follow from topology rather
than from a dynamical equation, and hence the ancestor of every closed
form on the circle side of this paper's anatomy. The same paper proposed
the anti-electron, which makes it the founding document of Problems~10--11
as well. What this paper takes from it is the argument's shape rather than
its content: an integer that cannot vary continuously has nowhere to move
to, which is why circle-side quantities in the ledger $(M,N,L;J,K)$ do not
run and cannot be dialed (Problem~4). Dirac's own comment that the
argument explains charge quantization ``if a single monopole exists
anywhere'' is the historical model for the cohort's habit of paying for
exactness with a topological hypothesis rather than a fitted parameter.
First cited in \S\ref{intro:p1011}.

\bibitem[Sakharov(1967)]{Sakharov1967}
\textbf{A.~D.~Sakharov.} \emph{Violation of CP invariance, C
asymmetry, and baryon asymmetry of the universe}, JETP Lett.\
\textbf{5} (1967) 24.
\par\noindent\textbf{Relevance.}
Three conditions --- baryon-number violation, C and CP violation, and
departure from thermal equilibrium --- without which no dynamical
mechanism can produce a matter--antimatter asymmetry from a symmetric
start. Sakharov's list has organized every baryogenesis proposal for half
a century, and its persistence is precisely the difficulty of
Problems~10--11: the conditions are necessary, they are individually
achievable, and the Standard Model satisfies all three too weakly by ten
orders of magnitude. This paper reads the list as an anatomy statement
rather than a checklist: the first two conditions are circle-side
questions about what the phase structure permits, the third is an
interval-side question about whether the descent is fast enough that the
composition has no inverse. That the three conditions sort cleanly onto
the two fibers is one of the paper's recurring pieces of evidence that the
sorting is doing real work. First cited in \S\ref{intro:p1011}.

\bibitem[Kobayashi and Maskawa(1973)]{KM1973}
\textbf{M.~Kobayashi and T.~Maskawa.} \emph{CP-violation in the
renormalizable theory of weak interaction}, Prog.\ Theor.\ Phys.\
\textbf{49} (1973) 652.
\par\noindent\textbf{Relevance.}
The observation that CP violation in the quark sector requires at least
three generations, because only then does the mixing matrix retain a
physical complex phase after all rephasings are exhausted. It is a
counting argument of exactly the kind this paper's census sections use,
and it earned its authors the Nobel Prize when the third family duly
appeared. The Kobayashi--Maskawa phase is also the reason Problems~10--11
are hard rather than merely unsolved: the phase exists, it is measured,
and \citet{Gavela1994} showed that the asymmetry it can generate falls
short of the observed value by some ten orders of magnitude. In this
paper's reading the quark phase is a chiral bystander --- a genuine
circle-side phase that is nevertheless not the one that pays the toll ---
and the leptonic phase of the cohort's pentagon is. First cited in
\S\ref{intro:p9}.

\bibitem[Fukugita and Yanagida(1986)]{FukugitaYanagida1986}
\textbf{M.~Fukugita and T.~Yanagida.} \emph{Baryogenesis without
grand unification}, Phys.\ Lett.\ B \textbf{174} (1986) 45.
\par\noindent\textbf{Relevance.}
Leptogenesis: the proposal that the baryon asymmetry is manufactured first
as a lepton asymmetry, in the CP-violating out-of-equilibrium decays of
heavy right-handed neutrinos, and then converted to baryon number by
electroweak sphalerons \citep{KRS1985}. It has become the leading
mechanism precisely because it recycles machinery introduced for another
purpose --- the seesaw that makes the light neutrinos light --- rather
than adding new sectors. The standing difficulty, and the reason
Problems~10--11 remain open, is that the mechanism is a framework with
free parameters rather than a prediction: the heavy masses and phases are
not measured. This paper's interest is in the geometric realization the
cohort supplies, in which the phases are ledger objects rather than
inputs, so that the toll acquires a computed value ($\eta_B \approx 7 \times
10^{-10}$, central $7.34$, $+20\%$ from the measured number inside its
factor-$\sim3$ band) with its one postulated exponent flagged in place. First cited in \S\ref{intro:p1011}.

\bibitem[Wilson(1974)]{Wilson1974}
\textbf{K.~G.~Wilson.} \emph{Confinement of quarks}, Phys.\ Rev.\ D
\textbf{10} (1974) 2445.
\par\noindent\textbf{Relevance.}
The paper that put gauge theory on a lattice and, in doing so, identified
the gauge-invariant content of the theory as the set of holonomies around
closed loops. Wilson's area law for the loop expectation value in the
strong-coupling limit is still the cleanest statement of what confinement
\emph{means}, and his reconstruction --- the field is recoverable from its
loop functionals --- is the technical fact behind this paper's central
sentence. When \S\ref{sec:statement} says that holonomy requires closed
loops, it is Wilson's variables that make the requirement more than a
definition: they are what a gauge theory is made of, and an open interval
supplies none of them. Problem~12's treatment is built on the same
foundation, with the $\mathbb{Z}_3$ center read as a winding label on the
circle and the registration rule written as a projection selection rule.
First cited in \S\ref{intro:p12}.

\bibitem[ATLAS Collaboration(2012)]{ATLAS2012}
\textbf{ATLAS Collaboration.} \emph{Observation of a new particle in
the search for the Standard Model Higgs boson with the ATLAS detector
at the LHC}, Phys.\ Lett.\ B \textbf{716} (2012) 1.
\par\noindent\textbf{Relevance.}
One of the two discovery papers for the scalar at $125$~GeV. It is
cited in \S\ref{sec:w2-record} for what it retired rather than for
what it found: a generation of model-building had rested on the
structural instinct that an elementary scalar cannot exist because
nothing protects its mass, and the instinct was wrong. This paper's own
account of the Higgs as a Wilson-line modulus is a structural argument
of the same genre run in the opposite direction, which is why the
episode is printed here rather than omitted. First cited in
\S\ref{sec:w2-record}.

\bibitem[CMS Collaboration(2012)]{CMS2012}
\textbf{CMS Collaboration.} \emph{Observation of a new boson at a mass
of 125~GeV with the CMS experiment at the LHC}, Phys.\ Lett.\ B
\textbf{716} (2012) 30.
\par\noindent\textbf{Relevance.}
The independent confirmation, on a different detector with different
systematics, of the particle reported by ATLAS in the same issue. Cited
alongside it in \S\ref{sec:w2-record} because the honest-record
argument turns on the falsification being beyond dispute. First cited
in \S\ref{sec:w2-record}. First cited in \S\ref{sec:w2-record}.

\bibitem[Hatanaka, Inami \& Lim(1998)]{HatanakaInamiLim1998}
\textbf{Hatanaka, H., Inami, T. \& Lim, C.~S. (1998).} The gauge
hierarchy problem and higher-dimensional gauge theories.
\emph{Mod.\ Phys.\ Lett.\ A}, \textbf{13}, 2601.
\par\noindent\textbf{Relevance.} The result that makes
\S\ref{ssec:three-tiers}'s tier~2 a theorem rather than a hope: the
effective potential of a Wilson-line phase is finite, because no local
counterterm for such a phase exists. This paper's account of why the
electroweak scale survives its own quantum corrections is inherited
from this line of work and is not claimed as new; what the anatomy
adds is the classification --- that the Higgs is the circle's one
continuous modulus, and that the naturalness problem is the error of
reading it as an interval-side address. First cited in
\S\ref{ssec:three-tiers}.

\bibitem[Hosotani(1983)]{Hosotani1983}
\textbf{Y.~Hosotani.} \emph{Dynamical mass generation by compact
extra dimensions}, Phys.\ Lett.\ B \textbf{126} (1983) 309.
\par\noindent\textbf{Relevance.}
The mechanism by which a gauge field's Wilson line around a compact extra
dimension acquires a vacuum expectation value and breaks the gauge
symmetry --- symmetry breaking with no elementary scalar, driven purely by
the holonomy of a circle. It is the most direct precedent in the
literature for the reading this paper generalizes: a physical, measurable
vacuum label that is a holonomy and nothing else, protected against local
corrections because it is not a local quantity --- the tier-2
protection of \S\ref{ssec:three-tiers}. Problem~2's account of why
the four interactions look so unalike leans on it directly: the circle's
Wilson-line label is what sets the electroweak address, while the
interval's warp sets the scale, and the apparent dissimilarity of the
forces is the two fibers doing two different jobs at once. The historical
lesson --- that a compact direction can supply a vacuum structure a
four-dimensional theory would have to postulate --- is one this paper
repeats in six other places. First cited in \S\ref{sec:weak}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}
\textbf{M.~B.~Green and J.~H.~Schwarz.} \emph{Anomaly cancellations
in supersymmetric $D=10$ gauge theory}, Phys.\ Lett.\ B \textbf{149}
(1984) 117.
\par\noindent\textbf{Relevance.}
The anomaly-cancellation mechanism that made ten-dimensional string theory
viable: a two-form field whose modified Bianchi identity $dH_3 = X_4$
absorbs an otherwise fatal anomaly, at the cost of a rigid relation among
the theory's group-theoretic coefficients. This paper cites it for the
mechanism rather than the setting. \AThree{} constructs the
six-dimensional, non-supersymmetric instance in which the same identity
closes local and global anomaly freedom for the bulk $\mathcal{M}_4 \times
I_r \times S^1_\theta$, and it is that closure which fixes the coefficient
arithmetic every closed form in Problem~4 spends. The general moral is the
one Problem~3 draws on the anatomy: anomaly coefficients are integers
computed by a topological index, they are therefore circle-side objects,
and quantities of that kind do not run with scale --- which is why the
cancellation is a constraint on the theory rather than a condition to be
maintained order by order. First cited in \S\ref{sec:stepA}.

\bibitem[Hopkins and Singer(2005)]{HopkinsSinger2005}
\textbf{M.~J.~Hopkins and I.~M.~Singer.} \emph{Quadratic functions in
geometry, topology, and M-theory}, J.\ Diff.\ Geom.\ \textbf{70}
(2005) 329.
\par\noindent\textbf{Relevance.}
The mathematical terminus of \PaperWOne{}'s arc: differential cohomology, in
which a quantity can be protected by an integer invariant with no symmetry
group anywhere in sight. The paper matters here as an existence proof for
a category this framework needs --- exactness that is not Noetherian ---
and as the reason the cohort's language distinguishes \emph{protected}
from \emph{conserved}. A conserved quantity is the output of a symmetry
and can be spoiled by breaking it; a protected quantity is the output of a
topological classification and can only change by a discontinuous jump
that the geometry forbids. That distinction is the whole content of the
circle side of this paper's anatomy, and the two white papers are shown
there to be reading the same structure from opposite ends. First cited in \S\ref{sec:w1-w2}.

\bibitem[Guth(1981)]{Guth1981}
\textbf{A.~H.~Guth.} \emph{Inflationary universe}, Phys.\ Rev.\ D
\textbf{23} (1981) 347.
\par\noindent\textbf{Relevance.}
The paper that turned three separate cosmological embarrassments --- the
horizon problem, the flatness problem, and the monopole problem --- into
one dynamical proposal: a period of accelerated expansion early enough to
place today's causally disconnected patches inside one past light cone.
Its success is not in dispute here; Problem~5 accepts the necessity of an
inflationary epoch and its evidence, and Problem~8's account of why
distant strangers agree is inflation's account. What Problem~5 disputes is
the \emph{menu}: after four decades the mechanism remains a class of
models with hundreds of members and no principle selecting one, so that
inflation predicts the pattern but not the parameters. This paper's
contribution is to let the census decide --- one non-compact non-spatial
coordinate is permitted, hence one field of the relevant kind --- so that
the potential's shape follows from the warp rather than from a choice.
First cited in \S\ref{intro:p5}.

\bibitem[Linde(1982)]{Linde1982}
\textbf{A.~D.~Linde.} \emph{A new inflationary universe scenario},
Phys.\ Lett.\ B \textbf{108} (1982) 389.
\par\noindent\textbf{Relevance.}
The new inflationary scenario, which rescued Guth's proposal from its
graceful-exit problem by having the field roll slowly along a flat plateau
rather than tunnel out of a false vacuum. Slow roll is what makes
inflation calculable: the spectral observables become derivatives of the
potential, and a model becomes a prediction. It is also, on this paper's
reading, the clearest dynamical portrait of what an interval traverse
\emph{is} --- a motion along a non-compact direction, damped by Hubble
friction, that happens once and does not return. Problem~5 identifies the
field doing the rolling with the interval's own position coordinate, which
is why the framework has one inflaton rather than a choice of inflatons,
and why $N^*$ becomes a length rather than a fitted number. First cited in
\S\ref{intro:p5}.

\bibitem[Starobinsky(1980)]{Starobinsky1980}
\textbf{A.~A.~Starobinsky.} \emph{A new type of isotropic
cosmological models without singularity}, Phys.\ Lett.\ B
\textbf{91} (1980) 99.
\par\noindent\textbf{Relevance.}
The original inflationary solution, predating the word: a de Sitter phase
driven by the $R^2$ correction to the gravitational action, decaying into
radiation. Its potential, written in the Einstein frame, is the plateau
whose spectral predictions --- a red tilt near $n_s \simeq 0.96$ and a
small tensor ratio --- the data have since favored over every competitor
with a steep potential. This paper cites it as the shape rather than the
model: the \GDOS{} anomaly argument forces a plateau class, and
Eq.~\eqref{eq:ns-r}'s parent potential is a member of it. The
discriminating claim in Problem~5 is not the tilt, on which many models
agree, but the pair $(n_s, r)$ read from a \emph{derived} $N^* = 52.6$,
which fixes $r = 0.0043$ with nothing left to adjust. First cited in
\S\ref{intro:p5}.

\bibitem[Peccei and Quinn(1977)]{PecceiQuinn1977}
\textbf{R.~D.~Peccei and H.~R.~Quinn.} \emph{CP conservation in the
presence of pseudoparticles}, Phys.\ Rev.\ Lett.\ \textbf{38} (1977)
1440.
\par\noindent\textbf{Relevance.}
The proposal that the strong CP problem is solved by a new global
symmetry, broken spontaneously, whose associated angle relaxes dynamically
to zero. Together with \citet{Weinberg1978} and \citet{Wilczek1978} --- who
pointed out that the broken symmetry leaves a light pseudoscalar behind
--- it created the axion, and with it the most economical dark-matter
candidate in the literature. The unsolved part, and the reason Problem~7
is still a problem, is the decay constant: the mechanism fixes the axion's
mass only once $f_a$ is chosen, and nothing in the construction chooses
it. The cohort's Kaxion is the same object with the choice removed ---
$f_K = f_{\rm bulk}/\Xi_{\rm cap}$ with $\Xi_{\rm cap} = K + N + L = 43$
--- which converts a parametrized family into a single line at
$8.1\,\mu$eV ($1.95$~GHz) with a computed doublet spacing. First cited in
\S\ref{intro:p7}.

\bibitem[Preskill, Wise, and Wilczek(1983)]{PreskillWiseWilczek1983}
\textbf{J.~Preskill, M.~B.~Wise, and F.~Wilczek.} \emph{Cosmology of
the invisible axion}, Phys.\ Lett.\ B \textbf{120} (1983) 127.
\par\noindent\textbf{Relevance.}
The relic-abundance calculation that made the invisible axion a
dark-matter candidate rather than a curiosity: the misalignment mechanism,
in which the field begins at a random angle, stays frozen until
$3H \simeq m_a$, and then oscillates as pressureless matter whose density
scales as $a^{-3}$. The calculation is the template Problem~7 follows
exactly --- the oscillation condition, the entropy dilution, the
initial-angle dependence --- and its structure is why the abundance is
predictive only once $f_a$ is fixed. This paper's interest is that every
input to the calculation except the misalignment angle becomes a ledger
quantity in the cohort's treatment, so the surviving uncertainty is
astrophysical rather than parametric. First cited in \S\ref{intro:p7}.

\bibitem[Weinberg(1989)]{Weinberg1989CC}
\textbf{S.~Weinberg.} \emph{The cosmological constant problem},
Rev.\ Mod.\ Phys.\ \textbf{61} (1989) 1.
\par\noindent\textbf{Relevance.}
The review that fixed the cosmological-constant problem in its modern
form, cataloged the failed approaches, and stated the discrepancy with a
precision that has embarrassed the field ever since: the vacuum energy
estimated from quantum field theory exceeds the observational bound by
some $120$ orders of magnitude, and no symmetry known will set it to zero
without setting it to zero exactly. Weinberg's own survey of mechanisms
--- adjustment, symmetry, anthropic selection --- is the map of the
territory Problem~6 and Problem~15 traverse. This paper's position is
stated as a repricing rather than a solution: the loop's reach is bounded
by the same $e^{-K}$ that sets the lock scale, the residual scale
$m_\Lambda = M_{\rm lock}^2/M_{\rm Pl, red} = 6.1$~meV is computed rather
than canceled, and the master table grades that row \emph{repriced}, not
\emph{solved}. First cited in \S\ref{intro:p6}.

\bibitem[Riess et al.(1998)]{Riess1998}
\textbf{A.~G.~Riess et al.} \emph{Observational evidence from
supernovae for an accelerating universe}, Astron.\ J.\ \textbf{116}
(1998) 1009.
\par\noindent\textbf{Relevance.}
One of the two supernova surveys that discovered the acceleration of the
cosmic expansion, and with it the observational necessity of a dark-energy
component. The measurement converted the cosmological constant from a
problem of principle --- why is it not enormous? --- into a problem of
value: why is it small but not zero, and why is it comparable to the
matter density now? Problem~6 quotes the result for the second question,
which is the harder one, and Problem~15 quotes it against
\citet{Weinberg1987}: the anthropic bound was an upper limit some two
orders of magnitude above what was subsequently measured, so the
measurement neither confirms nor refutes the selection argument. This
paper's inventory instead computes the shares directly, with $\Omega_m =
1/\pi$ and $\Omega_\Lambda = 1 - 1/\pi$ as fiber-volume ratios. First
cited in \S\ref{intro:p6}. First cited in \S\ref{sec:cat2}.

\bibitem[Perlmutter et al.(1999)]{Perlmutter1999}
\textbf{S.~Perlmutter et al.} \emph{Measurements of $\Omega$ and
$\Lambda$ from 42 high-redshift supernovae}, Astrophys.\ J.\
\textbf{517} (1999) 565.
\par\noindent\textbf{Relevance.}
The companion supernova survey, independent in team, instrument, and
analysis chain, reaching the same conclusion from $42$ high-redshift
objects. The pair is cited together throughout Part~I because the
agreement of two independent measurements is what made the acceleration
credible immediately rather than after a decade --- a methodological point
this paper makes repeatedly about its own two-route derivations
(Problem~6's ledger, Problem~12's $N = 3$). What the surveys established
is the target the cohort's inventory has to hit, and the honesty
discipline of \S\ref{sec:grades} requires the distinction to be kept
visible: the framework's closed forms are compared against these numbers,
not fitted to them. First cited in \S\ref{intro:p6}.

\bibitem[Planck Collaboration(2020)]{Planck2020}
\textbf{Planck Collaboration.} \emph{Planck 2018 results. VI.
Cosmological parameters}, Astron.\ Astrophys.\ \textbf{641} (2020)
A6.
\par\noindent\textbf{Relevance.}
The cosmological parameters from the full Planck mission --- the reference
dataset against which every quantitative claim in Problems~5--8 is
checked. The paper is cited for the inventory shares, the spectral index,
the tensor bound, the acoustic scale, and the decoupling redshift, and it
is the source of the numbers that make the cohort's closed forms falsifiable
rather than decorative. Two of the paper's disciplines are visible in how
it is used. First, the \emph{two-quantity} $z^*$ convention: the corridor
switch $z_{\rm sw} = e^7 = 1096.6$ is a topological quantity of the
framework, while the atomic decoupling redshift $z_{\rm dec} \approx 1090$
is the measured quantity Planck reports, and the two must never be
conflated even though they agree to a fraction of a per cent. Second, the
raw-versus-calibrated distinction: agreements are quoted in both registers,
with the zero-parameter number given first. First cited in
\S\ref{intro:p5}.

\bibitem[Gibbons and Hawking(1977)]{GibbonsHawking1977}
\textbf{G.~W.~Gibbons and S.~W.~Hawking.} \emph{Cosmological event
horizons, thermodynamics, and particle creation}, Phys.\ Rev.\ D
\textbf{15} (1977) 2738.
\par\noindent\textbf{Relevance.}
The extension of black-hole thermodynamics to cosmological horizons: de
Sitter space has a horizon, the horizon has a temperature and an entropy,
and an observer inside sees thermal radiation with no matter present
anywhere. It is the result that makes the accelerating universe a
thermodynamic object and ties Problem~16 to Problem~6 --- the same
horizon-entropy bookkeeping that governs a black hole governs the terminus
of the cosmic descent. This paper uses it at the end of Problem~16 to
place the information question in its cosmological setting: if a horizon
spends $\hbar$, $G_N$, and $k_B$ at once, the accounting cannot be a
property of collapsed stars in particular, and the two-fiber reading has
to work at both ends of the descent or at neither. First cited in
\S\ref{sec:w2-prob16}.

\bibitem[Renou et al.(2021)]{Renou2021}
\textbf{M.-O.~Renou et al.} \emph{Quantum theory based on real
numbers can be experimentally falsified}, Nature \textbf{600} (2021)
625.
\par\noindent\textbf{Relevance.}
The theoretical result that quantum theory formulated over the real
numbers is not merely inelegant but experimentally distinguishable from
the standard complex theory --- a Bell-type scenario in which the two
formulations predict different correlations. It is cited here for a
methodological reason that matters to this paper's whole enterprise: a
question that had been treated for ninety years as interpretational was
converted into a measurement. The framework's own claim about the origin
of the imaginary unit --- the custody principle, in which the dictionary
owns the sectors and the geometry owns the $i$ --- sits in exactly this
category, which is why the episode is used to explain what this paper
means by a falsifiable structural claim. The subsequent
experimental exclusions are \citet{Li2022} and \citet{Chen2022}. First
cited in \S\ref{sec:howtoread}. First cited in \S\ref{sec:howtoread}.

\bibitem[Wilson and Kogut(1974)]{WilsonKogut1974}
\textbf{K.~G.~Wilson and J.~Kogut.} \emph{The renormalization group
and the $\epsilon$ expansion}, Phys.\ Rep.\ \textbf{12} (1974) 75.
\par\noindent\textbf{Relevance.}
The review that made the renormalization group a picture rather than a
technique: couplings flowing under a change of resolution, fixed points
organizing the flow, and irrelevant operators dying out --- the modern
understanding that a quantum field theory is a trajectory rather than a
point. This is the single most important external input to Problem~3,
because the flow it describes is, on this paper's anatomy, a motion along
the interval. Kadanoff blocking composes forward and has no inverse; the
information discarded at each step is not recoverable; the flow is
therefore a semigroup, and $\partial_r = -\mu\partial_\mu$ identifies the
scale direction with the interval's coordinate. What the framework adds is
endpoints: an interval has two, so the flow has a beginning and an end
rather than running forever in both directions. First cited in
\S\ref{intro:p3}.

\bibitem[Lees et al.(2012)]{BaBar2012}
\textbf{J.~P.~Lees et al.\ (BaBar Collaboration).}
\emph{Observation of time-reversal violation in the $B^0$ meson
system}, Phys.\ Rev.\ Lett.\ \textbf{109} (2012) 211801.
\par\noindent\textbf{Relevance.}
The first direct observation of time-reversal violation as such --- not
inferred from CP violation plus the CPT theorem, but measured by comparing
the rates of $B^0$ transitions in one time order against the reverse, using
entangled meson pairs to prepare the initial states. It matters to
Problem~9's account for a reason of logical hygiene: with this measurement
in hand, all three of C, P, and T are known experimentally to fail
individually, and the survival of the combination CPT is a separate,
independently checked fact rather than a definitional convenience. That is
exactly the shape of the anatomy's claim --- three orientations that a
bundle can reverse individually, one total orientation that it cannot ---
so the experimental record and the geometric statement are testing the same
structure. First cited in \S\ref{intro:p9}.

\bibitem[Li et al.(2022)]{Li2022}
\textbf{M.-C.~Li et al.} \emph{Testing real quantum theory in an
optical quantum network}, Phys.\ Rev.\ Lett.\ \textbf{128} (2022)
040402.
\par\noindent\textbf{Relevance.}
One of the two 2022 experiments that closed the real-quantum-theory
question opened by \citet{Renou2021}, ruling out the real-valued
formulation in an optical quantum network. Cited alongside
\citet{Chen2022} as the completion of the episode: a structural question about the mathematical furniture of the
theory --- whether the complex numbers are essential or merely convenient
--- moved from philosophy to laboratory in under two years. The paper
holds this up as the standard its own structural claims should be held to,
and the falsifier table of \S\ref{sec:kill} is the attempt to meet
it. First cited in \S\ref{sec:howtoread}.

\bibitem[Chen et al.(2022)]{Chen2022}
\textbf{M.-C.~Chen et al.} \emph{Ruling out real-valued standard
formalism of quantum theory}, Phys.\ Rev.\ Lett.\ \textbf{128}
(2022) 040403.
\par\noindent\textbf{Relevance.}
The companion exclusion of real-valued quantum theory, performed on a
superconducting processor, independent of \citet{Li2022} in platform and
systematics. The pair is cited together for the same reason the two
supernova surveys are: two independent instruments reaching one conclusion
is what converts a result into a fact. In this paper's use, the episode
supplies the template for how a claim about structure --- rather than
about a value --- can be made to face an instrument, which is the
methodological demand made of the sixteen problem treatments that
follow. First cited in \S\ref{sec:howtoread}.

\bibitem[Arndt et al.(1999)]{Arndt1999}
\textbf{M.~Arndt, O.~Nairz, J.~Vos-Andreae, C.~Keller, G.~van der
Zouw, and A.~Zeilinger.} \emph{Wave--particle duality of $C_{60}$
molecules}, Nature \textbf{401} (1999) 680.
\par\noindent\textbf{Relevance.}
The fullerene interferometry experiment: $C_{60}$ molecules, roughly a
kilodalton of mass and seventy nuclei, sent through a grating and observed
to interfere. It is the standing benchmark for how large an object can be
and still show its wave nature, and it is cited in Problem~1 as the
empirical anchor of the decoherence account --- above this scale the
environmental monitoring rate exceeds any achievable coherence time, and
the wave nature becomes operationally inaccessible rather than absent.
The experiment matters to this paper's reading because it shows the
transition is quantitative and continuous, set by a rate, not by a
boundary between two kinds of matter: the interval's irreversible
composition is doing more of the work as the object grows, and nothing
switches on. First cited in \S\ref{sec:w2-prob1}.

\bibitem[Landauer(1961)]{Landauer1961}
\textbf{R.~Landauer.} \emph{Irreversibility and heat generation in
the computing process}, IBM J.\ Res.\ Dev.\ \textbf{5} (1961) 183.
\par\noindent\textbf{Relevance.}
The result that erasing one bit of information costs at least $k_B T
\ln 2$ of dissipated heat --- the first quantitative bridge between
information and thermodynamics, and the resolution of Maxwell's demon once
the demon's memory is included in the accounting. Problem~14 uses it as
the pivot of the whole treatment: entropy is not an analogy for missing
information but the same quantity, and the cost is charged at the moment
of \emph{erasure}, that is, at the moment a many-to-one map is applied.
On this paper's anatomy that is the interval's signature --- a composition
with no inverse --- which is why the demon is closed by the same structure
that supplies the arrow of Problem~13 rather than by a separate argument.
First cited in \S\ref{intro:p14}.

\bibitem[Gavela et al.(1994)]{Gavela1994}
\textbf{M.~B.~Gavela, P.~Hern\'andez, J.~Orloff, and O.~P\`ene.}
\emph{Standard model CP-violation and baryon asymmetry}, Mod.\
Phys.\ Lett.\ A \textbf{9} (1994) 795.
\par\noindent\textbf{Relevance.}
The calculation that quantified the shortfall: with the
Kobayashi--Maskawa phase as the only source of CP violation, the baryon
asymmetry generated at the electroweak transition falls below the observed
value by roughly ten orders of magnitude. It is the reason Problems~10--11
are hard rather than merely unfinished --- the Standard Model satisfies all
three Sakharov conditions and still fails, so the deficit is quantitative
and cannot be argued away. This paper cites it as the size of the debt any
proposal must actually pay, and as the reason the cohort's account routes
the asymmetry through the leptonic phase rather than the quark phase: the
quark phase is measured, its contribution is computed, and it is not
enough. First cited in \S\ref{intro:p1011}.

\bibitem[Wald(1984)]{Wald1984}
\textbf{R.~M.~Wald.} \emph{General Relativity}, University of
Chicago Press (1984).
\par\noindent\textbf{Relevance.}
The standard graduate text on general relativity, and the one this paper
recommends for the geometric machinery rather than for the applications:
manifolds, connections, curvature, and the causal structure of spacetime
developed with the care a reader of Appendix~A will need. Its treatment of
Killing vectors and stationarity is the background for the appendix's
discussion of what a static arena buys, and its chapters on horizons and
black-hole thermodynamics underwrite Appendix~I's four laws. Wald is
recommended over its rivals for this paper's purposes because it is
uncompromising about the distinction between a structure that exists on a
manifold and a structure that exists only in a chart --- the very
distinction the two-fiber reading turns into physics. First cited in
\S\ref{app:reading}.

\bibitem[Baumann(2022)]{Baumann2022}
\textbf{D.~Baumann.} \emph{Cosmology}, Cambridge University Press
(2022).
\par\noindent\textbf{Relevance.}
The modern graduate cosmology text, written by an inflationary theorist,
and the reference of first resort for Appendices~E and~F. Its derivations
of the Friedmann equations, the thermal history, the horizon problem, slow
roll, and the mode-freezing origin of the primordial spectra are the ones
Appendix~E's Step~1 follows in structure, and its treatment of the CMB
transfer functions is what a student needs before the inventory
comparisons of Problem~7 will make sense. It is recommended first among
the cosmology texts because it is the most recent and the most explicit
about which parts of the standard story are measured and which are
modeled --- a distinction this paper's grade vocabulary depends on.
First cited in \S\ref{app:reading}.

\bibitem[Weinberg(2008)]{WeinbergCosmology2008}
\textbf{S.~Weinberg.} \emph{Cosmology}, Oxford University Press
(2008).
\par\noindent\textbf{Relevance.}
Weinberg's cosmology, recommended for the analytic backbone that the more
pedagogical texts compress: the perturbation theory done in full, the
transfer functions derived rather than quoted, and the anisotropy
calculation assembled from first principles. Appendix~E cites it where a
reader wants to see the machinery rather than the result, and Problem~6's
discussion of the vacuum-energy problem quotes its framing alongside
\citet{Weinberg1989CC}. The book is also useful here as a model of tone:
it is scrupulous about the difference between what the standard framework
derives and what it accommodates, which is the discipline the honest
ledger of \S\ref{sec:honest} tries to keep. First cited in
\S\ref{app:reading}.

\bibitem[Kolb and Turner(1990)]{KolbTurner1990}
\textbf{E.~W.~Kolb and M.~S.~Turner.} \emph{The Early Universe},
Addison-Wesley (1990).
\par\noindent\textbf{Relevance.}
The early-universe reference: thermal history, relic abundances, freeze-out,
entropy accounting, and the $g_*$ counting that Appendix~E's reheating
discussion and Appendix~F's relic-density derivation both use. It remains
the cleanest source for the calculation Problem~7's Kaxion abundance
follows step for step --- the oscillation condition, the comoving number
density, the entropy dilution --- and for the effective-degrees-of-freedom
tables the descent's e-fold accounting requires. Its age shows in the
data it quotes and nowhere else; the machinery is unchanged. First cited
in \S\ref{app:reading}. First cited in \S\ref{app:reading}.

\bibitem[Sakurai and Napolitano(2020)]{SakuraiNapolitano2020}
\textbf{J.~J.~Sakurai and J.~Napolitano.} \emph{Modern Quantum
Mechanics}, 3rd ed., Cambridge University Press (2020).
\par\noindent\textbf{Relevance.}
The graduate quantum-mechanics text this paper recommends for Appendix~B's
prerequisites: the postulates stated cleanly, the measurement formalism
developed without interpretive commitment, and --- crucially for the
appendix's Step~1 --- density matrices, the reduced state, and the partial
trace treated as physics rather than as bookkeeping. A reader who can
follow its chapter on the interaction picture and its treatment of the
Berry phase has everything Appendix~B assumes. The Berry-phase material is
also the natural entry point to the holonomy language of Appendix~A, since
the geometric phase is the one holonomy most physicists meet before they
meet the word. First cited in \S\ref{app:reading}.

\bibitem[Zee(2010)]{Zee2010}
\textbf{A.~Zee.} \emph{Quantum Field Theory in a Nutshell}, 2nd
ed., Princeton University Press (2010).
\par\noindent\textbf{Relevance.}
Quantum field theory taught by physical argument rather than by
axiomatics, and the text recommended for a reader who wants the ideas of
Appendix~C before the technology: why the path integral organizes the
subject, what an effective field theory is, where the renormalization
group comes from, and why anomalies exist at all. Its treatment of gauge
theories and of the anomaly is the level of detail Appendix~C's Step~1
targets, and its chapters on condensed-matter analogies are useful
preparation for the semigroup reading of the flow --- Kadanoff blocking
appears there in its original setting, where the loss of information is
obvious rather than formal. First cited in \S\ref{app:reading}.

\bibitem[Georgi(1999)]{Georgi1999}
\textbf{H.~Georgi.} \emph{Lie Algebras in Particle Physics}, 2nd
ed., Perseus (1999).
\par\noindent\textbf{Relevance.}
The group theory a reader of this paper actually needs, taught as a
working tool: representations, roots and weights, tensor methods, and the
branching rules that decide how a large group's representations decompose
when it is broken to a subgroup. Appendix~D's parameter-counting theorem
and Appendix~C's discussion of $SU(5)$ content both assume this material,
and the anomaly-coefficient arithmetic of \AThree{} is unreadable without
it. Georgi is recommended over the mathematicians' texts because it keeps
the physics question --- what does this representation contain, and what
does it become --- in front of the formalism throughout. First cited in
\S\ref{app:reading}.

\bibitem[Dodelson and Schmidt(2020)]{DodelsonSchmidt2020}
\textbf{S.~Dodelson and F.~Schmidt.} \emph{Modern Cosmology}, 2nd
ed., Academic Press (2020).
\par\noindent\textbf{Relevance.}
The cosmology text that treats the observational chain in the most
practical detail: what a CMB experiment actually measures, how the
likelihood is built, where the degeneracies lie, and how a parameter
constraint is extracted. Appendix~E cites it where the question is not
what the theory predicts but how the comparison is made --- the question
Problem~7's inventory discussion and the objections about covariances turn
on. A reader who wants to check for himself whether $\Omega_m = 1/\pi$ is
a fair comparison against a marginalized posterior should start here.
First cited in \S\ref{app:reading}.

\bibitem[Nakahara(2003)]{Nakahara2003}
\textbf{M.~Nakahara.} \emph{Geometry, Topology and Physics}, 2nd
ed., IOP Publishing (2003).
\par\noindent\textbf{Relevance.}
The bridge text between physics and the geometry this paper is written in:
fiber bundles, connections, curvature, characteristic classes, homotopy
and homology, index theorems, and anomalies, all developed for physicists
and all used in Appendix~A. It is the single most useful reference for the
technical content of the central sentence, because it makes explicit what
a holonomy group \emph{is}, why $\mathrm{Hom}(\pi_1(M), G)/G$ is the right
moduli space for flat connections, and why that space is a circle for
$M = S^1$ and a point for $M = I$. A reader who works through its bundle
chapters will find that this paper's anatomy claim is not exotic
mathematics but the first consequence of the definitions. First cited in
\S\ref{app:reading}.

\bibitem[Misner et al.(1973)]{MTW1973}
\textbf{C.~W.~Misner, K.~S.~Thorne, and J.~A.~Wheeler.}
\emph{Gravitation}, W.~H.~Freeman (1973).
\par\noindent\textbf{Relevance.}
\emph{Gravitation}: the encyclopaedic text whose geometric pedagogy ---
tell the reader what the object is before writing its components ---
this paper's exposition tries to imitate. It is recommended for
Appendix~A's parallel-transport material, where its pictures of a vector
carried around a closed circuit on a sphere are still the clearest
illustration in print of the fact that holonomy is a property of a loop
and of nothing else. Its discussion of the difference between a coordinate
artifact and a geometric invariant is the background a reader needs before
the interval-versus-circle distinction can carry weight. First cited in
\S\ref{app:reading}.

\bibitem[Wu and Yang(1975)]{WuYang1975}
\textbf{T.~T.~Wu and C.~N.~Yang.} \emph{Concept of nonintegrable
phase factors and global formulation of gauge fields}, Phys.\ Rev.\
D \textbf{12} (1975) 3845.
\par\noindent\textbf{Relevance.}
The paper that identified the physically meaningful content of a gauge
field as the non-integrable phase factor --- the holonomy --- rather than
the potential or the field strength, and gave the global formulation in
bundle language that makes the Aharonov--Bohm effect and the Dirac
monopole two aspects of one statement. It is the historical hinge between
gauge theory as written by physicists and gauge theory as understood by
geometers, and it is therefore the most directly relevant external paper
to this one's central sentence. Appendix~A's Step~1 follows its dictionary
--- bundle, connection, holonomy on the mathematical side; gauge field,
potential, phase factor on the physical side --- and Problem~3's
readability theorem is that dictionary applied to the question of what a
measurement of a gauge theory can return. First cited in
\S\ref{app:reading}.

\bibitem[Nielsen and Chuang(2000)]{NielsenChuang2000}
\textbf{M.~A.~Nielsen and I.~L.~Chuang.} \emph{Quantum Computation
and Quantum Information}, Cambridge University Press (2000).
\par\noindent\textbf{Relevance.}
The standard reference for quantum information, cited here for the parts
Appendix~B builds on: completely positive trace-preserving maps, the Kraus
representation, the operator-sum decomposition, and the entropies. The
technical claim that a channel composes forward but does not in general
invert is stated there in its cleanest form, and that claim is precisely
what the interval side of the anatomy asserts about the physical
projection. A reader who wants to test whether this paper's use of
``semigroup'' is metaphorical or literal should check it against the
composition law for channels given here; the answer is that it is
literal. First cited in \S\ref{app:reading}.

\bibitem[Breuer and Petruccione(2002)]{BreuerPetruccione2002}
\textbf{H.-P.~Breuer and F.~Petruccione.} \emph{The Theory of Open
Quantum Systems}, Oxford University Press (2002).
\par\noindent\textbf{Relevance.}
The reference text on open quantum systems, and the source for the
Born--Markov derivation, the Lindblad generator, and the decoherence
models that Appendix~B works through in detail. Its treatment of the
environment correlation functions --- and of why their Fourier transforms
must be positive for the dynamics to be a legitimate channel --- is the
technical basis for the appendix's claim that irreversibility requires a
continuous spectrum, and hence that a compact fiber cannot supply it. That
single point carries a large share of the anatomy's weight in Problems~1,
13, and~14, so a reader who doubts it should check it here first. First
cited in \S\ref{app:reading}. First cited in \S\ref{app:reading}.

\bibitem[Peskin and Schroeder(1995)]{PeskinSchroeder1995}
\textbf{M.~E.~Peskin and D.~V.~Schroeder.} \emph{An Introduction to
Quantum Field Theory}, Addison-Wesley (1995).
\par\noindent\textbf{Relevance.}
The field-theory text most readers of this paper will have been trained
on, cited for the computations Appendix~C's Step~1 reproduces in outline:
the beta function with its two competing contributions, the
Callan--Symanzik equation, the chiral anomaly, and the renormalization of
gauge theories. It is recommended for the calculational spine; the
conceptual framing is better found in \citet{Zee2010} and the axiomatic
grounding in \citet{WeinbergQTF1}. Appendix~C's discussion of why anomaly
coefficients are integers that cannot run is written against its
presentation of the triangle diagram. First cited in \S\ref{app:reading}.

\bibitem[Weinberg(1995)]{WeinbergQTF1}
\textbf{S.~Weinberg.} \emph{The Quantum Theory of Fields, Vol.~I:
Foundations}, Cambridge University Press (1995).
\par\noindent\textbf{Relevance.}
Weinberg's first volume: quantum field theory derived from Lorentz
invariance, unitarity, and cluster decomposition rather than assumed as a
formalism. It is recommended for readers who want to know which features
of the standard framework are forced and which are conventional --- the
question this paper asks of every structure it inherits. Appendix~C cites
it for the derivation of the particle content from representation theory,
and Problem~3's account of gauge invariance as a redundancy rather than a
symmetry is closest in spirit to Weinberg's own presentation, in which the
gauge field appears as the price of a massless helicity-one field rather
than as a postulate. First cited in \S\ref{app:reading}.

\bibitem[Srednicki(2007)]{Srednicki2007}
\textbf{M.~Srednicki.} \emph{Quantum Field Theory}, Cambridge
University Press (2007).
\par\noindent\textbf{Relevance.}
A field-theory text organized so that a reader can take the pieces in the
order the argument needs them, which makes it the most convenient
reference for the specific results Appendices~C and~H quote: the
functional methods, the background-field treatment of the beta function,
and the operator analysis behind the dimension-six proton-decay operators.
It is recommended alongside \citet{PeskinSchroeder1995} rather than
instead of it --- the two are complementary in emphasis, and a student
working through Appendix~H will want both. First cited in
\S\ref{app:reading}.

\bibitem[Born(1926)]{Born1926}
\textbf{M.~Born.} \emph{Zur Quantenmechanik der Sto\ss vorg\"ange},
Z.\ Phys.\ \textbf{37} (1926) 863.
\par\noindent\textbf{Relevance.}
The paper in which the probability interpretation of the wave function
appears --- famously in a footnote added in proof, the modulus squared
corrected from the modulus. A century later the rule remains the one
element of the quantum formalism that has never been derived from the
others within the standard framework: unitary evolution is a theorem of
the postulates, the measure attached to outcomes is not. That is the whole
of Problem~1's difficulty, and it is why this paper's status claim is
carefully limited: the Born rule is closed at the relaxation level, with
the field-grade representation step named as the single analytic residue
rather than waved past. Everything the framework says about measurement
is downstream of what Born wrote in that footnote. First cited in
\S\ref{intro:p1}.

\bibitem[von Neumann(1932)]{vonNeumann1932}
\textbf{J.~von Neumann.} \emph{Mathematische Grundlagen der
Quantenmechanik}, Springer, Berlin (1932).
\par\noindent\textbf{Relevance.}
The book that gave quantum mechanics its Hilbert-space formulation and
separated its dynamics into two processes: the unitary evolution that
holds when nobody looks, and the projection that occurs when somebody
does. Naming the second process is what created the measurement problem
in its modern form, because nothing in the formalism says when it
applies. The book's celebrated no-hidden-variables theorem, later shown by
Bell to rest on an assumption stronger than necessary, is a cautionary
example this paper cites for its own sake: an impossibility proof is only
as strong as the class of theories it quantifies over, which is why
\S\ref{sec:grades} insists that every conditional result name its
conditions. Problem~1 reads von Neumann's two processes as the two
fibers' two behaviors rather than as two laws. First cited in
\S\ref{intro:p1}.

\bibitem[Einstein et al.(1935)]{EPR1935}
\textbf{A.~Einstein, B.~Podolsky, and N.~Rosen.} \emph{Can
quantum-mechanical description of physical reality be considered
complete?}, Phys.\ Rev.\ \textbf{47} (1935) 777.
\par\noindent\textbf{Relevance.}
The argument that quantum mechanics, if it is local, must be incomplete
--- from the perfect correlations of an entangled pair, together with a
locality assumption, to the conclusion that the theory omits elements of
reality. Its logic is often reported backwards; what EPR argued was a
conditional, and Bell's later achievement was to make the antecedent
testable. This paper cites it as the origin of the dispute
\S\ref{sec:philosophy} calls the oldest in physics, and as the point at which
a metaphysical preference became a physical claim with consequences. The
framework's own position on that dispute --- deterministic in the bulk,
probabilistic on the boundary, with 4D non-locality conceded openly --- is
a way of accepting the EPR conditional and denying its locality premise
where the premise is cheapest to deny. First cited in \S\ref{intro:p1}.

\bibitem[Bell(1964)]{Bell1964}
\textbf{J.~S.~Bell.} \emph{On the Einstein Podolsky Rosen paradox},
Physics \textbf{1} (1964) 195.
\par\noindent\textbf{Relevance.}
The inequality that converted the EPR dispute into an experiment: any
theory in which outcomes are determined by local hidden variables must
satisfy a bound that quantum mechanics violates. It is the most important
single result in the foundations of quantum mechanics, and its subsequent
experimental confirmation is what forces any realist framework to be
explicit about where its non-locality lives. This paper's answer is
geometric and stated without hedging: the correlations are local in the
six-dimensional bulk and non-local only in their four-dimensional shadow,
with the circle's winding label as the carrier. Problem~1 treats this as
a cost openly paid rather than as a difficulty finessed, and Part~X
returns to it whenever an objection asks what the framework gives up.
First cited in \S\ref{intro:p1}.

\bibitem[Everett(1957)]{Everett1957}
\textbf{H.~Everett III.} \emph{``Relative state'' formulation of
quantum mechanics}, Rev.\ Mod.\ Phys.\ \textbf{29} (1957) 454.
\par\noindent\textbf{Relevance.}
The relative-state formulation: keep the unitary evolution, drop the
collapse, and read the appearance of definite outcomes as a fact about
branches rather than about the world. It is cited in Problem~1 as one of
the three standard responses to the measurement problem --- alongside
dynamical collapse \citep{GRW1986} and the decoherence program
\citep{Zurek2003} --- and as the one that pays for its economy of
postulates with an ontological extravagance, together with a well-known
difficulty in recovering the measure. This paper's route is a different
trade: the outcomes are single because the projection is single-valued,
and the measure is what the relaxation supplies. Whether that trade is
better is a judgment the paper states rather than assumes. First cited
in \S\ref{intro:p1}. First cited in \S\ref{sec:cat1}.

\bibitem[Ghirardi et al.(1986)]{GRW1986}
\textbf{G.~C.~Ghirardi, A.~Rimini, and T.~Weber.} \emph{Unified
dynamics for microscopic and macroscopic systems}, Phys.\ Rev.\ D
\textbf{34} (1986) 470.
\par\noindent\textbf{Relevance.}
Dynamical collapse: modify the Schr\"odinger equation itself with a
stochastic localization term, tuned so that microscopic systems evolve
essentially unitarily while macroscopic superpositions collapse fast. It
is the honest rival to every interpretive move, because it makes the
measurement problem into a question about a rate that experiments can
bound, and successive interferometry results have been closing the
parameter space ever since. This paper cites it in Problem~1 both as the
second standard response and as a methodological model: a proposal that
buys definiteness at the price of new parameters and, in exchange, becomes
falsifiable. The framework's own claim is that the definiteness comes from
the interval's structure at no parametric cost, which is a stronger claim
and correspondingly more exposed. First cited in \S\ref{intro:p1}.

\bibitem[Zurek(2003)]{Zurek2003}
\textbf{W.~H.~Zurek.} \emph{Decoherence, einselection, and the
quantum origins of the classical}, Rev.\ Mod.\ Phys.\ \textbf{75}
(2003) 715.
\par\noindent\textbf{Relevance.}
The review that assembled decoherence and einselection into a complete
account of the emergence of classicality: the environment monitors the
system, the pointer basis is the one that survives the monitoring, and
interference terms decay at rates that can be computed and measured.
Everything in it is used here and none of it is disputed. What Problem~1
observes is the well-known residue --- decoherence explains why the
off-diagonal terms vanish and which basis is selected, but not why one
diagonal entry is realized rather than another, and not why the weights
are what they are. On this paper's anatomy the part decoherence explains
is the interval's part, which is why the program works so well and stops
exactly where it does. First cited in \S\ref{intro:p1}.

\bibitem[Fermi(1934)]{Fermi1934}
\textbf{E.~Fermi.} \emph{Versuch einer Theorie der
$\beta$-Strahlen}, Z.\ Phys.\ \textbf{88} (1934) 161.
\par\noindent\textbf{Relevance.}
The first quantum field theory of a force other than electromagnetism: beta
decay as a four-fermion contact interaction, with a coupling constant
$G_F$ carrying dimensions of inverse mass squared. It worked, and its
failure mode taught the field its most valuable lesson --- a dimensionful
coupling means the theory is an effective description with a ceiling, and
the ceiling turned out to be the $W$ boson. Problem~2 cites it as the
origin of the puzzle it addresses: the interactions look unalike partly
because they are described at different depths of the same effective
tower, and the framework's claim is that the depth is a position on the
interval and the label at that depth is a phase on the circle. First
cited in \S\ref{intro:p2}. First cited in \S\ref{sec:cat1}.

\bibitem[Yang and Mills(1954)]{YangMills1954}
\textbf{C.~N.~Yang and R.~L.~Mills.} \emph{Conservation of isotopic
spin and isotopic gauge invariance}, Phys.\ Rev.\ \textbf{96} (1954)
191.
\par\noindent\textbf{Relevance.}
The generalization of the gauge principle to a non-abelian group, written
for isotopic spin and rejected at the time for predicting massless charged
vector bosons that nobody had seen. It is now the mathematical spine of
three of the four interactions, and its history is the standard example of
a structure being right before its physical realization was understood.
Problem~2 uses it as the point at which force ceased to be a list and
became a representation-theoretic question, and Problem~3 uses it as the
first place where a local symmetry was recognized as a redundancy imposed
on the description rather than an observable property --- which is exactly
the reading the anatomy makes geometric. First cited in \S\ref{intro:p2}.

\bibitem[Weinberg(1967)]{Weinberg1967}
\textbf{S.~Weinberg.} \emph{A model of leptons}, Phys.\ Rev.\ Lett.\
\textbf{19} (1967) 1264.
\par\noindent\textbf{Relevance.}
The three-page paper that unified the electromagnetic and weak
interactions for leptons, with the Higgs mechanism supplying the masses
and the weak mixing angle as the one new parameter. It was almost
uncited for four years, until renormalizability was established, and then
became the most cited paper in particle physics --- a history this paper
notes in Problem~2 because it illustrates how much of the field's
confidence in a structure depends on being able to compute with it. The
mixing angle is where this paper's interest lies: the standard framework
measures $\sin^2\theta_W$, and the cohort computes it as $25/108 =
0.23148$ from the ledger, a comparison that is either a coincidence or
the point. First cited in \S\ref{intro:p2}.

\bibitem[Gross and Wilczek(1973)]{GrossWilczek1973}
\textbf{D.~J.~Gross and F.~Wilczek.} \emph{Ultraviolet behavior of
non-abelian gauge theories}, Phys.\ Rev.\ Lett.\ \textbf{30} (1973)
1343.
\par\noindent\textbf{Relevance.}
Asymptotic freedom: the discovery that a non-abelian gauge theory's
coupling weakens at short distances, which made QCD possible and explained
why quarks behave as free particles inside a nucleon while being
permanently bound. It is cited in Problem~2 as the resolution of the
apparent contradiction between the parton picture and confinement, and in
Problem~12 as the reason the confinement problem is hard: the strong
coupling regime where confinement lives is the regime perturbation theory
cannot reach. The two competing contributions to the beta function ---
fermion screening against gluon antiscreening --- are worked in
Appendix~C's Step~1, because the sign of that competition is what the
whole picture turns on. First cited in \S\ref{intro:p2}.

\bibitem[Politzer(1973)]{Politzer1973}
\textbf{H.~D.~Politzer.} \emph{Reliable perturbative results for
strong interactions?}, Phys.\ Rev.\ Lett.\ \textbf{30} (1973) 1346.
\par\noindent\textbf{Relevance.}
The independent discovery of asymptotic freedom, published back to back
with \citet{GrossWilczek1973} and sharing the Nobel Prize with it. Cited
alongside its companion throughout Problem~2 and Problem~12, and noted
here for the same methodological reason this paper keeps returning to:
two independent routes to one result is the strongest evidence a field
ever gets, which is why the cohort's genuine two-route derivation ---
$N = 3$, from the constitutional chain and from
\citeauthor{ArisakaP2N3}'s capacity bound, which share no step --- is
presented as load-bearing, while $K$ is presented as what it is: one
derivation, spent four ways. First cited in \S\ref{intro:p2}.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}
\textbf{H.~Georgi and S.~L.~Glashow.} \emph{Unity of all
elementary-particle forces}, Phys.\ Rev.\ Lett.\ \textbf{32} (1974)
438.
\par\noindent\textbf{Relevance.}
Minimal $SU(5)$: the proposal that the three Standard Model gauge groups
are the low-energy remnants of one simple group, which explains charge
quantization, predicts the weak mixing angle, and requires proton decay.
The prediction was its virtue and its undoing --- the minimal version's
proton lifetime has been excluded by \citet{SuperK2020} and its
descendants --- but the idea that the gauge structure is the shadow of
something simpler has never been abandoned. This paper cites it as the
program it belongs to and departs from: the cohort's unification is
geometric rather than group-theoretic, the proton is protected by a double
guard rather than by a single mass scale, and the resulting lifetime is a
consequence rather than a fitted upper bound. First cited in
\S\ref{intro:p2}.

\bibitem[Weyl(1918)]{Weyl1918}
\textbf{H.~Weyl.} \emph{Gravitation und Elektrizit\"at},
Sitzungsber.\ Preuss.\ Akad.\ Wiss.\ (1918) 465.
\par\noindent\textbf{Relevance.}
The first attempt to unify electromagnetism with gravity by making the
local scale of length a gauge freedom, and the first geometric
unification to be killed by an experiment it had not anticipated:
Einstein's objection that a locally gauged length makes a clock's rate
depend on its path, so that atoms with different histories would show
different spectra. They do not. It is cited here, in
\S\ref{sec:w2-record}, against this paper's own interest: Weyl gauged
the scale direction, this framework's fifth dimension \emph{is} the
scale direction, and the whole difference between the two is that here
it carries no gauge-invariant content. The reader is entitled to weigh
how thin a margin that is. First cited in \S\ref{sec:w2-record}.

\bibitem[Weyl(1929)]{Weyl1929}
\textbf{H.~Weyl.} \emph{Elektron und Gravitation. I}, Z.\ Phys.\
\textbf{56} (1929) 330.
\par\noindent\textbf{Relevance.}
The paper that gave the gauge principle its modern form: invariance of the
Dirac field under a local change of phase requires the electromagnetic
potential, and the potential's transformation law is fixed by that
requirement. Weyl had tried the same idea a decade earlier with a local
change of \emph{scale}, which failed; the survival of the name ``gauge''
from the failed version is a historical accident the field has never
corrected. Problem~3 cites it as the origin of the question it asks ---
why should nature be organized around a redundancy? --- and reads Weyl's
own instinct as vindicated in an unexpected way: the phase is a coordinate
on a compact fiber, and the scale is a coordinate on an open one, so both
of his ideas were geometry, but only one of them was a symmetry. First
cited in \S\ref{intro:p3}. First cited in \S\ref{sec:cat1}.

\bibitem[Dyson(1949)]{Dyson1949}
\textbf{F.~J.~Dyson.} \emph{The radiation theories of Tomonaga,
Schwinger, and Feynman}, Phys.\ Rev.\ \textbf{75} (1949) 486.
\par\noindent\textbf{Relevance.}
The paper that showed the Tomonaga, Schwinger, and Feynman formulations to
be the same theory, and organized renormalization into a systematic
procedure rather than a sequence of rescues. It is the moment
renormalization became a technique physicists could trust without knowing
what it meant --- a state of affairs that lasted twenty-five years until
\citet{WilsonKogut1974} explained it. Problem~3 cites the pair together
deliberately, because the gap between them is the paper's point: a
procedure that works long before it is understood is a signal that a
structure is being used without being seen, and the framework's claim is
that the structure is the interval. First cited in \S\ref{intro:p3}.

\bibitem['t Hooft and Veltman(1972)]{tHooftVeltman1972}
\textbf{G.~'t~Hooft and M.~Veltman.} \emph{Regularization and
renormalization of gauge fields}, Nucl.\ Phys.\ B \textbf{44} (1972)
189.
\par\noindent\textbf{Relevance.}
Dimensional regularization and the renormalization of gauge theories --- the
technical achievement that made the electroweak model calculable and
therefore believable, and the reason \citet{Weinberg1967} moved from
obscurity to ubiquity. It is cited in Problem~3 as the completion of the
program \citet{Dyson1949} began, and as a specific illustration of the
point Appendix~C's Step~1 develops: the regulator has to respect the gauge
structure or the anomaly cancellation is destroyed by the arithmetic of the
calculation rather than by physics. That the consistent regulator exists,
and that anomaly coefficients come out as integers regardless of how the
calculation is organized, is the fact Problem~3 reads as circle-side
bookkeeping. First cited in \S\ref{intro:p3}.

\bibitem[Elitzur(1975)]{Elitzur1975}
\textbf{S.~Elitzur.} \emph{Impossibility of spontaneously breaking
local symmetries}, Phys.\ Rev.\ D \textbf{12} (1975) 3978.
\par\noindent\textbf{Relevance.}
The theorem that a local gauge symmetry cannot be spontaneously broken:
gauge-non-invariant order parameters have vanishing expectation values,
and what the Higgs mechanism does is not break a symmetry but remove a
redundancy in a particular gauge. The result is standard, uncontroversial,
and routinely ignored in the language physicists actually use, which is
why Problem~3 gives it a section of its own. On this paper's reading
Elitzur's theorem is a corollary of a more basic statement --- what a
measurement can return is a holonomy, and a holonomy is gauge invariant by
construction --- so the impossibility is not a technical obstruction but a
statement about what is written on the circle and what is merely a chart.
First cited in \S\ref{intro:p3}.

\bibitem[Particle Data Group(2024)]{PDG2024}
\textbf{S.~Navas et al.\ (Particle Data Group).} \emph{Review of
Particle Physics}, Phys.\ Rev.\ D \textbf{110} (2024) 030001.
\par\noindent\textbf{Relevance.}
The Review of Particle Physics: the reference against which every
particle-sector number in this paper is checked, and the source of the
parameter inventory Problem~4 is about. It is cited for the masses, the
mixing matrices, the coupling constants at their quoted scales and
schemes, and the experimental limits that make the falsifier table's kill
conditions concrete. The Review also supplies a discipline this paper
tries to keep: every quoted value carries a scheme, a scale, and an
uncertainty, and a comparison that drops any of the three is not a
comparison. Appendix~D's Step~1 spends several pages on exactly this
point before any closed form is compared to anything. First cited in
\S\ref{intro:p4}.

\bibitem[Cahn(1996)]{Cahn1996}
\textbf{R.~N.~Cahn.} \emph{The eighteen arbitrary parameters of the
standard model in your everyday life}, Rev.\ Mod.\ Phys.\
\textbf{68} (1996) 951.
\par\noindent\textbf{Relevance.}
The essay that made the parameter problem vivid by asking what the
Standard Model's arbitrary numbers actually do --- which of them set the
size of an atom, the lifetime of a star, the stability of a nucleus ---
and showing how narrowly the world depends on values the theory cannot
compute. It is cited in Problem~4 as the clearest statement of why the
count matters: the numbers are not bookkeeping, they are the settings on
which everything else rests, and a framework that computes them is making
a claim of a different kind from one that fits them. The count itself is
convention-dependent, which is why Appendix~D's Step~1 works through the
$36 - (27 - 1) = 10$ theorem and the full sector-by-sector tally before
the codebook comparison is attempted. First cited in \S\ref{intro:p4}.

\bibitem[Lema\^itre(1931)]{Lemaitre1931}
\textbf{G.~Lema\^itre.} \emph{The beginning of the world from the
point of view of quantum theory}, Nature \textbf{127} (1931) 706.
\par\noindent\textbf{Relevance.}
The primeval-atom proposal: the universe began in a single quantum of
enormous energy whose disintegration produced everything else --- the first
statement that the cosmos has a beginning that physics might describe
rather than assume. Problem~5 cites it as the origin of its question and
as an early instance of the pattern the paper keeps noticing: a structural
proposal, made before any instrument could address it, that turned out to
be answerable. Lemaitre's own caution --- that the beginning would need a
quantum treatment --- is the position the framework still occupies, with
the entrance corridor supplied as a work order rather than as a completed
derivation. First cited in \S\ref{intro:p5}.

\bibitem[Hawking and Penrose(1970)]{HawkingPenrose1970}
\textbf{S.~W.~Hawking and R.~Penrose.} \emph{The singularities of
gravitational collapse and cosmology}, Proc.\ R.\ Soc.\ A
\textbf{314} (1970) 529.
\par\noindent\textbf{Relevance.}
The singularity theorems in their most general form: given reasonable
energy conditions and a trapped surface or its cosmological analogue,
geodesic incompleteness is unavoidable, and no amount of symmetry-breaking
cleverness in the initial data will remove it. The theorems are cited in
Problem~5 as the reason the beginning is a problem of principle rather
than of modeling --- classical general relativity predicts its own
failure --- and as the standing constraint on any proposal about what
preceded the descent. This paper's position is that the incompleteness is
real and that the interval's endpoint is where it lives, which converts
the singularity from a breakdown into a boundary with a work order
attached. First cited in \S\ref{intro:p5}.

\bibitem[Borde et al.(2003)]{BordeGuthVilenkin2003}
\textbf{A.~Borde, A.~H.~Guth, and A.~Vilenkin.} \emph{Inflationary
spacetimes are incomplete in past directions}, Phys.\ Rev.\ Lett.\
\textbf{90} (2003) 151301.
\par\noindent\textbf{Relevance.}
The theorem that inflationary spacetimes are past-incomplete: an expanding
region, however long it inflates, cannot be extended indefinitely into the
past along every geodesic. It removed the most popular escape from the
beginning --- eternal inflation as a way of never having to start --- and
is cited in Problem~5 for exactly that reason. The result matters to the
anatomy because it says something about the shape of the scale direction:
a coordinate along which the physical evolution runs one way and cannot be
continued backwards indefinitely is an interval with an endpoint, not a
line. What the framework adds is that the endpoint is the arena's, not an
artifact of the matter content. First cited in \S\ref{intro:p5}.

\bibitem[Penrose(1979)]{Penrose1979}
\textbf{R.~Penrose.} \emph{Singularities and time-asymmetry}, in
\emph{General Relativity: An Einstein Centenary Survey}, Cambridge
(1979).
\par\noindent\textbf{Relevance.}
The Weyl curvature hypothesis: the universe began in a state of
extraordinarily low gravitational entropy, and the improbability of that
state --- Penrose's celebrated estimate is one part in $10^{10^{123}}$ ---
is the real content of the second law. It is the sharpest available
statement of why the arrow of time is a cosmological problem rather than a
statistical one, and it is why Problem~13 cannot be answered by
coarse-graining arguments alone. This paper's response is structural
rather than statistical: if the physical composition along the descent has
no inverse, the arrow does not need a past hypothesis to explain it, and
Appendix~I's Step~3 shows explicitly why Loschmidt's and Zermelo's
objections both fail against a semigroup. First cited in
\S\ref{intro:p5}.

\bibitem[Dirac(1937)]{Dirac1937}
\textbf{P.~A.~M.~Dirac.} \emph{The cosmological constants}, Nature
\textbf{139} (1937) 323.
\par\noindent\textbf{Relevance.}
The large-numbers hypothesis: Dirac's observation that several
dimensionless ratios built from the fundamental constants come out near
$10^{40}$ or its powers, and his conjecture that the coincidences reflect
a law rather than an accident --- with the consequence, since excluded,
that $G_N$ varies with cosmic time. It is cited in Problem~6 as the
ancestor of every hierarchy argument and as an object lesson in both
directions: the instinct that unexplained large ratios demand explanation
was right, and the specific mechanism proposed for it was wrong and was
killed by measurement. The cohort's own large-number claims are held to
the standard that episode set, which is why $M_{\rm lock} =
\frac{1}{2}M_{\rm Pl}e^{-K}$ is stated with a resonance to look for
rather than as an interpretive flourish. First cited in \S\ref{intro:p6}.

\bibitem[Susskind(1979)]{Susskind1979}
\textbf{L.~Susskind.} \emph{Dynamics of spontaneous symmetry
breaking in the Weinberg--Salam theory}, Phys.\ Rev.\ D \textbf{20}
(1979) 2619.
\par\noindent\textbf{Relevance.}
The paper that stated the gauge hierarchy problem in the form the field has
worried about ever since: the Higgs mass receives quadratically divergent
corrections, so keeping the electroweak scale far below the Planck scale
requires a cancellation to many decimal places unless some structure
protects it. Technicolor was the proposed structure; supersymmetry became
the more popular one; neither has been found. Problem~6 cites it as the
statement of the debt and then declines both standard payments: on this
paper's reading the loop's reach is bounded by the same warp factor
$e^{-K}$ that sets the lock scale, so $\delta m_H^2/m_H^2$ is of order
unity and no cancellation is invoked --- at the price of a completion that
must exist at $3.849$~TeV. First cited in \S\ref{intro:p6}.

\bibitem['t Hooft(1980)]{tHooft1980}
\textbf{G.~'t~Hooft.} \emph{Naturalness, chiral symmetry, and
spontaneous chiral symmetry breaking}, in \emph{Recent Developments
in Gauge Theories}, NATO ASI Series B \textbf{59} (1980) 135.
\par\noindent\textbf{Relevance.}
The naturalness criterion: a small dimensionless parameter is natural if
setting it to zero enhances the symmetry of the theory, because then the
smallness is protected against radiative corrections by that symmetry. It
is the most influential single idea in model-building since the Standard
Model was completed, and its predictions for the LHC did not come true,
which is why Problem~6 treats it with care rather than dismissal. This
paper's position is that the criterion is correct and that its symmetry
clause is too narrow: protection can also come from topology, with no
symmetry anywhere \citep{HopkinsSinger2005}, and the circle side of the
anatomy is exactly where that second kind of protection lives. First
cited in \S\ref{intro:p6}. First cited in \S\ref{sec:cat2}.

\bibitem[Zwicky(1933)]{Zwicky1933}
\textbf{F.~Zwicky.} \emph{Die Rotverschiebung von extragalaktischen
Nebeln}, Helv.\ Phys.\ Acta \textbf{6} (1933) 110.
\par\noindent\textbf{Relevance.}
The Coma cluster virial analysis: the galaxies are moving too fast for the
luminous mass to hold them, so most of the mass is dark. The observation
was ignored for four decades, which makes it the field's standard example
of a correct result arriving before the community was ready to use it. It
is cited in Problem~7 as the beginning of the dark-matter question and as
a reminder that the evidence for the dark sector is dynamical and
gravitational --- the sector's existence is far better established than any
proposal for what it is made of, which is why the framework's Kaxion is
stated as a line to look for rather than as a claim already banked. First
cited in \S\ref{intro:p7}. First cited in \S\ref{sec:cat2}.

\bibitem[Rubin and Ford(1970)]{RubinFord1970}
\textbf{V.~C.~Rubin and W.~K.~Ford.} \emph{Rotation of the Andromeda
nebula from a spectroscopic survey of emission regions},
Astrophys.\ J.\ \textbf{159} (1970) 379.
\par\noindent\textbf{Relevance.}
The spectroscopic rotation curve of the Andromeda nebula: emission-region
velocities that stay flat far outside the luminous disc instead of falling
as Kepler requires. It is one galaxy, studied carefully, and the survey of
many galaxies that made the result decisive came later in the decade --- a
distinction this paper keeps because its own reproducibility standard
demands it. Problem~7 cites it as the observation that made dark matter
unavoidable rather than merely available, and as the second half of the
pairing with \citet{Zwicky1933}: cluster dynamics and galactic dynamics,
two independent scales, one conclusion. First cited in
\S\ref{intro:p7}.

\bibitem[Weinberg(1978)]{Weinberg1978}
\textbf{S.~Weinberg.} \emph{A new light boson?}, Phys.\ Rev.\ Lett.\
\textbf{40} (1978) 223.
\par\noindent\textbf{Relevance.}
One of the two papers that recognized the consequence of the Peccei--Quinn
proposal: a spontaneously broken global symmetry leaves a light
pseudoscalar behind, and the strong CP solution therefore predicts a new
particle whose mass and couplings are fixed by the symmetry-breaking
scale. The axion has been searched for ever since, and the search is now
the most sensitive dark-matter program running in the microwave band.
Problem~7 cites this and \citet{Wilczek1978} for the mechanism the cohort's
Kaxion inherits, with the one modification that matters: the breaking
scale is not chosen but computed as $f_K = f_{\rm bulk}/\Xi_{\rm cap}$,
which turns a family of candidates into a line. First cited in
\S\ref{intro:p7}.

\bibitem[Wilczek(1978)]{Wilczek1978}
\textbf{F.~Wilczek.} \emph{Problem of strong $P$ and $T$ invariance
in the presence of instantons}, Phys.\ Rev.\ Lett.\ \textbf{40}
(1978) 279.
\par\noindent\textbf{Relevance.}
The independent identification of the axion, published alongside
\citet{Weinberg1978} and cited together with it throughout Problem~7. The
paper's framing --- that the strong $P$ and $T$ problem is solved by a
dynamical field rather than by a tuned angle --- is the template the
cohort follows, and it is also the historical origin of the naming
convention the framework's own particle inherits. What the pair leaves
open is the decay constant, and that is exactly the gap the Kaxion's
$\Xi_{\rm cap} = K + N + L = 43$ closes, at the cost of a benchmark loading
that the honest ledger lists as an open integer with a discrete search
window rather than as a derived value. First cited in \S\ref{intro:p7}.

\bibitem[Misner(1968)]{Misner1968}
\textbf{C.~W.~Misner.} \emph{The isotropy of the universe},
Astrophys.\ J.\ \textbf{151} (1968) 431.
\par\noindent\textbf{Relevance.}
The paper that made the horizon problem quantitative: the microwave
background is isotropic to a part in $10^5$ across regions that, in a
decelerating universe, were never in causal contact, and Misner's
mixmaster proposal was the first serious attempt to explain the agreement
dynamically rather than assume it. The attempt failed and the problem
survived, to be answered a decade later by inflation. Problem~8 cites it
as the statement of the puzzle in its cleanest pre-inflationary form,
because the puzzle is what the framework's corridor has to solve: distant
strangers agree because they were compiled from the same bulk data along
a descent that had one traverse, not because a mixing dynamics equalized
them. First cited in \S\ref{intro:p8}.

\bibitem[Lee and Yang(1956)]{LeeYang1956}
\textbf{T.~D.~Lee and C.~N.~Yang.} \emph{Question of parity
conservation in weak interactions}, Phys.\ Rev.\ \textbf{104} (1956)
254.
\par\noindent\textbf{Relevance.}
The paper that noticed nobody had ever tested parity in the weak
interaction, surveyed the existing data to confirm the gap, and proposed
the experiments that would close it. It is the field's best example of a
symmetry assumed for so long that its assumption became invisible, and
Problem~9 opens with it for that reason: the mirrors were taken to be
exact because no one had looked. On this paper's reading the episode is
also a lesson in what kind of thing a discrete symmetry is --- an
orientation the geometry either admits or does not, rather than a
principle the world is obliged to respect --- and the anatomy's business
is to say which orientations exist and which fiber carries each. First
cited in \S\ref{intro:p9}. First cited in \S\ref{sec:cat3}.

\bibitem[Wu et al.(1957)]{Wu1957}
\textbf{C.~S.~Wu, E.~Ambler, R.~W.~Hayward, D.~D.~Hoppes, and
R.~P.~Hudson.} \emph{Experimental test of parity conservation in
beta decay}, Phys.\ Rev.\ \textbf{105} (1957) 1413.
\par\noindent\textbf{Relevance.}
The cobalt-60 experiment: polarized nuclei emitting beta electrons
preferentially against their spin direction, in maximal violation of
parity. The result arrived within months of the proposal and overturned an
assumption that had organized physics for decades. Problem~9 cites it as
the first of the three mirrors to break, and Appendix~G's Step~1 works
through the bilinear transformation table that makes the violation
precise. What the anatomy adds is a reason the breaking is maximal rather
than partial: the chirality that parity exchanges is a label on the
compact fiber, and a projection that keeps one chirality and discards the
other is what an interval does to a circle-side pair. First cited in
\S\ref{intro:p9}.

\bibitem[Christenson et al.(1964)]{Christenson1964}
\textbf{J.~H.~Christenson, J.~W.~Cronin, V.~L.~Fitch, and
R.~Turlay.} \emph{Evidence for the $2\pi$ decay of the $K_2^0$
meson}, Phys.\ Rev.\ Lett.\ \textbf{13} (1964) 138.
\par\noindent\textbf{Relevance.}
The discovery of CP violation in neutral kaon decay: a two-pion final
state at the level of $2 \times 10^{-3}$, where CP conservation forbids it
entirely. The result was unwanted, small, and correct, and it made the
matter--antimatter asymmetry of the universe a question physics could
address rather than a brute fact. Problem~9 cites it as the second mirror
and Problems~10--11 cite it as the enabling condition for the second
Sakharov requirement. Its size is also part of the difficulty: CP
violation exists, it is measured, and \citet{Gavela1994} showed that the
amount available in the quark sector is nowhere near enough. First cited
in \S\ref{intro:p9}. First cited in \S\ref{sec:cat3}.

\bibitem[L\"uders(1954)]{Luders1954}
\textbf{G.~L\"uders.} \emph{On the equivalence of invariance under
time reversal and under particle-antiparticle conjugation for
relativistic field theories}, Kgl.\ Danske Vid.\ Selskab, Mat.-fys.\
Medd.\ \textbf{28}, no.~5 (1954) 1.
\par\noindent\textbf{Relevance.}
The CPT theorem: any local, Lorentz-invariant quantum field theory with a
Hermitian Hamiltonian is invariant under the combined operation, whatever
happens to C, P, and T individually. It is the most robust discrete
statement in physics --- the equality of particle and antiparticle masses
and lifetimes has been checked to parts in $10^{18}$ --- and it is the fact
Problem~9 has to explain rather than assume. The anatomy's answer is that
the three individual operations are orientation reversals a bundle may or
may not admit, while their product is the total orientation of the whole
six-dimensional structure, which the geometry cannot reverse without
ceasing to be itself. Appendix~G's Step~1 gives the standard proof in
outline; Step~3 gives the geometric restatement. First cited in
\S\ref{intro:p9}.

\bibitem['t Hooft(1976)]{tHooft1976}
\textbf{G.~'t~Hooft.} \emph{Symmetry breaking through Bell--Jackiw
anomalies}, Phys.\ Rev.\ Lett.\ \textbf{37} (1976) 8.
\par\noindent\textbf{Relevance.}
The paper that showed the QCD vacuum has a topological structure --- the
$\theta$ angle --- and that baryon and lepton number are violated by the
electroweak anomaly, both consequences of instanton configurations that
perturbation theory cannot see. It creates two of this paper's problems at
once. The $\theta$ angle creates the strong CP problem of Problem~9,
because nothing in QCD explains why the measured value is smaller than
$10^{-10}$ \citep{Abel2020}. The anomalous $B + L$ violation supplies the
first Sakharov condition of Problems~10--11 by way of sphalerons
\citep{KRS1985}. Both are circle-side statements in the anatomy: winding
numbers, integer-valued, immune to the running that the interval governs.
First cited in \S\ref{intro:p9}.

\bibitem[Abel et al.(2020)]{Abel2020}
\textbf{C.~Abel et al.} \emph{Measurement of the permanent electric
dipole moment of the neutron}, Phys.\ Rev.\ Lett.\ \textbf{124}
(2020) 081803.
\par\noindent\textbf{Relevance.}
The current best limit on the neutron's permanent electric dipole moment,
$|d_n| < 1.8 \times 10^{-26}\,e\,$cm at $90\%$ confidence --- the
measurement that bounds the QCD vacuum angle below $10^{-10}$ and thereby
makes the strong CP problem quantitative. It stands here as the
model of what this paper means by an instrument closing a question: a structural puzzle about a parameter that
theory permits to be of order one, converted by an experiment into a
ten-order-of-magnitude gap that has to be explained by something. The
cohort's response is Problem~9's color-triangle account, and the same
instrument is one of the falsifier table's live kills. First cited in
\S\ref{sec:howtoread}.

\bibitem[Dirac(1928)]{Dirac1928}
\textbf{P.~A.~M.~Dirac.} \emph{The quantum theory of the electron},
Proc.\ R.\ Soc.\ A \textbf{117} (1928) 610.
\par\noindent\textbf{Relevance.}
The relativistic wave equation for the electron, which produced spin and
the correct magnetic moment as consequences rather than inputs, and left
negative-energy solutions that Dirac initially and incorrectly identified
with protons. The correction --- a new particle of the same mass and
opposite charge --- came in 1931 \citep{Dirac1931}, and the particle was
found in 1933 \citep{Anderson1933}. Problems~10--11 cite the sequence
because it is the cleanest instance in the record of the pattern this
paper is claiming for itself: a structure forced by consistency, resisted
because its consequence seemed absurd, and confirmed by measurement within
five years. First cited in \S\ref{intro:p1011}.

\bibitem[Anderson(1933)]{Anderson1933}
\textbf{C.~D.~Anderson.} \emph{The positive electron}, Phys.\ Rev.\
\textbf{43} (1933) 491.
\par\noindent\textbf{Relevance.}
The cloud-chamber photograph that established the positron: a track of
electron mass curving the wrong way in a magnetic field. Antimatter
stopped being an interpretive difficulty and became a particle. Problems
10--11 cite it as the completion of the Dirac sequence and as the
beginning of the asymmetry question in its modern form --- if the laws
produce matter and antimatter on equal terms, and the early universe was
hot enough to make both, the survival of one and not the other is
something that has to have happened rather than something that can be
assumed. First cited in \S\ref{intro:p1011}.

\bibitem[Kuzmin et al.(1985)]{KRS1985}
\textbf{V.~A.~Kuzmin, V.~A.~Rubakov, and M.~E.~Shaposhnikov.}
\emph{On anomalous electroweak baryon-number non-conservation in
the early universe}, Phys.\ Lett.\ B \textbf{155} (1985) 36.
\par\noindent\textbf{Relevance.}
Electroweak sphalerons: the demonstration that anomalous baryon-number
violation, exponentially suppressed at zero temperature, becomes rapid at
temperatures above the electroweak scale, so that $B + L$ is efficiently
erased in the early universe while $B - L$ survives. It is the fact that
makes leptogenesis work --- a lepton asymmetry generated at high scale is
partially converted to baryon number by the same process --- and therefore
the load-bearing input to Problems~10--11. Appendix~G's Step~1 works
through the barrier energetics; the anatomy reads the sphaleron as a
circle-side transition between vacua labeled by winding number, executed
because the interval's descent supplied the thermal energy to cross the
barrier. First cited in \S\ref{intro:p1011}.

\bibitem[Gell-Mann(1964)]{GellMann1964}
\textbf{M.~Gell-Mann.} \emph{A schematic model of baryons and
mesons}, Phys.\ Lett.\ \textbf{8} (1964) 214.
\par\noindent\textbf{Relevance.}
The quark model: hadrons as composites of three fractionally charged
constituents, proposed as a classification scheme with an explicit
agnosticism about whether the constituents were real. They were, and their
non-appearance as free particles became the confinement problem of
Problem~12. This paper cites the proposal for the shape of the puzzle it
created: a theory whose fundamental fields are never observed in
isolation, which forces a distinction between what the theory contains and
what an experiment can return --- exactly the distinction the anatomy
draws between the description and the holonomy that is readable from it.
First cited in \S\ref{intro:p12}.

\bibitem[Greenberg(1964)]{Greenberg1964}
\textbf{O.~W.~Greenberg.} \emph{Spin and unitary-spin independence
in a paraquark model of baryons and mesons}, Phys.\ Rev.\ Lett.\
\textbf{13} (1964) 598.
\par\noindent\textbf{Relevance.}
The observation that the quark model's baryon wave functions were
symmetric where Fermi statistics required antisymmetry, and that a hidden
threefold degree of freedom would fix it --- parastatistics then, color
now. It is the origin of the number three in the strong interaction, and
Problem~12's why-three-colors section is written against it. The
framework's claim is that the same integer arrives twice by routes with no
shared step: as the $\mathbb{Z}_3$ center of the winding structure and as
the count that \citet{ArisakaP2N3}'s open-system bound returns, which is the kind of
overdetermination this paper treats as evidence rather than as
decoration. First cited in \S\ref{intro:p12}.

\bibitem[Jaffe and Witten(2006)]{JaffeWitten2006}
\textbf{A.~Jaffe and E.~Witten.} \emph{Quantum Yang--Mills theory},
in \emph{The Millennium Prize Problems}, Clay Mathematics Institute
(2006).
\par\noindent\textbf{Relevance.}
The Clay Millennium Prize description of the Yang--Mills problem, which is
worth stating precisely because it is routinely misquoted: the prize is
for constructing a rigorous quantum Yang--Mills theory on
$\mathbb{R}^4$ with a positive mass gap, not for proving confinement.
Problem~12 cites it for that precision and takes the standard seriously:
the framework's account of confinement is graded \emph{reread}, not
\emph{solved}, because a geometric reading of why the registration rule
holds is not a proof of the mass gap and the paper says so in both the
master table and the honest ledger. First cited in \S\ref{intro:p12}.

\bibitem[Takenaka et al.(2020)]{SuperK2020}
\textbf{A.~Takenaka et al.\ (Super-Kamiokande).} \emph{Search for
proton decay via $p \to e^+\pi^0$ and $p \to \mu^+\pi^0$ with an
enlarged fiducial volume}, Phys.\ Rev.\ D \textbf{102} (2020)
112011.
\par\noindent\textbf{Relevance.}
The Super-Kamiokande proton-decay search with the enlarged fiducial volume,
setting partial lifetime limits beyond $10^{34}$ years for the $e^+\pi^0$
and $\mu^+\pi^0$ channels --- the measurement that killed minimal $SU(5)$
\citep{GeorgiGlashow1974} and constrains every grand unified successor.
Problem~12 cites it as the standing bound the framework's double guard must
clear, and the falsifier table lists proton decay as a live kill: the
cohort's two integrality conditions predict a lifetime, and the next
generation of detectors will test it. This is one of the entries that
makes the framework killable on a decade timescale rather than in
principle. First cited in \S\ref{intro:p12}.

\bibitem[Boltzmann(1877)]{Boltzmann1877}
\textbf{L.~Boltzmann.} \emph{\"Uber die Beziehung zwischen dem
zweiten Hauptsatze der mechanischen W\"armetheorie und der
Wahrscheinlichkeitsrechnung}, Wien.\ Ber.\ \textbf{76} (1877) 373.
\par\noindent\textbf{Relevance.}
The paper that gave entropy its probabilistic meaning --- $S = k \log W$,
the logarithm of the number of microstates compatible with a macrostate ---
and with it the first mechanical account of the second law. It also
created the difficulty Problem~13 exists to address, since the underlying
mechanics is time-symmetric and the conclusion is not: Loschmidt's
reversibility objection and Zermelo's recurrence objection are both
correct as stated against the original derivation. Appendix~I's Step~1
works through the H-theorem with both objections in place, and Step~3
shows why they fail against a composition law that has no inverse --- which
is the anatomy's reason for the arrow, requiring no past hypothesis to
supplement it. First cited in \S\ref{intro:p13}.

\bibitem[Eddington(1928)]{Eddington1928}
\textbf{A.~S.~Eddington.} \emph{The Nature of the Physical World},
Cambridge University Press (1928).
\par\noindent\textbf{Relevance.}
The book that named the arrow of time and stated its status with a
bluntness that has not been improved on: the second law holds, in
Eddington's judgment, the supreme position among the laws of nature.
Problem~13 cites it for the naming and for the framing --- that the
one-wayness of time is not a detail of thermodynamics but the feature of
the world most in need of a foundation. This paper's answer places the
one-wayness in the arena rather than in the initial conditions or the
statistics, which is a stronger claim than the standard one and is graded
accordingly. First cited in \S\ref{intro:p13}.

\bibitem[DeWitt(1967)]{DeWitt1967}
\textbf{B.~S.~DeWitt.} \emph{Quantum theory of gravity. I. The
canonical theory}, Phys.\ Rev.\ \textbf{160} (1967) 1113.
\par\noindent\textbf{Relevance.}
The canonical quantization of general relativity, and the origin of the
problem of time: the Hamiltonian constraint annihilates the wave function
of the universe, so the fundamental equation contains no time parameter at
all. Half a century of proposals --- relational time, thermal time,
emergent time --- have not produced consensus, which is why Problem~13
treats the question as open in the standard framework rather than as
awkward. The cohort's position is that the tick is derived rather than
postulated: $E = h\nu$ read as an angular velocity on the compact fiber
makes the rate of a clock a geometric quantity, and Appendix~I's Step~3
carries out the derivation. First cited in \S\ref{intro:p13}.

\bibitem[Feynman et al.(1963)]{Feynman1963}
\textbf{R.~P.~Feynman, R.~B.~Leighton, and M.~Sands.} \emph{The
Feynman Lectures on Physics}, Vol.~I, Addison--Wesley (1963).
\par\noindent\textbf{Relevance.}
The Lectures, cited for one sentence and one lesson. The sentence is
Feynman's admission that physics has no knowledge of what energy is --- it
computes a number that does not change, and the constancy is the whole of
what is known. The lesson is Problem~14's: the three most-used quantities
in physics, energy, mass, and entropy, are defined operationally and
understood structurally not at all, which is a strange state of affairs for
concepts so central. This paper's answers are geometric --- energy as the
circle's angular rate, mass as the cost of the interval's position, entropy
as the count of what the composition discarded --- and they are offered in
the register Feynman used, as an attempt to say what the quantity is rather
than how to compute it. First cited in \S\ref{intro:p14}.

\bibitem[Bekenstein(1973)]{Bekenstein1973}
\textbf{J.~D.~Bekenstein.} \emph{Black holes and entropy}, Phys.\
Rev.\ D \textbf{7} (1973) 2333.
\par\noindent\textbf{Relevance.}
The proposal that a black hole carries entropy proportional to its horizon
area, forced by the requirement that the second law survive an object
falling in. It is the beginning of black-hole thermodynamics and of the
holographic instinct that has organized the subject since, and its
strangeness is the point Problem~16 begins from: the entropy scales with
area rather than volume, so the information capacity of a region is set by
its boundary. Problem~14 cites it as the third of its three quantities
made geometric, and Appendix~I's Step~1 derives $S_{\rm BH}$ from the
Schwarzschild data as a worked problem so that a student can see the
factor of four arrive. First cited in \S\ref{intro:p14}.

\bibitem[Hawking(1975)]{Hawking1975}
\textbf{S.~W.~Hawking.} \emph{Particle creation by black holes},
Commun.\ Math.\ Phys.\ \textbf{43} (1975) 199.
\par\noindent\textbf{Relevance.}
The calculation that made black-hole thermodynamics literal rather than
analogical: a black hole radiates with a thermal spectrum at temperature
$T_H = \hbar c^3/8\pi G_N M k_B$, so the horizon entropy is entropy and the
surface gravity is temperature. The result is the single most important
input to Problem~16, and its consequence --- that the hole evaporates ---
is what turns an accounting curiosity into a paradox. Appendix~I's Step~1
sketches the Bogoliubov derivation and Step~3 asks what the framework has
to say about a formula that spends $\hbar$, $G_N$, and $k_B$ in one
expression: on the anatomy's reading no single fiber can produce it,
which is why the horizon is where both fibers are visible at once. First
cited in \S\ref{intro:p16}. First cited in \S\ref{sec:cat4}.

\bibitem[Hawking(1976)]{Hawking1976}
\textbf{S.~W.~Hawking.} \emph{Breakdown of predictability in
gravitational collapse}, Phys.\ Rev.\ D \textbf{14} (1976) 2460.
\par\noindent\textbf{Relevance.}
The paper that stated the information paradox: if a black hole forms from a
pure state and evaporates completely into thermal radiation, unitarity
fails, and quantum mechanics as we have it is wrong somewhere. Hawking's
own conclusion --- that predictability breaks down --- was resisted for
thirty years and eventually withdrawn, but the argument's structure has
never been faulted, only answered. Problem~16 cites it as the statement of
the debt and reads the resolution as a billing error rather than a new
physical process: the circle keeps what the hole must return, the interval
prices the return, and the two books reconcile on the schedule
\citet{Page1993} computed. First cited in \S\ref{intro:p16}.

\bibitem[Page(1993)]{Page1993}
\textbf{D.~N.~Page.} \emph{Information in black hole radiation},
Phys.\ Rev.\ Lett.\ \textbf{71} (1993) 3743.
\par\noindent\textbf{Relevance.}
The Page curve: if evaporation is unitary, the entanglement entropy of the
radiation must rise until roughly half the hole's entropy has been emitted
and then fall to zero, and the turning point is fixed by counting alone.
It is the most useful single constraint in the subject because it is
kinematic --- any unitary account must reproduce it, whatever the
mechanism --- which makes it a test rather than a model. Problem~16 uses
it as the reconciliation schedule of the two books and Appendix~I's Step~1
shows why the curve is forced rather than derived from dynamics. The
recent island computations \citep{Penington2020} recover it, and this paper
places that achievement intact rather than in competition. First cited in
\S\ref{intro:p16}.

\bibitem[Susskind et al.(1993)]{STU1993}
\textbf{L.~Susskind, L.~Thorlacius, and J.~Uglum.} \emph{The
stretched horizon and black hole complementarity}, Phys.\ Rev.\ D
\textbf{48} (1993) 3743.
\par\noindent\textbf{Relevance.}
Black hole complementarity: the proposal that the infalling and distant
descriptions are both correct and never comparable, because no single
observer can check them against each other --- a consistency argument
rather than a mechanism. It held the field for two decades until
\citet{AMPS2013} showed that the postulates it combined were mutually
inconsistent. Problem~16 cites it as the most sophisticated version of the
move this paper declines: rather than declaring two descriptions
incomparable, the framework says which fiber each one reads, and the two
books reconcile in a single accounting instead of being kept apart. First
cited in \S\ref{intro:p16}. First cited in \S\ref{sec:cat4}.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{J.~M.~Maldacena.} \emph{The large-$N$ limit of superconformal
field theories and supergravity}, Adv.\ Theor.\ Math.\ Phys.\
\textbf{2} (1998) 231.
\par\noindent\textbf{Relevance.}
The AdS/CFT correspondence: a gravitational theory in the bulk of anti-de
Sitter space is equivalent to a conformal field theory on its boundary,
with the radial coordinate of the bulk identified with the renormalization
scale of the boundary theory. It is the most successful bulk--boundary
dictionary in physics and the closest structural relative of this
framework's own projection, which is why Part~VII treats the comparison
explicitly rather than in passing. The differences are stated as plainly
as the debt: this bulk is six-dimensional with two fibers of different
topology rather than asymptotically AdS, the boundary is our spacetime
rather than a fictitious screen, and the radial--scale identification is a
theorem here rather than a conjecture supported by evidence. First cited
in \S\ref{intro:p16}. First cited in \S\ref{sec:cat4}.

\bibitem[Strominger and Vafa(1996)]{StromingerVafa1996}
\textbf{A.~Strominger and C.~Vafa.} \emph{Microscopic origin of the
Bekenstein--Hawking entropy}, Phys.\ Lett.\ B \textbf{379} (1996) 99.
\par\noindent\textbf{Relevance.}
The exact statistical-mechanical count of the microstates of an
extremal black hole, reproducing the Bekenstein--Hawking area law with
no adjustable factor. It is cited in \S\ref{ssec:w2-steelman} on the
opposing program's behalf: whatever one concludes about the
landscape, this result stands, and it is the strongest existing
evidence that horizon entropy counts something rather than measuring
an observer's ignorance. Problem~16's two-book reading has to be
consistent with it, and is. First cited in \S\ref{ssec:w2-steelman}.

\bibitem[Almheiri et al.(2013)]{AMPS2013}
\textbf{A.~Almheiri, D.~Marolf, J.~Polchinski, and J.~Sully.}
\emph{Black holes: complementarity or firewalls?}, JHEP \textbf{02}
(2013) 062.
\par\noindent\textbf{Relevance.}
The firewall argument: black hole complementarity, the equivalence
principle, and unitary evaporation cannot all hold, and something dramatic
must fail at the horizon of an old black hole. It is the sharpest
statement of what the information paradox actually costs, and it is why
Problem~16 cannot be treated as settled by the island computations alone.
This paper's account claims no firewall, because on the two-fiber reading
nothing needs to be destroyed at the horizon --- the information is on the
circle's books throughout, and the interval's descent controls only when
it is returned. That is a strong claim and it is stated as one, with the
grade attached. First cited in \S\ref{intro:p16}.

\bibitem[Penington(2020)]{Penington2020}
\textbf{G.~Penington.} \emph{Entanglement wedge reconstruction and
the information paradox}, JHEP \textbf{09} (2020) 002.
\par\noindent\textbf{Relevance.}
The entanglement-wedge computation that recovered the Page curve from a
gravitational path integral, with the quantum extremal surface jumping at
the Page time --- the result that, with its companions, convinced the field
that Hawking radiation is unitary after all. It is cited in Problem~16 with
its achievement intact: the calculation is the strongest evidence available
that the information comes out, and this framework's account has to agree
with it rather than replace it. What the paper adds is a reading of
\emph{why} the entropy turns over --- two sets of books with different
closure properties, reconciling --- and a claim that the island's
appearance is what that reconciliation looks like in the gravitational
description. First cited in \S\ref{intro:p16}.

\bibitem[Weinberg(1987)]{Weinberg1987}
\textbf{S.~Weinberg.} \emph{Anthropic bound on the cosmological
constant}, Phys.\ Rev.\ Lett.\ \textbf{59} (1987) 2607.
\par\noindent\textbf{Relevance.}
The anthropic bound on the cosmological constant, and the entry most often
misreported in the literature. Weinberg argued that if $\Lambda$ were
larger than a certain value, structure would never form and no observers
would exist, and he derived an \emph{upper bound} that lies roughly two
orders of magnitude above the value subsequently measured. That is a
successful bound, not a successful prediction, and the distinction matters
to Problem~15's whole argument: an anthropic constraint explains why we do
not observe a much larger value, and says nothing about why we observe the
value we do. This paper's alternative is to compute the number instead ---
the residual scale $m_\Lambda = 6.1$~meV from the lock and the reduced
Planck mass --- and to accept the exposure that computing it creates.
First cited in \S\ref{intro:p15}.

\bibitem[Bousso and Polchinski(2000)]{BoussoPolchinski2000}
\textbf{R.~Bousso and J.~Polchinski.} \emph{Quantization of
four-form fluxes and dynamical neutralization of the cosmological
constant}, JHEP \textbf{06} (2000) 006.
\par\noindent\textbf{Relevance.}
The mechanism that turned anthropic reasoning from a philosophical stance
into a concrete construction: many four-form fluxes, each quantized, whose
combinations produce an enormous discretuum of vacuum energies dense enough
that some member sits near the observed value. It is the technical
foundation of the landscape, and Problem~15 cites it as the strongest
version of the position the paper argues against. The disagreement is
structural and admits no compromise: a framework whose census permits
exactly one bulk of exactly this shape has no room to accommodate a
different answer, so it can be killed by a single measurement, whereas a
discretuum absorbs any result. Which of those is a virtue is the question
Problem~15 puts to the reader. First cited in \S\ref{intro:p15}.

\bibitem[Susskind(2003)]{Susskind2003}
\textbf{L.~Susskind.} \emph{The anthropic landscape of string
theory}, arXiv:hep-th/0302219 (2003).
\par\noindent\textbf{Relevance.}
The essay that named the anthropic landscape and argued that string
theory's enormous vacuum count, combined with eternal inflation, makes
environmental selection the correct explanation for the constants rather
than a failure of explanation. It is the clearest statement of the position
Problem~15 is written against, and this paper takes it seriously enough to
state its strength: if the constants really are environmental, then a
framework that computes them is attempting something that cannot be done,
and the framework will fail. The two positions make different bets and
both are exposed --- which is why Part~VII stakes this one on measurements
with dates rather than on argument. First cited in \S\ref{intro:p15}.

\bibitem[Wigner(1960)]{Wigner1960}
\textbf{E.~P.~Wigner.} \emph{The unreasonable effectiveness of
mathematics in the natural sciences}, Comm.\ Pure Appl.\ Math.\
\textbf{13} (1960) 1.
\par\noindent\textbf{Relevance.}
The essay on the unreasonable effectiveness of mathematics in the natural
sciences: Wigner's observation that the fit between mathematical structures
invented for their own sake and the physical world is a gift we neither
understand nor deserve. It stands behind this paper's
central claim, which is a specific instance of Wigner's puzzle turned
around --- if the effectiveness is unreasonable when mathematics is applied
to physics from outside, it is expected when the physics \emph{is} the
geometry and the integers are its topology. Whether that dissolves Wigner's
question or merely relocates it is a matter this paper raises honestly and
does not claim to settle; Part~X returns to it among the curious reader's
questions. First cited in \S\ref{sec:claim}.

\end{thebibliography}
\endgroup

\end{document}

