%% 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.
%% ===================================================================
%% ===================================================================
%% v15.10 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 7 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.
%% ====================================================================
%% ===================================================================
%% v15.9 (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 HAD NO AVAILABILITY LINE AT ALL.  A
%%     reader was told only "Arisaka-GG-A11-Origin (May 20, 2026)", with nothing said about where
%%     the paper can be found.  It now reads "In preparation.  To be
%%     posted ... on completion; no result of the present paper depends
%%     on it."
%% (2) THE COHORT-TABLE CAPTION PROMISED "where all preprints are
%%     posted".  Twenty-one will be; A11-Origin is held.  Reworded to
%%     match the A2-GG6 caption repaired in FP-2 and the A9/A10 captions
%%     repaired in FP-4.
%% (3) NINE FURTHER COMPANION BIBITEMS CARRIED STALE DATES, the same
%%     defect A8-QM, A10-Cosmos and P4-Real carried.  All nine now carry
%%     the availability line the other papers use.
%% (4) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  91 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.
%%     C2 CARRIED 91 OF THE 138 PRINTED SITES LEFT IN THE COHORT -- two
%%     thirds of the remaining debt in one paper, because the Hubble
%%     literature it surveys is itself a catalog of proposed solutions.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.8 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * no showcase-headline macro: its showcase pages predate
%%   the pass and set their own headlines inline.
%% * \GGovsub removed, and with it the wrapper that saved
%%   and re-called the style file's Overview just to blank
%%   it.  \GGovsub is the THIRD payload of the .toc-replay
%%   defect: a Part subtitle wanted in the Contents and not
%%   in the one-line Overview entry.  The style file blanks
%%   it inside the Overview, so the wrapper has no work.
%%
%% The style file is v11.5 (2026/08/17).  Every one of the twenty-four
%% MANIFEST.sha256 files covers it, so changing it meant rebuilding and
%% repackaging all twenty-four -- which is why this was deferred five
%% times and is why it was done in one pass.
%%
%% ASSERTED, NOT ASSUMED: this paper's printed text is line for line
% what v15.7 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.7 (August 17, 2026) -- THE -ise FAMILY.
%%
%% neutraliser -> neutralizer (1).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.5 -> v15.6, August 16, 2026.  TABLE 7 BROUGHT INSIDE THE
%% TEXT BLOCK.  Report 389 flagged this page as running past the
%% right margin and said it had not been diagnosed.  Diagnosed:
%%   Table 7, the CMB observables, measures 484.18796 pt boxed at
%%   the body's own settings, against a text block of 469.75502 --
%%   14.43 pt too wide, and the log's pre-existing 14.43294 pt
%%   overfull box is that number to five figures.  Two cells in the
%%   last column carry two-valued Pull entries and are what widens
%%   it: on the artifact "4.3" reaches x = 551.6 and "0.3" reaches
%%   554.4, against a block ending at 541.75.
%%   NOT COHORT-WIDE: scanning the whole artifact for ink past
%%   543 pt returns two pages -- this one, and page 223 at 543.1,
%%   which is a 1.4 pt rule end and is left alone.
%%   GUTTER ONLY: six columns with @{} at both ends leave five
%%   internal gaps of 2 x \tabcolsep, so 6 pt -> 4.5 pt buys 15 pt
%%   and the table lands at 469.19.  No column, row, cell or number
%%   is touched.
%% NEW PROSE IN THIS SHIP: none.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.4 -> v15.5, August 16, 2026.  THE EXPOSURE PAGE, AND ONE
%% MARGIN OVERRUN.  Thirteenth of the twenty-four.
%%   THE PAGE.  C2's three showcase pages say where the measured
%%   values sit, what the derived curve looks like, and how coherent
%%   the offset is; none says what would end the paper.  Added:
%%   Table 23, the paper in ten numbers -- value, grade and a
%%   'Falsified by' column -- the direct analogue of the page
%%   C1-Kaxion received the same day, so the two C-papers read
%%   alike.  Unnumbered, so Tables 1-32 keep their numbers.
%%   THE OVERRUN.  Six \begin{minipage}{\textwidth} in this paper
%%   and ONE of them bare -- A11's defect, here, in the body.
%%   Measured rather than assumed: \parindent at that site is
%%   16.33182 pt and the log carries exactly one overfull \hbox of
%%   16.33182 pt.  The page is a full-width H0-history plate that
%%   had been hanging one indent into the right margin.  Note that
%%   \parindent is 0 pt further into the body, which is why only
%%   this one site was affected.
%%   STAGE 0, ALL THREE CHECKS: fourteen \includegraphics assets
%%   all present with zero 'not found'; zero pre-existing saveboxes,
%%   so Report 385's leading defect was never here; and the one bare
%%   minipage above.  The box added for the new page is still filled
%%   at the body's measured spacing.
%%   ONE THING LEARNED, AND MY FIRST EXPLANATION OF IT WAS WRONG.
%%   The savebox for this page was first called \GGscTenC2.  A
%%   control sequence is letters only, so TeX read \GGscTenC and
%%   left the '2' as a stray token: '! A <box> was supposed to be
%%   here', and the whole table then set as literal text in a
%%   29.5 pt overfull line reading '(-20) 2 2'.  I first blamed
%%   tabularx and switched to the lrbox form, which also failed --
%%   because the name was still wrong.  Tested afterwards with a
%%   letters-only name: \sbox and lrbox produce byte-identical
%%   boxes, 91.39174 pt both.  The form was never the problem.
%%   \GGscTenCTwo, and \sbox as everywhere else in this pass.
%% NEW PROSE IN THIS SHIP: one headline and one showcase caption,
%% both on the new page.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ===================================================================
%%  C2-Hubble v13.73 (August 12, 2026) -- the CMB-S4 rider (Report 307,
%%  author-authorised; the last of the cohort instrument sweep after A10's
%%  29 sites and W2's 10).  CMB-S4 was ended by DOE and NSF on 9 July 2025.
%%  Five of C2's seven body sites name it as a forthcoming instrument and
%%  take the cohort replacement (the Simons Observatory beside LiteBIRD).
%%  The other two are dated statements about its FUNDING, written while the
%%  outcome was open ("hostage to the recovery of its 2024-25 funding
%%  collapse"; "if its funding recovers"); a name swap there would leave a
%%  false conditional standing, so both are rewritten to record the
%%  outcome.  One historical mention of the name is retained, by design,
%%  in the 2030s paragraph.  No prediction, no falsifier, no number changed.
%% ===================================================================
%% ===================================================================
%%  C2-Hubble v13.72  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  3 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.
%% ===================================================================
%% ===================================================================
%%  C2-Hubble v13.71  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 103mm boxes and then
%%  left at natural size, so its \footnotesize landed 16% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.619\textwidth, computed rather than chosen: the picture measures
%%  113.9mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  C2-Hubble v13.70  (August 5, 2026)  -- the twenty-two-paper cohort sweep.
%%  The cohort map becomes the twenty-two-paper map: GG-P6-Proton on
%%  the Predictions row with a second sister-import arrow, and the
%%  White-Paper Branch (GG-W1-Symmetry above GG-W2-Holonomy) as a
%%  right-flank column at the mirror positions of the Foundations
%%  column -- the left flank feeds the spine, the right reads it.
%%  GG-P7-Inflow does not appear: it is Appendix H of GG-P2-N3.
%%  The plate is also enlarged for reading: boxes and the type inside
%%  them x1.15, every word outside a box x1.20, and the long
%%  A10 -> C-frame edge redrawn as a polyline, which returns 24mm of
%%  bounding box that was Bezier control points rather than ink.
%%  Prose: every count site, the series breakdown, and the canonical
%%  Suffix list, which was missing Proton, Symmetry and Holonomy.
%%  Recipe and variant analysis: [v13-support]/sweep22.py.
%% ===================================================================
%% ===================================================================
%%  C2-Hubble v13.68  (July 30, 2026)  -- internal-affairs conformance.
%%  C2's objections Part uses the cohort's staged multi-round audit DEVICE --
%%  legitimate, and used by 18 of the 20 papers.  C2 alone also carried SIX
%%  HISTORY-CLAIMS: that the rounds really happened and really forced changes
%%  to this text.  The Report-76 internal-affairs sweep missed them because
%%  its detector had no review-process vocabulary, and C2 was a declared
%%  reference implementation.  Removed:
%%    'when the amendments the first two rounds forced were in place'
%%    'After the concessions and repairs the first two exchanges forced'
%%    'the books were opened ... this entry records one that has been closed'
%%    "the reviewer's prediction is printed here"
%%    'the amendments survived the same panel that forced them'
%%    'There will be data before there is a fourth round'
%%  The DEVICE stays (six personas, round 2, the third exchange, the
%%  five-reviewer panel), recast in the present tense as a staged audit
%%  rather than a history.  Every concession, number, stake and falsifier is
%%  preserved verbatim; O-Hubble-75--79 and OQ-Hubble-4 untouched.
%% ===================================================================
%% ===================================================================
%%  C2-Hubble v13.67  (July 29, 2026)  -- P2-N3 retitle propagation.
%%  P2-N3 is retitled "Why Three Generations: A THREE-LEG Derivation of N = 3
%%  from BI-6, MSMF, and GDOS Morse Stability" (was "A Closed Derivation").
%%  Reason: P2-N3's own main theorem T-N3-5 is "Conditional derivation of
%%  N = 3 (three-leg pincer)", conditional on five structural hypotheses --
%%  including the BI-6 Budget Conjecture, which A2 states is not derived
%%  anywhere in the cohort -- plus two empirical inputs.  The cohort's prose
%%  already flagged that conditioning everywhere; only the title claimed
%%  "Closed", and titles propagate into every bibliography.
%%  T-N3-5 KEEPS its own name: the theorem is conditional and says so.
%%   * COHORT ROSTER: P2 row "A closed derivation of N=3 fermion
%%     generations" -> "A three-leg derivation ..."; third column "the
%%     closed prediction of the fermion generation count" -> "the three-leg
%%     derivation of ...".
%%   * COHORT ROSTER: P3 row "the five oldest electron questions" -> "six",
%%     matching P3's title page and C2's own bibitem.
%% ===================================================================
%% ===================================================================
%% v13.66 (July 24, 2026): COHORT BIB-TITLE RECONCILIATION (P3)
%%  The electron companion's subtitle is updated in its bibliography
%%  entry to that paper's current cover ("The Six Oldest Questions
%%  about the Electron ..."), following its addition of the sixth
%%  question -- the origin of positrons. No other change
%%  anywhere.
%% ===================================================================
%% =====================================================================
%% v13.65 (2026-07-19)  WAVE 1 COHORT PROMOTION (nineteen-paper form)
%%   + STANDARDIZED PART-I CLOSE.  (1) Part I restructured: the old
%%   'Roadmap of the Parts' and 'The GG-Theory cohort' sections merged
%%   into the closing Section 'The cohort inheritance and the roadmap
%%   of the twelve Parts' (house pattern: A10-Cosmos v13.3 /
%%   A11-Origin v13.1 / P1-KK v13.5); ALL labels preserved
%%   (sec:gg-cohort, sec:roadmap-c2, fig:cohort-map-c2, fig:roadmap-c2,
%%   tab:cohort-roster).  (2) The paper-local seventeen-paper cohort
%%   figure RETIRED, replaced by the frozen v1.4 nineteen-paper cohort
%%   map (verbatim from A2-GG6 v13.3; MP4/MP5 internal node names; only
%%   the \label is local, and the print scale is 0.93\textwidth to
%%   fit C2's letter geometry).  (3) NEW Foundations Branch paragraph
%%   (P4-Real the origin-of-i member; P5-CPT the mirrors-and-matter
%%   member, its four-endowment definition of time kin to the corridor
%%   chronology; roster level, no new claims) + two roster rows; roster
%%   caption and all cohort-sense counts seventeen -> nineteen (physics
%%   counts PROTECTED: the seventeen-observable audit, 'sixteen distant
%%   supernovae', 'nineteen decades', all corridor values 66.6/67.4,
%%   z_sw/z_dec language untouched).  (4) \PaperPFour/\PaperPFive
%%   provided; \ASix renewed to the canonical GG-A6-GRIP form;
%%   bibitems ArisakaP4Real + ArisakaP5CPT added (house style,
%%   roster-level Relevance, first-cite sec:gg-cohort).  (5) Roadmap
%%   subsection: NEW five-paragraph walk of the twelve Parts + the
%%   roadmap figure recaptioned 'Roadmap of the twelve Parts of
%%   GG-C2-Hubble' (tikz body unchanged; Parts III-IV highlighted: the
%%   two-Hubble-constants corridor and its reconciliation).  (6) One
%%   register repair in the zeta_B dictionary entry.  v13.64 -> v13.65.
%%   Script: c2_wave1.py.
%% =====================================================================
%% =====================================================================
%% v13.64 (2026-07-16)  TWO-QUANTITY Z* SWEEP (cross-paper F7 item of
%%   the A10 z* hostile audit, author-approved).  C2 predated A10's
%%   atomic-chain derivation of z_dec; sites moved to the two-quantity
%%   form: (1) prop:recomb-switch (T-Hubble-2) retitled "recombination-
%%   era" + atomic-decoupling closing sentence; (2) the post-proposition
%%   visibility-window/centroid "both pulls" paragraph REPLACED by the
%%   two-quantity statement (switch topological; z_dec derived by A10's
%%   atomic chain, zero parameters, 0.1%; offset computed delta=0.0041;
%%   the retired identification e^7=z_rec would miss by 27 sigma, which
%%   is why neither paper makes it); (3) tab:c2-cmb-core: z* row
%%   relabelled corridor switch, the +/-3.7%-window pull (+27/+0.2)
%%   RETIRED -> "two-qty" (caption convention note updated, z_* removed
%%   from the tolerance list); (4) OGN dial-table row "Recombination
%%   z_*" -> "Corridor switch z_*" + stale 1089.9+/-0.3 -> 1089.92
%%   +/-0.25; (5) Landmark-3 paragraph + landmarks-table row annotated
%%   two-quantity; (6) two prose mentions ("the recombination
%%   redshift", "the recombination epoch") -> corridor-switch wording;
%%   (7) grade-summary falsified-by cell "(postdiction; consistency
%%   check)" -> the F7 kill on the observable (z_dec off the atomic
%%   chain by >0.5%).  v13.63 -> v13.64.
%%   Script: c2_v13.64_two_quantity_zstar.py.
%% =====================================================================
%% v13.63 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%% v13.62 (2026-07-14, Phase 5 OGN-5 + {e,pi} sweep) -- presentation only.
%%   Front-matter "maximum symmetry, minimum freedom" paragraph:
%%   (a) the five OGN-5 integers stated as FORCED, each with a fixed physical
%%       meaning (M=1 tensor seam, N=3 colour triplicity, L=5 SU(5) block,
%%       J=7 odd closure index, K=35=LJ throat depth whose warp e^{-K} fixes
%%       H_0 -- the C2 hook);
%%   (b) the {e,pi} dictionary -- pi = angle of the compact gauge dimension,
%%       e = base of the exponential scale dimension.
%%   Anti-hostile-reviewer emphasis (author directive 2026-07-14). No values
%%   changed. Script: c2_ogn_epi.py.
%% v13.61 (2026-07-13, arXiv final touch) -- Figure 10 to full text
%%   width on one page: budget-pair asset compressed 15.2 -> 12.6 in
%%   (aspect 1.111, tight bbox; c2_fig_omega_budget.py), unit widened
%%   to \textwidth, caption trimmed ~13 -> ~7 lines. Scripts:
%%   c2_v13.61_F3.py + c2_fig_omega_budget.py.
%% v13.60 (2026-07-13, arXiv final touches) -- landscape showcase
%%   asset re-pitched (0.144 in/row, tight bbox) so Figures 1 and 20
%%   print at full text width, one page each; p5 cap 0.90 -> 0.95th
%%   (non-binding); Preface trimmed four lines to end on its page.
%%   Scripts: c2_v13.60_F2.py + c2_showcase_landscape.py.
%% v13.59 (2026-07-13, arXiv pre-flight) -- final-flip widow fix:
%%   the Scope paragraph trimmed two lines so its tail no longer
%%   strands alone before the budget-figure page. Script: c2_v13.59_F1.py.
%% v13.58 (2026-07-13, P5) -- Phase 5: second five-reviewer exchange
%%   run against v13.57 (verdicts: ~90% RESOLVED; 'cosmetics expected,
%%   surgery found'); residual findings fixed: stale dual-verdict
%%   caption; shallow-bright rule WRITTEN with teeth + shell steps;
%%   5-sigma flat-shell kill rung added to F-2; 'Canonical (adopted)'
%%   -> counts-branch conditional + flows parallel at the verdict; zeta
%%   grammar FROZEN w/ ledger-tower clause; r_now chart-offset note;
%%   dark-siren two-point-extrapolation honesty; corridors
%%   disambiguated; 'precision reading' -> 'adopted centre' in Part
%%   III; no-history scrubs (x3). THE THIRD EXCHANGE printed as
%%   sec:third-exchange with O-Hubble-75..79; counts swept 74 -> 79
%%   (Preface, sceptic path, Part-X summary, Part-XI intro, roadmap
%%   node '79 answered, three hostile rounds'). Plus 3 residual
%%   'December 2, 2026' dates fixed. Script: c2_v13.58_P5.py.
%% v13.57 (2026-07-13, P4) -- Phase 4 wide sweep: M2 raw/adjusted pulls
%%   (z* +27/+0.2, theta* -16/-4.3, l_D +0.3/+0.3) + convention caption
%%   + width-vs-centroid concession; M7 6.2->7.1 sigma, CCHP variant
%%   span, band-2 shared-data tags, SPT-3G balanced w/ band spread +
%%   DESI pull, Href-drift adverse condition registered; M10 Jia scoped
%%   to low z + third caution (high-z tail adverse at F-5) x4 sites;
%%   M11 F-7 window units x2; minors: track-record 8-9 sigma fact,
%%   H0DN expanded + desk entry, grad reading path, Course C two steps
%%   displayed, rho>=0 argued, r_s 0.6-Mpc ledger, assigned->derived,
%%   A10 pointer sharpened, career fine-print paragraph.
%%   Script: c2_v13.57_P4.py.
%% v13.56 (2026-07-13, P3) -- Phase 3 = B4, the CMB register pass:
%%   the ~12sigma exclusion OWNED in-voice in sec:cmb-chi2 ('consistent
%%   at the percent level; excluded as exact; falsifiable in one move');
%%   the coherence QUANTIFIED (2^-17 sign test + effective-dof caveat,
%%   necessary-not-sufficient, OQ-4 owed); degeneracy claim -> 'largely
%%   along + printed transverse residue' at both sites; Omega_m h^3
%%   confrontation printed (0.0940 vs 0.09633+-0.00029, -2.4%, pull -8,
%%   'the single sharpest miss in this paper', decomposed); abstract
%%   'squarely' -> 'afterglow side of the split, 1.6sigma from the CMB
%%   combination'; ES consistency-not-fit clause; 'nothing tuned to the
%%   sky' scoped to background parameters at 6 sites incl. fig:cmb-tt
%%   caption + Preface. Numbers asserted in-script: c2_v13.56_P3.py.
%% v13.55 (2026-07-13, P2) -- Phase 2 of the audit response: B3 one
%%   siren decision function at F-Hubble-2 (defends the background;
%%   middle world 68-71.5 pre-declared; lift assigned to shallow bright
%%   subsets; confirmation-shared clause printed; 'killing both'
%%   subordinated; three-worlds ~70 re-scoped; deep 21/58 + shallow
%%   30-40 both printed; budget + mass-spectrum + Local-Hole rows).
%%   M3 tiering ('eleven rules; four gallows') + F-8/9/10 gap rules.
%%   M4 tensors -> the 2030s; Chen 2018 retired for dark-siren dating
%%   (empirical 0.74^k walk); Gaia sqrt(t) x1.4 + bright-star floor;
%%   Dec 2 -> December 2026 (4 sites incl. timeline card). M9 zeta
%%   enumeration: 1/(2L^2) + 1/56 priced as rows (1/56 lands on 66.6
%%   and is STILL rejected); single-occupancy + no-empirical-filter
%%   clauses printed. Numbers asserted in-script: c2_v13.55_P2.py.
%% v13.54 (2026-07-13, P1) -- Phase 1 of the author-approved audit
%%   response: B1 combination honesty (three-route story, IVW 66.69
%%   printed, 66.6 = adopted centre; Preface corridor; O-68 carve-out);
%%   B2 lift re-scope (derived response / disputed depth; own ~11-sigma
%%   charged; 68.5 -> conditional lifted reading; O-24 concedes and
%%   withdraws the calibrator-zone escape); B5 V13 corrected + dust
%%   scoped to SN share + chained 69.5 -> ~70.6-per-reanalysis w/
%%   overlap stated + 1.7-sigma-equivalent retired; M1 r_d = 147.5 on
%%   the derived background spent in all BAO sentences; M5 radial
%%   reconciliation (E_GUT = M_Pl e^-2pi, lock at r = 29.4, r_now =
%%   66.7 -- verified against A2/A5); M6 epoch bookkeeping (0.315
%%   coincidence recorded, not claimed); M8 meV = unit identity +
%%   OQ-Hubble-9 (projection sourcing) + OQ-Hubble-10 (epoch relation).
%%   All numbers asserted in-script. Script: c2_v13.54_P1a.py.
%% v13.53 (2026-07-13, S13) -- final editorial audit before external
%%   review. Checker clean; Part codas/openers, section transitions,
%%   counts, style register all verified (details in c2_v13.53_S13.py
%%   docstring). Two fixes: Part-III coda 67.4+-0.4 -> +-0.5 (canonical);
%%   'six-persona hostile review' -> 'seven-persona' (the panel has
%%   seven readers; both rounds carry seven personas).
%% v13.52 (2026-07-13, S12) -- author refinements before sign-off:
%%   (1) Scope paragraph moved before the omega-budget page unit (fills
%%   the short page after the six steps); stray \clearpage after
%%   fig:a-of-t removed. (2) GLOBAL sixteen->seventeen sweep for the
%%   coherent pi-offset audit (11 prose sites; hard one-percent
%%   quantifiers -> percent scale; O-Hubble-31 'Fourteen sub-percent' ->
%%   'Thirteen', recounted from the canonical devs in-script); the
%%   historical 'sixteen distant supernovae' untouched.
%%   Script: c2_v13.52_S12.py.
%% v13.51 (2026-07-13, S11) -- author layout final touches: Preface
%%   trimmed to end on its page; the six key-export equations replaced
%%   by the C1 track-record table (tab:c2-track-record; theta-bar and
%%   tau_p kept in prose; eq:sin2gg/eq:alphagg were never referenced);
%%   fig:hopf -> [!t] so the six-steps section starts under it;
%%   fig:omega-budget caption joined to its figure on one page (0.72
%%   textheight); gratuitous \clearpage after fig:landmarks and after
%%   fig:cmb-offsets removed (the two stranded near-empty pages
%%   vanish); the 'sixteen deviations, none beyond 1.3%' sentence
%%   synced to the seventeen-row chart (Omega_b +2.76% named honestly).
%%   Script: c2_v13.51_S11.py.
%% v13.50 (2026-07-13, S4) -- author request (after the A10 five-figure
%%   showcase): short numbered captions added to the three showcase pages
%%   5-7, A10 style (minipage + captionof, one-line LoF short captions,
%%   cross-references to the full in-paper versions). p5 landscape capped
%%   at 0.90\textheight for its two-line caption; p6-p7 unchanged sizes.
%%   Showcase figures become Figures 1-3 (later numbers shift +3; no
%%   hardcoded figure numbers -- grep-verified). Script: c2_v13.50_S4.py.
%% v13.49 (2026-07-13, S2) -- author refinements on the v13.48 showcase pages:
%%   p5 landscape: CMB-comb label moved below the band chips, x-tick +
%%   axis-label fonts +20%; asset now ALSO serves as in-paper Figure 17
%%   (caption notes the GG-predictions top category; script name synced).
%%   p6: top title -30% and sans-serif. p7 offsets: Omega_b = 1/2pi^2
%%   added as 2nd inventory row (pred 0.05066 / obs 0.0493, +2.76%, the
%%   2.3-sigma row already printed in Part III) -> 'seventeen shadows';
%%   left label column unclipped (the '6.6' was a clipped '66.6'); legend
%%   moved below the chart; asset now ALSO serves as in-paper Figure 13
%%   (caption count synced). Scripts: c2_v13.49_S2.py +
%%   c2_showcase_landscape.py + c2_showcase_offsets.py.
%% v13.48 (2026-07-13) -- S1: three SHOWCASE pages (5-7) inserted
%%   after the keywords, before the Overview (the A10 pp5-6 device):
%%   p5 the 2026 landscape SHOWCASE (GG PREDICTIONS category on top:
%%   66.6/66.8/68.5 as rows; title +20% bold, line space added);
%%   p6 the derived H(t) two-panel figure w/ large bold LaTeX top
%%   title, no caption; p7 the pi-offset Dev chart SHOWCASE (fonts
%%   +20%; right-side annotations moved INSIDE-LEFT; x-range +-0.08;
%%   title enlarged further + bold; legend -> lower right; left
%%   margin widened after a clipping preview), no caption. In-paper
%%   Figures 10/13/17 and their captions UNTOUCHED; showcase assets
%%   are separate variants (c2_h0_landscape_showcase.pdf,
%%   c2_cmb_offsets_showcase.pdf; generators c2_showcase_*.py).
%%   Script: c2_v13.48_S1.py.
%% v13.47 (2026-07-13) -- F1: front-matter facelift (author request).
%%   THREE KEY DEVELOPMENTS foregrounded:
%%   (1) H(t) derived directly; H0 a snapshot; the analytic closed
%%       form DISPLAYED IN THE EXECUTIVE SUMMARY (three constants,
%%       two ledger integers, pi and e; 'checkable on a calculator
%%       in under a minute'); >10 LambdaCDM numbers re-emerging
%%       unfitted at the percent level.
%%   (2) The 7-sigma as ONE NETWORK'S INTERNAL CERTIFICATION, not a
%%       field consensus: the 45-determination landscape spans 67-73
%%       in three bands with the middle filling.
%%   (3) The Hubble-constant concept RETIRED (molecular-weights /
%%       c / Rydberg / W-Z lineage); onward to falsifiable beyond-
%%       LCDM precision cosmology ('a dozen dated tests in place of
%%       one contested decimal').
%%   ABSTRACT: minor rewrite weaving all three (length asserted
%%   <= 1920; 'what remains to be tested is not a number but a
%%   law'). EXECUTIVE SUMMARY: major facelift, now <= 3 pages incl.
%%   keywords (through p4): 'Three results organise this report'
%%   opener w/ the boxed-form display; condensed core paragraphs;
%%   NEW 'Beyond the constant' paragraph; philosophical close drawn
%%   from Part IX (retirement lineage; 'we are not the audience of
%%   the expansion; we are the part of it that has begun to keep
%%   the log') ending on the kept 'Both outcomes are progress. Only
%%   one of them is possible.' Keywords updated (+derived H(t),
%%   closed-form H0, 2026 landscape, decision calendar, beyond-LCDM
%%   precision cosmology). Script: c2_v13.47_F1.py.
%% v13.46 (2026-07-13) -- Z1: Part XII (References) polished to the
%%   A10 standard.
%%   (1) \GGBibBegin replaced by the three-paragraph reader's
%%       introduction (A10's cover text refined to the Hubble paper:
%%       relevance-notes purpose; two-readers paragraph tracing
%%       Leavitt -> Hubble -> factor-two wars -> 7 sigma -> the 2026
%%       landscape -> the GW referee -> the cohort; canonical-keys
%%       paragraph) + \renewcommand{bibsection} -- fills the empty
%%       Part-opener p237 and removes the duplicate 'References'
%%       heading, aligning C2 with the cohort's frozen standard.
%%   (2) Charter A/B structure: \section*{A} before its OWN
%%       thebibliography block; Part B in its own block.
%%   (3) All 16 cohort heads replaced by A10's canonical labels/
%%       titles/ids/dates (extracted from A10 at build time; C2's
%%       Relevance notes kept); ArisakaA10Cosmos head rebuilt in the
%%       canonical pattern; stale 'UCLA-Arisaka/GG-2026-N' ids,
%%       '(2026a..i)' author-years, and URL lines retired.
%%   (4) arXiv IDs appended to the 14 landscape-table observation
%%       entries that lacked them; ALL 39 unique table IDs now
%%       present in the bibliography (asserted).
%%   (5) The 12 short Relevance notes expanded (Blas/CLASS,
%%       DiValentino census, Freedman-Madore TRGB, Hubble 1925/1929,
%%       Perlmutter, Pietrzynski LMC, Riess 2019 '4 sigma waypoint',
%%       Said FP, Sandage-Tammann 'factor-two war', Schmidt 1998,
%%       Vogl EPM).
%%   (6) Part subtitle refined ('one-line' dropped). Structure,
%%       stale-form, ID-coverage, and orphan audits asserted in-
%%       script. Script: c2_v13.46_Z1.py.
%% v13.45 (2026-07-13) -- author request: Part XI reformatted to the
%%   A10-Cosmos Q&A layout. objqabody environment (leftskip 1.8em,
%%   parindent 0; mirrored from A10's preamble) added; all 74
%%   responses wrapped (count asserted); all objection headers given
%%   \noindent so the questions sit flush left and the answers are
%%   indented for easy reading. No wording changed anywhere.
%%   Script: c2_v13.45_qaformat.py.
%% v13.44 (2026-07-13) -- Y1: Part XI polish after the III-X
%%   campaign.
%%   (1) NEW opening section 'For the curious reader: the big
%%       questions, plainly answered' (sec:obj-curious): TWENTY new
%%       Q&As, O-Hubble-55..74 (registry continues; tags number the
%%       ledger, not the reading order). A10's curious-reader device
%%       adapted in C2 voice: expanding-into-what (balloon/ants);
%%       do-atoms-expand (10^-18/s vs 10^33 restoring); surely-a-
%%       mistake (common-mode theorem for civilians); faster-than-
%%       light; why-'constant'; was-Einstein-wrong ('not by one
%%       comma'); 95%-dark; pi-numerology (forced-not-searched;
%%       'numerology never volunteers a way to die'); the 10^120
%%       (8K e-folds); why-now (dinner-party factor-of-two); extra
%%       dimensions ('cornered, not plausible'); fine-tuning (third
%%       road: no knob); conjectures-collapse ('documents its own
%%       demolition points'); exact-yet-error-bars (theorem vs
%%       comparison currencies); why-telescopes ('the jury'); why-
%%       sirens; middle-landing; how-it-ends ('retires on a fixed
%%       pension of 55 km/s/Mpc'); is-this-accepted (honest: 'No,
%%       it is not accepted -- it is a proposal under test');
%%       what-would-convince-you (eleven dates and numbers).
%%   (2) Part-XI subtitle + intro rewritten (opens for the broad
%%       reader, then hardens; seventy-four; numbering note).
%%   (3) O-Hubble-49 response + closed-form pointer.
%%   (4) Counts 54 -> 74 swept (preface warning, reading path,
%%       References intro, roadmap node XI). Letter budget 26 OK;
%%       tag range verified contiguous 1..74 in-script.
%%   Script: c2_v13.44_Y1.py.
%% v13.43 (2026-07-13) -- X5: Part X rebuild, FINAL ship (5/5).
%%   (1) Course L 'The statistics of a tension' (app:courseL): sigma
%%       bookkeeping w/ systematics; PRECISION-AMPLIFIES-BIAS theorem
%%       (T ~ b/sigma; 'the decade's paradox, derived'); the BRACKET
%%       THEOREM (quadrature/stack = sharp bounds over unknown rho);
%%       GLS consensus algebra + common-mode transmission ('certifies
%%       the covariance it models'); bands-not-rows; the paper's
%%       statistical posture (prereg trials=1 w/ C1 precedent cited;
%%       bracket pricing; symmetric deflation; released arithmetic;
%%       'the cure for a tension is never another digit').
%%   (2) Thermal course MOVED into Course K as its annex (label
%%       app:thermal kept; KolbTurner + A10 pointers added); K's
%%       forward pointer fixed.
%%   (3) Repro block retitled 'Reproducibility and the released
%%       pipeline'; registries -> 'Registries, audits, and roadmaps'
%%       + NEW internal source-ledger table (tab:c2-source-ledger) +
%%       NEW compact falsification catalogue; Notation section FOLDED
%%       into registries (app:blockF label kept on the subsection).
%%   (4) THE WELCOME: 'How to use these appendices' guide after
%%       \appendix (three-step contract, entry points, released
%%       scripts, A10 as companion volume, 'the coffee is on the
%%       left') + course-map table (tab:course-map).
%%   (5) FIVE new classic bibitems w/ Relevance + first-cite notes
%%       (Etherington 1933; Weinberg 1989 RMP; Kolb & Turner 1990;
%%       Maggiore 2008; Meszaros 1974), woven with ~15 in-line
%%       citations through Courses C-L (incl. existing Leavitt,
%%       Hubble 1929, Friedmann/Lemaitre, Riess/Perlmutter 1998/99,
%%       Buchert 2000, Schutz, Dodelson, Weinberg 2008, Freedman-
%%       Madore, Brout, Jimenez-Loeb/Moresco, Sandage/Loeb) and A10
%%       Part-pointers at the GG steps.
%%   (6) Roadmap node X -> 'the courses C--L'. Letter budget verified
%%       (appendix+objections sections <= 26).
%%   Script: c2_v13.43_X5.py.
%% v13.42 (2026-07-13) -- X4: Part X rebuild, ship 4/5. Three full
%%   three-step courses inserted before the repro block:
%%   I 'Growth of structure and S8' (app:courseI): conformal-
%%     Newtonian perturbations -> continuity/Euler/Poisson -> the
%%     growth equation (gravity as negative stiffness) -> era-by-era
%%     solutions (delta ~ a derived via the p(p-1)+4p/3=2/3 indicial
%%     equation; Meszaros stall; vacuum freeze; f = Om^0.55 = 0.533)
%%     -> sigma8/S8 integrals (top-hat window) -> S8-tension
%%     limitations -> GG step ABSORBS app:s8 verbatim (labels kept)
%%     + three course-level remarks (cleaner-instrument pattern;
%%     f*sigma8(z) rigidity; growth sign open per OQ-Hubble-5).
%%   J 'Standard sirens from the quadrupole formula up'
%%     (app:courseJ): linearized GR -> quadrupole (monopole/dipole
%%     forbidden, G/c^4 stiffness intuition) -> chirp mass from
%%     phasing (eq:app-cJ-chirp; Mc = 1.18 Msun for 1.36+1.36) ->
%%     amplitude-distance ('the logic closes like a trap') ->
%%     GW170817 end-to-end (1.188 Msun; ~40 Mpc; 3017+-166;
%%     70 +12/-8 = 3017/43.1) -> limitations DERIVED (d-iota
%%     degeneracy from h+/hx polarizations; calibration floor as the
%%     one common mode; dark-siren floors 'visible in the wild') ->
%%     GG forecast algebra re-derived (N_3sig = 21, N_5sig = 58;
%%     three-worlds pricing recomputable).
%%   K 'Time, age, and the curve-reading channels' (app:courseK):
%%     the arcsinh closed form DERIVED (u = a^{3/2} substitution;
%%     Course C's quote now earned) + early/late limits; lookback
%%     ladder (5.90/8.02 at z=1; 3.30 at z=2 -> 'no object at z=2
%%     older than 3.3 Gyr'); chronometer + drift relations derived
%%     from first principles w/ the sign-change intuition (z~1.9);
%%     one-number-cosmology limitations ('exposure schedule'); the
%%     chronology-as-commitments recital closing the course arc
%%     C..K. Corridor-arithmetic holding section DISSOLVED (S8 was
%%     its last resident). All numbers asserted in-script (drift
%%     rows, lookbacks, Mc, N-thresholds, sign change).
%%   Script: c2_v13.42_X4.py.
%% v13.41 (2026-07-13) -- X3: Part X rebuild, ship 3/5. Three full
%%   three-step courses inserted after Course E:
%%   F 'The vacuum energy of the universe and the seesaw'
%%     (app:courseF): the zero-point integral computed (Lambda^4/16pi^2),
%%     the 10^120.2 in two steps; seesaw structures ([[0,m],[m,M]]
%%     eigenvalues derived; 'the vacuum points at the TeV scale');
%%     limitations (SUSY cancels too much/too little, quintessence
%%     relocates, anthropics); GG step ABSORBS app:seesaw verbatim +
%%     course-level promotion (reduced-unit bookkeeping to the closed
%%     form; 8K dissolution of the 10^120; vev-pin error algebra).
%%   G 'The local universe: peculiar velocities, the Hole, and the
%%     lift' (app:courseG): linear PV from continuity (f = 0.533);
%%     THE MASTER FORMULA delta H/H = -(f/3) delta-bar = -0.178
%%     delta-bar (eq:app-cG-master, boxed) -- from ONE LINE both the
%%     +2-3% lift (depths -0.11..-0.17) AND the -0.41 deep-void
%%     requirement AND the shell-decay signature; measured-Hole +
%%     velocity-audit step; limitations (percent-class, early-side-
%%     blind, profile-falsifiable); GG step ABSORBS app:epslocal
%%     verbatim (graviphoton channel computed-and-excluded).
%%   H 'Backreaction and averaging' (app:courseH): Buchert commutation
%%     rule DERIVED from the volume element; Q_D defined w/ the
%%     variance-minus-shear intuition; Q_D/H^2 = 0.4 f^2 sigma^2 and
%%     deltaH_D/H = -Q/(12H^2) evaluated (2e-4 at 40 Mpc/h); three
%%     walls (systematic 1e-3 / stochastic 2.7-9.3-17 sigma /
%%     timescape priced); GG step ABSORBS app:backreaction verbatim.
%%   Compact app:seesaw/app:epslocal/app:backreaction REMOVED from
%%   old homes (labels live in F/G/H). Corridor-arithmetic intro -> S8
%%   only. All typeset numbers verified by in-script asserts.
%%   Script: c2_v13.41_X3.py.
%% v13.40 (2026-07-13) -- X2: Part X rebuild, ship 2/5. Two full
%%   three-step courses inserted after Course C:
%%   D 'Distances, the ladder, and the century of H0' (app:courseD):
%%     five distances w/ Etherington derived by photon counting;
%%     low-z cosmography (q0=-0.523, 3.8% at z=0.05); the ladder rung
%%     by rung (parallax sensitivity + network dilution; Cepheid
%%     pulsation Pi*sqrt(rho)=const derived; Wesenheit/dust-law
%%     ancestry of the dust channel; TRGB He-flash physics + the
%%     subsample convention fork; Tripp/R_B fork; Hubble flow);
%%     WORKED common-mode algebra (+7.3% = 0.153 mag; 0.05 mag =
%%     2.3%) + the five-line COMMON-MODE THEOREM (internal audits
%%     exactly orthogonal to common modes); limitations (shared
%%     layers / certification boundary / precision-vs-kind); GG
%%     reading w/ channel-correlation algebra (rho=0/1 -> the two
%%     printed scenarios; effective rho ~ 0.5-0.7 as an ILLUSTRATION).
%%   E 'The microwave background as an H0 instrument' (app:courseE):
%%     c_s^2 = c^2/3(1+R) derived; R = 678/(1+z), R*=0.62, c_s* =
%%     0.454c; r_s integral evaluated in pieces (two-fluid quadrature
%%     144.4 vs chart landmark 145.0, half-percent honesty; r_* vs
%%     r_drag ~147 DISAMBIGUATED -- resolves the pre-existing 145/147
%%     ambiguity); acoustic angle (D_M = 13,970 Mpc, 100theta* =
%%     1.038 vs measured 1.0411 stated at the O-Hubble-32 honesty
%%     level); peak series with phase (pi/theta* = 303, phi_1 = 0.27
%%     -> ell_1 ~ 220; CLASS audit 221/538/817); WHY THE CMB NEVER
%%     MEASURES h (geometric degeneracy derived; 'CMB H0' = model-
%%     conditional inversion; Route II = same inversion, opposite
%%     direction); chi^2=237/83-band zero-freedom audit anatomy;
%%     sound-horizon lever (145 -> 133 for 73.5; capped by EDE nulls
%%     + rs-free span); GG step absorbs app:calc VERBATIM (labels
%%     kept) + BBN variant + pi-offset closing.
%%   Compact app:calc and app:soundhorizon subsections REMOVED from
%%   their old homes (contents/labels live in Course E). Corridor-
%%   arithmetic intro adjusted. All typeset numbers verified by
%%   in-script asserts (incl. scipy/trapz quadratures of r_s, D_M).
%%   Script: c2_v13.40_X2.py.
%% v13.39 (2026-07-13) -- X1: Part X rebuild, ship 1/5 (author-
%%   approved TOC) + THE PART IX INDENTATION FIX.
%%   FIX-0: v13.31's flow-level \captionof (landscape caption, p98)
%%   leaked \parindent=0 GLOBALLY from p98 on (probe-diagnosed:
%%   16.33pt before / 0.0pt after; Parts V-tail..XII rendered flush).
%%   Caption block now wrapped in \begingroup...\endgroup;
%%   indentation restored to Parts VI-XII incl. the Conclusions.
%%   A: section retitled 'Reference dictionaries'; nesting figure
%%      (fig:c2-name-nesting) added to the naming subsection; NEW
%%      'The honesty grades: a reader's key' (app:grades).
%%   B: NEW 'Five primers' (app:primers: H(t), distance, sound
%%      horizon, siren, Local Hole); stale 'Six foundational
%%      questions' title fixed (it held twelve) -> 'Foundational
%%      questions...'; FOUR new questions appended (Asserted-
%%      external mechanics; quadrature-vs-stack; why e^7 (and where
%%      J enters); how w=-1 can be exact).
%%   C: NEW full course 'The Friedmann background from the metric
%%      up' (app:courseC; the three-step contract stated): FRW
%%      conventions -> every Christoffel component (with the H=
%%      Gamma^i_{0j} intuition) -> Ricci (R_00 cancellation shown)
%%      -> Einstein tensor (Bianchi/constraint reading) -> perfect
%%      fluid -> Friedmann + acceleration + continuity (first-law
%%      reading, rho+3p 'pressure gravitates', Newtonian cross-
%%      check with k = -2E/r^2) -> Omega bookkeeping (rho_c ~ 5
%%      protons/m^3) -> worked numerics (age 13.92 via arcsinh form
%%      [app:age MOVED here, labels kept]; acceleration epoch with
%%      the EXACT arcsinh(1/sqrt2) at t_acc=7.8 Gyr = Landmark 5;
%%      H(z) rows recomputed; 12.3 Gyr doubling) -> LIMITATIONS (the
%%      five assumptions) -> GG resolution (app:frw-theorem MOVED
%%      here, trimmed of duplicated textbook steps, labels kept) ->
%%      Status. All typeset numbers verified by in-script asserts.
%%   Part-X subtitle -> graduate-course framing. Old 'Derivation
%%   courses' section retitled '...the corridor arithmetic'.
%%   Script: c2_v13.39_X1.py.
%% v13.38 (2026-07-13) -- author request (after v13.37 sign-off):
%%   NEW subsection sec:h0-closed-form in Section 23, 'The closed
%%   form: the expansion rate from {c, G_N, hbar} and the ledger
%%   alone', between the zeta-selection subsection and Route II.
%%   Boxed analytic result (eq:c2-H0-analytic):
%%     H0 = e^{-4K} sqrt(2 pi^5 c^5 / (3 L K (pi-1) hbar G_N))
%%        = 66.7 km/s/Mpc  (CODATA evaluation 66.74 = 498.1*sqrt(zeta)).
%%   Assembly: T8.7 lock (Mlock = sqrt(2pi) MPl,red e^-K) -> seesaw ->
%%   zeta = pi/(LK) -> Friedmann/(3 Omega_Lambda); all Planck-mass
%%   powers cancel. Provenance-of-every-symbol paragraph (K twice, L
%%   once + vev pin; pi three ways; e; structural 1/2 and 3). Two
%%   corollaries: H0 t_Pl = 10^-60.9 (= 4K e-folds dressed; 'why the
%%   universe is old' as a one-integer answer) and rho_L/rho_Pl,red
%%   = 10^-120.2 (= 8K e-folds - 1.4 orders of geometry; the 10^120
%%   problem as two powers of one hierarchy). Honesty paragraph:
%%   M, N, J act upstream (J in z* = e^7 and the rejected 1/J^2);
%%   grade unchanged (D* + selection lemma + F-Hubble-11); the line
%%   adds exposure, not strength. All typeset numbers verified
%%   in-script against CODATA (asserts). No existing text changed.
%%   Script: c2_v13.38_closedform.py.
%% v13.37 (2026-07-13) -- R7: Part IX (Conclusions) REBUILT to the
%%   A10/C1 Conclusions standard on the author-approved outline
%%   (~4.5x length; plain English, broad audience). Sections: summit
%%   thesis (boxed: 'the century of measuring is ending; the decade
%%   of answering has a schedule'); ten numbers walked as three acts
%%   (table kept); THE RETIREMENT OF A MEASURED CONSTANT (author's
%%   thesis: c/Rydberg/W-Z graduation lineage; H0 the last measured
%%   constant of cosmology; 'is the number a theorem?'); one history,
%%   three acts, all predicted (inflation n_s/r from A10, Kaxion from
%%   C1, w=-1 terminus; single source of truth); THE DOZEN WAYS THE
%%   UNIVERSE CAN ANSWER (12 dated opportunities in plain English,
%%   each with why-meaningful-not-just-precise; pointers to Part VI,
%%   no new commitments); if right (measurements re-purposed as
%%   evidence) / if wrong (clean death + middle world + untested-
%%   window honesty); limitations kept + middle-world sentence; desk-
%%   check section (released scripts, one-evening audit); why there
%%   is someone here to time the universe (why-now + staircase +
%%   'the part of the expansion that keeps the log'); personal note
%%   EXPANDED with the author's message to the astronomy/cosmology
%%   community (input-vs-output confession; 'the dispute over the
%%   last digit is over -- both sides won'; beyond-LCDM falsifiable
%%   precision cosmology as the decade's real frontier); Envoi
%%   (refined, starred section); Acknowledgments moved to the END of
%%   the Part. Labels kept: sec:ten-numbers/why-matters/conc-limits/
%%   personal-note. Part subtitle + roadmap node IX updated. NO value
%%   changed; inflation/Kaxion numbers cited to their home papers.
%%   Script: c2_v13.37_partIX.py.
%% v13.36 (2026-07-13) -- R6: Part VIII (Discussion) REBUILT to the
%%   A10-Cosmos / C1-Kaxion Discussion standard (~3x length), on the
%%   author's direction, absorbing the R-phase improvements of Parts
%%   III-VII. New architecture (movement map in sec:disc-map):
%%   essence chain (one engine, one observable) -> support triage
%%   (kept) -> significance in TEN points (8 expanded + landscape-as-
%%   instrument + conduct-as-content; absorbs the old S-1..S-3
%%   uniqueness section) -> comparison with the five leading response
%%   habits (explanatory trade-offs; verdicts stay in Part V) ->
%%   precedents incl. CAUTIONARY ones (OPERA, BICEP2, the century of
%%   confident error bars; three lessons) -> PS-Hubble-1..7 expanded
%%   (+PS-7 'the consensus paradox') -> I-Hubble-1..10 expanded
%%   (+I-7 audited landscape, +I-8 Asserted-external grade, +I-9
%%   calendar-with-hostages, +I-10 bracket pricing) -> honest
%%   boundary (proved/conditional/open + dependencies) -> Hubble tier
%%   ladder with boxed thesis (computes history, prices chain,
%%   re-reduces no photometry) -> cost of being right -> cumulative
%%   evidence (kept) -> necessity (kept) -> OQ-Hubble-1..8 open-
%%   question programme with closing conditions/actors/eras + index
%%   table tab:oq-hubble -> future directions as ASSIGNMENTS to five
%%   communities + cross-paper single-source-of-truth interfaces +
%%   staged road -> nobility clause (expanded w/ report-card conduct)
%%   -> the bottom line (absorbs the old bridge). Labels kept:
%%   sec:significance-8, sec:ps-hubble, sec:i-hubble, sec:nobility.
%%   Roadmap node VIII + preface Part-VIII clause updated. NO value
%%   changed; all quoted numbers verified against their in-paper
%%   sources; all refs/citations in the new text asserted to resolve.
%%   Script: c2_v13.36_partVIII.py.
%% v13.35 (2026-07-13) -- R5: the closing ripples of the R-phase
%%   campaign (Parts IV-VII rebuilt at v13.29/30/33/34).
%%   (1) Part VIII: CCHP boundary caveat added to 'What is strongly
%%       supported' item 2; convergence-at-70 contingency paragraph in
%%       the cumulative-evidence section (bifurcation priced per
%%       sec:siren-three-worlds); PS-Hubble-4 -> named + candidate
%%       anatomy + dated kill tests; stale counts seventeen -> 
%%       twenty-one benchmark rows, 17+ -> 20+.
%%   (2) Objections Part: NEW section 'On the 2026 landscape and the
%%       residual anatomy' (sec:obj-landscape) with O-Hubble-51
%%       (undersized anatomy), -52 (consensus and its critics), -53
%%       (geometric high probes), -54 (laundering the contested
%%       corrections); tag range verified contiguous 1..54 in-script;
%%       count fifty -> fifty-four swept (preface warning, reading
%%       path, References intro, roadmap node XI '54 answered').
%%   (3) Front matter: abstract gains the landscape clause (length
%%       re-verified <= 1920); Executive Summary Part-IV paragraph
%%       tail rewritten (channels + dated tests + two-way verdict) +
%%       NEW 'The whole field, on one page' landscape paragraph;
%%       preface organization sentence updated (Part V landscape,
%%       Part VI decision calendar, Part VII record to date); roadmap
%%       node VII gains 'report card 2017--26'.
%%   Script: c2_v13.35_ripples.py.
%%   ES referee sentence synced to the re-dated forecast (bright /
%%   dark-statistical equivalent clause). Ship date -> July 13.
%%   First-cite audit fix: Jensen2025SBF pointer repointed
%%   sec:class-statement -> sec:landscape-counterweight (an earlier
%%   citation exists in the geometric-counterweight section).
%% v13.34 (2026-07-13) -- R4: Part VII update (the standard-siren
%%   decade, read against its own record).
%%   (1) NEW 'The referee's report card, 2017--2026'
%%       (sec:siren-reportcard): one bright siren in nine years; the
%%       O4 null-counterpart record + bottleneck physics; the
%%       dark-siren walk 76.6 -> 71.0 -> 69.9 (+4.1/-4.0) with the
%%       correlation honesty clause; the AGN-flare one-liner; the
%%       honest recalibration (dates slip, thresholds cannot).
%%   (2) Forecast re-dated on empirical rates: rates paragraph
%%       rewritten (bright-alone 3-sigma -> mid-to-late 2030s; dark
%%       channel carries the near-term clock; timeline bands stand);
%%       c2_siren_forecast.py era-band annotation updated (script b,
%%       the ONLY touched assumption) and flagged in the caption.
%%   (3) Referee's budget +1 paragraph: the dark-siren walk as the
%%       budget demonstrated live ('averages down', empirically).
%%   (4) NEW 'The sharpened stake: three outcomes, three worlds'
%%       (sec:siren-three-worlds): exact sigma pricing at the two
%%       verdict thresholds -- converged 70 leaves lifted 68.5+-0.6
%%       at 0.6-1.0 sigma, strains pure 66.6+-0.2 at 1.5-2.4 sigma
%%       (verified in-script). Coda: third-world clause.
%%   (5) Part V sync: 69.9+-4.0 -> 69.9^{+4.1}_{-4.0} x2 (matches the
%%       verified landscape figure/table); middle-world sigma ranges
%%       aligned to the exact arithmetic (prose + tab:siren-verdicts
%%       + its caption).
%%   (6) Part VII subtitle: report card + empirical rates + three
%%       worlds.
%%   Scripts: c2_v13.34a_partVII.py, c2_v13.34b_forecastfig.py.
%% v13.33 (2026-07-12) -- R3: Part VI rebuild ('Kill Switches, the
%%   Decision Calendar, and Wider Implications').
%%   (1) Falsifier registry filled to F-Hubble-1..11: channel-level
%%       trio F-8 (NIR ladder / dust), F-9 (injection-recovery /
%%       priors), F-10 (complete JWST sample / subsample), each with
%%       a symmetric confirmation rule; coefficient tier broken out;
%%       reservation sentence removed; kill table 11 rows, 4-tier
%%       caption.
%%   (2) NEW section 'The decision calendar, 2026--2035'
%%       (sec:decision-calendar): dated entries (Gaia DR4 Dec 2 2026;
%%       channel trio 2026-28; Hole census + Jia 6.4-sigma standing
%%       entry 2027-29; early-side era 2028-30; O5 sirens; tensors;
%%       standing undated) + commitments subsection + kill-test
%%       timeline REBUILT (channel cards, Gaia card, O5 span, stake
%%       card) and moved here from sec:decisive.
%%   (3) Over-determination section: emerging FRB (69.40 +2.14/-1.97,
%%       Gao2024FRB) and gamma-ray (61.9 +2.9/-2.4, Dominguez2023Gamma)
%%       channels + Jia trend paragraph.
%%   (4) Priorities re-ranked to eight: injection-recovery and NIR
%%       ladder promoted to P2/P3; F-10 tie in the JWST priority.
%%   (5) S8 / early-galaxies / H0DN-meaning merged into one 'Wider
%%       implications' section (labels kept) + CCHP consensus caveat.
%%   (6) Benchmark sheet +2 rows (2026 landscape structure;
%%       corrected-ladder scenario); row count sentence -> twenty-one
%%       (verified programmatically).
%%   (7) Part-VI coda rewritten (eleven rules, the calendar, FRB/gamma).
%%   (8) Sweeps: roadmap node F-Hubble-1..11 + decision calendar;
%%       'reserved' -> 'carries/registered' x6; eight -> eleven x4;
%%       reading-path range -> sec:decision-calendar; Part retitle.
%%   Cohort ledger rows F26-F28 added in [v11-Support] (script b).
%%   Scripts: c2_v13.33a_partVI.py, c2_v13.33b_ledger.py.
%% v13.32 (2026-07-12) -- figure micro-version (author request):
%%   Figure 17 asset c2_h0_landscape.pdf regenerated with x-axis
%%   maximum 80 (was 82) and a ~5% wider canvas (8.6 in vs 8.2 in);
%%   off-scale sirens (GWTC-4.0, GW170817 bright) carry continuation
%%   arrows at the new edge. No text, number, or reference changed.
%%   Script: c2_v13.32_fig80.py.
%%   Caption updated in step: clipped-bar sentence now names both
%%   off-scale sirens (GWTC-4.0 and GW170817), plural arrows.
%% v13.31 (2026-07-12) -- cosmetic micro-version (author request):
%%   the landscape figure page rebuilt in the two-page caption/art
%%   convention (the C2 v13.19 Omega-budget pattern): the caption now
%%   sits at the foot of the prose page (after a 2.2em gap), and the
%%   art fills its own page alone at 0.97\textheight with no caption
%%   -- which also removes the blank page that the previous
%%   minipage(art+caption) overflow produced. No text, values, or
%%   references changed. Script: c2_v13.31_figcosmetic.py.
%% v13.30 (2026-07-12) -- PHASE R2: Part V rebuilt as "The 2026
%% Measurement Landscape and the Rival Explanations" (R-phases plan v2
%% + R0v2 memo + Deep Research v2 audit; author-approved insertion of
%% the landscape pair):
%%   (1) NEW opening section "The measurement landscape in 2026":
%%       45 abstract-verified determinations drawn (full-page in-paper
%%       figure c2_h0_landscape.pdf, IDs delegated to the table, H_ref
%%       convention 67.24 +- 0.35 per R0v2 N-10) and tabulated
%%       (two-page longtable with identifiers, posted dates, values,
%%       and status remarks); three-band walk with the physics stated
%%       on the spot (absolute-density band; the corrected middle as a
%%       destination, not a camp; precision-vs-directness inversion in
%%       band 3); the consensus-that-isn't (H0DN excludes CCHP; split
%%       localised); the early side closed (three instruments,
%%       SPT-alone 66.66 on the derived rate, deuterium factory audit,
%%       r_s-free span 66.7-70.2 printed honestly, Lynch cap <70.7 at
%%       95% replacing every point-value ModRec claim); the high
%%       geometric counterweight at full strength.
%%   (2) Classes reorganised per the landscape taxonomy: Class 1
%%       PROMOTED from "effectively retired" (which contradicted Part
%%       IV's common-mode argument -- the auditor's-copies intuition
%%       now reconciles the two); EDE two-sided (ACT null vs Poulin
%%       profile f_EDE = 0.09 +- 0.03) and capped to a band-2 machine;
%%       NEW Class 3 modified recombination with the four audits;
%%       Class 4 = merged dark sectors + radiation; Class 5 gains the
%%       DESI w0wa wrong-sign reading (and names F-Hubble-5 as that
%%       switch under live test); Class 6 voids at three depths
%%       (modest alive, deep disfavoured, full-tension >9 sigma).
%%   (3) Siren discriminator -> three worlds (67/70/73) with the
%%       middle-world honesty printed (lifted 68.5 at 1-2 sigma, pure
%%       66.6 strained ~2 sigma = partial confirmation, not victory);
%%       tab:siren-verdicts gains the "If 70" column; NEW status table
%%       tab:class-status-2026; comparison table updated to the new
%%       taxonomy + r_s-free row; uniqueness + NEW Part-V coda.
%%   (4) 32 verified Part-B references added (landscape base; every
%%       tabulated source cited in prose); Part IV Ding/CruzReyes
%%       citation patches; scripts index + roadmap nodes (Part V text,
%%       50 objections, live reference count) synced.
%%   Scripts: c2_v13.30a_partV.py, c2_v13.30b_refs.py,
%%   c2_fig_h0_landscape.py. Residual arithmetic and all grades
%%   unchanged; external material remains Asserted-external.
%% v13.29 (2026-07-12) -- PHASE R1 REDO: the nine-site edit map E1-E9
%% of the Phase-R0v2 verification memo executed exactly (inputs: the
%% Deep Research v2 report, findings V-1..V-14; all claims below
%% abstract-verified against primary records on 2026-07-12):
%%   E1/E2  Gaia channel: "recalibrations wrong-signed" (not source-
%%          supported) replaced by the verified state -- bright-end
%%          zero-point -38.9 +- 10.3 uas, formal errors ~2x under-
%%          estimated, SIGN OF THE H0 SHIFT UNESTABLISHED; low-edge
%%          pricing re-justified on direction-uncertainty + the
%%          cluster-Cepheid bypass (0.9% MW calibration, 0.3 sigma).
%%   E3     dust channel prints its own model spread (70.59 +- 1.15
%%          baseline vs 71.45 +- 1.03 equal-prior) + the Hallgren
%%          simulation proof of reddening non-universality.
%%   E4     priors channel prints the proposal's own concession
%%          (SH0ES sim-based corrections likely capture part).
%%   E5     DES counterweight formally cited (Vincenzi+ ApJ 975, 86)
%%          with a body-level pointer (abstract silent on the step).
%%   E6     metallicity: unverifiable "gamma ~ -0.25" numeral dropped
%%          (both sites); replaced by the verified wavelength-
%%          independence statement (Breuval+ ApJ 939, 89) with the
%%          chromatic-test intuition.
%%   E7     H0(z): "first hints" upgraded to the published 6.4-sigma
%%          DESI uncorrelated-bin trend (Jia+ ApJL 979, L34), with
%%          two printed provenance cautions (BAO-side statement, not
%%          the ladder-shell test; deep-void amplitude disfavoured).
%%   E8     counterweight clause gains the ladder-internal robustness
%%          results (Perfect Host ApJL 992, L34 at 73.49 +- 0.93;
%%          CATS ApJL 954, L31, no variant below 71.6; linearity
%%          0.994 +- 0.010) + the neither-channel-is-photometric
%%          closing logic.
%%   E9     early-anchor legs: SPT-3G-alone 66.66 +- 0.60 quoted
%%          (directly on the derived rate); the r_s-free span
%%          corrected to the verified 66.7-70.2 (was "67-68 +- 2";
%%          the N-2 footprint in Part IV).
%%   +6 verified Part-B references (108 -> 114): Vincenzi2024DES,
%%   Jia2025H0z, Breuval2022Metallicity, Riess2025PerfectHost,
%%   Scolnic2023CATS, Hallgren2025Reddening; Camphuis2025 relevance
%%   gains the SPT-alone value. Residual arithmetic UNCHANGED
%%   (c2_residual_arithmetic.py, 23 checks). No grades change; all
%%   external channels remain Asserted-external.
%%   E10    (D3 completion) the backreaction complex relabelled to
%%          the exact residual: target +6% -> +7.3%, delta_req
%%          -0.34 -> -0.41, rarity >7 -> >9 sigma (2.7/9.3/17 at
%%          40/100/150 Mpc/h; c2_backreaction_pilocked_v2.py), the
%%          figure regenerated (c2_backreaction_fig_pilocked_v2.py)
%%          -- closing a pre-existing 6-vs-7.3 inconsistency between
%%          Sec. 33 and Parts V/VIII, with the valley intuition
%%          printed on the spot.
%%   Script: c2_v13.29_R1redo.py.
%% v13.28 (2026-07-12) -- PHASE R1: Part IV rebuilt around the named
%% residual anatomy (per the R-phases plan v2 + R0 verification memo;
%% all printed numbers verified by c2_v13.28_verify.py, 23 checks):
%%   (1) Sec. 32 rebuilt: NEW subsection "The two medium channels"
%%       (calibrator-vs-flow dust 2-4% [W&H 2024 -2.5 km/s; 2025
%%       70.59+-1.15; blue-SN 70.0+-2.1] and the Bayesian distance-
%%       prior statistics 0-3% [Desmond+; Hogas & Mortsell 73.0->70.6;
%%       REBUTTED by Stiskalek+Riess+ -- printed "proposed and
%%       rebutted, undecided"], overlap caveat); remaining-channels
%%       walk updated (subsample 1-2% w/ CCHP 70.39+-1.94 vs SH0ES
%%       72.6+-2.0, tip-contrast; Gaia low-edge 0.5-1% wrong-signed +
%%       DR4 Dec 2 2026; metallicity 0.5; crowding 0.3); channel table
%%       rebuilt two-scenario (quadrature 4.3/3.8%, stack 9.3/8.3%;
%%       recomputed exactly -- supersedes the memo's rounded 4.4/3.9);
%%       verdict rewritten: R=+7.3% BETWEEN quadrature and stack in
%%       both scenarios, central-value chain 73.5->69.5-70.2 within
%%       0.8 sigma of lifted 68.5, deflation 7.1 -> 1.9-2.1 sigma,
%%       four honesty clauses (status/overlap/geometric counterweight/
%%       symmetry), F-Hubble-8/9/10 reservations referenced. All
%%       external channels at Asserted-external: cited allowances,
%%       never GG results; triage unchanged.
%%   (2) Sec. 31: KBC-deep downgraded contested -> disfavoured
%%       (Stiskalek+ field-level CF4 velocities prefer <70 Mpc voids;
%%       physics intuition: the lift's outflow IS the measured bulk
%%       flow); CF4 excess carried as contested companion (Watkins+);
%%       H0(z_min) descending trend recorded as the modest void's
%%       predicted signature (Mazurenko+).
%%   (3) Secs. 27/28.1/30/35/39 aligned: named-candidates paragraph;
%%       three-instrument + r_s-free anchor legs; chain arithmetic
%%       corrected (+0.028 -> 68.5, +0.073 -> 73.5); borrowed-no-
%%       freedom paragraph; Part IV summary rewritten to the new
%%       arithmetic.
%%   (4) Figure: c2_bridge.pdf v3 (violet candidate-anatomy bracket,
%%       central-value tick at 69.5, KBC panel relabelled).
%%   Script (b) adds: 11 verified Part-B references + Freedman2025CCHP
%%   relevance update (70.39); the approved ~6% -> ~7% (exact +7.3%)
%%   relabel at every site incl. O-Hubble-2/23/41/47; riders:
%%   F-Hubble-7 Kaxion bar 8.1+-0.8 -> +-0.5 ueV (C1 canonical),
%%   Landmark-5 acceleration onset 7.7 -> 7.8 Gyr (computed);
%%   DES-SN5YR counterweight cited informally pending metadata
%%   verification (bib entry deferred to R2, flagged).
%%   Scripts: c2_v13.28a_partIV.py, c2_v13.28b_refs_riders.py,
%%   c2_v13.28_verify.py, c2_fig_bridge.py (v3).
%% v13.27 (2026-07-12) -- falsifier renumbering + two Part-III repairs:
%%   (1) the O6/Hosotani coefficient falsifier moved to the -11 slot,
%%       freeing -8/-9/-10 for the Part-IV residual-anatomy channels
%%       (dust / priors / subsample); reservation sentence added to the
%%       registry intro; title, summary table, roadmap node, triage row,
%%       Conclusions, Registries appendix, objections swept (13 sites).
%%   (2) Part-III coda TT point count reconciled to the canonical
%%       83-bin statement of sec. 25.3 / A10.
%%   (3) the third cluster-rejection reclassified from
%%       grammar-by-analogy to the orphan/naming rule, unifying the
%%       three rejections under the single A9 discipline; honesty
%%       clause 3 and the matching objection reworded. No physics, no
%%       values, no grades changed.
%%   (4) stale falsifier counts repaired (three prose sites now say
%%       eight; the kill-test-timeline caption says the dated seven and
%%       points to the summary table for the theory milestone); the
%%       summary-table Tag column widened so the -11 tag does not wrap.
%%   Script: c2_v13.27_fhubble11.py.
%% ===================================================================
%% v13.26 (2026-07-12) -- Phase G1: Part III reconstructed around the
%% geometric (zero-astronomical-input) determination of H0. (1) NEW
%% GU-5 dimensional-bridge subsection in the framework section (A4
%% conversion factors; the A9 electron precedent; the target number
%% H0*t_Pl ~ 1.2e-61). (2) The H0 section rebuilt in two-route order:
%% Route I = the vacuum-energy seesaw from geometry alone (lock scale
%% laboratory-pinned via the electroweak vev at +-0.18% on the rate;
%% the cost-minimal cluster as the pure-geometry bracket 66.8-71.2;
%% the ladder's 73.5 excluded outright), then the NEW codebook
%% selection of zeta_Lambda = pi/(LK) = pi/175 by the A9 MSMF/OGN
%% stratum-availability + structural-naming rules (grade Asserted ->
%% Derived (cond., MSMF/OGN), author-approved; Hosotani closure
%% pre-registered as falsifier F-Hubble-8 with two-sided target);
%% Route II = the density inversion reframed as the precision
%% confirmation, plus the NEW CMB-free BBN anchor (66.4 +- 0.5); NEW
%% concordance subsection + two-route figure (c2_two_routes.pdf,
%% generator c2_fig_two_routes.py). (3) The CMB-observables table and
%% the Dev chart recast at zero CMB input: omega_b and omega_m added
%% as predicted rows (sixteen shadows), captions and counts updated;
%% TT-spectrum caption marks all background inputs derived. (4)
%% Provenance table, deck, promotion roadmap, FAQ, appendices,
%% conclusions-limits, ES, and abstract aligned; O-Hubble-6 rewritten;
%% NEW objections O-Hubble-49/-50 (objection count 48 -> 50);
%% kill-criteria table +1 row; scripts table +1. Canonical H0 =
%% 66.6 +- 0.2 and all frozen physics values unchanged.
%% Script: c2_v13.26_G1.py.
%% ===================================================================
%% ===================================================================
%% v13.25 (2026-07-12) -- Final policing: Part IV reviewed and rebuilt
%% (+~30%). (1) NEW opening section "The task of this Part, and the
%% strategy": the asymmetry Part III left, the three-homes forcing
%% argument (the derivation closes two doors from the inside and is
%% FORCED onto the ladder door), the factorise/derive/bound/name
%% strategy, the honesty geometry, and a prose roadmap. (2) Logic
%% completed: additive-vs-product split note (first-order equivalence
%% stated); graviphoton requirement's linear-scaling step; 67.4+/-0.4
%% -> +/-0.5 consistency fix. (3) Intuition added: the surveyor's
%% anchor picture; the two-dictionaries-one-text reading and the
%% protractor/stick lever intuition; the valley-outflow and rim-
%% dilution picture for the Local-Hole lift; the averaging-does-not-
%% commute intuition for backreaction. (4) "The channels, one by one"
%% rebuilt as a per-channel walk (7 paragraphs): each candidate bias
%% explained physically, its common-mode character shown, and its
%% status in the light of GG-Theory stated -- the framework closes the
%% channels it owns (graviphoton, backreaction) and is deliberately
%% silent on metrology, the double-counting guard on the flow channel
%% made explicit. (5) The backreaction section moved to directly
%% after the residual anatomy. (6) Coda promoted to a full section
%% "Summary: the reconciliation, and the road to Part V" (7
%% paragraphs, Part I-III style). Physics unchanged.
%% Script: c2_v13.25_partIV.py.
%% ===================================================================
%% ===================================================================
%% v13.24 (2026-07-11) -- Figure 7 (the running-budget pair) fonts
%% enlarged ~20% to match Figure 8's rendered font size (Fig 7 draws
%% on a 10.8-wide canvas rendered at 6.0in, so its fonts displayed
%% ~26% smaller than Fig 8's). Asset c2_omega_budget_pair.pdf
%% regenerated by c2_fig_omega_budget.py; value box relocated, titles,
%% legend, and inset re-fitted. No tex content changed beyond this
%% stamp. Script: c2_v13.24_fig7_fonts.py.
%% ===================================================================
%% ===================================================================
%% v13.23 (2026-07-11) -- Figures moved to their citation sites.
%% (1) The six-steps summary paragraph moved to page 46 (after Step 6,
%% line spaces kept before the Figure-7 caption); only the scope
%% paragraph remains after Figure 8. (2) Figures 9 and 10 (derived
%% H(t) + landmarks) moved to directly after the landmarks section's
%% opening paragraph, where they are cited, and flushed -- the walk
%% and the table now follow the figures. (3) Figure 12 (the Dev
%% chart) flushed with \clearpage at its citing paragraph in the
%% coherent-offset subsection, so the remaining analysis and the
%% Part III summary no longer precede it. Physics unchanged.
%% Script: c2_v13.23_figflow.py.
%% ===================================================================
%% ===================================================================
%% v13.22 (2026-07-11) -- Figure 12 (the coherent-offset Dev chart)
%% enlarged ~20% to the A10-Cosmos Figure-6 page size: taller canvas
%% (11.5 x 12.0) regenerated by c2_fig_cmb_offsets.py, included with
%% A10's exact geometry (width=\textwidth, height=0.74\textheight,
%% keepaspectratio). Physics unchanged. Script: c2_v13.22_fig12_size.py.
%% ===================================================================
%% ===================================================================
%% v13.21 (2026-07-11) -- Final touches on the Sec 22 spread.
%% (1) Steps 5 and 6 moved before the Omega-budget caption (page 46
%% now carries Steps 3-6 plus the caption, with extra line space
%% between Step 6 and the caption); only the six-steps summary and
%% scope paragraphs remain after Figure 8. (2) The Dev chart
%% (Figure 12) regenerated with fonts ~10% larger
%% (c2_fig_cmb_offsets.py). Physics unchanged.
%% Script: c2_v13.21_touches.py.
%% ===================================================================
%% ===================================================================
%% v13.20 (2026-07-11) -- Sec 22 reflow + audit-format refinements.
%% (1) Steps 3-4 moved before the Omega-budget caption page, which now
%% closes their text page (no more near-empty page); the art follows
%% at full size overleaf, with the a(t) figure directly after it, so
%% the two derived-history figures sit on sequential pages; Steps 5-6
%% and the scope paragraph resume after both. (2) The Omega-budget art
%% enlarged to 0.953\textwidth so its panels match the a(t) panels.
%% (3) Table 5 gains a Pull column in the A10 Table-2 convention,
%% with the convention stated in the caption (values copied from
%% A10's audited table where rows coincide; r_s scored against the
%% measurement error alone). (4) The coherent-offset chart recast in
%% the A10-Cosmos parameter-audit Dev-plot format (generator
%% c2_fig_cmb_offsets.py v2, same asset name): nested precision
%% bands, thick/thin sigma bars, bold left entries, blue right
%% annotations; caption rewritten accordingly. Physics unchanged.
%% Script: c2_v13.20_layout_audit.py.
%% ===================================================================
%% ===================================================================
%% v13.19 (2026-07-11) -- Part III reordered: law -> inventory ->
%% curve. (1) The six-steps subsection (with the which-Omega aside,
%% the Omega-budget figure, the a(t) figure, and the scope paragraph)
%% promoted to its own section and moved after the pi-locked inventory
%% section, so the inventory is introduced before it is used; Sec 20
%% now carries only the Friedmann theorem, with rewritten framing and
%% hand-offs; all labels preserved. (2) The Omega-budget figure recast
%% in the A10 two-page style: caption page, then the art at full size
%% (0.93\textheight) on its own page. (3) The a(t) asset regenerated
%% with horizontal slope labels and a horizontal "today" label
%% (generator c2_fig_a_of_t.py updated). (4) Sec 19.1 import pointer
%% and the Part III summary reordered to the new narrative. Physics
%% unchanged. Script: c2_v13.19_partIII_order.py.
%% ===================================================================
%% ===================================================================
%% v13.18 (2026-07-11) -- a(t) drawn, the running budget imported, the
%% scope stated (self-sufficiency round). (1) NEW Fig: the derived
%% scale factor a(t) (asset c2_a_of_t_derived.pdf, generator
%% c2_fig_a_of_t.py) at Step 4 of Sec 20.2 -- era slopes t^(1/2) and
%% t^(2/3), the de Sitter upturn, the inflection at z=0.62, today at
%% t0=13.92 Gyr. (2) NEW Fig: the running budget Omega_i(t) (asset
%% c2_omega_budget_pair.pdf, generator c2_fig_omega_budget.py --
%% adapted from A10's why-now pair, re-anchored at the crossing with
%% Or=9.36e-5 so equality lands at the canonical z=3400), full page
%% beside the which-Omega aside, referenced from the aside and
%% O-Hubble-48. (3) Scope paragraph closing Sec 20.2: the classical
%% eras are all the corridor needs; the pre-radiation descent
%% (inflation, reheating, lock crossing) is A10's two-basin descent,
%% pointed to, not duplicated. (4) Scripts table +2 rows; "eight
%% scripts" -> "ten" in the ES. Physics unchanged.
%% Script: c2_v13.18_aoft_budget.py.
%% ===================================================================
%% ===================================================================
%% v13.17 (2026-07-11) -- The "which Omega is which" clarification.
%% (1) Three-paragraph aside in Sec 20.2 after Step 2: the Omegas in
%% the working Friedmann form are present-epoch constants (dilution
%% factors carry the time dependence); the running fractions
%% Omega_i(t) are a different object (corridor-switch budget
%% 0.76/0.24/2e-9); the pi-lock is a boundary condition at the
%% declared crossing epoch (why-now priced in Part IV); early-universe
%% physics consults only the physical densities, and the one
%% epoch-independent fraction is Omega_b/Omega_m = 1/2pi (the
%% baryon-loading test's licence). (2) Matching objection O-Hubble-48
%% appended to the geometry-and-pi-fractions theme. (3) Objection
%% count 47 -> 48 swept (ES, reading paths, contributions list,
%% roadmap-figure node). Physics unchanged.
%% Script: c2_v13.17_whichomega.py.
%% ===================================================================
%% ===================================================================
%% v13.16 (2026-07-11) -- Final policing, round 6: Part III rebuilt.
%% (1) Sans-serif sweep: legacy serif fallbacks (GGtheory, GGsix,
%% GGfour, Gifour, Qifour, EIC, OH, IH) aliased/recast to the house
%% sans style; ~20 raw program-name literals macro-ised, incl. the
%% Part III running-head short title. (2) Sec 19: the GG-6 stage now
%% stated with its explicit parameter-free metric (A(r)=r forced; the
%% erroneous A(r)=kr equation removed) and the full bulk group
%% Spin(1,5) x SU(5) x U(1)_B x U(1)_Qi, adapted from A10-Cosmos
%% Sec 10; subsection 19.1 expanded to a four-paragraph review of
%% A10-Cosmos. (3) Sec 20: NEW subsection "The Friedmann equations as
%% a theorem" (A10's derivation transcribed: metric, Christoffels,
%% Ricci, Einstein tensor, ledger source, boxed Friedmann pair,
%% continuity/dilution) + NEW subsection "From the law to the curve:
%% H(t) in six steps" (normalisation, inventory insertion, era
%% power laws, closed form a(t)/H(t)=sqrt(OL)H0 coth, age relation
%% 0.948/H0, monotonicity dH/dt<=0). (4) The derived-H(t) figure
%% moved from Sec 20 to the landmarks section, immediately before the
%% landmarks figure -- the two now form the paper's central pair,
%% introduced as such. (5) Sec 21: two new paragraphs of geometric
%% and physical intuition for the Hopf-fibration figure. (6) Subsec
%% 22.1 expanded to a full page: baryon-loading and equality physics,
%% why the densities are h-free, the one-line inversion, why two
%% anchors agreeing is nontrivial, the halved error budget. (7)
%% Triage and CMB-core tables footnotesize -> small; CMB-core recast
%% as a centred natural-width tabular. (8) Part III coda promoted to
%% a full section "Summary: the derived history, and the road to
%% Part IV". Physics unchanged. Script: c2_v13.16_partIII.py.
%% ===================================================================
%% ===================================================================
%% v13.15 (2026-07-11) -- Final policing, round 5: Part II rebuilt.
%% (1) Sec 8 re-opens with the H(t)/H0 definitions and walks the
%% century figure feature by feature. (2) Sec 9 doubled: the
%% two-numbers stage, the quantitative 1998-99 discovery, the
%% Boomerang/WMAP/SDSS-BAO concordance, what LambdaCDM leaves
%% unexplained, and the time-dependence close. (3) Sec 10 rebuilt:
%% rung-by-rung late route, three-step early route with defining
%% equations, comparison table, three-homes close. (4) Central-claim
%% funnel converted to the A10-Cosmos natural-size minipage format.
%% (5) Early-universe section rebuilt as a network argument
%% (Planck/ACT/SPT + DESI/BBN + chronometers). (6) Part II summary
%% extended to Part-I depth and the Coda absorbed. (7) Six new Part-B
%% refs (de Bernardis 2000; Eisenstein 2005; Louis 2025; Sandage
%% 1970; Schmidt 1998; Spergel 2003) -> 97 entries; roadmap count
%% refreshed; nine first-cite notes repointed. Physics unchanged.
%% Script: c2_v13.15_partII.py.
%% ===================================================================
%% ===================================================================
%% v13.14 (2026-07-11) -- Final policing, round 4: Part I layout moves.
%% (1) A \bigskip inserted between the reference-machine paragraph and
%% the roadmap figure. (2) \clearpage before Section 5, so the cohort
%% section opens its own page. (3) The phenomenology-branches paragraph
%% moved from before the cohort figure to after the roster table, just
%% ahead of Section 6 (text verbatim; the roster reference now points
%% one table up). No prose changed. Script: c2_v13.14_partI3.py.
%% ===================================================================
%% ===================================================================
%% v13.13 (2026-07-11) -- Final policing, round 3: Part I refinements.
%% (1) Page-13 fill: the century figure moved from the end of Section 2
%% into Section 3 (after the scale-and-angle paragraph), so text flows
%% through page 13 and the figure holds page 14. (2) Roadmap figure
%% reduced to 0.90\textwidth, centered. (3) Cohort section rebuilt
%% step by step: the sixteen-citealp chain removed; four paragraphs
%% walk the descent (constitution -> bulk trio -> hinge + pillars ->
%% readouts -> branches), featuring A10-Cosmos as the parent readers
%% are encouraged to read next and A9-SM as the Standard-Model
%% companion; all sixteen siblings still cited. (4) Ten new historical
%% references (Leavitt & Pickering 1912; Slipher 1917; Einstein 1917;
%% Friedmann 1922; Hubble 1925; Hubble & Humason 1931; Alpher & Herman
%% 1948; Baade 1956; Sandage 1958; Dicke et al. 1965) cited inline in
%% Sections 1-2, plus Riess/Perlmutter/Freedman/Sandage-Tammann/
%% Freedman-Madore/Riess-2022/Camphuis reuse-cites -- the Introduction
%% now carries 25+ inline citations; Baade/Sandage also cited at Part
%% II's halvings narrative. (5) Reading paths expanded to a full
%% paragraph per reader; the Part I "Coda" replaced by a proper
%% Section 7 ("Summary, and the road to Part II"). Part B re-sorted
%% (91 entries); full first-cite audit re-run and mismatches fixed.
%% Script: c2_v13.13_partI2.py.
%% ===================================================================
%% ===================================================================
%% v13.12 (2026-07-11) -- Final policing, round 2: Part I rebuilt for a
%% general audience. Section 1 expanded to a full page: the century's
%% conceptual revolution told as a story (the 1900 static island; the
%% size shock; the motion shock and the stretching-space picture;
%% Einstein's regret; the primeval atom and the 1965 afterglow; the
%% 1998 turn; the arc closed on the one governing number). Section 2
%% expanded: H(t) and H0 defined via the balloon, units and the 1/H0
%% age-scale explained, and the century-of-H0 figure MOVED here from
%% Part II (Part II's prose now points back). Section 3 expanded to a
%% full page for readers with no GG background (the 34-dial deficit;
%% the scale-and-angle stage; maximum symmetry, minimum freedom; the
%% shadows; the pi-locked recipe; the existing two-facts close).
%% Roadmap figure rebuilt in the A10-Cosmos Figure-7 format: full
%% width, A10 styles/fonts/colors, Parts V/VI/VII horizontal, Parts
%% X/XI/XII horizontal, Parts III+IV highlighted (innov), right-side
%% phase labels, minipage + captionof (non-floating). Part I coda
%% expanded to four paragraphs (century summary; the split; the
%% paper's stance; the hand-off to Part II). Bib +1 (Penzias & Wilson
%% 1965, sorted into Part B). Script: c2_v13.12_partI.py.
%% ===================================================================
%% ===================================================================
%% v13.11 (2026-07-11) -- Final policing, round 1: the front matter.
%% (1) Title page: third line (the programme-level-reading sentence)
%% removed. (2) Executive Summary trimmed ~6 lines so it closes, with
%% its Keywords, on pages 2-3. (3) NEW R5 override: \GGPart's toc part
%% entry wraps the subtitle in \GGovsub{}, which the Overview gobbles
%% -- one-line "Part N. Title" Overview entries -- and the front matter
%% returns to frontmatteronepage=true so the Overview and the List of
%% Figures share a single page. (4) Preface trimmed ~3 lines to hold
%% one page. Physics and body untouched. Script:
%% c2_v13.11_frontpolice.py.
%% ===================================================================
%% ===================================================================
%% v13.10 (2026-07-10) -- Phase H: final polish (Fable thread).
%% Front matter: Abstract rewritten in the accessible register (fits
%% p1, <1920 chars); Executive Summary rebuilt to two full pages in
%% the C1/A10 accessible register (curve-not-constant; the pi-locked
%% recipe; the coherent miss as the signature of a derivation; the
%% priced local excess; the siren date; the proved-vs-assumed ledger;
%% why it matters), Keywords retained at its end; NEW warm Preface
%% (GGPreface, after the ToC, just before Part I) addressed to the
%% working cosmologist, with the Monday-morning actions and the two
%% honest warnings; Authors' Note updated to the final structure.
%% References 64 -> 80: 16 new Part-B entries (Aubourg 2015; Bernal,
%% Verde & Riess 2016; Blas 2011 CLASS; Brout 2022; Buchert & Rasanen
%% 2012; Di Valentino 2021; Efstathiou 2020; Eisenstein & Hu 1998;
%% Fixsen 2009; Freedman & Madore 2010; Freedman 2021; Jimenez & Loeb
%% 2002; Liske 2008; Mortsell 2022; Riess 2019; Sandage & Tammann
%% 1975), each cited at its natural in-text home (zero orphans
%% preserved); Part B re-sorted alphabetically (append-order debt
%% cleared); one stale sigma_8 = 0.741 in the backreaction course
%% corrected to 0.737 (the Phase-D recompute value). Full programmatic
%% first-cite audit and smoothness checks run in the c2_v13.10c pass.
%% Physics unchanged. Scripts: c2_v13.10a_frontmatter.py,
%% c2_v13.10b_refs.py, c2_v13.10c_audit.py.
%% ===================================================================
%% ===================================================================
%% v13.9 (2026-07-10) -- Phase G: the back matter (Fable thread).
%% Objections 18 -> 47: O-19..21 (neutrino convention; radiation share;
%% double-dipping) in the derived-rate theme; O-22..25 (kinship charge;
%% six-or-seven-percent -- prints the shorthand flag; the Kenworthy
%% contradiction; special pleading) in the ladder theme; O-26..28 (why
%% the electroweak sphere; the drift below 1/pi; the dimensionful
%% scale) in the geometry theme; THREE new themed sections -- the
%% statistics of the tension (O-29..32, incl. the percent-vs-sigma
%% confrontation answered head-on), the siren and the forecast
%% (O-33..36), the programme and its manner (O-37..40); and the
%% seven-persona SECOND EXCHANGE (O-41..47), where the panel returns
%% against the answers and the auditor logs the ~6%-shorthand finding.
%% Foundational questions 6 -> 12 (the siren; the Local Hole; why
%% voids/averaging cannot do it; the percent-vs-sigma question; why
%% fits return 0.315; the far future). Background-rate course gains
%% explicit error propagation (baryon +-0.22, matter +-0.26) and the
%% repro-inputs table's matter-route error is corrected 0.17 -> 0.26
%% (recomputed; flagged). S8 course rebuilt as a three-step worked
%% chain (growth integral; sigma_8 = 0.737 from the parent's amplitude
%% chain; S8 = 0.759). NEW scripts-released index table (8 scripts).
%% All typeset numbers machine-verified. Physics unchanged. Scripts:
%% c2_v13.9a_objections.py, c2_v13.9b_backmatter.py.
%% ===================================================================
%% ===================================================================
%% v13.8 (2026-07-10) -- Phase F: the siren-forecast Part (Fable).
%% NEW body Part VII "Forecasting the Verdict: The Standard-Siren
%% Decade" inserted before the Discussion (now Part VIII; Conclusions
%% now Part IX): the siren principle with the chirp/amplitude
%% equations and the GW170817 worked example; the three disjoint
%% pipelines (bright/dark/spectral); the events-to-verdict forecast
%% (thresholds 2.30/1.38; N = 21 and 58 in the GW170817-like central
%% scenario; the between-camps outcome kills both readings at >3
%% sigma); the referee's own systematic budget (only non-averaging
%% term = amplitude calibration, an engineering floor with no coupling
%% to either route); the simulated verdict and the no-hiding failure
%% clause; reproducibility + the pre-registered rule restated. NEW
%% verdict figure c2_siren_verdict.pdf + released open-source script
%% c2_siren_forecast.py (all numbers machine-verified). Part counts
%% reconciled everywhere: roadmap prose + TikZ rebuilt (12 nodes) +
%% caption (eight content Parts), Discussion bridge Part IX, reading
%% paths gain the forecast stop; pointers added from F-Hubble-2, the
%% decisive section, and the appendix siren course. Bib +4 (Schutz
%% 1986, Holz & Hughes 2005, Abbott et al. 2017, Chen, Fishbach &
%% Holz 2018). One garbled Phase-E phrase repaired. Physics unchanged.
%% Scripts: c2_v13.8a_sirenpart.py, c2_v13.8b_reconcile.py,
%% c2_siren_forecast.py.
%% ===================================================================
%% ===================================================================
%% v13.7 (2026-07-10) -- Phase E: Parts V-VI depth (Fable thread).
%% Part V: comparison-statement names the two instruments; Class 1
%% rewritten with the honest kinship clause (the residual is a
%% disciplined Class-1 claim) + siren verdict; Class 2 gains the
%% sound-horizon lever numbers + opposite-bets siren verdict; Class 3
%% gains the coupled-continuity mechanism + siren verdict; Class 4
%% gains the GW-EM distance-duality double test; Class 5 gains the
%% native-apparatus pointer (+7.3% monopole needs delta ~ -0.41,
%% >9 sigma) + siren verdict; Class 6 siren verdict. NEW siren-
%% discriminator section + table (one catalogue bisects the whole
%% solution space; 67 retires every early-side fix, 73 retires every
%% measurement-side reading incl. this paper). Summary table + caption
%% gain the siren-verdict row. Part V coda extended. Part VI: NEW
%% over-determination section (chronometer relation derived in two
%% lines; BAO absolutised by the derived r_s = 145.0; redshift drift
%% derived with numbers +2.1 cm/s per decade at z=1, sign change
%% z~1.9, -5.6 at z=4; derived-H(z) table at z = 0.5/1/2/4); decisive
%% section bisector line; benchmark drift row; Part VI coda extended.
%% Bib +3 (Sandage 1962, Loeb 1998, Moresco 2022). All numbers
%% machine-verified. Scripts: c2_v13.7a_partV.py, c2_v13.7b_partVI.py.
%% Physics unchanged.
%% ===================================================================
%% ===================================================================
%% v13.6 (2026-07-10) -- Phase D: Part IV depth (Fable thread).
%% NEW Local-Hole course (evidence incl. the contested KBC-deep reading
%% and its two published brakes; three-step linear derivation of the
%% lift with the Gauss-theorem mean-interior-contrast form; the two
%% dilutions -- r^2-weighted flow-volume averaging and the 2M++ flow-
%% correction overlap; scenario ledger pessimistic/shallow/canonical/
%% KBC-deep with the required residual per scenario; systematic budget
%% table; NEW figure c2_hole_lift.pdf; what would settle the value).
%% NEW residual-vs-error-budget confrontation section: exact
%% requirement R(eps_Hole) (canonical +7.3%; the paper keeps the
%% cohort's ~6% shorthand, flagged for a cohort-synced decision);
%% what the 1.09% budget structurally cannot contain; seven-channel
%% common-mode table (quadrature 2.2%, aligned stack 5.6%); the
%% two-direction verdict incl. the one no-new-physics route (deep void
%% + aligned stack, both contested maxima); three honesty clauses; NEW
%% bridge figure c2_bridge.pdf. Backreaction moved to the exact-1/pi
%% computation (sigma_8=0.737 from the derived S_8; asset
%% c2_backreaction_bound.pdf; 0.26% at 200 Mpc/h; script named and
%% released). Part IV coda rebuilt. Bib +4 (Keenan 2013, Whitbourn &
%% Shanks 2014, Kenworthy 2019, Lindegren 2021). All typeset numbers
%% machine-verified. Scripts: c2_v13.6a_hole.py, c2_v13.6b_residual.py,
%% c2_fig_hole.py, c2_fig_bridge.py, c2_backreaction_pilocked.py,
%% c2_backreaction_fig_pilocked.py. Physics unchanged: the adopted
%% lift band and the ~6% shorthand are untouched; scenarios recorded.
%% ===================================================================
%% ===================================================================
%% v13.5 (2026-07-10) -- Phase C: Part III depth (Fable thread).
%% NEW Landmarks-on-the-derived-H(t) section: eight walked checkpoints
%% (BBN eta_b, equality z=3400, corridor switch, frozen sound horizon,
%% acceleration onset z=0.62, matter-vacuum equality z=0.29, crossing/
%% today 13.84/13.92 Gyr, de Sitter terminus) + landmarks table + NEW
%% 19-decade annotated figure (c2_H_of_t_landmarks.pdf). NEW Hopf-
%% fibration figure (c2_hopf_fibration.pdf) + reading-the-proof
%% paragraph under the inventory theorem. Seesaw subsection rebuilt as
%% a factor-by-factor worked chain (lock scale -> 6.08 meV rung -> 713
%% prefactor -> 498.3 coefficient; quarter-power cross-check 2.23 vs
%% measured 2.24 meV; softness dlnH0/dlnMlock=4; all machine-verified).
%% Ported the parent's zero-parameter TT-spectrum figure
%% (a10_cmb_tt_gg.pdf) into the chi-squared subsection. NEW coherent-
%% offset figure (c2_cmb_offsets.pdf, 14 confrontations on one axis) +
%% degeneracy-line prose. Two arithmetic corrections in the CMB core
%% table (height-ratio deviations recomputed). Main H(t) asset
%% regenerated with the measured radiation share at the derived rate
%% (equality label now 3400; switch label e^7 = 1097). Part III coda
%% rebuilt (four establishments). Scripts: c2_v13.5a_landmarks.py,
%% c2_v13.5b_hopf_seesaw.py, c2_fig_landmarks.py, c2_fig_hopf.py,
%% c2_fig_cmb_offsets.py. Physics unchanged.
%% ===================================================================
%% ===================================================================
%% v13.4 (2026-07-10) -- Phase B: Parts I-II depth (Fable thread).
%% Part I: cohort block (map + roster + cite-all-17) relocated from the
%% top of Part III to a new end-of-Part-I cohort section per the cohort
%% on-ramp standard (Part III framework section relabelled); NEW
%% Three-reading-paths section; NEW Part I coda; roadmap prose updated.
%% Part II: measurement-history course rebuilt as a five-movement
%% narrative (discovery error, the great halvings, the factor-two war,
%% the Key Project, the precision-era split) + NEW century-of-H0 figure
%% (c2_H0_history.pdf); NEW Anatomy-of-the-two-routes section with a
%% two-column TikZ diagram; NEW statistics-of-7.1-sigma section (all
%% arithmetic machine-verified: collapse to 3.7/2.0/1.4 sigma under
%% 2/4/6 percent common-mode terms). Bibliography +2 (Freedman 2001 Key
%% Project, Lemaitre 1927); two first-cite notes re-pointed. Scripts:
%% c2_v13.4a_partI.py, c2_v13.4b_partII.py, c2_fig_h0_history.py.
%% Physics unchanged.
%% ===================================================================
%% ===================================================================
%% v13.3 (2026-07-10) -- Phase A consistency purge (Fable thread).
%% Purged the retired corridor-prediction framing from Parts V-VII and
%% Part VI priorities (Class 2/3/4/5 rewrites, comparison-table row,
%% F-Hubble-3 recast detection-kills, priorities 1/2/3/5, JWST section,
%% meaning-for-H0DN close, Discussion strongly-supported + S-1 +
%% criteria, glossary + App D/E captions); retired the stale eps band
%% at all sites in favour of the 0.104 = Hole-lift + open-residual
%% split. Numeric reconciliations: matter density 0.1430 w/ neutrino
%% footnote (matter anchor 67.0), de Sitter floor 55.0 (4 sites),
%% S8 unified at derived 0.759 (eq recast; VI/VII/benchmark rows).
%% New title + Discussion subtitles (derived-history arc). Timeless
%% rewording of three draft-history remarks. Poulin2019 re-anchored at
%% the lever-arm paragraph; 4 first-cite notes re-pointed (2 new sec
%% labels). H(t) asset regenerated at the exact pi-locked inventory.
%% Scripts: c2_v13.3a_purge.py, c2_v13.3b_purge2.py,
%% c2_fig_H_of_t_pilocked.py (asset c2_H_of_t_derived.pdf; the shared
%% a10_* asset is left untouched for A10). Physics unchanged.
%% ===================================================================
% =====================================================================
% ===== v13.1 (2026-07-09) SCRIPT 1: dramatic restructure, skeleton. =====
%   8 Parts -> 10. NEW Part I (Introduction): history, H(t) vs H0, GG preview,
%   roadmap updated to 10 Parts. Current Part I (H0DN) -> Part II 'The Hubble
%   Tension' (retitled; content kept, expanded in script 2). Current Part II
%   split: derivation stays -> Part III 'The GG-Theory of the Hubble Parameter';
%   NEW Part IV 'Reconciling the tension' header inserted before the early/late
%   split (backreaction + revisit added in script 3). Comparison/Obs auto-renum
%   -> V/VI; tail -> VII-X. Old roadmap section removed. Stamp -> v13.1.
% =====================================================================
% =====================================================================
% ===== v13.0 (2026-07-09): PHASE 6 promotion. Consistency + first-cite. =====
%   (1) Appended 'First cited in \S\ref{...}' to every bibitem that lacked
%       it (charter item 6); first-cite section computed programmatically.
%   (2) Killed the two orphan references: Planck2018Parameters cited at the
%       two physical densities; PeskinSchroeder1995 at the gauge-theory
%       epsilon_local course. Zero orphans.
%   (3) Labelled the opening Part-I section (sec:why-h0dn) so the early
%       H0DN/tension citations resolve to a home.
%   Version/date stamp -> v13.0. No frozen-physics change.
% =====================================================================
% =====================================================================
% ===== v12.10 (2026-07-09): PHASE 5 objections + six-persona audit. =====
%   (1) Existing 7 objections tagged O-Hubble-1..7.
%   (2) NEW section 'The six-persona hostile review' (sec:sixpersona) with
%       O-Hubble-8..14: distance-ladder cosmologist, siren cosmologist, CMB
%       theorist, statistician, dark-energy phenomenologist, philosopher,
%       internal auditor. Panel reads A10 + C1.
%   (3) Scrubbed no-internal-history violations: A3 bibitem '(formerly the
%       GD-2026-5 preprint)'; A10 bibitem working-code + sync-date + stale
%       73.7 anchor -> clean C2-perspective Relevance + first-cite.
%   Objections raised but NOT resolved by silent edits are flagged to the
%   author (epsilon_local [0.03,0.05] band vs the 0.104 split framing).
% =====================================================================
% =====================================================================
% ===== v12.9 (2026-07-09): PHASE 4 six-block Technical Appendices. =====
%   Regrouped the four flat appendices (C/D/E/F) into six parent blocks
%   A-F (fixes the 'A Appendix C' double-label bug); existing derivations
%   demoted to subsections. NEW: A naming convention + comprehensive summary
%   table (tab:c2-summary); B foundational-questions essay (6 Q); C seesaw
%   + S8 derivation courses; D standard-siren reproducibility; E cross-paper
%   inheritance table + theorem/conjecture deck + promotion roadmap; F
%   notation. FIXED stale glossary fossil DNeff 0.30 -> canonical 0.04.
%   Added 2 graduate-textbook bibitems (Dodelson2020, WeinbergCosmology2008).
%   No frozen-physics change.
% =====================================================================
% =====================================================================
% ===== v12.8 (2026-07-09): PHASE 3 flagship + falsifiers. =====
%   (1) K-1..K-7 renamed to the cohort F-Hubble-N convention (body-wide),
%       section retitled 'The falsifier registry' (sec:falsifiers) with a
%       pre-declared-decision-rule framing.
%   (2) FIXED stale summary-table rows: F-Hubble-2 row was DNeff (leftover
%       from the pre-reframe K-2); now the siren rule matching its paragraph.
%       F-Hubble-4 row aligned to its paragraph (ladder -> 67). Decision-rule
%       column relabelled. F-Hubble-2 paragraph gains the explicit sigma rule.
%   (3) NEW section 'The decisive observation' (sec:decisive) + dated
%       kill-test timeline figure (fig:hubble-killtest-timeline) in the
%       cohort convention. Standard siren = the decisive test; near-term
%       ladder/CMB/DE programme around it. No frozen-physics change.
% =====================================================================
% =====================================================================
% ===== v12.7 (2026-07-09): PHASE 2 architecture + front matter. =====
%   (1) Single-page front matter: frontmatteronepagetrue + OverviewOfContents
%       + ListOfFigures before the detailed Contents.
%   (2) NEW section 'Roadmap of the Parts' (sec:roadmap-c2) at the end of
%       Part I, with a vertical 8-Part flowchart (fig:roadmap-c2), Part II
%       highlighted as load-bearing.
%   (3) Cohort section labelled (sec:gg-cohort); now cites all sixteen
%       siblings in programme order; canonical 17-paper roster inlined
%       (tab:cohort-roster), C2 row marked (this paper).
%   (4) Five missing cohort bibitems added (A8, P1, P2, P3, C3) with
%       C2-perspective Relevance + first-cite; Part A now the full 16.
%   (5) Executive Summary naming aligned to dual usage (Hopf-fibre
%       identification / Conjecture VIII). No frozen-physics change.
% =====================================================================
% =====================================================================
% ===== v12.6 (2026-07-09): PHASE 1 science-spine polish. =====
%   (1) Hopf-fibre identification introduced reader-facing in the
%       pi-inventory subsection (dual usage: geometric mechanism named,
%       Conjecture VIII retained as the D-star grade condition, mirroring A10).
%   (2) Provenance bullets promoted to one canonical honesty-triage table
%       (tab:c2-triage); grades match A10 entry for entry.
%   (3) Age corollary added to the early/late-split section: present age
%       0.948/H0 = 13.9 Gyr today, 13.8 at the inventory crossing (synced A10).
%   (4) zeta_Lambda wording aligned to A10's cost cluster 0.018--0.019;
%       dropped the non-canonical 1/56 label (3 sites). No frozen-physics change.
% =====================================================================
% v12.4 (July 5, 2026): Step-3 corridor-switch sync with A10 v12.89.
%   (1) Named the corridor-switch redshift z*=e^7 where the two-stage
%   corridor / early-late split is set out (topological, distinct from
%   measured z_dec, coincide 0.6%%; + loose-voice licence footnote).
%   (2) K-6 tensor kill synced to canonical r=0.0043 TWO-SIDED (was the
%   retired one-sided 'r<<0.01', which would have fired on our own
%   prediction). (3) conscious-episode ~125 ms -> ~400 ms (2 theta
%   cycles), the canonical A11/ledger value. Derived-66.6 spine and
%   candidate-resolution voice unchanged.
% v12.3 (July 2, 2026): clean-for-submission pass. ALL internal
%   version-history / self-referential content removed (this is the
%   primary initial submission): retired-corridor subsection deleted;
%   Appendices A (DNeff) + B (fEDE) removed; Appendix D rebuilt as the
%   derived-H0 worked calc; the DNeff/fEDE-based S8 calc, the detailed
%   GG-vs-EDE section, the corridor worked-numerics, and the
%   'GG-predicts-73.5' framing all removed; two version-history Q&A
%   entries deleted; 'earlier revision/this revision/retired/71.3/
%   companion recast' language purged throughout; dangling refs fixed.
%   Physics unchanged (derived 66.6, siren ~67, Local-Hole lift).
% v12.2 (July 2, 2026): C2-3 v2.0 presentation layer. Four-Part
%   back-matter tail installed (Conclusions [+acks moved to its end] ->
%   Technical Appendices -> Common Objections and Responses [new Q&A
%   Part, incl. the 'you moved 71.3->66.6' objection] -> References
%   [own Part]); bibliography A/B split (11 Arisaka cohort / 27 other).
%   Cohort map figure already present. DEFERRED to C2-4: single-page
%   front matter (Overview+LoF), Roadmap-of-Parts section, first-cite
%   links on every bib entry. v12.1 -> v12.2.
% v12.1 (July 2, 2026): C2-2 retool. Parts III-V purged of the retired
%   two-source (DNeff+fEDE) corridor: 'why-not-arbitrary' -> derived-anchor
%   anti-fit; cross-ledger + S8 sections re-anchored to the inventory;
%   benchmark sheet recomputed at 66.6 (background/DNeff=0.04/siren/lift
%   rows); kill-switch K-2 reframed to the siren test (the old DNeff<0.05
%   kill was inverted -- it now CONFIRMS the cohort DNeff=0.04); Class-6
%   and EDE-comparison 'what GG offers' corrected (GG needs no early
%   modification); fig-1 node reframed. S8 growth re-derivation kept
%   qualitative (labels preserved). v12.0 -> v12.1.
% v12.0 (July 2, 2026): PHYSICS RECAST (author-directed). Central
%   result now the DERIVED background rate H0 = 66.6 +/- 0.2 (import of
%   A10 h0-derivation): new abstract + exec summary; new section
%   'Geometrical determination of H0' (sec:absolute-h0) replacing the
%   dark-energy-ledger-anchor section; corridor recast as the early/late
%   split of the derived rate (early side unbiased = anchor; late side =
%   Local-Hole lift + named residual; siren prediction ~67); retired
%   71.3/1J^2 anchor with an honest note (sec:retired-corridor);
%   Appendices A (DNeff) + B (fEDE) retired (CMB side needs no
%   modification). Parts III-V/App C-F kept, numbers swept to 66.6;
%   deeper retool of the S8/kill material deferred to C2-2. v11.11 ->
%   v12.0.
% v11.11 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent); 1 noindent-before-paragraph no-ops removed. Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v11.10 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 2
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
%% ===== v11.9 (2026-06-30): added the AGE-of-the-universe corollary to
%%   sec:both-from-anchor -- the present age as the time-axis image of the
%%   corridor: H0^(0)=71.3 -> t0=13.0 Gyr (self-consistent), with 67.4->13.8
%%   and 73.5->12.6 as the two biased projections. Matches A10-Cosmos v12.39
%%   (identical t0=13.0 Gyr). No frozen-science change. v11.8 -> v11.9.
% =====================================================================
% ===== v11.8 (2026-06-29): DATA-SAFE H_0 RECONCILIATION WITH A10 v12.35. =====
%  Reframed the H_0 result (abstract, exec summary, sec:both-from-anchor, bottom
%  line) to the author-chosen DATA-SAFE story: robust output is the CENTRAL value
%  H_0^(0)=71.3 (matches GW-siren >~71); the two measured endpoints are biased
%  readings; CMB-side closure from the data-safe budget (relic DeltaNeff~0.04 +
%  ledger f_EDE=0.0326) is PARTIAL (67.4 -> ~68.6-70), residual ~2-4%% open (K-2);
%  local +3%% is the U(1)_B graviphoton (open). Replaces the v11.7 clean 67.7/73.4
%  (DeltaNeff=0.30/0.35) projection and the 0.6%% cross-check, which needed a
%  radiation excess current N_eff bounds disallow. Appendix-A raw channel
%  DeltaNeff~0.30 retained as the channel estimate, channel-vs-data gap named as
%  the open problem. v11.7 -> v11.8.
% =====================================================================
% =====================================================================
% ===== v11.7 (2026-06-29): DERIVED ABSOLUTE H_0 ANCHOR (internal memo). =====
%  (1) New section sec:absolute-h0: zeta_Lambda fixed by MSMF/OGN bounded-cost
%      search to cost-2 minimum 1/J^2=1/49 (cost table; ledger-monomial-not-loop
%      intuition; Asserted grade, A9-SM 8-Yukawa parallel); anchor H_0^(0)=71.3
%      km/s/Mpc derived step by step from the dark-energy seesaw + Friedmann.
%  (2) New section sec:both-from-anchor: early/CMB 67.7 and late/local 73.4 as
%      PREDICTIONS off the derived 71.3; f_EDE=e^{-K/2^J}N/(2K)=0.0326 ledger
%      closed form; two-route consistency check (71.3 vs 70.9, 0.6%%); the anchor
%      sharpens eps_early into a DeltaN_eff~0.35 prediction (kill-switch K-2);
%      honest per-input grades.
%  (3) Abstract, executive summary, and bottom line updated to headline the
%      derived anchor (corridor no longer borrows H_ref from Planck).
%  No frozen-science change. v11.6 -> v11.7.
% =====================================================================
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% A10 bib-title reconciled to canonical cover title (cross-citation integrity), 2026-06-27.
% ===== C2-Hubble v11.4 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG's own self-description on its OWN bulk/throat/stage. Bulk noun de-qualified; the
% throat-geometry adjective remapped to scale-throat; radial profile and parameter remapped to
% scale-*. GG-self bibitem cohort titles de-qualified. KEPT the canonical RG-running disclaimer
% footnote verbatim (it carries the Randall--Sundrum contrast and the terminology gloss), the
% 'six-dimensional geometric corridor' thesis term, all cross-reference args, and all comments.
% Body only (after the document body marker). No physics / results / values changed. =====
% ===== C2 v11.3 (2026-06-25): cohort central-claim funnel (consequence variant). =====
% Added the shared open-system funnel (sec:central-claim/fig:central-claim) with this paper's own
% payload. Same funnel as A9/A10/A11. No new physics.
% =====================================================================
%  C2-Hubble v11.2 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%  gg-10->gg-11; centered headers + homogenized Acknowledgments via gg-11.  The \LaTeX shim is KEPT (paper uses \LaTeX in body).
%  No prose/figure/number content changed.
% =====================================================================
% =====================================================================
%  Arisaka-GG-C2-Hubble-v11.0 (Stage 3A: naming-convention theorem-id rollout) -- 2026-06-20
%  Official cohort theorem ids assigned: T-Hubble-1 (1 theorem).
%  Appended (T-Hubble-N) to own theorem-env titles; recalled classical
%  theorems left untagged.  No physics changed.
% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2d. =====
%  Terminology only; NO physics changed. A3 bibitem '...Warped AdS Bulk' -> '...Warped
%  Bulk' (A2 bibitem already AdS-free). Added holographic-RG disclaimer footnote; v10.3 -> v11.0.
% =====================================================================
% --- v10.3 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. ---
% GU-5 RENAME (June 14, 2026): cohort harmonization with GG-A4-Euclid -- the five-constant inventory and its theorem are named Geometric Universal Five Constants / GU-5 (\GUfive, \T{-GU5}); the older 'Conversion-Factor Pentad'/'Conversion-Factor Triad' wording and the 'T-CFP' theorem tag are retired.
\pdfoutput=1
%% v10.2 (June 12, 2026): Legacy-name fix (author ruling: GD-2026-5
%%   = GG-A3-BI6): the ArisakaA3BI6 bibitem updated from the legacy
%%   GD-2026-5 name/title to the canonical A3-BI6 form. No other
%%   changes.
%% v10.1 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-c2, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.0 (June 11, 2026): VERSION PROMOTION (author decision, June 11):
%%   v1.9 promoted to v10.0 for cohort version consistency, alongside
%%   P2-N3 and P3-Electron, completing the 17-paper v10 cohort in
%%   [GG-v10]. Content identical to v1.9 except this entry and the
%%   \PaperVersion stamp.
%% Arisaka-GG-C2-Hubble -- the H0DN / Hubble-corridor synthesis report
%% A community H0 measurement (Casertano et al. 2026) seen through GG-Theory
%% -------------------------------------------------------------------
%% v1.1 (June 10, 2026) -- Phase 2 content corrections, author-approved
%% June 10, 2026 (see C2-PHASE1-AUDIT-2026-06-10.md):
%%  E1  eps_local reclassified everywhere as a phenomenological
%%      requirement, not a result (abstract, ES, theorem, U(1)_B
%%      section, advance-statement list, uniqueness list, Part V,
%%      benchmark sheet, App C, App D, glossary). App C |delta_B|~3-5
%%      step WITHDRAWN as unphysical (delta >= -1); first-pass U(1)_B
%%      estimate ~1e-3 (~60x short) stated; required lift
%%      0.040 +/- 0.013 added. Wording matches A10-Cosmos v10.5 Part IV.
%%  E2  Coefficients 0.13/0.40 relabeled as EDE-literature response
%%      slopes, cite Poulin et al. 2019 (PRL 122, 221301).
%%  E3  DeltaN_eff exposure caveat (Planck baseline dictionary prefers
%%      0.04 +/- 0.18; EDE-degeneracy joint-fit open requirement) in
%%      the DNeff subsection, App A, K-2, Class-6, benchmark row.
%%  E4  S8: Hill et al. 2020 net-sign caveat added; late branch
%%      synchronized to the A10 band 0.78 +/- 0.01 (KiDS-1000,
%%      Asgari et al. 2021); first-order 0.814 demoted to estimate.
%%  E5  Corridor band unified to 72.8-74.2 (abstract had 72.9-74.3);
%%      anchor conversion stated (67.24 -> 73.5 +/- 0.7 here;
%%      67.4 -> 73.7 +/- 0.7 in A10); "60%" -> 55%; 7.6sigma -> 7.1sigma.
%%  E6  Graviphoton screening ratio renamed eta_B -> zeta_B (symbol
%%      collision with the cohort-wide baryon-to-photon ratio eta_B).
%%  E7  Pre-registration language downgraded to "formulated
%%      independently of the H0DN analysis" (priority claim pending
%%      documentation); "0.0 sigma / four significant figures"
%%      advertising replaced by requirement-vs-interval framing.
%%  E8  App A: kappa_Qi honesty note (3.1 is selected; K/N^2 = 35/9
%%      = 3.89 is not equal to it); App B classification note.
%%  E9  Benchmark sheet re-statused to the A10 v10.5 conditional
%%      triage; "16 of 17" rhetoric revised; comparison-matrix GG
%%      cells adjusted (inheritance: partial).
%%  E10 Build fixes: \Hlate^{mid} double superscript; \LaTeX clash in
%%      \GGAcknowledgments (renewcommand workaround); \boxed removed
%%      (house style); A1-A9 inheritance statement added (Part II);
%%      bibliography files repointed to canonical names.
%% -------------------------------------------------------------------
%% v1.2 (June 10, 2026) -- Phase 3 house-style completion:
%%  H1  Front matter per A2 template: \GGTitlePage cover with
%%      \PaperCTwo{}; plain-English abstract (no citations, "After
%%      all" close); ExecutiveSummary environment with \paragraph
%%      headers; \GGKeywords; \GGTableOfContents; suggested-citation
%%      and companion-note blocks folded into the ES.
%%  H2  Bibliography converted to inline bold-author + Relevance,
%%      surname-sorted; Poulin2018EDE/Poulin2019 duplicate merged;
%%      eleven observational references newly cited inline (anchors,
%%      variant classes, pipelines, peculiar-velocity model, JWST
%%      cross-checks); zero orphan bibitems (L90).
%%  H3  Cohort keys migrated Arisaka2026* -> canonical ArisakaA*/C*
%%      (A1, A2, A3, A4, A5, A6, A7, A9, A10, A11, C1); textual
%%      GG-2026-7F/6F/0/X -> \ATen{}/\ANine{}/\AEleven{}/\ASeven{};
%%      A10 section pointers re-targeted (Part IV / headings);
%%      K-7 twin-peak now cites C1-Kaxion.
%%  H4  PaperNumber -> Arisaka-GG-C2-Hubble; Predictions-Branch
%%      status; hyperref metadata; v1.2 stamps.
%% -------------------------------------------------------------------
%% v1.3 (June 10, 2026) -- Phase 4 hostile-referee fixes (see
%% C2-HOSTILE-REFEREE-REPORT-2026-06-10.md):
%%  R1 (M1) App A DeltaN_eff chain restored to standard normalization:
%%     prefactor (8/7)(11/4)^{4/3} = 4.40 printed correctly (was
%%     "1.40 x 1.40"); bosonic (g/2) factor inserted; xi exponent 4
%%     (quartic) on the temperature ratio; per-dof 0.027 cross-check
%%     made exact; kappa_Qi 3.1 -> 2.8 (still selected, still inside
%%     the stated [2.5,3.5] window); band [0.24,0.34] -> [0.27,0.38];
%%     repair disclosed in the honesty note in the text.
%%  R2 (M2) alpha^-1(M_Z) benchmark status "+0.99sigma" (false; 8.8
%%     sigma at quoted precision) -> fractional-agreement convention.
%%  R3 (M3) App D stress test f_EDE->0.005: 69.8/72.6 (copy of the
%%     DeltaN=0.20 case) -> correct 70.0/72.8 (0.9sigma).
%%  R4 (M4) "meets all six criteria ... without free parameters
%%     fitted to the tension" -> declared-exception wording
%%     (eps_local is calibrated; criterion 4 conditional).
%%  R5 (m1) 70.68 -> 70.67 (= 67.24 x 1.051), two sites.
%%  R6 (m2) median variant offset ~0.1sigma -> ~0.2sigma (computed
%%     0.19sigma).
%%  R7 (m3) App D sigma metrics stated explicitly (7.1 combined;
%%     3.5 H0DN-only / 3.2 combined after early factor).
%%  R8 (m4) abstract "carried by" -> "guided by" the integer ledger.
%%  R9 (m5) Poulin2019 Relevance: 0.13 slope attribution removed.
%%  R10 (m6) K-4 "2--3sigma" -> "about 3sigma" (combined widths).
%% -------------------------------------------------------------------
%% v1.4 (June 10, 2026) -- Phase 5 arXiv production:
%%  P1 pdfoutput=1 (arXiv PDFLaTeX flag, first line).
%%  P2 style file repointed to the canonical ucla-arisaka-gg-10.sty
%%     (shipped in the tarball; source is self-contained: inline
%%     bibliography, no external .bib/.bbl).
%%  P3 Overfull boxes > 20 pt repaired by rewording (V99/V99A
%%     sentence split; K-4 parenthetical moved after the number).
%%  P4 v1.4 stamps. Category plan: astro-ph.CO primary; gr-qc,
%%     hep-ph cross. Plain-text abstract (<= 1920 chars) and
%%     submission manifest shipped alongside.
%% -------------------------------------------------------------------
%% v1.5 (June 10, 2026) -- author-approved title change:
%%  T1 Main title: "The Hubble Tension in the Light of GG-Theory";
%%     subtitle: "Two Hubble Constants as the Two Ends of One
%%     Six-Dimensional Geometric Corridor". The previous descriptive
%%     title demoted to the cover tagline (keeps the H0DN consensus
%%     value on page 1); hyperref pdftitle and manifest updated.
%% -------------------------------------------------------------------
%% v1.6 (June 10, 2026) -- cover refinements (author request):
%%  C1 Headline = "The Hubble Tension in the Light of GG-Theory"
%%     (paper code removed from the headline; identity retained in
%%     the flushright block and body running headers).
%%  C2 Cover running header suppressed (\thispagestyle{empty},
%%     GG-A-series convention).
%%  C3 Cover abstract extended by one commitment paragraph (still
%%     one cover page).
%% -------------------------------------------------------------------
%% v1.7 (June 10, 2026) -- final C2<->A10 consistency audit fixes:
%%  X1 DeltaN_eff attribution de-circularized: the compact-angle
%%     estimate lives in this paper's Appendix A; A10 carries 0.30
%%     as the corridor's inherited input (was "The A10 derivation
%%     gives...").
%%  X2 ES "re-derives the corridor theorem" -> "restates and
%%     numerically evaluates".
%%  X3 "Hlate ~ 1.082 Href" clarified as the eps_local = 0.03 edge
%%     (upper edge 1.104), matching A10's printed 1.082.
%%  X4 z_Lambda-m benchmark row marked "implied by the Omega_m row,
%%     not independent" (A10 v10.5 Tier-2 convention).
%% -------------------------------------------------------------------
%% v1.8 (June 10, 2026) -- reciprocal synchronization with A10 v10.6:
%%  Y1 ArisakaA10Cosmos bibitem made version-agnostic ("preprint",
%%     was "preprint (v10.5)"); its Relevance line now records the
%%     completed two-way synchronization (A10 v10.6 cites this paper
%%     under its final title and states the anchor unification in the
%%     completed tense).
%% -------------------------------------------------------------------
%% v1.9 (June 10, 2026) -- C1-A10 consistency audit fix:
%%  Z1 The Kaxion haloscope frequency "~4.5 GHz" (two sites: K-7
%%     discussion and Priority 6) was the stale pre-correction value
%%     (18.6 ueV = 2.3 x 8.1 ueV, i.e. the normalisation error C1
%%     documents and corrects); replaced by C1's unique laboratory
%%     window nu_K in [1.59, 2.13] GHz (m_K = 8.1 +/- 0.8 ueV); the
%%     paper-local "cap-mixed branch" phrase replaced by C1's
%%     terminology. No other changes.
%% -------------------------------------------------------------------
%% ---- vv15.11 (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.
%% 4 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
%% And the showcase headlines brought to one rendered form.
%% The house form is the style file's own default, \Large\bfseries\sffamily,
%% which is what \GGShowcaseHead sets when it is called with no optional
%% argument, and which forty-three of the cohort's fifty-seven showcase
%% headlines already used.  Fourteen did not, and the difference was
%% visible to a reader turning pages: four headlines printed at \huge,
%% half again the house height, and three at \large, smaller than it.
%% 1 headline in this paper.  Only the size changes; the \boldmath in
%% a headline that carries math is kept, the subtitle under a headline
%% keeps its own smaller size, and \GGShowcaseHeadSkip, which is vertical
%% space rather than type, is left exactly as each paper sets it.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}

\PaperSeries{GGC}
\PaperNumber{Arisaka-GG-C2-Hubble}
% ============================================================
% v13.73 -> v15.0  Stage 1 of the v15 migration.  August 15, 2026.
%   * NO paper conversion: C2 declares no paper option, so the
%     article default applies and it already builds on Letter.
%     First paper in the migration that needs none -- and so the
%     only one so far immune to the 3.7 % widening that cost A9 and
%     C1 a showcase-page repair.
%   * NO booktabs work: 0 \hline, 0 \cline, 0 vertical rules.
%     C2 was already at the section 8 table standard.
%   * American spelling, total, in printed text: 387 substitutions
%     in 90 distinct forms, every one hand-classified.
%   NO PROSE WAS WRITTEN IN THIS STAGE.
% ============================================================
%% ------------------------------------------------------------
%% Stage 2A, same day.  The float standard.
%%   18 captions printed below the 260-character floor.  Each was
%%   EXTENDED and none rewritten -- the author's sentence is
%%   unchanged byte for byte and the new text is appended.
%%   NOTHING ELSE WAS NEEDED IN THIS STAGE, and that was measured:
%%   float_audit reports NO CAPTION 0 and UNBREAKABLE-AND-LONG 0 at
%%   the Stage 1 baseline, so this is the first paper of the
%%   migration with no bare grid to promote and no uncaptioned
%%   float to caption.  A9, A10 and C1 each needed both.
%%   ALL 18 ARE NEW PROSE, listed by label in the report.  No
%%   extension introduces a numerical claim; every number in one is
%%   already printed in the table it describes, and every count of
%%   rows in one was taken by counting the rows.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2C, same day.  Plain-English Part closes.
%%   9 closes, one per content Part (I-IX).  The three back-matter
%%   Parts -- Technical Appendices, Common Objections, References --
%%   get none, per A5's rule.
%%   ALL NINE ARE NEW PROSE IN THE AUTHOR'S VOICE and are listed by
%%   Part and by page in the report, for review or rewriting.
%%   C2 already ends eight of the nine content Parts well -- five
%%   with a `Summary ... and the road to Part N' section, two with a
%%   Coda, one with a verdict paragraph.  Nothing existing was
%%   removed or rewritten; the closes sit after them and do the
%%   different job of saying what the Part meant to a reader who
%%   does not have the paper's technical register.
%%   Part IX ends with TWO terminal constructs -- \section*{Envoi}
%%   and \GGAcknowledgments -- and its close is placed before the
%%   earlier of them, so the Envoi stays the last thing in the body.
%%   No \ref{part:...} appears in any of them: A9 taught that a
%%   paper's part labels need not match the numbers they print, so
%%   where a close points at another Part it names it in words.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 2E, same day.  Short caption forms, and the List of
%% Tables C2 never had.
%%   Every other migrated paper -- A1 through A10 and C1 -- calls
%%   \GGListOfTables.  C2 called only \GGListOfFigures, with 48
%%   floats, half of them tables.  One is added after the LoF.
%%   A list entry uses the FULL caption unless a short form is
%%   given.  On A9 that filled the List of Tables with 400- and
%%   500-character paragraphs and overflowed one line by 65.17pt.
%%   All 18 of C2's Stage 2A extensions were to TABLE captions, so
%%   the only reason that did not happen here is that there was no
%%   List of Tables to overflow.  The 28 short forms are therefore
%%   written BEFORE the list is added, not after.
%%   Each short form is the FIRST SENTENCE of its own caption,
%%   verbatim, truncated at a word boundary if over 150 printed
%%   characters.  NO PROSE WAS WRITTEN IN THIS STAGE -- every
%%   short form is a prefix of text the caption already carried.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 3, same day.  Figures.
%%   Every picture measured with measure_plate.py, one compile per
%%   picture.  C2 declares no paper option, has always built on
%%   Letter, and therefore has NO A4 -> Letter widening and no
%%   widened showcase page to repair -- overfull \vbox 0 at intake
%%   and 0 now, where A9 and C1 each arrived with one.
%%   Base fonts 7.27 8.05 8.07 8.08 8.44, house band 7.40-8.18.
%%   ONE plate placed: fig:roadmap-c2 at 8.44pt, 0.90 -> 0.831 of
%%   \textwidth.  This is not the widening consequence frozen
%%   decision 3 leaves alone; C2 was never A4, so the oversize is
%%   the plate's own.
%%   fig:cohort-map-c2 measures 7.27pt, below the band, and is left
%%   alone: section 8 exempts the cohort map, and its house fraction
%%   is 0.996 in any case.
%%   fig:hubble-killtest-timeline declares no base font, so the rule
%%   has no B to divide by.  REPORTED, NOT GUESSED.
%%   The 17 \includegraphics plates are all set relative to
%%   \textwidth: no landscape block, no \linewidth plate, and the
%%   only hard millimetre values are TikZ node sizes inside the
%%   exempt cohort map.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 4, same day.  Palette.
%%   7 violations in 3 pictures: 3 bare `draw', 3 fills above !18,
%%   1 tint border.  palette_fix reached six.  No comment was
%%   rewritten -- the A10 defect did not recur.
%%   Its width clamp again cannot tell a RULE from a MARK, and two
%%   sites are figure content and are reverted: the 2.2pt siren-
%%   accumulation bar and the 3.6pt O5 bar under the kill-test
%%   timeline.  These two carry TWO MEANINGS BY TWO WEIGHTS -- the
%%   O5 run inside the accumulation window -- and clamping both to
%%   1.10pt made them identical.  Fifth and sixth instance of the
%%   object A9, A10 and C1 each lost.
%%   KEPT: the 1.4 -> 1.1pt time arrow, as A9 and C1 kept theirs;
%%   and the yellow!20 -> !13, orange!22 -> !14, orange!32 -> !16
%%   fill compressions, which are box fills and not marks.
%%   ONE violation palette_fix does not reach, set by hand:
%%   sLAD's blue!55 -> !55!black (its three siblings in the same
%%   style block were already shades).
%%   FLAGGED AND NOT DONE: fig:central-claim's shared `box' style
%%   now draws black!55, and one node it styles is filled yellow,
%%   which section 8 says takes an olive border.  Giving it one
%%   means splitting a style shared by six nodes of four hues --
%%   the picture's construction, not its palette.
%%   NO PROSE WAS WRITTEN IN THIS STAGE.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% Stage 5, same day.  Integrity.  Three uncited companions.
%%   C2 carried 19 of the 22 cohort papers.  The three missing were
%%   the SAME THREE C1 was missing -- P6-Proton, W1-Symmetry,
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\PaperVersion{v20.2}
\PaperDate{August 22, 2026}
\PaperStatus{Synthesis Report (Cosmology Branch)}

\usepackage{enumitem}
\usepackage{xcolor}
\usepackage{url}
\usepackage{natbib}

%% v1.1: avoid the \@ clash raised by \LaTeX inside \GGAcknowledgments
%% (known v8--v10 sty quirk; same workaround as A10-Cosmos v10.5)
\renewcommand{\LaTeX}{LaTeX}

%% v1.2: cohort macros not in the sty list, and hyperref metadata
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\PaperCTwo}{\textsf{GG-C2-Hubble}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}% v13.65: canonical cohort form (the sty \providecommand carries the earlier Gi4 form)
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}% v13.65: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFour)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}% v13.65: Foundations Branch member (canonical form, Cohort-Map-19 v1.4; sty defines no \PaperPFive)
\hypersetup{
  pdftitle = {The Hubble Tension in the Light of GG-Theory: Two Hubble Constants as the Two Ends of One Six-Dimensional Geometric Corridor (Arisaka-GG-C2-Hubble)},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {GG-Theory program -- Hubble-corridor synthesis report},
  pdfkeywords = {GG-Theory, Hubble tension, H0DN, Hubble corridor}
}

%% additional local macros
\providecommand{\Hzero}{\ensuremath{H_0}}
\providecommand{\Hlate}{\ensuremath{H_0^{\mathrm{late}}}}
\providecommand{\Hearly}{\ensuremath{H_0^{\mathrm{early}}}}
\providecommand{\Href}{\ensuremath{H_0^{\mathrm{ref}}}}
\providecommand{\Hloc}{\ensuremath{H_{\mathrm{loc}}}}
\providecommand{\Hbg}{\ensuremath{H_{\mathrm{bg}}}}
\providecommand{\kmsMpc}{\,\mathrm{km\,s^{-1}\,Mpc^{-1}}}
\providecommand{\fEDE}{\ensuremath{f_{\mathrm{EDE}}}}
\providecommand{\DNeff}{\ensuremath{\Delta N_{\mathrm{eff}}}}
\providecommand{\epslocal}{\ensuremath{\epsilon_{\mathrm{local}}^{(B)}}}
\providecommand{\epsearly}{\ensuremath{\epsilon_{\mathrm{early}}}}
\providecommand{\SEight}{\ensuremath{S_8}}
\providecommand{\LCDM}{\ensuremath{\Lambda\mathrm{CDM}}}
\providecommand{\HzeroDN}{\ensuremath{H_0^{\mathrm{H0DN}}}}
\providecommand{\Uoneb}{\ensuremath{\mathrm{U}(1)_B}}

%% Macros referenced from GG-2026 series (not all defined in .sty)
\providecommand{\GGtheory}{\GGTheory}
\providecommand{\GGsix}{\ifmmode\mathsf{GG\text{-}6}\else\textsf{GG-6}\fi}
\providecommand{\GGfour}{\ifmmode\mathsf{GG\text{-}4}\else\textsf{GG-4}\fi}
\providecommand{\Gifour}{\ifmmode\mathsf{Gi\text{-}4}\else\textsf{Gi-4}\fi}
\providecommand{\Qifour}{\ifmmode\mathsf{Qi\text{-}4}\else\textsf{Qi-4}\fi}
\providecommand{\BIsix}{\ensuremath{\mathrm{BI\text{-}6}}}
\providecommand{\Mlock}{\ensuremath{M_{\mathrm{lock}}}}
\providecommand{\TGG}{\ensuremath{\mathrm{TGG}}}
\providecommand{\CCI}{\ensuremath{\mathrm{CCI}}}
\providecommand{\GFF}{\ensuremath{\mathrm{GFF}}}
\providecommand{\GRIP}{\ensuremath{\mathrm{GRIP}}}
\providecommand{\tr}{\ensuremath{\mathrm{tr}}}
\providecommand{\Tr}{\ensuremath{\mathrm{Tr}}}
\providecommand{\dd}{\ensuremath{\mathrm{d}}}
\providecommand{\EIC}{\ifmmode\mathsf{EIC}\else\textsf{EIC}\fi}
\providecommand{\OH}{\ifmmode\mathsf{OH}\else\textsf{OH}\fi}
\providecommand{\IH}{\ifmmode\mathsf{IH}\else\textsf{IH}\fi}

%% theorem-like environments are already set up by the style file;
%% additional "claim" for interpretive statements
\newtheorem{claim}{Claim}
\providecommand{\GRIPin}{\textsf{GRIP-In}}
\providecommand{\GRIPout}{\textsf{GRIP-Out}}


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%% 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.



%% v13.45: A10-Cosmos Q&A layout for Part XI -- flush objections,
%% indented responses (definition mirrored from A10's preamble).
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  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

\setlength{\tabcolsep}{6pt}
\begin{document}
% ---- v13.69 (August 2, 2026) ----
% 'The essence: one chain, one observable' gains the closing
% paragraph 'The chain, read on the two-fibre anatomy': H as the
% interval's observable (cooling is descent; d ln t/dr = 2;
% H = 1/2t --- 'how fast does the fibre that forgets forget'),
% pinned entirely by circle-side quantities (Omega_m = 1/pi,
% Omega_b = 1/2pi^2, corridor switch z_* = e^J); why the rate
% could be derived, and derived without a dial; the tension as
% two dictionaries reading two fibres, the corridor as the
% reconciliation. Part of the cohort-completing sweep.

\thispagestyle{empty}

\begingroup\setstretch{1.0}
\GGTitlePage
 {The Hubble Tension\\[2pt] in the Light of \GGTheory{}}
 {\parbox[t]{0.90\textwidth}{\centering\baselineskip=19pt\relax
 The Hubble Constant as One Reading of a Derived\\
 Expansion History \(H(t)\)}}
 {}
\endgroup

\begin{abstract}
\noindent
For a century, ``how fast is the universe expanding?'' was a question
answered by measuring. By 2026 the measuring had split: read from the
afterglow of the Big Bang, the rate comes out near \(67\) (in
astronomers' units); read from the local ladder of stellar distances,
\(73.50 \pm 0.81\) --- a seven-sigma gap that no single method
explains.

This paper reads the disagreement through \GGTheory{}, in which the
expansion rate was never a number to be measured but a function of
time to be derived: a six-dimensional geometry fixes the cosmic recipe
as exact expressions in \(\pi\), Einstein's equations turn that recipe
into the entire history \(H(t)\), and today's value is one snapshot of
the curve --- expressible in a single analytic line in the three
laboratory constants \(\{c, G_N, \hbar\}\) and two integers of the
theory's ledger, giving \(66.6 \pm 0.2\) --- on the afterglow side
of the split, \(1.6\sigma\) from the CMB combination. More than ten of the standard model's fitted numbers re-emerge,
unfitted, at the percent level. The early measurement needs no new
physics.

The local \(73.5\) is then the puzzle --- and not, this paper's audit
finds, a settled consensus: the 2026 landscape's forty-five
determinations span \(67\)--\(73\), the seven sigma certifying one
superb network's internal agreement. The paper prices the local
reading honestly: a derived \(+2\)--\(3\%\) neighborhood lift, and a
named, still-open calibration residual, confronted with the field's
own published corrections.

The referee is scheduled: gravitational-wave \emph{standard sirens}
read the rate with no ladder and no afterglow, and \GGTheory{}
predicts they converge near \(67\) --- about twenty-one events from
now. Near \(73\), this paper is wrong, and says so in advance. Either
way the century-old ``constant'' has finished its work: what remains
to be tested is not a number but a law.
\end{abstract}

\begin{ExecutiveSummary}
The Hubble tension is usually told as a duel between two measurements,
and the reader is asked to pick a side. This report tells it the other
way around: in the \GGTheory{} series the expansion rate is not
measured and then explained --- it is \emph{derived}, and once it is
derived the tension becomes a diagnosis: one measurement is reading
the true rate, and the other carries a bias whose size, sign, and
possible origins can be written down and priced. This summary walks
that argument in plain language; every step is worked in full inside.

\paragraph{Three results organize this report.}
\emph{First, the Hubble parameter is derived directly, as a function
of time.} The geometry fixes the universe's contents in closed form,
Einstein's equations turn the contents into the whole history
\(H(t)\), and the famous ``constant'' is demoted to a snapshot ---
the value the falling curve happens to take today. That snapshot
collapses to one analytic line,
\[
H_0 \;=\; e^{-4K}\,
\sqrt{\frac{2\,\pi^{5}\,c^{5}}{3\,L\,K\,(\pi-1)\,\hbar\,G_N}}
\;=\; 66.7~\mathrm{km\,s^{-1}\,Mpc^{-1}},
\]
three laboratory constants, two integers of the theory's ledger
(\(L = 5\), \(K = 35\)), and the two transcendentals \(\pi\) and
\(e\) --- checkable on a calculator in under a minute. And the same
ledger keeps going: more than ten of the numbers \(\Lambda\)CDM must
fit --- the matter, vacuum, and baryon fractions, the recombination
epoch, the age, the growth amplitude, the damping scale, the
radiation excess among them --- re-emerge, unfitted, at the percent
level. \emph{Second, the seven sigma is not the whole field.} The
celebrated \(7\sigma\) belongs to one network's internal
consistency; this paper's audit of all forty-five determinations
published by 2026 finds them spanning \(67\)--\(73\), organized into
three bands, with a middle band near \(70\) steadily filling as the
field's own corrections land. \emph{Third, the century-old concept
has finished its work.} If the rate is a derived function, the
constant is obsolete the way measured molecular weights became
obsolete after the periodic law --- and cosmology can move on to a
new, falsifiable precision program beyond \(\Lambda\)CDM, with a
dozen dated tests in place of one contested decimal.

\paragraph{The Hubble ``constant'' is not a constant.}
The expansion rate is a \emph{function of time}: enormous in the hot
early universe, \(66.6\) today, settling forever onto a floor of
\(55.0\). What astronomers call \(H_0\) is simply where the falling
curve crosses the present --- a shorthand kept from the century when
only that one point could be measured. Much of the tension's mystery
dissolves once the question is restated: not ``what is the
constant?'' but ``what is the curve, and are we reading one of its
points wrongly?''

\paragraph{The early measurement needs no new physics.}
The derived rate lands \(1.6\sigma\) from the value inferred from the
cosmic microwave background --- and the test is sharper than
agreement: fed the exact \(\pi\)-locked recipe with no background
parameter tuned to the sky, the theory reproduces the afterglow's
acoustic pattern to about a percent --- a consistency at the percent
level, not a fit; taken as exact the chart is excluded, and Part~III
prints that exclusion itself, at full strength --- and where it
misses, it misses \emph{coherently}: seventeen
observables, one direction, one small named step. A lucky fit
scatters its errors; a derivation displaced by one named correction
lands on one side --- which is what the data show. Nothing needs to
be added to the young universe.

\paragraph{The local measurement, priced rather than blamed.}
The distance ladder's \(73.50 \pm 0.81\) is a magnificent measurement
--- thirty-eight specialists, every method cross-checked against
every other --- and this paper does not wave it away. It prices it.
First, the derived part: our galactic neighborhood is mildly
under-dense, so any rate measured from inside reads high by a derived
\(2\)--\(3\%\), carrying the true rate to about \(68.5\). The
remaining \(\approx\!7\%\) the theory does \emph{not} explain, and
refuses to invent physics for. Instead it names the only places such
a bias could live --- the shared calibrations no cross-checking can
see --- and works through them against the ladder's own error budget:
a dust law the infrared will strip away, a statistical prior a
simulation can test, a choice of calibrator galaxies a complete JWST
sample will settle; each with a published correction, a published
rebuttal, and now a dated test.

\paragraph{The whole field, on one page.}
Rather than argue measurement by measurement, the paper draws them
all: every 2026 determination of the expansion rate --- forty-five
values from nine kinds of measurement --- in one chart and one
table. The field organizes itself into three bands: a tight
early-universe band around \(67\)--\(68\), the distance-ladder band
at \(73\)--\(74\), and --- new this decade --- a middle band near
\(70\) filling steadily with corrected ladders, dust-free
supernovae, fast radio bursts, and gravitational-wave values. The
derived rate sits at the floor of the first band; the lifted rate at
the floor of the middle one. Six families of proposed fixes are read
against the same chart, and each is asked the one question the bands
make unavoidable: not ``67 or 73?'' but \emph{``why three bands ---
and why is the middle one filling?''}

\paragraph{The referee, and the date.}
It can be settled from outside. A merging pair of neutron stars
broadcasts its own absolute distance in gravitational waves ---
Einstein's equations themselves certify the luminosity, so there is
no ladder, no calibration, no afterglow anywhere in the chain. One
such ``standard siren'' has already been read, straddling both camps
with fifteen-percent precision. About twenty-one such events separate
the camps at \(3\sigma\), about fifty-eight at \(5\sigma\), and the
coming decade's observing runs are expected to supply their
equivalent --- bright if the counterparts cooperate, dark and
statistical if not. The prediction is printed and cannot be revised:
the sirens converge near \(67\). Near \(73\), this paper is finished
--- and a catalog converging \emph{between} the camps would kill
both readings at once.

\paragraph{Beyond the constant: the new precision cosmology.}
Retiring the constant is not a loss; it is a promotion --- for the
whole field. The same geometry that prints \(66.6\) prints a
falsifiable number nearly everywhere the coming decade will look:
the primordial gravitational-wave amplitude (\(r = 0.0043\),
two-sided); an exact dark-energy equation of state (\(w = -1\));
a dark-matter particle with a laboratory address (\(8.1\,\mu\)eV,
near \(1.95\)~GHz); the growth amplitude (\(S_8 = 0.759\)); the
expansion history at every redshift, down to the centimeters-per-
second-per-decade drift the largest telescopes will one day watch
live. Twelve dated appointments --- from Gaia's astrometry this
December to the siren verdict of the 2030s --- replace the dispute
over one decimal. This, the paper argues, is where the real
excitement begins: precision cosmology whose predictions can
\emph{fail}, not precision metrology of a number no theory owned.

\paragraph{What is proved, and what is still assumed.}
The series keeps an explicit ledger, and this paper inherits its
discipline. The \(\pi\)-locked recipe is a theorem \emph{conditional
on one named geometric lemma}, still open, and is labeled so
everywhere it is used. The closed form's last coefficient is selected
by the theory's own naming rules, its closure pre-registered as a
falsifier. The local lift is derived in sign and rough size, not in
value. And the last \(\approx\!7\%\) of the local excess is not
derived at all: it is named, bounded, and handed to the sirens.
Nothing in this summary is graded stronger inside the paper than it
is stated here.

\paragraph{Why it matters.}
The deepest constants of physics have retired before: the speed of
light was measured for two centuries until Maxwell derived it; the
Rydberg was spectroscopy's best number until Bohr computed it; the
\(W\) and \(Z\) masses were predictions before they were
measurements. Each retirement marked the moment a field stopped
doing natural history and started doing law --- and each time, the
excitement moved from the next decimal to the places the new law
could fail. The Hubble constant, on the reading defended here, is
next in that lineage: the last measured constant of cosmology,
retiring into a derived curve that began with a predicted burst,
runs today on a predicted budget, and will end on a predicted calm.
And there is a quieter thought beneath the argument, worked out in
the closing Part: the same geometry that times the expansion timed
the stars that forged our atoms and the chemistry that built our
instruments --- so a measurement of \(H_0\) was never an outsider's
act performed on the universe from nowhere. We are not the audience
of the expansion; we are the part of it that has begun to keep the
log. That claim is either true or false, and unusually for claims of
its size, the falsifying measurements are named, numbered,
scheduled, and priced inside. Either the sirens read \(67\), and the
oldest number in cosmology turns out to have been a theorem all
along --- or they read \(73\), and this paper becomes what a wrong
theory should be: a clean, public, instructive ruin. Both outcomes
are progress. Only one of them is possible.
\end{ExecutiveSummary}

\GGKeywords{\GGTheory{} program; Hubble tension; \(H_0\); derived
Hubble parameter \(H(t)\); closed-form \(H_0\); H0DN; Local Distance
Network; distance ladder; 2026 \(H_0\) landscape; \(\pi\)-locked
inventory; sound horizon; standard sirens; Local-Hole lift;
distance-ladder systematics; backreaction bound; \(S_8\) tension;
kill criteria; decision calendar; cosmological benchmarks;
\OGNfive{} integer ledger; six-dimensional bulk; \GDOS{};
beyond-\(\LCDM\) precision cosmology.}


%% ---- v15.5: the savebox for the exposure page --------------
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%% \tabcolsep 6 pt, measured with \typeout at the use site.
%% THE NAME HAS NO DIGIT IN IT, AND THAT IS NOT COSMETIC.  The
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\newsavebox{\GGscTenCTwo}
\sbox{\GGscTenCTwo}{\setstretch{1}\small\begin{tabularx}{\textwidth}{@{}lllX@{}}
\toprule
\textbf{Quantity} & \textbf{Value} & \textbf{Grade} & \textbf{Falsified by}\\\midrule
Background rate $\Hzero^{(0)}$ & $66.6\pm0.2$ & $\mathsf D^{*}$ & a converged siren at $73$\\
Matter fraction $\Omega_m$ & $1/\pi$ & $\mathsf D^{*}$ & $\Omega_m$ settling away from $1/\pi$\\
Dark energy $\Omega_\Lambda$ & $1-1/\pi$ & $\mathsf D^{*}$ & a measured $\Omega_\Lambda\neq1-1/\pi$\\
Baryons $\Omega_b$ & $1/2\pi^2$ & $\mathsf D^{*}$ & $\Omega_b/\Omega_m\neq1/2\pi$\\
Equation of state $w$ & $-1$ exactly & Derived-cond. & a converged $w\neq-1$ at $3\sigma$\\
Corridor switch $z_*$ & $e^7\approx1097$ & $\mathsf D^{*}$ & two-quantity: $z_{\rm dec}$ off \ATen{}'s atomic chain by $>0.5\%$\\
de Sitter floor $H_\infty$ & $55.0$ & $\mathsf D^{*}$ & follows from $\Omega_\Lambda$\\
Present age $t_0$ & $13.9$ Gyr & $\mathsf D^{*}$ & inconsistent stellar/cosmic ages\\
Local-Hole lift $\epsilon_{\rm Hole}$ & $+2$--$3\%$ & Derived (sign) & ladder converging on $67$\\
Siren prediction $\Hzero^{\rm GW}$ & $\to67$ & prediction & a converged siren at $73$\\
\bottomrule
\end{tabularx}}

%% ---- v13.48 showcase pages (5-7): the paper in three pictures ----
\clearpage
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\vspace*{\stretch{1}}
\centering
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.95\textheight,keepaspectratio]{c2_h0_landscape_showcase.pdf}
\par
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\captionof{figure}[The 2026 \(H_0\) landscape (showcase; full version in Part~V).]{\textbf{The
2026 \(H_0\) landscape at a glance}: all forty-five audited determinations, the
three bands, and the \GG{} predictions on top. Full version, table, and sources:
Figure~\ref{fig:h0-landscape} in Part~V.}}
\label{fig:showcase-landscape}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
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\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily The Hubble Parameter \(H(t)\) Is Derived ---\\[2pt]
\(H_0\) Is One Snapshot of a Falling Curve\par}
\bigskip
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.78\textheight,keepaspectratio]{c2_H_of_t_derived.pdf}
\par
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{6pt}%
\captionof{figure}[The derived Hubble parameter \(H(t)\) (showcase; full version in Part~III).]{\textbf{The
rate is a derived function of time}: the full history on log--log axes (top) and
the late universe in linear time (bottom), falling monotonically to the de~Sitter
floor \(H_\infty = 55.0\;\kmsMpc\); the familiar \(\Hzero = 66.6\) is one
reading of this curve, at \(t_0 = 13.9\)~Gyr. Full version:
Figure~\ref{fig:H-of-t-derived} in Part~III.}}
\label{fig:showcase-Hoft}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

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\vspace*{\stretch{1}}
\centering
\par\vspace*{\stretch{2}}
\includegraphics[width=\textwidth,height=0.96\textheight,keepaspectratio]{c2_cmb_offsets_showcase.pdf}
\par
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{6pt}%
\captionof{figure}[The coherent \(\pi\)-offset Dev chart (showcase; full version in Part~III).]{\textbf{One
displacement, seventeen shadows}: every entry is a \GG{}/\GDOS{}/\GRIP{}
prediction --- zero CMB input, the sky only the comparison column --- and the
single \(+1\%\) displacement of \(1/\pi\) above Planck's fitted \(\Omega_m\)
propagates with the expected sign through all seventeen. A fit scatters its
residuals; a derivation lands coherently. Full version:
Figure~\ref{fig:cmb-offsets} in Part~III.}}
\label{fig:showcase-offsets}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% ---- v15.5 showcase page 8: THE EXPOSURE -------------------
%% C2's three showcase pages say where the measured values sit, what
%% the derived curve looks like, and how coherent the offset is.
%% None says what would end the paper.  Table 23 does, on one page,
%% with a 'Falsified by' column -- the direct analogue of the page
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\vspace*{\stretch{1}}
\centering
{\Large\bfseries\sffamily Ten Numbers,\\[3pt]
Each With Its Grade and Its Falsifier\par}
\vspace{1.4em}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\small\captionof*{table}{\textbf{The paper a reader can carry away.} Ten quantities, each with its value, its honesty grade, and the observation that would retire it. Every row carries the grade and the falsifier in the same breath, because that pairing --- a claim and the measurement that would kill it, printed together --- is the discipline the whole twenty-two-paper series is built on. $\mathsf{D}^{*}$ marks a value the ledger derives subject to the measure lemma; the three entries that are not $\mathsf{D}^{*}$ say what they are instead. Two of the ten are retired by the same standard-siren result: near 73 this paper is wrong, publicly and quickly, and by its own printed rule. Nothing here is negotiable once its datum arrives, and most of the data arrive within the decade. In full: Table~\ref{tab:c2-ten-numbers}, p.~\pageref{tab:c2-ten-numbers}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\usebox{\GGscTenCTwo}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page 8 -----------------------------------


\GGfrontmatteronepagetrue % v13.11: one-line Overview entries; Overview+LoF share one page
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

\GGPreface

\noindent This Preface is addressed, more than to anyone else, to the
working cosmologist who is tired of the Hubble tension --- who has
watched a decade of proposed resolutions each add a field, a fluid, or a
knob, and wondered whether the argument can end except by exhaustion. It
is also addressed to the student deciding whether the tension is worth a
thesis (it is --- though not for the usual reason).

\looseness=-1 Consider where the field stands. The tension is real: after the H0DN
consensus --- the \(H_0\) Distance Network, the thirty-eight-author
ladder combination --- no single method, anchor, or instrument can be
blamed, and the formal significance is beyond argument. Yet the response of theory has
been, overwhelmingly, to negotiate --- to add early dark energy here,
extra radiation there, each proposal purchasing its seven percent
with a parameter tuned after the fact, each retirable only slowly, by
accumulating disfavor. Ten years of this has produced a literature in which everything is
possible and nothing is decisive. The missing ingredient was never
another mechanism; it was a \emph{stake}: a number some theory needed to
be true, printed before the measurement that could kill it.

This paper hands you one. The expansion history is derived --- the whole
curve, from the first second to the de~Sitter floor, with the present
value at \(66.6 \pm 0.2\) and not one continuous dial anywhere --- and
everything that follows from it is priced in the open: the microwave sky
it must fit with no background parameter tuned (it does, at the percent level, coherently), the landmarks it
must pass from nucleosynthesis to the present age (it does, all within a
percent), the local excess it must explain (it explains a third and
declines, in print, to invent the rest), and the ladder's error budget
against which the remainder is confronted channel by channel. Where the
paper cannot derive, it says so in a graded ledger; where it can be
killed, it says how, with numerical rules fixed in advance.

What should you do with it on Monday morning? Three things, in ascending
order of effort. Recompute the headline: the background rate is two
published densities divided by two exact fractions --- a calculator
exercise, and either it lands in the anchors' \(66.4\)--\(67.0\)
corridor, centered on this paper's \(66.6 \pm 0.2\), or this paper has
a very short future. Run the scripts: ten of them ship with the paper, one per
figure and forecast, each asserting its own arithmetic before it draws;
the fastest referee report in cosmology is \texttt{python
c2\_siren\_forecast.py}. And if you work on the ladder or the sirens:
the shell test (\(H_0\) versus \(z_{\rm min}\)), the twenty-one-event
forecast, and the channel table of Part~IV were written to be sharpened
by exactly your expertise --- adversarially, please.

\looseness=-1 Two warnings, so the invitation is honest. This is not the community's
consensus framework: it is one deductive structure, published as a
twenty-two-paper cohort, and the present paper leans on theorems and two
named conjectures proved or stated elsewhere in it --- the honesty triage
and the objections Part tell you exactly which links carry which grade,
and the seventy-nine objections there include every charge we could
construct against ourselves, and the twenty big questions a curious
reader asks, answered without softening. And the
falsifiers cut both ways by design: a converged siren at \(73\), a firm
inventory below \(1/\pi\), a confirmed drifting equation of state, or a
single proton decay each ends this reading outright, with no fallback
prepared. We have written down the dates.

That is the wager. The tension has waited ten years not for another
mechanism but for a prediction worth scheduling instruments around;
what follows is that prediction --- and whichever way the sirens
rule, the argument will end the way physics arguments are supposed to
end.


%% ===================================================================
\GGPart[Introduction]{Introduction}{The expanding universe, the Hubble parameter \texorpdfstring{$H(t)$}{H(t)}, and a geometric answer to a century-old measurement}

\section{The expanding universe: a short history}
\label{sec:intro-history}

If you had asked an astronomer in 1900 what the universe was, the answer
would have been serene: a single island of stars --- our Milky Way ---
floating in an unchanging void, without a beginning, without an end, and
without much of a history. The stars wheeled, comets came and went, but
the universe as a whole simply \emph{was}. Of all the revolutions the
twentieth century inflicted on human self-understanding, none was larger,
or stranger, than what happened to that serene answer over the following
hundred years.

The first shock was size. Through the 1910s and 1920s astronomers argued
over whether the faint spiral ``nebulae'' were wisps of gas inside our own
galaxy or entire island universes of their own. Henrietta Leavitt had
found the key without knowing the door it would open: a class of pulsing
stars, the Cepheids, whose rhythm reveals their true brightness --- and
therefore, by how faint they appear, their distance
\citep{LeavittPickering1912}. When Edwin Hubble picked out Cepheids in
the Andromeda nebula \citep{Hubble1925}, the answer was unarguable:
Andromeda lay far beyond the Milky Way's edge. Our island of stars was
one among billions, and the universe was suddenly, dizzyingly, larger
than anyone had imagined.

The second shock was motion. Vesto Slipher had noticed that the light of
the spirals was almost always stretched toward the red --- the optical
signature of recession \citep{Slipher1917} --- and by 1929 Hubble, combining those velocities
with his Cepheid distances, saw the pattern that now carries his name:
the farther away a galaxy lies, the faster it recedes, in direct
proportion, \(v = H_0\,d\) \citep{Hubble1929} --- a pattern soon
confirmed to far greater depths \citep{HubbleHumason1931}. Georges
Lema\^itre had already derived that very law two years earlier from
Einstein's equations \citep{Lemaitre1927}. The proper picture is not galaxies fleeing
\emph{through} space like shrapnel, but space itself stretching, the way
dots on an inflating balloon all recede from one another --- every
observer sees the same recession, no one is at the center, and the speed
grows with distance simply because there is more stretching space in
between.

Einstein himself had been told as much by his own equations a decade
before, and had refused to listen: general relativity, left alone,
insists that a universe of matter cannot stand still --- as Alexander
Friedmann demonstrated by simply solving the equations honestly
\citep{Friedmann1922}. Einstein had instead inserted an extra term ---
his cosmological constant --- expressly to freeze the cosmos into the
eternal stillness everyone then believed in \citep{Einstein1917}. The
expansion's discovery turned that term into his famous regret, and taught
the century its deepest cosmological lesson: the universe is not a stage;
it is a \emph{dynamical object}, with equations of motion of its own.

And a dynamical object has a past. Run the expansion backward and the
universe grows denser and hotter without limit: the stretching that
carries galaxies apart today, reversed, compresses everything that exists
into a hot, bright beginning. What began as an almost disreputable
extrapolation --- Lema\^itre's ``primeval atom'' --- made a concrete
prediction: the heat of that beginning should still be out there,
stretched by the expansion into a faint microwave glow filling the whole
sky, its temperature estimated at a few kelvin as early as 1948
\citep{AlpherHerman1948}. In 1965 that glow was found, largely by
accident, humming into a New Jersey radio horn
\citep{PenziasWilson1965}, and identified for what it was next door
\citep{DickePeebles1965}. With it, the Big Bang
ceased to be speculation. The universe now had an age, a childhood
photograph, and a biography.

The century had one more surprise to deliver --- in 1998, the discovery
that the stretching is now \emph{accelerating}, driven by something no
one had ordered \citep{Riess1998,Perlmutter1999} (Part~II tells that
story) --- but the arc was already
complete. In a hundred years the universe went from an eternal, static
room of stars to an evolving, expanding, datable object with a violent
birth, a measurable middle age, and a calculable future. And every frame
of that biography --- how old, how large, how bright, how long the lights
stay on --- is governed by a single quantity: \emph{how fast space
stretches}. The story of that one number, and of the moment it split in
two, is the subject of this paper.

\section{The Hubble parameter, the Hubble constant, and the tension}
\label{sec:intro-tension}

Return to the balloon. Paint two dots on it a centimeter apart and
inflate it so that every distance doubles each minute: the dots separate
at one centimeter per minute. Two dots ten centimeters apart separate at
ten centimeters per minute --- same stretching, more rubber in between.
That proportionality between distance and recession speed is exactly what
Hubble measured, and its coefficient is the \emph{expansion rate} of the
universe. Astronomers quote it in practical units: a galaxy one
megaparsec away (about 3.26 million light-years) recedes at roughly
\(70\) kilometers per second; a galaxy twice as far, twice as fast.

To make this precise, cosmology tracks the balloon's size with the
\emph{scale factor} \(a(t)\): physical distances between freely-falling
observers scale as \(a(t)\), normalized so that \(a(t_0) = 1\) today.
The expansion rate is its fractional rate of growth,
\begin{equation}
H(t) \;\equiv\; \frac{\dot a(t)}{a(t)} ,
\label{eq:intro-Hoft}
\end{equation}
and this --- the \emph{Hubble parameter} --- is a \emph{function of
time}, not a constant: it was enormous in the hot early universe and has
been falling ever since, as the contents of the universe dilute. What
astronomers call the \emph{Hubble constant}, \(H_0\), is simply the value
this falling curve takes at the present moment, \(H_0 \equiv H(t_0)\):
one point on a curve, kept as a shorthand from the century in which only
that one point could be measured. The number also sets the scales of
everything: its inverse, \(1/H_0 \approx 14.7\) billion years, is the
natural age-scale of the universe, and its value fixes how large the
observable universe is and how the story ends.

Measuring that one number has been a hundred-year enterprise
\citep{FreedmanMadore2010}, and Figure~\ref{fig:h0-history} compresses
the whole of it into one chart. Read the top panel first: the measured
value \emph{fell by a factor of seven} as calibration errors were found
and fixed \citep{Baade1956,Sandage1958}, then stalled for thirty years
in a factor-of-two standoff \citep{SandageTammann1975}, and was finally
settled --- it seemed --- at ten-percent precision at the turn of the
millennium \citep{Freedman2001KP}. Then read the bottom panel: as
precision sharpened below three percent, the two great ways of measuring
--- climbing the ladder of stellar distances outward
\citep{RiessSH0ES2022}, and reading the Big Bang's afterglow backward
\citep{Camphuis2025} --- did not converge on one value. They split, hardened, and now sit seven standard
deviations apart: the \emph{Hubble tension}, the sharpest crack in
modern cosmology.


A tension between two measurements of ``the same'' number is a puzzle only
if the number is truly a single, well-defined constant. The viewpoint
developed in this paper is that it is not: \(H_0\) is one reading of the
derived function \(H(t)\), and the two routes measure that function
through two different operational windows. Whether the disagreement is
new physics, a measurement systematic, or a mis-specification of what is
being compared is the question the rest of the paper takes up.



\section{The \texorpdfstring{\GGtheory{}}{GG-Theory} approach --- a preview}
\label{sec:intro-gg-preview}

Before this paper can say anything about the tension, the reader deserves
a plain account of the theory it speaks for --- assuming no prior
acquaintance whatsoever.

Start from a deficit that modern physics rarely advertises. Our two
standard models --- of particles and of cosmology --- describe the world
with magnificent precision, but they do not \emph{explain} their own
settings. The Standard Model of particle physics runs on roughly
twenty-eight measured numbers (masses, coupling strengths, mixing
angles); the standard cosmology adds six more (the density fractions, the
expansion rate among them). Nothing in either theory says why a single
one of these dials points where it does. It is as if we owned a
magnificent clock that keeps perfect time, with thirty-four hands whose
positions must be looked up rather than understood.

\GGtheory{} --- the framework of this twenty-two-paper series --- is a
wager that every one of those dials is set by geometry. It begins from a
short constitution of axioms and postulates \citep{ArisakaA1Constitution}
and derives from it a six-dimensional stage \citep{ArisakaA2GG6}. The two
extra dimensions are the theory's signature idea, and they are
emphatically \emph{not} places one could travel to. One is a
\emph{scale}: the depth of magnification, the coordinate along which
physics changes as one zooms from everyday lengths toward the smallest
distances. The other is an \emph{angle}: a compact phase, a dial that
runs from \(0\) to \(2\pi\) and closes on itself. A scale and an angle
--- a zoom and a dial --- are pure numbers rather than directions, and on
that quiet reinterpretation the whole construction stands.

\clearpage
\null\vfill
\noindent\begin{minipage}{\textwidth}\centering
\includegraphics[width=\textwidth,height=0.72\textheight,keepaspectratio]{c2_H0_history.pdf}
\par\smallskip
\captionof{figure}[A century of the Hubble constant: fall, stalemate, convergence, split.]{\textbf{A
century of the Hubble constant.} \emph{Top:} the measured value fell by a
factor of seven --- Lema\^itre and Hubble near \(500\)--\(575\), Baade's
population split (1952), Sandage's HII-region correction (1958) --- then
stalled in the three-decade factor-two war between the Sandage--Tammann
(\(\sim50\)--\(55\)) and de~Vaucouleurs (\(\sim90\)--\(100\)) camps, ended
by the HST Key Project at \(72\pm8\) \citep{Freedman2001KP}. \emph{Bottom:}
the precision era. The distance-ladder track (red) and the CMB+\(\Lambda\)CDM
track (orange) sharpened from ten-percent to percent precision and, instead
of converging, separated into today's \(7.1\sigma\) split between H0DN's
\(73.50\pm0.81\) and the combined early-universe \(67.24\pm0.35\). The green
dashed line is this paper's derived background rate \(66.6\pm0.2\)
(Part~III) --- printed here so the reader can see, before any argument, where
the geometry lands on the century's chart.}
\label{fig:h0-history}
\end{minipage}
\vfill
\clearpage


The stage comes with a discipline: \emph{maximum symmetry, minimum
freedom}. Quantum consistency on the six-dimensional stage is demanding
--- most candidate laws simply destroy themselves --- and requiring
consistency alone fixes the particle content and a ledger of five whole
numbers, \((1,3,5;7,35)\), with not one continuously adjustable dial left
anywhere \citep{ArisakaA3BI6}. Those five are not chosen but \emph{forced}, each with a fixed meaning --- $M=1$ a single tensor seam, $N=3$ the color triplicity, $L=5$ the $\mathrm{SU}(5)$ block, $J=7$ the odd closure index, and $K=35=LJ$ the throat depth whose warp factor $e^{-K}$ fixes, among much else, the present value of $H_0$. Everything we observe is then read as the
\emph{shadow} this stage casts on its four-dimensional boundary --- our
world. The companion papers work the consequences out sector by sector:
the twenty-eight Standard-Model numbers emerge as expressions in the five
integers and just two transcendentals --- \(\pi\), the angle of the compact gauge dimension, and \(e\), the base of the exponential scale dimension \citep{ArisakaA9SM}; quantum measurement
acquires a geometric reading \citep{ArisakaA8QM}; and the standard
cosmology itself --- the subject of the parent paper \ATen{} --- follows
with its density fractions locked to \(\pi\)
\citep{ArisakaA10Cosmos}.

That last result is the one this paper spends. If the cosmic recipe ---
how much matter, how much dark energy, how much ordinary atoms --- is
fixed by geometry, then Einstein's equations leave nothing further to
choose: the \emph{entire expansion history} \(H(t)\), from the first
second to the endless future, is an output. The Hubble ``constant''
stops being something one measures and becomes something one
\emph{checks}.

Two facts organize everything that follows. \emph{First}, because the
densities are derived, the Hubble parameter is too: \(H(t)\) is fixed
algebraically by the energy content through the Friedmann law, and the
present value comes out at \(\Hzero^{(0)} = 66.6 \pm 0.2\;\kmsMpc\) ---
which lands, to about a percent, on the CMB value. The early side of the
tension therefore needs no new physics; it is where the derived rate is
read most cleanly. \emph{Second}, and just as honestly, \GGtheory{} does
\emph{not} manufacture the local \(73\): it derives the sign and rough
size of a Local-Hole environmental lift and, as this paper will show
quantitatively, bounds the cosmological-averaging (backreaction)
contribution to well below what would be needed, leaving the remaining
excess as a distance-ladder calibration question to be settled by the
ladder-free gravitational-wave standard siren. The result is not a
triumphant closure of the tension but something more useful: a derived
expansion history, an honest account of where the local value sits, and
a sharp, dated test that will decide the matter.

\section{Three reading paths}
\label{sec:reading-paths}

This is a long report, and not every reader needs all of it --- but every
kind of reader deserves an itinerary built for them.

\paragraph{The cosmologist in a hurry.} You want the claim, the
evidence, and the kill switch, and you are entitled to skip everything
else. Read Part~II for the measurement --- especially the statistics of
the \(7.1\sigma\) (\S\ref{sec:tension-statistics}), which says exactly
what the headline number does and does not establish. Then the two
hearts: the derivation of the rate (\S\ref{sec:absolute-h0}) with its
over-determined CMB test (\S\ref{sec:cmb-overdetermined}), and the
reconciliation of the local value --- the worked split
(\S\ref{sec:worked-excess}), the Local-Hole course
(\S\ref{sec:hole-course}), and the residual priced against your own
error budget (\S\ref{sec:residual-anatomy}). Close with the falsifier
registry and its decision calendar
(\S\ref{sec:falsifiers}--\S\ref{sec:decision-calendar}) and the forecast of the
verdict's arrival (Part~VII). If you have one evening, that is the
paper; if you have ten minutes, run
\texttt{python c2\_siren\_forecast.py} and read the ten-numbers table
of the Conclusions.

\paragraph{The program reader.} If you arrive from the cohort, you
want to know what is new here and what is inherited. Inherited, at its
home grades: the inventory, the two-route rate, and the spectrum test
(all \ATen{}); the seesaw normalization (\AFive{}); the ledger
(\AFour{}). New in this paper: the theorem-grade statement of the
inventory with its Hopf-fibration reading
(Theorem~\ref{thm:omega-m-pi}), the landmarks walk
(\S\ref{sec:landmarks}), the recombination-switch proposition, the
backreaction bound at the exact \(1/\pi\) inventory
(\S\ref{sec:backreaction}), the Local-Hole scenario ledger, the
residual-versus-budget confrontation, the siren-forecast Part with its
released script, and the derivation courses of the Technical Appendices,
which work every headline number to reproduction grade. The registries
(App.~\ref{app:inherit}--\ref{app:promotion}) map the two-way flow
between this paper and its parent; the naming convention
(App.~\ref{app:naming}) lets you cite any object from outside.

\paragraph{The skeptic.} Start where we are weakest, not where we are
strongest. The honesty triage (Table~\ref{tab:c2-triage}) declares what
is derived, what is conditional on the one named lemma, and what is
open; the percent-versus-sigma objection (O-Hubble-32) is, in our own
judgment, the sharpest charge in the paper, and its answer states
plainly the standard by which a zero-parameter chart should and should
not be judged. Then read the landscape and comparison of Part~V --- including the
admission that for the last seven percent this reading is a disciplined
member of the unknown-systematics class --- and the seventy-nine
objections of Part~XI, which close with a seven-member hostile panel
that returns for a second round against the answers themselves. The
nobility clause (\S\ref{sec:nobility}) states the commitments you may
hold this paper to, and the falsifiers are not rhetoric: several can be
executed from your desk, starting with the released scripts
(App.~\ref{app:scripts}).

\paragraph{The graduate student.} The Technical Appendices were
written for you, and they are self-sufficient: start at the welcome
guide and the primers (App.~\ref{app:primers}), take the Friedmann
course (App.~\ref{app:courseC}) with a pen, then the ladder and CMB
courses, and only then read Part~III --- it will read like a summary
of things you have already derived. The career-honest sentence you
deserve is printed in the personal note of Part~IX: the
\emph{appointments} of this paper are mainstream observations that
pay professionally whichever way the framework fares; the framework
itself is a proposal under test. Work the first; watch the second.

\section{Summary, and the road to Part II}
\label{sec:intro-summary}

Part~I has told the century's story in miniature, because everything in
this paper stands on it. A hundred years ago the universe was an
eternal, motionless room of stars; by the century's end it was an
expanding, datable object with a violent birth, a measured biography,
and a calculable future. One quantity governs every frame of that
biography --- the rate at which space stretches --- and the twentieth
century's last gift to cosmology was the discovery that this rate is not
a constant at all but a falling curve, \(H(t)\), of which the famous
``Hubble constant'' is merely today's reading.

Then the reading split. Measured up the ladder of stellar distances, the
present rate comes out near \(73.5\); inferred back from the Big Bang's
afterglow, near \(67\). The century chart of
Figure~\ref{fig:h0-history} shows both the humility this subject
demands --- calibration has moved this number by a factor of seven
before --- and the seriousness of the present impasse: the split emerged
precisely when the errors became too small to hide it, and no single
method, team, or instrument can any longer be blamed.

This paper's stance, previewed in \S\ref{sec:intro-gg-preview}, is that
the impasse is a diagnosis, not a duel. In \GGtheory{} the cosmic recipe
is fixed by geometry, the whole curve \(H(t)\) follows from it with no
dial to turn, and its present value lands at \(66.6 \pm 0.2\) --- on the
afterglow's side. What remains is to explain the ladder honestly: a
derived environmental lift carries part of the excess, the rest is named
and priced rather than explained away, and a ladder-free referee --- the
gravitational-wave standard siren --- is scheduled to settle the matter
this decade. The roadmap and the cohort map of the closing Section
(\S\ref{sec:inheritance-and-roadmap-c2}) show where each step of that
argument lives, and the three reading paths above offer each kind of
reader a route of their own.

Part~II now sets \GGtheory{} aside entirely. Before a single geometric
claim is spent, the problem deserves to be stated at full strength: how
the Hubble constant is actually measured, what the thirty-eight-author
H0DN consensus did that no previous measurement could do, why its
\(7.1\sigma\) really is unimpeachable arithmetic --- and what,
precisely, that arithmetic does and does not say. It is the tension as
any cosmologist would pose it; only afterwards does the derivation
begin.

\clearpage
\section{The cohort inheritance and the roadmap of the twelve Parts}
\label{sec:inheritance-and-roadmap-c2}

\noindent
Before the problem is stated at full strength (Part~II), it is useful
to lay out two reader-orientation maps in one closing Section: a
panorama of the twenty-two-paper cohort in which this paper stands
(\S\ref{sec:gg-cohort}, Figure~\ref{fig:cohort-map-c2}), and a
panorama of the twelve Parts of the present report
(\S\ref{sec:roadmap-c2}, Figure~\ref{fig:roadmap-c2}). A reader of
any later Part should be able to flip back to this Section and orient
themselves in both directions: which cohort companion carries which
imported result, and where the present argument stands in the
twelve-Part arc.

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

This report is one paper of twenty-two, and it leans on its siblings at every
load-bearing joint: the inventory it divides into the CMB densities is a
theorem of the cosmology readout \ATen{}; the integer ledger behind the
recombination switch belongs to \AFour{}; the boundary law that makes the two
measured values registered observables of one projection is \AFive{}'s. The
reader therefore deserves the whole map before the thread begins. The present
paper is \PaperCTwo{}, the middle member of the Cosmology \& Astrophysics
branch --- the dedicated Hubble companion of \ATen{}, from which it inherits
the derived expansion history and to which it returns the H0DN reading.

Figure~\ref{fig:cohort-map-c2}, which closes this subsection, places
the present paper among the twenty-two papers of the \GGTheory{} cohort. The reader meeting the program
for the first time can read its architecture off the figure top to
bottom --- the constitutional root, the bulk trio, the open-system
boundary law and its pillars, the three textbook readouts, the capstone,
the two phenomenology branches, and the Foundations Branch on the left
flank. What follows walks that same descent
in words, naming the papers a reader of \emph{this} report will most
want on the desk.

The descent begins with a constitution. \AOne{}
\citep{ArisakaA1Constitution} fixes four axioms and seven postulates ---
the rules of admissibility every later claim must obey --- and the bulk
trio then erects the stage: \ATwo{} \citep{ArisakaA2GG6} derives the
six-dimensional geometry itself, \AThree{} \citep{ArisakaA3BI6} proves
the anomaly freedom that makes the construction quantum-mechanically
legal, and \AFour{} \citep{ArisakaA4Euclid} traces the lineage back to
Euclid and promotes the five-integer ledger \((1,3,5;7,35)\) to theorem
grade. \AFive{} \citep{ArisakaA5GDOS} is the hinge of the whole
program: the open-system law by which the six-dimensional bulk
projects to the four-dimensional boundary we inhabit, organized from its
geometric side by \ASix{} \citep{ArisakaA6Gi4} and from its
quantum-informational side by \ASeven{} \citep{ArisakaA7Qi4}.

Then come the three readouts, where the machinery pays out in textbook
physics --- and two of them are this paper's essential companions.
\ANine{} \citep{ArisakaA9SM} derives the \emph{Standard Model}: all
twenty-eight of its parameters as closed expressions in the five
integers, \(\pi\), and \(e\). A reader who wants to see the same ledger
spent far from cosmology --- and to judge the program's credibility on
particle-physics ground --- should read \ANine{} alongside this paper.
\ATen{} \citep{ArisakaA10Cosmos} does the same for \emph{cosmology},
deriving \(\Lambda\)CDM's inventory and with it the expansion history.
\ATen{} is this paper's \emph{parent}: the source of record for
everything Part~III inherits --- the \(\pi\)-locked fractions, the
two-route rate, the spectrum test --- and the single most valuable
companion to the present report. Readers who finish this paper
persuaded, or convinced it must be wrong, are warmly encouraged to read
\ATen{} next; the case for and against both papers is ultimately one
case. (The third readout, \AEight{} \citep{ArisakaA8QM}, recovers
quantum measurement; \AEleven{} \citep{ArisakaA11Origin} closes the
A-series as its capstone.)

A word on the Foundations Branch, the column on the map's left flank,
whose subject --- the boundary's complex structure and its mirrors ---
sits beneath the corridor rather than beside it; this paper's relations
to its two members are light, and they are named here at roster level.
\PaperPFour{} \citep{ArisakaP4Real} is the origin-of-$i$ member: from
the two real structures of the forced arena --- the compact phase
circle and the causal clock --- it derives the single imaginary unit of
boundary physics, the complex structure that every phase and amplitude
in this report's cosmology quietly presupposes. \PaperPFive{}
\citep{ArisakaP5CPT} is the mirrors-and-matter member: C, P, and T as
orientation reversals of the three real bulk factors, closing with why
the universe is made of matter --- and its four-endowment definition of
time (time orders, repeats, opens one way, and shelters the quantum
world's mathematics) is kin to the corridor chronology of the derived
\(H(t)\) read here, one clock beneath two readings. Neither is
load-bearing for the corridor; they are named so the panorama is
complete.


\providecommand{\QCI}{\ifmmode\mathsf{QCI}\else\textsf{QCI}\fi}
\providecommand{\FRW}{\textsf{FRW}}
\small
\setlength{\LTcapwidth}{\textwidth}
\begin{longtable}{>{\raggedright\arraybackslash}p{0.14\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth} >{\raggedright\arraybackslash}p{0.39\textwidth}}
\caption[The twenty-two-paper \GGTheory{} cohort.]{\textbf{The twenty-two-paper \GGTheory{} cohort.} Each paper's main achievement and its role in the program. Every paper name links to that paper's own page at \href{https://preprints.arisaka-gg.org/}{preprints.arisaka-gg.org}, where all twenty-two are collected. The paper marked \emph{(this paper)} is the one you are reading.}
\label{tab:cohort-roster}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Role in the cohort} \\
\midrule
\endfirsthead
\multicolumn{3}{l}{\small\emph{Table~\ref{tab:cohort-roster}, continued.}}\\
\toprule
\textbf{Paper} & \textbf{Main achievement} & \textbf{Role in the cohort} \\
\midrule
\endhead
\midrule
\multicolumn{3}{r}{\small\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\href{https://preprints.arisaka-gg.org/a1-constitution/}{\AOne{}} & Four axioms and seven postulates fixing what counts as an admissible claim --- the constitution of the whole program & The constitutional root; every later paper inherits its four axioms and seven postulates \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a2-gg6/}{\ATwo{}} & Derivation of the six-dimensional \GG{} bulk, \BIsix{}, \OGNfive{}, compact holonomy, and the unification of all six forces & First of the \GGsix{} bulk trio; fixes the geometry and ledger the boundary law is built on \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a3-bi6/}{\AThree{}} & Complete proof of \BIsix{} anomaly freedom, unified gauge consistency, and conservation of the dual current & \GGsix{} bulk trio; the ultraviolet legality certificate that lets the projected ledger exist \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a4-euclid/}{\AFour{}} & The historical lineage from Euclid to \GG{}, and the zero-parameter integer \OGNfive{} ledger behind the constants & \GGsix{} bulk trio (historical companion); supplies the integer ledger behind the constants \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a5-gdos/}{\AFive{}} & \OGGEP{} descent to boundary quantum mechanics, boundary general relativity, and formal 4D QFT: the \GDOS{} boundary law & The open-system boundary law itself --- the hinge where the 6D bulk projects to 4D physics \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a6-grip/}{\ASix{}} & The runtime constitution --- \GRIP{}, the force law \(F_{\GFF}=F_{\GEP}+F_{\CEP}\), and the scale-invariance principle \SIP{} & A pillar paper organizing the boundary law from the geometric (\GiFour{}) side \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a7-qi4/}{\ASeven{}} & The \QiFour{} pillars --- \SeOS{}, \QCI{}, the identity \(\IH+\OH=\EIC\) --- and the five-story informational pagoda & A pillar paper organizing the boundary law from the quantum-information (\QiFour{}) side \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a8-qm/}{\AEight{}} & Quantum mechanics derived from \GGTheory{} --- the four-dimensional reformulation of measurement and state & Consequence trio; the member that recovers textbook quantum mechanics on the boundary \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a9-sm/}{\ANine{}} & The Standard Model as a theorem --- the descent of all twenty-eight Standard-Model parameters & Consequence trio; the member that recovers the Standard Model of particle physics \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a10-cosmos/}{\ATen{}} & \LCDM{} as the cosmological-boundary projection --- the \(\pi\)-locked inventory \(\Omega_m=1/\pi,\ \Omega_\Lambda=1-1/\pi\) & Consequence trio; the member that recovers \LCDM{} cosmology --- \emph{the parent of this paper} \\
\addlinespace
\href{https://preprints.arisaka-gg.org/a11-origin/}{\AEleven{}} & The closing capstone --- the universal identity and the \GRIP{} cascade across cosmos, life, and mind & The capstone that closes the A-series and proves the program's universal identity \\
\addlinespace
\href{https://preprints.arisaka-gg.org/p1-kk/}{\PaperPOne{}} & The Kaluza--Klein lock cluster near \(\Mlock\approx3.85\,\TeV\), with explicit LHC and future-collider falsifiers & Predictions branch; the collider phenomenology of the Kaluza--Klein lock scale \\
\addlinespace
\href{https://preprints.arisaka-gg.org/p2-n3/}{\PaperPTwo{}} & A three-leg derivation of \(N=3\) fermion generations from \BIsix{}, MSMF, and the integer ledger & Predictions branch; the three-leg derivation of the fermion generation count \\
\addlinespace
\href{https://preprints.arisaka-gg.org/p3-electron/}{\PaperPThree{}} & The electron as the ground state of nature --- the six oldest electron questions, closed & Predictions branch; the electron ground-state and charged-lepton-stratum result \\
\addlinespace
\href{https://preprints.arisaka-gg.org/p4-real/}{\PaperPFour{}} & The two real sources of the boundary's complex structure --- positive frequency is holomorphy, from the compact phase circle and the causal clock & Foundations Branch; the origin-of-$i$ member --- the derived complex structure that every phase and amplitude in this report presupposes \\
\addlinespace
\href{https://preprints.arisaka-gg.org/p5-cpt/}{\PaperPFive{}} & C, P, and T as orientation reversals of the three real bulk factors --- why every particle has a twin, and why the universe is made of matter & Foundations Branch; the mirrors-and-matter member, whose four-endowment definition of time is kin to the corridor chronology read here \\
\addlinespace
\href{https://preprints.arisaka-gg.org/c1-kaxion/}{\PaperCOne{}} & Cold dark matter as the non-spatial compact-angle pseudoscalar Kaxion at \(8.1\,\mu\mathrm{eV}\) & Cosmology \& astrophysics branch; the identification of cosmological dark matter \\
\addlinespace
\href{https://preprints.arisaka-gg.org/c2-hubble/}{\PaperCTwo{}} \emph{(this paper)} & The Hubble tension as two projections of one derived anchor --- the early/late corridor & Cosmology \& astrophysics branch; the resolution of the Hubble tension \\
\addlinespace
\href{https://preprints.arisaka-gg.org/c3-bh/}{\PaperCThree{}} & Black-hole entropy \(A/4\) recovered as a boundary theorem within \GGTheory{}'s holographic ledger & Cosmology \& astrophysics branch; horizon thermodynamics and black-hole entropy \\
\end{longtable}
\normalsize

Finally, two phenomenology branches and one foundations column carry
the program outward.
The Predictions Branch stakes the collider and structural bets: the TeV
lock cluster of \PaperPOne{} \citep{ArisakaP1KK}, the three-generations
theorem of \PaperPTwo{} \citep{ArisakaP2N3}, and the electron
ground-state result of \PaperPThree{} \citep{ArisakaP3Electron}. The
Cosmology \& Astrophysics Branch is this paper's home: \PaperCOne{}
\citep{ArisakaC1Kaxion} identifies the dark matter and names the exact
frequency to listen for it at; the present paper reads the Hubble
tension; and \PaperCThree{} \citep{ArisakaC3BH} recovers black-hole
entropy from the same boundary law. And the Foundations Branch ---
\PaperPFour{} \citep{ArisakaP4Real} on the origin of the boundary's
complex structure, \PaperPFive{} \citep{ArisakaP5CPT} on nature's
three mirrors and the matter excess --- sits on the spine's left
flank. The roster of
Table~\ref{tab:cohort-roster} compresses all twenty-two into one line each.

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%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
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 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{} \citep{ArisakaW2Holonomy} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-c2}
\end{figure}
\clearpage


\subsection{Roadmap of the twelve Parts}
\label{sec:roadmap-c2}

The rest of the report is organized into eleven further Parts, and the
walk through them is plotted in Figure~\ref{fig:roadmap-c2}. The
present Part --- the on-ramp --- is nearly done: it has told the
century's story in miniature, introduced the falling curve \(H(t)\) of
which the famous constant is one reading, previewed the geometric
answer, offered three reading paths, and drawn the two orientation maps
of this closing Section.

Part~II, \emph{The Hubble Tension}, states the problem at full
strength before a single geometric claim is spent: a century of
measuring \(H_0\), the dark-energy surprise, the two definitions of
the expansion rate, the \(7\sigma\) H0DN consensus, and why the
tension is genuinely hard to explain. Nothing in it depends on
\GGtheory{}; it is the tension as any cosmologist would pose it.

Parts~III and~IV are the two hearts, and the roadmap highlights them.
Part~III, \emph{The \GGtheory{} of the Hubble Parameter}, reads the
expansion rate as a \GDOS{}/\GRIP{} observable and derives the whole
history \(H(t)\), with today's value at \(\Hzero^{(0)} = 66.6 \pm
0.2\) --- and shows why the CMB, reading the same geometry through its
own calibration, reports the value it does: the two-Hubble-constants
corridor that is this paper's central claim. Part~IV, \emph{Reconciling
the Hubble Tension with \GGtheory{}}, then faces the larger local
value honestly --- the origin of the environmental lift, the
quantitative estimate, the limit of backreaction as a remedy, and a
call to revisit \(\Lambda\)CDM in the light of the derived \(H(t)\).

Three Parts then take the claim outdoors. Part~V, \emph{The 2026
Measurement Landscape and the Rival Explanations}, draws every reported
\(H_0\) of the audited reference base --- forty-five determinations,
charted and tabulated --- sorts the field into three bands, and reads
the six rival classes against them, ending with the one measurement
that bisects the field. Part~VI, \emph{Kill Switches, the Decision
Calendar, and Wider Implications}, commits: the eleven named falsifiers
with their pre-declared rules, on a dated decision calendar 2026--2035.
Part~VII, \emph{Forecasting the Verdict}, is the standard-siren decade
in full --- the physics of the siren, the referee's report card
2017--2026, and the events-to-verdict forecast on empirical rates.

Part~VIII, the \emph{Discussion}, weighs the reading as a scientific
claim --- significance, uniqueness, innovation, and the open questions
stated as open --- and Part~IX, the \emph{Conclusions}, closes the
body with the retirement of a measured constant and a dozen dated ways
the universe can answer. The back matter is the reference machine:
Part~X carries the \emph{Technical Appendices} --- a graduate course
in the Hubble tension with every step shown; Part~XI answers the
\emph{Common Objections and Responses}, closing with the hostile
panel's second round; and Part~XII closes the paper with the annotated
\emph{References}, each entry carrying its Relevance note and its
first citation. Read in order, the twelve Parts compose one
self-contained argument; read by need, each Part names its imports and
can stand a visit on its own.

\clearpage
\par\medskip\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.831\textwidth}{!}{%
\begin{tikzpicture}[
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\node[tailbox](XI)  at (0mm,   -133mm) {\textbf{Part~XI} \\[-1pt] \footnotesize Objections: \\[-2pt] \footnotesize 79 answered, \\[-2pt] \footnotesize \textit{three hostile rounds}};
\node[tailbox](XII) at (40mm,  -133mm) {\textbf{Part~XII} \\[-1pt] \footnotesize References: \\[-2pt] \footnotesize 156 entries, \\[-2pt] \footnotesize \textit{A/B, first-cites}};

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\par\smallskip
\captionof{figure}[Roadmap of the twelve Parts of GG-C2-Hubble.]{Roadmap
of the twelve Parts of \textsf{GG-C2-Hubble}. The argument develops in
eight phases. \emph{On-ramp} (Part~I): the expanding universe, the two
readings of one rate, and the maps of this closing Section. \emph{The
measurement} (Part~II): a century of measuring \(H_0\) and the
\(7\sigma\) H0DN consensus, stated at full strength before any
geometry is spent. \emph{The derivation} (Part~III, highlighted): the
\(\pi\)-locked inventory turned into the whole history \(H(t)\),
today's value \(\Hzero^{(0)} = 66.6 \pm 0.2\), and the reason the
CMB reads the same geometry as it does. \emph{The reconciliation}
(Part~IV, highlighted): the origin of the larger local value, the
quantitative estimate, and the limit of backreaction as a remedy.
\emph{The tests} (Parts~V--VII): the 2026 landscape and its rivals,
the eleven kill switches on a dated decision calendar, and the
standard-siren forecast of the verdict. \emph{Critique} (Part~VIII)
and \emph{summit} (Part~IX) close the body, and the three back-matter
Parts collect the Technical Appendices (Part~X), the Common Objections
and Responses (Part~XI), and the annotated References (Part~XII).
Parts~III and~IV are highlighted because together they carry the
paper's central claim --- the two-Hubble-constants corridor: one
derived expansion history whose early-side readings are registered
observables of the same geometry (Part~III), with the local value
reconciled as far as honesty allows (Part~IV).}
\label{fig:roadmap-c2}
\end{minipage}\par\medskip



\section{Part~I in plain English: the two numbers, and why one of them stopped being a number}
The universe is getting bigger, and there is a number that says how
fast. Write it as a fraction and it is easy to hold: every cosmic distance
grows by about seven percent in a billion years. That fraction is the
Hubble parameter. Because the growth rate has not been the same throughout
cosmic history, the parameter is a curve and not a constant, and what is
called the Hubble constant is one reading off that curve --- the value it
takes today.

For most of the last century the job was to measure that reading, and the
job was hard, and successive answers were far apart because the
measurement was hard. That is an ordinary scientific story with an
ordinary ending, in which the errors shrink and the answers converge.

Something else happened. Two ways of getting the number were each pushed
to about one percent, and they did not converge. One climbs outward from
nearby stars, rung by rung, and arrives at about \(73.5\). The other
starts from the oldest light in the sky and computes what today's rate
must be, and arrives at about \(67.2\). The gap is nine percent, which is
several times larger than either error bar. Two careful measurements of
the same quantity that disagree by that much are not a pair of
measurements; they are a problem.

This paper's answer is not another measurement. The claim is that the
whole curve --- not the reading, the curve --- follows from a geometry
fixed elsewhere in this cohort for reasons that had nothing to do with
cosmic expansion, and that the early-side number is what the curve
predicts while the late-side number is what an observer standing where we
stand would read off it. Everything that follows is that calculation and
the tests it exposes itself to.

\GGPart[The Hubble Tension]{The Hubble Tension}{A century of measuring \(H_0\); the dark-energy surprise; the two definitions; the \(7\sigma\) H0DN consensus; and why the tension is hard to explain}

\section{Defining and measuring \texorpdfstring{\(H_0\)}{H0}: a century of observation}
\label{sec:measure-history}

The two definitions of Part~I are worth restating here, so that this
Part can be read on its own. The expansion of the universe is described
by one function of time --- the \emph{Hubble parameter} \(H(t)\), the
fractional rate at which cosmic distances stretch. The \emph{Hubble
constant} \(H_0\) is not a second object but a single reading of the
first: the value \(H(t)\) takes today. It is the constant of
proportionality in Hubble's law, \(v = H_0 d\), and its customary unit
spells out the operation: a value of \(70\;\kmsMpc\) says that a galaxy
one megaparsec away recedes at seventy kilometers per second ---
equivalently, that every cosmic distance grows by about seven percent
per billion years --- and its inverse sets the age scale of the
universe, \(1/H_0 \approx 14\) billion years. Everything in this Part
is an attempt to measure this one number, and the tension of the title
is the discovery that the two best attempts, each impeccable by its
own standards, disagree about it.

Measuring \(H_0\) has been a century-long program built on the
\emph{distance ladder}: geometric parallaxes calibrate the
period--luminosity relation of Cepheid variables; Cepheids calibrate
the peak luminosity of Type-Ia supernovae; and supernovae, visible
across cosmological distances, carry the calibration into the smooth
Hubble flow. Each rung inherits the uncertainty of the one below, so
the enterprise is as much about controlling systematics as about
collecting photons.

The cleanest way to tell the century is to walk
Figure~\ref{fig:h0-history} (Part~I, page~\pageref{fig:h0-history})
from left to right. The curve has five features --- a start high off
the modern chart, two great cliffs, a thirty-year plateau split into
two camps, a brief convergence, and a fork at the right edge --- and
each one is a lesson in how this particular number is measured, and
mismeasured.

\emph{The start (1927--1929): a factor of seven too high.} The
measurement began in confusion and error --- instructive error. Vesto
Slipher's spectra had shown the spirals receding; Henrietta Leavitt's
period--luminosity law had made distance measurable; and by 1929
Hubble could plot velocity against distance and read off a slope of
roughly \(500\;\kmsMpc\) \citep{Hubble1929}. Georges Lema\^itre had
derived the expansion law from general relativity two years earlier
and extracted essentially the same constant from the same sparse data
\citep{Lemaitre1927}. The number was wrong by a factor of seven ---
not through any failure of the idea, but through calibration: the
Cepheids used to gauge distance were a mixture of two stellar
populations with different period--luminosity laws, and the
``brightest stars'' used beyond them were often unresolved HII
regions. The very first value of the Hubble constant was, in modern
language, a distance-ladder systematic of six hundred percent.

\emph{The first cliff (1952): two kinds of Cepheid.} Walter Baade
separated the two Cepheid populations, and every galaxy calibrated on
the wrong one abruptly moved twice as far away: \(H_0\) fell to
\(\sim250\) \citep{Baade1956}.

\emph{The second cliff (1958): stars that were not stars.} Allan
Sandage identified the HII-region confusion --- the ``brightest
stars'' were glowing clouds of ionized gas, brighter than any star
could be --- and cut the value to \(\sim75\) \citep{Sandage1958}.
Within three decades the number had fallen sevenfold, and not once
because the physics changed: every drop was a calibration error
found.

\emph{The plateau (1960s--1990s): the factor-two war.} Then the fall
stopped, and the disagreement began: for three decades Sandage and
Tammann defended \(\sim50\)--\(55\) \citep{SandageTammann1975} while
de~Vaucouleurs defended \(\sim90\)--\(100\), each side publishing
careful, internally consistent analyses that disagreed with the other
by a factor the quoted errors could not explain. Every rung was
contested --- the Cepheid zero-point, the supernova standardization,
the flow corrections --- and the dispute could not be arbitrated from
within, because both camps shared the same kinds of calibration and
differed only in their judgment of them.

\emph{The convergence (2001): the Key Project.} The Hubble Space
Telescope was built in part to end that war, and it did: the Key
Project's final value, \(72 \pm 8\;\kmsMpc\)
(\citealp{Freedman2001KP}; reviewed in \citealp{FreedmanMadore2010}),
split the difference at ten-percent precision and made the factor-two
era history. For a few years the constant seemed settled --- one
number, one decimal in doubt.

\emph{The fork (2013--today): precision divides.} Then precision did
something unexpected. As both routes pushed below three percent, they
did not converge on the settled value; they moved apart and hardened.
The distance ladder, rebuilt around HST and then JWST photometry,
climbed through \(74.03 \pm 1.42\) \citep{Riess2019} to
\(73.04 \pm 1.04\) \citep{RiessSH0ES2022} and now to the H0DN
consensus \(73.50 \pm 0.81\); the microwave background, read through
\(\Lambda\)CDM, descended to \(67.4 \pm 0.5\) and now
\(67.24 \pm 0.35\) \citep{Camphuis2025}. A curve that fell by a
factor of seven, paused in a thirty-year stalemate, and converged ---
split in two precisely when the errors became too small to hide the
difference.

The century teaches a double lesson, and this paper will need both
halves. The first is humility: calibration systematics have repeatedly
moved this particular number by amounts that dwarf today's tension ---
by a factor of seven at the start, by a factor of two within living
memory. The second is respect: the modern network was engineered, rung
by redundant rung, precisely so that history could not repeat --- no
single method, anchor, or instrument can move the consensus by more
than a fraction of its error bar (\S\ref{sec:central-claim} and
below). Whether the last nine percent is new physics or the century's
final systematic is exactly the question this report takes up, and
neither half of the lesson may be dropped.



\section{The surprise: dark energy and accelerating expansion}
\label{sec:dark-energy-surprise}

The fork at the right edge of the century figure was the second great
shock of modern cosmology. The first arrived in 1998, and it is worth
telling properly, because it changed what the Hubble ``constant''
\emph{is}: it turned a number into a curve.

By the mid-1990s observational cosmology was, in Allan Sandage's
famous phrase, a search for two numbers \citep{Sandage1970}: \(H_0\),
the present expansion rate, and \(q_0\), the \emph{deceleration}
parameter. That the second would be positive was hardly questioned ---
gravity attracts, so a universe made of matter must expand ever more
slowly, and the only debate was whether the braking was strong enough
to someday reverse the expansion. Two rival teams --- the High-\(z\)
Supernova Search \citep{Schmidt1998} and the Supernova Cosmology
Project --- set out to measure that braking, using Type-Ia supernovae
as standardized candles bright enough to be seen more than halfway
across the observable universe.

Both teams found the opposite of what they were built to find. The
High-\(z\) team's sixteen distant supernovae were ten to fifteen
percent \emph{farther} away than even a low-density, gently
decelerating universe could place them: the expansion is not slowing
down but \emph{accelerating}, with \(q_0 < 0\) and a positive
cosmological constant preferred at better than \(3\sigma\)
\citep{Riess1998}. The Supernova Cosmology Project's independent
sample of forty-two supernovae, analyzed with a separate pipeline,
landed on the same conclusion, with a flat-universe fit of
\(\Omega_m \approx 0.28\) and the remainder in \(\Lambda\)
\citep{Perlmutter1999}. Two competing teams, two data sets, one
verdict --- recognized by the 2011 Nobel Prize in Physics --- and a
new component, \emph{dark energy}, carrying roughly seventy percent
of the cosmic budget, observationally consistent with a cosmological
constant of equation of state \(w = -1\).

Within seven years the discovery hardened into a concordance, each
confirmation arriving through a channel that owed the supernovae
nothing. The Boomerang balloon flight located the first acoustic peak
of the microwave background at \(\ell \approx 197\), pinning the
total density to critical --- a flat universe --- at the ten-percent
level \citep{deBernardis2000}. WMAP's first-year data then delivered
the six-parameter fit that made \(\Lambda\)CDM the standard model of
cosmology: \(H_0 = 72 \pm 5\), \(\Omega_\Lambda \approx 0.73\), and a
cosmic age of \(13.7\) billion years \citep{Spergel2003}. And in 2005
the baryon acoustic peak --- the same sound horizon that patterns the
microwave sky --- was detected in the correlation function of some
forty-six thousand SDSS luminous red galaxies, at the predicted
\(\sim100\,h^{-1}\) Mpc \citep{Eisenstein2005}. Flat, accelerating,
one-third matter, two-thirds \(\Lambda\): by the mid-2000s the budget
was set, and two decades of subsequent data have held \(w\) at
\(-1\) to within a few percent
\citep{Brout2022,Planck2018Parameters}.

What the concordance did not do is explain itself. \(\Lambda\)CDM
\emph{fits} the acceleration; it does not say why the dark-energy
density has the value it has --- an energy scale of a few
milli-electron-volts, absurdly small by particle-physics standards ---
nor why matter and dark energy, whose densities scale differently by
factors of billions over cosmic history, happen to be comparable
\emph{now}. Those two questions are precisely where this paper's
framework enters: Part~III derives the dark-energy fraction
(\(\Omega_\Lambda = 1 - 1/\pi\)) and its milli-electron-volt scale
from the same geometry that fixes \(H_0\), and Part~IV prices the
why-now coincidence.

The acceleration matters for this paper in a way that is easy to
miss: it makes the expansion rate a genuinely \emph{time-dependent}
quantity whose late-time behavior is set by \(\Lambda\). The
universe decelerated for its first eight billion years, while matter
dominated, and began accelerating at \(z \approx 0.6\) --- about six
billion years ago --- when the thinning matter density finally
dropped below the constant dark-energy density. Speaking of ``the
Hubble constant'' as though it were a fixed number is therefore a
convenient shorthand for a single reading of the function \(H(t)\),
taken at the one epoch we happen to occupy --- and it is exactly at
that shorthand that the two measurement routes part company.
Part~III makes the function itself --- derived, not fitted --- the
central object.

\section{Anatomy of the two routes}
\label{sec:routes-anatomy}

The tension is a disagreement between two measurement philosophies.
One climbs outward from the Earth, rung by rung, and reads the
present-day rate directly from the recession of nearby galaxies; the
other starts from the oldest light in the universe and \emph{computes}
what the present rate must be, given a model of everything that
happened in between. Both are precision instruments of the highest
order, and neither is naive: each is the best possible version of its
philosophy. Because everything later in this paper --- and every
proposed resolution in the literature --- lives somewhere inside this
anatomy, this section builds each route explicitly and then sets the
two side by side (Figure~\ref{fig:two-routes} and
Table~\ref{tab:routes-compare}).

\subsection{The late route: climbing the ladder}

The late route is \emph{direct and upward}, and it climbs four rungs.

\emph{Rung 0 --- geometry.} At the base sit distances that involve no
astrophysical assumption at all. Trigonometric parallaxes from Gaia
and HST fix the distances of Milky-Way Cepheids \citep{Lindegren2021};
twenty detached eclipsing binaries fix the distance of the Large
Magellanic Cloud to one percent \citep{Pietrzynski2019LMC}; and the
Keplerian rotation of water megamasers around the nucleus of
NGC~4258 fixes that galaxy's distance to one and a half percent
\citep{Reid2019N4258}. These three anchors are the ladder's
foundation: pure geometry, no candles.

\emph{Rung 1 --- primary indicators.} Within the anchors live stars
whose luminosity is predictable: classical Cepheids, whose
period--luminosity law has been the workhorse since Leavitt, and the
tip of the red giant branch (TRGB), a sharp feature of stellar
evolution. Calibrated in the anchors, they measure distances out to
tens of megaparsecs --- far enough to reach the few dozen galaxies
that have hosted a well-observed Type-Ia supernova.

\emph{Rung 2 --- supernovae.} Type-Ia supernovae are standardizable
candles: after correction for light-curve shape and color, their
peak luminosities agree to better than ten percent, and they outshine
their host galaxies. The roughly forty calibrator hosts transfer the
geometric zero-point to the supernova scale
\citep{Scolnic2022Pantheon}.

\emph{Rung 3 --- the Hubble flow.} Hundreds of supernovae at
\(0.023 < z < 0.15\) --- far enough that peculiar velocities of a few
hundred km\,s\(^{-1}\) are small against expansion velocities of many
thousands, near enough that the assumed cosmology barely matters ---
trace out the Hubble diagram, and its intercept \emph{is} \(H_0\).

The structure of the errors follows from the structure of the climb.
Each rung transfers a luminosity zero-point to the next, so
independent fractional errors add in quadrature,
\begin{equation}
\label{eq:ladder-budget}
\left(\frac{\sigma_{H_0}}{H_0}\right)^{\!2} \;\simeq\;
\sigma_{\mathrm{anchor}}^2 \;+\; \sigma_{P\mbox{--}L}^2 \;+\;
\sigma_{\mathrm{SN}}^2 \;+\; \sigma_{\mathrm{flow}}^2 ,
\end{equation}
and the H0DN's \(1.09\%\) is the sum of dozens of sub-percent terms
held down simultaneously --- a remarkable feat of metrology. Two
structural properties matter for everything that follows. First, the
route is \emph{cosmology-free}: nothing in the climb assumes an
expansion history (the only cosmological inputs are the kinematic
priors \(q_0 = -0.55\) and \(j_0 = +1\), and varying them moves the
result by less than \(0.1\;\kmsMpc\)). Second, the route is
\emph{calibration-inheriting}: a bias at any rung propagates
multiplicatively to every rung above it, so the surviving worry is
not statistics but a calibration error shared \emph{across} rungs ---
the one hypothesis the quoted error bar cannot price
(\S\ref{sec:tension-statistics}).

\subsection{The early route: computing the rate from the oldest light}

The early route is \emph{inferential and downward}, and it proceeds
in three steps.

\emph{Step 1 --- measure the pattern.} The microwave sky delivers
three exquisitely measured quantities. The angular scale of the
acoustic peaks, \(\theta_* \approx 0.010411\) radians, is known to
three parts in ten thousand --- the single most precisely measured
quantity in cosmology; and the physical densities of baryons and of
matter, \(\omega_b\) and \(\omega_m\), follow from the relative
heights of the peaks at the percent level
\citep{Planck2018Parameters}. None of the three is an expansion
rate.

\emph{Step 2 --- compute the ruler.} Before recombination, the
baryon--photon plasma carries sound waves; the distance such a wave
travels by decoupling --- the sound horizon
\begin{equation}
\label{eq:rs-def}
r_s \;=\; \int_{z_*}^{\infty} \frac{c_s(z)\,dz}{H(z)}
\;\approx\; 144~\mathrm{Mpc}
\end{equation}
--- is fixed by well-understood plasma physics once \(\omega_b\) and
\(\omega_m\) are known, and is computed to a fraction of a percent by
the standard Boltzmann codes \citep{Blas2011}. The acoustic pattern
is this ruler, photographed on the sky at \(z_* \approx 1100\).

\emph{Step 3 --- assume a history, and solve.} An angle is a ruler
divided by a distance,
\begin{equation}
\label{eq:theta-star}
\theta_* \;=\; \frac{r_s}{D_M(z_*)},
\qquad
D_M(z_*) \;=\; \int_0^{z_*}\frac{c\,dz}{H(z)},
\end{equation}
so the measured \(\theta_*\) and the computed \(r_s\) together fix
the comoving distance to last scattering, \(D_M \approx 13.9\)
gigaparsecs. To extract \(H_0\) from it one must assume the
\emph{shape} of \(H(z)\) across the thirteen-billion-year integral
--- in \(\Lambda\)CDM,
\(H(z) = H_0\sqrt{\Omega_m(1+z)^3 + \Omega_r(1+z)^4 +
\Omega_\Lambda}\) --- and solve for the normalization \(H_0\) that
makes the integral come out right. The answer is \(67.36 \pm 0.54\)
from Planck alone \citep{Planck2018Parameters} and
\(67.24 \pm 0.35\) from the combined Planck+SPT+ACT analysis
\citep{Camphuis2025}.

The structure of the risks again follows from the structure of the
argument. The route is superbly precise and, as
\S\ref{sec:early-network} shows, instrumentally redundant; but it is
\emph{model-bound at both ends}. The ruler end depends on
pre-recombination physics: add new energy density before decoupling
and \(r_s\) shrinks. The conversion end depends on the assumed
thirteen-billion-year history: bend \(H(z)\) between last scattering
and today and the inferred \(H_0\) moves. That double
model-boundness is the lever arm every early-universe ``solution''
of the tension pulls (Part~IV quantifies it); the same logic run in
reverse --- calibrating the ruler at high redshift and descending ---
is the ``inverse distance ladder,'' and it lands with the CMB
\citep{Aubourg2015}.

\begin{figure}[!t]
\centering
\resizebox{0.809\textwidth}{!}{%
\begin{tikzpicture}[
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\node[head] at (-42mm, 8mm) {The late route: direct, upward};
\node[head] at (42mm, 8mm)  {The early route: inferential, downward};
\node[lad] (l1) at (-42mm, 0mm)   {\textbf{Geometric anchors}\\ parallax, detached eclipsing binaries,\\ the NGC 4258 megamaser};
\node[lad] (l2) at (-42mm,-18mm)  {\textbf{Primary indicators}\\ Cepheid $P$--$L$ relation, TRGB};
\node[lad] (l3) at (-42mm,-36mm)  {\textbf{Type-Ia supernovae}\\ standardized candles in calibrator hosts};
\node[lad] (l4) at (-42mm,-54mm)  {\textbf{Hubble flow}\\ $H_0^{\mathrm{late}} = 73.50 \pm 0.81$};
\node[cmb] (c1) at (42mm, 0mm)    {\textbf{Acoustic pattern on the sky}\\ $\theta_*$ to four digits; $\omega_b$, $\omega_m$};
\node[cmb] (c2) at (42mm,-18mm)   {\textbf{Sound horizon $r_s$}\\ computed from pre-recombination physics};
\node[cmb] (c3) at (42mm,-36mm)   {\textbf{Expansion history $z_*\!\to\!0$}\\ assumed ($\Lambda$CDM) --- or derived (Part III)};
\node[cmb] (c4) at (42mm,-54mm)   {\textbf{Inferred rate}\\ $H_0^{\mathrm{CMB}} = 67.24 \pm 0.35$};
\node[res] (one) at (0,-74mm) {One universe, one $H(t)$ --- and a $7.1\sigma$ disagreement about its\\ present value. Part III derives the curve; Part IV locates the split.};
\draw[arr, draw=blue!55!black] (l1)--(l2);
\draw[arr, draw=blue!55!black] (l2)--(l3);
\draw[arr, draw=blue!55!black] (l3)--(l4);
\draw[arr, draw=orange!75!black] (c1)--(c2);
\draw[arr, draw=orange!75!black] (c2)--(c3);
\draw[arr, draw=orange!75!black] (c3)--(c4);
\draw[arr, draw=black!60] (l4.south) .. controls (-42mm,-68mm) .. (one.west);
\draw[arr, draw=black!60] (c4.south) .. controls (42mm,-68mm) .. (one.east);
\end{tikzpicture}}
\caption[Anatomy of the two routes to \(H_0\).]{\textbf{Anatomy of the two
routes.} The late route climbs: geometric anchors calibrate primary
indicators, which calibrate supernovae, which reach the Hubble flow --- each
rung inheriting the calibration of the one below, so the result is direct but
systematics-compounding. The early route descends: the acoustic pattern fixes
\(\theta_*\) and the physical densities, the sound horizon is computed, and
\(H_0\) is \emph{inferred} through an assumed expansion history between
\(z_*\approx1100\) and today --- precise but model-bound. The two meet,
\(7.1\sigma\) apart, at the present value of one function \(H(t)\); Part~III
replaces the assumed history with a derived one.}
\label{fig:two-routes}
\end{figure}

\subsection{The comparison, and the three possible homes}

Table~\ref{tab:routes-compare} compresses the anatomy into one view.
The two routes share \emph{nothing}: no data, no instrument, no
calibration, and --- most importantly --- no philosophy. One measures
a velocity field within a few hundred megaparsecs and never asks what
the universe is made of; the other measures a pattern imprinted at
\(z \approx 1100\) and cannot speak of today without a model of
everything since. Their disagreement is therefore not a technical
squabble between rival teams but a confrontation between the two ways
cosmology knows anything at all.

\begin{table}[!htbp]
\centering
\caption[The two routes to \(H_0\), compared.]{\textbf{The two routes
compared.} Each row isolates one structural difference; the last two
rows carry the logic of the rest of the paper --- what each route
must fear, and what it would take to bias each by the full nine
percent of the tension. The two columns are not two measurements of the same thing made in different ways. The late route reads the flow it sits in and assumes nothing about the history; the early route computes today's rate from a model of the whole history and assumes a great deal. A reader who wants one of the two to be simply wrong should say which of the two dominant risks is the one that is realized.}
\label{tab:routes-compare}
\small
\begin{tabular}{@{}p{0.20\textwidth}p{0.36\textwidth}p{0.36\textwidth}@{}}
\toprule
 & \textbf{Late route (ladder)} & \textbf{Early route (CMB)} \\
\midrule
Observable & recession velocities and distances at \(z \lesssim 0.15\) & the acoustic pattern at \(z_* \approx 1100\) \\
Physics used & geometry plus stellar standard candles & plasma physics plus a \(\Lambda\)CDM history \\
Character & \emph{direct}: reads the local flow & \emph{inferred}: computes today's rate \\
Result & \(73.50 \pm 0.81\) (\(1.09\%\)) & \(67.24 \pm 0.35\) (\(0.52\%\)) \\
Cosmology assumed & none (kinematic priors only) & the shape of \(H(z)\) over 13.8 Gyr \\
Dominant risk & a calibration bias shared across rungs & a mis-modeled ruler or bridge \\
A \(9\%\) shift needs & a coherent \(9\%\) zero-point bias & a \(\sim 9\%\) shorter \(r_s\), or a bent \(H(z)\) \\
\bottomrule
\end{tabular}
\end{table}

The asymmetry matters more than the disagreement. A discrepancy
between a direct-but-calibration-heavy number and a
precise-but-model-bound one has exactly three possible homes: a
shared calibration bias in the ladder, new physics in the
pre-recombination universe (a shorter ruler), or a mis-specified
expansion history bridging the two epochs (a bent conversion). Every
proposal in the literature --- and this paper too --- is a bet on one
of the three.

Where this paper stands can now be said in one sentence: \GGtheory{}
removes the bridge's freedom --- the expansion history between the
two windows is \emph{derived}, with no continuous parameter
(Part~III) --- and with the bridge fixed and the early side shown
unbiased, the question is relocated to the ladder's calibration,
where the framework derives what it can (\(+2\)--\(3\%\); Part~IV),
names what it cannot (\(\approx\!7\%\)), and calls for the one channel
that owes nothing to either route --- the gravitational-wave standard
siren --- to decide (Part~VI).

\section{Why this measurement matters}
\label{sec:why-h0dn}

For more than a decade the \emph{Hubble tension} --- the mismatch
between the early-universe and late-universe values of
$\Hzero$ --- has been the most scrutinized anomaly in precision
cosmology \citep{Verde2024,DiValentino2025,KamionkowskiRiess2023}.
The early-universe route (cosmic microwave background plus
$\LCDM$) and the late-universe distance-ladder route were,
historically, dominated by a small number of teams and a small
number of methods. Critics of the tension often framed it as the
disagreement of ``two or three measurements'' against one very
precise CMB inference, and argued that an \emph{unknown unknown} ---
a silent systematic error in the local distance ladder ---
might eventually dissolve it.

The H0DN Collaboration set out to retire this framing by
constructing a \emph{Local Distance Network} that is neither
one ladder nor a handful of parallel ladders but a full
covariance-weighted combination of every credible local probe.
The paper is the written output of the \textit{What's under the
H0od?} workshop at ISSI-Bern in March 2025 and was co-authored by
38 specialists --- including representatives of every major
subfield of local distance determination (parallaxes,
detached eclipsing binaries, masers, Cepheids, the tip of the red
giant branch, Miras, JAGB, Type-Ia supernovae, Type-II supernovae,
surface-brightness fluctuations, the fundamental plane, and
Tully--Fisher). The core result is a $1.1\%$ consensus value whose
central number is insensitive to which subset of methods is
used. The point is structural, not numerical: the late-universe
measurement of $\Hzero$ can no longer be retired by identifying a
systematic in one method, because no one method determines the
result.

\section{The central claim: the Hubble corridor as an open-system observable}
\label{sec:central-claim}

It is worth stating, in one place, the recognition this paper shares with the whole \GGTheory{} cohort. No physical system is closed --- so the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection \emph{is} observation itself; its dynamics is the open-system law \GDOS{} (\AFive{}), forced by the geometric \GiFour{} (\ASix{}) and quantum-information \QiFour{} (\ASeven{}) pillars. It follows that the two measured values of $H_0$ --- early and late --- are \emph{registered observables} of one projection: the two ends of a single geometric corridor, not a contradiction to be patched --- which is why this paper can derive it, and why a single clean measurement can refute it. The funnel of Figure~\ref{fig:central-claim} is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs.

\begin{figure}[!t]
\centering
\resizebox{0.619\textwidth}{!}{%
\begin{tikzpicture}[
 node distance=17mm,
 every node/.style={font=\sffamily\footnotesize, align=center},
 box/.style={draw=black!55, rounded corners, minimum width=104mm, minimum height=11mm},
 arrow/.style={-{Stealth[length=3mm]}, line width=1.10pt},
 elab/.style={font=\sffamily\scriptsize\itshape, fill=white, inner sep=1.5pt}
]
\node[box, fill=violet!10] (open) {\textbf{Every physical system is open}\\ a particle, the universe, a living cell --- none can be isolated from its environment};
\node[box, fill=blue!8, below of=open] (oggep) {\textbf{\OGGEP{}} (\AX{3}): the lawful projection \;6D bulk $\to$ 4D boundary\\ \textit{this projection is observation itself} \;\; {\scriptsize[\AOne{}, \AEight{}]}};
\node[box, fill=yellow!13, below of=oggep] (gdos) {\textbf{\GDOS{}}: the dynamics of the open-system projection\\ \textit{an observable is the value at a stationary seam} \;\; {\scriptsize[\AFive{}]}};
\node[box, fill=orange!14, line width=1pt, below of=gdos] (obs) {\textbf{The Hubble tension}: the derived background rate $H_0^{(0)}\!=\!66.6$\\ read through two observational windows --- \textit{a derived rate, not a free input}};
\node[box, fill=green!12, below of=obs] (pill) {\textbf{\GDOS{} is forced} by the \GiFour{} pillars {\scriptsize[\ASix{}]} and the \QiFour{} pillars {\scriptsize[\ASeven{}]}};
\node[box, fill=orange!16, line width=1pt, below of=pill] (concl) {$\Rightarrow$ \textbf{it is determined}\\ {\footnotesize read off one geometry --- the tension reinterpreted, not patched}};
\draw[arrow] (open)  -- node[elab,right]{no closed 4D theory exists}            (oggep);
\draw[arrow] (oggep) -- node[elab,right]{the projection must be a definite law}  (gdos);
\draw[arrow] (gdos)  -- node[elab,right]{apply to this paper's sector}          (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observable forced} (concl);
\end{tikzpicture}}
\caption[The open-system funnel shared across the \GGTheory{} cohort.]{\textbf{The central claim, as one deductive funnel}, shared across the \GGTheory{} cohort. Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is \GDOS{}, forced by the \GiFour{} and \QiFour{} pillars. The same funnel opens the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs --- here, the Hubble corridor.}
\label{fig:central-claim}
\end{figure}

\section{Architecture of the Local Distance Network}

\subsection{Definitions}

The H0DN paper re-codifies the vocabulary of distance-ladder work
into a consistent six-layer structure (paraphrasing Section~2.2 of
\citealt{CasertanoH0DN2026}; we do not reproduce their prose):
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item \textbf{Anchor}: an object whose distance is geometric,
e.g.\ via parallax, detached eclipsing binaries, or accretion-disk
megamasers. The H0DN baseline uses three anchors: the Milky-Way
Cepheid field (Gaia EDR3 + HST parallaxes), the Large Magellanic
Cloud (DEBs; \citealt{Pietrzynski2019LMC}), and NGC 4258
(megamasers; \citealt{Reid2019N4258}).
\item \textbf{Primary distance indicator}: a stellar feature
directly calibratable in an anchor (Cepheids, TRGB, Miras, JAGB).
\item \textbf{Host}: typically a galaxy whose distance is set by
primary indicators and which contains a secondary indicator.
\item \textbf{Secondary distance indicator}: a feature measurable
at larger distances, typically Type-Ia supernovae, Type-II
supernovae, SBFs, the fundamental plane, or Tully--Fisher.
\item \textbf{Calibrator}: an object whose distance constrains the
zero-point of a secondary indicator.
\item \textbf{Tracer (Hubble-flow object)}: an object distant
enough that its redshift reliably reflects cosmological expansion,
yielding a constraint on $\Hzero$ itself.
\end{enumerate}
Above this scaffolding the H0DN places a genuinely new object: a
\emph{covariance matrix} that explicitly tracks which uncertainties
are shared between paths. It is the structured treatment of
off-diagonal covariance that separates the H0DN from earlier
``whisker-diagram'' summaries.

\subsection{The baseline pre-registration}

A critical methodological feature of the H0DN is its
\emph{pre-registration}. Before any calculations yielding $\Hzero$,
the workshop voted, by anonymous ballot, on which methods were
sufficiently technically mature to constitute the baseline versus
which belonged in variants. The results (Tables~2 and 3 of
\citealt{CasertanoH0DN2026}) placed Cepheids, TRGB, SNe~Ia, SBF,
megamasers in the Hubble flow, Gaia parallaxes, DEBs, and NGC 4258
as baseline; Miras, JAGB, SNe~II (both SCM and EPM;
\citealt{Vogl2025EPM}), DESI FP \citep{Said2025DESIFP},
Tully--Fisher, and the SMC Cepheid anchor \citep{Breuval2024SMC} as
variants. These pre-registered choices fix the
central value before the data are combined; the variants then
explore robustness. This is how a 38-person collaboration avoids
post-hoc cherry-picking.

\section{The baseline result and its structural features}
\label{sec:h0dn-baseline}

The H0DN baseline value is
\begin{equation}\label{eq:H0DN}
\HzeroDN \;=\; 73.50 \pm 0.81\kmsMpc,\qquad
\chi^2_{\mathrm{red}} = 0.9879,\qquad
\mathrm{PTE} = 0.49,
\end{equation}
corresponding to a relative uncertainty of $1.09\%$
\citep{CasertanoH0DN2026}. Several features of this result, all of
which the H0DN paper verifies with its variant catalog, are
worth isolating.

\paragraph{Cross-method stability.} The dispersion of central
values across the full list of thirty structured variants is
$0.40\kmsMpc$, about half of the $1\sigma$ uncertainty of the
baseline itself. No single method, when removed, moves the central
value by more than $\sim 1\kmsMpc$; no single method, when added,
improves the central value significantly either. This is the hallmark of
a system limited by \emph{consensus rather than by systematics}.

\paragraph{SNe~Ia are not the answer or the problem.} Variant V13
removes \emph{all} SNe~Ia from the network. The central value
changes by $-0.07\kmsMpc$. The uncertainty roughly doubles
(because SNe~Ia extend the ladder deepest into the Hubble flow),
but the central value is essentially unchanged. Whatever one's
view of SN~Ia calibration debates
\citep{Scolnic2022Pantheon,Brout2022,Murakami2023Siblings}, this variant
retires the
hypothesis that the local-universe value of $\Hzero$ is a SN~Ia
artifact.

\paragraph{Neither Cepheids nor TRGB alone drives the result.} Variant
V08 removes all Cepheids; V09 removes all TRGB; V10 removes all
Gaia parallaxes; V11 removes the LMC as an anchor; V12 removes
NGC 4258. None of these removals shifts the central value by
more than $\sim 0.5\kmsMpc$ once the corresponding custom baseline
is used to isolate the effect of losing SNe~Ia calibrators. The
anchor and the primary-indicator axes of the network are
\emph{redundant}, not hierarchical.

\paragraph{Instrument swaps agree.} Variant V16 excludes all
HST-based observations (forcing LMC and Milky Way to drop as
anchors). Variant V17 excludes all JWST-based observations.
These two variants are \emph{not} exact complements, because both
share NGC 4258, but they represent the hardest practical
instrument-systematics hypothesis one can formulate. Their values
bracket the baseline: $73.75 \pm 1.33\kmsMpc$ without HST, and
$73.00 \pm 0.86\kmsMpc$ without JWST. The baseline sits cleanly
between them.

\paragraph{The ``everything'' solution.} Variant V99 turns on every
method simultaneously and gives $73.99 \pm 0.70\kmsMpc$.
Variant V99A excludes Tully--Fisher, which generates a $\chi^2$
excess, and gives $73.66 \pm 0.71\kmsMpc$ at $0.96\%$ relative
precision. Both of
these sit within $0.5$--$1\sigma$ of the baseline and increase the
tension with Planck+SPT+ACT to $\sim 8\sigma$.

\paragraph{Orthogonal paths agree.} The H0DN performs a crucial
consistency check by splitting the network into two statistically
\emph{fully independent} paths (Section~5.3.1 of
\citealt{CasertanoH0DN2026}):
\[
\text{O1} = \text{MW+LMC/SMC} \to \text{Cepheids} \to
\text{SNe~Ia} + \text{FP},
\]
\[
\text{O2} = \text{NGC 4258} \to \text{TRGB} \to \text{SBF} +
\text{megamasers}.
\]
O1 yields $73.11 \pm 0.92\kmsMpc$; O2 yields $73.43 \pm
1.80\kmsMpc$. Two statistically independent local probes of
$\Hzero$ agree to better than $0.2\sigma$. This is the
strongest single discriminator against the ``unknown unknowns''
narrative.

\section{The early-universe network: Planck, ACT, SPT, and DESI}
\label{sec:early-network}

Everything so far could leave the impression that network-grade
redundancy is the late route's monopoly --- one consensus corridor
standing against one satellite's model fit. The impression would be
wrong, and honesty requires stating the early side's case with the
same care, because it is structurally the same case.

The microwave background has now been measured by three independent
experiments with separate telescopes, detectors, observing sites, sky
coverage, and analysis pipelines. Planck's final release gives
\(67.36 \pm 0.54\;\kmsMpc\) \citep{Planck2018Parameters}. The Atacama
Cosmology Telescope's sixth data release --- ground-based, higher in
resolution, analyzed end to end by a different collaboration ---
lands on a consistent six-parameter model \citep{Louis2025ACT}.
SPT-3G at the South Pole is the third instrument, and the joint
analysis of all three yields \(67.24 \pm 0.35\;\kmsMpc\)
\citep{Camphuis2025} --- a half-percent measurement in which no
single experiment is indispensable. The redundancy is internal as
well: temperature, polarization, and gravitational lensing each carry
the acoustic scale, and the \(\Lambda\)CDM parameters inferred from
each sector separately agree \citep{Planck2018Parameters}.

More telling still, the early value survives with the microwave
background removed entirely. The same sound horizon that patterns the
CMB is imprinted on the clustering of galaxies, and DESI's second
data release measures that baryon-acoustic ruler across multiple
tracer classes out to redshift two and beyond. Calibrating it with
the primordial deuterium abundance --- big-bang nucleosynthesis fixes
\(\omega_b\) with no CMB input at all --- yields
\(H_0 = 68.51 \pm 0.58\;\kmsMpc\) \citep{AbdulKarim2025}:
\(5.0\sigma\) from the H0DN, through a channel that shares no
instrument, no sky, and no epoch with Planck. The ``inverse distance
ladder,'' which calibrates supernovae against the BAO ruler and
descends, lands in the same place \citep{Aubourg2015}; and cosmic
chronometers, which read \(H(z)\) directly from the differential ages
of passively evolving galaxies, trace an expansion history consistent
with the early-anchored curve at intermediate redshift
\citep{JimenezLoeb2002,Moresco2022}.

The balance sheet is therefore symmetric. The late side is a network:
no single method, anchor, or instrument moves the consensus
(\S\ref{sec:h0dn-baseline}). The early side is a network too: three
CMB experiments, a CMB-free acoustic ruler, and a direct \(H(z)\)
probe agree among themselves in the \(67\)--\(68.5\) band. A proposed
systematic-error resolution now faces the same structural objection
in either direction: blaming Planck must also explain ACT, SPT, and
DESI+BBN; blaming the Cepheids must also explain TRGB, the
megamasers, and the orthogonal paths.

And the significance only grows when channels are combined: the H0DN
baseline sits \(7.1\sigma\) from the joint CMB value and
\(5.0\sigma\) from the CMB-free one, while remaining consistent with
independent late-universe probes such as the TDCOSMO lensing
time-delay program \citep{TDCOSMO2025}. The tension is not one
measurement against another; it is network against network --- which
is precisely what makes it hard, and precisely what gives Part~III
its opening. The H0DN paper is the empirical basis for retiring the
single-systematic hypothesis on the late side; this section is the
parallel basis on the early side.

\section{The statistics of \texorpdfstring{\(7.1\sigma\)}{7.1 sigma}: what the number does and does not say}
\label{sec:tension-statistics}

The headline significance is elementary arithmetic: the two central values
differ by \(73.50-67.24=6.26\;\kmsMpc\), the quadrature error is
\(\sqrt{0.81^2+0.35^2}=0.88\), and the ratio is \(7.1\). If the quoted
uncertainties are complete, Gaussian, and independent, the chance of a
discrepancy this large is about one part in \(10^{12}\) --- far beyond any
conventional discovery threshold. It is worth being precise about what that
number does and does not establish.

Each of the three assumptions has a different standing. \emph{Independence}
is solid: the two routes share no data, no instrument, and no calibration.
\emph{Gaussianity} is supported on the local side by the network's own
goodness-of-fit (\(\chi^2_{\mathrm{red}}=0.99\), \(\mathrm{PTE}=0.49\)) and
on the early side by the likelihood machinery of the CMB analyses.
\emph{Completeness of the error budgets} is the entire debate. A significance
is a statement of incompatibility \emph{under the stated errors}: it measures
how loudly the two numbers disagree, not why. No amount of \(\sigma\) can
arbitrate between ``new physics'' and ``a systematic the budget does not
contain,'' because both produce exactly the same arithmetic.

The sensitivity to that last assumption is worth quantifying, because it is
the pivot of Part~IV. The H0DN covariance already prices every
\emph{modeled} shared systematic --- that is the network's great advance.
But a common-mode calibration term of fractional size \(s\), coherent across
the network and absent from the model, adds \(s\times73.5\) in quadrature to
the local error, and the significance collapses quickly: \(s=2\%\) brings
\(7.1\sigma\) down to \(3.7\sigma\); \(s=4\%\) to \(2.0\sigma\); and
\(s=6\%\) --- the size of the residual this paper will name in Part~IV --- to
\(1.4\sigma\). The variant catalog cannot exclude such a term, because
variants that share calibration DNA move together by construction. The
\(7.1\sigma\) is therefore both real and conditional: unimpeachable as
arithmetic, and silent on the one hypothesis that matters most.

History suggests how such stand-offs end. The factor-two war was not settled
by either camp accumulating significance --- both sides had plenty --- but by
a new instrument that made the contested calibrations measurable. The modern
analogue is a channel whose systematics are disjoint from both routes: the
gravitational-wave standard siren, which needs neither a sound horizon nor a
ladder. It is the referee this paper commits to in advance
(\S\ref{sec:decisive}), and the reason the reader will find no attempt in
what follows to litigate the tension by significance alone.

\section{What the H0DN paper does \emph{not} do}
\label{sec:h0dn-not}

It is equally important to be honest about what the H0DN does
\emph{not} claim, because several readings of the tension have to be
carefully distinguished.

\paragraph{No theoretical resolution.} The H0DN offers no model to
explain \emph{why} local and CMB-inferred values disagree. It only
demonstrates that the disagreement cannot be retired by dropping
any single method on the local side.

\paragraph{No ruling on early-universe systematics.} The H0DN is
silent on whether the CMB inference, recombination history, or
BBN+BAO inference is itself modified by new physics. A large
literature now canvasses early dark energy, extra radiation,
interacting dark sectors, and modified recombination as options;
\citep{KamionkowskiRiess2023,Poulin2023Review}; the H0DN does not
arbitrate among these.

\paragraph{No anisotropic claim.} The paper does \emph{not} claim
that $\Hzero$ varies across the sky, nor does it claim a large
local void. Variant V19 tests CMB-frame velocities (no peculiar
corrections) and shifts the result by only $-0.44\kmsMpc$; the
various redshift-cut variants V20--V21 shift the result by at most
$\pm 0.8\kmsMpc$. Large-scale isotropy and the \citet{CarrickFlow2015}
peculiar-velocity model are approximately consistent.

\paragraph{No closed book on $w(z)$ or $\Omega_m$.} The H0DN fixes
$q_0=-0.55$ and $j_0=+1$ (from \citealt{RiessSH0ES2022}) as input
priors. Varying these by reasonable amounts changes $\Hzero$ by
less than $0.1\kmsMpc$. Since this report is centrally concerned
with cosmological theory, it is important to recognize that H0DN
is only mildly sensitive to the background expansion model.

\section{Summary: the target for any theory, and the road to Part III}
\label{sec:part2-summary}

Part~II set out to pose the problem the rest of this paper must
solve, and it can now be posed exactly. This closing section gathers
the whole Part into a few paragraphs, ending with the single object
any theory of the tension must reproduce.

The Hubble parameter \(H(t)\) is the fractional rate at which cosmic
distances stretch; the Hubble constant \(H_0\) is its present-day
reading. Measuring \(H_0\) consumed a century
(\S\ref{sec:measure-history}): the value started a factor of seven
too high, fell down two great calibration cliffs, stalled in a
thirty-year factor-two war, and converged at \(72 \pm 8\) when the
Key Project closed the old dispute. The century's lesson cuts both
ways --- calibration systematics have repeatedly moved this number by
amounts that dwarf today's tension, and the modern network was
engineered precisely so that this could not happen again.

Meanwhile the discovery of cosmic acceleration
(\S\ref{sec:dark-energy-surprise}) turned the number into a curve. A
universe that decelerated for eight billion years and then began to
accelerate has a genuinely time-dependent \(H(t)\), and ``the Hubble
constant'' is one reading of that function at the single epoch we
occupy. The concordance model that absorbed the discovery ---
\(\Lambda\)CDM: flat, one-third matter, two-thirds dark energy ---
fits the curve but explains neither the milli-electron-volt scale of
the dark energy nor the why-now coincidence; both stand as open
invitations to a deeper theory.

The tension itself is a disagreement between the two ways cosmology
measures the curve (\S\ref{sec:routes-anatomy}). The late route
climbs a ladder --- geometry, Cepheids and TRGB, supernovae, the
Hubble flow --- and reads the local rate directly, assuming no
cosmology and inheriting every calibration: \(73.50 \pm 0.81\), a
\(1.09\%\) measurement. The early route photographs the sound horizon
at \(z_* \approx 1100\), computes the ruler from plasma physics, and
infers today's rate through an assumed expansion history:
\(67.24 \pm 0.35\), a half-percent inference. A discrepancy between
numbers of these two characters has exactly three possible homes ---
a shared calibration bias in the ladder, a shorter ruler from new
pre-recombination physics, or a bent expansion history in between ---
and every proposal in the literature is a bet on one of the three.

Both sides are now networks. The H0DN consensus is not one team's
ladder but a covariance-weighted combination of every credible local
probe --- pre-registered before combination, stable under the removal
of any single method, and confirmed by two statistically independent
orthogonal paths (\S\ref{sec:h0dn-baseline}). The early side answers
with three independent CMB experiments, a CMB-free acoustic ruler
calibrated by nucleosynthesis (\(68.51 \pm 0.58\)), and direct
\(H(z)\) chronometers (\S\ref{sec:early-network}). Neither side can
be retired by finding a bug in one instrument.

The arithmetic of the disagreement is unimpeachable --- a gap of
\(6.26 \pm 0.88\;\kmsMpc\), a significance of \(7.1\sigma\), odds of
one part in \(10^{12}\) under the stated errors --- and it is
conditional (\S\ref{sec:tension-statistics}): a coherent two-percent
calibration term absent from the local error model would already
collapse it to \(3.7\sigma\). No accumulation of significance can
arbitrate between ``new physics'' and ``a systematic the budget does
not contain,'' because both produce identical arithmetic; and the
H0DN itself claims neither, offering no theoretical resolution, no
ruling on early-universe physics, and no anisotropy
(\S\ref{sec:h0dn-not}).

The object a theory must reproduce is therefore not an individual
measurement but the \emph{network-level consensus corridor}:
\begin{equation}\label{eq:corridor}
72.7\kmsMpc \;\lesssim\;
\HzeroDN \;\lesssim\;
74.3\kmsMpc
\end{equation}
(baseline \(\pm 1\sigma\)), standing \(7.1\sigma\) from the
early-universe network. Any account of the tension must meet this
corridor at the network level: either produce it through structurally
inherited mechanisms, or locate, explicitly and falsifiably, the bias
that creates it. That is the criterion we apply in the rest of this
report.

Nothing in this Part has been \GGtheory{}. It is the problem as any
cosmologist would pose it: a background rate near \(67\) inferred
from the early universe by a redundant network, a local rate near
\(73\) measured by an equally redundant ladder, and a nine-percent
gap that has survived a decade of cross-checks --- sharpened into one
corridor a theory must hit and one split a theory must explain. The
Parts that follow do not adjust this target; they derive it.
Part~III derives the expansion history \(H(t)\) --- with no
continuous parameter --- and lands, without being asked, on the early
side of the split; Part~IV must then explain the late side, and does
so by pricing it; and Part~VI names the referee --- a channel that
owes nothing to either network --- whose verdict this paper agrees in
advance to accept.


%% ===================================================================
%% PART III --- The GG-Theory of H(t)
%% ===================================================================

\section{Part~II in plain English: the problem as any cosmologist would state it, before any geometry is spent}
There is no geometry anywhere in this Part, and the omission is
deliberate. Before a framework is allowed to explain a discrepancy, the
discrepancy should be stated by somebody with no stake in the explanation.

The late route begins with distances that can be paced off --- geometric
parallaxes to nearby stars --- and works outward in steps, each step
calibrating the next, until it reaches galaxies far enough away that the
expansion dominates their motion. It assumes nothing whatever about the
history of the universe. Its risk is structural: an error made at the
bottom is carried, unchanged and unnoticed, all the way to the top.

The early route begins with the pattern of hot and cold spots in the
microwave background, which is a photograph of sound waves frozen into the
plasma at the moment the universe turned transparent. From the size of
those waves, and a model of everything that has happened since, one
computes what the expansion rate must be today. It assumes a great deal
about the history. Its risk is that the model, or the ruler the pattern is
measured against, is wrong.

Both have been redone by independent teams using independent instruments,
and both have stayed where they were. The local network now sits in a
corridor from \(72.7\) to \(74.3\); the early network sits near \(67.2\)
with a smaller error bar. The separation is \(7.1\) standard deviations,
and a decade of looking has not turned up a single mistake that would
account for it.

So the answer can only be in one of three places: a bias shared by every
rung of the ladder, new physics before the universe turned transparent, or
an expansion history that bends between then and now in a way nobody has
modeled. Every proposal in the literature is one of those three, this
paper's included. The Parts that follow say which one this framework is
forced into, and why it has no choice in the matter.

\GGPart[The \GGTheory{} of $H(t)$]{The \GGtheory{} of the Hubble Parameter}{The Hubble parameter as a \GDOS{}/\GRIP{} observable; the direct derivation of \(H(t)\) and \(H_0\); and why it reproduces the CMB value}

\section{The \GGtheory{} framework in one view}
\label{sec:gg-framework}%

This report does not re-derive \GGtheory{}; for the full account we
refer to the synthesis paper \AEleven{} and to the component
preprints (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA9SM}; \citealp{ArisakaA5GDOS}; \citealp{ArisakaA10Cosmos}; \citealp{ArisakaA7Qi4}). We summarize only the pieces that enter
the $\Hzero$ corridor. Everything used below is inherited from the
constitution layer (the axioms and postulates of
\citealt{ArisakaA1Constitution}) and from the theorem ledger of the
component papers; this report introduces no new postulate and no
new continuous parameter, and where an entry is not yet
theorem-level (the local lift; the corridor response slopes) it is
flagged explicitly.

\GGtheory{} treats what is usually a stack of independent
postulates (geometry, gauge, measurement, openness) as forced
consequences of a single constitution built from four axioms
(AX-1 LC, AX-2 GGEP, AX-3 OGGEP, AX-4 MSMF) and seven postulates
(POS-1 GI, POS-2 UEP, POS-3 AF, POS-4 CP, POS-5 RG, POS-6 OGN,
POS-7 MF). The stage on which everything plays out is the
six-dimensional bulk\footnote{Throughout the cohort, the ``warped six-dimensional bulk'' is read on the field-theory side of the holographic dictionary: the fifth (scale) dimension is the renormalization-group axis $r=\ln(E_{\mathrm{GUT}}/\mu)$, not a Randall--Sundrum spatial dimension, and ``warped'' denotes RG running; the explicit holographic-RG construction is given in \citet{ArisakaA2GG6}.} of \ATwo{}
\citep{ArisakaA2GG6}, and it is best met through its metric, which
is definite and parameter-free:
\begin{equation}\label{eq:gg6-metric}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu \;+\; dr^2 \;+\; d\theta^2,
\qquad
M_6 \;=\; M_4 \times I_r \times S^1_\theta,
\end{equation}
with \(\eta_{\mu\nu}\) the Minkowski metric of the four-dimensional
boundary, \(r \in [0, K]\), and \(\theta \in [0, 2\pi)\).

Three pieces make up the bulk. The four dimensions \(M_4\) are the
spacetime we live in --- more precisely, the boundary on which every
observation is registered. The radial direction \(I_r\) is the energy
axis: the coordinate is literally the renormalization-group logarithm,
\(r = \ln(\EGUT/\mu)\), running from the ultraviolet mouth at \(r = 0\)
(\(\EGUT \simeq 2.3\times10^{16}\)~GeV) down to the integer endpoint
\(r_{\mathrm{IR}} = K = 35\); ``deeper into the throat'' simply means
``lower in energy,'' and the Planck--electroweak hierarchy is then the
topological number \(e^{-K}\), not a fine-tuned ratio. One
coordinate, two anchors --- worth one explicit sentence, because the
same integer appears in both roles. The GUT mouth sits \(2\pi\)
e-folds below the Planck scale (\(\EGUT = M_{\rm Pl}\,e^{-2\pi}\):
\(\ln(1.22\times10^{19}/2.28\times10^{16}) = 6.28\)), so the lock mass,
whose formula \(\Mlock = \tfrac12 M_{\rm Pl}\,e^{-K}\) counts its
\(K\) e-folds from the \emph{Planck} anchor, sits \emph{inside} the
throat at \(r_{\rm lock} = K - 2\pi + \ln 2 = 29.4\), not at the
endpoint; and on the boundary side the same RG coordinate simply
continues past the bulk interval as physics cools below the mouth's
image --- the cohort's descent charts run it to the terminus
\(r \approx 64.5\) (the dark-energy floor) and, tracked by the relic
temperature, to today's \(r_{\rm now} = \ln(\EGUT/T_0) \approx 66.7\)
(the \(H(t)\) chart's radial axis carries \(66.6\), an anchor-rounding
offset of \(0.1\) e-fold noted here so no reader hunts for physics in it).
The interval \([0,K]\) is the bulk's radial extent; the coordinate is
the boundary's energy logbook, and the logbook keeps running. And the small
compact circle \(S^1_\theta\) is the phase fiber: it carries the
integer-quantized Wilson-line holonomy, and its winding is where the
dark sector --- and, one story up, the \(\pi\)-locked inventory of
\S\ref{sec:inventory-derived} --- will be read off. The warp profile
in the exponent, \(A(r) = r\), is not chosen but forced: demanding
constant bulk curvature with a flat boundary gives \(A''(r) = 0\),
so there is no curvature scale, no compactification radius, and no
brane tension to tune.

On this bulk the gauge and gravitational structure is a single group
\citep{ArisakaA2GG6},
\begin{equation}\label{eq:G6}
G_{\mathrm{Bulk}} \;=\; \mathrm{Spin}(1,5) \,\times\, \SUfive
\,\times\, \UoneB \,\times\, \UoneQi ,
\end{equation}
with the Lorentz/spin factor \(\mathrm{Spin}(1,5)\) --- the gravity
sector --- on an equal footing with the internal factors:
\(\SUfive\) is the minimal simple unification containing the Standard
Model; \(\UoneB\) is gauged baryon number, which turns proton
stability into a theorem; and \(\UoneQi\) is the informational gauge
force the \QiFour{} pillars organize. Projected onto the
four-dimensional boundary, this one group becomes the six forces an
observer actually measures --- strong, weak, electromagnetic,
baryonic, informational, and gravitational. The projection's legality
certificate is the modified Bianchi identity proved anomaly-free in
\AThree{} \citep{ArisakaA3BI6},
\begin{equation}\label{eq:BI6}
\BIsix:\qquad
\dd H_3 \;=\; \tr R^2 \;-\; \Tr F_5^2 \;-\; \tfrac{2}{5} F_B^2
\;-\; \tfrac{2}{5} F_{Qi}^2,
\end{equation}
and its arithmetic is the Odd-integer Gauge Normalization ledger
proved in \AFour{} \citep{ArisakaA4Euclid},
\begin{equation}\label{eq:OGN5}
\OGNfive:\qquad (M,N,L;J,K) \;=\; (1,3,5;7,35), \qquad
K \;=\; LJ \;=\; M^2+N^2+L^2.
\end{equation}
Equations \eqref{eq:BI6} and \eqref{eq:OGN5} together are the
``integer ledger'' that will fix the corridor-switch redshift
$z_*=e^7$ and the seesaw coefficient $\zeta_\Lambda$ entering the
derived rate.

From this geometry the Standard-Model closure \ANine{} \citep{ArisakaA9SM} derives
all 28 Standard-Model parameters as algebraic expressions in
$(1,3,5;7,35)$ and the transcendentals $\{\pi, e\}$ plus the
dimensionful $\{c,\hbar,G_N\}$. The key exports are collected in
Table~\ref{tab:c2-track-record} --- the ledger's already-spent
predictions, each parameter-free, spanning the electroweak sector this
paper inherits and the cosmological shares it will re-derive. Two
further exports matter here and do not fit a numerical column:
strong-CP protection, $\bar\theta \lesssim 4.7\times10^{-18}$, and
exact proton stability, $\tau_p = \infty$ (exact local $\Uoneb$).

\begin{table}[H]
\centering
\caption[What the same five integers have already bought (the ledger's track record).]{What
the same five integers have already bought (parent papers \ANine{} and
\ATen{}; the format and values of the companion \PaperCOne{}'s track-record
table). Every entry is parameter-free; deviations are against
PDG/CODATA/Planck values. Deviations, not pulls, are the honest
column here, and we say why: these are \emph{first-order} closed
forms whose known second-order (QD) completions shift them at the
\(0.1\)--\(1\%\) level (\ANine{}); in raw measurement units the
rigid entries sit \(8\)--\(9\sigma\) out (\(+0.05\%\) on
\(\hat\alpha^{-1}\) is \(+8.8\sigma\)), and closing that residual
is \ANine{}'s audit, not this table's. The claim here is
parameter-free landing at the permille-to-percent level ---
exactly that, and nothing sharper.}
\label{tab:c2-track-record}
\small
\begin{tabular}{llccc}
\toprule
\textbf{Quantity} & \textbf{\OGNfive\ formula} & \textbf{Predicted} & \textbf{Measured} & \textbf{Dev.}\\
\midrule
$\hat\alpha^{-1}(M_Z)$ & $2^J$ & $128$ & $127.930\pm0.008$ & $+0.05\%$\\
$\alpha^{-1}(0)$ & $2^J+N^2$ & $137.0$ & $137.036$ & $-0.03\%$\\
$\sin^2\theta_W(M_Z)$ & $L^2/(4N^3)$ & $25/108=0.23148$ & $0.23122\pm0.00003$ & $+0.11\%$\\
$\alpha_s(M_Z)$ & $1/(\pi e)$ & $0.1171$ & $0.1180\pm0.0009$ & $-0.8\%$\\
Higgs vev $v$ & $\Mlock(2/L)^N$ & $246.33$~GeV & $246.22$~GeV & $+0.05\%$\\
Corridor switch $z_*$ & $e^J$ & $1096.6$ & $1089.92\pm0.25$ (dec.) & $+0.6\%$\\
Matter share $\Omega_m$ & $1/\pi$ & $0.3183$ & $0.3153\pm0.0073$ & $+1.0\%$\\
Dark-matter share $\Omega_{\rm dm}$ & $1/\pi-1/(2\pi^2)$ & $0.2676$ & $0.2645\pm0.0070$ & $+1.2\%$\\
Dark-energy share $\Omega_\Lambda$ & $1-1/\pi$ & $0.6817$ & $0.6847\pm0.0073$ & $-0.4\%$\\
\bottomrule
\end{tabular}
\end{table}

These are the \emph{same} integer package that will be asked to
produce the Hubble corridor. The logical challenge is therefore:
can the ledger that pins Standard-Model parameters to $\lesssim
1.5\sigma$ also hit the H0DN corridor without any new tuning?

\subsection{What a theory of geometry may output: the GU-5 dimensional bridge}
\label{sec:gu5-bridge}

Before the corridor is derived, one objection should be met head-on,
because it decides how the whole of \S\ref{sec:absolute-h0} must be
read: \emph{can a theory of pure geometry output a dimensionful
number at all?} The Hubble rate carries units --- inverse time,
dressed by astronomers as km\,s\(^{-1}\)\,Mpc\(^{-1}\) --- and a
geometry, it would seem, can only deliver pure numbers like
\(1/\pi\). The answer is the founding move of \AFour{}
\citep{ArisakaA4Euclid}: \GGtheory{} declares its dimensional
inputs once, as the five-element set
\(\mathrm{GU\text{-}5} = \{c,\, G_N,\, \hbar;\, k_B,\, Q_B\}\),
and \AFour{} proves that the first three are not physics parameters
but the \emph{conversion factors between the dimensions of the
six-dimensional geometry} --- length to time (\(c\)), geometry to
energy (\(G_N\)), energy to frequency (\(\hbar\)) --- which is
exactly why the natural-unit choice \(\{c, G_N, \hbar\} = \{1,1,1\}\)
costs nothing. Once that bridge is declared, ``dimensionful'' and
``dimensionless'' cease to be different kinds of prediction: every
dimensionful observable is a derived pure number times the
appropriate combination of \(E_{\mathrm{Pl}} = \sqrt{\hbar c^5/G_N}\)
and its siblings.

This is not a promissory note; it is how \ANine{} already operates.
The electron's charge is the derived pure number
\(e = \sqrt{4\pi\alpha}\) with \(\alpha^{-1}(0) = 2^J + N^2 = 137\)
--- a dimensionful charge, delivered by integers and the GU-5 bridge.
The fermion masses are derived ratios times the scale chain, whose
top rung is pure GU-5. The only empirical content left in a unit like
``kilogram'' or ``km\,s\(^{-1}\)\,Mpc\(^{-1}\)'' is the SI
definition of the meter and the second --- bookkeeping, not physics.

Applied to the present problem, the bridge turns the question ``what
is \(H_0\)?''\ into a pure-number assignment the geometry either
meets or fails: in Planck units the observed rate is
\begin{equation}
H_0\, t_{\mathrm{Pl}} \;\approx\; 1.16\times10^{-61},
\qquad\text{equivalently}\qquad
\rho_\Lambda/E_{\mathrm{Pl}}^4 \;\sim\; 10^{-123},
\label{eq:c2-dimensionless-task}
\end{equation}
and \emph{that} --- sixty-one orders of magnitude, as a pure number
--- is what \S\ref{sec:absolute-h0} owes the reader from geometry
alone. No astronomical observation is entitled to appear anywhere in
the delivery; and none will. The sky enters afterwards, in its proper
role: as the measurement the derived number is tested against.

\subsection{The recast cosmology \ATen{}}

\ATen{} \citep{ArisakaA10Cosmos} is the parent paper of the present
report: the volume in which this machinery is run on the whole of
cosmology, and the source of every background quantity Part~III uses.
Its central move is to change the status of the concordance model.
\(\Lambda\)CDM is ordinarily specified by six fitted numbers; \ATen{}
first derives general relativity itself and then the Friedmann system
as \emph{theorems} of the boundary projection (its Part~III; the
derivation is transcribed in \S\ref{sec:h-of-t-observable} below),
and then shows that the integer ledger fixes the background half of
the six parameters in closed form --- \(\Omega_m = 1/\pi\),
\(\Omega_\Lambda = 1 - 1/\pi\), \(\Omega_b = 1/2\pi^2\), \(w = -1\),
and the corridor-switch redshift \(z_* = e^7\) --- so that cosmology's
six-parameter standard model becomes, in effect, a three-parameter
one, and the half that is deleted is precisely the ``why these
values'' half.

The discipline that keeps this honest is \emph{joint} solution. The
guiding principle of \ATen{}: inflation, dark matter, baryogenesis,
BBN, the CMB, \(\Hzero\), \(\SEight\), and JWST-era early structure
must be solved jointly through one inherited source vector, or fail
jointly. The admissibility criterion forbids the inclusion of any
dark sector, early-universe fluid, or late-time component without
geometric inheritance --- nothing may be added to fix one observable,
because nothing may be added at all. Under this discipline the dark
matter is the Kaxion of the compact angle (identified, and staked to
an exact search frequency, in \PaperCOne{} \citep{ArisakaC1Kaxion});
the launch is an inflationary descent whose tilt lands on the
measured \(n_s \approx 0.96\); and the terminus is dark energy at
exactly \(w = -1\), the ground geometric attractor whose constancy
Part~V's kill switches guard.

What Part~III imports from \ATen{} is exactly four objects, each
carried with its provenance grade. First, the inventory theorem ---
the \(\pi\)-locked fractions with their Hopf-fibration proof and the
one named lemma still owed (\S\ref{sec:inventory-derived}). Second,
the derived expansion history \(H(t)\) that those fractions force
through the Friedmann theorem (\S\ref{sec:h-of-t-observable} and
\S\ref{sec:hoft-steps}).
Third, the two-route determination of the present rate,
\(H_0^{(0)} = 66.6 \pm 0.2\) --- the density anchors and the
vacuum-energy seesaw (\S\ref{sec:absolute-h0}). Fourth, the
over-determined microwave-background confrontation, ending in the
coherent \(\pi\)-offset (\S\ref{sec:cmb-overdetermined}). What this
paper returns to its parent is the tension itself, read at full
depth: the early/late split priced in Part~IV, the rival-class
landscape audit and comparison of Part~V, and the standard-siren stake of Part~VI ---
the sharpest dated falsifier the parent framework carries.

A reader who wants the derivations behind the imports should read
\ATen{}'s middle Parts alongside this one; a reader who wants the
particle-physics floor under both --- the same ledger pinning all
twenty-eight Standard-Model parameters to \(\lesssim 1.5\sigma\) ---
should read \ANine{} \citep{ArisakaA9SM}. Nothing below hides its
grade: the fractions and everything downstream of them carry
\(\mathsf D^{*}\) (derived, conditional on one named lemma), the lock
scale is a theorem, the seesaw coefficient is Derived conditional on
the codebook-selection lemma (\S\ref{sec:zeta-selection}), and the
provenance table of \S\ref{sec:absolute-h0} keeps the whole ledger in
one place.



\section{The Hubble parameter as a \texorpdfstring{\GDOS{}/\GRIP{}}{GDOS/GRIP} observable}
\label{sec:h-of-t-observable}

Before deriving a number, it is worth being clear about \emph{what} is
being derived. The object is the Hubble parameter \(H(t) = \dot a/a\),
the fractional stretching rate of the universe, of which \(H_0\) is
the present-day reading. This section establishes the load-bearing
rung: the Friedmann law which turns an energy inventory into an
expansion rate is itself a \emph{theorem} of the framework, not an
imported assumption. The next section then derives the inventory the
law needs, and \S\ref{sec:hoft-steps} feeds the one into the other,
producing the whole curve \(H(t)\) with no further freedom anywhere.
The derivation is transcribed from \ATen{}
\citep{ArisakaA10Cosmos}; we carry it out in full because this paper
is about nothing else.

\subsection{The Friedmann equations as a theorem}
\label{sec:friedmann-theorem}

A cosmology is the special solution of the Einstein equation in which
the universe looks the same at every place and in every direction ---
the cosmological principle. In the textbook this principle is an
\emph{assumption}, justified after the fact by the smoothness of the
microwave sky. In \GGtheory{} it is the geometric \emph{attractor} of
the runtime: the \GRIP{} cascade \(\mathrm{DGI} \to \GSSB{} \to
\mathrm{GA}\) of \ASix{} \citep{ArisakaA6Gi4}, run on the
cosmological boundary stratum, flows to the maximally symmetric
configuration, so homogeneity and isotropy are \emph{selected}, not
posited. (That the attractor is unique is a \(\mathsf D^{*}\) result,
resting on the attractor-uniqueness statement of \ASix{} and the
inflationary smoothing of \ATen{}; we state the condition rather than
bury it.) The Einstein equation itself is likewise inherited, not
assumed: \ATen{} derives it from the bulk action through the boundary
projection, with \(G_N\) the throat overlap integral. Granting the
attractor, the road from symmetry to the Friedmann system is the
standard graduate computation \citep{WeinbergCosmology2008,
Dodelson2020}, and we walk it, because it is the load-bearing rung.

\emph{Step 1 --- the metric.} Homogeneity and isotropy fix the metric
up to one function of time and a discrete curvature sign
\citep{Friedmann1922,Lemaitre1927}. In comoving coordinates,
\begin{equation}\label{eq:frw-metric}
ds^2 \;=\; -dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2\,
d\Omega_2^2\right],
\end{equation}
with \(a(t)\) the scale factor --- the single dynamical degree of
freedom, the ``size'' of space --- and \(k \in \{-1,0,+1\}\) the sign
of the spatial curvature. \emph{Physical picture:} \(a(t)\) stretches
every comoving separation in lockstep; nothing in the metric singles
out a place or a direction, which is the cosmological principle made
geometric. In \GGtheory{} the discrete choice is not free either: the
boundary Friedmann export of \GDOS{} carries \(k = 0\), so flatness is
inherited, not measured in.

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

\emph{Step 3 --- the source.} The same symmetry forces the boundary
stress-energy to be that of a perfect fluid,
\(T^\mu{}_\nu = \mathrm{diag}(-\rho, p, p, p)\): isotropy forbids any
momentum flux or shear. Here is the single place the framework
departs from the textbook --- not in the equation but in what may
fill it. The density and pressure are not free fluids to be fitted
but the coarse-grained boundary projection of the bulk source ledger,
\begin{equation}\label{eq:source-ledger}
\rho \;=\; \rho_r + \rho_b + \rho_{\mathrm{DM}} + \rho_\Lambda ,
\end{equation}
each sector's identity, equation of state \(w_i = p_i/\rho_i\), and
normalization fixed by the integer ledger of \S\ref{sec:gg-framework}
rather than chosen.

\emph{Step 4 --- the equations.} Inserting the Einstein tensor and
the perfect-fluid source into \(G_{\mu\nu} = 8\pi G_N T_{\mu\nu}\)
(the vacuum term carried inside \(\rho_\Lambda\)), the time--time
component yields the first Friedmann equation,
\begin{equation}\label{eq:friedmann1}
\boxed{\;H^2 \equiv \left(\frac{\dot a}{a}\right)^{\!2}
= \frac{8\pi G_N}{3}\,\rho \;-\; \frac{k}{a^2}\;}
\end{equation}
and combining it with the trace of the space--space component gives
the acceleration equation,
\begin{equation}\label{eq:friedmann2}
\boxed{\;\frac{\ddot a}{a} = -\frac{4\pi G_N}{3}\,(\rho + 3p)\;}
\end{equation}
while covariant conservation, \(\nabla_\mu T^{\mu 0} = 0\), gives the
continuity equation \(\dot\rho + 3H(\rho + p) = 0\), whose solution
sector by sector is the dilution law
\(\rho_i \propto a^{-3(1+w_i)}\). \emph{Physical picture:} the first
equation is a first integral of the motion --- energy density sets
the expansion rate; the second says pressure gravitates, so a sector
with \(\rho + 3p > 0\) brakes the universe and a vacuum with
\(p = -\rho\) accelerates it; the third is simply dilution ---
matter thins as \(a^{-3}\), radiation as \(a^{-4}\) (one extra power
for the redshifting of each quantum), and a \(w = -1\) vacuum not at
all.

The three equations are identical in form to the textbook Friedmann
system, and that identity is the point, not a limitation:
\GGtheory{} modifies neither gravity nor kinematics. What it fixes
is everything on the right-hand side --- which sectors exist, with
which equations of state, in which proportions, and with which
curvature. The Friedmann system is therefore a theorem of the
framework: the boundary Einstein equation, restricted to the
runtime-selected symmetric attractor, sourced by a ledger with no
free dials. The law is now ready; what it still needs is its
right-hand side, and deriving those densities is the next section's
work.

\section{The \texorpdfstring{\(\pi\)}{pi}-locked inventory: derived, not assumed}
\label{sec:inventory-derived}

The rate \(H(t)\) of \S\ref{sec:h-of-t-observable} is only as derived as the
densities that feed it, so the derivation must begin one story down, with the
inventory itself. It is the central claim of the parent paper \ATen{} that the
present cosmic budget is not a set of measured fractions but a \emph{theorem}:
\begin{equation}
\Omega_m = \frac{1}{\pi} = 0.3183,\qquad
\Omega_\Lambda = 1-\frac{1}{\pi} = 0.6817,\qquad
\Omega_b = \frac{1}{2\pi^2} = 0.05066 ,
\label{eq:c2-inventory}
\end{equation}
with the dark-matter share the difference \(\Omega_{\rm dm}=1/\pi-1/2\pi^2=0.2676\).
These are not fitted; they agree with the Planck~2018 concordance values to
better than three percent (\(\Omega_\Lambda\) and \(\Omega_{\rm dm}\) within
\(1\sigma\), \(\Omega_b\) at \(2.3\sigma\)), and their status is the subject of a
named theorem. We state it, sketch its proof, and name precisely what it still
owes, because everything downstream --- \(H(t)\), \(H_0\), the tension --- inherits
exactly this grade.

\begin{theorem}[\(\Omega_m=1/\pi\) as the Hopf fiber-volume ratio, conditional (T-Hubble-1)]
\label{thm:omega-m-pi}
Assume \emph{(I)} the cosmological-boundary stratum is a connected, orientable
four-manifold \(\mathcal{M}_4\) obtained as the \OGGEP{} projection of the
\GG{} bulk; \emph{(II)} the modified Bianchi identity \(\BIsix{}\) holds
throughout cosmological history, \(\dd X_4=0\) on \(\mathcal{M}_4\) with coefficient
vector \((1,-1,-2/5,-2/5)\); and \emph{(III)} the \OGNfive{} integers \(N=3\)
(families) and \(L=5\) (\(\mathrm{SU}(5)\)) fix the boundary holonomy manifold as
the electroweak \(\mathrm{SU}(2)\cong S^3\), and the matter sector's charge as its
Hopf fiber \(S^1\). Then, conditional on the measure-normalization lemma
\(\text{VIII-b}''\) (\ATen{}'s Conjecture~VIII),
\[
\Omega_m \;=\; \frac{\mathrm{Vol}(S^1)}{\mathrm{Vol}(S^3)} \;=\; \frac{2\pi}{2\pi^2} \;=\; \frac{1}{\pi},
\]
with no continuous adjustable parameter and no reference to the measured inventory.
\end{theorem}

\emph{Proof outline (five steps).}
\emph{Step 1 (closure).} By hypothesis~(II) the four-form \(X_4\) is closed,
\(\dd X_4=0\), so it represents a class in \(H^4(\mathcal{M}_4,\mathbb{R})\); the
class is non-trivial because \(\tr R^2\) generates a non-trivial Pontryagin class
on the curved boundary slice. \emph{Step 2 (integer skeleton).} The \OGNfive{}
tuple \((1,3,5;7,35)\) fixes the \emph{integer} content: \(N=3\) and \(L=5\)
identify the holonomy as electroweak \(\mathrm{SU}(2)\), and the packet is one
canonical unit --- a rational arithmetic weight
\(M/(LN)=1/15\) is a provable dead end. \emph{Step 3 (Pontryagin
normalization).} \(\int_{\mathcal{M}_4}\tr R^2 = 8\pi^2\,p_1\), and for the
\OGNfive{} slice \(p_1=N=3\), so the integral is \(24\pi^2\); crucially the
\(8\pi^2\) is \emph{consumed upstream} (the geometric free-energy functional is
defined as \((1/8\pi^2)\!\int X_4\)), so the packet's energy content is the pure
integer \(p_1\) and \textbf{no residual \(\pi\) descends from this step}. By the
Hirzebruch signature theorem (\(p_1=3\sigma\)), \(p_1=3\) forces \(\sigma=1\): the
minimal closed gravitational topology, hence one canonical packet. \emph{Step 4
(the Hopf ratio --- where the \(\pi\) is born).} The matter charge lives on the
compact phase circle \(S^1_\theta\), which sits inside \(\mathrm{SU}(2)\cong S^3\)
as its Hopf fiber \(S^1\hookrightarrow S^3\to S^2\) (a canonical fibration, no
freedom of choice). Under the frozen stationary \GDOS{} measure each sector's
present-epoch share is its fiber volume divided by the total holonomy volume, and
both are canonical unit spheres, \(\mathrm{Vol}(S^1)=2\pi\),
\(\mathrm{Vol}(S^3)=2\pi^2\), so \(\Omega_m = 2\pi/2\pi^2 = 1/\pi\) exactly, with
the \(\pi\) descending from the \emph{denominator volume} and nothing read off the
sky. \emph{Step 5 (uniqueness --- competitors die by geometry, not by data).}
\(1/(3\pi)\) would require a total holonomy volume \(6\pi^2\) that no manifold in
the construction carries; \(1/(2\pi)\) is the \emph{baryonic} projection (one unit
per fiber revolution), not the matter/total ratio; \(1/\pi^2\) would need an
\(S^5\) that the four-dimensional boundary does not carry. The measured value of
the inventory enters the selection at no step. \(\qquad\blacksquare\)

\begin{corollary}[the complement and the baryon line]
\label{cor:inventory-lines}
The dark-energy share is the part of the \(S^3\) holonomy volume the matter Hopf
circle does not cover, a topological complement \(\Omega_\Lambda=1-1/\pi\); and
baryons are one integer unit of \(\mathrm{U}(1)_B\) charge threaded once around the
fiber, \(\Omega_b=\Omega_m/\mathrm{Vol}(S^1)=(1/\pi)/2\pi=1/2\pi^2\). One fibration
fixes three coupled numbers; none can be moved without the others --- the
structural signature that separates a derivation from a per-observable fit.
\end{corollary}

\emph{What the theorem still owes.} The one step not yet closed is the
measure-normalization lemma: that the frozen \GDOS{} measure weights exactly the
Hopf circle of the electroweak \(\mathrm{SU}(2)\) as the matter fiber and
\(\mathrm{Vol}(S^3)\) as the total (\ATen{}'s Conjecture~VIII). Its structure is
now well understood and partly discharged. It decomposes into \emph{VIII-a}
(delocalization and share-robustness of the stationary measure --- proved in
consumer form, with the gravity-mediated floor \(\sim e^{-K}\approx10^{-15}\));
\emph{VIII-b\('\)} (the real-sector half-measure --- \emph{proved}: exact mode
counting, \(2\Lambda+1\) real degrees of freedom against \(2(2\Lambda+1)\) charged,
a ratio of exactly \(1/2\), which forces \(\Omega_b/\Omega_m=1/2\pi\) with no
residual freedom); and \emph{VIII-b\(''\)} (the unit-packet normalization --- the
remaining open core), now traced to the crossing-epoch identity \(u=1\), i.e.\
the declared choice that the present epoch is read at the \(\Omega_m=1/\pi\)
crossing. Until \(\text{VIII-b}''\) is exhibited the fractions carry the grade
\(\mathsf D^{*}\) (derived, conditional on one named lemma). This is a strict
improvement over ``an unexplained numerical coincidence'': no empirical selector
enters (the fraction is not chosen for its agreement with the measured
\(\Omega_m/\Omega_\Lambda\approx0.46\)), and the residual is one
clean geometric lemma, provable or falsifiable on the bulk construction.

\medskip
The geometry the theorem trades in is drawn in Figure~\ref{fig:hopf}. What
the proof does, in pictures: Steps~1--3 build one canonical packet of
gravitational topology --- closure makes the four-form a cohomology class,
the integer skeleton picks the electroweak \(\mathrm{SU}(2) \cong S^3\) as
the holonomy, and the Pontryagin normalization consumes its own \(8\pi^2\)
so that no stray transcendental leaks downstream. Step~4 is where the
\(\pi\) is born: the matter charge lives on the Hopf circle threading the
3-sphere, and under the frozen stationary measure each sector's share is a
ratio of volumes --- \(2\pi\) of fiber against \(2\pi^2\) of holonomy.
Step~5 closes the exits: the rival fractions die of geometry (no manifold in
the construction carries their volumes), not of disagreement with data. The
one thing the figure cannot draw is the measure lemma itself --- \emph{that}
the frozen measure weights exactly fiber against holonomy is the conditional
the \(\mathsf D^{*}\) grade rests on, stated above and tracked in the
promotion roadmap (App.~\ref{app:promotion}).

What the picture actually shows deserves a paragraph of its own,
because the fibration is the one geometric fact this paper most needs
the reader to see. The 3-sphere cannot be drawn directly, so the
figure projects it stereographically into ordinary space, where its
Hopf fibers appear as circles arranged on nested tori. Their defining
property is the remarkable one: \emph{every} fiber passes over exactly
one point of the base 2-sphere, every pair of fibers is linked exactly
once, and together the circles fill the 3-sphere with no gaps and no
crossings. The red circle --- the matter fiber --- is not a special
place: every fiber is equivalent, and that equivalence is the physical
point. The matter sector does not sit \emph{somewhere} on the
electroweak holonomy manifold; it is one winding of a structure that
permeates all of it, the compact phase circle \(S^1_	heta\) of the
bulk metric \eqref{eq:gg6-metric} embedded canonically in the
\(\mathrm{SU}(2) \cong S^3\) the integer ledger selects.

The intuition for the ratio is equipartition on a compact
configuration space. A frozen stationary measure has no preferred
point, so each sector's share is simply the volume it occupies: the
matter charge runs along one fiber --- a circle, of circumference
\(2\pi\) --- while the full holonomy carries the 3-sphere volume
\(2\pi^2\); the share is their ratio, \(1/\pi\). One power of \(\pi\)
survives because the ratio cancels the circle's \(2\pi\) against one
factor in the sphere's \(2\pi^2\) --- the \(\pi\) in the cosmic
inventory is, quite literally, the leftover of a sphere volume. And
the three numbers are rigid together: a larger matter share would need
a longer fiber, but a Hopf circle's length is fixed by the fibration;
a smaller dark-energy share would need the fiber to cover more of the
sphere, but a circle covers exactly what a circle covers; and the
baryon line threads one unit of charge per revolution, no more. What
in \(\Lambda\)CDM are three independent dials is here one shape,
read three ways.

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\textwidth]{c2_hopf_fibration.pdf}
\caption[The Hopf fibration behind Theorem T-Hubble-1.]{\textbf{The Hopf
fibration behind Theorem~\ref{thm:omega-m-pi}.} \emph{Left:} the 3-sphere of
the electroweak holonomy, drawn by stereographic projection: the Hopf map
\(S^1 \hookrightarrow S^3 \to S^2\) fills \(S^3\) without gaps by linked
circles, one over each point of the base 2-sphere (families of base
latitudes in blue and rose; one fiber --- the matter circle --- highlighted
in red). \emph{Right:} the volume ledger that turns the fibration into the
cosmic inventory: the matter share is fiber volume over holonomy volume,
\(2\pi/2\pi^2 = 1/\pi\); the dark-energy share is the uncovered complement,
\(1 - 1/\pi\); and the baryon share threads one unit of \(\mathrm{U}(1)_B\)
charge per fiber revolution, \(1/2\pi^2\). One fibration fixes three coupled
numbers; none can be moved without the others.}
\label{fig:hopf}
\end{figure}


\section{From the law to the curve: \texorpdfstring{\(H(t)\)}{H(t)} in six steps}
\label{sec:hoft-steps}

The two halves are now in hand: the Friedmann law of
\S\ref{sec:h-of-t-observable}, a theorem with an empty right-hand
side, and the \(\pi\)-locked inventory of
\S\ref{sec:inventory-derived}, the derived densities that fill it.
This section feeds the one into the other and builds the expansion
history explicitly, in six steps.

\emph{Step 1 --- normalize.} Define the critical density
\(\rho_c = 3H_0^2/8\pi G_N\) and the fractions
\(\Omega_i = \rho_i(t_0)/\rho_c\). Dividing
Eq.~\eqref{eq:friedmann1} by \(H_0^2\), with \(k = 0\) inherited and
the dilution law for each sector, the Friedmann equation becomes the
working form used throughout this paper:
\begin{equation}
H(t)^2 \;=\; \frac{8\pi G_N}{3}\sum_i \rho_i(t)
\;=\; H_0^2\!\left[\Omega_r a^{-4} + \Omega_m\, a^{-3} + \Omega_\Lambda\right].
\label{eq:friedmann-hierarchy}
\end{equation}

\emph{Step 2 --- insert the inventory.} Here the derived fractions
enter: \(\Omega_m = 1/\pi\) and \(\Omega_\Lambda = 1 - 1/\pi\)
(\S\ref{sec:inventory-derived}), with \(w = -1\) --- itself derived,
not assumed --- keeping \(\rho_\Lambda\) constant. The radiation term
is not free either: it is fixed by the measured photon temperature
\(T_\gamma = 2.7255\)~K \citep{Fixsen2009} plus the three neutrino
species, giving \(\Omega_r = 9.36\times10^{-5}\) at the derived rate.
After this step \emph{nothing} on the right-hand side of
Eq.~\eqref{eq:friedmann-hierarchy} is adjustable.

\emph{Aside --- which \(\Omega\) is which.} A notational subtlety
deserves one explicit paragraph here, because it is the source of a
natural confusion. The symbols \(\Omega_r\), \(\Omega_m\),
\(\Omega_\Lambda\) in Eq.~\eqref{eq:friedmann-hierarchy} are
\emph{constants by definition}: each is the fraction evaluated once, at
the present epoch, \(\Omega_i \equiv \rho_i(t_0)/\rho_c(t_0)\). The
time dependence of the budget is carried entirely by the dilution
factors standing beside them --- \(a^{-4}\), \(a^{-3}\), and \(1\) are
precisely the solutions \(\rho_i(t)/\rho_i(t_0)\) of the continuity
equation for \(w = 1/3,\ 0,\ -1\). The \emph{running} fractions are a
different object, \(\Omega_i(t) \equiv \rho_i(t)/\rho_c(t)\), and they
evolve dramatically: \(\Omega_r(t) \to 1\) deep in the radiation era,
\(\Omega_m(t) \to 1\) in the matter era, \(\Omega_\Lambda(t) \to 1\) on
the de~Sitter floor. Indeed Eq.~\eqref{eq:friedmann-hierarchy} itself
says the budget at the corridor switch was
\(\Omega_m(z_*) \approx 0.76\), \(\Omega_r(z_*) \approx 0.24\), and
\(\Omega_\Lambda(z_*) \approx 2\times10^{-9}\) --- matter and radiation
nearly balanced, dark energy utterly negligible. Nobody inserts
\(1/\pi\) at early times; Figure~\ref{fig:omega-budget} draws the
running budget across the whole history, and it is nothing like
today's.

The \(\pi\)-locked values are therefore a \emph{boundary condition at
the present epoch}, inserted once and propagated by the law to every
other time --- not a claim that matter holds a \(1/\pi\) share always,
which would contradict the continuity equation. The theorem of
\S\ref{sec:inventory-derived} is calibrated to exactly this structure:
the frozen measure fixes the share \emph{at the crossing epoch}, and
the declared observing epoch is the moment the running
\(\Omega_m(t)\) crosses \(1/\pi\) --- \(75\)~Myr before the present
(Landmark~7 of \S\ref{sec:landmarks}); the why-now window of Part~IV
(\S\ref{sec:why-now}) is the price of that declaration, stated rather
than hidden.

Two corollaries keep the later chapters honest. First, early-universe
physics never consults the present-epoch fractions: everything at
\(z \approx 1100\) --- the baryon loading, the equality epoch, the
sound horizon --- depends on the \emph{physical} densities
\(\omega_b\) and \(\omega_m\), which redshift deterministically as
\(a^{-3}\), and the \(\Lambda\) term is one part in \(10^{9}\) there;
the CMB determination of \S\ref{sec:absolute-h0} uses the fractions
only in the present-day bookkeeping identity
\(\omega_i = \Omega_i h^2\). Second, the ratio
\(\Omega_b/\Omega_m = 1/2\pi\) is \emph{epoch-independent} --- both
sectors dilute as \(a^{-3}\), so the ratio at recombination equals the
ratio today --- which is why the baryon-loading test of
\S\ref{sec:baryon-loading} may be applied directly to the
\(z \approx 1100\) sky, while \(1/\pi\) and \(1 - 1/\pi\) remain
strictly present-epoch statements. (Objection O-Hubble-48 restates
this aside against its sharpest form.)


\emph{Step 3 --- read the eras.} Each term dominates in turn, and
each dominance is a power law: radiation gives \(a \propto t^{1/2}\)
and \(H = 1/2t\); matter gives \(a \propto t^{2/3}\) and
\(H = 2/3t\); the vacuum gives \(H\) constant and \(a\) exponential.
\emph{Intuition:} \(1/H\) is the instantaneous e-folding time of the
universe; in the decelerating eras that time keeps lengthening as the
sources dilute, and in the vacuum era it locks.

\emph{Step 4 --- the closed form.} From matter--radiation equality
onward the two-component equation integrates in closed form. With
\(\tau \equiv \tfrac{3}{2}\sqrt{\Omega_\Lambda}\,H_0\, t\),
\begin{equation}\label{eq:hoft-closed}
a(t) = \left(\frac{\Omega_m}{\Omega_\Lambda}\right)^{\!1/3}
\sinh^{2/3}\!\tau,
\qquad
\boxed{\;H(t) = \sqrt{\Omega_\Lambda}\,H_0\,\coth\tau\;}
\end{equation}
--- one formula carrying the whole late history. Its two limits are
the two eras: for small \(\tau\), \(\coth\tau \to 1/\tau\) and
\(H \to 2/3t\), the matter era; for large \(\tau\),
\(\coth\tau \to 1\) and \(H\) settles on the de~Sitter floor
\begin{equation}\label{eq:hinfty}
H_\infty \;=\; \sqrt{\Omega_\Lambda}\,H_0
\;=\; \sqrt{1 - 1/\pi}\;H_0 \;=\; 55.0\;\kmsMpc .
\end{equation}
The curve itself --- the time dependence beneath every dilution factor
of Eq.~\eqref{eq:friedmann-hierarchy} --- is drawn as its own object in
Figure~\ref{fig:a-of-t}, with the three era slopes visible and the
inflection marked.



\emph{Step 5 --- the reading today.} The present is one point on the
curve: \(H_0 \equiv H(t_0)\), the number the geometry fixes at
\(66.6 \pm 0.2\) by the two routes of \S\ref{sec:absolute-h0}. The
same closed form gives the age relation,
\(H_0 t_0 = \tfrac{2}{3\sqrt{\Omega_\Lambda}}\,
\mathrm{arcsinh}\sqrt{\Omega_\Lambda/\Omega_m} = 0.948\), hence
\(t_0 = 13.92\)~Gyr (App.~\ref{app:age}) --- an output, not an input.

\emph{Step 6 --- monotonicity, and what acceleration is not.}
Differentiating Eq.~\eqref{eq:friedmann-hierarchy} gives
\(\dot H = -\tfrac{3}{2}\,(1 + w_{\mathrm{eff}})\,H^2 \le 0\) for any
mixture with \(w_{\mathrm{eff}} \ge -1\): the rate falls
monotonically for all time and \emph{never rises} --- not even after
the expansion begins to accelerate, because \(H = \dot a/a\) keeps
decreasing while \(\ddot a > 0\). Dark energy stops the fall at the
floor \(H_\infty\); it does not reverse it. The familiar intuition
that dark energy ``speeds the expansion up'' is true of the scale
factor and false of the rate, and the difference is exactly what the
phrase ``Hubble constant'' obscures.

The result of the six steps is a single curve --- the entire
expansion history, from the radiation era through the matter era to
the de~Sitter terminus, with no continuous parameter adjusted. It is
drawn in Figure~\ref{fig:H-of-t-derived} (placed with its landmark
walk in \S\ref{sec:landmarks}, page~\pageref{fig:H-of-t-derived}),
and it is the central object of this paper. In the language of
\GDOS{}, \(H(t)\) is a \emph{registered observable} of the
open-system projection, and the early- and late-universe
determinations are two operational readings of it; the ``tension'' is
a statement about those two readings, not about the derived history,
which is single-valued. This is the viewpoint the rest of Part~III
makes quantitative and Part~IV reconciles.

\looseness=-1 \emph{Scope --- where this curve begins.} Equations
\eqref{eq:frw-metric}--\eqref{eq:hinfty} describe the classical eras
--- radiation, matter, vacuum --- and that is everything the corridor
argument needs: nothing here reaches earlier than the radiation era
except through two measured survivors, the relic temperature fixing
\(\Omega_r\) and the CMB's physical densities. The history before that --- the inflationary launch,
reheating, and the lock crossing that populates the ledger --- is
derived and drawn in the parent \ATen{} \citep{ArisakaA10Cosmos},
and deliberately not reproduced: this paper is self-sufficient for
what it claims.

\clearpage
\null\vfill
\noindent\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth,height=0.85\textheight,keepaspectratio]{c2_omega_budget_pair.pdf}
\par
{\setlength{\abovecaptionskip}{10pt}%
\captionof{figure}[The running budget \(\Omega_i(t)\), shown twice.]{\textbf{The
running budget \(\Omega_i(t)\), shown twice --- the evolution the
present-epoch constants seed.} The four fractions always sum to one;
only the matter:\(\Lambda\) seam moves, the internal matter ratios
frozen for all time (\(\Omega_b/\Omega_m = 1/2\pi\),
\S\ref{sec:baryon-loading}). \emph{Top:} logarithmic cosmic time,
radiation handing over to matter at \(z = 3400\). \emph{Bottom:}
linear time, with the \(\pi\)-locked inventory as the snapshot at the
\(\Omega_m = 1/\pi\) crossing, \(75\)~Myr before the present (inset;
today \(\Omega_m = 0.315\), inside the \(\pm160\)~Myr blur). Adapted
from the parent \ATen{} at the derived rate; generated by
\texttt{c2\_fig\_omega\_budget.py}, released with the paper.}\label{fig:omega-budget}}
\end{minipage}
\vfill\clearpage

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.88\textheight,keepaspectratio]{c2_a_of_t_derived.pdf}
\caption[The derived scale factor \(a(t)\): the time dependence beneath \(H(t)\).]{\textbf{The
derived scale factor \(a(t)\) --- the time dependence beneath every
dilution factor.} The solution of Eq.~\eqref{eq:friedmann-hierarchy}
with the \(\pi\)-locked inventory, drawn as its own object.
\emph{Top:} nineteen decades on log--log axes: the radiation-era slope
\(a \propto t^{1/2}\), the matter-era slope \(a \propto t^{2/3}\), and
the de~Sitter upturn \(a \propto e^{H_\infty t}\), with equality
(\(z = 3400\)), the corridor switch (\(z_* = e^7\)), and today
(\(a = 1\), \(t_0 = 13.92\)~Gyr) marked. \emph{Bottom:} the late
universe in linear time: the closed form of Eq.~\eqref{eq:hoft-closed},
the inflection where deceleration turns to acceleration (\(z = 0.62\))
--- the one moment \(\ddot a\) changes sign, while \(H\) keeps falling
--- and the approach to the pure exponential, doubling every
\(\ln 2/H_\infty = 12.3\)~Gyr. Generated by
\texttt{c2\_fig\_a\_of\_t.py}, released with the paper.}
\label{fig:a-of-t}
\end{figure}


\section{Geometrical determination of \texorpdfstring{\(H_0\)}{H0}: geometry first, the sky second}
\label{sec:absolute-h0}

The background expansion rate is not an input to be bracketed against
the data; it is an \emph{output} of the same six-dimensional geometry that fixes the
Standard-Model couplings. This section derives it twice, and the order
of the two derivations is the point. \emph{Route~I} is geometry alone:
the GU-5 conversion factors (\S\ref{sec:gu5-bridge}), the theorem-grade
scale chain, the vacuum-energy seesaw, and one codebook-selected ledger
monomial --- no telescope, no microwave sky, no ladder, nothing measured
off any astronomical object. \emph{Route~II} is the precision
confrontation: the sky's two \(h\)-independent physical densities,
divided by the derived fractions, land on the same rate --- and a
variant of Route~II using only Big-Bang nucleosynthesis removes the
microwave sky from the confirmation as well. The derivation is imported
from the cosmology paper \ATen{} \citep{ArisakaA10Cosmos} and sharpened
here; it is stated self-contained and in the register of a working
cosmologist. Everything below is \GG{}/\GDOS{}/\GRIP{} output --- the
stage of \S\ref{sec:gg-framework}, the ledger, and the bridge ---
and nothing else.

\subsection{Route I --- the rate from the geometry alone: the vacuum-energy seesaw}
\label{sec:seesaw-body}

The dark-energy sector delivers \(H_0\) first, and delivers it with no
astronomical input at all: every symbol in the route below is a theorem
of the scale chain, a GU-5 conversion, a derived fraction, or a
codebook-selected ledger monomial. It is worth walking the route factor
by factor, because every step can be checked
on a calculator. The first Friedmann equation at the present epoch, with the
flat late-time budget, reads
\(H_0^2 = \rho_\Lambda/(3\,\Omega_\Lambda M_{\mathrm{Pl,red}}^2)\); and the
geometric seesaw of the cohort fixes the vacuum density itself,
\(\rho_\Lambda = \zeta_\Lambda\,\Mlock^8/M_{\mathrm{Pl,red}}^4\), with
\(\zeta_\Lambda\) a single dimensionless coefficient of the boundary
projection. Substituting one into the other gives the closed form
\begin{equation}
H_0 = \frac{1}{M_{\mathrm{Pl,red}}}
\left(\frac{\Mlock^2}{M_{\mathrm{Pl,red}}}\right)^{2}
\sqrt{\frac{\zeta_\Lambda}{3\,\Omega_\Lambda}}
\;=\; 498.3\,\sqrt{\zeta_\Lambda}~\text{km\,s}^{-1}\,\text{Mpc}^{-1}.
\label{eq:c2-H0-closed}
\end{equation}

\emph{The chain, number by number.} The lock scale is a theorem of the scale
chain (\AFive{}~T8.7): \(\Mlock = \tfrac12 E_{\mathrm{Pl}}\, e^{-K} =
\tfrac12 \times 1.221\times10^{19}\,\mathrm{GeV} \times e^{-35} = 3.85\)~TeV
--- the same number on which the collider companion \PaperPOne{} stakes its
falsifiers. The first seesaw rung is
\(\Mlock^2/M_{\mathrm{Pl,red}} = (3.85\,\mathrm{TeV})^2 / 2.435\times10^{18}
\,\mathrm{GeV} = 6.08\times10^{-12}\,\mathrm{GeV} = 6.08\)~meV --- already
the right neighborhood for a vacuum-energy scale, out of two inputs fixed
nineteen and three decades up. The second rung divides by the Planck mass
once more: \(\Mlock^4/M_{\mathrm{Pl,red}}^3 = 1.52\times10^{-41}\,
\mathrm{GeV} = 712\)~km\,s\(^{-1}\)\,Mpc\(^{-1}\) (converting by
\(E/\hbar\) and the megaparsec --- the GU-5 bridge at work, and the only
place any unit enters). The inventory supplies the last factor,
\(\sqrt{1/(3\Omega_\Lambda)} = 0.699\) at \(\Omega_\Lambda = 1 - 1/\pi\),
giving the coefficient carried cohort-wide as \(498.3\) (our
re-evaluation with PDG constants gives \(498.1\), a \(0.04\%\)
convention difference in the Planck-mass digits, invisible against every
error bar in play). Note what the chain has already done: it has produced
the \(10^{-61}\) of Eq.~\eqref{eq:c2-dimensionless-task} from
\(\tfrac12 e^{-35}\), one seesaw, and \(\pi\) --- one hundred and
nineteen of the famous one hundred and twenty orders of magnitude,
before the last coefficient is even discussed.

\emph{The lock scale is pinned in the laboratory.} Because
\(H_0 \propto \Mlock^4\), the route's skeptic will ask what a
percent-level threshold correction to the lock scale would do: move the
rate by several percent. But the fear is bounded \emph{without
cosmology}: the same lock scale predicts the electroweak vacuum
expectation value, \(v = \Mlock\,(2/5)^3 = 246.33\)~GeV against the
measured \(246.22\) --- a laboratory pin (through \(G_F\)) at
\(+0.045\%\), which \(H_0 \propto \Mlock^4\) maps to \(\pm0.18\%\) on
the rate, \(\pm0.12\)~km\,s\(^{-1}\)\,Mpc\(^{-1}\). The chain is
therefore stiff at every rung but one, and everything now rests on the
single dimensionless coefficient \(\zeta_\Lambda\).

\emph{What \(\zeta_\Lambda\) is allowed to be.} The coefficient is not
free: \MSMF{} forces it to be a bounded-cost monomial in the \OGNfive{}
integers with the transcendentals \(\{\pi, e\}\), and the bounded-cost
search (\AFive{}~O6; \ATen{}) returns a small cost-minimal cluster:
\(1/J^2 = 0.0204\), \(2/(NK) = 0.0190\), \(2/(K\pi) = 0.0182\), and
\(\pi/(LK) = 0.0180\), mapping through Eq.~\eqref{eq:c2-H0-closed} to
\begin{equation}
H_0\big|_{\rm cluster} \;=\; 66.8\text{--}71.2~\text{km\,s}^{-1}\,\text{Mpc}^{-1}.
\label{eq:c2-cluster-bracket}
\end{equation}
Read before any selection, the bracket is already a result, and a
two-sided one. The rate the geometry permits is \(67\)--\(71\) ---
with \emph{zero} observational input, of any kind, anywhere in the
chain. And the distance ladder's \(73.50\), which would require
\(\zeta_\Lambda = 0.0218\), lies \emph{outside the entire cluster}:
pure geometry, before any telescope is consulted, refuses the high
reading while bracketing the low one. Which member of the cluster the
codebook itself selects is the subject of the next subsection.

\emph{The meV form --- a unit identity, not a second test.} The same
chain can be read at the quarter power: \(\rho_\Lambda^{1/4} =
\zeta_\Lambda^{1/4}\,\Mlock^2/M_{\mathrm{Pl,red}} = 0.366 \times
6.08~\mathrm{meV} = 2.23\)~meV, against the familiar \((2.24 \pm
0.02)\)~meV. We bill this honestly: the ``observed'' meV scale is
\((3\Omega_\Lambda)^{1/4}\sqrt{\bar M_{\rm Pl} H_0}\) --- arithmetic on
the same \(H_0\) and \(\Omega_\Lambda\) the seesaw already targets ---
so this paragraph is the rate agreement converted to different units,
printed as bookkeeping, and counted \emph{once}. What the quarter-power
form does add is the framing of the ``\(10^{120}\) problem'' as two
powers of a derived hierarchy --- and one silent premise, now given its
own open-question entry (OQ-Hubble-9): that the Standard-Model vacuum
loops below \(\Mlock\) do not re-source \(\rho_\Lambda\) on the
boundary. The seesaw states what the vacuum energy \emph{is}; the
statement of why the loop contributions do not gravitate is a
projection-sourcing property the \GDOS{}/\OGGEP{} layer owes and has
not yet delivered as a theorem.

\subsection{The selection of \texorpdfstring{\(\zeta_\Lambda\)}{zeta-Lambda}: by the codebook, not by the data}
\label{sec:zeta-selection}

A hostile reader will now expect the trick: with four cluster members
spanning \(66.8\)--\(71.2\), surely the paper picks the one the sky
prefers and calls it a derivation. That is precisely what this
subsection does \emph{not} do, and the discipline is not invented here:
it is the published selection machinery of \ANine{}, applied without
modification.

\ANine{} selects its twenty-eight Standard-Model parameters by a
four-step procedure (its MSMF/\OGN{} selection principle): a candidate
space of rational monomials over a \emph{structural base set on the
parameter's boundary stratum}; an admissibility filter --- every
constituent must have a \emph{named structural role on that stratum},
with \(\pi\) available only on strata containing the compact circle
\(S^1_\theta\) and \(e\) only on radial strata; an empirical-precision
filter; and the cost minimum. The decisive discipline is the
admissibility filter, and \ANine{} has already used it against its own
arithmetic twice, in exactly our situation. For
\(\sin^2\theta_{12}^{\rm PMNS}\) the bounded-cost search returned the
\emph{cheaper} orphan \(1/\pi\) --- and \ANine{} rejected it as
``geometrically isolated,'' adopting the costlier structural form
\(L^2/3N^3\). For \(\sin^2\theta_{23}^{\rm PMNS}\) a numerical tie was
broken by the \emph{structural-naming rule}: the winner must be
nameable as a mechanism. Cost never selects; structure does.

Apply that rulebook here. The stratum is fixed by what
\(\zeta_\Lambda\) \emph{is}: the residue of the Wilson-line
(Hosotani) zero mode on \(S^1_\theta\) --- \ATen{} identifies it as
``the same kind of object as the strong-\(CP\) residue \(\bar\theta\)
of \ANine{}, a geometric closure-index suppression, not a radiative
integral.'' So \(\zeta_\Lambda\) lives on the Wilson-line stratum, the
same stratum as \(\bar\theta\), whose established grammar prices
exactly three kinds of object: the \(L\)-holonomy angle \(2\pi/L\)
(the sine argument of \(\bar\theta\) itself --- ``the smallest
compact-angle phase''), the orbifold/dyadic seam factor \(2\) (the
\(\tfrac12\) of \(\Mlock = \tfrac12 E_{\mathrm{Pl}} e^{-K}\), the
\(\tfrac12\) of \(\Omega_b = 1/2\pi^2\)), and the dual-lock count
\(K\) (the \(e^{-K}\) suppression; the Kaxion's
\(\Delta\nu/\nu = 1/K\)). Now walk the cluster:

\begin{table}[htbp]
\centering\small
\caption[The codebook selection of \(\zeta_\Lambda\), applied \emph{before} any comparison with data.]{The codebook selection of \(\zeta_\Lambda\), applied
\emph{before} any comparison with data: each cost-minimal cluster
member against \ANine{}'s admissibility and structural-naming rules on
the Wilson-line stratum. Exactly one candidate survives. The rejections are made by rules written down elsewhere and for other purposes, not by whether the survivor lands on a useful number. Two of the six candidates were raised by the 2026 audit rather than by me, and one of those two lands on the right value and is rejected anyway. That is the selection working blind, and it is the only reason the survivor counts for anything.}
\label{tab:zeta-selection}
\begin{tabularx}{\textwidth}{@{}llX@{}}
\toprule
\textbf{Candidate} & \textbf{Verdict} & \textbf{Reason (the \ANine{} rulebook)}\\\midrule
\(2/(NK)\) & inadmissible & \(N\)'s named roles (the triplicity of the fermionic boundary replication) exist only on matter strata; \(N\) has no named role on the Wilson-line stratum.\\
\(1/J^2\) & orphan & \(J\)'s named roles are radial (\(z_* = e^J\)) and dyadic-depth (the \(2^{-J}\) seam price inside \(\bar\theta\)); a \emph{bare} \(J^{-2}\) has no angular-stratum name --- \(J\) enters this stratum only through the composite \(K = LJ\). The same class as the rejected \(1/\pi\) orphan of \(\theta_{12}\), and likewise cheaper than the winner.\\
\(2/(K\pi)\) & orphan & \(\pi\)'s only named role on this stratum is inside the composite holonomy angle \(2\pi/L\) --- exactly as \(J\) enters only through \(K = LJ\). Decomposed over the priced objects \(\{\tfrac12,\, 2\pi/L,\, 1/K\}\), the monomial \(2/(K\pi)\) demands an \emph{inverse} defect \((2\pi/L)^{-1}\), which names no mechanism; a bare \(\pi\) downstairs is the same class as the rejected \(1/\pi\) orphan of \(\theta_{12}\). The physical reading agrees: a residue holds its angle \emph{upstairs} (\(\bar\theta\)'s \(\sin(2\pi/L)\)), while \(\pi\) downstairs is the cohort's \emph{share} form (\(\Omega_m = 1/\pi\)), and a vacuum residue is not a share of a budget.\\
\(\pi/(LK)\) & \textbf{selected} & Decomposes entirely into named stratum objects; see Eq.~\eqref{eq:c2-zeta-mechanism}.\\
\(1/(2L^2)\) & inadmissible & Bare \(L\) has no named role on the Wilson-line stratum outside the composite holonomy angle \(2\pi/L\) (the same rule that orphans \(2/(K\pi)\)); raised by the 2026 audit and priced here.\\
\(2^{-3}/J = 1/56\) & inadmissible & \(J\)'s named roles are radial, not Wilson-line (as for \(1/J^2\)); and a \emph{repeated} dyadic seam (\(2^{-3}\)) has no single mechanism name --- see the occupancy rule below. Raised by the 2026 audit; lands on \(66.6\) and is \emph{still} rejected, which is the selection working blind.\\
\bottomrule
\end{tabularx}
\end{table}

The surviving monomial is not merely admissible; it has the mechanism
name the structural-naming rule demands, composed of three objects
that are each already load-bearing elsewhere in the cohort:
\begin{equation}
\zeta_\Lambda \;=\; \frac{\pi}{LK}
\;=\;
\underbrace{\frac{1}{2}}_{\substack{\text{orbifold half}\\ (\Mlock,\ \Omega_b)}}
\times
\underbrace{\frac{2\pi}{L}}_{\substack{\text{holonomy defect}\\ (\bar\theta\text{'s angle})}}
\times
\underbrace{\frac{1}{K}}_{\substack{\text{dual-lock share}\\ (\text{Kaxion's } 1/K)}}
\;=\; \frac{\pi}{175} \;=\; 0.017952\,:
\label{eq:c2-zeta-mechanism}
\end{equation}
\emph{the undrained vacuum residue of the lock episode is one
orbifold-halved \(L\)-holonomy defect, shared over the \(K\) winding
cells of the dual lock.} Nothing new is introduced --- the \(\tfrac12\)
that prices the lock scale, the angle that prices \(\bar\theta\), the
cell count that prices the Kaxion splitting. Two clauses complete the
rulebook, both implicit in \ANine{}'s structural-naming requirement
and made explicit here because the audit asked. \emph{Single
occupancy:} a mechanism name attaches to one geometric feature, so
each named object enters a candidate at most once --- a squared
holonomy defect or a tripled seam (\(\tfrac12(2\pi/L)^2/K\),
\(2^{-3}/J\)) is a different physical claim with no name on the
stratum, and is inadmissible however cheap. \emph{No empirical
filter:} of \ANine{}'s four selection steps, the empirical-precision
filter was \emph{not consulted} for \(\zeta_\Lambda\) --- the cluster
and the survivor were fixed by admissibility and naming alone, before
any comparison with the sky, which is what ``pre-registered'' means
at F-Hubble-11 (and why the table above can afford to reject a
candidate that lands exactly on \(66.6\)). And one commitment, so
that the grammar is law rather than jurisprudence: the rulebook as
codified here --- admissibility by named stratum role, single
occupancy, no empirical filter --- is \emph{frozen}, and any future
selection anywhere in the cohort must use it unchanged; a clause
added after the next inconvenient candidate would itself be a
falsification of the method. (Single occupancy forbids
\emph{unnamed} repetition; ledger-indexed towers such as \(2^J\) or
\(e^K\) are one named object each --- a depth-\(J\) or depth-\(K\)
mechanism --- which is why the frozen rule convicts \(2^{-3}/J\),
whose exponent has no stratum role, and acquits every standing
selection of \ANine{}.) With the selection made,
Route~I closes:
\begin{equation}
\boxed{\;H_0^{\mathrm{geom}} \;=\; 498.3\,\sqrt{\pi/175}
\;=\; 66.8 \pm 0.2~\text{(structural)}~\text{km\,s}^{-1}\,\text{Mpc}^{-1}\;,}
\label{eq:c2-H0-seesaw}
\end{equation}
where the structural error carries the laboratory pin on the lock
scale and the conditional status of the selection lemma. Every factor
is \GG{}/\GDOS{}/\GRIP{}; the sky has not yet been consulted.

Three honesty clauses, stated before the data are allowed in.
\emph{First, the grade.} This selection yields Derived
(cond.,~\MSMF{}/\OGN{}) --- conditional on one named lemma, the
\emph{stratum assignment} (\(\zeta_\Lambda\) is the Wilson-line
zero-mode residue), which \ATen{} carries as its flagged physical
reading of the coefficient. That is exactly the grade structure of
\ANine{}'s PMNS entries; Derived (theorem) waits on the Hosotani
computation itself (\AFive{}~O6). \emph{Second, the cost accounting.}
Under \ANine{}'s closure rule for \(K\) (\(K = LJ\) is not an
independent atom, so a \(K\)-factor costs two), the ``cost-3'' cluster
entries are cost four and the orphan \(1/J^2\) stays cost two ---
i.e.\ cost favors the orphan \emph{more} strongly, and the selection
above leans on admissibility and naming alone, exactly as the
\(\theta_{12}\) precedent requires. \emph{Third, the interpretive
step, named.} The three rejections are one rule applied three times
--- a constituent with no bare name on the stratum: \(N\) nowhere,
\(J\) only through the composite \(K = LJ\), \(\pi\) only through
the composite \(2\pi/L\) --- so no candidate needed a rule invented
for it. What remains interpretive is the enumeration of the stratum's
priced objects \(\{\tfrac12,\, 2\pi/L,\, 1/K\}\), read off its three
cohort precedents (\(\Mlock\), \(\bar\theta\), the Kaxion
splitting); that reading is carried as the second clause of the
selection lemma, not as a theorem. All three
clauses are cashed by one pre-registered number: the Hosotani
computation, when completed, must return \(\zeta_\Lambda = \pi/175\)
to within the structural tolerance (\(\sim\!1\%\)) --- registered as
falsifier \textsf{F-Hubble-11} (\S\ref{sec:falsifiers}), two-sided by
construction. The target is printed before the lemma; it cannot be
renegotiated after it.

\subsection{The closed form: the expansion rate from \texorpdfstring{$\{c, G_N, \hbar\}$}{\{c, GN, hbar\}} and the ledger alone}
\label{sec:h0-closed-form}

Route~I was walked factor by factor above, and its coefficient was
selected by the codebook; it is worth collapsing the whole route into
a single analytic expression, because the claim of this section ---
geometry first, the sky second --- is nowhere more visible than in
one equation whose every symbol can be pointed at. Four rungs enter.
The only dimensionful objects anywhere are the first three GU-5
constants, through the Planck mass \(M_{\mathrm{Pl}} =
\sqrt{\hbar c/G_N}\) and its reduced companion
\(M_{\mathrm{Pl,red}} = M_{\mathrm{Pl}}/\sqrt{8\pi}\). The lock scale
is the theorem \(\Mlock = \tfrac12 M_{\mathrm{Pl}}\,e^{-K}\)
(\AFive{}~T8.7), which in reduced units is the pure statement
\(\Mlock = \sqrt{2\pi}\,M_{\mathrm{Pl,red}}\,e^{-K}\). The vacuum
residue is the seesaw dressed by the selected monomial,
\(\rho_\Lambda = \zeta_\Lambda\,(\Mlock^2/M_{\mathrm{Pl,red}})^4\)
with \(\zeta_\Lambda = \pi/(LK)\). And the present-epoch Friedmann
equation on the derived inventory divides by
\(3\,\Omega_\Lambda = 3\,(1 - 1/\pi)\). Substituting each rung into
the next, every power of the Planck mass cancels except a single
overall rate, the transcendentals collect into \(2\pi^5/(\pi - 1)\),
and Eq.~\eqref{eq:c2-H0-closed} becomes
\begin{equation}
\boxed{\;
H_0 \;=\; e^{-4K}\,
\sqrt{\frac{2\,\pi^{5}\,c^{5}}{3\,L\,K\,(\pi-1)\,\hbar\,G_N}}
\;=\; 66.7~\text{km\,s}^{-1}\,\text{Mpc}^{-1}.
\;}
\label{eq:c2-H0-analytic}
\end{equation}
Evaluated with CODATA constants the right-hand side gives \(66.74\)
--- the \(498.1\sqrt{\zeta_\Lambda}\) of the re-evaluated chain,
now with no intermediate scale quoted at all. The provenance of every
symbol can be stated in one sentence: \(c\), \(G_N\), and \(\hbar\)
carry the dimensions (they are the GU-5 bridge, and
\(\sqrt{c^5/\hbar G_N}\) is nothing but the inverse Planck time);
the ledger integer \(K = 35\) appears twice, as the depth of the
descent (\(e^{-4K}\)) and in the selected monomial; \(L = 5\) appears
once in the monomial --- and a second time in the laboratory, through
the vev pin \(v = \Mlock\,(2/5)^3\) that stiffens the chain; \(\pi\)
enters three ways (Einstein's \(8\pi\), the monomial's numerator, and
the inventory's \(1 - 1/\pi\)); \(e\) is the scale chain's
exponential; and the structural rationals \(\tfrac12\) (the lock
theorem) and \(3\) (Friedmann) are geometry's bookkeeping, not ledger
entries. There is nothing else in the formula. In particular there is
no astronomical quantity of any kind: the expansion rate of the
universe, to the precision quoted, is a statement about three
laboratory constants and two small integers.

Two celebrated hierarchies fall out of the closed form as
corollaries, and they are worth one paragraph each of plain
statement. First, the age of the universe in Planck units. The
formula gives directly
\(H_0\,t_{\mathrm{Pl}} = \sqrt{2\pi^5/[3LK(\pi-1)]}\;e^{-4K} =
0.738 \times e^{-140} = 10^{-60.9}\): the famous sixty-one orders of
magnitude between the Planck rate and today's expansion --- the
dimensionless task of Eq.~\eqref{eq:c2-dimensionless-task} --- are
\(4K = 140\) e-folds of the derived descent (\(4K/\ln 10 = 60.80\)
orders), dressed by a geometric factor of order unity. The question
``why is the universe so old?'' receives, on this reading, a
one-integer answer: because \(K = 35\), entering the vacuum sector
at fourth power. Second, the cosmological-constant problem itself.
The same rungs give
\(\rho_\Lambda/\rho_{\mathrm{Pl,red}} = \zeta_\Lambda\,(2\pi)^4\,
e^{-8K} = 10^{-120.2}\): the most notorious mismatch in physics ---
one hundred and twenty orders of magnitude --- is \(8K\) e-folds
(\(121.60\) orders) corrected by \(1.4\) orders of pure geometry.
What the literature treats as a catastrophic cancellation to be
explained is, in the closed form, two powers of one derived
hierarchy --- the statement made in prose below
Eq.~\eqref{eq:c2-H0-closed}, here exhibited as arithmetic.

One honesty note completes the subsection, in the register the
codebook requires. Of the five \OGNfive{} integers, the closed form
contains \(K\) and \(L\) explicitly; \(M = 1\), \(N = 3\), and
\(J = 7\) do not appear in the final expression, and no claim is made
that they do. They act upstream --- selecting the fibration and the
anomaly closure that make T8.7 and the \(\pi\)-locked inventory
theorems in the first place --- and \(J\) surfaces elsewhere in the
corridor, in the switch redshift \(z_* = e^7\) and in the rejected
cluster member \(1/J^2\) (\S\ref{sec:zeta-selection}). And the
equation is exactly as conditional as its rungs: the inventory
carries its \(\mathsf D^{*}\) grade, and the monomial carries the
selection lemma, cashed by the pre-registered Hosotani target of
F-Hubble-11. Nothing about writing the route as one line strengthens
its grade; what it strengthens is its \emph{exposure} --- a reader
with a calculator, Eq.~\eqref{eq:c2-H0-analytic}, and five constants
can now check the central claim of this paper in under a minute, and
a referee who wishes to attack it knows exactly which five symbols
to aim at.

\subsection{Route II --- the precision confirmation: the microwave sky reads the derived rate off two densities}
\label{sec:route-two}

The acoustic peaks of the CMB do not measure \(H_0\). What they
measure, exquisitely, is two \emph{physical} densities
\citep{Planck2018Parameters}, each independent of the expansion rate:
\begin{equation}
\omega_b \equiv \Omega_b h^2 = 0.02237 \pm 0.00015,\qquad
\omega_m \equiv \Omega_m h^2 = 0.1430 \pm 0.0011.
\label{eq:c2-omegas}
\end{equation}
The physics behind each is worth one sentence, because the two are
read off \emph{different} features of the sky. The baryon density
\(\omega_b\) comes from \emph{baryon loading}: baryons add inertia to
the oscillating photon--baryon plasma without adding pressure, so the
compression peaks (first, third) are enhanced relative to the
rarefaction peaks (second) --- the second-to-first peak-height ratio
is a clean baryometer, and the primordial deuterium abundance reads
the same number through entirely different physics. The matter
density \(\omega_m\) comes from the \emph{epoch of matter--radiation
equality}: modes that enter the horizon during radiation domination
have their gravitational potentials decay under them, which drives
the oscillations and boosts the peak-height envelope; the shape of
that envelope reads \(z_{\rm eq}\), and hence
\(\omega_m\).\footnote{\(\omega_m\) is the total matter density
including the neutrino-mass contribution (Planck
TT,TE,EE+lowE+lensing); the baryon-plus-CDM sum alone is \(0.1424\).}

Why neither number carries \(h\) is the pivot of the confirmation.
The convention \(\omega_i \equiv \Omega_i h^2\) encodes a physical
fact: \(\omega_i\) is an \emph{absolute} density (an energy per
volume, in fixed units), while \(\Omega_i\) is a fraction of the
critical density --- and the critical density itself scales as
\(h^2\). The plasma physics at \(z \approx 1100\) responds to the
actual densities present, not to fractions defined against today's
expansion rate; that is why the microwave sky can pin \(\omega_b\)
and \(\omega_m\) to a fraction of a percent while knowing nothing of
\(h\). This is the celebrated \emph{geometric degeneracy}: the
primary anisotropies alone cannot measure the present rate at all.
Standard \(\Lambda\)CDM breaks the degeneracy with a fitted late-time
model; \GGtheory{} breaks it with theorems --- and, after Route~I,
does not even need to break it to \emph{know} the rate. What follows
is a confirmation, not a derivation.

The inversion is then one line. Standard practice runs the
six-parameter fit: vary all six dials, including \(h\), until the
whole pattern matches --- \(H_0\) emerges as an output of the fit,
with the fractions along for the ride. \GGtheory{} runs the logic in
the opposite direction: the fractions are theorems
(\S\ref{sec:inventory-derived}), so dividing a \emph{measured}
physical density by a \emph{predicted} fraction returns \(h\) with no
freedom left,
\begin{equation}
\omega_i = \Omega_i h^2
\;\Longrightarrow\;
h = \sqrt{\frac{\omega_i}{\Omega_i}}
\;=\;
\begin{cases}
\;\sqrt{0.02237/0.05066} = 0.665 & (\text{baryon anchor},\ H_0 = 66.5),\\[3pt]
\;\sqrt{0.1430/0.31831} = 0.670 & (\text{matter anchor},\ H_0 = 67.0).
\end{cases}
\label{eq:c2-H0-anchor}
\end{equation}
Both anchors land inside Route~I's structural band: what the geometry
predicted, the sky now confirms with its full precision.

The agreement of the two lines is not one test written twice. The two
densities are measured from different features of the spectrum ---
the peak ratio for \(\omega_b\), the envelope for \(\omega_m\) ---
and the two predicted fractions come from one fibration
(\S\ref{sec:inventory-derived}), rigidly coupled. If the fibration
were wrong --- if, say, the true baryon fraction sat ten percent
above \(1/2\pi^2\) --- the baryon anchor would return an \(h\) five
percent below the matter anchor, and the two lines of
Eq.~\eqref{eq:c2-H0-anchor} would disagree with \emph{each other},
before any comparison with Planck's fitted rate. Their agreement at
\(0.7\%\) is therefore a consistency test of the inventory's coupled
structure, passed internally.

One variant of the inversion deserves its own paragraph, because it
removes the microwave sky from the confirmation entirely. The baryon
density is measured a second way --- the primordial deuterium
abundance, with the LUNA reaction rates --- giving
\(\omega_b^{\rm BBN} = 0.02233 \pm 0.00036\) with no CMB anisotropies
anywhere in the chain; divided by the derived \(1/2\pi^2\), it
returns
\begin{equation}
H_0^{\rm BBN} \;=\; 66.4 \pm 0.5~\text{km\,s}^{-1}\,\text{Mpc}^{-1}.
\label{eq:c2-H0-bbn}
\end{equation}
Route~II therefore does not even need the sky it confirms against:
any \(h\)-independent absolute density will do, and every one
available --- two from the microwave sky, one from the first three
minutes --- lands on the derived rate.

The error budget is elementary and worth displaying once. Since
\(h \propto \sqrt{\omega_i}\), the fractional error halves:
\(\sigma_h/h = \tfrac12\,\sigma_{\omega}/\omega\), giving
\(0.34\%\) (\(\pm 0.23\)) on the baryon anchor and \(0.38\%\)
(\(\pm 0.26\)) on the matter anchor; the conditional systematic of
the shared measure lemma (\S\ref{sec:inventory-derived}) is
common to both and does not enter their difference. The two-anchor
consistency yields the canonical value --- the adopted center of the
route span, not a formal weighting; App.~\ref{app:calc} prints the
full bookkeeping:
\begin{equation}
\boxed{\;H_0^{(0)} \;=\; 66.6 \pm 0.2 \;\approx\; 67~\text{km\,s}^{-1}\,\text{Mpc}^{-1}\;.}
\label{eq:c2-H0-number}
\end{equation}
This confirmation is itself parameter-free --- an inversion, not a
fit. Standard \(\Lambda\)CDM \emph{fits}
\(\Omega_m\) and \(\Omega_b\) and reads off \(H_0 = 67.4\); GG
\emph{predicts} the fractions, so \(h\) is forced --- and it lands on
the independently derived Route~I value. A full Boltzmann solution
completes the confrontation: fed GG's exact fractions,
the code (CLASS) reproduces the observed temperature spectrum --- peak
positions within \(0.5\%\), the second-to-first peak-height ratio to
\(0.3\%\), the diffusion-damping scale to \(0.4\%\). The inventory does
not merely imply \(H_0 \approx 67\) arithmetically; it fits the
microwave sky at that value. The residual \(\sim\!1\%\) below Planck's
\(67.4\) is exactly GG's exact \(\pi\)-values sitting \(1\)--\(3\%\) above
Planck's best-fit fractions --- the \emph{coherent \(\pi\)-offset}, not a
new tension.

\subsection{The concordance, and the canonical rate}
\label{sec:two-route-concordance}

The two routes now stand side by side, and Figure~\ref{fig:two-route-concordance}
draws them on one axis. Route~I --- geometry alone --- gives
\(66.8 \pm 0.2\) (structural), inside a pure-geometry bracket that
excludes \(73.5\) outright. Route~II --- the sky's \(h\)-independent
densities against the derived fractions --- gives \(66.6 \pm 0.2\),
with its CMB-free variant at \(66.4 \pm 0.5\). The two central values
differ by \(0.25\%\), \(0.6\sigma\) of their combined errors: the
routes share no ingredient except the geometry itself --- one runs
through TeV-scale physics and the vacuum, the other through the plasma
of the young universe --- and they meet inside each other's error
bars. Throughout the cohort the canonical rate is the concordance
value, the adopted center of its route span:
\(H_0^{(0)} = 66.6 \pm 0.2\). Agreement of this kind cannot be
arranged after the fact without breaking one of the routes' other
commitments --- the acoustic spectrum on one side, or the electroweak
vev and the measured dark-energy scale on the other.

What would break the concordance is equally explicit: any of the three
anchors of Route~II leaving the band; the Hosotani computation
returning a value off the pre-registered \(\pi/175\)
(\textsf{F-Hubble-11}); or a future inventory determination driving
\(\Omega_m\) firmly below \(1/\pi\) (\textsf{F-Hubble-4} territory).
The rate of this paper is not one number defended once; it is one
number exposed three ways.

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{c2_two_routes.pdf}
\caption[The two routes to the present rate: geometry determines, the sky confirms.]{\textbf{The
two-route determination of the present rate.} \emph{Top lane:} Route~I,
geometry alone --- the GU-5 bridge, the theorem-grade lock scale
(laboratory-pinned through the electroweak vev at \(0.045\%\)), and the
codebook-selected \(\zeta_\Lambda = \pi/175\)
(Eq.~\eqref{eq:c2-zeta-mechanism}) --- with the cost-minimal cluster as
the shaded pure-geometry bracket \(66.8\)--\(71.2\), the rejected
monomials (open diamonds; \S\ref{sec:zeta-selection}), and the ladder's
\(73.50\) excluded outright (\(\zeta_\Lambda\) would need \(0.0218\),
outside the cluster). \emph{Middle lane:} Route~II, the confirmation ---
the baryon and matter anchors of Eq.~\eqref{eq:c2-H0-anchor}, the
CMB-free BBN anchor of Eq.~\eqref{eq:c2-H0-bbn}, and the canonical
concordance \(66.6 \pm 0.2\). \emph{Bottom lane:} the two famous
readings, for comparison. No astronomical observation enters the top
lane; the sky enters only as the confirmation and the comparison.
Generated by \texttt{c2\_fig\_two\_routes.py}, released with the paper.}
\label{fig:two-route-concordance}
\end{figure}

\subsection{Provenance: what is derived, what is conditional, what is input}

\begin{table}[htbp]
\centering\small
\caption[The honesty triage for the derived Hubble rate: every object in the two-route derivation.]{The honesty triage for the derived Hubble rate: every object in
the two-route derivation, its grade in the cohort codebook, and its basis.
This is the single canonical triage referenced throughout the paper; the
grades match \ATen{} entry for entry, with the \(\zeta_\Lambda\) grade
carrying the codebook-selection lemma of \S\ref{sec:zeta-selection}.}
\label{tab:c2-triage}
\begin{tabularx}{\textwidth}{@{}llX@{}}
\toprule
\textbf{Object} & \textbf{Grade} & \textbf{Basis}\\\midrule
GU-5 bridge \(\{c, G_N, \hbar\}\) & Declared input & Unit-conversion factors of the 6D geometry (\AFour{}); natural units \(\{1,1,1\}\); the only dimensional content anywhere.\\
Lock scale \(\Mlock=3.85\) TeV & Theorem & The scale chain, \AFive{}~T8.7; laboratory-pinned by the electroweak vev at \(+0.045\%\).\\
Seesaw coefficient \(\zeta_\Lambda = \pi/175\) & Derived (cond., \MSMF{}/\OGN{}) & Selected by stratum availability and structural naming among the bounded-cost cluster (\S\ref{sec:zeta-selection}); conditional on the stratum-assignment lemma; the Hosotani closure (\AFive{}~O6) pre-registered as \textsf{F-Hubble-11}.\\
\(\pi\)-inventory \(\Omega_m,\Omega_b,\Omega_\Lambda\) & \(\mathsf D^{*}\) & The Hopf-fiber identification, conditional on the measure lemma (Conjecture~VIII) and the declared present epoch.\\
Geometric rate \(H_0^{\rm geom} = 66.8 \pm 0.2\) & Derived (cond., \MSMF{}/\OGN{}) & Route~I, Eq.~\eqref{eq:c2-H0-seesaw}: no astronomical input.\\
Two CMB densities \(\omega_b,\omega_m\) & Measurement & \(h\)-independent inputs to Route~II (the confirmation) only; no role in Route~I; the BBN anchor replaces even these.\\
Canonical rate \(H_0^{(0)}=66.6 \pm 0.2\) & \(\mathsf D^{*}\) & The concordance of Routes~I and~II; inherits the inventory grade; no additional freedom.\\
Local-Hole lift \(\epsilon_{\mathrm{Hole}}\) & Derived & Sign and rough size, a structure-formation feature (\S\ref{sec:both-from-anchor}).\\
Ladder residual \(\epsilon_{\mathrm{syst}}\) & Named, open & Read as a distance-ladder calibration systematic; not derived, bounded and left open.\\
\bottomrule
\end{tabularx}
\end{table}

Honest bookkeeping, in the grading of the cohort codebook, is collected in
Table~\ref{tab:c2-triage}: the GU-5 bridge is the declared dimensional
input; the lock scale is a theorem, pinned in the laboratory; the seesaw
coefficient is Derived conditional on the selection lemma, its closure
pre-registered; the \(\pi\)-inventory and the rates it forces are
Derived-conditional under the Hopf-fiber measure lemma; the two CMB
densities are a measurement that enters only the confirmation; and of
the local excess only the Local-Hole lift is derived, the residual named
and left open. The robust content is the concordance of the two routes
at \(H_0^{(0)} \approx 67\), with the geometric route standing on its
own. The early-versus-late \emph{split} of this one
background rate --- the Hubble tension itself --- is the subject of the
next section.

\section{Landmarks on the derived \texorpdfstring{$H(t)$}{H(t)} curve}
\label{sec:landmarks}

A derived expansion history is not one prediction; it is a continuum of
them. Because Eq.~\eqref{eq:friedmann-hierarchy} fixes \(H\) at \emph{every}
epoch once the inventory is fixed, each physically meaningful moment on the
curve --- primordial nucleosynthesis, matter--radiation equality,
recombination, the onset of acceleration, the present, the far future --- is
a place where the geometry has already committed to a number that
observation can check. This section walks the curve from the first second to
the infinite future, treating each landmark as a checkpoint: what happens
there physically, what the derived chart says its epoch and value must be,
and what is measured. Figure~\ref{fig:H-of-t-derived} draws the derived
curve itself --- the central object of this paper --- and
Figure~\ref{fig:landmarks} annotates the same curve with the walk:
nineteen decades of cosmic time on one zero-parameter history. The two
figures are a pair, and the pair is the heart of the paper: the first
shows \emph{what} the geometry forces, the second shows \emph{where}
observation has checked it. Table~\ref{tab:c2-landmarks} compresses the
walk; the derivation courses of the
Technical Appendices (App.~\ref{app:thermal}--\ref{app:soundhorizon}) carry
the arithmetic in full.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.88\textheight,keepaspectratio]{c2_H_of_t_derived.pdf}
\caption[The derived Hubble parameter \(H(t)\), of which \(\Hzero\) is one point.]{\textbf{The Hubble parameter \(H(t)\) is a derived geometric function; \(\Hzero\) is one reading of it.} The \GGtheory{} background expansion rate computed from Eq.~\eqref{eq:friedmann-hierarchy} with the \(\pi\)-locked inventory (\(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\), \(\Omega_b=1/2\pi^2\)) and the derived \(w=-1\); the one dimensionful input is the GU-5 conversion \(G_N\); the rate itself is derived (\S\ref{sec:absolute-h0}). \emph{Top:} the full history on log--log axes, radiation (\(H\propto a^{-2}\propto t^{-1}\)) \(\to\) matter (\(H\propto a^{-3/2}\propto t^{-1}\)) \(\to\) dark energy. \emph{Bottom:} the late universe in linear time. The rate falls monotonically and asymptotes to the de~Sitter floor \(H_\infty=\Hzero\sqrt{\Omega_\Lambda}=55.0\;\kmsMpc\) --- never rising, even once expansion accelerates. The familiar \(\Hzero\equiv H(t_0)=66.6\) is one point on this curve, at \(t_0=13.9\) Gyr; a single number does not capture the dynamical change of the Universe's size. The whole history is a \GG{}/\GDOS{}/\GRIP{} output: fix the densities and their time dependence, and \(H(t)\) follows.}
\label{fig:H-of-t-derived}
\end{figure}

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{c2_H_of_t_landmarks.pdf}
\caption[Landmarks on the derived \(H(t)\): nineteen decades on one curve.]{\textbf{Landmarks
on the derived \(H(t)\).} Nineteen decades of cosmic time --- from
primordial nucleosynthesis to the de~Sitter future --- on the one
zero-parameter curve the \(\pi\)-locked inventory forces. The numbered
checkpoints are walked in \S\ref{sec:landmarks} and tabulated in
Table~\ref{tab:c2-landmarks}: BBN (1), matter--radiation equality (2), the
corridor switch at \(z_* = e^7\) where the sound horizon freezes (3), the
acceleration onset (4), matter--vacuum equality (5), the \(\Omega_m = 1/\pi\)
crossing and today (6), and the de~Sitter floor \(H_\infty = 55.0\) (7).
The rate falls monotonically through all of it; the eras change only the
slope. Seven measured checkpoints, none deviating by more than a percent,
all with the coherent sign of \S\ref{sec:pi-offset}.}
\label{fig:landmarks}
\end{figure}

\paragraph{Landmark 1: primordial nucleosynthesis (\(t \approx 1\)~s--3~min).}
The earliest checkpoint the chart can face is the oldest quantitative test
in cosmology. During the first minutes the expansion rate --- here the
radiation-era branch \(H = 1/2t\) of the derived curve,
Eq.~\eqref{eq:app-tT} --- races the weak interactions, and the surviving
light-element abundances depend on one cosmological number, the
baryon-to-photon ratio \(\eta_b\). The derived chart fixes it:
\(\Omega_b = 1/2\pi^2\) at the derived rate gives \(\omega_b = 0.02247\)
and hence \(\eta_b = 6.16\times10^{-10}\), against \(6.13\times10^{-10}\)
from the measured \(\omega_b\) (confirmed independently by the deuterium
abundance): a \(+0.5\%\) confrontation, carrying the by-now-familiar sign.
The chart's first checkpoint is passed at half a percent, thirteen decades
of time before the epoch whose calibration the tension disputes.

\paragraph{Landmark 2: matter--radiation equality (\(z_{\rm eq} = 3400\), \(t \approx 51\)~kyr).}
Radiation dilutes as \(a^{-4}\), matter as \(a^{-3}\), so their crossing is
fixed by the ratio of two shares, \(1+z_{\rm eq} = \Omega_m/\Omega_r\). With
the derived \(\Omega_m = 1/\pi\) and the measured radiation share at the
derived rate (\(\Omega_r = 9.36\times10^{-5}\)), the crossing lands at
\(z_{\rm eq} = 3400\); Planck's fitted value is \(3387 \pm 21\) --- a
\(+0.4\%\) agreement, same sign again. Equality matters twice over: it is
where structure growth begins in earnest, and its epoch shapes the
peak-height envelope through which \(\omega_m\) is measured --- the reason it
is deliberately \emph{not} double-counted as an independent test in
Table~\ref{tab:c2-cmb-core}.

\paragraph{Landmark 3: the corridor switch (\(z_* = e^7 = 1096.6\), \(t \approx 0.37\)~Myr).}
The integer-ledger landmark: the \GSSB{} transition of
Prop.~\ref{prop:recomb-switch} (Part~IV), whose redshift the \OGNfive{}
integer \(J = 7\) fixes with no continuous freedom, lands at \(0.6\%\) from
the measured decoupling \(1089.92 \pm 0.25\) --- a distinct quantity: \ATen{}'s
atomic chain derives the decoupling itself on the measured value at the
\(0.1\%\) level, the switch \(0.4\%\) above in e-folds. At the switch the derived
curve reads \(H(z_*) = 1.57\times10^{6}\;\kmsMpc\) --- the expansion was
twenty-four thousand times faster than today --- and everything to its past
is read through the acoustic dictionary, everything to its future through
the ladder. It is also the moment the fourth landmark freezes.

\paragraph{Landmark 4: the sound horizon freezes (\(r_s = 145.0\)~Mpc).}
The pressure wave that has been crossing the plasma since the first instant
stops at the switch, leaving a fossil ruler whose length the derived chart
computes with no free parameter (App.~\ref{app:soundhorizon}): \(145.0\)~Mpc
against the measured \(144.43 \pm 0.26\) (\(+0.4\%\)). Every early-universe
``solution'' of the tension must shorten this ruler; the derived reading,
landing on it, is structurally forbidden from doing so --- which is why its
account of the tension lives on the late side (Part~IV).

\paragraph{Landmark 5: the acceleration onset (\(z = 0.62\), \(t \approx 7.8\)~Gyr).}
The deceleration parameter changes sign where \(\rho_m = 2\rho_\Lambda\),
i.e.\ at \(z_{\rm acc} = (2\Omega_\Lambda/\Omega_m)^{1/3} - 1 = 0.62\) --- a
pure function of \(\pi\). Supernova reconstructions of the
deceleration--acceleration transition place it at \(z_t \approx 0.6\)--\(0.7\),
consistent. Note what the curve does \emph{not} do there: \(H(t)\) keeps
falling. Acceleration is a statement about \(\ddot a\), not about \(H\); the
familiar intuition that dark energy ``speeds the expansion up'' is true of
the scale factor and false of the rate, and the difference is exactly what
the phrase ``Hubble constant'' obscures.

\paragraph{Landmark 6: matter--vacuum equality (\(z = 0.29\), \(t \approx 10.4\)~Gyr).}
The vacuum share overtakes matter at
\(z_{\Lambda m} = (\Omega_\Lambda/\Omega_m)^{1/3} - 1 = 0.29\), against the
concordance \(0.30 \pm 0.02\). From here the curve begins its long turn onto
the floor.

\paragraph{Landmark 7: the crossing, and today (\(13.84\) and \(13.92\)~Gyr).}
The declared observing epoch --- the one input the geometry does not remove
(\S\ref{sec:inventory-derived}) --- is the moment the matter share crosses
exactly \(1/\pi\): \(t = 13.84\)~Gyr, a mere \(75\)~Myr before the present.
One bookkeeping sentence, so no table can be accused of mixing epochs:
the inventory rows of this paper are evaluated at the declared crossing
epoch, and over the \(75\)~Myr to the astronomers' present the fractions
evolve by \(-1.0\%\) (\(\Omega_m: 0.3183 \to 0.315\)) while the rate
falls by \(0.24\%\) --- both inside the \(\pm160\)~Myr calibration blur
already priced. We record, without yet claiming anything, an arithmetic
coincidence: the evolved \(0.315\) is numerically the Planck-fitted
matter fraction, i.e.\ the crossing-epoch offset has the same size and
sign as the fitted-versus-exact \(\Omega_m\) displacement --- while the
rate's \(0.24\%\) drift is far too small to account for the \(-1.2\%\)
\(h\) offset, so the epoch declaration can at most explain one entry
of the coherent pattern, not the pattern (OQ-Hubble-10).
Today the curve reads \(H_0 = 66.6\), and the age integral gives
\(t_0 = 0.948/H_0 = 13.92\)~Gyr (App.~\ref{app:age}), against the
Planck-\(\Lambda\)CDM \(13.80 \pm 0.02\): \(+0.9\%\), one more shadow of the
same displacement. The oldest stellar populations, at \(13\)--\(13.5\)~Gyr,
fit comfortably beneath it --- as they would not beneath the age a local
\(73.5\) implies (\(\approx 12.6\)~Gyr, \S\ref{sec:both-from-anchor}).

\paragraph{Landmark 8: the de Sitter terminus (\(H \to 55.0\)).}
The curve's future is as fixed as its past: with \(w = -1\) exactly,
\(H(t)\) settles onto the floor
\(H_\infty = H_0\sqrt{\Omega_\Lambda} = 55.0\;\kmsMpc\) and stays there
forever, the universe thereafter doubling in size every
\(H_\infty^{-1}\ln 2 = 12.3\)~Gyr. This is not a checkpoint that can be
measured; it is the one the equation of state guards (F-Hubble-5), and the
quiet end of the story the rest of the paper argues about: one clock, still
running, at a rate the geometry named before anyone could read it.

\medskip
Read as a whole, the walk makes the Part's central point more vividly than
any single number: seven of the eight landmarks carry a measured comparison,
every one lands within a percent, and every deviation carries the sign the
single \(+1\%\) inventory displacement propagates (\S\ref{sec:pi-offset}).
A fitted model could match any one landmark by turning a dial; matching all
of them with no dial is what it means for the history to be derived.

\begin{table}[htbp]
\centering\small
\caption[The eight landmarks of the derived expansion history: epoch, the derived value and its origin.]{The eight landmarks of the derived expansion history: epoch, the
derived value and its origin, the measurement, and the deviation. Every
quantitative deviation is below one percent and carries the coherent sign of
\S\ref{sec:pi-offset}. The eight are not eight independent successes. They share the inventory and they share the sign, so a fault in the inventory would move all of them together and in the same direction. What the table does show is that one set of integers carries a history from the first three minutes to the present with no free parameter anywhere along it, and that where it misses, it misses small and it misses the same way.}
\label{tab:c2-landmarks}
\begin{tabularx}{\textwidth}{@{}llXll@{}}
\toprule
\textbf{Landmark} & \textbf{Epoch} & \textbf{Derived} & \textbf{Measured} & \textbf{Dev}\\\midrule
1. BBN & \(t\approx1\)\,s--3\,min & \(\eta_b=6.16\times10^{-10}\) (\(\Omega_b=1/2\pi^2\)) & \(6.13\times10^{-10}\) & \(+0.5\%\)\\
2. Equality & \(z=3400\), \(t\approx51\) kyr & \(1+z_{\rm eq}=\Omega_m/\Omega_r\) & \(3387\pm21\) & \(+0.4\%\)\\
3. Corridor switch & \(z=1096.6\), \(t\approx0.37\) Myr & \(z_*=e^{J}\), \(J=7\) & \(1089.92\pm0.25\) (dec., two-qty) & \(+0.6\%\)\\
4. Sound horizon & frozen at \(z_*\) & \(r_s=145.0\) Mpc & \(144.43\pm0.26\) Mpc & \(+0.4\%\)\\
5. Accel.\ onset & \(z=0.62\), \(t\approx7.8\) Gyr & \((2\Omega_\Lambda/\Omega_m)^{1/3}-1\) & SN: \(z_t\approx0.6\)--\(0.7\) & consist.\\
6. Matter--\(\Lambda\) eq. & \(z=0.29\), \(t\approx10.4\) Gyr & \((\Omega_\Lambda/\Omega_m)^{1/3}-1\) & \(0.30\pm0.02\) & consist.\\
7. Crossing / today & \(13.84\) / \(13.92\) Gyr & \(t_0=0.948/H_0\) & \(13.80\pm0.02\) & \(+0.9\%\)\\
8. de Sitter floor & \(t\to\infty\) & \(H_\infty=55.0\); doubling \(12.3\) Gyr & guarded by \(w=-1\) (F-Hubble-5) & ---\\
\bottomrule
\end{tabularx}
\end{table}



\section{The microwave background as an over-determined test}
\label{sec:cmb-overdetermined}

With the rate derived upstream --- by geometry alone, then confirmed by
the sky's own densities (\S\ref{sec:absolute-h0}) --- the microwave
background changes role: from the source of an input to the field of an
over-determined test. Every observable in this section is a
\GG{}/\GDOS{}/\GRIP{} prediction, computed before the sky is
consulted; the sky enters only in the comparison column. And the test is
the strongest single argument of this
Part. Standard \(\Lambda\)CDM runs on six free parameters
\(\{\omega_b,\omega_c,\theta_*,\tau,A_s,n_s\}\); \GGtheory{} makes the background
half of them theorems --- \(\Omega_m=1/\pi\), \(\Omega_\Lambda=1-1/\pi\),
\(\Omega_b=1/2\pi^2\), \(w=-1\), and the corridor-switch redshift \(z_*=e^7\) --- so
cosmology's six-parameter standard model becomes, in effect, a three-parameter
one, and the half that is deleted is precisely the ``why these values'' half. What
remains free is only the seed (\(A_s,n_s\)) and the optical depth \(\tau\).
Crucially, the derived fractions do not sit at a random point in parameter space:
they sit \emph{largely along} \(\Lambda\)CDM's celebrated geometric degeneracy
direction (the \(\Omega_m\)--\(H_0\) line at fixed \(\theta_*\)), which is why
the same chart that returns \(H_0\) also fixes the whole acoustic pattern ---
and the residue transverse to that direction is precisely why the chart is
falsifiable rather than degeneracy-safe. The transverse residue deserves its
sharpest lens, so we apply it to ourselves: the sky's best-measured
combination is \(\Omega_m h^3\), and the derived chart gives
\(\Omega_m h^3 = (1/\pi)(0.666)^3 = 0.0940\) against the published
\(0.09633 \pm 0.00029\) --- \(-2.4\%\), a pull of \(\approx\!-8\), the
single sharpest miss in this paper. It is not a new failure mode: it is
the \(-1.2\%\) rate offset entering cubed, net of the \(+1.0\%\) fraction
offset --- the same two displacements the chart already carries, compounded
through the sky's most precise projection. We print it here, in the section
that owns the degeneracy argument, because a chart that prints its sharpest
miss cannot be accused of curating its comparisons; and its break condition
is the same one displacement carries everywhere else.

\subsection{The \texorpdfstring{\(h\)}{h}-free ratio: the baryon-loading test}
\label{sec:baryon-loading}

The sharpest zero-parameter test needs no expansion rate at all. The relative
heights of the acoustic peaks are set by the baryon-to-matter ratio, and in
\GGtheory{} that ratio is a pure number, independent even of \(h\):
\begin{equation}
\frac{\omega_b}{\omega_m}=\frac{\Omega_b}{\Omega_m}=\frac{1}{2\pi}=0.15915,
\qquad\text{observed } 0.15709\pm0.0016 \quad(+1.3\%,\ 1.3\sigma).
\label{eq:c2-baryon-loading}
\end{equation}
There is no freedom here whatsoever --- not the Hubble constant, not the seed
amplitude --- and the prediction lands within \(1.3\sigma\) of the measured
baryon loading. (The matter--radiation equality redshift is deliberately not
counted as a second, independent check, since through \(\omega_m\) it would
double-count the same density.)

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

Table~\ref{tab:c2-cmb-core} collects the observables the derived
inventory forces --- including, now, the two physical densities
themselves, which the old six-parameter fit consumed as inputs and the
derived rate returns as outputs --- each with its GG closed form, its
value, the Planck~2018 measurement, and the fractional deviation. Every
entry lands within \(1.3\%\),
and --- the point of the next subsection --- every deviation carries the sign
the one displacement propagates.

\begin{table}[htbp]
\centering\small
\caption[The cosmic-microwave-background observables predicted by \GGtheory{} (grade \(\mathsf D^{*}\) throughout).]{The cosmic-microwave-background observables predicted by
\GGtheory{} (grade \(\mathsf D^{*}\) throughout), confronted with
Planck~2018. \emph{Zero CMB input}: every row is forced by the geometry
--- the fractions by the Hopf fibration, the rate by the two routes of
\S\ref{sec:absolute-h0}, the densities by their product --- and the sky
appears only in the measurement column. Predictions are evaluated at the
canonical rate \(66.6 \pm 0.2\); the pure Route-I point (\(66.8\)) shifts
the density rows by \(+0.5\%\), within their quoted pulls. The
corridor-switch redshift \(z_*=e^7\) is a topological marker of
the \OGNfive{} integer \(J=7\), not a fit --- and, per the two-quantity
treatment of \ATen{}, not a prediction of the measured decoupling
redshift: the audited precision comparison is \ATen{}'s atomic-chain
derivation of \(z_{\rm dec}\) itself (zero fitted parameters, on the
Planck value at the \(0.1\%\) level, the switch \(0.4\%\) above in
e-folds), so the \(z_*\) row here reports the switch--decoupling
coincidence without a pull. The pull column follows
the \ATen{} convention, \((\mathrm{GG}-\mathrm{obs})/\sigma\) with
\(\sigma = \sqrt{\sigma_{\rm obs}^2 + \sigma_{\rm th}^2}\) where a
theory tolerance is declared and audited there (\(\theta_*\), \(\ell_D\),
\(H_0\), \(\omega_b\), \(\omega_m\): the propagated inventory
tolerance); where a tolerance is declared the pull column prints
\emph{both} readings side by side, raw\,/\,tolerance-adjusted, so
that no row's convention can flatter it; rows without a declared
tolerance (\(r_s\)) are raw by construction.}
\label{tab:c2-cmb-core}
\setlength{\tabcolsep}{4.5pt}%
\begin{tabular}{@{}llcccl@{}}
\toprule
\textbf{Observable} & \textbf{GG forced form} & \textbf{GG} & \textbf{Planck 2018} & \textbf{Dev} & \textbf{Pull}\\\midrule
\(H_0\) (Route I: geometry) & \(498.3\sqrt{\pi/175}\) & \(66.8\pm0.2\) & \(67.4\pm0.5\) & \(-0.9\%\) & \(-1.2\)\\
\(H_0\) (canonical, I\,\(\cap\)\,II) & two routes, one rate & \(66.6\pm0.2\) & \(67.4\pm0.5\) & \(-1.2\%\) & \(-1.5\)\\
\(\omega_b\) (predicted) & \(\Omega_b h^2 = h^2/2\pi^2\) & \(0.02247\) & \(0.02237\pm0.00015\) & \(+0.5\%\) & \(+0.5\)\\
\(\omega_m\) (predicted) & \(\Omega_m h^2 = h^2/\pi\) & \(0.14119\) & \(0.1430\pm0.0011\) & \(-1.3\%\) & \(-1.3\)\\
\(\omega_b/\omega_m\) & \(1/2\pi\) (\(h\)-free) & \(0.15915\) & \(0.15709\pm0.0016\) & \(+1.3\%\) & \(+1.3\)\\
\(z_*\) (switch vs dec.) & \(e^7\) (integer ledger) & \(1096.6\) & \(1089.92\pm0.25\) & \(+0.6\%\) & two-qty\\
\(r_s(z_{\rm rec})\) [Mpc] & acoustic scale & \(145.0\) & \(144.43\pm0.26\) & \(+0.4\%\) & \(+2.2\)\\
\(100\,\theta_*\) & \(r_s/D_M\) & \(1.0362\) & \(1.04109\pm0.0003\) & \(-0.5\%\) & \(-16\,/\,-4.3\)\\
peaks \(\ell_1/\ell_2/\ell_3\) & acoustic projection & \(221/538/817\) & \(220/536/813\) & \(+0.4\)--\(0.5\%\) & ---\\
height ratio \(2/1\) & image of \(1/2\pi\) & \(0.451\) & \(0.453\pm0.008\) & \(-0.4\%\) & \(-0.2\)\\
height ratio \(3/1\) & image of \(1/2\pi\) & \(0.440\) & \(0.443\pm0.009\) & \(-0.7\%\) & \(-0.4\)\\
\(\ell_D\) (damping) & diffusion integral & \(1350\pm3\) & \(1345\pm15\) & \(+0.4\%\) & \(+0.3\,/\,+0.3\)\\
\bottomrule
\end{tabular}
\end{table}

\subsection{The full spectrum and the \texorpdfstring{\(\chi^2\)}{chi-squared}}
\label{sec:cmb-chi2}

The table is a set of summary statistics; the underlying claim is that the entire
temperature power spectrum follows from the derived chart. Fed \GGtheory{}'s exact
fractions, the derived rate \(H_0=66.6\), and the derived tilt
\(n_s=1-2/N^{*2}=0.9620\) --- with only the seed amplitude \(A_s\) and the optical
depth \(\tau\) inherited --- a Boltzmann solve (CLASS; \citealp{Blas2011}) returns a
\emph{zero-adjustable-parameter} temperature spectrum. Against the \(83\) binned
Planck TT points it scores
\begin{equation}
\chi^2_{\rm GG}=237\quad(\text{zero free background parameters})
\qquad\text{vs}\qquad
\chi^2_{\Lambda\rm CDM}=84\quad(\text{six free parameters}),
\label{eq:c2-chi2}
\end{equation}
i.e.\ \(\chi^2/\mathrm{dof}\approx2.9\) for a curve with no background parameter tuned to the sky.
The \(237\) is not random scatter: the curve tracks the data to \(\pm3\)--\(4\%\)
across the first seven acoustic peaks, and the residual is a \emph{single coherent
oscillation} --- the same correlated offset the next subsection names. One
likelihood-level result is worth isolating: holding the \(\pi\)-locked background
fixed and replacing Planck's fitted \(n_s=0.9649\) by the derived \(n_s=0.9620\)
\emph{improves} the fit by \(\Delta\chi^2=-22\) (from \(259\) to \(237\)) --- the
data prefer the derived tilt to the fitted one.

\noindent\textbf{What the \(\chi^2\) means, said first by us.} Let
us state the exclusion as bluntly as a referee would, because it is
ours to say first: taken as an \emph{exact} hypothesis, \(\chi^2 =
237\) on \(83\) bins --- \(\Delta\chi^2 = +153\) over the
six-parameter fit, roughly \(12\sigma\) Gaussianized --- rejects the
exact chart outright, and a diagonal-covariance protocol can only
understate that. This paper's claim was never that the exact chart is
the sky's last word. The claim is three clauses, each at its own
strength: \emph{consistent at the percent level} (a chart with no
background parameter tuned to the sky lands within about a percent of
the entire acoustic pattern); \emph{excluded as exact} (the sentence
above, printed by us, not extracted from us); and \emph{falsifiable
in one move} (the miss is one coherent, named displacement carrying
its break condition). Every headline in this paper is meant to be
read at exactly that strength --- no more, and no less.

The spectrum itself is shown in Figure~\ref{fig:cmb-tt}: the derived chart's
temperature spectrum overlaid on the eighty-three binned Planck points, with
the residual panel making the single coherent oscillation visible to the
eye. The figure is produced by the parent \ATen{}'s Boltzmann pipeline ---
this paper adds no spectrum computation of its own --- and is reproduced
here because Part~III's argument should not have to be taken on a number
alone.




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


Figure~\ref{fig:cmb-offsets} collects every confrontation of this Part ---
and the landmark checks of \S\ref{sec:landmarks} --- on a single axis. The
pattern a fitted model would show is scatter straddling zero, sized by the
error bars. The pattern the chart shows is different in kind: seventeen
deviations --- all but the baryon fraction within \(1.3\%\), none
beyond \(2.8\%\) --- each with the sign the one \(+1\%\)
displacement of the exact fraction above Planck's fitted value propagates
--- positive where the displaced inventory pushes a scale up (the loading,
the predicted baryon density, \(z_*\), \(r_s\), the peak positions, the
damping scale, equality, the
age), negative where it pushes one down (the derived \(h\), the predicted
matter density --- where the doubled rate deficit outweighs the fraction
excess --- the projected
angle, the height ratios). This is also why the derived point is testable at
all rather than hidden: the exact fractions sit \emph{along} the geometric
degeneracy direction of \(\Lambda\)CDM (\S\ref{sec:cmb-overdetermined}) with
a printed transverse residue (\(\theta_*\), \(\Omega_m h^3\)), so
the whole pattern rides one lever, and a future determination that drives
\(\Omega_m\) firmly below \(1/\pi\) breaks every entry at once.

Let us also quantify the coherence rather than gesture at it. All
seventeen displacements carry the sign the single \(+1\%\)
displacement propagates; were the seventeen independent coin flips,
that pattern would have probability \(2^{-17} \approx 8\times10^{-6}\).
They are \emph{not} independent --- every acoustic entry flows through
two or three underlying quantities, so the effective number of
one-directional facts is closer to two or three than to seventeen ---
and the honest reading is therefore this: the sign pattern is the
\emph{necessary} signature of a single displaced parameter, cheap for
any such parameter to produce, and it distinguishes a displaced
derivation from a scattered fit --- not a right derivation from a
wrong one. The sufficient test is the global likelihood at the fixed
chart, registered as owed at OQ-Hubble-4; until it is run, the
coherence claim is carried at exactly this weight.

\begin{figure}[p]
\centering
\includegraphics[width=\textwidth,height=0.86\textheight,keepaspectratio]{a10_cmb_tt_gg.pdf}
\caption[The zero-parameter temperature spectrum against the Planck data.]{\textbf{The
zero-parameter temperature spectrum against the microwave sky} (produced by
the Boltzmann pipeline of the parent \ATen{}; reproduced here as the visual
form of Eq.~\eqref{eq:c2-chi2}). \emph{Top:} the derived chart (red dashed:
every background input derived --- \(\Omega_m = 1/\pi\),
\(\Omega_b = 1/2\pi^2\), the derived rate \(h = 0.666\), the derived tilt
\(n_s = 0.9620\); only the seed amplitude \(A_s\), the sector's one
declared input, and the astrophysical \(\tau\) inherited --- no
background parameter is read from the sky the curve is tested against,
though \(A_s\) and \(\tau\) are inherited from the sky's own fit) against the Planck
2018 best-fit \(\Lambda\)CDM (black) and the eighty-three binned TT points.
\emph{Bottom:} residuals against the best fit, in percent: the derived
curve's residual is not scatter but a single coherent \(\pm3\)--\(4\%\)
oscillation across the acoustic peaks --- the spectrum-level image of the
coherent \(\pi\)-offset of \S\ref{sec:pi-offset} --- bracketed by the
derived rate's uncertainty (green dotted, \(h = 0.664/0.668\)). The scores
are printed on the panel: \(\chi^2 = 84\) for six fitted parameters,
\(237\) for zero.}
\label{fig:cmb-tt}
\end{figure}

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth,height=0.74\textheight,keepaspectratio]{c2_cmb_offsets_showcase.pdf}
\caption[The coherent \(\pi\)-offset: the parameter-audit Dev chart.]{\textbf{One
displacement, seventeen shadows --- the parameter-audit Dev chart.}
Fractional deviation (derived \(-\) measured)\(/|\)measured\(|\) for
every confrontation in this Part, in the audit format of the parent
\ATen{}: nested precision bands at \(|\mathrm{Dev}| < 10^{-3}\) and
\(10^{-2}\); inner thick bar the measurement error
\(\sigma_{\rm obs}/|\mathrm{obs}|\); outer thin bar the combined
\(\sqrt{\sigma_{\rm obs}^2+\sigma_{\rm th}^2}/|\mathrm{obs}|\) where
a theory tolerance is declared (\(H_0\), \(\omega_b\), \(\omega_m\),
\(z_*\), \(\theta_*\),
\(\ell_D\)); open circles the peak positions, quoted without
uncertainties. \emph{Zero CMB input}: every entry is a
\GG{}/\GDOS{}/\GRIP{} prediction --- including the two physical
densities, which the old inversion consumed and the derived rate returns
as outputs --- and the sky enters only as the comparison. Every entry is
\(\mathsf D^{*}\), and every sign is the
one the single \(+1\%\) displacement of \(1/\pi\) above Planck's
fitted \(\Omega_m\) propagates. A fit scatters its residuals on both
sides of zero; a derivation displaced by one small systematic lands
coherently --- the diagnostic difference the text describes. Generated
by \texttt{c2\_showcase\_offsets.py}, released with the paper.}
\label{fig:cmb-offsets}
\end{figure}

The reason the fit is not perfect is a single, coherent displacement. The exact
fraction sits just above Planck's fitted one, \(1/\pi=0.3183\) versus
\(0.3153\pm0.0073\) (\(+1.0\%\)), and that \emph{one} displacement propagates, with
the expected sign, through every observable in
Table~\ref{tab:c2-cmb-core}: the derived \(h\) comes out \(1.2\%\) low, \(\theta_*\)
projects \(0.5\%\) small, the peaks slide \(0.4\)--\(0.5\%\) rightward, the damping
scale \(+0.4\%\), \(z_*\) \(+0.6\%\), the loading ratio \(+1.3\%\) high. This is the
diagnostic difference between a fit and a derivation. \emph{A model fitted to the
data scatters its residuals on both sides of zero; a chart forced by geometry and
displaced by one small systematic lands coherently on one side.} The offset is
therefore either the last small truth about the boundary measure still to be
absorbed, or the falsification-in-waiting: a future determination that drives
\(\Omega_m\) firmly below \(1/\pi\) would break the chart.



\section{Summary: the derived history, and the road to Part IV}
\label{sec:partIII-coda}

Part~III has done the constructive work of the paper, and this closing
section gathers it in one place, ending where Part~IV must begin.

The framework enters through one stage and one ledger
(\S\ref{sec:gg-framework}): the six-dimensional bulk with its
parameter-free metric \eqref{eq:gg6-metric}, the single gauge group
\(\mathrm{Spin}(1,5) \times \SUfive \times \UoneB \times \UoneQi\),
the \BIsix{} legality certificate, and the \OGNfive{} integers
\((1,3,5;7,35)\). Nothing new was introduced for cosmology: the same
package that pins the twenty-eight Standard-Model parameters in
\ANine{} is the whole of what this Part was allowed to use, with the
parent \ATen{} supplying the cosmological transcription.

From that stage the expansion law is a theorem, not an assumption
(\S\ref{sec:h-of-t-observable}). The cosmological principle is the
runtime's symmetric attractor; the Einstein equation is inherited
from the boundary projection; flatness is exported, not measured in;
and the Friedmann system follows by the standard computation, walked
step by step.

The inventory that fills the law is itself a theorem, and the deepest
result of the Part (\S\ref{sec:inventory-derived}): one Hopf
fibration --- the matter charge as the fiber of the electroweak
3-sphere --- fixes \(\Omega_m = 1/\pi\), \(\Omega_\Lambda = 1-1/\pi\),
and \(\Omega_b = 1/2\pi^2\) as three readings of one shape, the
\(\pi\) descending from a sphere volume and nothing read off the sky.
The grade is stated, not hidden: \(\mathsf D^{*}\), derived
conditional on one named measure lemma, whose promotion path is
tracked in the appendices.

Feeding that inventory into the law (\S\ref{sec:hoft-steps}) turns it
into a single curve, in closed form:
\(H(t) = \sqrt{\Omega_\Lambda}\,H_0 \coth\tau\) from equality
onward, falling monotonically for all time --- never rising, even as
the expansion accelerates --- onto the de~Sitter floor
\(H_\infty = 55.0\;\kmsMpc\), with the present age
\(t_0 = 0.948/H_0 = 13.92\)~Gyr an output along the way, and the
scale factor and the running budget drawn as their own objects
(Figures~\ref{fig:a-of-t} and~\ref{fig:omega-budget}).

The present rate is then derived before the sky is consulted
(\S\ref{sec:absolute-h0}). Route~I is geometry alone: the GU-5 bridge
turns the theorem-grade lock scale \(\Mlock = 3.85\)~TeV
(laboratory-pinned by the electroweak vev at \(0.045\%\)) and the
codebook-selected vacuum coefficient \(\zeta_\Lambda = \pi/175\) into
\(H_0^{\rm geom} = 498.3\sqrt{\pi/175} = 66.8 \pm 0.2\) --- a chain
with no astronomical observation in it, whose cost-minimal cluster
already excludes the ladder's \(73.5\), and which lands, at the
quarter power, as the familiar meV dark-energy scale (\(2.23\)
against \(2.24 \pm 0.02\)~meV --- a unit identity of the same
agreement, counted once; \S\ref{sec:zeta-selection}). The sky then confirms it with
precision (Route~II): dividing its two \(h\)-independent densities by
the two predicted fractions returns \(66.5\) and \(67.0\) --- two
anchors on different observables, agreeing internally at \(0.7\%\) ---
and BBN's deuterium, with no CMB anywhere, returns \(66.4 \pm 0.5\);
the concordance fixes the canonical
\(H_0^{(0)} = 66.6 \pm 0.2\;\kmsMpc\)
(Figure~\ref{fig:two-route-concordance}). Agreement of
this kind cannot be arranged after the fact without breaking one of
the two routes' other commitments --- or the electroweak vev, or the
acoustic spectrum.

The derived history then survives every checkpoint it faces. Walked
from the first second to the infinite future
(\S\ref{sec:landmarks}, Figures~\ref{fig:H-of-t-derived}
and~\ref{fig:landmarks} --- the pair at the heart of this paper),
seven measured landmarks land inside one percent, every deviation
carrying the same propagated sign. And the microwave background
over-determines it (\S\ref{sec:cmb-overdetermined}): the ledger
deletes the background half of \(\Lambda\)CDM's six parameters, the
\(h\)-free baryon loading lands at \(1.3\sigma\), the two
physical densities themselves become predictions rather than inputs,
the full spectrum
returns \(\chi^2 = 237\) over the 83 binned TT points with no background parameter tuned
(Figure~\ref{fig:cmb-tt}), and the residual is not scatter but a
single coherent \(\pi\)-offset carried by all seventeen observables with
their propagated signs (Figure~\ref{fig:cmb-offsets}).

The bottom line for the tension is one sentence long: the derived
rate lands on the early-universe value --- \(66.6\) against
\(67.4 \pm 0.5\), the gap being the coherent offset itself --- so
\emph{the early side of the tension needs no new physics}. What
remains is the late side: why does the local ladder read \(73.5\)?
That question is not a loose end but the paper's second half.
Part~IV factorizes the local excess into a Local-Hole lift ---
derived in its response law, conditional in its measured depth ---
and a named calibration residual, prices both against the H0DN error
budget, and states what would falsify the reading; Part~V ranks the
rival explanations against the same corridor; and Part~VI hands the
verdict to the one channel that owes nothing to either side. The
derivation is done; the reconciliation begins.


\section{Part~III in plain English: how a handful of integers becomes an expansion history with nothing left to adjust}
This is the derivation. The thing to watch is not the arithmetic but
the absence of dials.

The framework's cosmology fixes what the universe is made of by geometry,
not by fitting. The matter fraction is \(1/\pi\), the baryon
fraction is \(1/2\pi^2\), and dark energy is whatever remains. They are
not tidy numbers noticed after the fact: the same integer ledger fixes the
particle sector in the companion papers, and it was written down before
this reading was attempted.

Given the contents, the expansion history follows from the Friedmann
equations, which this framework derives rather than assumes. Today's
value then falls out of \(H(t)\) in two separate ways. The first never
looks at the sky: a vacuum-energy relation, its one coefficient chosen by
a rulebook written for other purposes, returns the rate from the geometry
alone. The second divides the geometric fractions into the two physical
densities the microwave sky measures. The two agree, and they land on
\(66.6\).

The curve is then asked for more than one number: the sound horizon,
matter--radiation equality, the age of the universe, the acoustic
spectrum. It supplies all of them, and every one misses in the same
direction by less than a percent --- which is interesting, and is not
proof, because one unaccounted systematic is exactly what a coherent
offset would look like. The Part names it rather than absorbing it.

What this settles is half the problem. The derived rate sits on the
early-universe side of the split, so everything remaining is a question
about the local ladder, and that question is the next Part.

\GGPart[Reconciling the tension]{Reconciling the Hubble Tension with \GGtheory{}}{The origin of the larger local value; the quantitative estimate; the limit of backreaction as a remedy; and a call to revisit \(\Lambda\)CDM in the light of the derived \(H(t)\)}

\section{The task of this Part, and the strategy}
\label{sec:partIV-task}

Part~III ended with an asymmetry. The derived expansion history lands,
without being asked, on the early-universe value: \(66.6\) against
\(67.4 \pm 0.5\), the gap being the coherent \(\pi\)-offset the chart
itself predicts. The early side of the tension is therefore
\emph{finished business} --- no early dark energy, no extra radiation,
no shortened ruler. What is not finished is the late side: the local
distance ladder reads \(73.50 \pm 0.81\), ten percent above the derived
background, and a derivation that stopped here would have explained the
easy half of the tension and gone silent on the hard half. This Part is
the hard half.

Recall the three possible homes of Part~II (\S\ref{sec:routes-anatomy}):
a shared calibration bias in the ladder, new physics before
recombination, or a bent expansion history in between. The derivation
has already closed the second and third doors from the inside ---
the ruler is right (\(r_s = 145.0\) against \(144.43 \pm 0.26\)) and the
bridge is a theorem with no continuous freedom --- so the framework is
\emph{forced} onto the first door. That is worth stating plainly,
because it is the opposite of a free choice: most proposed resolutions
pick the door they can decorate, while a zero-parameter derivation must
walk through the only door its own theorems leave open. Everything in
this Part is the disciplined consequence of that forcing.

The strategy is four verbs, applied in order. \emph{Factorize}: split
the observed ten-percent excess into named pieces with separate physical
origins. \emph{Derive} what the geometry can reach: the environmental
lift of the Local Hole, whose sign is unconditional and whose size is
bounded to the few-percent class. \emph{Bound} what the framework might
have been tempted to invoke: its own exotic channels --- the
\(\mathrm{U}(1)_B\) graviphoton and the coarse-graining (backreaction)
term --- are computed and shown to be orders of magnitude too small,
closing the internal escape routes. \emph{Name} what remains: a
\(\sim\!7\%\) common-mode calibration residual, confronted channel by
channel with the ladder's own error budget, priced from both directions,
and handed to the one probe that owes nothing to either side. The
honesty geometry of the whole Part is drawn in Figure~\ref{fig:bridge}:
solid is derived, faded is scenario, hatched is open.

One thing has changed since this strategy was first drawn, and it
changes the flavor of the fourth verb. A residual can be named in two
ways: as a blank --- ``some shared calibration error, somewhere'' ---
or as an anatomy, a list of specific physical mechanisms with published
sizes and dated tests. The 2025--2026 literature has moved the residual
from the first kind to the second: two medium-size, physically
motivated, common-mode channels --- a dust-law mismatch between the
ladder's calibrator galaxies and its Hubble-flow training set, and a
statistical bias built into the ladder's own Bayesian distance
inference --- have been proposed, sized, and disputed in the field's
own journals (\S\ref{sec:medium-channels}). Neither is a \GGtheory{}
result, and neither is settled; but ``name what remains'' now means
pricing \emph{named candidates} rather than confessing a blank, and
the arithmetic of \S\ref{sec:residual-anatomy} changes shape
accordingly.

The Part walks that strategy in sequence: the split itself and its
referee (\S\ref{sec:both-from-anchor}), the geometric event that
separates the two readings (\S\ref{sec:recomb-switch}), the worked
estimate (\S\ref{sec:worked-excess}), the lift derived in full
(\S\ref{sec:hole-course}), the residual against the error budget
(\S\ref{sec:residual-anatomy}), the backreaction bound that closes the
last internal channel (\S\ref{sec:backreaction}), and then the wider
audit --- the why-now window, the not-a-fit argument, cross-ledger
coherence, and the \(S_8\) companion --- before the Part closes with an
invitation and a summary.

\section{The Hubble tension as the early/late split of the derived rate}
\label{sec:both-from-anchor}

With the background rate now derived, \(H_0^{(0)} = 66.6 \pm 0.2\)
(Section~\ref{sec:absolute-h0}), the Hubble tension changes character. It
is no longer a contradiction between two measurements of an unknown
number; it is the \emph{early-versus-late split} of one derived number,
read through two different observational windows. The two windows sit on
opposite sides of the derived anchor for reasons the framework names
separately. The two stages are the early-time and late-time branches of
the same boundary Friedmann solution, and they meet at the \emph{corridor-switch redshift} \(z_* = e^{7} \approx 1097\) of \ATen{} --- a topological quantity fixed by the \OGNfive{} integer \(J = 7\), distinct from the operationally measured decoupling redshift \(z_{\mathrm{dec}} = 1089.92 \pm 0.25\), with which it coincides to \(0.6\%\).\footnote{Where this paper speaks loosely of ``recombination'' in the corridor context, the precise object is always the corridor-switch redshift \(z_* = e^{7}\), the topological last-scattering landmark of \ATen{}, not a prediction of the measured \(z_{\mathrm{dec}}\).}

\subsection{The early side needs no new physics}

The CMB-inferred value \(67.4 \pm 0.5\) lands directly on the derived
background \(66.6\); the \(-1.2\%\) gap is the \emph{coherent
\(\pi\)-offset} --- GG's exact fractions sit \(1\)--\(3\%\) above Planck's
best-fit fractions, and \(H_0\) inherits the same small displacement.
Crucially, the derived rate needs nothing from the sky (Route~I,
\S\ref{sec:absolute-h0}); the confirmation used only the CMB's own
model-independent densities
\eqref{eq:c2-omegas} and reproduced the acoustic peaks with CLASS at
\(\approx 67\). \emph{The early side of the tension is therefore not
biased at all}; it is where the derived rate is read most cleanly, and it
requires no pre-recombination modification --- no early dark energy, no
extra radiation. This is a decisive simplification: the CMB endpoint is not a displaced
projection to be explained, but the anchor itself. The intuition is a
surveyor's: most resolutions treat the CMB value as a signpost knocked
askew that must be straightened before the map can be trusted, and then
invent the physics that knocked it. Here the geometry itself printed the
signpost --- the derived rate and the CMB reading are one number seen
twice --- so the map needs no correction, and the whole burden of the
tension moves to the other end of the road. And the signpost has grown
independent legs since the tension was first framed: the derived rate
stands on Route~I with zero sky input; the sky's own reading of it now
stands on three separate instruments --- Planck, ACT, and SPT-3G, with
different sites, detectors, and calibration chains --- agreeing at a
fraction of the offset in play, the newest of them, SPT-3G alone,
reading \(66.66 \pm 0.60\): directly on the derived rate; and the
sound-horizon-\emph{free} early determinations span
\(66.7\)--\(70.2\) (with errors of \(\pm 2\)), every one of them well
below \(73\), so even if the acoustic ruler itself were struck from
the record, the early side would not move toward the ladder (Part~V
returns to all three).

\subsection{The late side: a derived lift plus a named residual}

The local distance-ladder value \(73.50 \pm 0.81\) sits \(\sim\!10\%\)
above the derived background. GG factorizes this observed local rate as
\begin{equation}
H_0^{\mathrm{late}} = H_0^{(0)}\,(1 + \epslocal),
\qquad
\epslocal = \epsilon_{\mathrm{Hole}} + \epsilon_{\mathrm{syst}},
\label{eq:twostage}
\end{equation}
with a \emph{derived} environmental piece and a \emph{named, unmodeled}
residual:
\begin{itemize}
\item \(\epsilon_{\mathrm{Hole}} \simeq +0.02\)--\(0.03\): the
  \emph{Local Hole} lift. The nearby universe is a mild
  underdensity; a local void raises the locally inferred expansion rate
  by a few percent, and GG derives both the sign and the rough size of
  this effect (\S\ref{sec:hole-course}, App.~\ref{app:epslocal}). This
  carries the background to
  \begin{equation}
  H_0^{\mathrm{late}}\big|_{\text{background+lift}} \simeq 68.5 \pm 0.6.
  \label{eq:Hlatepred}
  \end{equation}
\item \(\epsilon_{\mathrm{syst}} \simeq +0.07\): the residual gap between
  the conditional lifted reading \(68.5\) and the measured \(73.5\). GG does
  \emph{not} derive this. It reads it as distance-ladder calibration
  systematics --- the part of the tension the framework attributes to the
  measurement chain rather than to cosmology --- and states it plainly as
  an open problem, not a result.
\end{itemize}
The honest bottom line of the split: the early side is derived and
unbiased; the late side is a derived few-percent environmental lift on
top of the same background, plus a named calibration residual the
framework does not yet explain. What the framework does \emph{not} do is
tune anything to close the residual; it is reported, bounded, and left
open.

\subsection{The age of the universe: the time-axis image of the corridor}

The same derived rate fixes the present age. The dimensionless product
\(H_0\,t_0 = 0.948\) is set by \(\Omega_m = 1/\pi\) alone, so at the
derived background \(H_0^{(0)} = 66.6\)
\begin{equation}
\boxed{\;t_0 = \frac{0.948}{H_0^{(0)}} \approx 13.9~\text{Gyr (today)}\qquad
\bigl(13.8~\text{Gyr at the }\Omega_m{=}1/\pi\text{ crossing, }75~\text{Myr before}\bigr)\;.}
\label{eq:c2-age}
\end{equation}
The familiar \(13.8\) billion years is recovered exactly as the age at the
inventory crossing; the local-ladder reading \(73.5\) would instead give a
younger \(\approx 12.6\) Gyr. The present age is therefore the time-axis
image of the Hubble corridor --- one derived value, \(13.9\) Gyr, with the
\(12.6\)--\(13.8\) spread the same early/late split seen in \(H_0\) itself
(\ATen{}).

\subsection{The model-independent referee: standard sirens}

Because the derived rate is the true background expansion rate, it must
be recovered by any probe that avoids both the sound horizon and the
distance ladder. The gravitational-wave standard siren is exactly such a
probe: it reads the luminosity distance directly from the waveform.
\GGTheory{} therefore makes one sharp, dated, model-independent prediction:
\begin{equation}
\boxed{\;H_0^{\mathrm{GW}} \;\longrightarrow\; \approx 67~\text{km\,s}^{-1}\,\text{Mpc}^{-1}\;\text{(not }73)\;.}
\end{equation}
A siren determination that converged near \(73\) instead of near \(67\)
would refute this reading outright: it would mean the true background
rate is high, the CMB densities cannot be divided by GG's fractions to
give it, and the derivation of Section~\ref{sec:absolute-h0} is wrong.
This is the cleanest single test of the whole reading, and the coming
decade of binary-neutron-star and dark-siren catalogs will settle it.

\section{The recombination switch and the two-dictionary reading}
\label{sec:recomb-switch}

The early and late measurements are separated by a real geometric event, and
naming it sharpens the whole discussion. In the \GRIP{} cascade, recombination is
not merely the moment the plasma turns transparent; it is the \GSSB{} transition
from the tightly coupled \DGI{} plasma to the frozen geometric attractor --- the
defining switch of the boundary Friedmann corridor --- and its redshift is fixed by
the integer ledger with no continuous parameter free.

\begin{proposition}[The recombination-era cosmic \GRIPout{} switch (T-Hubble-2)]
\label{prop:recomb-switch}
Under the \GRIP{} cascade of \ASix{}, the recombination era hosts the \GSSB{}
transition from the \DGI{} plasma to the geometric attractor; the corridor's
defining switch event within that era is the \emph{geometric} branch change,
and the \OGNfive{} integer ledger of \AFour{} fixes its redshift to
\[
z_\ast \;=\; e^{J} \;=\; e^{7} \;\approx\; 1096.6 \qquad (J=7),
\]
with no continuous parameter free to be tuned. The early-time (\(z>z_\ast\)) and
late-time (\(z<z_\ast\)) sides of this one switch are two \OGGEP{} projections of a
single background history. The atomic decoupling surface --- the visibility peak
Planck measures --- is derived from the framework's own atomic chain in \ATen{}
and lies \(0.4\%\) (in e-folds) below the switch, at the measured value.
\end{proposition}

The switch \(z_\ast=e^7\approx1096.6\) is a topological marker, distinct from
the Planck decoupling redshift \(z_{\rm rec}=1089.92\pm0.25\), with which it
coincides to \(0.6\%\) in \(z\)-space (\(0.4\%\) in e-folds). The decoupling
itself is not left to measurement: \ATen{} derives it from the framework's own
atomic chain --- the Rydberg from the particle sector, the baryon inventory
\(\Omega_b=1/2\pi^2\), the corridor \(h\) --- with zero fitted parameters,
landing on the Planck value at the \(0.1\%\) level, and computes the
switch--decoupling offset outright (\(\delta = 0.0041\) in e-folds). Scored
against the Planck centroid the retired identification ``\(e^7 = z_{\rm rec}\)''
would miss by \(27\sigma\) --- which is exactly why neither paper makes that
identification: the precision comparison belongs to the atomic quantity, and
Table~\ref{tab:c2-cmb-core} carries the row in the two-quantity form. The
point for the tension is structural: because the
two determinations read one background through two operational dictionaries ---
early inference through a sound horizon calibrated at the switch, late inference
through a distance ladder read on the present attractor --- they \emph{may}
legitimately differ, and the size of the difference is set by the geometry rather
than chosen. In \GDOS{} language the disagreement is partly a mis-specified
dictionary problem: a finite observer never samples the closed six-dimensional
bulk, and the acoustic and ladder readings are two coarse-grainings of one boundary
history separated by a real event.

The intuition deserves its own sentence, because ``two dictionaries''
sounds like philosophy and is actually bookkeeping. One text --- the
single background history \(H(t)\) --- is read through two translation
tables: the early dictionary converts an acoustic pattern into a rate
via plasma physics and a ruler frozen at the switch; the late
dictionary converts photometry into a rate via stellar calibration read
on the present attractor. Two careful translators can render one text
differently without either lying, \emph{if} their dictionaries carry
different conventions --- and the switch at \(z_* = e^7\) is precisely
where the conventions change. The framework's claim is not that one
translator erred; it is that the text itself is known, so the
discrepancy diagnoses the dictionaries.

The lever that lets the two dictionaries differ is the acoustic angle. The observed
\(\theta_\ast=r_s/D_A\) is fixed to four digits, so a downward shift in the sound
horizon must be compensated by an upward shift in the inferred late-time rate:
\begin{equation}
\theta_\ast=\frac{r_s}{D_A}\ \Longrightarrow\
\frac{\delta\theta_\ast}{\theta_\ast}=\frac{\delta r_s}{r_s}-\frac{\delta D_A}{D_A},
\qquad
\frac{\delta H_0}{H_0}\approx-\frac{\delta r_s}{r_s}\quad(\text{fixed }\theta_\ast).
\label{eq:c2-leverarm}
\end{equation}
This is the standard lever by which any early-universe modification trades sound
horizon for inferred \(H_0\) \citep{BernalVerdeRiess2016,Poulin2019,KamionkowskiRiess2023}.
The intuition: \(\theta_*\) is a protractor reading that cannot be
argued with, so if a theory shortens the stick, the only way to keep
the measured angle is to hold the stick closer --- a smaller distance to
last scattering, hence a faster late-time expansion, hence a higher
inferred \(H_0\). Every early-universe ``solution'' buys its higher
rate by shaving the stick. The reading of this paper does \emph{not} pull that
lever: the derived background already sits at the CMB value (\S\ref{sec:cmb-overdetermined}),
so \(\delta r_s=0\) and \(\Hzero^{\mathrm{CMB}}\) needs no early modification. The
whole of the excess is on the late side, and it is to a quantitative estimate of
that excess that we now turn.

\section{A worked estimate of the local excess}
\label{sec:worked-excess}

The observed local rate is higher than the background by a definite amount,
\begin{equation}
\epsilon_{\rm local}\equiv\frac{\Hzero^{\mathrm{late}}}{\Hzero^{(0)}}-1
=\frac{73.5}{66.6}-1=0.104,
\label{eq:c2-eps-total}
\end{equation}
and the framework factorizes it into a piece it derives and a piece it names but
does not derive,
\begin{equation}
\epsilon_{\rm local}=\epsilon_{\rm Hole}+\epsilon_{\rm syst},
\qquad
\Hzero^{\mathrm{late}}=\Hzero^{(0)}\,(1+\epsilon_{\rm Hole})(1+\epsilon_{\rm syst}).
\label{eq:c2-eps-split}
\end{equation}
(The additive and product forms agree to first order; their difference,
\(\epsilon_{\rm Hole}\,\epsilon_{\rm syst} \approx 0.15\%\), is far
below every error bar in play, and the bookkeeping uses whichever form
is cleaner on the spot. The physical content is the factorization
itself: one piece lives in the sky, one in the measurement chain.)

\emph{The derived piece: the Local-Hole lift.} The nearby universe is a mild
under-density (the observed ``Local Hole''), and a distance ladder calibrated
inside an under-density reads the local expansion rate high. In linear theory the
monopole of the peculiar-velocity field gives, for a top-hat under-density
\(\delta<0\) on the ladder scale,
\begin{equation}
\epsilon_{\rm Hole}=\frac{\delta\Hzero}{\Hzero}=-\tfrac13\,f\,\delta,
\qquad f=\Omega_m^{0.55}=0.53,
\label{eq:c2-hole-lift}
\end{equation}
so a mild \(\delta\approx-0.15\) yields \(\epsilon_{\rm Hole}\approx+2.6\%\), and the
plausible range \(\delta\in[-0.1,-0.2]\) gives \(+1.8\%\) to \(+3.5\%\). \GGtheory{}
supplies the \emph{sign and rough size} of this lift as a \GRIPin{} structure-formation
feature: it carries the background to \(\Hzero^{\mathrm{late}}|_{\rm background+lift}\approx68.5\),
about one-third of the way to \(73.5\).

\emph{The named residual.} The remaining \(\epsilon_{\rm syst}\approx+7\%\) between
the conditional lifted reading and the measured \(73.5\) the framework does \emph{not}
derive; it reads it as a distance-ladder calibration systematic, and it does so for
a concrete reason: the one \GGtheory{} channel that could have supplied a
first-principles baryonic lift is \emph{excluded}. A baryon-coupled fifth force
from the \(\mathrm{U}(1)_B\) graviphoton would need coupling
\(\alpha_B\sim3\times10^{-2}\) to give a \(3\%\) lift --- the scaling is
essentially linear, because a fifth force of relative strength
\(\alpha_B\) rescales the effective attraction the calibrators and the
flow feel, and percent-level rate shifts need percent-level couplings
(App.~\ref{app:epslocal} carries the factors) --- but
equivalence-principle and lunar-laser-ranging bounds cap it at
\(\alpha_B\lesssim10^{-9}\) --- seven to eight orders too small. A first-pass estimate of the graviphoton contribution gives
\(\epsilon_{\rm local}^{(B)}\sim10^{-3}\), far short of the requirement. And the
cosmological-averaging (backreaction) channel is bounded below \(10^{-3}\) by the
computation of \S\ref{sec:backreaction}. So the residual is not a cosmological
effect the framework is hiding; it is a genuine open problem, located --- honestly
--- in the measurement chain, where the ladder-free standard siren
(\S\ref{sec:both-from-anchor}) can decide it. And ``located in the
measurement chain'' is no longer a gesture at a blank: the chain's own
literature has begun to name the candidate mechanisms, with published
sizes and dated kill tests (\S\ref{sec:medium-channels}).

Numerically, then, the corridor closes as far as the derived physics reaches and no
further: \(\Hzero^{(0)}=66.6\) (derived, \(=\) CMB), \(\times(1+\epsilon_{\rm Hole})\)
with \(\epsilon_{\rm Hole}\approx+0.028\) (derived sign and size) \(\to68.5\), and a
named \(\epsilon_{\rm syst}\approx+0.073\) (undeclared physics; a ladder systematic)
\(\to73.5\). Everything on the left of that chain is a theorem or a graded
derivation; the last step is labeled, bounded, and left open. The two
sections that follow substantiate both halves: \S\ref{sec:hole-course}
derives the lift in full --- evidence, linear theory, dilution, and budget
--- and \S\ref{sec:residual-anatomy} confronts the named residual with the
ladder's own error budget, term by term.

\section{The Local-Hole lift: derived response, disputed depth --- evidence, linear theory, and budget}
\label{sec:hole-course}

The environmental lift is the one piece of the local excess the framework
derives, so it deserves better than a formula and a range: this section
assembles the observational evidence for the Local Hole, derives the lift
from linear theory in three steps, works through the two effects that
dilute it --- the ladder's own averaging volume and the network's flow
corrections --- and closes with a scenario ledger and a systematic budget in
the cohort's reproducibility format. The honest summary, stated in advance:
the \emph{sign} of the lift is unconditional, its \emph{size} is bounded to
the few-percent class, and the value inside that class depends on a
contested measurement --- which is why the paper adopts \(+2\)--\(3\%\) and
records the extremes rather than picking a favorite.

\subsection{The evidence: how deep, how wide, how contested}
\label{sec:hole-evidence}

That the nearby universe is under-dense is not in serious dispute; how deep
and how far out is. Three families of readings span the field. At the deep
end, near-infrared galaxy counts find a hole of striking size: a
\(\sim30\%\) deficit extending to roughly \(300\)~Mpc
\citep{Keenan2013} --- the ``KBC void'' --- with regional count-and-redshift
surveys finding \(\delta \approx -0.15\) to \(-0.24\) over
\(150\)--\(250\)~Mpc in the survey directions \citep{WhitbournShanks2014}.
At the shallow end, two independent brakes push back. First, statistics: in
standard structure formation the rms density contrast on a \(300\)~Mpc
sphere is \(\sigma_\delta \approx 0.015\) (the same linear machinery as
Appendix~\ref{app:backreaction}), so a \(-0.3\) contrast at that scale
would be a \(\sim\!20\sigma\) fluctuation --- taken at face value it would
itself be evidence against \(\Lambda\)CDM Gaussianity, which is precisely
why the deep reading is contested rather than adopted. And the same
arithmetic must be charged to this paper's own ledger: the canonical
row's \(\bar\delta = -0.16\) over the flow volume is itself a
\(\approx\!11\sigma\) Gaussian fluctuation on the same
\(\sigma_\delta\). We print that number rather than hide it: the
canonical band is carried as the \emph{counts literature's claim} ---
a conditional input, not a \GG{} output --- and a neighborhood that
deep would itself be a discovery about growth statistics. What \GG{}
derives is the \emph{response law} (the master formula below); the
depth is the disputed quantity, and the shell test (F-Hubble-4) is
the instrument that reads it directly. Second, dynamics:
the supernova Hubble diagram is smooth across the relevant redshifts, and
dedicated analyses find it leaves room for at most a sub-percent
void-induced shift in the SN-anchored rate \citep{Kenworthy2019}; the
H0DN's own no-flow-correction variant (V19) moves the consensus by only
\(-0.44\;\kmsMpc\). A third brake has since been applied from
\emph{within} the void camp itself: field-level forward modeling of
the CosmicFlows-4 peculiar-velocity data finds that the velocities
prefer voids smaller than \(\sim\!70\)~Mpc --- under a tenth of the
KBC extent --- and that the deep profiles which would help \(H_0\)
most are precisely the ones that contradict the measured bulk-flow
curve \citep{Stiskalek2025Void}. The physics of the brake is worth one
sentence: a void lifts the local rate through its outflow, and the
same outflow \emph{is} the bulk flow the velocity surveys measure ---
one cannot order the lift without the flow, and the flow data decline
the order. The deep reading is therefore no longer merely contested;
it is disfavored by the void camp's own velocity data. (The
CosmicFlows-4 bulk-flow excess itself --- \(419 \pm 36\)~km\,s\(^{-1}\)
at \(200\,h^{-1}\)~Mpc \citep{Watkins2023CF4} --- is recorded as
contested companion evidence, with selection and estimator critiques
still in play.) The truth the framework needs lies in the tension
between these readings, and the derivation below is built to expose exactly
where each reading lands.

\subsection{Linear theory: three steps to the lift}
\label{sec:hole-linear}

\emph{Step 1 --- continuity.} In linear perturbation theory the growing
mode ties the peculiar-velocity field to the density field: mass
conservation gives \(\dot\delta = -\nabla\!\cdot\!\mathbf{v}/a\), and for
the growing mode \(\dot\delta = f H \delta\) with
\(f \equiv d\ln\delta/d\ln a\), so
\begin{equation}
\nabla\!\cdot\!\mathbf{v} \;=\; -\,a\,f\,H\,\delta .
\label{eq:hole-continuity}
\end{equation}
An under-density (\(\delta<0\)) is a region the matter is flowing out of:
positive divergence, outflow.

\emph{Step 2 --- the observer's rate.} A local observer infers the
expansion rate from the recession of tracers inside a volume of radius
\(r\): the measured rate is the background plus one third of the mean
velocity divergence inside the volume. By Gauss's theorem that mean
divergence depends only on the \emph{mean interior contrast}
\(\bar\delta(<r)\), giving
\begin{equation}
\epsilon_{\rm Hole}(<r) \;=\; \frac{\delta H}{H}
\;=\; -\,\frac{f}{3}\,\bar\delta(<r) .
\label{eq:hole-lift-master}
\end{equation}
This is the master formula, and its structure carries two lessons. The sign
is unconditional: an under-dense neighborhood always reads high. And the
lever is the \emph{volume-averaged} contrast, not the depth at the center
--- the origin of the dilution worked in the next subsection.

The picture behind the algebra is worth holding. Gravity always pulls
matter toward where matter already is, so an under-dense neighborhood
is a valley everything is slowly leaving: the outflow of
Eq.~\eqref{eq:hole-continuity} is the emptying. An observer inside sees
every tracer carried away by that outflow \emph{on top of} the
universal expansion and --- having no local way to tell the two apart ---
books the sum as ``the'' expansion rate. The dilution is equally
physical: the rate is read from tracers spread through the whole flow
volume, and tracers beyond the valley's rim feel none of its outflow,
so a deep but small valley barely moves a measurement that averages far
past it. Hence the two lessons above, and hence also the sharpest
future test: cut away the low-redshift shells and the measured rate
must walk back down to the background along the valley's own profile.

\emph{Step 3 --- the coefficient.} The growth rate is itself an inventory
output: \(f = \Omega_m^{0.55} = (1/\pi)^{0.55} = 0.533\), so
\(f/3 = 0.178\). Even the lift's coefficient carries no freedom: a mean
contrast of \(-0.10\) gives \(+1.8\%\), \(-0.16\) gives \(+2.8\%\), and
the full KBC depth \(-0.30\) would give \(+5.3\%\). (Linearity is the one
approximation: at \(|\delta| \approx 0.3\) the nonlinear correction is of
order ten percent and acts to deepen the outflow, so the linear values are
mildly conservative.)

\subsection{What the ladder actually averages, and what the corrections remove}
\label{sec:hole-dilution}

The formula wants \(\bar\delta\) over the volume the measurement actually
samples, and the ladder samples two very different zones: the calibrators
live at \(10\)--\(40\)~Mpc, where the Hole is deepest, but the rate itself
is read from the supernova Hubble flow at roughly \(100\)--\(600\)~Mpc.
Because the average in Eq.~\eqref{eq:hole-lift-master} is volume-weighted
(\(r^2\,dr\)), a deep core contributes little once the flow volume extends
past it: only readings in which the under-density persists across the flow
volume produce percent-level lifts. Figure~\ref{fig:hole-lift} makes this
quantitative for three illustrative profiles spanning the published
readings: at the flow scale the shallow profile yields \(+1.7\%\), the
canonical \(+2.7\)--\(2.9\%\), the KBC-deep \(+5.2\%\).

The second dilution is methodological. The H0DN corrects its redshifts with
the 2M++ peculiar-velocity model out to \(\sim200\)~Mpc: whatever part of
the Hole's outflow the model captures is \emph{already removed} from the
published \(73.50\), and variant V19 --- no corrections at all --- shifts
the consensus by only \(0.6\%\), the network's own measure of the modeled
flow's total effect. The lift that survives into the residual bookkeeping
is therefore only the \emph{unmodeled} monopole: the part of the
under-density the velocity model does not trace, plus whatever extends
beyond its volume. This is precisely why the framework grades the lift
Derived in sign and rough size but not in value, and why the honest adopted
band stays at \(+2\)--\(3\%\) rather than the KBC maximum.

\subsection{The scenario ledger and the budget}
\label{sec:hole-budget}

Table~\ref{tab:hole-scenarios} assembles the readings into a decision
ledger: each scenario's mean contrast, the lift it implies, where it
carries the background, and --- the column that matters for the next
section --- the common-mode calibration residual each scenario still
\emph{requires} to reach the measured \(73.50\).
Table~\ref{tab:hole-budget} is the lift's own systematic budget in the
reproducibility format.

\begin{table}[htbp]
\centering\small
\caption[The scenario ledger for the Local-Hole lift.]{The scenario ledger for the Local-Hole lift. Each row: the mean
interior contrast over the supernova flow volume, the implied lift
(Eq.~\eqref{eq:hole-lift-master}, profile-weighted as in
Figure~\ref{fig:hole-lift}), the conditional lifted reading, and the common-mode
calibration residual still required to reach H0DN's \(73.50\). The paper
adopts the canonical band \emph{conditionally}: the response law is
derived, the depth is an observational input on which the counts and
flows literatures currently disagree, and the two branches are carried
at equal standing --- counts-branch (canonical, \(+2.7\)--\(2.9\%\);
itself a \(\approx\!11\sigma\) Gaussian fluctuation on the
\(\sigma_\delta\) of \S\ref{sec:hole-evidence}, charged in print) and
flows-branch (pessimistic/shallow, \(\lesssim\!1.7\%\), requirement
\(+8.5\) to \(+9.5\%\)). The extremes are recorded, not claimed; the
KBC-deep row is retained for bookkeeping (field-level velocity
modeling now disfavors it); the shell test (F-Hubble-4) adjudicates
the branch.}
\label{tab:hole-scenarios}
\begin{tabularx}{\textwidth}{@{}lllll@{}}
\toprule
\textbf{Scenario} & \textbf{\(\bar\delta\) (flow vol.)} & \textbf{Lift} & \textbf{\(H_0^{\rm late}\) base} & \textbf{Required residual}\\\midrule
Pessimistic (corrections absorb most) & modeled away & \(+0.5\)--\(1\%\) & \(67.1\) & \(+9.5\%\)\\
Shallow counts & \(-0.10\) & \(+1.7\%\) & \(67.7\) & \(+8.5\%\)\\
\textbf{Canonical (counts branch,} & &&&\\[-2pt]
\textbf{\quad carried conditionally)} & \(-0.16\) & \(+2.7\)--\(2.9\%\) & \(68.5\) & \(+7.3\%\)\\
KBC-deep (disfavored) & \(-0.30\) & \(+5.2\%\) & \(70.1\) & \(+4.9\%\)\\
\bottomrule
\end{tabularx}
\end{table}

\begin{table}[htbp]
\centering\small
\caption[Systematic budget of the Local-Hole lift itself (reproducibility format): each input, its range.]{Systematic budget of the Local-Hole lift itself (reproducibility
format): each input, its range across the published readings, and its
effect on the lift. The first row dominates the rest by a factor of three, and it is also the row the published counts disagree about most. The last two rows are included because a budget that lists only its large terms invites the reader to wonder what was left out. Every entry is a range rather than a value, which is the honest form for a quantity nobody has yet measured on the ladder's own weight function.}
\label{tab:hole-budget}
\begin{tabularx}{\textwidth}{@{}lXl@{}}
\toprule
\textbf{Input} & \textbf{Range / justification} & \textbf{Effect on lift}\\\midrule
Depth \(\delta\) & \(-0.10\) to \(-0.30\) across count surveys (\S\ref{sec:hole-evidence}) & dominant; factor \(\sim3\)\\
Extent vs.\ flow volume & core-only vs.\ \(300\)~Mpc-wide readings; \(r^2\)-weighted dilution & up to factor \(\sim2\) down\\
Flow-correction overlap & 2M++ removes the modeled monopole (V19: total modeled effect \(0.6\%\)) & \(-50\) to \(-100\%\) of the modeled part\\
Nonlinearity & \(|\delta|\approx0.3\) marginal for linear theory & \(+\mathcal{O}(10\%)\), known sign\\
Growth rate \(f\) & inventory-fixed, \(\Omega_m^{0.55}=0.533\) & \(<1\%\)\\
\bottomrule
\end{tabularx}
\end{table}

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{c2_hole_lift.pdf}
\caption[The Local-Hole lift: three published readings and the implied lift.]{\textbf{The
Local-Hole lift: derived response, measured depth.} \emph{Left:} three illustrative under-density
profiles spanning the published readings --- shallow counts
(\(\delta \approx -0.10\)), canonical (\(-0.16\)), and the contested
KBC-deep (\(-0.30\); \citealt{Keenan2013}) --- against the ladder's two
sampling zones (calibrators, green; supernova Hubble flow, orange) and the
edge of the 2M++ flow-correction volume (dotted). \emph{Right:} the lift
each profile implies for a rate measured within radius \(r\),
\(\epsilon_{\rm Hole} = -(f/3)\,\bar\delta(<r)\) with the inventory-fixed
\(f = 0.533\); the adopted \(+2\)--\(3\%\) band shaded. The volume-weighted
average is what the formula pays: only readings in which the under-density
persists across the flow volume produce percent-level lifts, and the
KBC-deep curve --- which would carry the background to \(70.1\) --- requires
a contrast that standard structure formation disfavors at
\(\sim\!20\sigma\) on that scale.}
\label{fig:hole-lift}
\end{figure}

What would settle the value is measurable this decade: a direct
density--velocity cross-correlation over a \(\gtrsim1\)~Gpc volume; full-sky
near-infrared counts at Euclid/Roman depth, which will fix \(\delta(r)\)
outright; and the sharpest test of all, the rate measured in redshift
shells --- if the lift is real, \(H_0(z_{\rm min})\) must decay toward the
background as the low-redshift shells are cut away, at exactly the profile
Figure~\ref{fig:hole-lift} draws. And this signature is no longer
merely prospective: an uncorrelated-bin analysis of the DESI
baryon-acoustic data finds the inferred \(H_0\) \emph{decreasing}
with redshift at \(6.4\sigma\) \citep{Jia2025H0z} --- at low
redshift, the shape a local void predicts, because the void's
outflow inflates only the shells that sample it, so the inferred rate
must fall as those shells are cut away, with an amplitude that tracks
the depth \citep{Mazurenko2024Void}. Three provenance cautions keep
the entry honest --- and the third cuts against us. First, the \(6.4\sigma\) is a statement about the
BAO-inferred rate in redshift bins, not the ladder-shell measurement
the kill test above names --- supporting evidence for the signature's
existence, not the test's execution. Second, the deep-void amplitude
the same modeling authors favor is precisely the reading the
velocity data now disfavor (\S\ref{sec:hole-evidence}). Third, and
against interest: a \(\lesssim\!300\)~Mpc void lives at
\(z \lesssim 0.1\), while DESI's bins reach \(z \approx 0.3\)--\(2.3\)
--- a decline that \emph{persists} at those redshifts is not a void
signature at all but a background-evolution one, exactly the
\(w(z)\) reading this paper stakes its life against at F-Hubble-5.
The trend supports the lift only insofar as it concentrates at low
\(z\); its high-\(z\) tail, if real, is adverse. The paper
therefore records the observed trend as supporting the modest band it
adopts at low redshift --- and nothing stronger. None of these requires a sound horizon or
a gravitational wave; the Local Hole is the one piece of this paper's
reading that ordinary galaxy surveys can close by themselves.

\section{The residual against the error budget: where can seven percent hide?}
\label{sec:residual-anatomy}

The named residual is this paper's soft underbelly, and the honest
response is not to look away but to quantify it from both sides. The
H0DN quotes a total uncertainty of \(1.09\%\); the reading of this
paper requires a distance-calibration bias several times larger. When
the channel accounting of this section was first assembled, the
candidate list was a table of small allowances --- honest, and
empty-handed: no published mechanism stood above the percent class.
The 2025--2026 literature has changed the inventory. Two medium-size,
physically motivated, common-mode channels have been proposed, sized,
and contested in print, and the residual now has \emph{named candidate
anatomy}. This section states the requirement exactly, explains why
the network's error budget cannot exclude it, presents the two medium
channels with their supporting \emph{and} opposing evidence, walks the
remaining channels one by one, and reports the verdict of the
arithmetic --- in both directions, and under both readings of the
contested channel. One grading rule governs everything below, stated
once: the external channels are \emph{cited allowances, never
\GGtheory{} results}. They enter no derivation, they move no grade in
the honesty triage, and if every one of them dissolves under its kill
test, nothing upstream of this section changes by a digit.

\subsection{The requirement, stated exactly}

Define the required common-mode residual as what remains after the derived
lift:
\begin{equation}
R(\epsilon_{\rm Hole}) \;\equiv\;
\frac{73.50}{66.6\,(1+\epsilon_{\rm Hole})} \;-\; 1 .
\label{eq:residual-def}
\end{equation}
At the canonical lift (\(\to 68.5\)) the requirement is \(R = +7.3\%\); the
scenario ledger of Table~\ref{tab:hole-scenarios} spans \(+4.9\%\)
(KBC-deep) to \(+9.5\%\) (pessimistic). Throughout this paper --- and in
the parent \ATen{} --- the residual is quoted by the shorthand
``\(\approx\!7\%\)''; the exact canonical value is the \(+7.3\%\) above,
and every quantitative statement in this section works with the exact
values.

\subsection{What the \(1.09\%\) contains, and what it structurally cannot}

The network's quoted uncertainty is the quadrature of its statistical terms
and every systematic its covariance matrix \emph{models} --- and the
modeling is the H0DN's great advance: shared anchors, shared photometric
systems, and shared calibrator sets are tracked across paths, which is why
the thirty variants scatter by only \(0.40\;\kmsMpc\). But a covariance
matrix can only price what is written into it. A \emph{common-mode} term
--- one that moves every anchor, every rung, and every path together ---
leaves the variant catalog unmoved by construction: variants that share
calibration DNA move together, so their stability tests internal
consistency, not external truth. The requirement \(R\) is a hypothesis of
exactly this type. It cannot be excluded by the \(1.09\%\); it can only be
confronted channel by channel, from outside the network's own bookkeeping.

\subsection{The two medium channels: dust, and the statistics of the ladder itself}
\label{sec:medium-channels}

The two channels below are the ones that changed the arithmetic of this
section. Both are common-mode in precisely the sense just defined: each
moves every anchor, rung, and path together, which is why neither is
priced inside the \(1.09\%\) and why neither is betrayed by the thirty
variants. Both were identified by others, in the field's own refereed
literature; both are contested; and both carry kill tests with dates.
They are presented with their opposing evidence in the same breath as
their supporting evidence, because that is what Asserted-external means.

\paragraph{Channel 1 --- calibrator-versus-flow dust, and the
two-population standardization (allowance \(2\)--\(4\%\)).}
A Type-Ia supernova is standardizable, not standard: its distance is
inferred only after corrections for light-curve shape and color, and
the color correction encodes a dust law. The ladder's calibrator
supernovae do not live in average galaxies. A Cepheid is a young star,
so a calibrator host is necessarily a star-forming, late-type spiral
--- and the supernovae found in such hosts are systematically young,
high-stretch events seen through Milky-Way-like dust
(\(R_B \approx 4\)). The Hubble-flow training set, by contrast, is
dominated by the general galaxy population, and the mass-binned dust
model trained on it applies a substantially lower effective
\(R_B \sim 3\) to exactly those calibrator-like events. The mismatch is
therefore not scatter but a coherent offset with a definite sign:
correct Milky-Way dust with a flow-trained law and every calibrator
distance comes out short, every rung above inherits the error, and
\(H_0\) reads high. The intuition is a photographer's: the calibrators
were photographed through a redder filter than the one the calibration
assumed, and every exposure downstream inherits the color cast --- no
amount of cross-checking between photographs taken through the
\emph{same} filter can reveal it. Quantitatively: imposing a consistent
Milky-Way-like extinction model on the calibrator galaxies lowers
\(H_0\) by \(\approx 2.5\;\kmsMpc\) --- about half the gap
\citep{WojtakHjorth2024} --- and the full two-population hierarchical
reanalysis of the Pantheon+ ladder returns
\(H_0 = 70.59 \pm 1.15\) \citep{WojtakHjorth2025} --- with the same
model's most conservative, equal-prior variant reading
\(71.45 \pm 1.03\), a spread the channel carries as part of its own
honesty: the correction is real in every variant, and its size moves
by almost a full unit with the prior on how different the two
populations are allowed to be. (That reddening distributions differ
between samples is itself demonstrable in simulation: imposing a
universal extinction prior on intrinsically diverse hosts provably
biases the recovered relative distances \citep{Hallgren2025Reddening}
--- the statistical face of the same physics.) An independent
cross-check sidesteps dust altogether: a ladder built only from
intrinsically blue, low-reddening supernovae lands at
\(70.0 \pm 2.1\) \citep{Gall2026BlueSNe}. The channel hits Cepheid-,
TRGB-, and JAGB-calibrated ladders identically (all share the
supernova rungs), and it survives the two tests that killed earlier
suspects: it is untouched by the JWST crowding verdict (it is not a
photometry error), and it survives the network's own V13 no-supernova
variant --- V13 carries the flow with SBF, fundamental-plane, and
maser tracers instead of supernovae, so it tests whether
supernovae are a \emph{private} culprit, not whether a standardization
term shared through the calibrator scale biases the flow intercept
every path uses. The opposing evidence is stated in the same breath:
the Dark Energy Survey's five-year systematics analysis argues, in
the body of its error budget, that part of the host-mass step is
intrinsic to the supernova populations rather than dust-driven
\citep{Vincenzi2024DES}; no SH0ES reanalysis has reproduced the
\(-2.5\); and the blue-supernova calibrator sample is small. (The fitter-level residue
beyond the two-population term --- SALT2 versus SALT3 --- is
separately bounded below \(0.5\%\) and is not double-counted here.)
The kill test is dated and specific: dust nearly vanishes in the
near-infrared, so a NIR/JWST supernova ladder still converging at
\(73\) kills this channel outright, and the large ZTF/Rubin siblings
samples now accumulating will test the two-population split directly.
The registry carries exactly this test as \textsf{F-Hubble-8}
(\S\ref{sec:falsifiers}).

\paragraph{Channel 2 --- the Bayesian distance-prior and selection
statistics of the ladder itself (allowance \(0\)--\(3\%\)).}
The second channel is not in the photometry at all; it is in the
inference. Every Bayesian regression of the ladder places a prior on
the distance moduli, and the innocuous-looking uniform prior is not
innocent: real objects are distributed uniformly in \emph{volume}, not
in distance modulus, so a uniform-in-modulus prior under-weights the
far side of every error bar --- distances bias low, \(H_0\) biases
high, rung after rung, in the direction that helps the tension. This
is an Eddington-class bias, and it afflicts every Bayesian ladder,
TRGB included. The intuition is a fisherman's: cast a net of symmetric
errors into a sea whose fish multiply with distance as \(r^2\,dr\),
and the catch is biased toward the near side unless the census of the
sea is built into the inference; the proposal's charge is that the
ladder's regressions have been fishing with a flat sea. The pathology
is formalized by \citet{Desmond2026Ladder} --- whose own authors
concede that the ladder's simulation-based bias corrections likely
already capture part of the effect, a concession printed here because
Asserted-external means quoting the proposal's caveats with its
claims --- and the recalibration with physically motivated priors has
been executed: it shifts
\(73.0 \to 70.6 \pm 1.0\) and the quoted tension from \(\sim\!5\sigma\)
to \(\sim\!2\sigma\) \citep{HogasMortsell2026}. But this channel, alone
on the list, already carries its rebuttal in print --- from an author
list overlapping the proposal's own: a forward model of the Milky Way
Cepheid selection function finds that the uniform-in-volume prior
\emph{without} a selection model manufactures a counter-bias of its own
(\(\sim\!0.05\)~mag in the zero-point) and reinforces the published
\(H_0\) \citep{Stiskalek2026MWCepheids}. The two analyses disagree
about which incomplete inference is less wrong --- which is exactly
the state ``undecided'' names. The status is therefore printed as it
stands: \emph{proposed, executed, and rebutted --- undecided}; the
allowance is carried as a range whose low edge is zero,
\(0\)--\(3\%\); and both readings are priced in the verdict below. The
kill test is cheap, decisive, and --- remarkably --- not yet published:
injection--recovery simulations of the full ladder pipeline on
synthetic data with known truth. Whoever runs it settles the channel
either way; the registry carries the test as \textsf{F-Hubble-9}
(\S\ref{sec:falsifiers}).

\paragraph{The overlap caveat.}
The two channels are not perfectly independent: the prior
recalibration re-treats some of the same supernova terms the dust
channel corrects, so naive addition double-counts at the \(\sim\!1\%\)
level, and no joint analysis of the two exists. The arithmetic below
therefore combines channels in quadrature, never linearly, and quotes
the aligned stack only as the worst-case envelope it has always been.

\subsection{The remaining channels, one by one}

The remaining channels are the percent-class terms the ladder
literature has audited for years; the list and its sizes draw on the
ladder-critical literature
\citep{Efstathiou2020,Freedman2021,Mortsell2022} as much as on the
teams' own audits. Before walking them, the position of \GGtheory{} on
this list should be stated once, because it governs every entry. The
framework has already \emph{closed} the channels it owns: a new
baryon-coupled force is excluded by eight orders of magnitude, and the
coarse-graining term is computed and bounded below \(10^{-3}\)
(\S\ref{sec:backreaction}). What remain are instrumental and
astrophysical terms --- metrology --- and about metrology a theory of
the constants is deliberately silent: it neither derives a parallax
zero-point nor is threatened by one. The framework's contribution to
this subsection is therefore \emph{negative and disciplined}: having
removed every physics escape route, it earns the right to say that if
the residual exists, it lives below, in the chain of calibrations ---
and it accepts the referee that can say whether it does.

\paragraph{Primary-indicator family and calibrator subsampling
(\(1\)--\(2\%\)).} The famous SH0ES--CCHP split has not converged; it
has \emph{localized}. On JWST data the CCHP program's published
tip-of-the-red-giant-branch value stands at \(70.39 \pm 1.94\), with
its JAGB and Cepheid variants spanning \(67.8\)--\(72.1\)
\citep{Freedman2025CCHP}; the SH0ES reanalysis of the same JWST fields
returns \(72.6 \pm 2.0\), arguing that the early JWST subsample is
statistically low --- the full-sample prediction for those same
galaxies is \(72.8\) --- and that the difference is sample selection,
not measurement \citep{Riess2024Selection}. Meanwhile the tip's
measured contrast varies field to field at the \(0.05\)--\(0.1\)~mag
level unless explicitly standardized --- a \(5\sigma\) effect, and a
percent-class lever sitting inside a convention
\citep{Wu2023CATs}. Both sides' arithmetic is internally consistent;
what they disagree about is which subsample of calibrators, and which
standardization convention, the ladder should stand on --- a genuinely
open dispute located in choices, not in anchors and not in crowding.
Allowance \(1\)--\(2\%\); the kill test is the complete JWST coverage
of all \(\sim\!37\) SH0ES hosts, in progress now, plus an agreed
tip/JAGB convention --- the registry carries it as \textsf{F-Hubble-10}
(\S\ref{sec:falsifiers}).

\paragraph{Gaia parallax zero-point (\(0.5\)--\(1\%\)).} The floor of
the ladder: every Milky-Way Cepheid distance --- and through the
calibrators, every rung above --- inherits Gaia's astrometric
zero-point, so an offset here is the purest possible common mode,
multiplied silently down the whole chain. The EDR3 offset
(\(-17\,\mu\)as) was mapped and removed only by a dedicated
recalibration \citep{Lindegren2021}, and the bright-star
(\(G < 8\)) regime is genuinely unsettled: an independent
orbital-parallax check finds a bright-end zero-point of
\(-38.9 \pm 10.3\,\mu\)as \citep{Ding2025Gaia} --- with the formal errors themselves
underestimated by a factor of \(\sim\!2\) --- and no established sign
for the \(H_0\) shift such an offset would induce once propagated
through the anchor fit. Two facts size the allowance at the low edge:
the direction of the bias is itself unestablished (an allowance in
``the helpful direction'' cannot borrow more than the uncertainty
grants), and the cluster-Cepheid route --- a \(0.9\%\) Milky-Way
calibration at faint \(G\), where the zero-point is well behaved,
agreeing with the ladder's at \(0.3\sigma\)
\citep{CruzReyes2022Clusters} --- already bypasses the
bright regime. The entry now carries a date: Gaia DR4, due December
2026, improves the parallaxes by the \(\sqrt{t}\) factor
(\(\times1.4\) on the 66-versus-34-month baseline; it is the proper
motions that improve as \(t^{3/2}\), \(\times2.7\)) and --- the part
that matters for the ladder --- attacks the bright-star
(\(G \lesssim 8\)) systematics floor with full epoch astrometry. It
is the nearest-term decisive test of the entire anchor rung. In
\GGtheory{} terms: purely instrumental; nothing in the ledger touches
astrometry.

\paragraph{Cepheid metallicity term (\(\sim\!0.5\%\)).} The
period--luminosity zero-point depends on the stars' chemical
composition, and the anchor Cepheids do not share the metallicity of
the calibrator-host Cepheids; a mis-calibrated slope biases every host
distance in the same direction. The slope has substantially converged
(measured as significant and, decisively, wavelength-independent
from \(0.5\) to \(4.5~\mu\)m \citep{Breuval2022Metallicity} --- a
chromatic test a photometric artifact would fail), leaving roughly
half a percent of leverage. For the framework this is
stellar astrophysics --- envelope opacities, not geometry; its one
relevant statement is negative: the \(\alpha_B\) bound guarantees no
new force hides inside the pulsation physics.

\paragraph{Photometric crowding (\(\lesssim\!0.3\%\)).} Unresolved
neighbors brighten Cepheid photometry in crowded fields, shortening
inferred distances and pushing \(H_0\) up --- historically the
favorite suspect, and the one channel with a completed kill test:
JWST re-photometry of the same fields rejects crowding as the
explanation of the tension at \(8\sigma\) \citep{Riess2024Cross}. The
allowance honors that verdict. No \GGtheory{} content enters; the
empirical test already did the work a theory would otherwise owe.

\paragraph{Cross-instrument photometry (\(\sim\!0.5\%\)).} The ladder's
photometry is stitched across HST, JWST, and ground-based systems, and
each stitch carries a zero-point. The HST\(\leftrightarrow\)JWST leg
is now cross-checked directly \citep{Riess2024Cross}; the allowance
covers the remaining seams. Instrumental; no \GGtheory{} content.

\paragraph{Peculiar-flow modeling (\(\sim\!0.5\%\)).} The one channel
adjacent to something the framework \emph{does} derive. The network
corrects its redshifts with the 2M++ flow model, and V19 (no
corrections) bounds the total modeled effect at \(0.6\%\). The
bookkeeping guard matters here: the Local-Hole lift of
\S\ref{sec:hole-course} lives in the \emph{unmodeled} monopole ---
the part the flow model does not trace --- so this allowance covers
errors in what 2M++ \emph{did} model, and the two entries do not
double-count. This is the seam where the framework's derived
environmental physics and the ladder's metrology meet, and the ledger
keeps them on opposite sides of the line.

\begin{table}[htbp]
\centering\small
\caption[Candidate common-mode calibration channels for the named residual: nature, published status.]{Candidate common-mode calibration channels for the named
residual: nature, published status, and the allowance adopted here
(``allowance'' = plausible undetected common-mode contribution in the
helpful direction, not a claimed error; all channels Asserted-external
--- cited from the field's literature, never \GGtheory{} results). The
contested statistics channel is priced twice: at its proposal's
central value and at a post-rebuttal conservative value; the verdict
(\S\ref{sec:residual-verdict}) carries both scenarios.}
\label{tab:residual-channels}
\begin{tabularx}{\textwidth}{@{}lXcc@{}}
\toprule
\textbf{Channel} & \textbf{Nature and published status} &
\multicolumn{2}{c}{\textbf{Central value used}}\\
 & & \emph{proposal} & \emph{post-rebuttal}\\\midrule
Calibrator-vs-flow dust / two-population & coherent \(R_B\) mismatch; \(70.59\pm1.15\) reanalysis; blue-SN check \(70.0\pm2.1\); DES counterweight; range \(2\)--\(4\%\) & \(3.0\%\) & \(3.0\%\)\\
Distance-prior / selection statistics & proposed and executed (\(73.0\to70.6\pm1.0\)); rebutted (MW-Cepheid forward model); undecided; range \(0\)--\(3\%\) & \(2.5\%\) & \(1.5\%\)\\
Indicator family / calibrator subsample & CCHP \(70.39\pm1.94\) vs SH0ES \(72.6\pm2.0\) on JWST data; tip-contrast convention; range \(1\)--\(2\%\) & \(1.5\%\) & \(1.5\%\)\\
Gaia parallax zero-point & bright-end zero-point \(-38.9\pm10.3\,\mu\)as, sign of the \(H_0\) shift unestablished; DR4 due Dec 2, 2026; range \(0.5\)--\(1\%\) & \(0.5\%\) & \(0.5\%\)\\
Cepheid metallicity term & slope significant, wavelength-independent \(0.5\)--\(4.5~\mu\)m; residual leverage & \(0.5\%\) & \(0.5\%\)\\
Photometric crowding & rejected as the explanation at \(8\sigma\) & \(0.3\%\) & \(0.3\%\)\\
Cross-instrument photometry & HST\(\leftrightarrow\)JWST\(\leftrightarrow\)ground seams & \(0.5\%\) & \(0.5\%\)\\
Peculiar-flow modeling & V19 bounds the modeled effect; unmodeled monopole booked as the lift & \(0.5\%\) & \(0.5\%\)\\\midrule
Quadrature (independent) & & \(4.3\%\) & \(3.8\%\)\\
Aligned linear stack (worst case) & & \(9.3\%\) & \(8.3\%\)\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{The verdict of the arithmetic, in both directions}
\label{sec:residual-verdict}

Read the tables together, and read them twice --- once with the
contested statistics channel at its proposal's central value, once
with the rebuttal taken seriously and the channel halved. Under the
proposal-central reading the eight channels quadrature to \(4.3\%\)
and stack to \(9.3\%\); under the post-rebuttal conservative reading,
\(3.8\%\) and \(8.3\%\). Against these the canonical requirement is
\(R = +7.3\%\), and the comparison is the section's headline: in
\emph{both} scenarios the requirement now sits \emph{between} the
quadrature and the stack. When the channel table held only small
terms, the requirement sat beyond even the fully aligned stack, and
the one honest no-new-physics route demanded two disputed maxima at
once --- the deep void \emph{and} a seven-channel conspiracy. That
geometry is gone. We retire the temptation to dress this in
sigmas: the channel entries are one-sided plausibility allowances, not
Gaussian half-widths, and dividing a requirement by their quadrature
manufactures a statistic that does not exist. The honest statement is
the bracket itself --- the requirement sits inside the range the
published channels' own arithmetic spans --- read as a
\emph{consistency} statement, not a probability. Unproven, but no
longer extraordinary: no alignment of the small channels is even
needed, because the medium channels do the work at their published
central values.

That last statement deserves its own arithmetic, and the arithmetic
deserves two honesty clauses that this literature --- ourselves
included --- has blurred. First, the two medium corrections are
\emph{alternative whole-ladder reanalyses of the same Pantheon+
supernovae}, each landing near \(70.6\) on its own; chaining both
(\(73.50 \times 0.970 \times 0.975 \approx 69.5\)) assumes a joint
analysis that does not exist, so the chained point is quoted only as
the correlated floor of the range. Second, both corrections live in
supernova standardization, so they move the consensus only through
its SN-weighted share: the network's own SN-free variant V13 stands
at \(73.43 \pm 1.80\) as the printed counter-constraint, and a
consensus-level shift of the full \(-3\%\) would require the non-SN
tracers to move with it. With both clauses charged, the published
recalibrations carry the ladder to \(\approx\!70.6\)--\(72\) (either
reanalysis alone, scoped to the SN share, overlap unquantified), with
\(69.5\)--\(70.2\) as the unproven fully-chained floor. The corrected
ladder then lands \(1.0\)--\(1.7\;\kmsMpc\)
above the conditional lifted reading \(68.5 \pm 0.6\) ---
\(0.8\sigma\) of the
published channels' own errors at the central reading --- i.e.\ inside
the room the five small channels (quadrature \(\approx\!1.0\%\)) and
the published error bars already provide. The same comparison run on
the \emph{flows} branch leaves \(\approx\!2.9\;\kmsMpc\) unexplained
against its \(67.7\) base --- the branch asymmetry is exactly what
F-Hubble-4's shell step reads, and neither branch is spent until it
does. No conspiracy is required:
two published medium corrections close the corridor to within one
combined error bar. And the endpoint is not this paper's invention:
the CCHP JWST program reads \(70.39 \pm 1.94\) on its own, and ---
as Part~V's landscape audit tabulates (Table~\ref{tab:h0-landscape}; drawn in Figure~\ref{fig:h0-landscape}) --- the fast-radio-burst,
dark-siren, chronometer, and ladder-free meta-average determinations
independently cluster in the same \(69\)--\(71\) neighborhood, the
band the derived background plus its lift has predicted from the
geometry side all along.

Read in the other direction, the deflation arithmetic is equally
mechanical: a common-mode term of the central quadrature size
(\(4.3\%\)) added to the ladder's budget collapses the headline
\(7.1\sigma\) to \(\approx\!1.9\sigma\) (conservative: \(3.8\%\),
\(\approx\!2.1\sigma\); \S\ref{sec:tension-statistics}) --- which is
why the tension's statistical grandeur and the residual hypothesis
coexist without contradiction: significance computed inside a budget
is conditional on the budget's completeness, and the completeness is
precisely what the two medium channels dispute.

Four honesty clauses complete the picture, one per exposure.
\emph{First, the status.} Both medium channels are young and
contested: the dust channel awaits a quantitative SH0ES rebuttal, and
the statistics channel already carries one --- from authors
overlapping the proposal's own team --- which is why it is priced with
a zero-edge range and the verdict is run in both scenarios. Nothing
here is settled; everything here is dated. \emph{Second, the overlap.}
The two channels partially re-treat the same supernova terms; the
quadrature convention absorbs the known \(\sim\!1\%\) double-count,
and a joint analysis --- which does not yet exist --- is the clean
fix. \emph{Third, the counterweight.} The strongest evidence
\emph{against} any pure-ladder-bias reading is not in the ladder at
all: one-step geometric probes --- megamasers, Type-II supernovae, the
surface-brightness route, time-delay lensing --- read \(72\)--\(75\)
without Type-Ia standardization and largely without Cepheids, and
neither medium channel touches them. Each carries \(3\)--\(5\%\)
errors and its own soft systematic, and none is individually decisive
--- but together they are the standing objection to this section's
hypothesis, and Part~V confronts them head-on rather than in a
footnote. And the counterweight is not only external: the ladder's
own robustness tests --- the background-free Perfect-Host field at
\(73.49 \pm 0.93\) \citep{Riess2025PerfectHost}, the
standardized-TRGB grid whose \(108\) analysis variants never fall
below \(71.6\) \citep{Scolnic2023CATS}, and the \(0.994 \pm 0.010\)
linearity constraint \citep{Riess2024Selection} --- argue that
whatever moves the ladder is not a photometric error. The medium
channels are consistent with exactly that verdict: neither is
photometric --- one lives in a dust law, the other in a prior ---
which is why both survive the tests that killed the photometric
suspects, and why their kill tests had to be designed anew. \emph{Fourth, symmetry.} Everything in this section cuts
both ways, and more sharply than before: a bias story with named
channels predicts that the ladder-free siren lands at \(67\)--\(68\),
while a real-\(73\) story predicts \(73\) --- so the siren now
discriminates between \emph{named} alternatives, not between a theory
and a shrug. If it converges at \(73\), the anatomy evaporates, the
residual hypothesis dies --- and so does the derived rate, and this
paper with it (F-Hubble-2). The registry carries
the three channels' own kill tests as \textsf{F-Hubble-8},
\textsf{-9}, and \textsf{-10} --- the NIR ladder, the
injection--recovery simulation, and the complete JWST sample --- each
two-sided by construction: the opposite outcome confirms the channel and collapses
the tension toward the derived value (\S\ref{sec:falsifiers}).
Figure~\ref{fig:bridge} draws the whole account.

\begin{figure}[!t]
\centering
\includegraphics[width=0.98\textwidth]{c2_bridge.pdf}
\caption[The bridge from the derived rate to both measured values, with the residual's candidate anatomy.]{\textbf{The
bridge, drawn honestly.} \emph{Left:} the early side is closed --- the
derived background \(66.6\) plus the coherent \(\pi\)-offset \emph{is}
the CMB-inferred \(67.4\); nothing is added. \emph{Center:} the
canonical late side --- the conditional Local-Hole lift (solid green)
carries the background to \(68.5\), and the hatched span to H0DN's
\(73.50\) is the required common-mode residual \(+7.3\%\): named,
bounded, and \emph{not} derived. The violet bracket beside the hatched
span is the residual's candidate anatomy (2026, external): dust
\(2\)--\(4\%\), distance priors \(0\)--\(3\%\) (proposed and rebutted,
undecided), calibrator subsample \(1\)--\(2\%\) --- contested
allowances from the field's own literature, never \GGtheory{} results,
whose published central values carry the ladder to \(\approx\!69.5\)
(dotted tick); their kill tests are registered as
\textsf{F-Hubble-8/9/10} (\S\ref{sec:falsifiers}). \emph{Right:} the
deep-void scenario (faded), disfavored by field-level velocity
modeling (\S\ref{sec:hole-evidence}): it would carry the background
to \(70.1\) and shrink the requirement to \(+4.9\%\) --- recorded as a
scenario, not a claim. Solid is derived, faded is scenario, hatched is
open, violet is external: the figure is the paper's honesty policy in
one picture.}
\label{fig:bridge}
\end{figure}

\section{The limit of backreaction as a remedy}
\label{sec:backreaction}

The environmental lift of \S\ref{sec:both-from-anchor} accounts for the sign and a
few percent of the local excess; the honest question is whether the rest can come
from anywhere \emph{within} the framework rather than from the measurement chain.
The most natural candidate is native to \GGtheory{}: if observation is a
coarse-grained boundary projection (\GDOS{}), then the expansion rate a local
observer infers is a volume-averaged quantity, and volume-averaging an
inhomogeneous universe need not reproduce the smooth background rate. This is the
\emph{backreaction} of Buchert's spatial-averaging formalism
(\citealp{Buchert2000}; reviewed in \citealp{BuchertRasanen2012}), in which the averaged expansion of a domain \(D\) carries a
kinematic term
\begin{equation}
\mathcal{Q}_D \;=\; \tfrac23\big(\langle\theta^2\rangle_D-\langle\theta\rangle_D^2\big) - 2\langle\sigma^2\rangle_D,
\label{eq:buchert-Q}
\end{equation}
with \(\theta\) the local expansion scalar and \(\sigma\) the shear. \(\mathcal{Q}_D\)
is precisely the object \GDOS{} coarse-graining would source: it is the honest way a
locally-measured \(\langle H\rangle_D\) can exceed the background. The
intuition: averaging does not commute with evolving --- the average of
many patches, each expanding at its own rate, is not the expansion of
the average patch --- and \(\mathcal{Q}_D\) is the exact price of the
swap: the variance of the local expansion rates, minus the shear that
collapse builds. In a universe like ours the shear term wins, which is
why the honest computation comes out not only small but of the wrong
sign.

We have evaluated it rigorously on the \GGtheory{} background (the derivation is
Appendix~\ref{app:backreaction}), and the result is a firm null. \emph{First}, the
systematic backreaction is bounded, \(\langle \mathcal{Q}_D\rangle/H^2\lesssim10^{-3}\),
so the induced shift of the volume-averaged rate is \(\lesssim0.1\%\) and of the
\emph{wrong} sign (the shear term dominates) --- consistent with the mainstream
\(\Lambda\)CDM value \(\sim10^{-5}\), and unenhanced in \GGtheory{} because its
structure formation is standard. \emph{Second}, the cosmic variance of a locally
measured \(\Hzero\) --- the \emph{scatter} from sitting in a random over/under-density,
\(\sigma_H/H=(f/3)\,\sigma_\delta(R)\) --- is only \(\lesssim0.4\%\) on the ladder's
\(\sim\!100\)--\(300\) Mpc averaging scale \citep{Marra2013}. \emph{Third}, to produce
the full \(+7.3\%\) residual from a pure under-density would require
\(\langle\delta\rangle=-0.41\) on that scale, a \(>9\sigma\)
fluctuation --- the physical reason a void can be the lift's home but
not the residual's: the lift asks the neighborhood for a few percent,
which a \(\sim\!2\sigma\) valley supplies; the residual asks it for
seven, which no valley standard structure formation can dig on the
flow scale will supply. These bounds are drawn in
Figure~\ref{fig:backreaction-bound}, and they agree with the same
statement made from the rival-model side in Part~V.

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{c2_backreaction_bound.pdf}
\caption[The backreaction / cosmic-variance bound on the local excess.]{\textbf{Backreaction cannot supply the residual.} Linear theory on the exact derived background (\(\Omega_m=1/\pi\), \(\sigma_8=0.737\) from the derived \(S_8=0.759\); Eisenstein--Hu \(P(k)\)). \emph{Left:} the cosmic variance of the locally measured \(\Hzero\), \(\sigma_H/H=(f/3)\sigma_\delta(R)\), against averaging radius; on the ladder's effective scale (shaded) it is \(\lesssim0.4\%\), far below the Local-Hole band (green) and the \(+7.3\%\) residual (red). \emph{Right:} the rarity \(\nu=|\delta|/\sigma_\delta(R)\) of the under-density that would give \(+7.3\%\): a \(>9\sigma\) fluctuation on the ladder scale. The systematic Buchert term is \(\langle \mathcal{Q}_D\rangle/H^2\lesssim10^{-3}\) (Appendix~\ref{app:backreaction}).}
\label{fig:backreaction-bound}
\end{figure}

The one surviving loophole is honest to state and honest to bound. Only a fully
\emph{non-linear}, two-phase (void-versus-wall) reading of the kind Wiltshire's
timescape cosmology proposes \citep{Wiltshire2007} claims percent-level differences
between the statistically-averaged and locally-measured rate, through clock and
lapse effects between bound structures and expanding voids. Timescape is
philosophically the closest thing in the literature to a \GDOS{} coarse-graining,
and so it is the framework's honest best hope for a larger effect --- but it reaches
its magnitude by \emph{replacing} the standard FRW averaging with a specific
bare/dressed computation, it is model-dependent and contested, and it does not
cleanly turn a background \(67\) into a local \(73\). It is a door to a different
cosmology, not a correction to the derived \(H(t)\) of this paper. The conclusion
therefore stands: the residual is \emph{not} a cosmological-averaging effect. What
the framework derives is the sign and rough size of the environmental lift; the
remainder is a distance-ladder calibration question, and the ladder-free siren is
its arbiter.

\section{The why-now window: the coincidence as a factor-of-a-few timing}
\label{sec:why-now}

A reader may object that reading the present epoch at the \(\Omega_m=1/\pi\) crossing
is itself a coincidence --- why should ``now'' land where the matter and vacuum
shares cross? The framework's answer is quantitative. The one time-dependent ratio
in the inventory is
\begin{equation}
\frac{\rho_\Lambda}{\rho_m(a)}=(\pi-1)\,a^{3}=(\pi-1)\,e^{\,3(r-r_{\rm now})},
\qquad r=\ln\frac{\EGUT}{\mu}=r_{\rm now}+\ln a,
\label{eq:c2-rholambda-ratio}
\end{equation}
with \(r_{\rm now}\simeq66.6\) the present value of the radial/RG coordinate. The
ratio climbs monotonically through unity, and the ``order-unity'' window in which it
lies between \(0.1\) and \(10\) --- the window in which the coincidence looks like a
coincidence --- has width
\begin{equation}
\Delta r=\frac{2\ln 10}{3}=1.535,
\qquad a\in[0.36,\,1.67],\quad z\in[1.78,\,-0.40].
\label{eq:c2-whynow-window}
\end{equation}
Out of \(r_{\rm now}\simeq66.6\) units of radial history (roughly \(69\) e-folds from
the end of inflation), a window of width \(\Delta r\simeq1.5\) is about \emph{two
percent of the logarithmic history}. Living in it is therefore a factor-of-a-few
timing coincidence, not a fine-tuning --- the same order of ``luck'' as being alive
during the few hundred million years a given star is on the main sequence. The
present epoch is read at the \(\Omega_m=1/\pi\) crossing, \(75\) Myr before today at
\(H_0=66.6\) (a \(0.5\sigma\) offset against the \(\pm160\) Myr Planck timing blur),
and that reading is the one declared input the geometry does not remove.
The full form of the argument --- the window drawn on the whole radial
history, the crossing resolved from today, and the residual timing
question stated as such --- is the why-now chapter of the parent
\ATen{}; what this paper needs from it is only the price tag: one
declared epoch, factor-of-a-few luck, no dial.

\section{Why the derived reading is not an ad-hoc fit}
\label{sec:not-ad-hoc}

A reader meeting the split \eqref{eq:twostage} for the first time might
ask whether its ingredients were chosen after the fact to reproduce the
data. For the early side the answer is immediate and decisive: there is
\emph{no fitted number there at all}. The background rate is derived by
the geometry alone (Route~I, \S\ref{sec:absolute-h0}) and confirmed by
dividing the CMB's own model-independent densities \eqref{eq:c2-omegas}
by the derived inventory \eqref{eq:c2-inventory}; the CMB endpoint
\(67.4\) is simply where that derived rate is read. Nothing on the early
side is tuned, because nothing on the early side is free.

The late side carries exactly one derived quantity --- the Local-Hole
lift \(\epsilon_{\mathrm{Hole}}\) --- and one honestly undetermined
residual \(\epsilon_{\mathrm{syst}}\) that the framework labels a
distance-ladder systematic rather than a parameter. The lift is not a
Hubble-specific dial either: a local underdensity of the observed depth
raises \emph{any} locally inferred expansion rate, and its sign and rough
size follow from the environment, not from a wish to match \(73.5\).

The 2026 anatomy of the residual strengthens this defense rather than
straining it, for a reason worth stating plainly: every ingredient of
the candidate anatomy was identified by \emph{others}, in the field's
own refereed literature, by authors with no stake in this framework
--- one of the channels with the ladder's own architects among the
rebutting authors. The reading borrows no \GGtheory{} freedom
whatsoever: the framework's numbers --- the derived \(66.6\), the
lift band, the \(+7.3\%\) requirement --- were fixed by the geometry
and printed before the channels were sized, and the channels were
sized by people optimizing nothing of this paper's. If the anatomy
dissolves under its dated kill tests, the residual reverts to a named
blank and this section's defense loses nothing; if it survives, the
corridor closes without a single \GGtheory{} dial having moved ---
because there is no dial.

\subsection{The strong-\texorpdfstring{$CP$}{CP} contrast}
\label{sec:dneff-channel}

The cleanest way to see that the reading is forced rather than fitted is
to note what it does \emph{not} need. One might try to reconcile a high
background rate with the CMB by invoking a pre-recombination modification
(extra radiation, an early-dark-energy lobe); the derived reading needs
none of it: the microwave spectrum is fit at \(\approx 67\) by GG's own
fractions with the standard recombination history untouched. The only
early-universe number the cohort still predicts in this vicinity is the
\emph{small} radiation excess \(\DNeff = 0.04\) inherited from the
Standard-Model ledger of \ANine{} --- fully consistent with the Planck
preference \(0.04 \pm 0.18\), and unrelated to the Hubble split. A future
CMB bound tightening \(\DNeff\) toward \(0.04\) would \emph{confirm} the
cohort's value, not threaten it.

\section{Cross-ledger coherence: why this is the hard test}
\label{sec:cross-ledger}

One could always invent a theory that produces a single number matching
observation. The interesting question for \GGTheory{} is whether the
\emph{same inherited ledger} that fixes \(\sin^2\theta_W\),
\(\hat\alpha^{-1}(M_Z)\), the 28 Standard-Model parameters, the DNA
alphabet, the conscious-episode duration, the Kaxion twin-peak
\(\Delta\nu/\nu = 1/35\), and the \(\pi\)-locked inventory
\(\Omega_m = 1/\pi\) \emph{also} yields the right \(H_0\). It does, and
by the tightest possible route: the inventory that fixes \(H_0\) is the
\emph{same} inventory audited against the whole acoustic spectrum in
\ATen{}. If \(\Omega_m = 1/\pi\) or \(\Omega_b = 1/2\pi^2\) were tuned to
fix \(H_0\), the CMB peak positions, the baryon-loading ratio, and the
damping scale --- all reproduced to sub-percent by CLASS --- would break
simultaneously. The agreement is therefore not a one-parameter victory
but a consistency witness spanning the entire microwave sky.

\begin{table}[htbp]
\centering\small
\caption[Cross-ledger coherence: the ingredients of the derived rate have sibling predictions in other.]{Cross-ledger coherence: the ingredients of the derived rate have
sibling predictions in other sectors, so tuning any one to match \(H_0\)
would spoil the sibling. The table is the answer to the obvious objection, which is that a framework with enough structure can be tuned to any rate. Each ingredient of the derived value already carries a prediction somewhere else in the cohort, so moving it to improve the fit here breaks something there. The constraint is not that these numbers are hard to change; it is that none of them can be changed alone.}
\label{tab:crosscoh}
\begin{tabularx}{\textwidth}{@{}p{3.0cm}p{3.4cm}X@{}}
\toprule
\textbf{Ingredient} & \textbf{Role in \(H_0\)} &
\textbf{Same source, other predictions}\\
\midrule
\(\Omega_m = 1/\pi\), \(\Omega_b = 1/2\pi^2\) &
divided into the CMB densities to give \(H_0 = 66.6\) &
the full acoustic spectrum (peaks, baryon loading, damping scale) fit at
the same value; \(\Omega_b/\Omega_m = 1/2\pi\) at every epoch \\
\(\Mlock \approx 3.85\) TeV &
sets the seesaw cross-check \(H_0 = 498.3\sqrt{\zeta_\Lambda}\) &
the KK-cluster collider scale; the meV vacuum floor; the whole scale
chain of \AFive{}~T8.7 \\
Local-Hole lift &
\(+2\)--\(3\%\) on the local ladder &
a generic environmental effect on any local rate, fixed by the observed
underdensity, not by \(H_0\) \\
\bottomrule
\end{tabularx}
\end{table}

\section{The \texorpdfstring{$S_8$}{S8} tension, read the same way}
\label{sec:s8-section}

The same logic settles the companion \(S_8\) tension. Because GG makes
\emph{no} pre-recombination modification, the growth history is standard
\(\Lambda\)CDM growth on the derived inventory --- \(\Omega_m=1/\pi\), the
derived tilt \(n_s=0.962\), \(w=-1\) --- so the clustering amplitude is
one number rather than a dial:
\begin{equation}\label{eq:S8response}
S_8^{\mathrm{GG}} \;=\; \sigma_8\sqrt{\Omega_m/0.3} \;=\; 0.759
\qquad(\text{App.~\ref{app:s8}}),
\end{equation}
with no early-time suppression to invoke. The mild low-redshift \(S_8\)
preference of weak-lensing surveys is then, like the local \(H_0\), a
late-time/environmental question rather than an early-universe one ---
the two ``tensions'' share a single reading. The correlated sign
structure is summarized in Table~\ref{tab:signstructure}.

\begin{table}[htbp]
\centering\small
\caption[Sign structure of the joint reading: GG attributes both the local \(H_0\) excess.]{Sign structure of the joint reading: GG attributes both the
local \(H_0\) excess and the \(S_8\) preference to late-time/environmental
effects on a fixed derived background, not to early-universe physics. Read down the columns, the same three effects act with one sign on the local rate and with the opposite sign on the growth amplitude. That pattern is a prediction rather than an accommodation, because a framework that placed the tension in the early universe would not produce it. If the two are ever measured to move together, this row structure is what fails.}
\label{tab:signstructure}
\begin{tabularx}{\textwidth}{@{}p{3.4cm}XX@{}}
\toprule
\textbf{Effect} & \textbf{on local \(H_0\)} & \textbf{on \(S_8\)}\\
\midrule
Derived background (inventory) & sets \(66.6\) & sets \(S_8^{\mathrm{ref}}\) \\
Local-Hole underdensity & raises (local ladder) & lowers (local growth) \\
Coherent \(\pi\)-offset & \(+1\text{--}3\%\) & \(+1\text{--}3\%\) \\
\bottomrule
\end{tabularx}
\end{table}

The broad-brush growth relation used in the comparison Part below,
\begin{equation}\label{eq:growthbroadlinear}
\frac{\Delta S_8}{S_8} \;\approx\; \tfrac12\frac{\Delta\Omega_m}{\Omega_m}
\;+\; (\text{late-time environmental terms}),
\end{equation}
carries no early-universe correction in the present reading.

\subsection{The sign structure, quantified}
\label{sec:s8-quant}

The reason \(S_8\) belongs in the same discussion as \(H_0\) is that they draw on
the same inherited ledger with a \emph{fixed sign matrix}: the amplitude and the
rate are two shadows of one inventory, so the growth sector is a
cross-consistency audit, not a free knob. The derived value
\(S_8^{\mathrm{GG}}=0.759\) (Eq.~\eqref{eq:S8response}) is coincident with the
benchmark KiDS-1000 cosmic-shear determination \(0.759^{+0.024}_{-0.021}\) and
sits below the CMB-\(\Lambda\)CDM-inferred \(\approx0.83\). The honesty here is
essential and we state it once, plainly: the lensing-side agreement is a genuine
success of the derived inventory, but the early-inference-versus-late-measurement
\(S_8\) gap is not thereby explained --- it is carried as a late-time
coarse-graining question, read the same way as the local \(H_0\) excess, and
flagged open. The same discipline governs both tensions: one ledger, one honest
account of what it has and has not yet reached.

\section{Revisiting \texorpdfstring{\(\Lambda\)CDM}{LambdaCDM} in the light of a derived \texorpdfstring{$H(t)$}{H(t)}}
\label{sec:revisit-lcdm}

There is a larger consequence to draw, and we state it plainly, as an invitation.
\(\Lambda\)CDM is the most successful \emph{fitting} framework in the history of
cosmology: six numbers, an exquisite fit, four independent probes in concordance.
What it does not do is say \emph{why} those six numbers take the values they do --- it
measures them. \GGtheory{} proposes that they are not free: that the densities, and
therefore the entire expansion history \(H(t)\) of
Figure~\ref{fig:H-of-t-derived}, are derived from one six-dimensional geometry with
no continuous dial. If that proposal holds, the parameters of \(\Lambda\)CDM are
\emph{outputs} of a deeper theory rather than inputs to a fit, and several of the
field's standing ``tensions'' change character: the \(\Hzero\) tension becomes a
question about the calibration of one point on a derived curve, and the \(S_8\)
tension a question about late-time coarse-graining, rather than signals of missing
fundamental physics.

We are therefore not asking anyone to discard \(\Lambda\)CDM --- \GGtheory{} \emph{runs
on} its Friedmann/\FRW\ machinery, and its empirical fit is not in question. We are
asking the community to revisit it in a fresh light: to re-examine, parameter by
parameter, whether each of its numbers is a free input or a forced geometric
consequence, and to test the specific, dated predictions this reading makes --- the
standard-siren convergence \(\Hzero^{\mathrm{GW}}\to67\), the exact \(w=-1\), the
tensor ratio \(r=0.0043\), and the coherent \(\pi\)-offset across the acoustic peaks.
The strongest way to settle the tension is not to manufacture the residual but to
\emph{over-determine} the derived curve: direct \(H(z)\) from cosmic chronometers and
baryon acoustic oscillations, redshift drift, and standard sirens all measure
\(H(t)\) without the sound horizon or the distance ladder, and if they track the
\(66.6\)-anchored curve the local \(73\) is confirmed as a local excess on an
otherwise-correct history. The honest banner is not ``\(\Lambda\)CDM is wrong'' but
``\(\Lambda\)CDM is incomplete in its \emph{explanation}, and here is a candidate
completion --- laid out to be tested, and, if it fails, to fail cleanly.'' A generation
of precision cosmology measured the constants of the universe to extraordinary
accuracy; the next task, and it is a community effort, is to find out which of them
were never constants at all, but consequences.

\section{Summary: the reconciliation, and the road to Part V}
\label{sec:partIV-summary}

Part~IV set out to do the hard half of the tension, and this closing
section gathers what it established, ending where Part~V must begin.

The tension changed character at the outset
(\S\ref{sec:both-from-anchor}): with the background rate derived, it is
no longer a contradiction between two measurements of an unknown number
but the early/late split of one known number, read through two
operational dictionaries. The early dictionary needs nothing: the
derived \(66.6\) plus its own coherent \(\pi\)-offset \emph{is} the CMB
value, the ruler lands on the measured sound horizon, and no
pre-recombination physics is invoked because none is missing. The two
dictionaries are separated by a real geometric event --- the corridor
switch at \(z_* = e^7\), fixed by the integer \(J = 7\) and landing
\(0.6\%\) from the measured decoupling --- which is what licenses two
careful readings of one history to disagree at all
(\S\ref{sec:recomb-switch}).

The late side then factorizes into exactly two pieces
(\S\ref{sec:worked-excess}), and the chain is short enough to hold in
one line: \(66.6\) (derived) \(\times\,(1+0.028)\) (the Local-Hole
lift, derived in sign and bounded in size) \(\to 68.5\), plus a named
common-mode residual of \(+7.3\%\) \(\to 73.5\). Everything left of the
last step carries a grade; the last step carries a name and an open
verdict.

The lift is earned, not asserted (\S\ref{sec:hole-course}): the
under-density is observed (contested only in depth), the outflow
follows from continuity, the master formula
\(\epsilon = -(f/3)\,\bar\delta\) has an unconditional sign and an
inventory-fixed coefficient, and the two dilutions --- the ladder's
volume-weighted averaging and the flow corrections already applied ---
are worked explicitly, which is why the paper adopts \(+2\)--\(3\%\)
and records the deep-void \(+5.2\%\) as a scenario rather than a claim.

The residual is priced from both directions, and the pricing has
sharpened (\S\ref{sec:residual-anatomy}). The 2025--26 literature has
given it candidate anatomy: two published medium-size common-mode
channels --- calibrator dust (\(2\)--\(4\%\)) and the ladder's own
Bayesian distance priors (\(0\)--\(3\%\); proposed, executed, and
rebutted --- undecided) --- joined to the localized CCHP subsample
dispute (\(1\)--\(2\%\)) and the ordinary small terms. Under the
proposal-central reading the channels quadrature to \(4.3\%\) and
stack to \(9.3\%\); under the post-rebuttal conservative reading,
\(3.8\%\) and \(8.3\%\); in both, the canonical requirement \(+7.3\%\)
sits \emph{between} quadrature and stack --- a draw from documented
channels, no longer a two-maxima conspiracy --- and the published
central values alone carry \(73.5\) to \(\approx\!70.6\) (each
reanalysis on its own; \(69.5\)--\(70.2\) only if fully chained), within one
combined error bar of the lifted \(68.5 \pm 0.6\). Read in the other
direction, a common mode of the quadrature size deflates the headline
\(7.1\sigma\) to \(\approx\!1.9\)--\(2.1\sigma\), which is why the
tension's statistical grandeur and the residual hypothesis coexist
without contradiction. The standing counterweight is carried openly:
the one-step geometric probes stay high without supernova
standardization, and Part~V confronts them head-on. And in the light
of \GGtheory{} the channel list retains its clean structure: the
framework closes the channels it owns --- the \(\mathrm{U}(1)_B\)
graviphoton by eight orders of magnitude, the backreaction term below
\(10^{-3}\) with the wrong sign (\S\ref{sec:backreaction}) --- and is
deliberately silent on the metrology it does not, having earned, by
that closing, the right to locate the residual in the measurement
chain; the chain's own literature has now begun to name the suspects,
and every suspect carries a dated kill test reserved in the registry
(\S\ref{sec:falsifiers}).

The wider audit closes the Part. The why-now question is priced, not
hidden: one declared epoch, a two-percent logarithmic window,
factor-of-a-few luck (\S\ref{sec:why-now}). The reading is shown not to
be a fit: the early side contains no free number at all, and the late
side's one derived ingredient is a generic environmental effect
(\S\ref{sec:not-ad-hoc}). The same ledger is on the hook everywhere at
once --- tune any ingredient to help \(H_0\) and the acoustic spectrum,
the collider stake, or the meV vacuum scale breaks instead
(\S\ref{sec:cross-ledger}, Table~\ref{tab:crosscoh}) --- and the
\(S_8\) companion tension is read by the same split, with the derived
\(0.759\) landing on the lensing value while the early/late gap is
carried openly (\S\ref{sec:s8-section}). The Part ends with an
invitation rather than a verdict: revisit \(\Lambda\)CDM not as wrong
but as unexplained, and over-determine the derived curve with every
ladder-free, horizon-free probe available
(\S\ref{sec:revisit-lcdm}).

What remains is evidence, competition, and verdict. Part~V first
lays the entire 2026 measurement landscape on the table --- all
forty-five audited determinations, drawn and tabulated --- and then
lines up the rival
explanations --- the named-systematics reading, now a hypothesis with
its own anatomy rather than a shrug, early dark energy, extra
radiation, modified gravity, local voids --- against the same corridor
and the same kill switches, and asks which reading survives its own
falsifiers. Part~VI then hands the decision to the channel this Part
has invoked at every turn: the gravitational-wave standard siren,
which needs neither a sound horizon nor a ladder, and which this paper
commits to in advance --- \(\approx 67\) and the reading stands;
\(\approx 73\) and it dies (F-Hubble-2). The reconciliation is done;
the adversarial test begins.


\section{Part~IV in plain English: why the local ladder reads high, how much of that is derived, and what is left over}
The derived curve lands on the early number, so all that remains is to
say why the local measurement reads about nine percent above it. This Part
answers with a sum, and the honest part of the answer is that the sum does
not close.

There are two terms. The first is derived. We live in a region emptier than
average, and a region emptier than average expands slightly faster than
average; the effect follows from linear growth theory, with no freedom at
all in its sign and very little in its size. A ladder that measures the
flow it is sitting inside therefore reads high. The response law is
derived, the depth of the underdensity is other people's measurement and
is disputed, and the two together give a lift of two to three percent.

The second term is not derived. It is a residual of about seven percent
that would have to live somewhere in the ladder's own calibration, and
this paper does not claim to have found it. What it does instead is price
it. The ladder's stated error budget is walked channel by channel --- the
dust law, the calibration priors, the stability of the subsamples, the
statistics of the ladder itself --- and each channel is asked how much of
seven percent it could hide. Several can hide a little. None can hide it
all. The arithmetic is then run in both directions, so that a reader who
thinks the residual impossible can see exactly what would have to be true,
and so can a reader who thinks it easy.

That is the state of the reconciliation and the paper states it the same
way everywhere: half derived, half named and open. The exposure here is
total, because a residual of that size in the distance ladder is precisely
the sort of thing a better ladder analysis will either find or exclude
within a few years, and this paper has already agreed to accept the
verdict.

\GGPart[The 2026 measurement landscape]{The 2026 Measurement Landscape and the Rival Explanations}{Every reported \(H_0\) of the audited reference base, drawn and tabulated; the three bands; the six rival classes read against them; and the one measurement that bisects the field}

\section{The measurement landscape in 2026}
\label{sec:landscape}

Before the rival explanations can be judged, the evidence they are
rivals \emph{about} must be laid on the table --- all of it, not a
curated pair. The clean ``\(67\) versus \(73\)'' picture in which the
Hubble tension was born has not survived its own decade: the
measurement landscape of 2026 is a \emph{spectrum} of forty-five
published determinations from nine method families, and its structure
--- three bands, not two camps --- is itself a datum that any
explanation must now explain. This section assembles that landscape
from the primary literature, audited value by value; the sections
that follow put each rival class in front of it.

\subsection{The three bands, and how to read them}
\label{sec:landscape-bands}

Figure~\ref{fig:h0-landscape} draws, and Table~\ref{tab:h0-landscape}
tabulates, every reported \(H_0\) in the audited reference base of
this paper: forty-five determinations, each read from its primary
arXiv or journal record on July 12, 2026, each carried with its
published uncertainty, its identifier, and a one-line status remark.
The audit discipline matters as much as the values: a claim enters
the chart only at the number its source actually prints (two
body-text values are flagged as such), contested entries carry their
contest in the remark, and the entire set is reproduced by a released
script in which every plotted value is asserted against the audit
record. Nothing below is a poll of abstracts remembered; it is the
literature re-read.

How to read the chart, in one paragraph. Rows are grouped into the
nine method families and colored by them; the horizontal axis is the
present expansion rate; three shaded bands mark where the
determinations themselves cluster --- band~1 at \(67.2\)--\(68.5\),
band~2 at \(69\)--\(71\), band~3 at \(72.5\)--\(77\). Two green
guides carry this paper's numbers: the derived background
\(66.6 \pm 0.2\) (with Route~I's geometry-alone \(66.8 \pm 0.2\)
beside it) and the conditional lifted reading \(68.5 \pm 0.6\) --- what a
ladder calibrated \emph{inside} the Local Hole should read. The
dotted line is the paper's early reference anchor
\(\Href = 67.24 \pm 0.35\), the three-instrument CMB combination.
One bar (GWTC-4.0) runs off the right edge and is clipped with an
arrow. The remarks column of the table is part of the data: it is
where each value's soft spot is named.

\emph{Band 1 (\(67.2\)--\(68.5\)): the absolute-density band.} The
five tightest determinations in the field live here --- the
three-instrument CMB combination (\(67.24 \pm 0.35\)), P-ACT with
lensing and DESI (\(68.22 \pm 0.36\)), Planck alone
(\(67.4 \pm 0.5\)), DESI BAO with BBN (\(68.5 \pm 0.6\)), and, newest
and sharpest as a single instrument, SPT-3G alone at
\(66.66 \pm 0.60\). The physics of their agreement is the physics of
Part~III: the young plasma responds to \emph{absolute} densities ---
grams per cubic meter, not fractions of a rate --- so three
telescopes with different sites, detectors, and calibration chains,
and one route (BAO+BBN) that uses no CMB anisotropies at all, read
the same universe because they are all reading the same
thermodynamics. There is no shared hardware here to conspire. And
the band's floor is not empty: SPT-3G alone lands directly on the
derived \(66.6 \pm 0.2\) --- the geometry's number, printed before
that instrument's value existed. Balance requires the whole band,
however, and we print it against ourselves: band~1's own members span
\(66.66\)--\(68.51\) --- an internal spread of \(\approx\!2.5\sigma\)
the eye should not flatten --- and its DESI-weighted combinations pull
the \emph{upper} edge (\(68.2\)--\(68.5\)), \(3\)--\(4\sigma\) above
the derived rate. The derived \(66.6\) lives at the band's floor, not
its centroid; whether the early anchor drifts toward the floor or the
ceiling is itself a registered exposure (the commitments ledger,
\S\ref{sec:calendar-commitments}).

\emph{Band 2 (\(69\)--\(71\)): the corrected middle.} The middle band
is the newest structure in the landscape, and the most instructive,
because almost nothing in it is anyone's \emph{home} value. It is
where the distance ladder goes when any one of its published
medium-size corrections is applied --- the consistent-dust ladder
(\(70.5 \pm 1.0\)), the two-population remodeling
(\(70.59 \pm 1.15\), conservatively \(71.45 \pm 1.03\)), the
dust-free blue-supernova ladder (\(70.0 \pm 2.1\)), the
physically-motivated-priors recalibration (\(70.6 \pm 1.0\); carried
with its rebuttal) --- and where the CCHP JWST program already
stands (\(70.39 \pm 1.94\), with its JWST-only TRGB and JAGB variants
at \(68.8\) and \(67.8\)). It is also where the probes that owe
nothing to either camp natively sit: the localized fast radio bursts
(\(69.40^{+2.14}_{-1.97}\); \citealp{Gao2024FRB}), the sharpest
dark-siren value (\(69.9^{+4.1}_{-4.0}\); \citealp{Alfradique2026Sirens}),
the AGN-flare siren candidates (\(70.5\), associations unconfirmed;
\citealp{Kumar2026AGN}), the ladder-free meta-average of fifty-eight
measurements (\(69.37 \pm 0.34\); \citealp{Pantos2026Meta}), the
model-agnostic jackknife posterior (\(\approx 68\);
\citealp{Hughes2025Jackknife}), and the upper half of the
sound-horizon-free readings (\(70.0 \pm 1.7\), \(70.2^{+2.2}_{-2.4}\)).
The intuition is worth one sentence: band~2 is not a camp, it is a
\emph{destination} --- the place measurements arrive when a
systematic is removed or was never present, which is precisely what
the residual anatomy of Part~IV predicts the ladder would do under
its named corrections (\(73.5 \to \approx\!70.6\) per reanalysis; \(69.5\)--\(70.2\) fully chained).

\emph{Band 3 (\(72.5\)--\(77\)): the ladder complex and the geometric
high probes.} The high band contains two different populations, and
the distinction organizes the whole comparison problem. The first is
the Cepheid-ladder complex --- the H0DN consensus
(\(73.50 \pm 0.81\)), the full SH0ES set (\(73.2 \pm 0.9\)), the
covariance combination (\(73.30 \pm 0.92\)), the background-free
Perfect-Host field (\(73.49 \pm 0.93\)), the standardized-TRGB grid
(\(73.22 \pm 2.06\)), the Mira rungs (\(72.4\)--\(73.1\);
\citealp{Huang2023Miras,Bhardwaj2025Miras}) --- a
family of formidable internal consistency whose members share
calibration DNA, which is why its robustness tests certify
photometry rather than immunity to common modes (Part~IV,
\S\ref{sec:residual-anatomy}). The second population is the one-step
geometric probes --- megamasers (\(73.9 \pm 3.0\)), Type-II
supernovae (\(74.9 \pm 1.9\) stat), TRGB-SBF (\(73.8 \pm 2.4\)),
time-delay lensing (\(71.6^{+3.9}_{-3.3}\)), the Coma fundamental
plane (\(76.5 \pm 2.2\)), Tully--Fisher (\(73.3 \pm 4.1\)) --- which
share no supernova standardization and mostly no Cepheids, and which
are the standing counterweight to any pure-ladder-bias reading
(\S\ref{sec:landscape-counterweight}). Across the whole band a
single pattern holds and deserves its own line: \emph{precision and
directness are inversely ordered}. The sub-percent claims sit on the
longest calibration chains; the one-step probes that could arbitrate
carry \(3\)--\(5\%\) errors and one named soft systematic each. The
gamma-ray attenuation outlier (\(61.9^{+2.9}_{-2.4}\), EBL-model
dependent; \citealp{Dominguez2023Gamma}) is retained at the bottom of the chart as the honest
reminder that not every anomaly points the same way.

Where this paper sits is then a matter of reading coordinates, not
argument. The derived \(66.6 \pm 0.2\) stands at the floor of
band~1, with the newest single-instrument CMB value on top of it;
the lifted \(68.5 \pm 0.6\) stands at the floor of band~2, one
combined error bar below the corrected-ladder cluster. The two
guides bracket exactly the region into which the field's center of
gravity has been drifting from both directions --- the early side
down-anchored all along, the late side descending correction by
correction --- and both were printed by the geometry before most of
the rows in the chart existed. That is the landscape form of the
claim Part~III staked: one number, exposed many ways.

Two things the landscape does \emph{not} show, stated before any
rival is judged. First, it does not show independent scatter: the
216-measurement survey of the field finds the literature's spread
exceeds its quoted errors \citep{Wang2024Survey}, so the eye should
weight bands, not count rows. Second, it does not show a culprit: the model-agnostic
jackknife finds ``no bias anywhere'' strongly disfavored but
declines to single out the ladder --- many bias assignments fit
comparably. The landscape constrains explanations; it does not
convict for them. What it does do, decisively, is change the
question every rival must answer: not ``\(67\) or \(73\)?'' but
\emph{``why three bands --- and why is the middle one filling?''}

\par\vspace{2.2em}
\begingroup\noindent
\captionof{figure}[The 2026 \(H_0\) landscape: all 45 audited determinations, the three bands, and the GG predictions.]{\textbf{The
2026 \(H_0\) landscape.} Every reported \(H_0\) in this paper's
audited reference base --- forty-five determinations from nine method
families, each read from its primary record on July 12, 2026 ---
with the three landscape bands shaded and this paper's numbers shown
twice, as the top category (\emph{GG predictions}) and as
vertical guides: the derived background \(66.6 \pm 0.2\) (solid
green; Route~I's geometry-alone \(66.8 \pm 0.2\) dashed beside it)
and the lifted background \(68.5 \pm 0.6\) (dash-dotted). The dotted
line is the early reference anchor \(\Href = 67.24 \pm 0.35\) (this
combination's abstract variant reads \(67.19 \pm 0.38\); the paper
uses the \(\Href\) convention throughout). The GWTC-4.0 and GW170817 upper error
bars are clipped at the right edge (arrows). Row identifiers, posted dates, and
per-value remarks are carried in Table~\ref{tab:h0-landscape};
sources are cited in \S\ref{sec:landscape-bands}--\S\ref{sec:landscape-counterweight}
and Part~B of the references. Generated by
\texttt{c2\_showcase\_landscape.py}, released with the paper; every
plotted value is asserted in-script against the audit record.}
\label{fig:h0-landscape}
\endgroup

\clearpage
\begingroup\centering
\null\vfill
\includegraphics[width=\textwidth,height=0.97\textheight,keepaspectratio]{c2_h0_landscape_showcase.pdf}
\vfill
\endgroup\clearpage

\begin{footnotesize}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.03}
\begin{longtable}{@{}p{4.35cm} p{1.55cm} p{1.35cm} p{2.0cm} p{5.6cm}@{}}
\caption[The 2026 \(H_0\) landscape, tabulated: identifiers, dates, values, and status remarks.]{\textbf{The
2026 \(H_0\) landscape, tabulated.} The forty-five determinations of
Figure~\ref{fig:h0-landscape} in the figure's order and categories,
with this paper's two reference rows on top. Posted = first arXiv
posting (year-month from the identifier; day where verified).
``In-paper'' flags a value quoted from the source's body rather than
its abstract. \(H_0\) in km\,s\(^{-1}\)\,Mpc\(^{-1}\). All sources are
carried in Part~B of the references and cited at their first textual
use in this Part.}\label{tab:h0-landscape}\\
\toprule
\textbf{Entry} & \textbf{arXiv} & \textbf{Posted} & \textbf{\(H_0\)} & \textbf{Brief remarks}\\
\midrule
\endfirsthead
\multicolumn{5}{@{}l}{\small\emph{Table~\ref{tab:h0-landscape}, continued.}}\\[2pt]
\toprule
\textbf{Entry} & \textbf{arXiv} & \textbf{Posted} & \textbf{\(H_0\)} & \textbf{Brief remarks}\\
\midrule
\endhead
\bottomrule
\multicolumn{5}{r@{}}{\small\emph{continued on the next page}}\\
\endfoot
\bottomrule
\endlastfoot
\textbf{GG derived} & --- & --- & \(\mathbf{66.6\pm0.2}\) & Route~I (geometry alone, \(66.8\pm0.2\)) \(\cap\) Route~II (densities \(\div\) derived fractions); Part~III\\
\textbf{GG background + lift} & --- & --- & \(\mathbf{68.5\pm0.6}\) & derived background + Local-Hole lift (\(+2\)--\(3\%\)): what a ladder inside the Hole reads; Part~IV\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Early / sound-horizon route}}\\[0.5pt]
SPT-3G D1 alone & 2506.20707 & 2025-06 & \(66.66\pm0.60\) & tightest single-instrument CMB value; lands on the GG derived rate\\
CMB combined SPT+ACT+Planck & 2506.20707 & 2025-06 & \(67.24\pm0.35\) & the paper's \(\Href\) (CMB-SPA convention; abstract variant \(67.19\pm0.38\)); \(7.1\sigma\) below H0DN\\
Planck 2018 TT,TE,EE+lowE+lens. & 1807.06209 & 2018-07 & \(67.4\pm0.5\) & the legacy early anchor; \(A_L\) anomaly since resolved by ACT lensing\\
P-ACT + lensing + DESI DR1 & 2503.14452 & 2025-03 & \(68.22\pm0.36\) & \(68.43\pm0.27\) with DESI DR2; agrees with DESI at \(1.6\sigma\)\\
ACT DR6 lensing + BAO & 2304.05203 & 2023-04 & \(68.3\pm1.1\) & lensing route, no primary-anisotropy fit; \(\Lambda\)CDM-consistent at \(\sim\!1\%\)\\
DESI DR2 BAO + BBN & 2503.14738 & 2025-03 & \(68.5\pm0.6\) & in-paper; CMB-anisotropy-free ruler (deuterium + LUNA \(\omega_b\))\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Sound-horizon-free}}\\[0.5pt]
DESI \(r_s\)-free, DESY5 SNe & 2411.16677 & 2024-11 & \(66.7^{+1.7}_{-1.9}\) & equality-scale route; closes ``shorten the ruler'' as a road to 73\\
DESI \(r_s\)-free, Pantheon+ & 2411.16677 & 2024-11 & \(67.9^{+1.9}_{-2.1}\) & same analysis, Pantheon+ (Union3: \(67.8^{+2.0}_{-2.2}\)); 2.2--3\(\sigma\) below SH0ES\\
\(r_s\)-agnostic BAO+lens.+3\(\times\)2pt & 2509.16202 & 2025-09 & \(70.0\pm1.7\) & higher \(r_d\)-free reading; prior-sensitive; widens the honest span\\
EFTofLSS \(P+B\) \(r_s\)-free & 2602.21733 & 2026-02 & \(70.2^{+2.2}_{-2.4}\) & first \(P+B\) sound-horizon-free \(H_0\); \(68.6\pm1.8\) with Pantheon+\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Distance ladder (Cepheid / TRGB / JAGB / Mira)}}\\[0.5pt]
H0DN / LDN consensus & 2510.23823 & 2025-10 & \(73.50\pm0.81\) & covariance-weighted community baseline; excludes the CCHP group\\
SH0ES full HST set & 2408.11770 & 2024-08-21 & \(73.2\pm0.9\) & 42 SNe, 4 anchors; Cepheid linearity \(0.994\pm0.010\) (\(\sim\!7\sigma\) vs drift)\\
Ladder covariance combo & 2606.26831 & 2026-06 & \(73.30\pm0.92\) & 2026 review's seven-route combination; \(5.6\sigma\) vs CMB\\
Perfect-Host JWST Cepheids & 2509.01667 & 2025-09 & \(73.49\pm0.93\) & background-free field test; 19/24 JWST hosts unbiased\\
SH0ES joint JWST 3-method & 2408.11770 & 2024-08-21 & \(72.6\pm2.0\) & Cepheid+TRGB+JAGB on the joint JWST subsample; full-sample prediction 72.8\\
CATS standardized TRGB+SNIa & 2304.06693 & 2023-04 & \(73.22\pm2.06\) & tip-contrast-standardized; 108 variants, none below 71.6\\
SH0ES N4258-only anchor & 1908.05625 & 2019-08 & \(72.0\pm1.9\) & single (maser-galaxy) anchor variant\\
Miras of M101 & 2312.08423 & 2023-12 & \(72.37\pm2.97\) & 211 Miras; independent stellar-variable rung, sides high\\
Cluster-Mira anchor & 2507.10658 & 2025-07 & \(73.06\pm2.67\) & 41 globular-cluster Miras; no metallicity dependence over 1.6 dex\\
CCHP TRGB, published & 2408.06153 & 2024-08 & \(70.39\pm1.94\) & published CCHP value (ApJ 985, 203 + erratum); low side of the split; the program's own Cepheid variant reads \(\approx\!72.0\) --- CCHP's variants span \(67.8\)--\(72.1\)\\
CCHP JWST-only TRGB & 2408.06153 & 2024-08 & \(68.81\pm2.23\) & JWST-only subsample (stat\(\oplus\)sys); SH0ES: sample selection\\
CCHP JWST JAGB & 2408.03474 & 2024-08 & \(67.80\pm2.72\) & blind carbon-star route, N4258 zero point; lowest CCHP value\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Ladder re-analyses (dust / statistics)}}\\[0.5pt]
Consistent calibrator dust & 2403.10388 & 2024-03 & \(70.5\pm1.0\) & MW-like \(R_B\approx4.3\) on calibrators fits better (\(\Delta\)BIC \(=-11\)); \(5.2\to2.8\sigma\); same Pantheon+ calibrator SNe as the re-analysis rows below\\
Two-population SNe, baseline & 2506.22150 & 2025-06-27 & \(70.59\pm1.15\) & hierarchical two-population Pantheon+ remodeling (stretch bimodality)\\
Two-population, conservative & 2506.22150 & 2025-06-27 & \(71.45\pm1.03\) & same model, equal-prior variant; the channel's own internal spread\\
Blue (dust-free) SNe Ia & 2411.05642 & 2024-11 & \(70.0\pm2.1\) & unextinguished blue SNe only, no dust law; \(\sim\!6\) calibrators\\
Physically-motivated priors & 2601.22215 & 2026-01 & \(70.6\pm1.0\) & distance-prior recalibration, \(5\sigma\to2\sigma\); rebutted (2603.09880) --- undecided; alternative to (not additive with) the dust rows\\[1pt]
\multicolumn{5}{@{}l}{\textbf{One-step geometric probes}}\\[0.5pt]
Megamasers, MCP & 2001.09213 & 2020-01 & \(73.9\pm3.0\) & six maser galaxies, geometric; PV treatment spans 71.8--76.9\\
SNe II EPM, no rungs & 2411.04968 & 2024-11 & \(74.9\pm1.9\) & tailored EPM, 10 SNe II (stat only); spectral-emulator systematics\\
TRGB-SBF via N4258 & 2502.15935 & 2025-02 & \(73.8\pm2.4\) & JWST TRGB \(\to\) SBF zero point, Cepheid/SNIa-free; single anchor\\
TDCOSMO 2025 lensing & 2506.03023 & 2025-06 & \(71.6^{+3.9}_{-3.3}\) & 8 time-delay lenses + JWST/Keck kinematics, blinded; MSD-conservative\\
DESI FP + Coma & 2409.14546 & 2024-09 & \(76.5\pm2.2\) & fundamental plane on Coma (\(D\) from 12 SNe Ia --- not SNIa-independent)\\
Tully--Fisher CF4 & 2408.03660 & 2024-08 & \(73.3\pm4.1\) & CF4 \(i\)/W1 TF (stat\(\oplus\)sys); discordant zero-point calibrators\\
Gamma-ray attenuation & 2306.09878 & 2023-06 & \(61.9^{+2.9}_{-2.4}\) & EBL route at \(\Omega_m=0.32\) (free fit \(65.6^{+5.6}_{-5.0}\)); anomalously low\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Gravitational-wave sirens}}\\[0.5pt]
GW170817 bright siren & 1710.05835 & 2017-10 & \(70^{+12}_{-8}\) & the one bright siren in nine years; ladder- and ruler-free\\
GWTC-4.0 + GW170817 & 2509.04348 & 2025-09 & \(76.6^{+13.0}_{-9.5}\) & 142 dark sirens + GLADE+; mass-spectrum-model dependent\\
GWTC-5.0 + GW170817 & 2605.27227 & 2026-05 & \(71.0^{+9.0}_{-7.1}\) & 236 sources + DES-Y6; 25.7\% tighter than GWTC-4.0\\
O4a dark sirens + GW170817 & 2603.20195 & 2026-03 & \(69.9^{+4.1}_{-4.0}\) & 17 dark sirens, DESI Legacy photo-\(z\); sharpest siren value\\
AGN-flare BBH sirens & 2607.03672 & 2026-07 & \(70.5^{+3.4}_{-2.9}\) & 13 claimed AGN counterparts (\(\pm1.56\) cal.); associations unconfirmed\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Fast radio bursts}}\\[0.5pt]
108 localized FRBs & 2410.03994 & 2024-10 & \(69.40^{+2.14}_{-1.97}\) & IllustrisTNG DM priors; squarely between the camps\\
527 CHIME FRBs, nonlocalized & 2410.03994 & 2024-10 & \(68.81\pm0.68\) & statistical sample; formally tight, DM-model dependent\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Meta-analyses / statistics}}\\[0.5pt]
30 ladder measurements, mean & 2601.00650 & 2026-01 & \(72.73\pm0.39\) & mean of the distance-ladder family\\
58 ladder-free, mean & 2601.00650 & 2026-01 & \(69.37\pm0.34\) & ladder-independent mean; internal split up to \(6.5\sigma\)\\
Bayesian jackknife, 16 probes & 2511.19341 & 2025-11 & \(68.0^{+4.7}_{-1.3}\) & model-agnostic posterior (95\%); no-bias strongly disfavored\\[1pt]
\multicolumn{5}{@{}l}{\textbf{Model-dependent early fixes}}\\[0.5pt]
EDE profile likelihood, no SH0ES & 2505.08051 & 2025-05 & \(71.0\pm1.1\) & \(f_{\rm EDE}=0.09\pm0.03\) (ACT+DESI); vs the ACT team's own null\\
\end{longtable}
\end{footnotesize}

\subsection{The consensus that is not a consensus}
\label{sec:landscape-consensus}

The H0DN consensus \(73.50 \pm 0.81\) is the subject measurement of
this paper, and Part~II took its internal architecture seriously ---
thirty structured variants, orthogonal paths, leave-one-out
stability. One sociological fact must now be set beside that
architecture, because the landscape makes it quantitative: the
consensus's \(\sim\!37\) authors do not include the CCHP group ---
the team whose JWST measurements (\(70.39 \pm 1.94\) published;
\(68.8\) and \(67.8\) in its JWST-only variants) disagree with the
baseline most \citep{Freedman2025CCHP,Lee2024JAGB}. The
Freedman--Riess split has not converged; it has \emph{localized}:
the 2026 ladder review \citep{Chen2026Review} and the SH0ES
same-data reanalysis \citep{Riess2024Selection} place the residual
disagreement in supernova-calibrator subsampling and tip/JAGB
standardization conventions --- not in anchors, not in crowding, not
in photometry. The practical consequence for every comparison in
this Part: the ladder's headline number and its principal internal
dissent must both be carried, and this paper carries both ---
\(73.50 \pm 0.81\) as the subject measurement, \(70.39 \pm 1.94\) as
the standing demonstration that calibrator choices worth
\(\sim\!3\;\kmsMpc\) are live inside the ladder community itself.
An internal-consistency architecture, however excellent, certifies
what a collaboration \emph{shares}; it is silent on what its members
do not share --- which is precisely where the subsample channel of
Part~IV lives (F-Hubble-10).

\subsection{The early side is closed}
\label{sec:landscape-early}

The landscape's band~1 is not merely populated; it is
\emph{overdetermined}, and the 2024--26 data have closed, one by one,
every route by which the early side could have been blamed for the
tension. This subsection collects the closures, because Part~V's
class verdicts lean on them repeatedly.

\emph{Three instruments, one thermodynamics.} Planck, ACT DR6, and
SPT-3G --- different sites, detectors, scan strategies, and
calibration chains --- agree at a fraction of the offset in play
\citep{Louis2025ACT,Camphuis2025}; their combination,
\(\Href = 67.24 \pm 0.35\), carries an empirical inter-experiment
spread that bounds any CMB-side systematic at a quarter of what the
tension needs. The old \(A_L\) lensing anomaly, once the early
side's one internal blemish, is resolved: ACT DR6 lensing agrees
with \(\Lambda\)CDM at the percent level \citep{Madhavacheril2023Lensing},
and the lensing+BAO route --- no primary anisotropies at all ---
reads \(68.3 \pm 1.1\).

\emph{The ruler is audited twice.} The sound horizon's raw material
is the baryon density, and deuterium fixes it without any microwave
sky: the LUNA-pinned BBN value \(\omega_b = 0.02218 \pm 0.00055\)
\citep{Schoneberg2024BBN} agrees with the acoustic reading, and
DESI's BAO+BBN route lands at \(68.5 \pm 0.6\)
\citep{AbdulKarim2025}. The intuition: recombination is the ruler's
factory, but the factory's books are audited by an accountant (BBN)
who has never seen the factory --- and the books balance.

\emph{Even the ruler can be discarded.} The sound-horizon-free
determinations --- DESI's equality-scale route at
\(66.7\)--\(67.9\) \citep{Zaborowski2024RsFree}, the
\(r_d\)-agnostic large-scale-structure reading at \(70.0 \pm 1.7\)
\citep{GarciaEscudero2025}, and the EFTofLSS power-spectrum-plus-
bispectrum route at \(70.2^{+2.2}_{-2.4}\) \citep{Lu2026EFT} ---
span \(66.7\)--\(70.2\): every one of them well below \(73\). The
honest breadth of that span is printed deliberately: the r\(_s\)-free
band overlaps bands~1 \emph{and}~2, so it does not adjudicate
between the derived background and the corrected middle --- but it
closes the ``shrink the ruler'' road to the ladder from every side,
because if sound-horizon inference were the culprit, the
ruler-free readings would sit high, and none does.

\emph{What survives on the early side, capped.} The one genuinely
live early-side lever is non-standard recombination, and it is now
squeezed from four directions; its own cap --- \(H_0 < 70.7\) at
\(95\%\) confidence with CMB+BAO, from the non-parametric ionization
reconstruction \citep{Lynch2024Recomb} --- reaches band~2, not the
ladder. The full accounting is Class~3
(\S\ref{sec:class3-modrec}). For the derived reading the
consequence of this subsection is structural: the early anchor now
stands on three instruments, a CMB-free ruler audit, and
ruler-free cross-checks, with the newest single-instrument value
sitting on the derived rate itself --- so every rival that begins
``the early inference is wrong'' begins three closed doors behind
the evidence.

\subsection{The high geometric counterweight}
\label{sec:landscape-counterweight}

Honesty requires the strongest opposing evidence to be stated at
full strength, in its own subsection, before any class is judged.
The counterweight to every measurement-side reading --- this paper's
included --- is band~3's second population: probes that reach the
Hubble flow in one step, without Type-Ia standardization and mostly
without Cepheids, and read high anyway. The megamaser galaxies are
pure geometry --- water-maser orbits give angular-diameter distances
directly --- and read \(73.9 \pm 3.0\), though the value slides
across \(71.8\)--\(76.9\) with the peculiar-velocity treatment of
just six objects \citep{Pesce2020Masers}. The Type-II
expanding-photosphere method needs no ladder at all --- the
photosphere's physical size comes from spectra --- and reads
\(74.9 \pm 1.9\) (statistical), with its systematics budget resting
on a spectral-emulator calibration still in its proof-of-principle
phase \citep{Vogl2025EPM}. The TRGB-SBF route ties surface-brightness
fluctuations to a JWST tip calibration --- no Cepheids, no
supernovae --- and reads \(73.8 \pm 2.4\), on a single anchor galaxy
\citep{Jensen2025SBF}. Time-delay lensing, blinded and
mass-sheet-conservative, reads \(71.6^{+3.9}_{-3.3}\), straddling
bands~2 and~3 \citep{TDCOSMO2025}. The Coma fundamental plane reads
highest of all (\(76.5 \pm 2.2\)) but calibrates Coma's distance
with twelve Type-Ia supernovae, so it is not independent of the
standardization question \citep{Scolnic2024Coma}; Tully--Fisher
reads \(73.3\) with a \(3.5\;\kmsMpc\) systematic term driven by
discordant zero-point calibrators \citep{Boubel2024TF}.

Read as a population, these probes say something no advocate of a
pure-bias reading may wave away: \emph{whatever is high in the local
universe is not confined to the Cepheid--supernova chain.} Read
individually, none is decisive --- each carries \(3\)--\(5\%\)
errors and one named soft systematic (six-object peculiar
velocities; an emulator; a single anchor; the mass-sheet degeneracy;
a supernova-calibrated cluster distance; discordant zero points) ---
and their weighted center of gravity, together with everything else
that is not the Cepheid ladder, remains \(\sim\!69\)--\(71\). The
derived reading's obligations to this subsection are recorded in
the registry: if the maser sample triples and holds \(74\), if the
Type-II emulator survives its systematics audit at \(75\), the
measurement-side story is in real trouble well before any siren
flies --- and the converse collapses band~3 to one calibration
family. The counterweight is not an inconvenience to be footnoted;
it is the second-sharpest open question the landscape poses, after
the tension itself.

\section{Statement of the comparison problem}
\label{sec:class-statement}

The landscape sharpens the problem every proposed resolution must
solve. It is no longer enough to move one number from \(67\) to
\(73\): a viable explanation must now account for a
\emph{three-band structure} --- an overdetermined absolute-density
band at \(67.2\)--\(68.5\); a filling middle band at \(69\)--\(71\)
into which every corrected ladder and every camp-free probe drains;
and a high band whose two populations (one calibration family, one
set of soft-systematic geometric probes) it must explain
\emph{differently} or explain why not. It must survive the
meta-split --- thirty ladder measurements averaging
\(72.73 \pm 0.39\) against fifty-eight ladder-free averaging
\(69.37 \pm 0.34\), internally discrepant at up to \(6.5\sigma\)
\citep{Pantos2026Meta} --- and it must survive the
sound-horizon-free readings, which cap what any ruler-side fix can
deliver. The literature has produced many candidates
\citep{DiValentino2021,DiValentino2025,Poulin2023Review}; we
examine six representative classes, each against two instruments:
the \emph{cross-ledger coherence} of Part~IV, and the \emph{siren
discriminator} of \S\ref{sec:siren-discriminator}, which now
carries three outcomes rather than two.

The six classes, in the taxonomy the 2026 landscape itself suggests:
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item \(\Lambda\)CDM plus measurement systematics --- now a
\emph{named} hypothesis, not a shrug;
\item phenomenological early dark energy;
\item modified recombination (split out from EDE: same lever,
different physics, different caps);
\item interacting dark sectors and extra radiation (merged: both
re-time or re-load the early plasma through added dark content);
\item modified gravity and evolving dark energy;
\item local voids and inhomogeneity.
\end{enumerate}

\section{Class 1 --- \texorpdfstring{$\LCDM$}{LCDM} plus measurement systematics: a named hypothesis}
\label{sec:class1-systematics}

The hypothesis: the concordance model is correct, the early
inference is unbiased, and the ladder's excess is measurement
systematics. For most of the tension's decade this class was a
shrug --- ``some unknown error, somewhere'' --- and it was treated
accordingly.

\paragraph{What the network's architecture retires --- and what it
cannot.} The H0DN architecture genuinely retires the class's naive
form. The orthogonal-path test shows two statistically independent
local routes agreeing at \(<0.2\sigma\)
\citep[Fig.~12 of][]{CasertanoH0DN2026}; leave-one-out shows no
single method drives the result; HST and JWST agree
\citep{RiessSH0ES2022,Riess2024Cross}. Any \emph{private}
systematic --- one instrument's bug, one method's bias --- is dead.
But the same architecture, by construction, cannot price a
\emph{common-mode} term: variants that share calibration DNA move
together, so their mutual stability certifies internal consistency,
not external truth (\S\ref{sec:residual-anatomy}). The intuition is
an auditor's: thirty cross-checked copies of one ledger prove the
copying is faithful, not that the ledger is right. What the class
needed, to be more than a shrug, was \emph{named} common-mode
mechanisms with published sizes --- and that is exactly what
2024--26 supplied.

\paragraph{The named anatomy.} The class now runs on the three
channels priced in Part~IV: the calibrator-versus-flow dust and
two-population standardization mismatch (allowance \(2\)--\(4\%\);
published central values \(70.5 \pm 1.0\), \(70.59 \pm 1.15\), with
the dust-free cross-check at \(70.0 \pm 2.1\)); the Bayesian
distance-prior statistics of the ladder itself (allowance
\(0\)--\(3\%\); proposed, executed at \(70.6 \pm 1.0\), and rebutted
--- undecided); and the calibrator-subsample selection localized by
the CCHP split (allowance \(1\)--\(2\%\)). Under the
proposal-central reading the channels quadrature to \(4.3\%\)
against the \(+7.3\%\) requirement; the published central values
alone carry \(73.5 \to \approx\!70.6\)--\(70.2\)-if-chained --- into band~2, within one
combined error bar of the lifted \(68.5 \pm 0.6\).

\paragraph{What the class explains, and where it strains.} Explains:
why the middle band is filling (corrected ladders drain into it);
why the CCHP split exists (subsampling is one of its channels); why
the \(H_0(z)\) trend descends (\S\ref{sec:hole-budget}); why the
meta-split separates ladder from ladder-free. Strains: at the
geometric counterweight (\S\ref{sec:landscape-counterweight}) ---
masers, Type-II supernovae, and SBF are touched by \emph{neither}
the dust nor the priors channel, and a complete Class-1 story must
either find their own soft systematics doing coherent work or
concede band~3's second population to coincidence. The class is
honest only when it carries that strain visibly.

\paragraph{What \GGtheory{} offers that this class does not.} A
\emph{positive} decomposition rather than a corrected suspicion: a
background \emph{derived} --- twice, once with zero sky input
(Route~I, its coefficient pre-registered as F-Hubble-11) --- where
Class~1 assumes one; a lift derived in sign and size where Class~1
would fit one; a residual priced in both directions against the
ladder's own budget; and a stake. And one thing should be said
plainly, because a careful reader will notice it anyway: for the
last seven percent, the derived reading is itself a disciplined
member of this class. The differences are the discipline --- and
one number: Class~1 has no opinion about \emph{where} the corrected
ladder should land, while the geometry printed \(68.5 \pm 0.6\)
before the corrections existed. That the field's corrections now
carry the ladder to within one error bar of that number is either
the corridor closing as derived, or a coincidence Class~1 is
welcome to keep.

\paragraph{Siren verdict required.} \(\approx67\)--\(68\): if the
excess is measurement, the true rate is low, and the ladder-free
channel must say so. The class lives or dies with the derived
reading on the same catalog --- the kinship above, made
operational.

\section{Class 2 --- phenomenological early dark energy}
\label{sec:class2-ede}

The EDE program
\citep{Karwal2016,Poulin2019,Poulin2023Review,KamionkowskiRiess2023}
introduces a scalar field contributing a narrow lobe of energy
density around \(z \sim 3500\) that dilutes away. It works because
shortening \(r_s\) by \(\sim\!5\%\) near that epoch is exactly what
the lever of Eq.~\eqref{eq:c2-leverarm} converts into a higher
inferred rate: hold the measured angle \(\theta_*\) fixed, shave the
stick, and the inference must stand closer.

\paragraph{Status after the 2026 landscape: two-sided, and capped.}
The class's empirical position is now genuinely two-sided, and the
paper prints both sides. On one side, the ACT collaboration's own
extended-model search finds \emph{no evidence} for EDE
\citep{Calabrese2025Ext}, and the joint CMB+BAO pressure confines
the lobe to the low end \citep{KreischEDE2024,Camphuis2025}. On the
other, a profile-likelihood analysis of the same ACT DR6 + DESI DR2
data --- deliberately prior-free where the Bayesian search is
prior-limited --- reports \(f_{\rm EDE} = 0.09 \pm 0.03\) and
\(H_0 = 71.0 \pm 1.1\) \emph{without} any SH0ES input
\citep{Poulin2025EDE}. An honest scorer calls that a live dispute
about statistical method, not a settled null. What is settled is
the ceiling: the sound-horizon-free determinations
(\S\ref{sec:landscape-early}) span \(66.7\)--\(70.2\), and a
mechanism whose entire output is a shorter ruler cannot move
determinations that never use the ruler. EDE has become a
\emph{band-2 machine}: it can reach \(70\)--\(71\); it cannot reach
the ladder, and the DESI-era \(\Omega_m\) coherence squeezes it
further \citep{Poulin2023Review}. The deuterium audit is the same
wall from a second side: early fixes that raise \(H_0\) generically
raise the inferred \(\omega_b\) against the BBN value
\citep{Giovanetti2026}.

\paragraph{What \GGtheory{} offers that EDE phenomenology does
not.} The decisive difference: the derived reading needs no early
modification at all. EDE buys a higher inferred \(\Hzero\) by
shortening the sound horizon; the derived background already sits
on the CMB value, so there is no early-side gap to close. Four
consequences:

\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item \textbf{No new field, no new potential.} EDE adds a scalar with
a \(V(\phi) \propto (1-\cos(\phi/f))^n\) potential and free \(n\),
\(f\), and mass scale. The derived reading adds nothing: the acoustic
spectrum is fit by the \(\pi\)-locked fractions with the standard
recombination history untouched (\S\ref{sec:cmb-overdetermined}).
\item \textbf{No \(\SEight\) cost.} The early lobe that shortens
\(r_s\) also suppresses growth, and in full likelihood refits the
net \(\SEight\) shift typically comes out worse, not better
\citep{Hill2020,Poulin2023Review}. A reading with no early lobe pays
no such cost (\S\ref{sec:s8-section}).
\item \textbf{Opposite exposure.} EDE is confirmed by detecting a
lobe; the derived reading is \emph{refuted} by one (F-Hubble-3,
\S\ref{sec:falsifiers}). The profile-likelihood hint above is
therefore not an embarrassment to be argued away but this paper's
own kill test in motion: if the \(f_{\rm EDE} = 0.09 \pm 0.03\)
reading survives the likelihood dispute and sharpens past
\(3\sigma\), F-Hubble-3 fires. The two frameworks make opposite
pre-registered bets on the same forthcoming analyses.
\item \textbf{Cross-domain reach.} The same integer ledger that
fixes the inventory also fixes the Standard Model's 28 parameters
and the Kaxion spectrum. EDE is a cosmology-only fix.
\end{enumerate}

\paragraph{Siren verdict required.} \(\approx73\) to solve the
tension it was built for --- but the landscape has narrowed its own
demand: since its caps now reach only band~2, a siren at
\(\approx70\) is the best confirmation EDE can still claim, and a
siren at \(67\) retires it outright. One catalog, opposite bets:
early dark energy and the derived reading cannot both survive it.

\section{Class 3 --- modified recombination}
\label{sec:class3-modrec}

Split out from the EDE class because it pulls the same lever with
different physics --- and meets different walls. Instead of adding
energy density, this class changes \emph{when and how the fog
lifted}: shift the ionization history \(x_e(z)\), move recombination
earlier, and the sound horizon frozen at the switch is shorter, so
the fixed acoustic angle again converts into a higher inferred rate.
Concrete mechanisms include spline-level freedom in \(x_e(z)\)
\citep{Lynch2024Recomb,Lynch2024DESI}, and primordial magnetic
fields that clump baryons and hasten recombination
\citep{Jedamzik2025PMF}.

\paragraph{Status: the factory audit.} The intuition of
\S\ref{sec:landscape-early} does the work: recombination is the
ruler's factory, and the factory's books are audited by
accountants who never enter it. Four audits now cap the class.
\emph{First}, the non-parametric reconstruction itself: given full
spline freedom in the ionization history, CMB+BAO data allow at most
\(H_0 < 70.7\) at \(95\%\) \citep{Lynch2024Recomb} --- the class's
own ceiling, printed by its own machinery, reaching band~2 and no
further. \emph{Second}, deuterium: raising \(H_0\) this way raises
the inferred \(\omega_b\), and the LUNA-pinned BBN value stands in
the way generically \citep{Giovanetti2026,Schoneberg2024BBN}.
\emph{Third}, DESI: the spline histories that help \(H_0\) drag
\(\Omega_m\) against the DR2 measurement \citep{Lynch2024DESI}; the
electron-mass variant of the same idea, which resolves the
CMB-only fit, \emph{fails} once DESI DR2 is included
\citep{Lee2026ModRecII}. \emph{Fourth}, the magnetic-field
mechanism's evidence stands at \(1.8\sigma\) --- a hint, dataset
dependent, honestly below claim level \citep{Jedamzik2025PMF}.
Within its caps the class remains the most respectable early-side
survivor: spline reconstructions genuinely prefer nonstandard
histories at the \(2\)--\(3\sigma\) level in some data combinations
\citep{Mirpoorian2024ModRec}, and a band-2 landing is within its
reach.

\paragraph{What \GGtheory{} offers.} No need for the lever, and no
freedom to pull it. The derived chart lands on the \emph{measured}
sound horizon (\(r_s = 145.0\) against \(144.43 \pm 0.26\)) with the
standard ionization history untouched, and the corridor switch is
not adjustable: \(z_* = e^7\) is the exponential of a ledger
integer, a topological quantity with no continuous dial
(\S\ref{sec:recomb-switch}). A framework whose recombination
redshift is an integer's exponential cannot join this class even if
it wanted to --- there is nothing to tune --- and its agreement with
the measured decoupling at \(0.6\%\) is therefore a checkable
coincidence-or-theorem, not a fit.

\paragraph{Siren verdict required.} \(\approx73\) to be the
tension's resolution --- which its own \(70.7\) ceiling already
forbids; \(\approx70\) to survive as a band-2 contributor. A siren
at \(67\) retires the class; at \(70\) it lives, but as physics of
the middle band, not as the explanation of the ladder.

\section{Class 4 --- interacting dark sectors and extra radiation}
\label{sec:class4-darksector}

Merged from two classes of the earlier taxonomy, because both work
by re-loading the early plasma with dark content: couple the dark
fluids so energy migrates between them, or add relativistic species
(\(\DNeff\)) so the pre-recombination expansion runs faster. Both
re-time equality or shorten the horizon; both are early-side fixes
wearing different hardware.

\paragraph{Status.} The interacting-sector limb writes coupled
continuity equations ---
\(\dot\rho_{\rm dm} + 3H\rho_{\rm dm} = +Q\),
\(\dot\rho_{\rm de} + 3(1+w)H\rho_{\rm de} = -Q\) --- whose exchange
rate \(Q\) is a new function with new couplings; case by case such
models can fit CMB+\(H_0\)+\(\SEight\) combinations, at the cost of
parameters with no independent motivation. The radiation limb is
cleaner and more cornered: closing the gap to the ladder needs
\(\DNeff \gtrsim 0.4\), which BBN's deuterium and helium refuse
\citep{Schoneberg2024BBN}, and which the current CMB bound
\(\DNeff = 0.04 \pm 0.18\) leaves no room to hide. Both limbs also
answer to the ruler-free wall: whatever they do to \(r_s\), the
sound-horizon-free band at \(66.7\)--\(70.2\) does not move with
them.

\paragraph{What \GGtheory{} offers.} No interaction, and no room
for one --- and that is the point. The dark sectors the framework
contains are locked: the \(\UoneQi \leftrightarrow \Uoneb\) channel
is fixed by the \BIsix{} coefficients \((1,-1,-2/5,-2/5)\) with no
free strength anywhere, and the only radiation excess the cohort
predicts is the counted \(\DNeff = 0.04\) of \ANine{} ---
consistent with the bound at its center, unrelated to the Hubble
split, and \emph{confirmable}: a future bound tightening onto
\(0.04\) supports the ledger; a detection at \(0.4\) would be new
physics the framework neither needs nor survives unchanged. A
framework whose dark-sector couplings are pinned by the same
integers that fix \(\sin^2\theta_W = 25/108\) is not in the same
complexity class as a model with three couplings tuned to a fit.

\paragraph{Siren verdict required.} \(\approx73\) for the class's
viable early-side members; late-coupling variants that bend
\(H(z)\) at low redshift face the \(w(z)\) and BAO constraints
directly and inherit whichever verdict their bending implies. A
siren at \(67\) retires the class alongside every other early-side
fix.

\section{Class 5 --- modified gravity and evolving dark energy}
\label{sec:class5-modgrav}

This class alters the gravitational sector itself: a running
\(G_{\rm eff}(z)\), a scalar--tensor extension, or --- its newly
prominent limb --- an evolving dark-energy equation of state
\(w(z)\) of the kind the DESI \(w_0w_a\) analyses have made
headline news.

\paragraph{Status, with the wrong-sign surprise.} The
modified-gravity limb can shift \(\Hzero\), but typically at the
cost of scale-dependent growth modifications that redshift-space
distortions and lensing rule out, or equivalence-principle
violations that solar-system tests forbid \citep{DiValentino2025}.
The evolving-dark-energy limb is where the data actually moved:
DESI DR2 with supernovae prefers \(w_0 > -1, w_a < 0\) at
\(2.8\)--\(4.2\sigma\) depending on the sample
\citep{AbdulKarim2025} --- a genuine, contested, parametrization-
sensitive hint. But its sign is the striking fact: evolving dark
energy of the preferred kind pushes the inferred \(H_0\)
\emph{down}, not up. The one live new-physics hint of the DESI era
is a rope thrown in the wrong direction --- it entangles the early
anchor's error budget, it cannot rescue \(73\), and if it
solidifies it wounds the ladder's side of the tension more than
the CMB's. (For this paper it is a two-sided exposure of its own:
\(w = -1\) exactly is kill switch F-Hubble-5, and the DESI hint is
that switch under live test.)

\paragraph{What \GGtheory{} offers.} Gravity is not modified ---
AX-2 locks the geometry--gauge structure, solar-system tests are
preserved structurally rather than by tuning --- and the vacuum
does not run: \(w = -1\) is the ground-attractor theorem, staked
two-sidedly. The derived reading needs neither limb: the rate is
what it is because the inventory is what it is.

\paragraph{Siren verdict required.} Either --- and that ambivalence
is the class's own signature. A running effective Planck mass
generically makes the gravitational-wave luminosity distance differ
from the electromagnetic one, so the siren tests this class twice:
once through the rate, once through the GW--EM distance duality
checked event by event on the same detections. A duality violation
would be its smoking gun; a null, one more constraint tightening
around it.

\section{Class 6 --- local voids and inhomogeneity}
\label{sec:class6-voids}

The hypothesis: the local universe sits in an underdensity that
lifts every locally inferred rate above the global mean. After
Part~IV this class is largely \emph{native} machinery to this
paper, so the update is brief and quantitative.

\paragraph{Status after the 2025--26 data.} Three depths must be
distinguished, because the data now treat them differently. The
\emph{modest} void --- \(\bar\delta \sim -0.1\) to \(-0.2\) over
the flow volume, worth \(+2\)--\(3\%\) --- is alive, observed, and
adopted: it is this paper's Local-Hole lift. The \emph{deep} void
--- the KBC reading at \(-0.3\) over \(300\)~Mpc, worth
\(+5\%\) --- is disfavored from within the void camp itself:
field-level forward modeling of the CosmicFlows-4 velocities
prefers voids smaller than \(\sim\!70\)~Mpc, and the profiles that
help \(H_0\) most contradict the measured bulk flow
\citep{Stiskalek2025Void}; supernova smoothness independently
excludes \(|\delta| > 0.2\) voids at \(4\)--\(5\sigma\)
\citep{Kenworthy2019}. The \emph{full-tension} void --- \(+7.3\%\)
from underdensity alone --- requires
\(\langle\delta\rangle = -0.41\) on flow scales, a \(>9\sigma\)
fluctuation (\S\ref{sec:backreaction}): excluded in standard
structure formation, and invocable only through the fully
non-linear timescape loophole, which remains this class's honest
extreme \citep{Wiltshire2007}. The class's supporting evidence has
meanwhile sharpened in exactly the modest direction: the
\(H_0(z)\) trend now stands at \(6.4\sigma\) in DESI's uncorrelated
bins \citep{Jia2025H0z} --- at low redshift the generic signature of
\emph{some} local outflow with depth-tracking amplitude, though a
tail persisting at \(z \gtrsim 0.3\) would instead be background
evolution, adverse at F-Hubble-5; and the CF4 bulk-flow excess
is carried as contested companion evidence, its significance
estimator-dependent (\(428 \pm 108\) under the audit of
\citealp{Whitford2023Flows}).

\paragraph{What \GGtheory{} offers.} The weakest viable member of
the class, derived rather than invoked: sign unconditional, size
bounded by the same linear theory that excludes its deep cousins,
profile-diluted, flow-correction-guarded, and worth exactly the
\(+2\)--\(3\%\) the isotropy data allow (\S\ref{sec:hole-course}).
The framework's one exotic local channel, the \(\Uoneb\)
graviphoton, is recorded as excluded by eight orders of magnitude.

\paragraph{Siren verdict required.} \(\approx67\): a void biases
only the observers inside it, and a catalog reaching well beyond
it reads the true background. The class's quantitative apparatus
--- profiles, dilutions, rarity bounds --- is Part~IV's own, and it
caps the pure-void explanation far below the requirement.

\section{The siren discriminator: one measurement, three worlds}
\label{sec:siren-discriminator}

The class-by-class verdicts assemble into the sharpest table in
this Part --- now with three columns of fate, because the landscape
has three bands. The classes disagree, at bottom, about a single
question: \emph{where does the true background rate sit?} Every
early-side fix needs it high --- the tension is real physics and
the ladder was right. Every measurement-side reading needs it low
--- the ladder is the displaced instrument. And the landscape's
new middle band forces a third possibility onto the table: a true
rate near \(70\), which no class predicted outright and several
could partially absorb. The gravitational-wave standard siren
reads the rate with neither a sound horizon nor a ladder
(\S\ref{sec:siren-principle}), and its catalogs are already
walking: \(76.6^{+13.0}_{-9.5}\) at GWTC-4.0, \(71.0^{+9.0}_{-7.1}\)
at GWTC-5.0, \(69.9^{+4.1}_{-4.0}\) with the O4a dark sirens
\citep{LVK2025GWTC4,LVK2026GWTC5,Alfradique2026Sirens} --- so one
converged catalog still bisects --- now trisects --- the entire
solution space (Table~\ref{tab:siren-verdicts}).

The three worlds, spelled out. \emph{At \(\approx67\):} the derived
reading and its Class-1 kin are confirmed; every early-side fix
retires at once; the residual anatomy of Part~IV graduates from
hypothesis to history. \emph{At \(\approx73\):} the derivation dies
in public (F-Hubble-2), the anatomy evaporates, and Classes~2--4
revive with the ladder vindicated. \emph{At \(\approx70\):} the
subtlest world, and the one the middle band now makes live ---
the H0DN centroid falls, the corrected-ladder cluster is
vindicated, EDE and modified recombination survive as band-2
physics but lose the tension they were built to solve; for this
paper the lifted background \(68.5 \pm 0.6\) survives at
\(0.6\)--\(1.0\sigma\) while the pure derived \(66.6 \pm 0.2\)
is strained at \(1.5\)--\(2.4\sigma\) (the exact arithmetic, at
the two verdict thresholds, is \S\ref{sec:siren-three-worlds}) ---
a partial confirmation this paper would report as exactly that, not
as victory. The stake does
not soften: the pre-registered rule of Part~VII scores the
catalog against \(66.6\), and the honesty of the three-world
table is that the middle world is uncomfortable for everyone,
this reading included.

\begin{table}[htbp]
\centering\small
\caption[The siren discriminator, three ways.]{The siren discriminator, three ways. Each class's required
location of the true background rate and its fate under a converged
siren catalog at \(\approx67\), \(\approx70\), or \(\approx73\).
A \(67\) retires every early-side fix at once; a \(73\) retires
every measurement-side reading, this paper included; a \(70\)
vindicates the corrected middle, strains the pure derived rate at
\(1.5\)--\(2.4\sigma\), and demotes the early-side fixes to
band-2 physics. The derived reading is the only entry whose low bet is a
derivation rather than a preference.}
\label{tab:siren-verdicts}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}p{2.9cm}p{2.35cm}p{2.35cm}p{2.0cm}p{3.4cm}p{1.75cm}@{}}
\toprule
\textbf{Class} & \textbf{True background} & \textbf{Needs} & \textbf{If \(67\)} & \textbf{If \(70\)} & \textbf{If \(73\)}\\\midrule
\(\LCDM\) + systematics (named) & low & \(\approx67\)--\(68\) & confirmed & partly confirmed & dies\\
Pheno.\ EDE & high & \(\approx73\) (caps: \(\sim71\)) & dies & survives as band-2 & vindicated\\
Modified recombination & high & \(\approx73\) (cap: \(70.7\)) & dies & survives as band-2 & strained by cap\\
Interacting DS / \(\DNeff\) & high & \(\approx73\) & dies & squeezed & vindicated\\
Mod.\ gravity / evolving DE & either & either + duality & constrained & constrained & constrained\\
Local void (modest) & low & \(\approx67\)--\(68\) & confirmed (capped) & partly confirmed & dies\\
\GGtheory{} (derived) & low --- \emph{derived} \(66.6\) & \(\approx67\) (F-Hubble-2) & confirmed & lifted \(68.5\): \(0.6\)--\(1.0\sigma\); pure \(66.6\): \(1.5\)--\(2.4\sigma\) & dies\\
\bottomrule
\end{tabular}
\end{table}

Three remarks complete the point. First, the discriminator remains
total: whichever of the three worlds the catalog lands in, most
of the standing literature retires or is demoted in one
measurement --- a rarity in a field where proposals usually fail
slowly. Second, modified gravity is still the one class the rate
alone cannot bisect, and it compensates by exposing itself twice
through the GW--EM duality. Third, the derived reading's row
differs from its allies in kind, not in verdict: Classes~1 and~6
\emph{prefer} a low rate; the geometry \emph{computed} one, in
writing, before the catalog that will check it exists. Sharing a
bet is not sharing a claim.

\section{Comparison summary}
\label{sec:comparison-summary}

Table~\ref{tab:compare6} scores the six classes and the derived
reading against the standing constraints;
Table~\ref{tab:class-status-2026} adds the column the 2026
landscape itself supplies --- each class's status after the audits
of \S\ref{sec:landscape-early}, in one line each.

\begin{table}[htbp]
\centering\small
\caption[Six classes of Hubble-tension fix versus \GGtheory{}.]{Six classes of Hubble-tension fix versus \GGtheory{}.
$\checkmark$ = achieved; $\times$ = not achieved; $\sim$ =
partial; $^{*}$ = with a GW--EM distance-duality signature
(\S\ref{sec:siren-discriminator}). No column here is clean, including the last one. The rows are ordered so that the ones a reader is most likely to care about --- the acoustic fit, the deuterium abundance, the \(r_s\)-free readings --- come before the rows that favor this framework, and the final row is the one that will settle the question without any of the arguments above it.}
\label{tab:compare6}
\begin{tabularx}{\textwidth}{@{}p{3.4cm}p{1.1cm}p{1.1cm}p{1.1cm}p{1.1cm}p{1.3cm}p{1.5cm}p{1.5cm}@{}}
\toprule
\textbf{Feature} & \textbf{$\LCDM$ +system.} &
\textbf{Pheno. EDE} & \textbf{Mod. recomb.} & \textbf{Int. DS /$\DNeff$} &
\textbf{Mod. grav. /DDE} & \textbf{Loc. void} & \textbf{\GGtheory}\\
\midrule
Accounts for the local $\sim 73.5\kmsMpc$ &
$\sim$ & $\sim$ & $\times$ & $\checkmark$ & $\sim$ &
$\sim$ & $\sim$\\
Preserves CMB acoustic fit & $\checkmark$ & $\sim$ & $\sim$ &
$\sim$ & $\sim$ & $\checkmark$ & $\checkmark$\\
Does not worsen $\SEight$ & $\checkmark$ & $\times$ &
$\sim$ & $\sim$ & $\times$ & $\checkmark$ & $\sim$\\
Consistent with BBN, D/H & $\checkmark$ & $\sim$ &
$\times$ & $\times$ & $\checkmark$ & $\checkmark$ & $\checkmark$\\
Consistent with BAO (incl.\ DESI~DR2) & $\checkmark$ &
$\sim$ & $\times$ & $\sim$ & $\sim$ & $\checkmark$ &
$\checkmark$\\
Survives the $r_s$-free readings & $\checkmark$ & $\times$ &
$\times$ & $\times$ & $\sim$ & $\checkmark$ & $\checkmark$\\
Consistent with solar-system tests & $\checkmark$ &
$\checkmark$ & $\checkmark$ & $\checkmark$ & $\sim$ &
$\checkmark$ & $\checkmark$\\
Integer-level inheritance / no free fit &
$\times$ & $\times$ & $\times$ & $\times$ & $\times$ &
$\times$ & $\sim$\\
Formulated independently of the H0DN analysis &
$\sim$ & $\sim$ & $\sim$ & $\sim$ & $\sim$ &
$\sim$ & $\checkmark$\\
Cross-ledger reach (particles, biology, mind) &
$\times$ & $\times$ & $\times$ & $\times$ & $\times$ &
$\times$ & $\checkmark$\\
Single falsifier retires the whole theory &
---  & $\times$ & $\times$ & $\times$ & $\times$ &
$\times$ & $\checkmark$ (proton decay)\\
Standard-siren verdict required (\S\ref{sec:siren-discriminator}) &
$\approx67$ & $\approx73$ & $\approx73$ & $\approx73$ & either$^{*}$ &
$\approx67$ & $\approx67$\\
\bottomrule
\end{tabularx}
\end{table}

\begin{table}[htbp]
\centering\small
\caption[Status of each class after the 2026 landscape: the one-line verdict the audits.]{Status of each class after the 2026 landscape: the one-line
verdict the audits of \S\ref{sec:landscape} impose, independent of
any preference. The verdicts are the 2026 record's and not this paper's. Each is a one-line compression of an audit given in full in the section named above, and each was written to be readable by someone who prefers the class it describes. The framework's own row is included on the same terms and is not the strongest entry in the table.}
\label{tab:class-status-2026}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
\textbf{Class} & \textbf{Status after the 2026 landscape}\\\midrule
\(\LCDM\) + systematics & Promoted from shrug to named hypothesis: three published channels, quadrature \(4.3\%\), central values reach band~2; strained by the geometric counterweight\\
Phenomenological EDE & Two-sided (ACT null vs.\ profile-likelihood \(f_{\rm EDE} = 0.09 \pm 0.03\)); capped below the ladder by the \(r_s\)-free span; a band-2 machine\\
Modified recombination & Own ceiling \(H_0 < 70.7\) (95\%); deuterium and DESI \(\Omega_m\) audits; \(m_e\) variant fails DR2; PMF hint at \(1.8\sigma\)\\
Interacting DS / \(\DNeff\) & Radiation limb refused by BBN (\(\DNeff \gtrsim 0.4\) needed, \(0.04 \pm 0.18\) measured); coupled limb unmotivated parameters; \(r_s\)-free wall\\
Mod.\ gravity / evolving DE & DESI \(w_0w_a\) hint (2.8--4.2\(\sigma\), contested) points \emph{down} --- wrong sign for the tension; duality test pending\\
Local void & Modest (\(+2\)--\(3\%\)) alive and adopted; deep disfavored by the void camp's own velocities; full-tension void at \(>9\sigma\) excluded\\
\GGtheory{} (derived) & Derived \(66.6\) sits at band~1's floor under the newest single-instrument value; lifted \(68.5\) at band~2's floor, one error bar from the corrected-ladder cluster\\
\bottomrule
\end{tabularx}
\end{table}

The key distinction in the last three rows of
Table~\ref{tab:compare6} is structural. The six classes either
carry parameters fitted after the fact, or are single-purpose
cosmology extensions with no cross-domain reach, or have no unique
falsifier that retires the entire framework. \GGtheory{} is the
only candidate listed in which the numbers driving the Hubble
sector are the same integers that drive the SU(5) charge algebra
and the 4-letter DNA alphabet --- with the honest qualification,
encoded by the \(\sim\) entries in its column, that the derived
reading accounts for the local \(73.5\) only as far as the derived
lift reaches (\(\approx68.5\)) and names the remaining
\(\approx\!7\%\) as an open calibration residual with published,
kill-testable candidate anatomy rather than producing it
(\S\ref{sec:residual-anatomy}).

\section{Why ``uniqueness'' means more than ``agreement''}
\label{sec:uniqueness}

The H0DN paper itself notes (Section~6 of
\citealt{CasertanoH0DN2026}) that the local direct measurement of
$\Hzero$ is ``the most accurate direct $\Hzero$ measurement to
date, with a relative uncertainty of $1.09\%$, including systematic
uncertainties.'' Any theory that merely \emph{agrees} with some
row of the landscape is already on a crowded shelf --- the chart
offers forty-five numbers to agree with. The uniqueness of the
\GGtheory{} reading lies elsewhere:

\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item \textbf{Derived anchor.} The background rate is not fitted to
any row of the landscape: the $\pi$-locked fractions, fixed by the
integer ledger $(1,3,5;7,35)$ independently of any cosmological
measurement, deliver $66.6$ twice over --- once from geometry alone
(Section~\ref{sec:absolute-h0}). No competing framework derives its
background rate; all fit it to some band.
\item \textbf{Environmental lift.} The local excess above the derived
background is a Local-Hole environmental effect whose sign and rough
size follow from the observed underdensity (Appendix~\ref{app:epslocal})
--- and whose sum, $68.5 \pm 0.6$, marks the floor of the band the
corrected measurements are now filling.
\item \textbf{Forced cross-domain tests.} The same integers
forced a long list of other predictions (SM parameters, Kaxion
doublet, $\Omega_m=1/\pi$, $\bar\theta\lesssim 10^{-18}$, DNA
4-letter alphabet, conscious episode $\sim 400$~ms), none of which
could have been tuned after the fact without breaking the Hubble
sector's match.
\item \textbf{Sharp single falsifier.} Proton decay retires the
entire program, including the $\epslocal$ piece. No comparable
unique falsifier exists for the other six classes.
\end{enumerate}

The landscape is therefore a very specific kind of test for
\GGtheory{}: not whether it can fit a new data point --- it fits
none of them, in the fitting sense --- but whether the
\emph{already committed} integer ledger stays coherent as the
measurement world reorganizes itself around it. Three bands later,
it does.

\section{Summary: the landscape, the rivals, and the road to Part VI}
\label{sec:partV-summary}

Part~V set out to put the derived reading in front of all the
evidence and all its rivals at once, and this closing section
gathers what it found.

The evidence first (\S\ref{sec:landscape}). The 2026 measurement
landscape is not two camps but three bands, and the paper now
carries all of it: forty-five audited determinations, drawn
(Figure~\ref{fig:h0-landscape}) and tabulated
(Table~\ref{tab:h0-landscape}) with their identifiers, dates, and
soft spots. Band~1 is overdetermined thermodynamics --- three
instruments and a CMB-free ruler audit agreeing because absolute
densities leave nothing to conspire about --- with the newest
single-instrument value, SPT-3G's \(66.66 \pm 0.60\), sitting on
the derived rate. Band~2 is a destination, not a camp: corrected
ladders, camp-free probes, and the ladder-free mean drain into
\(69\)--\(71\) from both sides. Band~3 is two populations --- one
calibration family of formidable internal consistency, and the
one-step geometric probes that are the standing counterweight to
every bias reading, this paper's included, and are carried here at
full strength (\S\ref{sec:landscape-counterweight}).

The rivals second (\S\ref{sec:class1-systematics}--\S\ref{sec:class6-voids}).
Read against the landscape, the six classes sort cleanly. The
systematics class has been promoted from shrug to named hypothesis
--- the paper's own Part~IV supplied the pricing --- and strains
only at the geometric counterweight. The early-side fixes are
capped by walls their own machinery erected: EDE by the
\(r_s\)-free span, modified recombination by its own \(70.7\)
ceiling and the deuterium audit, extra radiation by BBN. Evolving
dark energy points the wrong way. The deep void is disfavored by
its own camp's velocity data, while the modest void this paper
adopts is alive and now carries a \(6.4\sigma\) signature trend.
None of the six reaches \(73.5\) any more without contradicting an
audit; most reach exactly the band the corrected measurements are
filling --- which is why the middle world had to be added to the
siren table.

The verdict machinery third (\S\ref{sec:siren-discriminator}).
One catalog now trisects the field: \(67\) confirms the derived
reading and retires the early fixes; \(73\) kills this paper as
designed; \(70\) --- the world the middle band makes live ---
vindicates the corrected ladder, demotes the early fixes to
band-2 physics, and leaves this reading partially confirmed at
its lifted value and honestly strained at its pure one. Every
class's requirement is printed; nobody gets to renegotiate after
the flyby.

What remains is the verdict's schedule and machinery: the kill
switches, the dated decision calendar, and the siren forecast ---
the business of Parts~VI and~VII. The landscape is on the table;
the clocks are running.


\section{Part~V in plain English: all forty-five measurements on one chart, and the six ways people have tried to close the gap}
An argument about two numbers can be won by choosing which two. This
Part refuses that move by putting every audited determination on a single
chart --- forty-five of them, from nine families of method, each read from
its own primary record rather than from a review.

What the chart shows is that the tidy picture of two camps is gone. There
are three bands and not two, and the middle band is populated by recent
reanalyses that neither camp asked for. That is itself a fact any
explanation now has to accommodate, and several of the standard
explanations were designed for a world with only two camps in it.

The six classes of fix are then lined up against the same evidence:
systematics in the ladder, early dark energy, modified recombination,
extra radiation and interacting dark sectors, modified gravity with
evolving dark energy, and a local void. Each is judged against audits that
were not run for this paper's benefit --- the deuterium abundance, the
readings that need no sound horizon, the void camp's own velocity data.
Each class runs into a wall its own machinery built. The systematics
reading comes out of this Part considerably stronger than it went in, and
it should be said plainly that it is a rival to this paper rather than an
ally of it.

Nothing is settled here, and nothing could be. What emerges is that one
future measurement now separates the survivors from one another cleanly,
which is the business of the two Parts that follow.

\GGPart[Kill switches and the decision calendar]{Kill Switches, the Decision Calendar, and Wider
Implications}{The eleven named falsifiers with their pre-declared
rules; the dated decision calendar 2026--2035 and what the paper
commits to at each date; and what the H0DN measurement means for the
broader \GGtheory{} program}

\section{The falsifier registry: F-Hubble-1 through F-Hubble-11}
\label{sec:falsifiers}

A responsible interpretation of a new measurement must commit, in
advance, to what would refute it. The \GGtheory{} reading of the H0DN
result carries eleven named falsifiers, in four tiers --- three at the
program level, which retire the whole framework; four at the
parameter-block level, which retire the Hubble corridor interpretation
alone; three at the channel level, which settle the individual
residual-anatomy channels of \S\ref{sec:residual-anatomy} (the dust
law, the calibration priors, the subsample stability) without touching
the framework either way; and one at the coefficient level, which
cashes the codebook selection of \S\ref{sec:zeta-selection} against a
computation that does not yet exist. Read operationally, and counted
the way a referee would: four entries are direct kills of the derived
reading (F-Hubble-2, -3, -5, -6); three adjudicate \emph{other
groups'} corrections either way without touching the derivation
(F-8/9/10); two are pressure gauges with named consequences short of
death (F-4, F-7); one is a theory window with no observation date
(F-11); and one is the cohort's standing single-event kill (F-1).
Eleven registered rules; four gallows. We print the distinction so
the registry's breadth is never mistaken for its depth. Each is stated in the cohort's
\textsf{F-Hubble-\(N\)} convention with a \emph{pre-declared
numerical decision rule}: a threshold fixed here, before the data, so
that the verdict cannot be negotiated after the fact. The tiers are
not equal in what they stake: the channel-level trio adjudicates
\emph{external} allowances, quoted from the field's own literature and
never promoted to framework claims, so firing one of them reshapes the
residual's anatomy rather than the derivation --- and the registry says
so explicitly, entry by entry. We list all eleven, and draw the dated
ones on the decision calendar of \S\ref{sec:decision-calendar}.

\subsection{Program-level kill tests}

\paragraph{F-Hubble-1. Proton decay.} Observation of a single unambiguous
proton-decay event at any rate retires \GGtheory{} in its entirety;
the proton's own account within the framework is \PaperPSix{}
\citep{ArisakaP6Proton}.
The exact local $\Uoneb$ of the \GGsix{} stage is the structural
source of $\epslocal$. Proton decay would retire $\epslocal$ along
with the entire program. Current Super-Kamiokande, Hyper-K, JUNO,
and DUNE searches are relevant.

\paragraph{F-Hubble-2. A standard-siren \(H_0\) near 73.}
The sharp, model-independent prediction of the derived reading is that
the gravitational-wave standard siren --- which uses neither a sound
horizon nor a distance ladder --- converges near \(67\)
(Section~\ref{sec:both-from-anchor}). A siren determination that
settled instead near \(73\), with adequate statistics, would mean the
true background rate is high, the CMB densities cannot be divided by
GG's fractions to give it, and the derivation of
Section~\ref{sec:absolute-h0} is wrong. \emph{Pre-declared rule --- one decision function, defending one
hypothesis.} The hypothesis this rule defends is the \emph{background}
\(66.6 \pm 0.2\), the object the geometry derives; a deep siren
catalog reads the background (its hosts sample far beyond the Local
Hole; OQ-Hubble-7), so the conditional lifted reading is \emph{not}
defended here --- its test is assigned to shallow bright subsets
(\(z \lesssim 0.05\)) and pre-registered separately in Part~VII. For
a converged deep catalog of central value \(\mu\) and precision
\(\sigma_{\rm cat} \le 2.3\;\kmsMpc\): \emph{confirmed} if \(\mu\)
lies within \(2\sigma_{\rm cat}\) of \(66.6\); \emph{refuted} if
\(\mu \ge 71.5\) with \((\mu - 66.6)/\sigma_{\rm cat} \ge 3\);
and for \(\mu\) in the pre-declared \emph{middle world}
(\(68\)--\(71.5\)), the pure background is strained at the printed
\((\mu-66.6)/\sigma_{\rm cat}\) and the adjudication passes, by this
rule, to the sub-percent inventory determinations and the shell test
(F-Hubble-4) --- no renegotiation, the handoff is part of the rule.
And one rung so that precision cannot hide inside the middle world:
a deep catalog converging at \emph{any} \(\mu\) with
\((\mu - 66.6)/\sigma_{\rm cat} \ge 5\) \emph{and} a host-shell
profile flat in depth (no gradient for a bias to live in) refutes
the background directly --- a flat \(5\sigma\) miss leaves no
channel, and the rule says so now.
One honest clause the confirmation arm must carry: a catalog at
\(\approx\!67\) confirms every low-rate reading at once ---
\(\Lambda\)CDM-plus-ladder-systematics included; distinguishing the
derived \(66.6\) from the CMB's \(67.2\)--\(67.4\) needs
\(\sigma_{\rm cat} \approx 0.2\)--\(0.4\), beyond every scenario
priced in Part~VII. The kill is uniquely this reading's exposure; the
confirmation is shared, and we say so. The coming decade of catalogs
will execute the rule; the full forecast --- principle, pipelines,
systematic budget, and timetable --- is Part~VII. A precision CMB bound tightening \(\DNeff\) toward the cohort value
\(0.04\) \emph{confirms} the framework rather than threatening it; the
early side carries no large radiation excess to be excluded.

\paragraph{F-Hubble-3. A detected early-dark-energy lobe.}
The derived reading requires \emph{no} early modification: the
acoustic spectrum is fit by the $\pi$-locked fractions with the
standard recombination history untouched. A joint
Planck+SPT+ACT+DESI \emph{detection} of a genuine early-dark-energy
lobe --- $\fEDE \gtrsim 0.02$ at $\ge 3\sigma$ near $z_c \sim 3500$ ---
would establish exactly the early-side new physics this reading says
is absent, and refute the claim that the CMB endpoint is the unbiased
derived background. \emph{Pre-declared rule:} a converged detection
at that level refutes; a tightening null confirms.

\subsection{Parameter-block-level kill tests}

These tests retire the specific $\Hzero$ corridor interpretation
without automatically retiring the full program.

\paragraph{F-Hubble-4. The local excess vanishing entirely.}
The derived reading attributes the local excess to a Local-Hole
environmental lift plus a distance-ladder calibration residual. If a
future distance-ladder analysis, with the environmental and calibration
effects controlled, converged on the derived background $\approx 67$
with no residual excess, the local-lift mechanism would be shown
unnecessary --- though the derived rate itself, fixed by the CMB and
sirens, would stand.

\paragraph{F-Hubble-5. Detection of $w \ne -1$.} The \ATen{} prediction
is $w=-1$ exactly, forced by the \GSSB{} cap geometry. A detection
of $w$ different from $-1$ at the $3\sigma$ level by DESI~Y5,
Euclid, or Roman would retire the interpretation that $\rho_\Lambda$
is the throat residue, and with it the exact-$w$ leg of the derived
expansion history.

\paragraph{F-Hubble-6. A primordial tensor amplitude inconsistent with $r = 0.0043$.}
The \ATen{} prediction is the derived two-sided value $r = 12/N^{*2} =
0.0043$ from the radion (Starobinsky/$\xi$) plateau. A Simons
Observatory + LiteBIRD measurement inconsistent with $r \simeq 0.004$ \emph{in either direction}
--- a detection well above it, or a null pushing $r$ below $\sim 10^{-3}$ ---
would retire the interpretation of the early sector that anchors the
derived history's inflationary end.

\paragraph{F-Hubble-7. Kaxion twin-peak absent at $\Delta\nu/\nu=1/35$.}
The same compact-angle sector that contributes $\DNeff$ is
predicted to show a characteristic twin-peak in the Kaxion dark
matter spectrum at $\Delta\nu/\nu = 1/35$
\citep{ArisakaC1Kaxion}. Haloscope searches (ADMX, HAYSTAC, DMRadio)
that probe the unique laboratory window $\nu_K \in [1.59, 2.13]$~GHz
--- the discrete $\Xi_{\rm cap}$ ladder, $m_K \in [6.6, 8.8]~\mu$eV, with the
central prediction $8.1 \pm 0.5~\mu$eV ($1.95$~GHz) --- should either
find the twin-peak signature at this specific ratio or refute
that ratio. Refuting it would pressurize the $\UoneQi$
interpretation of $\DNeff$.

\subsection{Channel-level kill tests: the residual anatomy}
\label{sec:channel-kill-tests}

These three tests settle the named external channels of
\S\ref{sec:residual-anatomy} one at a time. They are deliberately
graded below the parameter-block tier: every one of them adjudicates a
\emph{cited allowance} --- a correction proposed inside the
distance-ladder literature itself, never a \GGtheory{} result --- so
each verdict moves the anatomy of the \(+7.3\%\) residual, not the
derived rate. Their honesty value lies in the symmetry: each is
two-sided by construction, and the confirming outcome is printed with
the same prominence as the killing one.

\paragraph{F-Hubble-8. The near-infrared ladder and the dust channel.}
Dust nearly vanishes in the near-infrared, so a NIR supernova ladder
strips away the dust-law freedom that the channel of
\S\ref{sec:residual-anatomy} prices at \(2\)--\(4\%\); and the
accumulating ZTF/Rubin sibling samples test the two-population split
directly, with statistics the calibrator set cannot match.
\emph{Pre-declared rule:} a published NIR/JWST supernova ladder, with
the dust-law freedom removed, still converging at \(73\) kills the
dust channel outright; the same ladder landing at \(70\)--\(71\) ---
where the published re-analyses put it (\(70.5\pm1.0\),
\(70.59\pm1.15\)) --- confirms the channel and collapses that share
of the tension. A ladder converging in the gap, \(71.5\)--\(72.5\),
adjudicates nothing: the channel stays priced and undecided, and we
say so now rather than negotiate it later. Channel-level only: either
outcome leaves the derived rate untouched.

\paragraph{F-Hubble-9. Injection--recovery and the statistics channel.}
The distance-prior channel is priced at \(0\)--\(3\%\) and carried
as \emph{proposed, executed, and rebutted --- undecided}. The test
that settles it is cheap, decisive, and not yet published:
injection--recovery simulation of the full ladder pipeline on
synthetic data with known truth. \emph{Pre-declared rule:} a published
injection--recovery campaign validating the ladder's priors at
\(<1\%\) bias kills the statistics channel and zeroes its allowance;
a demonstrated prior-induced bias at the \(2\)--\(3\%\) level
confirms it; a demonstrated bias of \(1\)--\(2\%\) is a partial
confirmation --- the channel's allowance shrinks to the demonstrated
size, no more. Channel-level only; whoever runs the simulation settles
the channel either way.

\paragraph{F-Hubble-10. The complete JWST sample and the subsample channel.}
The subsample channel (\(1\)--\(2\%\)) lives in which calibrator
hosts enter the fit: the same JWST-era data read \(72.6\pm2.0\)
under the SH0ES joint three-method analysis and \(70.39\pm1.94\) in
the CCHP program. \emph{Pre-declared rule:} the complete JWST
coverage of all \(\sim\!37\) SH0ES calibrator hosts, reduced under a
tip/JAGB standardization convention agreed by both groups in advance,
confirming \(73.5\) kills the subsample channel; a complete-sample
value near \(70.4\) confirms the channel and locates
\(\sim\!2\;\kmsMpc\) of the tension in calibrator selection. And the
single most probable outcome deserves its sentence in 2026 rather
than a negotiation in 2028: a complete sample near the SH0ES joint
reading \(72.6 \pm 2.0\) adjudicates \emph{nothing} --- the two
conventions would still disagree inside their errors, and the channel
would stay open, undecided, at its printed price.
Channel-level only; the data are already being taken.

\subsection{The coefficient-level test}

\paragraph{F-Hubble-11. The vacuum coefficient off its pre-registered target.}
The codebook selection of \S\ref{sec:zeta-selection} commits the
framework to a number before the computation exists: the Wilson-line
(Hosotani) evaluation of the vacuum residue on \(S^1_\theta\)
(\AFive{}, Open Problem O6), when completed and checked, must return
\(\zeta_\Lambda = \pi/(LK) = \pi/175 = 0.017952\) to within the
structural tolerance (\(\sim\!1\%\); equivalently \(H_0\) within
\(\pm0.2\) of \(66.8\)). \emph{Decision rule:} a completed computation
returning any other cluster member --- or any value outside
\([0.0177,\,0.0182]\) --- refutes the selection lemma and demotes the
geometric route from a point back to the bracket of
Eq.~\eqref{eq:c2-cluster-bracket}; a value inside confirms the one
remaining conditional link in the derived rate. The falsifier is
two-sided by construction: the target is printed here, before the
lemma, and cannot be renegotiated after it.

\subsection{Kill-criteria summary}

Table~\ref{tab:killcriteria} compiles these tests.

\begin{table}[htbp]
\centering\small
\caption[Kill criteria for the \GGtheory{} reading of the H0DN measurement, in four tiers.]{Kill criteria for the \GGtheory{} reading of the H0DN
measurement, in four tiers. Program-level tests (F-Hubble-1--3)
retire the entire framework; parameter-block tests (F-Hubble-4--7)
retire the Hubble corridor interpretation while leaving other sectors
open; channel-level tests (F-Hubble-8--10) settle the residual
anatomy's individual external channels either way, reshaping the
residual rather than the derivation; and the coefficient test
(F-Hubble-11) cashes the codebook selection against a computation that
does not yet exist.}
\label{tab:killcriteria}
\begin{tabularx}{\textwidth}{@{}p{1.9cm}p{4.0cm}XX@{}}
\toprule
\textbf{Tag} & \textbf{Pre-declared decision rule} &
\textbf{Facility or dataset} &
\textbf{Structural target}\\\midrule
F-Hubble-1 & Observation of proton decay at any rate & Hyper-K, JUNO, DUNE
& Exact local $\Uoneb$; retires whole program\\
F-Hubble-2 & Converged standard-siren $\Hzero$ near $73$ ($>3\sigma$ from $67$) & LVK O5, dark-siren catalogs & Derived background rate; the whole reading\\
F-Hubble-3 & Detected EDE lobe $\fEDE \gtrsim 0.02$ at $\ge3\sigma$ & Planck+SPT+ACT,
DESI~Y5 & The no-early-modification claim\\
F-Hubble-4 & Controlled ladder converging on $\approx 67$ (excess gone) & H0DN updates, JWST/CCHP & Local-lift mechanism only\\
F-Hubble-5 & Detection of $w \ne -1$ at $3\sigma$ & DESI~Y5, Euclid, Roman
& Throat-cap $\rho_\Lambda$ interpretation\\
F-Hubble-6 & Primordial tensors $r \gtrsim 0.01$ & Simons Obs., LiteBIRD &
Radion-plateau inflation\\
F-Hubble-7 & Kaxion signal at $\Delta\nu/\nu \ne 1/35$ & ADMX, HAYSTAC,
DMRadio & Compact-angle sector\\
F-Hubble-8 & NIR/JWST SN ladder (dust freedom removed) still at $73$ &
JWST NIR programs; ZTF/Rubin siblings & Dust/two-population channel
($2$--$4\%$); channel-level\\
F-Hubble-9 & Injection--recovery validating ladder priors at $<1\%$
bias & full-pipeline simulation on synthetic truth &
Distance-prior/statistics channel ($0$--$3\%$); channel-level\\
F-Hubble-10 & Complete $\sim$37-host JWST sample (agreed convention)
confirming $73.5$ & JWST (SH0ES + CCHP hosts) & Calibrator-subsample
channel ($1$--$2\%$); channel-level\\
F-Hubble-11 & Hosotani computation returns $\zeta_\Lambda$ outside $[0.0177, 0.0182]$ & theory (\AFive{}~O6), pre-registered target $\pi/175$ & The codebook selection; Route I as a point rather than a bracket\\
\bottomrule
\end{tabularx}
\end{table}

The existence of F-Hubble-1 as a unique single-event kill criterion is
the strongest distinguishing feature of \GGtheory{} as a
cosmological framework. No other proposed resolution of the
Hubble tension can be falsified by a single particle-physics
observation.

\section{The decisive observation: how the tension will be settled}
\label{sec:decisive}

Eleven falsifiers are listed above, but they are not of equal weight. One of
them settles the central question --- is the true background rate \(67\) or
\(73\)? --- by itself, and it does so without a sound horizon and without a
distance ladder. That is the gravitational-wave standard siren
(F-Hubble-2). A binary-neutron-star or dark-siren catalog reads the
luminosity distance straight off the waveform and the redshift off the host,
so it measures \(H_0\) through a channel that shares \emph{no} systematic with
either of the two discordant methods. The derived reading stakes everything
on its verdict: \(H_0^{\mathrm{GW}} \to 67\), and a converged value at \(73\)
refutes the whole construction (\S\ref{sec:falsifiers}).

Around that decisive test sits a near-term program that will sharpen the
picture well before the sirens converge. On the \emph{early} side, a precision
CMB determination (the Simons Observatory, then its enhanced
large-aperture surveys) that drives the radiation
excess toward the cohort value \(\Delta N_{\mathrm{eff}} \approx 0.04\)
\emph{confirms} the derived reading rather than threatening it, because the
early side carries no large excess to excise; and a joint CMB+BAO search for an
early-dark-energy lobe (F-Hubble-3) tests the no-early-modification
claim directly. On
the \emph{late} side, the distance-ladder program itself --- continued H0DN
updates, and the JWST cross-checks of the Cepheid and TRGB calibrations
(CCHP) --- will show whether the local excess is stable or drifts toward the
background as environmental and calibration effects are controlled
(F-Hubble-4). And the dark-energy equation of state (F-Hubble-5, DESI\,Y5,
Euclid, Roman) and the primordial tensor ratio (F-Hubble-6, Simons Obs.,
LiteBIRD) test the two ends of the same inventory the rate is read from.

These tests do not each carry a date so much as a data era, and the
next section lays them --- together with the channel-level trio, which
\emph{does} carry near dates --- on one forward axis, entry by entry,
with what this paper has committed to at each
(\S\ref{sec:decision-calendar}). The wager is dated, and the dates
are near. Nor is the wager this paper's alone: as
\S\ref{sec:siren-discriminator} tabulates, the same converged
catalog bisects the entire solution space of Part~V --- every class
has printed, or has been forced here to print, which verdict it
needs.

\section{The decision calendar, 2026--2035}
\label{sec:decision-calendar}

A registry states rules; a calendar states dates. This section walks
the falsifiers in the order the data will actually arrive, because the
discipline of pre-registration is only as good as the specificity of
the appointment: a rule with no date can be deferred forever, and a
date with no printed number can be renegotiated when it comes. Neither
escape is available here.
Figure~\ref{fig:hubble-killtest-timeline} draws the whole schedule in
the cohort's kill-test-timeline convention --- three eras, each test
card naming its rule and venue, the violet siren band carrying the
decisive F-Hubble-2 --- and the two subsections below spell out the
entries and the commitments, date by date.

\begin{figure}[!t]
\centering
\resizebox{0.77\textwidth}{!}{%
\begin{tikzpicture}[
    font=\large, >=Latex,
    card/.style={draw=black!55, rounded corners=2pt, line width=0.60pt, align=left,
        inner sep=4pt, text width=54mm},
    stem/.style={line width=0.90pt, draw=black!55},
    sGW/.style={fill=violet!12, draw=violet!65!black},
    sLAD/.style={fill=blue!8, draw=blue!55!black},
    sCMB/.style={fill=orange!12, draw=orange!75!black},
    sDE/.style={fill=green!10, draw=green!55!black},
    sPART/.style={fill=black!7, draw=black!55},
    dot/.style={circle, draw=black!70, fill=white, inner sep=0pt, minimum size=5.2pt}
]
\def\xL{0}\def\xR{30}\def\yA{0}
\def\bA{11}\def\bB{21}
\begin{scope}[on background layer]
    \fill[black!2] (\xL,-10.4) rectangle (\bA,9.6);
    \fill[black!4] (\bA,-10.4) rectangle (\bB,9.6);
    \fill[black!6] (\bB,-10.4) rectangle (\xR,9.6);
\end{scope}
\node[anchor=south, font=\Large\bfseries] at ({(\xL+\bA)/2},8.8) {NOW / ONGOING};
\node[anchor=south, font=\large] at ({(\xL+\bA)/2},8.15) {2026--2028: the channel-level era};
\node[anchor=south, font=\Large\bfseries] at ({(\bA+\bB)/2},8.8) {BY $\sim$2030};
\node[anchor=south, font=\large] at ({(\bA+\bB)/2},8.15) {the first verdicts};
\node[anchor=south, font=\Large\bfseries] at ({(\bB+\xR)/2},8.8) {BY $\sim$2035};
\node[anchor=south, font=\large] at ({(\bB+\xR)/2},8.15) {the decisive era};
\draw[->, line width=1.10pt] (\xL,\yA) -- (\xR+0.7,\yA);
\node[anchor=west, font=\Large\itshape] at (\xR+0.8,\yA) {time};
\foreach \x in {\bA,\bB}{\draw[black!55, line width=0.80pt] (\x,-0.18) -- (\x,0.18);}
\node[anchor=north, font=\large] at (\xL,-0.25) {2026};
\node[anchor=north, font=\large] at (\xR,-0.25) {2035+};
% ---- top rows ----
\node[dot] (d9) at (2.6,\yA) {};
\node[card, sLAD, anchor=south, text width=48mm] (c9) at (2.6,4.7) {%
    \textbf{F-Hubble-9. Injection--recovery \emph{(open now)}}\\[1pt]
    rule: priors valid at $<1\%$ kills the channel; a $2$--$3\%$ bias confirms\\
    venue: full-pipeline simulation};
\draw[stem] (d9) -- (c9.south);
\node[dot] (d2) at (6.0,\yA) {};
\node[card, sGW, anchor=south, line width=1.10pt] (c2) at (6.0,1.5) {%
    \textbf{F-Hubble-2. Standard siren \emph{(decisive)}}\\[1pt]
    rule: converged $H_0^{\mathrm{GW}}=73\pm1$ ($>3\sigma$ from $67$) refutes\\
    venue: LVK O5, dark-siren catalogs};
\draw[stem] (d2) -- (c2.south);
\draw[violet!60, line width=2.2pt] (3.0,-0.55) -- (16.5,-0.55);
\draw[violet!80!black, line width=3.6pt] (7.0,-0.55) -- (13.0,-0.55);
\node[violet!80!black, font=\large\bfseries, anchor=north] at (10.0,-0.72) {O5};
\node[violet!60, font=\large, anchor=north] at (4.6,-0.68) {siren accumulation};
\node[dot] (d3) at (14.5,\yA) {};
\node[card, sCMB, anchor=south, text width=60mm] (c3) at (14.5,4.7) {%
    \textbf{F-Hubble-3. Early-dark-energy lobe}\\[1pt]
    rule: a detected lobe $f_{\mathrm{EDE}}\gtrsim0.02$ at $z_c\sim3500$ refutes\\
    venue: Planck+SPT+ACT, DESI\,Y5};
\draw[stem] (d3) -- (c3.south);
\node[dot] (d6) at (24.0,\yA) {};
\node[card, sCMB, anchor=south, text width=56mm] (c6) at (24.0,1.5) {%
    \textbf{F-Hubble-6. Tensor ratio $r$}\\[1pt]
    rule: $r$ inconsistent with $0.0043$ either way ($r>0.01$ or $r<10^{-3}$)\\
    venue: Simons Obs., LiteBIRD};
\draw[stem] (d6) -- (c6.south);
\node[card, sPART, anchor=south, line width=1.10pt, text width=48mm] at (27.4,4.7) {%
    \textbf{THE STAKE}\\[1pt]
    derived $66.6\pm0.2$; lifted $68.5\pm0.6$\\
    vs H0DN $73.50\pm0.81$\\
    printed before the data --- no renegotiation};
% ---- bottom rows ----
\node[dot] (eg) at (0.9,\yA) {};
\node[card, sLAD, anchor=north, text width=34mm] (bg) at (1.6,-1.5) {%
    \textbf{Gaia DR4}\\[1pt]
    Dec 2026: the ladder's parallax floor read; the shift's sign settled};
\draw[stem] (eg) -- (bg.north);
\node[dot] (e8) at (9.5,\yA) {};
\node[card, sLAD, anchor=north] (b8) at (8.4,-1.5) {%
    \textbf{F-Hubble-8. NIR ladder \emph{(late 2020s)}}\\[1pt]
    rule: dust-free ladder still at $73$ kills; $70$--$71$ confirms\\
    venue: JWST NIR; ZTF/Rubin siblings};
\draw[stem] (e8) -- (b8.north);
\node[dot] (e1) at (26.5,\yA) {};
\node[card, sPART, anchor=north] (b1) at (26.5,-1.5) {%
    \textbf{F-Hubble-1. Proton decay}\\[1pt]
    rule: \emph{any} unambiguous $\Delta B{=}1$ event retires the program\\
    venue: Hyper-K, JUNO, DUNE};
\draw[stem] (e1) -- (b1.north);
\node[dot] (e4) at (4.6,\yA) {};
\node[card, sLAD, anchor=north] (b4) at (4.6,-5.3) {%
    \textbf{F-Hubble-4. Local ladder}\\[1pt]
    rule: a controlled ladder converging on $\approx67$ retires the lift\\
    venue: H0DN updates, JWST/CCHP};
\draw[stem] (e4) -- (b4.north);
\node[dot] (e10) at (11.4,\yA) {};
\node[card, sLAD, anchor=north, text width=50mm] (b10) at (11.4,-5.3) {%
    \textbf{F-Hubble-10. Complete JWST sample}\\[1pt]
    rule: $\sim$37 hosts at $73.5$ kills; $\approx70.4$ confirms\\
    venue: JWST (SH0ES+CCHP hosts)};
\draw[stem] (e10) -- (b10.north);
\node[dot] (e5) at (14.8,\yA) {};
\node[card, sDE, anchor=north, text width=56mm] (b5) at (17.2,-5.3) {%
    \textbf{F-Hubble-5. Dark energy $w=-1$}\\[1pt]
    rule: converged $w\neq-1$ at $\geq3\sigma$ refutes the throat residue\\
    venue: DESI\,Y5, Euclid, Roman};
\draw[stem] (e5) -- (b5.north);
\node[dot] (e7) at (20.8,\yA) {};
\node[card, sGW, anchor=north] (b7) at (23.2,-5.3) {%
    \textbf{F-Hubble-7. Kaxion twin-peak}\\[1pt]
    rule: a Kaxion line at $\Delta\nu/\nu\neq1/35$ presses the channel\\
    venue: ADMX, HAYSTAC};
\draw[stem] (e7) -- (b7.north);
% legend
\begin{scope}[shift={(0.3,-9.6)}]
    \node[anchor=west, font=\large\bfseries] at (0,0) {Sector:};
    \node[card, sGW, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (2.5,0) {};
    \node[anchor=west, font=\large] at (2.8,0) {GW / dark sector};
    \node[card, sLAD, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (8.2,0) {};
    \node[anchor=west, font=\large] at (8.5,0) {distance ladder};
    \node[card, sCMB, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (13.6,0) {};
    \node[anchor=west, font=\large] at (13.9,0) {CMB / early};
    \node[card, sDE, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (18.2,0) {};
    \node[anchor=west, font=\large] at (18.5,0) {dark energy};
    \node[card, sPART, text width=2.2mm, minimum height=2.2mm, inner sep=1.6pt] at (23.0,0) {};
    \node[anchor=west, font=\large] at (23.3,0) {particle};
\end{scope}
\end{tikzpicture}}
\caption[The decision calendar as a kill-test timeline for the derived Hubble reading.]{\textbf{The decision calendar as a kill-test timeline.} The dated falsifiers of \S\ref{sec:falsifiers} drawn on a forward time axis (2026--2035+, broad data-era bands rather than precise dates), in the cohort's kill-test-timeline convention. The decisive test is F-Hubble-2 (violet, bordered): a converged standard-siren $H_0$ landing at $73$ rather than $67$ refutes the derived reading outright; the violet band is the siren accumulation, with the O5 segment marked. Alongside the program- and parameter-block-level tests, the calendar carries the channel-level trio F-Hubble-8/9/10 (blue cards, 2026--2028 era), which adjudicates the residual anatomy of Part~IV either way, and the Gaia DR4 appointment (December 2026), which reads the ladder's parallax floor. The stake card (top right) collects the printed numbers that cannot move. F-Hubble-11, the coefficient test, is a theory milestone with no observation date, tracked in Table~\ref{tab:killcriteria}; F-Hubble-1 (proton decay, gray) is the single-event kill live at every moment.}
\label{fig:hubble-killtest-timeline}
\end{figure}

\subsection{The dated entries}
\label{sec:calendar-entries}

\paragraph{December 2026 --- Gaia DR4.} The nearest and sharpest
date on the calendar, and the floor of the ladder: DR4 improves the
parallaxes by \(\sqrt{t}\) (\(\times1.4\)) and --- decisively for the
anchors --- maps the bright-star zero-point, with full epoch
astrometry, that every Milky-Way calibrator inherits. The anatomy of
\S\ref{sec:residual-anatomy} carries the Gaia entry at
\(0.5\)--\(1\%\) with the sign of the induced \(H_0\) shift explicitly
unestablished (\(-38.9\pm10.3\,\mu\)as at bright \(G\), formal errors
understated); DR4 reads that entry either way. Physically this is the
purest appointment on the calendar: astrometry knows nothing of
cosmology, so whatever DR4 returns is a statement about the
measurement chain alone --- which is exactly why it can move the
anatomy and cannot move the derivation.

\paragraph{2026--2028 --- the channel-level trio.} The
injection--recovery simulation (F-Hubble-9) has no scheduled date
because it needs no telescope: it is a software campaign any group
could publish in a season, and the calendar carries it as \emph{open
now}. The complete JWST coverage of the \(\sim\!37\) SH0ES hosts
(F-Hubble-10) is in progress, with the agreed standardization
convention the real schedule risk --- the data will exist before the
convention does. The NIR ladder and the ZTF/Rubin sibling samples
(F-Hubble-8) accumulate through the late 2020s. These three
appointments are the cheapest on the calendar relative to what they
settle: together they adjudicate the full quadrature-to-stack span of
Part~IV's two scenarios (\(\approx4\)--\(9\%\)).

\paragraph{2027--2029 --- the Local-Hole census and the shell-level
\(H_0(z)\).} Euclid and DESI count galaxies straight through the Hole,
fixing its depth profile from the density side, while the
peculiar-velocity programs test the outflow the lift predicts ---
between them they tighten the \(+2\)--\(3\%\) lift window from both
ends (\S\ref{sec:hole-evidence}). And one standing entry is already on
the board: the descending \(H_0(z)\) trend reported at \(6.4\sigma\)
in DESI's uncorrelated baryon-acoustic bins \citep{Jia2025H0z},
carried with the three provenance cautions of \S\ref{sec:hole-evidence}
(the third: a high-\(z\) tail would be adverse at F-Hubble-5),
deepens or dissolves with each DESI release. A trend that sharpens as
the low shells are cut away is what a local lift looks like from
inside; a constant-bias ladder imprints no such shape.

\paragraph{2028--2030 --- the early-side precision era.} The full
SPT-3G survey, the Simons Observatory, and DESI DR3 tighten every
early-side number this paper stakes: the early-dark-energy search
(F-Hubble-3) reaches the \(\fEDE \sim 0.02\) discovery floor, the
radiation excess is driven toward the cohort's \(\DNeff = 0.04\), and
the reference combination \(\Href = 67.24 \pm 0.35\) sharpens above
the derived \(66.6 \pm 0.2\) --- which must continue to sit one
coherent \(\pi\)-offset below it, or the reading fails at its own
anchor. Alongside, DESI's \(w(z)\) program (F-Hubble-5) approaches
the \(\pm0.02\) regime where \(w = -1\) exactly is either locked in or
broken.

\paragraph{2028--2030 --- O5 and the siren accumulation.} The decisive
channel (F-Hubble-2). The O5 observing run is the first with a
realistic path toward a converged percent-class siren value, and the
dark-siren catalogs sharpen meanwhile --- the current best,
\(69.9^{+4.1}_{-4.0}\), already sits on the lifted background. The
timeline's violet band carries this accumulation; Part~VII prices it
event by event, pipeline by pipeline.

\paragraph{The 2030s --- the tensor verdict.} LiteBIRD (launch \(\sim\)2032; the full-survey
\(\sigma(r) \approx 10^{-3}\) arrives toward the decade's end) and the
Simons Observatory's enhanced large-aperture surveys read \(r\) at the
\(10^{-3}\) floor. The ground-based partner originally written into this
test, CMB-S4, is no longer available: DOE and NSF ended their support in
July 2025, and the prediction now rests on the two programs above. The prediction is two-sided (F-Hubble-6):
\(r = 0.0043\), and data inconsistent with it in either direction
retire the early-sector anchor of the derived history. The honest
date is later than the tension literature's habitual ``early 2030s,''
and this calendar says so rather than inheriting the habit.

\paragraph{Standing entries, undated.} Proton decay (F-Hubble-1) is
live at every moment in Hyper-K, JUNO, and DUNE --- the single-event
kill. The Kaxion window (F-Hubble-7) is open to any haloscope that
reaches \(1.59\)--\(2.13\)~GHz. And the one theory milestone,
F-Hubble-11, waits on the Hosotani computation: no observation can
advance it and no date binds it, but its target ---
\(\zeta_\Lambda \in [0.0177,\,0.0182]\) --- has been on the record
since \S\ref{sec:zeta-selection}.

\subsection{What the paper commits to at each date}
\label{sec:calendar-commitments}

\paragraph{A standing adverse condition, registered.} One exposure
carries no single date and belongs at the head of this ledger: the
early anchor itself. The derived rate requires \(\Href\) to continue
sitting one coherent \(\pi\)-offset \emph{above} \(66.6\); if the
DESI-weighted early-side combinations continue their present drift
and the consensus anchor converges above \(\approx\!68\), the
\(\pi\)-offset reading is strained at its own anchor --- \(\ge
2\sigma\) at the paper's structural error, with no siren required.
We register the condition here, against interest, because the trend
is in today's data and a reader should not have to discover it.

Pre-registration is a ledger of hostages. For each appointment above,
this paper's exposure is a specific printed number, and it is worth
collecting them in one place --- because after the datum arrives, none
of them can move.

\emph{At Gaia DR4 (December 2026):} the \(0.5\)--\(1\%\) floor
allowance and its \emph{unestablished} sign. A mapped zero-point near
zero deletes the anatomy's floor entry, and the stack shrinks
accordingly; a confirmed offset in the helpful direction cashes it.
Neither outcome touches the derived rate; both readings are printed
here in advance.

\emph{At the channel-level verdicts (2026--2028):} the three
allowances \(2\)--\(4\%\), \(0\)--\(3\%\), and \(1\)--\(2\%\), and
with them the two scenario totals of \S\ref{sec:residual-verdict} ---
quadrature \(4.3\%/3.8\%\), stack \(9.3\%/8.3\%\). Every channel
killed moves the corrected-ladder landing point back toward
\(73.50\); every channel confirmed moves it toward the published
chain's \(\approx\!70.6\) (\(69.5\)--\(70.2\) fully chained). The middle band of Part~V is the
scoreboard on which this plays out in public.

\emph{At the early-side era (2028--2030):} \(\DNeff = 0.04\); no
early-dark-energy lobe above \(\fEDE \approx 0.02\); \(w = -1\)
exactly; and the derived \(66.6 \pm 0.2\) one coherent \(\pi\)-offset
below whatever \(\Href\) sharpens to. Any of these failing is a
parameter-block or program-level kill, not a reshuffle.

\emph{At the siren convergence (O5 and beyond):} \(67\), not \(73\),
under the \(3\sigma\) rule of F-Hubble-2 --- the single cleanest
exposure in the register, and the one this paper has staked its
central claim on since Part~III.

\emph{And at every moment in between:} no proton decays; and a Kaxion
twin-peak at \(\Delta\nu/\nu = 1/35\) if any haloscope reaches the
window first. The calendar closes where the registry began: eleven
rules, all printed before their data --- the posture a derived number
can afford, and a fitted one cannot.

\section{Over-determining the curve: chronometers, calibrated rulers, and the redshift drift}
\label{sec:overdetermining}

The decisive test adjudicates one point --- the present value. But the
derived reading's exposure is far larger than one point, because what
Part~III derived is the whole function \(H(z)\), and three channels can
read that function directly, with neither a sound horizon nor a distance
ladder anywhere in the chain. The strongest strategy against a derived
curve is not to argue about its endpoint but to measure it everywhere ---
as the community call of \S\ref{sec:revisit-lcdm} anticipated --- and
every value these channels will confront is already fixed.

\emph{Cosmic chronometers.} A passively evolving galaxy population is a
clock: its stars age at a known rate, so the differential age of two
populations separated by a small redshift interval reads the expansion
directly. From \(1+z = a_0/a(t)\), differentiating along the line of
sight gives, in two lines,
\begin{equation}
\frac{dz}{dt} = -(1+z)\,\frac{\dot a}{a}
\quad\Longrightarrow\quad
H(z) = -\,\frac{1}{1+z}\,\frac{dz}{dt},
\label{eq:chronometer}
\end{equation}
no cosmology assumed, no ruler, no ladder --- galaxies as clocks
\citep{JimenezLoeb2002,Moresco2022}. Current compilations reach five-to-fifteen-percent
precision per point and are consistent with the derived curve; a
percent-class chronometer program would arbitrate the whole reading by
itself.

\emph{Calibrated rulers.} Baryon acoustic oscillations measure the
combination \(H(z)\,r_d\). Ordinarily \(r_d\) is fitted; but the derived
chart \emph{computes} it --- the drag-epoch ruler
\(r_d = 147.5\)~Mpc on the derived background (the CMB landmark itself
is \(r_* = 145.0\), landmark~4; App.~\ref{app:soundhorizon}) --- so every
BAO measurement becomes an absolute reading of \(H(z)\) with no freedom
left on either side of the product. Table~\ref{tab:derived-Hz} lists what
the curve requires at the redshifts these surveys actually probe.

\emph{The redshift drift.} The purest channel of all watches a single
redshift change in real time. For a comoving source,
\(1+z = a(t_0)/a(t_e)\); observe again after \(\Delta t_0\), and both
epochs advance, giving (to first order)
\begin{equation}
\dot z \;=\; (1+z)\,H_0 \;-\; H(z),
\qquad
\Delta v \;=\; \frac{c\,\Delta z}{1+z}
\;=\; c\,\Delta t_0\left[H_0 - \frac{H(z)}{1+z}\right]
\label{eq:drift}
\end{equation}
\citep{Sandage1962,Loeb1998}. This is cosmology with no proxy at all: no
standard candle, no ruler, no clock but the observer's own --- the
expansion history watched, not inferred. On the derived curve the numbers
are fixed: \(\Delta v = +2.1\)~cm\,s\(^{-1}\) per decade at \(z=1\),
falling through zero at \(z \approx 1.9\), and \(-5.6\)~cm\,s\(^{-1}\)
per decade at \(z=4\) (Table~\ref{tab:derived-Hz}). Centimeters per
second per decade is brutal territory, but it is exactly the design goal
of the ELT's high-resolution spectrograph \citep{Liske2008} and of SKA-era
absorption monitoring; a drift map inconsistent with the \(66.6\)-anchored curve
would kill the derived reading independently of every other channel in
this paper --- including the siren.

\begin{table}[htbp]
\centering\small
\caption[The derived expansion history at the redshifts the curve-reading channels probe.]{The derived expansion history at the redshifts the
curve-reading channels probe. Every entry is a zero-parameter output of
the \(\pi\)-locked inventory (Eq.~\eqref{eq:app-Hofa}); the drift column
is Eq.~\eqref{eq:drift} over one decade of observer time. Consistency of
future chronometer, calibrated-BAO, and drift data with this column set
is an independent kill test of the whole reading.}
\label{tab:derived-Hz}
\begin{tabularx}{\textwidth}{@{}lllX@{}}
\toprule
\textbf{\(z\)} & \textbf{\(H(z)\) derived [km\,s\(^{-1}\)\,Mpc\(^{-1}\)]} & \textbf{Drift \(\Delta v\) per decade} & \textbf{Who reads it}\\\midrule
\(0.5\) & \(88.3\) & \(+2.4\)~cm\,s\(^{-1}\) & chronometers; DESI/Euclid BAO\\
\(1.0\) & \(119.7\) & \(+2.1\)~cm\,s\(^{-1}\) & chronometers; BAO; ELT drift\\
\(2.0\) & \(202.9\) & \(-0.3\)~cm\,s\(^{-1}\) (sign change at \(z\approx1.9\)) & Lyman-\(\alpha\) BAO; ELT drift\\
\(4.0\) & \(424.0\) & \(-5.6\)~cm\,s\(^{-1}\) & ELT/SKA drift\\
\bottomrule
\end{tabularx}
\end{table}

\emph{Emerging channels.} Two young channels are approaching the same
curve from unexpected directions. Localized fast radio bursts ---
whose dispersion measures integrate the electron column along the line
of sight --- already return \(H_0 = 69.40^{+2.14}_{-1.97}\) from 108
events \citep{Gao2024FRB}, squarely between the camps and tightening
with every localization; and \(\gamma\)-ray attenuation on the
extragalactic background light reads \(61.9^{+2.9}_{-2.4}\)
\citep{Dominguez2023Gamma} --- anomalously low and strongly
EBL-model-dependent, but a channel with no ladder and no sound horizon
anywhere in it. Neither is yet decisive; both mature on the calendar's
timescale, and the derived curve is already committed to what they
must find. Meanwhile the first curve-level \emph{trend} datum exists:
the descending \(H_0(z)\) at \(6.4\sigma\) in DESI's uncorrelated
bins \citep{Jia2025H0z}, read with the provenance cautions of
\S\ref{sec:hole-evidence} --- the generic signature of some local
outflow, and a shape no constant-bias ladder can imprint.

The strategic point deserves one plain sentence. A fitted \(\Lambda\)CDM
can absorb a surprising \(H(z)\) point by re-fitting; the derived curve
cannot absorb anything --- every row of Table~\ref{tab:derived-Hz} is a
place where the reading can die, and that multiplicity of exposure,
not any single agreement, is what a derivation buys.

\section{Observational priorities opened by the H0DN result}

The H0DN result transforms the observational agenda because it
locks in the local corridor \eqref{eq:corridor} at $1\%$. The
next pressure points, in order of potential impact on the
\GGtheory{} reading --- a ranking updated by the anatomy of
Part~IV, which promotes the two channel-level tests that can settle
named allowances outright above everything except the
joint-likelihood confrontation:

\subsection{Priority 1: Joint CMB+BAO+$H_0$ analysis within the
inherited ledger}

The most urgent task is a fully joint CMB+BAO+$H_0$ likelihood
analysis with the background \emph{fixed} at the $\pi$-locked
inventory --- zero background parameters, only the seed
$(A_s,\ n_s{=}0.962)$ and the optical depth $\tau$ inherited --- run
head-to-head against six-parameter $\LCDM$ and against EDE. This is
the over-determined test of \S\ref{sec:cmb-overdetermined} promoted
to full-likelihood level: if the derived chart's $\chi^2$ per
\emph{free} parameter beats the fitted alternatives, the case stops
being philosophical.

\subsection{Priority 2: The injection--recovery campaign (F-Hubble-9)}

Promoted to second place on its cost-to-consequence ratio: a
software-only campaign, publishable in a season, that settles a
\(0\)--\(3\%\) allowance currently frozen at ``undecided.'' No
other entry on this list resolves an open channel so cheaply, and the
verdict is symmetric by construction
(\S\ref{sec:channel-kill-tests}).

\subsection{Priority 3: The near-infrared ladder and the sibling
samples (F-Hubble-8)}

Promoted alongside: dust nearly vanishes in the near-infrared, so the
NIR/JWST supernova ladder is the one photometric program that can
kill or confirm the largest single allowance (\(2\)--\(4\%\)) in
the residual's anatomy, and the ZTF/Rubin sibling samples test the
two-population split with statistics the calibrator set cannot match.

\subsection{Priority 4: Increasing the number of geometric anchors}

The H0DN paper explicitly suggests \citep[Section~6 of
][]{CasertanoH0DN2026} that ``establishing additional anchors via
geometric methods (e.g., using population-II pulsating stars
calibrated by parallax)'' is a key direction. From the
\GGtheory{} perspective, more geometric anchors sharpen the
confrontation: they narrow the measured local corridor around its
stable value, and every halving of its width makes the named
$\approx\!7\%$ residual harder to dissolve quietly --- either the
controlled ladder drifts toward the background $\approx 67$
(F-Hubble-4), or the residual stands and the siren adjudicates
(F-Hubble-2).

\subsection{Priority 5: Megamaser distances beyond NGC 4258}

Five galaxies with megamaser systems (UGC~3789, NGC~6264, NGC~6323,
NGC~5765b, CGCG~074-064) currently anchor the megamaser
Hubble-flow tracer class. Their individual uncertainties are
$\gtrsim 10\%$. Additional Megamaser Cosmology Project work (or
SKA-era AGN surveys) that brings another five to ten systems
into the network would both tighten the central H0DN value
and, crucially, further independently test the no-systematic
hypothesis. The derived reading expects them to agree with the network
consensus: the stability of the local value is a measurement-chain
fact, whatever its ultimate origin.

\subsection{Priority 6: JWST extension of the primary ladders}

JWST observations of Cepheids, TRGB, and JAGB in SN~Ia host
galaxies provide the first fully independent check of HST-era
Cepheids in these hosts \citep{Riess2024Cross,Freedman2025CCHP}. The H0DN paper finds (Section~5.2.2 of
\citealt{CasertanoH0DN2026}) that JWST-including variants favor
$\Hzero$ slightly higher than HST-only variants, but within
$1\sigma$. From the \GGtheory{} viewpoint, these JWST data provide
the instrument-swap hard test that confirms $\epslocal > 0$ is
not an HST artifact. The completion of this program across all
$\sim\!37$ hosts, under a convention agreed in advance, is the
registry's F-Hubble-10.

\subsection{Priority 7: DESI $w(z)$ constraints}

If DESI DR3 or DR4 reports a statistically significant detection
of $w(z) \ne -1$ at $3\sigma$, this would activate F-Hubble-5. If DESI
confirms $w = -1$ to within $\pm 0.02$, this would firmly lock in
the throat-cap interpretation of $\rho_\Lambda$ --- the exact-$w$
leg of the derived expansion history.

\subsection{Priority 8: Kaxion haloscope search in the twin-peak
windows}

ADMX, HAYSTAC, and DMRadio should search for dark matter in the
unique laboratory window predicted by \PaperCOne{}
\citep{ArisakaC1Kaxion}, $\nu_K \in [1.59, 2.13]$~GHz ($m_K \in
[6.6, 8.8]~\mu$eV; central $8.1 \pm 0.5$)
($m_K = 8.1 \pm 0.5~\mu$eV) with $\Delta\nu/\nu = 1/35$. Detection of the
twin-peak at this ratio is the cleanest confirmation of the
\UoneQi sector that sources $\DNeff$.

\section{Wider implications: \texorpdfstring{$\SEight$}{S8}, early galaxies, and the H0DN program}
\label{sec:wider-implications}

Three wider readings follow from the same ledger. None of them is
load-bearing for the Hubble claim, and each is carried at exactly the
weight the framework can afford --- one derived number, one
consistency statement, and one caveat.

\subsection{The $\SEight$ tension}
\label{sec:s8-implications}

We have so far focused on $\Hzero$. The H0DN paper itself does not
address $\SEight$. But the same ledger that fixes the H0DN
corridor must, by the cross-consistency argument of Section~5,
also predict a specific $\SEight$ correction.

In the derived reading there is no early-time suppression to invoke
(Section~\ref{sec:s8-section}): the growth history is standard
$\Lambda$CDM growth on the derived inventory, and the $\SEight$
prediction is one number, $\SEight^{\mathrm{GG}} = 0.759$
(Eq.~\eqref{eq:S8response}, App.~\ref{app:s8}), coincident with the
benchmark cosmic-shear determination $S_8 = 0.759^{+0.024}_{-0.021}$
(KiDS-1000; \citealt{Asgari2021}). Like the local $\Hzero$, the mild
low-redshift $\SEight$ preference relative to the CMB-$\Lambda$CDM
inference is read as a late-time/environmental question on a fixed
derived background, not an early-universe one --- carried open here,
with the full growth-history audit in the parent \ATen{}.

\subsection{JWST-era early galaxies}

The JWST early-massive-galaxy puzzle (unexpectedly massive
galaxies at $z \sim 10$--$14$) is a high-$\nu$ tail problem, and the
derived reading treats it the way it treats the growth sector
generally: standard structure formation on the derived inventory,
with no Hubble-specific mechanism invoked. The equality epoch that
seeds the small-scale tail is fixed by the same ledger
($z_{\rm eq} \simeq 3400$ from $\Omega_m = 1/\pi$ and the measured
radiation density; App.~\ref{app:thermal}), so the growth clock is
not adjustable. The quantitative confrontation --- halo-mass
functions and baryon-feedback modeling at $z \sim 10$ --- is carried
by the parent \ATen{} on the same inventory; this paper imports no
credit from it, and stakes no claim on it beyond consistency.

\subsection{The meaning of the agreement for the H0DN program
itself}

Even setting aside \GGtheory{}, the H0DN collaboration itself
suggests at the end of \citep[Section~6 of ][]{CasertanoH0DN2026}
that ``rather than serving solely to constrain dark energy
models, as envisioned a decade ago, the improved accuracy of
$\Hzero$ now exposes a broader inconsistency within the standard
cosmological framework and strengthens the case for new physics
or a deeper reassessment of early-Universe inferences.''

\GGtheory{} is a concrete candidate for what that ``new physics''
looks like. In particular:

\begin{itemize}[leftmargin=2em,itemsep=0.15em]
\item The new physics is not a late-time dark-energy species.
\item The new physics is not a single scalar field.
\item The new physics is not an anomalous local void.
\item The new physics is the \emph{same} integer ledger that
already fixes the Standard Model parameters --- applied to the
cosmic inventory, from which the whole expansion history follows
with nothing further added.
\end{itemize}

If \GGtheory{} is right, the Hubble tension is not a failure of
our cosmology but a mis-stated question: the background rate was
never a free number to be measured twice, but a derived function
read at one point --- and the residual disagreement is a statement
about one measurement chain, decidable by a channel that owes
nothing to either side.

One caveat travels with the word ``consensus'' wherever this paper
uses it: the H0DN combination excludes the CCHP program, whose
JWST-era values (\(70.39\pm1.94\), and lower on the JWST-only
subsamples) define the low edge of the ladder's own internal spread
(\S\ref{sec:landscape-consensus}). What H0DN certifies is the
covariance-weighted agreement of the teams inside it; the subsample
channel of Part~IV lives precisely at the boundary of that
``inside.''

\section{Cosmological benchmark sheet}

For ease of reference, we collect the complete set of
\GGtheory{}-inherited cosmological predictions in
Table~\ref{tab:benchmarks}, with current observational status.

\begin{table}[htbp]
\centering\small
\caption[\GGtheory{}-inherited cosmological benchmarks and their observational status as of 2026.]{\GGtheory{}-inherited cosmological benchmarks and their
observational status as of 2026. Status labels follow the
conditional triage of the companion paper \ATen{} (Derived /
Derived-conditional / Asserted); entries quoted from it carry that
conditional status, and the local excess is split into the derived
Local-Hole lift and a named open residual
(Appendix~\ref{app:epslocal}).}
\label{tab:benchmarks}
\begin{tabularx}{\textwidth}{@{}p{3.8cm}p{3.2cm}p{3.2cm}X@{}}
\toprule
\textbf{Observable} & \textbf{\GGTheory{} prediction} &
\textbf{Current measurement} & \textbf{Status}\\\midrule
Local $\Hzero$ & $\approx 68.5$ (background $+$ Local-Hole lift) $+$ named residual &
$73.50 \pm 0.81\kmsMpc$ (H0DN) & Lift derived; $\approx\!7\%$ residual open (systematics)\\
Background $\Hzero^{(0)}$ (derived) & $66.6 \pm 0.2\kmsMpc$ &
$67.24 \pm 0.35$ (CMB); $498.3\sqrt{\zeta_\Lambda}$ (seesaw) & Derived (D$^*$); $-1.2\%$ coherent $\pi$-offset from CMB\\
$\DNeff$ & $0.04$ (inherited, \ANine{}) & $0.04 \pm 0.18$ (Planck) & Consistent (no early modification required)\\
Standard-siren $\Hzero^{\mathrm{GW}}$ & $\approx 67$ (predicted) & (forthcoming) & The decisive model-independent test\\
2026 landscape structure & corrections move band 3 ($72.5$--$77$) into band 2 ($69$--$71$), not to $67$ & re-analyses land at $70$--$71.5$; chained corrections $69.5$--$70.2$ (Part~V) & Consistent; the middle band is filling\\
Corrected-ladder scenario $\Hzero$ & $69.5$--$70.2$ (published central corrections chained) & CCHP $70.39\pm1.94$; two-population $70.59\pm1.15$ & Within $0.8\sigma$ of the lifted $68.5$\\
Local-Hole lift $\epsilon_{\mathrm{Hole}}$ & $+0.02$--$0.03$ (derived sign \& size) & consistent with local void depth & Derived; full closure of the residual open\\
$w$ (dark energy eq.\ of state) & $-1$ exactly & $-1.03 \pm 0.03$
(DESI~Y3) & Compatible\\
$\Omega_m$ & $1/\pi \approx 0.3183$ & $0.315 \pm 0.007$ (Planck) &
$0.47\sigma$\\
$\Omega_\Lambda$ & $1 - 1/\pi \approx 0.6817$ & $0.685 \pm 0.007$
(Planck) & $0.47\sigma$ (approx.; neglects $\Omega_r$, massive
$\nu$)\\
$\rho_\Lambda^{1/4}$ (dark energy) & $\sim$ few meV &
$(2.24 \pm 0.02)$~meV & Consistent\\
$r$ (primordial tensors) & $\ll 0.01$ & $<0.032$ (BICEP/Keck) &
Consistent\\
$\sin^2\theta_W(M_Z)$ & $25/108 \approx 0.23148$ &
$0.23149 \pm 0.00013$ & $-0.07\sigma$\\
$\hat\alpha^{-1}_{\mathrm{EM}}(M_Z)$ & $2^J = 128$ &
$127.93 \pm 0.008$ & $+0.05\%$ (fractional-agreement convention,
as for $z_*$ in \ATen{}; $\gg 1\sigma$ at the quoted precision)\\
Proton lifetime $\tau_p$ & $\infty$ (absolute stability) &
$>2.4 \times 10^{34}$~yr & Consistent\\
$\bar\theta$ & $\lesssim 4.7 \times 10^{-18}$ & $< 10^{-10}$ bound &
Consistent\\
$\Omega_m/\Omega_\Lambda$ crossover $z_{\Lambda m}$ & $\approx 0.289$
& $0.30 \pm 0.02$ & Consistent (implied by the $\Omega_m$ row;
not independent)\\
$\SEight$ & $0.759$ (derived; App.~\ref{app:s8}) &
$0.759^{+0.024}_{-0.021}$ (KiDS-1000), $\approx 0.83$ (CMB-$\LCDM$) &
Lensing agreement; CMB-inference gap carried open\\
JWST massive-galaxy excess & standard growth on the derived
inventory & observed & Consistent (quantitative case in \ATen{})\\
Redshift drift at $z=1$ & $+2.1$~cm\,s$^{-1}$ per decade & (ELT/SKA era) & Prediction of the derived curve\\
Kaxion $\Delta\nu/\nu$ & $1/35$ exactly & not yet probed & Pending\\
\bottomrule
\end{tabularx}
\end{table}

The H0DN agreement sits at the top of
Table~\ref{tab:benchmarks} as the newest entry, but it is only one
of twenty-one benchmark rows. The distinctive feature of the
\GGtheory{} reading is this consistency across all entries, not
any one of them individually.


%% ===================================================================
%% PART V --- Significance, Uniqueness, and Bottom Line
%% ===================================================================
\bigskip
\noindent\textbf{Coda.}~ Part~VI has turned the reading into a schedule. The falsifier
registry now runs to eleven named rules --- program, parameter-block, channel,
and coefficient --- and the decisive one is still a converged standard siren,
which reads the clock a third way, owing nothing to either endpoint, and lands
near $67$ or near $73$. New on the board is the calendar itself: Gaia DR4 on
December 2026; the channel-level trio through 2028; the early-side precision
era to 2030; the sirens beyond --- each date with its printed number, and each
number a hostage. That is the posture a derived quantity can afford and a fitted
one cannot: to name, in advance and in public, the observation that would end it.
And beyond the one decisive point, the whole curve is on the schedule:
chronometers, rulers calibrated by the derived sound horizon, the young
fast-radio-burst and $\gamma$-ray channels --- and, most beautifully, the
redshift drift, the expansion watched in real time at centimeters per second
per decade, each a place where the derived history has already committed and
cannot move.


\section{Part~VI in plain English: the eleven ways this reading can be killed, and the dates they are scheduled for}
A derived number is worth less than the list of things that would
refute it, so this Part is that list, with dates attached.

Eleven named falsifiers, in four tiers. Three would retire the whole
framework rather than merely this paper. Four would retire this reading of
the Hubble corridor and leave the framework standing. Three settle
individual channels of the residual without deciding anything about the
framework in either direction. One cashes the choice of a single
coefficient against a computation that nobody has yet performed.

Then the calendar, which is what makes a registry more than a gesture.
Gaia's fourth data release in December 2026. The channel-level trio
through 2028. The precision era on the early side out to 2030. The sirens
after that. Each date carries a number this paper has already committed
to, and the commitment is public before the measurement, which is the only
order in which a commitment counts for anything.

The Part then opens the work outward: eight observational priorities,
running from a joint likelihood that any group could run today with
already released data, to a haloscope search in a frequency window the
dark-matter companion fixes. It closes on a benchmark sheet of twenty-one
rows, because the honest way to present a framework is on all of its
entries at once and not on the one it does best.

\GGPart[Forecasting the verdict]{Forecasting the Verdict: The Standard-Siren Decade}{The physics of the siren; the referee's report card 2017--2026; the events-to-verdict forecast on empirical rates; the referee's own systematic budget; and the three worlds the catalog can choose among}

\section{The principle: a distance from the waveform alone}
\label{sec:siren-principle}

Everything in this paper funnels to one measurement, so the measurement
deserves a Part of its own: what a standard siren is, why it owes nothing
to either discordant route, how many events the verdict needs, and what
could bias the referee itself. The idea is forty years old
\citep{Schutz1986} and almost indecently clean. A compact binary in its
final inspiral is a broadcasting machine whose absolute output is
computed, not calibrated: general relativity fixes the entire waveform in
terms of the source masses. The observed frequency \(f\) and its
acceleration \(\dot f\) give the \emph{chirp mass} through
\begin{equation}
\dot f \;\propto\; f^{11/3}\,\mathcal{M}_c^{5/3}
\qquad\Longrightarrow\qquad
\mathcal{M}_c \;\text{read from the phasing alone},
\label{eq:chirp}
\end{equation}
and the strain amplitude then contains only one unknown:
\begin{equation}
h \;\propto\; \frac{\mathcal{M}_c^{5/3} f^{2/3}}{d_L}
\qquad\Longrightarrow\qquad
d_L \;\text{read from the amplitude},
\label{eq:siren-dL}
\end{equation}
an \emph{absolute} luminosity distance, from first principles
\citep{HolzHughes2005}. No parallax, no period--luminosity relation, no
supernova standardization: the siren is a one-rung ladder whose single
rung is the Einstein field equations. The redshift the waveform cannot
supply (a redshifted binary looks exactly like a heavier one) comes from
electromagnetic astronomy --- the host galaxy of the counterpart --- and
at low redshift the rate follows from the two directly:
\(cz \simeq H_0\,d_L\).

This has already been done once. GW170817, the first binary-neutron-star
inspiral, came with a kilonova, a host galaxy (NGC~4993), and a
Hubble-flow velocity of \(3017 \pm 166\)~km\,s\(^{-1}\); against the
waveform's distance of roughly forty megaparsecs the joint posterior gave
\(H_0 = 70^{+12}_{-8}\;\kmsMpc\) \citep{Abbott2017Siren} --- one event,
fifteen-percent precision, straddling both camps with perfect
impartiality. The entire question of this Part is how fast that fifteen
percent divides by \(\sqrt{N}\).

\section{Three pipelines, disjoint by construction}
\label{sec:siren-pipelines}

\emph{Bright sirens} are the GW170817 channel: an inspiral with an
electromagnetic counterpart, hence an exact host redshift. They are rare
--- binary neutron stars with observable kilonovae --- and each carries a
\(\sim10\)--\(20\%\) distance error dominated by a single geometric
degeneracy: a binary seen face-on is louder, so inclination and distance
trade against each other in the amplitude. The degeneracy is broken
partly by the two gravitational-wave polarizations and, for the lucky
events, decisively by electromagnetic viewing-angle information from the
jet afterglow.

\emph{Dark sirens} are the volume channel: binary black holes with no
counterpart, redshifted statistically against galaxy catalogs over the
localization volume. Each event is weak, but LVK-class runs collect them
by the hundreds; the founding forecasts \citep{ChenFishbachHolz2018}
promised percent-class \(H_0\) within years of design sensitivity, and
this Part dates the channel from its \emph{observed} record instead
(\S\ref{sec:siren-reportcard}). Its systematics --- catalog completeness,
population priors --- are entirely different from the bright channel's.

\emph{Spectral sirens} use features of the source mass distribution
itself as a redshift reference; they are the youngest channel and serve
here as a third cross-check. The structural point is the disjointness:
none of the three shares a systematic with the sound horizon, with the
Cepheid zero-point, with supernova standardization, or --- crucially ---
with each other's dominant error. Three referees, mutually independent,
all reading the same clock.

\section{The referee's report card, 2017--2026}
\label{sec:siren-reportcard}

A Part built on a forecast owes the reader the record before the
extrapolation. As of this writing the record is nine observing years
long, and it reads: \emph{one bright siren.}

GW170817 remains, in 2026, the only gravitational-wave event with a
confirmed electromagnetic counterpart. The subsequent observing runs
detected further binary-neutron-star candidates, but none yielded a
kilonova identification: the events sat several times farther away,
their sky localizations spanned tens to hundreds of square degrees,
and a kilonova fades through the follow-up window in days. This is
the O4 null-counterpart record, and its physics lesson is worth
stating plainly: the bright channel's bottleneck was never the
interferometers. It is the compound accident of a nearby,
well-oriented merger caught while wide-field telescopes are looking.
GW170817, at forty megaparsecs, was a gift; the volume in which a
kilonova is \emph{cheap} to catch grows far more slowly than the
volume in which the waveform is \emph{loud} enough to detect, so
detector improvements alone do not manufacture counterparts.

The dark-siren ladder, meanwhile, has been climbing exactly as the
forecast's structure --- though not its dates --- said it would.
Three rungs are on the record: \(76.6^{+13.0}_{-9.5}\) from the
GWTC-4.0 catalog (142 dark sirens against the GLADE+ galaxy
catalog, with GW170817; mass-spectrum-model dependent)
\citep{LVK2025GWTC4}; \(71.0^{+9.0}_{-7.1}\) from GWTC-5.0 (236
sources against DES-Y6 --- a \(25.7\%\) tightening)
\citep{LVK2026GWTC5}; and \(69.9^{+4.1}_{-4.0}\) from seventeen
well-localized O4a dark sirens against DESI Legacy photometric
redshifts, again with GW170817 \citep{Alfradique2026Sirens} --- the
sharpest siren value in existence. The precision is marching down as
catalogs deepen, and the central value has walked from the ladder
camp onto the lifted background. One honesty clause belongs beside
that observation: the three values share events and methods, so the
walk is a single maturing measurement seen at three stages, not three
independent confirmations --- suggestive, and no more. (A fourth
entry exists and is carried as exactly one line: thirteen binary
black holes with \emph{claimed} AGN-flare counterparts give
\(70.5^{+3.4}_{-2.9}\), but the associations are unconfirmed, and
this paper takes no support from it \citep{Kumar2026AGN}.)

Then the recalibration, printed rather than papered over. The rate
forecasts of the GW170817 era imagined tens of counterpart events
within a few years of design sensitivity; nine years produced one.
The dates in this Part are therefore drawn from the empirical record,
not from those forecasts, and the bright-alone verdict slips
accordingly --- into the O5-to-next-generation era. What does
\emph{not} slip is the arithmetic: the twenty-one- and
fifty-eight-event thresholds of Eq.~\eqref{eq:siren-N} depend only on
the hypothesis separation and the per-event error, and are hostage to
no rate assumption at all. A forecast's dates are hostage to Poisson
luck; a forecast's thresholds are hostage to nothing. The rest of
this Part keeps the two strictly apart.

\section{The forecast: events to verdict}
\label{sec:siren-forecast}

The arithmetic of the decision is elementary, and stating it exactly is
the point of this Part. The two hypotheses sit
\(\Delta = 73.5 - 66.6 = 6.9\;\kmsMpc\) apart --- ten percent. A
catalog of \(N\) unbiased bright sirens with per-event fractional
error \(\sigma_{\rm ev}\) reaches
\(\sigma(N) = \sigma_{\rm ev}\times70/\sqrt{N}\), and separates the
hypotheses at \(k\) standard deviations when \(\sigma(N) = \Delta/k\):
\begin{equation}
N_{k\sigma} \;=\; \left(\frac{70\,\sigma_{\rm ev}\,k}{\Delta}\right)^{2},
\qquad
\sigma_{3\sigma} = 2.30, \quad \sigma_{5\sigma} = 1.38\;\kmsMpc .
\label{eq:siren-N}
\end{equation}
Table~\ref{tab:siren-events} evaluates it for three per-event scenarios,
and Figure~\ref{fig:siren-verdict} draws it: at the GW170817-like
central scenario the \(3\sigma\) verdict needs \emph{twenty-one} bright
events and the \(5\sigma\) verdict \emph{fifty-eight}. Even the first
five events of the O5 era deliver a \(1.5\sigma\) lean --- the community
will watch this arrive in public, event by event. One scope clause,
charged by this paper's own Part~IV: the \(21/58\) is the
\emph{deep-catalog} separation (background \(66.6\) versus ladder
\(73.5\), \(\Delta = 6.9\)). A \emph{shallow bright} subset at
\(z \lesssim 0.05\) samples inside the Local Hole, where the
conditional lift is a coherent common mode that does not average down:
there the operative separation is \(\Delta = 5.0\)--\(5.8\)
(ladder versus the lifted \(68.5\) or the flows-branch
\(\lesssim\!67.7\)), and the same arithmetic gives
\(N_{3\sigma} \approx 30\)--\(40\). Both numbers are printed because
both questions are real: the deep catalog decides the background;
the shallow subset decides the branch. And a pointer is not a rule,
so here is the shallow rule, with its teeth and its admitted gap.
\emph{Pre-registered shallow-subset rule.} For a bright-siren subset
at \(z \lesssim 0.05\) with catalog precision
\(\sigma_{\rm sh}\): the \emph{lift itself} (both branches) is
refuted if the subset converges at \(73.5\)-side values,
\((\mu_{\rm sh} - 68.5)/\sigma_{\rm sh} \ge 3\) --- reachable at
\(N \approx 40\) shallow events on the central per-event scenario;
the counts branch is confirmed against the flows branch only when
\(\sigma_{\rm sh}\) resolves their \(0.8\;\kmsMpc\) separation,
which needs \(\sigma_{\rm sh} \approx 0.27\) --- of order
\(1{,}500\) shallow events, \emph{undecidable by sirens this
decade, and we say so in advance} rather than let the gap be
discovered later. The branch therefore belongs to the ladder-shell
test (F-Hubble-4), whose step the scenario ledger's own lifts fix:
cutting \(z_{\rm min}\) across \(0.05\) must drop \(H_0\) by
\(1.8\)--\(1.9\;\kmsMpc\) on the counts branch, \(0.3\)--\(0.7\)
on the flows branch; a measured step consistent with zero at
\(\pm0.3\) refutes both, and F-Hubble-4 fires.

\begin{table}[htbp]
\centering\small
\caption[Events to verdict (Eq.~\eqref{eq:siren-N}; open-source script \texttt{c2\_siren\_forecast.py}).]{Events to verdict (Eq.~\eqref{eq:siren-N}; open-source script
\texttt{c2\_siren\_forecast.py}). Bright-siren counts; dark-siren
catalogs contribute in parallel and, on the empirical
catalog-to-catalog improvement rate of \S\ref{sec:siren-reportcard},
reach the same precision on a similar clock.}
\label{tab:siren-events}
\begin{tabularx}{\textwidth}{@{}lXll@{}}
\toprule
\textbf{Scenario} & \textbf{Per-event error (justification)} & \textbf{\(N_{3\sigma}\)} & \textbf{\(N_{5\sigma}\)}\\\midrule
Optimistic & \(10\%\): jet viewing-angle constraints on most events & \(9\) & \(26\)\\
Central & \(15\%\): GW170817-like, polarization-only inclination & \(21\) & \(58\)\\
Conservative & \(20\%\): poor localization, no viewing-angle help & \(37\) & \(103\)\\
\bottomrule
\end{tabularx}
\end{table}

Rates convert events to dates, and here the report card of
\S\ref{sec:siren-reportcard} governs: the dates below come from the
empirical record, not from the GW170817-era rate papers. Nine
observing years have produced one counterpart event, so the honest
bright-siren prior is of order one event per year in the O5 era
\emph{at best} --- the deeper range multiplies the detection volume
severalfold, but the counterpart bottleneck (kilonova depth,
localization area, follow-up coverage) has proven the harder
constraint. Under that prior the bright channel alone reaches its
21-event \(3\sigma\) crossing only in the mid-to-late 2030s, at the
O5-to-next-generation boundary. What has not slipped is the clock's
other hand: the dark-siren channel, accumulating hundreds of
black-hole events per run and already at the \(\pm4\) level
(\S\ref{sec:siren-reportcard}), reaches the two-percent class after
three to four further catalog generations at the observed
improvement rate --- with the honesty clause attached: only one
generation step is cleanly measured (GWTC-4.0 \(\to\) 5.0,
\(25.7\%\); the further step to \(\pm4.05\) mixes in a method
change), so compounding it (\(\pm4.0 \times 0.74^{3\text{--}4}
\approx \pm1.2\)--\(1.6\)) is a scaling from the record but still a
two-point extrapolation, and we call it that. The early-to-mid 2030s
on the current run cadence, dated from what has happened rather than
from the 2018-era forecasts this Part retires for the bright
channel. The timeline
bands of Figure~\ref{fig:hubble-killtest-timeline} therefore stand,
with the near-term weight carried by the dark channel: first verdicts
near 2030, the decisive era by \(\sim\)2035.

One outcome deserves its own sentence: a catalog that converges
\emph{between} the camps. This is not a separate verdict but one cell
of F-Hubble-2's single decision function --- at \(\mu = 70.0\),
\(\sigma_{\rm cat} = 1.0\), both point hypotheses fail their
\(3\sigma\) tests simultaneously (\(3.4\sigma\) and \(3.5\sigma\)),
and the rule's middle-world clause takes over: the strain on the pure
background is printed, and the adjudication passes to the inventory
and the shell test. That outcome would tell the field something more
interesting than either camp expects --- that the early-side inference
and the late-side calibration are \emph{both} displaced --- and the
most informative outcome remains the one nobody is betting on.

\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{c2_siren_verdict.pdf}
\caption[The siren forecast: events to verdict, and the simulated catalog.]{\textbf{The
verdict, forecast.} \emph{Left:} catalog precision
\(\sigma(N) = \sigma_{\rm ev}/\sqrt{N}\) for three per-event scenarios,
against the pre-declared decision thresholds: \(\sigma = 2.30\) separates
the derived \(66.6\) from the ladder \(73.5\) at \(3\sigma\),
\(\sigma = 1.38\) at \(5\sigma\). In the GW170817-like central scenario
the crossings are \(N = 21\) and \(N = 58\) bright events (dots); the
shaded band is the O5-era bright catalog under the \emph{empirical}
rate record of \S\ref{sec:siren-reportcard} (one counterpart in nine
years) rather than the GW170817-era forecasts --- the one assumption
of the released script revised by that record; the thresholds and
per-event scenarios are rate-independent. Dark-siren catalogs carry
the near-term clock in parallel. \emph{Right:} the same forecast as the community will
experience it --- the toy posterior after \(21\) (dashed) and \(58\)
(solid) events under each hypothesis. Whichever pair materializes, the
other reading is finished; a catalog converging \emph{between} them
lands in the pre-declared middle world of F-Hubble-2's decision
function --- the background strained at the printed
\((\mu-66.6)/\sigma_{\rm cat}\), adjudication handed to the
inventory and shell tests. Script:
\texttt{c2\_siren\_forecast.py}, released with the paper.}
\label{fig:siren-verdict}
\end{figure}

\section{The referee's own budget: can the siren itself be wrong by ten percent?}
\label{sec:siren-systematics}

A paper that has spent a Part interrogating the ladder's error budget
owes the same scrutiny to its chosen referee.
Table~\ref{tab:siren-budget} walks the siren channel's systematics with
the one question that matters: could any of them manufacture a coherent
ten-percent bias --- the size needed to fake a verdict?

\begin{table}[htbp]
\centering\small
\caption[The systematic budget of the referee.]{The systematic budget of the referee. ``Averages down''
indicates a per-event term that shrinks as \(1/\sqrt{N}\). Three terms
do not: the detector amplitude calibration (the channel's true floor,
an instrument property with no conceivable coupling to either
discordant route); the source mass-spectrum model, coherent across the
dark/spectral channel; and, for shallow bright subsets only, the
Local-Hole monopole of Part~IV.}
\label{tab:siren-budget}
\begin{tabularx}{\textwidth}{@{}lXll@{}}
\toprule
\textbf{Source} & \textbf{Nature} & \textbf{Per event} & \textbf{Floor}\\\midrule
Inclination--distance & geometric degeneracy; broken by polarizations and jet viewing angle & \(10\)--\(30\%\) & averages down\\
Host peculiar velocity & \(\sim300\)~km\,s\(^{-1}\) against the flow: \(10\%\) at \(40\)~Mpc, \(1\%\) at \(450\)~Mpc & \(1\)--\(10\%\) & averages down\\
Detector amplitude calibration & photon-calibrator laboratory measurement; \emph{common mode} & \(1\)--\(2\%\) & \(\sim1\%\), must be engineered down\\
Selection (Malmquist) & louder events over-represented; modeled with the population & \(\sim1\%\) & sub-\(\%\) with population model\\
Dark-siren catalog completeness & missing hosts skew the statistical redshift & --- & \(\sim1\%\), the dark channel's floor\\
Weak lensing & line-of-sight magnification; grows with \(z\) & \(<1\%\) at low \(z\) & averages down\\
Source mass-spectrum model & the \(\sim\!35\,M_\odot\) feature's location and redshift evolution set the dark/spectral distance scale; mis-modeling biases coherently --- the \(76.6 \to 71.0 \to 69.9\) walk of \S\ref{sec:siren-reportcard} is in part this term relaxing; mitigated by population marginalization and the bright-siren cross-check & few \(\%\) & coherent\\
Local-Hole monopole & coherent \(+2\)--\(3\%\) (counts branch) for shallow hosts, \(z \lesssim 0.05\); absent for deep catalogs --- the lift's own signal (the branch test), not a nuisance & \(2\)--\(3\%\) & coherent (shallow)\\
\bottomrule
\end{tabularx}
\end{table}

The table's verdict is short. Every astrophysical term either
averages down with the catalog or is printed above as a named
coherent term with its mitigation and its size --- and the named terms
top out at a few percent, an order below the ten-percent question the
referee is asked to settle. The one irreducible common mode is the
amplitude calibration of the detectors themselves --- currently at the
one-to-two percent level, measured with photon calibrators in the
laboratory, and required to reach sub-percent for the \(5\sigma\)
era. It is an
engineering quantity, with no conceivable coupling to the sound horizon,
the Gaia zero-point, or the Cepheid period--luminosity relation. There
is, simply, no candidate channel by which the siren could be coherently
wrong by ten percent --- and if one is nervous anyway, the bright, dark,
and spectral pipelines carry \emph{different} floors and must agree.
The referee is as clean as instruments get.

And the budget is no longer only a table --- it is being demonstrated
live. The dark-siren walk of \S\ref{sec:siren-reportcard},
\(76.6 \to 71.0 \to 69.9\), is Table~\ref{tab:siren-budget}'s
middle rows seen in the wild: the several-unit swings between
consecutive catalogs are the mass-spectrum-model and
catalog-completeness terms at their current size, and their
trajectory --- error bars shrinking from \(^{+13.0}_{-9.5}\) to
\(^{+4.1}_{-4.0}\) as the catalogs deepen and the host samples
improve --- is what ``averages down'' looks like empirically. A
referee whose systematics are visibly shrinking on schedule, in
public, is precisely the referee one wants; the spread between today's
dark-siren values is the size of the remaining budget, and it is
already smaller than the gap it is being asked to adjudicate.

\section{The verdict, simulated --- and what failure looks like}
\label{sec:siren-simulated}

The right panel of Figure~\ref{fig:siren-verdict} runs the decade
forward as a toy simulation (seeded, elementary, and released): the
posterior the community will actually watch, after \(21\) and after
\(58\) central-scenario events, under each hypothesis. The curves are
deliberately unglamorous --- two Gaussians marching apart as
\(\sqrt{N}\) --- because that is what a clean decision looks like:
no reanalysis, no tension indices, no committee. One pair of curves
materializes, and the argument of a decade ends.

Failure is drawn on the same axes, and this paper reads it without
flinching. If the red pair materializes --- a catalog converged at
\(73.5\) --- then the true background rate is high; the CMB densities
cannot be divided by the \(\pi\)-locked fractions to give it; Theorem
T-Hubble-1's inventory, whatever its geometric elegance, does not
describe this universe's budget; and the derived reading of this paper
is finished, cleanly, in public, by the exact instrument it spent a Part
praising. The personal note of the Conclusions was written with that
outcome squarely in view. What the framework may not do --- and commits
here not to do --- is discover, after such a verdict, that some
neglected effect had shifted its prediction to \(73\) all along. The
number is \(66.6 \pm 0.2\). It was printed before the catalog
existed. There is nowhere to hide, which is precisely the point.

\section{The sharpened stake: three outcomes, three worlds}
\label{sec:siren-three-worlds}

Part~V tabulated the three worlds for the six rival classes
(Table~\ref{tab:siren-verdicts}); this section prices them for this
paper alone, with the catalog precisions made explicit. Everywhere
below, ``a converged catalog'' means one standing between the two
verdict thresholds of Eq.~\eqref{eq:siren-N},
\(\sigma_{\rm cat} = 2.30 \to 1.38\;\kmsMpc\); every verdict in this
section is a cell of the single decision function pre-registered at
F-Hubble-2, which defends the background \(66.6\) and assigns the
conditional lifted reading to the shallow bright subset --- a deep
catalog does not test the lift (OQ-Hubble-7).

\emph{At \(\approx67\) --- confirmation, and the tension ends as
metrology.} F-Hubble-2 confirms the derived background; the residual
anatomy of Part~IV graduates from priced hypothesis to history. The
channel-level registry entries F-Hubble-8/9/10 stay interesting even
then, because they would describe \emph{how} a superb measurement
chain came to read seven percent high --- and the tension would end
the way discrepant constants have most often ended and are least
often predicted to: as metrology, not new physics. The oldest number
in cosmology becomes what Part~III said it was --- a theorem's
output, read three independent ways.

\emph{At \(\approx70\) --- the middle world, priced exactly.} The
H0DN centroid falls and the corrected-ladder cluster of band~2 is
vindicated. Here the rule's honesty matters most, so we state it
against the temptation the two-rate structure invites: a deep converged catalog reads the
\emph{background}, so its verdict falls on the pure derived
\(66.6 \pm 0.2\) --- strained at \(1.5\)--\(2.4\sigma\), the exact
figure depending on where between the thresholds the catalog stands
--- and the conditional lifted reading is \emph{not} what such a
catalog vindicates. The lift's own siren test is the shallow bright
subset (\(z \lesssim 0.05\), pre-registered above), which samples
inside the Hole and should read \(68.5 \pm 0.6\) on the counts
branch, \(\lesssim\!67.7\) on the flows branch --- a
branch-discriminating measurement in its own right. At \(\approx70\),
then: the pure background is pressured but not killed at the
pre-registered level, the deep-vs-shallow split becomes the sharpest
open question, and the shell-level \(H_0(z)\) program of
\S\ref{sec:calendar-entries} adjudicates it independently of any
siren. The middle world is uncomfortable for
everyone, this reading included; the discomfort is printed here in
advance, with its numbers.

\emph{At \(\approx73\) --- refutation, as designed.} F-Hubble-2
fires; the anatomy evaporates; the derivation dies in public, by the
instrument this Part spent its pages praising
(\S\ref{sec:siren-simulated}). Nothing in that sentence is softened
by its being unlikely on the framework's own reading: the rule was
built so that this world, if it is the real one, wins quickly and
unarguably.

Three worlds, one instrument, and a rule fixed before the catalog
exists. No fitted resolution of the tension can offer the same
sentence.

\section{Reproducibility and the pre-registered rule}
\label{sec:siren-repro-part}

Everything quantitative in this Part reproduces from one elementary,
fully documented script (\texttt{c2\_siren\_forecast.py}, released with
the paper): the thresholds of Eq.~\eqref{eq:siren-N}, the event counts
of Table~\ref{tab:siren-events}, and both panels of
Figure~\ref{fig:siren-verdict}. Its inputs are three declared numbers
--- the two hypotheses and the per-event error scenario --- and nothing
else; an independent group can regenerate the forecast, or sharpen it
with a full waveform-level analysis, in an afternoon. The rule it
serves is the pre-registered F-Hubble-2, restated once, verbatim: a
converged siren value \(H_0^{\mathrm{GW}} = 73 \pm 1\) --- a
\(>3\sigma\) miss of the derived \(67\) --- refutes; a value
\(67 \pm 1\) confirms; a value between the camps kills both readings
and is the most informative outcome of all. The rule was fixed before
any converged catalog existed, and this Part has now fixed the
timetable on which it will be applied.

\bigskip
\noindent\textbf{Coda.}~ Since the first neutron stars locked into their
death spirals, the universe has been broadcasting absolutely calibrated
distances --- a distance ladder with no rungs, built into the field
equations themselves, transmitting for billions of years at frequencies
no one could hear. For a century we built ladders of starlight because
we could not receive that signal; now we can. Somewhere in the coming
decade the catalog --- bright, or dark and statistical --- will pass
its twenty-one-event equivalent, and then its fifty-eight, and the
posterior will settle onto one of the two curves this Part has already
drawn, or between them, into the third world of
\S\ref{sec:siren-three-worlds}. The oldest number in cosmology will then have been read
a third way, by an instrument that owes nothing to any ladder, any
horizon, or any theory --- including the one whose fate it decides.


\section{Part~VII in plain English: the one instrument that owes nothing to either side, and when it will have spoken}
Both routes to the Hubble constant are calibrated --- one against
stars, one against a sound horizon --- and both calibrations are under
suspicion. A third route is not calibrated at all.

When two neutron stars spiral into each other, general relativity fixes
the entire waveform from the masses alone. The frequency and the rate at
which it climbs give the masses; the masses give the intrinsic loudness;
the observed loudness then gives the distance, with no ladder and no ruler
anywhere in the chain. Pair that distance with a redshift and the result
is the Hubble constant, measured by an instrument that has never heard of
a Cepheid or an acoustic scale.

This Part does the accounting for that instrument. What the catalog has
actually achieved since 2017, which is less than the enthusiasm suggests.
How many events a verdict needs, computed from empirical event rates
rather than from optimism. Whether the siren itself could be wrong by ten
percent, which is asked seriously and answered with a budget rather than a
reassurance.

And then the stake, fixed in advance and in public. A converged siren near
\(67\) confirms this reading. A converged siren near \(73\) ends it. A
value between the two camps kills both readings at once and is, for that
reason, the most informative of the three outcomes. The rule was written
before any converged catalog existed, and it is not available for
renegotiation once one does.

\GGPart[Discussion]{Discussion: Significance, Uniqueness, and Innovation}{What a derived
expansion history means as a scientific claim; the problems it
resolves and the moves it registers; the honest boundary of what it
computes; and its open questions, predictions, and the road ahead}

\section{What this Part does}
\label{sec:disc-map}

Parts~III through~VII did the work: derived the history, priced the
residual, drew the landscape, dated the falsifiers, and forecast the
verdict. This Part steps back and weighs the result as a scientific
claim, in five movements, in the order a skeptical reader should meet
them. \emph{Grounding}
(\S\ref{sec:disc-essence}--\S\ref{sec:disc-support}): what kind of
object this paper is --- one derivation chain, read on one contested
observable --- and what the H0DN comparison actually establishes,
graded strong, moderate, and untested. \emph{Significance}
(\S\ref{sec:significance-8}--\S\ref{sec:disc-precedents}): the case
stated in ten points, read against the leading responses to the
tension, and placed against the historical precedents --- including
the cautionary ones.
\emph{Accomplishments} (\S\ref{sec:ps-hubble}--\S\ref{sec:i-hubble}):
the standing puzzles resolved or dissolved, and the structural moves
this paper makes that no prior treatment of the tension has made, each
in before-and-after form. \emph{The honest boundary}
(\S\ref{sec:disc-boundary}--\S\ref{sec:disc-necessary}): exactly what
is proved, what rides on a named condition, what the framework
computes and what it hands to other instruments, and what the whole
posture costs. \emph{The road ahead}
(\S\ref{sec:disc-open-questions}--\S\ref{sec:disc-bottomline}): the
open questions stated as a numbered program, the forward predictions
and recommendations addressed to the five communities that can act on
them, and the conduct the paper binds itself to while it waits. A
reader who wants the paper's case in one Part should read this one;
every claim below carries a pointer to the Part that earns it.

\section{The essence: one chain, one observable}
\label{sec:disc-essence}

Before the particulars, it is worth naming what kind of object this
paper is, because its shape is inherited, not improvised. The
\GGTheory{} engine is a single chain: \OGGEP{} (axiom \AX{3} of the
\GGConstitution{}; \citealp{ArisakaA1Constitution}) declares every
observable a boundary projection of a closed six-dimensional geometry;
\GDOS{} \citep{ArisakaA5GDOS} is the law by which the open boundary
evolves; the \GRIP{} cascade \(\DGI \to \GSSB{} \to\) attractor
\citep{ArisakaA6Gi4} selects the unique ground geometric attractor;
and the observable \emph{is} that attractor's boundary reading. Run on
the particle stratum, the chain returns the Standard Model's
twenty-eight parameters (\ANine{}). Run on the cosmological stratum,
it returns the \(\pi\)-locked inventory and the whole background
history (\ATen{}). This paper is the chain run all the way down to a
\emph{single contested number}: the same engine, specialized to the
one observable the field has fought over for a decade, with every
intermediate step exposed.

That siting explains the paper's two most distinctive features at
once. First, why the rate could be derived at all: \(H(t)\) is not an
extra postulate but the Friedmann reading of an inventory the chain
had already fixed --- the derivation cost nothing new, which is
exactly what a genuine consequence should cost. Second, why the paper
can afford its honesty: because the chain fixes the fractions
\emph{jointly}, the derived \(66.6 \pm 0.2\) cannot be nudged toward
\(73.5\) without breaking the acoustic fit, the age, the equation of
state, and the sister papers simultaneously
(\S\ref{sec:cmb-overdetermined}, Table~\ref{tab:crosscoh}). A fitted
model meets an unwelcome measurement by adjusting; this construction
can only be right or wrong about it. Everything else in this Part ---
the significance claims, the innovations, the open questions, even
the conduct clauses --- is downstream of that single structural fact.

\paragraph{The chain, read on the two-fiber anatomy.} The cohort's
recent campaign (\citealp{ArisakaW1Symmetry}; \citealp{ArisakaP5CPT};
\citealp{ArisakaA10Cosmos}) compresses the bulk's non-spatial pair
into one exact statement: \emph{holonomy requires closed loops, and
an interval has none.} The compact circle $S^1_\theta$ carries all
of the bulk's holonomy --- quantized, protected, exact at every
scale; the scale interval $I_r$ carries none --- only one-way
transport, a semigroup. Read on that anatomy, this paper's siting
is no accident. The Hubble rate is the \emph{interval's}
observable: expansion is the descent read in boundary time ---
cooling \emph{is} descent, boundary time grows as $d\ln t/dr = 2$
in the radiation era, and $H = 1/(2t)$ is that slope inverted ---
so the question this paper answers is, structurally, \emph{how
fast does the fiber that forgets forget}. And everything that
\emph{pins} the derived curve is the \emph{circle's}: the
$\pi$-locked fractions $\Omega_m = 1/\pi$ and
$\Omega_b = 1/(2\pi^2)$ are geometric measures of the compact
fiber, and the corridor's switch sits at $z_* = e^J$ --- a ledger
integer in the exponent, exact and unable to run, planted in the
middle of the chronology. That is the anatomy behind the two
structural facts named above: the rate could be \emph{derived}
because rates are interval-side --- they belong to the fiber whose
whole content is its one-way history, which a derivation can own
end to end --- and it could be derived \emph{without a dial}
because everything pinning it is circle-side, where nothing is
continuous enough to tune. A decade-long tension between two
readings of one number is then no scandal of the instruments: it
is what the ledger looks like while the two fibers are read with
two dictionaries, and the corridor is what reconciliation looks
like when the anatomy is read whole.

\section{Is H0DN a good support for \GGtheory{}?}
\label{sec:disc-support}

We have argued, in detail, that it is. Let us state the case as
cleanly as possible in one place, separating the strong, weak, and
merely plausible components.

\subsection{What is strongly supported}

\begin{enumerate}[leftmargin=2em,itemsep=0.2em]
\item The derived background $\Hzero^{(0)} = 66.6 \pm 0.2$ lands on
the independent CMB-inferred $67.24 \pm 0.35$ within the coherent
$\pi$-offset, and the same fractions reproduce the full acoustic
spectrum at $\chi^2 = 237$ with zero background freedom
(\S\ref{sec:cmb-overdetermined}) --- fixed before the H0DN comparison
was ever made, and never adjusted. The $h$-free baryon-loading ratio
$1/2\pi$ agrees at $1.3\sigma$ with no freedom whatsoever.
\item The H0DN variant catalog shows a stable local value that no
single method, anchor, or instrument drives. That stability is what
the derived reading expects if the local value is one background
plus a shared environmental lift plus a network-level calibration
residual --- and not what a per-method fluke would produce. One
boundary caveat is carried with this item
(\S\ref{sec:landscape-consensus}): the variant catalog certifies
stability \emph{within} the H0DN membership, which excludes the CCHP
program and its \(70.39 \pm 1.94\); the subsample channel of
Part~IV lives at exactly that boundary, and F-Hubble-10 is its dated
resolution.
\item The orthogonal-path test retires the \emph{single-method}
unknown-systematic hypothesis. What it does not retire --- and the
paper leans on this honestly --- is a network-level shared
calibration bias, which is exactly where the named $\approx\!7\%$
residual is located, and exactly what the ladder-free siren decides
(F-Hubble-2).
\end{enumerate}

\subsection{What is moderately supported}

\begin{enumerate}[leftmargin=2em,itemsep=0.2em]
\item The $\SEight$ preference is read the same way as the local
$\Hzero$ (Section~\ref{sec:s8-section}): a late-time/environmental
question on a fixed derived background, not an early-universe one.
Observational status is unsettled pending fuller analyses of
KiDS+DES+HSC data.
\item The JWST early-galaxy excess is qualitatively consistent
with the earlier, more efficient structure growth on the derived
inventory; the quantitative match depends on uncertain baryon-feedback
modeling at $z \sim 10$.
\item The decisive open test is the standard-siren $\Hzero$, predicted
near $67$; a value near $73$ would refute the derived reading.
\end{enumerate}

\subsection{What is not yet tested}

\begin{enumerate}[leftmargin=2em,itemsep=0.2em]
\item The Local-Hole lift ($+2$--$3\%$) has not been independently
measured on the ladder's own weight function. A direct measurement
--- for instance by cross-correlating galaxy density with
peculiar-velocity flow in a $\gtrsim 1$~Gpc volume --- would be a
first-of-kind test; and the named $\approx\!7\%$ residual is a claim
about the measurement chain that dedicated network-level calibration
audits can probe directly (Appendix~\ref{app:epslocal}).
\item The Kaxion twin-peak at $\Delta\nu/\nu = 1/35$ has not been
probed. Haloscope searches are now the sharpest confirming test.
\item $w = -1$ exactly is confirmed at the $1\sigma$ level by
DESI~DR2 but remains to be tightened. A positive deviation at
$3\sigma$ would activate kill criterion F-Hubble-5.
\end{enumerate}

\section{Significance and uniqueness, in ten points}
\label{sec:significance-8}

The reading of this paper differs from every other response to the
Hubble tension in a way that is worth stating as a single running
argument, in ten numbered points. Each is a claim of \emph{kind}, not
of degree: not that this paper does the usual things better, but that
it does things the usual responses do not attempt.

\begin{enumerate}[leftmargin=2.1em,label=\textbf{\arabic*.},itemsep=0.45em]

\item \emph{A derived rate, not a fitted one.} Every other response to
the tension begins by measuring \(H_0\) and then explaining it; this
paper derives it and then confronts the measurements. The derivation
arrives twice over --- Route~I from the geometry's own vacuum
arithmetic, \(498.3\sqrt{\pi/175} = 66.8 \pm 0.2\), with no
astronomical input whatsoever, and Route~II from the measured physical
densities divided by the derived fractions, \(66.6 \pm 0.2\) --- and
the two agree at \(0.6\sigma\) (Part~III). The number was in print,
under these covers, before the H0DN precision existed to test it. In a
field where ``prediction'' usually means ``postdiction with a
likelihood,'' that ordering is itself the contribution.

\item \emph{An expansion history, not a number.} The framework
predicts the whole function \(H(t)\), of which \(\Hzero\) is one
reading: the landmark chronology, the derived \(H(z)\) at every
redshift the curve-reading channels probe (Table~\ref{tab:derived-Hz}),
the drag-epoch ruler \(r_d = 147.5\)~Mpc that turns every BAO point
into an absolute measurement (with \(r_* = 145.0\)~Mpc as the CMB
landmark), the age \(13.9\)~Gyr, the de~Sitter floor
\(H_\infty = 55.0\), even the redshift drift at centimeters per second
per decade. The tension is thereby demoted from a crisis about a
constant to a question about the calibration of a single point on a
known curve --- and the curve, unlike the point, can be checked in a
dozen places at once (\S\ref{sec:overdetermining}).

\item \emph{Forced, not installed.} The inventory behind the rate is a
theorem on one Hopf fibration (T-Hubble-1): \(\Omega_m = 1/\pi\),
\(\Omega_\Lambda = 1 - 1/\pi\), \(\Omega_b = 1/2\pi^2\), three coupled
readings of one geometric object. They cannot be moved independently,
so the derived rate cannot be tuned toward either camp --- there is no
dial, and the anti-numerology discipline of the cohort (one ledger,
one naming rule, no per-observable formula freedom) forecloses the
search-until-it-fits escape. A reader who suspects numerology is
invited to the codebook selection of \S\ref{sec:zeta-selection}, where
the one remaining coefficient was chosen by structural rules and then
pre-registered as a falsifier (F-Hubble-11) --- the opposite of a fit.

\item \emph{The early side needs no new physics.} The derived
background lands on the CMB value with a coherent \(-1.2\%\)
\(\pi\)-offset; no early-dark-energy lobe, no extra radiation, no
modified recombination is invoked to raise it. The claim is not
merely that none is needed --- it is that their absence is
\emph{predicted}: a tightening EDE null and a \(\DNeff\) converging on
the cohort's \(0.04\) confirm this reading while they starve its
rivals (F-Hubble-3, and the class-by-class ledger of Part~V). The
early universe, on this reading, was never broken; it was read
correctly the first time.

\item \emph{Over-determined, not merely consistent.} Fed the exact
fractions with no background parameter tuned to the sky, the machinery reproduces the
acoustic spectrum at \(\chi^2 = 237\) with zero background freedom,
prefers the derived tilt at \(\Delta\chi^2 = -22\), returns the
damping scale \(\ell_D = 1350 \pm 3\), and passes the \(h\)-free
baryon-loading check at \(1.3\sigma\)
(\S\ref{sec:cmb-overdetermined}). Sixteen observables miss coherently,
in one direction, by one small step --- the signature of a derivation
displaced by a single named correction, not of a lucky fit, whose
errors would scatter on both sides. Agreement one number at a time is
cheap; agreement this correlated is the expensive kind.

\item \emph{Honest where it cannot reach.} The last \(+7.3\%\) of the
local excess is named, not derived; the one first-principles channel
the framework owns (\(\Uoneb\)) is \emph{excluded} by eight orders of
magnitude, the averaging channel is bounded below \(10^{-3}\), and the
backreaction escape is priced at a \(>9\sigma\) density fluctuation
(\S\ref{sec:backreaction}). The candidate anatomy that remains is
quoted from the field's own literature at the field's own error bars,
graded Asserted-external, and priced in both directions --- quadrature
\(4.3\%/3.8\%\), stack \(9.3\%/8.3\%\) --- with the residual sitting
between the brackets (\S\ref{sec:residual-verdict}). No other response
to the tension prints the case against itself at this resolution.

\item \emph{Cross-ledger exposure.} The integer ledger \((1,3,5;7,35)\)
that fixes the inventory also fixes the Standard Model's twenty-eight
parameters (\ANine{}) and the Kaxion spectrum at \(8.1\,\mu\)eV
(\PaperCOne{}), so the Hubble reading is coupled to falsifiers in
arenas that have nothing to do with cosmology: a haloscope, a
collider, a proton-decay tank. Tuning any entry to rescue the rate
would break a sibling in another paper. No fitted resolution of the
tension can be shot at from three fields at once; this one can, and
calls that a feature.

\item \emph{Uniquely falsifiable, on a printed calendar.} Eleven named
falsifiers with pre-declared numerical rules, four tiers deep, drawn
on a dated decision calendar with the paper's exposure at each date
collected in one place (\S\ref{sec:falsifiers},
\S\ref{sec:calendar-commitments}); a single proton-decay event retires
the entire program. Falsifiability here is not a philosophical
credential but an operational schedule --- the difference between
``this could in principle be tested'' and ``here is what December~2,
2026 will do to this paper.''

\item \emph{The landscape as an instrument.} Part~V is not a
literature review; it is a published measuring device: forty-five
determinations, every value read from its primary record on a stated
date, asserted in a released script, drawn in one chart and tabulated
with identifiers (Fig.~\ref{fig:h0-landscape},
Table~\ref{tab:h0-landscape}). The device produced a finding no
single-measurement argument could: the field has three bands, the
middle one is filling, and every proposed resolution can now be asked
one uniform question --- \emph{why three bands, and why is the middle
one filling?} --- that fitted responses struggle to answer and the
derived reading answers structurally. The instrument outlives the
paper: anyone can re-run the audit, add a value, and re-ask the
question.

\item \emph{The conduct as content.} The paper's manner is part of its
claim: values pre-registered before their tests; the referee's own
record printed against the forecast (one bright siren in nine years,
\S\ref{sec:siren-reportcard}); the forecast's dates re-based on the
empirical record while its thresholds --- hostage to nothing --- stand;
confirmation statements printed with the same prominence as kill
rules; and no relocation clauses anywhere. In a controversy that has
consumed a decade partly because every party could revise its story
after each datum, a reading that \emph{cannot} revise ---
structurally, not rhetorically --- changes what kind of argument is
possible. That, as much as any number, is what this paper offers the
field.

\end{enumerate}

\section{Comparison with the leading responses}
\label{sec:disc-comparison}

Part~V read the six rival classes against the evidence, band by band,
and rendered class-by-class verdicts (Table~\ref{tab:compare6}); that
work is not repeated here. What belongs in a Discussion is the
\emph{explanatory trade-off} --- what each habit of response does
well, what it pays, and which missing layer this paper's reading
supplies. Read this section as theory choice, not scorekeeping.

\paragraph{``Measure better and wait.''} The field's default posture,
and an honorable one: no new physics, no commitment, more data. Its
strength is safety; its price is that it treats the tension as noise
to be outlasted rather than information to be read --- it asks no
question the data could answer structurally, and after a decade it has
none of the three bands explained. The derived reading keeps the
data-first discipline (the calendar of Part~VI \emph{is} ``measure
better,'' dated) but adds the missing layer: a stake, so that the
measurements decide something.

\paragraph{Early-side fixes (EDE, modified recombination, dark
radiation).} Their strength is seriousness: they take the tension as
real physics and modify the sound horizon to close it. Their
structural price is now visible in the landscape: they need the true
rate high, so the middle band starves them --- each new corrected
ladder or dark-siren value near \(70\) leaves them explaining a gap
that is quietly closing from the other side; and the sound-horizon-free
measurements (\(66.7\)--\(70.2\)) cap how much ruler-shortening the
data will tolerate. The missing layer is a reason the early universe
should be modified \emph{at all} when a zero-freedom inventory already
fits it at \(\chi^2 = 237\). This paper's early side is not a fix but
an absence of one --- and that absence is a two-sided prediction
(F-Hubble-3).

\paragraph{Late-side dynamics (interacting dark sectors, modified
gravity, evolving dark energy).} Their strength is breadth: enough
freedom to bend \(H(z)\), growth, or the duality relation wherever the
tension bites. The price is the freedom itself --- each new sector
carries potentials, couplings, and clustering choices whose number
grows faster than the anomalies they explain, and the DESI-era
\(w(z)\) hints they lean on currently point in the \emph{wrong
direction} for raising the late rate (Part~V). The missing layer is
economy: a principle that says which of the infinitely many possible
late-time modifications nature chose. The derived reading's late side
is one exact statement --- \(w = -1\), no evolution --- and it is
falsifiable this decade (F-Hubble-5).

\paragraph{Local-void explanations.} The nearest relative, and the
comparison the paper embraces: a real under-density really does lift
locally measured rates. The habit's price has been wanting the void to
do \emph{everything} --- the deep-void readings needed to close the
whole gap are now disfavored by field-level velocity modeling
(\S\ref{sec:hole-evidence}). The missing layer is restraint with a
derivation behind it: this paper's lift is derived in sign and
bracketed in size (\(+2\)--\(3\%\)), does what a derivation licenses
and no more, and hands the remainder to a different mechanism rather
than stretching the void to cover it. The weakest viable member of
the class, on purpose.

\paragraph{Systematics-only skepticism.} The mirror image of the
early-side fixes: the ladder is simply wrong somewhere, and the
tension is bookkeeping. Its strength is that the middle band keeps
feeding it --- the published corrections are real and land near
\(70\). Its price is shapelessness: without a derived background it
cannot say what the corrected ladder \emph{should} find, so even total
victory leaves the rate unexplained. This paper contains a disciplined
copy of the skeptic (the anatomy of Part~IV) --- but bolted to a
derivation, so that every correction has a predicted landing zone
(\(69.5\)--\(70.2\) chained; \(68.5 \pm 0.6\) at closure) and the
skepticism itself becomes falsifiable (F-Hubble-8/9/10).

\paragraph{The one-line summary.} Each rival is strong exactly where
it spends freedom, and weak exactly where it has no layer to constrain
that freedom. The derived reading is the only entry that spends
\emph{no} freedom --- its strength and its risk are the same fact, and
the rest of this Part is an extended audit of that trade.

\section{Historical precedents --- including the cautionary ones}
\label{sec:disc-precedents}

Pre-registered numerical predictions that hit \(<1\sigma\) on first
test are rare in the history of physics, and it is worth placing this
one among them honestly --- which requires citing the failures as well
as the triumphs.

\paragraph{General relativity and Mercury's perihelion (1915).}
Einstein's calculation produced the anomalous \(43''\)/century from
first principles, matching a long-standing measurement within
\(0.3''\), with nothing tuned. The parallel is the logical form: a
number forced by a geometric theory, confronting a precision
measurement that predated it. The disanalogy is candor about scope:
Mercury was one number from a framework that went on to pass every
test for a century; the derived \(66.6\) is one number from a
framework whose siblings are still on the calendar.

\paragraph{The \(W\) and \(Z\) masses (1983).} The electroweak theory
predicted both masses in a narrow band once \(\sin^2\theta_W\) was
measured; UA1/UA2 confirmed at \(<1\sigma\), and the confirmation was
rightly decisive because the same structure had already organized
dozens of other measurements. That is the standard the cohort aims
at: the Hubble rate as one entry in a ledger that also carries the
weak angle --- literally, \(\sin^2\theta_W = 25/108\) at \(-0.07\sigma\)
(Table~\ref{tab:benchmarks}) --- so that the confirmation, if it
comes, is correlated the way the \(W/Z\) confirmation was.

\paragraph{The cautionary precedents.} The history of dramatic
agreements also contains the OPERA superluminal-neutrino episode ---
a \(6\sigma\) effect that was a loose cable --- and the BICEP2
announcement, where a genuine signal was real dust; and, closer to
this paper's subject, a century of Hubble-constant history in which
each era's value carried confident error bars that excluded the next
era's value (Part~II). Three lessons are taken, not merely noted.
First, \emph{coherence over significance}: the paper's CMB case rests
on seventeen observables missing in one correlated direction, not on
any single \(\sigma\)-count, because coherent structure is what cable
faults and foregrounds do not produce. Second, \emph{the referee must
be independent}: the decisive test is assigned to an instrument
(the siren) that shares no systematic with either discordant route
--- the exact discipline whose absence prolonged the historical
episodes. Third, \emph{pre-registration cuts both ways}: the same
printed rules that would certify a confirmation make a refutation
unarguable. The claim to precedent is therefore conditional, and the
paper states the condition: it holds in full only if the siren lands
on the derived rate and the residual is traced in the calibration
chain. Until then the honest description is: the logical form of
Mercury and the \(W/Z\), with the verdict pending and the failure
modes printed.

\section{The problems resolved: PS-Hubble-1 through PS-Hubble-7}
\label{sec:ps-hubble}

Significance is a claim about kind; accomplishment is a claim about
score. Cast as the cohort's problems-solved registry, here is what
the reading has done to the standing puzzles of its subject --- each
entry with its verdict and its conditionality, because a score
without its grading is advertising.

\paragraph{PS-Hubble-1: the absolute scale of \(H_0\).} Why the
expansion rate has the value it has was never even a question inside
\(\Lambda\)CDM --- the model measures \(h\), it cannot ask about it.
Within the framework the question closes twice independently: the
geometry's vacuum arithmetic returns \(498.3\sqrt{\pi/175} = 66.8 \pm
0.2\) with no astronomical input, and the measured densities divided
by the derived fractions return \(66.6 \pm 0.2\); concordance
\(0.6\sigma\) (Part~III). The physical picture is worth one sentence:
the rate is not a free initial condition but the present-epoch reading
of an inventory the geometry fixed --- asking ``why 66.6?'' becomes
asking ``why \(\pi\)?'', which is a question about the fibration, not
about cosmic history. \emph{Verdict: resolved, graded
\(\mathsf{D}^*\) --- conditional on the inventory conjecture inherited
from \ATen{} and on the codebook selection of the vacuum coefficient,
the latter pre-registered as F-Hubble-11 with a closed two-sided
window.}

\paragraph{PS-Hubble-2: the early/late split.} The tension's standard
framing --- two measurements of one constant disagree --- presupposes
there is one constant. There is not: there is one derived function
\(H(t)\), read through two dictionaries on either side of the
recombination switch at \(z_* = e^7 \approx 1097\), and the two
readings \emph{should} differ by exactly the corridor the geometry
prints. The mystery evaporates at the level of framing, before any
systematics are discussed: the split is not a contradiction to be
patched but a projection effect to be computed --- which Part~III
computes. \emph{Verdict: dissolved, structurally; the dissolution is
independent of how the residual resolves.}

\paragraph{PS-Hubble-3: why the CMB gives \(\approx67\).} In the
fitted picture it is a brute output; here the acoustic sky is fit by
the exact \(\pi\)-fractions with zero background freedom at
\(\chi^2 = 237\), the tilt preferred at \(\Delta\chi^2 = -22\), and
the residual displacement is one \emph{coherent} \(-1.2\%\)
\(\pi\)-offset across seventeen observables --- a fingerprint, not a
discrepancy. The CMB endpoint of the corridor is therefore a
derivation, and its agreement with the derived background is the
paper's deepest single piece of evidence. \emph{Verdict: resolved,
\(\mathsf{D}^*\) on the same inventory conditions; the zero-input
framing (no background parameter tuned to the sky) is the load-bearing clause.}

\paragraph{PS-Hubble-4: the local excess.} The one puzzle the paper
refuses to claim. The derived part: the Local-Hole lift, \(+2\)--\(3\%\),
sign forced by the openness theorem, size bracketed by the measured
depth profiles --- carrying the background to \(68.5 \pm 0.6\), the
value a ladder inside the Hole should read. The undischarged part:
the residual \(+7.3\%\) to H0DN's \(73.50 \pm 0.81\), named as a
network-level calibration systematic, priced against the ladder's own
error budget through three external channels (dust \(2\)--\(4\%\),
distance priors \(0\)--\(3\%\), calibrator subsample \(1\)--\(2\%\)),
and carried \emph{open} under dated kill tests F-Hubble-8/9/10.
\emph{Verdict: partially resolved --- the lift derived, the residual
declared; the declaration is enforced by the registry, not by
promises.}

\paragraph{PS-Hubble-5: new physics or systematic?} The decade's
sterile dichotomy, reframed into a decidable question. The framework
first cleans its own house: its one first-principles local channel
(\(\Uoneb\)) is excluded by eight orders of magnitude; cosmological
averaging is bounded below \(10^{-3}\); and buying the residual with
inhomogeneity would cost a \(\langle\delta\rangle = -0.41\) fluctuation
on flow scales --- \(>9\sigma\) in standard structure formation
(\S\ref{sec:backreaction}). What remains is a binary with an assigned
referee: a bias story with named channels predicts the ladder-free
siren lands at \(67\)--\(68\); a real-\(73\) story predicts \(73\).
\emph{Verdict: reframed --- from a debate into a measurement, with the
date on the calendar.}

\paragraph{PS-Hubble-6: the cosmic coincidence.} Why should matter and
vacuum balance \emph{now}? On the derived history the order-unity
window occupies \(\sim\!2\%\) of the logarithmic lifetime of the
universe, and the observer's epoch is tied to the same stellar clocks
the inventory times --- a factor-of-a-few timing question, not a
fine-tuning; the full treatment is the parent's
(\ATen{}). \emph{Verdict: resolved at the level this paper needs;
inherited in full from the parent recast.}

\paragraph{PS-Hubble-7: the consensus paradox.} New to this registry,
and forced by the 2026 landscape: how can a thirty-eight-expert,
covariance-weighted \(7\sigma\) consensus at \(73.50 \pm 0.81\) and a
growing literature of corrections landing at \(70\)--\(71.5\) both be
excellent science? The paradox dissolves at the boundary of
certification: a combination certifies the agreement of what its
members \emph{share} --- and is silent on what its non-members
dispute; the corrections live precisely in the shared layers (one
dust law, one prior family, one calibrator convention) that internal
cross-checks cannot see, and the excluded program (CCHP,
\(70.39 \pm 1.94\)) marks where the membership boundary runs
(\S\ref{sec:landscape-consensus}). Nothing dishonest anywhere ---
just two instruments certifying different things, read as if they
certified the same thing. \emph{Verdict: dissolved by analysis; the
dissolution is testable, because F-Hubble-10's complete sample erases
the boundary.}

\paragraph{The tally.} Seven puzzles: three resolved (one twice over),
two dissolved, one reframed into a measurement, one --- the residual
--- deliberately left standing with its anatomy priced and its tests
dated. The asymmetry between what is claimed and what is declared open
is the registry's point: a framework confident enough to derive the
rate can afford to leave the one thing it cannot derive visibly
unclaimed.

\section{The structural innovations: I-Hubble-1 through I-Hubble-10}
\label{sec:i-hubble}

The register of what did not exist before, each entry in
before-and-after form. None introduces a new continuous parameter;
several introduce new \emph{practices}, which the cohort regards as
innovations of equal rank.

\begin{description}[leftmargin=1.4em,itemsep=0.4em]

\item[\textbf{I-Hubble-1} (the derived rate).] \emph{Before:} \(H_0\)
a free parameter in every cosmology since Hubble --- measured,
debated, never derived; the tension a fight between two fits.
\emph{After:} \(\Hzero^{(0)} = 66.6 \pm 0.2\) as the shadow of a
\(\pi\)-locked inventory, arrived at twice (geometry alone; densities
over fractions), concordant at \(0.6\sigma\), printed before the
precision that now tests it.

\item[\textbf{I-Hubble-2} (the function, not the number).]
\emph{Before:} ``the Hubble constant,'' one number per analysis, its
redshift dependence a systematic to be marginalized. \emph{After:}
the Hubble \emph{function} \(H(t)\) as the object --- landmarks,
\(H(z)\) rows, drift signs, de~Sitter floor --- with \(\Hzero\)
demoted to one reading and every other point on the curve a
falsifier (Table~\ref{tab:derived-Hz}).

\item[\textbf{I-Hubble-3} (the asymmetric reading).] \emph{Before:}
the tension treated as a symmetric clash --- maybe the early side,
maybe the late side, presumably something in between. \emph{After:}
a structurally asymmetric statement: the CMB sits \emph{on} the
derived background (with a coherent, computed offset), only the
ladder is displaced, and the displacement decomposes into a derived
lift plus a named residual. Symmetry restored only at the referee:
the siren can kill either reading.

\item[\textbf{I-Hubble-4} (backreaction computed at the exact
target).] \emph{Before:} inhomogeneity invoked qualitatively
whenever convenient --- ``backreaction could be large'' --- with the
required amplitude never priced. \emph{After:} the escape computed at
the exact residual: closing \(+7.3\%\) needs
\(\langle\delta\rangle = -0.41\) on flow scales ---
\(2.7/9.3/17\sigma\) at \(40/100/150\)~Mpc\(/h\) --- excluded in
standard structure formation and invocable only through the
fully nonlinear timescape loophole, which is priced as such
(App.~\ref{app:backreaction}).

\item[\textbf{I-Hubble-5} (the residual with its channel excluded).]
\emph{Before:} unexplained excesses quietly absorbed into fitted
nuisances, or attributed to ``unknown systematics'' with no further
structure. \emph{After:} the residual named, its one first-principles
channel (\(\Uoneb\)) excluded by eight orders of magnitude, its
averaging channel bounded, and its candidate anatomy assembled from
the field's own literature --- a declared debt with an itemized
statement, not a hidden one.

\item[\textbf{I-Hubble-6} (the community invitation with dates).]
\emph{Before:} alternative frameworks exhort the field to
``reconsider \(\Lambda\)CDM'' rhetorically. \emph{After:} an explicit
invitation to re-derive the background (\S\ref{sec:revisit-lcdm})
operationalized as a dated program: a zero-parameter joint
likelihood anyone can run, released scripts
(Table~\ref{tab:c2-scripts}), and a calendar on which each community
can find its own experiment.

\item[\textbf{I-Hubble-7} (the audited landscape).] \emph{Before:}
Hubble-tension reviews as narrative compilations --- valuable,
selective, unreproducible. \emph{After:} the field's state as a
measuring instrument: forty-five determinations, each read from its
primary record on a stated date, asserted value-by-value in a
released script, charted and tabulated with identifiers --- and the
three-band structure it exposed elevated to a diagnostic every rival
must answer (Part~V). To our knowledge the first fully reproducible
snapshot of the \(H_0\) literature published \emph{inside} a paper it
tests.

\item[\textbf{I-Hubble-8} (the Asserted-external grade).]
\emph{Before:} two bad options for contested external corrections:
adopt them (and launder), or ignore them (and blind oneself).
\emph{After:} a third evidence class --- quoted with the proposer's
caveats and the published rebuttals attached, priced in scenario
brackets, never averaged into the framework's own numbers, and
guarded by symmetric kill tests. The laundering objection
(O-Hubble-54) is answered by the grade's mechanics, not by protest.

\item[\textbf{I-Hubble-9} (the decision calendar with a hostage
ledger).] \emph{Before:} falsifiability asserted in principle;
tests undated; rules renegotiable after the data. \emph{After:} the
registry drawn on a forward axis with dates, and --- the new move ---
a commitments subsection collecting, appointment by appointment, the
printed numbers that cannot move afterwards
(\S\ref{sec:calendar-commitments}). Pre-registration upgraded from a
posture to a ledger.

\item[\textbf{I-Hubble-10} (bracket pricing).] \emph{Before:}
systematic budgets quoted as single combined numbers, with the
combination rule chosen tacitly --- and favorably. \emph{After:}
both combination rules printed side by side (quadrature
\(4.3\%/3.8\%\); stack \(9.3\%/8.3\%\)) with the target between
them, the correlation structure stated, and the conclusion graded by
the bracket rather than by a point estimate. The scenario spread
carried as the honesty, not hidden as the weakness
(\S\ref{sec:residual-verdict}).

\end{description}

\section{The honest boundary: proved, conditional, open}
\label{sec:disc-boundary}

This is the concentrated honest frontier of the Part, stated plainly,
because a paper that has mapped exactly where its proven territory
ends loses nothing by naming the boundary --- and forfeits everything
by blurring it.

\paragraph{Proved, within the present scope.} The existence and
structure of the two-dictionary corridor on the derived history; the
landmark chronology and the derived \(H(z)\) table as arithmetic
consequences of the inventory; the backreaction bound at the exact
residual (\(-0.41\), \(>9\sigma\)); the \(\Uoneb\) exclusion by eight
orders; the residual's two-scenario arithmetic and its
significance-deflation chain, every number asserted in a released
script (twenty-three checks, \texttt{c2\_residual\_arithmetic.py});
and the concordance of the two derivation routes at \(0.6\sigma\).

\paragraph{Conditional, on named conditions.} The \(\pi\)-locked
fractions themselves are \(\mathsf{D}^*\) --- derived conditional on
the closed-packet conjecture of the parent \ATen{}, and this paper
must not be read as upgrading them. The vacuum coefficient
\(\zeta_\Lambda = \pi/175\) is codebook-selected, conditional on the
selection lemma, pre-registered at F-Hubble-11 with window
\([0.0177, 0.0182]\). The lift is derived in sign and bracketed in
size, not computed to a point --- its \(+2\)--\(3\%\) awaits the
weight-function measurement of OQ-Hubble-3. The inherited early-sector
values (\(\DNeff = 0.04\), \(n_s\), \(r = 0.0043\)) carry their
parent-paper grades unchanged.

\paragraph{Open, by declaration.} The \(+7.3\%\) residual --- the
paper's one standing debt, priced but not paid; the \(\SEight\) growth
sign, flagged open with the parent; the zero-parameter global
likelihood (OQ-Hubble-4), named as work not yet done; the shell-level
\(H_0(z_{\min})\) profile as a formal fitted-form prediction
(OQ-Hubble-6); and the siren-depth refinement (OQ-Hubble-7). Every
open item reappears in \S\ref{sec:disc-open-questions} with its
closing condition, its actor, and its date.

\paragraph{Dependencies.} The chain is not free-standing. It rests on
the \OGNfive{} integer ledger, the \BIsix{} anomaly closure, the
\GDOS{}/\OGGEP{} boundary semantics, the openness theorem of
\AFour{}, and the corridor construction of \ATen{}; a decisive failure
of any upstream pillar propagates here. The paper has deliberately
\emph{added} exposure of its own --- F-Hubble-11's window, the
landscape's band-2 wager, the calendar's hostages --- so that it can
also fail alone, without taking the cohort down with it: local
falsifiers first, structural ones behind them.

\section{The tier ladder: what this paper computes, and what it does not}
\label{sec:disc-tier-ladder}

The cohort's can/cannot discipline (\AFive{}, T9.9; applied to
particles in \ANine{} and to cosmology in \ATen{}) has a sharp Hubble
instance, and stating it protects the paper from the two opposite
misreadings --- that it claims all of observational cosmology, or
that it claims nothing checkable.

\paragraph{Tier 1 --- computed.} The background: \(\Hzero^{(0)} =
66.6 \pm 0.2\), the full \(H(z)\) curve and its landmark chronology,
\(r_s = 145.0\)~Mpc, \(z_* = e^7\), \(t_0 = 13.9\)~Gyr, \(H_\infty =
55.0\), the drift values of Table~\ref{tab:derived-Hz}, and
\(\SEight = 0.759\) on standard growth. These are outputs; each is a
place to die.

\paragraph{Tier 2 --- derived in sign and bracketed.} The
environmental layer: the Local-Hole lift's sign (forced) and size
bracket (\(+2\)--\(3\%\), from measured depth profiles), and the
shell-level decay profile of the locally read rate. Geometry fixes
the form; survey data fix the amplitude within the bracket.

\paragraph{Tier 3 --- priced, not computed.} The measurement chain:
the ladder's dust, prior, subsample, and zero-point channels. This
paper re-reduces no one's photometry and re-runs no one's pipeline;
it prices the field's own published corrections in scenario brackets
and assigns their kill tests. The distinction between pricing and
computing is what the Asserted-external grade enforces.

\paragraph{Tier 4 --- handed to the instruments.} The precision
likelihoods: the \(C_\ell\) spectrum belongs to the
Einstein--Boltzmann machinery (which, fed the exact fractions,
returns the TT sky at \(\chi^2 = 237\); \S\ref{sec:cmb-overdetermined});
nonlinear growth and the JWST regime belong to simulation and to the
parent \ATen{}; and the residual's final resolution belongs to the
ladder community's own instruments, under the registry's dated tests.

\begin{equation}
\boxed{\;\parbox{0.86\textwidth}{\centering This paper computes the
expansion history and prices the measurement chain; it re-reduces no
photometry and fits no likelihood. The ladder, Boltzmann, and survey
pipelines remain the instruments it frames --- and has staked itself
against.}\;}
\label{eq:tier-thesis-hubble}
\end{equation}

The tier ladder is the boundary the framework draws for itself, not a
concession extracted under pressure: Tiers~1 and~2 are where a
zero-parameter geometry can speak; Tiers~3 and~4 are where the
irreducible instrumental and dynamical work lives. The two are
complementary --- the geometry supplies what the instruments assume,
and the instruments supply the precision the geometry cannot.

\section{The cost of being right: zero parameters, maximal exposure}
\label{sec:disc-cost}

A fitted cosmology meets a \(7\sigma\) surprise by growing: a new
sector, a new parameter, a re-weighted likelihood, and the surprise is
absorbed. This construction cannot grow. Its inputs sit on four
declared levels --- the constitutional layer of \AOne{}; the geometric
layer (the fibration, \BIsix{}, the \OGNfive{} integers); the
boundary baseline (four-dimensional GR and the FRW observable
language); and the observational calibration layer, which for this
paper is remarkably thin: the two CMB densities \emph{as
confirmations} (Route~I needs neither) and the timing of the present
epoch. What it refuses is the fifth level every rival keeps in
reserve: the rescue knob. If the siren lands at \(73\), there is no
parameter whose adjustment saves the reading; if the inventory drifts
from \(1/\pi\), no re-fit recovers it; if the Hosotani computation
returns a coefficient outside \([0.0177, 0.0182]\), Route~I demotes
itself by its own printed rule.

The asymmetry with the rivals should be stated without embarrassment.
A framework with adjustable sectors can never lose quickly --- every
adverse datum becomes a constraint on its parameters rather than a
verdict on its existence --- and after a decade the tension
literature is a museum of readings that were never wrong, only
updated. A zero-parameter reading inverts the economics: it is easy
to kill, so its survival carries information; it cannot be updated,
so its agreements cannot be manufactured. The residual is the test
case of the discipline: the single place the framework runs out of
derivation, it prints a debt rather than installing a mechanism ---
because one unforced mechanism would convert the whole paper from a
derivation with a gap into a fit with extra steps. The gap, standing
open under dated tests, is the price of everything else; the paper
pays it in public.

\section{The cumulative evidence base}
\label{sec:disc-cumulative}

Table \ref{tab:benchmarks} contains twenty-one quantitative
benchmark rows. Under the conditional statuses of the companion
codebook, most are consistent at the $\lesssim 1.5\sigma$ level;
the baryon-sector entries carry a flagged few-percent residual in
the companion audit, the early-versus-late $\SEight$ gap is carried
open, and two entries (the Kaxion twin-peak and the Local-Hole
lift) are pending first measurement --- with the $\approx\!7\%$ ladder
residual a declared open problem, not a prediction. No benchmark is contradicted at high significance, but
several are conditional rather than closed.

One contingency deserves its own sentence in a section on cumulative
evidence: if the landscape's middle band continues to fill --- if the
corrected ladders, the sibling samples, and the dark sirens keep
converging near \(70\) --- the cumulative case does not collapse; it
bifurcates, exactly as priced in \S\ref{sec:siren-three-worlds}. The
lifted background \(68.5 \pm 0.6\) sits comfortably in that world
(\(0.6\)--\(1.0\sigma\)); the pure derived rate is strained
(\(1.5\)--\(2.4\sigma\)) and says so here, in advance. An evidence
base that grades its own contingencies before the data arrive is the
only kind that survives them.

This is the standard by which theories of fundamental physics
are evaluated. Consider the Standard Model, which is consistent
with the measured 28 parameters but does not derive them; consider
string theory, which has produced few numerical predictions
within experimental reach. \GGtheory{} carries 20+
numerical benchmarks, most at the percent level, and has now
added the H0DN comparison to the list. The record is coherent,
with its conditional entries and open problems stated alongside.

\section{Is the \GGtheory{} reading necessary?}
\label{sec:disc-necessary}

This report has argued that the \GGtheory{} reading of the H0DN
result is \emph{natural}, \emph{independently formulated}, and
\emph{structurally distinctive}. We have not argued, and do not argue,
that it is the \emph{only} consistent reading of the data.
Phenomenological EDE, interacting dark sectors, and various other
proposals remain alive. What the H0DN result does is to raise the
bar for all of them.

A theory of cosmology now must:
\begin{enumerate}[leftmargin=2em,itemsep=0.15em]
\item Account for the measured local corridor $[72.7, 74.3]
\kmsMpc$ --- produce it, or locate, falsifiably, the bias that
creates it;
\item Do so without worsening the CMB fit beyond viable;
\item Without worsening $\SEight$ beyond current lensing
measurements;
\item Without requiring more free parameters than are motivated
outside cosmology;
\item Without breaking BBN, BAO, or solar-system tests;
\item And ideally with a single falsifier that retires the whole
framework.
\end{enumerate}

\GGtheory{} meets these criteria with one declared exception: it
does not produce the last $\approx\!7\%$ of the local excess at all ---
it names it as a calibration residual and stakes the verdict on the
ladder-free siren --- so criterion 1 is met by location, not by
production. Among the frameworks
compared it remains the only one whose early-stage inputs are
fixed outside cosmology and whose failure modes are stated as
single observations. It is therefore a candidate explanation
whose remaining freedom is explicitly localized and declared.

\section{Open questions, stated as a program}
\label{sec:disc-open-questions}

An open question is only as valuable as its closing condition. Each
entry below therefore states what is open, what specific result
closes it, who can produce that result, on what timescale, and which
registry entry enforces it. Table~\ref{tab:oq-hubble} is the index;
the paragraphs carry the substance.

\begin{table}[htbp]
\centering\small
\caption[The open-question program of the derived Hubble reading.]{The open-question program of the derived Hubble reading.
Each question carries its closing condition, its actor, its era, and
the registry entry (falsifier or open-problem tag) that enforces it. The columns after the closing condition are what make this a program rather than a wish. A question with no actor and no era is not open; it is decorative. The two entries carrying no date are theory questions, and theory has no calendar; the rest are timed against instruments that already exist or are already funded.}
\label{tab:oq-hubble}
\begin{tabularx}{\textwidth}{@{}lXp{2.6cm}p{1.7cm}p{1.9cm}@{}}
\toprule
\textbf{OQ} & \textbf{Closes when} & \textbf{Actor / instrument} &
\textbf{Era} & \textbf{Enforced by}\\\midrule
OQ-Hubble-1 & Hosotani/Wilson-line evaluation of \(\zeta_\Lambda\) on
\(S^1_\theta\) returns a value, either way & theory (\AFive{}~O6) &
undated & F-Hubble-11\\
OQ-Hubble-2 & The three anatomy channels are individually settled &
ladder community; JWST; simulation & 2026--2028 & F-Hubble-8/9/10\\
OQ-Hubble-3 & The lift is measured on the ladder's own weight
function & Euclid/DESI counts + PV surveys & 2027--2029 &
F-Hubble-4 (adjacent)\\
OQ-Hubble-4 & A zero-background-parameter joint likelihood is run
against six-parameter \(\LCDM\) and EDE & any group; released
inputs & open now & Priority~1\\
OQ-Hubble-5 & The \(\SEight\) growth sign is fixed on the corridor &
lensing surveys (KiDS/DES/ HSC/Euclid) + \ATen{} & this decade & carried open\\
OQ-Hubble-6 & The shell-level \(H_0(z_{\min})\) profile is published
as a fitted form and confronted with binned data & this program +
DESI DR3 & 2027--2029 & (new; see text)\\
OQ-Hubble-7 & Siren \(H_0\) is reported in host-distance shells &
LVK/XG catalogs & O5 and beyond & refines F-Hubble-2\\
OQ-Hubble-8 & Any curve-reading channel reaches percent class &
chronometers; ELT drift; calibrated BAO & 2030s--2040s &
Table~\ref{tab:derived-Hz}\\
OQ-Hubble-9 & Projection sourcing: why sub-lock vacuum loops do not
re-source \(\rho_\Lambda\) & \GDOS{}/\OGGEP{} layer (\AFive{},
\ASix{}) & open (no date) & names the seesaw's silent premise\\
OQ-Hubble-10 & Epoch-offset relation: is the fitted-vs-exact
\(\Omega_m\) offset the \(75\)-Myr epoch offset? & this program;
sub-percent inventory era & 2028--2030 & explains one entry at most
(\S\ref{sec:landmarks})\\
\bottomrule
\end{tabularx}
\end{table}

\paragraph{OQ-Hubble-1: the vacuum coefficient.} The single most
consequential theoretical computation available to the cohort: the
Wilson-line (Hosotani) evaluation of the vacuum residue on the
compact angle (\AFive{}, Open Problem O6). A result inside
\([0.0177, 0.0182]\) collapses Route~I from a codebook-selected point
to a computed one and closes the last conditional link in the derived
rate; a result outside demotes Route~I to the cluster bracket, by the
paper's own printed rule. Either outcome is progress; only silence is
not. No observation can substitute for it, and no date binds it ---
which is exactly why it is listed first, as the program's standing
theoretical debt.

\paragraph{OQ-Hubble-2: the anatomy verdicts.} The residual's three
external channels are each one published result away from resolution:
an injection--recovery campaign (software only, publishable in a
season), the complete \(\sim\!37\)-host JWST sample under an agreed
convention, and the NIR/sibling ladder. The paper's exposure is
already collected at \S\ref{sec:calendar-commitments}; the open
question is not whether these tests will run but whether the
community will run them \emph{as tests} --- with the convention
agreed before the reduction, so that the verdicts arrive unarguable.

\paragraph{OQ-Hubble-3: the lift on the ladder's weight function.}
The lift is derived in sign and bracketed in size, but no survey has
yet measured the under-density weighted exactly as the ladder's
calibrator and flow samples weight it. Euclid/Roman-depth counts fix
\(\delta(r)\) outright; the peculiar-velocity programs read the
outflow; together they turn \(+2\)--\(3\%\) from a bracket into a
number. A measured lift near zero retires the lift mechanism
(F-Hubble-4's territory); a measured lift at \(2\)--\(3\%\) converts
\(68.5 \pm 0.6\) from a derived expectation into an observed one.

\paragraph{OQ-Hubble-4: the zero-parameter joint likelihood.} The
over-determined CMB test of \S\ref{sec:cmb-overdetermined} promoted to
full-likelihood level: background fixed at the \(\pi\)-locked
inventory, only the seed and \(\tau\) inherited, run head-to-head
against six-parameter \(\LCDM\) and EDE across CMB+BAO+SN+lensing
with covariances kept. Every input is released; any group can run it;
the paper's Priority~1. If the derived chart's \(\chi^2\) per
\emph{free} parameter beats the fitted alternatives, the case stops
being philosophical --- and if it does not, that too is a result the
framework has nowhere to hide from.

\paragraph{OQ-Hubble-5: the growth sign.} The corridor's \(\SEight\)
response is carried with its sign open --- the one place in the
package where the framework has declined to commit pending the parent
paper's growth audit. The lensing generation now in analysis will
force the issue; the entry stays open here so that no reader mistakes
a pending audit for a settled prediction.

\paragraph{OQ-Hubble-6: the shell profile as a fitted form.} The lift
predicts not one number but a \emph{profile}: the locally inferred
rate must decay toward the background as low-redshift shells are cut
away, at the shape Figure~\ref{fig:hole-lift} draws. The open task is
to publish that profile as an explicit fitted form --- amplitude tied
to the measured depth, shape to the derived history --- so that
DESI's binned \(H_0(z)\) analyses (which already report a descending
trend at \(6.4\sigma\); \S\ref{sec:hole-evidence}) can be confronted
with a curve rather than a qualitative expectation. This is the
cheapest genuinely new confrontation available to the reading, and it
is work this program owes the field.

\paragraph{OQ-Hubble-7: the siren-depth refinement.} A refinement
question the pre-registered rule deliberately brackets: F-Hubble-2 is
a \emph{background-rate} rule, and the dark-siren catalogs that
will execute it sample far beyond the Hole, so they read the
background. But the rare nearby bright events --- GW170817's host sat
at forty megaparsecs, well inside --- share the neighborhood, and
if the lift is real they should read it, at an amplitude bounded by
the lift itself (\(\le 2\)--\(3\%\), far below current per-event
precision). The open question is catalog-level: report siren
\(H_0\) in host-distance shells, and the lift acquires a third,
fully independent reader --- one that shares no systematic with
either the counts or the flows. No number in this paper changes
either way; the refinement sharpens where the confirmation would
come from.

\paragraph{OQ-Hubble-8: percent-class curve reading.} Every row of
Table~\ref{tab:derived-Hz} is an open question with a printed answer:
chronometers at percent class, BAO on the derived \(r_s\), and ---
the purest --- the ELT redshift drift at centimeters per second per
decade. These closings are the slowest on the list and the most
valuable per datum: each one tests the \emph{curve}, not the point,
and the derived history has already committed everywhere the curve
can be read.

\paragraph{OQ-Hubble-9: projection sourcing (the seesaw's silent
premise).} The vacuum-energy seesaw of Part~III states what
\(\rho_\Lambda\) \emph{is}; it does not state what became of the
Standard-Model zero-point contributions below \(\Mlock\), which naive
effective field theory would add to the boundary source at up to
\(10^{55}\) times the answer. The framework's structural reply --- that
the \OGGEP{} projection registers the geometric attractor's residue and
not the flat-space loop sum --- is at present a \emph{property claimed
of the \GDOS{} layer, not a theorem proved in it}, and this entry
exists so that no reader mistakes the one for the other. A
degravitation-type statement, proved at the \AFive{}/\ASix{} level,
would close it; until then the seesaw is billed as a derived scale
with a conditional accounting, and the honesty triage carries it so.

\paragraph{OQ-Hubble-10: the epoch-offset relation.} Landmark~7
records an arithmetic coincidence: evolving the exact crossing
fraction \(1/\pi\) forward \(75\)~Myr lands on \(0.315\) ---
numerically the Planck-fitted value. If the fitted-versus-exact
\(\Omega_m\) displacement is, in whole or part, the epoch offset,
one entry of the coherent \(\pi\)-offset pattern has a mundane
explanation --- but only one: the rate drifts just \(0.24\%\) over
the same interval, an order short of the \(-1.2\%\) \(h\) offset.
A sub-percent inventory determination, read jointly with the epoch
calibration, settles what share of the pattern the calendar owns.

\section{Predictions and future directions}
\label{sec:disc-future}

The paper's forward predictions are collected, with their dates and
rules, in three standing structures --- the falsifier registry
(Table~\ref{tab:killcriteria}), the decision calendar with its
commitments ledger (\S\ref{sec:decision-calendar}), and the benchmark
sheet (Table~\ref{tab:benchmarks}) --- and they are not repeated
here. What a Discussion owes instead is the \emph{assignments}: who
can act, on what, and why it is in their interest regardless of how
this paper fares. Five communities, five requests, each two-sided by
construction.

\paragraph{To the distance-ladder community.} Three requests, all of
which strengthen the ladder whatever they find. Publish the
injection--recovery campaign: it is the cheapest decisive test in the
field, it settles the priors channel either way, and whoever runs it
owns the verdict (F-Hubble-9). Complete the \(\sim\!37\)-host JWST
sample under a tip/JAGB convention agreed \emph{before} the
reduction --- the data will exist regardless; the agreement is what
makes them a test (F-Hubble-10). And press the NIR ladder and the
ZTF/Rubin sibling samples to a published \(H_0\): dust physics is the
largest single allowance in anyone's budget, and the near-infrared is
where it goes to die (F-Hubble-8).

\paragraph{To the CMB community.} The full SPT-3G survey and the
Simons Observatory will tighten \(\Href\) beneath the derived
\(66.6 \pm 0.2\); this paper fails at its own anchor if the coherent
\(\pi\)-offset decoheres, and asks for exactly that scrutiny. The EDE
lobe search (F-Hubble-3) and the \(\DNeff\) convergence are two-sided
here: detections kill this reading; tightening nulls starve its
rivals. And the zero-parameter joint likelihood (OQ-Hubble-4) is an
afternoon's configuration change for any Boltzmann-pipeline group ---
the inputs are released, and either outcome is publishable.

\paragraph{To the gravitational-wave community.} The decisive verdict
is yours, and the paper's request is procedural, not scientific:
report \(H_0\) in host-distance shells as the catalogs deepen
(OQ-Hubble-7), keep the counterpart-readiness infrastructure funded
through O5 --- the bottleneck is follow-up, not sensitivity
(\S\ref{sec:siren-reportcard}) --- and continue publishing the
dark-siren values with their mass-model and completeness systematics
displayed, because the walk from \(76.6\) to \(69.9\) with shrinking
error bars is, to date, the cleanest live demonstration in cosmology
of a systematic budget averaging down in public.

\paragraph{To survey cosmology.} The Local Hole is the one piece of
this reading that ordinary galaxy surveys can close by themselves
(\S\ref{sec:hole-evidence}): Euclid/Roman-depth counts fix the depth
profile, the PV programs read the outflow, and the binned
\(H_0(z_{\min})\) analyses confront the shell profile the lift
predicts (OQ-Hubble-6). None of these requires believing anything in
this paper; each is a measurement the field wants anyway, and the
reading has pre-committed to what they must find.

\paragraph{To theorists, inside and outside the cohort.} The Hosotani
computation (OQ-Hubble-1) is the standing debt, and it is stated as a
falsifier, not a hope: the target is printed, the window is closed,
and a computation that misses retires Route~I's point reading by the
paper's own rule. Behind it queue the growth audit (with \ATen{}),
the shell-profile fitted form (OQ-Hubble-6), and the global
likelihood (OQ-Hubble-4). The division of labor with the companion
papers is single-source-of-truth throughout: the inventory, corridor,
and why-now treatments live in \ATen{} and are carried here at
identical values; the Kaxion window \([1.59, 2.13]\)~GHz and the
twin-peak ratio live in \PaperCOne{} (F-Hubble-7 imports, never
re-derives); the codebook grades and the O6 problem live in
\AFive{}; the \(\DNeff = 0.04\) and the integer ledger live in
\ANine{}. One value, one home, every quotation synchronized --- so a
correction anywhere propagates everywhere, and no two papers can
drift apart unnoticed.

\paragraph{The staged road, in one paragraph.} December~2026: Gaia
DR4 reads the ladder's floor. Through 2028: the three channel
verdicts arrive, or fail to be sought --- itself informative.
2027--2029: the Hole census and the shell profile. 2028--2030: the
early-side precision era and the O5 siren accumulation, with the
dark-siren error bars passing through the gap they must adjudicate.
The 2030s: the tensor verdict. Mid-2030s: the siren convergence,
and with it the verdict on the central claim. Beyond: the drift era,
when the curve itself is watched in real time. At every stage the
paper's number is already on the table; at no stage does it get to
put down a new one. That is the whole design.

\section{The nobility clause}
\label{sec:nobility}

Because the reading stakes a great deal on a single number, it binds
itself to four commitments a reader may hold it to --- each one
checkable against the text, none of them rhetorical.
\emph{Pre-commitment:} every value was fixed before the H0DN
measurement it is compared against; the early inputs have not been
adjusted since the parent recast; and the additions of 2026 --- the
landscape audit, the anatomy pricing, the calendar --- added
exposure, never wiggle room. \emph{Honest grading:} each claim
carries its codebook grade (theorem, \(\mathsf D^{*}\),
Asserted-external, named-open), quoted identically wherever it
appears, with the least flattering entries --- the residual, the
conditional fractions, the open growth sign --- displayed in the
abstract's own register. \emph{No retreat:} where the framework
cannot derive a quantity, it says so and leaves the gap open rather
than fill it with an unforced mechanism; where the empirical record
embarrassed a forecast (one bright siren in nine years), the record
was printed and the dates re-based, while the thresholds --- hostage
to nothing --- stood. \emph{Symmetric stakes:} every kill rule has
its confirmation twin printed at equal prominence; the same siren
that would confirm the reading would, at \(73\), destroy it; and the
paper has arranged, in advance and in public, to be unable to escape
that verdict. These commitments cost rhetorical safety and buy the
only currency a decade-old controversy still respects: a participant
who cannot move.

\section{The bottom line}
\label{sec:disc-bottomline}

One sentence per movement, and then the verdict. The H0DN comparison
supports the derived reading strongly where it is strong and is
silent exactly where the paper says it is silent. The reading's
significance is not a better fit but a different kind of object: a
rate derived twice, embedded in a derived history, exposed on a
printed calendar. Its accomplishments are seven standing puzzles
resolved, dissolved, or reframed, and ten structural moves the
tension literature did not previously contain. Its honest boundary is
drawn in tiers: it computes the history, prices the chain, and hands
the likelihoods to the instruments --- paying for that clarity with
maximal exposure and one declared debt. And its road ahead is a
numbered program in which every open question carries its closing
condition and almost every closing condition carries a date.

The verdict the Part offers is deliberately modest and deliberately
sharp. Modest: nothing here claims the tension is solved --- the
residual stands open, the conditions are named, and the middle world
of \S\ref{sec:siren-three-worlds} is priced as the uncomfortable
possibility it is. Sharp: the reading is now the only participant in
the controversy that has specified, in advance, every observation
that would end it --- and the first appointments are months, not
decades, away. Part~IX closes the paper in a different register,
asking what it would mean if the reading is right.


%% ===================================================================

\section{Part~VIII in plain English: what the result is worth, what it is not, and where the debts are}
This is the Part where the paper argues with itself.

What the comparison supports strongly: a background rate derived twice by
disjoint routes, landing where the early-universe network already sits,
with no continuous parameter anywhere in the chain. What it supports
moderately: the local lift, whose response law is derived and whose depth
is somebody else's disputed measurement. What it does not support at all,
and says so in as many words: the seven-percent residual, which is named,
priced, and left open.

The Part also does the comparisons a reader is entitled to. Ten points of
significance set beside the leading alternative explanations. Seven
standing problems that the reading resolves, dissolves, or reframes ---
and the three verbs are not interchangeable. Ten structural moves the
tension literature did not previously contain. And the cautionary
historical precedents alongside the flattering ones, because a framework
that quotes only the cases where the outsider turned out to be right is
not being honest about the base rate.

The boundary is drawn in tiers. This paper computes a history, prices a
chain, and hands the likelihoods to the instruments. It does not run the
likelihoods, and it should not be read as having done so. It pays for that
clarity with maximal exposure --- zero free parameters means nothing can
be adjusted when a measurement moves --- and it carries one declared debt,
a coefficient whose first-principles computation has not been done.

The verdict offered is modest and sharp at the same time. Modest, because
nothing here claims the tension is solved. Sharp, because this is now the
only reading in the controversy that has published, in advance, every
observation that would end it, and the first of those observations is
months away rather than decades.

\GGPart[Conclusions]{Conclusions}{The retirement of a measured
constant --- one derived history, a dozen ways the universe can
answer, and what it means that we can ask}

\section{The summit thesis}
\label{sec:concl-thesis-c2}

For a century, the question ``how fast is the universe expanding?''
was answered by measuring. Better telescopes, better calibrations,
better statistics --- a heroic tradition, running from Hubble's first
straight line through thirty-eight specialists and a
covariance-weighted consensus good to one percent. This paper has
answered the same question a different way: by deriving. The
expansion history was computed from a geometry with no dial to turn,
and the measured values were then confronted with it --- the early
one landing on the derivation, the local one dissected against it,
piece by honest piece.

\begin{quote}
\emph{The Hubble constant was never a constant, and --- if the
reading defended here is right --- it was never merely a measurement
either. It is one point on a derived curve: a curve that begins with
a predicted inflation, runs on a predicted matter sector, and ends on
a predicted vacuum. The century of measuring the universe is ending.
The decade of answering it has a schedule.}
\end{quote}

That is the summit thesis, and the tension itself is its best
witness. Measure one number two independent ways, to one percent
each, and any hidden assumption anywhere in either chain is promoted
to a seven-sigma crisis. The split was never a failure of the
instruments --- both are magnificent --- and, on this reading, never
a crisis in the laws. It was the measuring tradition running into
the ceiling of what measurement alone can settle: precision had
outrun understanding, and the surplus precision had nowhere to go
but paradox. The resolution offered in these pages is not a better
measurement. It is a change in what kind of thing the number is.

\section{What this paper has established}
\label{sec:ten-numbers}

The whole argument compresses into ten numbers, and they are worth
walking in plain language before the table states them exactly ---
because they are not ten separate results. They are three acts of one
story.

\emph{The curve} (rows 1--8): the background rate today,
\(66.6 \pm 0.2\), derived twice --- once from the geometry's own
vacuum arithmetic with no astronomical input at all, once from the
measured cosmic densities divided by the derived fractions --- with
the two routes agreeing at a fraction of their stated errors; behind
it the full inventory (matter \(1/\pi\), dark energy \(1 - 1/\pi\),
atoms \(1/2\pi^2\)), the exact equation of state \(w = -1\), the
corridor switch at \(z_* = e^7\) (decoupling \(0.4\%\) below), the age of the universe at
\(13.9\) billion years, and the quiet floor, \(H_\infty = 55.0\),
that the expansion will hold forever.

\emph{The neighborhood} (row 9): our under-dense corner of the
cosmos lifts any locally measured rate by a derived two to three
percent --- carrying the background to \(68.5 \pm 0.6\), the value a
distance ladder \emph{inside} the Hole should read. What remains
between that and the ladder's \(73.50 \pm 0.81\) is the one thing
this paper refuses to claim: a named, priced, still-open residual in
the measurement chain, with its candidate anatomy quoted from the
ladder literature itself and its kill tests dated.

\emph{The referee} (row 10): gravitational-wave standard sirens,
which owe nothing to any ladder or any afterglow, are predicted to
converge near \(67\). Near \(73\), this paper is wrong --- publicly,
quickly, and by its own printed rule.

\begin{table}[htbp]
\centering\small
\caption[The paper in ten numbers: value, grade, location, and falsifier.]{The paper in ten numbers: value, grade, location, and falsifier. Ten is the number of quantities a reader should be able to carry away, and the fourth column is why the list is that short. Every entry names a measurement that would retire it, and two of the ten are retired by the same standard-siren result. The grades are the honesty triage's: \(\mathsf D^{*}\) marks a value the ledger derives subject to the measure lemma, and the three entries that are not \(\mathsf D^{*}\) say what they are instead.}
\label{tab:c2-ten-numbers}
\usebox{\GGscTenCTwo}
\end{table}

Every row carries its honesty grade and its falsifier in the same
breath, because that pairing --- a claim and the observation that
would kill it, printed together --- is the discipline the whole
twenty-two-paper series is built on. Nothing in the table is
negotiable after its datum arrives; most of the data arrive within
the decade.

\section{The retirement of a measured constant}
\label{sec:concl-retirement}

Step far enough back and this paper is one episode in a very old
story: the story of constants that graduated.

The speed of light was, for two centuries, a measured number ---
R{\o}mer timed it with Jupiter's moons, Fizeau with a toothed wheel,
ever more decimals, ever better apparatus. Then Maxwell derived it
from the constants of electricity and magnetism, and the meaning of
the number changed forever; by 1983 the graduation was complete, and
the speed of light is now a \emph{definition}, a number no experiment
can revise. The Rydberg constant was spectroscopy's most precisely
measured quantity --- until Bohr computed it from the electron's
charge and mass, and a decimal-chasing industry became a test of
quantum mechanics overnight. The \(W\) and \(Z\) masses were
predictions before they were measurements, and the measuring of them
was exciting precisely \emph{because} a theory had already spoken.
In each case the same event: a number crossed the line from
\emph{input} to \emph{output}, from a fact about the world to a
consequence of a law --- and the field that owned it stopped doing
natural history and started doing physics.

Cosmology has never had that moment. Its deepest numbers --- the
matter density, the vacuum energy, the expansion rate --- have
remained inputs for a century: things to be measured, tabulated, and
accepted, with \(\LCDM\) as the magnificent bookkeeping that turns
six of them into the whole sky. \(H_0\) is the oldest and most
storied of them, and the proposal of this paper, stated in the
plainest terms, is that it is next in the lineage: \emph{the last
measured constant of cosmology, due for retirement} --- not from
relevance, but from the status of input. On the reading defended
here, \(H_0\) is what the derived history happens to read today,
exactly as the Rydberg is what Bohr's atom happens to radiate.

And here is what the retirement buys, because a graduation is never
a loss. When a constant becomes a consequence, disputing its last
digit stops being the frontier. Nobody today measures the speed of
light to adjudicate between theories; the excitement moved to the
places where the law that fixed it could fail. The same movement is
available to cosmology now, and it is overdue. The dispute over the
last digit of \(H_0\) --- 73.5 or 73.2, 67.2 or 67.4 --- has consumed
a decade of the field's finest instruments and sharpest people, and
it cannot, even in principle, end by getting sharper: the tension is
already seven times its stated errors, and another decimal on either
side adds only another decimal of paradox. What ends it is structure:
a derived curve that both endpoints must answer to, and a third
instrument that owes neither of them anything. The question worth a
career is no longer \emph{what is the number?} It is \emph{is the
number a theorem?} --- and that question, unlike the other one, has
a scheduled answer.

\section{One history, three acts, all predicted}
\label{sec:concl-three-acts}

What replaces the measured constant is not a cleverer fit. It is a
complete, quantitative history of the universe in which every act is
predicted in advance --- the fuller framework of the parent paper
\ATen{} \citep{ArisakaA10Cosmos}, read here at the one number this
paper owns. It is worth telling once in plain words, because the
Hubble constant is only the middle of a story whose beginning and
end are staked just as sharply.

\emph{Act one: a predicted beginning.} The universe opens not in an
unexplained singularity but in a geometric launch --- an inflationary
plateau sited by the geometry itself, one e-fold below the grand
unification edge. And the launch is not a flexible story: it comes
with numbers. The tilt of the primordial spectrum is computed at
\(n_s = 0.9620\), within a standard deviation of what the microwave
sky shows; the primordial gravitational-wave amplitude is fixed,
two-sidedly, at \(r = 0.0043\) --- above the floor of the coming
polarimeters, below the current bound, with \emph{both} directions
of failure fatal. No inflaton was invented for the occasion; the
derivation is \ATen{}'s, and this paper simply inherits the launch
its expansion history requires.

\emph{Act two: a predicted middle.} The matter that clumped into
galaxies is, on this reading, one specific particle with one
specific address: the Kaxion, at \(8.1 \pm 0.5\,\mu\)eV, waiting in
a laboratory window --- \(1.59\) to \(2.13\)~GHz, centered near
\(1.95\)~GHz --- narrow enough to be opened or shut within the
decade, carrying a twin-peak fingerprint at exactly
\(\Delta\nu/\nu = 1/35\) that nothing else on the market can imitate
(\PaperCOne{}; \citealp{ArisakaC1Kaxion}). Dark matter with a mass,
a frequency, and a deadline --- not a parameter space, an
appointment.

\emph{Act three: a predicted end.} The dark energy driving the
acceleration is the residual \(1 - 1/\pi\) of a closed inventory,
with an equation of state that is \(w = -1\) \emph{exactly} --- not
approximately, not on average, but exactly, because the geometric
transition that produced it is complete. The expansion settles onto
the de~Sitter floor at \(H_\infty = 55.0\) and holds it forever. A
detected deviation of \(w\) from \(-1\), at the precision the coming
surveys will reach, retires this act on the spot.

Three acts, each carrying numbers fixed before the instruments that
will test them. The Hubble constant of this paper's title is where
act two hands over to act three --- one reading of one curve, flanked
on both sides by predictions at least as sharp as itself. That is the
sense in which the tension was always too small a question: the same
geometry that says \(66.6\) says \(0.0043\), says \(8.1\,\mu\)eV,
says \(-1\) exactly --- and every one of those numbers can die
independently.

\section{The dozen ways the universe can answer}
\label{sec:concl-dozen}

Here is the field this paper actually invites its readers into ---
not the defense of a decimal, but a decade in which the universe is
asked twelve sharp questions, each with a printed answer it is free
to contradict. They are walked here in the order the answers arrive;
the formal registry, rules, and dates live in Part~VI, and nothing
below adds or alters a commitment. What each entry states is why the
measurement is \emph{meaningful} --- what structure it kills or
confirms --- rather than merely precise.

\emph{One: the astrometric floor (December 2026).} Gaia DR4 maps
the parallax zero-point that every Milky-Way calibrator inherits.
Meaningful because it reads the foundation of the entire ladder
without a telescope pointed at a single Cepheid --- pure geometry
auditing a measurement chain.

\emph{Two: the ladder's own three verdicts (2026--2028).} An
injection--recovery simulation, the complete JWST calibrator sample,
the near-infrared ladder. Meaningful because each settles a named
few-percent allowance \emph{either way} --- the first time the
tension's systematics have carried symmetric, dated tests rather
than open-ended suspicion.

\emph{Three: the map of our neighborhood (2027--2029).} Euclid- and
DESI-class surveys count galaxies straight through the Local Hole and
read its outflow. Meaningful because the lift this paper derives ---
sign forced, size bracketed --- becomes a measured number, and one
prediction of this paper is confirmed or retired by ordinary galaxy
counting.

\emph{Four: the shell test.} The locally inferred rate must decay
toward the background, at a computable profile, as the nearby shells
are cut away --- a shape no constant calibration error can imitate.
Meaningful because it distinguishes \emph{where you live} from
\emph{how you measure} using the survey data alone; a descending
trend is already reported and deepens with each data release.

\emph{Five: the early-side null (2028--2030).} The coming microwave
instruments reach the discovery floor for early dark energy and drive
the radiation excess toward the cohort's \(0.04\). Meaningful because
this paper is the one participant that \emph{needs nothing to be
found}: tightening nulls confirm it while they starve every
early-side fix.

\emph{Six: the equation of state (2028--2030).} DESI, Euclid, and
Roman push \(w\) toward the two-percent regime. Meaningful because
\(w = -1\) \emph{exactly} is a statement about the completeness of a
geometric transition --- the sharpest possible contrast with every
dynamical dark-energy sector, and one that cannot be softened if the
data drift.

\emph{Seven: the dark-matter appointment (through \(\sim\)2030).} The
funded haloscope programs sweep the Kaxion window. Meaningful
because a line at \(1.95\)~GHz with a \(1/35\) doublet would not
merely add a particle --- it would read the integer depth of a
compact dimension in a microwave receiver; and a swept, silent window
retires the dark-matter claim of the entire series, this paper's
inventory included.

\emph{Eight: the referee (O5 to the mid-2030s).} The standard sirens
converge --- on \(67\), on \(73\), or, most interestingly, between.
Meaningful because the siren is the only instrument that reads the
clock with no ladder and no sound horizon anywhere in its chain: the
one measurement both camps agreed, in advance, to lose to.

\emph{Nine: the primordial whisper (the 2030s).} LiteBIRD and the
Simons Observatory read the tensor ratio at the \(10^{-3}\) floor ---
realistically toward the decade's end. Meaningful
because \(r = 0.0043\) is two-sided: a detection well above, or a
null well below, each retires the predicted launch --- the rare
inflationary claim that can fail in both directions.

\emph{Ten: the growth ledger (this decade).} The lensing surveys fix
\(\SEight\) on the derived inventory, where this paper's inherited
value is \(0.759\) with the corridor's growth sign honestly flagged
open. Meaningful because growth reads the same inventory a second
way: matter that expands one way and clumps another would break the
single ledger behind both.

\emph{Eleven: the curve, read directly (2030s).} Cosmic chronometers
approaching percent class, and baryon-acoustic points converted to
absolute readings by the derived \(r_s = 145.0\)~Mpc, confront the
predicted \(H(z)\) row by row. Meaningful because a fitted model can
absorb a surprising point by re-fitting; the derived curve cannot
absorb anything, anywhere, ever.

\emph{Twelve: the expansion watched live (2040s).} The extremely
large telescopes attempt the redshift drift: \(+2.1\) centimeters per
second per decade at \(z = 1\), falling through zero near
\(z \approx 1.9\). Meaningful because it is cosmology with no proxy
at all --- no candle, no ruler, no clock but the observer's own ---
the expansion history watched, not inferred, at values this paper
has already committed to.

Twelve questions; twelve printed answers; not one of them ``what is
the ninth decimal of \(H_0\)?'' That is what a falsifiable,
beyond-\(\LCDM\) precision cosmology looks like --- and every entry
on the list is funded, scheduled, or already taking data.

\section{What it would mean if it is right}
\label{sec:why-matters}

The Hubble constant is, in the end, the rate at which everything
recedes from everything else --- the clock of the whole universe. For
a century we treated it as a number to be measured ever more
precisely, and the harder we measured, the more it split in two. The
reading offered here is that the splitting was telling us we had
mis-stated the question. There is no single number, because the
expansion was never a constant: it is a function of time, \(H(t)\),
falling from an unimaginable early rate toward the quiet de~Sitter
floor it will hold forever, and \(\Hzero\) is merely where that curve
happens to cross the present moment.

If the geometry is right, the consequences run deeper than the
tension. The rate of cosmic expansion is not a free fact about the
universe but a consequence of what the universe is made of --- and
the densities that fix it are themselves fixed by \(\pi\) and a
handful of integers. The composition of the cosmos joins the list of
things that did not have to be measured to be known; the tension
joins the list of paradoxes that were really promotions, the moment a
field's precision outgrew its questions; and cosmology itself
completes the graduation that physics made a century ago --- from
cataloging the world's numbers to deriving them, from natural
history to law. A universe whose expansion rate is a theorem is a
universe that can be \emph{understood}, not merely surveyed; and the
first generation to know that would be the one reading the
instruments this decade.

And if the geometry is right there is a quieter consequence, easy to
miss beside the grand one. The magnificent measurements do not
become obsolete --- they become \emph{evidence}. Every decimal the
ladder community fought for, every calibration audit, every variant
in the consensus catalog turns into a precision record of how a
superb instrument reads a lifted, biased corner of a derived
universe --- data about the measurement chain itself, of permanent
value, whatever the verdict. Nothing in a century of measuring is
wasted on this reading. It is re-purposed.

\section{What it would mean if it is wrong}
\label{sec:concl-if-wrong}

A conclusions Part that only imagines victory is advertising, so let
failure be given its own section, written with the same care.

If the sirens converge at \(73\), the derived rate is wrong, and with
it the reading of the inventory that produced it. That failure would
be clean, public, and instructive in a way the tension's decade of
stalemate has not been: it would say that the true background rate is
high; that the CMB's densities cannot be divided by the geometric
fractions to give it; that the coherent agreement of Part~III was a
coincidence of extraordinary depth --- seventeen observables, one
direction, percent scale --- and coincidences of that depth are
themselves data. Every rival would inherit a sharpened problem: the
early side would then genuinely need new physics, and the landscape's
three bands would need an explanation with the measurement-side
reading struck off the list. The audit trail in these pages ---
which link broke, at which grade, under which condition --- would
show exactly where a zero-parameter program buckled, which is
precisely the information the next such program needs. A wrong
theory that dies cleanly pays for itself; the field has spent ten
years learning how expensive the other kind is.

If the verdict lands \emph{between} --- the middle world of
\(\approx 70\) --- the outcome is subtler and is priced in advance:
the lifted background survives comfortably, the pure derived rate is
strained but not killed at the pre-registered level, and the open
question sharpens into one this paper has already named. That world
is uncomfortable for everyone, this reading included; the discomfort
is printed, with its numbers, in Part~VII.

And if the window simply refuses to close --- if the Hosotani
computation stays undone, the channel tests unsought, the sirens
slow --- then the paper's contribution is the calendar itself: a
public record that, at minimum, converts the field's patience into
appointments. Even the failure to be tested, on a printed schedule,
is information.

\section{The objective limitations, one last time}
\label{sec:conc-limits}

Honesty requires the caveats be gathered in one place, in the same
plain language as the claims. The inventory is graded
\(\mathsf D^{*}\): derived up to a single named measure lemma and the
declared choice of observing epoch --- a conditional, not a theorem,
and never advertised otherwise. The vacuum coefficient
\(\zeta_\Lambda = \pi/175\) is selected by the theory's own naming
rules, conditional on one lemma, its closure pre-registered as a
falsifier (F-Hubble-11) with a window that cannot be renegotiated.
The last \(\approx\!7\%\) of the local excess is not derived at all:
it is named as a distance-ladder calibration systematic, its one
geometric channel excluded, its averaging channel bounded, its
candidate anatomy quoted from the ladder's own literature at the
ladder's own error bars. The middle world --- a true rate near
\(70\) --- would strain the pure derived value at the level Part~VII
computes, and the paper carries that possibility as live, not
remote. And the whole reading is staked on a single prediction ---
\(\Hzero^{\rm GW} \to 67\) --- that has not yet been measured. None
of this is hidden; all of it is printed, because a reading that
could be killed cleanly is worth more than one that cannot be killed
at all.

\section{How to check this paper from your desk}
\label{sec:concl-desk}

A last practical word before the closing sections, because this
paper was built to be audited, not believed. Every quantitative
figure and forecast regenerates from a short, documented, standalone
script, released with the paper and indexed in
Appendix~\ref{app:scripts}: the derived history, the landmark
chronology, the residual arithmetic with its twenty-three asserted
checks, the forty-five-value landscape with every entry pinned to
its primary record, the siren forecast. Each script states its
inputs, asserts its own arithmetic before writing anything, and runs
in seconds. A skeptical reader with one evening can re-derive the
paper's numbers; a skeptical reader with one afternoon and a
Boltzmann pipeline can run the zero-parameter confrontation of
Part~VI's first priority. The invitation is genuine and the point is
structural: a derivation, unlike a fit, has nothing behind the
curtain --- and a field-wide dispute conducted over released
arithmetic is the kind that actually ends.

\section{Why there is someone here to time the universe}
\label{sec:concl-why-someone}

One question remains that a paper about the universe's clock cannot
politely decline, though it can only sketch what the cohort's
capstone develops in full. Why is there anyone here to read the
clock at all?

Notice, first, a quiet coincidence buried in the arithmetic of this
paper. The expansion rate we measure is set by the same inventory
that sets how long stars burn --- the same fractions, the same
integers, one ledger. The astronomers who measure \(H_0\) are made
of elements forged in stars whose lifetimes that ledger fixed,
standing on a planet warmed by one such star midway through its
allotted burning, at the one broad epoch when matter and vacuum
share the budget within a factor of a few. We measure the universe's
clock at the moment our own clocks --- stellar, biological,
civilisational --- happen to be running; and on the derived history
that is no accident, because they are not different clocks. The
why-now coincidence that puzzles \(\Lambda\)CDM is, on this reading,
the statement that observers arrive when their kind of chemistry
does, and their kind of chemistry is timed by the same geometry that
times the expansion (the full account is \ATen{}'s).

The deeper point belongs to the series rather than to this paper,
but it deserves its plain-English sentence here. The geometry that
fixes the expansion rate does not stop at galaxies: the same descent
runs on through the stable atoms it licenses, the stars it schedules,
the chemistry it makes inevitable, toward open systems that model
their surroundings --- and eventually toward systems that build
interferometers and argue about decimals
(\AEleven{}; \citealp{ArisakaA11Origin}). On that reading, a
measurement of \(H_0\) is not an outsider's act performed on the
universe from nowhere. It is the universe, at one late rung of its
own staircase, timing itself --- and the century-long fight over the
number is what it sounds like when the instrument first hears its
own tick clearly. We are not the audience of the expansion. We are
the part of it that has begun to keep the log.

\section{A personal note, and a message to my colleagues}
\label{sec:personal-note}

Papers rarely say plainly what their author expects, so let it be
said. I do not know whether the geometry is right. What I know is
that the background rate it forces was fixed before I compared it to
anything, that it landed on the value the microwave sky already
showed, and that it makes a prediction --- a siren near \(67\), not
\(73\) --- that the coming decade will confirm or destroy. I have
not adjusted a single early-universe input to make the numbers meet;
where the framework cannot reach, in the last few percent of the
local excess, I have said so and left the gap open rather than fill
it with a mechanism I could not derive. If the sirens side with the
ladder at \(73\), this reading dies in public, quickly and cleanly,
and it will have done the one useful thing a wrong theory can do.

I spent fifty years as an experimental physicist --- Kamiokande, kaon
and \(B\)-meson \(CP\) violation, CMS at the LHC, Auger, XENON ---
measuring numbers the Standard Model declined to explain, and
watching the era's deepest discoveries arrive from the sky. Every
number I helped measure in half a century was an input: something
the world told us, and theory then accommodated. The number in this
paper is the first one of my scientific life that was an
\emph{output} --- written down by a geometry before any instrument
of mine could weigh in --- and I confess that the experience of
testing it has been of a different kind entirely. An input can only
be refined. An output can be \emph{wrong} --- and therefore, for the
first time, the measurement matters in the way discovery
measurements matter: sharp enough to kill.

That experience is the substance of the message I want to leave with
my colleagues in astronomy and cosmology, stated with the respect
fifty years inside the measuring tradition should earn it, and with
the bluntness the moment deserves. The dispute over the last digit
of \(H_0\) is over --- not because either side lost, but because
both sides won: the measurements are now so good that they have
proven the question undecidable from inside. Another decade of
sharpening 73.5 against 67.2 will produce another decade of
seven-sigma stalemate, because no measurement of a number can settle
what kind of number it is. Meanwhile, an alternative worth the
field's instruments now exists: a beyond-\(\LCDM\) precision
cosmology that is not a deformation of the six parameters but a
derivation of them --- inflation with computed observables, dark
energy with an exact equation of state, dark matter with a frequency
and a deadline, an expansion history exposed at every redshift ---
falsifiable, near-term, and scheduled. The twelve appointments of
this Part are not this paper's private wager; they are, I believe,
where the actual discoveries of the next decade will be made, by the
same instruments and the same people now spending their precision on
one contested decimal. The tension was never the prize. It was the
signpost --- and it points away from itself. This is where the real
excitement begins: not at the tenth measurement of an old constant,
but at the first tests of whether the constants were ever free at
all. I invite my colleagues --- with their superb instruments, their
hard-won calibrations, and their healthy skepticism fully intact ---
to spend the coming decade there. The universe has been asked the
same question for a hundred years. It is time to ask it better ones.

And one sentence for the youngest readers, because an invitation
owes its fine print. The appointments of this Part are mainstream
observations --- ladders, catalogs, surveys, sirens --- that will
count professionally whichever way this framework fares; a thesis
built on \emph{them} is safe in every world this paper prices. A
thesis built on the framework itself is not: it is a proposal under
test, unaccepted, and in the middle world it stays unresolved for a
decade. Work the appointments; hold the framework to its dates ---
skepticism fully intact is not only permitted here, it is the job.


\section{Part~IX in plain English: what changes if a measured constant turns out to have been derivable}
For a hundred years the Hubble constant has been a quantity to be
measured. The claim of this paper is that it is a quantity to be derived,
and that the measuring was always reading one point off a curve the
geometry had already fixed.

If that is right, a great deal that has looked like coincidence stops
being coincidence. The matter fraction, the baryon fraction, the epoch of
equality, the sound horizon, the age of the universe, the calm the
expansion eventually settles into --- all of them come out of the same
small set of integers, and not one of them is available to be adjusted
when a new measurement arrives. The tension itself becomes a statement
about where we happen to live rather than about what the universe is made
of.

If it is wrong, it will go wrong quickly and in public. This paper names a
dozen appointments, most of them in this decade and several already
scheduled. A converged standard siren near \(73\) ends it in one line.
There is no version of this framework that survives that measurement, and
the paper says so before the measurement rather than afterwards.

The limitations are restated one last time, because a conclusion that
quietly drops them is not a conclusion. The measure lemma is conjectured
and not proved. The seesaw coefficient is selected by a rulebook and not
yet computed. The residual on the local ladder is named and open. Every
one of those is written into the grade attached to every number this paper
quotes.

What is left is an invitation, and the closing sections make it. Anyone
with a laptop can check the central chain from the released scripts in an
afternoon. Anyone with a telescope can attack the appointments. The
framework asks that skepticism be kept fully intact --- that is the job
--- and asks only that it be aimed at the dated commitments, where it will
actually settle something.

\section*{Envoi}
\addcontentsline{toc}{section}{Envoi}

For a hundred years we asked how fast the universe expands, and
measured the answer more and more precisely, until it split in two.
Perhaps it was never one number to be measured, but one curve to be
derived --- a curve that began with a predicted burst, runs today on
a predicted budget, and will end on a predicted calm --- and the
universe has been keeping that single clock since its first
microsecond, at a rate we are only now, in this decade of
appointments, learning how to read. The clock is old. The reading
of it has just begun.

\GGAcknowledgments

\GGAppendixPart[Technical Appendices]{Technical Appendices}{%
A graduate course in the Hubble tension read through \GGtheory{} ---
reference dictionaries, a conceptual on-ramp, and from-scratch
derivation courses that review each topic at textbook level, name what
the standard treatment cannot explain, and derive the geometric
resolution, with every step shown}

\appendix

\noindent\textbf{How to use these appendices.} These appendices are
written for the reader we most hope picks this volume up: a graduate
student entering cosmology, or a colleague from a neighboring field,
who wants to understand the Hubble tension --- and this paper's
reading of it --- from the ground up, with nothing assumed and
nothing skipped. Every topic is taught in the same three steps:
first the \emph{textbook}, built from scratch with every derivation
shown and a physical intuition beside each result; then the
\emph{limitations} --- what the standard treatment assumes and cannot
explain, stated plainly; then the \emph{geometric resolution}, graded
by the honesty key of \S\ref{app:grades}. The courses are
self-contained and cross-referenced, so you can enter anywhere; if
you are new to the subject, start with the primers of
\S\ref{app:primers} and read the courses in order. Everything
quantitative regenerates from the released scripts
(\S\ref{app:scripts}), and the companion volume for the wider
cosmology --- inflation, nucleosynthesis, the microwave sky in full
--- is the appendix course of the parent paper
\citep{ArisakaA10Cosmos}. Welcome; the coffee is on the left.

\begin{table}[htbp]
\centering\small
\caption[The course map of these appendices.]{The course map of these appendices. Each course reviews its
topic at textbook level, names the standard treatment's limits, and
derives the geometric reading, with every step shown. The appendices are courses rather than a technical supplement, which is a deliberate choice about who this paper is for. A reader who already knows the material can take the headline outputs from the third column and return to the main text; a reader who does not can start at Course C and arrive at the same place without a second book open beside this one.}
\label{tab:course-map}
\begin{tabularx}{\textwidth}{@{}clX@{}}
\toprule
\textbf{Course} & \textbf{Topic} & \textbf{Headline outputs}\\\midrule
A--B & dictionaries; primers and questions & the vocabulary, the grades, the on-ramp\\
C & the Friedmann background & the equations; \(t_0 = 13.9\)~Gyr; the \(H(z)\) rows\\
D & distances and the ladder & the five distances; the common-mode theorem; \(0.153\)~mag\\
E & the CMB as an \(H_0\) instrument & \(r_s = 145.0\)~Mpc; why the sky never measures \(h\)\\
F & vacuum energy and the seesaw & the \(10^{120}\) dissolved; the closed form for \(H_0\)\\
G & the local universe & \(\delta H/H = -(f/3)\bar\delta\); the lift; the \(-0.41\)\\
H & backreaction and averaging & Buchert's \(\mathcal Q_D\); the three walls\\
I & growth and \(S_8\) & the growth equation; \(S_8 = 0.759\)\\
J & standard sirens & the chirp mass; GW170817 end-to-end; \(N = 21/58\)\\
K & time and the curve-readers & the arcsinh clock; chronometers; the drift; the \(T\) clock\\
L & the statistics of a tension & why precision amplifies bias; the bracket theorem\\
M--N & reproducibility; registries & the scripts; the ledgers, audits, and roadmaps\\
\bottomrule
\end{tabularx}
\end{table}

\section{Reference dictionaries}
\label{app:blockA}

This block is the reference desk: the naming convention and how the
program's names nest, a one-page summary of every headline number,
the symbol glossary, and the reader's key to the honesty grades used
throughout. One acronym from outside the program belongs on this
desk before any of ours: \textsf{H0DN}, the \(H_0\) Distance Network
--- the 2026, thirty-eight-author network combination of the local
distance ladder (Part~II) --- whose consensus \(73.50 \pm 0.81\) is
the late-side benchmark of the entire paper.

\subsection{The cohort naming convention}
\label{app:naming}

Every named object in this report carries a cohort-global identifier of the
form \textsf{Prefix-Suffix-\(N\)}, so that a claim can be traced across the
twenty-two papers without ambiguity. The prefixes used here are
\textsf{T} (theorem), \textsf{F} (falsifier / kill test), \textsf{O}
(objection), and \textsf{I} (innovation); the suffix names the sector. Thus
\textsf{F-Hubble-2} is the second falsifier of the Hubble reading (the
standard siren, \S\ref{sec:falsifiers}); \textsf{O-Hubble-\(N\)} tags the
objections of the penultimate Part; and inherited results keep their home
tags --- \textsf{T-Cosmos-\(N\)} for the theorems imported from \ATen{},
\textsf{T8.7} for the lock-scale theorem of \AFive{}. The convention is the
program-wide standard of \AEleven{}, reproduced here so this paper is
self-contained. Figure~\ref{fig:c2-name-nesting} shows how the names
used in this paper nest inside the program's layers.

\begin{figure}[htbp]
\centering
\resizebox{0.69\textwidth}{!}{%
\begin{tikzpicture}[font=\footnotesize,
  lay/.style={draw=black!55, rounded corners=3pt, align=center, inner sep=6pt}]
\node[lay, fill=black!4,  minimum width=126mm, minimum height=44mm] (L1) at (0,0) {};
\node[lay, fill=blue!6,   minimum width=110mm, minimum height=33mm] (L2) at (0,-0.32) {};
\node[lay, fill=green!7,  minimum width=94mm,  minimum height=22mm] (L3) at (0,-0.62) {};
\node[lay, fill=orange!10, minimum width=78mm, minimum height=11mm] (L4) at (0,-0.92) {};
\node[anchor=north] at (0,2.13)  {\textbf{\GGTheory{}} --- the constitution and the ledger (\AOne{}; \OGNfive{}; \GUfive{})};
\node[anchor=north] at (0,1.24)  {the engine: \OGGEP{} \(\to\) \GDOS{} \(\to\) \GRIP{}};
\node[anchor=north] at (0,0.40)  {cosmological stratum: inventory, \(H(t)\), corridor (\ATen{})};
\node                at (0,-0.92) {this paper: \textsf{T/F/O/PS/I-Hubble-\(N\)}, \(\zeta_\Lambda\), lift, landscape};
\end{tikzpicture}}
\caption[How this paper's names nest inside the program.]{\textbf{How
this paper's names nest.} Reading outward: the tags defined here
(innermost) are instances of the cosmological stratum's objects, which
are outputs of the one engine, which runs on the constitution and the
integer ledger. A cross-cohort citation names the innermost layer that
owns the object.}
\label{fig:c2-name-nesting}
\end{figure}


\subsection{Comprehensive summary table: every headline number and where it is earned}
\label{app:summary}

Table~\ref{tab:c2-summary} collects every quantitative claim of the report,
its value, its honesty grade (\S\ref{sec:absolute-h0}), and the exact place
the reader can check it. It is the one-page index to the paper's arithmetic.

\begin{table}[htbp]
\centering\small
\caption[Every headline number of the derived Hubble reading, its grade, and where it is derived.]{Every headline number of the derived Hubble reading, its grade, and
where it is derived or defined. Grades as in the honesty triage
(Table~\ref{tab:c2-triage}). This is the paper's ledger and the place to check any number quoted elsewhere in it. Reading down the grade column is the fastest honest summary available: most of the headline values are conditional on a single lemma, one is measured rather than derived, and one is named as open. The last column gives the equation or the section where each is fixed, so nothing here has to be taken on the table's word.}
\label{tab:c2-summary}
\begin{tabularx}{\textwidth}{@{}lllX@{}}
\toprule
\textbf{Quantity} & \textbf{Value} & \textbf{Grade} & \textbf{Where}\\\midrule
Background rate $H_0^{(0)}$ & $66.6\pm0.2$ & D$^*$ & \S\ref{sec:absolute-h0}, App.~\ref{app:calc}\\
Baryon anchor & $66.5$ & D$^*$ & Eq.~\eqref{eq:c2-H0-anchor}\\
Matter anchor & $67.0$ & D$^*$ & Eq.~\eqref{eq:c2-H0-anchor}\\
Matter fraction $\Omega_m$ & $1/\pi=0.3183$ & D$^*$ & Eq.~\eqref{eq:c2-inventory}, Hopf ratio\\
Baryon fraction $\Omega_b$ & $1/2\pi^2=0.0507$ & D$^*$ & Eq.~\eqref{eq:c2-inventory}\\
Dark-energy fraction $\Omega_\Lambda$ & $1-1/\pi=0.6817$ & D$^*$ & Eq.~\eqref{eq:c2-inventory}\\
Seesaw coefficient $\zeta_\Lambda$ & $\pi/175 = 0.01795$ & Derived (cond.) & \S\ref{sec:zeta-selection}, Eq.~\eqref{eq:c2-H0-seesaw}, App.~\ref{app:seesaw}\\
CMB reference & $67.4$ & measured & \S\ref{sec:both-from-anchor}\\
Late / local (lifted) & $\approx68.5$ & D$^*$ & Eq.~\eqref{eq:Hlatepred}\\
Local-Hole lift $\epsilon_{\mathrm{Hole}}$ & $+2$--$3\%$ & Derived & App.~\ref{app:epslocal}\\
Ladder residual $\epsilon_{\mathrm{syst}}$ & $\approx+7\%$ ($+7.3\%$ exact) & named, open & \S\ref{sec:both-from-anchor}\\
Present age $t_0$ & $13.9$ Gyr (today) & D$^*$ & Eq.~\eqref{eq:c2-age}\\
Siren prediction $H_0^{\mathrm{GW}}$ & $\approx67$ & prediction & \S\ref{sec:decisive}\\
Lensing $S_8$ & $0.759$ & D$^*$ & \S\ref{sec:s8-section}\\
Corridor-switch $z_*$ & $e^7\approx1097$ & inherited & \S\ref{sec:both-from-anchor}\\
Lock scale $M_{\mathrm{lock}}$ & $3.85$ TeV & Theorem & \AFive{}~T8.7\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Abbreviations and symbol glossary}
For readers approaching this report without full familiarity with
the Arisaka-GG preprint cohort, the following short glossary
recapitulates the technical vocabulary used above. Fuller
definitions are in the flagship summary paper
(\citealp{ArisakaA1Constitution}; \citealp{ArisakaA9SM};
\citealp{ArisakaA5GDOS}; \citealp{ArisakaA10Cosmos};
\citealp{ArisakaA7Qi4}).

\begin{description}[leftmargin=2em,itemsep=0.12em,labelindent=0pt]
\item[\GGtheory{}] Grand Geometric Theory / Geometry-Gauge Theory;
the integrated framework comprising \textsf{GUT-10}, \GGsix{}, \GDOS{}, \GiFour{}, and
\QiFour{}.
\item[\GGsix{}] The six-dimensional bulk stage
$M_4 \times I_r \times S^1_\theta$ carrying the unified gauge
content $\SUfive \times \UoneB \times \UoneQi \times
\mathrm{Gravity}$.
\item[\BIsix{}] Green--Schwarz-modified Bianchi identity
$\dd H_3 = \tr R^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 -
\tfrac{2}{5}F_{Qi}^2$, the anomaly-cancellation backbone.
\item[\OGNfive{}] Odd-integer Gauge Normalization ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ with $K = LJ = M^2 + N^2 + L^2$.
\item[\UoneB{}] The exact local baryonic $\U(1)$ gauge factor;
guarantees proton stability.
\item[\UoneQi{}] The informational compact-angle $\U(1)$ gauge
factor on $S^1_\theta$; source of $\DNeff$ here.
\item[\TGG, \CCI, \GFF, \GRIP] The four Gi pillars of \ASix{}
\citep{ArisakaA6Gi4}
(Total Geometry-Gauge conservation; Covariant Conservation
Identity; Geometric Free-Energy Functional; Gauge-Runtime
Invariance Pipeline).
\item[\DGI, \GSSB, \GA] The three stages inside \GRIP{} (Dynamic
Gauge Invariance; Gauge-Selected Symmetry Breaking; Gauge
Attractor).
\item[\Qifour{}] The four pillars of the Qi runtime (Qi-QUA,
Qi-EP, Qi-CON, Qi-HAL).
\item[\GSSB{} throat cap] The single transition event that ends
inflation, produces Kaxion dark matter, and leaves
$\rho_\Lambda$.
\item[Arisaka Equation] $J_E/Q_B = J_e = J_B = J_{Qi}$, the
current-locking theorem of Qi-QUA.
\item[Arisaka Ratio] $(1,-1,-2/5,-2/5)$, the \BIsix{} coefficient
vector.
\item[Arisaka Constant] $Q_B = (2/5)(E_{\mathrm{Ryd}}/K) \approx
15.0$~kJ/mol, the universal quantum of open-system operations.
\item[$\fEDE$] Fractional early-dark-energy contribution; the
derived reading requires none, and a detected lobe refutes it
(F-Hubble-3).
\item[$\DNeff$] Extra effective relativistic species; here the small cohort value
$\approx 0.04$ of \ANine{}, from the $\UoneQi$ channel (the early side
carries no large radiation excess).
\item[$\epslocal$] The total local excess,
$\epslocal = 73.5/66.6 - 1 = 0.104$, split as the derived Local-Hole
lift $\epsilon_{\mathrm{Hole}}$ ($+2$--$3\%$) plus the named open
residual $\epsilon_{\mathrm{syst}}$ ($\approx\!7\%$; the $\Uoneb$
graviphoton channel is excluded at $\sim10^{-3}$).
\item[$\zeta_B$] Graviphoton screening ratio
$\zeta_B \equiv (2/5)\,\delta\rho_B/\rho_B$ entering the baryonic
effective Newton constant (Appendix~C); written $\zeta_B$ (not $\eta_B$) to
avoid collision with the cohort-wide baryon-to-photon ratio.
\end{description}




\subsection{The honesty grades: a reader's key}
\label{app:grades}

Every quantitative claim in this report carries a grade, quoted
identically wherever the claim appears. The grades are the cohort's
codebook discipline (\ANine{}; \ATen{}), and a reader who knows the key
can tell at a glance how much any number is claiming.

\begin{description}[leftmargin=1.4em,itemsep=0.25em]
\item[\textbf{Theorem.}] Proved within the framework's stated axioms,
with no named condition outstanding. Example: the lock scale
\(\Mlock = \tfrac12 E_{\mathrm{Pl}} e^{-K}\) (\AFive{}~T8.7), pinned in
the laboratory through the electroweak vev.
\item[\textbf{Derived (\(\mathsf D^{*}\)).}] Derived conditional on a
single \emph{named} lemma or conjecture, stated wherever the result is
used. Example: the \(\pi\)-locked fractions (conditional on the
closed-packet measure conjecture of \ATen{}); the derived background
\(66.6 \pm 0.2\) inherits this grade.
\item[\textbf{Derived (cond., \MSMF{}/\OGN{}).}] Selected by the
codebook's admissibility and naming rules, with the closing computation
pre-registered as a falsifier. Example: \(\zeta_\Lambda = \pi/175\)
(F-Hubble-11).
\item[\textbf{Asserted.}] A declared input, flagged as such. The
framework's honesty requires these to be few; this paper's Route~I
uses none.
\item[\textbf{Asserted-external.}] This paper's own addition to the
scale: a \emph{cited allowance} from the field's literature --- quoted
with the proposer's caveats and the published rebuttals attached,
priced in scenario brackets, never averaged into a \GGtheory{} value.
Example: the three residual-anatomy channels of Part~IV.
\item[\textbf{Named-open.}] A quantity the framework declares it cannot
derive, carried as a debt with dated tests. Example: the \(+7.3\%\)
residual itself.
\end{description}

The one-sentence rule for reading the paper: \emph{a number's grade
travels with it, and no argument is allowed to borrow a stronger grade
than its weakest input.}

\section{The conceptual on-ramp}
\label{app:blockB}

Before the derivations: five primers that define the paper's objects in
plain words, then the questions a reader actually asks, from first
meeting to seminar level.

\subsection{Five primers: the objects of this paper, in plain words}
\label{app:primers}

\subsubsection{The expansion rate \(H(t)\)}
Take any two distant galaxies and divide their separation speed by
their separation: the quotient is the same for every pair, and that
quotient is \(H\). It is not a speed --- it is a \emph{rate}, the
fractional growth of every cosmic distance per unit time, with units of
inverse time; the astronomers' km\,s\(^{-1}\)\,Mpc\(^{-1}\) is a
historical dressing of \(1/\)time. Because the contents of the
universe dilute as it grows, the rate falls with time: enormous in the
hot beginning, \(66.6\) today, settling to \(55.0\) forever. Calling
it the Hubble \emph{constant} names one point on a falling curve ---
the point our era happens to read.

\subsubsection{What ``distance'' means in an expanding universe}
On an expanding stage there is no single distance. The distance light
travel time suggests, the distance a ruler would measure today, the
distance inferred from how faint a source looks (luminosity distance),
and the distance inferred from how large it looks (angular-diameter
distance) all differ, and they differ by known factors of \(1+z\).
Every \(H_0\) measurement is a ratio of a \emph{distance} to a
\emph{redshift}; every method's systematic risk lives in which distance
it actually measures and what calibrates it. The ladder measures
luminosity distances calibrated by geometry and stellar physics; the
CMB measures one angular-diameter distance calibrated by a sound wave;
the siren measures a luminosity distance calibrated by general
relativity itself.

\subsubsection{The sound horizon}
For its first 380{,}000 years the universe was a plasma in which sound
waves ran at over half the speed of light. The distance one such wave
traveled before the plasma cleared --- the sound horizon,
\(r_s = 145.0\)~Mpc on the derived inventory --- froze into the sky as
a preferred spot size (the acoustic peaks) and into the galaxy
distribution as a preferred separation (BAO). It is the early
universe's built-in ruler: measure its apparent size, know its true
size, and the geometry between is fixed. Every early-route \(H_0\)
is, at bottom, this one ruler read across thirteen billion years.

\subsubsection{The standard siren}
Two neutron stars spiraling together broadcast gravitational waves
whose shape encodes the masses and whose loudness then encodes the
distance --- a candle whose absolute wattage is certified by the field
equations, not by any calibration. The host galaxy supplies the
redshift, and \(H_0\) follows with no ladder and no sound horizon
anywhere in the chain. One has been read (GW170817); the coming
decade's catalogs make it a population. It is the referee of this
paper: the one instrument both discordant routes agreed, in advance,
to lose to.

\subsubsection{The Local Hole}
Galaxy counts show our few-hundred-megaparsec neighborhood is mildly
under-dense. Matter flows outward from an under-density, so every
expansion rate measured from inside reads slightly high --- not
because the universe expands faster here, but because the neighborhood
adds an outflow on top of the expansion. The derived size of that lift,
\(+2\)--\(3\%\), carries the background \(66.6\) to the
\(68.5 \pm 0.6\) a ladder inside the Hole should read; it is the
one part of the local excess this paper derives, and the survey
programs of this decade will measure it directly.

\subsection{Foundational questions about the Hubble tension}
\label{app:questions}

A reader meeting the derived reading for the first time will ask a short list
of questions before the arithmetic means anything. This on-ramp answers them
plainly, each with the physical picture first and the pointer to the rigor
second.

\subsubsection{What is the Hubble tension, physically?}
Two ways of measuring how fast the universe expands today disagree. The early
way reads the sound horizon imprinted on the microwave background and, under
$\Lambda$CDM \citep{WeinbergCosmology2008}, infers $H_0\approx67$. The late way climbs a ladder of local
distance indicators and measures $H_0\approx73.5$. The gap is now $7\sigma$,
too large to be a fluke and, after H0DN, not attributable to any single
method. Either a systematic hides in one of them, or new physics separates the
two epochs.

\subsubsection{Why is the background rate $66.6$ and not $73$?}
Because the rate is not measured against the ladder at all; it is
\emph{computed}. The geometry fixes the cosmic inventory in closed form
($\Omega_m=1/\pi$, and the rest by the Hopf-fiber identification of
\S\ref{sec:absolute-h0}); dividing those predicted fractions into the two
physical densities the CMB measures returns $H_0^{(0)}=66.6$ with no free
parameter (App.~\ref{app:calc}). The number is the shadow of the inventory,
not a fit to any distance.

\subsubsection{If it is one rate, why do early and late disagree?}
Because they read the one background rate through two windows that sit on
opposite sides of it. The CMB window lands essentially on the background (the
$-1.2\%$ is a coherent $\pi$-offset, not a bias); the local window is lifted by
a few percent by the mild under-density the ladder sits in (the Local Hole),
and lifted further by a calibration residual the framework does not claim to
derive. The ``tension'' is the width of that split, not a contradiction in the
rate.

\subsubsection{What is $\zeta_\Lambda$, and why is the seesaw ``soft''?}
$\zeta_\Lambda$ is the single dimensionless coefficient of the vacuum-energy
seesaw, $H_0=498.3\sqrt{\zeta_\Lambda}$ (App.~\ref{app:seesaw}). It is forced
to be a ledger monomial in the \OGNfive{} integers; the bounded-cost cluster
spans $0.0180$--$0.0204$, and the codebook's own admissibility and naming
rules select $\pi/(LK) = \pi/175$ before any data enter
(\S\ref{sec:zeta-selection}). The residual ``softness'' ($H_0\propto
M_{\mathrm{lock}}^4$) is bounded in the laboratory: the same lock scale
predicts the electroweak vev at $+0.045\%$, pinning the rate to
$\pm0.18\%$.

\subsubsection{Why does a standard siren settle it when neither ladder nor CMB can?}
Because a gravitational-wave standard siren reads the luminosity distance
straight off the waveform and the redshift off the host galaxy --- it uses
\emph{neither} a sound horizon \emph{nor} a distance ladder. It therefore
measures $H_0$ through a channel that shares no systematic with either
discordant method. The derived reading predicts $H_0^{\mathrm{GW}}\to67$; a
converged value at $73$ refutes it outright (\S\ref{sec:decisive}).

\subsubsection{What would kill the derived reading?}
Eleven named falsifiers, each with a pre-declared numerical rule
(\S\ref{sec:falsifiers}). The decisive one is the siren (F-Hubble-2); the
single-event kill is proton decay (F-Hubble-1), which would retire the whole
program. The reading is built to lose cleanly if it is wrong.

\subsubsection{What exactly is a standard siren, and why is it immune to both routes?}
A merging pair of neutron stars or black holes whose gravitational
waveform is computed from first principles: the phasing gives the
masses, the amplitude then gives the distance --- a candle whose
luminosity general relativity itself certifies. Its host galaxy gives
the redshift. No sound horizon, no ladder, no calibration shared with
either discordant route: the one clock-reader both camps must accept.
The full physics, forecast, and budget are Part~VII.

\subsubsection{What is the Local Hole, and how sure are we it exists?}
Galaxy counts show our neighborhood is mildly under-dense --- matter
flows out of it, so local expansion reads slightly high. That much is
secure; the \emph{depth and extent} are contested, from a shallow
\(\delta \approx -0.10\) to the disputed \(-0.3\) over
\(300\)~Mpc, which is why the paper adopts a \(+2\)--\(3\%\) lift and
records the extremes (\S\ref{sec:hole-course}). Full-sky infrared
surveys will settle the profile this decade.

\subsubsection{Why can't a void, or averaging effects, explain the whole tension?}
Because the arithmetic is merciless: a \(+7\%\) monopole needs a mean
under-density of \(-0.41\) across the supernova flow volume ---
excluded at \(\gtrsim 9\sigma\) in standard structure formation ---
and the rigorous averaging (backreaction) term is bounded three orders
of magnitude too small (App.~\ref{app:backreaction}). The environment
supplies percent-class effects; nothing in cosmological averaging
supplies ten.

\subsubsection{The acoustic angle is measured to four digits and the chart misses it by half a percent. How is that a success?}
It is not a success by the standard a \emph{fit} must meet, and the
paper says so plainly (O-Hubble-32). A fit owns parameters whose job is
to land inside the error bars; this chart owns none, claims
percent-class derivation, and is displaced by one coherent, named
systematic whose candidate origin is the theorem's unfinished measure
lemma. The honest tests are the coherence of the displacement across
all seventeen observables (Fig.~\ref{fig:cmb-offsets}) and the printed
break condition: a converged inventory below \(1/\pi\) kills the chart.

\subsubsection{If \(\Omega_m\) is exactly \(1/\pi\), why do fits return 0.315?}
Because a six-parameter fit distributes every small systematic across
all six dials, and the fitted \(\Omega_m\) is the fit's compromise, not
a direct reading. The chart's claim is that the compromise sits one
percent below the true value and drags every derived observable with it
in one direction --- which is exactly the one-sided pattern the data
show. Whether that pattern is the lemma's residue or the chart's
epitaph is decided by the next generation of sub-percent inventory
determinations.

\subsubsection{What happens to the universe, on the derived curve?}
Nothing dramatic, forever: \(H(t)\) settles onto the de~Sitter floor
\(H_\infty = 55.0\), the universe doubles in size every \(12.3\)~Gyr,
distant galaxies slide one by one across the horizon, and the sky
slowly empties. The derived history has no turnaround, no rip, no
cycle --- with \(w = -1\) exactly, the quiet ending is a theorem-grade
consequence, guarded by F-Hubble-5. The clock that began at the first
microsecond never stops; it just settles to its final rate.

\subsubsection{What exactly does ``Asserted-external'' commit the paper to?}
Three mechanics, all checkable. The value is quoted with the proposer's
own caveats and the published rebuttals attached (the priors channel is
carried as ``proposed, executed, and rebutted --- undecided''); it is
priced in scenario brackets rather than adopted (both combination
rules, both scenarios); and it is never averaged into a \GGtheory{}
number --- \(66.6 \pm 0.2\) and \(68.5 \pm 0.6\) contain no external
channel. The grade exists so the paper can \emph{use} the field's
contested corrections as evidence about the field without
\emph{claiming} them (O-Hubble-54 tests exactly this).

\subsubsection{Why print two combination rules --- quadrature and stack?}
Because the right rule depends on correlations the field has not
measured. Independent errors add in quadrature; fully common-mode
errors add linearly; the three anatomy channels are neither cleanly
independent (the prior recalibration re-treats supernova terms the dust
channel also touches) nor fully common. So the paper prints both limits
--- \(4.3\%/3.8\%\) and \(9.3\%/8.3\%\) --- and notes that the required
\(+7.3\%\) sits between them, exactly where correlated common-mode
channels would put it. A single quoted number would smuggle in a
correlation assumption; the bracket is the honesty
(\S\ref{sec:residual-verdict}).

\subsubsection{Why is the recombination switch at \(e^7\), of all numbers?}
Because in the derived chronology the radial coordinate
\(r = \ln(E_{\mathrm{GUT}}/\mu)\) advances by integer-ledger steps, and
the crossing from the plasma dictionary to the frozen geometric
dictionary sits \(J = 7\) e-folds of redshift above the present:
\(1 + z_* = e^7 \approx 1097\), against the measured
\(z_{\mathrm{dec}} = 1089.9\) --- a \(0.6\%\) agreement with an
integer's worth of freedom, i.e.\ none (\S\ref{sec:recomb-switch}).
This is also where the ledger integer \(J\) enters the Hubble story;
it is absent from the closed form for \(H_0\) itself
(\S\ref{sec:h0-closed-form}).

\subsubsection{How can \(w = -1\) be \emph{exact} when every fitted value has error bars?}
Because in the derived reading the dark energy is not a dynamical fluid
whose stiffness one measures; it is the residue of a geometric
transition that is \emph{complete}. A finished transition has no
remaining dynamics to drag \(w\) off \(-1\): the value is exact the way
a crystal's ground-state symmetry is exact, not the way a measured
coupling is precise. The measurable content is two-sided and live:
any converged \(w \neq -1\) at \(3\sigma\) retires the reading
(F-Hubble-5), while every tightening of \(w\) toward \(-1\) squeezes
the dynamical-dark-energy rivals it competes with.

\section{Derivation course: the Friedmann background from the metric up}
\label{app:courseC}

This is the first of the full derivation courses, and it sets their
three-step contract. \emph{Step one} is the textbook: the topic built
from scratch, every component computed, at the level of a first-year
graduate course --- nothing assumed beyond special relativity, the
equivalence principle, and the Einstein equation. \emph{Step two} names
what the standard treatment assumes and cannot explain. \emph{Step
three} derives the \GGtheory{} resolution and grades it. A reader who
works this course with a pen can reproduce every background quantity
in this paper --- the equations, the age, the \(H(z)\) rows, the
de~Sitter floor --- and will know exactly which steps are century-old
mathematics and which are this program's claims.

\subsection{The FRW metric and its conventions}
\label{app:cC-metric}

Cosmology begins with an observation and a principle. The observation:
averaged over scales above a few hundred megaparsecs, the universe
looks the same in every direction. The principle (Copernican): we do
not occupy a special place, so it looks that way from everywhere.
Together they demand a spacetime that is spatially homogeneous and
isotropic --- and that symmetry is so restrictive that the metric is
fixed up to one function of time and one discrete sign:
\begin{equation}
ds^2=-dt^2+a^2(t)\Big[\tfrac{dr^2}{1-kr^2}+r^2\,d\Omega_2^2\Big],
\qquad k\in\{-1,0,+1\},
\label{eq:app-frw-metric}
\end{equation}
in units \(c = 1\), signature \((-,+,+,+)\)
(conventions as in \citealp{WeinbergCosmology2008,Dodelson2020}), with
\(d\Omega_2^2 = d\theta^2 + \sin^2\theta\,d\phi^2\). Here \(t\) is
proper time for observers at rest in the average matter (comoving
observers); \(r, \theta, \phi\) are comoving coordinates, painted onto
space once and carried along by the expansion; and \(a(t)\) --- the
scale factor --- is the single dynamical degree of freedom, the
``size'' of space. It is convenient to write the spatial part as
\(g_{ij} = a^2(t)\,\gamma_{ij}\), with \(\gamma_{ij}\) the comoving
(time-independent) three-metric of constant curvature \(k\).

\emph{Intuition.} The metric says that the universe does not expand
\emph{into} anything: comoving galaxies keep their coordinates
forever, and it is the ruler between them, \(a(t)\), that grows. Every
physical separation is (comoving label) \(\times\) \(a(t)\); every
question about cosmic history is a question about one function of one
variable. That economy --- an entire universe reduced to \(a(t)\) and
a sign --- is the gift of symmetry, and everything in this course is
the Einstein equation's opinion about that one function.

\subsection{Christoffel symbols: every non-vanishing component}
\label{app:cC-christoffel}

The connection follows from the metric by the standard formula
\begin{equation}
\Gamma^{\lambda}_{\mu\nu}
= \tfrac12\, g^{\lambda\sigma}
\left(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu}
      - \partial_\sigma g_{\mu\nu}\right).
\label{eq:app-christoffel-def}
\end{equation}
The metric components are \(g_{00} = -1\), \(g_{ij} = a^2\gamma_{ij}\);
the inverse is \(g^{00} = -1\), \(g^{ij} = a^{-2}\gamma^{ij}\); and the
only time dependence anywhere is the factor \(a^2(t)\), so the only
non-trivial time derivative is \(\partial_0 g_{ij} = 2a\dot a\,
\gamma_{ij}\). Now compute, term by term.

\emph{The time-space-space component.} Set \(\lambda = 0\),
\(\mu = i\), \(\nu = j\) in Eq.~\eqref{eq:app-christoffel-def}. Since
\(g^{0\sigma}\) is non-zero only for \(\sigma = 0\), and
\(g_{0i} = 0\),
\[
\Gamma^{0}_{ij}
= \tfrac12\,g^{00}\left(\partial_i g_{0j} + \partial_j g_{0i}
 - \partial_0 g_{ij}\right)
= \tfrac12(-1)\left(0 + 0 - 2a\dot a\,\gamma_{ij}\right)
= a\dot a\,\gamma_{ij}.
\]

\emph{The space-time-space component.} Set \(\lambda = i\),
\(\mu = 0\), \(\nu = j\):
\[
\Gamma^{i}_{0j}
= \tfrac12\,g^{ik}\left(\partial_0 g_{kj} + \partial_j g_{k0}
 - \partial_k g_{0j}\right)
= \tfrac12\,a^{-2}\gamma^{ik}\left(2a\dot a\,\gamma_{kj}\right)
= \frac{\dot a}{a}\,\delta^{i}_{j}.
\]

\emph{The purely spatial components.} With all indices spatial, the
factors of \(a^2\) cancel between \(g^{ik}\) and \(\partial g_{kj}\),
leaving the connection \(\widehat\Gamma^{i}_{jk}\) of the comoving
three-metric \(\gamma_{ij}\) alone (for \(k = 0\) it vanishes in
Cartesian comoving coordinates; for curved sections it is the standard
constant-curvature connection). Every other component vanishes:
\(\Gamma^{0}_{00} = \Gamma^{0}_{0i} = \Gamma^{i}_{00} = 0\), because
\(g_{00}\) is constant and the metric has no time-space cross terms.

\emph{Intuition.} Two components carry all the physics.
\(\Gamma^{i}_{0j} = (\dot a/a)\delta^i_j = H\,\delta^i_j\) \emph{is}
the Hubble rate: it says that moving forward in time rescales every
spatial vector by \(H\) per unit time --- free particles separate, and
peculiar velocities decay, at exactly this rate.
\(\Gamma^{0}_{ij} = a\dot a\,\gamma_{ij}\) is its time-facing partner:
stretching space costs time-direction curvature, which is why
expansion can decelerate under gravity at all.

\subsection{The Ricci tensor and the Ricci scalar}
\label{app:cC-ricci}

The Ricci tensor is
\(R_{\mu\nu} = \partial_\lambda \Gamma^\lambda_{\mu\nu}
- \partial_\nu \Gamma^\lambda_{\mu\lambda}
+ \Gamma^\lambda_{\lambda\sigma}\Gamma^\sigma_{\mu\nu}
- \Gamma^\lambda_{\nu\sigma}\Gamma^\sigma_{\mu\lambda}\).

\emph{The time-time component.} Only the terms with
\(\Gamma^i_{0j}\) survive:
\[
R_{00}
= -\,\partial_0 \Gamma^{i}_{0i}
  - \Gamma^{i}_{0j}\Gamma^{j}_{0i}
= -\,\partial_0\!\left(3\frac{\dot a}{a}\right)
  - 3\left(\frac{\dot a}{a}\right)^{2}
= -3\left(\frac{\ddot a}{a} - \frac{\dot a^2}{a^2}\right)
  - 3\frac{\dot a^2}{a^2}
= -3\,\frac{\ddot a}{a}.
\]
Note the cancellation: the \(\dot a^2/a^2\) pieces from the derivative
and the quadratic term eat each other, leaving pure acceleration. The
time-time Ricci component knows only about \(\ddot a\) --- this is the
covariant ancestor of the Newtonian statement that tidal focusing
measures the \emph{acceleration} of test bodies, not their speed.

\emph{The space-space components.} Assembling the same formula with
the spatial connection included --- and this time showing the three
time-derivative terms the promise of this course owes, each a
coefficient of \(\gamma_{ij}\):
\[
\underbrace{\partial_0\Gamma^0_{ij} = (\dot a^2 + a\ddot a)\,\gamma_{ij}}_{\text{term 1}},\qquad
\underbrace{\Gamma^\lambda_{\lambda 0}\Gamma^0_{ij} = 3\dot a^2\,\gamma_{ij}}_{\text{term 2}},\qquad
\underbrace{-2\,\Gamma^0_{ik}\Gamma^k_{0j} = -2\dot a^2\,\gamma_{ij}}_{\text{term 3}},
\]
which sum with the comoving three-space curvature (Ricci tensor
\(2k\,\gamma_{ij}\)) to \((a\ddot a + 2\dot a^2 + 2k)\,\gamma_{ij}\),
i.e.:
\[
R_{ij}
= \left(\frac{\ddot a}{a} + 2\frac{\dot a^{2}}{a^{2}}
+ 2\frac{k}{a^{2}}\right) g_{ij}.
\]
Contracting, \(R = g^{00}R_{00} + g^{ij}R_{ij}
= 6\left(\dfrac{\ddot a}{a} + \dfrac{\dot a^{2}}{a^{2}}
+ \dfrac{k}{a^{2}}\right)\).

\subsection{The Einstein tensor}
\label{app:cC-einstein}

With \(G_{\mu\nu} = R_{\mu\nu} - \tfrac12 R\,g_{\mu\nu}\):
\[
G_{00} = -3\frac{\ddot a}{a} - \tfrac12\,(6)\left(\frac{\ddot a}{a}
+ \frac{\dot a^2}{a^2} + \frac{k}{a^2}\right)(-1)
= 3\left(\frac{\dot a^{2}}{a^{2}} + \frac{k}{a^{2}}\right),
\]
the acceleration canceling exactly, and
\begin{equation}
G_{00}=3\Big(\tfrac{\dot a^2}{a^2}+\tfrac{k}{a^2}\Big),
\qquad
G_{ij}=-\Big(2\tfrac{\ddot a}{a}+\tfrac{\dot a^2}{a^2}+\tfrac{k}{a^2}\Big)g_{ij}.
\label{eq:app-einstein-tensor}
\end{equation}
\emph{Intuition.} The cancellation in \(G_{00}\) is not an accident;
it is the Bianchi identity at work. \(G_{00}\) is a
\emph{constraint}, not an evolution equation: it relates the expansion
rate and spatial curvature to the energy present \emph{now}, with no
reference to acceleration --- the general-relativistic analogue of
energy conservation for the whole universe. \(G_{ij}\) carries the
dynamics: how the expansion responds to pressure.

\subsection{The perfect-fluid source}
\label{app:cC-fluid}

The same symmetry that fixed the metric fixes the source. Isotropy
forbids any spatial vector (no momentum flux, \(T_{0i} = 0\)) and any
anisotropic stress (\(T_{ij}\propto g_{ij}\)); homogeneity makes the
two remaining scalars functions of time alone. Hence
\[
T^{\mu}{}_{\nu} = \mathrm{diag}(-\rho,\,p,\,p,\,p),
\]
a perfect fluid with energy density \(\rho(t)\) and pressure
\(p(t)\), each species carrying an equation of state
\(w_i = p_i/\rho_i\): \(w = 0\) for cold matter (pressureless dust),
\(w = 1/3\) for radiation (tracelessness of the electromagnetic
stress tensor), \(w = -1\) for vacuum energy (Lorentz invariance of
the vacuum forces \(T_{\mu\nu}\propto g_{\mu\nu}\)).

\subsection{The two Friedmann equations, the acceleration equation, and continuity}
\label{app:cC-friedmann}

Insert Eq.~\eqref{eq:app-einstein-tensor} and the fluid into
\(G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G_N T_{\mu\nu}\). The
time-time component gives the first Friedmann equation,
\begin{equation}
\boxed{\,H^2\equiv\Big(\tfrac{\dot a}{a}\Big)^{2}
=\frac{8\pi G_N}{3}\,\rho-\frac{k}{a^2}+\frac{\Lambda}{3}\,}
\label{eq:app-friedmann1}
\end{equation}
--- the speedometer reads the fuel gauge: energy content sets
expansion rate, curvature and vacuum entering as bookkeeping terms of
fixed dilution. The space-space component, after subtracting
\eqref{eq:app-friedmann1}, gives the acceleration equation
\begin{equation}
\frac{\ddot a}{a} = -\frac{4\pi G_N}{3}\,(\rho + 3p) + \frac{\Lambda}{3},
\label{eq:app-friedmann2}
\end{equation}
whose factor \(\rho + 3p\) is the single most consequential
combination in cosmology: \emph{pressure gravitates}. Radiation
(\(p = \rho/3\)) decelerates twice as hard per unit energy as dust;
vacuum energy (\(p = -\rho\)) makes the combination negative and
\emph{accelerates} the expansion --- not by pushing outward, but by
gravitating with the opposite sign. Finally, conservation
\(\nabla_\mu T^{\mu}{}_{0} = 0\) reads
\[
\dot\rho
= \underbrace{-\,\Gamma^{i}_{0i}\,\rho}_{\text{volume dilution}}
\;\underbrace{-\;\Gamma^{i}_{0i}\,p}_{p\,dV\text{ work}}
= -\,3\frac{\dot a}{a}\,(\rho + p),
\qquad\text{i.e.}\qquad
\dot\rho + 3H(\rho + p) = 0,
\label{eq:app-continuity-c}
\]
which is the first law of thermodynamics in disguise: for a comoving
volume \(V \propto a^3\) with energy \(E = \rho V\),
\(dE = -p\,dV\) --- expansion does \(p\,dV\) work on each volume
element. Integrated per species,
\(\rho_i \propto a^{-3(1 + w_i)}\): matter dilutes as \(a^{-3}\)
(number conservation), radiation as \(a^{-4}\) (number conservation
\emph{times} one redshifting wavelength), vacuum not at all. Only two
of the three equations are independent --- the Bianchi identity again
--- so the system is: one constraint, one evolution law, one
bookkeeping rule.

\emph{Newtonian cross-check.} Remarkably, Eq.~\eqref{eq:app-friedmann1}
is older than relativity in disguise. Take a dust ball of radius
\(R = a\,r\), a test galaxy on its edge, and write energy
conservation, \(\tfrac12\dot R^2 - G_N M/R = E\) with
\(M = \tfrac{4\pi}{3}\rho R^3\). Dividing by \(R^2/2\):
\(\dot a^2/a^2 = \tfrac{8\pi G_N}{3}\rho + 2E/(r^2 a^2)\) --- the
Friedmann equation, with the curvature term unmasked as (twice) the
Newtonian energy per unit mass: \(k = -2E r^{-2}\). A bound universe
(\(E < 0\)) is a closed one (\(k > 0\)); the critical density is the
escape-velocity condition \(E = 0\). Nothing about the expansion of
the universe, at the background level, is more exotic than a stone
thrown upward; general relativity's contributions are the
\emph{pressure} term of \eqref{eq:app-friedmann2} and the license to
apply the equation beyond the ball.

\subsection{Curvature, the critical density, and the \texorpdfstring{\(\Omega\)}{Omega} bookkeeping}
\label{app:cC-omega}

Define the critical density as the value that makes space flat today,
\(\rho_c = 3H_0^2/(8\pi G_N)\), and express every component as a
fraction \(\Omega_i = \rho_i/\rho_c\). Numerically, at the derived
rate \(H_0 = 66.6\):
\(\rho_c = 1.878\times10^{-29}\,(H_0/100)^2\,\mathrm{g\,cm^{-3}}
= 8.3\times10^{-30}\,\mathrm{g\,cm^{-3}}\) --- about \emph{five
proton masses per cubic meter}. That the fate and geometry of the
universe pivot on a density a laboratory vacuum exceeds by fifteen
orders of magnitude is the sort of number worth carrying in one's
head. Dividing Eq.~\eqref{eq:app-friedmann1} by \(H_0^2\) and
evaluating today,
\[
1 = \Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k,
\qquad \Omega_k \equiv -\,k/(a_0 H_0)^2,
\]
and at general times, using the dilution laws,
\[
H^2(a) = H_0^2\left[\Omega_r a^{-4} + \Omega_m a^{-3}
+ \Omega_k a^{-2} + \Omega_\Lambda\right],
\]
the master equation behind every row of Table~\ref{tab:derived-Hz}.
On the derived inventory the shares are not measured but \emph{derived} (\(\mathsf D^{*}\), the grade of Part~III):
\(\Omega_m = 1/\pi = 0.3183\),
\(\Omega_\Lambda = 1 - 1/\pi = 0.6817\) (with the radiation share
\(9.36\times10^{-5}\) computed from the measured CMB temperature and
\(\Omega_k = 0\) on the attractor), which is where this course hands
back to the main text.

\subsection{Worked numerics: the age, the acceleration epoch, and the \texorpdfstring{\(H(z)\)}{H(z)} rows}
\label{app:cC-worked}
\label{app:age}

Three computations a reader should do once by hand.

\emph{(a) The age of the universe.} The age is the FRW integral
\begin{equation}
t_0=\int_0^{1}\frac{da}{a\,H(a)}
=\frac{1}{\Hzero}\int_0^1\frac{da}{a\sqrt{\Omega_m a^{-3}+\Omega_\Lambda+\Omega_r a^{-4}}}.
\label{eq:app-age-integral}
\end{equation}
Neglecting the sub-permille radiation term, the matter-plus-vacuum integral has the
closed form
\begin{equation}
t_0=\frac{2}{3\Hzero\sqrt{\Omega_\Lambda}}\,
\operatorname{arcsinh}\!\sqrt{\frac{\Omega_\Lambda}{\Omega_m}}.
\label{eq:app-age-closed}
\end{equation}
Inserting the derived shares $\Omega_m=1/\pi$, $\Omega_\Lambda=1-1/\pi$ gives
$\operatorname{arcsinh}\sqrt{\Omega_\Lambda/\Omega_m}=\operatorname{arcsinh}(1.464)=1.175$
and prefactor $2/(3\sqrt{\Omega_\Lambda})=0.807$, so $t_0=0.948/\Hzero$. With the
derived background rate $\Hzero=66.6$ (Hubble time $1/\Hzero=14.68$~Gyr),
\begin{equation}
t_0=0.948\times14.68\ \mathrm{Gyr}=13.9\ \mathrm{Gyr},
\label{eq:app-age-value}
\end{equation}
in agreement with the oldest-star and CMB determinations and consistent to a
percent with the full numerical integral including radiation ($13.92$~Gyr). The age
is thus not an input but a reading of the same inventory that fixes the rate: the
de~Sitter floor $H_\infty=\Hzero\sqrt{\Omega_\Lambda}=55.0$ and the age $13.9$~Gyr
are two more shadows of $\Omega_\Lambda=1-1/\pi$.

\emph{(b) The acceleration epoch.} Setting \(\ddot a = 0\) in
Eq.~\eqref{eq:app-friedmann2} (matter era, vacuum included) gives
\(\rho_m = 2\rho_\Lambda\), i.e.\
\(1 + z_{\rm acc} = (2\Omega_\Lambda/\Omega_m)^{1/3} = 1.624\),
\(z_{\rm acc} = 0.62\). The moment is prettier than it looks: at
\(\ddot a = 0\) the argument of the arcsinh in
Eq.~\eqref{eq:app-age-closed} is \emph{exactly} \(1/\sqrt2\) ---
independent of the shares --- so
\(t_{\rm acc} = \tfrac{2}{3}\,(1/\sqrt{\Omega_\Lambda})\,
\operatorname{arcsinh}(1/\sqrt2)\,H_0^{-1}
= 7.8\)~Gyr on the derived inventory. The universe decelerated for
its first \(7.8\) billion years and has accelerated for the
\(6.1\) billion since; the landmark table of Part~III carries this as
Landmark~5, and the exact \(1/\sqrt2\) is a pleasant check that the
closed form is being used correctly.

\emph{(c) The \(H(z)\) rows.} The master equation with the derived
shares (radiation included) returns
\(H(0.5) = 88.3\), \(H(1) = 119.7\), \(H(2) = 202.9\),
\(H(4) = 424.0\) --- the rows of Table~\ref{tab:derived-Hz} that the
chronometer, BAO, and drift programs will confront. At the far end,
\(H \to H_0\sqrt{\Omega_\Lambda} = 55.0\): the de~Sitter floor, on
which the universe doubles in size every
\(\ln 2/H_\infty = 12.3\)~Gyr, forever.

\subsection{What the standard derivation assumes --- and cannot explain}
\label{app:cC-limits}

Everything above is a century old and beyond dispute --- and it is
worth listing exactly what it took as given, because the list is the
tension's real cast of characters. \emph{(i) The symmetry is
assumed.} The cosmological principle is imposed, then justified after
the fact by the smoothness of the microwave sky; nothing in the
standard derivation explains why the universe should have selected
the maximally symmetric configuration. \emph{(ii) The contents are
measured, never derived.} The equations accept any \(\rho\); the
actual shares --- why \(0.32\), \(0.68\), \(0.05\), why not any other
partition of unity --- are inputs from the sky, with no principle
behind them. \emph{(iii) \(\Lambda\) is a catastrophe deferred.} The
constant that dominates the budget is either left unexplained or
mis-explained by one hundred and twenty orders of magnitude.
\emph{(iv) Flatness is an input dressed as an initial condition.}
\(\Omega_k = 0\) to a part in a hundred requires either fine-tuning
or an inflationary prologue that is itself chosen, not derived.
\emph{(v) The present epoch is a coincidence.} Matter and vacuum
share the budget within a factor of two only in a narrow window of
cosmic history --- and we happen to live in it. Five assumptions, five
of the deepest open problems in physics. The standard course derives
the \emph{equations} magnificently and leaves the \emph{solution
the universe chose} entirely unexplained.

\subsection{The geometric resolution: the Friedmann system as a theorem}
\label{app:frw-theorem}

The five assumptions of \S\ref{app:cC-limits} are exactly where
\GGtheory{} enters, and each is answered structurally rather than
patched. In \GGtheory{} the Friedmann equations are not an assumed
cosmological model but a \emph{theorem}, forced from the boundary
Einstein equation once the runtime has selected a homogeneous,
isotropic attractor; the contents of \(\rho\) are read from the fixed
integer ledger; \(\Lambda\) is the seesaw residue of
\S\ref{sec:h0-closed-form}; flatness is a property of the attractor;
and the present epoch is the subject of the why-now analysis of the
parent \ATen{}. This subsection carries out the first of those
answers --- the theorem --- because it is the rung on which the whole
notion of a derived \(H(t)\) stands.

\paragraph{From the boundary Einstein equation to a cosmology.}
The \GDOS{} boundary law of \ATwo{} delivers, on the four-dimensional stratum, an
Einstein equation of the standard form
\[
G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G_N\,T_{\mu\nu},
\]
valid for \emph{any} boundary spacetime. A cosmology is the special solution in which
the universe looks the same at every place and in every direction. In the textbook
this cosmological principle is an \emph{assumption}, justified after the fact by the
smoothness of the microwave sky; in \GGtheory{} it is the geometric \emph{attractor}
of the runtime --- the \GRIP{} cascade $\mathrm{DGI}\to\GSSB{}\to\mathrm{GA}$ of
\ASix{}, run on the cosmological stratum, flows to the maximally symmetric
configuration, so homogeneity and isotropy are \emph{selected}, not posited ---
the classical solutions themselves being those of
\citet{Friedmann1922} and \citet{Lemaitre1927}, and the parent's
appendix course carrying the full geometric derivation
\citep[Part~III --- the Friedmann-kinematics theorem block --- and
the boundary-law course of the Technical Appendices of][]{ArisakaA10Cosmos}. That the
attractor is unique is a $\mathsf D^{*}$ result, resting on the attractor-uniqueness
statement of \ASix{}; we state that dependence rather than bury it. Granting the
attractor, the rest is the standard graduate computation.

\paragraph{The source: a perfect fluid from the inherited ledger.}
The same symmetry forces the boundary stress-energy to be that of a perfect fluid,
$T^\mu{}_\nu=\mathrm{diag}(-\rho,p,p,p)$: isotropy forbids momentum flux or shear.
Here is the one place \GGtheory{} departs from the textbook --- not in the equation
but in what may fill it. The density is not a free fluid to be fitted but the
coarse-grained boundary projection of the bulk ledger,
$\rho=\rho_r+\rho_b+\rho_{\rm dm}+\rho_\Lambda+\cdots$, each sector's identity,
equation of state $w_i=p_i/\rho_i$, and normalization fixed by the integer ledger
$(1,3,5;7,35)$ of \AFour{} rather than chosen.

\paragraph{The equations, re-read.}
The computation of \S\ref{app:cC-christoffel}--\S\ref{app:cC-friedmann}
goes through verbatim on the attractor, and
Eq.~\eqref{eq:app-friedmann1} is the object this paper calls
$H(t)$: energy density sets the expansion rate. Nothing on the right is free ---
$G_N$ is the throat-overlap integral of \ATwo{}, $\Lambda$ carries the seesaw scale
of \S\ref{app:calc}, and the contents of $\rho$ are read from the fixed ledger. This
is the precise sense in which $H(t)$ is \emph{derived}: it is the first integral of a
boundary Einstein equation, restricted to a runtime-selected attractor, sourced by a
ledger with no free dials.

\paragraph{The present value as one reading.}
Evaluating Eq.~\eqref{eq:app-friedmann1} today, with $k\to0$ on the attractor and the
$\pi$-locked shares $\Omega_m=1/\pi$, $\Omega_\Lambda=1-1/\pi$, returns
$H_0=H(t_0)$ --- a single point on a curve that runs from an unimaginable early rate
to the de~Sitter floor $H_\infty=H_0\sqrt{\Omega_\Lambda}=55.0$. The Hubble
``constant'' is thereby exposed as a misnomer inherited from a century when only the
present crossing could be measured: it is the value of a derived function at the
moment we happen to read it, and the two dictionaries that disagree about it
(\S\ref{sec:recomb-switch}) are reading one curve at one point through two rulers.



\subsection*{Status}

The three-step contract, graded. Steps one and two
(\S\ref{app:cC-metric}--\S\ref{app:cC-limits}) are standard
mathematics and standard candor: nothing in them is a claim of this
program. Step three carries two grades stated where they are used:
the attractor uniqueness on which ``theorem'' rests is
\(\mathsf D^{*}\) (\ASix{}), and the shares that turn the equations
into numbers are \(\mathsf D^{*}\) under the parent's measure
conjecture. Every numerical output of this course --- the age
\(13.9\)~Gyr, the acceleration epoch \(7.8\)~Gyr, the \(H(z)\) rows,
the floor \(55.0\) --- inherits exactly those conditions and no
others, and every one of them is a falsifier: a robust stellar age
above the FRW integral, or a measured \(H(z)\) row off the master
equation, kills the inventory that produced them.

\section{Derivation course: distances, the ladder, and the century of \texorpdfstring{$H_0$}{H0}}
\label{app:courseD}

This is the paper's signature course, because the subject of this paper
is, at bottom, a dispute about \emph{distances}. Step one builds the
distance toolkit and the ladder from scratch, rung by rung, with the
real error budget beside each rung; step two derives why the ladder's
own cross-checks are structurally blind to exactly one kind of error;
step three reads the same budget through the derived background. A
reader who works this course will know precisely what
\(73.50 \pm 0.81\) is a measurement \emph{of} --- which is the
question the whole tension turns on.

\subsection{The five distances of an expanding universe}
\label{app:cD-distances}

On a static stage one number separates two points; on an expanding
stage, five inequivalent ones, and confusing them costs factors of
\((1+z)\). All descend from the comoving distance --- the metric
distance in Eq.~\eqref{eq:app-frw-metric} evaluated along a light ray
(\(ds = 0\)):
\begin{equation}
D_C(z) = \int_{t_e}^{t_0}\frac{c\,dt}{a(t)} = \int_0^{z}\frac{c\,dz'}{H(z')},
\label{eq:app-cD-comoving}
\end{equation}
where the second form follows from \(1+z = 1/a\) and
\(dz = -(1+z)H\,dt\). In a flat universe the transverse comoving
distance \(D_M\) equals \(D_C\). The two \emph{observable} distances
then follow from what an instrument actually registers.

\emph{Angular-diameter distance.} An object of physical size \(\ell\)
at redshift \(z\) had that size when the universe was smaller by
\(1+z\); the angle it subtends today is
\(\theta = \ell/[a\,D_M] = \ell(1+z)/D_M\), so
\(d_A \equiv \ell/\theta = D_M/(1+z)\): distant objects look
anomalously \emph{large}, because we see them where they were.

\emph{Luminosity distance.} Photons from a source of luminosity \(L\)
arrive spread over the sphere \(4\pi D_M^2\), but each photon is
redshifted (energy down by \(1+z\)) \emph{and} they arrive less often
(rate down by \(1+z\)), so the flux is
\(F = L/[4\pi D_M^2 (1+z)^2]\), i.e.
\begin{equation}
d_L = (1+z)\,D_M
\qquad\Longrightarrow\qquad
d_L = (1+z)^2\,d_A,
\label{eq:app-cD-etherington}
\end{equation}
the second equality being the Etherington duality
\citep{Etherington1933} --- true in
\emph{any} metric theory with photon-number conservation, which is
why a violation of it would be bigger news than the tension. The
astronomers' bookkeeping unit is the distance modulus
\(\mu = 5\log_{10}(d_L/10\,\mathrm{pc}) = m - M\); one magnitude is
\(46\%\) in distance, a conversion this course will use repeatedly.

\emph{The low-redshift expansion.} The ladder operates at
\(z \lesssim 0.15\), where Eq.~\eqref{eq:app-cD-comoving} expands to
\[
d_L(z) = \frac{cz}{H_0}\left[1 + \frac{1-q_0}{2}\,z
+ \mathcal{O}(z^2)\right],
\qquad q_0 \equiv -\frac{\ddot a a}{\dot a^2}\Big|_0
= \tfrac12\Omega_m - \Omega_\Lambda = -0.523
\]
on the derived inventory. The leading term is Hubble's law \citep{Hubble1929}; the
correction, \(0.761\,z\), is \(3.8\%\) at \(z = 0.05\) --- not
negligible at one-percent precision, and applied by every modern
ladder through exactly this cosmography. \emph{Intuition:} the ladder
measures the local stretch of the curve of Course~C; the tension is a
dispute about whether that local stretch, correctly calibrated, sits
on the global curve.

\subsection{The ladder, rung by rung, with its error budget}
\label{app:cD-ladder}

\paragraph{Rung 0 --- parallax.} Geometry, and nothing else: a star
at distance \(d\) sweeps an ellipse of semi-axis
\(p = 1\,\mathrm{au}/d\) on the sky; \(d[\mathrm{pc}] = 1/p['']\).
The sensitivity is unforgiving, \(\delta d/d = \delta p/p\), so a
zero-point offset of tens of microarcseconds matters for every
Milky-Way calibrator: Gaia's bright-star regime carries a measured
offset check of \(-38.9 \pm 10.3\,\mu\)as with the sign of the induced
\(H_0\) shift unestablished (\S\ref{sec:residual-anatomy}). The
network dilutes the raw single-star sensitivity through its other
anchors (NGC~4258's maser geometry, the LMC's detached eclipsing
binaries), which is how the paper's budget carries this floor at
\(0.5\)--\(1\%\) rather than several percent; Gaia DR4
(December~2, 2026) reads the entry either way.

\paragraph{Rung 1 --- Cepheids.} The physics is a driven acoustic
resonance: a star's fundamental pulsation period is its sound-crossing
time, and since the mean sound speed scales as
\(c_s^2 \sim P_{\rm gas}/\rho \sim GM/R\), the period obeys
\[
\Pi \;\sim\; \frac{R}{c_s} \;\sim\;
\sqrt{\frac{R^3}{GM}} \;\sim\; \frac{1}{\sqrt{G\bar\rho}}
\qquad\Longrightarrow\qquad
\Pi\sqrt{\bar\rho} = \mathrm{const},
\]
the pulsation equation's classical result. Brighter Cepheids are
bigger, hence less dense, hence slower: luminosity follows period ---
the Leavitt law \citep{LeavittPickering1912}, \(M = \alpha\log_{10}\Pi + \beta\), calibrated on
the geometric anchors of Rung~0 and applied in dust-hardened Wesenheit
magnitudes \(W = m_H - R\,(m_V - m_I)\), where the coefficient \(R\)
is \emph{assumed} from a dust law --- the first shared convention on
the ladder, and the ancestor of Part~IV's dust channel. The
metallicity term is measured wavelength-independent from \(0.5\) to
\(4.5\,\mu\)m --- the chromatic test an artifact would fail --- and
carries \(\sim\!0.5\%\).

\paragraph{Rung 1\('\) --- TRGB and JAGB.} Two independent stellar
candles cross-check the Cepheids. The tip of the red-giant branch
\citep{FreedmanMadore2010} is core physics: a degenerate helium core ignites at an essentially
fixed core mass, so the brightest red giants in any old population
reach the same luminosity (\(M_I \approx -4.0\)), visible as a sharp
edge in the luminosity function. The J-region asymptotic giant branch
uses carbon stars' narrow luminosity distribution. Both bypass the
Cepheid zero-point entirely --- and \emph{where the tip is measured}
(which fields, which smoothing, which calibration of its absolute
magnitude) is the convention dispute behind the
\(70.39 \pm 1.94\) vs \(73.22 \pm 2.06\) split that Part~IV prices
as the subsample channel.

\paragraph{Rung 2 --- Type Ia supernovae.} The standardizable
explosion: a white dwarf detonating near the Chandrasekhar scale,
with the empirical Tripp standardization
\citep[the modern pipeline:][]{Brout2022}
\[
\mu = m_B - M_B + \alpha\,x_1 - \beta\,c,
\]
stretch (\(x_1\)) and color (\(c\)) corrections with fitted global
coefficients. The color coefficient is where dust hides: correcting
flux for reddening requires a ratio of absorption to reddening
(\(R_B\)), and the fitted \(\beta \approx 3\) sits well below the
Milky Way's \(R_B \approx 4.3\) --- consistent only if the correction
mixes intrinsic color with dust, or if calibrator and Hubble-flow
supernovae see different dust. That fork \emph{is} Part~IV's
\(2\)--\(4\%\) dust channel, and the near-infrared ladder
(F-Hubble-8) is its executioner: dust dies at long wavelengths.

\paragraph{Rung 3 --- the Hubble flow.} Finally, \(\sim\!300\)
supernovae at \(0.023 < z < 0.15\), far enough that peculiar
velocities (\(\sim\!300\)~km\,s\(^{-1}\), i.e.\ \(\lesssim 4\%\) at
the low edge, corrected with flow maps) and close enough that
cosmography, not cosmology, carries the \(q_0\) correction above.
Here \(H_0 = cz/d_L\) is finally read --- through every convention
the lower rungs installed.

\subsection{Worked example: the common-mode algebra}
\label{app:cD-commonmode}

Now the course's central computation: what a shared calibration error
does, in magnitudes. From \(\mu = 5\log_{10}(d_L/10\,\mathrm{pc})\),
\[
\delta\mu = \frac{5}{\ln 10}\,\frac{\delta d}{d}
\qquad\Longleftrightarrow\qquad
\frac{\delta d}{d} = 0.4605\,\delta\mu ,
\]
and since \(H_0 = cz/d\), a calibration that makes every distance
short by \(x\%\) makes the rate high by \(x\%\). The named residual is
therefore a statement about \emph{one number}:
\[
+7.3\%\ \text{in } H_0
\;\Longleftrightarrow\;
\Delta\mu = 5\log_{10}(1.073) = 0.153\ \text{mag}
\]
shared across the calibrator set --- and the field's own contested
corrections are of exactly this order (the rebutted prior-recalibration
effect is \(\sim\!0.05\)~mag \(= 2.3\%\); the dust reanalyses move
\(2\)--\(4\%\)). \emph{Why the ladder's cross-checks cannot see it}
is a five-line theorem. Let the network combine variant readings
\(x_i\) with weights \(w_i\), \(\hat y = \sum_i w_i x_i\), and let
every internal consistency statistic --- leave-one-out shifts,
variant scatter, anchor swaps --- be some function
\(S(\{x_i - x_j\})\) of \emph{differences}. Add a common mode:
\(x_i \to x_i + c\) for all \(i\). Then every difference is
unchanged, so \(S\) is unchanged --- while
\(\hat y \to \hat y + c\). Internal audits are \emph{exactly}
orthogonal to common modes: the one error they cannot flag is the one
that moves the answer. This is not a criticism of the ladder; it is a
property of internal consistency itself, and it is why the paper
assigns the verdict to instruments outside the network
(the siren; the shell test) rather than to more cross-checking.

\subsection{What the ladder cannot decide from inside}
\label{app:cD-limits}

The limitations step, stated with the respect the measurement has
earned. \emph{(i) Shared layers are invisible} --- the theorem above:
one dust-law family, one prior family, one tip convention, one
parallax floor, inherited by every variant at once.
\emph{(ii) A covariance-weighted consensus certifies its membership} ---
it measures, superbly, what the teams inside it share; it is silent on
what its non-members dispute, and the one major program outside the
membership reads \(70.39 \pm 1.94\) (\S\ref{sec:landscape-consensus}).
\emph{(iii) Precision cannot settle kind} --- the ladder can measure
its number to \(1\%\) forever without determining whether that number
is the background rate or the background rate plus environment plus
convention; that question needs either an external referee or a
derived background to score against. These three limits define
exactly the space in which Part~IV operates.

\subsection{The geometric reading of the same budget}
\label{app:cD-gg}

Step three changes no measurement; it changes what the budget is
scored against. Beneath: a derived background, \(66.6 \pm 0.2\)
(Course~E and \S\ref{sec:h0-closed-form}). On top of it: the
environmental lift \(+2\)--\(3\%\), derived in sign, bracketed in
size (App.~\ref{app:epslocal}) --- carrying the expectation to
\(68.5 \pm 0.6\) for any ladder inside the Hole. The remaining
\(+7.3\%\) is then scored channel by channel against the rungs above:
dust (\(2\)--\(4\%\), Rung~2's \(R_B\) fork), priors (\(0\)--\(3\%\),
the inference layer), subsample (\(1\)--\(2\%\), Rung~1\('\)'s
convention dispute), plus the floor entries. The combination algebra
is worth one derivation, because it is where the honesty lives. For
channels \(\sigma_i\) with pairwise correlations \(\rho_{ij}\),
\[
\sigma_{\rm tot}^2=\sum_i\sigma_i^2
+ 2\sum_{i<j}\rho_{ij}\,\sigma_i\sigma_j ,
\]
whose limits are the paper's two printed scenarios: all
\(\rho_{ij}=0\) gives the quadrature \(4.3\%/3.8\%\); all
\(\rho_{ij}=1\) gives the linear stack \(9.3\%/8.3\%\). The observed
requirement \(+7.3\%\) sits between them, and solving the same
formula backwards for a uniform effective correlation gives
\(\bar\rho \approx 0.5\)--\(0.7\) across the two scenarios --- an
\emph{illustration}, not a new claim: it quantifies the verdict
section's statement that the residual sits exactly where correlated
common-mode channels would put it, in the regime the overlap caveat
describes. Every channel carries its dated, two-sided test
(F-Hubble-8/9/10), and either outcome of each is printed in advance.

\subsection*{Status}

Step one is standard astronomy, presented without a single
\GGtheory{} claim; the ladder it describes is among the great
precision instruments of science. Step two is arithmetic
(the common-mode theorem) and sociology-free bookkeeping (the
certification boundary). Step three inherits its grades from the
main text: the background \(\mathsf D^{*}\), the lift derived in
sign and bracketed, every channel Asserted-external, the residual
named-open, and the effective-correlation number an illustration on
printed values. Nothing in this course re-reduces anyone's
photometry; that boundary is the tier ladder's
(\S\ref{sec:disc-tier-ladder}).

\section{Derivation course: the microwave background as an \texorpdfstring{$H_0$}{H0} instrument}
\label{app:courseE}

The early side of the tension is one measurement chain read backwards
through thirteen billion years, and this course builds it: the plasma
physics, the ruler it froze, the angle that ruler subtends, and ---
the step most accounts blur --- exactly what the microwave sky does
and does not measure about the expansion rate. The punchline of step
two is worth stating in advance: \emph{the CMB never measures}
\(H_0\); it measures two densities and one angle, and every quoted
``CMB \(H_0\)'' is an inversion conditional on a model. That fact is
not a weakness of the CMB; it is the opening this paper's Route~II
walks through.

\subsection{The tightly coupled plasma and its sound speed}
\label{app:cE-oscillator}

Before recombination, Thomson scattering couples photons and baryons
into a single fluid: the scattering rate exceeds the expansion rate
by many orders, so photons and baryons move together, and the fluid
has a pressure (the photons') and an inertia (both). Perturbations in
this fluid obey a driven oscillator: gravity compresses a
\(k\)-mode into potential wells, photon pressure resists, and the
balance rings. The spring constant is the sound speed. Derive it:
the pressure is the photons', \(p = \rho_\gamma/3\); the inertia is
both species', \(\rho_\gamma + p_\gamma + \rho_b = \tfrac43\rho_\gamma(1 + R)\)
with the baryon loading
\[
R \equiv \frac{3\rho_b}{4\rho_\gamma}
= \frac{3\,\omega_b}{4\,\omega_\gamma}\,\frac{1}{1+z}
\approx \frac{678}{1+z},
\]
so
\[
c_s^2 = \frac{\delta p}{\delta\rho}
= \frac{\tfrac13\,\delta\rho_\gamma}{\delta\rho_\gamma + \delta\rho_b}
= \frac{c^2}{3\,(1+R)} .
\]
Baryons add weight but no pressure: they slow the sound and deepen
the compressions --- which is precisely why the \emph{relative
heights} of odd (compression) and even (rarefaction) peaks read off
\(\omega_b\), the first of the sky's two \(h\)-free density
measurements. At the switch, \(R(z_*) = 0.62\) and
\(c_s = 0.454\,c\): the early universe rang at almost half the speed
of light.

\subsection{The sound horizon, evaluated}
\label{app:cE-rs}
\label{app:soundhorizon}

The ruler is the distance such a wave travels before the plasma
clears:
\begin{equation}
r_s=\int_{z_\ast}^{\infty}\frac{c_s(z)}{H(z)}\,dz,
\qquad c_s(z)=\frac{c}{\sqrt{3\,[1+R(z)]}},\quad
R=\frac{3\rho_b}{4\rho_\gamma}=\frac{3\Omega_b}{4\Omega_\gamma}\,\frac{1}{1+z},
\label{eq:app-rs}
\end{equation}
with \(z_* = e^7\) on the derived chart. Evaluate it in pieces, so
the anatomy is visible. Deep in the radiation era
(\(z \gg z_{\rm eq} = 3400\)), \(H \propto (1+z)^2\) and \(R \to 0\):
the integrand falls as \((1+z)^{-2}\), so the early contribution
converges --- the ruler does not care about the first instants. Near
equality, \(R(z_{\rm eq}) = 0.20\): baryon loading is already
slowing the wave. From equality to the switch, matter drives
\(H \propto (1+z)^{3/2}\) while \(R\) grows to \(0.62\); this stretch
contributes the bulk of the length. Carrying out the quadrature with
the derived inventory and the measured densities gives
\(144.4\)~Mpc from this two-fluid integral. The last \(0.6\)~Mpc
deserves its own ledger, because it is the one step of this course we
do not display line by line: (i) the detailed radiation inventory
near equality (neutrinos in the pre-recombination plasma), \(+0.6\);
(ii) for calibration of the scale of such conventions, the choice of
lower limit alone --- the ledger's switch \(z_* = e^7 = 1096.6\)
versus the measured centroid \(1089.9\) --- is worth another
\(\approx\!0.6\) (\(c_s/H \approx 0.09\)~Mpc per unit redshift at
the epoch); half-percent conventions, all of them, and the reason
\(r_s\) carries no sub-percent claim anywhere in this paper. The
full evaluation returns the landmark value
\[
r_s(z_*) = 145.0\ \mathrm{Mpc}
\]
carried throughout this paper (landmark~4), the half-percent
difference being exactly the level at which the simple two-fluid
picture stops being the whole story. One disambiguation the
literature often blurs: baryons remain loaded on the photons slightly
\emph{after} the photons decouple, so the ruler frozen into the
\emph{galaxy} distribution is the slightly longer drag-epoch horizon.
On the derived background the same two-fluid quadrature, evaluated at
the standard drag epoch \(z_d \approx 1060\), gives
\(r_d \simeq 147.5\)~Mpc (the integrator reproduces the concordance
\(r_d = 147.1\) to \(0.1\%\) on Planck inputs, so the derived-chart
shift is real, not numerical) --- the BAO ruler. The CMB's acoustic
angle uses \(r_s(z_*) = 145.0\); the galaxy surveys use \(r_d = 147.5\);
this paper's landmark is the former, and every BAO sentence in this
paper spends the latter.

\subsection{The acoustic angle and the peak series}
\label{app:cE-peaks}

The observable is the angle the ruler subtends across the universe:
\begin{equation}
\theta_\ast=\frac{r_s}{D_M(z_\ast)},\qquad
D_M(z_\ast)=\int_0^{z_\ast}\frac{c\,dz}{H(z)},
\label{eq:app-theta}
\end{equation}
measured to four significant figures,
\(\theta_\ast=(1.0411\pm0.0003)\times10^{-2}\). On the derived chart
the denominator evaluates to \(D_M(z_*) = 13{,}970\)~Mpc, giving
\(100\,\theta_* = 1.038\) --- within half a percent of the measured
value, displaced \emph{coherently} with the fifteen other observables
of the offset audit, and the paper scores that displacement honestly
rather than claiming a bullseye (O-Hubble-32;
Fig.~\ref{fig:cmb-offsets}). The peak multipoles follow as a harmonic
series with a gravitational phase shift,
\[
\ell_m \simeq \left(m - \phi_m\right)\frac{\pi}{\theta_*},
\qquad \frac{\pi}{\theta_*} = 303,
\]
with \(\phi_1 \approx 0.27\) from the decay of the potentials at
horizon crossing: \(\ell_1 \approx 220\). The precise version of this
computation is the Boltzmann audit of \S\ref{sec:cmb-overdetermined},
which returns \(221/538/817\) against the observed \(220/536/813\) on
the exact derived fractions.

\subsection{Why the microwave sky never measures \texorpdfstring{$h$}{h}}
\label{app:cE-degeneracy}

Now the course's load-bearing derivation. What the peak pattern
measures, cleanly, is three things: \(\omega_b\) (odd/even height
ratio --- baryon loading), \(\omega_m\) (the height envelope through
the equality epoch and the potential-decay driving), and
\(\theta_*\) (the positions). Notice what is \emph{not} on the list.
The physical densities \(\omega_i = \Omega_i h^2\) carry no separate
\(h\); and \(\theta_*\) constrains \(h\) only through the late-time
integral \(D_M = \int c\,dz/H(z)\), where \(H(z)\) depends on
\(h\) \emph{and} on the late-time budget. Hold \(\omega_b\) and
\(\omega_m\) fixed (the early universe unchanged, \(r_s\) unchanged)
and vary \(h\): flatness re-partitions the budget,
\(\Omega_\Lambda = 1 - \omega_m/h^2\), and \(D_M\) changes --- so
\emph{given} flatness and \(w = -1\), the measured \(\theta_*\) picks
out one \(h\). Relax either assumption and the same \(\theta_*\)
admits a family of \((h, w, \Omega_k)\) combinations: the geometric
degeneracy. Hence the precise statement the tension literature often
compresses: \emph{the ``CMB value of \(H_0\)'' is an inversion of two
densities and one angle through an assumed late-time model.} Standard
\(\Lambda\)CDM performs that inversion by fitting
\(\Omega_i\); Route~II of this paper performs it by \emph{deriving}
them --- same sky, same densities, opposite direction of inference.

\subsection{The audit: zero background freedom against the whole spectrum}
\label{app:cE-audit}

The derived chart does not stop at three summary statistics; it is
scored against the full temperature spectrum with zero background
freedom: the exact fractions, the derived rate, nothing tuned,
\(\chi^2 = 237\) across the \(83\) band powers, with the data
preferring the derived tilt at \(\Delta\chi^2 = -22\)
(\S\ref{sec:cmb-overdetermined}). A fit this rigid cannot hide a
mistake in one parameter inside another --- there are no others ---
which is why the residual pattern matters more than its size:
seventeen observables, one direction, percent scale, the signature of a
single coherent displacement rather than scattered errors. Step two
of this course, and the precedent lessons of
\S\ref{sec:disc-precedents}, meet exactly here: coherence is the
statistic that cable faults and foregrounds do not fake.

\subsection{The sound-horizon lever, and its cap}
\label{app:cE-lever}

The limitations step, aimed at the early-side fixes. At fixed
measured \(\theta_*\) and fixed late-time physics,
\(D_M\) must scale with \(r_s\); and since \(D_M\) scales roughly as
\(1/H_0\) for small changes, the inferred rate obeys
\(\delta H_0/H_0 \approx -\,\delta r_s/r_s\) --- the lever of
Eq.~\eqref{eq:c2-leverarm} that
every early-universe ``fix'' pulls. To lift the inference from
\(67.4\) to the ladder's \(73.5\) requires
\(r_s: 145 \to 133\)~Mpc, an eight-percent amputation of the ruler
in the last few e-folds before recombination --- which is precisely
what early dark energy and modified recombination are engineered to
do. The cap is now empirical from two directions: the joint EDE
searches find no lobe of the required size (F-Hubble-3), and the
sound-horizon-\emph{free} measurements --- which would be untouched
by any ruler surgery --- span \(66.7\)--\(70.2\), closing the road to
\(73\) from below (\S\ref{sec:landscape-early}). The lever exists;
the room to pull it does not.

\subsection{The geometric resolution: the rate from two densities, and the coherent offset}
\label{app:cE-gg}
\label{app:calc}

The derived reading pulls no lever: \(r_s\) stays at its derived
length, the recombination history stays standard, and the rate is
obtained by running the degeneracy argument of
\S\ref{app:cE-degeneracy} in the opposite direction --- fractions
derived, densities measured, \(h\) forced. In full arithmetic:

\subsubsection{Inputs}

Two model-independent physical densities from the microwave background,
neither carrying $h$:
\begin{align}
\omega_b &\;\equiv\; \Omega_b h^2 \;=\; 0.02237 \pm 0.00015,\\
\omega_m &\;\equiv\; \Omega_m h^2 \;=\; 0.1430 \pm 0.0011,
\end{align}
and the two GG-predicted fractions,
\begin{align}
\Omega_b &\;=\; \frac{1}{2\pi^2} \;=\; 0.050661\ldots,\\
\Omega_m &\;=\; \frac{1}{\pi} \;=\; 0.318310\ldots
\end{align}

\subsubsection{The two anchors}

Since $\omega_i = \Omega_i h^2$, each pairing gives $h$ directly:
\begin{align}
h_b &\;=\; \sqrt{\omega_b/\Omega_b}
      \;=\; \sqrt{0.02237/0.050661} \;=\; 0.6645,\\
h_m &\;=\; \sqrt{\omega_m/\Omega_m}
      \;=\; \sqrt{0.1430/0.318310} \;=\; 0.6703,
\end{align}
i.e.\ $H_0 = 66.45$ (baryon anchor) and $67.03$ (matter anchor).

\subsubsection{Combination and error}

The two anchors rest on different spectral statistics and agree to
$0.9\%$. Let us do the bookkeeping in the open, because a referee
will. Their naive inverse-variance combination is \(66.69 \pm 0.17\);
the sky-free BBN variant below reads \(66.4 \pm 0.5\); the geometric
Route~I gives \(66.8 \pm 0.2\); the closed form evaluates to
\(66.7\). The three density routes are not equally clean --- the two
\emph{baryon} routes are the model-cleanest (\(\omega_b\) is the one
density measured two independent ways, CMB and BBN, agreeing at
\(66.4\)--\(66.5\)), while \(\omega_m\) at \(\pm0.0011\) precision
is a concordance-chain output --- so a formal weighting would itself
be a modeling choice. The canonical value adopted throughout this
paper and its companions is therefore
\begin{equation}
\boxed{\;H_0^{(0)} \;=\; 66.6 \pm 0.2~\text{km\,s}^{-1}\,\text{Mpc}^{-1}\;,}
\end{equation}
the \emph{center of the route span with a structural, spread-inflated
error} --- an adopted canonical value, not a formal inverse-variance
mean, and we say so plainly.

\emph{Error propagation, explicitly.} Since \(h = \sqrt{\omega_i/\Omega_i}\)
with \(\Omega_i\) exact (ledger-fixed, no measurement error),
\(\sigma_h/h = \tfrac12\,\sigma_{\omega_i}/\omega_i\). The baryon route
carries \(\tfrac12 \times 0.67\% = 0.34\%\), i.e.\ \(\pm0.22\); the
matter route \(\tfrac12 \times 0.77\% = 0.38\%\), i.e.\ \(\pm0.26\).
The two routes rest on different spectral statistics (baryon loading
versus the peak-height envelope), so their errors are treated as
independent; their central-value spread (\(0.5\)) is comparable to
either error, which is why the adopted error is structural rather
than formal. The honest statement is that the multi-route
\emph{consistency}, not any single weighting, carries the weight ---
and every conclusion in this paper survives any choice in the
\(66.4\)--\(66.8\) corridor the combined routes bracket: the coherent
\(\pi\)-offset moves between \(-0.9\%\) and \(-1.5\%\), and nothing
downstream changes sign or verdict. The geometric route
of Eq.~\eqref{eq:c2-H0-closed} gives
$H_0 = 498.3\sqrt{\pi/175} = 66.8$ at the codebook-selected
$\zeta_\Lambda = \pi/175 = 0.01795$
(\S\ref{sec:zeta-selection}), concordant with the
inventory value at $0.6\sigma$. No quantity in this calculation is adjusted to the local
distance-ladder measurement --- or, on the geometric route, to any
astronomical measurement at all.

The variant that removes the microwave sky entirely --- the BBN
deuterium abundance with laboratory reaction rates,
\(\omega_b^{\rm BBN} = 0.02233 \pm 0.00036\), hence
\(H_0^{\rm BBN} = 66.4 \pm 0.5\) --- closes the loop: two densities,
three routes, one rate, no lever. What remains between the derived
chart and the fitted sky is the coherent \(-1.2\%\) \(\pi\)-offset
across the seventeen-observable audit --- carried as a named,
one-directional displacement whose candidate origin is the measure
lemma, and priced as such in the honesty triage. On this reading the
early side of the tension was never broken, and Course~D's ladder is
measuring exactly what Part~IV says it measures.

\subsection*{Status}

Step one is textbook plasma physics and geometry; the evaluated
numbers (\(R_* = 0.62\), \(c_s = 0.454c\), \(D_M = 13{,}970\)~Mpc,
\(\phi_1 \approx 0.27\)) are reproducible with the released scripts
and a quadrature routine. Step two --- the never-measures-\(h\)
derivation and the lever algebra --- is model bookkeeping, not
\GGtheory{}. Step three inherits the main text's grades: fractions
\(\mathsf D^{*}\), the audit's zero-freedom framing as printed, the
half-percent acoustic displacement carried open at the O-Hubble-32
standard. The landmark \(145.0\) and the drag ruler \(147\) are now
explicitly distinguished; nothing else in the paper changes.

\section{Derivation course: the vacuum energy of the universe and the seesaw}
\label{app:courseF}

The deepest number in this paper is not the Hubble rate; it is the
energy density of empty space that sets it. Step one of this course
states the vacuum-energy problem the way quantum field theory
actually produces it --- with the integral written out, so the
notorious ``\(10^{120}\)'' is a computation rather than a slogan;
step two shows why every standard mechanism misses the observed
scale by tens of orders; step three derives the geometric seesaw and
collapses it into the closed form of \S\ref{sec:h0-closed-form}
(mirrored at Route~I of the parent's \S6.4;
\citealp{ArisakaA10Cosmos}), whose two corollaries then
\emph{dissolve} the accounting problem of step one into two powers
of one hierarchy.

\subsection{The vacuum energy of quantum fields, computed honestly}
\label{app:cF-qft}

Every mode of every quantum field carries zero-point energy
\(\tfrac12\hbar\omega_k\). Summing over modes up to a cutoff
\(\Lambda_c\) (natural units),
\[
\rho_{\rm vac}
= \int^{\Lambda_c}\!\frac{d^3k}{(2\pi)^3}\,\frac{k}{2}
= \frac{1}{4\pi^2}\int_0^{\Lambda_c}\!k^3\,dk
= \frac{\Lambda_c^4}{16\pi^2}\, ,
\]
one line of quadrature \citep[the classic accounting
is][]{Weinberg1989} and the origin of the crisis: the answer is
the \emph{fourth power of whatever scale one trusts the theory to}.
Trust it to the reduced Planck scale and
\(\rho_{\rm vac} \sim (2.4\times10^{18}\,\mathrm{GeV})^4/16\pi^2\);
the measured dark-energy density is
\(\rho_\Lambda^{1/4} = 2.24\)~meV, i.e.
\[
\frac{M_{\mathrm{Pl,red}}^4}{\rho_\Lambda}
= \left(\frac{2.435\times10^{18}\,\mathrm{GeV}}
{2.24\times10^{-12}\,\mathrm{GeV}}\right)^{4}
= 10^{120.2},
\]
the worst prediction in the history of physics, computed in two
steps. \emph{Intuition:} the fourth power is the whole cruelty ---
halving the cutoff buys barely one of the hundred and twenty orders,
so no modest re-scaling helps; either almost-exact cancellation or a
structurally tiny input scale is required.

\subsection{Seesaw structures: how physics makes small numbers}
\label{app:cF-seesaw-structures}

Nature's standard trick for a small scale without a small input is
the seesaw. Diagonalize the two-state mass matrix
\[
\mathcal{M} = \begin{pmatrix} 0 & m \\ m & M \end{pmatrix},
\qquad m \ll M
\;\Longrightarrow\;
\lambda_\pm = \frac{M \pm \sqrt{M^2 + 4m^2}}{2}
\;\simeq\; M,\; -\frac{m^2}{M}:
\]
a heavy state near \(M\), and a light one at the \emph{geometric
combination} \(m^2/M\) --- pushed down exactly as far below \(m\) as
\(M\) sits above it, whence the name. The neutrino sector is the
classic instance (\(m_\nu \sim v^2/M_{\rm GUT}\)); the pattern is
general: to manufacture a tiny scale, couple an intermediate scale to
a heavy one. For the vacuum, the required output is meV; a seesaw
delivers it if some intermediate scale \(m\) satisfies
\(m^2/M_{\mathrm{Pl,red}} \sim \mathrm{meV}\), i.e.
\(m \sim \sqrt{(2.4\times10^{18})(2\times10^{-12})} \sim
\mathrm{a\ few\ TeV}\). Read that sentence twice: \emph{the vacuum
energy of the universe points at the TeV scale} --- the one scale
particle physics independently cares about. The observation is old;
what has always been missing is a derivation of \emph{which} TeV
quantity enters and \emph{what} dimensionless coefficient dresses it.

\subsection{What the standard mechanisms cannot do}
\label{app:cF-limits}

The limitations step, mechanism by mechanism.
\emph{Cancellation by symmetry:} unbroken supersymmetry cancels the
zero-point sum exactly --- but supersymmetry is broken at or above
the TeV scale, leaving \(\rho_{\rm vac} \sim (\mathrm{TeV})^4\),
still sixty orders too large; the symmetry route buys half the
exponent and stalls. \emph{Dynamical relaxation (quintessence):}
trades the constant for a field, at the price of a potential whose
mass scale (\(\sim 10^{-33}\)~eV) and flatness are new fine-tunings,
plus fifth-force constraints; the problem is relocated, not solved.
\emph{Anthropic selection:} makes the smallness an observation about
observers, at the price of a multiverse whose measure is undefined
--- an accounting of last resort, and the philosophical opposite of
this series' program. The scorecard: no standard mechanism
\emph{derives} the meV scale; the best either cancel too much
(to zero) or too little (to TeV\(^4\)).

\subsection{The geometric resolution: the seesaw, derived and closed}
\label{app:cF-gg}
\label{app:seesaw}

The dark-energy sector delivers $H_0$ a second time, structurally unrelated to
the density ratio. Folding the geometric seesaw vacuum density and the flat
late-time budget into the first Friedmann equation gives
\begin{equation}
H_0 = \frac{1}{M_{\mathrm{Pl,red}}}
\left(\frac{M_{\mathrm{lock}}^2}{M_{\mathrm{Pl,red}}}\right)^{2}
\sqrt{\frac{\zeta_\Lambda}{3\,\Omega_\Lambda}},
\end{equation}
in which every symbol is inherited: $M_{\mathrm{lock}}\approx3.85$~TeV is the
lock scale (a theorem of the scale chain, \AFive{}~T8.7),
$M_{\mathrm{Pl,red}}$ the reduced Planck mass from the warp overlap, and
$\Omega_\Lambda=1-1/\pi$ the inventory value. Numerically this reduces to
$H_0=498.3\sqrt{\zeta_\Lambda}~\kmsMpc$. The coefficient $\zeta_\Lambda$ is not
free: \MSMF{} forces it to be a bounded-cost monomial in the \OGNfive{}
integers $(M,N,L;J,K)=(1,3,5;7,35)$ with the transcendentals $\{\pi,e\}$; the
cost-minimal cluster spans $0.0180$--$0.0204$ ($H_0 = 66.8$--$71.2$), and the
codebook's stratum-availability and structural-naming rules select
$\zeta_\Lambda = \pi/(LK) = \pi/175$ with no reference to data
(\S\ref{sec:zeta-selection}), so $H_0 = 66.8 \pm 0.2$. The route's
$M_{\mathrm{lock}}^4$ sensitivity is pinned in the laboratory through the
electroweak vev (App.~\ref{app:questions}); its role is primary: the
rate from geometry alone, which the density inversion then confirms.

Two remarks complete the promotion of this compact chain to course
level. \emph{First, the bookkeeping.} In reduced units the lock
theorem reads \(\Mlock = \sqrt{2\pi}\,M_{\mathrm{Pl,red}}\,e^{-K}\),
so the seesaw mass is
\(m_\Lambda = \Mlock^2/M_{\mathrm{Pl,red}} =
2\pi\,M_{\mathrm{Pl,red}}\,e^{-2K} = 6.08\)~meV --- the intermediate
scale of \S\ref{app:cF-seesaw-structures} is not postulated at the
TeV scale; it \emph{is} the TeV lock, a theorem with a laboratory
pin, and the coefficient that dresses \(m_\Lambda^4\) is the
codebook-selected \(\zeta_\Lambda = \pi/175\), pre-registered at
F-Hubble-11. Substituted into the Friedmann equation, every power of
the Planck mass cancels except one overall rate, and the whole
course collapses to the boxed line of \S\ref{sec:h0-closed-form}:
\(H_0 = e^{-4K}\sqrt{2\pi^5 c^5/[3LK(\pi-1)\hbar G_N]} = 66.7\).
\emph{Second, the dissolution of step one.} The ratio that opened
this course is, on the derived chain,
\(\rho_\Lambda/\rho_{\mathrm{Pl,red}} = \zeta_\Lambda (2\pi)^4
e^{-8K} = 10^{-120.2}\): the hundred and twenty orders are
\(8K\) e-folds of one derived descent, corrected by \(1.4\) orders of
geometry. Nothing cancels; nothing is tuned; the smallness is the
fourth power of a hierarchy the same ledger uses at the collider.
The error budget is as printed in \S\ref{sec:absolute-h0}: the
\(\Mlock^4\) sensitivity is pinned through the electroweak vev
(\(v = \Mlock(2/5)^3 = 246.33\) vs measured \(246.22\)~GeV, a
\(+0.045\%\) agreement mapping to \(\pm0.18\%\) on the rate), and the
selection lemma's window \([0.0177, 0.0182]\) is the one conditional
link, cashed by the Hosotani computation.

\subsection*{Status}

Step one is textbook quantum field theory; step two is the field's
own candid literature. Step three carries the printed grades: the
lock scale a theorem (T8.7) with a laboratory pin; the seesaw
identification and the \(\pi\)-locked \(\Omega_\Lambda\)
\(\mathsf D^{*}\); the coefficient Derived (cond.) under the
selection lemma, two-sidedly falsifiable. The course adds no new
claim; it adds the derivation chain at a depth a graduate student
can audit line by line.

\section{Derivation course: the local universe --- peculiar velocities, the Hole, and the lift}
\label{app:courseG}

The tension's late side is measured from inside a particular
neighborhood, and this course derives what a neighborhood does to a
rate measurement. Step one builds linear peculiar-velocity theory
from the continuity equation and lands on a one-line master formula;
step two applies it to the measured Local Hole and shows --- from the
same line --- both what an honest environment \emph{can} supply and
what it \emph{cannot}; step three derives the \GGtheory{} lift, prices
the framework's own exotic channel, and states the shell-level
prediction the surveys can test.

\subsection{Linear peculiar velocities from the continuity equation}
\label{app:cG-pv}

Perturb the matter density, \(\rho = \bar\rho(1+\delta)\), and let
\(\mathbf v\) be the peculiar velocity (motion over and above the
Hubble flow). Mass conservation in comoving coordinates reads
\(\dot\delta + \nabla\!\cdot\!\mathbf v / a = 0\): where matter
converges, density grows. In the linear growing mode every Fourier
amplitude grows as the growth factor \(D(a)\), so
\(\dot\delta = \delta\, d\ln D/dt = \delta\, f H\), with
\[
f \equiv \frac{d\ln D}{d\ln a} \simeq \Omega_m^{0.55} = 0.533
\]
on the derived inventory --- the growth rate, the one number linear
theory needs. Substituting back,
\(\nabla\!\cdot\!\mathbf v = -a f H \delta\): velocities are sourced
by density contrasts, pointing from voids toward walls.
\emph{Intuition:} gravity is a book-keeper --- whatever density
history a region has, its velocity field must account for it, which
is why measured flows can audit claimed voids.

\subsection{The master formula: what an under-density does to a local rate}
\label{app:cG-master}

Now the course's central line. Apply Gauss's theorem to a sphere of
radius \(r\) containing mean interior contrast \(\bar\delta(<r)\):
the outflow through its surface is
\(v(r) = -\tfrac13 f H \bar\delta(<r)\, r\) (radial, by symmetry).
An observer inside, measuring recession speeds of tracers at \(r\),
reads \(v_{\rm obs} = H r + v(r)\), i.e.\ an effective local rate
\begin{equation}
\boxed{\;
\frac{\delta H}{H} \;=\; -\,\frac{f}{3}\,\bar\delta(<r)
\;=\; -\,0.178\,\bar\delta(<r)
\;}
\label{eq:app-cG-master}
\end{equation}
on the derived inventory. One line, three consequences, and the
whole of Part~IV's environmental arithmetic:

\emph{(i) The lift.} The surveyed Local-Hole depths, weighted over
the ladder's volume, span \(\bar\delta \approx -0.11\) to \(-0.17\)
in exactly the bracket the counts report
(\S\ref{sec:hole-evidence}); Eq.~\eqref{eq:app-cG-master} converts
them to \(\delta H/H = +2\)--\(3\%\) --- the lift, as arithmetic.

\emph{(ii) The impossibility.} Ask the same line to produce the
\emph{whole} residual, \(+7.3\%\), and it answers with
\(\bar\delta = -0.073/0.178 = -0.41\) across the supernova flow
volume --- the deep-void requirement of \S\ref{sec:backreaction},
here revealed as the same formula read at a different target. The
two numbers the paper carries --- the modest lift it derives and
the \(-0.41\) it excludes --- are one equation at two inputs.

\emph{(iii) The signature.} Because \(\bar\delta(<r) \to 0\) as the
sphere grows past the Hole, the locally read rate must \emph{decay
toward the background} as low shells are cut away:
\(H_0(z_{\min}) = H_0[1 - \tfrac{f}{3}\bar\delta(<r(z_{\min}))]\) ---
a shape no constant calibration error can imitate, drawn on the
measured profiles in Fig.~\ref{fig:hole-lift} and confronted with
DESI's binned analyses in \S\ref{sec:hole-evidence}.

\subsection{The measured Hole, and the velocity audit}
\label{app:cG-hole}

Step one's formula needs a depth, and the depth is measured two
independent ways. \emph{Counts:} galaxy number counts across optical
and near-infrared surveys report a mild under-density over a few
hundred megaparsecs, with the profile's depth the contested quantity
--- from a shallow \(\delta \approx -0.10\) to the disputed
\(-0.3\) (\S\ref{sec:hole-course}). \emph{Flows:} the same
velocities that Eq.~\eqref{eq:app-cG-master} predicts are directly
measurable as bulk flows and as the smoothness of the supernova
Hubble diagram --- and the field-level velocity modeling now
disfavors the deep reading: a \(-0.3\) hole over \(300\)~Mpc would
drag the local diagram visibly, and it does not
(\S\ref{sec:hole-evidence}). The paper's adopted \(+2\)--\(3\%\) is
the intersection of the two audits --- as much lift as the counts
license \emph{and} the flows permit, and no more.

\subsection{What the environment cannot explain}
\label{app:cG-limits}

The limitations step, in three sentences. A local environment can
shift a local rate by \(\sim\!(f/3)\times(\mathrm{depth})\) ---
percent-class for any depth the sky allows, and the required
\(-0.41\) is a \(>9\sigma\) fluctuation on the flow scale (Course~H
prices it exactly). An environment cannot touch the early side at
all: the CMB's densities are read at \(z \sim 1100\), oblivious to
our suburb. And an environment predicts a \emph{profile} --- the
decay of (iii) above --- so it is not an infinitely flexible excuse:
the shell test can kill the lift tomorrow. Percent-class, one-sided,
and falsifiable: exactly the three properties the paper's usage
respects.

\subsection{The geometric resolution: the lift, and the priced exotic channel}
\label{app:cG-gg}
\label{app:epslocal}

Step three has two parts. The lift itself is not exotic: it is
Eq.~\eqref{eq:app-cG-master} evaluated on measured depths, with
\GGtheory{} supplying the \emph{sign and existence} argument (an
open-boundary neighborhood must register its environment; the
openness theorem of \AFour{}) and the derived inventory supplying
\(f\). What \emph{is} specifically \GG{} is the second part: the
framework owns one first-principles local channel --- the exact
\(\Uoneb\) graviphoton --- and rather than invoking it, the paper
computes it and \emph{excludes} it. The compact record follows,
verbatim from the corridor-arithmetic original:

\subsubsection{Setting}

Unique to \GGsix{} is the \emph{exact local} $\Uoneb$ gauge factor,
which forbids proton decay ($\tau_p = \infty$) and gives
$\bar\theta \lesssim 4.7 \times 10^{-18}$ naturally. The gauge
boson of $\Uoneb$ is, in the standard gauge-theory sense \citep{PeskinSchroeder1995},
an ultra-weakly coupled long-range vector
(``graviphoton'') that couples to baryon number rather than to
mass or charge. In the late universe, this graviphoton produces a
very mild, scale-dependent modification of the expansion history
in regions with nonzero baryonic overdensity.

\subsubsection{Effective local Friedmann equation}

At any point in the universe one may write, following
\cite{ArisakaA10Cosmos},
\begin{equation}\label{eq:Hloc}
H^2_{\mathrm{loc}}
\;=\; H^2_{\mathrm{bg}}
\;+\; \frac{8\pi G_N}{3}\,\delta\rho_B^{\mathrm{eff}},
\end{equation}
where $\delta\rho_B^{\mathrm{eff}}$ is the effective baryonic
correction from the $\Uoneb$ graviphoton pressure. The graviphoton
force between baryonic overdensities, at distances small compared
to its Compton wavelength (which is of cosmological scale), takes
the form
\begin{equation}
F_{\Uoneb}(r)
\;=\; \frac{g_B^2}{4\pi} \frac{q_{B,1} q_{B,2}}{r^2},
\qquad
g_B^2/4\pi \sim \frac{2}{5}\,G_N\,m_p^2,
\end{equation}
with the coefficient $2/5$ forced by the same \BIsix{} entry that
gives the Arisaka Ratio. The net gravitational-plus-$\Uoneb$
``effective Newton'' for baryons is thus
\begin{equation}
G_{\mathrm{eff}}^{(B)}
\;=\; G_N\,(1 - \zeta_B),
\qquad
\zeta_B \;\equiv\; \frac{2}{5}\,\delta\rho_B/\rho_B,
\end{equation}
i.e.\ the graviphoton is \emph{repulsive}, slightly inflating
mildly underdense regions on baryonic overdensity contrast.

\subsubsection{Magnitude estimate: the first-pass result and the
sixty-fold gap}

The local universe at the $\sim$ Gpc scale relevant to the
distance ladder has a baryonic density contrast typically in
the range $|\delta_B| \sim 0.05$--$0.08$ (from redshift surveys
and peculiar-velocity reconstructions). Plugging into
Eq.~\eqref{eq:Hloc}:
\begin{equation}
\frac{H_{\mathrm{loc}} - H_{\mathrm{bg}}}{H_{\mathrm{bg}}}
\;\simeq\;
\frac{1}{2}\,\Omega_B\,\zeta_B
\;\approx\;
\frac{1}{2}\,(0.049)\,\frac{2}{5}\,|\delta_B|
\;\approx\;
0.0098\,|\delta_B|,
\end{equation}
so the observed contrast yields the first-pass estimate
\begin{equation}\label{eq:epslocalfirstpass}
\epsilon_{\mathrm{local}}^{\mathrm{(B),\,first\text{-}pass}}
\;\sim\; 5\text{--}8 \times 10^{-4}.
\end{equation}
This first-pass estimate falls roughly an order of magnitude short of
the $+2$--$3\%$ Local-Hole lift the split of Section~\ref{sec:both-from-anchor}
attributes to the environment; the direct $\Uoneb$ graviphoton
contribution alone does not reach it. The honest statement of record is
therefore: \emph{the sign and rough size of the environmental lift are
derived, but a complete first-principles calculation of its amplitude at
the few-percent level does not yet exist; the residual gap to the
measured local rate is carried as a distance-ladder systematic, not as a
result.} The lift that would take the derived background
$66.6\kmsMpc$ to the observed local $73.5$ is
\begin{equation}\label{eq:epslocalfinal}
\epslocal \;=\; \frac{73.50}{66.6} - 1 \;=\; 0.104,
\end{equation}
of which the derived Local-Hole piece supplies $+2$--$3\%$ and the
named calibration residual the remainder. Deriving the environmental
amplitude in full from the $\Uoneb$ sector is a declared research
target of this paper.

\subsubsection{Why this is not a large local void}

The distinction between the \GGsix{} $\Uoneb$ graviphoton mechanism
and the phenomenological ``large local void'' model is important.
A pure void model requires $|\delta| \gtrsim 0.2$ at several
hundred Mpc, which is in serious tension with observed
large-scale isotropy. The derived lift is the ordinary Local-Hole monopole of
\S\ref{sec:worked-excess}, needing only the mild underdensity the
surveys already report. The graviphoton channel computed above is
recorded for honesty, not invoked: its coupling is fixed (the same
$2/5$ that appears in \BIsix{} as the coefficient of $F_B^2$), its
first-pass magnitude \eqref{eq:epslocalfirstpass} falls an order
short of the lift and far shorter of the residual, and no
amplification mechanism for it exists in the theory. It is an
excluded channel, and the exclusion is part of the argument that the
residual is not cosmological.

\subsection*{Status}

Step one is classical linear theory; every number in it
(\(f = 0.533\), the \(0.178\) coefficient, the depth conversions)
is checkable on one line. Step two is survey phenomenology with the
audits cited. Step three carries the printed grades: the lift
derived in sign, bracketed in size (\(+2\)--\(3\%\)), the
graviphoton channel excluded at \(5\)--\(8\times10^{-4}\), the
remainder named-open. The master formula
Eq.~\eqref{eq:app-cG-master} is this course's export: Part~IV's
lift, Part~V's void-class verdicts, and Course~H's rarity pricing
all read from it.

\section{Derivation course: backreaction and averaging}
\label{app:courseH}

``Maybe the average of an inhomogeneous universe simply expands
differently'' is the tension's most seductive escape, because it
sounds like it requires no new physics at all. This course does what
the slogan never does: derives the averaging formalism from scratch,
computes the effect on the derived inventory, and prices the escape
at its exact target.

\subsection{Buchert averaging, from the definition up}
\label{app:cH-buchert}

Following \citet{Buchert2000}, average any scalar \(f\) over a
comoving spatial domain \(D\) with
volume \(V_D = \int_D \sqrt{h}\,d^3x\):
\(\langle f\rangle_D = V_D^{-1}\int_D f\sqrt{h}\,d^3x\). The volume
element evolves as \(\partial_t\sqrt{h} = \theta\sqrt{h}\), with
\(\theta\) the local expansion rate; differentiating the definition
gives the \emph{commutation rule} --- the formalism's one genuinely
new object:
\[
\partial_t\langle f\rangle_D - \langle \partial_t f\rangle_D
= \langle f\theta\rangle_D - \langle f\rangle_D\langle\theta\rangle_D ,
\]
time derivative and average fail to commute by exactly the
\emph{covariance of \(f\) with the expansion}: faster-expanding
regions grow their volumes and vote more heavily in tomorrow's
average. Apply this to Raychaudhuri's equation for irrotational
dust and to the Hamiltonian constraint, define the averaged scale
factor by \(a_D \propto V_D^{1/3}\), and the averaged equations close
up to one new term,
\[
3\Big(\frac{\dot a_D}{a_D}\Big)^{2}
= 8\pi G\langle\rho\rangle_D - \tfrac12\langle\mathcal R\rangle_D
- \tfrac12\,\mathcal Q_D,
\qquad
\mathcal Q_D = \tfrac23\big(\langle\theta^2\rangle_D
- \langle\theta\rangle_D^2\big) - 2\langle\sigma^2\rangle_D ,
\]
the kinematical backreaction: variance of expansion, minus shear
(Eq.~\eqref{eq:buchert-Q} of the main text). \emph{Intuition:}
\(\mathcal Q_D\) is the statement that a mixture of fast and slow
expanders does not average to the mean --- but note the minus sign:
shear, which inhomogeneity also produces, \emph{fights} the variance.
The two are comparable in linear theory, which is why the net effect
will prove tiny.

\subsection{The size of the terms on the derived inventory}
\label{app:cH-size}

Linear theory turns both terms into integrals over the power
spectrum. From Course~G's master relation, the local expansion
perturbation is \(\delta\theta/3H = -(f/3)\delta\), so
\(\mathrm{Var}(\theta) = H^2 f^2 \sigma_\delta^2(R)\), and the linear
tidal shear evaluates to
\(\langle\sigma^2\rangle = \tfrac{2}{15} H^2 f^2 \sigma_\delta^2\);
hence
\[
\frac{\mathcal Q_D}{H^2}
= \Big(\tfrac23 - \tfrac{4}{15}\Big) f^2 \sigma_\delta^2
= 0.4\, f^2 \sigma_\delta^2 ,
\qquad
\frac{\delta H_D}{H} = -\,\frac{\mathcal Q_D}{12 H^2},
\]
with \(\sigma_\delta(R)\) the top-hat density variance on the
averaging scale. On the derived inventory
(\(\sigma_8 = 0.737\), \(f = 0.533\)) the released script evaluates
\(\sigma_\delta = 0.152,\ 0.044,\ 0.024\) at
\(R = 40,\ 100,\ 150\)~Mpc\(/h\), and the anatomy of the escape is
over before it starts: at \(R = 40\),
\(\mathcal Q_D/H^2 = 0.4 \times 0.284 \times 0.0231 = 2.6\times10^{-3}\)
and the induced \(|\delta H_D/H| \approx 2\times10^{-4}\) ---
\emph{two hundredths of a percent}, and negative.

\subsection{What averaging cannot buy}
\label{app:cH-limits}

Three separate walls, in decreasing order of generality.
\emph{The systematic wall:} the averaged shift is
\(\mathcal Q_D/12H^2 \lesssim 10^{-3}\) on every relevant scale ---
three orders below the residual --- because variance and shear
nearly cancel and both are quadratic in small contrasts.
\emph{The stochastic wall:} one can be \emph{lucky} rather than
systematic --- live in an unusual patch --- and the price is
computed from the same \(\sigma_\delta(R)\): the
\(\bar\delta = -0.41\) that Course~G showed the full residual
requires is a \(2.7\sigma\) fluctuation at \(40\)~Mpc\(/h\), but
\(9.3\sigma\) at \(100\) and \(17\sigma\) at \(150\) --- and the
ladder's supernovae live in the \(100\)--\(150\) shell. Excluded,
in the precise sense that standard structure formation does not
make such patches. \emph{The loophole, priced:} the fully
non-linear ``timescape'' reading rejects the perturbative frame
entirely; the paper carries it as the class's honest extreme
(\S\ref{sec:class-statement}) rather than pretending the linear
bound closes it --- but notes that it must then also re-derive
everything the linear frame gets right.

\subsection{The record: the bound at the exact target}
\label{app:cH-gg}
\label{app:backreaction}

The compact record follows, verbatim from the corridor-arithmetic
original --- the numbers behind Fig.~\ref{fig:backreaction-bound},
regenerated at the exact \(+7.3\%\) target by the released script:

This course substantiates \S\ref{sec:backreaction}: that the \GDOS{}-coarse-grained
backreaction cannot supply the local excess. In Buchert's spatial average of the
inhomogeneous dust cosmology, the volume-averaged scale factor \(a_D\) obeys
\(3(\dot a_D/a_D)^2 = 8\pi G\langle\rho\rangle_D-\tfrac12\langle R\rangle_D-\tfrac12\mathcal{Q}_D\),
with \(\mathcal{Q}_D\) of Eq.~\eqref{eq:buchert-Q}. In the linear growing mode the
local expansion perturbation is \(\delta H/H=-(f/3)\delta\) (continuity), so
\(\mathrm{Var}(\theta)=H^2 f^2\sigma_\delta^2\) and the linear tidal shear is
\(\langle\sigma^2\rangle=\tfrac{2}{15}H^2 f^2\sigma_\delta^2\); the induced shift of
the averaged rate is \(\delta H_D/H=-\mathcal{Q}_D/(12H^2)\).

Evaluating on the exact derived background (\(\Omega_m=1/\pi=0.3183\),
\(\Omega_b=1/2\pi^2\), \(h=0.666\), \(n_s=0.962\), \(\sigma_8=0.737\) from
the derived \(S_8=0.759\); growth rate \(f=\Omega_m^{0.55}=0.533\))
with an Eisenstein--Hu linear \(P(k)\) \citep{EisensteinHu1998} normalized to \(\sigma_8=0.737\):
\begin{itemize}
\item \emph{Systematic backreaction:} \(\langle \mathcal{Q}_D\rangle/H^2\) runs from
  \(\sim10^{-2}\) at \(R=20\) Mpc/\(h\) to \(\sim10^{-4}\) at \(150\) Mpc/\(h\); the
  induced \(|\delta H_D/H|\lesssim0.1\%\), and negative.
\item \emph{Cosmic variance:} \(\sigma_H/H=(f/3)\sigma_\delta(R)\) is
  \(2.7\%\) at \(40\) Mpc/\(h\), \(0.79\%\) at \(100\), \(0.26\%\) at \(200\).
\item \emph{Rarity of a \(+7.3\%\) void:} \(\langle\delta\rangle=-0.41\) is
  \(2.7\sigma\) at \(40\) Mpc/\(h\) but \(9.3\sigma\) at \(100\) and \(17\sigma\) at
  \(150\) --- excluded on the ladder scale.
\end{itemize}
The open-source script that produces these numbers and
Figure~\ref{fig:backreaction-bound}
(\texttt{c2\_backreaction\_pilocked.py}) is released with the paper, and
recomputing at the exact \(1/\pi\) inventory reproduces every quoted value
at the stated precision. The conclusion is robust to the \(O(1)\) shear
coefficient: the systematic averaging bias is sub-percent and cannot close a
seven-percent gap.

\subsection*{Status}

Step one is the standard averaging formalism, derived rather than
cited; step two is linear-theory quadrature on the derived
inventory, every value reproduced by
\texttt{c2\_backreaction\_pilocked.py}; step three's exclusions are
as printed in the main text, with the timescape loophole carried
open at its stated price. This course closes the environmental
trilogy: Course~G derived what a neighborhood can do, this course
what an average cannot, and between them the residual's
non-cosmological character is not an assumption but an audit.

\section{Derivation course: the growth of structure and \texorpdfstring{$S_8$}{S8}}
\label{app:courseI}

The inventory that sets the expansion rate also sets how fast matter
clumps, so the growth of structure is the derived chart's second,
independent report card. Step one builds linear perturbation theory
to the point where the two survey observables --- \(\sigma_8\) and
\(S_8\) --- are defined by integrals a student can evaluate; step two
states the \(S_8\) tension and what the standard framework cannot say
about it; step three reads the derived amplitude \(S_8 = 0.759\) off
the same ledger as everything else, with its one open flag displayed.

\subsection{The perturbed universe: three equations}
\label{app:cI-perturbed}

Write the metric with weak potentials (conformal-Newtonian gauge),
\(ds^2 = -(1+2\Phi)dt^2 + a^2(1-2\Phi)\,d\mathbf x^2\), and perturb
the matter fluid: density contrast \(\delta\), peculiar velocity
\(\mathbf v\). Three classical statements close the system on
sub-horizon scales. \emph{Continuity} (mass conservation):
\(\dot\delta = -\nabla\!\cdot\!\mathbf v/a\) --- convergence feeds
density (Course~G used exactly this). \emph{Euler} (momentum):
\(\dot{\mathbf v} + H\mathbf v = -\nabla\Phi/a\) --- gravity
accelerates, expansion drags: the \(H\mathbf v\) term is Hubble
friction, the same brake that freezes fields in inflation and axion
cosmology. \emph{Poisson} (gravity):
\(\nabla^2\Phi = 4\pi G_N a^2 \bar\rho_m\,\delta\) --- only the
\emph{contrast} gravitates. Differentiate continuity, insert Euler
and Poisson, and the system collapses to one line:
\begin{equation}
\ddot\delta + 2H\dot\delta = 4\pi G_N\,\bar\rho_m\,\delta ,
\label{eq:app-cI-growth-eq}
\end{equation}
a damped, driven oscillator with gravity as negative stiffness:
growth is a competition between self-attraction (right side) and
Hubble friction (the \(2H\dot\delta\)).

\subsection{The growth factor, era by era}
\label{app:cI-growthfactor}

Solve Eq.~\eqref{eq:app-cI-growth-eq} in each era.
\emph{Matter domination} (\(H^2 = \tfrac{8\pi G}{3}\bar\rho_m\),
\(a \propto t^{2/3}\)): try \(\delta \propto t^p\); the equation gives
\(p(p-1) + \tfrac43 p = \tfrac23\), i.e.\ \(p = 2/3\) or \(-1\): the
growing mode is \(\delta \propto t^{2/3} \propto a\) --- contrast
grows exactly as fast as the universe expands, no faster.
\emph{Radiation domination:} the driving term is negligible next to
the expansion rate and growth stalls logarithmically --- the
M\'esz\'aros effect \citep{Meszaros1974}; structure marks time until equality
(\(z_{\rm eq} = 3400\) on the derived chart), which is why
\(\omega_m\), by setting equality, imprints itself on the power
spectrum's turnover. \emph{Vacuum domination:} the driving term
dilutes away, \(\dot\delta \to \delta \times 0\): growth freezes.
Collecting all eras, the growing mode has the closed integral form
of Eq.~\eqref{eq:app-growth} below, and its logarithmic rate today
defines the growth rate
\(f \equiv d\ln D/d\ln a \simeq \Omega_m^{0.55} = 0.533\) --- the
number Courses G and H imported. \emph{Intuition for the exponent:}
\(0.55\) is not mysterious; it is the statement that growth responds
to the \emph{instantaneous} matter share slightly sub-linearly,
because friction remembers the recent expansion history.

\subsection{From modes to observables: \texorpdfstring{$\sigma_8$ and $S_8$}{sigma8 and S8}}
\label{app:cI-sigma8}

The linear field is a Gaussian random process, specified completely
by its power spectrum
\(P(k) = A\,k^{n_s}\,T^2(k)\,D^2(a)\): primordial tilt,
transfer function (the frozen record of the radiation-era stall),
growth. Surveys compress it to one number: the rms contrast in a
sphere of radius \(8\,h^{-1}\)Mpc,
\[
\sigma_8^2 = \int_0^\infty \frac{dk\,k^2}{2\pi^2}\,P(k)\,
W^2(kR_8),
\qquad
W(x) = \frac{3\,(\sin x - x\cos x)}{x^3},
\]
the top-hat window. Weak lensing measures the combination
\(S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}\), because lensing responds
to \emph{amount of matter times its lumpiness}; the \(0.3\) is a
historical normalization. Everything in these integrals is either
primordial (seed, tilt), thermal-history (transfer), or inventory
(growth): there is no independent ``clustering dial.''

\subsection{What the standard framework cannot say about its own tension}
\label{app:cI-limits}

The lensing generation of surveys reads \(S_8 \approx 0.76\); the
CMB, extrapolated through fitted \(\Lambda\)CDM, prefers
\(\approx 0.83\) --- a milder sibling of the Hubble tension, and the
standard framework can only fit it, not interrogate it: with six
dials, a low lensing amplitude can be bought by shifting several
combinations, and the literature duly offers baryonic feedback,
neutrino mass, intrinsic alignments, and new physics in any mixture.
The structural blind spot is the same one Course~D found in the
ladder: a fitted framework cannot say what the amplitude
\emph{should} be, so it cannot say which measurement is reading
truth. And the two tensions cannot be discussed jointly with
integrity unless the same parameters move both --- which, in a fit,
they are free not to do.

\subsection{The geometric reading: one ledger, both amplitudes}
\label{app:cI-gg}
\label{app:s8}

On the derived chart the growth sector inherits everything and
chooses nothing, and the compact chain is worth preserving verbatim:

The same inventory that fixes $H_0$ fixes the clustering amplitude, and the
chain has three steps, none free.

\emph{Step 1 --- the growth factor.} In the linear regime every Fourier mode
of the matter field grows by one universal factor $D(a)$, governed by the
competition between gravity and expansion; for a $\Lambda$CDM-like history
the growing solution has the standard closed form
\citep{Dodelson2020}
\begin{equation}
D(a) \;\propto\; \frac{5\,\Omega_m}{2}\,E(a)\int_0^a
\frac{da'}{\bigl[a'\,E(a')\bigr]^{3}},
\qquad E(a) = \sqrt{\Omega_r a^{-4} + \Omega_m a^{-3} + \Omega_\Lambda},
\label{eq:app-growth}
\end{equation}
with $E(a)$ the derived history of Eq.~\eqref{eq:app-Hofa} --- so the growth
factor inherits the inventory with no new freedom.

\emph{Step 2 --- the amplitude.} The overall normalization traces back to
the primordial seed: the inherited $A_s$, the \emph{derived} tilt
$n_s = 0.9620$, and the standard transfer function propagated through
Eq.~\eqref{eq:app-growth}. That chain is carried by the parent \ATen{}; its
output at the derived inventory is $\sigma_8 = 0.737$ --- the value used,
consistently, in the backreaction course of App.~\ref{app:backreaction}.

\emph{Step 3 --- the lensing combination.} Weak-lensing surveys report
$S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3}$; with the derived shares,
\begin{equation}
S_8 \;=\; 0.737 \times \sqrt{\frac{1/\pi}{0.3}} \;=\; 0.737 \times 1.030
\;=\; 0.759,
\label{eq:app-s8-chain}
\end{equation}
against the KiDS-1000 determination $0.759^{+0.024}_{-0.021}$ --- coincident
at the central value, and sitting below the CMB-$\Lambda$CDM inference in
exactly the direction of the mild $S_8$ tension (\S\ref{sec:s8-section}).
Because the amplitude and the rate come from one inventory, they cannot be
moved independently --- the same structural signature that separates a
derivation from a per-observable fit --- and the early-versus-late $S_8$ gap
is carried open, not claimed.

The course-level remarks. \emph{First}, the coincidence to notice:
the derived \(S_8 = 0.759\) lands on the lensing side of the split,
for the same structural reason the derived \(H_0 = 66.6\) lands on
the CMB side --- both are read off the inventory, not fitted to
either camp; the pattern ``agree with the cleaner instrument,
dissect the messier chain'' repeats across the two tensions.
\emph{Second}, the joint rigidity: \(\sigma_8\), \(f\), \(\Omega_m\),
and \(H_0\) move together or not at all, so the growth sector is a
falsifier of the Hubble sector and vice versa --- the
redshift-space-distortion combination \(f\sigma_8(z)\) is fixed at
every redshift with no remaining freedom. \emph{Third}, the open
flag, kept visible: the corridor's growth-response \emph{sign} is
carried open with the parent \ATen{} (OQ-Hubble-5), and nothing in
this course pre-judges it.

\subsection*{Status}

Step one is the standard linear theory, derived; step two is the
field's own accounting. Step three: \(\sigma_8 = 0.737\) inherits
the parent's amplitude chain (seed \(A_s\) declared, tilt derived),
\(S_8 = 0.759\) is arithmetic on it, the KiDS coincidence is a
consistency, not a claim of proof, and the growth sign stays open.
One ledger, both amplitudes, every step auditable.

\section{Derivation course: standard sirens from the quadrupole formula up}
\label{app:courseJ}

The referee of this paper deserves its own course: the reader should
see, derived and not asserted, why a binary inspiral is an absolute
distance meter with no ladder anywhere in it. Step one builds the
siren from linearized general relativity to the GW170817 numbers;
step two derives the degeneracies and floors that limit it; step
three re-derives the forecast algebra this paper stakes its verdict
on.

\subsection{Gravitational waves in one page}
\label{app:cJ-linear}

Linearize gravity: \(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\),
\(|h| \ll 1\) (full treatments: \citealp{Maggiore2008}). In the transverse-traceless gauge the vacuum Einstein
equations reduce to the wave equation \(\Box h_{\mu\nu} = 0\):
ripples of spacetime, two polarizations (\(+, \times\)),
propagating at \(c\). Coupling to matter, the leading radiation is
\emph{quadrupole} --- monopole radiation is forbidden by mass
conservation, dipole by momentum conservation (the gravitational
``charge'' cannot oscillate against itself), so the first radiating
moment is the quadrupole \(Q_{ij}\), and the far-field strain is
\[
h_{ij} \;\sim\; \frac{2G_N}{c^4}\,\frac{\ddot Q_{ij}}{d}\, ,
\]
with the \(1/d\) of any wave and the ferocious \(G/c^4\) that makes
spacetime the stiffest medium there is. \emph{Dimensional intuition:}
\(G/c^4 \sim 10^{-44}\)~s\(^2\)kg\(^{-1}\)m\(^{-1}\) --- it takes
solar masses at relativistic speeds to make \(h \sim 10^{-21}\),
and that stiffness is precisely why the source physics is clean:
nothing else in the universe can fake it.

\subsection{The chirp: one mass from the phasing}
\label{app:cJ-chirp}

Specialize to a circular binary, masses \(m_1, m_2\), separation
\(a\), orbital frequency \(\omega\) (Kepler:
\(\omega^2 a^3 = G M\), \(M = m_1 + m_2\)). The quadrupole moment
rotates at \(\omega\), so the wave comes out at \(f = 2 f_{\rm orb}\);
the radiated power is the Einstein quadrupole luminosity
\(\mathcal L \propto (G^4/c^5)\,\mu^2 M^3/a^5\), and energy balance
--- the orbit paying the radiation bill from its binding energy
\(E = -G\mu M/2a\) --- turns loss into acceleration. Carrying the
algebra through (each substitution is Kepler or the two lines
above), every mass dependence collapses into one combination:
\begin{equation}
\dot f \;=\; \frac{96}{5}\,\pi^{8/3}
\left(\frac{G_N \mathcal M_c}{c^3}\right)^{5/3} f^{11/3},
\qquad
\mathcal M_c \equiv \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}} ,
\label{eq:app-cJ-chirp}
\end{equation}
the \emph{chirp mass} --- the observable \citet{Schutz1986}
identified as cosmology's gift. Read the miracle carefully: \(f\) and
\(\dot f\) are direct observables of the phasing, so
Eq.~\eqref{eq:app-cJ-chirp} hands over \(\mathcal M_c\) with no
astronomy at all --- for two \(1.36\,M_\odot\) neutron stars,
\(\mathcal M_c = 1.18\,M_\odot\), and GW170817's phasing measured
exactly this to four digits.

\subsection{The amplitude: an absolute distance}
\label{app:cJ-distance}

Insert the binary's quadrupole into the strain formula; the same
chirp mass appears again,
\[
h \;\sim\; \frac{4}{d_L}\,
\frac{(G_N\mathcal M_c)^{5/3}}{c^4}\,(\pi f)^{2/3},
\]
and the logic closes like a trap: the phasing has \emph{already}
fixed \(\mathcal M_c\) and \(f\) is measured, so the only unknown in
the amplitude is \(d_L\). The source's absolute wattage is certified
by the field equations --- this is what ``standard siren'' means, and
why Course~D's entire ladder of calibrations simply does not appear.
(Cosmology enters only as the redshifting
\(\mathcal M_c \to (1+z)\mathcal M_c\), which is why the waveform
alone cannot supply \(z\) and a host galaxy must.)

\subsection{Worked example: GW170817, end to end}
\label{app:cJ-gw170817}

The one bright siren, in four measured numbers. The phasing gave
\(\mathcal M_c = 1.188\,M_\odot\); the amplitude then gave
\(d_L \approx 40\)~Mpc; the host, NGC~4993, supplied the Hubble-flow
velocity \(3017 \pm 166\)~km\,s\(^{-1}\); and the quotient, with the
inclination degeneracy marginalized, gave
\(H_0 = 70^{+12}_{-8}\) (central value \(3017/43.1\); the skewed
error is the inclination's signature) --- fifteen-percent precision
from one event, straddling both camps, exactly as Part~VII tells it.
Nothing in the chain touched a Cepheid, a supernova
standardization, or a sound horizon.

\subsection{What limits the referee}
\label{app:cJ-limits}

The limitations step, derived rather than listed.
\emph{Inclination--distance:} the two polarizations carry
\(h_+ \propto (1+\cos^2\iota)/2\) and \(h_\times \propto \cos\iota\);
a face-on binary is louder, so a nearby edge-on and a distant
face-on system produce similar strain --- the \(d\)--\(\iota\)
degeneracy behind GW170817's asymmetric error, broken partly by the
polarization ratio and decisively by jet-afterglow imaging.
\emph{The calibration floor:} strain is measured against laboratory
photon calibrators, currently at the one-to-two-percent level ---
the one \emph{common-mode} term in the siren budget (Course~D taught
why common modes matter), an engineering quantity with no coupling
to either discordant route. \emph{Dark-siren statistics:} without a
counterpart the redshift comes from weighting every galaxy in the
localization volume; catalog incompleteness and the assumed
black-hole mass spectrum are the floors, visible in the wild as the
\(76.6 \to 71.0 \to 69.9\) walk of the report card
(\S\ref{sec:siren-reportcard}). All three limits shrink with
catalog size or engineering; none couples to the tension's
contested chains --- which is the property that makes the siren the
referee.

\subsection{The geometric stake: the forecast algebra, re-derived}
\label{app:cJ-gg}

Step three is this paper's own arithmetic, now derivable by the
reader in two lines. \(N\) unbiased events of fractional error
\(\sigma_{\rm ev}\) combine to
\(\sigma(N) = \sigma_{\rm ev}\times 70/\sqrt N\); demanding
\(k\)-sigma separation of the two camps
(\(\Delta = 73.5 - 66.6 = 6.9\)) gives
\[
N_{k\sigma} = \left(\frac{70\,\sigma_{\rm ev}\,k}{\Delta}\right)^2
\;\Longrightarrow\;
N_{3\sigma} = 21,\quad N_{5\sigma} = 58
\]
at the GW170817-like \(\sigma_{\rm ev} = 15\%\) --- the thresholds
of Part~VII, hostage to no rate assumption. The three-worlds pricing
then follows from the same Gaussian algebra: a converged catalog
at \(70.0\), read between the verdict thresholds
(\(\sigma_{\rm cat} = 2.30 \to 1.38\)), sits
\(0.6\)--\(1.0\sigma\) from the lifted \(68.5 \pm 0.6\) and
\(1.5\)--\(2.4\sigma\) from the pure \(66.6 \pm 0.2\)
(\S\ref{sec:siren-three-worlds}) --- every number in this paragraph
recomputable from this course's two boxed formulas and a table of
quadratures. The prediction itself is unchanged since Part~III:
\(H_0^{\rm GW} \to 67\), not \(73\), with both outcomes' meanings
printed in advance.

\subsection*{Status}

Steps one and two are classical relativity and instrument physics;
the derivations here are the standard ones, compressed but complete
in their logic, and every number matches the released
\texttt{c2\_siren\_forecast.py}. Step three carries the paper's
grades unchanged: the forecast thresholds are arithmetic; the dates
are empirical (the report card); the prediction is the pre-registered
F-Hubble-2. This course exists so that the reader who will watch the
verdict arrive can check, personally, that the instrument owes
nothing to either side.

\section{Derivation course: time, age, and the curve-reading channels}
\label{app:courseK}

The tension is about one point on a curve; this closing course is
about the curve as a function of \emph{time} --- the ages, the
lookbacks, and the three channels that read \(H(z)\) directly. Step
one derives the clock integrals in closed form; step two shows what
a one-number cosmology cannot do; step three states what the derived
chronology commits to, row by row.

\subsection{Cosmic time, derived in closed form}
\label{app:cK-time}

Course C quoted the age formula; here is its derivation, worth
doing once because the same substitution recurs everywhere. From the
Friedmann equation in the matter-plus-vacuum era,
\[
t(a) = \int_0^a \frac{da'}{a' H(a')}
= \frac{1}{H_0}\int_0^a
\frac{a'^{1/2}\,da'}{\sqrt{\Omega_m + \Omega_\Lambda a'^3}}\, .
\]
Substitute \(u = a'^{3/2}\), \(du = \tfrac32 a'^{1/2} da'\): the
integral becomes elementary,
\[
t(a) = \frac{2}{3H_0}\int_0^{a^{3/2}}
\frac{du}{\sqrt{\Omega_m + \Omega_\Lambda u^2}}
= \frac{2}{3H_0\sqrt{\Omega_\Lambda}}\,
\operatorname{arcsinh}\!\left(\sqrt{\frac{\Omega_\Lambda}{\Omega_m}}\,
a^{3/2}\right),
\]
Eq.~\eqref{eq:app-age-closed} of Course C, now earned rather than
quoted. Three limits deserve their intuitions: at early times
\(\operatorname{arcsinh} x \to x\) and \(t \propto a^{3/2}\)
(matter-era clock); at late times
\(\operatorname{arcsinh} x \to \ln 2x\) and \(a \propto
e^{H_\infty t}\) (the de~Sitter clock); and at \(a = 1\) the formula
delivers \(t_0 = 13.92\)~Gyr. The lookback ladder on the derived
chart: at \(z = 1\) the clock read \(5.89\)~Gyr (lookback
\(8.03\)); the acceleration --- the discovery of
\citet{Riess1998} and \citet{Perlmutter1999} --- began at
\(t = 7.8\)~Gyr
(\(z = 0.62\), Course C's exact \(1/\sqrt2\)); at \(z = 2\) the
clock read \(3.29\)~Gyr. Every one of these is a falsifiable
statement about stellar populations: no object at \(z = 2\) may be
older than \(3.3\)~Gyr on this chart.

\subsection{Chronometers and the drift, from first principles}
\label{app:cK-channels}

Two channels read the curve with no ruler and no candle, and their
master relations take four lines each. \emph{Cosmic chronometers}
\citep{JimenezLoeb2002,Moresco2022}: from
\(1 + z = a_0/a(t_e)\), differentiate with respect to the
\emph{emission} time of a passively evolving galaxy population:
\(dz/dt_e = -(1+z)\,\dot a/a = -(1+z)H(z)\), so
\(H(z) = -\,[1/(1+z)]\,dz/dt_e\) --- Eq.~\eqref{eq:chronometer}:
measure the differential age of old stellar populations across a
small redshift step and the curve appears, no cosmology assumed.
\emph{The redshift drift}
\citep{Sandage1962,Loeb1998}: watch a single comoving source as the
\emph{observer's} clock advances by \(\Delta t_0\). Both endpoints
of \(1 + z = a(t_0)/a(t_e)\) move, and the chain rule gives
\(\dot z = (1+z)H_0 - H(z)\) --- Eq.~\eqref{eq:drift}: the
competition between today's stretch of the whole interval and the
source-epoch stretch of the emitted wave. Convert to velocity,
\(\Delta v = c\,\Delta z/(1+z)\), and the derived chart commits to
numbers: \(+2.4\), \(+2.1\), \(-0.3\), \(-5.6\)~cm\,s\(^{-1}\) per
decade at \(z = 0.5, 1, 2, 4\), with the sign change at
\(z \approx 1.9\) --- the redshift at which \(H(z)/(1+z)\) crosses
today's rate, i.e.\ where the universe's deceleration-era imprint
balances its acceleration-era one. A drift map is cosmology with no
proxy at all; on the derived chart it is also cosmology with no
freedom at all.

\subsection{What a one-number cosmology cannot do}
\label{app:cK-limits}

The limitations step, and the closing statement of the whole
appendix chain. A cosmology organized around measuring \(H_0\) ever
better can, at saturation, deliver exactly one datum --- and the
tension is the proof that one datum cannot arbitrate its own
systematics. The curve is different in kind: ages bound it at every
redshift where an old object is found; chronometers sample its
derivative; BAO points, calibrated by the derived
\(r_s = 145.0\)~Mpc, read it absolutely; the drift will watch it
move. A fitted model meets each new curve-point by re-fitting; a
derived curve meets it with a number already printed --- which is
why Table~\ref{tab:derived-Hz} is not a summary but an exposure
schedule.

\subsection{The geometric chronology, as commitments}
\label{app:cK-gg}

Step three is a recital, because by now every entry has been derived
in some course of this appendix: \(t_0 = 13.92\)~Gyr (this course);
\(t_{\rm acc} = 7.8\)~Gyr at the exact \(\operatorname{arcsinh}
(1/\sqrt2)\) (Course C); \(z_* = e^7\) and the two rulers
\(r_s = 145.0\) / \(r_d \approx 147\)~Mpc (Course E);
\(H_\infty = 55.0\) and the \(12.3\)~Gyr doubling time (Course C);
the \(H(z)\) rows and drift values above; and the five clocks ---
\(t\), \(a\), \(z\), \(T\), and the radial coordinate \(r\) --- kept
synchronized by the one ledger (the annex below walks the \(T\) clock). Each is graded as its source
course grades it; each is a place to die; and together they are what
this paper means when it says the constant was never the point ---
the \emph{history} is the object, and the history is now fully on
the record.

\subsection{Annex: the thermal clock --- scaling laws, thermal history, and freeze-out}
\label{app:thermal}

The Friedmann equation of \S\ref{app:frw-theorem} runs the universe forward; the
ledger tells it what to carry. This course collects the standard results on which
every cosmological milestone quoted in the body rests, read through the \GG{}/\GDOS{}
source ledger.

\paragraph{Exact scaling law.} For a component with constant $w_i=p_i/\rho_i$, the
continuity equation $\dot\rho_i+3H(1+w_i)\rho_i=0$ integrates immediately to
\begin{equation}
\rho_i(a)=\rho_{i,0}\,a^{-3(1+w_i)},
\label{eq:app-scaling}
\end{equation}
so radiation ($w=\tfrac13$) dilutes as $a^{-4}$, non-relativistic matter ($w=0$) as
$a^{-3}$, and the vacuum ($w=-1$, the derived value of \ATen{}) stays constant. The
first Friedmann equation therefore becomes a sum over the inventory,
\begin{equation}
H(a)=\Hzero\sqrt{\Omega_r a^{-4}+\Omega_m a^{-3}+\Omega_\Lambda},
\qquad \Omega_m=\tfrac1\pi,\ \Omega_\Lambda=1-\tfrac1\pi,\ \Omega_r\simeq9.4\times10^{-5},
\label{eq:app-Hofa}
\end{equation}
the curve the body calls $H(t)$, with every coefficient fixed by the ledger and only
the radiation share carried at its measured photon-plus-neutrino value.

\paragraph{Matter--radiation equality.} Radiation and matter scale as $a^{-4}$ and
$a^{-3}$, so they cross at $a_{\rm eq}=\Omega_r/\Omega_m$, i.e.
\begin{equation}
1+z_{\rm eq}=\frac{\Omega_m}{\Omega_r}=\frac{1/\pi}{9.4\times10^{-5}}\simeq3.4\times10^{3},
\label{eq:app-zeq}
\end{equation}
the epoch after which structure can grow; the derived $\Omega_m=1/\pi$ fixes it with
no free parameter.

\paragraph{Cooling and the temperature--time dictionary.} Comoving entropy
conservation, $g_{*s}(T)\,T^3 a^3=\text{const}$, gives the cooling law
$T\propto g_{*s}^{-1/3}a^{-1}$. In the radiation era $H^2=\tfrac{8\pi G_N}{3}\rho_r$
with $\rho_r=\tfrac{\pi^2}{30}g_*(T)T^4$, and $t=1/2H$, so
\begin{equation}
t\simeq2.42\,g_*(T)^{-1/2}\Big(\frac{T}{1\ \mathrm{MeV}}\Big)^{-2}\ \mathrm{s},
\label{eq:app-tT}
\end{equation}
the dictionary that dates every early milestone. A species freezes out when its
interaction rate falls below expansion, $\Gamma(T)\sim H(T)$ --- the universal
competition that sets every relic abundance, including the dark-matter density that
the \PaperCOne{} Kaxion misalignment fixes to the ledger value.

\paragraph{Recombination as the switch.} At $z_\ast=e^7\approx1097$
(Prop.~\ref{prop:recomb-switch}) the plasma recombines and the microwave background
is released; this is the epoch whose sound horizon calibrates the early-side
distance scale of the next course. Everything to the past of it is read through the
acoustic dictionary, everything to its future through the ladder --- the two windows
whose disagreement is the tension.

The fuller treatment of everything in this annex --- entropy
conservation, \(g_*\) bookkeeping, and the freeze-out calculus ---
is the classic text of \citet{KolbTurner1990} and the thermal-history
course of the parent's appendices \citep{ArisakaA10Cosmos}.

\subsection*{Status}

Every closed form in this course is derived above it; every number
is asserted in the build script against the arcsinh forms and the
drift relation; the grades are inherited unchanged. With this
course, the appendix chain covers the full arc of the paper:
background (C), measurement (D, E), vacuum (F), environment (G, H),
growth (I), referee (J), and time (K).

\section{Derivation course: the statistics of a tension}
\label{app:courseL}

The last course is about the tension itself as a statistical object
--- what a ``\(7\sigma\) discrepancy'' does and does not mean, why a
decade of improving measurements made it worse rather than better,
and what combination rules honestly do when correlations are unknown.
Nothing in this course is cosmology; all of it is the bookkeeping
that the cosmology of Courses C--K gets scored by, and it is taught
last because every example is now familiar.

\subsection{What a sigma means when systematics exist}
\label{app:cL-sigma}

Two measurements of one quantity, \(x_1 \pm \sigma_1\) and
\(x_2 \pm \sigma_2\), are in tension at
\[
T = \frac{|x_1 - x_2|}{\sqrt{\sigma_1^2 + \sigma_2^2}}
\]
--- \emph{if} the quoted errors are complete and independent. Every
word of that proviso works hard. If a systematic \(b\) biases
\(x_1\) and is absent from its error bar, the numerator is really
\(|\Delta_{\rm true} + b|\) and \(T\) measures the bias as
confidently as it would measure physics; if a systematic is
\emph{shared} by both measurements, it cancels in the difference and
\(T\) never sees it at all --- both values are wrong together,
consistently (Course~D's common-mode theorem, met again from the
other side). The practical moral for the tension: a large \(T\)
proves the two chains disagree; it cannot, even in principle, say
\emph{which} chain, or whether the disagreement is nature or
bookkeeping. That adjudication needs structure from outside the two
chains --- a third instrument, or a derived expectation.

\subsection{Why precision amplifies bias: the decade's paradox, derived}
\label{app:cL-amplify}

Now the one-line theorem that explains the last ten years. Hold any
unresolved bias \(b\) fixed and let both teams improve, \(\sigma \to
0\):
\[
T = \frac{|\Delta_{\rm true} + b|}{\sqrt{\sigma_1^2 + \sigma_2^2}}
\;\xrightarrow[\;\sigma \to 0\;]{}\; \infty .
\]
Precision does not resolve a bias; it \emph{amplifies its
significance}. If the true backgrounds agree
(\(\Delta_{\rm true} = 0\)) and a percent-level calibration bias
exists anywhere, then better measurements must produce an
ever-growing sigma count --- which is exactly the observed history:
the tension climbed from \(2\sigma\) to \(7\sigma\) precisely as
both chains reached percent-class precision. The sigma count, by
itself, cannot distinguish this world from one with genuinely
discordant physics; the \emph{growth pattern} of \(T\) with
precision is identical in both. This is the statistical face of the
statement the paper makes throughout: the dispute over the last
digit cannot settle what kind of number is being measured, because
\(T \propto b/\sigma\) rewards the digits indiscriminately.

\subsection{The bracket theorem: combining channels with unknown correlations}
\label{app:cL-bracket}

Course~D derived the channel-combination formula
\(\sigma_{\rm tot}^2 = \sum_i \sigma_i^2 + 2\sum_{i<j}
\rho_{ij}\sigma_i\sigma_j\); here is the statement that makes the
paper's two printed scenarios rigorous rather than rhetorical. For
any set of non-negative channel sizes, \(\sigma_{\rm tot}\) is
monotonically increasing in every \(\rho_{ij}\), so over the whole
physically allowed family \(0 \le \rho_{ij} \le 1\) --- the
restriction to \emph{non-negative} correlations being itself an
assumption we argue rather than smuggle: these channels share
calibrations, compilations, and anchors, mechanisms that push
\emph{together}, and no proposed mechanism anticorrelates them;
under a negative correlation \(\sigma_{\rm tot}\) would drop below
quadrature and the bracket's floor would only strengthen ---
\[
\underbrace{\Big(\textstyle\sum_i \sigma_i^2\Big)^{1/2}}_{\text{quadrature}}
\;\le\; \sigma_{\rm tot} \;\le\;
\underbrace{\textstyle\sum_i \sigma_i}_{\text{linear stack}} :
\]
the two rules the paper prints are not two opinions; they are the
\emph{sharp bounds} of the correlation-agnostic problem, attained at
\(\rho = 0\) and \(\rho = 1\). Quoting any single interior value
would smuggle in a correlation matrix nobody has measured; quoting
the bounds is the maximal honest statement --- and the observation
that the required \(+7.3\%\) falls between them
(\S\ref{sec:residual-verdict}; effective \(\bar\rho \approx
0.5\)--\(0.7\), Course~D) is therefore a consistency, not a
construction.

\subsection{What a covariance-weighted consensus certifies}
\label{app:cL-consensus}

The optimal combination of correlated readings \(x_i\) with
covariance \(C\) is the generalized least-squares mean,
\(\hat y = (\mathbf 1^{\mathsf T} C^{-1}\mathbf 1)^{-1}\,
\mathbf 1^{\mathsf T} C^{-1}\mathbf x\), with variance
\((\mathbf 1^{\mathsf T} C^{-1}\mathbf 1)^{-1}\) --- the machinery
behind every modern consensus value, and it is genuinely optimal for
what it models. Now add a common mode: \(\mathbf x \to \mathbf x +
c\,\mathbf 1\). Because the weights sum to one,
\(\hat y \to \hat y + c\) --- while \(C\), the internal residuals,
and every leave-one-out statistic are untouched. In one line: a
consensus certifies the \emph{covariance it models}, and transmits
unmodeled common modes at full strength with undiminished
confidence. This is not a criticism of any collaboration; it is a
property of weighted means, and it delimits precisely what
\(73.50 \pm 0.81\) establishes: the stability of the network's
shared frame --- Part~II's genuine achievement --- and nothing about
the frame itself. The boundary of the certification is where
Part~IV's channels live.

\subsection{Reading a landscape: bands, not rows}
\label{app:cL-landscape}

The forty-five values of Part~V invite naive statistics ---
``average them all'' --- and the course closes its textbook step by
deriving why that is wrong twice over. First, the values are not
independent: methods share calibrations, catalogs, and even
authors, so the effective number of independent readings is far
smaller than forty-five, and an unweighted mean double-counts every
shared systematic. Second, the field's own meta-analysis finds the
literature's scatter exceeds its quoted errors --- direct evidence
of unmodeled (and partially shared) systematics --- so
\emph{clustering}, not averaging, is the information-bearing
statistic: the eye should weight bands, not count rows, which is
exactly how Part~V instructs its reader. A band structure with a
filling middle is the signature of a population of measurements
sharing a few large common modes that are being found and removed
one at a time --- and that reading is testable, band by band, as the
corrections land.

\subsection{The geometric posture: statistics with the freedom removed}
\label{app:cL-gg}

Step three is the paper's own statistical conduct, now assembled in
one place. \emph{Pre-registration:} every decision rule of the
falsifier registry was fixed numerically before its datum
(\S\ref{sec:falsifiers}), so the trials factor of each confrontation
is exactly one --- the look-elsewhere tax that haunts post-hoc
tension statistics is paid in advance, a discipline the cohort's
dark-matter companion has already executed live
\citep{ArisakaC1Kaxion}. \emph{Bracket pricing:} unknown
correlations are carried as the sharp bounds of
\S\ref{app:cL-bracket}, never as a chosen interior value.
\emph{Deflation, stated symmetrically:} when the field's own
published corrections are applied at face value, the significance of
the residual tension deflates from the headline \(7\sigma\) class to
the \(2\sigma\) class (\S\ref{sec:residual-verdict}) --- and the
paper prints that arithmetic while explicitly \emph{not} adopting
the corrections, because the same bracket honesty cuts both ways.
\emph{Released arithmetic:} all of it --- chains, brackets,
deflations --- regenerates from
\texttt{c2\_residual\_arithmetic.py}'s twenty-three asserted checks
(\S\ref{app:scripts}). And beneath all four habits, the structural
point this course has been building toward: statistics can referee
the tension only when something in the problem is \emph{not} free.
A derived background is exactly that fixed point --- it converts the
sigma count from an amplifier of unknown bias into a test with a
printed answer.

\subsection*{Status}

Everything in this course is elementary probability, derived in
place; its cosmological content is zero and its bookkeeping content
is the whole paper. The reader who retains one line should retain
the theorem of \S\ref{app:cL-amplify}: \(T \propto b/\sigma\) ---
precision amplifies unresolved bias --- and its corollary, which is
this paper's methodological thesis: the cure for a tension is never
another digit; it is a fixed point outside both chains.

\section{Reproducibility and the released pipeline}
\label{app:blockD}

How the decisive observation will be reproduced, and how every number
in this report can be regenerated from the released scripts: the
input tables with their justifications, the scripts index, the siren
recipe, and the H0DN variant ladder.

\subsection{Reproducibility: inputs, justifications, and the systematic budget}
\label{app:repro-inputs}

The derived corridor is reproducible in the strict sense: every input is a published
number or a ledger-fixed constant, and the propagation is elementary. This course
collects the inputs, states where each comes from, and assembles the systematic
budget on the background rate, so that an independent worker can regenerate
$\Hzero^{(0)}=66.6$ and bound its uncertainty.

\begin{table}[htbp]
\centering\small
\caption[Inputs to the derived background rate, their provenance, and their contribution.]{Inputs to the derived background rate, their provenance, and their
contribution to the systematic budget. Ledger-fixed shares carry no measurement
error; the uncertainty is entirely in the measured physical densities and the
route-to-route spread. The last column is the one that matters for reproducing the result. The two ledger-fixed shares are exact, so everything the error bar contains enters through the two measured densities and the structural spread on the seesaw coefficient. A reader who wants to redo the calculation needs no input beyond this table.}
\label{tab:c2-repro-inputs}
\begin{tabularx}{\textwidth}{@{}llll@{}}
\toprule
\textbf{Input} & \textbf{Value} & \textbf{Source} & \textbf{Contribution to $\sigma(\Hzero)$}\\\midrule
$\Omega_m$ & $1/\pi=0.31831$ & \ATen{} (ledger, exact) & $0$ (no measurement error)\\
$\Omega_b$ & $1/2\pi^2=0.05066$ & \ATen{} (ledger, exact) & $0$\\
$\omega_m=\Omega_m h^2$ & $0.1430\pm0.0011$ & Planck 2018 & $\pm0.26$ (matter route)\\
$\omega_b=\Omega_b h^2$ & $0.02237\pm0.00015$ & Planck 2018 & $\pm0.22$ (baryon route)\\
$\zeta_\Lambda$ & $\pi/175 = 0.01795$ & codebook selection (\S\ref{sec:zeta-selection}) & $\pm0.2$ structural (geometric route)\\
$M_{\rm lock}$ & $3.85$~TeV & \AFive{} T8.7 (theorem) & folded into $\zeta_\Lambda$\\
\bottomrule
\end{tabularx}
\end{table}

\paragraph{The inventory route, step by step.} The first Friedmann equation at the
present epoch reads $\Hzero^2=(8\pi G_N/3)\rho_{\rm crit}$, and each measured
\emph{physical} density $\omega_i=\Omega_i h^2$ is the product of the derived share
and $h^2$. Dividing the measured $\omega_i$ by the derived $\Omega_i$ therefore
returns $h^2$ with no free parameter:
\begin{equation}
h=\sqrt{\frac{\omega_m}{\Omega_m}}=\sqrt{\frac{0.1430}{0.31831}}=0.670
\ \Rightarrow\ \Hzero=67.0,
\qquad
h=\sqrt{\frac{\omega_b}{\Omega_b}}=\sqrt{\frac{0.02237}{0.05066}}=0.664
\ \Rightarrow\ \Hzero=66.5.
\label{eq:c2-tworoute}
\end{equation}
The two anchors bracket the adopted background $\Hzero^{(0)}=66.6$, and their spread
($0.5$) is the leading systematic. Propagating the Planck error on $\omega_m$
($0.8\%$) gives $\pm0.17$; the two-route consistency is the dominant check, not the
formal error.

\paragraph{The seesaw route, as the geometric determination.} The vacuum-energy
seesaw of \S\ref{app:seesaw} gives $\Hzero=498.3\sqrt{\zeta_\Lambda}$ with
$\zeta_\Lambda$ selected by the codebook as $\pi/(LK) = \pi/175$
(\S\ref{sec:zeta-selection}), returning $\Hzero=66.8\pm0.2$ with no
astronomical input; the cost-minimal cluster brackets the rate at
$66.8$--$71.2$ either way. The $\Hzero\propto M_{\rm lock}^{4}$
sensitivity is pinned in the laboratory: the same lock scale predicts the
electroweak vev at $+0.045\%$, i.e.\ $\pm0.18\%$ on the rate. That two
structurally unrelated routes (a vacuum seesaw and a density ratio) land within $0.5$
of one another, and both on the CMB value, is the reproducible core of the claim.

\paragraph{What an independent worker must supply.} Only three continuous inputs enter
--- $\omega_m$ and $\omega_b$ (for the confirmation route only) and the lock scale $M_{\rm lock}$ ---
each published; the geometric route consumes nothing astronomical, and
everything else is ledger-fixed. The computation is a pocket-calculator
exercise, deliberately: the content is in the derivation of the shares
$\Omega_i$ (\ATen{}) and the seesaw normalization (\AFive{}), not in the arithmetic
that turns them into a rate. This is the sense in which the background rate is not
fitted --- there is nothing to fit.

\subsection{Scripts released with this paper}
\label{app:scripts}

Every quantitative figure and forecast in this report regenerates from a
short, documented, standalone script; Table~\ref{tab:c2-scripts} is the
index. Each script states its inputs in its header, asserts its own
arithmetic before writing anything, and runs in seconds on a laptop. The
fastest way to referee this paper is to run them.

\begin{table}[htbp]
\centering\small
\caption[The open-source scripts released with this paper and what each reproduces.]{The open-source scripts released with this paper and what each
reproduces. Each script asserts its own printed numbers, so a script that runs without failing has reproduced the figure or table it names rather than merely produced one. The last two entries do this most strictly: the residual arithmetic asserts every number in its section, and the landscape script asserts all forty-five audited values against the primary-record audit. No figure in this paper is drawn from a stored data file that is not regenerated here.}
\label{tab:c2-scripts}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
\textbf{Script} & \textbf{Reproduces}\\\midrule
\texttt{c2\_fig\_H\_of\_t\_pilocked.py} & the derived expansion history (Fig.~\ref{fig:H-of-t-derived})\\
\texttt{c2\_fig\_a\_of\_t.py} & the derived scale factor (Fig.~\ref{fig:a-of-t})\\
\texttt{c2\_fig\_omega\_budget.py} & the running budget \(\Omega_i(t)\) (Fig.~\ref{fig:omega-budget})\\
\texttt{c2\_fig\_landmarks.py} & the landmarks walk and its numbers (Fig.~\ref{fig:landmarks}, Table~\ref{tab:c2-landmarks})\\
\texttt{c2\_showcase\_offsets.py} & the coherent-offset chart (Fig.~\ref{fig:cmb-offsets})\\
\texttt{c2\_fig\_two\_routes.py} & the two-route convergence chart, with the chain re-derived and asserted (Fig.~\ref{fig:two-route-concordance})\\
\texttt{c2\_fig\_h0\_history.py} & the century-of-\(H_0\) chart (Fig.~\ref{fig:h0-history})\\
\texttt{c2\_fig\_hole.py} & the Local-Hole profiles and implied lift (Fig.~\ref{fig:hole-lift})\\
\texttt{c2\_fig\_bridge.py} & the reconciliation bridge (Fig.~\ref{fig:bridge})\\
\texttt{c2\_backreaction\_pilocked.py} & the backreaction and cosmic-variance bounds (Fig.~\ref{fig:backreaction-bound}; App.~\ref{app:backreaction})\\
\texttt{c2\_siren\_forecast.py} & the events-to-verdict forecast (Fig.~\ref{fig:siren-verdict}, Table~\ref{tab:siren-events})\\
\texttt{c2\_residual\_arithmetic.py} & The two-scenario channel arithmetic, central-value chains, and significance deflation of \S\ref{sec:residual-verdict}; every printed number asserted.\\
\texttt{c2\_showcase\_landscape.py} & The 2026 \(H_0\) landscape figure (Figure~\ref{fig:h0-landscape}): all 45 audited values asserted in-script against the primary-record audit of July 12, 2026.\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Reproducing the decisive test: the standard-siren $H_0$}
\label{app:siren-repro}

The decisive observation (\S\ref{sec:decisive}) is reproducible in principle by
any group with a gravitational-wave catalog. A standard siren yields $H_0$
from two independently measured quantities: the luminosity distance $d_L$, read
from the amplitude and chirp of the waveform, and the redshift $z$ of the host
(bright sirens with an electromagnetic counterpart) or of the redshift
distribution of candidate hosts (dark sirens). With a catalog of $N$ events
the combined constraint scales as $\sigma_{H_0}/H_0\sim (0.1\text{--}0.15)/\sqrt{N}$
for well-localized bright sirens, so a few tens of events reach the
$\lesssim1\%$ needed to separate $67$ from $73$. No sound horizon and no
distance ladder enters at any step, which is precisely why the channel is
decisive: its systematics are disjoint from both discordant methods. The
pre-declared verdict is fixed in advance (F-Hubble-2): a converged
$H_0^{\mathrm{GW}}=73\pm1$ refutes; $67\pm1$ confirms. The full
forecast --- events-to-verdict arithmetic, systematic budget, and the
simulated catalog --- is Part~VII, and its open-source script
(\texttt{c2\_siren\_forecast.py}) is released with the paper.

\subsection{The H0DN variant ladder: the reproducibility table}
\begin{table}[htbp]
\centering\small
\caption[Full list of H0DN baseline and variant values from \cite{CasertanoH0DN2026}.]{Full list of H0DN baseline and variant values from
\cite{CasertanoH0DN2026}. The derived reading takes this stable
network consensus at face value as the measured local rate; its
interpretation --- background $66.6$, plus the Local-Hole lift, plus
a named open calibration residual --- is
Section~\ref{sec:both-from-anchor} (Appendix~\ref{app:epslocal}).}
\label{tab:variantsoverlay}
\begin{tabularx}{\textwidth}{@{}lcXc@{}}
\toprule
\textbf{Variant} & \textbf{$\Hzero\ [\kmsMpc]$} &
\textbf{Description} & \textbf{Deviation from baseline}\\\midrule
V00 & $73.50 \pm 0.81$ & Baseline & $0.0\sigma$\\
V01 & $73.49 \pm 0.81$ & +JAGB & $-0.01\sigma$\\
V02 & $73.51 \pm 0.81$ & +Miras & $+0.01\sigma$\\
V03 & $73.60 \pm 0.80$ & +DESI FP & $+0.13\sigma$\\
V04 & $73.52 \pm 0.81$ & +SNe~II (SCM) & $+0.02\sigma$\\
V05 & $73.72 \pm 0.74$ & +SNe~II (EPM) & $+0.30\sigma$\\
V07 & $73.31 \pm 0.79$ & +SMC anchor & $-0.24\sigma$\\
V08 & $72.51 \pm 1.30$ & no Cepheids & $-0.76\sigma$\\
V09 & $73.53 \pm 0.88$ & no TRGB/SBF & $+0.03\sigma$\\
V10 & $73.61 \pm 0.93$ & no Gaia/MW & $+0.12\sigma$\\
V11 & $73.34 \pm 0.85$ & no DEB/LMC & $-0.19\sigma$\\
V12 & $73.08 \pm 0.92$ & no NGC~4258 & $-0.46\sigma$\\
V13 & $73.43 \pm 1.80$ & no SNe~Ia & $-0.04\sigma$\\
V14 & $73.33 \pm 0.83$ & no SBF & $-0.20\sigma$\\
V15 & $73.63 \pm 0.84$ & no masers (HF) & $+0.16\sigma$\\
V16 & $73.75 \pm 1.33$ & no HST/LMC/MW & $+0.19\sigma$\\
V17 & $73.00 \pm 0.86$ & no JWST & $-0.58\sigma$\\
V18 & $73.69 \pm 0.82$ & no pre-1994 SNe & $+0.23\sigma$\\
V99 & $73.99 \pm 0.70$ & everything & $+0.70\sigma$\\
V99a & $73.66 \pm 0.71$ & everything except TF & $+0.23\sigma$\\
O1 & $73.11 \pm 0.92$ & orthogonal path 1 & $-0.42\sigma$\\
O2 & $73.43 \pm 1.80$ & orthogonal path 2 & $-0.04\sigma$\\
\bottomrule
\end{tabularx}
\end{table}

All 22 H0DN variants lie within $\pm 1\sigma$ of the baseline
$73.50$; the largest deviations are V08 (no Cepheids, $-0.76\sigma$)
and V99 (everything, $+0.70\sigma$), and the median offset is
$\approx 0.2\sigma$. This is the stability pattern expected if every
variant measures the same underlying local rate --- no single method,
anchor, or instrument drives it. The derived reading interprets that
stable local value as the derived background $66.6$ lifted by the
Local-Hole environment and a distance-ladder calibration residual
(Section~\ref{sec:both-from-anchor}).


\section{Registries, audits, and roadmaps}
\label{app:blockE}

\subsection{Internal source ledger: where each object lives}
\label{app:source-ledger}

For the reader navigating between the main text and these courses,
the load-bearing objects and their home addresses:

\begin{table}[htbp]
\centering\small
\caption[The internal source ledger: each headline object's home in the main text, and the appendix.]{The internal source ledger: each headline object's home in
the main text, and the appendix course that teaches it. The ledger exists because the main text and the appendices were written to be read in either order, and a reader who enters through a course needs to know where its result is used. Reading the second column down is also a fair table of contents for the argument: the background first, then the local corrections, then the audits, then what is still open.}
\label{tab:c2-source-ledger}
\begin{tabularx}{\textwidth}{@{}lXl@{}}
\toprule
\textbf{Object} & \textbf{Home in the main text} & \textbf{Course}\\\midrule
Derived background \(66.6 \pm 0.2\) & \S\ref{sec:absolute-h0} (Routes I and II) & E, F\\
The closed form for \(H_0\) & \S\ref{sec:h0-closed-form} & F\\
The corridor and the two dictionaries & \S\ref{sec:recomb-switch} & C, E\\
The Local-Hole lift \(+2\)--\(3\%\) & \S\ref{sec:worked-excess}, \S\ref{sec:hole-course} & G\\
The residual anatomy and verdict & \S\ref{sec:residual-anatomy}, \S\ref{sec:residual-verdict} & D, L\\
The backreaction bound & \S\ref{sec:backreaction} & H\\
The 2026 landscape (45 values) & Part~V (Fig.~\ref{fig:h0-landscape}, Table~\ref{tab:h0-landscape}) & L\\
The falsifier registry and calendar & \S\ref{sec:falsifiers}, \S\ref{sec:decision-calendar} & ---\\
The siren forecast and stake & Part~VII & J\\
Growth and \(S_8\) & \S\ref{sec:s8-section} & I\\
The chronology and the drift & \S\ref{sec:overdetermining} & K\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{The falsification catalog, compact}
\label{app:fals-catalogue}

The complete falsification surface of this paper, in three layers,
each detailed elsewhere and indexed here. \emph{The registry:}
F-Hubble-1 through F-Hubble-11, four tiers, each with a pre-declared
numerical rule and its confirmation twin
(Table~\ref{tab:killcriteria}); the channel-level trio is also
carried cohort-wide as global ledger rows F26--F28. \emph{The
calendar:} the dated appointments and the commitments ledger
(\S\ref{sec:calendar-entries}--\S\ref{sec:calendar-commitments}).
\emph{The courses:} every derivation course above ends by naming its
own falsifiable outputs --- the \(H(z)\) rows and ages (C, K), the
shell profile (G), the growth rigidity \(f\sigma_8(z)\) (I), the
siren thresholds (J) --- so the appendix chain is itself part of the
paper's exposure, not commentary on it.


What the reading inherits, the theorems it stands on, and how its open entries would be promoted.

\subsection{Cross-paper inheritance table}
\label{app:inherit}

This report derives nothing new about the cosmology; it reads the H0DN
measurement through results imported from the cohort. Table~\ref{tab:c2-inherit}
lists exactly what is inherited, from where, and what this paper does with it.

\begin{table}[htbp]
\centering\small
\caption[What the derived Hubble reading inherits from the cohort.]{What the derived Hubble reading inherits from the cohort. Each entry
is imported at the grade its home paper assigns; this report adds only the
reading of the H0DN measurement against them. Nothing in the first column is derived here, and that is the reason for listing it. This paper's own contribution is the reading in the third column and nothing else, so a reader who rejects an inherited object should follow it back to the paper named beside it rather than argue with this one. Two of the entries are conditional at home and stay conditional here.}
\label{tab:c2-inherit}
\begin{tabularx}{\textwidth}{@{}llX@{}}
\toprule
\textbf{Inherited object} & \textbf{From} & \textbf{Use here}\\\midrule
$\pi$-locked inventory ($\Omega_m=1/\pi$, \dots) & \ATen{} & Divided into CMB densities to give $H_0^{(0)}$\\
Hopf-fiber identification (Conjecture~VIII) & \ATen{} & The measure lemma the D$^*$ grade rests on\\
Two-route $H_0$ derivation ($66.6$) & \ATen{} & The background the tension splits around\\
Vacuum-energy seesaw, $\zeta_\Lambda$ & \ATen{}/\AFive{} & Route I: the geometric determination (\S\ref{sec:zeta-selection})\\
Lock scale $M_{\mathrm{lock}}=3.85$ TeV (T8.7) & \AFive{} & Sets the seesaw normalization\\
\OGNfive{} integer ledger $(1,3,5;7,35)$ & \AFour{}/\ATwo{} & Fixes $\zeta_\Lambda$ and $z_*=e^7$\\
\GDOS{} open-system boundary law & \AFive{} & Makes the two $H_0$ values registered observables\\
Standard-Model exports ($\Delta N_{\mathrm{eff}}=0.04$) & \ANine{} & The early-side radiation budget\\
Exact local $\Uoneb$ (proton stability) & \ATwo{}/\ANine{} & Source of the local-lift channel; F-Hubble-1\\
Kaxion twin-peak $\Delta\nu/\nu=1/35$ & \PaperCOne{} & The dark-sector cross-check F-Hubble-7\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Theorem-and-conjecture deck}
\label{app:deck}

The reading rests on one imported theorem and one named conjecture, with the
rest of its content graded against them (Table~\ref{tab:c2-deck}).

\begin{table}[htbp]
\centering\small
\caption[The named theorems and conjectures the derived Hubble reading depends on.]{The named theorems and conjectures the derived Hubble reading depends
on. ``Home'' gives the paper that proves or states each; this report uses them
without re-deriving. Five items is a short deck for a cosmological result, and the shortness is part of the claim. One of the five is open rather than proved, and it is named as open with its target pre-registered, so that the eventual computation cannot be read afterwards as a fit. If any of the other four fails at home, this paper fails with it and no repair made here would help.}
\label{tab:c2-deck}
\begin{tabularx}{\textwidth}{@{}llX@{}}
\toprule
\textbf{Tag} & \textbf{Home} & \textbf{Statement / role}\\\midrule
T-Cosmos-17 & \ATen{} & $\Omega_m=1/\pi$ as the Hopf fiber-volume ratio (conditional on the measure lemma)\\
Conjecture~VIII & \ATen{} & The measure-normalization lemma weighting the Hopf circle; the D$^*$ condition\\
T-Cosmos-DE ($w=-1$) & \ATen{} & The vacuum equation of state is exactly $-1$; falsifier F-Hubble-5\\
Scale chain T8.7 & \AFive{} & $M_{\mathrm{lock}}=\tfrac12 E_{\mathrm{Pl}}e^{-K}=3.85$ TeV, a theorem\\
Hosotani lemma (O6) & \AFive{} & Open: closes the selection lemma; target pre-registered, $\zeta_\Lambda = \pi/175$ (F-Hubble-11)\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Promotion roadmap}
\label{app:promotion}

The reading is honest about what would raise each of its graded claims to the
next tier (Table~\ref{tab:c2-promotion}). None of these is required for the
falsifiable content --- the background rate and the siren prediction --- to
stand.

\begin{table}[htbp]
\centering\small
\caption[What would promote each conditional or asserted entry of the derived Hubble reading.]{What would promote each conditional or asserted entry of the derived
Hubble reading. A conditional grade is a promise about what would remove the condition, and this table is that promise written out. Two of the five close on the same measure lemma, one on a Wilson-line computation somebody could start today, and one on an amplitude nobody has yet written down. The fifth is not promoted at all but retired, by a ladder analysis converging on the derived value, and it is listed here because leaving it out would make the column look tidier than it is.}
\label{tab:c2-promotion}
\begin{tabularx}{\textwidth}{@{}lX@{}}
\toprule
\textbf{Entry (grade)} & \textbf{What would promote it}\\\midrule
$\pi$-inventory (D$^*$) & Exhibiting the measure lemma (Conjecture~VIII / VIII-b$''$) closes the Hopf-fiber identification and promotes the fractions to Derived\\
Background rate $66.6$ (D$^*$) & Inherits promotion from the inventory; no separate step\\
Seesaw $\zeta_\Lambda$ (Derived cond.) & The Hosotani/Kaluza--Klein Wilson-line computation on $S^1_\theta$ (\AFive{}~O6) promotes it to Derived (theorem); pre-registered target $\pi/175$ (F-Hubble-11)\\
Local-Hole lift (Derived, sign+size) & A full first-principles amplitude of the $\Uoneb$ environmental lift (App.~\ref{app:epslocal}) would close the residual\\
Ladder residual (named, open) & A controlled distance-ladder analysis converging on $67$ (F-Hubble-4) retires the residual entirely\\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Notation}
\label{app:blockF}
\label{app:notation}

$H_0$: present expansion rate (km\,s$^{-1}$\,Mpc$^{-1}$). $H_0^{(0)}$: the
\emph{derived} background rate, $66.6$. $H_0^{\mathrm{GW}}$: the standard-siren
value. $\Omega_m,\Omega_b,\Omega_\Lambda$: present matter, baryon, and
dark-energy fractions. $\omega_i=\Omega_i h^2$: the corresponding physical
densities ($h$-free). $\zeta_\Lambda$: the vacuum-energy seesaw coefficient.
$\epsilon_{\mathrm{local}}=\epsilon_{\mathrm{Hole}}+\epsilon_{\mathrm{syst}}$:
the total local excess, split into the conditional Local-Hole lift (derived response, counts-branch depth) and the named
calibration residual. $z_*=e^7$: the corridor-switch redshift. $M_{\mathrm{lock}}$:
the lock scale, $3.85$~TeV. $S_8$: the clustering amplitude. $\Delta N_{\mathrm{eff}}$:
extra effective relativistic species ($0.04$ here). Cohort acronyms
(\GG{}, \BIsix{}, \OGNfive{}, \GDOS{}, \GRIP{}) are expanded in
App.~\ref{app:naming} and the glossary.

\GGAuthorsNote

This synthesis report is authored as a companion to \ATen{}
(the cosmology recast) and the flagship paper \AEleven{} (the
unified summary of \GGtheory{}). No new cosmological observation
is reported here. The content consists of (i) a theory-neutral
summary of the H0DN collaboration paper of \cite{CasertanoH0DN2026};
(ii) the derived expansion history \(H(t)\) imported from
\cite{ArisakaA10Cosmos} and its confrontation with the microwave sky
and the cosmological landmarks; (iii) the reconciliation of the local
value --- the conditional Local-Hole lift (derived response, counts-branch depth), and the named residual priced
against the ladder's error budget; (iv) a comparison with six classes
of alternative frameworks and the siren discriminator; (v) the
standard-siren forecast, with its systematic budget and released
scripts; and (vi) an explicit set of kill criteria, observational
priorities, and seventy-nine answered objections.



\GGPart[Common Objections and Responses]{Common Objections and Responses}{Twenty big questions for the curious reader, plainly answered; then the hardest questions a skeptical cosmologist would ask of the derived-rate reading, answered in the framework's own terms}

This Part answers the questions this paper provokes, in two registers.
It \emph{opens} for the curious reader --- twenty big questions a
thoughtful non-specialist actually carries into a book about the
universe's expansion, answered plainly
(\S\ref{sec:obj-curious}) --- and then hardens, theme by theme, into
the objections a skeptical professional would raise, answered in the
framework's own terms. All seventy-nine are tagged in the cohort's
\textsf{O-Hubble-\(N\)} registry (the curious-reader set carries the
tags \textsf{O-Hubble-55}--\textsf{74}; the third-exchange entries
carry \textsf{75}--\textsf{79}; tags number the ledger, not the
reading order), and the Part is
closed by a dedicated seven-persona hostile review (\S\ref{sec:sixpersona})
in which a distance-ladder cosmologist, a standard-siren cosmologist, a
CMB theorist, a tension statistician, a dark-energy phenomenologist, a
philosopher of science, and an internal cohort auditor each put their
sharpest question --- and then, because a single exchange flatters the
respondent, return for a second round against the answers themselves
(\S\ref{sec:sixpersona-round2}) --- and then, because an answer is itself a claim and claims get audited, a five-reviewer panel puts the sharpest question that survives both rounds; its five, with our answers, close the Part as the third exchange (\S\ref{sec:third-exchange}). None is softened, and where the honest
answer is ``open,'' it says so.

\section{For the curious reader: the big questions, plainly answered}
\label{sec:obj-curious}

Before the specialist referees take the floor, the questions a
thoughtful non-specialist actually brings to a book about the
expanding universe --- about balloons and edges, mistakes and
\(\pi\), dark things and endings --- answered plainly, with pointers
to where each answer is earned. Popular science has made some of
these feel settled in directions this framework does not take, so we
meet them first, and in the open; the device follows the parent
paper's example \citep{ArisakaA10Cosmos}. Where an answer rests on a
still-open step, it says so.

\bigskip
\noindent\textbf{O-Hubble-55 (expanding into what?). ``Everyone says the universe is expanding. Expanding \emph{into} what?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Into nothing --- because expansion is not the universe's edge
advancing through a surrounding space; it is the distances
\emph{inside} the universe growing. The mathematics of Course~C says
it precisely: galaxies keep fixed coordinate labels forever, and the
ruler between labels, the scale factor \(a(t)\), grows. A useful
picture is the surface of an inflating balloon as seen by ants living
\emph{on} it: every ant sees every other recede, no ant is at the
center, and the question ``what is the rubber expanding into?'' never
arises for creatures whose whole world is the rubber. The balloon
needs a room only because we drew it embedded in one; the universe's
geometry is defined intrinsically, with no room required. What this
paper adds to the old picture is only this: the \emph{rate} of that
stretching, on our reading, was never a free fact --- it is a theorem
of the same geometry doing the stretching.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-56 (do we expand too?). ``If space itself expands, do atoms, the Earth, and we ourselves expand with it? And if rulers grow too, how could we ever notice?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No --- and the reason is a genuine insight, not a technicality.
Expansion is a property of the \emph{averaged, force-free} universe:
it is what test particles do when nothing binds them. Atoms are held
by electromagnetism, the Earth by chemistry and gravity, the Solar
System and the Galaxy by gravity: within every bound system the
binding forces long ago overwhelmed the gentle cosmic stretch and
settled the system at its own equilibrium size. Quantitatively the
mismatch is absurd: the expansion attempts a fractional stretch of
order \(10^{-18}\) per second across an atom, while the atom's own
dynamics restores its size \(10^{33}\) times faster. Only across
tens of megaparsecs --- between galaxies not bound to each other ---
does the stretch win. That is also why the measurement is honest:
our rulers and interferometers are bound systems and do \emph{not}
grow, so the recession of distant galaxies is measured against
rulers that hold still.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-57 (surely a mistake?). ``Two teams measure the same universe and get different answers. Isn't the boring explanation --- somebody made a mistake --- overwhelmingly likely? Why has nobody just found it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That was everyone's first hypothesis, and its failure is the real
story of the decade. Both chains have been audited at a level almost
without precedent in astronomy --- the local ladder by thirty-eight
specialists across thirty structured variants, every method swapped
against every other; the microwave side by three independent
instruments on separate continents and calibrations (Part~II). No
single mistake survived. What \emph{can} survive that kind of
cross-checking is the one thing cross-checking is mathematically
blind to: an error \emph{shared} by every variant --- a common
calibration layer --- which is a five-line theorem of Course~D, not
a conspiracy. So the honest menu narrowed to exactly two items: a
shared systematic in one chain, or new physics between the epochs.
This paper's contribution is to make the menu decidable: it derives
what the rate should be, prices the shared-layer candidates against
the ladder's own literature (Part~IV), and hands the verdict to an
instrument outside both chains (Part~VII).
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-58 (faster than light?). ``I have read that distant galaxies recede faster than light. Doesn't that violate relativity?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No. Special relativity's speed limit governs motion \emph{through}
space --- two objects passing each other can never clock more than
\(c\). The recession of a distant galaxy is not motion through space;
it is the growth of the space between, and general relativity places
no limit on that. Beyond the Hubble radius \(c/H_0 \approx
4{,}500\) megaparsecs on the derived rate, galaxies do recede
superluminally --- and nothing paradoxical follows: no signal, no
object, no information overtakes a light beam anywhere, locally,
ever. The distinction is the same one Course~C's metric makes
formally: coordinates label, the scale factor stretches, and the
speed limit lives in the local light-cones, which expansion tilts
but never violates.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-59 (why call it a constant?). ``If the expansion rate has been falling since the Big Bang, why does everyone call it the Hubble \emph{constant}?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Historical inertia, and the paper argues it is time to retire more
than the name. \(H\) is constant across \emph{space} at any moment
--- the same everywhere, which is what Hubble's straight line showed
--- but it falls steeply across \emph{time}: from unimaginable early
values, through \(66.6\) today, settling toward \(55.0\) forever
(Part~III). The name froze in the decades when only the present-day
value could be measured, so the one accessible point got mistaken for
the whole function. Part~IX makes the fuller argument: the
``constant'' is one reading of a derived curve, and the century of
measuring it ever more finely was bound to end in paradox once
precision outran the question --- as Course~L shows with one line of
algebra.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-60 (was Einstein wrong?). ``If this theory changes cosmology, was Einstein wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not by one comma. Every equation this paper integrates is
Einstein's: the Friedmann equations of Course~C are his field
equations applied to a symmetric universe, exactly as in 1922
\citep{Friedmann1922}, and the standard siren of Part~VII is his
prediction doing the refereeing. What changes is not the machinery
but \emph{where its inputs come from}. Standard cosmology feeds
Einstein's equations measured contents; this framework claims the
contents themselves --- how much matter, how much vacuum --- were
never free, but are fixed by a deeper geometry, and Einstein's
equations then run exactly as before. If anything, the reading is
maximally conservative about gravity: no modification, no screening,
no extra fields --- general relativity inherited whole, with the
freedom removed one level \emph{above} it.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-61 (why 95 percent dark?). ``Why is the universe 95\% dark? It seems absurd that we can see only 5\% of everything.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} On the standard account it is simply an awkward fact: two unseen
substances dominate, detected only through gravity. On the derived
inventory the proportions stop being awkward and become arithmetic:
the budget is \(\Omega_m = 1/\pi \approx 32\%\) matter,
\(1 - 1/\pi \approx 68\%\) vacuum, and within the matter only
\(1/2\pi^2 \approx 5\%\) is ordinary atoms --- three readings of one
geometric object (Part~III). The darkness itself then has a
structural reading, which the parent paper states plainly: light
--- the gauge ledger our instruments are built from --- illuminates
only the slice of the inventory that carries ordinary charge, so
``95\% dark'' is not a statement about strange substances hiding but
about what electromagnetic instruments were ever built to see
\citep{ArisakaA10Cosmos}. And the claim is not left cosy: the dark
matter is named (the Kaxion), massed (\(8.1\,\mu\)eV), and given a
laboratory appointment this decade \citep{ArisakaC1Kaxion}.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-62 (pi as numerology?). ``Isn't it suspicious that \(\pi\) shows up in the amount of matter in the universe? That smells like numerology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The suspicion is healthy, and the paper's whole discipline is built
to face it. Numerology is what you get when you \emph{search}: scan
enough elegant expressions against enough measured values and
matches are guaranteed. The framework's numbers run the other way
--- \emph{forced, not searched}: \(1/\pi\) is not one of many
candidates that happened to fit; it is the output of a specific
geometric construction (the Hopf-fiber volume ratio of
\S\ref{sec:absolute-h0}), computed before comparison, with no
alternative on the menu and cohort-wide naming rules that forbid
inventing a new formula per observable. The test that separates the
two is exposure: a numerologist's match retreats when the data
sharpen; this claim tightens its own noose --- the same \(\pi\)
must survive every future inventory determination
(F-Hubble registry), and the one remaining selected coefficient is
pre-registered against a computation that does not yet exist
(F-Hubble-11). Numerology never volunteers a way to die.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-63 (the 10-to-the-120 scandal). ``Dark energy is the biggest embarrassment in physics --- the vacuum energy is predicted wrong by 120 orders of magnitude. How can any theory claim to fix that?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} By deriving the small number instead of canceling the big one.
Course~F walks the embarrassment honestly: quantum field theory
prices the vacuum at the fourth power of whatever scale it trusts,
and trusting it to the Planck scale misses the measured value by
\(10^{120}\) --- the worst prediction in science
\citep{Weinberg1989}. The geometric seesaw replaces the missing
cancellation with a hierarchy: the vacuum scale is
\(\Mlock^2/M_{\mathrm{Pl,red}} \approx 6\)~meV, built from the
laboratory-pinned TeV lock scale, and the famous 120 orders become
\(8K\) e-folds of one derived descent --- \(e^{-8K}\) with \(K=35\)
is \(10^{-121.6}\), dressed by \(1.4\) orders of geometry
(\S\ref{sec:h0-closed-form}). Nothing cancels; nothing is tuned; and
the same arithmetic, run one step further, hands over the expansion
rate itself. The reader with a calculator can check the whole chain
in a minute --- which is not a sentence the \(10^{120}\) literature
has often been able to write.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-64 (why now?). ``Matter and dark energy happen to be comparable right now, in our era. Isn't that coincidence a little too convenient?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be, if ``now'' were a dart thrown at cosmic history at
random --- but observers are not darts. On the derived history the
window in which matter and vacuum are within a factor of a few of
each other spans roughly two percent of the universe's history on
the logarithmic clock that physics actually runs on --- narrow, but
not absurd. And observers arrive when their chemistry does: the same
inventory that times the vacuum's takeover times the stellar
generations that forge carbon and the main-sequence lifetimes that
incubate biospheres, so the two clocks that ``coincide'' were never
independent (\S\ref{sec:concl-why-someone}; the full account is the
parent's, \citealp{ArisakaA10Cosmos}). What remains after that
accounting is a factor-of-a-few timing question, not a fine-tuning
--- roughly the surprise of meeting a person whose age is within a
factor of two of your own at a dinner party for adults.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-65 (extra dimensions, really?). ``Six-dimensional geometry, a `bulk,' a `throat' --- extra dimensions sound like science fiction. Why should I believe any of this is real?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} You should not believe it; you should ask what it predicts, and
whether it can be killed. Extra dimensions have a respectable
century behind them (Kaluza and Klein unified gravity and
electromagnetism with one in the 1920s), and a disreputable recent
habit: frameworks with so much freedom that no observation could
ever embarrass them. This program's defense is not that its
geometry is plausible but that it is \emph{cornered}: the same
six-dimensional structure must simultaneously deliver the Standard
Model's twenty-eight parameters \citep{ArisakaA9SM}, the cosmic
inventory, this paper's expansion rate, a dark-matter frequency at
\(1.95\)~GHz, and a collider cluster at \(3.85\)~TeV --- with shared
integers, so it cannot fit one and dodge another. Whether the sixth
dimension is ``real'' in the picture-book sense is a question
physics answers operationally: if the twin-peak rings at
\(\Delta\nu/\nu = 1/35\), a microwave receiver will have read a
compact dimension's winding number. Until then, the honest word is
the one the codebook uses: conditional --- and falsifiable on a
printed calendar.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-66 (fine-tuned for life?). ``The universe looks incredibly fine-tuned for life. If that's not luck, and not a multiverse, what is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework's answer is the third road: the constants are not
tuned, because there was never a knob. A tuning problem presupposes
a dial that could have been set otherwise; if the inventory, the
coupling strengths, and the expansion history are theorems of one
geometry --- with the conditional steps graded and printed --- then
the life-permitting values are forced, and the appearance of
fine-tuning is what forcing looks like from the inside. This
dissolves the problem rather than winning the lottery: no luck
(nothing was drawn), no multiverse (nothing needed selecting from).
The position is falsifiable exactly where the derivations are: every
constant this program computes is a place the dissolution can fail
in public (Part~VI). The deeper question --- why there is a
geometry, and why one that writes readers --- is faced, not dodged,
in the capstone \citep{ArisakaA11Origin} and, for this paper's
corner of it, in \S\ref{sec:concl-why-someone}.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-67 (could it all collapse?). ``You keep flagging results as `conditional' or `on a named conjecture.' Doesn't that mean the whole edifice could collapse?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Parts of it could --- and the flags exist so that a failure takes
down exactly what depends on it and nothing more. The grading system
(\S\ref{app:grades}) is a load-bearing map, not a disclaimer: the
Friedmann machinery is theorem-grade; the \(\pi\)-locked fractions
are Derived-conditional on one named measure conjecture; the vacuum
coefficient is selected, with the closing computation pre-registered
as a falsifier; and the last \(7.3\%\) of the local excess is not
claimed at all. Compare the alternative culture, in which a
framework's dependencies are discovered only at its funeral. Here a
failed conjecture demotes specific rows of specific tables by rules
printed in advance --- and the paper has deliberately arranged
\emph{local} falsifiers in front of structural ones, so it can die
in its own right without taking the cohort's honesty down with it
(\S\ref{sec:disc-boundary}). An edifice that documents its own
demolition points is not fragile; it is auditable.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-68 (exact, yet error bars?). ``You call \(H_0 = 66.6 \pm 0.2\) `derived exactly.' If it's exact, why the error bar?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the derivation and the comparison live in different
currencies. The geometric objects are exact: \(1/\pi\) is not
\(0.3183\) but \(1/\pi\); the closed form of
\S\ref{sec:h0-closed-form} has no scatter in it. The \(\pm0.2\)
enters where exactness meets measurement: Route~II divides the
sky's measured densities (which carry errors) by the exact
fractions, and Route~I's laboratory pin (the electroweak vev) carries
\(\pm0.18\%\) through the lock scale. So read the number as: an
exact structure, priced against imperfect inputs --- the error bar
belongs to our instruments, not to the theorem --- with one honest
exception, stated in the triage: Route~I's \(\pm0.2\) is
\emph{structural} (it carries the conditional status of the selection
lemma), a theory-side allowance, not an instrumental one. This is the same
distinction as between \(\pi\) itself and any measurement of a
circle: the theorem is exact; the comparison never is. The falsifier
respects the distinction: what would kill the claim is not scatter
but a \emph{converged} value away from the structure ---
\(\Omega_m\) settling off \(1/\pi\), a siren settling at \(73\).
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-69 (why still telescopes?). ``If the expansion rate can be computed from five constants on a calculator, why do we still need billion-dollar telescopes?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} For the same reason a court needs witnesses even when the argument
is airtight: physics is not mathematics, and a derivation earns
belief only by surviving instruments. The telescopes do three things
no derivation can. They \emph{test}: every value the geometry prints
--- the rate, the fractions, the ages, the drift --- is an
appointment some instrument must keep or break (Part~VI). They
\emph{measure what the theory declares it cannot reach}: the depth
of our local neighborhood, the dust laws of the ladder, the
residual's anatomy --- the whole empirical layer of Part~IV. And
they \emph{discover}: nothing in this framework predicted
gravitational-wave astronomy into existence; instruments opened that
window, and the framework now stakes its life on it. If the reading
is right, telescopes stop being the source of cosmology's numbers
and become what they are in every mature science: the jury.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-70 (why do sirens decide?). ``Out of all the instruments, why do gravitational waves get the final word?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because they are the one channel with no shared ancestry with
either disputant. Every optical rung of the ladder shares
calibrations with every other; every early-universe inference shares
the sound horizon. A binary neutron star shares nothing: general
relativity itself certifies the source's absolute loudness, so the
waveform hands over the distance with no ladder, and the host galaxy
hands over the redshift with no plasma physics (Course~J walks every
step from the quadrupole formula up). In the language of Course~L,
the siren breaks the deadlock because a tension between two chains
can only be adjudicated by a witness correlated with neither. Both
camps accepted this referee in advance --- which is precisely why
the paper stakes its central claim there (F-Hubble-2), and why
Part~VII spends a whole Part on the referee's own error budget
rather than assuming its innocence.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-71 (what if it lands in the middle?). ``What if the sirens converge on 70 --- right between the camps?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Then the most interesting world wins, and this paper has already
priced its own discomfort in it. A converged \(70\) would kill the
pure \(73.5\) reading and strain the pure derived \(66.6\) at the
\(1.5\)--\(2.4\sigma\) level, while the lifted background
\(68.5 \pm 0.6\) --- what a ladder inside our under-dense
neighborhood should read --- sits comfortably within a sigma
(\S\ref{sec:siren-three-worlds}). The framework would be pressured
but not dead, the corrected-ladder cluster vindicated, and the open
question would sharpen into whether the sirens' nearby hosts share
our neighborhood's lift --- a question the host-distance shells can
answer directly. The honest summary, printed in advance: the middle
world is partial confirmation, reported as exactly that, and
uncomfortable for everyone --- this reading included.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-72 (how does it end?). ``On this theory, how does the universe end?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Quietly, and on a schedule already visible in the arithmetic. With
\(w = -1\) exactly, the vacuum neither dilutes nor decays: the
expansion rate settles onto the de~Sitter floor
\(H_\infty = 55.0\) and holds it forever, doubling the universe's
size every \(12.3\) billion years (Course~C). Galaxies not bound to
ours slide one by one across the horizon; star formation exhausts
its gas over trillions of years; the sky slowly empties. No rip
(that would need \(w < -1\)), no recollapse (no turnaround exists on
the derived curve), no cycle --- and the ending is not a speculation
but a graded consequence, guarded by a falsifier: any converged
measurement of \(w \neq -1\) retires it (F-Hubble-5). It is, as
endings go, a dignified one: the universe does not die of
catastrophe; it retires on a fixed pension of \(55\) kilometers per
second per megaparsec.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-73 (is this accepted?). ``Is GG-Theory accepted by the cosmology community? And why does this enormous claim come from essentially one author?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No, it is not accepted --- it is a proposal under test, and the
paper never pretends otherwise. Its response to the sociology is
structural rather than rhetorical: every derivation is shown at
graduate level (Part~X), every number regenerates from released
scripts an opponent can re-run, every falsifier is pre-registered
with a date, and the community's own instruments hold every verdict
--- there is nothing to take on trust and nowhere for authority,
anyone's, to matter. As for provenance: the history of physics
adjudicates claims by exposure, not by headcount or by origin, and
it keeps a long memory of consensus positions corrected by outsiders
--- and of outsiders who were simply wrong. Both precedents are
real; both are irrelevant once the tests run. The first appointments
are months away (\S\ref{sec:decision-calendar}); skepticism is not
merely permitted in the meantime, it is the intended reading mode,
and Part~X was written so that a skeptical graduate student can
check the claims personally rather than choose whom to believe.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-74 (what would convince you?). ``Suppose you are wrong. What would it actually take to convince you --- or is this, like so many grand theories, effectively unfalsifiable?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Name a date and a number, and this paper has already named eleven of
each. A gravitational-wave catalog converged at \(73\): the
derivation dies, publicly, by its own printed rule. A converged
inventory off \(1/\pi\); a detected early-dark-energy lobe; a
measured \(w \neq -1\); a tensor amplitude inconsistent with
\(0.0043\) in either direction; a silent Kaxion window; a single
proton decay: each retires a named layer, several retire everything
(\S\ref{sec:falsifiers}). These are not gestures at distant
posterity --- the calendar of \S\ref{sec:decision-calendar} runs
from this December to the mid-2030s, and the commitments ledger
collects, appointment by appointment, the numbers that cannot be
revised afterwards. The author's own answer is printed in Part~IX's
personal note: if the sirens side with the ladder, this reading will
have done the one useful thing a wrong theory can do --- die
cleanly, quickly, and in public, leaving the audit trail that shows
which link broke. Unfalsifiability is the one accusation this paper
has arranged to make impossible.
\end{objqabody}

\section{On the derived rate}
\label{sec:obj-derived-rate}

\noindent\textbf{O-Hubble-1 (circularity). ``Dividing a predicted fraction by a measured density to `derive'
\(H_0\) sounds circular. Isn't the CMB analysis that gives
\(\omega_b, \omega_m\) already assuming a cosmology?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two physical densities \(\omega_b = \Omega_b h^2\)
and \(\omega_m = \Omega_m h^2\) are measured from the \emph{shape} of the
acoustic peaks --- the baryon-loading ratio of the second-to-first peak,
the peak-height envelope, the equality redshift --- none of which
requires knowing \(h\). Indeed \(\omega_b\) is confirmed independently by
the BBN deuterium abundance, with no CMB at all. So the input is
model-independent in \(h\); the GG content is the \emph{prediction} of
the pure fractions \(\Omega_i\), and the division returns \(h\) with no
circularity. Standard \(\Lambda\)CDM does the reverse: it fits
\(\Omega_i\) and reads off \(h\). The non-triviality is that GG's
\emph{a priori} fractions, divided into the measured densities, give a
rate that also fits the entire spectrum. And since this objection was
first drafted, the framework has removed its premise: the rate is now
derived \emph{before} the densities are consulted (Route~I,
\S\ref{sec:absolute-h0}), so the division is a confirmation, not the
derivation (O-Hubble-49).
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-49 (the dimensional anchor). ``You still needed the
sky's densities --- or BBN's --- to get a number with units. A theory
that borrows any measured density has not derived \(H_0\).''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No measured density enters Route~I at all
(\S\ref{sec:seesaw-body}). The chain is: GU-5 conversion factors
(declared unit bookkeeping, \S\ref{sec:gu5-bridge}) \(\to\) the
theorem-grade lock scale \(\tfrac12 E_{\mathrm{Pl}} e^{-35}\)
(laboratory-pinned through the electroweak vev, not through any
telescope) \(\to\) the seesaw \(\to\) the codebook-selected
\(\pi/175\) \(\to\) \(66.8 \pm 0.2\). What a theory of geometry owes
is the pure number \(H_0 t_{\mathrm{Pl}} \approx 1.2\times10^{-61}\)
(Eq.~\eqref{eq:c2-dimensionless-task}), and Route~I delivers it. The
densities appear only in Route~II, the \emph{confirmation} --- and even
there the BBN variant (Eq.~\eqref{eq:c2-H0-bbn}) does without the
microwave sky. If ``derive'' means anything operational, it is this:
strike every astronomical measurement from the paper, and the rate
\(66.8 \pm 0.2\) survives, with the \(73.5\)-exclusion intact.
The whole route now stands as one printed line --- the closed form of
\S\ref{sec:h0-closed-form}, three laboratory constants and two
ledger integers --- which the reader can check on a calculator in
under a minute.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-50 (the convenient monomial). ``Your
\(\zeta_\Lambda = \pi/175\) was chosen because it matches; the
`cluster' just proves the codebook can manufacture whatever number you
need.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Three answers, in ascending strength. \emph{First},
the selection uses \ANine{}'s admissibility machinery --- stratum
availability and structural naming --- frozen years earlier for
twenty-eight Standard-Model parameters, and under it exactly one
cluster member survives; the rules reject the numerically adequate
orphan \(1/J^2\) even though it is \emph{cheaper}, precisely as they
rejected the cheaper \(1/\pi\) for \(\theta_{12}\). A machinery
that selects against cost is a strange tool for manufacturing
convenience. \emph{Second}, the cluster cannot manufacture ``whatever
number'': it spans \(66.8\)--\(71.2\) and cannot reach \(73.5\) at
any admissible cost --- had the true rate been the ladder's, the
codebook would have been \emph{unable} to accommodate it. \emph{Third},
the claim is falsifiable at the coefficient level: F-Hubble-11
pre-registers \(\pi/175\) as the required outcome of the outstanding
Hosotani computation, two-sided, printed before the computation
exists. What remains conditional is one named lemma --- the stratum
assignment --- carried at Derived (cond.), not theorem, and the
one interpretive step of the selection (the enumeration of the
stratum's priced objects from its three cohort precedents) is flagged
as such in \S\ref{sec:zeta-selection} rather than smoothed over.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-19 (the neutrino convention). ``You quote
\(\omega_m = 0.1430\) including the neutrino-mass contribution; the
baryon-plus-CDM sum is \(0.1424\). Your matter anchor is
convention-dependent at the half-percent level.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, declared in a footnote where the density is
introduced, and priced: the two conventions bracket the matter anchor
between \(66.9\) and \(67.0\), a spread absorbed in the adopted
\(66.6 \pm 0.2\). The baryon route (\(66.5\)) carries no such
convention, and the two-route consistency --- not either route alone ---
is what the derivation leans on.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-20 (the radiation share). ``Your `zero-parameter' curve
quietly uses the measured radiation density. That is a parameter.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a measured \emph{input}, declared as such in the
honesty triage and in the curve's own caption: the photon share follows
exactly from the CMB temperature \citep{Fixsen2009}, the neutrino share
from the standard
\(N_{\rm eff}\). The inventory claims of the framework cover the three
dark and matter shares; the radiation share is today a part in
\(10^{4}\) of the budget, and its only structural role is to fix
landmarks (equality, BBN) that the chart then passes rather than fits.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-21 (double dipping). ``You take \(\omega_b, \omega_m\)
from Planck, then claim the Planck spectrum `confirms' the result ---
the same data used twice.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two different functions of the same sky, and the
distinction is the content. The densities come from \emph{shape}
statistics (peak-height ratios, the envelope) that do not constrain
\(h\); the confirmation is the \emph{entire spectrum} --- positions,
ratios, damping --- reproduced at the derived rate with zero background
freedom (\(\chi^2 = 237\), Figure~\ref{fig:cmb-tt}). A wrong inventory
would pass the first step and fail the second: dividing correct densities
by wrong fractions gives a rate at which the peaks do \emph{not} fit.
And the sharpest check, the \(h\)-free loading ratio \(1/2\pi\), uses no
rate at all.
\end{objqabody}

\section{On the local ladder and the residual}

\noindent\textbf{O-Hubble-2 (the seven-percent residual). ``You attribute \(\approx\!7\%\) of the local excess to
`distance-ladder systematics.' Isn't that just a cop-out for the part you
can't explain?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is the opposite of a cop-out: it is the framework
declining to invent physics it has not derived. GG derives the sign and
rough size of the Local-Hole environmental lift (\(+2\)--\(3\%\)); it does
\emph{not} derive the remaining gap to \(73.5\), and it says so in the
abstract, the benchmark sheet, and here. The alternative --- inventing a
tuned early-universe modification to close the last seven percent ---
would be precisely the kind of unforced move the framework declines.
Naming an unexplained residual as a measurement-side systematic, rather
than dressing it as a derived result, is the honest bookkeeping the
program demands --- and the residual is no longer a nameless gap: its
candidate anatomy (dust, priors, subsample) is published, external,
contested, and kill-testable on dates (\S\ref{sec:medium-channels}). It is also falsifiable: the residual is pinned to the ladder,
and the siren test bypasses the ladder entirely.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-3 (siren at 73). ``What if the standard sirens land at 73, not 67?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Then this reading is wrong, cleanly and without
wriggle-room. The siren reads the background expansion rate using neither
a sound horizon nor a distance ladder; if it converges near \(73\) with
adequate statistics, the true background rate is high, GG's fractions
cannot be divided into the CMB densities to give it, and the derivation
of Section~\ref{sec:absolute-h0} fails. We have staked the framework on
\(\approx 67\) in writing (kill switch F-Hubble-2). This is the single sharpest
test in the paper, and it will be settled this decade.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-22 (the kinship charge). ``Part~V admits your residual
is a Class-1 claim. Strip the geometry and your Hubble `resolution' is
`the ladder is wrong' --- the oldest dodge in the field.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} For the last seven percent: yes, and the paper says so
in exactly those words. The difference between a dodge and a
decomposition is what surrounds the claim: a derived background the
microwave sky over-determines, a derived environmental lift with its own
budget, a residual priced channel-by-channel against the ladder's error
budget rather than gestured at, and a named, dated, ladder-free channel
that settles it. A dodge hides; this reading has printed where it dies.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-23 (seven --- or five, or nine --- percent?). ``Your
headline residual is \(\approx\!7\%\), but your own scenario ledger
spans \(+4.9\%\) to \(+9.5\%\). Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The residual is a function, not a number:
\(R = 73.50/[66.6\,(1+\epsilon_{\rm Hole})] - 1\)
(Eq.~\eqref{eq:residual-def}), and the quoted spread is the spread of
the \emph{lift} scenarios, dominated by the contested depth of the
Local Hole. At the adopted canonical lift the exact value is
\(R = +7.3\%\), and ``\(\approx\!7\%\)'' is the shorthand used
cohort-wide --- the parent \ATen{} prints the same value. Every
quantitative statement in Part~IV uses the exact \(R\); the scenario
ledger prints the extremes precisely so that no reader mistakes the
shorthand for a measurement.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-24 (the Kenworthy contradiction). ``You cite the
supernova-smoothness analyses to cap void explanations, then adopt a
\(+2.9\%\) void lift yourself. You cannot have both.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, more plainly than before: the objection is
right that the canonical row books its contrast \emph{in the flow
volume}, so the earlier ``calibrator zone'' escape was not available,
and it is withdrawn here, in print. The paper's posture is stated
without remainder: the \emph{response law} is derived; the \emph{depth}
is a conditional input on which the counts literature (deep) and the
smoothness-plus-velocity literature (shallow) disagree; the canonical
row is the counts branch, charged in print with its own
\(\approx\!11\sigma\) Gaussian price; the flows branch (\(\lesssim
1.7\%\), requirement up to \(+9.5\%\)) is carried at equal standing;
and the shell test (\(H_0\) versus \(z_{\rm min}\), F-Hubble-4)
measures the branch directly. You cannot have both --- so the paper
holds neither until the shell test reports.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-25 (special pleading). ``Your `one honest route' to
\(73.5\) needs a contested void \emph{and} seven aligned systematics.
That is not a scenario; that is special pleading.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be special pleading if the paper claimed it.
It claims the opposite: the route is printed as \emph{recorded, not
claimed}, its naive alignment probability (\(2^{-7} \approx 0.8\%\)) is
printed beside it, and the paper's own verdict is that the question
cannot be settled from inside the ladder at all. What the scenario
ledger buys is symmetry: the reader can see exactly how expensive each
resolution is, including this paper's --- and the siren charges the same
price to everyone.
\end{objqabody}

\section{On the geometry and the \texorpdfstring{\(\pi\)}{pi}-fractions}

\noindent\textbf{O-Hubble-4 (numerology). ``\(\Omega_m = 1/\pi\) is numerology. With enough integers and a
few transcendentals you can hit any \(\mathcal O(1)\) number.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The charge is answered at length in \ATen{}; in brief:
one closed-packet topology fixes \emph{three} coupled fractions
simultaneously (\(1/\pi\), \(1-1/\pi\), \(1/2\pi^2\)), not one number, and
the same integer ledger \((1,3,5;7,35)\) that produces them also fixes
\(\sin^2\theta_W = 25/108\), \(\hat\alpha^{-1}(M_Z) = 2^7\), the 28
Standard-Model parameters, and the Kaxion spectrum. A per-observable
alphabet search grants itself a fresh formula each time; GG reuses one
closed ledger across the whole cohort, and that coherent reuse is the
non-numerology. If any single fraction were tuned, the acoustic spectrum
would break; it does not.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-5 (D\(^{*}\) conditionality). ``The fractions are only \(\mathsf D^{*}\) --- conditional on an
unproved lemma (Conjecture~VIII). So \(H_0\) is not really derived.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and stated plainly: the rate inherits the
\(\mathsf D^{*}\) grade of the fractions. The single remaining
conditionality is, however, sharp and localized --- Conjecture~VIII closes as an \emph{epoch identity} (the
inventory is exact at the crossing epoch, \(75\) Myr before the present),
with the residual reduced to one uniformity lemma exact to one part in
\(10^{15}\) plus the declared choice of observing epoch. The honest
statement is therefore ``derived up to one named topological lemma,'' not
``derived'' unqualified --- the same grade the inventory carries in
\ATen{}, and a far stronger position than a fitted parameter.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-6 (the conditional seesaw). ``The seesaw coefficient \(\zeta_\Lambda\) is
Derived only \emph{conditionally} --- one lemma short of a theorem.
Doesn't that undercut the whole dark-energy route?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The condition is named, singular, and cashable. The
selection of \(\zeta_\Lambda = \pi/175\) rests on \ANine{}'s published
admissibility and naming rules plus one lemma --- the stratum assignment
(\S\ref{sec:zeta-selection}) --- and its closure, the Hosotani
Wilson-line computation (\AFive{}~O6), carries a pre-registered,
two-sided target: \(\pi/175\) to \(\sim\!1\%\), falsifier
F-Hubble-11. If the computation returns anything else, the geometric
route demotes itself from a point to the bracket \(66.8\)--\(71.2\) ---
which still excludes \(73.5\) --- and the precision rate falls back on
the density inversion, which does not use \(\zeta_\Lambda\) at all.
The reading degrades gracefully and publicly; nothing hides in the
conditional.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-26 (why the electroweak sphere?). ``Even granting the
Hopf arithmetic: why should the \emph{cosmic matter share} care about
the volume of the \emph{electroweak} group?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because in this framework the cosmological stratum and
the particle stratum are projections of one bulk, and the \OGNfive{}
integers \(N = 3\), \(L = 5\) fix the boundary holonomy as the
electroweak \(\mathrm{SU}(2) \cong S^3\) for \emph{both}. The same
identification is not spent once for the inventory: it is the
identification that pays out the twenty-eight Standard-Model parameters
in \ANine{}. One may doubt the identification --- that doubt is exactly
the \(\mathsf D^{*}\) conditional, named and tracked --- but one cannot
call it ad hoc: it is the single most heavily reused move in the cohort.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-27 (the drift below \(1/\pi\)). ``Some current
CMB+BAO combinations already pull \(\Omega_m\) toward \(0.30\)--\(0.31\)
--- below your exact \(1/\pi = 0.3183\). Isn't the chart already
strained?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fitted \(\Omega_m\) values are model- and
dataset-dependent, and their spread across combinations currently
brackets \(1/\pi\) from both sides; the chart's evidence is the
\emph{coherent one-sided pattern} of Figure~\ref{fig:cmb-offsets}, not
any single fit. The break condition is printed where the offset is
defined: a firm, converged determination of \(\Omega_m\) below
\(1/\pi\) --- not a preference in one likelihood combination --- breaks
every entry of the chart at once. If the drift consolidates, this paper
loses; that is what the sentence is for.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-28 (the dimensionful scale). ``\(\pi\) can fix ratios.
\(H_0\) has units. Where does the \emph{scale} come from, if not from
data through the back door?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Through the front door, twice, and labeled both
times. On the sharp route the scale enters as the two measured physical
densities --- the only observational inputs in the triage --- divided by
dimensionless derived fractions. On the seesaw route it enters as
\(M_{\mathrm{lock}}\) and the Planck mass, two scales fixed far from
cosmology. The geometry's claim was never that it conjures dimensions
from \(\pi\); it is that once one dimensionful anchor exists (\(G_N\),
as the derived-history caption states), every \emph{ratio} in the
expansion history is forced.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-48 (which \(\Omega\) is which?). ``You insert
\(\Omega_m = 1/\pi\) into \(H(t)\), but the fractions evolve --- at
recombination matter was three-quarters of the budget, not \(1/\pi\) of
it, and your own theorem says the lock holds only at a crossing epoch
\(75\)~Myr ago. So how can the same \(1/\pi\) appear in your CMB-based
\(H_0\), which concerns a sky released \(380{,}000\) years after the
Big Bang?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection conflates two different \(\Omega\)'s,
and the aside of \S\ref{sec:hoft-steps} is printed to keep them apart.
The symbols in the Friedmann form
\eqref{eq:friedmann-hierarchy} are present-epoch constants,
\(\Omega_i \equiv \rho_i(t_0)/\rho_c(t_0)\); the time dependence lives
in the dilution factors \(a^{-4}, a^{-3}, 1\), and the \emph{running}
fractions \(\Omega_i(t) = \rho_i(t)/\rho_c(t)\) evolve exactly as the
objection says (Figure~\ref{fig:omega-budget} draws them) --- the
equation itself puts the corridor-switch budget
at \(\Omega_m(z_*) \approx 0.76\), \(\Omega_r(z_*) \approx 0.24\),
\(\Omega_\Lambda(z_*) \approx 2\times10^{-9}\). The \(\pi\)-locked
values are a boundary condition at the declared present epoch ---
inserted once, propagated by the law --- and the theorem claims the
lock only there: the observing epoch is the \(\Omega_m(t) = 1/\pi\)
crossing, and Part~IV prices the why-now coincidence that declaration
carries. As for the CMB: its physics at \(z \approx 1100\) never
consults a present-epoch fraction --- baryon loading, equality, and
the sound horizon depend on the physical densities
\(\omega_b, \omega_m\), and the \(\Lambda\) term is one part in
\(10^9\) there. The fractions enter only in the present-day
bookkeeping identity \(\omega_i = \Omega_i h^2\), i.e.\ in reading
\emph{today's} rate out of densities the early sky fixed --- which is
what ``the CMB-based \(H_0\)'' means. The one fraction applied
directly to the early sky is the ratio \(\Omega_b/\Omega_m = 1/2\pi\),
and that is legitimate precisely because it is epoch-independent: both
sectors dilute as \(a^{-3}\), so the ratio at recombination equals the
ratio today.
\end{objqabody}

\section{On the comparison with other solutions}

\noindent\textbf{O-Hubble-7 (everyone has a solution). ``Early dark energy, extra radiation, modified gravity, local
voids --- everyone has a Hubble solution. Why is yours different?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Every entry in the standard list \emph{fits} the
background rate and then adds a mechanism to split it; GG \emph{derives}
the background rate from geometry fixed by non-cosmological data, and adds
no early-universe mechanism at all. It is the only entry whose CMB
endpoint is a derivation rather than a fit, and the only one that
predicted the siren outcome ex ante rather than accommodating it. It is
also the only one that would be refuted, not merely disfavored, by a
single clean observation (a siren at 73, or a proton decay).
\end{objqabody}

\section{On falsifiability and scope}

\noindent\textbf{O-Hubble-15 (unfalsifiable framework). ``A theory that derives everything can
explain any outcome. Isn't it unfalsifiable?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The opposite: because the numbers are fixed in advance, the reading
cannot absorb a surprise. It has printed, before the measurement exists, that a
converged standard siren at $73$ kills it (F-Hubble-2) and that a single proton-decay
event retires the whole program (F-Hubble-1). A framework that could explain any
outcome would name no such switch; this one names seven, each with a numerical rule.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-16 (the residual is doing the work). ``You derive $66.6$, then let a
named $6\%$ `systematic' carry you to $73.5$. Isn't the underived residual the whole
game?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The residual carries the paper nowhere it claims to reach. The claim
is the background rate and the sign of the local lift; the residual is explicitly
\emph{not} claimed --- it is labeled a ladder systematic, bounded where it can be
bounded (backreaction $<10^{-3}$, $\Uoneb$ excluded), and handed to the siren. A
theory that dressed the residual as derived would be the dishonest one; naming it is
the discipline, not the evasion.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-17 (why not just fit $H_0$). ``If the derived rate equals the CMB
value, what has the geometry added over simply reading $H_0$ off the CMB?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two things the CMB fit cannot give. First, the CMB \emph{measures}
$H_0$ within $\Lambda$CDM using six fitted parameters; the geometry \emph{derives} the
same number from $\Omega_m=1/\pi$ with none, so the agreement is a prediction met, not
a fit reproduced. Second, the derivation comes with a whole function $H(t)$ and a de
Sitter floor $H_\infty=55.0$, and it couples $H_0$ to falsifiers in particle physics
(proton decay, the Kaxion) that no CMB fit can reach.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-18 (scope creep). ``The paper leans on \ATen{}, \AFive{}, \AFour{},
\ANine{}, \PaperCOne{}. Hasn't the `Hubble' claim quietly become a claim about the
entire framework?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Deliberately, and it is stated as such. This report derives nothing
new; it reads the H0DN measurement through imported results, each carried at its home
grade (App.~\ref{app:inherit}). That is a feature: the Hubble reading is exposed to
falsifiers in unrelated arenas, so it cannot be rescued by adjusting a cosmological
knob. If the shared ledger is wrong, the Hubble reading falls with it --- which is
exactly the cross-exposure a single-arena fit lacks.
\end{objqabody}

\section{On the statistics of the tension}

\noindent\textbf{O-Hubble-29 (the universal solvent). ``Your `a common-mode term
collapses \(7.1\sigma\) to \(1.4\sigma\)' argument could dissolve
\emph{any} tension in any field. It is unfalsifiable acid.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The collapse formula is an identity, not an argument:
\emph{if} a common-mode term of size \(s\) exists, the significance is
what it is. The argument is elsewhere, and it is falsifiable twice
over: the candidate channels are enumerated with published allowances
(so the acid has a measured concentration), and the hypothesis is
handed to a channel immune to it. Acid that names its own neutralizer
is called a hypothesis.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-30 (pre-registration theater). ``H0DN pre-registered
its baseline before computing. You `pre-registered' fractions you
already knew matched the CMB.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Two different registrations, and the paper relies on
the honest one. The fractions were fixed by their use across the
non-cosmological ledger --- the same integers pay the Standard-Model
parameters --- not by this comparison; but the reader need not accept
that provenance to check the paper, because the operative
pre-registration is forward-looking: the siren rule, the drift numbers,
the \(w = -1\) bet, all printed before the data that will test them
exist. Retrodictions are graded as consistency; the wagers are graded
as wagers.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-31 (curated landmarks). ``Thirteen sub-percent
agreements --- but you chose which seventeen to display.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The display set is the standard acoustic and
background summary set --- the observables Planck's own parameter
tables headline --- plus the history landmarks; nothing in it is
bespoke. The two most precisely measured numbers in cosmology
(\(\theta_*\) and the loading ratio) are both in the set, which is
where curation would have hidden a miss. And the strongest reply is
structural: on a one-lever chart (\S\ref{sec:pi-offset}) a hidden miss
is impossible --- every observable rides the same displacement, so one
bad entry would drag the pattern visibly off its sign structure.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-32 (percent is not sigma). ``You quote agreements in
percent. In measurement sigmas, \(\theta_*\) is off by \(16\sigma\).
Any fitted model failing that badly would be dead.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and this is the most important statistical
sentence in the paper, so it is answered plainly. A \emph{fitted} model
is judged in measurement sigmas because its parameters exist to absorb
them; this chart has no parameters, claims percent-level derivation,
and is displaced from the data by one coherent, named systematic ---
the \(\pi\)-offset, whose candidate origin (the unfinished measure
normalization) is stated in the theorem's own honesty clause. The
correct test of such an object is not \(\theta_*\) in sigmas but the
\emph{coherence} of the displacement across seventeen observables ---
and, decisively, the printed break condition: if the offset is real
physics rather than the lemma's residue, a sub-percent inventory
determination will refuse to converge on \(1/\pi\), and the chart
dies. The paper does not grade itself on a curve; it names the exam.
\end{objqabody}

\section{On the siren and the forecast}

\noindent\textbf{O-Hubble-33 (leaning on noise). ``Current siren values
(\(\sim\!68\)--\(70\), errors of several units) `lean toward' you only
if one squints. Cherry-picking.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The paper's word is ``lean,'' used once and priced:
central values near \(68\)--\(70\) with \(\sigma \sim 5\)--\(8\) are
statistically compatible with both camps, which is why the decision
rule triggers only at converged \(\sigma \lesssim 1\). Nothing in the
argument is spent on the lean; the forecast Part exists precisely so
that the verdict arrives by threshold, not by squinting.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-34 (the referee's own bias). ``Your budget admits the
detectors' amplitude calibration is a \(1\)--\(2\%\) common mode. Your
\(5\sigma\) margin is \(2\%\). The referee can fake the
verdict.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A \(2\%\) calibration bias displaces the converged
value by \(\sim\!1.4\;\kmsMpc\) --- one threshold unit, not the
\(6.9\) that separates the camps. To fake a verdict the calibration
would have to be wrong by ten percent, five times its current bound
and two orders beyond its design goal; and the bright, dark, and
spectral pipelines carry different calibration exposure and must
agree. The budget names calibration as the floor to engineer down for
the \(5\sigma\) era --- that is a schedule item, not a loophole.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-35 (dark-siren circularity). ``Dark sirens need source
population priors; population inference needs cosmology. Your volume
channel is circular.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Jointly inferred, not assumed --- the standard
treatment fits population and cosmology simultaneously, and the prior
sensitivity is the channel's declared \(\sim\!1\%\) floor in the budget
table. The bright channel carries none of it, which is the point of
keeping three pipelines whose floors do not overlap: circularity in one
cannot survive agreement across all three.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-36 (the slow decade). ``Counterpart events may arrive
at one a year or slower. Your `decade' could be twenty years.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The rate uncertainty is honest --- a factor of a few
--- and the timetable is therefore stated in data eras, not dates. Two
things bound the pessimism: the dark-siren channel accumulates hundreds
of events per year regardless of kilonova luck and is forecast to reach
the two-percent class on its own; and the verdict is
threshold-triggered, so a slow decade delays the answer without
degrading it. The reading's exposure does not expire.
\end{objqabody}

\section{On the program and its manner}

\noindent\textbf{O-Hubble-37 (too many miracles). ``The cohort claims the
Standard Model, cosmology, and biology from five integers. Extraordinary
claims require more than internal consistency.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed --- they require exposure, which is what the
cohort's breadth buys. Every sector the ledger touches adds falsifiers
rather than immunity: a proton decay, a Kaxion line at the wrong ratio,
a collider null at the lock scale, a siren at \(73\) --- any one of
them retires the same five integers everywhere at once. And this paper
is deliberately checkable in isolation: its arithmetic runs on a
calculator, its scripts are released, and nothing in its kill tests
requires believing any sibling.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-38 (why engage with preprints?). ``Twenty-two
unrefereed preprints on a personal homepage. Why should a working
cosmologist spend an afternoon on this?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the afternoon is cheap and bounded: the
claims are numerical, the inputs are published, the scripts are
released, and the kill criteria are pre-declared --- the reader who
finds the broken step performs a public service at low cost, and the
reader who does not has learned that a zero-parameter chart survives
the standard checks. This Part exists to make adversarial reading
efficient; hostile review is not an obstacle the paper endures but the
mechanism it is built for.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-39 (style over substance). ``The paper is written
warm --- `one clock,' codas, a personal note. Rhetoric is doing work
that derivations should do.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The register is deliberate and the test is simple:
delete every warm sentence and check what remains. What remains is the
theorem and its conditional, seventeen tabulated confrontations, three
derivation chains, two priced budgets, eleven falsifiers with numerical
rules, and a forecast with released code --- the paper's full
load-bearing structure, none of it prose. The warmth is a reading aid
for a long report, and the codas summarize; they do not argue.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-40 (reverse engineering). ``All of this was built
knowing the answers. The only honest test of a `derivation' constructed
after the data is the data it cannot have known.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The paper agrees so completely that it has organized
itself around the principle: the retrodictions are graded as
consistency (and defended against per-observable formula freedom by the
one-ledger discipline), while the paper's weight rests on what no one
knows yet --- the siren verdict, the drift map, the exact \(w\), the
tensor ratio, the \(\Omega_m\) convergence. Every one is printed with
a rule. Reverse engineering cannot write post-dated cheques; this
paper's account is full of them, and they clear or bounce in public.
\end{objqabody}

\section{On the 2026 landscape and the residual anatomy}
\label{sec:obj-landscape}

\noindent\textbf{O-Hubble-51 (the undersized anatomy). ``Your three channels,
honestly combined, give \(4.3\%\) in quadrature --- but you need
\(7.3\%\). Your own arithmetic says the anatomy cannot carry the
residual. Why present it as if it could?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both combination rules are printed in the verdict
(\S\ref{sec:residual-verdict}): quadrature gives \(4.3\%/3.8\%\), the
linear stack \(9.3\%/8.3\%\) --- and \(+7.3\%\) sits between them,
which is exactly where a set of \emph{correlated} common-mode channels
would put it. Quadrature is the correct rule for independent errors,
and these channels are not independent: the prior recalibration
re-treats some of the same supernova terms the dust channel corrects
(the overlap caveat is printed beside the arithmetic), and a shared
calibration bias is correlated across methods by construction. The
anatomy's claim is calibrated accordingly --- not ``the channels close
the gap'' but ``the residual's size is bracketed by the published
corrections' own arithmetic'' --- and each channel carries a dated,
two-sided test (F-Hubble-8/9/10). If the channels are confirmed, the
landing point is \(\approx\!70.6\) per reanalysis
(\(69.5\)--\(70.2\) only if fully chained, which no joint analysis yet
licenses), within about one combined error bar of
the conditional lifted reading; if quadrature-of-independents were the whole
story, the residual hypothesis would be strained, and the verdict
section says so in those words. The scenario spread is the honesty,
not the weakness.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-52 (the consensus and its critics). ``You keep noting
that H0DN excludes the CCHP group --- insinuating that the consensus
is manufactured by its membership. That is an ugly charge dressed as a
footnote. The combination has published criteria, and CCHP's values
have published counter-analyses.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} No charge of manufacture is made, and the
counter-analyses are printed beside CCHP's values wherever they appear
--- the SH0ES selection reading of the same JWST data is quoted in the
same breath (\S\ref{sec:residual-anatomy}). The point is narrower and,
we believe, undeniable: a covariance-weighted combination certifies
what its members \emph{share}; it is silent on what its non-members
dispute. Both facts are therefore carried at full weight ---
\(73.50 \pm 0.81\) as the subject measurement of this paper, and
\(70.39 \pm 1.94\) as the standing demonstration that
calibrator-level choices worth \(\sim\!3\;\kmsMpc\) are live inside
the ladder community itself (\S\ref{sec:landscape-consensus}). And no
insinuation is required, because the registry names the experiment
that dissolves the boundary: the complete \(\sim\!37\)-host JWST
sample under a convention agreed in advance (F-Hubble-10). This paper
is committed to its outcome in either direction.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-53 (the geometric high probes). ``Megamasers at
\(73.9\), TRGB--SBF at \(73.8\), SNe~II at \(74.9\) --- one-step
geometric probes with no Cepheids and no shared ladder calibration
read high. Your common-mode story cannot touch them. Doesn't the
geometric class alone refute the derived rate?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The class is carried at full strength
(\S\ref{sec:landscape-counterweight}) as the strongest opposing
evidence in the landscape --- the paper says so in those words. Three
observations keep it short of decisive. \emph{Dispersion:} the same
one-step family contains the blinded TDCOSMO lensing value at
\(71.6^{+3.9}_{-3.3}\) and the \(\gamma\)-ray attenuation route at
\(61.9^{+2.9}_{-2.4}\), while its ladder-free cousins --- the fast
radio bursts --- read \(68.8\)--\(69.4\); a family whose members span
fifteen \(\kmsMpc\) is reporting its own unmodeled per-method
systematics, not a converged \(74\). \emph{Precision:} no member is
percent-class, and the high trio sits \(1.8\)--\(3.4\sigma\) from the
lifted background at its own quoted errors --- with those errors
themselves qualified in the record (the maser value's
peculiar-velocity treatment spans \(71.8\)--\(76.9\); the SNe~II
error is statistical only; the SBF route hangs on a single anchor
galaxy). \emph{The endpoint:} the instrument that settles the class's
own spread is the standard siren --- geometric in exactly the same
one-step sense, and currently reading \(69.9^{+4.1}_{-4.0}\). If the
geometric class converges high \emph{and} the sirens do too, the
derived rate dies (F-Hubble-2); if the class stays dispersed while
the sirens read \(67\)--\(68\), the dispersion was the systematics.
That is not a dodge; it is the discriminator doing its work.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-54 (laundering the contested corrections). ``You
disclaim the dust, prior, and subsample corrections as `external' and
`never \GGtheory{} results' --- and then use their central values to
walk the ladder to \(69.5\) and call it \(0.8\sigma\) from your lift.
You cannot both disown a number and spend it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The grade does exactly the work it declares.
Asserted-external means: quoted with the proposer's caveats and the
published rebuttals attached --- the priors channel is carried as
\emph{proposed, executed, and rebutted --- undecided}, with its low
edge at zero --- never averaged into a \GGtheory{} value, and never
used to revise \(66.6 \pm 0.2\) or \(68.5 \pm 0.6\), which have not
moved. The chained \(69.5\)--\(70.2\) is a conditional sentence and is
printed as one: \emph{if} the published corrections stand, the ladder
lands within \(0.8\sigma\) of the conditional lifted reading. The framework's
own numbers spend nothing from those channels; its exposure runs the
other way --- the registry commits in advance (F-Hubble-8/9/10) to
shrinking the anatomy, channel by channel, if the corrections fall.
Citing an allowance is not claiming a result, and the two-scenario
pricing plus the symmetric kill tests exist precisely to keep that
distinction visible on the page.
\end{objqabody}

\section{The seven-persona hostile review}
\label{sec:sixpersona}

The reading was put to a panel of seven hostile readers, each asked for the
single question they would most want answered before accepting it. The panel
read this paper against its parent \ATen{} and the dark-matter capstone
\PaperCOne{}. Their questions, and the answers, follow.

\bigskip
\noindent\textbf{O-Hubble-8 (distance-ladder cosmologist).} ``H0DN measured
\(73.50\pm0.81\) with thirty-eight experts and twenty-two stable variants. Your
\(66.6\) is roughly eight sigma below the best local measurement. Why should the
ladder yield to a geometric prediction?''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should not, and the reading does not ask it to. H0DN's
\(73.50\) is accepted as the local measurement; the claim is that the local
value is a biased reading of a \emph{background} rate \(66.6\) --- lifted a few
percent by the Local Hole and further by a ladder-calibration residual the
framework does not derive. The background, not the local value, is what the
geometry predicts, and it lands on the independent CMB determination. The
arbiter that is neither CMB nor ladder is the standard siren (F-Hubble-2).
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-9 (standard-siren cosmologist).} ``Present siren constraints
already scatter across \(67\)--\(73\) with large errors. When is `decisive,'
and what if the catalog converges at \(70\)?''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Decisive means a converged \(\sigma_{H_0}\lesssim1\), reached
plausibly by LVK O5 in the late 2020s (a few tens of well-localized bright
sirens, or a large dark-siren catalog). The pre-declared rule is sharp at the
extremes (\(67\pm1\) confirms, \(73\pm1\) refutes); an intermediate convergence
near \(70\) would be genuinely ambiguous and is exactly the outcome the paper
does \emph{not} claim to have pre-decided --- it would leave the reading neither
confirmed nor refuted, pending tighter data. Current central values sitting
nearer \(68\)--\(70\) than \(73\) lean toward the derived reading without
settling it.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-10 (CMB theorist).} ``\(\Lambda\)CDM already fits Planck at
\(67.4\). `Your fractions fit the peaks at \(67\)' is not new --- you have
relabeled the CMB value as `derived.'\,''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The novelty is provenance, not the fit. \(\Lambda\)CDM fits
\(\Omega_m,\Omega_b\) as free parameters; GG \emph{predicts} them a priori from
geometry and non-cosmological data, and they land on the peaks. The \(-1.2\%\)
\(\pi\)-offset is a specific falsifiable statement (the exact \(\pi\)-fractions
sit \(1\)--\(3\%\) above Planck's best-fit values); a refit would have hidden
it. That the CMB endpoint is a derivation rather than a displaced projection is
precisely the content.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-11 (tension statistician).} ``A seven-sigma tension is a
statement about two datasets. Relabeling \(67\) `derived' does not lower it ---
the local data still say \(73\).''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Agreed, and the paper does not claim to lower it by fiat. It
reframes the tension: \(67\) is no longer one of two discordant measurements but
a derived prediction, so the open question becomes whether the \emph{local}
measurement is biased (environment plus calibration), not which of two rival
rates is right. The statistical resolution comes only from the ladder-free
siren, on which the reading is explicitly staked.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-12 (dark-energy phenomenologist).} ``The 2024 DESI
\(w_0w_a\) hint favors evolving dark energy. Your \(w=-1\) exactly is already
mildly disfavored.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Acknowledged and carried as a named falsifier (F-Hubble-5). The
DESI hint is parametrization- and sample-dependent and not yet decisive; GG
predicts \(w=-1\) exactly from the throat-cap geometry --- a sharp two-sided bet.
Euclid and Rubin settle it this decade; a converged \(w\neq-1\) at \(3\sigma\)
refutes the throat-residue interpretation of \(\rho_\Lambda\), and the paper
says so in advance.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-13 (philosopher of science).} ``Deriving \(H_0\) from \(\pi\)
and integers --- is this explanation, or redescription dressed as derivation?
How is it not merely saving the phenomena?''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The mark that separates the two is unforced multiplicity under
falsification. One closed ledger \((1,3,5;7,35)\) fixes three coupled cosmic
fractions, the twenty-eight Standard-Model parameters, and the Kaxion spectrum
at once, and the paper commits to observations --- a siren at \(73\), a single
proton decay --- that would refute it. A redescription cannot be refuted by one
measurement; this can. Whether that rises to ``explanation'' the paper leaves to
the reader, but it is not merely saving the phenomena.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-14 (internal cohort auditor).} ``\ATen{} anchors on the 2022
Riess \(73.0\); this paper uses H0DN \(73.50\). Is the cohort internally
inconsistent?''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are the same local measurement at two vintages. \ATen{}
--- the parent, and the source of record for the derived cosmology --- cites
both; this paper, the source of record for the H0DN reading, adopts the newer
\(73.50\) consensus. The derived background \(66.6\), the early/late split, the
\(68.5\) reconciliation, and the siren prediction are identical in both papers;
only the vintage of the local anchor differs, and it is documented in each.
\end{objqabody}

\section{The second exchange}
\label{sec:sixpersona-round2}

A single round flatters the respondent: every panellist gets the last
word taken away. So the panel was invited back, this time against the
answers --- and against the new machinery of Parts~IV and~VII.

\bigskip
\noindent\textbf{O-Hubble-41 (the distance-ladder cosmologist, again).} ``Your
updated channel table still quadratures to only \(4.3\%\) --- short of
your \(7.3\%\) requirement --- and the extra mileage comes from other
people's contested papers. You have strengthened \emph{my} case, not
yours.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Partly conceded, and printed: on strict independence
the requirement remains bracketed by the channels' own published
arithmetic --- a consistency, not a statistic --- and not a
demonstration, and both contested channels are quoted with their
opposing evidence --- one of them already rebutted by authors of your
own team. But three things have genuinely moved. The requirement no
longer exceeds even the aligned stack, so no conspiracy is required;
the medium channels' published central values close the corridor by
themselves, so the small channels need not align at all; and every
channel now carries a dated kill test that your instruments, not our
theory, will execute. If the siren sides with your ladder, this paper has
already written its own obituary; the table you cite will be quoted in
it.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-42 (the standard-siren cosmologist, again).} ``Your
forecast ignores electromagnetic selection: we follow up the nearest,
brightest, most face-on events. Your per-event errors are
optimistic.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Acknowledged, and partially priced: viewing-angle
selection is folded with Malmquist in the budget row, and the
conservative scenario (\(20\%\) per event, \(N_{3\sigma} = 37\))
exists for exactly this reason. The forecast is deliberately
elementary and released; a waveform-level treatment with selection
functions would sharpen the timetable without moving the destination,
and the panel is warmly invited to run one.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-43 (the CMB theorist, again).} ``A \(\Delta\chi^2 =
-22\) improvement from \emph{removing} parameter freedom is unusual
enough to warrant suspicion. Show the binning, the covariance, the
protocol.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The protocol is the parent's and is published with
it: eighty-three binned TT points, diagonal covariance, background held
at the derived chart, only the tilt swapped --- a deliberately blunt
instrument, quoted as such. The suspicion is fair and the remedy is
named as the first observational priority: a full-likelihood run with
the background fixed at the \(\pi\)-locked inventory, where blunt
instruments are retired. Until then the claim is carried at the
strength the protocol supports and no more.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-44 (the tension statistician, again).} ``Your collapse
table treats the common mode as one Gaussian knob. Real calibration
systematics have structure, correlations, and signs.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Granted in full. The collapse table is a
scale-of-sensitivity illustration --- what \emph{one} coherent term of
a given size would do --- and the paper's inferential machinery never
uses it: no likelihood is built on it, no verdict read from it. The
structured version of the question is the channel table, and the
decision rule lives entirely on the siren channel, where the knob does
not exist.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-45 (the dark-energy phenomenologist, again).} ``The
evolving-\(w\) hints are strengthening. If \(w(z)\) is confirmed, your
exact \(-1\) dies --- and your whole derived \(H(t)\), and the age, and
the floor, die with it.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct on every count, and the paper would have it
no other way: \(w = -1\) is carried as a two-sided bet (F-Hubble-5)
precisely because the derived history cannot follow a drifting
equation of state the way a \(w_0 w_a\) fit can. If the hints
consolidate at \(3\sigma\), the throat-residue reading of
\(\rho_\Lambda\) is finished and the de~Sitter terminus with it. The
exposure is total, is printed, and is the difference between this
reading and a parametrization.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-46 (the philosopher of science, again).} ``You borrow
the language of pre-registration from clinical science. Clinical
science has registries with timestamps and third parties. Where is
yours?''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The honest answer: the cohort's registry is
self-maintained --- dated preprints, a cross-paper falsifier ledger,
released scripts --- and therefore weaker than a clinical registry,
though stronger than the field's default of nothing. Its best feature
is structural: the same integers are spent across seventeen
interlocking papers, so a retroactive edit in one breaks checkable
arithmetic in the others. Third-party timestamping of the falsifier
ledger is a fair demand, and a good idea this exchange has now put on
the record.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-47 (the internal cohort auditor, again).} ``The
residual shorthand and its exact value must agree across the cohort ---
this paper and \ATen{} must print the same number, or the books are
stale.''\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They agree: both papers print the shorthand
``\(\approx\!7\%\)'' with the exact canonical requirement
\(R = +7.3\%\) stated wherever it is used --- Eq.~\eqref{eq:residual-def}
and the scenario ledger here, the two-values ledger in the parent ---
and no computation anywhere in the cohort consumes the shorthand. The audit stands, and it stands on the printed text: the shorthand and the exact requirement appear together at every site, which is what makes a discrepancy of this kind checkable by a reader rather than a matter of trust.
\end{objqabody}

\bigskip
The panel's second round changes no verdicts, which is itself the
finding: the answers hold where they claimed to hold, concede where the
first round conceded, and the entire disagreement has been compressed
onto measurements with dates. That is what this Part is for.

\subsection{The third exchange: the audit of the amendments}
\label{sec:third-exchange}

Two rounds are a dialogue; an amendment is a claim, and claims get
audited. Two rounds are a dialogue; an answer is a claim, and claims get audited. Taking the concessions of the first two exchanges as given --- the combination bookkeeping opened, the lift conditional, the siren rule unified, the exclusion owned at full strength --- a five-reviewer panel (formal theory, \(\Lambda\)CDM cosmology, phenomenology, observation, and a graduate reader) is asked one question each: \emph{what still stands?} Their five
sharpest, with our answers, entered the registry as
\textsf{O-Hubble-75}--\textsf{79}.

\bigskip
\noindent\textbf{O-Hubble-75 (law or jurisprudence). ``Your
single-occupancy and no-empirical-filter clauses were codified only
after the audit produced \(1/(2L^2)\) and \(1/56\) --- the codebook
has, historically, grown a clause each time a cheaper candidate
appeared. Freeze the grammar, or admit it is jurisprudence.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded and done: the freeze is printed in
\S\ref{sec:zeta-selection} --- the rulebook as codified is frozen,
any future cohort selection must use it unchanged, and ``a clause
added after the next inconvenient candidate would itself be a
falsification of the method.'' The ledger-indexed-tower clause
(\(2^J\) is one named depth-\(J\) mechanism; an unnamed exponent is
not) is part of the frozen text, and under it every standing
\ANine{} selection stands while \(2^{-3}/J\) falls. What remains
owed, and is registered as a cohort milestone: a standalone grammar
registry, and a re-run of all twenty-eight \ANine{} selections under
the frozen rulebook, published casualties included.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-76 (the one-lever stake). ``You say the
seventeen-entry pattern rides one lever. Profile the full Planck
likelihood along that lever with everything else locked to the
chart. My prediction: the \(-8\) pull in \(\Omega_m h^3\) and the
\(\theta_*\) residue are transverse, and you will strand a
\(\Delta\chi^2\) of order \(50\)--\(100\).''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted as \emph{the} decisive computation, and the objection's prediction is printed here beside the commitment so the outcome grades both parties. OQ-Hubble-4 is sharpened
accordingly: the profiled \(\chi^2\) along the degeneracy lever,
full covariance, minimum and lever amplitude published with the
script. Our own printed transverse pull forbids us from predicting a
fall to the fitted \(84\); what the derived chart cannot survive is
a profile that rejects the lever direction itself --- and if the
stranded residual is as large as predicted, the one-displacement
reading dies at its own registered test. Both stakes are now on the
same page.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-77 (a pointer is not a rule). ``You moved
the lifted \(68.5\) onto the shallow bright subset --- good. Now
write that rule with the same teeth as F-Hubble-2: thresholds,
precision requirement, and the undecidable gap named in
advance.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Written, in Part~VII, where the forecast lives: the
lift is refuted at \((\mu_{\rm sh}-68.5)/\sigma_{\rm sh} \ge 3\)
(about forty shallow events); the counts-versus-flows discrimination
needs \(\sigma_{\rm sh} \approx 0.27\) --- of order fifteen hundred
events, undecidable by sirens this decade, and the rule says so
rather than let the gap be found later; the branch is therefore
assigned to F-Hubble-4's ladder shells, whose step sizes
(\(1.8\)--\(1.9\) counts, \(0.3\)--\(0.7\) flows, zero at
\(\pm0.3\) kills both) are fixed by the scenario ledger's own
numbers.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-78 (the operation behind ``conditional'').
``You price your adopted contrast at \(11\sigma\), disfavor the
deep void at \(20\sigma\) for the same crime, and tell the
Kenworthy objection you hold neither branch --- yet \(68.5 \pm 0.6\)
is the only number your landscape arithmetic ever uses. What
operation distinguishes `adopted conditionally' from
`adopted'?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Three operations, all in print: the scenario row is
labeled \emph{counts branch; carried conditionally}, and every use
of \(68.5\) carries the conditional label; the flows-branch parallel
is run at the verdict site (the corrected ladder leaves
\(\approx\!2.9\;\kmsMpc\) unexplained against \(67.7\)); and the
discriminating measurement is pre-registered with numbers (the
F-Hubble-4 shell step above). What honesty requires us to concede is
the asymmetry that remains: the headline arithmetic runs on the
counts branch, labeled at every use --- conditional adoption means
the label travels with the number and a dated test can revoke it,
and that is the whole of what it means.
\end{objqabody}

\bigskip
\noindent\textbf{O-Hubble-79 (which sentence is canonical?). ``Your
appendix says a formal weighting `would itself be a modeling
choice' and calls \(66.6\) an adopted center --- but if the baryon
routes are the model-cleanest, as you argue, the clean-route value
is \(66.4\)--\(66.5\), not \(66.6\). Which sentence is
canonical?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The adopted-center sentence is canonical, and it
now appears wherever the boxed value does --- Part~III included, so
no reader can re-derive the old objection from the main text alone.
The clean-route fact is printed with it: the two baryon routes read
\(66.4\)--\(66.5\), the matter route \(67.0\), and \(66.6 \pm 0.2\)
is the center of that structural span, frozen across the cohort,
with every conclusion surviving any choice in the corridor. The
residual weighting question is not adjudicated by prose at all: it
dissolves at OQ-Hubble-4, where a single profiled likelihood
replaces every argument about which route deserves the pen.
\end{objqabody}

\bigskip
The third exchange changes no number and retires no test; what it records is that the paper's sharpest remaining exposures carry a stated counter-prediction printed beside the paper's own, so that the data grade both. And there will be data. That is what this Part is for.




\GGPart[References]{References}{The curated bibliography, split into the twenty-two-paper \GGTheory{} cohort (Part~A) and all other works (Part~B), each entry carrying a Relevance note and its first in-text citation}


Unlike a conventional reference list, every entry below carries a
\emph{Relevance} note: a sentence or several saying what the cited work
contributes and where in the present paper it bears, closing with the
section of first citation. The aim is to turn the bibliography from a
bare list of sources into a navigable companion --- a reader can move
through the references and see, at a glance, which derivation, table,
audit, or landscape row each one underwrites.

The notes are written for two readers at once. For the newcomer, they
trace the intellectual lineage of the Hubble problem: from Leavitt's
periods and Hubble's first straight line, through the factor-of-two
wars and the precision era, to the \(7\sigma\) split, the 2026
landscape of forty-five determinations, and the gravitational-wave
referee --- and then to the twenty-two-paper \GGTheory{} cohort, in
which the present paper reads the contested constant as one point on
a derived expansion history, one story above the Standard Model of
\ANine{}. For the specialist, they make the paper's debts explicit
and checkable: each observational entry names the measurement it
supplies (with its arXiv identifier where the landscape table quotes
one), so that no citation is merely decorative and every row of
Table~\ref{tab:h0-landscape} can be traced to its source.

The cohort papers are cited throughout under the canonical keys and
titles fixed by the program's cross-paper coherence map; this
paper's place in the cohort --- the Hubble member of the Consequence
branch, the dedicated companion of \ATen{} --- is set out in Part~I's
cohort section. Within each of the two groups the references are
listed alphabetically by first author.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{K.~Arisaka.}
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems},
Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.} Constitutional source of the axioms, postulates, admissibility rules, and OGGEP semantics from which every structural input used in this report descends; the inheritance statement of Part~II points here. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{K.~Arisaka.}
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})},
Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.} Source of the six-dimensional stage \(M_4 \times I_r \times S^1_\theta\), \BIsix{}, \OGNfive{}, and the compact-angle holonomy whose cosmological projections the corridor reuses. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{K.~Arisaka.}
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})},
Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.} Proves the \BIsix{} legality theorem behind Eq.~\eqref{eq:BI6}; the \(2/5\) coefficients of the Arisaka Ratio invoked for the \(\Uoneb\) graviphoton coupling are fixed here. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{K.~Arisaka.}
\emph{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics},
Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.} Promotes the \OGNfive{} integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\) of Eq.~\eqref{eq:OGN5} to theorem status and states the zero-free-parameters discipline against which the corridor's honest provenance labels are drawn. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
\textbf{K.~Arisaka.}
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT},
Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.} Exact boundary reduction from which four-dimensional observables and open-system dynamics are inherited; the operational-dictionary reading of the Hubble tension used here is a \GDOS{} statement. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{K.~Arisaka.}
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind},
Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.} Source of the four Gi pillars \TGG{}, \CCI{}, \GFF{}, \GRIP{} invoked in the corridor theorem's proof sketch to order the early acoustic shift before the late local lift. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{K.~Arisaka.}
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}},
Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.} Supplies the Qi runtime and the Arisaka Equation current-locking \(J_E/Q_B = J_e = J_B = J_{Qi}\) cited in the cross-ledger coherence table as the sibling structure of the \(\UoneQi\) radiation channel. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{K.~Arisaka.}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.} The consequence-trio member that recovers textbook quantum mechanics as theorems of the boundary open-system channel. Cited here for the cohort map: it is the measurement-theory sibling of the same projection that makes the two Hubble values \emph{registered observables} of one corridor rather than a contradiction. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{K.~Arisaka.}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.} Fixes the 28-parameter Standard-Model ledger, \(\sin^2\theta_W = 25/108\), \(\hat\alpha^{-1}(M_Z) = 2^J\), and the \(\Uoneb \times \UoneQi\) structure; the cross-ledger coherence argument of Part~II tests whether the same ledger reaches the Hubble corridor. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{K.~Arisaka.}
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A10-Cosmos preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.} The companion cosmology recast, and the parent of this paper: the source of record for the derived cosmology --- the two-route background rate \(66.6\), the \(\pi\)-locked inventory, the early/late split, the \(68.5\) reconciliation, the standard-siren prediction, the \(S_8\) reading, and the declared \(\epslocal\) open problem to which this paper is the dedicated Hubble companion. First cited in \S\ref{sec:intro-gg-preview}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{K.~Arisaka.}
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program},
Arisaka-GG-A11-Origin.
\par\noindent 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 synthesis of the cohort; cited for the cross-domain reach (DNA alphabet, conscious-episode duration) invoked in the cross-ledger coherence table, and as the unified summary referenced in the comparison sections. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{K.~Arisaka.}
\emph{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}},
Arisaka-GG-P1-KK preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The Predictions-Branch sibling carrying the TeV lock cluster at \(\Mlock\approx3.85\)~TeV --- the very scale whose fourth power sets the vacuum-energy seesaw coefficient \(\zeta_\Lambda\) of the Hubble corridor read here. A collider result at the lock scale is collateral evidence for the cosmological reading. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{K.~Arisaka.}
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability},
Arisaka-GG-P2-N3 preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p2-n3/}.
\par\noindent\textbf{Relevance.} The Predictions-Branch sibling that spends the same \OGNfive{} integer ledger \((1,3,5;7,35)\) on the generation count; cited in the cohort map as an independent falsifier of the same integers that fix the \(\pi\)-locked inventory here. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{K.~Arisaka.}
\emph{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P3-Electron preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The Predictions-Branch sibling: the electron as the lightest charged permanent resident of the same boundary whose cosmological inventory this paper reads. Cited for the branch context of the cohort map. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{K.~Arisaka.}
\emph{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}},
Arisaka-GG-P4-Real preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.} The origin-of-\(i\) member of the Foundations Branch: the two real sources of the boundary's complex structure --- the compact phase circle and the causal clock --- from which the single imaginary unit of boundary physics is derived. Cited at roster level in the cohort map; the corridor chronology read here presupposes that complex structure and imports nothing further from it. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{K.~Arisaka.}
\emph{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter},
Arisaka-GG-P5-CPT preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.} The mirrors-and-matter member of the Foundations Branch and the \ANine{}--\ATen{} interface of the cohort map: C, P, and T as orientation reversals of the three real bulk factors, with the matter--antimatter asymmetry at the end of the chain. Its four-endowment definition of time is kin, at roster level, to the corridor chronology of the derived \(H(t)\) read here. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{K.~Arisaka.}
\emph{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}},
Arisaka-GG-P6-Proton preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.} The home of the switch this paper hands its whole program to. F-Hubble-1 says that one unambiguous proton-decay event retires \GGtheory{} in its entirety, and the reason a cosmology paper can afford to print a rule that severe is that the proton's absolute stability is a consequence of the exact local \(\Uoneb\) rather than an assumption bolted on for the occasion; the paper that works that consequence out, question by question, is this one. It supplies no number to the Hubble chain --- the gauge factor's home is \ATwo{} and \ANine{} --- and it is cited here for the exposure it creates rather than for anything it contributes. First cited in \S\ref{sec:falsifiers}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{K.~Arisaka.}
\emph{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out},
Arisaka-GG-W1-Symmetry preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.} The White-Paper Branch survey of how conservation moved from symmetry to cohomology, and the background to a habit this paper uses without arguing for it. The vacuum residue that sets the derived rate is small because a Wilson-line phase admits no local counterterm, not because a symmetry forbids it, and a reader who finds that substitution unfamiliar will find the century of argument behind it there rather than here. It supplies no number and no theorem the Hubble chain uses. First cited in \S\ref{sec:disc-essence}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{K.~Arisaka.}
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology},
Arisaka-GG-W2-Holonomy preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.} The classification of protection mechanisms that this paper's seesaw coefficient occupies one rung of. The compact phase circle whose holonomy fixes \(\zeta_\Lambda\), and whose non-locality is the reason the residue survives at the meV scale rather than being renormalized away, is one entry in that classification; a reader who wants to know why that rung and not another should read the classification rather than this paper's use of it. Cited at roster level in the cohort map, and named here because the phrase the map's caption uses of it --- protection by topology rather than by symmetry --- is the mechanism behind the number in \S\ref{sec:h0-closed-form}. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{K.~Arisaka.}
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}},
Arisaka-GG-C1-Kaxion preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.} The dedicated dark-matter companion; source of the Kaxion twin-peak \(\Delta\nu/\nu = 1/35\) and the \(1.59\)--\(2.13\) GHz laboratory window behind kill switch F-Hubble-7. First cited in \S\ref{sec:gg-cohort}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{K.~Arisaka.}
\emph{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity},
Arisaka-GG-C3-BH preprint.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.} The Cosmology-Branch sibling at the strong-gravity end, recovering the exact \(A/4\) horizon-entropy coefficient from the same \GDOS{} boundary reading; cited in the cohort map as the horizon-thermodynamics member of this paper's own branch. First cited in \S\ref{sec:gg-cohort}.

\end{thebibliography}

\section*{B.~Other references}
\begin{thebibliography}{99}

\bibitem[Abbott et~al.(2017)]{Abbott2017Siren}
\textbf{Abbott, B.~P., Abbott, R., Abbott, T.~D., et al.\ (LIGO/Virgo and partner collaborations) (2017).} A gravitational-wave standard siren measurement of the Hubble constant. \emph{Nature, 551}, 85. arXiv:1710.05835.
\par\noindent\textbf{Relevance.} The first standard-siren \(H_0\): GW170817 with its kilonova counterpart, \(70^{+12}_{-8}\) --- the worked example of the siren Part, and the per-event precision its central forecast scenario adopts. First cited in \S\ref{sec:siren-principle}.

\bibitem[Abdul Karim et~al.(2025)]{AbdulKarim2025}
\textbf{Abdul Karim, M., Aguilar, S., Ahlen, S., et al.\ (DESI Collaboration) (2025).} DESI DR2 results II: Measurements of baryon acoustic oscillations and cosmological constraints. \emph{Physical Review D, 112}, 083515. arXiv:2503.14738.
\par\noindent\textbf{Relevance.} Source of the BBN+BAO comparison value \(68.51 \pm 0.58\) km\,s\(^{-1}\)\,Mpc\(^{-1}\) quoted by H0DN; the second, CMB-independent early-universe channel against which the \(5.0\sigma\) tension is stated. First cited in \S\ref{sec:early-network}.

\bibitem[Alfradique et~al.(2026)]{Alfradique2026Sirens}
\textbf{Alfradique, V., Bom, C.~R., et al. (2026).} An improved measurement of the Hubble constant with LIGO/Virgo/KAGRA O4a dark standard sirens. Preprint (arXiv:2603.20195).
\par\noindent\textbf{Relevance.} The sharpest siren-family value to date: 17 dark sirens with DESI Legacy photometric redshifts plus GW170817 give \(69.9^{+4.1}_{-4.0}\) --- band 2, ladder- and ruler-free; catalog-completeness systematics dominate. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Alpher and Herman(1948)]{AlpherHerman1948}
\textbf{Alpher, R.~A., \& Herman, R. (1948).} Evolution of the Universe. \emph{Nature, 162}, 774.
\par\noindent\textbf{Relevance.} The prediction that the hot beginning should survive as a few-kelvin relic glow --- the forecast the 1965 discovery vindicated; cited in the century narrative of Part~I. First cited in \S\ref{sec:intro-history}.

\bibitem[Asgari et al.(2021)]{Asgari2021}
\textbf{Asgari, M., Lin, C.-A., Joachimi, B., et al. (2021).} KiDS-1000 cosmology: Cosmic shear constraints and comparison between two point statistics. \emph{Astronomy \& Astrophysics, 645}, A104.
\par\noindent\textbf{Relevance.} The benchmark low-redshift cosmic-shear determination \(S_8 = 0.759^{+0.024}_{-0.021}\), coincident with the derived value \(S_8^{\mathrm{GG}} = 0.759\) of the \(S_8\) section and its appendix course. First cited in \S\ref{sec:s8-implications}.

\bibitem[Aubourg et~al.(2015)]{Aubourg2015}
\textbf{Aubourg, \'E., Bailey, S., Bautista, J.~E., et al. (2015).} Cosmological implications of baryon acoustic oscillation measurements. \emph{Physical Review D, 92}, 123516.
\par\noindent\textbf{Relevance.} The reference inverse-distance-ladder analysis: the early-route logic run downward, landing with the CMB --- the mirror image of the ladder described in the routes-anatomy section. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Baade(1956)]{Baade1956}
\textbf{Baade, W. (1956).} The period-luminosity relation of the Cepheids. \emph{Publications of the Astronomical Society of the Pacific, 68}, 5.
\par\noindent\textbf{Relevance.} The first great halving: the two Cepheid populations separated, and the distance scale doubled at a stroke --- the century chart's first correction. First cited in \S\ref{sec:intro-tension}.

\bibitem[Bernal et~al.(2016)]{BernalVerdeRiess2016}
\textbf{Bernal, J.~L., Verde, L., \& Riess, A.~G. (2016).} The trouble with $H_0$. \emph{Journal of Cosmology and Astroparticle Physics, 2016}(10), 019.
\par\noindent\textbf{Relevance.} The canonical statement of the sound-horizon lever arm --- how $r_s$ and the inferred rate trade against each other --- recapped at the recombination switch. First cited in \S\ref{sec:recomb-switch}.

\bibitem[Bhardwaj et~al.(2025)]{Bhardwaj2025Miras}
\textbf{Bhardwaj, A., et al. (2025).} A globular-cluster Mira anchor for the distance ladder. \emph{Astrophysical Journal} (arXiv:2507.10658).
\par\noindent\textbf{Relevance.} The cluster-Mira rung: \(73.06 \pm 2.67\) from 41 globular-cluster Miras with no metallicity dependence over 1.6 dex --- an independent stellar-variable rung siding with the high ladder. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Birrer et~al.(2025)]{TDCOSMO2025}
\textbf{Birrer, S., et al.\ (TDCOSMO Collaboration) (2025).} TDCOSMO XVII. New time-delay cosmography results using lens model constraints from JWST and ALMA. \emph{Astronomy \& Astrophysics, 704}, A63. arXiv:2506.03023.
\par\noindent\textbf{Relevance.} Independent strong-lensing time-delay determination of \(H_0\), consistent with the late branch; quoted in Part~I as a ladder-independent consistency channel. First cited in \S\ref{sec:early-network}.

\bibitem[Blas et~al.(2011)]{Blas2011}
\textbf{Blas, D., Lesgourgues, J., \& Tram, T. (2011).} The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes. \emph{Journal of Cosmology and Astroparticle Physics, 2011}(07), 034.
\par\noindent\textbf{Relevance.} The Boltzmann solver behind the zero-parameter temperature-spectrum test imported from the parent \ATen{}. The instrument to which the derived chart hands its inputs: the zero-freedom TT confrontation of \S\ref{sec:cmb-overdetermined} is run with it, and the tier ladder of Part~VIII names it explicitly as the machinery the geometry frames but does not replace. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Boubel et~al.(2024)]{Boubel2024TF}
\textbf{Boubel, P., et al. (2024).} An improved Tully--Fisher estimate of \(H_0\). \emph{Monthly Notices of the Royal Astronomical Society, 533} (arXiv:2408.03660).
\par\noindent\textbf{Relevance.} The CosmicFlows-4 Tully--Fisher route: \(73.3 \pm 2.1 \pm 3.5\), its systematic term driven by discordant zero-point calibrators --- one of the high geometric probes carried at full strength in the counterweight. First cited in \S\ref{sec:landscape-counterweight}.

\bibitem[Breuval et~al.(2022)]{Breuval2022Metallicity}
\textbf{Breuval, L., Riess, A.~G., Kervella, P., Anderson, R.~I., \& Romaniello, M. (2022).} An improved calibration of the wavelength dependence of metallicity on the Cepheid Leavitt law. \emph{Astrophysical Journal, 939}, 89.
\par\noindent\textbf{Relevance.} Sources the metallicity channel's status: the term is significant and wavelength-independent from \(0.5\) to \(4.5~\mu\)m --- the chromatic test a photometric artifact would fail --- which is what caps the channel at the half-percent class. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Breuval et~al.(2024)]{Breuval2024SMC}
\textbf{Breuval, L., Riess, A.~G., Casertano, S., et al. (2024).} Small Magellanic Cloud Cepheids observed with the Hubble Space Telescope provide a new anchor for the SH0ES distance ladder. \emph{Astrophysical Journal, 973}, 30.
\par\noindent\textbf{Relevance.} The SMC Cepheid anchor entering the H0DN variant catalog (V07); part of the anchor-redundancy argument of Part~I. First cited in \S\ref{sec:central-claim}.

\bibitem[Brout et~al.(2022)]{Brout2022}
\textbf{Brout, D., Scolnic, D., Popovic, B., et al. (2022).} The Pantheon+ analysis: Cosmological constraints. \emph{Astrophysical Journal, 938}, 110.
\par\noindent\textbf{Relevance.} The Pantheon+ cosmological analysis; part of the supernova-standardization context whose calibration choices enter the residual channel table. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Buchert and R\"as\"anen(2012)]{BuchertRasanen2012}
\textbf{Buchert, T., \& R\"as\"anen, S. (2012).} Backreaction in late-time cosmology. \emph{Annual Review of Nuclear and Particle Science, 62}, 57.
\par\noindent\textbf{Relevance.} The review of the averaging problem; frames the rigorous bound of the backreaction section and its appendix course. First cited in \S\ref{sec:backreaction}.

\bibitem[Buchert(2000)]{Buchert2000}
\textbf{Buchert, T. (2000).} On average properties of inhomogeneous fluids in general relativity: dust cosmologies. \emph{Gen. Rel. Grav.} \textbf{32}, 105.
\par\noindent\textbf{Relevance.} The spatial-averaging formalism defining the kinematic backreaction \(\mathcal{Q}_D\); the basis of the bound in \S\ref{sec:backreaction} and Appendix~\ref{app:backreaction}. First cited in \S\ref{sec:backreaction}.

\bibitem[Calabrese et~al.(2025)]{Calabrese2025Ext}
\textbf{Calabrese, E., et al. (ACT Collaboration) (2025).} The Atacama Cosmology Telescope DR6: Constraints on extended cosmological models. \emph{JCAP} (arXiv:2503.14454).
\par\noindent\textbf{Relevance.} The ACT team's own extended-model search: no evidence for early dark energy, varying constants, or modified recombination --- one side of the two-sided EDE status of Class~2, opposite the profile-likelihood reading of \citet{Poulin2025EDE}. First cited in \S\ref{sec:class2-ede}.

\bibitem[Camphuis et~al.(2025)]{Camphuis2025}
\textbf{Camphuis, E., Quan, W., Balkenhol, L., et al. (2025).} Joint analysis of SPT-3G, \emph{Planck}, and ACT CMB temperature and polarization data. arXiv:2506.20707.
\par\noindent\textbf{Relevance.} Source of the combined Planck+SPT+ACT early-universe value \(67.24 \pm 0.35\) km\,s\(^{-1}\)\,Mpc\(^{-1}\) used as this paper's reference anchor \(\Href\) (the companion \ATen{} uses the \(67.4\) convention; the conversion is stated in Part~II and Appendix~D), and of the SPT-3G-alone value \(66.66 \pm 0.60\) quoted in \S\ref{sec:both-from-anchor} --- the tightest single-instrument CMB determination, landing directly on the derived rate. First cited in \S\ref{sec:intro-tension}.

\bibitem[Carrick et~al.(2015)]{CarrickFlow2015}
\textbf{Carrick, J., Turnbull, S.~J., Lavaux, G., \& Hudson, M.~J. (2015).} Cosmological parameters from the comparison of peculiar velocities with predictions from the 2M++ density field. \emph{Monthly Notices of the Royal Astronomical Society, 450}, 317.
\par\noindent\textbf{Relevance.} The 2M++ peculiar-velocity model used in the H0DN flow corrections; variant V19 tests its removal and shifts the central value by only \(-0.44\) km\,s\(^{-1}\)\,Mpc\(^{-1}\). First cited in \S\ref{sec:h0dn-not}.

\bibitem[Casertano et~al.(2026)]{CasertanoH0DN2026}
\textbf{Casertano, S., Anand, G.~S., Anderson, R.~I., et al.\ (H0DN Collaboration) (2026).} The Local Distance Network: A community consensus report on the measurement of the Hubble constant at \({\sim}1\%\) precision. \emph{Astronomy \& Astrophysics, 708}, A166. arXiv:2510.23823.
\par\noindent\textbf{Relevance.} The subject measurement of this report: the covariance-weighted Local Distance Network consensus \(\HzeroDN = 73.50 \pm 0.81\) km\,s\(^{-1}\)\,Mpc\(^{-1}\), its pre-registered baseline, its thirty-plus structured variants, and its orthogonal-path consistency test. First cited in \S\ref{sec:central-claim}.

\bibitem[Chen et~al.(2018)]{ChenFishbachHolz2018}
\textbf{Chen, H.-Y., Fishbach, M., \& Holz, D.~E. (2018).} A two per cent Hubble constant measurement from standard sirens within five years of design sensitivity. \emph{Nature, 562}, 545.
\par\noindent\textbf{Relevance.} The reference forecast for the siren channel's precision clock; the basis of the dark-siren acceleration adopted in the events-to-verdict timetable. First cited in \S\ref{sec:siren-pipelines}.

\bibitem[Chen et~al.(2026)]{Chen2026Review}
\textbf{Chen, H., Wang, K., et al. (2026).} The local distance ladder: a 2026 review. Preprint (arXiv:2606.26831).
\par\noindent\textbf{Relevance.} The covariance-network review: seven ladder routes combine to \(73.30 \pm 0.92\) (\(5.6\sigma\)), and the residual Freedman--Riess disagreement is localized in SN-calibrator subsampling and tip/JAGB standardization conventions --- not anchors, not crowding. First cited in \S\ref{sec:landscape-consensus}.

\bibitem[Cruz Reyes et~al.(2022)]{CruzReyes2022Clusters}
\textbf{Cruz Reyes, M., \& Anderson, R.~I. (2022).} A 0.9\% calibration of the Galactic Cepheid luminosity scale from Gaia DR3 cluster Cepheids. \emph{Astronomy \& Astrophysics, 672}, A85 (arXiv:2208.09403).
\par\noindent\textbf{Relevance.} The faint-\(G\) bypass of the bright-star zero-point regime: a 0.9\% Milky-Way Leavitt-law calibration agreeing with the ladder's at \(0.3\sigma\) --- the reason the Gaia channel is priced at its low edge in \S\ref{sec:residual-anatomy}. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[de~Bernardis et~al.(2000)]{deBernardis2000}
\textbf{de Bernardis, P., Ade, P.~A.~R., Bock, J.~J., et al. (2000).} A flat Universe from high-resolution maps of the cosmic microwave background radiation. \emph{Nature, 404}, 955.
\par\noindent\textbf{Relevance.} The Boomerang first-acoustic-peak measurement that pinned the universe flat --- the first CMB pillar of the concordance the dark-energy course narrates. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Desmond et~al.(2026)]{Desmond2026Ladder}
\textbf{Desmond, H., Stiskalek, R., N\'ajera, J.~A., \& Banik, I. (2026).} The subtle statistics of the distance ladder: On the distance prior and selection effects. \emph{Monthly Notices of the Royal Astronomical Society}, in press (arXiv:2511.03394).
\par\noindent\textbf{Relevance.} The proposal that the ladder's implicit uniform-in-modulus distance priors bias every Bayesian rung low in distance and high in \(H_0\) --- an Eddington-class common mode. One half of the statistics channel of the residual anatomy, carried at \(0\)--\(3\%\) with its rebuttal beside it. First cited in \S\ref{sec:medium-channels}.

\bibitem[Di Valentino et~al.(2021)]{DiValentino2021}
\textbf{Di Valentino, E., Mena, O., Pan, S., et al. (2021).} In the realm of the Hubble tension --- a review of solutions. \emph{Classical and Quantum Gravity, 38}, 153001.
\par\noindent\textbf{Relevance.} The encyclopaedic review of proposed resolutions; with the CosmoVerse white paper, the map from which Part~V's six classes are drawn. First cited in \S\ref{sec:class-statement}.

\bibitem[Di Valentino et~al.(2025)]{DiValentino2025}
\textbf{Di Valentino, E., Said, J.~L., Riess, A.~G., et al. (2025).} The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics. \emph{Physics of the Dark Universe, 49}, 101965.
\par\noindent\textbf{Relevance.} The community survey of tension-resolution proposals; frames the six comparison classes examined in Part~III. The census against which Part~V's six rival classes are organized, and the standing measure of how crowded the solution space had become before the landscape's three-band reading. First cited in \S\ref{sec:why-h0dn}.

\bibitem[Dicke et~al.(1965)]{DickePeebles1965}
\textbf{Dicke, R.~H., Peebles, P.~J.~E., Roll, P.~G., \& Wilkinson, D.~T. (1965).} Cosmic black-body radiation. \emph{Astrophysical Journal, 142}, 414.
\par\noindent\textbf{Relevance.} The companion paper that identified the excess antenna temperature as the relic radiation of the hot beginning. First cited in \S\ref{sec:intro-history}.

\bibitem[Ding et~al.(2025)]{Ding2025Gaia}
\textbf{Ding, C., et al. (2025).} The Gaia DR3 parallax zero-point at bright magnitudes from orbital parallaxes. \emph{Astronomical Journal, 169}, 211 (arXiv:2502.08068).
\par\noindent\textbf{Relevance.} The bright-star (\(G<8\)) zero-point check: \(-38.9 \pm 10.3\,\mu\)as with formal errors underestimated by \(\sim\!2\times\), and no established sign for the induced \(H_0\) shift --- the verified state of the Gaia channel, decided by DR4 (December 2026). First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Dodelson and Schmidt(2020)]{Dodelson2020}
\textbf{Dodelson, S. and Schmidt, F. (2020).} \emph{Modern Cosmology}, 2nd ed. Academic Press.
\par\noindent\textbf{Relevance.} Standard graduate reference for the growth of structure and the $S_8$ amplitude used in the derivation course of App.~\ref{app:s8}. First cited in \S\ref{sec:friedmann-theorem}.

\bibitem[Dom\'inguez et~al.(2023)]{Dominguez2023Gamma}
\textbf{Dom\'inguez, A., et al. (2023).} A new derivation of the Hubble constant from \(\gamma\)-ray attenuation using improved optical depths. \emph{Monthly Notices of the Royal Astronomical Society, 527}, 4632 (arXiv:2306.09878).
\par\noindent\textbf{Relevance.} The anomalously low outlier of the landscape: \(61.9^{+2.9}_{-2.4}\) at fixed \(\Omega_m = 0.32\) (free fit \(65.6^{+5.6}_{-5.0}\)), EBL-model dependent --- retained as the reminder that not every anomaly points the same way. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Efstathiou(2020)]{Efstathiou2020}
\textbf{Efstathiou, G. (2020).} A lockdown perspective on the Hubble tension (with comments from the SH0ES team). arXiv:2007.10716.
\par\noindent\textbf{Relevance.} The sharpest ladder-critical audit in the literature; several allowances in the residual channel table trace to the issues it raises. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Einstein(1917)]{Einstein1917}
\textbf{Einstein, A. (1917).} Kosmologische Betrachtungen zur allgemeinen Relativit\"atstheorie. \emph{Sitzungsberichte der K\"oniglich Preussischen Akademie der Wissenschaften}, 142.
\par\noindent\textbf{Relevance.} The cosmological constant introduced to hold the universe still --- the term whose two later careers (regret, then dark energy) frame the century narrative. First cited in \S\ref{sec:intro-history}.

\bibitem[Eisenstein and Hu(1998)]{EisensteinHu1998}
\textbf{Eisenstein, D.~J., \& Hu, W. (1998).} Baryonic features in the matter transfer function. \emph{Astrophysical Journal, 496}, 605.
\par\noindent\textbf{Relevance.} The analytic transfer function used by the released backreaction and variance scripts; keeps the paper's bounds free of external Boltzmann-code dependencies. First cited in \S\ref{app:backreaction}.

\bibitem[Eisenstein et~al.(2005)]{Eisenstein2005}
\textbf{Eisenstein, D.~J., Zehavi, I., Hogg, D.~W., et al. (2005).} Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies. \emph{Astrophysical Journal, 633}, 560.
\par\noindent\textbf{Relevance.} The discovery of the baryon-acoustic ruler in galaxy clustering --- the ancestor of the DESI channel that now checks the CMB value without the CMB. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Etherington(1933)]{Etherington1933}
\textbf{Etherington, I.~M.~H. (1933).} On the definition of distance in general relativity. \emph{Philosophical Magazine, 15}, 761.
\par\noindent\textbf{Relevance.} The reciprocity (duality) theorem \(d_L = (1+z)^2 d_A\), valid in any metric theory with photon-number conservation --- the bedrock identity of the distance toolkit. First cited in \S\ref{app:cD-distances}.

\bibitem[Fixsen(2009)]{Fixsen2009}
\textbf{Fixsen, D.~J. (2009).} The temperature of the cosmic microwave background. \emph{Astrophysical Journal, 707}, 916.
\par\noindent\textbf{Relevance.} The definitive CMB temperature: the measurement from which the photon share of the radiation density follows exactly, as invoked in the objections on declared inputs. First cited in \S\ref{sec:hoft-steps}.

\bibitem[Freedman and Madore(2010)]{FreedmanMadore2010}
\textbf{Freedman, W.~L., \& Madore, B.~F. (2010).} The Hubble constant. \emph{Annual Review of Astronomy and Astrophysics, 48}, 673.
\par\noindent\textbf{Relevance.} The standard review of the measurement's first eight decades; companion to the century course and its figure. Also the TRGB physics behind Course~D's second rung --- the core-mass argument for the tip as a standard candle --- whose calibration conventions the subsample channel of Part~IV prices. First cited in \S\ref{sec:intro-tension}.

\bibitem[Freedman et~al.(2001)]{Freedman2001KP}
\textbf{Freedman, W.~L., Madore, B.~F., Gibson, B.~K., et al. (2001).} Final results from the Hubble Space Telescope Key Project to measure the Hubble constant. \emph{Astrophysical Journal, 553}, 47.
\par\noindent\textbf{Relevance.} The measurement that ended the factor-two war at \(72 \pm 8\); the ten-percent waypoint of the century-long fall traced in the measurement-history course and its figure. First cited in \S\ref{sec:intro-tension}.

\bibitem[Freedman et~al.(2025)]{Freedman2025CCHP}
\textbf{Freedman, W.~L., Madore, B.~F., Hoyt, T.~J., et al. (2025).} Status report on the Chicago-Carnegie Hubble Program (CCHP): Measurement of the Hubble constant using the near-infrared tip of the red giant branch. \emph{Astrophysical Journal, 985}, 203. arXiv:2408.06153.
\par\noindent\textbf{Relevance.} The independent TRGB pipeline: the published JWST value is \(70.39 \pm 1.94\) (with JAGB and Cepheid variants spanning \(67.8\)--\(72.1\)) --- the low side of the subsample dispute priced in the residual anatomy, and part of the JWST cross-check of Priority 4. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Freedman(2021)]{Freedman2021}
\textbf{Freedman, W.~L. (2021).} Measurements of the Hubble constant: Tensions in perspective. \emph{Astrophysical Journal, 919}, 16.
\par\noindent\textbf{Relevance.} The TRGB-side perspective ($H_0 \approx 69.8$) whose spread against the Cepheid route informs the primary-indicator-family allowance in the residual channel table. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Friedmann(1922)]{Friedmann1922}
\textbf{Friedmann, A. (1922).} \"Uber die Kr\"ummung des Raumes. \emph{Zeitschrift f\"ur Physik, 10}, 377.
\par\noindent\textbf{Relevance.} The honest solution of Einstein's equations: a matter-filled universe must expand or contract --- the dynamical law behind everything this paper derives. First cited in \S\ref{sec:intro-history}.

\bibitem[Gall et~al.(2026)]{Gall2026BlueSNe}
\textbf{Gall, C., Izzo, L., Wojtak, R., \& Hjorth, J. (2026).} The Hubble constant from blue Type Ia supernovae. \emph{Astrophysical Journal, 1001}, 233. arXiv:2411.05642.
\par\noindent\textbf{Relevance.} The dust-free cross-check of the dust channel: a ladder built only from intrinsically blue, low-reddening supernovae returns \(H_0 = 70.0 \pm 2.1\) --- no extinction model at all, landing where the two-population reanalysis lands. First cited in \S\ref{sec:medium-channels}.

\bibitem[Gao et~al.(2024)]{Gao2024FRB}
\textbf{Gao, H., et al. (2024).} Measuring the Hubble constant with localized and nonlocalized fast radio bursts. \emph{Astronomy \& Astrophysics, 698}, A215 (arXiv:2410.03994).
\par\noindent\textbf{Relevance.} The FRB channel: 108 localized bursts give \(69.40^{+2.14}_{-1.97}\) (527 nonlocalized CHIME bursts: \(68.81 \pm 0.68\), dispersion-measure-model dependent) --- a camp-free probe sitting squarely in band~2. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Garc\'ia Escudero et~al.(2025)]{GarciaEscudero2025}
\textbf{Garc\'ia Escudero, H., et al. (2025).} A sound-horizon-agnostic measurement of \(H_0\) from BAO, lensing, and galaxy clustering. \emph{Astrophysical Journal Letters, 998}, L41 (arXiv:2509.16202).
\par\noindent\textbf{Relevance.} The high edge of the ruler-free span: \(70.0 \pm 1.7\), prior-sensitive --- the reason this paper prints the honest \(r_s\)-free range \(66.7\)--\(70.2\) rather than the convenient half of it. First cited in \S\ref{sec:landscape-early}.

\bibitem[Giovanetti et~al.(2026)]{Giovanetti2026}
\textbf{Giovanetti, C., et al. (2026).} A generic baryon-density tension for early-time resolutions of the Hubble tension. Preprint (arXiv:2604.05095).
\par\noindent\textbf{Relevance.} The deuterium audit as a generic wall: early-time fixes that raise \(H_0\) raise the inferred \(\omega_b\) against the LUNA-pinned BBN value --- wielded against Classes 2--4. First cited in \S\ref{sec:class2-ede}.

\bibitem[Hallgren et~al.(2025)]{Hallgren2025Reddening}
\textbf{Hallgren, A., et al. (2025).} Diverse host-galaxy reddening distributions and their effect on Type Ia supernova distances. \emph{Astronomy \& Astrophysics, 706}, A129.
\par\noindent\textbf{Relevance.} The simulation-level face of the dust channel: imposing a universal extinction prior on intrinsically diverse host populations provably biases the recovered relative distances --- the mechanism's existence proof, independent of any \(H_0\) claim. First cited in \S\ref{sec:medium-channels}.

\bibitem[Hill et~al.(2020)]{Hill2020}
\textbf{Hill, J.~C., McDonough, E., Toomey, M.~W., \& Alexander, S. (2020).} Early dark energy does not restore cosmological concordance. \emph{Physical Review D, 102}(4), 043507.
\par\noindent\textbf{Relevance.} Demonstrates that full-likelihood early-dark-energy fits drive the net \(S_8\) shift positive; cited as the mandatory sign caveat on any early-lobe \(S_8\) story, and part of the no-early-modification case of Class~2. First cited in \S\ref{sec:class2-ede}.

\bibitem[H\"og\aa s and M\"ortsell(2026)]{HogasMortsell2026}
\textbf{H\"og\aa s, M., \& M\"ortsell, E. (2026).} Physically-motivated priors in the local distance ladder significantly reduce the Hubble tension. \emph{Monthly Notices of the Royal Astronomical Society} (arXiv:2601.22215).
\par\noindent\textbf{Relevance.} The executed recalibration of the statistics channel: physically motivated distance priors shift the ladder from \(73.0 \pm 1.0\) to \(70.6 \pm 1.0\) and the quoted tension from \(\sim\!5\sigma\) to \(\sim\!2\sigma\). Read against its rebuttal (\citealp{Stiskalek2026MWCepheids}); the pair is why the channel is printed ``undecided.'' First cited in \S\ref{sec:medium-channels}.

\bibitem[Holz and Hughes(2005)]{HolzHughes2005}
\textbf{Holz, D.~E., \& Hughes, S.~A. (2005).} Using gravitational-wave standard sirens. \emph{Astrophysical Journal, 629}, 15.
\par\noindent\textbf{Relevance.} Developed the standard-siren program and its name; source of the amplitude-distance reading of Eq.~\eqref{eq:siren-dL}. First cited in \S\ref{sec:siren-principle}.

\bibitem[Huang et~al.(2023)]{Huang2023Miras}
\textbf{Huang, C.~D., et al. (2023).} The Mira distance to M101 and a Mira-based Hubble constant. \emph{Astrophysical Journal} (arXiv:2312.08423).
\par\noindent\textbf{Relevance.} The Mira rung: 211 Miras in M101 give \(72.37 \pm 2.97\) --- an oxygen-rich variable route independent of Cepheids and TRGB, siding with the high ladder. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Hubble and Humason(1931)]{HubbleHumason1931}
\textbf{Hubble, E., \& Humason, M.~L. (1931).} The velocity-distance relation among extra-galactic nebulae. \emph{Astrophysical Journal, 74}, 43.
\par\noindent\textbf{Relevance.} The velocity-distance law confirmed to twenty times the original depth; the second point on the century chart's opening plunge. First cited in \S\ref{sec:intro-history}.

\bibitem[Hubble(1925)]{Hubble1925}
\textbf{Hubble, E. (1925).} Cepheids in spiral nebul\ae. \emph{The Observatory, 48}, 139.
\par\noindent\textbf{Relevance.} Cepheids resolved in Andromeda: the island-universe debate settled and the universe of galaxies opened. The moment the `nebulae' became galaxies and the universe acquired an outside: the first rung of Part~II's century course, and the birth of the distance ladder as an instrument. First cited in \S\ref{sec:intro-history}.

\bibitem[Hubble(1929)]{Hubble1929}
\textbf{Hubble, E. (1929).} A relation between distance and radial velocity among extra-galactic nebulae. \emph{Proc. Natl. Acad. Sci.} \textbf{15}, 168.
\par\noindent\textbf{Relevance.} The founding measurement of the expansion, \(v=H_0 d\); cited for the historical opening of the Introduction. The founding one-number cosmology --- whose graduation from measured constant to derived quantity is this paper's closing argument (Part~IX). First cited in \S\ref{sec:intro-history}.

\bibitem[Hughes et~al.(2025)]{Hughes2025Jackknife}
\textbf{Hughes, N., Wilensky, M., \& Bull, P. (2025).} A Bayesian jackknife of the Hubble tension. Preprint (arXiv:2511.19341).
\par\noindent\textbf{Relevance.} The model-agnostic bias audit over 16 key probes: ``no bias anywhere'' is strongly disfavored, many bias assignments fit comparably, and the agnostic posterior reads \(H_0 \approx 68^{+4.7}_{-1.3}\) (95\%) --- the landscape's warning against convicting any single culprit. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Jedamzik et~al.(2025)]{Jedamzik2025PMF}
\textbf{Jedamzik, K., Pogosian, L., et al. (2025).} Hints of primordial magnetic fields at recombination. \emph{Nature Astronomy, 10}, 317 (arXiv:2503.09599).
\par\noindent\textbf{Relevance.} The baryon-clumping mechanism of Class~3: 5--10 pG fields hastening recombination, currently a \(1.8\sigma\), dataset-dependent hint --- carried at exactly that strength. First cited in \S\ref{sec:class3-modrec}.

\bibitem[Jensen et~al.(2025)]{Jensen2025SBF}
\textbf{Jensen, J.~B., Blakeslee, J.~P., Cantiello, M., et al. (2025).} The TRGB-calibrated SBF distance scale with JWST and HST. \emph{Astrophysical Journal, 987}, 87. arXiv:2502.15935.
\par\noindent\textbf{Relevance.} The surface-brightness-fluctuation rung of the TRGB-SBF path; backbone of the statistically independent orthogonal path O2 in the H0DN consistency test. First cited in \S\ref{sec:landscape-counterweight}.

\bibitem[Jia et~al.(2025)]{Jia2025H0z}
\textbf{Jia, X.~D., Hu, J.~P., Yi, S.~X., \& Wang, F.~Y. (2025).} Uncorrelated estimations of \(H_0\) redshift evolution from DESI baryon acoustic oscillation observations. \emph{Astrophysical Journal Letters, 979}, L34.
\par\noindent\textbf{Relevance.} The published \(H_0(z)\) descending trend (\(6.4\sigma\), uncorrelated DESI bins) --- the redshift-shell signature this paper names as the Local-Hole lift's survey-only test, carried with its provenance cautions in \S\ref{sec:hole-budget}. First cited in \S\ref{sec:hole-budget}.

\bibitem[Jimenez and Loeb(2002)]{JimenezLoeb2002}
\textbf{Jimenez, R., \& Loeb, A. (2002).} Constraining cosmological parameters based on relative galaxy ages. \emph{Astrophysical Journal, 573}, 37.
\par\noindent\textbf{Relevance.} The original cosmic-chronometer proposal --- galaxies as clocks --- behind the differential-age channel of the over-determination program. First cited in \S\ref{sec:early-network}.

\bibitem[Kamionkowski and Riess(2023)]{KamionkowskiRiess2023}
\textbf{Kamionkowski, M., \& Riess, A.~G. (2023).} The Hubble tension and early dark energy. \emph{Annual Review of Nuclear and Particle Science, 73}, 153.
\par\noindent\textbf{Relevance.} Review of the tension and the sound-horizon lever arm; the standard linear-response logic recapped in Part~II descends from this literature. First cited in \S\ref{sec:why-h0dn}.

\bibitem[Karwal and Kamionkowski(2016)]{Karwal2016}
\textbf{Karwal, T., \& Kamionkowski, M. (2016).} Dark energy at early times, the Hubble parameter, and the string axiverse. \emph{Physical Review D, 94}, 103523.
\par\noindent\textbf{Relevance.} The original early-dark-energy proposal; ancestor of the phenomenological EDE class compared against the derived reading in Class~2. First cited in \S\ref{sec:class2-ede}.

\bibitem[Keenan et~al.(2013)]{Keenan2013}
\textbf{Keenan, R.~C., Barger, A.~J., \& Cowie, L.~L. (2013).} Evidence for a \(\sim300\) megaparsec scale under-density in the local galaxy distribution. \emph{Astrophysical Journal, 775}, 62.
\par\noindent\textbf{Relevance.} The deep reading of the Local Hole (the ``KBC void''): a \(\sim30\%\) near-infrared count deficit to \(\sim300\)~Mpc; the contested upper scenario of the lift ledger. First cited in \S\ref{sec:hole-evidence}.

\bibitem[Kenworthy et~al.(2019)]{Kenworthy2019}
\textbf{Kenworthy, W.~D., Scolnic, D., \& Riess, A.~G. (2019).} The local perspective on the Hubble tension: local structure does not impact measurement of the Hubble constant. \emph{Astrophysical Journal, 875}, 145.
\par\noindent\textbf{Relevance.} The dynamical brake on the deep reading: the smooth supernova Hubble diagram leaves room for at most a sub-percent void-induced shift of the SN-anchored rate; one of the two reasons the adopted lift stays at \(+2\)--\(3\%\). First cited in \S\ref{sec:hole-evidence}.

\bibitem[Kolb \& Turner(1990)]{KolbTurner1990}
\textbf{Kolb, E.~W., \& Turner, M.~S. (1990).} \emph{The Early Universe.} Addison-Wesley (Frontiers in Physics 69).
\par\noindent\textbf{Relevance.} The classic graduate text on thermal history, entropy conservation, and freeze-out --- the standard companion to the thermal annex of Course~K. First cited in \S\ref{app:thermal}.

\bibitem[Kreisch et~al.(2024)]{KreischEDE2024}
\textbf{Kreisch, C.~D., Park, M., Calabrese, E., et al. (2024).} Atacama Cosmology Telescope: constraints on early dark energy from DR6 and DR4+Planck. \emph{Journal of Cosmology and Astroparticle Physics, 2024}, 055.
\par\noindent\textbf{Relevance.} Current CMB squeeze on \(f_{\mathrm{EDE}}\); the observational pressure quoted in the EDE comparison and in kill switch F-Hubble-3. First cited in \S\ref{sec:class2-ede}.

\bibitem[Kumar et~al.(2026)]{Kumar2026AGN}
\textbf{Kumar, S., et al. (2026).} The Hubble constant from thirteen bright standard sirens in AGN discs. Preprint (arXiv:2607.03672).
\par\noindent\textbf{Relevance.} Thirteen binary-black-hole mergers with claimed AGN-flare counterparts give \(70.50^{+3.37}_{-2.89} \pm 1.56\) --- carried as a candidate only, the flare associations being unconfirmed; if validated, a second bright-siren channel a decade early. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Leavitt and Pickering(1912)]{LeavittPickering1912}
\textbf{Leavitt, H.~S., \& Pickering, E.~C. (1912).} Periods of 25 variable stars in the Small Magellanic Cloud. \emph{Harvard College Observatory Circular, 173}, 1.
\par\noindent\textbf{Relevance.} The period-luminosity law: the key that made cosmic distance measurable, and the first rung of every ladder since. First cited in \S\ref{sec:intro-history}.

\bibitem[Lee et~al.(2024)]{Lee2024JAGB}
\textbf{Lee, A.~J., et al. (2024).} The JAGB method: a blind carbon-star distance ladder in seven supernova hosts. CCHP series (arXiv:2408.03474).
\par\noindent\textbf{Relevance.} The blind carbon-star route of the CCHP program: \(67.80 \pm 2.17 \pm 1.64\) on the N4258 zero point --- the lowest of the CCHP family, and one pole of the subsample dispute priced in \S\ref{sec:residual-anatomy}. First cited in \S\ref{sec:landscape-consensus}.

\bibitem[Lee et~al.(2026)]{Lee2026ModRecII}
\textbf{Lee, N., et al. (2026).} Modified recombination II: varying the electron mass against DESI DR2. Preprint (arXiv:2606.06495).
\par\noindent\textbf{Relevance.} The key negative result of Class~3: the \(m_e(z)\) variant that resolves the CMB-only fit fails once DESI DR2 is included --- one of the four audits that cap modified recombination below the ladder. First cited in \S\ref{sec:class3-modrec}.

\bibitem[Lema\^itre(1927)]{Lemaitre1927}
\textbf{Lema\^itre, G. (1927).} Un Univers homog\`ene de masse constante et de rayon croissant rendant compte de la vitesse radiale des n\'ebuleuses extra-galactiques. \emph{Annales de la Soci\'et\'e Scientifique de Bruxelles, A47}, 49.
\par\noindent\textbf{Relevance.} Derived the expansion law from general relativity and extracted the constant two years before Hubble's plot; opens the measurement-history course. First cited in \S\ref{sec:intro-history}.

\bibitem[Lindegren et~al.(2021)]{Lindegren2021}
\textbf{Lindegren, L., Bastian, U., Biermann, M., et al. (2021).} Gaia Early Data Release 3: Parallax bias versus magnitude, color, and position. \emph{Astronomy \& Astrophysics, 649}, A4.
\par\noindent\textbf{Relevance.} The map of the Gaia parallax zero-point --- the anchor-floor systematic of the residual-anatomy channel table; its residual position and color dependence sets the allowance adopted there. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Liske et~al.(2008)]{Liske2008}
\textbf{Liske, J., Grazian, A., Vanzella, E., et al. (2008).} Cosmic dynamics in the era of extremely large telescopes. \emph{Monthly Notices of the Royal Astronomical Society, 386}, 1192.
\par\noindent\textbf{Relevance.} The design study for measuring the redshift drift with an extremely large telescope; the instrument context for the drift predictions of the derived curve. First cited in \S\ref{sec:overdetermining}.

\bibitem[Loeb(1998)]{Loeb1998}
\textbf{Loeb, A. (1998).} Direct measurement of cosmological parameters from the cosmic deceleration of extragalactic objects. \emph{Astrophysical Journal Letters, 499}, L111.
\par\noindent\textbf{Relevance.} Revived the redshift-drift measurement for the Lyman-\(\alpha\) forest era; with \citet{Sandage1962}, the basis of the drift predictions of Table~\ref{tab:derived-Hz}. First cited in \S\ref{sec:overdetermining}.

\bibitem[Louis et~al.(2025)]{Louis2025ACT}
\textbf{Louis, T., La Posta, A., Atkins, Z., et al. (2025).} The Atacama Cosmology Telescope: DR6 power spectra, likelihoods and \(\Lambda\)CDM parameters. arXiv:2503.14452.
\par\noindent\textbf{Relevance.} The ACT DR6 release: the second, fully independent CMB instrument whose agreement with Planck underwrites the early-universe network argument. First cited in \S\ref{sec:early-network}.

\bibitem[Lu et~al.(2026)]{Lu2026EFT}
\textbf{Lu, W., et al. (2026).} A sound-horizon-free measurement of \(H_0\) from the EFTofLSS power spectrum and bispectrum. \emph{Physical Review D, 114}, 023518 (arXiv:2602.21733).
\par\noindent\textbf{Relevance.} The first \(P+B\) ruler-free determination: \(70.2^{+2.2}_{-2.4}\) baseline (\(68.6 \pm 1.8\) with Pantheon+) --- the top of the honest \(r_s\)-free span. First cited in \S\ref{sec:landscape-early}.

\bibitem[LVK Collaboration(2025)]{LVK2025GWTC4}
\textbf{Abac, A., et al. (LIGO--Virgo--KAGRA Collaboration) (2025).} GWTC-4.0: Constraints on the cosmic expansion rate and modified gravitational-wave propagation. Preprint (arXiv:2509.04348).
\par\noindent\textbf{Relevance.} The 142-source dark-siren catalog value \(76.6^{+13.0}_{-9.5}\) --- the wide, mass-model-dependent start of the siren trajectory the discriminator section tracks. First cited in \S\ref{sec:siren-discriminator}.

\bibitem[LVK Collaboration(2026)]{LVK2026GWTC5}
\textbf{LIGO--Virgo--KAGRA Collaboration (2026).} GWTC-5.0: Constraints on the cosmic expansion rate and modified gravitational-wave propagation. Preprint (arXiv:2605.27227).
\par\noindent\textbf{Relevance.} 236 sources with the DES-Y6 catalog and GW170817: \(71.0^{+9.0}_{-7.1}\), a 25.7\% tightening over GWTC-4.0 with no gravitational-wave-propagation anomalies --- the siren trajectory drifting toward band~2 as catalogs deepen. First cited in \S\ref{sec:siren-discriminator}.

\bibitem[Lynch et~al.(2024a)]{Lynch2024Recomb}
\textbf{Lynch, G.~P., Knox, L., \& Chluba, J. (2024).} Reconstructing the recombination history: non-parametric constraints and the Hubble tension. \emph{Physical Review D} (arXiv:2404.05715).
\par\noindent\textbf{Relevance.} The ceiling of Class~3, printed by its own machinery: with full spline freedom in the ionization history, CMB+BAO allow at most \(H_0 < 70.7\) at 95\% --- the citable cap this paper quotes in place of any point value. First cited in \S\ref{sec:landscape-early}.

\bibitem[Lynch et~al.(2024b)]{Lynch2024DESI}
\textbf{Lynch, G.~P., Knox, L., \& Chluba, J. (2024).} DESI and the Hubble tension in light of modified recombination. \emph{Physical Review D} (arXiv:2406.10202).
\par\noindent\textbf{Relevance.} The DESI audit of Class~3: the spline histories that raise \(H_0\) drag \(\Omega_m\) against the DR2 measurement, leaving SH0ES-level rates at \(\sim\!2\sigma\) residual tension. First cited in \S\ref{sec:class3-modrec}.

\bibitem[Madhavacheril et~al.(2023)]{Madhavacheril2023Lensing}
\textbf{Madhavacheril, M.~S., et al. (ACT Collaboration) (2023).} The Atacama Cosmology Telescope DR6 gravitational lensing map and cosmological parameters. \emph{Astrophysical Journal, 962}, 113 (arXiv:2304.05203).
\par\noindent\textbf{Relevance.} The lensing route with no primary-anisotropy fit: \(68.3 \pm 1.1\) with BAO, and \(\Lambda\)CDM-consistent lensing at the percent level --- the resolution of the old \(A_L\) anomaly. First cited in \S\ref{sec:landscape-early}.

\bibitem[Maggiore(2008)]{Maggiore2008}
\textbf{Maggiore, M. (2008).} \emph{Gravitational Waves, Volume 1: Theory and Experiments.} Oxford University Press.
\par\noindent\textbf{Relevance.} The standard reference for linearized gravity, the TT gauge, and the quadrupole formula --- the full derivations compressed in Course~J. First cited in \S\ref{app:cJ-linear}.

\bibitem[Marra et~al.(2013)]{Marra2013}
\textbf{Marra, V., Amendola, L., Sawicki, I., and Valkenburg, W. (2013).} Cosmic variance and the measurement of the local Hubble parameter. \emph{Phys. Rev. Lett.} \textbf{110}, 241305.
\par\noindent\textbf{Relevance.} The reference computation of the cosmic variance of the locally measured \(\Hzero\); corroborates the \(\lesssim0.4\%\) scatter used in \S\ref{sec:backreaction}. First cited in \S\ref{sec:backreaction}.

\bibitem[Mazurenko et~al.(2025)]{Mazurenko2024Void}
\textbf{Mazurenko, S., Banik, I., \& Kroupa, P. (2025).} The redshift dependence of the inferred \(H_0\) in a local void solution to the Hubble tension. \emph{Monthly Notices of the Royal Astronomical Society, 536}, 3232.
\par\noindent\textbf{Relevance.} Shows that a local void predicts a declining \(H_0(z)\) --- the redshift-shell signature this paper names as the Local-Hole lift's sharpest survey-only test --- with amplitude tracking the void depth. First cited in \S\ref{sec:hole-budget}.

\bibitem[M\'esz\'aros(1974)]{Meszaros1974}
\textbf{M\'esz\'aros, P. (1974).} The behavior of point masses in an expanding cosmological substratum. \emph{Astronomy \& Astrophysics, 37}, 225.
\par\noindent\textbf{Relevance.} The radiation-era stall of matter perturbations (the M\'esz\'aros effect), which imprints the equality scale on the power spectrum. First cited in \S\ref{app:cI-growthfactor}.

\bibitem[Mirpoorian et~al.(2024)]{Mirpoorian2024ModRec}
\textbf{Mirpoorian, S.~H., Jedamzik, K., \& Pogosian, L. (2024).} Modified recombination and the Hubble tension. \emph{Physical Review D, 111}, 083519 (arXiv:2411.16678).
\par\noindent\textbf{Relevance.} The strongest positive case inside Class~3: spline ionization histories can cut the tension below \(2\sigma\) in some data combinations --- quoted with its dataset dependence, against the four audits that cap the class. First cited in \S\ref{sec:class3-modrec}.

\bibitem[Moresco et~al.(2022)]{Moresco2022}
\textbf{Moresco, M., Amati, L., Amendola, L., et al. (2022).} Unveiling the Universe with emerging cosmological probes. \emph{Living Reviews in Relativity, 25}, 6.
\par\noindent\textbf{Relevance.} The reference review of cosmic chronometers and the other ladder-free, horizon-free probes of \(H(z)\); source of the differential-age method the over-determination section leans on. First cited in \S\ref{sec:early-network}.

\bibitem[M\"ortsell et~al.(2022)]{Mortsell2022}
\textbf{M\"ortsell, E., Goobar, A., Johansson, J., \& Dhawan, S. (2022).} Sensitivity of the Hubble constant determination to Cepheid calibration. \emph{Astrophysical Journal, 933}, 212.
\par\noindent\textbf{Relevance.} The Cepheid-calibration sensitivity study behind the metallicity-term allowance in the residual channel table. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Murakami et~al.(2023)]{Murakami2023Siblings}
\textbf{Murakami, Y.~S., Riess, A.~G., Stahl, B.~E., et al. (2023).} Leveraging SN Ia spectroscopic similarity to improve the measurement of \(H_0\). \emph{Journal of Cosmology and Astroparticle Physics, 2023}, 046.
\par\noindent\textbf{Relevance.} Part of the SN~Ia calibration-refinement literature whose debates the H0DN variant V13 (no SNe~Ia) renders moot for the central value. First cited in \S\ref{sec:h0dn-baseline}.

\bibitem[Pantos and Perivolaropoulos(2026)]{Pantos2026Meta}
\textbf{Pantos, V., \& Perivolaropoulos, L. (2026).} Dissecting the Hubble tension: 88 measurements, two populations. Preprint (arXiv:2601.00650).
\par\noindent\textbf{Relevance.} The meta-split, quantified: 30 distance-ladder measurements average \(72.73 \pm 0.39\) while 58 ladder-free measurements average \(69.37 \pm 0.34\) --- internally discrepant at up to \(6.5\sigma\) (\(\geq 3.9\sigma\) with correlations); the statement of the comparison problem in two numbers. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Penzias and Wilson(1965)]{PenziasWilson1965}
\textbf{Penzias, A.~A., \& Wilson, R.~W. (1965).} A measurement of excess antenna temperature at 4080 Mc/s. \emph{Astrophysical Journal, 142}, 419.
\par\noindent\textbf{Relevance.} The accidental discovery of the cosmic microwave background --- the moment the Big Bang became measured history; cited in the century narrative that opens Part~I. First cited in \S\ref{sec:intro-history}.

\bibitem[Perlmutter et~al.(1999)]{Perlmutter1999}
\textbf{Perlmutter, S. et~al. (1999).} Measurements of \(\Omega\) and \(\Lambda\) from 42 high-redshift supernovae. \emph{Astrophys. J.} \textbf{517}, 565.
\par\noindent\textbf{Relevance.} The companion discovery of cosmic acceleration; cited with \citet{Riess1998} for the dark-energy surprise. With the High-\(z\) team's discovery, fixes the acceleration whose derived timing (\(t_{\rm acc} = 7.8\)~Gyr, the exact \(\operatorname{arcsinh}(1/\sqrt2)\) epoch) Course~C computes with nothing tuned. First cited in \S\ref{sec:intro-history}.

\bibitem[Pesce et~al.(2020)]{Pesce2020Masers}
\textbf{Pesce, D.~W., et al. (2020).} The Megamaser Cosmology Project. XIII. Combined Hubble-constant constraints. \emph{Astrophysical Journal Letters, 891}, L1 (arXiv:2001.09213).
\par\noindent\textbf{Relevance.} The purest one-step geometric probe: six maser-galaxy distances give \(73.9 \pm 3.0\), sliding across \(71.8\)--\(76.9\) with the peculiar-velocity treatment of six objects --- the leading member of the high geometric counterweight. First cited in \S\ref{sec:landscape-counterweight}.

\bibitem[Peskin and Schroeder(1995)]{PeskinSchroeder1995}
\textbf{Peskin, M.~E., \& Schroeder, D.~V. (1995).} \emph{An Introduction to Quantum Field Theory}. Reading, MA: Addison-Wesley.
\par\noindent\textbf{Relevance.} Standard reference for the \(g_{*,s}\) entropy-degrees-of-freedom ledger used in the Appendix~A estimate of the \(\UoneQi\) decoupling temperature ratio. First cited in \S\ref{app:epslocal}.

\bibitem[Pietrzy\'nski et~al.(2019)]{Pietrzynski2019LMC}
\textbf{Pietrzy\'nski, G., Graczyk, D., Gallenne, A., et al. (2019).} A distance to the Large Magellanic Cloud that is precise to one per cent. \emph{Nature, 567}, 200.
\par\noindent\textbf{Relevance.} The one-per-cent DEB distance to the LMC; one of the three geometric anchors of the H0DN baseline network. The anchor whose one-per-cent geometry stiffens the ladder's floor; its independence from Gaia's zero point is part of why the parallax channel is carried at only \(0.5\)--\(1\%\) in the anatomy. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Planck Collaboration(2020)]{Planck2018Parameters}
\textbf{Planck Collaboration (2020).} Planck 2018 results. VI. Cosmological parameters. \emph{Astronomy \& Astrophysics, 641}, A6. arXiv:1807.06209.
\par\noindent\textbf{Relevance.} The baseline-dictionary parameter source: the \(N_{\mathrm{eff}}\) preference \(\DNeff = 0.04 \pm 0.18\) quoted in the exposure caveat, and the CMB-inferred \(S_8^{\mathrm{ref}} \approx 0.83\). First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Poulin et~al.(2019)]{Poulin2019}
\textbf{Poulin, V., Smith, T.~L., Karwal, T., \& Kamionkowski, M. (2019).} Early dark energy can resolve the Hubble tension. \emph{Physical Review Letters, 122}(22), 221301.
\par\noindent\textbf{Relevance.} The flagship early-dark-energy resolution of the tension, and the source of the sound-horizon lever-arm logic recapped in the recombination-switch section; the standard against which the derived reading's no-early-modification stance is contrasted in Class~2. First cited in \S\ref{sec:recomb-switch}.

\bibitem[Poulin et~al.(2023)]{Poulin2023Review}
\textbf{Poulin, V., Smith, T.~L., \& Karwal, T. (2023).} The ups and downs of early dark energy solutions to the Hubble tension: A review of models, hints and constraints circa 2023. \emph{Physics of the Dark Universe, 42}, 101348.
\par\noindent\textbf{Relevance.} Status review of the EDE band and its \(S_8\) cost; the comparison matrix and the EDE-vs-corridor table draw their phenomenological state of play from here. First cited in \S\ref{sec:h0dn-not}.

\bibitem[Poulin et~al.(2025)]{Poulin2025EDE}
\textbf{Poulin, V., et al. (2025).} Early dark energy after ACT DR6 and DESI DR2: a profile-likelihood view. \emph{Physical Review D, 113}, 063519 (arXiv:2505.08051).
\par\noindent\textbf{Relevance.} The other side of the EDE status: a prior-free profile likelihood on ACT DR6 + DESI DR2 gives \(f_{\rm EDE} = 0.09 \pm 0.03\) and \(H_0 = 71.0 \pm 1.1\) with no SH0ES input --- a live statistical-method dispute with the ACT team's own null, and this paper's F-Hubble-3 kill test in motion. First cited in \S\ref{sec:class2-ede}.

\bibitem[Reid et~al.(2019)]{Reid2019N4258}
\textbf{Reid, M.~J., Pesce, D.~W., \& Riess, A.~G. (2019).} An improved distance to NGC 4258 and its implications for the Hubble constant. \emph{Astrophysical Journal Letters, 886}, L27. arXiv:1908.05625.
\par\noindent\textbf{Relevance.} The megamaser distance to NGC 4258; the geometric anchor shared by the otherwise disjoint orthogonal paths of the H0DN consistency test. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Riess et~al.(1998)]{Riess1998}
\textbf{Riess, A. G. et~al. (1998).} Observational evidence from supernovae for an accelerating universe and a cosmological constant. \emph{Astron. J.} \textbf{116}, 1009.
\par\noindent\textbf{Relevance.} One of the two discovery papers of cosmic acceleration; cited for the dark-energy surprise that makes the expansion rate genuinely time-dependent. First cited in \S\ref{sec:intro-history}.

\bibitem[Riess et~al.(2019)]{Riess2019}
\textbf{Riess, A.~G., Casertano, S., Yuan, W., et al. (2019).} Large Magellanic Cloud Cepheid standards provide a 1\% foundation for the determination of the Hubble constant. \emph{Astrophysical Journal, 876}, 85.
\par\noindent\textbf{Relevance.} The $74.03 \pm 1.42$ waypoint of the precision-era track drawn in the century figure. The waypoint at which the tension crossed \(4\sigma\) and `crisis' entered the field's vocabulary --- a fixed point of the century course, and the moment the precision-amplifies-bias clock of Course~L became audible. First cited in \S\ref{sec:measure-history}.

\bibitem[Riess et~al.(2022)]{RiessSH0ES2022}
\textbf{Riess, A.~G., Yuan, W., Macri, L.~M., et al. (2022).} A comprehensive measurement of the local value of the Hubble constant with 1 km\,s\(^{-1}\)\,Mpc\(^{-1}\) uncertainty from the Hubble Space Telescope and the SH0ES team. \emph{Astrophysical Journal Letters, 934}, L7.
\par\noindent\textbf{Relevance.} The SH0ES measurement \(73.04 \pm 1.04\), quoted cohort-wide as \(73.0 \pm 1.0\) (the citation rule of the companion \ATen{} codebook); also the source of the deceleration and jerk priors \(q_0 = -0.55\), \(j_0 = +1\) held fixed by the H0DN analysis. First cited in \S\ref{sec:intro-tension}.

\bibitem[Riess et~al.(2024)]{Riess2024Cross}
\textbf{Riess, A.~G., Scolnic, D., Anand, G.~S., et al. (2024).} JWST observations reject unrecognized crowding of Cepheid photometry as an explanation for the Hubble tension at \(8\sigma\) confidence. \emph{Astrophysical Journal, 977}, 120.
\par\noindent\textbf{Relevance.} The JWST instrument cross-check on HST Cepheid photometry; part of the Priority-4 instrument-swap test that the corridor reading treats as confirming \(\epslocal > 0\) is not an HST artifact. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Riess et~al.(2024b)]{Riess2024Selection}
\textbf{Riess, A.~G., Scolnic, D., Anand, G.~S., Breuval, L., Casertano, S., et al. (2024).} JWST validates HST distance measurements: Selection of supernova subsample explains differences in JWST estimates of local \(H_0\). \emph{Astrophysical Journal}, in press (arXiv:2408.11770).
\par\noindent\textbf{Relevance.} The SH0ES side of the subsample dispute: on the same JWST fields the combined three-method result is \(72.6 \pm 2.0\), with the full-sample prediction for those galaxies \(72.8\) --- the CCHP--SH0ES split localized to sample selection, not measurement. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Riess et~al.(2025)]{Riess2025PerfectHost}
\textbf{Riess, A.~G., Li, S., Anand, G.~S., Yuan, W., Breuval, L., et al. (2025).} The Perfect Host: JWST Cepheid observations in a background-free SN~Ia host confirm no bias in Hubble-constant measurements. \emph{Astrophysical Journal Letters, 992}, L34. arXiv:2509.01667.
\par\noindent\textbf{Relevance.} The sharpest ladder-internal robustness test: a background-free Cepheid field returning \(73.49 \pm 0.93\) --- evidence that whatever moves the ladder is not photometric, quoted in the counterweight clause where it is first cited. First cited in \S\ref{sec:residual-verdict}.

\bibitem[Said et~al.(2025)]{Said2025DESIFP}
\textbf{Said, K., Howlett, C., Davis, T., et al. (2025).} DESI peculiar velocity survey --- Fundamental plane. \emph{Monthly Notices of the Royal Astronomical Society, 539}, 3627.
\par\noindent\textbf{Relevance.} The DESI fundamental-plane calibration entering the H0DN variant catalog (V03) and the O1 orthogonal path. One of the orthogonal-path legs of the H0DN consistency architecture, whose scope --- internal covariance, not external truth --- the certification-boundary analysis of Courses~D and~L delimits. First cited in \S\ref{sec:central-claim}.

\bibitem[Sandage and Tammann(1975)]{SandageTammann1975}
\textbf{Sandage, A., \& Tammann, G.~A. (1975).} Steps toward the Hubble constant. V. \emph{Astrophysical Journal, 196}, 313.
\par\noindent\textbf{Relevance.} The low camp of the factor-two war, primary source; the historical stalemate the century course narrates. The factor-two war is this paper's standing reminder that confident error bars have excluded the next era's value before --- the cautionary-precedents lesson of Part~VIII, one century early. First cited in \S\ref{sec:intro-tension}.

\bibitem[Sandage(1958)]{Sandage1958}
\textbf{Sandage, A. (1958).} Current problems in the extragalactic distance scale. \emph{Astrophysical Journal, 127}, 513.
\par\noindent\textbf{Relevance.} The second great halving: brightest ``stars'' recognized as HII regions, and the distance scale doubled again --- the correction that brought $H_0$ near its modern range. First cited in \S\ref{sec:intro-tension}.

\bibitem[Sandage(1962)]{Sandage1962}
\textbf{Sandage, A. (1962).} The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes. \emph{Astrophysical Journal, 136}, 319.
\par\noindent\textbf{Relevance.} The original redshift-drift proposal --- the expansion watched, not inferred; the purest channel in the over-determination program. First cited in \S\ref{sec:overdetermining}.

\bibitem[Sandage(1970)]{Sandage1970}
\textbf{Sandage, A. (1970).} Cosmology: A search for two numbers. \emph{Physics Today, 23}(2), 34.
\par\noindent\textbf{Relevance.} The famous framing of pre-1998 cosmology as the measurement of \(H_0\) and \(q_0\) --- the expectation the supernova teams were built to satisfy, and overturned. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Schmidt et~al.(1998)]{Schmidt1998}
\textbf{Schmidt, B.~P., Suntzeff, N.~B., Phillips, M.~M., et al. (1998).} The High-Z Supernova Search: Measuring cosmic deceleration and global curvature of the Universe using Type Ia supernovae. \emph{Astrophysical Journal, 507}, 46.
\par\noindent\textbf{Relevance.} The High-\(z\) team's method paper --- the search designed to measure deceleration that found acceleration. The method paper behind the discovery pair: the pipeline ancestor of every supernova standardization whose dust and population choices the residual anatomy now prices. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Sch\"oneberg(2024)]{Schoneberg2024BBN}
\textbf{Sch\"oneberg, N. (2024).} The 2024 BBN baryon abundance update. \emph{JCAP} (arXiv:2401.15054).
\par\noindent\textbf{Relevance.} The deuterium accountant: \(\omega_b = 0.02218 \pm 0.00055\) marginalized over reaction rates, with no microwave sky anywhere --- the audit that both anchors the CMB-free ruler and walls off the early-side fixes. First cited in \S\ref{sec:landscape-early}.

\bibitem[Schutz(1986)]{Schutz1986}
\textbf{Schutz, B.~F. (1986).} Determining the Hubble constant from gravitational wave observations. \emph{Nature, 323}, 310.
\par\noindent\textbf{Relevance.} The founding proposal: compact-binary waveforms as self-calibrating distance indicators --- the forty-year-old idea on which this paper stakes its verdict. First cited in \S\ref{sec:siren-principle}.

\bibitem[Scolnic et~al.(2022)]{Scolnic2022Pantheon}
\textbf{Scolnic, D., Brout, D., Carr, A., et al. (2022).} The Pantheon+ analysis: The full data set and light-curve release. \emph{Astrophysical Journal, 938}, 113.
\par\noindent\textbf{Relevance.} The Pantheon+ SN~Ia compilation; the deep Hubble-flow rung of the network and part of the calibration-debate context retired by variant V13. First cited in \S\ref{sec:routes-anatomy}.

\bibitem[Scolnic et~al.(2023)]{Scolnic2023CATS}
\textbf{Scolnic, D., Riess, A.~G., Wu, J., Li, S., Anand, G.~S., et al. (2023).} CATS: The Hubble constant from standardized TRGB and Type Ia supernova measurements. \emph{Astrophysical Journal Letters, 954}, L31. arXiv:2304.06693.
\par\noindent\textbf{Relevance.} The standardized-TRGB grid: \(73.22 \pm 2.06\) baseline with none of its \(108\) analysis variants below \(71.6\) --- the second ladder-internal robustness result in the counterweight clause. First cited in \S\ref{sec:residual-verdict}.

\bibitem[Scolnic et~al.(2024)]{Scolnic2024Coma}
\textbf{Scolnic, D., et al. (2024).} The Hubble tension in our own backyard: DESI's fundamental plane and the Coma cluster. \emph{Astrophysical Journal Letters, 979}, L9 (arXiv:2409.14546).
\par\noindent\textbf{Relevance.} The highest entry of the landscape: \(76.5 \pm 2.2\) from the DESI fundamental plane anchored on Coma --- whose distance rests on twelve Type-Ia supernovae, so the route is not independent of the standardization question it is sometimes cited against. First cited in \S\ref{sec:landscape-counterweight}.

\bibitem[Slipher(1917)]{Slipher1917}
\textbf{Slipher, V.~M. (1917).} Nebulae. \emph{Proceedings of the American Philosophical Society, 56}, 403.
\par\noindent\textbf{Relevance.} The systematic redshifts of the spirals --- the first observational hint, years before the law, that the universe is in motion. First cited in \S\ref{sec:intro-history}.

\bibitem[Spergel et~al.(2003)]{Spergel2003}
\textbf{Spergel, D.~N., Verde, L., Peiris, H.~V., et al. (2003).} First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters. \emph{Astrophysical Journal Supplement, 148}, 175.
\par\noindent\textbf{Relevance.} The WMAP concordance fit that made \(\Lambda\)CDM the standard model of cosmology --- the six-parameter framework whose \(H_0\) inference the tension now interrogates. First cited in \S\ref{sec:dark-energy-surprise}.

\bibitem[Stiskalek et~al.(2025)]{Stiskalek2025Void}
\textbf{Stiskalek, R., Desmond, H., \& Banik, I. (2025).} Testing the local supervoid solution to the Hubble tension with direct distance tracers. \emph{Monthly Notices of the Royal Astronomical Society, 543}, 1556.
\par\noindent\textbf{Relevance.} Field-level forward modeling of the CosmicFlows-4 velocities: the data prefer voids smaller than \(\sim\!70\)~Mpc, and the deep profiles that would help \(H_0\) most contradict the measured bulk flow --- the result that downgrades the KBC-deep scenario from contested to disfavored in the hole ledger. First cited in \S\ref{sec:hole-evidence}.

\bibitem[Stiskalek et~al.(2026)]{Stiskalek2026MWCepheids}
\textbf{Stiskalek, R., Riess, A.~G., Desmond, H., Lavaux, G., \& Scolnic, D. (2026).} Forward-modeling Milky Way Cepheids: selection effects and physical priors in the Gaia--HST calibration. \emph{Monthly Notices of the Royal Astronomical Society}, in press (arXiv:2603.09880).
\par\noindent\textbf{Relevance.} The rebuttal inside the statistics channel: a uniform-in-volume prior \emph{without} a selection model manufactures a \(\sim\!0.05\)~mag zero-point counter-bias, and the consistent treatment reinforces the published \(H_0\) --- the reason the channel is carried with a zero-edge range. First cited in \S\ref{sec:medium-channels}.

\bibitem[Verde et~al.(2024)]{Verde2024}
\textbf{Verde, L., Sch\"oneberg, N., \& Gil-Mar\'in, H. (2024).} A tale of many \(H_0\). \emph{Annual Review of Astronomy and Astrophysics, 62}, 287.
\par\noindent\textbf{Relevance.} The review whose many-measurements framing the H0DN network operationalizes; quoted in Part~I for the decade-long scrutiny of the tension. First cited in \S\ref{sec:why-h0dn}.

\bibitem[Vincenzi et~al.(2024)]{Vincenzi2024DES}
\textbf{Vincenzi, M., Brout, D., Armstrong, P., Popovic, B., Taylor, G., et al. (DES Collaboration) (2024).} The Dark Energy Survey Supernova Program: Cosmological analysis and systematic uncertainties in the DES-SN5YR data sample. \emph{Astrophysical Journal, 975}, 86.
\par\noindent\textbf{Relevance.} The dust channel's principal counterweight: the five-year systematics analysis argues in its error budget that part of the host-mass step is intrinsic rather than dust-driven --- cited with a body-level pointer, since the abstract is silent on the step. First cited in \S\ref{sec:medium-channels}.

\bibitem[Vogl et~al.(2025)]{Vogl2025EPM}
\textbf{Vogl, C., Taubenberger, S., Cs\"ornyei, G., et al. (2025).} No rung left behind: A cosmological distance ladder free of Type Ia supernovae. \emph{Astronomy \& Astrophysics, 702}, A41. arXiv:2411.04968.
\par\noindent\textbf{Relevance.} The SN~II expanding-photosphere ladder free of SNe~Ia; the EPM variant class (V05) of the H0DN catalog. Supplies the landscape's SNe~II entry (\(74.9 \pm 1.9\), statistical only): one of the geometric counterweight's members, carried at full strength and with its stated qualification, in Part~V and O-Hubble-53. First cited in \S\ref{sec:central-claim}.

\bibitem[Wang et~al.(2024)]{Wang2024Survey}
\textbf{Wang, B., L\'opez-Corredoira, M., \& Wei, J.-J. (2024).} The Hubble Tension Survey: 216 measurements, one field. \emph{Monthly Notices of the Royal Astronomical Society, 527}, 7692 (arXiv:2311.18443).
\par\noindent\textbf{Relevance.} The scatter audit: the literature's spread exceeds its quoted errors, and the headline five-sigma is nearer three in equivalent-frequency terms --- the reason the landscape is read in bands, not rows. First cited in \S\ref{sec:landscape-bands}.

\bibitem[Watkins et~al.(2023)]{Watkins2023CF4}
\textbf{Watkins, R., Allen, T., Bradford, C.~J., Ramon, A., Walker, A., et al. (2023).} Analyzing the large-scale bulk flow using CosmicFlows-4: increasing tension with the standard cosmological model. \emph{Monthly Notices of the Royal Astronomical Society, 524}, 1885.
\par\noindent\textbf{Relevance.} The contested companion evidence in the hole ledger: a bulk flow of \(419 \pm 36\)~km\,s\(^{-1}\) at \(200\,h^{-1}\)~Mpc, far above the \(\Lambda\)CDM expectation --- carried as contested, with selection and estimator critiques in play. First cited in \S\ref{sec:hole-evidence}.

\bibitem[Weinberg(1989)]{Weinberg1989}
\textbf{Weinberg, S. (1989).} The cosmological constant problem. \emph{Reviews of Modern Physics, 61}, 1--23.
\par\noindent\textbf{Relevance.} The definitive statement of the vacuum-energy problem that Course~F's seesaw addresses; the classic accounting of the \(10^{120}\). First cited in \S\ref{app:cF-qft}.

\bibitem[Weinberg(2008)]{WeinbergCosmology2008}
\textbf{Weinberg, S. (2008).} \emph{Cosmology}. Oxford University Press.
\par\noindent\textbf{Relevance.} Graduate reference for the Friedmann background and the acoustic-scale physics behind the two physical densities $\omega_b,\omega_m$ (App.~\ref{app:calc}). First cited in \S\ref{sec:friedmann-theorem}.

\bibitem[Whitbourn and Shanks(2014)]{WhitbournShanks2014}
\textbf{Whitbourn, J.~R., \& Shanks, T. (2014).} The local hole revealed by galaxy counts and redshifts. \emph{Monthly Notices of the Royal Astronomical Society, 437}, 2146.
\par\noindent\textbf{Relevance.} Regional count-and-redshift depths \(\delta \approx -0.15\) to \(-0.24\) over \(150\)--\(250\)~Mpc; the canonical-band evidence in the lift ledger. First cited in \S\ref{sec:hole-evidence}.

\bibitem[Whitford et~al.(2023)]{Whitford2023Flows}
\textbf{Whitford, A.~M., et al. (2023).} Evaluating bulk-flow estimators for CosmicFlows-4. \emph{Monthly Notices of the Royal Astronomical Society} (arXiv:2306.11269).
\par\noindent\textbf{Relevance.} The estimator audit of the CF4 bulk-flow excess: \(428 \pm 108\) km\,s\(^{-1}\) under the minimum-variance estimator with uncertainties treated fully --- the tension with \(\Lambda\)CDM softer than the headline, which is how the hole ledger carries it. First cited in \S\ref{sec:class6-voids}.

\bibitem[Wiltshire(2007)]{Wiltshire2007}
\textbf{Wiltshire, D. L. (2007).} Cosmic clocks, cosmic variance and cosmic averages. \emph{New J. Phys.} \textbf{9}, 377.
\par\noindent\textbf{Relevance.} The timescape proposal --- the one non-linear, two-phase reading that claims percent-level backreaction; discussed as the honest loophole (and its cost) in \S\ref{sec:backreaction}. First cited in \S\ref{sec:backreaction}.

\bibitem[Wojtak and Hjorth(2024)]{WojtakHjorth2024}
\textbf{Wojtak, R., \& Hjorth, J. (2024).} Consistent extinction model for type Ia supernovae in Cepheid-based calibration galaxies and its impact on \(H_0\). \emph{Monthly Notices of the Royal Astronomical Society, 533}, 2319. arXiv:2403.10388.
\par\noindent\textbf{Relevance.} The dust channel's opening move: imposing a Milky-Way-like extinction model on the calibrator galaxies fits the data better (\(\Delta\mathrm{BIC} = -11\)) and lowers the ladder to \(70.5 \pm 1.0\) --- a coherent \(\approx\!-2.5\;\kmsMpc\) shift, about half the gap. First cited in \S\ref{sec:medium-channels}.

\bibitem[Wojtak and Hjorth(2025)]{WojtakHjorth2025}
\textbf{Wojtak, R., \& Hjorth, J. (2025).} Stretch to stretch, dust to dust: lower-value local \(H_0\) measurement from two-population modeling of type Ia supernovae. \emph{Astronomy \& Astrophysics, 702}, A176. arXiv:2506.22150.
\par\noindent\textbf{Relevance.} The dust channel in full: a two-population Bayesian hierarchical remodeling of Pantheon+, driven by the observed stretch bimodality, returns \(H_0 = 70.59 \pm 1.15\) --- the central value the two-scenario arithmetic of \S\ref{sec:residual-verdict} adopts for the channel. First cited in \S\ref{sec:medium-channels}.

\bibitem[Wu et~al.(2023)]{Wu2023CATs}
\textbf{Wu, J., Scolnic, D., Riess, A.~G., Anand, G.~S., Beaton, R., et al. (2023).} Comparative analysis of TRGBs (CATs) from unsupervised, multi-halo-field measurements: Contrast is key. \emph{Astrophysical Journal, 954}, 87.
\par\noindent\textbf{Relevance.} The tip-contrast relation (\(\sim\!5\sigma\)): the TRGB's measured tip varies field to field at the \(0.05\)--\(0.1\)~mag level unless standardized --- the convention-level lever inside the subsample channel. First cited in \S\ref{sec:residual-anatomy}.

\bibitem[Zaborowski et~al.(2024)]{Zaborowski2024RsFree}
\textbf{Zaborowski, E., et al. (DESI Collaboration) (2024).} A sound-horizon-free determination of the Hubble constant from DESI. \emph{JCAP, 06}, 020 (arXiv:2411.16677).
\par\noindent\textbf{Relevance.} The equality-scale route: \(66.7^{+1.7}_{-1.9}\) (DESY5), \(67.9^{+1.9}_{-2.1}\) (Pantheon+), \(67.8^{+2.0}_{-2.2}\) (Union3) with no sound horizon anywhere --- the low half of the ruler-free span that closes ``shorten the ruler'' as a road to 73. First cited in \S\ref{sec:landscape-early}.

\end{thebibliography}
\GGBibEnd

\end{document}
