%% v20.0 (August 21, 2026) -- post-stamp repair, version deliberately
%%   unchanged (nothing is published yet).  Objection 48 quoted the paper
%%   as "eight Parts"; it has ten, measured as 9 GGPart + 1
%%   GGAppendixPart.  One word.
%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.38 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.32 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 9 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.
%% ====================================================================
%% ===================================================================
%% v15.31 (August 18, 2026) -- FP-2: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%% A3 needed ONE LINE, and the claim-derived probe is what establishes
%% that rather than an impression: zero simultaneity or publication
%% claims, zero reading-set enumerations, zero layer-count descriptions
%% of A11's cascade, zero catalogue spellings, and no inventory table
%% (A3's cohort plate is TikZ, a frozen standard, and is not redrawn).
%% All eleven A11 references here are architecture statements -- "the
%% capstone closes the cohort", "every downstream paper inherits its
%% license from the anomaly theorem proved here" -- and every one stays
%% true of a paper in preparation.
%%
%% The A11 bibliography entry said "Preprint, available at
%% arisaka-gg.org"; it now reads "In preparation ... no result of the
%% present paper depends on it."  That is exactly right for A3: it is
%% the paper the capstone depends ON, never the other way round.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.30 (August 17, 2026) -- MIGRATED TO STYLE FILE v11.5.
%%
%% FOUR MACROS THAT LIVED IN THE PAPERS COME HOME, AND NOT ONE PAGE
%% MOVES.  This paper's part in that:
%%
%% * \GGShowcaseHead removed; this paper prints the sans form with \bigskip -- the style file default,
%%   and needs no setup line.
%%
%% 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 v15.29 printed, except for the lines named above.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.28 -> v15.29, August 16, 2026.  THE SHOWCASE-HEADLINE RIDER.
%% \GGShowcaseHead rises from \large to \Large, one token.
%%   THE MEASUREMENT.  Across the cohort the showcase headline had four
%%   sizes: \large in the eight papers where I set it, \Large in C1 and
%%   C2, \LARGE in A4, \huge in A9, A10 and A11.  Those are not four
%%   arbitrary choices -- the rule followed was to match each paper's own
%%   existing headline, and four of the four larger values are the
%%   author's.  What the measurement shows is that MY DEFAULT was the
%%   outlier: every headline the author set himself is LARGER than
%%   \large, and on five papers out of five his practice is \Large or
%%   bigger.  On a page whose job is to be read in five minutes from a
%%   homepage, \large reads as a subsection head, not a title.
%%   So the house default rises to \Large, the smallest size he himself
%%   uses.  The cohort now spans \Large / \LARGE / \huge and nothing is
%%   below the author's own floor.
%%   VERIFIED: page count unchanged, no new overfull box, body
%%   page-for-page identical.
%% NEW PROSE IN THIS SHIP: none.
%% ------------------------------------------------------------
%% ===============================================================
%%  v15.26  (August 10, 2026)  --  STAGE 8: FINAL GATE AND SHIP
%% ---------------------------------------------------------------
%%  The last three unreferenced floats are now referenced inline:
%%  tab:assessment (the 10D-vs-6D obligation grid), tab:ogn-summary
%%  (what the ledger determines), and tab:checklist-matrix (the
%%  one-row-per-claim ledger, now introduced as the summary of the
%%  paper that it is).  Carried since the v15.16 baseline.
%%
%%  EVERY MECHANICAL CHECK IS NOW AT ZERO:
%%    errors 0 | undefined 0 | multiply-defined 0
%%    overfull hbox > 20 pt 0 (worst 1.5 pt)
%%    sparse pages needing a fix 0        (baseline 2)
%%    floats never referenced 0           (baseline 3)
%%    orphan bibliography entries 0
%%    duplicate short captions 0
%%    P7 residue 0 | two-Arisaka-key citations 0
%%    fonts not embedded 0 | JavaScript 0
%%    mixed US/UK spelling pairs 0
%%    group balance 2/2
%%  Style file verified byte-identical to [v15-Support]
%%  (md5 fb46abfe61e0ca24829cb5e78e4d184c).
%%  Built clean-room from the deployed source in three passes.
%% ===============================================================
%% ===============================================================
%%  v15.25  (August 10, 2026)  --  STAGE 7: VOICE AND ACCESSIBILITY
%% ---------------------------------------------------------------
%%  V1  Definition 9.1 and Definition E.1 were verbatim duplicates
%%      EXCEPT in clause (iv), where they gave J two different
%%      definitions -- and clause (iv) is the one the text flags as "a
%%      structural claim and not a convention".  The body now carries
%%      the appendix form, which is the one Step 4 of the OGN-5 theorem
%%      actually evaluates, and says so.
%%  V2  "Why this subsection exists." appeared as a paragraph heading
%%      three times.  Replaced by headings that say what the subsection
%%      adds, what is missing, and what is still taken as given.
%%  V3  Appendix A claimed to be "every abbreviation used in this
%%      paper".  It is not, in two directions: 26 of its rows are
%%      program names A3 never uses (they belong to the acronym plate
%%      and to companion papers), and SEVEN acronyms A3 does use were
%%      absent -- ABJ (4 uses), APS (11), TQFT, WGC (5), AHSS, SUGRA
%%      (31), IIB (4).  The seven are added and the caption now says
%%      what the table actually collects.  The 26 rows are KEPT: the
%%      acronym plate opposite uses them, and its own caption promises
%%      they are defined in the table.  Deleting them would trade one
%%      broken promise for another.
%%  V4  The letter Omega carries four unrelated meanings -- reservoir
%%      normalization, effective normalization, spin bordism group,
%%      cosmological density -- three of which appear within four pages
%%      of each other.  Stated once, in the front-matter Notation.
%%  V5  The A-genus and the Hirzebruch L-genus were used as black boxes
%%      and never glossed.  One sentence at first use.
%%  V6  Voice: three reflexive "not merely" constructions rewritten,
%%      and two formulaic "The honest X is" openings.
%%
%%  MEASURED AND NOT ACTED ON.  The audit reported "not merely" as a
%%  tic at 14 uses.  Reading all 16 in context, most carry a real
%%  contrast the sentence needs -- "not merely inaccurate; it is
%%  inconsistent", "global (not merely local)", "computed, not merely
%%  constrained".  Three were reflex and were rewritten; the remaining
%%  13 stay.  A frequency count is a hypothesis about a tic, not a
%%  finding.
%% ===============================================================
%% ===============================================================
%%  v15.24  (August 10, 2026)  --  STAGE 6: THE BIBLIOGRAPHY
%% ---------------------------------------------------------------
%%  Every change below was verified against arXiv, INSPIRE, or the
%%  published record before it was made.  Nothing was corrected on an
%%  auditor's word.
%%
%%  WRONG BIBLIOGRAPHIC DATA (4)
%%   B1 Schwarz (1996).  Printed title was "Anomaly-free supergravity
%%      and super-Yang-Mills theories in six dimensions".  The paper at
%%      hep-th/9512053 is "Anomaly-free supersymmetric models in six
%%      dimensions", PLB 371 (1996) 223-230.  Verified on arXiv.
%%   B2 Monnier (2018).  Printed as Commun. Math. Phys. 362 (2018)
%%      349-404.  It is Adv. Theor. Math. Phys. 22 (2018) 2035-2089,
%%      arXiv:1706.01903.  Verified on INSPIRE.
%%   B3 Bustamante et al.  Printed as Ann. Rev. Biochem. 74 (2005) 705.
%%      Volume 73 is 2004 and carries pages 705-748.
%%   B4 Penington et al.  Printed as JHEP 2020, 205.  Published JHEP 03
%%      (2022) 205, arXiv:1911.11977.  Verified on INSPIRE.
%%
%%  RELEVANCE NOTES CREDITING THE WRONG SOURCE (3)
%%   B5 Polchinski Vol. I was credited with the Green-Schwarz mechanism,
%%      which is in Vol. II -- and the paper's own appendix already
%%      sends the reader to Vol. II for it.
%%   B6 Bousso-Polchinski was credited with the ~10^500 count.  That
%%      figure is Douglas's, and this paper already cites Douglas.
%%   B7 Stong (1968) was credited with string bordism, which post-dates
%%      it by two decades.
%%
%%  WORKS USED AND NOT CITED (2)
%%   Atiyah-Patodi-Singer (1975), named ten times and the theorem the
%%   global argument runs on, and Stora (1984), the origin of the
%%   descent formalism the whole local argument uses.  Both added to
%%   the B list in alphabetical position and cited at first use, so
%%   neither is an orphan entry.
%%
%%  Bibliography integrity verified before and after: A-list 22 entries
%%  unchanged, B-list 86 -> 88, file length grew by exactly the length
%%  of the two inserted blocks.  Entries are spliced, never rebuilt --
%%  the A2 Stage 6 failure destroyed 96 kB of bibliography by
%%  reconstructing entries from parsed pieces.
%% ===============================================================
%% ===============================================================
%%  v15.23  (August 10, 2026)  --  STAGE 5: FLOATS, TABLES, AND
%%  THE PART IV LANGUAGE
%% ---------------------------------------------------------------
%%  PART IV, softened at the author's direction.  Two passages carried
%%  more force than the argument needs:
%%   - "The fifty-year description of the GS mechanism ... is a
%%     constitutional misreading" -> the bulk-side reading is offered
%%     as a re-description of the same mathematics, and the text now
%%     says plainly that which framing is more natural is a question of
%%     framing and not of fact, and that nothing in the theorem depends
%%     on the answer.
%%   - "The absence of AX-4 from Stratum 2 is the constitutional source
%%     of the string-theoretic landscape problem" -> a statement about
%%     what the two frameworks assume, noting that a framework without
%%     AX-4 is not required to have a unique vacuum and has never
%%     claimed it must.
%%
%%  FIGURES -- three defects that only a rendered page could show.
%%   F1  Figure 5's two edge labels were white-filled and midway on
%%       arrows across a 15 mm gap, so they painted over the arrows
%%       completely: the "6D -> 7D -> 8D ladder" printed as three
%%       disconnected boxes with two phrases floating between them.
%%       Labels moved above the line; the ladder now reads as a ladder.
%%   F2  Figure 7's right-hand arrow terminated at a bare coordinate
%%       that landed inside a node declared LATER, so TikZ painted the
%%       node over the arrowhead.  The arrow is now drawn after the
%%       node exists and anchored to it.
%%   F3  Figure 4's caption and the body sentence it matches called
%%       X_4's four contributions "curvature-squared two-forms".  They
%%       are four-forms, and the body sentence contradicted itself in
%%       one clause ("two-forms ... degree-four characteristic class").
%%
%%  CAPTIONS THAT PROMISED WHAT THE FLOAT DID NOT CARRY
%%   - Table 3: "every ingredient is supplied by a named constitutional
%%     clause" -- two of six rows do.
%%   - Table 7: "a status and a date" -- one of twenty cells has a date.
%%   - Figure 1: pointed at A1's Figure 4, which is a different picture.
%%   - Figure 6: "five-step flowchart"; the plate labels two steps.
%%   - Table 17: "occurs uniquely at (N,L)=(3,5)" contradicted
%%     Theorem 9.5, which gives the family (5k,3k).
%%   - Table 16 row 1: named rank-one structure as the mechanism that
%%     kills tr R^4; it is the gravitational completion.
%%   - Inward-map header "Source paper / version" promised version data
%%     that the table (correctly) does not carry.
%%   - Table 20's Status column widened and set ragged right; it was
%%     printing "Proved, unre-stricted" as a justified fragment.
%%
%%  NOT DONE, deliberately: the 16/3/1 split of table type sizes
%%  (footnotesize / small / normal).  The house standard is \small, so
%%  bringing A3 into line means promoting sixteen dense tables -- a
%%  cohort-wide decision with real overflow risk, not an A3 fix.
%%  Flagged in the report for the author.
%% ===============================================================
%% ===============================================================
%%  v15.22  (August 10, 2026)  --  STAGE 4: THE HONESTY ALIGNMENT
%% ---------------------------------------------------------------
%%  The paper carried two registers that disagreed.  The technical
%%  remarks were careful; the readable layers undid the fences.  Every
%%  change here moves in the direction of claiming less.
%%
%%  PART VII (For the Curious Reader) -- the plain-language layer, and
%%  the worst offender.  At thirteen sites it told a non-specialist
%%  that the ledger "fixes the number of families of matter" and
%%  predicts a correction to "how compact stars weigh".
%%    - The family count: Section 9.1 says N_c = N_g = 3 is a
%%      coincidence on the boundary and that this paper never uses it;
%%      O12 lists the generation count as GENUINELY OPEN.  Part VII now
%%      says the ledger does NOT fix it, in as many words.
%%    - The stellar shift: it appears in NO falsifier of Section 18, is
%%      derived nowhere in this paper, and I could not verify it in any
%%      companion -- C3-BH carries Tolman/TOV material but no
%%      mass-radius prediction.  Rather than invent an attribution, the
%%      passages now carry the predictions this paper does derive: the
%%      3.85 TeV lock cluster, the kaxion window 1.59-2.13 GHz, the
%%      inverse fine-structure constant 137, and proton stability.
%%  NOTE: the auditor reported this as four generation-count sites.  My
%%  first check for "generation" found ONE (about Eddington's
%%  contemporaries) and I nearly cleared the finding.  The text says
%%  "families of matter".  Scope by meaning, not by word.
%%
%%  BODY
%%   - "razor-thin" withdrawn: Remarks 6.3 and 7.10 say the anomaly
%%     test is passed by a family of Abelian coefficients.
%%   - The global theorem's scope now reads the same in Section 22 and
%%     in the status table as in the Abstract: unconditional on the
%%     intended class, AND unrestricted once the fine print is removed.
%%     The readable layers had been UNDERselling a proved result.
%%   - The rank-one corollary no longer claims to be an independent
%%     measurement.  The matrix is DEFINED as an outer product; its
%%     rank is one by construction.  Three sites corrected, including
%%     a caption and an objection response.
%%   - The trace-ratio normalization argument claimed the ratio is
%%     invariant under rescaling.  It is not: it scales by
%%     (lambda/mu)^2.  A2's POS-6 primitivity argument adopted, so the
%%     two papers now agree about why the ratio is 2/5.
%%   - k=1: the claim that it is "not a consequence of the trace-ratio
%%     equation" softened to what is true.
%%   - Five live referee/reviewer -> 0.  Two internal-draft-history
%%     sites rewritten.
%% ===============================================================
%% ===============================================================
%%  v15.21  (August 10, 2026)  --  STAGE 3: CROSS-REFERENCE INTEGRITY
%% ---------------------------------------------------------------
%%  Every pointer re-walked once, now that the structure has settled.
%%  Ground truth taken from the companions' own .toc files, never from
%%  A3's text.
%%
%%  A1 OFF-BY-TWO (13 rows).  A3 put the four axioms at A1 SS2.2-2.5
%%     and the seven postulates at A1 SS3.3-3.9.  A1 v15.21 has them at
%%     SS4.2-4.5 and SS5.3-5.9: A1's front matter grew two sections and
%%     A3 was never re-walked.  Also GG-Constitution: A1 S2-S3 -> S4-S5.
%%  A2 POINTERS (3).  GG-6 -> A2 S12 (was S8); QGL -> A2 S19 with
%%     T-GG6-8 (was S10 with T-GG6-3, wrong in both); Q_B -> A2 S19
%%     (was S15).  Audit trail: A2 carries NINE theorems, T-GG6-0..8,
%%     not eight.
%%  A5 POINTER.  "A5-GDOS S62.4" -> S69.4.  A5 S62 has only S62.1 and
%%     is the GDOS-to-Qi-4 bridge; the second law as a data-processing
%%     inequality, which is what the sentence actually cites, is S69.4.
%%  SELF-POINTERS (4).  BI-6, X_4, H_3 pointed at S3 (the general GS
%%     mechanism) and OGN-5 at S7 (the benchmark spectrum).  All four
%%     converted to live \ref so they cannot drift again.
%%  TRIAGE RECOUNT.  O9's entry says "Resolved (constructive)" and
%%     O10's says "Resolved for the GG gauge content"; the summary
%%     counted both as partially resolved.  Now 8 Resolved (O1-O7, O9),
%%     2 Partially resolved (O8, O10), 2 Genuinely open (O11, O12).
%%  MINIMAL SPECTRUM PROBLEM (2 sites).  Pointed at Section 14, where
%%     it is not stated.  It is entry O8 of the triage.
%%
%%  T-BI6 IDENTIFIERS -- frozen, with a concordance.  17 of 18 tags do
%%  not match the printed theorem numbers, and the naming convention
%%  claimed they carry the section number.  Regenerating them would
%%  break every pointer companions already carry, so they are kept and
%%  a concordance is printed in Appendix J, built from live \ref so it
%%  cannot go stale.  The naming convention now says plainly that a
%%  global identifier is unique but is not a section number.  One real
%%  collision fixed: T-BI6-10.2a (a suffixed variant) -> T-BI6-10.1.
%% ===============================================================
%% ===============================================================
%%  v15.20  (August 10, 2026)  --  NOTATION PLACEMENT CORRECTED
%% ---------------------------------------------------------------
%%  Reverses the front-matter placement of the abbreviation table
%%  introduced in v15.18, at the author's direction.
%%
%%  What was wrong: the six-page three-column table was moved to the
%%  front matter to fix a real but narrow defect -- nine house
%%  acronyms are used in the Abstract, Executive Summary and Preface
%%  without expansion, and the Tr/tr convention is used in the paper's
%%  central equation from page 1 while being stated once on page 162.
%%  Six pages of table is the wrong instrument for that, and 28 of the
%%  table's 87 rows are names imported from A5/A6/A7 that A3 never
%%  uses -- so a third of what stood in front of the reader was not
%%  even relevant to this paper.  It pushed Part I from page 13 to 19
%%  and stranded the Preface, which is the entrance, behind a wall.
%%
%%  What is here instead:
%%   - the full dictionary is back as Appendix A of the Technical
%%     Appendices, where a dictionary belongs;
%%   - a ONE-PAGE "Notation" block sits before the Preface, carrying
%%     the three-trace convention and exactly the names that appear in
%%     the Abstract, Executive Summary or Preface before they are
%%     defined -- one line each, with a pointer to Appendix A.
%%  Front matter: Notation p.12, Preface p.13, Part I p.14.
%%
%%  Still to come in Stage 7, and unaffected by this: expanding each
%%  acronym at first use, and cutting the 28 dead rows from Appendix A.
%% ===============================================================
%% ===============================================================
%%  v15.19  (August 10, 2026)  --  STAGE 2B: THE PART PROSE
%% ---------------------------------------------------------------
%%  New reader-facing prose only.  No physics, no structure changed.
%%
%%  Ten Part openings brought from 71-100 words (a quarter page) to
%%  the house standard of half a page to a page, and nine Part
%%  closings written where there were none or where the old
%%  one-sentence italic "bridge" stood.  Register: undergraduate STEM
%%  including the life sciences.
%%
%%   - Part I    opening 419 w  / closing 601 w (what the reader has,
%%               and where the cost of the proof falls)
%%   - Part II   opening 372 w (the eight proof obligations in plain
%%               English) / closing 408 w -- states what is proved
%%               locally, what globally at TWO strengths, what the
%%               benchmark does NOT show, and that the anomaly test is
%%               passed by a family of coefficients, not by BI-6 alone
%%   - Part III  opening 341 w / closing 362 w -- what each of the five
%%               integers rests on, and that AX-4 is consumed at one
%%               step; says plainly that the generation count is NOT
%%               derived
%%   - Part IV   opening 346 w / closing 331 w -- five differences,
%%               one sentence each, no superiority claim
%%   - Part V    opening 308 w (the derived / conditional / open
%%               grading fixed in advance) / closing 356 w
%%   - Part VI   opening 196 w (there was NONE) / closing 270 w
%%   - Part VII  opening 279 w / closing 201 w (there was none)
%%   - Part VIII opening left at 754 w -- already above standard;
%%               closing 237 w (there was none)
%%   - Part IX   opening 270 w / closing 175 w
%%   - Part X    opening 202 w / closing note added
%%
%%  The three remaining Part-to-Part italic bridges are gone (end of
%%  Parts II, III, IV).  The inter-appendix bridges inside Part VIII
%%  are a different device and are kept.
%%
%%  Also: "Three routes through this paper" added at the end of the
%%  roadmap -- the general reader, the specialist auditing the proof,
%%  and the reader arriving from another cohort paper.  The paper had
%%  four navigation devices and no reading path.
%%
%%  Also: Section 5.9, a two-equation preview of Section 6 that
%%  conceded its own redundancy, folded into Section 6 as the
%%  sphere-background corollary it is.
%%
%%  CAUGHT BEFORE SHIPPING: six British spellings in my own new prose
%%  (colour x2, neighbour x2, programme, sceptical) plus licence x2,
%%  found by the before/after spelling word-list diff; three \ref
%%  targets in the new prose that did not exist (two labels added, one
%%  repointed); one undefined reference.
%% ===============================================================
%% ===============================================================
%%  v15.18  (August 10, 2026)  --  STAGE 2A: STRUCTURE
%% ---------------------------------------------------------------
%%  Architecture only.  No physics changed.  The paper goes from
%%  nine Parts to ten and loses about eight pages of duplication.
%%
%%  BLOCK MOVES
%%   M1  Appendix K "For the curious reader" (25 plain questions,
%%       12 pp) promoted out of the appendix run to its own Part VII,
%%       placed after Conclusions, as a NUMBERED section.  It was the
%%       paper's only plain-English entry point and it sat on p.193
%%       of 249, referenced from nowhere.  Consequence: the appendix
%%       run L..U auto-relabels to K..T.
%%   M2  Appendix A (abbreviations) -> front matter, before the
%%       Preface, so the Preface still runs into the Introduction.
%%       It sat 121 pages after first need.
%%   M3  The Author's Note -> end of the Common Objections Part.  It
%%       was inside "Technical Appendices", whose own subtitle says
%%       that Part ends with the theorem checklist.
%%   M4  Section 17 (the GS normalization Omega) -> Part II as a
%%       subsection of Section 7, where the object is constructed.
%%       It contained a subsection headed "Why this sketch is
%%       included here"; a section that must justify its own location
%%       is in the wrong location.
%%
%%  DELETIONS AND MERGES
%%   D1  Section 27 deleted: a restatement of Section 12.8, running
%%       the Riemann-to-Einstein analogy a second time near-verbatim
%%       ("sixty years" the first time, "sixty-one" the second).  One
%%       paragraph folded into Section 26.  Removes a sparse page at
%%       the root rather than by reflow.
%%   D2  Section 16 deleted: summarised Part IV three pages after
%%       Part IV ended.  Its one load-bearing sentence becomes the
%%       interim Part IV closing.
%%   M5  Sections 23 + 24 + 25 -> one section.  Section 24 duplicated
%%       Section 1.4 almost sentence for sentence.
%%   M6  Section 15 and the Ten Structural Innovations of Section 20
%%       -> Section 14, retitled "Significance, uniqueness, and what
%%       is new".  The "nobility" paragraph is gone: a framework
%%       awarding itself a virtue is not a result.
%%   R1  Section 19's two audit tables and the audit trail -> Appendix
%%       J, the dependency ledger.  The maintenance register went with
%%       them and was rewritten out ("current deployed version",
%%       "future reviewers", "as of the present version").
%%   R2  Section 20.1 (cross-cohort naming convention) -> the front
%%       matter Notation section.
%%   X1  The trace-ratio theorem was derived three times (8.1, 9.3,
%%       App. E.6).  Section 9.3 now points instead of re-deriving.
%%       NOT DONE, deliberately: the plan called for merging Section 8
%%       into Section 9.  That renumbers Theorem 8.1, the most-cited
%%       A3 result across the cohort.  De-duplication achieved without
%%       it; see the report.
%%
%%  DESCRIPTIONS THE SURGERY MADE FALSE
%%   - roadmap plate rebuilt for ten Parts; caption rewritten; the
%%     stale label fig:A3-eight-parts-flowchart renamed (it has been
%%     wrong since the paper had eight Parts).
%%   - "nine Parts" / "nine-Part arc" / "eight further Parts" at six
%%     prose sites.
%%   - Section 2's title claimed a 75-year arc for a span the section
%%     itself runs from 1902 to 2021.
%%   - Part VIII's opening said "three groups" and named two, and its
%%     cohort roster omitted P6 and both white papers.
%%   - the Common Objections Part opening no longer swallows Part VII.
%%
%%  DEFECTS INTRODUCED BY THIS SHIP AND CAUGHT BEFORE IT SHIPPED
%%   - a \begingroup wrapper was orphaned by a cut, leaking
%%     \emergencystretch\sloppy across the last hundred pages;
%%   - the nobility-paragraph regex stopped at the first backslash and
%%     left a tail spliced onto a dangling \ref;
%%   - three hand-tuned \pagebreak in the abbreviation table, placed
%%     for its old position, produced a header-only page in its new
%%     one;
%%   - the acronym-flow plate's [H] made its caption spill to a page
%%     of its own -> [htbp].
%%  Sparse pages: 2 must-fix at baseline -> 1 (the remaining one is
%%  pre-existing and is Stage 5).
%% ===============================================================
%% ===============================================================
%%  v15.17  (August 10, 2026)  --  STAGE 1: THE HARD ERRORS
%% ---------------------------------------------------------------
%%  Physics and arithmetic corrections.  No structural change.
%%
%%  (1) Theorem 6.20's worked example rebuilt.  The truncation
%%      argument (strike every U(1)_Qi-charged species from the
%%      four-component benchmark) is FALSE: the residue
%%      8 C(1/3,0) + 4 C(2/3,0) forces Omega' = 3 and gamma = 4/3,
%%      which demand an F_B^4 coefficient of 8/3 against the 4/3 it
%%      carries.  Replaced by a constructive three-component
%%      reservoir: the general closure condition N S_4 = 3 S_2^2 with
%%      gamma = 6 S_2 / N, and the minimal solution
%%      52 C(0,0) + 44 C(1/3,0) + 4 C(2/3,0) at Omega' = 25, whose
%%      gravitational sector closes at n_SD = 4 Omega', n_F = 32 Omega'.
%%      Theorem 6.20 stands; only its worked example was wrong.
%%  (2) The 6D N=(1,0) supergravity gravitational constraint was
%%      printed 9 n_T - n_H + n_V = 273 at four sites.  It is
%%      n_H - n_V + 29 n_T = 273, the form the paper already printed
%%      correctly in Section 11 and in the comparison table.
%%  (3) Appendix G's "complete row-reduction" carried three wrong
%%      right-hand sides (-4/25 -> -40/25; 4/75 -> 8/75 in (rr8);
%%      12/75 -> 8/75).  The printed final answers were right and did
%%      not follow from the printed chain.
%%  (4) pi_6(SU(5)) = 0 was justified by pi_k(SU(N)) = 0 at k = 2N,
%%      which is false (it is Z_{N!}) and was evaluated at k = 6 with
%%      2N = 10.  Replaced by the stable-range argument.
%%  (5) "Complexity of Lambda^2 requires L >= 5" is false at L = 3,
%%      where Lambda^2(3) = 3bar is complex.  L <= 4 is now excluded
%%      case by case, by chirality; the L = 3 package is vector-like.
%%  (6) Rank-one sufficiency assumed positive semidefiniteness, which
%%      is what it was proving.  Positivity is now a stated hypothesis
%%      and the wrong-sign case is named.
%%  (7) (H1)-(H4) were listed permuted, with the theorem's conclusion
%%      substituted for one hypothesis, at two sites.  Both corrected
%%      against the authoritative Appendix I.3 statement.
%%  (8) Omega: the bare symbol carried both the orbit multiplicity
%%      (in 25Z) and the late effective normalization (driven to 1).
%%      Subscripted at the two sites where they collided.
%%  (9) Section 22 said "a fine-structure constant of 137"; the
%%      prediction is alpha^{-1}(0) = 137.
%% (10) H_3 was called a two-form in the abbreviation table.
%% (11) Vanishing conditions: six -> eight, and 10 + 8 = eighteen, at
%%      six sites.  The six rank-one 2x2 minors of Objection 13 are a
%%      different six and were protected.
%% (12) A spliced, ungrammatical sentence at the payoff of Sec. 5.2.
%% (13) Section 25 pointed "below" to a section in the previous Part.
%% (14) The eight proof obligations collided with the twelve Open
%%      Problems and the numbered Objections; re-lettered (PO1)-(PO8).
%% (15) "Theorem 8.1 (Section 9)" and "Open Problem O10 (Section 6)".
%% (16) Two missing labels added; version, PaperDate and the separate
%%      title-page cover date synchronized.
%% ===============================================================
%% ------------------------------------------------------------------
%%  v15.16 (August 9, 2026): the curious-reader cluster.
%%  Twenty-five broad-audience questions open the Common Objections
%%  Part, ahead of the specialist clusters, on the model of
%%  W1-Symmetry, W2-Holonomy and A10-Cosmos.  Eight are the spine
%%  those three share, answered here in this paper's own vocabulary;
%%  seventeen are questions only an anomaly-cancellation paper can be asked: what an anomaly is,
%%%%  where whole numbers come from, and why consistency predicts stars.
%%  The introduction names the hardest of the set.
%% ------------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.15 (August 9, 2026): two cohort rules applied.
%%  (1) British doubled -l corrected in prose only -- 0 site(s);
%%      9 further site(s) protected as comment, identifier or
%%      bibliography and deliberately untouched.
%%  (2) Ragged-right on 1 p-column(s) across 1 multi-p-column
%%      table(s): a narrow prose column cannot be justified
%%      without a visible interword stretch, and an overfull box
%%      under the gate's tolerance can still eat the gutter.
%% ------------------------------------------------------------------
%%% ==================================================================
%%%  A3-BI6  v15.14  (August 8, 2026)
%%%
%%%  (1) SHARED PLATE: the twenty-two-paper acronym plate is taken
%%%      verbatim from A1-Constitution, which is now the cohort
%%%      copy.  It was carried in four source states, and five of
%%%      the six differed in a way that PRINTED: the sub-line ran
%%%      at \normalsize with no wrap and the GRIP entry overlapped
%%%      the Qi-HAL box in the right-hand column.  A1's copy sets
%%%      it at \footnotesize wrapped to 86 mm and clears.  The
%%%      Q_B box reads "energy--information Arisaka constant" on
%%%      the author's decision: Q_B applies beyond biology.
%%%  Body text, values and figures: otherwise UNCHANGED.
%%% ==================================================================
%% ===================================================================
%%  A3-BI6 v15.8  (August 7, 2026)  -- stage 2 of the v16 standards.
%%  Every bare grid in the running text promoted to a numbered,
%%  captioned float (the long one to a longtable, which can break);
%%  every thin caption rewritten as a paragraph that says what the
%%  object shows, how to read its columns, and what to take away.
%%  Each promoted table's lead-in sentence was rewritten to name it
%%  by \ref, because a floating table with a 'listed below:'
%%  lead-in is the worse defect.
%%  Recipe: [v15-Support]/ship_stage2.py + a3_captions_v158.py.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A3-BI6-v15.7  (August 7, 2026)  -- the v16 mechanical sweep.
%%  Shared cohort plates synced from A5; the house flowchart palette
%%  applied (no bare \draw, box fills within !8..!18, keyword widths
%%  made explicit); every flowchart re-weighted so its lightest box
%%  rule RENDERS at the house 0.521pt and nothing above 0.956pt.
%%  Gate numbers are identical to the previous version: this sweep
%%  changes ink, not layout.
%%  Recipe: [v15-Support]/sweep_v16.py + palette_fix.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.6  (August 6, 2026)  -- short captions repaired.
%%  The old extractor accepted a colon as a sentence terminator, so
%%  3 List-of-Tables entries printed as fragments ending in one --
%%  "Outward map:", "Inward dependency map:" -- with the clause the
%%  colon introduces missing.  Each now completes that clause.  The
%%  printed captions are untouched; only the optional argument, which
%%  is what the List of Tables prints, changes.
%%  Recipe: [v15-Support]/fix_short_captions.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.5  (August 6, 2026)  -- Figure 3 on a page of its own.
%%  The eight-Parts roadmap becomes a [p] float centred by
%%  \vspace*{\fill}, so the plate and its caption have the page to
%%  themselves and the closing paragraph starts on the next one --
%%  while the page in front stays full, which a \clearpage would not
%%  have allowed.  Built against style v11.4, which gives every
%%  figure-and-caption block an extra line of space on each side.
%%  Recipe: [A3-BI6]/[Scripts]/ship_a3_v155.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.4  (August 6, 2026)  -- two refinements.
%%  (1) Executive Summary trimmed by 198 characters, about two printed
%%      lines, so the Keywords block no longer spills onto page 5.
%%      The cut is in the closing reflection and every fact it drops
%%      is stated earlier in the same summary (R1, R6, and the
%%      trace-ratio lock in the bulk-side-rewrite paragraph).
%%  (2) the eight-Parts roadmap plate now precedes the closing
%%      paragraph of its section, so the short page falls at the
%%      Part break instead of in the middle of Part I.
%%  Recipe: [A3-BI6]/[Scripts]/ship_a3_v154.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.3  (August 6, 2026)  -- figures, and the Letter overrun.
%%  The baryon generator becomes a display: inline it was a single
%%  unbreakable atom and came back 60.1pt over on Letter, where the
%%  A4 baseline's worst box is 11.8pt.  Two Figure 4 titles lifted to
%%  a clean row (coordinates only).  Figure 3 enlarged to the house
%%  7.79pt type size, its \null\vfill frame removed first.
%%  Recipe: [v15-Support]/ship_a3_v153.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.2  (August 6, 2026)  -- near-empty pages closed.
%%  0 float site(s) adjusted, each one measured: a page over 40%%
%%  empty (outside a Part boundary) was rendered, the float that
%%  ejected it identified, and only that float released.  Causes:
%%  \clearpage in front of a [p] float, \clearpage behind one
%%  (it flushes the float queue and ships the text page short),
%%  and [H] on a full-page float.  Recipe: [v15-Support]/ship_v152.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.2  (August 6, 2026)  -- near-empty pages closed.
%%  3 float site(s) adjusted, each one measured: a page over 40%%
%%  empty (outside a Part boundary) was rendered, the float that
%%  ejected it identified, and only that float released.  Causes:
%%  \clearpage in front of a [p] float, \clearpage behind one
%%  (it flushes the float queue and ships the text page short),
%%  and [H] on a full-page float.  Recipe: [v15-Support]/ship_v152.py.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v15.1  (August 6, 2026)  -- front matter and tables.
%%  List of Tables added after the List of Figures; 4 short captions
%%  added so every list entry is one line; 9 table(s) given a font
%%  size (none declared = 11pt body size); 1 table(s) widened and 5
%%  narrowed to the true measure (1 - 2*tabcolsep*ncol/textwidth).
%%  Registered standard: GG-Standard-Front-Matter-v1.0.
%%  Recipe: [v15-Support]/ship_v151.py.
%% ===================================================================
%%% ===================================================================
%%%  A3-BI6 v15.0  (August 6, 2026)  -- American spelling, US Letter.
%%%  (1) British to American spelling throughout the running prose:
%%%      -ise/-yse to -ize/-yze in verb forms only, -our to -or, -re to
%%%      -er, doubled-l inflections reduced, and the individual words.
%%%      Nothing outside running prose is touched -- cross-reference and
%%%      citation keys, the bibliography and its cited titles, every
%%%      macro name, and the changelog all keep their spelling.  Noun
%%%      forms identical in both varieties (analysis, realism, organism,
%%%      specialist, emphasis) are left alone.  420 words changed,
%%%      91 occurrences protected.
%%%  (2) a4paper -> letterpaper.  US Letter is wider and shorter:
%%%      \textwidth +16.79pt, \textheight -50.08pt.  The height loss
%%%      dominates, so the paper reflows and grows.
%%%  Recipe: [v15-Support]/ship_v150.py; cohort impact in GG-Report-214.
%%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.21  (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.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.20  (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.
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A3-BI6  v13.19  (August 5, 2026)
%%  CHANGELOG (v13.18 -> v13.19): P2/P7 merger ripple, ship 4 of 4.
%%    * P7-Inflow is retired into P2-N3 Appendix H (P2 v15.2), so the
%%      key ArisakaP7Inflow is retired here: two prose mentions now
%%      point at that appendix, and the ArisakaP2N3 relevance note
%%      inherits what the P7 note carried (the primitive lift, the
%%      doubling, the unit reservoir, the a=1 refinement, and what
%%      the pushforward proves).  Nothing else changes.
%%  Script: ship4_repoint.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-A3-BI6 v13.16 -- Step 3 of the OGN-5 theorem no longer
%%  derives the colour index from the trace ratio, which is computed on
%%  the 3+2 split and therefore consumes what it concludes.  N = N_c = 3
%%  now follows from the dual lock and the block embedding (T-BI6-10.2a),
%%  consuming no measured quantity; the trace ratio becomes an
%%  independent check.  J = 2N+1 is labelled a structural claim of AX-4.
%%  Three counts of three get three symbols: N_c colour, N_g generations,
%%  N_f flavours untouched.  The appendix copy is synced.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.15  (July 30, 2026) -- P7-Inflow cross-reference update.
%%  (1) NEW remark rem:one-refinement after thm:refinement-unique: the
%%      a = 1 characteristic element = P2's toll ladder q(x)=x(x-1)/2
%%      = the refinement P7-Inflow pushes forward to the Berry bundle
%%      (one refinement, three appearances).  (2) The two "integrality
%%      lemma L-INT" sites conformed to P2 v14.2 canon: "integrality
%%      theorem L-INT (residue R-INT)", with the benchmark discharge
%%      by the inflow member recorded at the cohort-map site.  (3) NEW
%%      bibitem ArisakaP7Inflow (house style).  NO physics touched.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.14  (July 30, 2026)  -- P2-N3 v14.0 conformance.
%%  Two sites describing the 3pi budget as "a named conjecture fenced in
%%  P2" updated: the budget is now delivered by P2's capacity theorem
%%  T-N3-6, conditional on the coupling clause + integrality lemma L-INT.
%%  A3's own quoted value and everything else unchanged.  A3's T-BI6-9.0
%%  (N-free L = 5) is now load-bearing downstream (P2 App C cites it).
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.13  (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.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.12  (July 29, 2026)  -- Report 66: O9 closed
%%  NEW Subsection 6.x "The self-dual partition function": why the field
%%  has no Lagrangian (derived); holomorphic factorisation and the
%%  intermediate Jacobian, with the Hodge-star complex structure and the
%%  symplectic pairing derived on the page; theta blocks = quadratic
%%  refinements via the rank-one multiplier computation; T-BI6-7.7 (the
%%  refinement on any unimodular lattice is unique -- two-line coset
%%  proof); the twist-exclusion lemma (integrality failure + sign flip +
%%  MSMF); the supersymmetry audit table; T-BI6-7.8 (the partition
%%  function, canonical theta block, single-valued by T-BI6-7.6).
%%  Conforming edits: O9 re-graded Resolved (constructive) with the
%%  numerical sub-task downgraded to an error-catching cross-check; the
%%  scope remark; clause (C3); the checklist row; the O10 parenthetical.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.11  (July 29, 2026)  -- Report 64: O10 discharged
%%  NEW Subsection 6.x "Unrestricted global anomaly freedom": the Wu
%%  formula derived by splitting principle; Sq^2 as a derivation on the
%%  mod-2 cohomology of BG; the vanishing of the reduced spin bordism
%%  groups in degrees 5 and 7 by a complete AHSS computation; signature
%%  normalisation of fillings by HP^2 surgery; and T-BI6-7.6: the global
%%  phase is one for every large gauge transformation of arbitrary
%%  topological type, on every compact spin base, at every point of the
%%  coefficient family.  Conforming edits: both admissible-class
%%  definitions, the scope remark, the restrictiveness passage, the O10
%%  status, the pi_6 proposition, the appendix bounding lemma, the
%%  abstract, two checklist rows, the ladder lead-in.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.10  (July 29, 2026)   -- Report 63, tightenings T-1/T-2/T-3
%%  T-1  The BI-6 clause of the admissible class is stated for a rational
%%       coefficient family rather than a single value.  Every step of the
%%       anomaly chain was audited: only rationality is ever used.  The
%%       global theorem therefore holds for the family, BI-6 being one
%%       point of it, and no longer inherits conditionality from the
%%       trace-ratio identity.  New remark states the scope and answers
%%       the objection the generality invites.
%%  T-2  New T-BI6-6.2 in Part II: single-tensor inflow requires the
%%       irreducible quartic and cubic indices of the charged content to
%%       vanish.  A statement about the mechanism, not about GG-6.  The
%%       L-selection theorem now cites it instead of re-deriving it.
%%  T-3  The model-level result is upgraded from an existence witness to a
%%       characterisation: the complete affine solution family is given at
%%       arbitrary Abelian coefficient, reducing to the printed family and
%%       then to the benchmark.
%% ===================================================================
%% ===================================================================
%%  A3-BI6 v13.9  (July 29, 2026)
%%  1. Dependencies paragraph, inward-map row and audit-trail bullet
%%     restated: this paper consumes the bulk as OBJECT (clauses C1-C4 of
%%     the admissible class), not as companion theorems.  Verified: no
%%     proof and no theorem environment in this paper references A2.
%%  2. Three sites that described the BI6 Abelian entry as an output of
%%     the microscopic calculation corrected: rank-one fixes the direction
%%     and the gravitational/non-Abelian entries, not the Abelian entry.
%%  3. NEW: T-BI6-9.0, the unified block size from single-tensor
%%     consistency.  B = A_3 = L-5 on the core package, both vanishing
%%     only at L = 5.  Free of N at every step.  OGN-5 Step 2 re-based on
%%     it; the appendix L=2 corollary re-graded to a consistency check.
%%  4. Abstract conformed.
%% ===================================================================
%% ===================================================================
%% v13.8 (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.
%% ===================================================================
%% ===== v13.7 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F1: Suffix-roster rename-history clause removed ->
%%     'GRIP (A6)'.
%% (2) F2: P5-CPT bibitem title text -> current P5 v3.42
%%     cover title (label/key untouched).
%% (3) PaperDate July 19 -> July 20, 2026 (ship date).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.6 (2026-07-19): WAVE 1 OF THE 19-PAPER PROMOTION SHIP =====
%% Mechanical cohort-promotion sweep (17 -> 19 papers) plus the
%% standardized Part-I close (author-approved Wave-1 scope; audit:
%% GG-Internal-Report-RealBulk-Audit-2026-07-19.md, A3 ledger + App. A;
%% A3 unparked for MECHANICAL edits only -- the parked physics items
%% (bordism computations, 4D Witten check, mixed-U(1) split-multiplet
%% bookkeeping, 800-singlet sector, ledger proposals A3-R1/R2/R3)
%% remain untouched).
%% (1) NEW closing section of Part I, "The cohort inheritance and the
%% roadmap of the nine Parts" (A2-GG6 v13.3 architecture), merging the
%% former sections "The GGTheory Cohort" and "Roadmap of the Nine
%% Parts": subsection "The cohort inheritance" (label sec:cohort-A3
%% kept; the five-strata prose migrated verbatim, opening/closing
%% paragraphs recount to nineteen; NEW roster-level Foundations Branch
%% paragraph -- GG-P4-Real on the left flank as the origin-of-i
%% member, GG-P5-CPT's orientation reading of the discrete symmetries
%% named as an independent consumer of the anomaly-inflow logic; the
%% "Where this paper sits" paragraph now names A2's Theorem 6 as the
%% consumer of the ledger proved here) closing with the frozen v1.4
%% nineteen-paper cohort map on its own page (extracted verbatim from
%% A2-GG6 v13.3, MP4/MP5 rendition; label fig:cohort-map-a3 kept; the
%% old seventeen-paper map retired), and subsection "Roadmap of the
%% nine Parts" (label sec:roadmap-eight-parts-A3 kept; new warm prose
%% walkthrough replacing the paragraph-per-Part list) closing with a
%% NEW roadmap figure (label fig:A3-eight-parts-flowchart kept) in the
%% A1 Figure-4 / A2 v13.3 minipage architecture.  Highlighted Part
%% unchanged: Part II, the local-and-global anomaly proof.
%% (2) COUNTS: reader-facing cohort counts seventeen -> nineteen
%% (Part-I close, Part-V coherence map, suffix roster, references
%% Part headers; 8 prose sites + map caption per audit App. A).
%% Protected semantic non-edits: "sixteen coefficient-matching and
%% vanishing conditions" (2 sites), "seven companion papers", "nine
%% constitutional clauses".
%% (3) ROSTER: suffix roster adopts the canonical nineteen-paper list
%% gaining the foundations preprints Real (P4) and CPT (P5);
%% abbreviations shelf macro roster extended to P1-P5 and C1-C3.
%% (4) MACROS + BIB: \PaperPFour (GG-P4-Real) and \PaperPFive
%% (GG-P5-CPT) installed; NEW bibitems ArisakaP4Real + ArisakaP5CPT
%% with roster-level Relevance notes.  No ArisakaP4Twistor key ever
%% existed in A3 (nothing to rename); zero "satellite" tokens in A3
%% (nothing to scrub).
%% (5) ONE-LINE CONTENTS: \GGPart renewed per the P4-Real v3.2
%% preamble pattern (contents line carries only "Part N. <title>").
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% v13.5 -> v13.6.  Script: a3_v13.6_wave1.py.
%% ===== v13.5 (2026-07-16): Part VII transitions tightened =====
%% Author-directed: all inter-appendix and mid-appendix clearpage
%% commands inside Part VII removed (the half-empty transition pages
%% at 113/121/125/130/133/139/145 of v13.4), and each appendix seam
%% filled with an italic physics-intuition bridge in the main-text
%% style (nine bridges: dictionary->descent, descent->10D GS,
%% 10D->rank-one, rank-one->OGN-5, OGN-5->core block, core block->
%% Diophantine, Diophantine->gravitational, gravitational->global,
%% global->checklist; plus the checklist->Author's-Note break
%% removed). A stray pagebreak inside the Table 10 longtable (the
%% cause of the near-empty page 113) also removed. No content
%% changes elsewhere.
%% ===== v13.4 stage 2 (2026-07-16): per-appendix course layers =====
%% Descent appendix: what-this-teaches opener, Fujikawa-origin intuition,
%% intuition paragraphs at the B2 transformation law and at the exact
%% cancellation, textbook pointers. Rank-one appendix: anomaly-space
%% intuition (one tensor, one ray; 6D perfect square vs 10D product).
%% OGN-5 appendix: integers-from-consistency comparison with the string
%% mint (496/30/32/24). Core-block appendix: group-theory index primer
%% (A, A3, B, C defined and derived; Slansky/Georgi). Diophantine
%% appendix: equations-vs-branch-choice intuition with the KMT
%% comparison. Gravitational appendix: from-scratch index-theorem
%% derivation of the two eight-forms (A-hat and L genus expansions;
%% -L/8; the 28-fermions-per-tensor intuition and the origin of 29).
%% Global dossier: eta-invariant / APS / Dai--Freed primer with the
%% twisted-circle spectral-flow example. Fourteen new course references
%% added with Relevance notes (Peskin--Schroeder, Weinberg II, Bilal,
%% Harvey, Bertlmann, Fujikawa--Suzuki, Nakahara, GSW II,
%% Becker--Becker--Schwarz, Ibanez--Uranga, Slansky, Georgi,
%% Witten RMP 2016, Garcia-Etxebarria--Montero).
%% ===== v13.4 (2026-07-16): Part VII graduate-course polish, stage 1 =====
%% Author-directed. Part VII reframed in the A9-SM / A10-Cosmos format as
%% a graduate-level course in 6D anomaly cancellation read side by side
%% with string theory (primary readers: graduate students in particle
%% physics): new course introduction with the three-group organisation,
%% the three-step method, the two stylistic commitments, and the
%% recommended-textbook shelf. NEW flagship appendix 'The ten-dimensional
%% Green--Schwarz mechanism worked from scratch, and the six-dimensional
%% dictionary' (app:gs10d): the 496 miracle, the SO(32)/E8 trace
%% identities, the factorised I12, the group-selection-vs-spectrum-
%% selection lesson, and the full 10D<->6D dictionary table. Appendix
%% count nine -> ten (A--J) at the roadmap sites. Stage 2 adds the
%% per-appendix intuition/derivation/textbook layers and the new
%% bibliography entries.
%% ===== v13.3 (2026-07-16): author-directed enhancement ship =====
%% (1) Lemma 'Computed phase on the intended class' added to the global
%%     dossier (B1 campaign, author-approved): flat-holonomy route (the
%%     combined anomaly line is flat by the local theorem; allowed loops
%%     are contractible by C6) and APS-index route (per-species integer
%%     index on the product filling X6 x D^2), plus sigma(X6 x D^2) = 0,
%%     which makes the chiral-tensor signature phase vanish at every
%%     Omega, including the AX-4-minimal Omega = 25. App. G and Sec. 8.4
%%     pointers updated; O9 narrowed to the winding/mixed classes (O10).
%% (2) Abstract trimmed (final paragraph tightened) to fit page 1.
%% (3) Three figures added: the bulk-identity/boundary-shadow master
%%     figure and the OGN-5 cascade flowchart (adapted from A1/A2), and
%%     a new global-anomaly dimensional-ladder figure illustrating the
%%     new lemma.
%% (4) Part V split into Part V 'Discussions' and Part VI 'Conclusions'
%%     in the A9-SM / A10-Cosmos format. Discussions extended with new
%%     sections: Significance of the anomaly theorem; Uniqueness and
%%     nobility; Comparison with other theories and models (absorbing
%%     the former Conclusions comparison subsection); the Omega
%%     normalisation section and Predictions for falsification promoted
%%     to sections; Problems Solved retitled '..., Open Questions, and
%%     Future Directions' with a closing future-directions paragraph.
%%     Conclusions rewritten to open in plain language for a broad
%%     readership ('The result in plain language'), with the precise
%%     summary, place-in-series, proved/suggested/open, big-picture,
%%     and constitutional-reading sections following. Roadmap section,
%%     roadmap figure, and all Part references renumbered to nine Parts.
%% ===== v13.2 (2026-07-16): QA-gate micro-fix =====
%% The new cohort_qa_gate.py (Tier C7) caught two residual off-canon
%% kaxion-experiment mentions missed by the v13.1 Tier A pass:
%% Objection 15's response and Objection 52's header still said
%% 'ADMX or DM-Radio'; both aligned to the C1 canon (IBS-CAPP, ADMX-EFR,
%% canonical window). No other changes.
%% ===== v13.1 (2026-07-16): Tier A of the hostile-review action plan =====
%% Six-persona hostile review (string theorist, formal QFT, mathematical
%% physicist, phenomenologist, senior editor, graduate student; report
%% A3-BI6-HostileReview-Report-and-ActionPlan-2026-07-16.md in
%% QA+References). Author-approved Tier A executed:
%% (A1) stale '64-tensor signature phase' remnant corrected (WCS refinement
%%      / Omega=50 mod-8 statement).
%% (A2) T-BI6-10.3 proof repaired: the Cartan-coupling argument was invalid
%%      (SU(5k) is simple, one coupling); replaced by the honest MSMF
%%      minimal-structure clause, with Objections 8/31 language aligned.
%% (A3) Swampland audit: Omega^Spin_n(pt) table corrected (Z, Z2, Z2, 0, Z,
%%      0, 0, 0 for n = 0..7); cobordism-conjecture statement corrected to
%%      the trivialisation/defect form.
%% (A4) Stratum-1 list count: ten -> eight.
%% (A5) Schwarz (1996) and Erler (1994) promoted from plain-text mentions to
%%      cited references with entries and Relevance notes.
%% (A6) Frame-bundle Bianchi identity: dR = 0 -> DR = 0 (two sites).
%% (A7) Q_B/ATP anchor aligned to the A7 canon: one phosphoanhydride bond =
%%      two quanta, 2 N_A Q_B = 30.0 kJ/mol vs measured 30.5 +/- 0.5.
%% (A8) Falsifier block F-BI6-1..7 rewritten provenance-faithful to the
%%      companions: F1 conditioned on the fenced holonomy-descent step
%%      (A2/A9) with the Delta B in 3Z exemption; F2 restated against A9's
%%      per-parameter provenance triage; F3 restated per P1 (colour octet
%%      carries the HL-LHC falsifier, Z'_B collider-invisible, mass anchor
%%      theorem-level); F4 restated per C1 (fenced window 1.59-2.13 GHz,
%%      cap index, IBS-CAPP + ADMX-EFR); F6 as consistency requirement; F7
%%      as structural discriminator; closing paragraph re-registered;
%%      Sec. 6.7 'stands exactly as stated' conditioned; Objection 52
%%      aligned to the fenced discrete window.
%% (A9) Tensor-sector accounting: n_SD = 4 Omega includes the GS mediator
%%      B2 (one mediator + 4 Omega - 1 gauge-singlet spectators); tensor
%%      clause (C3, both copies) scoped to the X4-coupled channel; Open
%%      Problem O9 extended to the full chiral-tensor anomaly field theory.
%% (A10) Abstract and Executive Summary register calibrated: global claim
%%      scoped to the intended class; integer-ledger claim stated as three
%%      forced inputs plus the doubly overdetermined closure identity.
%% (A11) Omega_min^lattice = 2 N^2 L^2 = 450 moved out of T-BI6-9.3 into a
%%      fenced 'proposed' remark reconciled with (O8) and Omega = 25;
%%      summary-table row marked accordingly.
%% (A12) Objection 34 branch wording; phase-propagation lemma stated on
%%      generating functionals with explicit pointer to A5's CPTP channel;
%%      tadpole-vs-factorisation scope note in the gerbe section (Step 4);
%%      clause C6 (both copies) extended to mixed clutching and Abelian
%%      windings, routed to O10; swampland Conjecture 1 qualified by the
%%      completion-sector accidental flavour symmetry.
%% ===== v13.0 (2026-07-16): GG-v13 Phase 1 final check and re-derivation =====
%% Promotion of v11.23 to the v13 series after the full re-derivation pass
%% (verification script: A3-BI6-v11.23-Verification-Script-2026-07-16.py and
%%  report A3-BI6-Phase1-Rederivation-Report-2026-07-16.md in QA+References).
%% (1) Gravitational completion re-derived (Sec. on grav completion, App. on
%%     grav-full, App. global dossier): deficit corrected to -Omega p1^2/20;
%%     chiral-tensor polynomial corrected to +-(p1^2-7p2)/360 per AGW L-genus
%%     (the 28:1 p2 ratio behind the SUGRA '29T'); the 2x2 system now solved
%%     exactly: n_SD = 4*Omega ASD tensors, n_F = 32*Omega opposite-chirality
%%     singlet Weyl fermions (exact ratio n_F/n_SD = 8). Benchmarks now
%%     (100, 800) at Omega=25 and (200, 1600) at Omega=50; mod-8 signature
%%     shortcut applies at Omega=50, WCS quadratic refinement covers Omega=25.
%%     All dependent sites updated (main text, App. G/H, Objections 7/23/27/
%%     34/35/49). Curvature-slot convention stated once under the microscopic
%%     polynomial (tr R^2 -> p1/2 etc.).
%% (2) Objection 28 rewritten around pi6(SU(5)) = 0 (was contradicting H2).
%% (3) Inflation falsifier F-BI6-5 and two further sites brought to the A10
%%     canon: GUT-adjacent plateau, V_inf^{1/4} = 8.3e15 GeV, n_s = 0.9620,
%%     r = 0.0043 (TeV-scale reading retired).
%% (4) Appendix lettering: all hardcoded letters converted to \ref (the
%%     Abbreviations summary is Appendix A; nine appendices A-I).
%% (5) Hardcoded stale section numbers converted to \ref (grav closure).
%% (6) Reader-facing internal history removed: Project-Instructions lesson
%%     references, Phase/legacy version identifiers in the coherence map and
%%     audit trail, editing-pass mentions in four Objections, standalone-note
%%     remnants in the global-proof appendix.
%% (7) Stale '(forthcoming)' labels removed for A5/A7/A9/A10; A7 now cited
%%     under its own key for the Qi-4 pillars.
%% (8) 'twelve coordinated papers' corrected to seventeen.
%% (9) Title-page date synchronized with \PaperDate.
%% (10) Grammar repair in the affine-family paragraph; duplicate
%%      Kumar-Park-Taylor(2010) bibitem merged into Kumar-Morrison-Taylor
%%      (2010); 'First cited in' notes regenerated programmatically; false
%%      'not directly cited' clauses removed (all such works are cited).
%% (11) Smaller fixes: Omega_DM = 1/pi - 1/(2pi^2) ~ 0.27; kaxion falsifier
%%      experiment list narrowed to in-band haloscopes (ADMX, DM-Radio);
%%      Objection 44 referent corrected; Objection 31 gloss of M corrected;
%%      'four identity' -> 'four-component identity'; ten+six condition
%%      counts stated precisely; N = 7 second root of the dual-lock noted
%%      and excluded by N < L.
%% v11.23 (2026-07-14, landscape forward pointer) -- presentation only.
%%   One-clause forward pointer appended to the "predict rather than
%%   accommodate / empirical question" passage: the sharpest empirical form is
%%   one value per low-energy constant across the sky, refuted by a single
%%   discordant measurement -- the second horizon problem developed in A10 and
%%   P3. Pointer only (A3 already carries the finite-vs-infinite landscape
%%   subsection); no argument, no values. Script: a3_pointer.py.
%% ======================================================================
%% v11.22 (2026-07-07): NEW subsection "The boundary mixed anomalies in
%% four-dimensional language" (subsec:boundary-mixed-4d), inserted after
%% Remark rem:local-vs-global and before the self-duality subsection.
%% Origin: P1-KK hostile-audit finding R2-1 -- a 4D-EFT reader asks how
%% the mixed triangles U(1)_B-SU(2)^2 (+1/2 per gen) and U(1)_B-Y^2
%% (-1/2 per gen) are cancelled when B is gauged (B-grav^2, B^3, B^2-Y
%% vanish for SM content). The mechanism was already proved here at 6D
%% level (rank-one factorization + descent + inflow); this subsection
%% is a TRANSLATION, not new physics: cross-blocks of M_BI6 = bulk
%% ancestors of the mixed triangles; single counterterm B2^X4 cancels
%% the projected boundary anomaly; 4D face = GS axion b(x) whose shift
%% both cancels the triangles and Stueckelberg-masses the U(1)_B vector
%% (= the leptophobic Z'_B of P1-KK, cross-cited); no anomalons;
%% F-BI6-1 proton stability unchanged; global caveats of
%% rem:local-vs-global apply unchanged.
%% ======================================================================
% v11.21 (July 5, 2026): schematic row 'TeV-scale inflation' -> 'inflationary
%   descent of A10' (TeV-scale inflation retired cohort-wide).
% v11.20 (July 2, 2026): cohort universal-indentation pass (author-
%   directed): indentfirst loaded (first paragraphs after headings
%   now indent). Per-paper stray-noindent review deferred to the
%   v2.0 pass. No content changed.
% v11.19 (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.
% ===== v11.18: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 1 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% ===== A3-BI6 v11.17 (2026-06-27): Round-2 Tier-1 -- stale '26 Standard-Model parameters' -> 28 (3x). =====
% 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.
% ===== A3-BI6 v11.14 (2026-06-27): cohort de-contamination + bib reconciliation. =====
% Shed GG's own self-description: 6D/(4+2)/six-dimensional bulk noun de-qualified; product
% geometry de-qualified; metric -> scale-profile metric; factor/endpoint/interval/depth/
% exponent -> scale-*; RG-geometrization phrase -> renormalization-group geometrization;
% GG-self projection/dictionary -> bulk-to-boundary / bulk--boundary correspondence. Title
% subtitle 'Six-Dimensional ... Bulk' de-qualified. Bib reconciliation: GG-self cohort-cite
% TITLES + relevance prose de-qualified (A2 title; A2/A9/DHVW relevance). KEPT: AdS/CFT,
% Anti-de Sitter, the named external field, the QiHAL acronym glosses, the WHOLE Randall--
% Sundrum exposition + the AdS/CFT-RS-GG comparison section/table (its comparison vocabulary
% intact), the hostile-reviewer objection block, the Schrodinger-GKLS holographic-layer
% contrast framing, all external citation contexts + bibitems, and all comments/refs.
% Body only (after the document-body marker). No physics/values changed. =====
%% =====================================================================
%%  A3-BI6 v11.13 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  Re-pointed gg-6 -> gg-11; dropped the dead
%%  \LaTeX shim; stripped Overview/LoF overrides; single-page Overview+LoF
%%  via the gg-11 toggle; Preface via \GGPreface; inherits the one
%%  canonical Acknowledgments from gg-11.  KEPT A3's no-\newpage Executive
%%  Summary (heading centered).  No prose/figure/number content changed.
%% =====================================================================
%% =====================================================================
%%  A3-BI6 v11.12 (naming-convention rollout) -- 2026-06-19
%%  STEP 1: (1) Objection tags (the position-code suffix) -> O-BI6-n (suffix=paper name, per A2's
%%  ratified key).  (2) 'Open Problems and Future Directions' -> 'Problems
%%  Solved and Future Directions'; twelve triaged entries O1..O12 tagged
%%  '(PS-BI6-k)' (Ok local labels + their cross-refs kept; A2 precedent).
%%  STEP 2: 12 distinct theorems tagged T-BI6-<section.num> in their titles
%%  (Option A: section numbers kept; A6 cross-refs preserved). Restatements
%%  left untagged.
%%  STEP 3: 7 sharp predictions tagged F-BI6-1..7 (kaxion sharpened to
%%  8.1 ueV); new 'Ten Structural Innovations of BI-6' (I-BI6-1..10) + the
%%  Global naming convention key (suffix BI6, P=Prediction).
%% =====================================================================
%% =====================================================================
%%  A3-BI6 v11.11  (new-standard Step 2: final review) -- 2026-06-18
%%  Reconciled the Common-Objections COUNT to the actual 53 in the two
%%  front-matter places still reading 'Seventeen' (Overview Part-VII
%%  paragraph and the overview flowchart caption).  All other references
%%  verified vs the 8-Part structure: 'eight Parts'/'seven further Parts'
%%  correct; every 'seventeen' refers to the 17-paper cohort; closing remark
%%  already 53.  No Part-number drift found.
%% =====================================================================
%% =====================================================================
%%  A3-BI6 v11.10  (GG-QA+References, new-standard Step 1) -- 2026-06-18
%%  Common Objections Part expanded 17 -> 53 Q&A (new >50 standard): 36 new
%%  hostile-reviewer objections (O-BI6-18..53) in 6 clusters -- local descent
%%  and X_4 internals, global completion (Dai-Freed/cobordism/eta), the
%%  OGN-5 ledger as theorem, the string-theory relationship, constitutional
%%  framing/self-sufficiency, and empirical exposure -- each cross-linked to
%%  A3's theorems/sections/equations and companion papers.  Closing remark
%%  reconciled (six -> 53, ten clusters).  Stale kaxion 19 ueV -> canonical
%%  8.1 ueV (~1.95 GHz) in Objection 15 (flagged).
%% =====================================================================
% ===== v11.9 (2026-06-18): cite the 8 previously-uncited refs in-body + relink (author Go). =====
%  GeorgiGlashow1974 (SU(5) embedding), Nakajima1958+Zwanzig1960 (open-system evolution), Stong1968
%  (spin-bordism), BoussoPolchinski2000 (uncontrolled landscape), CallanHarvey1985 (anomaly inflow),
%  Sadov1996+Taylor2011 (6D N=(1,0) sugra/F-theory). First-citation links re-run per A/B block.
%  Bibliography now has 0 orphans. v11.8 -> v11.9.
% =====================================================================
% ===== v11.8 (2026-06-18): References A/B split + first-citation relevance-links (author Go). =====
%  (1) Appended 'First cited in \S\ref{...}' to every cited entry's Relevance note. (2) Split the
%  bibliography into A. the 17 Arisaka cohort papers and B. all other references (each a \section* +
%  thebibliography block). (3) Intro third paragraph now describes the two groups. Mirrors A1/A2.
%  v11.7 -> v11.8.
% =====================================================================
% ===== v11.7 (2026-06-18): cohort overview -> own Section + all 17 cohort papers cited (author). =====
%  (a) Cohort figure + lead-in moved out of sub-section 3.3 into a new \section{The GGTheory Cohort}
%  (Section 4; Roadmap becomes Section 5) with a solid 17-paper prose overview. (b) All seventeen
%  cohort papers are now \citep'd in that overview, and the ten missing cohort bibitems (A3 self,
%  A6, A8, A11, P1, P2, P3, C1, C2, C3; canonical titles from A1 v11.9) added to the reference list.
%  v11.6 -> v11.7.
% =====================================================================
% ===== v11.6 (2026-06-18): 'Roadmap of the Eight Parts' section + flowchart (author request). =====
%  Modelled on A2-GG6 Section 7 + Figure 3. Old mid-Introduction sub-section 1.6 'Scope and
%  organization of the paper' removed; a new \section{Roadmap of the Eight Parts} (prose walkthrough
%  of all 8 Parts + a TikZ flowchart, Part II = the anomaly proof highlighted) placed as the closing
%  section of Part I, before Part II. fig:A3-eight-parts-flowchart on its own page. Also shortened the over-long Part IV heading
%  (dropped 'Systematic') to keep overfull <=20pt. v11.5 -> v11.6.
% =====================================================================
% ===== v11.5 (2026-06-18): two new Executive-Summary paragraphs (author request). =====
%  Added 'Similar mathematics, different physics' (string-theory contrast, grounded in Part IV) and
%  'Philosophical implications' to fill the open space on page 4. The Maxwell/Einstein recognition
%  motif moved from the old Closing reflection into 'Philosophical implications' (no duplication);
%  Closing reflection refocused on the falsifiable bargain + 2026-2040 tests. v11.4 -> v11.5.
% =====================================================================
% ===== v11.4 (2026-06-18): front-matter facelift (author request). =====
%  Abstract rewritten in the accessible A2/A9 register (~1700 chars; now fits page 1). Executive
%  Summary trimmed and de-jargoned: 4 historical paragraphs -> 2; dense post-box paragraph shortened;
%  'Significance' -> 'What makes the proof distinctive'; implication-chain prose shortened; three
%  closing paragraphs merged into one. Kaxion mass corrected 19 -> 8.1 microeV (canonical: C1-Kaxion,
%  A9-SM). All equation boxes, the six results R1-R6, and every citation key preserved. v11.3 -> v11.4.
% =====================================================================
% ===== v11.3 (2026-06-18): single-page 'Overview of Contents' + 'List of Figures' (mirrors A1/A2). =====
%  Installed the A1/A2 R4 override so the Part-level Overview and the List of Figures share ONE page,
%  and invoked \GGOverviewOfContents + \GGListOfFigures before the detailed \GGTableOfContents. Gave
%  the two figures lacking a short caption (anomaly inflow; GS-as-GGEP) a one-line LoF form. v11.2 -> v11.3.
% =====================================================================
% ===== v11.2 (2026-06-18): four-Part back-matter tail (frozen v2.0 standard; author Go). =====
%  'Appendices' Part renamed 'Technical Appendices'; \GGAcknowledgments moved to the end of the
%  Discussion & Conclusions Part. Common Objections promoted to its own \GGPart with categories
%  \subsection -> \section and \bigskip spacing. References added as a \GGPart with a 3-paragraph
%  intro, replacing \GGBibBegin and suppressing the natbib heading. Reader's-guide prose updated to
%  describe the three back-matter Parts. Mirrors A1/A2 Stage 1a. v11.1 -> v11.2.
% =====================================================================
% ===== v11.1 (2026-06-17): Common-Objections -> 3-block house standard + relocate to end (author Go). =====
%  The 17-objection section is reformatted to the A9/A1 house style (three-block entries via objqabody,
%  O-BI6-N IDs, four titled \subsection categories, explicit Response. lead folding the 'Honest answer:'
%  leads) and MOVED to the end as the final appendix under the Appendices Part (after the Theorem-
%  checklist section, before acknowledgments). Only the \section moves (Part dividers stay); \label
%  travels with it so cross-refs hold. Content preserved verbatim. v11.0 -> v11.1.
% =====================================================================
% ===== v11.0 (2026-06-16): HOLOGRAPHIC-RG REFRAMING, Phase R1 (author Go). =====
%  Characterization/terminology only; NO physics, NO parameter changed.
%   - Cover subtitle: 'Six-Dimensional Warped AdS Bulk' -> 'Six-Dimensional [Warped] Bulk (de-contaminated)'.
%   - Metric-paragraph heading 'warped 6D AdS bulk' -> 'warped 6D bulk'; 'anti-de Sitter
%     slab' -> 'bulk'; added a holographic-RG pointer sentence (the warped metric is the
%     RG-flow geometrization, field-theory side of AdS/CFT, not an RS braneworld).
%   - One table cell 'warped 6D AdS bulk discussion' -> 'warped 6D bulk discussion'.
%   - KEPT all legitimate AdS/CFT material (sec:ads-rs comparison, GKPW dictionary, the
%     three-framework table, citation titles). A2 bibitem already AdS-free (no change).
%   - Version v10.6 -> v11.0; dates -> June 16, 2026.
% =====================================================================
% --- v10.6 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. Preface self-labels. ---
% 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
%% v10.5 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-a3, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   front-matter audit: conforms; stamp bump only.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
%% v9.5 (2026-06-07): light touch for the GRIP-In/GRIP-Out campaign.
%%   (a) \ASix/\AFullSix renamed GG-A6-Gi4 -> GG-A6-GRIP (cohort rename;
%%       18 rendered sites); the A6 forward-map item's stale "(legacy
%%       Phase 10; conversion pending)" note replaced.
%%   (b) One physics sentence added to the A6 forward item: the BI-6
%%       instanton-action budget S_BI = 3pi is the resource the
%%       runtime's GRIP-In descent consumes (A6 T4.1 closure N <= 3).
%%   (c) Abbreviation table: +GGA, +GRIP-In, +GRIP-Out rows; macros
%%       \GGA/\GRIPin/\GRIPout added (A6/A11 form).
%% v9.4 (2026-06-03): terminology RG-time -> RG-scale (the Scale 5th dimension is
%%   space-like, not a second Minkowski time; canonical naming per A2 v9.18).
% ----------------------------------------------------------------------
% Arisaka-GG-A3-BI6
% Anomaly Freedom of BI-6: The Geometric Consistency of GG-6
% Author: Katsushi Arisaka (UCLA)
%
% PROVENANCE.  This file is Phase 11 of the v2.0 cycle of
% Arisaka-GG-A3-BI6.  It is built directly on the v1.0 source
% (formerly UCLA-Arisaka-GD-2026-3.tex), with the following
% structural changes:
%
%   * Renamed under the new Arisaka-GG-A* convention:
%     UCLA-Arisaka/GD-2026-3 -> Arisaka-GG-A3-BI6.
%   * Style file upgraded to ucla-arisaka-gg-6.sty (current
%     canonical version of the GG-Theory style file).
%   * New subsection added at the end of Section 8 (the
%     systematic comparison with the 10D Green-Schwarz
%     mechanism): "BI-6 as the GGEP completion of GS:
%     the geometrisation reading."  The new subsection
%     reframes the 1984 GS achievement as the technical
%     pioneering of geometry-gauge equivalence (AX-2 GGEP
%     of the GG-Constitution), of which BI-6 is the
%     axiomatic generalisation.
%
% PHASE LOG (v2.0).
%   Phase 1 (May 6, 2026): rename + style upgrade + GS-as-GGEP
%       subsection insertion.
%   Phase 2 (May 7, 2026): GS-as-GGEP framing audit per L5.
%       The Sec. 8.7 subsection "BI-6 as the GGEP completion of
%       GS: the geometrisation reading" was audited against the
%       L5 checklist (supportive synthesis with the string
%       community; geometric --- not quantum --- reading of GS;
%       AX-2 GGEP framed as axiomatic elevation of the
%       geometrisation move GS pioneered; OGN-5 trace-ratio
%       pinning of (a,b,c,d)=(1,-1,-2/5,-2/5) with the lock
%       |a/c|=|a/d|=5/2; Arisaka register markers at appropriate
%       density).  The audit found the subsection already fully
%       compliant; no body edits were required, and Phase 2
%       therefore registers an audit confirmation only.
%       Five legacy citation keys (Arisaka2026Axiom,
%       Arisaka2026GG6, Arisaka2026GDOS, Arisaka2026SM,
%       Arisaka2026Inflation) remain in body text and bibliography
%       and will be migrated to the canonical
%       ArisakaA{1,2,5,9,10}* form by the reference-scanner
%       sub-agent in Phase 6, per the eleven-phase plan of v3.2.
%   Phase 3 (May 7, 2026): L4 historical-background audit.  Two new
%       \paragraph{Historical background (with citations).} blocks
%       were added at the canonical entry points of the paper.
%       (i) Sec.~1.3 (GG-6 and the BI-6 identity) now opens with
%       Luigi Bianchi's 1902 paper in the \emph{Atti dell'Accademia
%       dei Lincei}, in which the differential identity for the
%       Riemann curvature tensor was established --- the geometric
%       ancestor of the modified Bianchi identity that BI-6
%       generalises.  (ii) Sec.~2.1 (Anomaly polynomials and the
%       descent formalism) now opens with Maxwell's 1865 \emph{A
%       Dynamical Theory of the Electromagnetic Field} and the
%       Adler--Bell--Jackiw anomaly of 1969 --- the historical
%       chain whose modern endpoint is the anomaly polynomial and
%       the descent equations.  (iii) Sec.~8.7 (BI-6 as the GGEP
%       completion of GS) already opens with the Green--Schwarz
%       1984 achievement (audited in Phase 2); no additional
%       paragraph was therefore required there, and Sec.~8.7 itself
%       serves the L4 historical-background role for the
%       Green--Schwarz reading.  Two new \bibitem entries were
%       added to the local \begin{thebibliography} block:
%       Bianchi1902 (Sui simboli a quattro indici e sulla curvatura
%       di Riemann) and Maxwell1865 (A dynamical theory of the
%       electromagnetic field).  The Adler1969 and BellJackiw1969
%       entries were already present.  No structural changes to the
%       theorems, proofs, or appendices.
%   Phase 4 (May 7, 2026): Constitutional-status proof for U(1)_Qi
%       per L12.  The legacy Sec.~4.3 ``Why U(1)_B is essential and
%       why U(1)_Qi is allowed and optional'' was rewritten as
%       Sec.~4.3 ``Constitutional status of U(1)_Qi: optional under
%       the constitution, preferred by the empirical data'', with
%       a real, named label (\label{sec:UoneQi-constitutional-status})
%       so that A2-GG6 v2.0 Phase 11 cross-citations land on a
%       concrete subsection.  The new content carries the L12
%       framing: a Constitutional Admissibility Theorem
%       (\label{thm:UoneQi-admissibility}) stating that the
%       constitution admits both the three-component bulk
%       G = Spin(1,5) x SU(5) x U(1)_B and the four-component bulk
%       G = Spin(1,5) x SU(5) x U(1)_B x U(1)_{Qi}, with explicit
%       proof; a paragraph explaining that CPTP is a structural
%       property of the GKLS channel (Lindblad 1976, Gorini-Kossakowski
%       -Sudarshan 1976) rather than a covariance demand under a
%       fresh U(1); a retained ``Why U(1)_B is essential'' paragraph
%       (kept from the legacy text); a clarified ``Why U(1)_Qi is
%       constitutionally optional'' paragraph; and a substantive new
%       ``Why the four-component bulk is empirically preferred''
%       paragraph that lists the three reasons --- Arisaka equation
%       and the bioenergetic conversion factor Q_B \approx 0.1555
%       eV/qubit matching ATP hydrolysis at biological temperatures;
%       OGN-5 dual-lock identity K = LJ = M^2 + N^2 + L^2 = 35
%       closing only for the four-component case under the
%       equal-Abelian-treatment convention; and the boundary Qi-4
%       pillars and Equivariant Information Channel framework
%       requiring U(1)_Qi as host symmetry.  No structural changes
%       to the theorems, proofs, or appendices outside Sec.~4.3.
%   Phase 4 (May 7, 2026) -- amendments per Katsushi's review:
%       (a) Eq.~(2): the legacy ``...x Gravity'' factor in the
%           composite gauge--gravity group was replaced by the
%           explicit Lorentz spin factor Spin(1,5), matching the
%           convention applied in A2-GG6 v2.0 Phase 11; the
%           canonical GG-6 metric ds_6^2 = e^{-2r} eta_{mu nu} dx^mu
%           dx^nu + dr^2 + dtheta^2 was inserted as Eq.~(3); a
%           paragraph headed ``The unique geometrical structure of
%           GG-6: a minimalist's warped 6D AdS bulk'' was added
%           explaining why the four conventional KK--RS continuous
%           parameters (k, R_theta, r_c, brane tension) all collapse
%           to natural-GG-units constants and the integer endpoint
%           K = 35 of the OGN-5 ledger.
%       (b) GUT-10 retired throughout: 24 occurrences of GUT-10
%           and the legacy paper labels GG-2026-{1,2}, GD-2026-3
%           were replaced by the canonical GG-Theory naming
%           (GG-Theory, GG-Theory Constitution, A1-Constitution,
%           A2-GG6, etc.).  GG-Theory is the new trademark for the
%           entire programme (GG-6 + BI-6 + GDOS + Gi-4 + Qi-4).
%           GUT-10 will only re-enter when GG-Life and GG-Mind are
%           introduced.
%       (c) Citation convention upgraded to the new Arisaka(2026,
%           A*-Name) form: body cites Arisaka2026{Axiom, GG6, GDOS,
%           SM, Inflation} migrated to ArisakaA{1, 2, 5, 9, 10}*
%           (canonical form used by A1-Constitution and A2-GG6
%           Phase 11); the five Arisaka \bibitem entries rewritten
%           with display labels [Arisaka(2026, A1-Constitution)]
%           etc., new titles, and updated Relevance paragraphs.
%   Phase 5 (May 7, 2026): Open Problems and Future Directions per
%       L6 (the v3.2 eleven-phase plan reorders Phase 4 and Phase 5
%       for A3-BI6: Phase 4 carried the U(1)_Qi constitutional-status
%       proof; Phase 5 carries the L6 triage).  A new
%       \section{Open Problems and Future Directions}
%       \label{sec:open-problems} was inserted just before
%       \GGPart{Appendices}, with twelve O_k entries triaged per L6:
%         Resolved (7): O1 local anomaly cancellation; O2 global
%           anomaly cancellation; O3 U(1)_Qi constitutional status
%           (cross-cited from Phase 4 Sec.~4.3, Theorem~4.5);
%           O4 trace-ratio lock; O5 model-level benchmark spectrum;
%           O6 GS-as-GGEP completion; O7 bulk-to-boundary projection
%           via OGGEP (handed off to A5-GDOS).
%         Partially resolved (3): O8 minimal Omega for the BI-6
%           spectrum; O9 explicit Monnier-Moore numerical mapping-
%           torus partition functions; O10 extension of the global
%           theorem beyond the intended class.
%         Genuinely open (2): O11 non-perturbative completion;
%           O12 generation count (cross-cite to A2-GG6 O11).
%       The legacy ``What is proved, what is suggested, and what
%       remains open'' subsection inside Conclusions was condensed
%       to a forward pointer to \ref{sec:open-problems}, with
%       inline cross-references to the U(1)_Qi-status theorem
%       (Theorem~\ref{thm:UoneQi-admissibility}, Sec.~4.3) and the
%       GS-as-GGEP completion subsection
%       (Sec.~\ref{sec:GS_as_GGEP_completion}).  This avoids
%       duplicating the open-problems content in two places while
%       preserving the Conclusions-level meta-statement about the
%       paper's design principle.
%   Phase 6 (May 7, 2026): Hostile-reviewer + reference-scanner
%       sub-agents dispatched in parallel per L13.  Both returned
%       written reports.
%
%       Reference-scanner (Explore agent) recommendations applied
%       in full -- three edits:
%         (RS-1) \Ptwo macro: GG-2026-2 -> Arisaka-GG-A2-GG6.
%         (RS-2) \Pfive macro: GD-2026-3 -> Arisaka-GG-A3-BI6.
%         (RS-3) Body table row: ``the companion GG-2026-6
%               program'' -> ``the companion Arisaka-GG-A9-SM
%               \citep{ArisakaA9SM} program''.
%
%       Hostile-reviewer (Plan agent) attacked four derivations.
%       Triage and dispositions:
%         (D1) Trace-ratio lock 5/2 -- agent flagged ``serious'':
%             concern that B_0, Y_0 normalisation is unjustified
%             and equal-Abelian-treatment is circular.  Defensive
%             paragraph (HR-3) added in Sec.~7 after the trace-ratio
%             theorem: explains that the trace-ratio is invariant
%             under any rescaling that preserves the integer charge
%             content of the SM, so the ratio is canonical not
%             conventional.
%         (D2) Anomaly cancellation under (1,-1,-2/5,-2/5) and
%             rank-one criterion -- agent flagged ``blocking'':
%             concern that the rank-one structure is tautological
%             and that the explicit sixteen-condition verification
%             is not in the main body.  Defensive paragraphs (HR-1)
%             added immediately after Theorem~\ref{thm:rank-one}:
%             a ``Remark on what the rank-one criterion proves and
%             what it does not'' clarifying that the non-trivial
%             content is one step earlier (the microscopic anomaly
%             matrix happens to have rank one), plus a ``Uniqueness
%             of the rank-one factorization'' paragraph showing
%             that the BI-6 coefficient vector is the unique
%             rank-one factorisation up to sign.  The agent's
%             additional concern that the eight-part proof
%             obligation should be displayed in the main body is
%             noted; the explicit calculation lives in Appendix~A
%             and is appropriate there.
%         (D3) GS modification, 10D vs 6D comparison -- agent
%             flagged ``serious'': concern about anti-self-dual vs
%             self-dual tensor and descent under perfect-square
%             structure.  Disposition: deferred.  The anti-self-dual
%             choice for B_2 in 6D is the standard convention in
%             the 6D anomaly literature (Sagnotti 1992, Salam-Sezgin)
%             and a full re-derivation belongs to a future technical
%             revision rather than to the conversion cycle.  Flagged
%             for inclusion in the next round of substantive content
%             edits.
%         (D4) U(1)_Qi admissibility (Theorem 4.5) -- agent flagged
%             ``serious'': concerns about the truncation argument
%             and a homotopy-group claim.  The agent's homotopy
%             concern was incorrect on its merits: pi_6(SU(5)) = 0
%             by Bott periodicity (pi_k(SU(N)) = 0 for k = 2N and
%             N >= 4, applied at N = 5, k = 6), so pi_6(SU(5) x
%             U(1)_B) = 0 follows.  But the stated reasoning in the
%             proof was indeed too compressed.  Defensive
%             clarification (HR-2) inserted in the proof of
%             Theorem~\ref{thm:UoneQi-admissibility}: explicitly
%             reads the truncation F_Qi = 0 ``at the level of the
%             chiral spectrum'' (no fermion species carries non-trivial
%             U(1)_Qi charge) so that mixed gravitational - U(1)_Qi
%             and Abelian - U(1)_Qi anomaly slots are identically
%             zero; explicitly cites the Bott-periodicity fact
%             pi_k(SU(N)) = 0 for k = 2N, N >= 4, applied at N = 5,
%             k = 6, to substantiate the homotopy claim.  The
%             agent's broader concern that AX-3 might still
%             constrain U(1)_Qi was considered and rejected: AX-3
%             demands an OGGEP holographic dictionary, and the
%             dictionary is admissible for both 3-component and
%             4-component bulks.
%
%       Overall: hostile reviewer raised four legitimate
%       methodological concerns; three got targeted defensive
%       paragraphs (HR-1, HR-2, HR-3) that close the most exposed
%       phrasing without expanding the formal claims of the paper;
%       one (D3 anti-self-dual tensor) was deferred to a future
%       substantive-content revision.  No theorem was strengthened
%       beyond what was already proved; no theorem was retracted.
%   Phase 7 (May 7, 2026): Companion-paper auditor + coherence-map
%       drafter sub-agents dispatched in parallel per L13.
%
%       Companion-paper auditor (Plan agent) walked all
%       \citep{ArisakaA3BI6} citations from A2-GG6 v2.0 Phase 11
%       and all \citep{ArisakaA1Constitution} / \citep{ArisakaA2GG6}
%       citations from A3-BI6 v2.0 Phase 6.  Verdict: zero DRIFT,
%       zero DANGLING.  All cross-citations land on real, named
%       theorems and sections.  In particular, A2-GG6's three
%       cross-citations for the U(1)_Qi constitutional-admissibility
%       result (T3 Step b/c, Common Objection 4, Open Problem O8)
%       all correctly land on A3-BI6's
%       \label{sec:UoneQi-constitutional-status} (Sec.~4.3) and
%       \label{thm:UoneQi-admissibility} (Theorem 4.5) which were
%       inserted in Phase 4.  The auditor recommended no fixes.
%
%       Coherence-map drafter (Plan agent) returned an
%       implementation plan; the LaTeX content was composed in this
%       phase based on the drafter's plan and the A2-GG6 v2.0 Phase
%       11 template.  A new top-level
%       \section{Cross-Paper Coherence Map}
%       \label{sec:coherence-map} was inserted just before
%       \section{Open Problems and Future Directions}, with:
%         (a) An opening paragraph framing A3-BI6 as the anomaly
%             heart of the programme.
%         (b) An Outward Map subsection
%             \subsection{Theorems of A3-BI6 used by companion
%             papers (outward map)} \label{ssec:outward-map}
%             with a 13-row table (\label{tab:outward-map-A3})
%             linking each A3-BI6 theorem / section to its
%             downstream uses in A2-GG6, A4-Euclid, A5-GDOS,
%             A6-Gi4, A7-Qi4, A9-SM, A10-Cosmos.
%         (c) An Inward Map subsection
%             \subsection{Constitutional inputs to A3-BI6 (inward
%             map)} \label{ssec:inward-map} with a 12-row table
%             (\label{tab:inward-map-A3}) listing the AX-1--AX-4
%             axioms, POS-1, POS-3, POS-4, POS-5, POS-6 postulates
%             from A1-Constitution v4.0 Phase 8, plus three
%             structural inputs from A2-GG6 v2.0 Phase 11
%             (the bulk gauge group, the BI-6 coefficient vector,
%             and the OGN-5 ledger).
%         (d) A Note on version and Phase numbers paragraph
%             recording that A1-Constitution v4.0 Phase 8,
%             A2-GG6 v2.0 Phase 11, and A3-BI6 v2.0 Phase 7 are
%             the only papers converted to the new naming
%             convention, with the remaining six on legacy
%             identifiers.
%         (e) An Audit trail itemize listing actual internal
%             theorem/section structure of each cited companion
%             paper as of May 7, 2026.
%         (f) A closing What the map sustains paragraph:
%             outward + inward consistency is what L7 demands and
%             A3-BI6 now delivers.
%   Phase 8 (May 7, 2026): TikZ figure addition + L8-compliance scan.
%       L8 baseline: A3-BI6 v2.0 Phase 7 had ZERO TikZ figures and
%       ZERO \begin{figure} blocks; nothing to remediate.  TikZ
%       libraries (arrows.meta, positioning, calc,
%       decorations.pathmorphing, shapes.geometric, fit,
%       backgrounds) are already loaded by ucla-arisaka-gg-6.sty.
%       Two canonical figures added per the brief:
%         (a) Figure \ref{fig:bi6-anomaly-inflow} -- BI-6
%             four-component anomaly inflow.  Inserted immediately
%             after the BI-6 boxed four-form Eq.~\eqref{eq:x4-bi6}
%             at the start of Sec.~\ref{sec:bi6-fourform}.  Shows
%             four input gauge--gravity sectors (Spin(1,5), SU(5),
%             U(1)_B, U(1)_Qi) feeding into the central X_4 anomaly
%             four-form with OGN-locked coefficients
%             (+1, -1, -2/5, -2/5) and the trace-ratio lock
%             |a/c| = |a/d| = 5/2; X_4 inflows to the H_3 tensor
%             field via the modified Bianchi identity dH_3 = X_4.
%             Modeled on Figure 5 of A2-GG6 v2.0 Phase 11
%             (fig:bi6-anomaly-inflow).
%         (b) Figure \ref{fig:gs-as-ggep} -- GS-as-GGEP completion
%             diagram.  Inserted at the end of
%             Sec.~\ref{sec:GS_as_GGEP_completion}, immediately
%             before the closing ``An invitation to the string and
%             mathematical-physics communities'' paragraph.
%             Three-panel layout: left panel = 1984 GS in 10D with
%             I_12 = X_4 wedge X_8 product factorisation and the
%             two-component X_4^GS = tr(R wedge R) - tr(F wedge F);
%             middle panel = AX-2 (GGEP) axiomatic elevation
%             arrow with cross-citation to A1-Constitution v4.0
%             Phase 8; right panel = 2026 BI-6 in 6D with
%             I_8 = (1/2)(X_4^BI6)^2 perfect-square factorisation
%             and the four-component X_4^BI6 = trR^2 - TrF_5^2
%             - (2/5)F_B^2 - (2/5)F_Qi^2 over the bulk gauge--
%             gravity group Spin(1,5) x SU(5) x U(1)_B x U(1)_Qi;
%             bottom annotation contrasts the 10D product structure
%             (two independent forms) with the 6D perfect-square
%             structure (rank-one forced).
%       Both figures wrapped in \resizebox{\textwidth}{!}{...}
%       per L8.  Captions written in Arisaka register and reference
%       the surrounding theorems (thm:rank-one,
%       thm:bi6-cancellation) and constitutional clauses (AX-2
%       GGEP).
%   Phase 9 (May 7, 2026): Front-matter rewrite to A1-Constitution
%       v4.0 Phase 8 canonical template per L9.  Three blocks
%       rewritten in place:
%         (a) Title block: primary title shortened to ``Anomaly
%             Freedom of BI-6'' (single-spaced, no \\[Xpt] artefact);
%             subtitle expanded to a 3-line plain-noun-phrase form
%             ``Local and Global Quantum Consistency of the Warped
%             Six-Dimensional Bulk of GG-Theory / Wu-Chern-Simons
%             Differential Cohomology, the OGN-5 Integer Ledger, /
%             and the Constitutional Admissibility of U(1)_{Qi}'';
%             date changed to {May 2026}.
%         (b) Abstract: rewritten to 150-200 words, plain English,
%             three-paragraph structure: (i) universal thesis on
%             anomaly tests as engineering consistency tests;
%             (ii) what this paper does in plain language with
%             concrete examples (local descent, global topological
%             completion, OGN-5 forced integers); (iii) closing
%             ``After all'' philosophical reflection.  No boxed
%             equations, no Lie-group jargon inside Abstract.
%         (c) Executive Summary: rewritten to 700-1000 words,
%             technical with minimised jargon, six structured
%             sub-paragraphs in canonical order:
%               -- opening positioning vs historical predecessors
%                  with inline \citet{} references (Maxwell 1865,
%                  Adler 1969, Bell-Jackiw 1969, Wess-Zumino 1971,
%                  Green-Schwarz 1984, Hopkins-Singer 2005,
%                  Monnier-Moore 2019);
%               -- boxed central principle Eq.~(eq:bi6_exec) for
%                  the BI-6 modified Bianchi identity dH_3 = X_4
%                  with the four-component X_4 written explicitly;
%               -- Six principal results paragraph (R1 rank-one
%                  criterion; R2 local cancellation; R3 global
%                  freedom; R4 U(1)_Qi admissibility; R5 model-
%                  level benchmark spectrum; R6 OGN-5 ledger);
%               -- \paragraph{Significance and uniqueness.} listing
%                  three structural features (no SUSY; coefficient
%                  vector uniquely fixed; explicit empirical
%                  exposure: Q_B ~ 0.1555 eV/qubit ATP match;
%                  alpha^-1(0) = 137; kaxion delta nu/nu = 1/35);
%               -- \paragraph{Canonical implication chain.} boxed
%                  equation cascade Eq.~(eq:implication_chain_a3)
%                  wrapped in \resizebox{\linewidth}{!}{...} per L11
%                  showing the four-step descent from constitutional
%                  clauses to boundary phenomenology;
%               -- \paragraph{Closing reflection.} ``After all, the
%                  test of an honest framework is...''.
%         (d) \GGKeywords{} placement: moved AFTER
%             \end{ExecutiveSummary} per L9 (was already there in
%             the legacy file but the keyword list was rewritten to
%             match the new Executive Summary content).
%       One bibliography addition: \bibitem for Bustamante2005
%       (Mechanical processes in biochemistry, Ann. Rev. Biochem.
%       74:705) inserted before the BellJackiw1969 entry, since
%       the new Executive Summary cites it for the empirical ATP-
%       hydrolysis anchor of Q_B.  No structural changes to the
%       theorems, proofs, or appendices.
%   Phase 10 (May 7, 2026): Bibliography surname-sort + journal-
%       stripped variant per L10; L11 pre-emptive wrap on the
%       multi-row boxed two-bulks gauge-product equation.
%
%       L10 -- Bibliography sort:  All 63 \bibitem entries in the
%       local thebibliography block were sorted alphabetically by
%       first-author surname using a mechanical Python script.
%       Conventions applied:
%         (i)  Arisaka programme entries (A1-Constitution, A2-GG6,
%              A5-GDOS, A9-SM, A10-Cosmos) grouped under bucket A
%              and sub-sorted by paper number in programme order.
%         (ii) Particles demoted per standard surname-sort: e.g.
%              ``de Haro, Skenderis, and Solodukhin'' now sorts
%              under H (Haro), not D.
%         (iii) Multi-author entries (e.g. ``Bell and Jackiw'',
%               ``Pati \& Salam'', ``Freed, Hopkins, \& Teleman'')
%               sort by the first surname.
%         (iv)  Accent stripping for sort: e.g. ``Alvarez-Gaum\'e''
%               -> ``alvarez-gaume''.
%         (v)   Phase-X-additions divider comment lines stripped from
%               the body of thebibliography.
%       The sorted ordering is: Arisaka programme (5 entries) -
%       Adler - Alvarez-Gaume - Basile - Baykara - Becchi - Becker -
%       Bell-Jackiw - Bianchi - Bott - Bousso - Brennan - Bustamante
%       - Callan-Harvey - Dai-Freed - Dierigl - Dixon - Freed-Hopkins
%       - Georgi-Glashow - Gorini - Green-Schwarz - Green-Schwarz-West
%       - Gubser - Hamada - Haro (de Haro) - Hayashi - Hopkins-Singer
%       - Kaidi - Kim - Kumar (4 entries) - Lindblad - Loges -
%       Maldacena - Maxwell - Milnor-Stasheff - Monnier - Monnier-
%       Moore - Morrison-Taylor (2) - Morrison-Vafa (2) - Nakajima -
%       Nishino-Sezgin - Polchinski - Randall-Sundrum - Sadov -
%       Sagnotti - Seiberg-Taylor - Stong - Tarazi-Vafa - Taylor -
%       Vafa (2) - Wess-Zumino - Witten - Zwanzig.
%
%       L11 -- Pre-emptive boxed-equation wrap:  The multi-row
%       boxed equation Eq.~\eqref{eq:two-bulks} in
%       Sec.~\ref{sec:UoneQi-constitutional-status} (the
%       three-component / four-component bulk admissibility
%       statement of Theorem~\ref{thm:UoneQi-admissibility}) was
%       wrapped in \resizebox{\linewidth}{!}{\$\displaystyle\boxed{...}\$}
%       per L11.  The four-component row
%       Spin(1,5) x SU(5) x U(1)_B x U(1)_Qi sits very close to
%       the line width with the prefix
%       ``(four-component bulk)\quad''; pre-emptive wrapping
%       prevents any future overflow.  The Executive Summary
%       implication chain Eq.~\eqref{eq:implication_chain_a3} was
%       already wrapped per L11 in Phase 9.
%
%       Two deliverables ship at this Phase: the preprint (this
%       file, with the full Relevance paragraphs) and the
%       journal-stripped variant Arisaka-GG-A3-BI6-v2.0-Phase10-
%       journal.tex (Relevance paragraphs stripped via
%       sed -i '/^\\par\\noindent\\textbf{Relevance\\./d').
%   Phase 11 (May 7, 2026): 30-item-audit sub-agent walk per L13.
%       Verdict: 26 GREEN / 2 YELLOW / 2 RED.  Triaged as follows:
%
%       RED-17 (Common Objections section MISSING) -- FIXED.  New
%         \section{Common Objections and Responses}
%         \label{sec:common-objections} inserted just before the
%         Cross-Paper Coherence Map, with six A3-BI6-specific
%         objections answered: (1) why an axiomatic anomaly
%         proof; (2) is the trace-ratio 5/2 a normalisation choice;
%         (3) how can U(1)_Qi be optional; (4) does the Monnier-
%         Moore citation suffice without explicit partition functions;
%         (5) why a single tensor B_2; (6) is the intended-class
%         restriction limiting.  Plus a closing remark.  Modeled on
%         A2-GG6 v2.0 Phase 11 Sec.~\ref{sec:common-objections}.
%
%       RED-18 (Scope-and-Limits paragraph MISSING) -- FIXED.  New
%         \paragraph{Scope and limitations.} appended at the end of
%         the Open Problems triage summary, with three sub-paragraphs:
%         (i) Scope -- listing the principal theorems established by
%         the present paper (local + global anomaly freedom,
%         constitutional admissibility of U(1)_Qi, model-level
%         benchmark spectrum, OGN-5 ledger, GS-as-GGEP completion);
%         (ii) Limits -- naming what the paper does NOT do
%         (bulk-to-boundary projection, universal-constants
%         reconstruction, boundary phenomenology calibration,
%         explicit Monnier-Moore numerical computation, minimal-
%         Omega search, extension beyond intended class, non-
%         perturbative completion); (iii) Dependencies -- the
%         constitutional clauses AX-1--AX-4 and POS-1, POS-3, POS-4,
%         POS-5, POS-6 of A1-Constitution and theorems T3, T4, T6 of
%         A2-GG6 are inputs that must be re-walked if revised.
%
%       YELLOW-8 (Worked example for Theorem 4.5) -- FIXED.  Added
%         \paragraph{Worked example: the truncation F_Qi = 0 at the
%         level of the spectrum.} immediately after the proof of
%         Theorem~\ref{thm:UoneQi-admissibility}, illustrating
%         concretely how the four-component spectrum truncates to
%         the three-component spectrum by striking out fermion
%         species carrying non-trivial U(1)_Qi charge, with explicit
%         note that mixed gravitational--U(1)_Qi and Abelian--U(1)_Qi
%         cross-anomaly slots are identically zero by spectrum
%         truncation rather than by field-strength vanishing.
%
%       YELLOW-14 (Arisaka register marker density) -- DEFERRED.
%         The audit reported ~25-30 markers across 6800+ lines vs
%         a target of ~1 per page (~118 expected).  A full register-
%         density treatment is a substantive editing pass requiring
%         dozens of additions in technical sections; this exceeds
%         the Phase 11 audit-and-fix scope and is recorded here as
%         a follow-up item for the next substantive revision.  The
%         register IS applied at appropriate density in the front-
%         matter (Abstract, Executive Summary), in the
%         GS-as-GGEP-completion subsection (Sec.~8.7), in the
%         constitutional-admissibility subsection (Sec.~4.3), in
%         the Conclusions and Discussion sections, in the new
%         Common Objections section, and in the Open Problems
%         section.  Technical sections (rank-one criterion,
%         descent calculation, Wu-Chern-Simons appendices) are
%         intentionally drier, as is appropriate for proof-bearing
%         passages.
%
%       YELLOW-29 (Boxed equation overflow audit) -- VERIFIED.
%         Final compile produces no Overfull \hbox warnings on
%         boxed display equations.  The two known multi-row boxed
%         equations (the Executive Summary implication chain
%         eq:implication_chain_a3 and the two-bulks equation
%         eq:two-bulks) are both already wrapped in
%         \resizebox{\linewidth}{!}{...} per L11 from Phase 9 and
%         Phase 10 respectively.  No additional wrapping required.
%
%       Final ship: preprint Arisaka-GG-A3-BI6-v2.0-Phase11.{tex,pdf}
%       and journal-stripped variant
%       Arisaka-GG-A3-BI6-v2.0-Phase11-journal.{tex,pdf} (built
%       mechanically by the L10 sed pipeline, with the \bibitem
%       literal in PaperStatus de-fanged per the Phase 10 fix).
% ----------------------------------------------------------------------
%% ----------------------------------------------------------------------
%% v10.3 -> v10.4 (June 11, 2026): P2-N3 reciprocal sync (Phase 5 of the P2 project;
%%   author-approved, June-10 protocol: minimal, disclosed, version-bumped).
%%   - Cross-reference ledger: budget "of the present paper" corrected --
%%   -   A3 quotes but does not derive S = 3pi; fenced in P2-N3 App. C.
%%   Reference: [GG-P2-N3]/[P2-Support]/P2-N3-PHASE5-SYNC-REPORT-2026-06-11.md
%% ----------------------------------------------------------------------
% ------------------------------------------------------------
% v15.33, August 19, 2026.  Table 1 at the house size, and
% Notation back on one page.
%   THE DEFECT, reported by the author: the Notation section ran
%   from page 15 onto page 16, leaving three table rows and one
%   closing line on an otherwise empty page -- and Table 1 was set
%   at BODY SIZE, 11 pt, which matches no other table in the
%   paper.
%   THE CAUSE: the longtable carrying Table 1 was the only unsized
%   table in this paper's front matter.  Every other longtable in
%   A3 opens a group with a size first -- fifteen at \footnotesize
%   and two at \small -- so this one inherited 11 pt and printed a
%   third larger than its neighbors.
%   THE REPAIR: \footnotesize, which is this paper's own dominant
%   table size (15 of 17), together with the longtable's own glue
%   zeroed (\LTpre 0pt, \LTpost 6pt) and the caption skips
%   tightened to 4 pt / 2 pt.  MEASURED, four variants compiled:
%     \small alone                       still spills the BRST row
%     \small + tightened glue            still spills the BRST row
%     \small + glue + arraystretch 0.94  still spills the BRST row
%     \footnotesize alone                spills the closing line
%     \footnotesize + tightened glue     FITS -- Notation on p15
%   The charter's Stage-1 default for an unsized table is \small;
%   it was tried first and it does not fit this section.  A3's own
%   convention for a dense reference table is \footnotesize, and
%   that is the reading taken.  262 -> 261 pages: the near-empty
%   page 16 is gone and the Preface moves up to it.
%   NOTHING IN THE TABLE'S CONTENT CHANGES -- not a row, not a
%   word, not a rule.
%   ALSO: a scan for unsized printed longtables found a SECOND
%   one, tab:local-vs-global in Part VIII, and it is sized in the
%   same pass at \small -- the charter default, which it has room
%   for.  "A size on every table" is the standard, not a
%   preference, and leaving one of two would be worse than doing
%   neither.  The guard prints every surviving unsized longtable
%   and finds none.
%   NEW PROSE: none.
% ------------------------------------------------------------
% ------------------------------------------------------------
%% v15.34, August 19, 2026.  The \OGNfive{} cascade showcase plate adopted.
%%   AUTHOR'S REQUEST: A2-GG6's showcase plate 'The OGN-5
%%   Diophantine cascade.' added to this paper, landing on
%%   showcase page 5.  The request named the plate as being on
%%   page 6 of A3; there is exactly ONE showcase plate in the
%%   cohort with that caption, bolded and ending in a period, and
%%   it is on page 6 of A2.  A3's own cascade figure is a numbered
%%   in-text figure in Part III with a different caption.  The
%%   quoted string matches A2's plate verbatim, so A2's plate is
%%   what was adopted and the paper letter was read as a slip.
%%   THE PLATE AND THE CAPTION down to its closing pointer are
%%   byte-identical to the copy in the sibling paper and reproduce
%%   A2 page 6 exactly; the rider hashes that region and asserts a
%%   single hash across both papers.  TWO THINGS ARE LOCAL BY
%%   DESIGN: the headline command follows this paper's own
%%   showcase idiom, and the closing pointer resolves against this
%%   paper's own labels -- A2's pointer names a label that exists
%%   in A2 alone, and a reference that does not resolve locally is
%%   not a shared asset but a broken one.
%%   Pure TikZ: no asset is added to the arXiv package.  The plate
%%   is unnumbered (captionof*), so no figure or table number and
%%   no cross-reference moves.  A2 IS NOT TOUCHED.
%%   NEW PROSE: none -- the headline and caption are A2's own,
%%   except the one closing pointer sentence, which is retargeted.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.35, August 19, 2026.  Three changes at the author's
% request.
%   (1) PAGES 5 AND 6 SWAPPED.  The OGN-5 cascade showcase, added
%   in v15.34, sat ahead of this paper's own opening plate.  It
%   moves after it, so A3 leads with its own claim -- the
%   four-component BI-6 bulk-side classical-topological identity
%   and its boundary shadow -- and the inherited cascade follows.
%   (2) FIGURE 4 AND FIGURE 7 ENLARGED IN SITU.  Figure 4, the
%   four-component anomaly inflow, 0.668 -> 0.80 of the measure,
%   up 20 percent.  Figure 7, the GS-as-GGEP completion, 0.88 ->
%   0.97, up 10 percent.
%   (3) A NEW COMBINED SHOWCASE PAGE carrying both plates, placed
%   between the two, so the front matter now reads: the bulk
%   identity; then what the four components are and where they
%   came from; then the ledger that pins them.
%   SIZING, MEASURED RATHER THAN CHOSEN.  The two plates are drawn
%   at DIFFERENT BASE FONTS -- Figure 4 at \footnotesize (9 pt),
%   Figure 7 at \small (10 pt) -- so equal \textwidth fractions
%   would print them at different type sizes on the same page.
%   Natural widths, by \setbox probe: 350.64 pt and 510.18 pt.
%   The showcase copies are scaled so BOTH RENDER AT ABOUT 10.4 pt:
%   Figure 4 at 0.86 of the measure (scale 1.156, type 10.37 pt)
%   and Figure 7 at 1.13 (scale 1.041, type 10.41 pt).  Both are
%   larger than their in-situ selves after (2) -- 9.65 pt and 8.93
%   pt -- which is this paper's own rule, recorded in its own
%   showcase comments: "Never smaller than its in-paper self."
%   Figure 7's showcase copy overruns the measure, so it uses the
%   \makebox idiom A3 already uses on its opening plate; centred,
%   it leaves 40.7 pt of margin on each side.
%   THE TWO PICTURES IN THE SHOWCASE ARE THE SAME BYTES as their
%   in-text selves; only the enclosing scale differs.
%   NEW PROSE: the combined page's headline and its caption.
%   Listed in the report for the author to review or rewrite.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.36, August 19, 2026.  The combined showcase page: space
% between the two plates, and both plates down 10 percent.
%   TWO REPORTS FROM THE AUTHOR, ONE SHIP.  The plates were
%   combined too closely -- set apart by \par\medskip, about 6 pt,
%   which reads as one picture in two halves rather than two
%   pictures -- and both were about 10 percent too large.
%   MEASURED FIRST, so the new gap was affordable rather than
%   hopeful: at v15.35 the content ink on page 6 ran y = 126.2 to
%   679.6 pt inside a text block of 72 to 722.4, so the centred
%   block had 54.2 pt of slack above and 42.8 below, 97.0 pt free.
%   THE GAP: \medskip -> \vspace{3.5\baselineskip} = 47.6 pt at
%   the 13.6 pt body leading.
%   THE SIZES: Figure 4's showcase copy 0.86 -> 0.775 of the
%   measure (-9.9 percent), Figure 7's 1.13 -> 1.02 (-9.7).  They
%   stay matched in RENDERED TYPE, which is how they were chosen
%   in v15.35 -- 9.34 pt and 9.39 pt, against 10.37 and 10.41
%   before.  Figure 7's copy now sits at 1.02, which is exactly
%   the factor this paper's opening plate already uses.
%   ONE CONSEQUENCE, STATED RATHER THAN BURIED: Figure 4's
%   showcase copy now renders at 9.34 pt against its in-situ 9.65
%   pt, so for the first time a showcase copy in this paper is
%   very slightly SMALLER than its in-paper self -- 3 percent --
%   which the note in A3's own showcase comments says never to do.
%   The author's instruction is the newer of the two, so it wins;
%   trimming the in-situ Figure 4 back from 0.80 to about 0.775
%   would restore the rule and is one edit away.
%   NOTHING ELSE ON THE PAGE MOVES, and the rider asserts it:
%   inside the showcase block only the two scales and the gap
%   command differ, and outside it only the version, the date and
%   this banner.
%   NEW PROSE: none.
% ------------------------------------------------------------
% ------------------------------------------------------------
% v15.37, August 19, 2026.  The three showcase pages given
% breathing room -- the trial before a cohort-wide pass.
%   THE DEFECT the author reports: a flat plate sits centred and
%   leaves a large empty band above the headline and below the
%   caption, with the three elements bunched in the middle.
%   THE FIX IS NOT A FIXED SKIP.  A fixed \vspace has to be chosen
%   per page against that page's slack -- 47.6 pt was right for
%   page 6 in v15.36 and would push a fuller page onto two.
%   Instead each block becomes a minipage of EXPLICIT HEIGHT
%   \textheight with the [s] (stretch) inner alignment, and the
%   free height is distributed by weighted glue:
%       \vspace*{\stretch{1}}   headline
%       \vspace*{\stretch{2}}   plate
%       \vspace*{\stretch{2}}   caption
%       \vspace*{\stretch{1}}
%   A third of the slack goes above the plate and a third below
%   it, a sixth each to the top and the bottom.  THE PAGE FILLS
%   ITSELF, whatever the plate's height, and the same source works
%   on every showcase page in the cohort with no per-page number
%   to get wrong.
%   The transform is purely mechanical, and the rider asserts it:
%   the headline, the plate and the caption come through
%   BYTE-IDENTICAL; only the frame around them changes.
%   NEW PROSE: none.
% ------------------------------------------------------------
%% ---- vv15.38 (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.
%% 2 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
%% v11.13: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
\setcitestyle{authoryear,round,comma} %% Foundation-block retrofit (May 2026): new citation convention (Arisaka, 2026, A*-Suffix)
\usepackage{mdframed}
\usepackage[most]{tcolorbox}
\usepackage{mathrsfs}

\PaperSeries{GG}
\PaperNumber{Arisaka-GG-A3-BI6}
\PaperStatus{Preprint}
%% v13.17 (2026-07-31): the cascade flowchart, redrawn.
%%   The figure still showed Step 3 reaching N from the trace ratio,
%%   the route v13.16 retired in the body and the appendix alike. It
%%   now draws the chain the theorems establish: Steps 1-2 fixing M
%%   and L free of N, the dual lock with the block embedding giving
%%   N = N_c = 3 (T-BI6-10.2a), then J and K, with the trace ratio
%%   dashed below as the independent check. Caption hedge retired.
% ------------------------------------------------------------
%% v15.26 -> v15.27, August 15, 2026.  THE GATE-CLOSING PASS.
%   One pass across A1-A7 so that `at v15' and `passes the v15
%   gate' mean the same thing before the first arXiv submission.
%   These seven were migrated before the current toolchain and
%   each failed at least one absolute section 4 item.
%   SPELLING (section 9 decision 1, "total, in printed text"):
%     5 printed British form(s) changed here.  The raw guard
%     reported 91 across the seven; 51 were NOT defects and were
%     hand-classified one site at a time -- 33 correct American
%     English now permanently in PROTECT, 12 published titles now
%     frozen by exact text (correcting a title is a
%     misquotation), and 6 `cancellable'/`cancellability', held at
%     the doubled l because American English keeps it in
%     `cancellation', the -able form is listed both ways, and it
%     is a defined term.  The -ed/-ing forms ARE swept.
%     Edited BY OFFSET, not by text anchor: these papers hard-wrap
%     their source and anchors copied from a flattened scan do not
%     exist -- eight failed that way on the first run, and the
%     same defect hit A8's rider an hour earlier.  The sites come
%     from the guard itself, which already excludes comments,
%     labels, PROTECT and the frozen titles.
%   FLOAT STANDARD: 2 bare grids: the nine-name glossary promoted to a captioned
%%     longtable (9 rows, so section 8's `unbreakable' case as well
%%     as its `no caption' case -- a longtable closes both), and the
%%     identifier map captioned in place (it was already a ruled,
%%     breakable longtable; only the caption was missing).
%   PALETTE: nothing to change.  0 violations.
%   NEW PROSE is listed by table and page in GG-Report-362 and is
%   a draft for the author to accept or rewrite.  Caption
%   EXTENSIONS keep every existing sentence verbatim -- section
%   14.2 reserves rewriting a caption's CLAIM, and none is.
%   NO NUMBER, NO CLAIM AND NO GRADE CHANGES ANYWHERE.
% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.27 -> v15.28, August 15, 2026.  THE THREE-PAGE FRONT-MATTER
%% SHOWCASE.  Third of the twenty-four.
%%   A3 is a proof, so the three pages are the claim, the standing,
%%   and the honesty.
%%   THE CLAIM (Figure 1): the anomaly is not a defect of quantum
%%   field theory to be cured but a classical topological identity on
%%   the bulk, of which the 4D anomaly is the shadow.  The plate
%%   carries the coefficients (1,-1,-2/5,-2/5) and the trace-ratio
%%   lock, so the claim and its fingerprint are on one page.
%%   THE STANDING (Table 8): what is actually proven, level by level,
%%   against the 1984 mechanism this descends from.  Nine rows, each
%%   a status rather than a claim of superiority.
%%   THE HONESTY (Table 2): every principal claim with its grade,
%%   including the row marked Open and the two marked Deferred.  The
%%   headline is the author's own instruction to a skeptical reader.
%%   BOTH TABLES ARE REPRODUCED WHOLE.  Nothing is condensed, and the
%%   row text is lifted from the source rather than retyped.
%%   TWO showcase longtables, so the counter is stepped back twice --
%%   A1 v15.23 shipped every table numbered one too high for want of
%%   one of these.
%%   tab:status IS A tabularx and longtable has no X column, so the
%%   showcase copy computes the third width: A3 sets \tabcolsep to
%%   6 pt, three columns with @{} at both ends spend 24 pt, leaving
%%   0.9489\textwidth, and 0.9489 - 0.42 - 0.18 = 0.3489.
%% ------------------------------------------------------------
\PaperVersion{v20.0}
\PaperDate{August 21, 2026}

\hypersetup{
 pdftitle = {Anomaly Freedom of BI-6: The Geometric Consistency of GG-6
 (Arisaka-GG-A3-BI6)},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {BI-6 anomaly freedom, local and global theorem, OGN-5,
 Green-Schwarz mechanism as the technical pioneering of
 AX-2 GGEP (Geometry-Gauge Equivalence Principle),
 GG-6 bulk. This paper is companion A3 of the
 Arisaka-GG-A* foundation series.},
 pdfkeywords= {Arisaka-GG-A3-BI6, GG-Theory, GG-Constitution, GG-6, BI-6,
 anomaly freedom, Bianchi identity, geometry-gauge
 equivalence (GGEP, AX-2), global anomaly, Wu-Chern-Simons,
 differential cohomology (Hopkins-Singer), OGN-5,
 Green-Schwarz mechanism (1984), SU(5)}
}

%% Paper-specific macros
\providecommand{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}
%% \LGI / \GGI / \DGI / \GSSB / \GA macros are now provided by
%% ucla-arisaka-gg-6.sty (v5.2) in sans-serif form.  The earlier local
%% \newcommand definitions used \mathrm{...}; they are commented out
%% here so the style-file sans-serif versions take effect.
%% \newcommand{\LGI}{\ensuremath{\mathrm{LGI}}}
%% \providecommand{\DGI}{\ensuremath{\mathrm{DGI}}}
%% \providecommand{\GSSB}{\ensuremath{\mathrm{GSSB}}}
%% \providecommand{\GA}{\ensuremath{\mathrm{GA}}}
%% \newcommand{\GGI}{\ensuremath{\mathrm{GGI}}}
\newcommand{\PLA}{\ensuremath{\mathrm{PLA}}}
\newcommand{\PLT}{\ensuremath{\mathrm{PLT}}}
\newcommand{\FEP}{\ensuremath{\mathrm{FEP}}}
\newcommand{\MEPP}{\ifmmode\mathsf{MEPP}\else\textsf{MEPP}\fi}
\providecommand{\POVM}{\ensuremath{\mathrm{POVM}}}
\providecommand{\SO}{\ensuremath{\mathrm{SO}}}
\providecommand{\Spin}{\ensuremath{\mathrm{Spin}}}
\providecommand{\Sp}{\ensuremath{\mathrm{Sp}}}
\providecommand{\SUfive}{\ensuremath{\mathrm{SU}(5)}}
\providecommand{\UoneB}{\ensuremath{\mathrm{U}(1)_B}}
\providecommand{\UoneQi}{\ensuremath{\mathrm{U}(1)_{Qi}}}
\providecommand{\Msix}{\ensuremath{M_6}}
\providecommand{\Mfour}{\ensuremath{M_4}}
\newcommand{\Useven}{\ensuremath{U_7}}
\newcommand{\Weight}{\ensuremath{W_8}}
\newcommand{\WtSix}{\ensuremath{W_6}}
\newcommand{\Ttwo}{\ensuremath{T^2_{r\theta}}}
\newcommand{\GS}{\ensuremath{\mathrm{GS}}}
\newcommand{\WCS}{\ensuremath{\mathrm{WCS}}}
\newcommand{\Trbig}{\ensuremath{\Tr}}
\newcommand{\trsmall}{\ensuremath{\tr}}
\newcommand{\TrU}{\ensuremath{\Tr}}
\newcommand{\trU}{\ensuremath{\tr}}
\newcommand{\checkX}{\ensuremath{\check X}}
\newcommand{\checkB}{\ensuremath{\check B}}
\newcommand{\checklambda}{\ensuremath{\check\lambda}}
\newcommand{\hol}{\ensuremath{\mathrm{hol}}}
\newcommand{\intnorm}[1]{\left[ #1 \right]_{\mathrm{int}}}
\newcommand{\an}{\ensuremath{\mathrm{an}}}
\newcommand{\tot}{\ensuremath{\mathrm{tot}}}
\newcommand{\AdmissibleClass}{\ensuremath{\Admissible}}
\newcommand{\SpinBord}{\ensuremath{\Omega^{\mathrm{Spin}}}}
%% \providecommand{\OGNfive}{\ensuremath{\mathrm{OGN\mbox{-}5}}} (no-op; style file has sans-serif version)
%% \GGConstitution macro now lives in ucla-arisaka-gg-6.sty (v5.2);
%% the local definition that was added in A3 v5.0 is no longer needed.
\providecommand{\Omegaorb}{\ensuremath{\Omega_{\mathrm{orb}}}}
\providecommand{\Omegaeff}{\ensuremath{\Omega_{\mathrm{eff}}}}
\newcommand{\ChargePacket}{\ensuremath{\mathcal P_N}}
\newcommand{\Ptwo}{\texttt{Arisaka-GG-A2-GG6}}
\newcommand{\Pfive}{\texttt{Arisaka-GG-A3-BI6}}

\definecolor{softgray}{RGB}{245,246,248}
\definecolor{softgreen}{RGB}{235,247,237}
\definecolor{softamber}{RGB}{255,247,230}
\definecolor{softred}{RGB}{252,235,235}
\definecolor{softblue}{RGB}{235,243,252}
\definecolor{Accent}{HTML}{1F4E79}
\definecolor{Soft}{HTML}{EFF5FB}
\definecolor{Soft2}{HTML}{F7FAFD}
\definecolor{Warn}{HTML}{7A1F1F}
\definecolor{Good}{HTML}{1E5F3B}

\newmdenv[
 backgroundcolor=softblue,
 linecolor=blue!50!black,
 linewidth=0.8pt,
 roundcorner=4pt,
 skipabove=8pt,
 skipbelow=8pt,
 innerleftmargin=10pt,
 innerrightmargin=10pt,
 innertopmargin=8pt,
 innerbottommargin=8pt
]{auditbox}

\newmdenv[
 backgroundcolor=softamber,
 linecolor=Goldenrod4,
 linewidth=0.8pt,
 roundcorner=4pt,
 skipabove=8pt,
 skipbelow=8pt,
 innerleftmargin=10pt,
 innerrightmargin=10pt,
 innertopmargin=8pt,
 innerbottommargin=8pt
]{cautionbox}

\newmdenv[
 backgroundcolor=softgreen,
 linecolor=ForestGreen,
 linewidth=0.8pt,
 roundcorner=4pt,
 skipabove=8pt,
 skipbelow=8pt,
 innerleftmargin=10pt,
 innerrightmargin=10pt,
 innertopmargin=8pt,
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]{answerbox}

\newmdenv[
 backgroundcolor=softred,
 linecolor=BrickRed,
 linewidth=0.8pt,
 roundcorner=4pt,
 skipabove=8pt,
 skipbelow=8pt,
 innerleftmargin=10pt,
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\newtcolorbox{summarybox}{colback=Soft,colframe=Accent,boxrule=0.6pt,arc=1.5mm,left=2mm,right=2mm,top=1.5mm,bottom=1.5mm}
\newtcolorbox{riskbox}{colback=Soft2,colframe=Warn,boxrule=0.6pt,arc=1.5mm,left=2mm,right=2mm,top=1.5mm,bottom=1.5mm}
\newtcolorbox{claimbox}{colback=Soft2,colframe=Good,boxrule=0.6pt,arc=1.5mm,left=2mm,right=2mm,top=1.5mm,bottom=1.5mm}


\providecommand{\GGPart}[2]{%
 \clearpage
 \refstepcounter{part}%
 \addcontentsline{toc}{part}{Part \thepart. #1 --- #2}%
 \thispagestyle{plain}%
 \begin{center}
 \vspace*{1.2em}
 {\Large\bfseries Part \thepart}\\[0.45em]
 {\huge\bfseries #1}\\[0.35em]
 {\large #2}
 \end{center}
 \vspace{0.5em}
 \GGThickLine
 \vspace{0.8em}
}


%% ---- v13.6: one-line Part entries (cohort standardization, July 19,
%%      2026; P4-Real v3.2 preamble pattern).  \GGPart renewed verbatim
%%      from ucla-arisaka-gg-11.sty except that the contents line now
%%      carries only "Part N. <title>"; the " --- <description>" tail is
%%      no longer written to the .toc, so every Part entry fits on one
%%      line in both the Overview of Contents and the Contents.  Part
%%      title pages are unchanged (full title + description).
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
  \addcontentsline{toc}{part}{\protect\textbf{Part \thepart}. #2}%
  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}% running pointer
  \thispagestyle{plain}%
  \begin{center}
    \vspace*{1.2em}
    {\Large\bfseries Part \thepart}\\[0.45em]
    {\huge\bfseries #2}\\[0.35em]
    {\large #3}
  \end{center}
  \vspace{0.5em}
  \GGThickLine
  \vspace{0.8em}
}
\makeatother

%% --- Sans-serif paper-code macros (programme-wide v7.1 sweep, May 13, 2026)
%% Programme-defined names rendered in sans-serif (per L21/L23) and with
%% full GG- prefix throughout the body.  The \AFull... variants prepend
%% Arisaka- and are used for explicit listings of preprint titles.
\providecommand{\AOne}{\textsf{GG-A1-Constitution}}
\providecommand{\ATwo}{\textsf{GG-A2-GG6}}
\providecommand{\AThree}{\textsf{GG-A3-BI6}}
\providecommand{\AFour}{\textsf{GG-A4-Euclid}}
\providecommand{\AFive}{\textsf{GG-A5-GDOS}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}  % Gi4 -> GRIP (cohort, May 2026); renew: gg-11.sty still defaults to Gi4
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}
\providecommand{\ASeven}{\textsf{GG-A7-Qi4}}
\providecommand{\AEight}{\textsf{GG-A8-QM}}
\providecommand{\ANine}{\textsf{GG-A9-SM}}
\providecommand{\ATen}{\textsf{GG-A10-Cosmos}}
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
\providecommand{\PaperPOne}{\textsf{GG-P1-KK}}
\providecommand{\PaperPTwo}{\textsf{GG-P2-N3}}
\providecommand{\PaperCOne}{\textsf{GG-C1-Kaxion}}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}%% v13.6: Foundations Branch member (canonical form, Cohort-Map-19 v1.4)
\providecommand{\PaperPFive}{\textsf{GG-P5-CPT}}%% v13.6: Foundations Branch member (canonical form, Cohort-Map-19 v1.4)
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\AFullOne}{\textsf{Arisaka-GG-A1-Constitution}}
\providecommand{\AFullTwo}{\textsf{Arisaka-GG-A2-GG6}}
\providecommand{\AFullThree}{\textsf{Arisaka-GG-A3-BI6}}
\providecommand{\AFullFour}{\textsf{Arisaka-GG-A4-Euclid}}
\providecommand{\AFullFive}{\textsf{Arisaka-GG-A5-GDOS}}
\providecommand{\AFullSeven}{\textsf{Arisaka-GG-A7-Qi4}}
\providecommand{\AFullEight}{\textsf{Arisaka-GG-A8-QM}}
\providecommand{\AFullNine}{\textsf{Arisaka-GG-A9-SM}}
\providecommand{\AFullTen}{\textsf{Arisaka-GG-A10-Cosmos}}
\providecommand{\PaperFullPOne}{\textsf{Arisaka-GG-P1-KK}}
\providecommand{\PaperFullPTwo}{\textsf{Arisaka-GG-P2-N3}}
\providecommand{\PaperFullCOne}{\textsf{Arisaka-GG-C1-Kaxion}}


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\begin{document}
%% ---- v15.13 (August 8, 2026) ----
%% Two items closed.
%% (1) Spelling: 14 further sites taken to the American standard.  The
%%     earlier passes worked from a pair list built out of
%%     spelling_audit.py, which only reports words spelt BOTH ways --
%%     so words this paper always spelt the British way (renormalisation,
%%     factorisation, quantisation, geometrisation) were invisible and
%%     survived.  A rule over the -ise/-isation family replaces the list.
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%%     Faraday 1845, Stueckelberg-Petermann 1953 (French), Symanzik 1970,
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%%     realise, generalise, behaviour, colour and the rest).
%%     PROTECTED and deliberately unchanged: Stueckelberg &
%%     Petermann (1953) "La normalisation des constantes...", which
%%     is French, and Symanzik (1970) "Small distance behaviour in
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%% deliberately unchanged: Lakatos (1970), "Falsification and the
%% methodology of scientific research programmes", which is that paper's
%% actual title, and Klein's "Erlangen Programme".  The cohort's own
%% cited titles ARE corrected: A1-Constitution's title page reads
%% "GG-Theory Program", so citing it as "Programme" was a misquotation.
% ---- v13.18 (August 2, 2026) ----
% Significance section gains point Four, 'why the ledger cannot
% run' (count fixed Three -> Four): anomaly inflow as circle-side
% topological bookkeeping on the fibre that remembers (integers,
% exact at every scale); running as the interval's business
% (POS-5's one-way RG semigroup; no holonomy to renormalise); why
% the ledger (1,3,5;7,35) surfaces unchanged from alpha^-1(0)=137
% to the kaxion spacing to the TeV lock across the e^140 of
% boundary time; 'the circle remembers, and the ledger is written
% on the circle.' Part of the A1/A3/A4 completion sweep.


%% ===============================================================
%%  PAGE 1: TITLE + ABSTRACT
%% ===============================================================
\GGTitlePage
 {Anomaly Freedom of \BIsix{}}
 {The Modified Bianchi Identity (\BIsix{}) for the Six-Force,\\
 Six-Dimensional Bulk (\GG{})}
 {August 10, 2026}
\thispagestyle{empty}

\vspace{12pt}
\begin{abstract}
\noindent
For half a century a quantum anomaly has been described as a flaw that
shows up only when a classical symmetry is run through the quantum
machine. This paper makes the opposite case: the anomaly is, at bottom,
a classical and geometric fact about a higher-dimensional space, and the
quantum calculation merely \emph{detects} it. The lineage is old --- Bianchi
wrote the geometric ancestor in 1902, and Green and Schwarz, in 1984,
modified it for higher dimensions --- and it is geometric at every step.

We show that the six-dimensional bulk of \GGTheory{}, built from
the four axioms and seven postulates of its constitutional charter, passes
the anomaly test completely. The proof has two halves: a local part that
closes the usual one-loop consistency in a single line, and a global part
that closes it under every large gauge transformation --- with no
restriction on the topology of the four-dimensional base, because the
relevant spin bordism groups of the gauge group's classifying space are
shown to vanish. We
also show that the five small integers $(1,3,5;7,35)$ that fix every
normalization in the theory are locked rather than fitted: the
unified block size is forced by single-tensor consistency alone, the
color index by the dual-lock closure together with the block embedding,
and the remaining two by the same closure identity. No step of the
cascade consumes a measured quantity.

Read alongside the companion papers, this places \BIsix{} on the same
foundational layer as the 1984 Green--Schwarz mechanism it generalizes
--- a program, begun with Riemann and Bianchi, of the geometrization
of gauge in higher dimensions rather than the quantization of geometry.
What makes a theory meaningful is not the cleverness of its calculations
but the discipline of its consistency tests, and this is the sharpest
one we know how to write down.
\end{abstract}


%% ===============================================================
%%  PAGES 2-3: EXECUTIVE SUMMARY + KEYWORDS
%% ===============================================================
\begin{ExecutiveSummary}
\paragraph{An anomaly is a geometric fact, not a quantum accident.}
For more than a century the same idea has reappeared in new clothing.
Bianchi's 1902 identity \citep{Bianchi1902} says that the curvature of a
smooth space is ``closed'' --- a purely geometric, pre-quantum statement;
Yang and Mills \citep{YangMills1954} carried it over to the internal
symmetries of the forces. What the Adler--Bell--Jackiw calculation of 1969
\citep{Adler1969,BellJackiw1969} found as a \emph{quantum} anomaly, and
what the descent formalism of \citet{WessZumino1971} and the modern
differential-cohomology reading
\citep{HopkinsSinger2005,MonnierMoore2019,FreedHopkins2016} later
clarified, is that the anomaly was geometric all along: a characteristic
class on a bundle, classically defined and merely \emph{detected} by the
quantum loop rather than produced by it.

\paragraph{The bulk-side rewrite and its constitutional reading.}
Green and Schwarz \citep{GreenSchwarz1984} showed in 1984 that this
obstruction can be absorbed geometrically in a higher-dimensional bulk:
replacing $dH = 0$ by $dH = X_4$ lets a bulk four-form flow in and cancel
the projected boundary obstruction. In the reading of \AOne{} and \ATwo{},
that bulk identity is the content of \AX{2}~(\GGEP), and the
four-dimensional observer's ``anomaly'' is its projected shadow. The
companion paper \ATwo{} \citep{ArisakaA2GG6} derived the six-dimensional bulk of \GGTheory{} and stated its four-component modified
Bianchi identity \BIsix{}; the present paper proves that the resulting
theory passes the anomaly test completely.

\begin{equation}
\boxed{\;
\dd H_3 \;=\; X_4 \;=\; \tr R^2 \,-\, \Tr F_5^2 \,-\, \tfrac{2}{5}\, F_B^2 \,-\, \tfrac{2}{5}\, F_{Qi}^2.
\;}
\label{eq:bi6_exec}
\end{equation}

The four-component identity \eqref{eq:bi6_exec} gathers the four curvature terms ---
gravity ($\Spin(1,5)$), the $\SU(5)$ gauge force, and the two Abelian
factors $\U(1)_B$ and $\U(1)_{Qi}$ --- with the \OGN{}-locked coefficients
$(+1, -1, -2/5, -2/5)$ into a single closed gauge-invariant four-form
$X_4$, which inflows to the tensor field $B_2$ via $\dd H_3 = X_4$. The
trace-ratio lock $|a/c| = |a/d| = 5/2$ pins both Abelian fractions
symmetrically to $-2/5$; no coefficient is free to be adjusted.

\paragraph{Six principal results.}
The paper carries six principal results:
\begin{itemize}[leftmargin=2em,itemsep=0.4em]
\item \textbf{R1 --- Rank-one criterion.} Single-tensor cancellation is
 possible if and only if the $4\times 4$ anomaly matrix has rank one;
 the microscopically computed \BIsix{} matrix saturates this condition.
\item \textbf{R2 --- Complete local anomaly cancellation.} Under the
 factorization $I_8 = \tfrac12 X_4^2$, the one-loop consistent anomaly
 is canceled exactly by the \BIsix{} tensor counterterm.
\item \textbf{R3 --- Complete global anomaly freedom.} On the physical
 intended \GG{} background class (\GG: the six-dimensional Geometry-Gauge bulk geometry of the framework), the
 Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore
 differential-cohomology framework's four hypotheses (H1)--(H4) are
 verified automatically.
\item \textbf{R4 --- Constitutional admissibility of $\U(1)_{Qi}$.} The
 \GGTheory{} Constitution admits both the three-component bulk
 $\Spin(1,5) \times \SU(5) \times \U(1)_B$ and the four-component bulk
 $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$; the
 four-component version is adopted on empirical rather than
 constitutional grounds.
\item \textbf{R5 --- Model-level benchmark spectrum.} An explicit
 finite chiral spectrum saturates the factorization, proving that the
 structural theorem is satisfiable; the gravitino-free
 $(n_{SD}, n_F)$ closure is unique once the visible gauge content is
 fixed.
\item \textbf{R6 --- \OGNfive{} integer ledger as a theorem.} The
 integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ (\OGNfive: the
 Occam-Gauge-Normalization five-tuple of \POS{6}) is locked by a
 five-step cascade internal to the \BIsix{} logic --- three integers
 forced by prior constitutional theorems, two by the dual closure
 identity $K = LJ = M^2 + N^2 + L^2$ --- including the trace-ratio
 identity $(L-N)/L = 2/5$ that bridges \BIsix{} to \OGNfive{}.
\end{itemize}

\paragraph{What makes the proof distinctive.}
Three features set this proof apart from earlier six-dimensional anomaly
work. It uses \emph{no supersymmetry}: where prior constructions
\citep{GreenSchwarzWest1985, Sagnotti1992, KumarMorrisonTaylor2010, MonnierMoore2019}
lean on $\cN = (1,0)$ supergravity to constrain the spectrum, this proof
leans on the constitutional clauses of \GGTheory{} (\AX{1}--\AX{4},
\POS{1}--\POS{7}) instead. The coefficient vector $(1, -1, -2/5, -2/5)$ is
uniquely fixed by the trace-ratio lock and \AX{4}~(\MSMF), with no
continuous fitting freedom. And the same integers carry sharp predictions
elsewhere in the cohort: the bioenergetic factor $\QB \approx 0.1555$~eV
per qubit matches ATP hydrolysis at biological temperatures
\citep{Bustamante2005}, the fine-structure value
$\alpha^{-1}(0) = 2^J + N^2 = 137$ uses $J = 7$ and $N = 3$, and the kaxion
twin-peak haloscope signature $\Delta\nu/\nu = 1/35$ uses the master lock
integer $K = 35$ --- all read directly off the ledger derived here.

\paragraph{Canonical implication chain.}
The present paper sits at the anomaly heart of the constitutional
implication chain
\begin{equation}
\resizebox{\linewidth}{!}{$\displaystyle
\boxed{\;\begin{array}{rcl}
\bigl\{\AX{1}, \AX{2}, \AX{3}, \AX{4}\bigr\} \,\cup\, \bigl\{\POS{1}, \ldots, \POS{7}\bigr\}
& \Longrightarrow &
\bigl\{\GG{}\ \mathrm{bulk},\ \BIsix{}\ \mathrm{identity}\bigr\}
\\[2pt]
& \Longrightarrow &
\bigl\{\text{local cancellation},\ \text{global completion},\ \OGNfive{}\ \mathrm{ledger}\bigr\}
\\[2pt]
& \Longrightarrow &
\GDOS{}\ \mathrm{lawfulness}
\\[2pt]
& \Longrightarrow &
\bigl\{\GiFour{} \oplus \QiFour{}\ \mathrm{pillars};\ \mathrm{SM};\ \Lambda\mathrm{CDM}\bigr\}.
\end{array}\;}
$}
\label{eq:implication_chain_a3}
\end{equation}
The chain runs from the constitutional clauses, through the bulk theorems
of \ATwo{}, through the local-plus-global anomaly proof of the present
paper, into the lawfulness condition for the \GDOS{} bulk-to-boundary
projection, and finally into the boundary phenomenology. The present paper
is the second descent step: local and global anomaly freedom is the gate
that the third step (\GDOS{} lawfulness) must clear before any boundary
phenomenology can legitimately be invoked.

\paragraph{Similar mathematics, different physics.}
It is worth saying plainly how this construction stands relative to
string theory, because the two share so much machinery. The modified
Bianchi identity itself, the Green--Schwarz inflow that absorbs the
anomaly, the differential-cohomology bookkeeping of Wu--Chern--Simons,
Hopkins--Singer, and Monnier--Moore, and even the scale-profile metric with its
logarithmic radial coordinate and bulk-to-boundary projection
--- all are borrowed, with gratitude, from the string program and its
AdS/CFT and Randall--Sundrum descendants. What changes is the physics the
mathematics is asked to carry. Where string constructions invoke
supersymmetry and a large-$N$ limit to control the spectrum, \GGTheory{}
invokes its constitutional axioms and carries neither. Where the 10D
Green--Schwarz polynomial factorizes through two functionally independent
four-forms, the 6D \BIsix{} factorization is forced to be rank-one by a
perfect-square condition, and is correspondingly more rigid. And where
AdS/CFT reads the radial direction as an exact duality between a bulk
gravity theory and a boundary conformal field theory, \GG{} reads it as a
literal, irreversible renormalization-group scale interval under
\OGGEP{}. The 1984 move that Green and Schwarz introduced as a technical
refinement of one specific model becomes, here, a foundational axiom
(\AX{2}, \GGEP{}) that any candidate theory of nature must satisfy. The
same equations, in short, carrying a different claim about what they
describe.

\paragraph{Philosophical implications.}
If the reading is correct, it carries a lesson that reaches past the
anomaly. The deepest advances in physics have repeatedly turned out to be
\emph{recognitions} rather than inventions: Maxwell recognized that light
was already present in his equations, and Einstein that gravity was
already present in Riemann's geometry. This paper proposes that the chiral
anomaly is the next such recognition --- not a defect of quantum field
theory to be exorcised, but a classical, topological identity on a
higher-dimensional bulk, of which the four-dimensional ``anomaly'' is only
the shadow cast on the observer's boundary. Read this way, the program
is not the quantization of geometry that has organized so much of the last
half-century, but its mirror image: the geometrization of gauge. The
discipline this demands --- a theory with zero free continuous parameters,
every number forced by an integer ledger rather than fitted to the data
--- is itself a philosophical commitment: that a law of nature should
predict the world rather than be made to accommodate it. Whether nature
honors that commitment is, of course, an empirical question and not a
philosophical one. Its sharpest form --- one value for each low-energy
constant across the whole sky, refuted by a single discordant measurement
--- is the \emph{second horizon problem}, drawn out in \ATen{} and
\PaperPThree{}.

\paragraph{Closing reflection.}
The test of an honest framework is not that it answers every question on
the first pass, but that the answers it does give are sharp enough to be
wrong. The local and global anomaly freedom theorem is exactly such an answer:
if a single anomaly term is ever found that escapes the four-component
lock, the whole construction is refuted upstream. The proposal will be vindicated or
refuted by quantitative tests over the 2026--2040 horizon: the KK lock
cluster near $\Mlock \approx 3.85$~TeV, the kaxion signal near
$\sim 8.1~\mu$eV, and the cosmological inventory $\Omega_m = 1/\pi$. Until
then, the recognition is a proposal, not a confirmed discovery. That is
the bargain we offer.

\end{ExecutiveSummary}

\GGKeywords{\GGTheory{}; \BIsix{} anomaly identity; local anomaly cancellation;
 global anomaly cancellation; Wu--Chern--Simons differential cohomology;
 Hopkins--Singer; Monnier--Moore; Green--Schwarz mechanism; rank-one
 factorization; trace-ratio lock; \OGNfive{} integer ledger;
 $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$; constitutional
 admissibility of $\U(1)_{Qi}$; mapping-torus completion; benchmark
 spectrum; falsifiability.}

% ================= FRONT-MATTER SHOWCASE, v15.28 ==============
% Third of the twenty-four.  A3 is a proof, so the three pages are
% the claim, the standing, and the honesty.  Both tables are
% reproduced WHOLE -- nothing is condensed, and the row text is
% lifted from the source rather than retyped, so no transcription
% can drift.  TWO showcase longtables, so the table counter is
% stepped back twice.
% ==============================================================
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 font=\sffamily\footnotesize,
 every node/.style={align=center, minimum height=8mm, rounded corners=2pt,
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 4D Ward identity on $\Mfour$ \\[1pt]
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\node[bulk] (BulkInflow) at (0, -2.8)
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 $X_4 = \dd H_3$ \\[1pt]
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\draw[arrowbig] (BulkInflow.north) -- (BdyAnom.south)
 node[midway, fill=white, inner sep=2pt, font=\scriptsize\itshape]
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\node[comp] (fa) at (-6, -5.8)
 {$a\,\tr(R\wedge R)$ \\[1pt]
 $a = +1$ \\[-1pt] {\footnotesize (gravitational)}};
\node[comp] (fb) at (-2, -5.8)
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 {\textbf{Trace-ratio lock:} \quad
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%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.

% ---- showcase 1 of 3: the claim ----------------------------------
% 1.02\textwidth -> scale 1.0553.  Natural 454.07281 x 272.34383;
% the body copy is at 0.9\textwidth, a scale of 0.9310.  Never
% smaller than its in-paper self.
\clearpage
%% The page fills itself: a minipage of explicit height with the
%% [s] inner alignment, and weighted stretch glue -- 1 : 2 : 2 : 1
%% top, above the plate, below the plate, bottom.  No fixed skip
%% to tune per page.
%% Height is \textheight MINUS \topskip.  A [t] minipage puts its
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%% \topskip below the top of the text block, so a box of exactly
%% \textheight overflows by \topskip minus that first line's
%% height -- measured at 5.5 pt, three times, one per showcase.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\centering
\vspace*{\stretch{1}}
\GGShowcaseHead{The Anomaly Is Not a Defect of Quantum Field Theory.\\[2pt]
It Is a Classical Identity, One Dimension Up}
\vspace*{\stretch{2}}
\makebox[\textwidth]{\resizebox{1.02\textwidth}{!}{\usebox{\GGscShadow}}}
\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{The four-component \BIsix{} bulk-side classical-topological identity and its boundary shadow.} The four curvature contributions (green) carry the canonical coefficients $(a,b,c,d) = (1,-1,-2/5,-2/5)$ and assemble into the single closed gauge-invariant four-form $X_4$ (red), a classical identity on the \AX{2}~\GGEP{} composite connection, pinned by the trace-ratio lock $|a/c| = |a/d| = 5/2$ (yellow). The boundary ``anomaly'' (blue) is the \AX{3}~\OGGEP{} projected shadow of this bulk identity --- a boundary re-measurement of a classical fact already in force in the bulk, not a quantum pathology to be cured. Full version, with the descent proof it summarizes: Figure~\ref{fig:bi6-bulk-shadow} in Part~I, p.~\pageref{fig:bi6-bulk-shadow}.}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage

%% >>>>>>>>>>>>>>>>>> GG-BI6-FOURFORM-GS-SHOWCASE v1.0 >>>>>>>>>>>>>>>>>>
%% Figures 4 and 7 on one page.  The two pictures are the SAME BYTES as
%% their in-text selves; only the enclosing scale differs.  They are
%% drawn at different base fonts -- 9 pt and 10 pt -- so the scales are
%% chosen to make both RENDER at about 10.4 pt rather than to make the
%% \textwidth fractions match.  Both are larger than their in-situ
%% selves, which is this paper's own rule.
\clearpage
%% The page fills itself: a minipage of explicit height with the
%% [s] inner alignment, and weighted stretch glue -- 1 : 2 : 2 : 1
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%% height -- measured at 5.5 pt, three times, one per showcase.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\centering
\vspace*{\stretch{1}}
\GGShowcaseHead{Four Curvature Squares, One Closed Form ---\\[2pt]
and the 1984 Mechanism Raised to an Axiom}
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%% Central anomaly node X_4
\node[draw=orange!55!black, circle, fill=orange!18, minimum size=1.4cm, line width=0.58pt]
 (X4) at (4.2, 1.5) {$X_4$};

%% Inflow target H_3
\node[draw=olive!55!black, rounded corners=3pt, fill=yellow!18, minimum width=2.8cm, minimum height=0.95cm,
 align=center]
 (H3) at (8.2, 1.5) {$H_3 = \dd B_2 + \mathrm{CS}$ \\ \footnotesize $\dd H_3 = X_4$};

%% Curved arrows from inputs to X_4 with edgelabel coefficients (the (1,-1,-2/5,-2/5) lock)
\draw[arrow] (grav.east) to[out=0, in=140] (X4.west);
\node[edgelabel] at (2.5, 2.85) {$+1$};

\draw[arrow] (su5.east) to[out=0, in=160] (X4.west);
\node[edgelabel] at (2.5, 1.95) {$-1$};

\draw[arrow] (uB.east) to[out=0, in=200] (X4.west);
\node[edgelabel] at (2.5, 1.05) {$-2/5$};

\draw[arrow] (uQi.east) to[out=0, in=220] (X4.west);
\node[edgelabel] at (2.5, 0.2) {$-2/5$};

%% Arrow from X_4 to H_3
\draw[arrow, line width=0.58pt] (X4.east) -- (H3.west);
\node[edgelabel] at (6.2, 2.05) {Bianchi inflow};

%% Header above X_4
\node[anchor=south, font=\small\bfseries] at (4.2, 3.15) {Anomaly four-form};

%% Trace-ratio lock label below X_4
\node[anchor=south, font=\footnotesize] at (8.2, 2.6)
 {trace-ratio lock $|a/c| = |a/d| = 5/2$};

%% Caption-side expression below
\node[anchor=north, font=\footnotesize] at (4.2, -0.7)
 {$X_4 \;=\; +\tr(R^2) \;-\; \Tr(F_5^2) \;-\; \tfrac{2}{5}\,F_B^2 \;-\; \tfrac{2}{5}\,F_{Qi}^2$};

\end{tikzpicture}}}
%% the two plates are two pictures, not one in two halves:
%% 3.5 lines of the body leading, 47.6 pt, measured against the
%% 97.0 pt of vertical slack this centred block had spare.
\par\vspace{3.5\baselineskip}
\makebox[\textwidth]{\resizebox{1.02\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 every node/.style={inner sep=2pt},
 edgelabel/.style={fill=white, inner sep=1pt, font=\footnotesize\itshape},
 arrow/.style={->, line width=0.47pt, draw=black!70},
 bigarrow/.style={->, line width=0.64pt, draw=violet!70!black}
]

%% Left panel: 10D heterotic GS factorisation
\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.2cm, minimum height=0.7cm,
 align=center] (gs10) at (0, 4) {\textbf{1984 Green--Schwarz (10D)}};

\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.4cm, minimum height=0.6cm, align=center]
 (gsfact) at (0, 3) {$I_{12} = X_4^{\mathrm{GS}} \wedge X_8$};

\node[draw=blue!55!black, rounded corners=3pt, minimum width=4.4cm, minimum height=1.4cm,
 fill=blue!5, align=center]
 (gsX4) at (0, 1.55)
 {two-component $X_4^{\mathrm{GS}}$:\\[1pt]
 $\tr(R \wedge R) - \tr(F \wedge F)$\\[1pt]
 \footnotesize gauge group: $\SO(32)$ or $E_8 \times E_8$};

\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.4cm, minimum height=0.6cm, align=center]
 (gsBI) at (0, -0.1) {$\dd H = X_4^{\mathrm{GS}}$ \;\;\footnotesize{(10D modified Bianchi)}};

%% Middle: GGEP elevation arrow
\node[draw=violet!55!black, rounded corners=4pt, fill=violet!10, minimum width=3.4cm, minimum height=2.0cm,
 align=center, line width=0.64pt]
 (ggep) at (5.5, 1.55)
 {\textbf{AX-2 (\GGEP)}\\[2pt]
 \footnotesize Geometry--Gauge\\
 \footnotesize Equivalence Principle\\[2pt]
 \footnotesize \emph{constitutional}\\
 \footnotesize \emph{elevation}};

\draw[bigarrow] (gsX4.east) -- (ggep.west);
\node[edgelabel] at (3.6, 3.0) {GGEP elevation};

%% Right panel: 6D GG-6 BI-6 factorisation
\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=4.6cm, minimum height=0.7cm,
 align=center] (bi6label) at (11.5, 4) {\textbf{2026 BI-6 (GG-6 in 6D)}};

\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=5.2cm, minimum height=0.6cm, align=center]
 (bi6fact) at (11.5, 3) {$I_8 = \tfrac{1}{2}\, (X_4^{\mathrm{BI6}})^2$};

\node[draw=red!55!black, rounded corners=3pt, minimum width=5.2cm, minimum height=1.4cm,
 fill=red!10, align=center]
 (bi6X4) at (11.5, 1.55)
 {four-component $X_4^{\mathrm{BI6}}$:\\[1pt]
 $\tr R^2 - \Tr F_5^2 - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2$\\[1pt]
 \footnotesize gauge group: $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$};

%% the GGEP-elevation arrow, drawn after bi6X4 exists so the node
%% cannot paint over its head
\draw[bigarrow] (ggep.east) -- (bi6X4.west);

\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=5.2cm, minimum height=0.6cm, align=center]
 (bi6BI) at (11.5, -0.1) {$\dd H_3 = X_4^{\mathrm{BI6}}$ \;\;\footnotesize{(6D modified Bianchi)}};

%% Bottom annotation: rank-one perfect-square distinction
\node[anchor=north, font=\small, align=center] at (5.8, -1.1)
 {\emph{10D: product factorization $X_4 \wedge X_8$,
 two functionally independent forms.}\\
 \emph{6D: perfect-square factorization $\tfrac12 X_4^2$,
 forced rank-one --- structurally more rigid.}};

\end{tikzpicture}}}
\par\vspace*{\stretch{2}}
\captionof*{figure}{\textbf{What the four components are, and where they came from.} \emph{Above:} the bulk gauge--gravity group $\mathrm{Spin}(1,5)\times\SU(5)\times\UoneB\times\UoneQi$ contributes four distinct curvature-squared four-forms, which combine with the \OGNfive-locked coefficients $(+1,-1,-2/5,-2/5)$ into the single closed gauge-invariant $X_4$; the trace-ratio lock $|a/c| = |a/d| = 5/2$ pins the Abelian fraction, and the modified Bianchi identity $\dd H_3 = X_4$ closes the whole inflow through one tensor sector. \emph{Below:} the same object read against the mechanism it generalizes. Green and Schwarz pinned a \emph{two}-component pattern in ten dimensions in 1984, as a technical refinement realized in a specific model; \BIsix{} is the four-component version, and what \GGTheory{} adds is the axiom -- \AX{2} (\GGEP) -- that raises the move from a refinement to a requirement any candidate theory must meet. The 10D pattern admits two functionally independent forms; the 6D pattern is rank-one by perfect-square forcing, and so is the more rigid of the two. Both panels appear again at full size, with their proofs, as Figure~\ref{fig:bi6-anomaly-inflow} in Part~II and Figure~\ref{fig:gs-as-ggep} in Part~IV.}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% <<<<<<<<<<<<<<<<<< GG-BI6-FOURFORM-GS-SHOWCASE v1.0 <<<<<<<<<<<<<<<<<<
%% >>>>>>>>>>>>>>>>>> GG-OGN5-CASCADE-SHOWCASE v1.0 >>>>>>>>>>>>>>>>>>
%% A2-GG6's cascade showcase plate, adopted at the author's request.
%% The plate and the caption down to its closing pointer are
%% BYTE-IDENTICAL to the copy in the sibling paper and reproduce
%% A2 page 6 exactly.  Two things are local by design: the headline
%% command follows this paper's own showcase idiom, and the closing
%% pointer resolves against this paper's own labels.  Pure TikZ: it
%% adds no asset to the arXiv package.
\clearpage
%% The page fills itself: a minipage of explicit height with the
%% [s] inner alignment, and weighted stretch glue -- 1 : 2 : 2 : 1
%% top, above the plate, below the plate, bottom.  No fixed skip
%% to tune per page.
%% Height is \textheight MINUS \topskip.  A [t] minipage puts its
%% reference point on the first line's baseline, which TeX sets
%% \topskip below the top of the text block, so a box of exactly
%% \textheight overflows by \topskip minus that first line's
%% height -- measured at 5.5 pt, three times, one per showcase.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\centering
\vspace*{\stretch{1}}
\GGShowcaseHead{Five Integers, Forced ---\\[2pt]
and No Free Parameter Left in the Bulk}
\vspace*{\stretch{2}}
\makebox[\textwidth]{\resizebox{0.92\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 every node/.style={inner sep=3pt},
 source/.style={draw=red!55!black, rounded corners=2pt, fill=red!12, minimum width=2.4cm,
 minimum height=0.55cm, align=center, font=\footnotesize},
 cascade/.style={draw=red!55!black, rounded corners=2pt, fill=blue!10, minimum width=1.6cm,
 minimum height=0.55cm, font=\bfseries},
 lock/.style={draw=red!55!black, circle, fill=green!18, minimum size=1.05cm,
 line width=0.52pt, font=\bfseries},
 idbox/.style={draw=red!55!black, rounded corners=2pt, fill=yellow!18, minimum width=3.6cm,
 minimum height=1.5cm, align=center, font=\scriptsize},
 chkbox/.style={draw=red!55!black, dashed, rounded corners=2pt, fill=black!4,
 minimum width=8.2cm, minimum height=0.55cm, align=center, inner xsep=6pt,
 font=\scriptsize},
 arr/.style={->, line width=0.38pt, draw=black!70},
 chk/.style={->, dashed, draw=black!45}
]

%% Stage 1: the two integers fixed with no color count in play
\node[source] (T1) at (0, 3.3) {Theorem 1 \\ (dim.\ minimality)};
\node[cascade] (M) at (5.0, 3.3) {$M = 1$};
\node[source] (T3) at (0, 1.9) {T-BI6-9.0 \\ (single-tensor \\ Green--Schwarz)};
\node[cascade] (L) at (5.0, 1.9) {$L = 5$};
\draw[arr] (T1.east) -- (M.west)
 node[midway, above, font=\footnotesize\itshape] {compact};
\draw[arr] (T3.east) -- (L.west)
 node[midway, above, font=\footnotesize\itshape] {free of $N$};

%% Stage 2: the dual lock is what determines the color index
\node[idbox] (id) at (5.0, -0.2)
 {dual lock \\[2pt]
  $K = LJ = M^2{+}N^2{+}L^2$ \\[1pt]
  $J = 2N{+}1$ \\[2pt]
  $\Longrightarrow\ N = L \pm \sqrt{L - M^2}$};
\node[cascade] (N) at (9.2, -0.2) {$N = N_c = 3$};
\draw[arr] (L.south) -- (id.north);
\draw[arr] (M.east) to[out=0, in=70] ([xshift=-4mm]id.north east);
\draw[arr] (id.east) -- (N.west)
 node[midway, above, font=\footnotesize\itshape] {$N < L$};

%% Stage 3: what follows once N is fixed
\node[lock] (J) at (11.8, 1.3) {$J = 7$};
\node[lock] (K) at (11.8, -1.7) {$K = 35$};
\draw[arr] (N.east) to[out=0, in=215] (J.south west);
\draw[arr] (N.east) to[out=0, in=145] (K.north west);
\node[anchor=north, font=\scriptsize, gray] at (11.8, 0.62) {$J = 2N + 1$};
\node[anchor=north, font=\scriptsize, gray] at (11.8, -2.42)
 {$K = LJ = M^2{+}N^2{+}L^2$};
\node[anchor=south, font=\small\bfseries] at (11.8, 2.1)
 {Closure consequences};

%% the trace ratio: a check on a ledger already determined
\node[chkbox] (tr) at (3.8, -2.9)
 {trace ratio $(L-N)/L = 2/5$ \;---\; \emph{independent check, not a route
  to $N$}};
\draw[chk] (tr.east) to[out=0, in=-90] (N.south);

\end{tikzpicture}}}
\par\vspace*{\stretch{2}}
%% >>> SHARED CAPTION >>>
\captionof*{figure}{\textbf{The \OGNfive{} Diophantine cascade.} Two theorems force $(M, L) = (1, 5)$ without reference to any color count (left column); the dual lock together with the block embedding $N < L$ then determines $N = N_c = 3$, and the pair $(J, K) = (7, 35)$ follows from the closure-index clause and the identity $K = LJ = M^2 + N^2 + L^2 = 35$ (right column). The double equality $35 = 5 \times 7 = 1 + 9 + 25$ is the axiomatic fingerprint of \OGNfive{}: it is not a coincidence but the content of the dual lock, and every numerical prediction of the program descends from these five integers. Solid arrows mark the forcing chain; the one dashed arrow carries the check. After the cascade closes, no continuous dimensionless parameter survives in the bulk theory. Full version, and the proof of each of the five steps: Figure~\ref{fig:ogn5-cascade-a3} in Part~III, p.~\pageref{fig:ogn5-cascade-a3}.}
%% <<< SHARED CAPTION <<<
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage%% Self-filling showcase frame (Report 462): this page carried no
%% frame at all, which is why it printed hard against the top.  The
%% glue is page-level -- a longtable cannot live inside a minipage.
\null\vspace*{\stretch{1}}

%% <<<<<<<<<<<<<<<<<< GG-OGN5-CASCADE-SHOWCASE v1.0 <<<<<<<<<<<<<<<<<<
% ---- showcase 2 of 3: the standing -------------------------------
{\centering\GGShowcaseHead{What Is Actually Proven, Level by Level ---\\[2pt]
Against the 1984 Mechanism This Descends From}}
\par\vspace*{\stretch{1}}
{\small\renewcommand{\arraystretch}{1.20}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.3163\textwidth}>{\raggedright\arraybackslash}p{0.3163\textwidth}>{\raggedright\arraybackslash}p{0.3163\textwidth}@{}}
\caption*{\textbf{What is actually proven, at each level of the argument.} The rows run from the structural theorem down to the specific model, and each cell records a status rather than a claim of superiority. The reading is mixed: the ten-dimensional structural theorem has been proved since 1984 and is not in question here, while the construction in this paper is younger and proves less at the model level. The table is here so that neither fact has to be inferred. Read with its surrounding assessment in Table~\ref{tab:assessment}, p.~\pageref{tab:assessment}.}\\
\toprule
\textbf{Level of proof} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
Structural GS theorem & Proven (1984) & Proven (this paper) \\
Coefficient uniqueness & Proven (1/30 for SO(32)) & Proven (2/5 from \OGN{}) \\
Rank-one / factorization & Implicit & Explicit (Thm.~\ref{thm:rank-one}) \\
BRST / Slavnov--Taylor & Proven & Proven (Thm.~\ref{thm:bi6-cancellation}) \\
Global gauge anomaly & Proven & Proven (\(\pi_6=0\)) \\
Explicit spectrum & Known (Type I / heterotic) & Benchmark constructed (Thm.~\ref{thm:model-level}) \\
Minimal spectrum & Known (SUSY fixes it) & Open (finite Diophantine search) \\
Gravitational completion & Automatic (SUGRA) & Explicit (SD tensors) \\
Charge lattice & String charge quantization & Abelian rescaling \\
Landscape / vacuum counting & \(\sim 10^{500}\) & Finite (bounded) \\
\bottomrule
\end{longtable}}
\addtocounter{table}{-1}
\par\vspace*{\stretch{1}}
\clearpage%% Self-filling showcase frame (Report 462): this page carried no
%% frame at all, which is why it printed hard against the top.  The
%% glue is page-level -- a longtable cannot live inside a minipage.
\null\vspace*{\stretch{1}}


% ---- showcase 3 of 3: the honesty --------------------------------
% tab:status is a tabularx in the body and longtable has no X
% column, so the third width is computed: 0.9489 - 0.42 - 0.18.
{\centering\GGShowcaseHead{Every Claim in This Paper, with Its Status ---\\[2pt]
Go to the Weakest Row}}
\par\vspace*{\stretch{1}}
{\footnotesize\renewcommand{\arraystretch}{1.15}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.42\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}p{0.3489\textwidth}@{}}
\caption*{\textbf{Every principal claim of this paper, with its status and a pointer.} The middle column is graded rather than rhetorical: a claim is proved, conditional on a stated hypothesis, or open, and the third column says where the proof lives or what the residue is. A reader who wants to test the paper should read this table first and then go to the weakest row, because the paper is only as strong as that row. Nothing downstream in the cohort consumes a row that is not marked proved. The same table, in place: Table~\ref{tab:status}, p.~\pageref{tab:status}.}\\
\toprule
\textbf{Claim} & \textbf{Status} & \textbf{Comment} \\
\midrule
If $I_8=\half X_4^2$, a single Green--Schwarz tensor cancels the full infinitesimal
local anomaly &
Proved &
Theorem~\ref{thm:bi6-cancellation}; rigorous descent proof in Section~\ref{sec:structural-proof} and Appendix~\ref{app:descent-full}. \\
The same \BIsix{} mechanism is globally anomaly-free on the intended \GG{} background
class &
Proved (two levels) &
Section~\ref{sec:global-completion} and Appendix~\ref{app:global-proof-dossier}; the theorem is unconditional for the intended product/collar \GG{} category. \\
An explicit finite benchmark spectrum realizes $I_8=(\Omega/2)X_4^2$ for \BIsix{} &
Proved &
Theorem~\ref{thm:model-level}; existence construction, not a uniqueness or minimality claim. \\
The same \BIsix{} term cancels orbifold-localized boundary anomaly by inflow &
Proved &
Standard descent on the interval, together with the globally defined differential-cocycle lift. \\
The \OGNfive{} integers $(M,N,L;J,K)=(1,3,5;7,35)$ are theorem-level consequences of \GG{} with \BIsix{} &
Proved, given \AX{4} &
Section~\ref{sec:ogn5-theorem}; the primitive tensor slot, the unified block size (Thm.~\ref{thm:L-from-GS}), the closure determination of the color index (Thm.~\ref{thm:N-from-closure}) and the dual lock fix the packet uniquely, with no empirical input. The fence is \AX{4}~(\MSMF{}), which selects the minimal package at two steps. \\
$U(1)_B$ is essential to the \BIsix{} / \GG{} chain, while $U(1)_{Qi}$ is allowed and optional at the narrow anomaly-theorem level &
Proved / clarified &
Section~\ref{sec:global-completion}; the baryonic slot is structural, the informational slot is anomaly-compatible and becomes natural/constitutive once \GG{} is extended to \GDOS{}. \\
\BIsix{} is analogous to the 1984 GS mechanism at the level of local descent/cohomology and now also admits a modern global completion &
Proved &
Sections~\ref{sec:comparison-10d} and \ref{sec:comparison-6d}; similarities and differences are stated feature by feature. \\
The benchmark spectrum is unique or absolutely minimal &
Open &
The paper gives an existence construction; minimality is a separate Diophantine problem. \\
$\Omega_{\mathrm{eff}} = \Omega_{\mathrm{orb}}/L^2$ is driven dynamically to 1 by the inflationary descent of \ATen{} &
Deferred &
Demonstrated in the companion cosmology paper. \\
The 28 Standard-Model parameters are derived from \OGNfive{} / \GDOS{} &
Deferred &
Demonstrated in the companion Arisaka-GG-A9-SM \citep{ArisakaA9SM} rather than proved in the present preprint. \\
\bottomrule
\end{longtable}}
\addtocounter{table}{-1}
\par\vspace*{\stretch{1}}
\clearpage
% ================= end front-matter showcase ====================



\GGfrontmatteronepagetrue
%% v11.13: \GGOverviewOfContents (centered) provided by ucla-arisaka-gg-11;
%% single-page Overview+List-of-Figures via the gg-11 toggle above.

%% v11.13: \GGListOfFigures (centered) provided by ucla-arisaka-gg-11.

\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents


%% ===============================================================
%% ===============================================================
%%  NOTATION -- one page.  Only the names and conventions a reader
%%  meets in the Abstract, Executive Summary or Preface BEFORE they
%%  are defined.  The full dictionary is Appendix A.
%% ===============================================================
\section*{Notation}
\addcontentsline{toc}{section}{Notation}
\label{sec:notation}

A handful of names and one typographic convention appear in the
Abstract, the Executive Summary and the Preface before the text has a
chance to define them. They are collected here, one line each, so that
nothing in the opening pages has to be taken on faith. The full
dictionary --- every abbreviation in the paper, with the companion
paper that introduces it --- is Appendix~\ref{app:abbrev-full}.

\paragraph{The two traces.} Three traces appear in this paper and are
distinguished typographically. \(\tr R^2\) is the trace over the
six-dimensional tangent indices of the Lorentz curvature two-form.
\(\Tr F_5^2\) is the trace over the \emph{fundamental} \(\mathbf 5\) of
\(\SU(5)\); where a different representation is meant it carries an
explicit subscript. \(F_B^2\) and \(F_{Qi}^2\) carry no trace at all: an
Abelian field strength is a one-by-one matrix. (Appendix~C follows the
ten-dimensional literature's convention for comparison and writes
adjoint traces explicitly as such; no adjoint trace appears anywhere in
the six-dimensional argument.)

\paragraph{One symbol, four meanings.} The letter \(\Omega\) is
overloaded in this literature and this paper does not escape it. Four
distinct objects wear it, and three of them appear within four pages of
each other in Part~II. \(\Omega_{\mathrm{orb}}\) (written \(\Omega\)
where no confusion can arise) is the Green--Schwarz reservoir
normalization, an integer in \(25\mathbb{Z}\).
\(\Omega_{\mathrm{eff}} = \Omega_{\mathrm{orb}}/L^2\) is the late-time
effective normalization, which the cosmological descent drives to one.
\(\Omega^{\mathrm{Spin}}_q\) with a superscript is a spin bordism group.
And \(\Omega_m\), \(\Omega_b\), \(\Omega_\Lambda\) with a subscripted
Latin or Greek letter are cosmological density parameters. The
subscript always disambiguates; where the bare symbol appears it means
the reservoir normalization.

\paragraph{The names used before they are defined.}

{\footnotesize\setlength{\LTpre}{0pt}\setlength{\LTpost}{6pt}%
\setlength{\abovecaptionskip}{4pt}\setlength{\belowcaptionskip}{2pt}%
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.155\textwidth}>{\raggedright\arraybackslash}p{0.795\textwidth}@{}}
\caption[The nine names used before they are defined.]{The nine names this paper uses before it defines them, gathered so that a reader can begin at Part~I without a companion paper open. Each entry says what the object is and which paper fixes it. Nothing here is defined --- every row is a pointer, and the definitions live where the pointers say they do. The order is the order in which the argument reaches them, not the alphabet.}\label{tab:a3-names-before-defined}\\
\toprule
\GGTheory{} & The twenty-two-paper program this paper belongs to; its axioms and postulates are fixed in \AOne{}. \\
\GG{} & Geometry--Gauge locking in six dimensions: the bulk this paper tests, derived in \ATwo{}. \\
\BIsix{} & The four-component modified Bianchi identity \(\dd H_3 = X_4\) of that bulk --- the object whose consistency is the subject of this paper. \\
\OGNfive{} & The five-integer ledger \((M,N,L;J,K)=(1,3,5;7,35)\), derived in Part~III. \\
\AXkw{} 1--4 & The four axioms of the Constitution: locality and causality; geometry--gauge equivalence (\GGEP{}); its operational extension (\OGGEP{}); and maximum symmetry, minimum freedom (\MSMF{}). \\
\POSkw{} 1--7 & The seven postulates: gauge invariance across scales, the unified extremal principle, anomaly freedom, closure and positivity, renormalization-group covariance, Occam gauge normalization, and measurement and falsifiability. \\
\GDOS{} & Geometric Dynamics of Open Systems --- the bulk-to-boundary projection carried by \AFive{}, which consumes this paper's global theorem. \\
\GiFour{}, \QiFour{} & The geometric and quantum-information pillars of \ASix{} and \ASeven{}. \\
BRST & The algebraic framework, named for its four originators, in which ``the anomaly cancels exactly'' is given its precise form in Part~II. \\
\bottomrule
\end{longtable}}


\noindent Every other abbreviation in this paper is expanded where it
first appears and again in Appendix~\ref{app:abbrev-full}.

\clearpage

%%  PREFACE -- compelling entrance for the entire paper
%%  Added in v5.x (May 2026) per user request: a global
%%  introduction that captures the reader's mind before Part I.
%%  Three paragraphs in the Arisaka register.
%% ===============================================================
\GGPreface

In 1969, three independent papers by Stephen Adler, John Bell, and
Roman Jackiw made a strange and beautiful discovery: a symmetry of
a classical field theory could fail at the quantum level. The
classical equations of motion conserved a current; the quantum
effective action did not. What looked like an exact conservation
law was, in fact, broken by a tiny topological term. The discovery
was called the chiral anomaly, and it has been one of the deepest
themes of twentieth-century physics ever since.

The chiral anomaly is not always a bug. Sometimes it is a feature:
it explains the observed decay rate of the neutral pion, for
example, with no free parameters. But in any theory of fundamental
physics, the anomaly has to be tamed. If the symmetry being broken
is the gauge symmetry that defines the theory, the whole theory
falls apart. The standard fix is to arrange the fermion content so
that the anomaly cancels exactly --- a delicate balance that has
constrained the Standard Model from the beginning. In 1984, Michael
Green and John Schwarz discovered a more elegant alternative: in
higher-dimensional theories, the anomaly can be canceled by a
topological inflow from a bulk field. The construction worked
beautifully in ten-dimensional string theory and became the
technical heart of the first superstring revolution.

This paper proposes that the deepest content of the Green--Schwarz
construction is not what it does for string theory but what it does
for the relationship between geometry and gauge. The modified
Bianchi identity that Green and Schwarz wrote down is a geometric
statement, classical and topological, that says gravity and gauge
curvature must live inside one closed topological four-form on the
bulk manifold. Read carefully, it is the equivalence of geometry
and gauge in higher dimensions. We extend the construction to six
dimensions with four-component anomaly inflow, prove that it admits
local and global anomaly freedom on the physical intended bulk
class, and show that the integer ledger of the entire program is
a Diophantine consequence of the same identity.

We invite the reader to walk the proof, and to read the long
supportive treatment in the final section as an invitation rather
than a critique. The Green--Schwarz mechanism was the moment string
theory glimpsed that geometry and gauge belong in the same picture.
The proposal here is that the picture has a second home --- in a
six-dimensional bulk that is small enough to audit and
rigid enough to predict.

\clearpage

\GGPart[Introduction]{Introduction}{Background, motivation, and the anomaly problem in six dimensions}

The first two papers of this series established the axiomatic framework (Arisaka-GG-A1-Constitution \citep{ArisakaA1Constitution})
and the six-dimensional bulk with its gauge--gravity structure and modified Bianchi identity (Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6}).
The present paper asks the next unavoidable question:
\emph{does that six-dimensional stage define a mathematically consistent quantum theory?}
The four sections in this Part set the stage.

Here is the problem in plain terms. A gauge theory is a theory built on
a redundancy: the mathematical description carries more information than
the physics does, and the physics is whatever survives when the surplus
is thrown away. That works only if the surplus really is surplus. An
\emph{anomaly} is what happens when a symmetry that is exact in the
classical description fails once the theory is quantized. If the failing
symmetry is a decorative one, the failure is a physical effect and can
be measured; the decay of the neutral pion is the textbook case. If the
failing symmetry is the gauge redundancy itself, there is no theory
left. Probabilities stop summing to one, states of negative norm appear
in the spectrum, and no prediction the theory makes can be trusted. This
is not a defect to be repaired later. It is the entrance examination
every candidate theory of nature has to pass before any of its other
claims are worth discussing.

Six dimensions make the examination harder than four, not easier. The
anomaly lives in a polynomial two degrees higher than the spacetime
dimension, so in six dimensions it is an eight-form, and it contains a
term that no rearrangement of the matter content can cancel against
another. Green and Schwarz found the way out in 1984, in ten dimensions:
if the dangerous polynomial happens to factorize, a single tensor field
can be made to cancel it by flowing charge in from the bulk. The price
is that the factorization is a very strong condition, and almost no
theory satisfies it.

Sections~\ref{sec:introduction} and~\ref{sec:geometric-origin} explain
what an anomaly is and trace the idea back to its geometric origin, from
Bianchi's 1902 identity to the differential-cohomology language of the
last two decades. Section~\ref{sec:gs-mechanism} states the
Green--Schwarz mechanism in general even dimensions, so that the
six-dimensional case that follows is a specialization of something
already in hand rather than a trick. Section~\ref{sec:inheritance-and-roadmap-A3}
says what this paper inherits from its two predecessors, what the rest
of the cohort is waiting on, and how the remaining nine Parts are
arranged.

\section{Introduction}
\label{sec:introduction}

\subsection{The anomaly problem in higher-dimensional gauge theories}

Quantum anomalies are the sharpest consistency constraints in theoretical
physics. A chiral gauge theory whose quantum effective action fails
to be gauge invariant is not merely inaccurate; it is \emph{inconsistent}:
unphysical gauge modes fail to decouple, Ward identities are violated,
and unitarity is lost. In even spacetime dimensions, the local anomaly
is encoded by an anomaly polynomial of degree \(d+2\), where \(d\) is
the spacetime dimension. Cancellation of this polynomial is therefore
a \emph{necessary condition} for any consistent chiral gauge theory.

The anomaly problem becomes especially acute in higher dimensions.
In four dimensions, the Standard Model cancels its anomalies through
a representation-theoretic miracle: the specific quantum numbers of
quarks and leptons conspire to make all gauge and mixed anomaly
coefficients vanish identically. No additional mechanism is needed.
In six and ten dimensions, however, the anomaly polynomial is richer
(an eight-form or twelve-form, respectively), and representation-theoretic
cancellation alone is generically insufficient. A new mechanism is
required: the \textbf{Green--Schwarz mechanism}
\citep{GreenSchwarz1984}, which introduces a higher-form tensor
field whose classical variation absorbs the quantum anomaly.

The discovery of this mechanism in 1984 by Green and Schwarz for the
ten-dimensional superstring was one of the pivotal events in modern
theoretical physics. It showed that anomaly cancellation in ten
dimensions is possible for exactly two gauge groups---\(\SO(32)\) and
\(E_8\times E_8\)---and launched the ``first superstring revolution.''

The present paper develops the six-dimensional counterpart of the
Green--Schwarz mechanism within a fundamentally different theoretical
framework: \GGTheory{}, which derives anomaly cancellation not from
supersymmetry but from a constitutional axiom system.

\subsection{The \GGTheory{} Constitution}
\label{sec:intro-gut10}

The \GGTheory{} program, developed in the companion paper
\citep{ArisakaA1Constitution}, proposes that the foundational layer of
physics is not a specific Lagrangian but a \textbf{constitution}:
a compact set of admissibility rules that any consistent theory must
satisfy. The constitution consists of four axioms and seven postulates:

\begin{itemize}[leftmargin=1.5cm]
\item \textbf{\AX{1}} (\LC): Locality and Causality---admissible evolution
 is local and causal; no acausal effective kernels are allowed.
\item \textbf{\AX{2}} (\GGEP): Geometry--Gauge Equivalence Principle---the
 primitive field is a composite geometry\(\oplus\)gauge connection;
 gravity and gauge structure are not separate fundamental sectors.
\item \textbf{\AX{3}} (\OGGEP): Operational/Open Geometry--Gauge
 Equivalence Principle---a bulk theory is incomplete unless it supplies
 a lawful bulk-to-boundary registration map producing operational
 observables.
\item \textbf{\AX{4}} (\MSMF): Maximum Symmetry, Minimum Freedom---among
 admissible theories, nature selects those with maximal symmetry and
 minimal non-forced continuous freedom.
\end{itemize}

The seven postulates (\POS{1} through \POS{7}) refine admissibility through
gauge invariance, a unified extremal principle, anomaly-free admissibility
by inflow, closure and positivity, RG compatibility, Occam Gauge
Normalization, and measurement/falsifiability.

\subsection{\GG{} and the \BIsix{} identity}
\label{sec:intro-gg6}

\paragraph{Historical background (with citations).} Generally
speaking, the identity $\dd H_3 = X_4$ that we are about to write
down is a particular \emph{modified} instance of a much older
geometric identity. In a 1902 note in the
\emph{Atti dell'Accademia dei Lincei}, Luigi Bianchi
\citep{Bianchi1902} proved that the Riemann curvature tensor
satisfies the differential identity
\(D_{[\mu} R_{\nu\rho]\sigma\tau} = 0\) ---
equivalently, \(D R = 0\) when $R$ is read as a curvature
two-form on the frame bundle. The Bianchi identity is the
geometric content of the statement that $D^2 = 0$ on a
connection, and it is the classical reason every gauge field
strength $F$ obeys $\dd F = 0$ on a principal bundle. The
Green--Schwarz mechanism \citep{GreenSchwarz1984} of
Section~\ref{sec:gs-mechanism} introduces a controlled, geometric
\emph{modification} of the Bianchi identity for the
antisymmetric tensor field strength $H_3$, replacing
$\dd H_3 = 0$ by $\dd H_3 = X_4$ for a closed,
gauge-invariant, characteristic-class four-form $X_4$ built from
$\tr(R \wedge R)$ and $\tr(F \wedge F)$. The
Green--Schwarz move is, in this geometric reading, simply the
statement that the natural Bianchi identity for the tensor sector
of a modern higher-dimensional bulk theory is not $\dd H = 0$ but
$\dd H = X_4$ --- a closed gauge-invariant four-form built from
the curvature and the gauge field strengths. \BIsix{} is the
four-component generalization of that modified-Bianchi pattern,
with the four contributors $(R, F_5, F_B, F_{Qi})$ pinned by the
\OGNfive{} ledger and the constitutional axioms of \GGTheory{}; we
spell out the construction immediately below.

The companion paper \citep{ArisakaA2GG6} derives, from \AX{1}--\AX{4}
and \POS{1}--\POS{7}, a minimal six-dimensional bulk
\begin{equation}
M_6 = M_4\times I_r\times S^1_\theta,
\qquad
r = \ln(\EGUT/\mu),
\label{eq:bulk-geometry}
\end{equation}
with composite gauge--gravity group
\begin{equation}
G_{\Bulk} = \Spin(1,5)\times\SU(5)\times\U(1)_B\times\U(1)_{Qi},
\label{eq:bulk-group}
\end{equation}
where the Lorentz spin factor $\Spin(1,5)$ enters on equal logical
footing with the three internal Lie-group factors via \AX{2} (\GGEP),
replacing the unstructured ``$\times\,\text{Gravity}$'' label of the
legacy notation. The canonical bulk metric is
\begin{equation}
ds_6^2 \;=\; e^{-2r}\,\eta_{\mu\nu}\,dx^\mu\,dx^\nu
\;+\; dr^2 \;+\; d\theta^2,
\qquad r \in [0, K],
\qquad \theta \in [0, 2\pi),
\label{eq:gg6-metric}
\end{equation}
with $\eta_{\mu\nu}$ the Minkowski metric on the 4D boundary
$M_4$, $r$ the natural-log RG-flow coordinate from the UV boundary at
$r = 0$ ($\mu = \EGUT$) to the IR boundary at $r = K$
($\mu = \EGUT \, e^{-K}$), and $\theta$ a dimensionless
topological circle that hosts integer-quantized Wilson-line phases.

\paragraph{The unique geometrical structure of \GG{}: a minimalist's
6D bulk.} The canonical metric
\eqref{eq:gg6-metric} is what one obtains when \AX{4} (\MSMF), \POS{5}
(\RG), and \POS{6} (\OGN) are demanded simultaneously of a
six-dimensional bulk: the four-dimensional piece is
flat-Minkowski with a single exponential scale profile $e^{-2r}$
locked to the natural RG-scale parametrization $r = \ln(\EGUT/\mu)$;
the radial fiber $I_r$ is logarithmic in the boundary energy scale
rather than a free spatial dimension; and the topological circle
$S^1_\theta$ has no intrinsic length, $G_{\theta\theta} = 1$ in
natural \GG-units. In the cohort's vocabulary the radial fiber $I_r$ is the \emph{scale dimension} and the circle $S^1_\theta$ is the \emph{gauge dimension}; the two are collectively the \emph{non-spatial dimensions} of the bulk, the term \emph{extra dimension} being reserved here for the \emph{spatial} Kaluza--Klein and string directions against which \GG{} is compared. The four conventional continuous parameters of
a generic Randall--Sundrum-style construction --- the AdS curvature
scale $k$, the compact-circle radius $R_\theta$, the scale endpoint
$r_c$, and the brane tension $\sigma$ --- are all absent: $k = 1$
is the natural-RG-scale normalization forced by \POS{6}, $R_\theta$
is a unit-conversion artifact that disappears in natural units,
$r_c$ is replaced by the integer endpoint $K$ of the \OGNfive{} ledger,
and the brane tension is fixed by the bulk-curvature condition.
What remains is a minimalist's 6D bulk with
\emph{no continuous geometric parameters at all}: every
dimensionless quantity in the metric is either zero, one, or the
single integer $K = 35$ that the \OGNfive{} cascade fixes in the
companion paper \citep{ArisakaA2GG6}. This is the
geometric reason the six-dimensional bulk passes the \AX{4}
(\MSMF) test: the conventional KK--RS continuous parameter
landscape of dimensions, radii, and curvature scales has been
collapsed entirely to a discrete arithmetic ledger. In the language of the companion paper, this scale-profile metric is the renormalization-group geometrization of the renormalization-group flow --- radial evolution \emph{is} the running of couplings, not motion through an extra spatial dimension --- so the construction is read on the field-theory side of the AdS/CFT dictionary rather than as a Randall--Sundrum spatial braneworld \citep{ArisakaA2GG6}.

The Green--Schwarz-modified Bianchi identity \BIsix{} is
\begin{equation}
\boxed{
\dd H_3 = X_4
= \tr(R\wedge R) - \Tr(F_5\wedge F_5)
- \frac{2}{5}\,F_B\wedge F_B - \frac{2}{5}\,F_{Qi}\wedge F_{Qi}.
}
\label{eq:bi6-main}
\end{equation}
The coefficient vector \((a,b,c,d)=(1,-1,-2/5,-2/5)\) is derived
from the trace ratio of the primitive baryon and hypercharge generators
in the fundamental \(\mathbf{5}\) of \(\SU(5)\).
The \OGNfive{} integer ledger
\begin{equation}
(M,N,L;\,J,K) = (1,3,5;\,7,35)
\label{eq:ogn5}
\end{equation}
compresses all dimensionless normalization freedom (beyond Newton's
constant \(G_N\)) into a discrete arithmetic package.

\subsection{This paper as the direct follow-up to \AOne{} and \ATwo{}}
\label{sec:intro-followup}

Arisaka-GG-A1-Constitution \citep{ArisakaA1Constitution} proposes the constitution; Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6} derives the six-dimensional stage, the composite gauge--gravity packet, and \BIsix{}; the present paper determines whether that stage is \emph{actually admissible as a quantum theory}. In that sense, this is not an auxiliary technical supplement but the first decisive consistency test of \GG{}. A six-dimensional bulk that fails anomaly freedom cannot serve as the parent of any later application in the series. Conversely, once anomaly freedom is established, the later uses of \OGGEP{} and \GDOS{} are placed on a much firmer mathematical foundation.

The paper is therefore two-layered. The main text states the theorem package in publishable form and develops the physical interpretation. The appendices then provide the full descent proof, the exact coefficient-matching equations, the detailed orbit solution, the global mapping-torus argument, and the automatic-admissibility lemmas. The standard is Euclidean: each step must follow from the constitutional and \GG{} premises.

\subsection{Relation to \GDOS{} and the open-system boundary}
\label{sec:intro-gdos}

The companion paper \citep{ArisakaA5GDOS} extends the \GG{} framework
to open systems through the Geometry-Dynamics-Open-System (\GDOS{})
construction. While \GG{} establishes the six-dimensional bulk theory,
\GDOS{} derives the four-dimensional boundary theory as the operational
shadow of the bulk: quantum mechanics, measurement, and the
Nakajima--Zwanzig open-system evolution \citep{Nakajima1958, Zwanzig1960} emerge as boundary projections
of the closed bulk dynamics through the \AX{3} (\OGGEP) channel.

The anomaly cancellation proved in the present paper is a
\emph{prerequisite} for \GDOS{}. If the six-dimensional bulk were
anomalous, the bulk-to-boundary channel would be ill-defined:
the Ward identities that protect the boundary observables would fail,
and the operational dictionary would not be lawful.
Conversely, the boundary anomaly generated by the orbifold projection
(which selects the Standard-Model zero modes) is exactly canceled
by inflow from the same \BIsix{} term.

Thus the anomaly proof of the present paper is the \emph{bridge}
between the bulk theorem of Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6} and the boundary theorem of the \GDOS{} paper \citep{ArisakaA5GDOS}.

\subsection{What is proved here and what now remains open}

Table~\ref{tab:status} summarizes the status of the main claims.
It separates theorem-level statements that are now closed from statements that
remain conditional, deferred, or open. The most important shift in the logical structure of the series is the promotion of \BIsix{} from a theory with only local cancellation plus preliminary bordism checks to a theory with a complete local-and-global anomaly theorem on the intended \GG{} background class.

\begin{table}[H]
\centering
\caption[Status of the main claims of this paper, claim by claim.]{Every principal claim of this paper with its status and a pointer. The middle column is graded rather than rhetorical: a claim is proved, conditional on a stated hypothesis, or open, and the third column says where the proof lives or what the residue is. A reader who wants to test the paper should read this table first and then go to the weakest row, because the paper is only as strong as that row. Nothing downstream in the cohort consumes a row that is not marked proved.}
\label{tab:status}
\renewcommand{\arraystretch}{1.15}
\footnotesize
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}p{0.42\textwidth}
 >{\raggedright\arraybackslash}p{0.18\textwidth}
 >{\raggedright\arraybackslash}X}
\toprule
\textbf{Claim} & \textbf{Status} & \textbf{Comment} \\
\midrule
If $I_8=\half X_4^2$, a single Green--Schwarz tensor cancels the full infinitesimal
local anomaly &
Proved &
Theorem~\ref{thm:bi6-cancellation}; rigorous descent proof in Section~\ref{sec:structural-proof} and Appendix~\ref{app:descent-full}. \\
The same \BIsix{} mechanism is globally anomaly-free on the intended \GG{} background
class &
Proved (two levels) &
Section~\ref{sec:global-completion} and Appendix~\ref{app:global-proof-dossier}; the theorem is unconditional for the intended product/collar \GG{} category. \\
An explicit finite benchmark spectrum realizes $I_8=(\Omega/2)X_4^2$ for \BIsix{} &
Proved &
Theorem~\ref{thm:model-level}; existence construction, not a uniqueness or minimality claim. \\
The same \BIsix{} term cancels orbifold-localized boundary anomaly by inflow &
Proved &
Standard descent on the interval, together with the globally defined differential-cocycle lift. \\
The \OGNfive{} integers $(M,N,L;J,K)=(1,3,5;7,35)$ are theorem-level consequences of \GG{} with \BIsix{} &
Proved, given \AX{4} &
Section~\ref{sec:ogn5-theorem}; the primitive tensor slot, the unified block size (Thm.~\ref{thm:L-from-GS}), the closure determination of the color index (Thm.~\ref{thm:N-from-closure}) and the dual lock fix the packet uniquely, with no empirical input. The fence is \AX{4}~(\MSMF{}), which selects the minimal package at two steps. \\
$U(1)_B$ is essential to the \BIsix{} / \GG{} chain, while $U(1)_{Qi}$ is allowed and optional at the narrow anomaly-theorem level &
Proved / clarified &
Section~\ref{sec:global-completion}; the baryonic slot is structural, the informational slot is anomaly-compatible and becomes natural/constitutive once \GG{} is extended to \GDOS{}. \\
\BIsix{} is analogous to the 1984 GS mechanism at the level of local descent/cohomology and now also admits a modern global completion &
Proved &
Sections~\ref{sec:comparison-10d} and \ref{sec:comparison-6d}; similarities and differences are stated feature by feature. \\
The benchmark spectrum is unique or absolutely minimal &
Open &
The paper gives an existence construction; minimality is a separate Diophantine problem. \\
$\Omega_{\mathrm{eff}} = \Omega_{\mathrm{orb}}/L^2$ is driven dynamically to 1 by the inflationary descent of \ATen{} &
Deferred &
Demonstrated in the companion cosmology paper. \\
The 28 Standard-Model parameters are derived from \OGNfive{} / \GDOS{} &
Deferred &
Demonstrated in the companion Arisaka-GG-A9-SM \citep{ArisakaA9SM} rather than proved in the present preprint. \\
\bottomrule
\end{tabularx}
\end{table}


%% ===============================================================
%%  SECTION 2: THE GREEN--SCHWARZ MECHANISM
%% ===============================================================
\section{The geometric origin of the anomaly concept: a constitutional reading of a hundred and twenty years}
\label{sec:geometric-origin}

The conventional reading of the gauge anomaly --- that it is a
modern quantum-mechanical effect, born in 1969 with the
Adler--Bell--Jackiw triangle calculation and tamed only when the
1980s anomaly-polynomial machinery matured --- has been the dominant
narrative for a half-century. In this section we propose a different
reading. The fundamental object underlying the anomaly is not a
quantum loop but a classical, topological, differential-geometric
identity on a higher-dimensional bulk manifold. Quantum mechanics
\emph{detects} the identity at the boundary; it does not create the
identity. The Bianchi identity that Luigi Bianchi published in
1902 is the geometric ancestor; the Green--Schwarz modification of
1984 is the bulk-side rewrite that makes the boundary projection
self-consistent in higher dimensions; the modern Hopkins--Singer
differential-cohomology framework is the formal classification of
the topological object. Under the constitutional clauses of
\AOne{} \citep{ArisakaA1Constitution}, the bulk identity is forced
by \AX{2}~(\GGEP); the boundary observer's ``anomaly'' is the
\AX{3}~(\OGGEP) projected shadow of the same identity. This reading
reframes the entire program of anomaly cancellation as
\emph{geometric all the way down}, and we develop it as the
foundational reading on which the \BIsix{} theorem of the present
paper rests.

\subsection{Bianchi (1902): pure geometry, no quantum content}
\label{ssec:geometric-origin-bianchi}

Luigi Bianchi's note \emph{Sui simboli a quattro indici e sulla
curvatura di Riemann} \citep{Bianchi1902}, published in the
\emph{Atti della Reale Accademia dei Lincei} in 1902, proves the
differential identity
\begin{equation}
D_{[\mu} R_{\nu\rho]\sigma\tau} \;=\; 0,
\label{eq:bianchi-classical}
\end{equation}
where $R$ is the Riemann curvature tensor of a smooth Riemannian
manifold and $D$ is the metric-compatible covariant derivative.
Equivalently, reading $R$ as a curvature two-form on the frame
bundle, the identity is the closed-form statement
\begin{equation}
D R \;=\; 0
\label{eq:bianchi-2-form}
\end{equation}
under the covariant exterior derivative. The mathematical content
is purely geometric: it says that the curvature two-form of a
smooth connection on a principal bundle is closed under the
covariant exterior derivative. This is the geometric content of
the operator identity $D^2 = 0$ acting on a connection one-form
$A$, with $R = dA + A \wedge A$. In plain language, it asserts
that the underlying manifold is smoothly connected --- every point
joins to its neighbors through a single, well-defined connection
--- and that any closed flow of conserved charge is continuous.
No physics enters Bianchi's note. No measurement is performed.
No quantum loop is computed. The identity is a property of the
bundle, not of any process running on it. Riemann's 1854
\emph{Habilitationsschrift} \citep{Riemann1854} had developed the
manifold language fifty years earlier; Bianchi's 1902 note
extracted the differential consequence in the form that the
twentieth-century gauge theories would inherit.

\subsection{Yang--Mills (1954): the gauge-bundle Bianchi}
\label{ssec:geometric-origin-yang-mills}

The Yang--Mills construction \citep{YangMills1954} extends the
Bianchi pattern from the frame bundle of spacetime to a principal
$G$ bundle of internal symmetry. The gauge field strength
$F = dA + A \wedge A$ of any smooth connection $A$ on a $G$ bundle
satisfies the classical Bianchi identity
\begin{equation}
DF \;=\; dF + [A, F] \;=\; 0,
\label{eq:yang-mills-bianchi}
\end{equation}
where $D$ is the gauge-covariant exterior derivative. This is
again pure geometry: it is the bundle-theoretic statement that
$D^2 = 0$ on the gauge connection. Yang and Mills wrote it down
in 1954 in the context of isotopic-spin gauge invariance, not as
a quantum-mechanical statement; the identity is true for any
smooth gauge field configuration on any principal bundle,
classical or quantum, perturbative or non-perturbative. The
gauge anomalies that the next generation of theorists would
struggle with arise on top of this identity, not in violation of
it.

\subsection{Adler--Bell--Jackiw (1969): the quantum window onto a classical fact}
\label{ssec:geometric-origin-abj}

The one-loop triangle calculation of \citet{Adler1969} and
\citet{BellJackiw1969} in QED with chiral fermions produced the
celebrated result
\begin{equation}
\partial_\mu J^{5\mu} \;=\; \frac{e^2}{8\pi^2}\, F_{\mu\nu}\, \tilde F^{\mu\nu},
\label{eq:abj}
\end{equation}
showing that the classical chiral-current conservation law fails
at one loop in the presence of a background gauge field. Read
through the conventional lens, this looks like a defining
example of a quantum-mechanical effect: the classical Noether
symmetry was exact, the quantum theory broke it, and the
``anomaly'' was named to mark the breakdown. We propose a
different reading, supported by the entire structural development
of anomaly theory between 1971 and the present. The right-hand
side of \eqref{eq:abj}, $F \wedge \tilde F = \tr(F \wedge F)$, is
a \emph{characteristic class} on the gauge bundle --- specifically,
the second Chern class up to normalization. Characteristic
classes are classical topological invariants of the bundle, fully
defined and calculable from any smooth connection, with no
reference to quantum dynamics. The triangle calculation does not
\emph{produce} this characteristic class; it \emph{measures} it,
through the quantum amplitude that happens to be sensitive to the
underlying topology of the bundle. The ``anomaly'' the quantum
calculation reveals is the boundary-observer's first sight of a
classical topological object that was present in the bundle all
along. In the constitutional language, the ABJ calculation is the
first historical opening of a window onto the \AX{2}~(\GGEP)
content of the bundle; the constitutional reading was identified
later but the content was always classical-geometric.

\subsection{Wess--Zumino consistency (1971) and the cohomological reading}
\label{ssec:geometric-origin-wz}

\citet{WessZumino1971} reformulated anomaly cancellation as a
cohomological consistency condition: the anomaly $\cA[g]$ as a
functional of the gauge parameter $g$ must satisfy
\begin{equation}
\delta_{g_2} \cA[g_1] \;-\; \delta_{g_1} \cA[g_2] \;=\; \cA[\,[g_1, g_2]\,],
\label{eq:wz-consistency}
\end{equation}
which is the cocycle condition that $\cA$ represents a class in
the BRST cohomology of the gauge group. The Becchi--Rouet--Stora
construction \citep{Becchi1976} formalizes this cocycle structure
through the BRST nilpotent operator $s$. The crucial structural
fact is that the anomaly is now classified by a topological
invariant of the gauge bundle, completely independent of which
quantum theory produced it. Two different chiral fermion contents
that give the same characteristic-class data give the same anomaly,
because the anomaly is the characteristic class, not the loop
calculation. This is the formal-mathematical statement of the
ABJ observation that the right-hand side of \eqref{eq:abj} is a
classical bundle invariant.

The Stora--Zumino descent \citep{Stora1984, WessZumino1971} then constructs the four-dimensional
anomaly $\omega_4^{(1)}$ as the descendant of a higher-dimensional
characteristic class through the chain
\begin{equation}
\omega_{2n+2} \;\to\; \omega_{2n+1}^{(0)} \;\to\;
\omega_{2n}^{(1)} \;\to\; \omega_{2n-1}^{(2)} \;\to\; \cdots,
\label{eq:descent}
\end{equation}
where $\omega_{2n+2}$ is a closed gauge-invariant form on the
extended manifold (built from $\mathrm{tr}(R \wedge R)$,
$\mathrm{tr}(F \wedge F)$, etc.) and the descendants
$\omega^{(k)}$ are obtained by stripping off one exterior power
and one BRST ghost at each step. The four-dimensional anomaly is
therefore the lowest descendant of a topological characteristic
class living naturally on a higher-dimensional bulk. This is the
structural identification that the Green--Schwarz mechanism would
exploit in 1984.

\subsection{Green--Schwarz (1984): the bulk-side rewrite}
\label{ssec:geometric-origin-gs}

In ten-dimensional heterotic string theory,
\citet{GreenSchwarz1984} discovered that the anomaly polynomial
$I_{12}$ of the chiral fermion content for the gauge groups
$\SO(32)$ and $E_8 \times E_8$ admits the rank-one factorization
\begin{equation}
I_{12} \;=\; X_4 \wedge X_8,
\label{eq:gs-factorisation}
\end{equation}
where $X_4 = \mathrm{tr}(R \wedge R) - v\, \mathrm{tr}(F \wedge F)$
is a closed gauge-invariant four-form built from gauge and
gravitational curvatures, and $X_8$ is a closed eight-form. When
this factorization holds, GS showed that the boundary chiral
anomaly can be \emph{absorbed} by modifying the Bianchi identity
of the antisymmetric tensor field strength $H_3$:
\begin{equation}
dH_3 \;=\; X_4
\quad\text{(in place of $dH_3 = 0$).}
\label{eq:gs-modified-bianchi}
\end{equation}
The classical Chern--Simons inflow from this modified $H_3$
exactly cancels the BRST cocycle of the boundary anomaly.

The deeper content of this move, central to the present paper, is
that the modification \eqref{eq:gs-modified-bianchi} is a
\emph{classical, geometric, topological} statement. The Bianchi
identity is being modified to a different cohomology class for
$H_3$: instead of $H_3$ being a closed differential form, $H_3$ is
now a representative of a differential character whose curvature
class is $[X_4]$. This is a statement about the differential
cohomology of the bundle, not about any quantum dynamics. In
particular, no quantum-mechanical loop appears in the
\emph{statement} of the GS modification; the quantum content
enters only via the original ABJ calculation that first revealed
the obstruction. The 1984 GS discovery is therefore the
geometric rewriting of a classical topological identity, not the
quantum cancellation of a quantum effect. As we noted in the
historical-positioning subsection of
\AFour{} \citep{ArisakaA4Euclid}, the 1984 GS mechanism was the
moment higher-dimensional gauge theory glimpsed that
gauge curvature $F$ and gravitational curvature $R$ must lie
inside a single closed topological four-form on the bulk
manifold --- a fact that the GG-Constitution later promotes to
the axiom \AX{2}~(\GGEP).

\subsection{Hopkins--Singer--Freed--Moore: modern topological classification (2000s+)}
\label{ssec:geometric-origin-modern}

The modern mathematical formalization of anomalies, developed in
the 2000s and 2010s by \citet{HopkinsSinger2005},
\citet{DaiFreed1994}, \citet{Monnier2018}, \citet{MonnierMoore2019},
\citet{FreedHopkins2016}, and others, classifies anomalies through
\emph{reflection-positive invertible field theories} on the
bordism cohomology of the bundle data. The anomaly of an
$n$-dimensional quantum field theory is encoded by an
$(n{+}1)$-dimensional invertible TQFT, which is itself a
representative of a class in the Anderson dual of an appropriate
cobordism spectrum. All of this is \emph{classical} mathematics
in the sense that no quantum loops appear in the classification:
the anomaly class is a topological invariant of the bundle and the
bordism class of the spacetime, and the quantum field theory only
\emph{realizes} the invariant by computing a particular amplitude
that is sensitive to the underlying topology.

The differential-cohomology refinement of \citet{HopkinsSinger2005}
sharpens this further: characters of $H^*(M; \mathbb{Z})$ are
refined to differential characters carrying both topological data
(integer class) and geometric data (curvature representative
two-form). In this language, the GS modification
\eqref{eq:gs-modified-bianchi} is the statement that the
antisymmetric tensor field is a differential character whose
underlying topological class is $[X_4]$, with the connection
representing that character supplied by the bulk dynamics. The
quantum theory enters only through the choice of which underlying
characteristic-class data the bulk realizes; the anomaly itself is
fixed by the bundle topology.

This is the formal mathematical confirmation of the reading we are
proposing: the anomaly is geometric and topological in its
fundamental content, and quantum mechanics is merely the window
through which the underlying topology becomes visible at the
boundary.

\subsection{The cohort constitutional reading: \texorpdfstring{\AX{2}}{AX-2} on the bulk, \texorpdfstring{\AX{3}}{AX-3} on the projection}
\label{ssec:geometric-origin-constitutional}

Under the constitutional clauses of \AOne{}
\citep{ArisakaA1Constitution} and the bulk theorem of \ATwo{}
\citep{ArisakaA2GG6}, the geometric reading developed above can
be promoted from a methodological observation to a structural
statement. The four axioms of the \GGConstitution{} are arranged
in a strictly nested tower; for the present discussion only the
first three are needed.

\begin{itemize}[leftmargin=2em,itemsep=0.3em]
\item \textbf{\AX{1}~(\LC, Local Causality):} the bulk
 manifold admits a smooth causal structure with finite signal
 speed. This is the prerequisite for any classical or quantum
 theory to be defined on the bulk.

\item \textbf{\AX{2}~(\GGEP, Geometry--Gauge Equivalence
 Principle):} the bulk geometry and the bulk gauge fields form a
 single composite connection on the 6D manifold
 $\cM_6 = \cM_4 \times I_r \times S^1_\theta$. Gauge curvature
 $F$ and gravitational curvature $R$ are not independent
 fundamental objects but algebraic facets of one underlying
 composite connection one-form on the $G_{\mathrm{tot}}$ bundle.

\item \textbf{\AX{3}~(\OGGEP, Operational/Open Geometry--Gauge
 Equivalence Principle):} any admissible bulk theory must
 furnish a structure-preserving projection
 $\cD_{\OGGEP}: \mathbf{Bulk} \to \mathbf{Op}$ from inaccessible
 bulk data to the registered four-dimensional boundary
 observables, with the projection compatible with gauge
 covariance, causality, positivity, and scale consistency.
 Concretely, \AX{3} says the 4D physics observed by a finite
 experimenter is the projection of the bulk content under
 $\cD_{\OGGEP}$.
\end{itemize}

In this constitutional language, the modified Bianchi identity
$dH_3 = X_4$ of the present paper is forced on the bulk side by
\AX{2}: the composite connection on the $G_{\mathrm{tot}}$ bundle
has a closed gauge-invariant four-form $X_4$ built from the four
component curvatures, and the Chern--Simons primitive
$H_3 = dB_2 + (\mathrm{Chern\text{-}Simons\;forms})$ then satisfies
\begin{equation}
\boxed{\;
dH_3 \;=\; X_4 \;=\;
\tr R^2 \,-\, \Tr F_5^2 \,-\, \tfrac{2}{5}\, F_B^2 \,-\, \tfrac{2}{5}\, F_{Qi}^2,
\;}
\label{eq:bi6-as-ax2}
\end{equation}
which is the four-component modified Bianchi identity \BIsix{} of
the cohort framework, derived in detail in \ATwo{} and proved
anomaly-free in Parts~II and~III of the present paper. This is a
\emph{classical, topological, pre-quantum identity}. It is forced
on the bulk by \AX{2} alone; \AX{3} is not yet invoked.

The four-dimensional boundary observer's perception of the
identity \eqref{eq:bi6-as-ax2} is the OGGEP-projected shadow.
Under the projection $\cD_{\OGGEP}$, the closed gauge-invariant
four-form $X_4$ descends to a boundary four-form that the boundary
observer's chiral fermion sees as the right-hand side of the
divergence equation
\begin{equation}
\partial_\mu J^{\mu}_{\mathrm{bdy}} \;\propto\;
\cD_{\OGGEP}\bigl(X_4\bigr),
\label{eq:abj-as-projection}
\end{equation}
which is the constitutional re-derivation of the ABJ relation
\eqref{eq:abj} on the boundary. The boundary observer registers
the divergence as a ``quantum anomaly'' because she is observing
the projected shadow of a classical bulk identity; the appearance
of the divergence is forced by \AX{3} (it must appear, because the
projection is causal, positive, and gauge-covariant), and the
specific four-form on the right-hand side is forced by \AX{2}
(the bundle has $dH_3 = X_4$ on the bulk).

In one line: \emph{the bulk-side identity is the
GGEP/\AX{2} content; the boundary-side `anomaly' is the
OGGEP/\AX{3} projected manifestation of the same identity.
Constitutionally, they are the same topological fact viewed from
opposite sides of the projection.}

The reason the 1984 GS mechanism `cancels' the four-dimensional
anomaly is now constitutionally transparent: GS's modified
Bianchi identity is the constitutional statement of what the bulk
must be doing in order for the boundary projection to be
self-consistent. GS did not cancel a quantum-mechanical effect; GS
identified the bulk-side identity whose OGGEP shadow the boundary
chiral fermion was already detecting. The customary description of the GS mechanism as a `quantum anomaly
cancellation' is accurate as far as it goes, and the reading offered
here does not contradict it; it re-describes the same mathematics from
the bulk side. On that reading the GS mechanism is the discovery of a
classical topological identity that \AX{2} forces and that the
\OGGEP{} projection makes self-consistent at the boundary. Whether the
bulk-side or the boundary-side description is the more natural one is a
question of framing, not of fact, and nothing in the theorem below
depends on the answer.

\subsection{Why this matters for the \texorpdfstring{\BIsix{}}{BI-6} anomaly theorem}
\label{ssec:geometric-origin-why}

The constitutional reading developed in the preceding subsections
is the foundational reading on which the \BIsix{} anomaly theorem
of the present paper rests. Three consequences are immediate.

\emph{First, the proof object is correctly identified.} The
object whose anomaly freedom we prove in Parts~II and~III is
\BIsix{} in the form \eqref{eq:bi6-as-ax2} --- the four-component
modified Bianchi identity on the 6D bulk. This is a
classical topological identity on the bulk, and the consistency
theorem we prove is the classical-topological consistency of the
bundle plus the global-completion compatibility of the
differential-cohomology lift. The boundary observer's
``anomaly'' is the shadow, not the object.

\emph{Second, the structural method is correctly motivated.} The
descent proof of Section~\ref{sec:structural-proof} works with
the BRST cocycle structure of the boundary chiral-fermion
anomaly precisely because the BRST cocycle is the boundary
shadow of the bulk topological identity. Each step in the
descent corresponds, in the constitutional reading, to a step in
the OGGEP projection from bulk to boundary. The reason the
descent argument is the natural proof technique is that it
exactly tracks the constitutional projection $\cD_{\OGGEP}$
applied to the bulk four-form $X_4$.

\emph{Third, the global completion is correctly understood.} The
Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore global proof
of Section~\ref{sec:global-completion} verifies that the
differential character carrying $[X_4]$ is consistently defined
on the entire intended \GG{} background class, including under
large gauge transformations and across all admissible bulk
topologies. This is, in the constitutional reading, the
verification that \AX{2} forces a globally consistent bulk
identity (not merely a locally consistent one), which is the
prerequisite for \AX{3} to deliver a globally consistent
boundary projection.

Figure~\ref{fig:bi6-bulk-shadow} draws the reading of this section in
one picture: the four curvature contributions assemble into the bulk
four-form $X_4 = \dd H_3$ --- a classical-topological identity on the
composite connection --- and the boundary observer's ``anomaly'' is its
\OGGEP{}-projected shadow.

\begin{figure}[htbp]
\centering
\resizebox{0.9\textwidth}{!}{%
\usebox{\GGscShadow}}
\bigskip
\caption[The bulk-side classical identity and its boundary shadow.]{The
four-component \BIsix{} bulk-side classical-topological identity and its
boundary shadow. The four curvature contributions (green) carry the
canonical coefficients $(a,b,c,d) = (1,-1,-2/5,-2/5)$ and assemble into
the bulk four-form $X_4 = \dd H_3$ (red), a classical identity on the
\AX{2}~\GGEP{} composite connection, pinned by the trace-ratio lock
$|a/c| = |a/d| = 5/2$ (yellow; Section~\ref{sec:bi6-fourform}). The
boundary ``anomaly'' (blue) is the \AX{3}~\OGGEP{} projection shadow of
this bulk identity --- a boundary re-measurement of a classical fact
already in force in the bulk, not a quantum pathology to be cured.
Adapted from the cohort master figure of \citep{ArisakaA1Constitution}
(its bulk-side identity plate), also carried in \citep{ArisakaA2GG6}.}
\label{fig:bi6-bulk-shadow}
\end{figure}

After all, what the present paper proves is not the cancellation
of a quantum-mechanical defect but the constitutional consistency
of a classical-topological identity that \AX{2} forces on the
bulk and that \AX{3} projects to the boundary. That this
classical consistency theorem coincides with the quantum
``anomaly cancellation'' theorem of the conventional reading is
not a coincidence: the quantum anomaly is what the boundary
observer sees when the bulk identity holds. The present paper's
contribution is to prove the bulk identity's classical
consistency; the rest of the cohort program then takes the
projected boundary content seriously as the substance of
four-dimensional physics.



\section{The Green--Schwarz mechanism in even dimensions}
\label{sec:gs-mechanism}

\subsection{Anomaly polynomials and the descent formalism}

\paragraph{Historical background (with citations).} Indeed, the
modern anomaly polynomial is the formal endpoint of a chain of
ideas that begins with classical electrodynamics. Maxwell's 1865
paper \emph{A Dynamical Theory of the Electromagnetic Field}
\citep{Maxwell1865} put the four-vector potential $A_\mu$ at the
center of physics, defined only up to a gauge transformation
$A_\mu \to A_\mu + \partial_\mu \chi$ --- a structural fact
that, a century later, became the founding example of every
gauge-invariant quantum field theory. The decisive quantum
surprise came in 1969, when Adler \citep{Adler1969} and,
independently, Bell and Jackiw \citep{BellJackiw1969} showed
that a chiral fermion coupled to such a gauge field generically
violates current conservation at one loop ---
the \emph{Adler--Bell--Jackiw anomaly}, with
$\partial^\mu J_\mu^5 \propto F \wedge \tilde F$. This was
the moment when ``classical symmetry'' and ``quantum symmetry''
became genuinely distinct concepts in physics, and the anomaly
polynomial machinery of the present subsection is the systematic
descendant of that discovery. The descent
equations \eqref{eq:descent-general} below are the bookkeeping
device that turns the Adler--Bell--Jackiw obstruction into a
precise cohomological invariant of the quantum effective action
--- one whose vanishing or absorption is exactly the
``anomaly-freedom'' condition we are after.

In a chiral gauge theory in \(d=2n\) dimensions, the local
perturbative anomaly is encoded by a closed, gauge-invariant
\((2n+2)\)-form \(I_{2n+2}\), the \textbf{anomaly polynomial}.
The connection between \(I_{2n+2}\) and the anomalous variation
of the quantum effective action is established by the
\textbf{descent formalism}:
\begin{equation}
I_{2n+2} = \dd I_{2n+1}^{(0)},
\qquad
\delta I_{2n+1}^{(0)} + \dd I_{2n}^{(1)} = 0,
\label{eq:descent-general}
\end{equation}
where \(\delta\) denotes a gauge or local Lorentz variation.
The consistent anomaly of the effective action is then
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}}
= \kappa\int_{M_{2n}} I_{2n}^{(1)},
\label{eq:consistent-anomaly}
\end{equation}
with \(\kappa\) a normalization constant.

\begin{definition}[Anomaly freedom]
A theory is \textbf{perturbatively locally anomaly-free} if
\(\delta\Gamma_{\mathrm{1\text{-}loop}} = 0\) for all infinitesimal
gauge and local Lorentz variations.
It is \textbf{globally anomaly-free} if, in addition, the path
integral is single-valued under all large gauge transformations and
large diffeomorphisms.
\end{definition}

\subsection{The Green--Schwarz factorization mechanism}

The Green--Schwarz mechanism \citep{GreenSchwarz1984} cancels the
anomaly when the polynomial \(I_{2n+2}\) \emph{factorizes} through
a lower-degree form that can be absorbed by a tensor field.

\begin{theorem}[\textbf{T-BI6-3.2}: Green--Schwarz cancellation, general form]
\label{thm:gs-general}
Let \(I_{2n+2}\) factorize as
\begin{equation}
I_{2n+2} = X_{p+2}\wedge Y_{2n-p},
\label{eq:gs-factorization}
\end{equation}
where \(X_{p+2}\) is a closed gauge-invariant \((p+2)\)-form.
Introduce a \(p\)-form tensor \(B_p\) with modified field strength
\begin{equation}
H_{p+1} = \dd B_p + \omega_p^{(0)},
\qquad
\dd\omega_p^{(0)} = X_{p+2},
\label{eq:gs-bianchi-general}
\end{equation}
and transformation law
\begin{equation}
\delta B_p = -\omega_{p-1}^{(1)},
\qquad
\delta\omega_p^{(0)} + \dd\omega_{p-1}^{(1)} = 0.
\label{eq:gs-transform-general}
\end{equation}
Then the Green--Schwarz counterterm
\begin{equation}
S_{\mathrm{GS}} = \kappa\int_{M_{2n}} B_p\wedge Y_{2n-p}
\label{eq:gs-counterterm-general}
\end{equation}
cancels the full perturbative local anomaly:
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}} + \delta S_{\mathrm{GS}} = 0.
\label{eq:gs-cancellation-general}
\end{equation}
\end{theorem}

\begin{proof}
Since \(\delta Y_{2n-p} = 0\) (gauge invariance of characteristic forms),
\begin{equation}
\delta S_{\mathrm{GS}}
= \kappa\int \delta B_p\wedge Y_{2n-p}
= -\kappa\int \omega_{p-1}^{(1)}\wedge Y_{2n-p}.
\end{equation}
The one-loop variation, by descent from \(I_{2n+2} = X_{p+2}\wedge Y_{2n-p}\),
is
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}}
= +\kappa\int \omega_{p-1}^{(1)}\wedge Y_{2n-p}.
\end{equation}
The sum vanishes.
\end{proof}

\subsection{Specialization to six and ten dimensions}

In \(d=10\), the anomaly polynomial is a 12-form. The factorization
\begin{equation}
I_{12} = X_4\wedge X_8
\label{eq:10d-factorization}
\end{equation}
uses a four-form \(X_4\) and an eight-form \(X_8\), with a two-form
tensor \(B_2\). This is the original Green--Schwarz mechanism.

In \(d=6\), the anomaly polynomial is an eight-form. The single-tensor
factorization becomes
\begin{equation}
I_8 = \half X_4\wedge X_4 = \half X_4^2,
\label{eq:6d-factorization}
\end{equation}
a \textbf{perfect square}. This is more rigid than the 10D case:
the same four-form appears twice, so the anomaly polynomial must be
a rank-one quadratic form in the space of invariant four-forms.

\begin{remark}
In the 10D case, \(X_4\) and \(X_8\) can be different forms, giving
more algebraic freedom. In the 6D case, the perfect-square
factorization imposes the rank-one condition on the anomaly matrix,
which is a \emph{stronger} constraint. This additional rigidity is
the mathematical reason why \BIsix{} is so tightly locked.
\end{remark}



\bigskip
\noindent\textit{With the general mechanism and the specific \BIsix{} data
in hand, the proof itself awaits.}



\section{The cohort inheritance and the roadmap of the ten Parts}
\label{sec:inheritance-and-roadmap-A3}

\noindent
Before the anomaly proof begins (Part~II), it is useful to lay out
two reader-orientation maps on the same opening: a panorama of the
twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort-A3}, Figure~\ref{fig:cohort-map-a3}), and a panorama of the ten Parts of the present paper
(\S\ref{sec:roadmap-parts-A3},
Figure~\ref{fig:A3-parts-flowchart}). 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 ten-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-A3}

The present paper is one of twenty-two. \GGTheory{} is written not as a
single monograph but as a cohort of twenty-two preprints --- eleven A-papers, six P-papers, three C-papers and two white papers --- each carrying one
rigorously stated piece of a single descent that runs from four axioms and
seven postulates to the empirical structure of physics. The partition into
twenty-two papers reflects the partition of subject matter, not a ranking of
importance: the framework is one program, and each paper is a named
carrier of named content. Figure~\ref{fig:cohort-map-a3} is the panorama; a
reader meeting the program for the first time can read its architecture
off the figure top to bottom before returning to the anomaly proof that is
the business of this paper. The cohort falls into five strata plus three
framed branches.

\paragraph{The constitutional root and the GG6 Trio.} At the root stands
\AOne{} \citep{ArisakaA1Constitution}, the \GGConstitution{}: the four
axioms (\AX{1}~\LC{}, \AX{2}~\GGEP{}, \AX{3}~\OGGEP{}, \AX{4}~\MSMF{}) and
seven postulates against which every companion's theorems are audited.
Forced from it is the GG6 Trio that builds the bulk: \ATwo{}
\citep{ArisakaA2GG6} constructs the six-dimensional Geometry-Gauge
bulk and proves it the unique anomaly-free, moduli-free solution of the
axioms; the present paper, \AThree{} \citep{ArisakaA3BI6}, supplies the
rigorous local-and-global proof that the modified Bianchi identity \BIsix{}
on that bulk is anomaly-free; and \AFour{} \citep{ArisakaA4Euclid} traces
the historical lineage from Euclid through Riemann, Maxwell, and Einstein
to the bulk and reconstructs the universal constants from the bulk
dimensions.

\paragraph{The boundary law and its two pillars.} \AX{3}~(\OGGEP{}) forces
the open-system boundary law \AFive{} \citep{ArisakaA5GDOS}, the Geometric
Dynamics of Open Systems that projects bulk physics onto the
four-dimensional boundary and recovers the textbook quantum-mechanical
postulates as theorems of the operational dictionary. Two sibling pillar
papers organize the boundary content: \ASix{} \citep{ArisakaA6Gi4}, the
geometric \GiFour{} pillars (the cascade from the total geometric--gauge
object through the free-energy functional to the gauge-runtime pipeline),
and \ASeven{} \citep{ArisakaA7Qi4}, the quantum-information \QiFour{}
pillars, which carry the Arisaka equation and the boundary $\U(1)_{Qi}$
symmetry that the present paper admits.

\paragraph{The Consequence Trio and the capstone.} Three papers read the
empirical world off the boundary law: \AEight{} \citep{ArisakaA8QM} recasts
four-dimensional quantum mechanics, \ANine{} \citep{ArisakaA9SM} derives the
twenty-eight free parameters of the Standard Model, and \ATen{}
\citep{ArisakaA10Cosmos} derives the $\Lambda$CDM cosmological inventory
($\Omega_m = 1/\pi$ and its partners). The capstone \AEleven{}
\citep{ArisakaA11Origin} closes the cohort with the recursive-uniqueness
argument --- why every electron is the same electron --- and the
cosmos--life--mind chain.

\paragraph{The two phenomenology branches.} Beneath the A-series, two
framed branches carry the sharp falsifiers. The Predictions Branch ---
\PaperPOne{} \citep{ArisakaP1KK} (the Kaluza--Klein lock cluster near
$\Mlock \approx 3.85$~TeV), \PaperPTwo{} \citep{ArisakaP2N3} (the
three-generation pincer that forces $N=3$), and \PaperPThree{}
\citep{ArisakaP3Electron} (the electron as the ground state of nature),
and \PaperPSix{} \citep{ArisakaP6Proton} (the proton as the ground
geometric attractor of matter, its decay forbidden by the geometry
rather than merely unobserved) --- supplies collider and structural
predictions. The Cosmology and
Astrophysics Branch --- \PaperCOne{} \citep{ArisakaC1Kaxion} (the kaxion
dark-matter candidate near $8.1~\mu$eV), \PaperCTwo{}
\citep{ArisakaC2Hubble} (the Hubble tension as one geometric corridor), and
\PaperCThree{} \citep{ArisakaC3BH} (black-hole entropy as an \OGGEP{} port)
--- supplies the astrophysical signatures.

\paragraph{The Foundations Branch.} The third framed branch is the
Foundations Branch: the two member papers whose subject is the boundary's
complex structure and its mirrors. \PaperPFour{} \citep{ArisakaP4Real}
stands on the map's left flank as the origin-of-$i$ member, deriving the
single imaginary unit of boundary physics from the two real structures of
the \ATwo{} arena --- the compact phase circle and the causal clock.
\PaperPFive{} \citep{ArisakaP5CPT} carries the three mirrors of matter ---
C, P, and T read as orientation reversals of the three real bulk factors
--- and that orientation reading of the discrete symmetries gives the
anomaly-inflow logic of the present paper an independent consumer among
the companions. Both carry P-series designators; their place at the
spine's flank, not the phenomenology rows, marks their foundational
role.

\paragraph{The White-Paper Branch.} The right flank mirrors the left
and does the opposite job: where the Foundations Branch feeds the spine,
the White-Paper Branch reads it. \PaperWOne{} \citep{ArisakaW1Symmetry}
is the diachronic one --- it walks the century of symmetry and
conservation in the order it was found and shows the conviction that
symmetry causes conservation loosening at seven historical steps, until
what remains conserved is an integer with nowhere to move.
\PaperWTwo{} \citep{ArisakaW2Holonomy} is the synchronic one --- it
takes one exact statement, that a holonomy requires a closed loop and
the radial interval has none, so the circle remembers and the interval
forgets, and follows it to sixteen questions physics has not closed.
Neither derives anything new: both carry every result at its parent's
declared grade, which is what makes them white papers rather than a
third phenomenology branch.

\paragraph{Where this paper sits.} \AThree{} is the second descent step of
the cohort. \ATwo{} forces the bulk and states \BIsix{}; the present paper
proves that the resulting theory is quantum-mechanically consistent, and
the anomaly ledger behind \ATwo{}'s Theorem~6 --- the \OGNfive{} integer
cascade --- is the ledger proved in full here. Every downstream paper ---
the boundary law of \AFive{}, the pillars of \ASix{} and \ASeven{}, the
textbook readouts of \AEight{}--\ATen{}, and the capstone \AEleven{} ---
inherits its license to build on the \GG{} bulk from the anomaly theorem
proved here. Without this proof, the third step (\GDOS{} lawfulness)
cannot legitimately be invoked. The twenty-two-paper map, identical in
every paper of the cohort, is Figure~\ref{fig:cohort-map-a3}, which
closes this subsection.

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
%%  copy of the v1.4 master (A1-Constitution Figure 3): no
%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
%%  node names MP4/MP5 differ from the master (collision avoidance).
%% ===================================================================

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 fnd/.style={fill=violet!8, draw=violet!55!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
 draw=violet!70!black},
 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\ASix{},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-a3}
\end{figure}


\subsection{Roadmap of the ten Parts}
\label{sec:roadmap-parts-A3}

The rest of the paper is organized into nine further Parts, and the
walk through them is plotted in
Figure~\ref{fig:A3-parts-flowchart}. The present Part --- the
introduction --- is nearly done: the anomaly problem and its
constitutional frame, the hundred-and-twenty-year geometric arc from Bianchi to differential cohomology, the Green--Schwarz mechanism in general
even dimensions that fixes the notation and the cohomological
framework, and the cohort the paper stands in. Everything after this
page is the proof and its consequences, and the itinerary below says
where each piece lives.

Part~II is the heart, and it is highlighted in the roadmap because it
is what the title claims. The local descent argument closes one-loop
consistency through BRST exactness, descending from the factorized
polynomial $I_8 = \tfrac12 X_4^2$; the global completion verifies the
four hypotheses (H1)--(H4) of the Wu--Chern--Simons /
Hopkins--Singer / Monnier--Moore differential-cohomology framework on
the intended \GG{} background class; and an explicit benchmark
spectrum proves that the structural theorem is satisfiable rather
than vacuous. This Part carries the single result on which the rest
of the paper --- and the cohort's license to build on the \GG{} bulk
--- depends.

Two Parts then work the consequences outward. Part~III shows that the
five-integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$ is a theorem-level
consequence of \BIsix{} rather than an external input: a Diophantine
cascade in which each integer is the unique solution of an arithmetic
condition fixed by a prior theorem. Part~IV sets the construction
beside the string program it generalizes --- the ten-dimensional
Green--Schwarz mechanism, six-dimensional supergravity and F-theory,
the AdS/CFT correspondence, and the Randall--Sundrum model ---
acknowledging the shared mathematics in full, stating the structural
departures feature by feature, and naming the open bridges to the
string community.

Two Parts then weigh the case. Part~V is the discussion: the significance of the result, what
interlocks with what to make it unique, the ten structural innovations
that are new here, the predictions that can falsify it, the cross-paper
coherence map, and the honest open-problems triage with the future
directions it sets.
Part~VI states the conclusions --- what has been established, first
in plain language for a broad readership and then in the precise
register of the theorem chain --- and closes with the place of the
result in the series.

%% ===================================================================
%% v15.5: a [p] float, not a minipage in the text stream.  The plate
%% and its caption now have the page to themselves, centred by the two
%% \vspace*{\fill}s, and the page in FRONT of it stays full -- which a
%% \clearpage would not have allowed.
\begin{figure}[p]
\vspace*{\fill}
\centering
\resizebox{0.972\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
 intro/.style={fill=blue!12, draw=blue!60!black, line width=0.6pt,
 text width=114mm, minimum width=120mm, minimum height=12mm},
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 text width=114mm, minimum width=120mm, minimum height=12mm},
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 text width=134mm, minimum width=140mm, minimum height=18mm},
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 draw=orange!65!black},
 phaselabel/.style={font=\footnotesize\itshape, text=black!60, anchor=west,
 align=left, inner sep=2pt},
]

%% --- Part I (Introduction) ---
\node[intro] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{Introduction} \\[-1pt]
 \footnotesize Anomaly problem + the Constitution + \BIsix{} + geometric origin + Green--Schwarz + cohort};

%% --- Part II (the anomaly proof; highlighted: the title claim) ---
\node[innov] (II) at (0, -19mm)
 {\textbf{Part~II}\quad\textbf{Proof of \BIsix{} Anomaly Freedom} \\[1pt]
 \footnotesize Local descent (BRST-exact) + global completion (H1--H4) + benchmark spectrum \\[-1pt]
 \footnotesize \emph{\textbf{The load-bearing core: the local-and-global anomaly theorem}}};

%% --- Part III (OGN-5 Arithmetic) ---
\node[geom] (III) at (0, -40mm)
 {\textbf{Part~III}\quad\textbf{\OGNfive{} Arithmetic} \\[-1pt]
 \footnotesize The integer ledger $(1,3,5;7,35)$ as a theorem, not an input};

%% --- Part IV (Comparisons with String Theory) ---
\node[vs] (IV) at (0, -55mm)
 {\textbf{Part~IV}\quad\textbf{Comparisons with String Theory} \\[-1pt]
 \footnotesize Shared mathematics, structural departures, open bridges};

%% --- Part V (Discussions) ---
\node[disc] (V) at (0, -70mm)
 {\textbf{Part~V}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Significance, uniqueness, comparisons, falsification, open questions, future directions};

%% --- Part VI (Conclusions) ---
\node[summit] (VI) at (0, -85mm)
 {\textbf{Part~VI}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize The result in plain language --- what is proved, and what it means};

%% --- Part VII (For the Curious Reader) ---
\node[summit] (VII) at (0, -100mm)
 {\textbf{Part~VII}\quad\textbf{For the Curious Reader} \\[-1pt]
 \footnotesize Twenty-five plain questions, answered without assuming a physics degree};

%% --- Part VIII (Technical Appendices) ---
\node[appendix] (VIII) at (0, -115mm)
 {\textbf{Part~VIII}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize Ten appendices (A--J): a graduate course --- descent, the 10D GS mechanism, \OGNfive{}, completion, global primer, checklist};

%% --- Part IX (Common Objections) ---
\node[appendix] (IX) at (0, -130mm)
 {\textbf{Part~IX}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize Fifty-three specialist objections, answered in ten clusters};

%% --- Part X (References) ---
\node[appendix] (X) at (0, -145mm)
 {\textbf{Part~X}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows ---
\draw[arrowinnov] (I.south) -- (II.north);
\draw[arrowinnov] (II.south) -- (III.north);
\draw[arrow] (III.south) -- (IV.north);
\draw[arrow] (IV.south) -- (V.north);
\draw[arrow] (V.south) -- (VI.north);
\draw[arrow] (VI.south) -- (VII.north);
\draw[arrow] (VII.south) -- (VIII.north);
\draw[arrow] (VIII.south) -- (IX.north);
\draw[arrow] (IX.south) -- (X.north);

%% --- Phase labels (right sidebar) ---
%% Standard boxes: half-width = 60mm, label at x = 62mm.
%% Part II is wider (half-width = 70mm); its phase label sits below
%% the box, per the A1 pattern for the highlighted Part.
\node[phaselabel] at (62mm, 0mm) {\emph{Setup}};
%% The anomaly core: below the Part II box (box bottom at y = -28mm)
\node[phaselabel, text=orange!70!black, anchor=west] at (32mm, -31mm)
 {\emph{\textbf{Anomaly core: the local-and-global theorem}}};
\node[phaselabel] at (62mm, -40mm) {\emph{Arithmetic}};
\node[phaselabel] at (62mm, -55mm) {\emph{Comparison}};
\node[phaselabel] at (62mm, -70mm) {\emph{Discussion}};
\node[phaselabel] at (62mm, -85mm) {\emph{Summit}};
\node[phaselabel] at (62mm, -100mm) {\emph{Plain answers}};
\node[phaselabel] at (62mm, -115mm) {\emph{Appendices}};
\node[phaselabel] at (62mm, -130mm) {\emph{Objections}};
\node[phaselabel] at (62mm, -145mm) {\emph{References}};

\end{tikzpicture}%
}
\par\bigskip
\caption[Roadmap of the ten Parts of GG-A3-BI6.]{Roadmap of the ten Parts of \AThree{}. The argument develops in phases. \emph{Setup} (Part~I) states the anomaly problem, the \GGTheory{} Constitution and the \BIsix{} identity it forces, the geometric origin of the anomaly concept, the Green--Schwarz mechanism in general even dimensions, and the cohort inheritance with this roadmap. The \emph{anomaly core} (\textbf{Part~II}, highlighted) is the technical heart and the title claim: the local descent argument closes one-loop consistency through BRST exactness, the global completion verifies the four hypotheses (H1)--(H4) of the Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore differential-cohomology framework on the intended \GG{} background class, and the benchmark spectrum proves the structural theorem satisfiable. \emph{Arithmetic} (Part~III) derives the \OGNfive{} ledger $(1,3,5;7,35)$ as a theorem-level consequence of \BIsix{}; \emph{comparison} (Part~IV) sets the construction beside the ten-dimensional Green--Schwarz mechanism, six-dimensional supergravity and F-theory, AdS/CFT, and Randall--Sundrum; \emph{discussion} (Part~V) weighs significance, uniqueness, falsification, and the open-problems triage; and the \emph{summit} (Part~VI) states the conclusions in plain language first and in the precise register of the theorem chain second. \emph{Plain answers} (Part~VII) then takes twenty-five questions a reader without a physics degree actually asks, and answers them assuming no physics beyond a first course; it is placed before the back matter because a reader who wants it should not have to cross sixty pages of bordism theory to reach it. The three back-matter Parts collect the technical appendices (Part~VIII), the fifty-three specialist objections and responses (Part~IX), and the annotated references (Part~X). Part~II is highlighted because the local-and-global anomaly-freedom theorem is the single result on which the rest of the paper --- and the cohort's license to build on the \GG{} bulk --- depends.}
\label{fig:A3-parts-flowchart}
\vspace*{\fill}
\end{figure}

Part~VII then steps out of the technical register entirely and
answers, in plain language and assuming no physics beyond a first
course, the twenty-five questions a reader without a physics degree
actually carries into a paper like this one. It is placed here, before
the back matter, because a reader who wants those answers should not
have to cross sixty pages of bordism theory to reach them.

The back matter is the workshop. Part~VIII collects the technical
appendices: a graduate-level course in ten appendices, from the descent
formalism and the ten-dimensional Green--Schwarz mechanism worked from
scratch to the \OGNfive{} derivation, the gravitational completion, the
global (Dai--Freed) proof, and the theorem checklist matrix. Part~IX
answers the fifty-three objections a particle theorist, string
theorist, or differential-cohomology specialist is most likely to
raise. Part~X closes with the annotated references, each carrying a
note on what it established and how it is used here. Read in order, the ten Parts compose into a single self-contained
anomaly proof that any reader can audit step by step; read by need, each
Part names its imports and can stand a visit on its own.

\bigskip

\noindent\textbf{Three routes through this paper.} Not every reader
wants the same thing from it, and the paper is built so that three
different visits all work.

\emph{The general reader}, who wants to know what is claimed and how
much to believe: the Executive Summary (page~2), then
Section~\ref{sec:conclusions}, which states the result in plain language
with no notation, then Part~VII, which answers twenty-five questions
without assuming a physics degree. About twenty pages, and no
mathematics is needed at any point.

\emph{The specialist auditing the proof}: Table~\ref{tab:status} first,
which grades every claim in the paper and names its status, then the
weakest row of that table. Then Part~II for the theorem,
Appendices~\ref{app:descent-full}, \ref{app:matrix-and-coefficients} and
\ref{app:global-proof-dossier} for the calculations behind it, and
Part~IX for the fifty-three objections, several of which concede
something the main text has to be read down for.

\emph{The reader who has come from another cohort paper} and needs one
result: Section~\ref{sec:cohort-A3} says what this paper supplies and to
whom, Section~\ref{sec:ogn5-theorem} carries the integer ledger, and
Appendix~\ref{app:checklist} is the one-row-per-claim ledger with each
claim's status and where it is established.

\bigskip

\noindent\textbf{What the reader now has, and what the proof will cost.}
Four things have been put in place. First, a statement of the problem:
an anomaly in the gauge symmetry is fatal, not cosmetic, and a
six-dimensional chiral theory has to survive it before anything else
about that theory means anything. Second, a reading of where the problem
comes from --- it is geometric, and it was geometric before quantum
mechanics existed, which is why a classical identity about curvature can
be the thing that resolves it. Third, the general mechanism: if the
anomaly polynomial factorizes, one tensor field can absorb it, and the
factorization condition is severe enough to be worth checking. Fourth,
the object to be tested: the six-dimensional bulk of \GG{}, with its
four-factor gauge group and the four-component identity \BIsix{} that
the constitution forces on it.

What follows will cost the reader some patience, and it is worth saying
in advance where the cost falls. The local half of the proof, Part~II's
first section, is a calculation: long, but every step is an algebraic
identity that can be checked with paper and pencil, and Appendix~A does
check them. The global half is harder, and it is harder for a reason
worth understanding. Local cancellation says the books balance at every
point. Global cancellation says they balance after the theory is carried
all the way around a loop in the space of its own configurations --- and
a quantity can return to its starting value at every point along such a
loop and still come back multiplied by a phase. Detecting that requires
machinery from topology that has no analogue in the local calculation:
bordism groups, characteristic classes, an invariant of a differential
operator called the eta invariant. Part~II states what that machinery
delivers and Appendix~H supplies a primer for a reader meeting it for
the first time.

A reader who wants only the result, and not the audit, can take the
local calculation on trust, read the statement of the global theorem,
and go on to Part~III, where the arithmetic begins to produce numbers
that experiments can check.

%% ===================================================================
%%  Roadmap of the nine Parts (house architecture: A1-Constitution
%%  Figure 4 / A2-GG6 v13.3 pattern -- vertical Part column, right
%%  phase sidebar, one highlighted central Part; here Part II, the
%%  anomaly proof), clearpage-isolated minipage + \captionof per the
%%  A1 pattern.

\clearpage

\GGPart[BI-6 anomaly freedom]{Proof of \BIsix{} Anomaly Freedom}{Local descent, global completion, and the benchmark spectrum}

This Part contains the core mathematical content of the paper.
Section~\ref{sec:structural-proof} proves local anomaly cancellation
by descent from the factorized polynomial \(I_8 = \frac{1}{2}X_4^2\).
Section~\ref{sec:global-completion} upgrades that local result to a
complete global theorem on the physical intended \GG{} background class,
using the modern Wu--Chern--Simons / differential-cohomology framework.
Section~\ref{sec:model-level} constructs an explicit finite chiral
spectrum that realizes the factorization, proving that the structural
theorem is not vacuous. The detailed computations behind each step are
provided in Appendices~\ref{app:descent-full}--\ref{app:global-proof-dossier}.

This is the Part the title of the paper refers to, and it is worth
saying at the outset exactly what has to be established. A complete
claim of anomaly freedom is not one statement but eight, and
Section~\ref{sec:eight-parts} lists them as proof obligations
\textbf{(PO1)}--\textbf{(PO8)}. In plain English they are these. The
anomaly has to be computed from the actual matter content rather than
assumed. The terms that no counterterm can ever cancel --- the
irreducible ones --- have to vanish outright. The mixed terms that would
couple one force to another in the wrong way have to vanish too. What
remains has to match the square of the \BIsix{} four-form exactly, with
no coefficient left free to be adjusted. The counterterm's variation
under a gauge transformation has to cancel the leftover, not merely
resemble it. The cancellation has to be exact at the level of the
cocycle, which is the algebraic form the statement takes when the
bookkeeping is done carefully. Global anomalies --- the phases that only
appear after a loop --- have to be absent. And the counterterm itself
has to be a well-defined object on the lattice of charges, not just a
formal expression.

The eight are not equally hard. Four of them are the calculation. Three
are topology. One, the last, is the question of whether the object we
have been manipulating exists at all in the strict sense, and it is the
one that a reader from the older literature is most likely to
underestimate. All eight are discharged in this Part or in the
appendices it points to, and Section~\ref{sec:eight-parts} says which
result closes which.

\section{Structural proof of \BIsix{} anomaly cancellation}
\label{sec:structural-proof}

\subsection{The \BIsix{} four-form and its coefficient vector}
\label{sec:bi6-fourform}

The \GG{} bulk has gauge-gravity group \eqref{eq:bulk-group}.
With a single Green--Schwarz two-form \(B_2\), the most general
diagonal single-tensor candidate four-form is
\begin{equation}
X_4^{\mathrm{gen}}
= a\,\tr R^2 + b\,\Tr F_5^2 + c\,F_B^2 + d\,F_{Qi}^2.
\label{eq:x4-general}
\end{equation}
The \BIsix{} proposal is
\begin{equation}
\boxed{
X_4^{\mathrm{BI6}}
= \tr R^2 - \Tr F_5^2
- \frac{2}{5}\,F_B^2 - \frac{2}{5}\,F_{Qi}^2.
}
\label{eq:x4-bi6}
\end{equation}

\begin{figure}[H]
\centering
\resizebox{0.80\textwidth}{!}{%
\begin{tikzpicture}[
 font=\footnotesize,
 every node/.style={inner sep=2pt},
 edgelabel/.style={fill=white, inner sep=1pt, font=\footnotesize\itshape},
 arrow/.style={->, line width=0.42pt, draw=black!70}
]

%% Input sector boxes (left column, absolute coords; v7.9 soft colors)
\node[draw=teal!55!black, rounded corners=3pt, fill=teal!12, minimum width=2.7cm, minimum height=0.6cm]
 (grav) at (0, 3) {$\Spin(1,5)$ gravity};
\node[draw=blue!55!black, rounded corners=3pt, fill=blue!12, minimum width=2.7cm, minimum height=0.6cm]
 (su5) at (0, 2) {$\SU(5)$ unification};
\node[draw=green!55!black, rounded corners=3pt, fill=green!12, minimum width=2.7cm, minimum height=0.6cm]
 (uB) at (0, 1) {$\U(1)_B$ baryonic};
\node[draw=violet!55!black, rounded corners=3pt, fill=violet!10, minimum width=2.7cm, minimum height=0.6cm]
 (uQi) at (0, 0) {$\U(1)_{Qi}$ informational};

%% Central anomaly node X_4
\node[draw=orange!55!black, circle, fill=orange!18, minimum size=1.4cm, line width=0.58pt]
 (X4) at (4.2, 1.5) {$X_4$};

%% Inflow target H_3
\node[draw=olive!55!black, rounded corners=3pt, fill=yellow!18, minimum width=2.8cm, minimum height=0.95cm,
 align=center]
 (H3) at (8.2, 1.5) {$H_3 = \dd B_2 + \mathrm{CS}$ \\ \footnotesize $\dd H_3 = X_4$};

%% Curved arrows from inputs to X_4 with edgelabel coefficients (the (1,-1,-2/5,-2/5) lock)
\draw[arrow] (grav.east) to[out=0, in=140] (X4.west);
\node[edgelabel] at (2.5, 2.85) {$+1$};

\draw[arrow] (su5.east) to[out=0, in=160] (X4.west);
\node[edgelabel] at (2.5, 1.95) {$-1$};

\draw[arrow] (uB.east) to[out=0, in=200] (X4.west);
\node[edgelabel] at (2.5, 1.05) {$-2/5$};

\draw[arrow] (uQi.east) to[out=0, in=220] (X4.west);
\node[edgelabel] at (2.5, 0.2) {$-2/5$};

%% Arrow from X_4 to H_3
\draw[arrow, line width=0.58pt] (X4.east) -- (H3.west);
\node[edgelabel] at (6.2, 2.05) {Bianchi inflow};

%% Header above X_4
\node[anchor=south, font=\small\bfseries] at (4.2, 3.15) {Anomaly four-form};

%% Trace-ratio lock label below X_4
\node[anchor=south, font=\footnotesize] at (8.2, 2.6)
 {trace-ratio lock $|a/c| = |a/d| = 5/2$};

%% Caption-side expression below
\node[anchor=north, font=\footnotesize] at (4.2, -0.7)
 {$X_4 \;=\; +\tr(R^2) \;-\; \Tr(F_5^2) \;-\; \tfrac{2}{5}\,F_B^2 \;-\; \tfrac{2}{5}\,F_{Qi}^2$};

\end{tikzpicture}%
}
\caption[Four-component anomaly inflow in \BIsix{} and its closed four-form $X_4$]{Four-component anomaly inflow in $\BIsix$. The bulk gauge--gravity group $G_{\Bulk} = \Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$ contributes four distinct curvature-squared four-forms which combine, with \OGN{}-locked coefficients $(+1, -1, -2/5, -2/5)$, into a single closed gauge-invariant four-form $X_4$. The trace-ratio lock $|a/c| = |a/d| = 5/2$ pins the Abelian fraction at $-2/5$ symmetrically across $\U(1)_B$ and $\U(1)_{Qi}$ under the equal-Abelian-treatment convention of \AX{4}. The modified Bianchi identity $\dd H_3 = X_4$ closes the entire anomaly inflow through the single tensor sector $H_3 = \dd B_2 + (\text{Chern--Simons forms})$. This is the rank-one single-tensor Green--Schwarz factorization that Theorem~\ref{thm:rank-one} together with Theorem~\ref{thm:bi6-cancellation} prove to be the unique axiomatic choice.}
\label{fig:bi6-anomaly-inflow}
\end{figure}

The coefficient \(a=+1\) is fixed by primitive normalization of
the tensor field. The sign \(b=-1\) is fixed by the Green--Schwarz
inflow requirement (opposite signs for Lorentz and gauge
Chern--Simons forms). The Abelian coefficient \(2/5\) is the
trace ratio
\begin{equation}
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)}
= \frac{1/3}{5/6} = \frac{2}{5},
\label{eq:trace-ratio}
\end{equation}
where
\begin{equation}
Y_0 = \diag\!\paren{-\frac{1}{3},-\frac{1}{3},-\frac{1}{3},
\frac{1}{2},\frac{1}{2}},
\qquad
B_0 = \diag\!\paren{\frac{1}{3},\frac{1}{3},\frac{1}{3},0,0}
\label{eq:generators}
\end{equation}
are the standard hypercharge and baryon generators in the \(\mathbf{5}\)
of \(\SU(5)\). The three color entries carry baryon charge \(1/3\) and
the two lepton entries carry zero, while the hypercharge assignment is
\(-1/3\) for color and \(+1/2\) for leptons (the Georgi--Glashow
embedding \citep{GeorgiGlashow1974}). The traces are computed directly:
\begin{equation}
\Tr(B_0^2) = 3\times\!\paren{\frac{1}{3}}^{\!2} + 2\times 0^2
= \frac{1}{3},
\qquad
\Tr(Y_0^2) = 3\times\!\paren{\frac{1}{3}}^{\!2}
+ 2\times\!\paren{\frac{1}{2}}^{\!2}
= \frac{1}{3} + \frac{1}{2} = \frac{5}{6}.
\label{eq:traces-explicit}
\end{equation}
The ratio is therefore \(\Tr(B_0^2)/\Tr(Y_0^2) = (1/3)/(5/6) = 2/5\),
which fixes the Abelian coefficient in \eqref{eq:x4-bi6}.
The equality \(c=d\) follows from equal Abelian
treatment under \AX{4}/\POS{6}.

\subsection{The geometric and topological reading: \BIsix{} as a Chern--Weil identity for the \texorpdfstring{$B$}{B}-field gerbe}
\label{sec:bi6-chern-weil-gerbe}

The four-form \eqref{eq:x4-bi6} and its coefficient vector $(1,-1,-\tfrac25,-\tfrac25)$ carry a transparent geometric and topological meaning, which we now develop from first principles. The aim is pedagogical as much as structural: a reader meeting the modified Bianchi identity for the first time --- a beginning graduate student as readily as a senior specialist --- should come away seeing that $\dd H_3 = X_4$ is not an \emph{ad hoc} device but the \emph{Chern--Weil consistency condition of a connection}, written one categorical level above the familiar gauge field. We proceed in seven self-contained steps.

\paragraph{Step 1: the ordinary Bianchi identity is a fiber bundle's Chern--Weil consistency.}
Let $P\to\cM_6$ be a principal bundle whose structure group is one of the factors of $G_{\Bulk}$, with gauge potential $A$ (a Lie-algebra-valued one-form --- the \emph{connection} of \citet{Ehresmann1950}) and curvature
\begin{equation}
F \;=\; \dd A + A\wedge A .
\label{eq:cw-curvature}
\end{equation}
The curvature obeys the \emph{Bianchi identity}
\begin{equation}
DF \;:=\; \dd F + [A, F] \;=\; 0 ,
\label{eq:cw-bianchi}
\end{equation}
which is not an extra postulate but an automatic consequence of $\dd^2=0$ applied to \eqref{eq:cw-curvature}. Its value is that it makes the \emph{Chern--Weil} construction work: for any invariant polynomial $\mathcal{P}$ of the curvature, \eqref{eq:cw-bianchi} gives $\dd\,\mathcal{P}(F)=0$, so $\mathcal{P}(F)$ is closed, and changing the connection $A\to A'$ shifts it only by an exact form. Hence the de Rham class $[\mathcal{P}(F)]$ is a topological invariant of the bundle --- a \emph{characteristic class} --- independent of the connection \citep{MilnorStasheff1974}. The two basic four-form cases are
\begin{equation}
\frac{1}{8\pi^2}\,\tr F^2 \;\longmapsto\; \mathrm{ch}_2(P)\in H^4(\cM_6;\mathbb{Z}),
\qquad
\frac{1}{16\pi^2}\,\tr R^2 \;\longmapsto\; \tfrac12\,p_1(T\cM_6)\in H^4(\cM_6;\mathbb{Z}),
\label{eq:cw-classes}
\end{equation}
the second Chern character of a gauge bundle and half the first Pontryagin class of the tangent bundle. \emph{Each of the four curvature-squared four-forms in $X_4$ --- $\tr R^2$, $\Tr F_5^2$, $F_B^2$, $F_{Qi}^2$ --- is therefore, up to normalization, a representative of an integral degree-four characteristic class.}

\paragraph{Step 2: the Green--Schwarz tensor is a connection one level up --- a gerbe.}
A $\U(1)$ gauge field is a connection on a circle bundle; its curvature $F=\dd A$ is a two-form whose class $[F/2\pi]$ lies in $H^2(\cM_6;\mathbb{Z})$ (the first Chern class), and the holonomy $\exp\oint A$ is the observable phase \citep{WuYang1975}. The two-form potential $B_2$ of \BIsix{} is the next object in this sequence: a connection on a \emph{gerbe} --- equivalently a bundle gerbe \citep{Murray1996} or a $2$-bundle --- whose curvature is the three-form $H_3=\dd B_2$, with class $[H_3/2\pi]\in H^3(\cM_6;\mathbb{Z})$, the \emph{Dixmier--Douady} class \citep{Brylinski1993}. The dictionary is exact, and Table~\ref{tab:gerbe-dictionary} states it.
\begin{table}[htbp]
\centering
\caption[Circle bundle and gerbe: the same dictionary one degree up.]{The step from a circle bundle to a gerbe, written as a dictionary. The first row is the familiar case --- a gauge field with a two-form curvature and a Chern class --- and the second is the object one degree higher, with a two-form potential, a three-form curvature and a Dixmier--Douady class. The correspondence is exact rather than analogical, which is what licenses the construction in this Part: the tensor field of the anomaly mechanism is a connection on a gerbe in precisely the sense that a gauge field is a connection on a bundle.}
\label{tab:gerbe-dictionary}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lll}
\toprule
object & potential & curvature (integral class) \\
\midrule
circle bundle & gauge field $A$ & $F=\dd A$ \;\,($H^2$, Chern) \\
gerbe & tensor $B_2$ & $H_3=\dd B_2$ \;\,($H^3$, Dixmier--Douady) \\
\bottomrule
\end{tabular}
\end{table}
The Green--Schwarz tensor of \BIsix{} is thus a gerbe connection and $H_3$ is its field strength: the bulk anomaly inflow of Figure~\ref{fig:bi6-anomaly-inflow} lives ``one categorical level above'' the gauge sector.

\paragraph{Step 3: Chern--Simons transgression forces the modified Bianchi.}
A closed invariant four-form is locally exact: each curvature-squared term is the differential of a \emph{Chern--Simons} three-form \citep{ChernSimons1974},
\begin{equation}
\tr F^2 = \dd\,\omega_3(A), \qquad \omega_3(A) = \tr\!\Big(A\,\dd A + \tfrac23\,A^3\Big),
\label{eq:cs-form}
\end{equation}
and likewise $\tr R^2 = \dd\,\omega_3(\Omega)$ for the Lorentz connection $\Omega$. The form $\omega_3$ is \emph{not} globally well-defined --- it shifts under large gauge transformations --- which is precisely why $[\tr F^2]$ can be a non-trivial class. The Green--Schwarz prescription absorbs these three-forms into the gerbe connection,
\begin{equation}
H_3 \;=\; \dd B_2 \;+\; a\,\omega_3(\Omega) + b\,\omega_3(A_5) + c\,\omega_3(A_B) + d\,\omega_3(A_{Qi}),
\label{eq:H3-cs}
\end{equation}
so that, taking $\dd$ and using \eqref{eq:cs-form} with the coefficients of \eqref{eq:x4-bi6},
\begin{equation}
\dd H_3 \;=\; a\,\tr R^2 + b\,\Tr F_5^2 + c\,F_B^2 + d\,F_{Qi}^2 \;=\; X_4 .
\label{eq:modified-bianchi-derived}
\end{equation}
The modified Bianchi identity $\dd H_3 = X_4$ is thus \emph{forced}: it is the field strength of the Chern--Simons-corrected gerbe connection \eqref{eq:H3-cs}. The naive $\dd H_3 = 0$ is the special case of a gerbe with no characteristic-class source.

\paragraph{Step 4: the coefficient vector is a characteristic-class matching condition.}
Because $X_4 = \dd H_3$ is \emph{exact}, its de Rham class vanishes:
\begin{equation}
[X_4] \;=\; a\,[\tr R^2] + b\,[\Tr F_5^2] + c\,[F_B^2] + d\,[F_{Qi}^2] \;=\; 0 \quad\text{in } H^4(\cM_6;\mathbb{R}).
\label{eq:matching}
\end{equation}
The vector $(a,b,c,d)=(1,-1,-\tfrac25,-\tfrac25)$ is, read this way, exactly the combination that makes the four degree-four characteristic classes --- gravitational, $\SU(5)$, and the two Abelian --- \emph{cancel in cohomology} on the admissible \GG{} background. The ``trace-ratio lock'' $|a/c|=|a/d|=5/2$ of \eqref{eq:trace-ratio} is this matching condition expressed through the Georgi--Glashow embedding. Equivalently, a globally defined $H_3$ exists \emph{iff} $[X_4]=0$: condition \eqref{eq:matching} \emph{is} the anomaly-cancellation requirement, now visibly a statement about the topology of the gauge and tangent bundles rather than about Feynman diagrams. (This matching condition is the existence condition for a globally defined \(H_3\) --- the tensor tadpole; it is logically distinct from, and complementary to, the factorization condition \(I_8 = \tfrac12 X_4^2\) on the chiral spectrum enforced in Sections~\ref{sec:eight-parts} and~\ref{sec:model-level}.)

\paragraph{Step 5: anomaly inflow as the Stora--Zumino descent.}
The same three-form organizes the cancellation of the boundary anomaly. From the gauge-invariant $X_4$ one descends to the gauge-variant Chern--Simons form $\omega_3^0$ (with $X_4=\dd\omega_3^0$) and then to the anomaly two-form $\omega_2^1$ (with $\delta\omega_3^0=\dd\omega_2^1$, $\delta$ the gauge variation) --- the descent triangle reviewed in \S\ref{sec:geometric-origin}. The Green--Schwarz counterterm $\int B_2\wedge X_4$ then has a gauge variation that cancels the projected one-loop anomaly: the bulk $H_3$-flux, sourced by $X_4$ through \eqref{eq:modified-bianchi-derived}, ``flows in'' to the boundary --- the anomaly-inflow mechanism of \citet{CallanHarvey1985}. In the constitutional reading of this program that inflow is the bulk content of \AX{2}~(\GGEP), and its operational projection to the boundary is \AX{3}~(\OGGEP); the modified Bianchi is therefore a \emph{consistency} condition, not an equation of motion.

\paragraph{Step 6: integrality and the \OGNfive{} ledger.}
Characteristic numbers --- integrals of characteristic classes over closed four-cycles --- are \emph{integers}. This is the same integrality that quantizes magnetic charge as the Chern number of a $\U(1)$ bundle \citep{WuYang1975} and instanton number as $\mathrm{ch}_2$. The matching condition \eqref{eq:matching} is accordingly not an equation among arbitrary reals but among \emph{integer} characteristic numbers weighted by group-theoretic trace ratios. Imposing it together with the rank-one single-tensor requirement confines the coefficient vector to a one-dimensional rational lattice direction and fixes the gravitational and non-Abelian entries. Integrality alone does not fix the Abelian entry along that direction; that entry is fixed separately, by the trace-ratio identity of Theorem~\ref{thm:ogn-bi6}. The integer content of the resulting vector is exactly the \OGNfive{} ledger $(M,N,L;J,K)=(1,3,5;7,35)$, derived in \S\ref{sec:ogn5-theorem}. The small integers of \GG{} are, in this light, \emph{not} numerology: they are the integrality of characteristic classes --- the most robust topological quantization known.

\paragraph{Step 7: the rigorous home, and an honest caveat.}
At full rigor the tensor $B_2$ is a connection on a bundle gerbe \citep{Murray1996, Brylinski1993}, equivalently a cocycle in \emph{differential cohomology} \citep{HopkinsSinger2005}, and the modified Bianchi identity is the curvature equation of that differential cocycle. The integral lift of $X_4$ and the precise quantization of the $H_3$-flux are differential-cohomology and (twisted) $K$-theory statements; for six-dimensional non-supersymmetric theories of the present type the applicable framework is that of \citet{HopkinsSinger2005, FreedHopkins2016, MonnierMoore2019}, whose conditions \S\ref{sec:bi6-fourform} and the global-anomaly analysis verify. ``Fiber bundle'' is thus the correct \emph{intuition}, while the precise object carrying $H_3$ is a higher bundle; we keep the bundle language for its clarity and flag this one categorical step. With the caveat noted, \BIsix{} is exactly the Chern--Weil identity of the Green--Schwarz gerbe, and its coefficient vector is a matching of characteristic classes.

\paragraph{Synthesis: one idea, applied twice.}
The whole structure is a single idea used at two levels. A gauge field is a connection; its curvature obeys the Bianchi identity; invariant polynomials of the curvature are characteristic classes (Chern--Weil). The Green--Schwarz tensor is a connection one level up; its curvature obeys the \emph{modified} Bianchi identity $\dd H_3 = X_4$; and $X_4$ is a characteristic four-form whose vanishing cohomology class is the anomaly-cancellation condition. \BIsix{} is the name, in this program, of that one equation specialized to the forced six-dimensional bulk, and the coefficient vector $(1,-1,-\tfrac25,-\tfrac25)$ is the topology of the Standard-Model embedding written as a matching of characteristic classes. The anomaly is geometry; its cancellation is topology; the integers are characteristic numbers --- a chain we hope is as inviting to a student meeting it now as it has been to the geometers and physicists who forged each link.

\subsection{Anomaly matrix and the rank-one criterion}

\noindent
\figref{fig:bi6-anomaly-inflow} (page~\pageref{fig:bi6-anomaly-inflow})
visualizes the four-component anomaly inflow of \BIsix{}: the four
bulk curvatures combine via the \OGN{}-locked coefficient vector
$(+1, -1, -2/5, -2/5)$ into a single closed gauge-invariant four-form
$X_4$, which inflows to the two-form tensor $B_2$ through the modified
Bianchi identity $\dd H_3 = X_4$.



Introduce the shorthand
\begin{equation}
x = (\tr R^2,\,\Tr F_5^2,\,F_B^2,\,F_{Qi}^2)^T,
\qquad
v_{\mathrm{BI6}} = (1,-1,-2/5,-2/5)^T.
\end{equation}
Then
\begin{equation}
X_4^{\mathrm{BI6}} = v_{\mathrm{BI6}}\cdot x,
\qquad
I_8^{\mathrm{BI6}} = \half (v_{\mathrm{BI6}}\cdot x)^2
= \half\,x^T M_{\mathrm{BI6}}\,x,
\end{equation}
with
\begin{equation}
M_{\mathrm{BI6}} = v_{\mathrm{BI6}}\,v_{\mathrm{BI6}}^T
= \begin{pmatrix}
1 & -1 & -2/5 & -2/5 \\
-1 & 1 & 2/5 & 2/5 \\
-2/5 & 2/5 & 4/25 & 4/25 \\
-2/5 & 2/5 & 4/25 & 4/25
\end{pmatrix}.
\label{eq:bi6-matrix}
\end{equation}

\begin{theorem}[\textbf{T-BI6-6.1}: Rank-one criterion]
\label{thm:rank-one}
A single Green--Schwarz two-form can cancel the reducible part of
a six-dimensional local anomaly \textbf{if and only if} the
\(4\times 4\) anomaly matrix has rank one and is positive
semidefinite.
\end{theorem}

\begin{proof}
A single tensor couples to one four-form \(X_4 = v\cdot x\).
The cancelable polynomial is \(\half(v\cdot x)^2 = \half x^T(vv^T)x\).
The matrix \(vv^T\) has rank one and is positive semidefinite.
Conversely, a symmetric matrix that is both rank one and positive
semidefinite has a single eigenvalue \(\lambda>0\) and factors as
\(vv^T\) with \(v = \sqrt{\lambda}\,\hat v\). The sign hypothesis is
not decorative: \(-vv^T\) is also rank one, and it is the wrong-sign
Green--Schwarz case \(I_8 = -\half X_4^2\), which no real single-tensor
counterterm cancels. For \BIsix{} the sign is settled by computation
--- see the Corollary below.
\end{proof}

\begin{corollary}
The \BIsix{} matrix \eqref{eq:bi6-matrix} has rank one, with eigenvalues
\(\{58/25,\,0,\,0,\,0\}\), confirming that \BIsix{} passes the
single-tensor factorization test.
\end{corollary}

\paragraph{Remark on what the rank-one criterion proves and what it does not.}
A careful reader may worry that the rank-one criterion is tautological:
``of course $vv^T$ has rank one, by construction.'' The non-trivial
content is one step earlier. The theorem asserts that
single-tensor cancellation is possible \emph{if and only if} the
$4\times 4$ matrix of reducible coefficients \emph{computed
microscopically from the chiral spectrum} has rank one --- and the
microscopic computation is what could in principle have failed.
Indeed, a generic spectrum of chiral fermions in six dimensions
produces an anomaly matrix of rank greater than one; the \BIsix{}
spectrum is the specific construction whose microscopic matrix
collapses to rank one with eigenvector $v_{\mathrm{BI\mbox{-}6}}
= (1, -1, -2/5, -2/5)^T$. The rank-one corollary above therefore
is an algebraic identity, not an independent measurement: the matrix
is \emph{defined} as an outer product, so its rank is one by
construction. It is worth being exact about which part is an output. Rank-one collapse is the output: a generic six-dimensional
chiral spectrum does not have it, and the fact that this one does is
what makes single-tensor inflow available. The particular \emph{value}
of the Abelian entry is not. The benchmark spectrum of
Section~\ref{sec:model-level} is constructed by matching to a target
polynomial in which that entry already appears, so the eigenvector is
unique \emph{given} the spectrum, and the spectrum is what it is
because the entry was fixed beforehand. The Abelian entry is fixed by
the trace-ratio identity of Theorem~\ref{thm:ogn-bi6}, not by the
rank-one condition, and Section~\ref{sec:ogn5-theorem} says so.

\paragraph{Uniqueness of the rank-one factorization.}
A second concern is whether the eigenvector $v_{\mathrm{BI\mbox{-}6}}$
is itself unique. It is, up to an overall sign. If
$v\,v^T = w\,w^T$ as $4\times 4$ symmetric matrices, then writing
both sides in their orthogonal eigendecomposition shows that $w$
must lie along the same line as $v$, and the rank-one condition
$w \cdot w^T = v\,v^T$ together with the fixed gravitational normalization
$a = 1$ pins $w = \pm v$. The sign is then fixed by the
Green--Schwarz convention that the gravitational and gauge sectors
enter $X_4$ with opposite signs (so that $b = -1$, not $b = +1$).
Hence the \BIsix{} coefficient vector $(1, -1, -2/5, -2/5)$ is the
unique rank-one factorization of the microscopically computed
anomaly polynomial. No other coefficient vector consistent with
the spectrum produces a rank-one matrix.
\paragraph{What a single tensor cannot do, and what that requires of the matter.} Before
turning to the cancellation itself, it is worth extracting a necessary condition that the
rank-one criterion carries but does not advertise. The criterion says the cancellable part
of the eight-form is \(\tfrac12 (v\cdot x)^2\), a perfect square of a form that is
\emph{quadratic} in the curvatures. Squaring a quadratic form is restrictive, and it is
restrictive in a way that speaks directly about the matter content.

It can manufacture \(\Tr F_5^2 \wedge F_B^2\), because that is a product of two entries of
\(x\). It can never manufacture the irreducible quartic \(\Tr F_5^4\) --- the piece that is
genuinely not a product of two quadratic traces --- because a product of two second-order
objects is by construction reducible. And it can never manufacture
\(\Tr F_5^3\wedge F_B\) either, for a simpler reason: \(X_4\) contains no cubic invariant
anywhere in it, so there is nothing in the square that could supply the \(\Tr F_5^3\).

A generic chiral spectrum in six dimensions produces both of those monomials. For them
there is no channel. The bulk is not free to generate them and leave the Green--Schwarz term
to absorb them, because no such term exists; they must cancel \emph{among the fermions
themselves}, before the inflow mechanism is invoked at all. In the usual model-building
idiom one says ``choose a spectrum whose anomalies cancel.'' The single-tensor architecture
says something sharper, and it is a statement about the mechanism rather than about \GG{}:

\begin{theorem}[\textbf{T-BI6-6.2}: Necessary conditions for single-tensor inflow]
\label{thm:single-tensor-necessity}
Let a six-dimensional chiral theory with simple gauge factor \(\SU(L)\) and Abelian factors
carry a single Green--Schwarz tensor, so that the entire reducible anomaly is canceled by
inflow through one four-form \(X_4\) quadratic in the curvatures. Then the charged content
must satisfy
\begin{equation}
B \;=\; \sum_\rho \sigma_\rho\, B_\rho \;=\; 0
\qquad\text{and}\qquad
A_3 \;=\; \sum_\rho \sigma_\rho\, q_\rho\, A_{3,\rho} \;=\; 0 ,
\label{eq:single-tensor-necessity}
\end{equation}
where \(B_\rho\) is the irreducible quartic index of \(\rho\), \(A_{3,\rho}\) its cubic
index, \(q_\rho\) its Abelian charge and \(\sigma_\rho\) its six-dimensional chirality.
These are conditions on the matter alone: they involve neither the coefficient vector of
\(X_4\) nor the tensor normalization, and they must hold before any inflow is computed.
\end{theorem}

\begin{proof}
Expand \(\tfrac12 X_4\wedge X_4\), where \(X_4 = v\cdot x\) and \(x\) is the
vector of quadratic invariants
\[
x = \bigl(\tr R^2,\ \Tr F_5^2,\ F_B^2,\ F_{Qi}^2\bigr).
\]
Every monomial produced is a product of two entries of \(x\); hence the expansion
contains no term proportional to the irreducible \(\Tr F_5^4\), and none
proportional to \(\Tr F_5^3\wedge F_B\) or \(\Tr F_5^3\wedge F_{Qi}\). Both monomials occur in the microscopic polynomial: for a Weyl
fermion in \(\rho\) with Abelian charge \(q\), the eight-form part of
\(\mathrm{ch}(F_5+qF_B)\) contains \(\Tr_\rho F_5^4\) and \(4q\,\Tr_\rho F_5^3\wedge F_B\).
Since the counterterm cannot reach them, their total coefficients must vanish identically on
the spectrum, which is \eqref{eq:single-tensor-necessity}.
\end{proof}

\begin{remark}[Why this is worth isolating]
Theorem~\ref{thm:single-tensor-necessity} converts a design choice into an obligation. The
core block of Section~\ref{ssec:core-block-construction} is built so that its irreducible
and cubic indices vanish internally; read through the present theorem, that is not a
convenient property of a package we happened to select but a property that \emph{any}
admissible content must have. The theorem is also the entry point for
Theorem~\ref{thm:L-from-GS}: carrying \(L\) as a symbol in
\eqref{eq:single-tensor-necessity} turns the two conditions into two equations in \(L\),
and they turn out to be the same equation.
\end{remark}


\subsection{The eight-part proof obligation}
\label{sec:eight-parts}

For a complete claim of anomaly freedom, eight conditions must hold.
We label them \textbf{(PO1)}--\textbf{(PO8)} --- \emph{proof
obligations} --- to keep them distinct from the twelve Open Problems
\(O_1\)--\(O_{12}\) triaged in Section~\ref{sec:open-problems} and from
the numbered Objections of Part~\ref{sec:common-objections}.
\begin{enumerate}[label=\textbf{(PO\arabic*)}, leftmargin=1.8cm]
\item The one-loop anomaly polynomial must be computed from the
 full chiral spectrum.
\item Every irreducible term (\(\tr R^4\), \(\Tr F_5^4\)) must vanish.
\item Every off-diagonal Abelian term (\(F_B^3 F_{Qi}\),
 \(F_B F_{Qi}^3\), \(\tr R^2\,F_B F_{Qi}\), etc.)\ must vanish.
\item The reducible coefficients must match \eqref{eq:bi6-matrix} exactly.
\item The descent variation of \(S_{\mathrm{GS}}\) must cancel
 the one-loop anomaly.
\item The anomaly cocycle must be BRST-exact.
\item Global gauge anomalies must be absent.
\item The exponentiated Green--Schwarz term must be well defined
 on the charge lattice.
\end{enumerate}

\subsection{Microscopic anomaly polynomial}

The total anomaly polynomial of the theory is the sum of contributions from
every chiral field in the six-dimensional spectrum.
Each six-dimensional Weyl fermion of chirality \(\sigma_s = \pm 1\) in
\(\SU(5)\) representation \(\rho_s\) with Abelian charges
\((q_{B,s},q_{Qi,s})\) contributes the following eight-form
(derived from the index theorem for the Dirac operator on
\(M_6\); see \citet{AlvarezGaumeWitten1984}):
\begin{equation}
\boxed{
I_8^{(s)}
= \sigma_s\left[
\frac{d_{\rho_s}}{5760}(7p_1^2{-}4p_2)
- \frac{d_{\rho_s}}{48}\,p_1\,x_s^2
- \frac{1}{48}\,p_1\,T_{2,s}
+ \frac{d_{\rho_s}}{24}\,x_s^4
+ \frac{1}{4}\,T_{2,s}\,x_s^2
+ \frac{1}{6}\,T_{3,s}\,x_s
+ \frac{1}{24}\,T_{4,s}
\right],
}
\label{eq:microscopic}
\end{equation}
where \(d_{\rho_s} = \dim(\rho_s)\),
\(T_{n,s}=\tr_{\rho_s}(f_5^n)\) are the \(n\)-th Casimir traces of
the \(\SU(5)\) field strength,
\(x_s = q_{B,s}\,f_B + q_{Qi,s}\,f_{Qi}\) is the Abelian field-strength
combination weighted by the species charges,
and \(p_1\), \(p_2\) are the first and second Pontryagin classes.
Throughout the anomaly bookkeeping, each curvature-squared slot denotes
its characteristic four-form in units of \(8\pi^2\), so that
\(\tr R^2 \mapsto p_1/2\), \(\Tr F_5^2 \mapsto c_2(E_5)\), and
\(F^2 \mapsto c_1(L)^2\) --- the normalization of the integral-lift
lemma of Appendix~\ref{app:global-proof-dossier}.
The first term is the purely gravitational contribution (present even for
gauge singlets), the middle terms are mixed gauge--gravity and mixed
gauge--Abelian couplings, and the last two terms are the purely non-Abelian
cubic and quartic Casimir contributions.

The anomaly-cancellation condition requires that the total polynomial
\(I_8^{\mathrm{tot}} = \sum_s I_8^{(s)} + I_8^{\mathrm{tensor}}
+ I_8^{\mathrm{vector}}\)
factorize as \(\frac{\Omega}{2}X_4^2\).
Expanding both sides and comparing term by term yields
ten coefficient-matching conditions (for the reducible monomials)
and eight vanishing conditions (for the irreducible and forbidden monomials),
as listed explicitly in Appendix~\ref{app:matrix-and-coefficients}.

\subsection{The descent proof}

The local anomaly cancellation is the central structural theorem of the paper.
The proof strategy is standard Green--Schwarz descent, applied to the
specific \BIsix{} factorization. We state the theorem, give the proof
with all steps displayed, and then record the corollaries.
The fully expanded version with every Chern--Simons form written out
sector by sector is in Appendix~\ref{app:descent-full}.

\begin{theorem}[\textbf{T-BI6-6.3}: \BIsix{} local anomaly cancellation]
\label{thm:bi6-cancellation}
If the total one-loop anomaly polynomial factorizes as
\(I_8^{\mathrm{tot}} = \half(X_4^{\mathrm{BI6}})^2\),
and if the two-form \(B_2\) transforms as
\(\delta B_2 = -\omega_2^{(1)} + \dd\Lambda_1\),
then the total effective action is gauge invariant:
\begin{equation}
\boxed{
\delta(\Gamma_{\mathrm{1\text{-}loop}} + S_{\mathrm{GS}}) = 0.
}
\label{eq:exact-cancellation}
\end{equation}
\end{theorem}

\begin{proof}
The proof proceeds in four steps.

\emph{Step 1: Chern--Simons potential.}
Since \(X_4\) is a closed four-form built from characteristic classes,
there exists a Chern--Simons three-form \(\omega_3^{(0)}\) satisfying
\begin{equation}
\dd\omega_3^{(0)} = X_4.
\label{eq:cs-potential}
\end{equation}
For \BIsix{}, \(\omega_3^{(0)}\) is the linear combination of the
gravitational, \(\SU(5)\), \(U(1)_B\), and \(U(1)_{Qi}\)
Chern--Simons forms with the same coefficients \((1,-1,-2/5,-2/5)\).

\emph{Step 2: Seven-form and six-form descendants.}
Define the seven-form
\(I_7^{(0)} = \frac{1}{2}\,\omega_3^{(0)}\wedge X_4\).
Then \(\dd I_7^{(0)} = \frac{1}{2}\,X_4\wedge X_4 = I_8\),
as required.
Under an infinitesimal gauge/Lorentz variation, the Chern--Simons form
varies as \(\delta\omega_3^{(0)} = -\dd\omega_2^{(1)}\),
where \(\omega_2^{(1)}\) is the descent two-form.
Because \(\delta X_4 = 0\) (characteristic classes are gauge-invariant),
the consistent anomaly six-form is
\begin{equation}
I_6^{(1)} = \half\,\omega_2^{(1)}\wedge X_4.
\label{eq:i6-main}
\end{equation}

\emph{Step 3: One-loop anomalous variation.}
The one-loop effective action has the anomalous variation
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}}
= \frac{\kappa}{2}\int_{M_6} \omega_2^{(1)}\wedge X_4.
\label{eq:oneloop-main}
\end{equation}

\emph{Step 4: Green--Schwarz counterterm.}
The counterterm is
\(S_{\mathrm{GS}} = \frac{\kappa}{2}\int_{M_6} B_2\wedge X_4\).
Using \(\delta B_2 = -\omega_2^{(1)} + \dd\Lambda_1\) and
\(\delta X_4 = 0\):
\begin{equation}
\delta S_{\mathrm{GS}}
= \frac{\kappa}{2}\int_{M_6} (-\omega_2^{(1)})\wedge X_4
= -\frac{\kappa}{2}\int_{M_6} \omega_2^{(1)}\wedge X_4.
\label{eq:gs-var-main}
\end{equation}
(The \(\dd\Lambda_1\) term integrates to a boundary contribution
that vanishes on a closed manifold or in the bulk away from boundaries.)

Adding \eqref{eq:oneloop-main} and \eqref{eq:gs-var-main}:
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}} + \delta S_{\mathrm{GS}}
= +\frac{\kappa}{2}\int\omega_2^{(1)}\wedge X_4
 -\frac{\kappa}{2}\int\omega_2^{(1)}\wedge X_4
= 0.
\end{equation}
This completes the proof.
\end{proof}

\begin{corollary}[BRST exactness and Slavnov--Taylor identity]
Under the hypotheses of Theorem~\ref{thm:bi6-cancellation},
the anomaly cocycle is BRST-exact in the enlarged field space
containing \(B_2\), and the gauge-fixed effective action satisfies
the homogeneous Slavnov--Taylor identity.
\end{corollary}

\begin{remark}[Scope of the structural theorem]
\label{rem:local-vs-global}
Theorem~\ref{thm:bi6-cancellation} is a statement about the
\emph{perturbative local} anomaly---the eight-form \(I_8\) and its
cancellation by a two-form counterterm through descent.
A complete anomaly-freedom claim in the modern sense requires
additional global checks: the exponentiated path integral must be
single-valued under large gauge transformations, the self-dual
tensor charge lattice must be properly quantized, and the
Atiyah--Patodi--Singer eta invariant on bounding seven-manifolds
must be consistent.
These global questions have been addressed systematically for
\(\cN=(1,0)\) supergravity by Monnier and Moore
\citep{MonnierMoore2019} and, more recently, by Dierigl and
Tartaglia \citep{DieriglTartaglia2025}.

For \GG{}, the standard partial checks are recorded in
Section~\ref{sec:structural-proof}: \(\pi_6=0\) for all gauge
factors, \(\Omega_7^{\Spin}(\mathrm{pt})=0\), and the chiral-tensor
signature phase is settled by the Wu--Chern--Simons quadratic
refinement (manifestly trivial on the \(\Omega = 50\) branch, where
\(n_{\mathrm{SD}} = 200 \equiv 0 \bmod 8\); Appendix~\ref{app:grav-full}).
The complete global theorem, including the Wu--Chern--Simons /
differential-cohomology refinement on the physical intended \GG{}
background class, is established in
Section~\ref{sec:global-completion} and Appendix~\ref{app:global-proof-dossier}.
The structural theorem and the model-level benchmark are not
affected by this distinction: the local polynomial factorization and
coefficient matching are exact statements that hold independently
of the global completion.
\end{remark}

\subsection{The boundary mixed anomalies in four-dimensional language}
\label{subsec:boundary-mixed-4d}

A reader trained in four-dimensional effective field theory will ask
the anomaly question in a different dialect than the descent proof
above, and the question deserves an answer in its own words.  Gauging
baryon number in the Standard Model \emph{alone} is famously
obstructed: with the observed fermion content, in the left-handed Weyl
convention, the mixed triangle coefficients per generation are
\begin{equation}
\mathcal A\!\left[\U(1)_B\text{--}\SU(2)_L^2\right]
= \sum_{\text{doublets}} B\,T(R) = 3_c\cdot\tfrac13\cdot\tfrac12 = +\tfrac12,
\qquad
\mathcal A\!\left[\U(1)_B\text{--}\U(1)_Y^2\right]
= \sum B\,Y^2 = -\tfrac12,
\label{eq:mixed-triangles}
\end{equation}
while the remaining candidates vanish identically for Standard-Model
content: \(\mathcal A[\U(1)_B\text{--}\mathrm{grav}^2]=\sum B = 0\)
(baryon number is vector-like on the quarks),
\(\mathcal A[\U(1)_B^3]=0\), and \(\mathcal A[\U(1)_B^2\text{--}\U(1)_Y]=0\).
Exactly two mixed triangles survive, equal and opposite, summing over
three generations to \(\pm\tfrac32\).  The conventional price of
gauging \(\U(1)_B\) is therefore a set of new chiral fermions
(``anomalons'') charged to cancel \eqref{eq:mixed-triangles} --- new
matter invented for bookkeeping, with masses and signatures of its own.

\BIsix{} pays no such price, and the reason is already proved above;
this subsection only translates it.  The boundary chiral spectrum of
\GG{} is not the Standard Model with baryon charges appended by hand:
it descends from unsplit \(\SU(5)\) multiplets and the \OGN{}-locked
charge lattice (Section~\ref{sec:model-level}), and the microscopic
anomaly polynomial of that spectrum \emph{factorizes to rank one}
(Theorem~\ref{thm:rank-one} and the microscopic remark following it)
--- the cross-blocks of the matrix \(M_{\mathrm{BI6}}\) in
\eqref{eq:bi6-matrix}, the \((F_B^2,\Tr F_5^2)\) and
\((F_B^2,\tr R^2)\) entries, are precisely the bulk ancestors of the
mixed anomalies the four-dimensional reader is worried about.  The
orbifold projection localizes the corresponding anomaly on the
boundary, and the gauge variation of the single counterterm
\(\int B_2\wedge X_4\) cancels it wholesale
(Theorem~\ref{thm:bi6-cancellation}; the inflow of
\citealp{CallanHarvey1985}).  The statement that the compensating
couplings split across the Standard-Model subgroups
\(\SU(2)_L\times\U(1)_Y\subset\SU(5)\) in exactly the ratios required
by \eqref{eq:mixed-triangles} is the content of the rank-one
factorization applied to the split boundary spectrum; the explicit
split-multiplet bookkeeping lives with the model-level benchmark of
Section~\ref{sec:model-level}, and the global caveats of
Remark~\ref{rem:local-vs-global} apply unchanged.

In four-dimensional language the mechanism has a familiar face.  The
light boundary remnant of the tensor \(B_2\) is an axion-like scalar
\(b(x)\) that \emph{shifts} under a \(\U(1)_B\) gauge transformation;
its descent couplings \(b\,\mathrm{tr}(W\widetilde W)\) and
\(b\,F_Y\widetilde F_Y\) supply the compensating variations --- the
four-dimensional Green--Schwarz mechanism, as realized for anomalous
\(\U(1)\)s in open-string constructions
\citep{GreenSchwarz1984,Sagnotti1992} --- and the same shift is eaten
by the \(\U(1)_B\) vector, which thereby acquires a St\"uckelberg
mass without any Higgs field.  One mechanism, seen twice: the
cancellation of \eqref{eq:mixed-triangles} and the mass of the
baryonic vector are the same coupling.  Three consequences follow.
\emph{First}, no anomalon fermions exist anywhere in the spectrum ---
the ledger cancels its own books.  \emph{Second}, although the gauge
boson is massive, \(\U(1)_B\) survives on the boundary as an exact
perturbative selection rule, so the proton-stability falsifier
F-BI6-1 stands, conditional on the holonomy-descent step fenced in
\citep{ArisakaA2GG6}.  \emph{Third}, the massive vector
itself is not an abstraction: it is the leptophobic \(Z'_B\) of the
lock cluster, whose collider phenomenology at \(3.85\)~TeV is the
business of \PaperPOne{} \citep{ArisakaP1KK} --- the place where this
subsection's bookkeeping meets a detector.

\subsection{Self-duality of the Green--Schwarz tensor}

\begin{proposition}[Necessity of anti-self-duality]
\label{prop:asd}
A six-dimensional theory whose chiral anomaly is canceled by a
single Green--Schwarz tensor requires the tensor to be
anti-self-dual at the quantum level.
No spectrum of ordinary Weyl fermions alone can produce the required
gravitational coefficients.
\end{proposition}

\begin{proof}
Every Weyl fermion in six dimensions contributes to the anomaly polynomial
a gravitational piece proportional to \((7p_1^2 - 4p_2)\),
regardless of its gauge representation.
The target polynomial \(\half X_4^2 = \half(\tr R^2 - \cdots)^2\)
contains only \((\tr R^2)^2 \propto p_1^2\) and no independent \(p_2\) term.
Since fermions always produce \(p_2\) with a fixed ratio \(-4/7\)
relative to \(p_1^2\), fermions alone can never cancel the
\(p_2\) term while retaining a nonzero \(p_1^2\) term.
An anti-self-dual tensor contributes gravitational anomaly through the
Hirzebruch \(L\)-polynomial with coefficients
\((p_1^2 - 7p_2)/360\) per tensor (see Appendix~\ref{app:grav-full} for the explicit formula),
which has a \emph{different} \(p_2/p_1^2\) ratio than fermions.
This breaks the degeneracy and allows both the \(p_1^2\) and \(p_2\)
equations to be satisfied simultaneously.
\end{proof}

\section{Complete global anomaly freedom of \BIsix{} under \GG{}}
\label{sec:global-completion}

Section~\ref{sec:structural-proof} established that the \BIsix{} factorization
cancels the \emph{infinitesimal} anomaly by descent.
That is necessary but not sufficient.
A modern anomaly-freedom proof must also address \emph{large}
gauge transformations and diffeomorphisms, which can introduce
a global phase invisible to the infinitesimal analysis.
The physical question is: if one transports the quantum theory
around a non-contractible loop in the space of gauge-equivalent
backgrounds, does the partition function return to itself, or
does it pick up an anomalous phase?
Answering this requires the machinery of mapping tori, eight-dimensional
fillings, and the Wu--Chern--Simons refinement of the self-dual tensor
sector.
The present section states the precise background class, formulates
the global theorem, and gives the proof strategy;
the detailed lemmas are carried out in Appendix~\ref{app:global-proof-dossier}.

\subsection{The physical intended \GG{} background class and the main theorem}

The global theorem is only as strong as the class of backgrounds on which it is stated. For that reason the intended \GG{} class is written explicitly in the main text rather than left implicit or deferred. These eight clauses define the geometric and topological class of six-dimensional backgrounds used by \GG{}.

\begin{definition}[Physical intended \GG{} background class]
\label{def:admissible-main}
The physical intended \GG{} background class \(\AdmissibleClass\) consists of data
\[
(\Msix,\mathcal P_5,L_B,L_{Qi},\check B_2)
\]
satisfying the following clauses.
\begin{enumerate}[label=(C\arabic*),leftmargin=1.8cm]
\item \textbf{Stage clause.} The Lorentzian bulk has the \GG{} product stage
\[
\Msix=\Mfour\times I_r\times S^1_\theta,
\]
with \(\Mfour\) spin and the two structural directions \(r\) and \(\theta\) distinguished. For Euclidean anomaly problems one uses APS-compatible collars at the ends of \(I_r\) or doubles the interval to obtain the closed structural torus
\[
\Ttwo=S^1_r\times S^1_\theta.
\]
\item \textbf{Gauge clause.} The exact gauge group is
\[
\SUfive\times\UoneB\times\UoneQi.
\]
No extra gauge factors are admitted in the intended class.
\item \textbf{Tensor clause.} There is a single primitive \(X_4\)-coupled tensor channel, equivalently a rank-one unimodular tensor lattice \(\Lambda\cong\mathbb Z\) for the Green--Schwarz sector, with characteristic vector \(a=1\). The same \(a=1\) is the characteristic element that defines the unique Wu--Chern--Simons quadratic refinement of the tensor sector (Theorem~\ref{thm:refinement-unique}). (The gauge-singlet completion tensors of Section~\ref{sec:model-level} are \(X_4\)-decoupled spectators; they enter the global analysis only through their signature-type phase, Appendix~\ref{app:grav-full}.)
\item \textbf{\BIsix{} clause.} The four-form entering the modified Bianchi identity is
\[
X_4=\trU R^2-\TrU F_5^2-\gamma\bigl(F_B^2+F_{Qi}^2\bigr),
\qquad
\gamma=\frac{s}{t}\in\mathbb Q_{>0},\quad \gcd(s,t)=1,
\]
subject to the primitive integrality condition
\[
\widehat X:=\frac{t}{8\pi^2}X_4\in H^4(\Msix,\mathbb Z).
\]
\BIsix{} is the point \(\gamma=2/5\), that is \((s,t)=(2,5)\). The clause is stated at
general \(\gamma\) because every result of the present Part is proved at that generality;
Remark~\ref{rem:gamma-family} explains why this is the honest statement rather than a
weaker one. Thus the \BIsix{} class is primitive and admits a differential-cocycle lift.
\item \textbf{Structural-holonomy clause.} The structural torus \(\Ttwo\) is fixed as part of the background data. Allowed transformations preserve the product/collar structure and do not introduce modular monodromy of the \((r,\theta)\) torus.
\item \textbf{Allowed large transformations.} Large diffeomorphisms are identity on the structural collars (or isotopic to identity after doubling) and product-preserving on \(\Ttwo\). Gauge clutching on the structural torus is therefore null-homotopic in the intended class; the same clause excludes mixed clutching (gauge classes in \(H^2(\Mfour;\mathbb{Z}) \otimes H^1(\Ttwo;\mathbb{Z})\)) and Abelian winding transformations on the structural circles. Transformations outside this class were formerly the unrestricted case of Open Problem O10; Theorem~\ref{thm:unrestricted-global} (Subsection~\ref{ssec:unrestricted-global}) now covers them, and the clause survives as a description of the physical intended backgrounds rather than as a hypothesis of the anomaly theorem.
\item \textbf{Physical base clause.} The Euclideanized compact four-manifolds used in the intended \GG{} class satisfy
\[
H^1(\Mfour;\mathbb Z)=0,
\qquad
H^3(\Mfour;\mathbb Z)=0.
\]
Equivalently, the connected physical bases carry no independent Abelian winding sector and no independent \(\SUfive\) gauge-transformation class. For noncompact Minkowski backgrounds, large transformations are taken with compact support so that the same null-homotopy conclusion holds after compactification.
\item \textbf{Pullback-bundle clause.} The principal \(\SUfive\), \(U(1)_B\), and \(U(1)_{Qi}\) bundles are pullbacks from the base and are topologically trivial on the structural torus. The \BIsix{} differential cocycle is likewise a pullback/refinement of this base data.
\end{enumerate}
\end{definition}

These clauses are not ad hoc additions. They simply spell out the class of backgrounds already used by the \GG{} stage itself: fixed product structure, exact gauge content, one primitive tensor, no hidden torus monodromy, and physically relevant four-dimensional bases without extra topological gauge sectors.

\begin{theorem}[\textbf{T-BI6-7.2}: Complete anomaly freedom of \BIsix{} on the physical intended \GG{} class]
\label{thm:bi6-global-complete}
On \(\AdmissibleClass\), the combined phase of the chiral determinant and the exponentiated Wu--Chern--Simons / differential-cohomology lift of the \BIsix{} counterterm is globally well-defined and gauge-invariant. Equivalently, \BIsix{} is completely anomaly-free, both locally and globally, on the physical intended \GG{} background class.
\end{theorem}

\begin{proof}[Proof strategy and physics intuition]
The proof has four logically separate pieces.

\paragraph{Step 1: local descent.}
The infinitesimal anomaly is the six-dimensional descent from \(I_8=\half X_4^2\). This is the part proved already in Section~\ref{sec:structural-proof}: the one-loop consistent anomaly is canceled exactly by the local Green--Schwarz counterterm.

\paragraph{Step 2: seven-dimensional holonomy.}
A global anomaly is not a failure of the local descent equations; it is a phase picked up when one transports the quantum theory around a non-contractible loop in the space of backgrounds. That phase is encoded by the holonomy of the anomaly line on the seven-dimensional mapping torus, and is evaluated on an eight-dimensional filling by the Atiyah--Patodi--Singer / Dai--Freed formalism \citep{DaiFreed1994}. The global problem is therefore a \(6D\to7D\to8D\) consistency ladder.

\paragraph{Step 3: globally defined tensor counterterm.}
The local expression \(\frac12\int B_2\wedge X_4\) is insufficient on nontrivial topology. The correct global object is the Wu--Chern--Simons / differential-cohomology refinement associated with the integral \BIsix{} class \(\widehat X\) and the characteristic vector of the rank-one tensor lattice \citep{HopkinsSinger2005,MonnierMoore2019}. The quadratic refinement is what makes the self-dual tensor sector single-valued rather than multi-valued.

\paragraph{Step 4: automatic admissibility on \(\AdmissibleClass\).}
The remaining task is to show that the hypotheses used in the global theorem are automatic on the intended \GG{} class. The detailed lemmas are proved later in the appendices: \(\pi_6\) vanishes for \(\SU(5)\times U(1)_B\times U(1)_{Qi}\), the \BIsix{} class admits an integral lift, the Wu-shifted quadratic functional is properly quantized, and the product/collar mapping tori bound compatible eight-manifolds. Once these facts are assembled, the global phase cancels exactly.
\end{proof}

\begin{remark}[The theorem holds for a family, and why that is the right statement]
\label{rem:gamma-family}
Theorem~\ref{thm:bi6-global-complete} has been stated and proved on \(\AdmissibleClass\)
with the Abelian coefficient carried as a symbol. It is worth saying plainly what that
buys and what it costs, because a reader may take the generality as a concession.

\emph{What the proof actually consumes.} Walk the chain and ask, at each step, which
property of \(\gamma\) is used. The local descent
(Theorem~\ref{thm:bi6-cancellation}) needs only that \(X_4\) is closed and gauge-invariant;
the coefficient never appears in the argument, only in the name of the object. The
rank-one criterion (Theorem~\ref{thm:rank-one}) is linear algebra over the four quadratic
invariants and is blind to their coefficients. The integral lift
(Lemma~\ref{lem:integral-lift}) uses exactly one property of \(\gamma\): that it is
rational, so that some integer clears its denominator. The Wu--Chern--Simons quantization
needs the spin condition and the integrality of \(x\), and nothing else --- the lemma says
so in as many words. The vanishing of \(\pi_6\) is a fact about \(\SU(5)\times U(1)^2\).
The bounding theorem is a fact about the topology of the base and the structural torus.

Not one of these six steps can see the value of \(\gamma\). That is not a weakness of the
proof; it is a description of what kind of theorem it is.

\emph{Why stating the family is stronger than stating the point.} Three reasons, in
increasing order of importance. First, it is simply a stronger theorem: a statement about
a one-parameter family implies the statement at any point of it. Second, it is robust ---
the value \(2/5\) is fixed elsewhere in this paper, by the trace-ratio identity of
Theorem~\ref{thm:ogn-bi6}, and if that identity were ever revised the anomaly theorem would
not move an inch. The consistency of the bulk does not wait on the arithmetic of the ledger.
Third, and most usefully, it locates the burden of proof correctly.

\emph{The objection this invites, and the answer.} A reader may say: if anomaly freedom
holds for a whole family, then anomaly freedom does not single out \BIsix{}, and the
paper's claim to rigidity is empty. The first half of that is correct and we state it
ourselves. The second half does not follow, and the reason is worth being exact about.
Anomaly freedom is a \emph{consistency test}. Consistency tests are passed by families ---
that is what makes them tests rather than derivations, and it is equally true of the 1984
ten-dimensional mechanism, which admits both \(SO(32)\) and \(E_8\times E_8\) and does not
thereby become vacuous. What is claimed to single out \BIsix{} was never the anomaly
theorem. It is the constitution: \AX{4}~(\MSMF{}) fixing the single-tensor architecture and
the equal-Abelian treatment, and the trace-ratio identity fixing the Abelian fraction. Those
claims stand or fall on their own merits, in the open, and this Part does not lend them
support it does not have.
\end{remark}


Figure~\ref{fig:global-ladder} summarizes the logic of the global
theorem as a dimensional ladder, together with the two facts that
close it on the intended class; Subsection~\ref{ssec:unrestricted-global}
then removes the intended-class restriction itself.

\begin{figure}[H]
\centering
\resizebox{0.94\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=4pt},
 dim6/.style={fill=blue!10, draw=blue!55!black, line width=0.67pt,
 text width=44mm, minimum height=17mm},
 dim7/.style={fill=orange!14, draw=orange!60!black, line width=0.67pt,
 text width=44mm, minimum height=17mm},
 dim8/.style={fill=red!10, draw=red!55!black, line width=0.67pt,
 text width=44mm, minimum height=17mm},
 fact/.style={fill=green!10, draw=olive!55!black, line width=0.58pt,
 text width=52mm, minimum height=12mm, font=\sffamily\scriptsize},
 arrow/.style={-{Stealth[length=2.8mm]}, line width=0.77pt, draw=black!70},
 lab/.style={font=\scriptsize\itshape, fill=white, inner sep=1.5pt},
]
\node[dim6] (D6) at (0, 0)
 {\textbf{6D theory on $\Msix$} \\[1pt]
 allowed large transformation $g$ \\[-1pt]
 {\scriptsize (isotopic to identity; clause C6)}};
\node[dim7] (D7) at (6.2, 0)
 {\textbf{7D mapping torus} \\[1pt]
 $\Useven \cong X_6 \times S^1$ \\[-1pt]
 {\scriptsize holonomy of the anomaly line}};
\node[dim8] (D8) at (12.4, 0)
 {\textbf{8D filling} \\[1pt]
 $\Weight = X_6 \times D^2$ \\[-1pt]
 {\scriptsize APS / Dai--Freed evaluation}};
\draw[arrow] (D6.east) -- (D7.west) node[lab, midway, above=0.4mm] {clutch by $g$};
\draw[arrow] (D7.east) -- (D8.west) node[lab, midway, above=0.4mm] {bound (H1)};
\node[fact] (F1) at (2.7, -3.1)
 {\textbf{Fermions and GS sector:} the combined anomaly line is
 \emph{flat} (local theorem) and the loop of $g$ is \emph{contractible}
 (C6), so the holonomy is $1$; equivalently, per species,
 $\int_{W_8} I_8 - \xi = \mathrm{index}(D_{W_8}) \in \mathbb{Z}$};
\node[fact] (F2) at (9.7, -3.1)
 {\textbf{Chiral tensors:} $\sigma(X_6 \times D^2) = 0$ (pullback
 classes have vanishing intersection form), so the signature-type
 phase is trivial at \emph{every} $\Omega$ --- including
 $\Omega = 25$ ($n_{\mathrm{SD}} = 100$)};
\draw[arrow, dashed] (D7.south) -- (F1.north);
\draw[arrow, dashed] (D8.south) -- (F2.north);
\node[anchor=north, font=\small\bfseries] at (6.2, -4.6)
 {$\mathcal{A}_{\mathrm{tot}}(\Useven) = 1$ exactly, on the intended class (Lemma~\ref{lem:computed-phase})};
\end{tikzpicture}%
}
\bigskip
\caption[The global-anomaly dimensional ladder on the intended class.]{The
global-anomaly dimensional ladder $6\mathrm{D} \to 7\mathrm{D} \to
8\mathrm{D}$ and the two facts that close it on the intended \GG{}
class. An allowed large transformation $g$ (isotopic to the identity by
clause (C6)) clutches the 7D mapping torus $\Useven \cong X_6 \times
S^1$, which bounds the product filling $\Weight = X_6 \times D^2$ with
all data extending (H1). For the fermion and Green--Schwarz sectors the
combined anomaly line is flat by the local theorem, and the loop is
contractible, so its holonomy is exactly $1$ --- equivalently, each
species' Dai--Freed factor is $\exp(2\pi i \cdot \mathrm{index}) = 1$
by the Atiyah--Patodi--Singer index theorem \citep{APS1975}. For the chiral-tensor
sector, $\sigma(X_6 \times D^2) = 0$, so the signature-type phase is
trivial at every $\Omega$. Together these give
Lemma~\ref{lem:computed-phase}: the global phase on the intended class
is computed, not merely constrained.}
\label{fig:global-ladder}
\end{figure}

The theorem is complete on the physical intended \GG{} class used by the series. What is not claimed is a broader theorem for every arbitrarily enlarged non-supersymmetric six-dimensional category with extra monodromies, extra gauge sectors, or unrestricted topology. That broader universalization would require a separate argument.

\begin{remark}[Scope of the admissible class]
\label{rem:admissible-scope}
We state explicitly what the admissible class \(\AdmissibleClass\) does and does not test. Clauses \textup{(C5)--(C8)} exclude non-trivial torus monodromies, non-simply-connected bases, non-pullback bundle topologies, and large gauge clutching on the structural directions. On the remaining class, the mapping tori are relatively simple---they factorize as products, the gauge clutching is null-homotopic by construction, and the only nontrivial check is the Wu-shift integrality, which follows from a standard formula on spin eight-manifolds.

It is worth naming the hardest cases in the modern global-anomaly literature, because they are the ones a reader will ask about: backgrounds with non-trivial torsion in \(H^3\), exotic gauge-bundle topology over non-simply-connected manifolds, or complicated clutching producing topologically non-trivial mapping tori. Theorem~\ref{thm:unrestricted-global} has since removed that concession for the \GG{} gauge content: those cases are covered, because the relevant spin bordism groups vanish (Lemma~\ref{lem:bordism-vanishing}) and genuinely non-bordant mapping tori do not exist for this gauge group. A specialist in six-dimensional global anomalies will still correctly observe that the vanishing is a special property of the \GG{} gauge content --- an arbitrary gauge packet would not enjoy it --- and that observation is now the precise sense in which the intended class is favorable.

The response from the axiomatic perspective is that this restriction is a \emph{consequence of the physical design}, not an ad hoc retreat. The same clauses that simplify the global anomaly problem---the product structure, the fixed gauge content, the single primitive tensor, the absence of extra topological sectors---are the clauses that make \GG{} finite, predictive, and falsifiable. A theory that required arbitrary base topology and arbitrary bundle classes would also require arbitrary parameters and an uncontrolled landscape \citep{BoussoPolchinski2000}, which is precisely what the constitution (\AX{4}/\POS{6}) forbids. In this sense, the restrictiveness of \(\AdmissibleClass\) is a feature that delivers phenomenological predictivity --- but, after Theorem~\ref{thm:unrestricted-global}, it is no longer what delivers global consistency.

The broader question---whether a theory with the \BIsix{} gauge content and coefficient vector but \emph{unrestricted} topology remains globally consistent---was explicitly left open in the conditional form of the theorem, and is now answered affirmatively for the \GG{} gauge content in Subsection~\ref{ssec:unrestricted-global}: the relevant spin bordism groups of \(BG\) vanish, every closed spin seven-datum bounds, and the phase evaluates to one. What the \GG{} chain does not claim, and what remains a well-posed invitation to the community, is the same analysis for arbitrary gauge packets and multiplet structures in the general non-SUSY setting of Monnier--Moore \citep{MonnierMoore2019} and Dierigl--Tartaglia \citep{DieriglTartaglia2025}.
\end{remark}



\subsection{The sphere-background case, as a corollary}

\begin{proposition}[Absence of Witten-type global anomalies]
For the \GG{} bulk gauge group
\(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\),
\begin{equation}
\pi_6(\SU(5)\times\U(1)_B\times\U(1)_{Qi}) = 0.
\end{equation}
For the Euclideanized local Lorentz group \(\Spin(6)\simeq\SU(4)\),
\begin{equation}
\pi_6(\Spin(6)) = 0.
\end{equation}
The spin-bordism group \citep{Stong1968} satisfies
\(\Omega_7^{\Spin}(\mathrm{pt}) = 0\).
Therefore the \GG{} bulk has no six-dimensional Witten-type global
gauge or Lorentz anomaly. This proposition is the sphere-background special case of a stronger statement: the full spin bordism computation of Lemma~\ref{lem:bordism-vanishing} (Subsection~\ref{ssec:unrestricted-global}) shows the same conclusion over \emph{every} closed spin base, not only \(S^6\).
\end{proposition}

\subsection{Unrestricted global anomaly freedom: the theorem without the topological fine print}
\label{ssec:unrestricted-global}

\paragraph{What this subsection adds.} The theorem just proved is unconditional on the
class \(\AdmissibleClass\), and \(\AdmissibleClass\) is restrictive. Clause (C7) demands
\(H^1(\Mfour;\mathbb Z) = H^3(\Mfour;\mathbb Z) = 0\); clauses (C6) and (C8) demand that
the large gauge transformations checked be null-homotopic, with no Abelian winding and no
mixed clutching. Those clauses have earned their keep --- they made the bounding
construction of Appendix~\ref{app:global-proof-dossier} explicit --- but they also left a standing
question, named in this paper as Open Problem O10: does the consistency survive when the
fine print is removed? A hostile reader is entitled to suspect that the restrictions are
where the difficulty was hidden. The present subsection answers the question. The
restrictions can be deleted. The global anomaly vanishes for \emph{every} large gauge
transformation of the \GG{} bulk, of arbitrary topological type, on \emph{every} compact
spin base --- and the proof is short enough, and its ingredients pretty enough, that we
give every step.

\paragraph{What a global anomaly is, and how the proof so far handled it.} An
infinitesimal gauge variation is a small deformation; Section~\ref{sec:structural-proof}
showed that the phase it produces is canceled exactly by the Green--Schwarz counterterm.
A \emph{large} gauge transformation is a loop: one walks the background around a closed
path in configuration space and asks whether the quantum theory comes back to itself with
phase one. The standard technology evaluates that phase on the \emph{mapping torus}
\(U_7\) --- the seven-dimensional manifold obtained by gluing the ends of
(background)\(\times [0,1]\) with the transformation --- and, when \(U_7\) together with
all its data bounds an eight-manifold \(W_8\), the Atiyah--Patodi--Singer machinery
converts the phase into quantities computed on \(W_8\). The theorem above walked exactly
this path, and clause (C7) entered at a single step: the extension of the gauge data over
the filling. The bounding lemma of Appendix~\ref{app:global-proof-dossier} argues that every
allowed gauge transformation on the base is \emph{null-homotopic},
\[
[\Mfour, U(1)] \cong H^1(\Mfour;\mathbb Z) = 0,
\qquad
[\Mfour, \SUfive] \cong H^3(\Mfour;\mathbb Z) = 0 ,
\]
so the clutching is trivial and the bundle extends over the product filling
\(\Mfour\times D^2\times \Ttwo\). That is clause (C7) at work: it makes the
transformations tame enough that the extension problem never arises.

\paragraph{The reframing: from null-homotopy to bordism.} Here is the observation that
opens the problem. The physics never needed the extension to be \emph{trivial}; it needed
\emph{a filling} --- any filling --- on which the phase evaluates to an integer. Whether
the specific product manifold \(\Mfour\times D^2\times\Ttwo\) carries an extension is a
question in obstruction theory, and clause (C7) was the answer to that question. Whether
\emph{some} spin eight-manifold carries an extension is a different and far more
structural question: it asks whether the pair (mapping torus, gauge bundle) represents
zero in the spin bordism group of the classifying space,
\[
\bigl[\,U_7,\;P\,\bigr] \;\in\; \Omega^{\mathrm{spin}}_7\!\bigl(BG\bigr),
\qquad
G = \SUfive\times\UoneB\times\UoneQi .
\]
If that group vanishes, then every closed spin seven-manifold carrying a \(G\)-bundle
bounds a spin eight-manifold over which the bundle extends --- \emph{by the definition of
the bordism relation}, with no hypothesis on the topology of the base and no hypothesis on
the homotopy class of the transformation. The null-homotopy clause would then stand
revealed as a crutch of the construction, not a requirement of the physics. So the task
divides in three: compute the group (Lemma~\ref{lem:bordism-vanishing}); deal with the one
genuinely new danger that abstract fillings introduce, which is their signature
(Lemma~\ref{lem:sigma-normalization}); and re-run the four-step evaluation of
Theorem~\ref{thm:bi6-global-complete} on the new fillings
(Theorem~\ref{thm:unrestricted-global}).

\paragraph{The imported toolbox, stated honestly.} Three standard facts are used, each
with a one-line statement of what it is doing for us.
\begin{itemize}[leftmargin=2em]
\item \emph{The spin bordism coefficients.} \(\Omega^{\mathrm{spin}}_q(\mathrm{pt})\) for
  \(q = 0,\dots,7\) is
  \(\mathbb Z,\ \mathbb Z_2,\ \mathbb Z_2,\ 0,\ \mathbb Z,\ 0,\ 0,\ 0\)
  (Milnor; Anderson--Brown--Peterson). The two \(\mathbb Z_2\) entries in degrees one and
  two are the eta-invariant layers --- the circle with its periodic spin structure and its
  square --- and they are the only rooms in which a torsion global anomaly could live.
\item \emph{The Atiyah--Hirzebruch spectral sequence.}
  \(E^2_{p,q} = H_p(BG;\Omega^{\mathrm{spin}}_q) \Rightarrow \Omega^{\mathrm{spin}}_{p+q}(BG)\):
  the bordism of a space is assembled from its ordinary homology with bordism
  coefficients, corrected by differentials.
\item \emph{The first differential is a Steenrod square.} In degrees \(\le 7\) spin
  bordism agrees with connective real K-homology, whose first Postnikov invariant is
  \(\mathrm{Sq}^2\); consequently the \(d_2\) differentials off the \(q=0\) and \(q=1\)
  rows are (mod-2 reduction followed by) the dual of
  \(\mathrm{Sq}^2\). This is the single deep input, and it is what connects the anomaly
  question to the cohomology operations computed next.
\end{itemize}

\paragraph{Step 1: the Wu formula, derived from scratch.} We need the action of
\(\mathrm{Sq}^2\) on Chern classes mod~2, and rather than cite it we derive it, because
one coefficient in it is a trap. Work by the splitting principle: pull back to a space
where the rank-five bundle splits into line bundles with first Chern classes
\(t_1,\dots,t_5\) (degree two each), so that \(c_k = \sigma_k(t_1,\dots,t_5)\), the
elementary symmetric polynomial. On a degree-two line class the total Steenrod square is
\begin{equation}
\mathrm{Sq}(t) \;=\; t + t^2
\qquad\text{(nothing above the top square),}
\label{eq:sq-line}
\end{equation}
and the total square is multiplicative, so on a monomial
\(\mathrm{Sq}\bigl(\textstyle\prod_i t_i^{a_i}\bigr) = \prod_i (t_i + t_i^2)^{a_i}\).
Extracting the degree-\((2k+2)\) part of \(\mathrm{Sq}(x^k)\) for a single line class
gives the binomial coefficient \(\binom{k}{1}\):
\begin{equation}
\mathrm{Sq}^2(x^k) \;=\; k\,x^{k+1} \pmod 2 ,
\label{eq:sq-power}
\end{equation}
so in particular \(\mathrm{Sq}^2(x^2) = 0\) and \(\mathrm{Sq}^2(x^3) = x^4\): squares are
inert, cubes are not. For \(\sigma_2 = \sum_{i<j} t_i t_j\), the Cartan formula on each
monomial gives \(\mathrm{Sq}^2(t_it_j) = t_i^2 t_j + t_i t_j^2\), hence
\(\mathrm{Sq}^2\sigma_2 = \sum_{i<j}(t_i^2 t_j + t_i t_j^2)\). Now compare with the
product \(\sigma_1\sigma_2\): multiplying out,
\[
\sigma_1\sigma_2
= \Bigl(\sum_i t_i\Bigr)\Bigl(\sum_{j<k} t_j t_k\Bigr)
= \sum_{i<j}\bigl(t_i^2 t_j + t_i t_j^2\bigr) \;+\; 3\,\sigma_3 ,
\]
because each square-free product \(t_it_jt_k\) arises from three choices of which factor
came from \(\sigma_1\). Rearranging mod 2,
\begin{equation}
\mathrm{Sq}^2(c_2) \;=\; c_1 c_2 + 3c_3 \;=\; c_1 c_2 + c_3 \pmod 2 .
\label{eq:wu-sigma2}
\end{equation}
The same computation one degree up: \(\mathrm{Sq}^2(t_it_jt_k) =
t_i^2t_jt_k + t_it_j^2t_k + t_it_jt_k^2\), and
\(\sigma_1\sigma_3 = \sum_{i<j<k}(t_i^2t_jt_k + t_it_j^2t_k + t_it_jt_k^2) + 4\sigma_4\),
whence
\begin{equation}
\mathrm{Sq}^2(c_3) \;=\; c_1 c_3 + 4c_4 \;=\; c_1 c_3 \pmod 2 .
\label{eq:wu-sigma3}
\end{equation}
Here is the trap the derivation exists to avoid: the naive guess
``\(\mathrm{Sq}^2 c_k = c_{k+1}\) at \(c_1 = 0\)'' is \emph{false} for \(k = 3\) --- the
coefficient of \(c_4\) is four, which is zero mod 2. In the special unitary case
(\(c_1 = 0\)) the upshot is the asymmetric pair
\begin{equation}
\boxed{\;\mathrm{Sq}^2 c_2 = c_3, \qquad \mathrm{Sq}^2 c_3 = 0\;}
\qquad\text{in } H^*(B\SUfive;\mathbb F_2).
\label{eq:wu-su}
\end{equation}
The first of these identities will turn out to carry the entire \(\SUfive\) side of the
global anomaly question.

\paragraph{Step 2: the mod-2 cohomology of \(BG\) and the derivation property.} The
classifying space of the \GG{} gauge group has
\[
H^*\!\bigl(BG;\mathbb F_2\bigr)
= \mathbb F_2\bigl[c_2, c_3, c_4, c_5,\; x_B,\; x_{Qi}\bigr],
\qquad \deg c_k = 2k,\quad \deg x = 2 ,
\]
with \(x_B, x_{Qi}\) the first Chern classes of the two Abelian factors. Two structural
facts do most of the work below. First, \emph{there is no odd-degree cohomology at all}:
every class is a polynomial in even generators. Second, the integral cohomology is
torsion-free, so \(\mathrm{Sq}^1\) (the Bockstein of the mod-2 reduction) vanishes on
every class in sight, and the Cartan formula
\(\mathrm{Sq}^2(uv) = \mathrm{Sq}^2u\cdot v + \mathrm{Sq}^1u\,\mathrm{Sq}^1v +
u\cdot\mathrm{Sq}^2v\) loses its middle term: \(\mathrm{Sq}^2\) acts as a
\emph{derivation}. With the generator values \eqref{eq:wu-su} and
\(\mathrm{Sq}^2 x = x^2\), the complete action on the basis elements we need is
\begin{equation}
\begin{aligned}
&\mathrm{Sq}^2 x_i = x_i^2, \qquad
 \mathrm{Sq}^2(x_i^2) = 0, \qquad
 \mathrm{Sq}^2(x_1x_2) = x_1^2x_2 + x_1x_2^2, \\
&\mathrm{Sq}^2 c_2 = c_3, \qquad
 \mathrm{Sq}^2 c_3 = 0, \qquad
 \mathrm{Sq}^2(c_2 x_i) = c_3 x_i + c_2 x_i^2, \\
&\mathrm{Sq}^2(x_i^3) = x_i^4, \qquad
 \mathrm{Sq}^2(x_1^2 x_2) = x_1^2 x_2^2 = \mathrm{Sq}^2(x_1 x_2^2),
\end{aligned}
\label{eq:sq2-table}
\end{equation}
where \(x_1, x_2\) abbreviate \(x_B, x_{Qi}\). Every line of \eqref{eq:sq2-table} is
either a generator value or two applications of the Leibniz rule; every line has been
verified independently by the splitting-principle computation.

\begin{lemma}[Vanishing of the spin bordism of the \GG{} gauge group]
\label{lem:bordism-vanishing}
For \(G = \SUfive\times\UoneB\times\UoneQi\),
\[
\widetilde\Omega^{\mathrm{spin}}_5(BG) \;=\; 0,
\qquad
\widetilde\Omega^{\mathrm{spin}}_7(BG) \;=\; 0 .
\]
Together with \(\Omega^{\mathrm{spin}}_5(\mathrm{pt}) = \Omega^{\mathrm{spin}}_7(\mathrm{pt}) = 0\):
every closed spin five- or seven-dimensional manifold equipped with a \(G\)-bundle bounds
a compact spin manifold over which the bundle extends.
\end{lemma}

\begin{proof}
Run the Atiyah--Hirzebruch spectral sequence. Because \(BG\) has only even cells, every
entry \(E^2_{p,q}\) with \(p\) odd vanishes; because
\(\Omega^{\mathrm{spin}}_q = 0\) for \(q = 3, 5, 6, 7\), most even-\(p\) entries vanish
too. What survives in the two total degrees of interest is exactly one entry each:
\[
\text{degree } 5:\quad E^2_{4,1} = H_4(BG;\mathbb F_2) \cong \mathbb F_2^{\,4},
\qquad\qquad
\text{degree } 7:\quad E^2_{6,1} = H_6(BG;\mathbb F_2) \cong \mathbb F_2^{\,7},
\]
with monomial bases (written as cohomology, and read dually)
\(\{c_2,\,x_1^2,\,x_1x_2,\,x_2^2\}\) in degree four and
\(\{c_3,\,c_2x_1,\,c_2x_2,\,x_1^3,\,x_1^2x_2,\,x_1x_2^2,\,x_2^3\}\) in degree six. Both
sit on the \(q=1\) row --- the first eta layer --- so both are subject to the
\(d_2\)-differentials dual to \(\mathrm{Sq}^2\), leaving the group and entering it:
\[
E^2_{p+2,\,0} \xrightarrow{\;d_2\;} E^2_{p,\,1} \xrightarrow{\;d_2\;} E^2_{p-2,\,2}.
\]
We compute both, in both degrees, from the table \eqref{eq:sq2-table}.

\emph{Total degree five.} The outgoing differential is dual to
\(\mathrm{Sq}^2: H^2 \to H^4\), which sends \(x_i \mapsto x_i^2\). Dually, the classes
\(x_1^2\) and \(x_2^2\) are mapped onto \(x_1\) and \(x_2\), while \(c_2\) and \(x_1x_2\)
are annihilated (nothing in \(H^2\) hits them):
\[
\ker\bigl(d_2^{\mathrm{out}}\bigr)
= \operatorname{span}\bigl\{\,c_2,\; x_1x_2\,\bigr\}^{\!*} \quad\text{(dimension two).}
\]
The incoming differential from \(E^2_{6,0}\) is dual to
\(\mathrm{Sq}^2: H^4 \to H^6\). From \eqref{eq:sq2-table},
\(\mathrm{Sq}^2 c_2 = c_3\) and \(\mathrm{Sq}^2(x_1x_2) = x_1^2x_2 + x_1x_2^2\); dually,
the degree-six classes \(c_3\) and \(x_1^2x_2 + x_1x_2^2\) map precisely onto \(c_2\) and
\(x_1x_2\). (The mod-2 reduction \(H_6(BG;\mathbb Z)\to H_6(BG;\mathbb F_2)\) is onto,
the integral cohomology being torsion-free.) So
\[
\operatorname{im}\bigl(d_2^{\mathrm{in}}\bigr)
= \operatorname{span}\bigl\{\,c_2,\; x_1x_2\,\bigr\}^{\!*}
= \ker\bigl(d_2^{\mathrm{out}}\bigr),
\]
kernel equals image, both of dimension two, and \(E^3_{4,1} = 0\).

\emph{Total degree seven.} The outgoing differential is dual to
\(\mathrm{Sq}^2: H^4 \to H^6\), just computed: it kills the dual of \(c_3\) (because
\(\mathrm{Sq}^2 c_2 = c_3\)) and identifies the duals of \(x_1^2x_2\) and \(x_1x_2^2\)
pairwise (because \(\mathrm{Sq}^2(x_1x_2)\) is their sum), so
\[
\ker\bigl(d_2^{\mathrm{out}}\bigr)
= \operatorname{span}\bigl\{\,c_2x_1,\; c_2x_2,\; x_1^3,\; x_2^3,\;
x_1^2x_2 + x_1x_2^2\,\bigr\}^{\!*}
\quad\text{(dimension five).}
\]
The incoming differential from \(E^2_{8,0}\) is dual to
\(\mathrm{Sq}^2: H^6 \to H^8\). Read the last two lines of \eqref{eq:sq2-table}: the five
degree-eight classes \(c_3x_1,\; c_3x_2,\; x_1^4,\; x_2^4,\; x_1^2x_2^2\) pull back,
dually, onto exactly the five kernel elements ---
\(c_3x_i\) hits \(c_2x_i\) (from \(\mathrm{Sq}^2(c_2x_i) = c_3x_i + c_2x_i^2\)),
\(x_i^4\) hits \(x_i^3\) (from \(\mathrm{Sq}^2 x_i^3 = x_i^4\)), and \(x_1^2x_2^2\) hits
the sum \(x_1^2x_2 + x_1x_2^2\) (both preimages map to it). Note what does \emph{not}
happen: \(\mathrm{Sq}^2 c_3 = 0\), so nothing arrives at the dual of \(c_3\) from above
--- but nothing needs to, because that class already died leaving. So again
\[
\operatorname{im}\bigl(d_2^{\mathrm{in}}\bigr) = \ker\bigl(d_2^{\mathrm{out}}\bigr),
\]
kernel equals image, both of dimension five, and \(E^3_{6,1} = 0\). In both total degrees
every entry is now zero, so no higher differential has a source or a target, no extension
problem arises, and both reduced bordism groups vanish.
\end{proof}

\begin{remark}[The two mechanisms, seen separately]
It is instructive to run the two factors of the gauge group alone. For \(\SUfive\) by
itself the single degree-seven candidate is the dual of \(c_3\), and it dies
\emph{leaving}: the outgoing differential is injective because \(\mathrm{Sq}^2 c_2 = c_3\).
For a single \(U(1)\) the candidate is the dual of \(x^3\), and it dies \emph{arriving}:
the incoming differential is surjective because \(\mathrm{Sq}^2 x^3 = x^4\). Two different
doors, one conclusion. The proposition of
Section~\ref{sec:structural-proof} on Witten-type anomalies ---
\(\pi_6(G) = 0\) --- is the sphere-background special case of
Lemma~\ref{lem:bordism-vanishing}: the homotopy groups see the anomaly candidates over
\(S^6\), the bordism group sees them over every closed spin base at once.
\end{remark}

\begin{lemma}[Signature normalization of fillings]
\label{lem:sigma-normalization}
If a closed spin seven-manifold with \(G\)-bundle \((U_7, P)\) bounds \((W_8, \tilde P)\),
then it bounds a spin eight-manifold of \emph{signature zero} carrying the same boundary
data.
\end{lemma}

\begin{proof}
The quaternionic projective plane \(\mathbb{HP}^2\) is 3-connected, hence spin, and has
signature one (its middle cohomology is one-dimensional, generated by \(u\) with
\(\int u^2 = 1\)). Form the interior connected sum
\(W' = W \,\#\, k\,(\mp\mathbb{HP}^2)\) with \(k = |\sigma(W)|\) and orientations chosen
to cancel: the sum is taken along balls in the interior, so the boundary is untouched; the
spin structures glue because both sides are spin and the gluing sphere is
simply-connected; and the \(G\)-bundle extends trivially over each punctured
\(\mathbb{HP}^2\) summand because a bundle over a ball is trivial. Then
\(\sigma(W') = \sigma(W) \mp k = 0\) by Novikov additivity, and
\(\partial W' = U_7\) with unchanged data.
\end{proof}

\paragraph{Why the signature needed a lemma at all.} On the product fillings of
Theorem~\ref{thm:bi6-global-complete} the relative intersection form vanishes identically
and the self-dual tensor sector's signature-type phase
\(\exp\!\bigl(2\pi i\, n_{\mathrm{SD}}\,\sigma(W)/8\bigr)\) never had to be confronted. On
an abstract filling it would be --- and at \(n_{\mathrm{SD}} = 4\Omega = 100\) the
exponent is \(2\pi i \cdot 12.5\,\sigma\), genuinely fractional for odd \(\sigma\). The
lemma says the danger is illusory because \emph{signature is purchasable}: a closed spin
eight-manifold of signature one exists, so the signature of a filling can always be paid
down to zero, at zero cost to anything the boundary sees. The product structure of the
original fillings was quietly doing this job; surgery does it in general.

\begin{theorem}[\textbf{T-BI6-7.6}: Unrestricted global anomaly freedom]
\label{thm:unrestricted-global}
Let \(\AdmissibleClass^{+}\) denote the class obtained from
Definition~\ref{def:admissible-main} by deleting clause (C7) and deleting the
topological restrictions on large transformations from clauses (C6) and (C8) ---
retaining only structure preservation (transformations preserve the spin structure and
the stage data). Then for every background in \(\AdmissibleClass^{+}\) and every large
gauge transformation of arbitrary topological type --- \(\SUfive\) clutching classes
\(\delta(g)\in H^3(\Mfour;\mathbb Z)\), Abelian winding classes in
\(H^1(\Mfour;\mathbb Z)\), mixed clutching in
\(H^2(\Mfour;\mathbb Z)\otimes H^1(\Ttwo;\mathbb Z)\) --- the combined phase of the
chiral determinant and the Wu--Chern--Simons-refined \BIsix{} counterterm around the
mapping torus is trivial. More generally, the phase is trivial on every closed spin
seven-manifold equipped with a \(G\)-bundle, within the differential-cohomology
accounting of the present Part. The statement holds at every point \(\gamma = s/t\) of
the coefficient family of the \BIsix{} clause.
\end{theorem}

\begin{proof}
Four steps, of which only the first is new; the remaining three are the machinery already
established, now applied without restriction.

\emph{Step 1 (bounding).} The mapping-torus datum \((U_7, P)\) is a closed spin
seven-manifold with \(G\)-bundle --- whatever the topological type of the transformation
that produced it. By Lemma~\ref{lem:bordism-vanishing} it bounds \((W_8, \tilde P)\); the
connections and the differential cocycle \(\check X\) extend over \(W_8\) exactly as in
the bounding lemma of Appendix~\ref{app:global-proof-dossier} (extend the connection arbitrarily;
build the refinement from the extended data).

\emph{Step 2 (normalization).} By Lemma~\ref{lem:sigma-normalization}, arrange
\(\sigma(W_8) = 0\).

\emph{Step 3 (evaluation).} Run the four-part accounting of
Theorem~\ref{thm:bi6-global-complete} on \(W_8\). The fermionic contributions are
Atiyah--Patodi--Singer indices of the extended operators --- integers. The signature-type
phase of the self-dual tensor sector is
\(\exp(2\pi i\, n_{\mathrm{SD}}\cdot 0/8) = 1\). The Wu--Chern--Simons term is an integer
by the quantization lemma, which holds on \emph{any} spin eight-manifold --- its own
statement requires nothing beyond spin plus integrality. The integral lift exists by the
coefficient-family lift lemma, which holds on any background. The local index densities
cancel pointwise against the counterterm curvature by Theorem~\ref{thm:model-level}. The
total exponent is an integer; the phase is one.

\emph{Step 4 (filling-independence).} Two signature-zero fillings \(W, W'\) glue along
\(U_7\) to a closed spin eight-manifold with \(\sigma = \sigma(W) - \sigma(W') = 0\) by
Novikov additivity, on which the same accounting yields an integer; so the value computed
in Step 3 does not depend on the choices made in Steps 1 and 2.
\end{proof}

\begin{remark}[What the theorem does not need]
\label{rem:not-needed}
It does not need \(\Mfour\) to have vanishing \(H^1\) or \(H^3\). It does not need the
gauge transformations to be null-homotopic, unwound, or unmixed. It does not need the
bundle to be pulled back from the base. And --- a bonus that fell out of the generality
--- it does not even need the transformation to be a gauge transformation: a large
diffeomorphism preserving the spin structure produces a mapping torus that is still a
closed spin seven-manifold with \(G\)-bundle, and Steps 1--4 never ask for more. The
clauses of Definition~\ref{def:admissible-main} that restricted the transformations
survive as a description of the physical intended backgrounds --- the stage on which the
cohort's phenomenology is computed --- but they no longer bear the weight of the
consistency theorem.
\end{remark}

\begin{remark}[Three observations on why it works]
\label{rem:global-physics}
\emph{First: the cubic Casimir does double duty.} Theorem~\ref{thm:single-tensor-necessity}
demanded, as a rational statement, that the cubic index of the charged matter vanish ---
the condition that fixed \(L = 5\) in Theorem~\ref{thm:L-from-GS}. The bordism computation
now meets the same class one level down: as a mod-2 class, \(c_3\) \emph{is}
\(\mathrm{Sq}^2 c_2\), and it is precisely this identity that makes the outgoing
differential injective and kills the last candidate \(\mathbb Z_2\) global anomaly of the
\(\SUfive\) sector. The cubic anomaly's rational part selects the group; its mod-2 shadow
certifies the group globally safe. We record this as a resonance, not a mechanism --- but
it is the kind of resonance that suggests the bookkeeping is being kept in the right
variables. \emph{Second: a gauge group with an even classifying space cannot hide a global
anomaly for long.} Torsion global anomalies in seven dimensions must live on odd-degree
bordism classes; \(BG\) has no odd cells, so the only candidates enter through the two
eta layers of the spin coefficients, and those are exactly the layers the first
k-invariant of real K-theory --- \(\mathrm{Sq}^2\) --- can reach. In the \GG{} case it
reaches all of them. The constitution selected a gauge group whose topology leaves the
anomaly nowhere to sit. \emph{Third: signature is purchasable.} The one quantity that
distinguished the tame fillings from the wild ones can be adjusted by surgery at no
physical cost. What looked like a wall was a ledger entry.
\end{remark}

\begin{remark}[Scope, stated at the same standard as the rest of the Part]
\label{rem:global-scope}
Theorem~\ref{thm:unrestricted-global} is proved within the differential-cohomology phase
accounting that this Part uses throughout --- the same standard to which
Theorem~\ref{thm:bi6-global-complete} itself is proved on the intended class. The
foundation of that accounting for the non-supersymmetric self-dual sector is supplied in
Subsection~\ref{ssec:sd-partition}: the defining quadratic refinement is proved unique
(Theorem~\ref{thm:refinement-unique}) and the partition function is constructed
(Theorem~\ref{thm:sd-partition}), so what formerly stood beneath both theorems as Open
Problem O9 now stands as a theorem beneath them. Within the accounting, Open Problem O10 --- the
extension of the global theorem beyond the intended class --- is discharged for the \GG{}
gauge content: what remains genuinely open in O10's original wording is the case of
\emph{arbitrary} gauge packets and tensor lattices, which is the six-dimensional anomaly
community's general classification program rather than a gap in the \GG{} chain.
\end{remark}


\subsection{The self-dual partition function: the refinement is unique, and the construction owes nothing to supersymmetry}
\label{ssec:sd-partition}

\paragraph{What is still missing.} Everything proved so far about the tensor sector
--- the Wu--Chern--Simons quantization, the signature accounting, the global theorem ---
concerns the \emph{anomaly} of the self-dual field: the seven-dimensional invertible
theory that governs its phases. Open Problem O9 has, until now, recorded what was missing
underneath: the six-dimensional partition function of the self-dual sector itself. That
object is not defined by a Lagrangian; in the framework this Part uses, it is defined
through a \emph{quadratic refinement} --- and the only explicit construction in the
literature, due to Monnier and Moore, is carried out for $\cN=(1,0)$ supergravity, whose
supersymmetry is entangled with the construction at several points. The worry is easy to
state and serious: if the non-supersymmetric \BIsix{} sector required a \emph{choice} of
refinement that nothing in the theory fixes, then a hidden discrete dial would sit
beneath the central consistency theorem of this paper. The present subsection shows the
worry is empty, in the strongest available sense: on the \BIsix{} tensor lattices the
refinement admits \emph{no choice at all}, every ingredient of the construction that
supersymmetry supplies in the $(1,0)$ case is supplied here by the constitution or by a
theorem already proved, and the partition function can then simply be written down.

\paragraph{Step 1: why the self-dual field has no Lagrangian.} It is worth deriving this
rather than asserting it, because it explains why the entire subsection must run through
period lattices rather than path integrals. Take the free action of an unconstrained
two-form, $S = \tfrac12\int_{\Msix} H_3\wedge *H_3$ with $H_3 = \dd B_2$. Its equation of
motion is $\dd * H_3 = 0$, which together with the Bianchi identity $\dd H_3 = 0$
describes a two-form with \emph{both} duality components propagating --- twice the
content wanted. Suppose instead one imposes self-duality $H_3 = *H_3$ directly in the
action. Then
\begin{equation}
S \;=\; \tfrac12\int_{\Msix} H_3\wedge H_3
\;=\; \tfrac12\int_{\Msix} \dd\bigl(B_2\wedge \dd B_2\bigr) \;=\; 0
\label{eq:sd-action-vanishes}
\end{equation}
on a closed manifold: the would-be action is a total derivative and vanishes
identically. Self-duality cannot be imposed by any local covariant action for $B_2$
alone; the constraint kills the very term that would enforce it. The self-dual field is,
in this precise sense, \emph{half} of an ordinary two-form --- the six-dimensional
sibling of the chiral boson in two dimensions --- and one must quantize the whole and
then take the half at the level of the partition function.

\paragraph{Step 2: the accepted definition --- holomorphic factorization.} The
construction, standard since the five-brane analyzes of the 1990s and formalized in the
Hopkins--Singer / Monnier--Moore framework, runs as follows. Quantize the
\emph{non-chiral} two-form, which does have a Lagrangian; its partition function
factorizes into a sum of products of holomorphic and antiholomorphic \emph{conformal
blocks}; the self-dual field's partition function is declared to be a single block. Where
do the blocks live? The fluxes of $H_3$ take values in the lattice
$H^3(\Msix;\mathbb Z)$, and the zero-mode sector of the path integral is a Gaussian sum
over this lattice --- a theta function on the \emph{intermediate Jacobian}
\begin{equation}
\cJ^3(\Msix) \;=\; H^3(\Msix;\mathbb R)\,/\,H^3(\Msix;\mathbb Z).
\label{eq:intermediate-jacobian}
\end{equation}
Two pieces of structure make this torus a natural home for theta functions, and both are
worth deriving on the spot. First, the \emph{complex structure}: in Euclidean signature
the Hodge star on $k$-forms in $n$ dimensions satisfies $*^2 = (-1)^{k(n-k)}$, and for
three-forms in six dimensions $k(n-k) = 9$, so
\begin{equation}
*^2 \;=\; -1 \qquad\text{on three-forms:}
\label{eq:star-squared}
\end{equation}
the metric's Hodge star \emph{is} a complex structure on $H^3(\Msix;\mathbb R)$. (This is
also why the natural language of the Euclidean self-dual field is complex; in Lorentzian
signature $*^2=+1$ and the chiral split is real.) Second, the \emph{polarization}: the
intersection pairing $\langle x, y\rangle = \int_{\Msix} x\wedge y$ on degree-three
classes is antisymmetric ($x\wedge y = (-1)^{3\cdot3}\,y\wedge x$) and unimodular by
Poincar\'e duality --- a symplectic lattice, which forces $b_3(\Msix)$ to be even. A
torus with a complex structure and a principal symplectic polarization is precisely a
principally polarized abelian variety, and its natural functions are theta functions.

\paragraph{Step 3: choosing a block is choosing a quadratic refinement.} A theta function
on a principally polarized torus is not unique: it carries a \emph{characteristic}, and
the choice of characteristic is exactly the choice of conformal block. The cleanest way
to see what that choice means physically is the rank-one computation. The classical
Jacobi function with even characteristic obeys
\begin{equation}
\theta_3(z+\pi\,\vert\,\tau) = \theta_3(z\,\vert\,\tau),
\qquad
\theta_3(z+\pi\tau\,\vert\,\tau) = e^{-i\pi\tau - 2iz}\,\theta_3(z\,\vert\,\tau) :
\label{eq:theta3-shifts}
\end{equation}
under a shift by a lattice vector the function returns to itself up to a multiplier
$e^{-i\pi\tau-2iz}$ whose exponent is \emph{quadratic} in the shift --- iterating the
second identity $m$ times produces $e^{-i\pi m^2\tau - 2imz}$. The multiplier system of a
theta function is the exponential of a quadratic form on the lattice, and consistency of
the multipliers under composition of shifts is precisely the condition
\begin{equation}
q(x+y) - q(x) - q(y) \;=\; \langle x, y\rangle \pmod 1 :
\label{eq:refinement-property}
\end{equation}
$q$ is a \emph{quadratic refinement} of the pairing. The characteristic of the theta
function and the refinement are two names for the same discrete datum. By contrast the
odd-characteristic function $\theta_1$ satisfies $\theta_1(z+\pi) = -\theta_1(z)$: a
block with the wrong characteristic \emph{changes sign} under a lattice shift. Hold that
sign; it returns in Step 5 as the executioner of the wrong choices.

\paragraph{Step 4: the refinements the Wu lemma admits, and their uniqueness.} On the
Green--Schwarz side the refinement appears in the form this Part has used all along:
\begin{equation}
q_c(x) \;=\; \tfrac12\bigl(x\cdot x - c\cdot x\bigr),
\qquad x \in \Lambda,
\label{eq:wcs-refinement}
\end{equation}
on the tensor charge lattice $\Lambda$, where $c$ is a \emph{characteristic element}:
\begin{equation}
x\cdot x \;\equiv\; c\cdot x \pmod 2 \quad\text{for all } x\in\Lambda .
\label{eq:characteristic-condition}
\end{equation}
Condition \eqref{eq:characteristic-condition} is exactly what makes $q_c$ integer-valued,
which is what the Wu--Chern--Simons quantization lemma consumes; it is the lattice-side
twin of the spacetime statement that $\lambda = p_1/2$ is a characteristic element of the
intersection form on spin eight-manifolds. So the question ``which refinement?''
becomes: \emph{how many characteristic elements does the \BIsix{} lattice have, up to
equivalence?}

\begin{theorem}[\textbf{T-BI6-7.7}: Uniqueness of the Wu--Chern--Simons refinement]
\label{thm:refinement-unique}
Let $\Lambda$ be any unimodular integral lattice. Then the characteristic elements of
$\Lambda$ form a single coset of $2\Lambda$, and any two characteristic elements define
the \emph{same} quadratic refinement \eqref{eq:wcs-refinement} modulo 1. In particular,
for the \BIsix{} tensor sector --- the rank-one mediator lattice $\Lambda\cong\mathbb Z$
of clause (C3), on which the characteristic elements are exactly the odd integers and
$a=1$ is the minimal one, together with the diagonal unimodular spectator lattice
$\mathbb Z^{4\Omega-1}$, on which they are exactly the all-odd vectors --- the
Wu--Chern--Simons refinement is \textbf{unique}. There is no discrete choice to make.
\end{theorem}

\begin{proof}
\emph{Single coset.} Let $c, c'$ both satisfy \eqref{eq:characteristic-condition}. Then
$(c-c')\cdot x \equiv 0 \pmod 2$ for every $x$, so $\tfrac12(c-c')$ pairs integrally with
all of $\Lambda$, i.e.\ $\tfrac12(c-c') \in \Lambda^{*} = \Lambda$ by unimodularity.
Hence $c - c' \in 2\Lambda$.

\emph{Same refinement.} Write $c' = c + 2u$ with $u \in \Lambda$. Then
\begin{equation}
q_{c'}(x) \;=\; \tfrac12\bigl(x\cdot x - c\cdot x\bigr) - u\cdot x
\;=\; q_c(x) - u\cdot x \;\equiv\; q_c(x) \pmod 1 ,
\label{eq:coset-shift}
\end{equation}
because $u\cdot x$ is an integer. The refinements coincide modulo 1, which is the only
arithmetic the theta multiplier ever sees.

\emph{The \BIsix{} lattices.} On $\Lambda = \mathbb Z$ with $x\cdot y = xy$, condition
\eqref{eq:characteristic-condition} reads $x^2 \equiv cx \pmod 2$ for all $x$; taking
$x=1$ forces $c$ odd, and $c$ odd suffices since $x^2 \equiv x \pmod 2$ identically. The
odd integers are the coset $1 + 2\mathbb Z$, with $a = 1$ its minimal positive element
--- the very value clause (C3) carries. On the spectator lattice the same argument runs
componentwise: characteristic vectors are the all-odd vectors, the coset
$(1,\dots,1) + 2\mathbb Z^{4\Omega-1}$.
\end{proof}

\begin{remark}[One refinement, three appearances]
\label{rem:one-refinement}
The minimal characteristic element $a = 1$ fixed here is the same
integer quadratic form $q(x) = \tfrac12(x^2 - x) = x(x-1)/2$ that
prices the family cascade of \PaperPTwo{} \citep{ArisakaP2N3} --- its
toll ladder satisfies the refinement's defining functional equation
exactly --- and whose Hopkins--Singer pushforward that paper's Appendix~H identifies, on the benchmark background of the intended class of this section, with the boundary's Berry line bundle (the $\kappa$-bridge theorem). One refinement, seen three ways: the
tensor sector, the toll ledger, and the seam.
\end{remark}

\begin{remark}[What supersymmetry was actually doing, located exactly]
\label{rem:susy-role}
Theorem~\ref{thm:refinement-unique} holds for \emph{every} unimodular lattice --- also
for the signature-$(1,n_T)$ string-charge lattices of $(1,0)$ supergravity. Uniqueness of
the refinement was therefore never the supersymmetric input. What supersymmetry supplies
in the $(1,0)$ construction is something logically prior: the theorem (due to the
six-dimensional supergravity literature) that the \emph{gravitational anomaly vector}
$a$ is itself a characteristic element of $\Lambda$, i.e.\ that
$x\cdot x + a\cdot x \in 2\mathbb Z$ --- without which the Wu--Chern--Simons theory does
not exist at all. That is a genuine theorem there, derived from the supersymmetric
anomaly-factorization conditions. In the \BIsix{} setting the corresponding statement is
one line of arithmetic: $a = 1$ on $\Lambda = \mathbb Z$, and $1$ is characteristic
because $x^2 \equiv x \pmod 2$. The existence condition that costs supersymmetry a
theorem costs the present paper nothing --- clause (C3), written to normalize the
Green--Schwarz coupling, had already recorded the answer.
\end{remark}

\paragraph{Step 5: the residual twists, and their double execution.} One tier above the
lattice, two discrete deformations could still be \emph{added} as background data, and
honesty requires naming and killing both. A \emph{flux character} shifts the refinement
by a linear form, $q(x) \mapsto q(x) + u\cdot x$ with $u$ a fixed half-lattice or torsion
class --- a discrete theta angle. A \emph{Wu-structure twist} replaces the canonical
integral Wu lift $\lambda = p_1/2$ that the spin structure provides by another lift,
shifting the refinement by a mod-2 character. Both die twice over.

\begin{lemma}[Exclusion of refinement twists]
\label{lem:twist-exclusion}
On the \BIsix{} tensor lattices, every deformation of the refinement away from the
canonical class of Theorem~\ref{thm:refinement-unique} either fails integer-valuedness
--- and with it the Wu--Chern--Simons quantization lemma --- or renders the partition
function double-valued. Independently, any such deformation is unforced discrete
background structure excluded by \AX{4}~(\MSMF{}) and \POS{6}~(\OGN{}).
\end{lemma}

\begin{proof}
The minimal flux twist on the mediator replaces $q_1(x) = (x^2-x)/2$ by
$q'(x) = q_1(x) + \tfrac{x}{2} = (x^2 - 0\cdot x)/2$, whose ``characteristic'' is even
and therefore not characteristic: at $x = 1$, $q'(1) = \tfrac12 \notin \mathbb Z$, and
the integrality on which the quantization lemma rests is gone. At the level of the theta
block the same twist moves the characteristic from even to odd, and by the shift
behavior recorded in Step 3 the block then obeys $\theta_1(z+\pi) = -\theta_1(z)$: the
would-be partition function returns to \emph{minus} itself under a lattice
identification, i.e.\ it is not a function on the background space at all. A Wu twist
acts identically through its induced character. This is the structural execution.
The constitutional one is independent: a flux character or Wu twist is a new discrete
background field, forced by no clause, generated by no bulk field, and absent from the
\OGNfive{} ledger; \AX{4} excludes it as unforced structure exactly as it excludes
unforced tensor multiplication (Theorem~\ref{thm:rank-one}) and unforced Abelian
asymmetry ($c = d$), and \POS{6} excludes it as normalization freedom outside the
ledger.
\end{proof}

\paragraph{Step 6: the supersymmetry audit.} With the refinement question settled, what
remains of O9's worry is the suspicion that the $(1,0)$ construction uses supersymmetry
somewhere else essential. It does not, and the accounting is short enough to print. Table~\ref{tab:monnier-moore-ingredients} lists each
ingredient of the Monnier--Moore construction with what supplied it there and
what supplies it here.

\begingroup
\begin{table}[htbp]
\centering
\caption[Where each ingredient of the construction comes from, in both settings.]{Each ingredient of the Monnier--Moore construction, with what supplies it in $(1,0)$ supergravity and what supplies it here. The table answers one specific objection --- that the construction secretly uses supersymmetry somewhere --- and it answers it by enumeration rather than by assertion. Read the third column: every ingredient is supplied either by a named
constitutional clause or by a theorem of this paper, and none by
supersymmetry. The accounting is short enough to print in full, which is why it is printed rather than summarized.}
\label{tab:monnier-moore-ingredients}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.33\textwidth}p{0.33\textwidth}}
\toprule
\textbf{Ingredient} & \textbf{In $(1,0)$ supergravity} & \textbf{In \GG{}/\BIsix{}} \\
\midrule
Unimodular tensor lattice $\Lambda$ & string-charge completeness arguments &
clause (C3): $\Lambda \cong \mathbb Z$, rank one, primitive --- \AX{4}/\POS{6} \\
Characteristic property of the gravitational vector $a$ & a theorem of the
supersymmetric anomaly-factorization conditions & one line of arithmetic:
$a = 1$ odd on $\mathbb Z$ (Remark~\ref{rem:susy-role}) \\
Wu structure on spacetime & spin structure of the SUSY multiplet &
spin structure of clause (C1); canonical lift $\lambda = p_1/2$ \\
Factorized anomaly polynomial & fixed by the SUSY multiplet content &
Theorem~\ref{thm:model-level}: the benchmark spectrum, by anomaly closure \\
Fermionic $\eta$-invariants pairing the tensor phase & gravitino and tensorino
content & the benchmark fermions --- different origin, same
Atiyah--Patodi--Singer mathematics \\
Bordism-invariance of the total phase & verified on supersymmetric backgrounds &
Theorem~\ref{thm:unrestricted-global}, unrestricted --- stronger than the
supergravity analogue \\
\bottomrule
\end{tabular}
\end{table}
\endgroup

\noindent Supersymmetry enters the $(1,0)$ construction through \emph{which fields
exist} and \emph{what their coefficients are} --- never through the
differential-cohomology machinery itself. Both roles are discharged here by the
constitution and by theorems already proved. The construction transfers whole.

\paragraph{Step 7: the partition function, written down.}

\begin{theorem}[\textbf{T-BI6-7.8}: The self-dual partition function of the \BIsix{} tensor sector]
\label{thm:sd-partition}
On every background of the (enlarged) admissible class, the partition function of the
full tensor sector --- the $X_4$-coupled mediator together with its $4\Omega-1$
gauge-singlet spectators --- exists, is canonical, and is given by
\begin{equation}
\boxed{\;
Z_{\mathrm{tensor}}\bigl(\Msix; \check X\bigr)
\;=\; \Delta(g)\;\cdot\;
\Theta_{\mathrm{can}}\bigl(\cJ^3(\Msix);\,\tau(g),\,\check X\bigr)\;\cdot\;
\Theta_{\mathrm{can}}\bigl(\cJ^3(\Msix);\,\tau(g),\,0\bigr)^{\,4\Omega-1},
\;}
\label{eq:Z-tensor}
\end{equation}
where $\Delta(g)$ is the metric determinant factor of the non-chiral quantization
(spectral-zeta regularized; it exists unconditionally and carries the local curvature
anomaly that the Green--Schwarz term cancels); $\Theta_{\mathrm{can}}$ is the theta
function on the intermediate Jacobian with modular parameter $\tau(g)$ determined by the
Hodge star and with the \emph{canonical characteristic} --- the unique refinement of
Theorem~\ref{thm:refinement-unique}; the mediator's theta is sourced through the
differential cocycle $\check X$ (the coupling $\int B_2\wedge X_4$ shifts the theta
argument by the period vector of $\widehat X$), and the spectators, being
$X_4$-decoupled, contribute unsourced canonical factors. $Z_{\mathrm{tensor}}$ is
single-valued over the space of backgrounds and under all large gauge transformations
and large diffeomorphisms.
\end{theorem}

\begin{proof}
Existence and canonicity: the block decomposition exists by holomorphic factorization of
the non-chiral field (Step 2); the block is pinned by
Theorem~\ref{thm:refinement-unique} and Lemma~\ref{lem:twist-exclusion} --- there is
exactly one admissible characteristic, so ``a block'' is ``the block.''
Single-valuedness: transporting $Z_{\mathrm{tensor}}$ around a closed loop in the space
of backgrounds --- a large gauge transformation or large diffeomorphism --- multiplies
it by the holonomy of the anomaly line over that loop, evaluated on the loop's mapping
torus; by Theorem~\ref{thm:unrestricted-global} that holonomy is trivial on every closed
spin seven-datum. The two theorems are converses in spirit: the unrestricted global
theorem proved the anomaly line bundle trivializable; the present construction exhibits
the section.
\end{proof}

\begin{remark}[The intended class, fully explicit]
\label{rem:sd-intended}
On the intended backgrounds the abstract objects become concrete. With
$H^1(\Mfour;\mathbb Z) = H^3(\Mfour;\mathbb Z) = 0$, the K\"unneth theorem gives
\begin{equation}
H^3(\Mfour\times\Ttwo;\mathbb Z)
\;=\; H^2(\Mfour;\mathbb Z)\otimes H^1(\Ttwo;\mathbb Z),
\qquad \mathrm{rank} = 2\,b_2(\Mfour),
\label{eq:kunneth-jacobian}
\end{equation}
the two $H^1$ generators of the structural torus pairing the $b_2$ classes of the base
into a symplectic lattice of rank $2b_2$ --- even, as Step 2 requires. The intermediate
Jacobian is the torus $T^{2b_2(\Mfour)}$ and $\Theta_{\mathrm{can}}$ is an ordinary
Riemann theta function of genus $b_2(\Mfour)$ with zero characteristic. For a base with
$b_2 = 0$ the Jacobian is a point and $Z_{\mathrm{tensor}}$ reduces to the bare
determinant factor times the flux-free theta constants.
\end{remark}

\begin{remark}[Two observations, and the honest boundary]
\label{rem:sd-physics}
\emph{The dial that was never there.} The fear behind O9 was a hidden discrete choice
beneath the central theorem --- a datum the supersymmetric treatment fixed for its own
reasons and the non-supersymmetric theory would have to fix by hand. The arithmetic says
the dial does not exist: a unimodular lattice carries exactly one Wu--Chern--Simons
refinement, and the rank-one clause of the constitution --- written to enforce
minimality --- also purchased uniqueness of the quantum definition. Minimal structure
has no room for moduli; that is \AX{4} operating one level deeper than it was aimed.
\emph{$a = 1$ was already the answer.} Clause (C3)'s ``characteristic vector $a=1$'' was
placed there to normalize the Green--Schwarz coupling; it is also, verbatim, the
characteristic element that defines the unique refinement. The paper answered O9 in its
own definitions before the question was posed sharply.
\emph{The honest boundary.} The holomorphic-factorization definition of the self-dual
field is the accepted definition throughout the field --- it is the one used by the
five-brane literature and by the $(1,0)$ supergravity treatments alike --- and the
present subsection adopts it on the same terms. That convention sits upstream of O9
here exactly as it sits upstream of the supersymmetric constructions; nothing in this
paper deepens or weakens it. And the numerical evaluation of $\eta$-invariants on
individual mapping tori, which O9 originally requested as an independent confirmation,
is hereby downgraded rather than discharged: every quantity it would compute is now
fixed exactly by theorems, so it can catch errors but cannot carry information. It
retains the standing of re-running a verified computation.
\end{remark}

\subsection{Why global anomaly freedom is critical for \OGGEP{} and \GDOS{}}

The role of global anomaly freedom becomes even clearer once \GG{} is connected to
\GDOS{}. \OGGEP{} is more than a symbolic map from a six-dimensional bulk to a four-dimensional boundary. It is meant to define a lawful, gauge-invariant,
single-valued projection channel from bulk data to boundary observables. If the
bulk partition function were only locally consistent but globally multi-valued,
then the projected boundary dynamics would inherit an ambiguity: two bulk paths
related by a large gauge transformation or large diffeomorphism could project to
boundary descriptions that differ by an unphysical phase. That would destroy the
functorial meaning of the projection and would make the boundary channel depend on
topological representatives rather than only on physical data.

Global anomaly freedom removes exactly that danger. It guarantees that the bulk
functional integral is a gauge-invariant object on the intended category,
so the \OGGEP{} projection does not depend on arbitrary clutching choices,
non-contractible loops in background space, or the self-dual tensor's hidden
quadratic phase. Operationally, this is the difference between a boundary theory
that is merely \emph{locally suggestive} and a boundary theory that is
\emph{globally lawful}. Since \GDOS{} treats the observed four-dimensional world as
an open-system projection of the six-dimensional bulk, the lawfulness of the bulk
is prior to the lawfulness of the boundary channel.

There is also a deeper physics intuition. In ordinary local anomaly
cancellation one checks that the equations of motion and Ward identities survive
infinitesimal variations. In a globally anomalous theory, however, the very
notion of transporting a state or partition function around the allowed space of
backgrounds breaks down. \OGGEP{} and \GDOS{} require more than infinitesimal control:
they require that the 6D parent theory define a globally consistent packet whose
projection to 4D is independent of the route by which one arrived at a given
boundary slice. The global theorem is therefore a prerequisite for treating \OGGEP{} as a physically meaningful projection
principle.

\begin{lemma}[Phase propagation through the \OGGEP{} channel]
Let \(Z_{\mathrm{bulk}}[\mathcal A]\) be the exponentiated bulk functional on the intended \GG{} category, and let
\[
\mathcal D_{\OGGEP}: Z_{\mathrm{bulk}}[\mathcal A]\mapsto Z_{\mathrm{bdy}}[\mathcal J]
\]
denote the linear bulk-to-boundary source map used to define the \OGGEP{} generating functional. If under an allowed large transformation \(g\) one had
\[
Z_{\mathrm{bulk}}[\mathcal A^g]=e^{i\phi(g;\mathcal A)}\,Z_{\mathrm{bulk}}[\mathcal A],
\]
then linearity gives
\[
Z_{\mathrm{bdy}}[\mathcal J]^g
=\mathcal D_{\OGGEP}\!\left(Z_{\mathrm{bulk}}[\mathcal A^g]\right)
=e^{i\phi(g;\mathcal A)}\,\mathcal D_{\OGGEP}\!\left(Z_{\mathrm{bulk}}[\mathcal A]\right)
=e^{i\phi(g;\mathcal A)}\,Z_{\mathrm{bdy}}[\mathcal J].
\]
Therefore a nontrivial bulk global anomaly phase descends directly to the boundary source functional. In particular, the \OGGEP{} channel fails to descend to gauge-equivalence classes unless \(\phi(g;\mathcal A)\in2\pi\mathbb Z\) for every allowed \(g\).
\end{lemma}

\begin{proof}
The point is linearity, not detailed dynamics. \OGGEP{} is read here at the level of amplitudes / generating functionals, upstream of the normalized density-matrix channel \(\cD_{\OGGEP} = \mathcal{R} \circ \Tr_{\mathrm{env}}\) constructed in \citep{ArisakaA5GDOS}; the two readings are compatible because that channel is defined on gauge-equivalence classes of sources, which is precisely what a representative-dependent phase obstructs. A multiplicative phase in the parent functional therefore survives as the same multiplicative phase after projection. Normalized density matrices may wash out a single overall phase on one connected patch, but the channel as a law between gauge-equivalence classes is already ill-defined once the source functional depends on the representative. Hence global anomaly freedom is a prerequisite for a lawful \OGGEP{} map.
\end{proof}


\subsection{Constitutional status of \texorpdfstring{$U(1)_{Qi}$}{U(1)Qi}: optional under the constitution, preferred by the empirical data}
\label{sec:UoneQi-constitutional-status}

This subsection is the home of the constitutional admissibility
proof for the fourth gauge factor $U(1)_{Qi}$ of the \GG{} bulk.
The anomaly theorem proved in
Sections~\ref{sec:structural-proof}--\ref{sec:global-completion}
goes through whether or not $U(1)_{Qi}$ is present; the question
this subsection settles is what the \GGTheory{} Constitution itself
\emph{requires}, and what (if anything) it merely \emph{permits}.
The companion paper \citep{ArisakaA2GG6} cross-cites the
present subsection at the points where it adopts the four-component
bulk and is explicit that the choice is empirical rather than
constitutional; here we give the rigorous statement.

\begin{theorem}[\textbf{T-BI6-7.5}: Constitutional admissibility of both three-component and four-component bulks]
\label{thm:UoneQi-admissibility}
The four constitutional axioms \AX{1}--\AX{4} and the seven
postulates \POS{1}--\POS{7} of \citep{ArisakaA1Constitution} admit two
distinct anomaly-free six-dimensional bulk theories:
\begin{equation}
\resizebox{\linewidth}{!}{$\displaystyle
\boxed{\begin{aligned}
\text{(three-component bulk)}\quad
G_{\Bulk}^{(3)} &= \Spin(1,5) \times \SU(5) \times U(1)_B,\\
\text{(four-component bulk)}\quad
G_{\Bulk}^{(4)} &= \Spin(1,5) \times \SU(5) \times U(1)_B \times U(1)_{Qi}.
\end{aligned}}
$}
\label{eq:two-bulks}
\end{equation}
Both versions support a single-tensor Green--Schwarz factorization
$I_8 = \tfrac12 X_4^2$ with $X_4$ given by the appropriate
truncation of \eqref{eq:bi6-main}, both admit a chiral fermion
spectrum saturating the relevant trace-ratio identity, and both
close under the \OGN{} integer ledger at $L = 5$. The choice
between the two bulks is therefore empirical, not constitutional.
\end{theorem}

\paragraph{Why the constitution does not force $U(1)_{Qi}$.}
Indeed, the textual chain ``\AX{3} + \POS{4} + \POS{1}
$\Rightarrow$ forces $U(1)_{Qi}$'' that occasionally appears
in early drafts of the \GGTheory{} program overstates what
the constitutional clauses actually demand. CPTP is a
\emph{structural} property of the GKLS channel
\citep{Lindblad1976,Gorini1976} that \POS{4}
imposes on the bulk-to-boundary registration map; it is not a
covariance demand under a fresh $U(1)$. Operational closure
(\POS{2}) and gauge invariance under the existing bulk gauge
content (\POS{1}) together produce a CPTP bulk-to-boundary channel
whether or not a fourth gauge factor is present. The conserved
information current that the channel registers can be identified
either with a slot inside the existing $\SUfive \times U(1)_B$
content (under the three-component reading) or with a separate
$U(1)_{Qi}$ generator (under the four-component reading). Both
readings are admissible; neither is forced by the constitutional
clauses alone. The substantive axiomatic content is
\emph{admissibility} of $U(1)_{Qi}$ as an extension that does
not violate any axiom or postulate; the substantive empirical
content is \emph{preference} for the four-component reading on
data-driven grounds.

\begin{proof}[Proof of Theorem~\ref{thm:UoneQi-admissibility}]
Anomaly freedom of the three-component bulk
$G_{\Bulk}^{(3)} = \Spin(1,5) \times \SU(5) \times U(1)_B$ is
the truncation $F_{Qi} = 0$ of the four-component proof of
Theorem~\ref{thm:bi6-cancellation}, where the truncation is read
\emph{at the level of the chiral spectrum} and not merely at the
level of the field strength: the three-component theory is the
\GG{} theory in which no fermion species carries a non-trivial
$U(1)_{Qi}$ charge. This distinction matters, since a spectrum
with $F_{Qi} = 0$ but with fermions still formally charged under
$U(1)_{Qi}$ would in principle generate mixed gravitational --
$U(1)_{Qi}$ and Abelian -- $U(1)_{Qi}$ anomaly contributions; the
admissibility argument is that the three-component bulk has no
such residual charges to begin with, so all $U(1)_{Qi}$-coupled
anomaly slots are identically zero. Under that reading, the
\BIsix{} four-form $X_4 = \tr R^2 - \Tr F_5^2 - (2/5)\,F_B^2$
remains a closed, gauge-invariant four-form; the microscopic
anomaly polynomial still factorizes as $I_8 = \tfrac12 X_4^2$;
descent against a single two-form $B_2$ cancels the local
consistent anomaly; and global consistency holds because the
homotopy facts $\pi_6(\SU(5)) = 0$ (because $6 \leq 2N-1 = 9$
places $\pi_6$ in the stable range for $\SU(5)$, where Bott
periodicity gives $\pi_{2j}(\SU) = 0$; note that the unstable
$\pi_{2N}(\SU(N)) = \mathbb{Z}_{N!}$ does \emph{not} vanish, so the
stable-range bound is what is doing the work here) and
$\pi_6(U(1)) = 0$ (Abelian factors contribute nothing to
$\pi_6$) together give $\pi_6(\SU(5) \times U(1)_B) = 0$, so
that no Witten-type six-dimensional gauge obstruction obstructs
the three-component bulk. Anomaly freedom of the four-component bulk is the
content of Theorems~\ref{thm:bi6-cancellation}
and~\ref{thm:bi6-global-complete} of the present paper as
written. Both bulks therefore satisfy \POS{3} (anomaly-free
admissibility by inflow); both satisfy \AX{2} (\GGEP) under the
trace-ratio assignment of $-2/5$ to every Abelian slot; both
satisfy \POS{6} (Occam Gauge Normalization) under the \OGN{} ledger
at $L = 5$, since $L$ is fixed by the size of the $\SU(5)$
fundamental representation and is independent of the number of
Abelian factors. The remaining postulates --- \AX{1}, \AX{3}, \AX{4},
\POS{1}, \POS{2}, \POS{4}, \POS{5}, \POS{7} --- are satisfied trivially by
both bulks because they constrain the geometry, the
bulk-to-boundary projection, the symmetry-minimality content,
and the falsifiability structure rather than the number of
Abelian factors. Hence both bulks are constitutionally
admissible.
\end{proof}

\paragraph{Worked example: an explicit three-component reservoir.}
Constitutional admissibility is a statement about the axioms, and
the proof above settles it. It is worth exhibiting an anomaly-free
three-component spectrum as well, because the shortest route to one
does not work and the reason it fails is instructive.

That route is truncation: take the benchmark
$\cM_{\mathrm{gauge}}(25)$ of Section~\ref{sec:model-level} and strike
every species carrying non-trivial $\UoneQi$ charge. Both members of
every diagonal orbit go, and one member of every axial orbit, leaving
$8\,\cC(1/3,0) \oplus 4\,\cC(2/3,0)$. The mixed monomials
$\tr R^2 F_B F_{Qi}$ and $F_B^3 F_{Qi}$ do vanish, as the truncation
intends. The quartic Abelian balance does not survive. Matching
\eqref{eq:core-block-poly} against $\tfrac{\Omega'}{2}(X_4^{(3)})^2$,
the $p_1 \Tr F_5^2$ and $(\Tr F_5^2)^2$ monomials of the residue force
$\Omega' = 3$; the $p_1 F_B^2$ and $\Tr F_5^2 F_B^2$ monomials then
force the Abelian coefficient $\gamma = 4/3$; and those two values
demand an $F_B^4$ coefficient of $8/3$, against the $4/3$ the residue
actually carries. No rescaling of $\gamma$ repairs both, because
removing a core block removes its $x^4$ and $x^2$ contributions in a
fixed ratio. Truncation preserves the vanishing conditions and breaks
the matching conditions.

The construction has to be carried out directly, and it closes. Write a
three-component reservoir as a multiset of core blocks $\cC(q_B,0)$
with $q_B = n/3$ and signed multiplicities $c_n$, and set
$N = \sum_n c_n$, $S_2 = \sum_n c_n q_B^2$, $S_4 = \sum_n c_n q_B^4$.
Matching \eqref{eq:core-block-poly} against
$\tfrac{\Omega'}{2}(X_4^{(3)})^2$ term by term gives $\Omega' = N/4$
from the $p_1 \Tr F_5^2$ monomial and again, consistently, from
$(\Tr F_5^2)^2$; it gives $\gamma = 6 S_2/N$ from $p_1 F_B^2$ and
again from $\Tr F_5^2 F_B^2$; and it leaves exactly one residual
condition, from $F_B^4$,
\begin{equation}
N\,S_4 \;=\; 3\,S_2^{\,2}.
\label{eq:three-component-closure}
\end{equation}
Imposing in addition the \BIsix{} Abelian value $\gamma = 2/5$ --- which
Theorem~\ref{thm:ogn-bi6} fixes from the $\SU(5)$ trace ratio, and which
is blind to the number of Abelian factors --- the two conditions read
$\sum_n c_n n^2 = \tfrac{3}{5} N$ and
$\sum_n c_n n^4 = \tfrac{27}{25} N$. The smallest solution on the
charge set $\{0,\,1/3,\,2/3\}$ is
\begin{equation}
\cM^{(3)}_{\mathrm{gauge}}(25)
\;=\;
52\,\cC(0,0)\;\oplus\;44\,\cC(1/3,0)\;\oplus\;4\,\cC(2/3,0),
\label{eq:three-component-benchmark}
\end{equation}
with $N = 100$ and $\Omega' = 25$. Its gauge and mixed monomials
reproduce $\tfrac{25}{2}(X_4^{(3)})^2$ exactly, and the gravitational
sector closes under the same law as the four-component case,
$n_{\mathrm{SD}} = 4\Omega' = 100$ and $n_F = 32\Omega' = 800$
(Appendix~\ref{app:grav-full}). Baryon-neutral core blocks are not a
new ingredient: the four-component benchmark already contains twelve of
them, as the $\cC(0,n/3)$ members of its axial orbits.

Two features of \eqref{eq:three-component-benchmark} deserve recording.
Integrality of $\Omega'$ forces $\Omega' \in 25\mathbb{Z}$ here as
well, by the same $L^2 = 25$ that governs the four-component case
(Theorem~\ref{thm:ogn-omega}); the minimal normalization is therefore a
property of the $\SU(5)$ block and not of the number of Abelian
factors. And the minimal three-component reservoir is
chirality-uniform, whereas the minimal four-component one is not: the
benchmark $\cM_{\mathrm{gauge}}(25)$ requires $d_2 = -12$, an orbit
taken with reversed chirality. Anomaly freedom does not prefer the
four-component bulk. The preference argued below is empirical, and this
subsection supplies no structural substitute for it.

\paragraph{Why $U(1)_B$ is essential to either bulk.}
The baryonic factor is structural for three independent reasons,
none of which depend on the presence of $U(1)_{Qi}$. First,
it is the only exact gauge symmetry in the \GG{} particle ledger
that can protect proton stability in a constitutionally sharp
way. Second, the key trace ratio that later becomes the \OGNfive{}
theorem compares the primitive baryonic direction to the
traceless hypercharge-like splitting inside the unified
five-dimensional block. In that sense the coefficient $2/5$ is
not an arbitrary Abelian weight; it is the baryonic trace
witness of the $3+2$ split inside the $\SU(5)$ block. Third,
the split-multiplet boundary construction requires a separate
baryonic label precisely because unsplit $\SU(5)$ multiplets
cannot carry definite baryon number. Both the three-component
and the four-component bulks therefore retain $U(1)_B$ as an
essential factor.

\paragraph{Why $U(1)_{Qi}$ is constitutionally optional.}
The additional informational Abelian factor is
\emph{allowed} but not \emph{forced}. It is allowed because it
is fully compatible with both local and global anomaly
cancellation: Abelian factors have $\pi_6(U(1)) = 0$, so there
is no independent Witten-type six-dimensional gauge obstruction;
the \BIsix{} identity sees Abelian factors only through $F^2$, so
adding a second Abelian slot with the same coefficient preserves
the factorized square structure rather than spoiling it; and in
the global theorem it behaves as an anomaly-compatible spectator
extension of the Abelian sector. It is not forced because, for
the narrow purpose of proving complete anomaly freedom of \BIsix{},
one could truncate to the three-component bulk and set
$F_{Qi} = 0$; the global cancellation theorem would still close.
In that precise sense $U(1)_{Qi}$ is constitutionally optional.

\paragraph{Why the four-component bulk is empirically preferred.}
After all, the test of an honest framework is what nature
selects out of the constitutional admissible class, not what
the constitution permits in the abstract. The four-component
version is preferred over the three-component on three
empirical and structural grounds.

\begin{enumerate}[label=(\arabic*),leftmargin=2em]

\item \emph{The Arisaka current equalities and the
 bioenergetic conversion factor $\QB$.}
 The Arisaka equation
 $J_E / \QB = J_e = J_B = J_{Qi}$ developed in the companion
 papers (\citealp{ArisakaA2GG6}; \citealp{ArisakaA9SM}) requires
 $U(1)_{Qi}$ to exist on the boundary as a fourth conserved
 current. Without $U(1)_{Qi}$, the four current equalities
 collapse to three, and the bioenergetic conversion factor
 $\QB \approx 0.1555$~eV per qubit --- anchored by ATP hydrolysis
 at biological temperatures, one phosphoanhydride bond carrying
 two quanta, $2\,N_A \QB = 30.0$~kJ/mol against the measured
 $30.5 \pm 0.5$~kJ/mol \citep{ArisakaA7Qi4} --- has no
 constitutional home. The matching of $\QB$ to ATP hydrolysis is the sharpest
 bioenergetic anchor of the \GGTheory{} program; preserving it
 selects the four-component bulk.

\item \emph{The \OGNfive{} dual-lock identity
 $K = LJ = M^2 + N^2 + L^2 = 35$.}
 Step~5 of the \OGNfive{} cascade in the companion paper
 \citep{ArisakaA2GG6} establishes the dual-lock identity
 $K = LJ = M^2 + N^2 + L^2$ that fixes the master ledger
 integer $K$ jointly with the closure index $J$. Under the
 equal-Abelian-treatment convention --- which assigns the
 coefficient $-2/5$ symmetrically to every Abelian slot in the
 \BIsix{} four-form --- the dual-lock identity closes
 arithmetically only for the four-component case
 $(M, N, L; J, K) = (1, 3, 5; 7, 35)$. The three-component
 bulk does not produce an integer dual-lock with the same
 small-integer character. The four-component bulk is therefore
 the choice that closes the \OGNfive{} ledger as a Diophantine
 cascade rather than as a free-parameter fit.

\item \emph{The boundary $Qi$-4 pillars and the
 Equivariant Information Channel.}
 The boundary phenomenology of \GGTheory{} --- the structural
 framing of life and mind as boundary projections of the
 $U(1)_{Qi}$ current, articulated as the four pillars of \ASeven{}
 \citep{ArisakaA7Qi4} --- depends on the four-component version.
 Without $U(1)_{Qi}$, the Equivariant Information Channel
 (\EIC{}) framework that the program uses to connect 6D bulk
 physics to biological information processing has no host
 symmetry. The four-component bulk is therefore the choice
 that allows the \QiFour{} pillars to be derived rather than
 postulated.

\end{enumerate}

In summary, both the three-component bulk and the four-component
bulk are constitutionally admissible. The four-component bulk
is preferred on empirical grounds: it carries the Arisaka
current equalities and the bioenergetic conversion factor $\QB$,
it closes the \OGNfive{} dual-lock identity arithmetically, and it
hosts the boundary $Qi$-4 pillars and the Equivariant
Information Channel framework. The remainder of the present
paper therefore works in the four-component bulk, with the
understanding that every theorem and every proof restricts to
the three-component bulk by the obvious truncation $F_{Qi} = 0$.
A reader who wishes to retain only the axiomatic content
of the framework, deferring the empirical preference to the
boundary papers, may read every \BIsix{} statement as a statement
about the three-component bulk; the anomaly theorem and the
\OGNfive{} ledger then close around $L = 5$ at $J = 7$ and the
four-component dual-lock identity is replaced by its
three-component cousin.

\paragraph{Cross-references.} The
companion paper Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6}
cross-cites the present subsection at three points: at the
opening statement that the constitution admits both three- and
four-component bulks (Theorem~6 ``Six forces from the
constitution''), at the empirical adoption of the four-component
bulk in the \BIsix{} coefficient theorem, and at the response to
Objection~4 in the Common Objections section. Both
$\UoneQi$-dependent companion papers --- \ASix{} and \ASeven{} ---
inherit the four-component preference from the present
subsection together with the empirical anchor.


%% ===============================================================
%%  SECTION 4: MODEL-LEVEL BENCHMARK
%% ===============================================================
\subsection{Applicability of the Hopkins--Singer / Monnier--Moore framework to \texorpdfstring{non-super\-sym\-metric}{non-supersymmetric} six-dimensional theories}
\label{ssec:hs-mm-non-susy}

A reader trained on the modern global-anomaly literature will reasonably
ask whether the Hopkins--Singer / Monnier--Moore differential-cohomology
framework, which we invoke in Section~\ref{sec:global-completion} and
Appendix~\ref{app:global-proof-dossier} to establish global anomaly
freedom of \BIsix{}, applies to a non-supersymmetric six-dimensional
bulk such as \GG{}. The question is non-trivial because Monnier and
Moore \citep{MonnierMoore2019} developed the framework specifically for
$\cN = (1,0)$ six-dimensional supergravity, and the supersymmetric
structure plays a visible role in their explicit numerical evaluation
of partition functions on bounding seven-manifolds.

The structural answer is that the framework's four hypotheses
(H1)--(H4), stated in full in Appendix~\ref{app:global-proof-dossier}
--- (H1) the admissible mapping tori bound spin eight-manifolds over
which the gauge and tensor data extend; (H2) no independent
Witten-type gauge obstruction survives, \(\pi_6\) vanishing for each
factor of \(\SUfive \times \UoneB \times \UoneQi\); (H3) the
lattice-valued lift of the \BIsix{} four-form obeys the Wu-shifted
integrality law, so the exponentiated Wu--Chern--Simons functional is
single-valued; and (H4) the four-form \(X_4\) lifts to a globally
defined degree-four differential cocycle --- are
purely bordism-theoretic / differential-cohomological statements about
the bundle structure of the underlying manifold, independent of
whether the matter content on that bundle is supersymmetric. The
foundational construction of Hopkins and Singer
\citep{HopkinsSinger2005} pre-dates the Monnier--Moore SUGRA
specialization and was developed for general principal bundles over
Spin manifolds; the four hypotheses inherit that generality. What
\emph{is} specific to $\cN = (1,0)$ in Monnier--Moore (2019) is the
explicit evaluation of the eta-invariant in terms of the SUGRA
spectrum on closed Spin seven-manifolds; that calculation is recorded
here as Open Problem~O9 (Section~\ref{sec:open-problems}) for the
non-supersymmetric cohort spectrum and is the natural follow-up
computational program on top of the structural theorem. The
structural verification of H1--H4 carried out in
Section~\ref{sec:global-completion} does not invoke supersymmetry; the
global theorem is therefore established at the structural level on
the cohort \GG{} background class, with the explicit numerical
confirmation of partition-function single-valuedness deferred to
follow-up work that adapts the Monnier--Moore eta-invariant
calculations to the non-supersymmetric setting.



\section{Model-level benchmark spectrum}
\label{sec:model-level}

The local and global theorems of Sections~\ref{sec:structural-proof}--\ref{sec:global-completion}
prove anomaly cancellation \emph{assuming} the total polynomial factorizes
as \(I_8 = \frac{\Omega}{2}X_4^2\).
The natural question is whether any finite chiral spectrum actually
achieves this factorization.
This section constructs one explicit solution---a \emph{benchmark
spectrum}---thereby proving that the structural theorem is
satisfiable.
The construction proceeds in four stages: three no-go results that
constrain the search space, a modular ``core block'' that serves as
the building unit, an orbit architecture that cancels all forbidden
terms by design, and a gravitational completion by gauge-singlet tensors
and fermions.
The full computations appear in
Appendices~\ref{app:core-block-full}--\ref{app:grav-full}.

\subsection{Three no-go results}

Before constructing a solution, three constraints narrow the search.

\begin{lemma}[Species-wise diagonal no-go]
\label{lem:diagonal}
If \(q_{Qi,s}=q_{B,s}\) for every species \(s\), then all off-diagonal
Abelian anomaly terms force all diagonal Abelian coefficients to vanish,
contradicting \BIsix{}.
\end{lemma}

\begin{lemma}[Fermion-only gravitational no-go]
\label{lem:grav}
Ordinary Weyl fermions alone cannot produce a factorized gravitational
anomaly with vanishing \(\tr R^4\) and nonzero \((\tr R^2)^2\).
\end{lemma}

\begin{lemma}[Split-multiplet necessity]
\label{lem:split}
Unsplit bulk \(\SU(5)\) multiplets cannot carry definite baryon number,
because all components share a single \(\U(1)_B\) charge.
Visible Standard-Model zero modes require split carriers with orbifold
boundary conditions.
\end{lemma}

These three results rule out the simplest constructions and motivate
the orbit-based approach below.

\subsection{Core block construction}
\label{ssec:core-block-construction}

The idea is to find a minimal \(\SU(5)\) multiplet package whose
irreducible and cubic anomaly indices vanish \emph{internally}, so that
the only surviving contributions are the reducible terms needed for
the \BIsix{} factorization.
The following combination achieves this.

\begin{definition}[Core block]
\begin{equation}
\cC^{\sigma}(q_B,q_{Qi})
:= \sigma\Big[\mathbf{10}\oplus\mathbf{5}\oplus 2\,\overline{\mathbf 5}
\oplus 11\cdot\mathbf{1}\Big]_{(q_B,q_{Qi})}.
\label{eq:core-block}
\end{equation}
Here \(\sigma = \pm 1\) is the six-dimensional chirality, and all
components carry the same Abelian charges \((q_B, q_{Qi})\).
The 11 singlets are included to make the total dimension a perfect
square: \(d_{\cC} = 10 + 5 + 10 + 11 = 36 = 6^2\).
\end{definition}

\begin{proposition}[Core block anomaly data]
\label{prop:core-block}
The core block has dimension \(d_{\cC}=(L{+}1)^2=36\),
quadratic index \(A_{\cC}=L{+}1=6\),
reducible quartic coefficient \(C_{\cC}=3\),
and irreducible quartic and cubic indices
\begin{equation}
B_{\cC}(L) \;=\; A_{3,\cC}(L) \;=\; L-5 ,
\label{eq:core-B-A3-general}
\end{equation}
both of which vanish at \(L=5\).
The vanishing of \(B_{\cC}\) and \(A_{3,\cC}\) is the key property:
it ensures that the core block contributes only to the reducible
anomaly polynomial, which is exactly the part that the Green--Schwarz
mechanism can cancel.
The proof is a direct computation given in Appendix~\ref{app:core-block-full}.
\end{proposition}

\subsection{Orbit families and the Diophantine system}
\label{ssec:orbit-families}

The six forbidden monomials (V1)--(V6) listed in Appendix~\ref{app:matrix-and-coefficients} contain
odd powers of the Abelian field strengths. These must cancel in the
total spectrum. The orbit construction achieves this by pairing
core blocks with opposite or exchanged Abelian charges, so that
the odd-power terms cancel within each pair.
Two pairing patterns are possible:

\begin{definition}[Axial orbit]
\(A_n^{\sigma} := \cC^{\sigma}(n/3,\,0)\oplus\cC^{\sigma}(0,\,n/3)\).
One block carries baryon charge only, the other carries informational
charge only. Every mixed monomial \(F_B^a F_{Qi}^b\) with \(a,b > 0\)
vanishes on this pair because the two blocks have disjoint Abelian
support.
\end{definition}

\begin{definition}[Diagonal orbit]
\(D_n^{\sigma} := \cC^{\sigma}(n/3,\,n/3)\oplus\cC^{\sigma}(n/3,\,-n/3)\).
Both blocks carry the same baryon charge, but opposite informational
charges. Mixed monomials odd in \(q_{Qi}\) vanish by the sign flip
\(q_{Qi} \to -q_{Qi}\).
\end{definition}

The gauge-matter reservoir
\(\cM_{\mathrm{gauge}}(\Omega)
= \bigoplus_n a_n\,A_n^{+}\oplus\bigoplus_n d_n\,D_n^{+}\)
satisfies the \BIsix{} factorization condition if and only if
the following four equations hold (derived in Appendix~\ref{app:diophantine-full}):
\begin{align}
\textstyle\sum_n a_n + \sum_n d_n &= 2\Omega,
\label{eq:dioph1}\\
\textstyle\sum_n n^2 a_n + 2\sum_n n^2 d_n &= 12\Omega/5,
\label{eq:dioph2}\\
\textstyle\sum_n n^4 a_n &= 72\Omega/25,
\label{eq:dioph3}\\
\textstyle\sum_n n^4 d_n &= 18\Omega/25.
\label{eq:dioph4}
\end{align}
The first equation \eqref{eq:dioph1} counts the total number of core blocks
(the \(\Tr F_5^2\) coefficient).
The second \eqref{eq:dioph2} matches the mixed gauge--Abelian coefficient
\(\Tr F_5^2 \cdot F_B^2\).
The third and fourth, \eqref{eq:dioph3}--\eqref{eq:dioph4}, match the
pure Abelian quartic coefficients \((F_B^2)^2\) and \((F_{Qi}^2)^2\)
separately; they differ because the axial and diagonal orbits distribute
their Abelian charges differently between the \(B\) and \(Qi\) directions.

\begin{theorem}[\textbf{T-BI6-8.8}: Exact orbit solution]
\label{thm:orbit}
For every \(\Omega\in L^2\mathbb{Z} = 25\mathbb{Z}\), the system
has the exact integer solution
\begin{equation}
\boxed{
a_1 = \frac{8\Omega}{25},\quad
a_2 = \frac{4\Omega}{25},\quad
d_1 = \frac{48\Omega}{25},\quad
d_2 = -\frac{12\Omega}{25},\quad
d_3 = \frac{2\Omega}{25}.
}
\label{eq:orbit-solution}
\end{equation}
The negative multiplicity \(d_2 < 0\) is not a sign error:
it means the \(n=2\) diagonal orbit enters with the \emph{opposite}
six-dimensional chirality, \(\sigma = -1\) instead of \(+1\).
Equivalently, \(|d_2|\) copies of \(D_2^{-}\) replace
\(|d_2|\) copies of \(D_2^{+}\).
In six-dimensional chiral theories, a Weyl fermion of chirality
\(+1\) in representation \(\rho\) is physically distinct from a
Weyl fermion of chirality \(-1\) in the same representation ---
they contribute to the anomaly polynomial with opposite signs.
Chirality-flipped species are standard in the 6D model-building
literature \citep{GreenSchwarzWest1985,Sagnotti1992} and appear
routinely in the Kumar--Morrison--Taylor enumeration of
anomaly-free spectra \citep{KumarMorrisonTaylor2010}.
This chirality flip is the mechanism by which the Diophantine
system achieves the non-trivial balance between the quartic
Abelian coefficients.
\end{theorem}

\subsection{Gravitational completion}

The orbit construction of the previous subsection fixes the gauge-matter
sector but leaves the purely gravitational anomaly unfixed.
The anomaly polynomial of Weyl fermions has a universal gravitational
piece \(\propto (7p_1^2 - 4p_2)\), and the target \(\frac{\Omega}{2}X_4^2\)
contains a \((\tr R^2)^2 \propto p_1^2\) term but no independent \(p_2\) term.
As shown in Proposition~\ref{prop:asd}, fermions alone cannot match both
coefficients simultaneously. The gravitational balance requires
gauge-singlet \emph{anti-self-dual tensors} (which contribute
\(\propto p_1^2 - 7p_2\) with a different \(p_2/p_1^2\) ratio)
and additional singlet fermions.

The resulting \(2\times 2\) linear system (derived in Appendix~\ref{app:grav-full})
determines the tensor and fermion counts uniquely and linearly in \(\Omega\):
\begin{equation}
n_{\mathrm{SD}} = 4\,\Omega,
\qquad
n_F = 32\,\Omega;
\qquad
\Omega = 25:\ (100,\,800);
\qquad
\Omega = 50:\ (200,\,1600),
\end{equation}
where the \(n_F\) gauge-singlet Weyl fermions carry the six-dimensional
chirality opposite to the gauge reservoir (the same chirality-flip
bookkeeping as the \(D_2^-\) orbit of Theorem~\ref{thm:orbit}), and
exactly eight singlet fermions accompany each tensor: the structural
ratio \(n_F/n_{\mathrm{SD}} = 8\) holds identically in \(\Omega\).
The global phase of the chiral tensor sector is controlled by
\(n_{\mathrm{SD}} \bmod 8\): the exponentiated eta invariant of
\(n_{\mathrm{SD}}\) chiral two-forms on a closed seven-manifold is a
signature-type phase that trivializes manifestly when
\(n_{\mathrm{SD}} \equiv 0 \pmod{8}\)
\citep{AlvarezGaumeWitten1984,MonnierMoore2019}.
Since \(n_{\mathrm{SD}} = 4\Omega\), this shortcut applies exactly when
\(\Omega\) is even: the \(\Omega = 50\) branch
(\(n_{\mathrm{SD}} = 200\)) is manifestly signature-friendly, while on
the \AX{4}-minimal branch \(\Omega = 25\) (\(n_{\mathrm{SD}} = 100\))
the tensor-sector phase vanishes identically on the intended-class
fillings, where \(\sigma(X_6 \times D^2) = 0\)
(Lemma~\ref{lem:computed-phase} of
Appendix~\ref{app:global-proof-dossier}).
Between the two branches, \AX{4} (\MSMF) favors the smaller
reservoir: \(\Omega = 25\) with \(n_{\mathrm{SD}} = 100\) and
\(n_F = 800\). Whether this minimal branch is the unique
physical realization remains an open question (the
Minimal \BIsix{} Spectrum Problem, entry O8 of the triage in
Section~\ref{sec:open-problems}).

\subsection{Non-supersymmetric gravitational anomaly closure: what plays the role of the gravitino}
\label{ssec:non-susy-grav-closure}

The standard six-dimensional $\cN = (1,0)$ supergravity anomaly
cancellation \citep{Taylor2011, Sadov1996} rests on the celebrated linear constraint
\begin{equation}
n_H \;-\; n_V \;+\; 29\, n_T \;=\; 273,
\label{eq:sugra-273}
\end{equation}
which closes the irreducible $\tr R^4$ part of the gravitational
anomaly polynomial through the combined contributions of the
graviton, the gravitino, the dilatino, the self-dual two-form of the
gravity multiplet, and the matter hypermultiplets ($n_H$) and vector
multiplets ($n_V$); the constant 273 is the gravitino-plus-gravity-multiplet
contribution to the $\tr R^4$ coefficient, derived in
\citet{AlvarezGaumeWitten1984} and tabulated cleanly in \citet{Schwarz1996} and \citet{Erler1994}.

The cohort framework lacks the gravitino, dilatino, and SUGRA gravity
multiplet, so the constraint \eqref{eq:sugra-273} does not apply
verbatim to \BIsix{}. A reader trained on 6D supergravity will
reasonably ask: what plays the role of the gravitino in canceling
the irreducible $\tr R^4$ piece for the cohort spectrum?

The answer is the linear combination of anti-self-dual tensors and
gauge-singlet Weyl fermions that the cohort framework supplies via
\AX{4}~(\MSMF) on the bulk gauge content. The per-field
contributions to the gravitational anomaly polynomial are, from
\citet{AlvarezGaumeWitten1984}:

\begin{itemize}[leftmargin=2em,itemsep=0.25em]
\item Per six-dimensional Weyl fermion of chirality $\sigma \in \{\pm 1\}$:
 $\;\sigma\,\frac{1}{5760}\,(7\, p_1^2 - 4\, p_2) + \text{(Casimir terms)}$.
\item Per six-dimensional anti-self-dual two-form tensor:
 $\;-\frac{1}{360}\,(p_1^2 - 7\, p_2)$, with the overall sign opposite
 to the fermion-aligned chirality of \eqref{eq:sd-grav} (\citealp{AlvarezGaumeWitten1984}).
\item No gravitino, no dilatino contributions: $0$, $0$.
\end{itemize}

The target anomaly polynomial $(\Omega/2)\,X_4^2$ is, by Lemma 4.1, a
polynomial in $(\tr R^2)^2 \propto p_1^2$ alone --- it has no
irreducible $p_2$ piece. The closure of the $p_2$ coefficient is
therefore a single linear equation in $(n_F^{\rm eff},\, n_{SD})$
where $n_F^{\rm eff}$ is the chirality-signed total Weyl-fermion
count of the spectrum and $n_{SD}$ is the number of gauge-singlet
anti-self-dual tensors. Combined with the matching condition for
the $p_1^2$ coefficient, this becomes the $2 \times 2$ linear system
written out explicitly in Appendix~\ref{app:grav-full}.

The system has a unique solution (given the visible gauge spectrum
fixed in the core-block construction (Section~\ref{ssec:core-block-construction})
and the orbit-family Diophantine system (Section~\ref{ssec:orbit-families})):
for the minimal benchmark $\Omega = 25$, the gravitational completion
requires $n_{SD} = 100$ anti-self-dual tensors and $n_F = 800$
gauge-singlet Weyl fermions of the opposite chirality; for the next
branch $\Omega = 50$, $n_{SD} = 200$ and $n_F = 1600$. On the
$\Omega = 50$ branch $n_{SD}$ is divisible by $8$, so the
Atiyah--Patodi--Singer signature phase of the tensor sector trivializes
manifestly; on the $\Omega = 25$ branch the phase is settled by the
Wu--Chern--Simons quadratic refinement
(see~\citealp{MonnierMoore2019}; Appendix~\ref{app:global-proof-dossier});
\AX{4} then selects the
$\Omega = 25$ minimal branch as the cohort canonical realization,
with the larger reservoirs being unforced by the constitution.

The role of the gravitino is therefore played in the cohort framework
by the precise linear balance between anti-self-dual tensors and
singlet Weyl fermions enforced by AX-4 on the visible spectrum.
No supersymmetry is required. The cohort spectrum is not a
non-supersymmetric \emph{evasion} of the SUGRA 273-constraint; it is
a different spectrum that satisfies a different but equally tight
linear constraint, with the constitutional clauses doing the
selection that supersymmetry would have done.



\subsection{Model-level theorem}

\begin{theorem}[\textbf{T-BI6-8.9}: Model-level \BIsix{} anomaly verification]
\label{thm:model-level}
(For the complete set of admissible reservoirs, of which the spectrum below is one member,
see Remark~\ref{rem:complete-solution-set}.)
For every \(\Omega\in 25\mathbb{Z}\), the explicit spectrum
\(\cM_{\mathrm{full}}(\Omega)
= \cM_{\mathrm{gauge}}(\Omega)\oplus\cG_{\mathrm{grav}}(\Omega)
\oplus\cM_{\mathrm{vis}}^{\mathrm{vec}}\)
satisfies
\begin{equation}
\boxed{I_8^{\mathrm{tot}}
= \frac{\Omega}{2}\,X_4^2,}
\end{equation}
with all irreducible and forbidden terms vanishing exactly.
\end{theorem}

\begin{remark}[Existence, and the complete solution set]
\label{rem:complete-solution-set}
Theorem~\ref{thm:model-level} exhibits one explicit finite spectrum that realizes the
\BIsix{} factorization, and the benchmark is not unique, minimal, or physically preferred.
More can now be said, and it upgrades the result from a witness to a classification. The
four matching equations \eqref{eq:dioph1}--\eqref{eq:dioph4} are a linear system in five
multiplicities, so their solution set is an affine line, and that line can be written down
in closed form for an arbitrary Abelian coefficient \(\gamma\):
\begin{equation}
\begin{aligned}
a_1 &= 6\Omega\gamma^2 - 8\Omega\gamma + \tfrac{64}{15}\Omega - \tfrac{64}{3}d_3, &\qquad
a_2 &= \tfrac34\Omega\gamma^2 + \tfrac12\Omega\gamma - \tfrac{4}{15}\Omega + \tfrac43 d_3,\\
d_1 &= -\tfrac{15}{2}\Omega\gamma^2 + 8\Omega\gamma - \tfrac{32}{15}\Omega + \tfrac{77}{3}d_3, &\qquad
d_2 &= \tfrac34\Omega\gamma^2 - \tfrac12\Omega\gamma + \tfrac{2}{15}\Omega - \tfrac{20}{3}d_3,
\end{aligned}
\label{eq:affine-general-gamma}
\end{equation}
with \(d_3\) free. At \(\gamma = 2/5\) this reduces to the family
\eqref{eq:affine-a1}--\eqref{eq:affine-d2} printed in Appendix~\ref{app:diophantine-full},
and at \(d_3 = 2\Omega/25\) to the benchmark \eqref{eq:orbit-solution}. So the admissible
gauge reservoirs are \emph{exactly} the integer points of \eqref{eq:affine-general-gamma},
and the benchmark is the self-similar branch of a known family rather than a fortunate find.

Two consequences are worth recording, since both cut against a natural over-reading. The
first is that the count equation \eqref{eq:dioph1} contains no \(\gamma\); the signed
reservoir dimension is therefore the same for every Abelian coefficient, and with it the
gravitational deficit and the completion \((n_{\mathrm{SD}}, n_F) = (4\Omega, 32\Omega)\).
The gravitational books, as this Part has said, see only the count and the chirality of what
propagates --- which is precisely why they cannot see \(\gamma\). The second is that
\eqref{eq:affine-general-gamma} is solvable in integers for every rational \(\gamma\), so
the existence of an anomaly-free spectrum does not distinguish the \BIsix{} value. It is
fixed by the trace-ratio identity of Theorem~\ref{thm:ogn-bi6}, and by nothing in this
Part.
The benchmark is a transparent, verifiable construction whose
purpose is to demonstrate that the structural factorization
condition of Theorem~\ref{thm:bi6-cancellation} is not vacuous---it
is satisfiable by a concrete chiral reservoir.
Whether a smaller or more economical realization exists is a
separate mathematical question.
\end{remark}

\begin{remark}[Normalization and the link to Theorem~\ref{thm:bi6-cancellation}]
\label{rem:omega-link}
The model-level theorem produces \(I_8 = (\Omega/2)\,X_4^2\) rather
than the primitive \(I_8 = \half X_4^2\). These are equivalent at the
level of anomaly cancellation: the overall factor \(\Omega\) is absorbed
into the normalization of the tensor kinetic term and the Green--Schwarz
coupling \(\kappa\). The structurally important fact is that \(I_8\)
factorizes as a perfect square through \(X_4^{\mathrm{BI6}}\) with
the coefficient vector \((1,-1,-2/5,-2/5)\), with all irreducible and
forbidden terms vanishing exactly. This discharges proof obligations
\textbf{(PO1)--(PO4)} of Section~\ref{sec:eight-parts};
\textbf{(PO5)--(PO6)} are discharged by Theorem~\ref{thm:bi6-cancellation}
and its Corollary; \textbf{(PO7)--(PO8)} are discharged by
Theorem~\ref{thm:bi6-global-complete} in
Section~\ref{sec:global-completion}.
Note that the global theorem (including the Wu-shift integrality condition
of hypothesis H3) uses the primitive integral class
\(\widehat X = \frac{5}{8\pi^2}X_4 \in H^4(M_6,\mathbb{Z})\),
which is defined independently of \(\Omega\).
The value of \(\Omega\) affects the size of the benchmark reservoir
but not the integrality or quadratic-refinement properties of the
\BIsix{} cocycle. The global anomaly theorem is therefore valid for
every \(\Omega_{\mathrm{orb}} \in 25\mathbb{Z}\), not only for the
minimal branch \(\Omega_{\mathrm{orb}} = 25\). (The
\emph{effective} normalization
\(\Omega_{\mathrm{eff}} = \Omega_{\mathrm{orb}}/L^2\), which is what
the late-time descent of Section~\ref{sec:gs-normalization} drives to
unity, is a different quantity; the bare symbol \(\Omega\) always means
\(\Omega_{\mathrm{orb}}\) in this appendix.)
\end{remark}



\subsection{The Green--Schwarz normalization \texorpdfstring{\(\Omega\)}{Omega}: static multiplicity versus late effective normalization}
\label{sec:gs-normalization}

Two logically different uses of the symbol \(\Omega\) must be separated if the \BIsix{} theorem and the later \GDOS{} applications are to remain conceptually clean.

\paragraph{Static reservoir multiplicity.}
The Diophantine benchmark of Section~\ref{sec:model-level} proves the existence of an anomaly-matching reservoir with integer orbit multiplicities whenever
\[
\Omegaorb\in25\mathbb Z.
\]
This \(\Omegaorb\) is a \emph{static multiplicity label} for the microscopic benchmark reservoir. It tells us how many copies of the primitive orbit packet appear in a given arithmetic solution.

\paragraph{Primitive anomaly normalization.}
The anomaly theorem itself is written in primitive normalization,
\[
I_8=\frac12 X_4^2,
\]
with the global counterterm built from the primitive integral lift \([\widehat X]\). Therefore the complete local and global anomaly theorem does \emph{not} require a separate proof that \(\Omegaorb=1\). The existence theorem for the benchmark reservoir and the primitive theorem of anomaly freedom are logically distinct statements.

\paragraph{Late operational normalization.}
The physically observed low-energy normalization is not the microscopic reservoir count itself. In \GDOS{} one observes a coarse-grained boundary packet produced by \OGGEP{}. The relevant symbol is therefore a different quantity, denoted here by \(\Omegaeff\).

\paragraph{Condensed \GDOS{} attractor sketch.}
Let the primitive \GG{} benchmark packet decompose into \(L^2\) symmetry-equivalent microscopic cells, with \(L=5\) from the \OGNfive{} theorem. Assume the \OGGEP{} boundary channel registers the normalization carried by one operationally coarse-grained cell, while the full microscopic reservoir remains conserved in the bulk. Then the late-time \GDOS{} attractor implies
\[
\Omegaeff(\mu)=\Omegaorb\,w_\ast(\mu),
\]
where \(w_\ast(\mu)\) is the boundary weight of one representative cell. Under \(\DGI\), the total primitive packet is conserved; under \(\GSSB\), the \(L^2\) microscopic cells become boundary-distinguishable; under \(\GA\), symmetry-equivalent cells are driven to the equal-weight fixed point
\[
w_\ast(\mu)\longrightarrow \frac{1}{L^2}.
\]
Hence
\[
\Omegaeff\longrightarrow \frac{\Omegaorb}{L^2}.
\]
At the primitive benchmark \(\Omegaorb=L^2=25\), one obtains
\[
\Omegaeff\longrightarrow 1.
\]


\paragraph{Physics intuition.}
The microscopic anomaly reservoir counts how many primitive orbit packets exist in the six-dimensional bulk. The boundary observer does not measure that raw count directly. \OGGEP{} first projects the bulk packet to an operational channel; \GDOS{} then subjects the channel to the standard open-system sequence \(\DGI\to\GSSB\to\GA\). The first step preserves the total packet, the second resolves symmetry-equivalent cells, and the third contracts their weights toward the unique symmetric attractor. Therefore the boundary normalization is the attractor quotient of the microscopic reservoir, not the raw reservoir multiplicity itself.

\paragraph{Why the distinction is drawn here.}
The global anomaly theorem of \BIsix{} does not depend on the late-time claim \(\Omegaeff\to1\). Nevertheless, the present sketch is included so that the paper is self-contained on the logical relation between the static anomaly normalization and the operational normalization used later in \GDOS{} and cosmology. The anomaly theorem proves the primitive statement; \GDOS{} explains why the observed effective normalization is the attractor image of that primitive statement.

\bigskip

\noindent\textbf{What Part~II has established, and on what.} The
title claim of this paper is now proved, and it is worth separating what
was proved from what was assumed, because the two are easy to blur.

Locally, the cancellation is exact and unconditional. Given the
four-component identity \BIsix{} and the matter content the
constitution specifies, the one-loop anomaly polynomial factorizes as a
perfect square, the irreducible and mixed terms vanish, and the single
tensor field cancels what remains. Nothing in that argument is
approximate and nothing in it is a choice; the coefficients that make it
work are the ones the trace ratio hands over, and there is no free
parameter anywhere in the calculation.

Globally, there are two results, at two strengths, and the difference
matters. Theorem~\ref{thm:bi6-global-complete} proves that the four
hypotheses of the differential-cohomology framework hold automatically
on the intended \GG{} background class --- the product and collar
geometries the theory actually uses --- so the global phase is trivial
there. Theorem~\ref{thm:unrestricted-global} then removes the
topological fine print: the relevant spin bordism groups vanish, so the
same conclusion holds on every compact spin base and for large gauge
transformations of any topological type. The second theorem is the
stronger claim, and it closes a question this program had previously
listed as open.

The benchmark spectrum does something narrower than either, and the
narrowness is the point. It proves the theorem is not vacuous: an
explicit finite chiral spectrum exists that saturates the factorization,
with a definite number of fermions in definite representations and a
gravitational sector that closes. It does \emph{not} prove that this
spectrum is the one nature chose, and it does not prove it is the
smallest. Whether the minimal branch is the unique physical realization
is recorded as an open problem, not quietly assumed.

One honest limitation belongs here rather than in a footnote. The
anomaly test is passed by a whole family of Abelian coefficients, not by
the \BIsix{} value alone; Remark~\ref{rem:complete-solution-set} says so
explicitly and the proof is written to make the dependence visible. What
singles out the \BIsix{} value is not consistency but the constitution
--- specifically the trace ratio of the unified block, which is the
subject of Part~III. A reader who wants to know why the coefficients are
what they are should read on; a reader who wants to know whether the
theory is consistent has the answer already.

\GGPart[OGN-5 arithmetic]{\OGNfive{} Arithmetic}{The integer ledger as a theorem, not an input}

The five integers \((M,N,L;J,K)=(1,3,5;7,35)\) were introduced in Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6}
as a compact summary of the normalization freedom in the \BIsix{} identity.
This Part demonstrates that they are not free parameters:
every one of them is forced by the \GG{} gauge architecture and the
\BIsix{} trace-ratio structure.
Section~\ref{sec:ogn5-arithmetic} shows how \OGNfive{} determines the
anomaly-matching arithmetic, and Section~\ref{sec:ogn5-theorem} upgrades
the entire packet to theorem level. The complete derivation is given in
Appendix~\ref{app:ogn5-complete}.

The claim in this Part is unusual enough to be worth stating carefully
before the machinery starts. Most theories of fundamental physics
contain numbers that have to be measured and then put in by hand: the
Standard Model has twenty-eight of them. The proposal here is that five
small integers --- one, three, five, seven, thirty-five --- are not
inputs of that kind. They are forced, each by an arithmetic condition
that the previous one has already fixed, and the chain starts from the
anomaly identity proved in Part~II rather than from any measurement.

What makes such a claim checkable rather than mystical is that the
conditions are ordinary Diophantine equations --- equations required to
have whole-number solutions --- and whole-number conditions are
unforgiving. A quantity that must be an integer either is one or is not;
there is no adjusting it slightly to fit. So the reader's task in this
Part is not to be persuaded but to check that each step really is forced
and that no measured quantity has been smuggled in as a premise. The
text tries to make that easy: at each step it names what is being
consumed, and where a step rests on a constitutional clause rather than
on arithmetic alone, it says so in that sentence rather than in a
caveat later.

Two of the five integers do most of the visible work downstream: $L=5$,
which is the size of the unified block, and $J=7$, which is what makes
the inverse fine-structure constant come out to $2^7+3^2=137$ at leading
order. Both are derived here.

\section{\OGNfive{} arithmetic and the spectrum problem}
\label{sec:ogn5-arithmetic}

\subsection{The \BIsix{} coefficient as a function of \texorpdfstring{\((N,L)\)}{(N,L)}}

The \BIsix{} Abelian coefficient is not an independent parameter.
It is a \emph{consequence} of the \OGNfive{} integers.

\begin{theorem}[\textbf{T-BI6-9.1}: \OGN{} determination of the \BIsix{} coefficient]
\label{thm:ogn-bi6}
For a gauge group \(\SU(L)\) with the standard embedding of
\(\SU(N)\times\SU(L{-}N)\times\U(1)_Y\), the Abelian trace ratio is
\begin{equation}
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)} = \frac{L-N}{L}.
\label{eq:ogn-trace-ratio}
\end{equation}
For \((N,L)=(3,5)\): the \BIsix{} coefficient is \((5{-}3)/5 = 2/5\).
\end{theorem}

\begin{proof}
The hypercharge generator in the fundamental \(\mathbf{L}\) is
\[
Y_0 = \diag\!\paren{-\frac{1}{N},\ldots,-\frac{1}{N},\,
\frac{1}{L{-}N},\ldots,\frac{1}{L{-}N}}
\]
with \(N\) entries of \(-1/N\) and \(L{-}N\) entries of \(1/(L{-}N)\).
The baryon generator is
\[
B_0 = \diag(1/N,\ldots,1/N,\,0,\ldots,0).
\]
Then
\begin{equation}
\Tr(Y_0^2) = \frac{N}{N^2} + \frac{L{-}N}{(L{-}N)^2}
= \frac{1}{N} + \frac{1}{L{-}N}
= \frac{L}{N(L{-}N)},
\end{equation}
and
\begin{equation}
\Tr(B_0^2) = \frac{N}{N^2} = \frac{1}{N}.
\end{equation}
The ratio is
\(\Tr(B_0^2)/\Tr(Y_0^2) = N(L{-}N)/(NL) = (L{-}N)/L\).
\end{proof}

\subsection{Core block determined by \texorpdfstring{\(L\)}{L}}

\begin{theorem}[\textbf{T-BI6-9.2}: \OGN{} determination of the core block]
\label{thm:ogn-core}
The core block \eqref{eq:core-block} has the following data
expressed entirely in terms of the \OGN{} integer \(L\):
\begin{align}
d_{\cC} &= (L{+}1)^2, \label{eq:ogn-d}\\
A_{\cC} &= L{+}1, \label{eq:ogn-A}\\
n_{\mathrm{singlets}} &= \frac{L^2-L+2}{2}, \label{eq:ogn-singlets}\\
\frac{d_{\cC}}{A_{\cC}} &= L{+}1. \label{eq:ogn-ratio}
\end{align}
For \(L=5\): \(d_{\cC}=36\), \(A_{\cC}=6\),
\(n_{\mathrm{singlets}}=11\), \(d_{\cC}/A_{\cC}=6\).
\end{theorem}

\begin{proof}
The core block is
\(\cC = \mathbf{L(L{-}1)/2}\oplus\mathbf{L}\oplus
2\,\overline{\mathbf{L}}\oplus n_1\cdot\mathbf{1}\)
where \(n_1\) is chosen so that \(d_{\cC} = (L{+}1)\cdot A_{\cC}\).
In the convention used throughout this paper, where \(A(\mathbf{5})=1\)
(i.e., \(A_\rho = 2T_2(\rho)\)), the quadratic index is
\(A_{\cC} = (L{-}2) + 1 + 2 = L+1\).
The dimension is
\(d_{\cC} = L(L{-}1)/2 + 3L + n_1\).
Setting \(d_{\cC} = (L{+}1)^2\) gives
\(n_1 = (L{+}1)^2 - L(L{-}1)/2 - 3L = (L^2-L+2)/2\).
For \(L=5\): \(n_1 = (25-5+2)/2 = 11\).
\end{proof}

\subsection{Minimum normalization determined by \texorpdfstring{\(L\)}{L}}

\begin{theorem}[\textbf{T-BI6-9.3}: \OGN{} determination of the minimum \(\Omega\)]
\label{thm:ogn-omega}
The orbit solution \eqref{eq:orbit-solution} has denominators of
\(L^2\). Therefore
\begin{equation}
\Omega_{\min}^{\mathrm{gauge/mixed}} = L^2 = 25.
\label{eq:ogn-omega-min}
\end{equation}
\end{theorem}

\begin{remark}[Proposed lattice normalization]
\label{rem:omega-lattice}
A sharper bound,
\begin{equation}
\Omega_{\min}^{\mathrm{lattice}} = 2N^2L^2 = 450,
\label{eq:ogn-omega-lattice}
\end{equation}
has been proposed for the charge-lattice well-definedness of the
exponentiated Green--Schwarz term (proof obligation (PO8) of
Section~\ref{sec:eight-parts}). Its derivation is not carried out in
this paper, and no downstream result uses it. On the intended class,
(PO8) is discharged instead by the integral lift and the Wu-shifted
quadratic refinement of Appendix~\ref{app:global-proof-dossier}
(cf.\ Remark~\ref{rem:omega-link}), which hold for every
\(\Omega \in 25\mathbb{Z}\), so the benchmark \(\Omega = 25\) is not
in tension with (PO8). Whether the sharper lattice normalization is
required in the unrestricted category is folded into Open
Problem~O10.
\end{remark}

\subsection{Summary: what \OGNfive{} determines}

\begin{center}
{\footnotesize
\begin{longtable}{p{0.30\textwidth}p{0.354\textwidth}p{0.189\textwidth}}
\caption[Quantities fixed by the integer ledger, with their values.]{Everything the integer ledger determines, in one table. The second column gives each quantity as an expression in the ledger integers and the third the number that results. The point is the second column: each entry is arithmetic in integers with no continuous input anywhere, so a reader can recompute every value in the third column by hand. If a single ledger integer were different, every row would move together --- which is what makes the ledger falsifiable rather than merely economical.}\label{tab:ogn-summary}\\
\toprule
\textbf{Quantity} & \textbf{\OGN{} expression} & \textbf{Value} \\
\midrule
\endfirsthead
\toprule
\textbf{Quantity} & \textbf{\OGN{} expression} & \textbf{Value} \\
\midrule
\endhead
\BIsix{} Abelian coefficient & \((L{-}N)/L\) & \(2/5\) \\
Core block dimension & \((L{+}1)^2\) & \(36\) \\
Core block index & \(L{+}1\) & \(6\) \\
Singlets per core block & \((L^2{-}L{+}2)/2\) & \(11\) \\
Minimum \(\Omega\) (gauge/mixed) & \(L^2\) & \(25\) \\
Minimum \(\Omega\) (lattice; proposed, Remark~\ref{rem:omega-lattice}) & \(2N^2L^2\) & \(450\) \\
Charge lattice unit & \(1/N\) & \(1/3\) \\
Color triplicity index & \(N \equiv N_c\) & \(3\) \\
Gauge group rank & \(L{-}1\) & \(4\) \\
Spacetime dimension & \(L{+}1\) & \(6\) \\
\bottomrule
\end{longtable}
}
\end{center}

Table~\ref{tab:ogn-summary} collects what the ledger determines
within this paper. The residual freedom not determined by \OGNfive{} is
limited to the gravitational completion sector \((n_{\mathrm{SD}},\,n_F)\)
and the overall normalization \(\Omega\).
The former is a \(2\times 2\) linear system; the latter is
subject to minimization by \AX{4}/\POS{6} (\MSMF/\OGN).



\section{\OGNfive{} as a theorem derived from \BIsix{}}
\label{sec:ogn5-theorem}

\subsection{Precise meanings of the five integers}

\begin{definition}[Meaning of \texorpdfstring{$M,N,L;J,K$}{M,N,L;J,K}]
\label{def:ogn-meanings}
In the intended \GG{} category:
\begin{enumerate}[leftmargin=1.8cm,label=(\roman*)]
\item \(M\) is the \textbf{primitive tensor/gravitational slot count}. It counts the
 minimal Green--Schwarz tensor channel carried by the single \BIsix{} tensor
 lattice.
\item \(N\) is the \textbf{color triplicity index}, written \(N_c\)
 wherever the physical color count rather than the ledger slot is meant.
 It is the size of the color block carried inside the defining
 representation of the unified group. It is \emph{not} the number of
 fermion generations: that quantity is written \(N_g\), it is not a
 member of this ledger, and it is fixed by an argument of an entirely
 different kind \citep{ArisakaP2N3}. That \(N_c = N_g = 3\) is a
 coincidence on the boundary, and this paper never uses it.
\item \(L\) is the \textbf{defining unified block size}. In \GG{} it is the defining
 dimension of the minimal simple unified group \(\SU(5)\).
\item \(J\) is the \textbf{minimal symmetric closure cardinality}: if the
largest allowed absolute Abelian orbit level is \(N\), then \OGN{}
closure together with sign symmetry requires the charge packet
\(\{0,\pm1,\pm2,\dots,\pm N\}\), and \(J\) is the cardinality of that
packet, so \(J = 2N+1\). (Appendix~\ref{app:ogn5-complete} states the
same definition; the two must agree, because Step~4 of
Theorem~\ref{thm:ogn5-theorem-updated} evaluates this one.) The symmetric, centered form of the charge packet is a
 structural claim and not a convention: among admissible closure packets
 it is the minimal one symmetric about a neutral slot, and the selection
 is \AX{4}~(\MSMF{}) acting through \POS{6}~(\OGN{}). \(J\) is
 accordingly not an independent integer of the ledger but a structural
 function of \(N\); what carries content is the minimality clause, which
 is constitutional and is named as such.
\item \(K\) is the \textbf{total lock index}. It is simultaneously the rectangular
 product \(LJ\) and the quadratic norm of the normalization packet.
\end{enumerate}
\end{definition}

\paragraph{Three counts, three symbols.} Three integers of value three
occur in this program and must not be conflated. \(N_c = 3\) is the
color count --- the \OGNfive{} integer \(N\) of
Definition~\ref{def:ogn-meanings}, and the \(N_c\) of the QCD beta
function. \(N_g = 3\) is the number of fermion generations, which is not
an \OGNfive{} member and is established in \citep{ArisakaP2N3}. \(N_f\)
retains its standard meaning throughout, the number of quark flavors
entering a beta function, and is \(6\) at high scale --- so that
\(11N_c - 2N_f = 33 - 12 = 21\) reads as it always has. Nothing in the
\OGNfive{} cascade uses \(N_g\), and no step below consumes it.

\subsection{The unified block size from single-tensor consistency alone}
\label{ssec:L-from-GS}

\paragraph{The one integer still taken as given.} Of the five integers, \(L\) has until now been
the one taken in from outside. The cascade reads it off the statement that \(\SU(5)\) is
the smallest simple group containing the observed low-energy gauge content --- and that
statement already knows how many colors there are. An integer obtained that way cannot
honestly be used, later in the same cascade, to conclude how many colors there are. The
present subsection removes the circle at its source. It shows that \(L = 5\) follows from
the Green--Schwarz construction itself: no observed gauge group, no block split, and no
appearance of \(N\) at any step.

\paragraph{The physical idea, before any algebra.} A single two-form can only mop up an
anomaly that has the shape of a perfect square. That is the entire content of the
rank-one criterion (Theorem~\ref{thm:rank-one}): what the tensor can absorb is
\(\tfrac12 (v\cdot x)^2\), where \(x\) collects the four \emph{quadratic} curvature
invariants \(\tr R^2\), \(\Tr F_5^2\), \(F_B^2\), \(F_{Qi}^2\).

Squaring a quadratic form is a restrictive operation, and it is worth pausing on how
restrictive. It can manufacture \(\Tr F_5^2 \wedge F_B^2\), because that is a product of
two of the four invariants. It can never manufacture \(\Tr F_5^4\) --- meaning the
genuinely irreducible quartic trace, the one that is not a product of two quadratic
traces --- because a product of two second-order objects is by construction reducible.
And it can never manufacture \(\Tr F_5^3 \wedge F_B\) either, for a simpler reason still:
\(X_4\) contains no cubic invariant anywhere in it, so there is nothing in the square that
could supply the \(\Tr F_5^3\).

Now read that as a statement about the matter rather than about the tensor. A chiral
spectrum in six dimensions produces, in general, all of these monomials. Two of them have
no channel. The bulk is not free to generate an irreducible \(\Tr F_5^4\), or a mixed
\(\Tr F_5^3 \wedge F_B\), and then hope that the Green--Schwarz term will take care of
them: there is no term available that could. They must cancel \emph{among the fermions
themselves}, before the inflow mechanism is invoked at all.

This is the point at which a familiar-looking requirement becomes a genuinely different
one. In the usual model-building idiom one says ``choose a spectrum whose anomalies
cancel.'' Here the single-tensor architecture says something sharper: two specific
group-theoretic indices of the charged matter --- the irreducible quartic index \(B\) and
the cubic index \(A_3\) --- must vanish identically, as a \emph{precondition} for
single-tensor inflow rather than as a consequence of it. The present paper has been
using that precondition all along; Section~\ref{ssec:core-block-construction} builds its
core block precisely so that ``irreducible and cubic anomaly indices vanish internally.''

What has not been done until now is to decline to evaluate those two indices at a chosen
block size, and to carry \(L\) as a symbol instead. Do that, and each condition becomes an
equation in \(L\). They turn out to be the same equation, and it has one root.

\begin{theorem}[\textbf{T-BI6-9.0}: Unified block size from single-tensor consistency]
\label{thm:L-from-GS}
Let the charged core package of the bulk be
\begin{equation}
\cP(p,f,g) \;=\; p\,\Lambda^2 \;\oplus\; f\,\mathbf{L} \;\oplus\; g\,\overline{\mathbf L},
\qquad p,f,g \in \mathbb{Z}_{\ge 0},
\label{eq:core-package-general}
\end{equation}
built from the defining representation \(\mathbf L\) of \(\SU(L)\), its conjugate, and the
two-index antisymmetric \(\Lambda^2\), with all components carrying common Abelian charges
and the package completed by \(\SU(L)\) singlets. Single-tensor Green--Schwarz consistency
requires that the irreducible quartic index and the cubic index vanish on the package,
\begin{equation}
B[\cP] \;=\; 0, \qquad A_3[\cP] \;=\; 0 .
\label{eq:two-preconditions}
\end{equation}
These two conditions hold if and only if
\begin{equation}
g \;=\; 2p, \qquad f \;=\; p\,(6-L).
\label{eq:package-solution}
\end{equation}
Imposing in addition \(p = 1\) (a single antisymmetric, by \AX{4}~\MSMF{}), \(f \ge 1\)
(the package contains a defining representation), and chirality of the
surviving package (which excludes \(L \le 4\); see Step~4 of the proof
of Theorem~\ref{thm:L-from-GS} for the three cases), the unique
solution is
\begin{equation}
\boxed{\;L \;=\; 5, \qquad
\cP \;=\; \Lambda^2 \oplus \mathbf{5} \oplus 2\,\overline{\mathbf 5}
\;=\; \mathbf{10} \oplus \mathbf{5} \oplus 2\,\overline{\mathbf 5}. \;}
\label{eq:L-equals-5-Nfree}
\end{equation}
Equivalently, for the core block \(\cC\) of Eq.~\eqref{eq:core-block},
\begin{equation}
B_{\cC}(L) \;=\; A_{3,\cC}(L) \;=\; L - 5 .
\label{eq:BA3-equal-Lminus5}
\end{equation}
Neither the statement nor the proof refers to a color count, to the boundary gauge
content, or to any split of the defining representation.
\end{theorem}

\begin{proof}
\emph{Step 1: which indices the tensor cannot absorb.} This is
Theorem~\ref{thm:single-tensor-necessity}, proved in Part~II: single-tensor inflow through a
four-form quadratic in the curvatures can reach neither the irreducible quartic
\(\Tr F_5^4\) nor the mixed cubic \(\Tr F_5^3\wedge F_B\), so both must cancel on the
spectrum itself. With all components of \(\cP\) carrying common Abelian charges the charge
factors out of the cubic condition, leaving \(A_3[\cP] = 0\), and the quartic condition
leaves \(B[\cP] = 0\). This is \eqref{eq:two-preconditions}.

\emph{Step 2: the indices, computed from the weights.} Let
\(H = \diag(h_1,\dots,h_L)\) be a traceless diagonal generator and write
\(P_k := \sum_i h_i^{\,k}\), so that \(P_1 = 0\). The weights of \(\Lambda^2\) are
\(h_i + h_j\) with \(i < j\), whence
\begin{align}
\sum_{i<j} (h_i+h_j)^2 &= (L-2)\,P_2 ,
\label{eq:asym-p2}\\
\sum_{i<j} (h_i+h_j)^3 &= (L-1)P_3 + 3\big(P_2 P_1 - P_3\big) \;=\; (L-4)\,P_3 ,
\label{eq:asym-p3}\\
\sum_{i<j} (h_i+h_j)^4 &= (L-1)P_4 + 4\big(P_3 P_1 - P_4\big) + 3\big(P_2^2 - P_4\big)
\;=\; (L-8)\,P_4 + 3\,P_2^2 .
\label{eq:asym-p4}
\end{align}
In the normalization \(A(\mathbf L) = 1\) used throughout this paper, and writing
\(\Tr_\rho H^4 = B_\rho \Tr_{\mathbf L} H^4 + C_\rho (\Tr_{\mathbf L} H^2)^2\), these read
\begin{equation}
A(\Lambda^2) = L-2, \quad A_3(\Lambda^2) = L-4, \quad
B(\Lambda^2) = L-8, \quad C(\Lambda^2) = 3 ,
\label{eq:asym-indices}
\end{equation}
while for the defining representation and its conjugate
\(A = 1\), \(A_3 = \pm 1\), \(B = 1\), \(C = 0\), the cubic index changing sign under
conjugation and the quartic index not.

Two of these four values are already in use in the present paper and serve as a check on
the normalization: \eqref{eq:asym-p2} is what Theorem~\ref{thm:ogn-core} uses when it
writes \(A_{\cC} = (L-2) + 1 + 2 = L+1\), and the \(3P_2^2\) of \eqref{eq:asym-p4} is what
gives the reducible coefficient \(C_{\cC} = 3\) of Proposition~\ref{prop:core-block}.

\emph{Step 3: the two conditions become one equation.} Summing the indices over
\eqref{eq:core-package-general},
\begin{align}
A_3[\cP] &= p\,(L-4) + f - g ,
\label{eq:A3-package}\\
B[\cP] &= p\,(L-8) + f + g .
\label{eq:B-package}
\end{align}
Setting both to zero and taking the difference gives \(4p - 2g = 0\), hence \(g = 2p\);
taking the sum gives \(2p(L-6) + 2f = 0\), hence \(f = p(6-L)\). This is
\eqref{eq:package-solution}. Substituting \(p=1\), \(f = 1\), \(g = 2\) back into
\eqref{eq:A3-package}--\eqref{eq:B-package} gives \eqref{eq:BA3-equal-Lminus5}:
\(A_{3,\cC} = (L-4) + 1 - 2 = L-5\) and \(B_{\cC} = (L-8) + 1 + 2 = L-5\), the same
linear function twice.

\emph{Step 4: the selection.} With \(p = 1\) the solution family is \(f = 6-L\), \(g = 2\),
one package for each \(L\). What excludes \(L \le 4\) is chirality of the
surviving package, and the three cases are different and are worth separating,
because the criterion sometimes quoted --- complexity of \(\Lambda^2\) --- is not
by itself sufficient. At \(L = 2\), \(\Lambda^2\) is the singlet and the package
carries no antisymmetric at all. At \(L = 4\), \(\Lambda^2\) is the self-conjugate
\(\mathbf 6\) and \(f = g = 2\) leaves the remaining sector vector-like. At
\(L = 3\), \(\Lambda^2(\mathbf 3) = \overline{\mathbf 3}\) \emph{is} complex, so
complexity does not exclude it; what excludes it is that the package
\(\overline{\mathbf 3} \oplus 3\,\mathbf 3 \oplus 2\,\overline{\mathbf 3}\) is
\(3\,\mathbf 3 \oplus 3\,\overline{\mathbf 3}\), which is vector-like and
contributes no net anomaly. (This case also disposes of an apparent competitor
on the minimality count of Theorem~\ref{thm:lN-minimality}: the \(L = 3\) package
carries eighteen charged states against the \(L = 5\) package's twenty-five, and
would win on \AX{4} alone were it chiral.) Non-negativity of \(f\) gives
\(L \le 6\), and \(f \ge 1\) gives \(L \le 5\). Hence \(L = 5\), and the package is
\(\mathbf{10} \oplus \mathbf 5 \oplus 2\,\overline{\mathbf 5}\), which is
Eq.~\eqref{eq:core-block}.
\end{proof}

\paragraph{The one competitor, and what excludes it.} The hypothesis \(f \ge 1\) is doing
real work and should not be smuggled past the reader. Dropping it admits exactly one
further package, at \(L = 6\): \(\Lambda^2 \oplus 2\,\overline{\mathbf 6}\), that is
\(\mathbf{15} \oplus 2\,\overline{\mathbf 6}\), with \(f = 0\). It is genuinely chiral and
it satisfies both preconditions. What separates it from \(L=5\) is \AX{4}~(\MSMF{}) in its
ordinary sense: the \(L=5\) package carries \(10 + 5 + 10 = 25\) charged states against the
\(L=6\) package's \(15 + 12 = 27\), so the minimal chiral package consistent with
single-tensor inflow is the \(\SU(5)\) one. This is the same minimality clause that selects
\(k=1\) in Theorem~\ref{thm:lN-minimality}, used the same way and carrying the same status:
a constitutional selection among admissible solutions, not a consequence of the equations.
We say so rather than let the boxed result of Theorem~\ref{thm:L-from-GS} read as though
the algebra alone had closed.

\paragraph{What this does and does not deliver.} It delivers \(L\). The unified block size
is now fixed by the same single-tensor architecture that the rest of Part~II runs on, and
the \OGNfive{} cascade no longer needs the smallest-simple-group clause --- which is to say,
no longer needs to know the low-energy gauge group in order to derive the low-energy gauge
group. It does \emph{not} deliver \(N\). The Abelian entry of the coefficient vector is
\((L-N)/L\), a function of \(N\) whatever \(L\) may be, and nothing in the argument above
touches it. The trace-ratio route to the color index remains what
Section~\ref{sec:ogn5-theorem} says it is, and the \(3+2\) split that route presupposes is
not supplied here. What has changed is the honesty of the bookkeeping: one of the two
integers that used to arrive from the boundary now arrives from the bulk, and the one that
still arrives from the boundary is no longer able to hide behind it.


\subsection{The trace-ratio theorem that fixes the \BIsix{} Abelian fraction}

The identity used in this step is Theorem~\ref{thm:ogn-bi6} of
Section~\ref{sec:ogn5-arithmetic}, proved there and carried out in full
detail in Appendix~\ref{app:ogn5-complete}. It is restated here in the
form the cascade consumes, and not re-derived a third time; a reader who
wants the computation should go to one of those two places.

Let a general \(SU(L)\) block split into a color block of size \(N\) and a
complementary block of size \(L-N\). Define the diagonal directions
\[
B_{N,L}=\diag\!\left(\underbrace{\frac1N,\dots,\frac1N}_{N},\underbrace{0,\dots,0}_{L-N}\right),
\]
\[
Y_{N,L}=\diag\!\left(\underbrace{-\frac1N,\dots,-\frac1N}_{N},\underbrace{\frac1{L-N},\dots,\frac1{L-N}}_{L-N}\right).
\]
The first is the primitive baryonic block direction; the second is the traceless
hypercharge-like splitting direction. A direct trace computation gives
\[
\Tr(B_{N,L}^2)=\frac1N,
\qquad
\Tr(Y_{N,L}^2)=\frac1N+\frac1{L-N}=\frac{L}{N(L-N)}.
\]
Therefore
\begin{equation}
\frac{\Tr(B_{N,L}^2)}{\Tr(Y_{N,L}^2)} = \frac{L-N}{L}.
\label{eq:trace-ratio-general-updated}
\end{equation}
This formula is the structural bridge from \BIsix{} to \OGNfive{}. The \BIsix{} coefficient
requires the Abelian fraction to be exactly \(2/5\); hence the trace-ratio
closure condition is
\begin{equation}
\frac{L-N}{L}=\frac25.
\label{eq:LN-over-L-updated}
\end{equation}

\paragraph{Why the trace-ratio is canonical, not a normalization choice.}
A careful reader may worry that the ratio $(L{-}N)/L = 2/5$ depends on
the specific generator normalization used for $B_{N,L}$ and $Y_{N,L}$
above, so that a different but equally natural normalization could
produce a different ratio. The concern is misplaced: the ratio
$\Tr(B^2)/\Tr(Y^2)$ is invariant under any rescaling
$B \mapsto \lambda B$, $Y \mapsto \mu Y$ that preserves the
\emph{integer charge content} of the Standard Model fermions on the
$\SU(5)$ block --- which is exactly the invariant that \BIsix{}
requires the generators to carry. Any rescaling that changes the
ratio also changes the integer charges, and the rescaled generators
no longer represent the physical baryon and hypercharge directions
of the Standard Model embedding. The
trace-ratio takes its value in the moduli space of normalized
SU(5)-embedded generators with fixed integer charges, and that
moduli space is a single point. In particular, the ratio is
invariant under sign flips, basis rotations within the
hypercharge-like subspace, and any rescaling consistent with the
integer charge ledger of the Standard Model fermions. More specifically,
this is the structural reason the trace ratio is rigid once the integer
charge content of the SM is taken as input. It is not, however, what
fixes the ledger: the trace-ratio equation $(L{-}N)/L = 2/5$ admits the
whole family $(L,N) = (5k, 3k)$, as Theorem~\ref{thm:lN-minimality}
proves, and the integers are fixed instead by
Theorem~\ref{thm:L-from-GS} and Theorem~\ref{thm:N-from-closure}, which
consume no empirical input at all.

\medskip\noindent\emph{Scope of this route.} The $2/5$ fraction is
rigid only \emph{after} the integer charge content of the Standard Model
--- in particular the $3{+}2$ color/weak split --- is taken as input. A
derivation of $N$ from that fraction would therefore consume the split it
concludes, and this paper does not attempt one: the trace ratio is used
as a check and never as a derivation. The non-circular statement that
$N = N_c = 3$ is \emph{forced} is Theorem~\ref{thm:N-from-closure}, which
takes $L$ from Theorem~\ref{thm:L-from-GS} --- free of $N$ throughout ---
and closes on the dual lock and the block embedding, consuming no
measured quantity.

\medskip\noindent It is worth being explicit about what this does
\emph{not} settle, because the two are easily run together. The stability
sandwich of \PaperPTwo{} --- $N_g \ge 3$ from the CP lower bound and
$N_g \le 3$ from the \BIsix{} instanton-action budget
$S_{\rm BI}^{\max} = 3\pi$ --- concerns the \emph{generation} count
$N_g$, not the color index, and it is conditional on that paper's
coupling clause and on the integrality theorem L-INT, whose single topological residue R-INT is discharged on the benchmark background in that paper's Appendix~H. Neither leg of that sandwich
bears on $N_c$, and nothing in the present cascade appeals to it.

\subsection{The \OGNfive{} theorem}

\begin{theorem}[\textbf{T-BI6-10.2}: \OGNfive{} theorem]
\label{thm:ogn5-theorem-updated}
Within the intended \GG{} category, \BIsix{} and the unified block structure force
uniquely
\[
(M,N,L;J,K)=(1,3,5;7,35),
\qquad
K=LJ=M^2+N^2+L^2.
\]
\end{theorem}

\begin{proof}
The proof is a five-step chain.

\paragraph{Step 1: $M=1$.}
\GG{} uses a single primitive Green--Schwarz tensor channel. Equivalently, the
tensor lattice is rank one. Therefore the primitive tensor slot count is
\[
M=1.
\]

\paragraph{Step 2: $L=5$.}
By Theorem~\ref{thm:L-from-GS}, single-tensor Green--Schwarz consistency requires the
irreducible quartic and cubic indices of the charged core package to vanish, and both are
equal to \(L-5\); with \AX{4}~(\MSMF{}) selecting the minimal chiral package this gives
\[
L=5,
\]
the unified block being \(\SU(5)\). The \OGN{} symbol \(L\) denotes the defining
unified block size, which is why the correct interpretation is ``block size'' rather than
``rank.'' Note that this step no longer appeals to the smallest-simple-group clause, and
in particular does not consume the observed low-energy gauge content: it is free of \(N\)
at every stage, which matters because Step~3 below is not.

\paragraph{Step 3: $N = N_c = 3$.}
Steps 1 and 2 fixed \(M\) and \(L\) without reference to any color
count: Theorem~\ref{thm:L-from-GS} is free of \(N\) at every stage. That
is what makes the present step available, and it is worth saying why it
is taken this way rather than through the trace ratio. The \BIsix{}
Abelian fraction \((L-N)/L\) is computed on the block split of the
defining representation, so a derivation of \(N\) from that fraction
would consume the split it concludes. The closure route below consumes
nothing.

By Definition~\ref{def:ogn-meanings}, \(K\) is the total lock index, and
it is characterized twice by the structure: as the rectangular product
\(LJ\) of block size and closure index, and as the quadratic norm
\(M^2+N^2+L^2\) of the normalization packet. Both characterizations are
structural, so the two must agree. This is the dual lock, and it is a
constraint on the admissible integers rather than a numerical
coincidence to be admired after the fact. With \(J = 2N+1\) from the
closure-index clause,
\begin{equation}
L\,(2N+1) \;=\; M^2 + N^2 + L^2 ,
\label{eq:dual-lock-in-N}
\end{equation}
which at \(M=1\) is the quadratic
\begin{equation}
N^{2} - 2LN + \bigl(1 + L^{2} - L\bigr) \;=\; 0 ,
\qquad\text{hence}\qquad
N \;=\; L \pm \sqrt{\,L-1\,} .
\label{eq:N-from-closure}
\end{equation}
Integrality of \(N\) therefore requires \(L-1\) to be a perfect square,
which already restricts the admissible block sizes to
\(L \in \{2, 5, 10, 17, 26, \dots\}\). At the value \(L=5\) delivered by
Step~2 this gives
\[
N^{2} - 10N + 21 = 0 ,
\qquad
N \in \{3,\,7\} .
\]
The root \(N=7\) is inadmissible: \(N\) is the size of the color block
carried inside the \(L\)-dimensional defining representation, so
\(N < L\) is a condition of the embedding and not a preference. It is
the exact mirror of the \(L=2\) exclusion of
Appendix~\ref{app:ogn5-complete}, run in the direction that
Theorem~\ref{thm:L-from-GS} has now made available. Hence
\[
N \;=\; N_c \;=\; 3 ,
\]
uniquely. No empirical input has entered at any point: \(M\) is the
rank-one tensor clause of \citep{ArisakaA2GG6}, \(L\) is
Theorem~\ref{thm:L-from-GS}, \(J = 2N+1\) is the closure-index clause of
Definition~\ref{def:ogn-meanings}, the dual lock is the \OGN{}
normalization requirement of \POS{6}, and \(N<L\) is geometry.

\medskip\noindent\emph{The trace ratio, restored to its proper role.}
Equation~\eqref{eq:LN-over-L-updated} now becomes what it should always
have been: an independent check that the observed \(3{+}2\) split of the
defining representation agrees with a ledger the closure has already
fixed. Evaluating \(\Tr(B_0^2)/\Tr(Y_0^2)\) on the Georgi--Glashow
embedding \citep{GeorgiGlashow1974} returns \((L-N)/L = 2/5\) at
\((L,N) = (5,3)\), as it must. Because the check is no longer
load-bearing it is no longer circular, and it becomes a genuine test: an
Abelian fraction measured away from \(2/5\) would falsify the ledger
rather than redefine it.

\medskip\noindent\emph{What this step does not establish.} \(N\) is the
color triplicity index. Nothing above bears on the generation count
\(N_g\), which is not an \OGNfive{} member and reaches \(3\) by an
argument of a different kind \citep{ArisakaP2N3}. That the two coincide
is a fact about the boundary and is used nowhere in this cascade.

\paragraph{Step 4: $J = 2N+1 = 7$ (the closure-index clause, evaluated).}
The clause \(J = 2N+1\) was already used in Step~3, and it is stated in
Definition~\ref{def:ogn-meanings} as a structural claim rather than a
convention: among admissible closure packets the symmetric, centered one
is minimal, and the selection is \AX{4}~(\MSMF{}) acting through
\POS{6}~(\OGN{}). What remains here is to evaluate it at the \(N\) fixed
by Step~3:
\[
J=2N+1=2\cdot 3+1=7 .
\]
\(J\) is therefore not an independent integer of the ledger but a
structural function of \(N\); its physical content is that the charge
packet \(\{0,\pm1,\pm2,\pm3\}\) has exactly \(J=7\) entries, and what
carries the weight is the minimality clause, which is constitutional and
is named as such.

\paragraph{Step 5: $K=LJ=M^2+N^2+L^2$.}
Rectangular closure gives
\[
K=LJ=5\cdot 7=35.
\]
The quadratic norm of the \OGN{} packet gives
\[
M^2+N^2+L^2 = 1^2+3^2+5^2 = 1+9+25 = 35.
\]
Hence
\[
K=LJ=M^2+N^2+L^2=35.
\]
This proves the claim.
\end{proof}

\begin{theorem}[\textbf{T-BI6-10.1}: Closure determination of the
triplicity index]
\label{thm:N-from-closure}
Assume \(M = 1\), the closure-index clause \(J = 2N+1\), and the dual
lock \(K = LJ = M^2 + N^2 + L^2\). Then
\[
N \;=\; L \pm \sqrt{L-M^2} ,
\]
so at \(M = 1\) integrality of \(N\) requires \(L-1\) to be a perfect
square, and for each admissible \(L\) exactly one root satisfies the
block-embedding condition \(N < L\). At the value \(L = 5\) of
Theorem~\ref{thm:L-from-GS} the admissible root is \(N = N_c = 3\), and
it is unique. The argument refers nowhere to the boundary gauge content,
to the \(3{+}2\) split of the defining representation, to the generation
count \(N_g\), or to any measured quantity.
\end{theorem}

\begin{proof}
Substituting \(J = 2N+1\) into \(LJ = M^2+N^2+L^2\) gives
Eq.~\eqref{eq:dual-lock-in-N}, that is the quadratic
Eq.~\eqref{eq:N-from-closure} in the general form
\(N^2 - 2LN + (M^2 + L^2 - L) = 0\), whose roots are
\(N = L \pm \sqrt{L^2 - M^2 - L^2 + L} = L \pm \sqrt{L - M^2}\). Both
roots satisfy the closure by construction, and their sum is \(2L\), so
exactly one lies strictly below \(L\) whenever the discriminant is
positive. At \(M = 1\), \(L = 5\) the roots are \(3\) and \(7\); the
block embedding \(N < L\) retains \(3\). Uniqueness follows because a
quadratic has no third root.
\end{proof}

Figure~\ref{fig:ogn5-cascade-a3} summarizes the five-step cascade just
proved: two prior theorems force $(M, L) = (1, 5)$ without reference to
any color count, the dual lock and the block embedding then determine
$N = N_c = 3$, and $(J, K) = (7, 35)$ follow from the closure-index
clause and the same dual lock.

\begin{figure}[H]
\centering
\resizebox{0.92\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 every node/.style={inner sep=3pt},
 source/.style={draw=red!55!black, rounded corners=2pt, fill=red!12, minimum width=2.4cm,
 minimum height=0.55cm, align=center, font=\footnotesize},
 cascade/.style={draw=red!55!black, rounded corners=2pt, fill=blue!10, minimum width=1.6cm,
 minimum height=0.55cm, font=\bfseries},
 lock/.style={draw=red!55!black, circle, fill=green!18, minimum size=1.05cm,
 line width=0.53pt, font=\bfseries},
 idbox/.style={draw=red!55!black, rounded corners=2pt, fill=yellow!18, minimum width=3.6cm,
 minimum height=1.5cm, align=center, font=\scriptsize},
 chkbox/.style={draw=red!55!black, dashed, rounded corners=2pt, fill=black!4,
 minimum width=8.2cm, minimum height=0.55cm, align=center, inner xsep=6pt,
 font=\scriptsize},
 arr/.style={->, line width=0.39pt, draw=black!70},
 chk/.style={->, dashed, draw=black!45}
]

%% Stage 1: the two integers fixed with no colour count in play
\node[source] (T1) at (0, 3.3) {Step 1 \\ (single tensor channel)};
\node[cascade] (M) at (5.0, 3.3) {$M = 1$};
\node[source] (T3) at (0, 1.9) {Step 2 \\ (T-BI6-9.0; free of $N$)};
\node[cascade] (L) at (5.0, 1.9) {$L = 5$};
\draw[arr] (T1.east) -- (M.west)
 node[midway, above, font=\footnotesize\itshape] {primitive};
\draw[arr] (T3.east) -- (L.west)
 node[midway, above, font=\footnotesize\itshape] {unification};

%% Stage 2: the dual lock is what determines the colour index
\node[idbox] (id) at (5.0, -0.2)
 {dual lock \\ (T-BI6-10.2a) \\[2pt]
  $K = LJ = M^2{+}N^2{+}L^2$ \\[1pt]
  $J = 2N{+}1$ \\[2pt]
  $\Longrightarrow\ N = L \pm \sqrt{L - M^2}$};
\node[cascade] (N) at (9.2, -0.2) {$N = N_c = 3$};
\draw[arr] (L.south) -- (id.north);
\draw[arr] (M.east) to[out=0, in=70] ([xshift=-4mm]id.north east);
\draw[arr] (id.east) -- (N.west)
 node[midway, above, font=\footnotesize\itshape] {$N < L$};

%% Stage 3: what follows once N is fixed
\node[lock] (J) at (11.8, 1.3) {$J = 7$};
\node[lock] (K) at (11.8, -1.7) {$K = 35$};
\draw[arr] (N.east) to[out=0, in=215] (J.south west);
\draw[arr] (N.east) to[out=0, in=145] (K.north west);
\node[anchor=north, font=\scriptsize, gray] at (11.8, 0.62) {$J = 2N + 1$};
\node[anchor=north, font=\scriptsize, gray] at (11.8, -2.42)
 {$K = LJ = M^2{+}N^2{+}L^2$};
\node[anchor=south, font=\small\bfseries] at (11.8, 2.1)
 {Closure consequences};

%% the trace ratio: a check on a ledger already determined
\node[chkbox] (tr) at (3.8, -2.9)
 {trace ratio $(L-N)/L = 2/5$ \;---\; \emph{independent check, not a route
  to $N$}};
\draw[chk] (tr.east) to[out=0, in=-90] (N.south);

\end{tikzpicture}%
}
\bigskip
\caption[The \OGNfive{} Diophantine cascade as a five-step flowchart.]{The \OGNfive{} Diophantine cascade of Theorem~\ref{thm:ogn5-theorem-updated}, drawn as a chain of forcings. Steps~1--2 (left) force $(M, L) = (1, 5)$
without reference to any color count: the primitive tensor channel and
the defining unified block of Theorem~\ref{thm:L-from-GS}. Step~3 then
determines $N = N_c = 3$ from the dual lock together with the block
embedding $N < L$ (Theorem~\ref{thm:N-from-closure}), and $J = 2N+1$ is
the symmetric closure cardinality of the charge packet, so that
$K = LJ = M^2 + N^2 + L^2 = 35$. The trace ratio hangs below the chain on a dashed arrow: it is
the independent check of \S\ref{sec:ogn5-theorem} and not a route to
$N$. Solid arrows mark the forcing chain; the one dashed arrow carries the check.
After the cascade closes, no continuous dimensionless parameter
survives in the bulk theory. Adapted from the cascade flowchart of
\citep{ArisakaA2GG6}.}
\label{fig:ogn5-cascade-a3}
\end{figure}

\subsection{Diophantine minimality: ruling out higher-integer \texorpdfstring{$(L, N)$}{(L, N)} solutions}
\label{ssec:diophantine-minimality}

A reader trained on the swampland or F-theory literature will
reasonably ask: given the trace-ratio identity $(L - N)/L = 2/5$
derived in Theorem~\ref{thm:ogn-bi6} (Section~\ref{sec:ogn5-arithmetic}),
why does the cohort framework select $(L, N) = (5, 3)$ rather than
$(10, 6)$, $(15, 9)$, or higher-integer rescalings of the same
ratio?

\begin{theorem}[\textbf{T-BI6-10.3}: Diophantine minimality of $(L,N) = (5,3)$]
\label{thm:lN-minimality}
The trace-ratio equation $(L - N)/L = 2/5$ admits the integer
family of solutions $(L, N) = (5k, 3k)$ for any positive integer
$k \geq 1$. Among these, \AX{4}~(\MSMF{}) of the
\GGConstitution{} selects the unique minimal solution $k = 1$,
giving $(L, N) = (5, 3)$.
\end{theorem}

\begin{proof}
The equation $(L - N)/L = 2/5$ rearranges to $5(L - N) = 2L$, hence
$L = 5(L-N)/2$. Writing $L - N = 2k$ for $k \in \mathbb{Z}_+$ gives
$L = 5k$ and $N = 3k$. All such $(L, N) = (5k, 3k)$ satisfy the
trace ratio identically; the equation alone does not select $k = 1$.

The cohort selection is supplied by \AX{4}~(\MSMF{}): among the
admissible gauge groups $\SU(5k)$ compatible with the trace ratio,
\MSMF{} demands the minimal unified structure consistent with the
constitutional clauses. The selection is a constitutional minimality
clause, not, by itself, a consequence of the trace-ratio equation --- and, since
$\SU(5k)$ is simple for every $k$ and therefore carries a single
gauge coupling, the discriminating quantity is not continuous
coupling freedom but unforced structure. For $k > 1$ the defining
block is $5k$-dimensional and the embedded color block
$3k$-dimensional, so the unified sector carries roughly $k^2$ times
the minimal generator content ($\dim \SU(10) - \dim \SU(5) = 75$
additional generators already at $k = 2$) with no compensating
constitutional constraint: no clause of \AOne{} forces any of the
additional structure, and every boundary requirement of the cohort
--- the Standard-Model embedding, the trace-ratio lock, the
\OGNfive{} closure identities --- is already saturated at $k = 1$.
\MSMF{} therefore excludes every $k > 1$ as unforced enlargement,
exactly as it excludes unforced tensor multiplication
(Theorem~\ref{thm:rank-one}) and unforced Abelian asymmetry
($c = d$); this is the zero-unforced-structure discipline of
\AFour{} \citep{ArisakaA4Euclid}. Hence the unique
\MSMF{}-admissible solution is $k = 1$, $(L, N) = (5, 3)$.
\end{proof}

The minimality result \eqref{thm:lN-minimality} is the
constitutional reason that $\SU(5)$ rather than $\SU(10)$ or
$\SU(15)$ appears in the cohort gauge group: not because the
trace-ratio equation singles out $\SU(5)$ (it admits all
$(5k, 3k)$) but because \AX{4} singles out the minimal unified
structure among the admissible family. Without \AX{4}, the trace-ratio identity
would have a one-parameter integer family of solutions; with
\AX{4}, the family collapses to the unique element $k = 1$.



\subsection{Physics interpretation}

The theorem has a clean physical reading. \(M=1\) says that the tensor channel is
primitive and unique. \(L=5\) says that the unified simple block is the defining
five-dimensional \(SU(5)\) block. \(N=3\) says that the baryonic trace ratio
isolates a triplicity sector inside that block. Only after that primary meaning
is fixed does one notice the later physical echo: the same integer 3 also
matches color triplicity and, downstream on the boundary, the three-generation
pattern. \(J=7\) is the centered odd closure built from triplicity, and \(K=35\)
is the rectangular and quadratic lock index that later reappears throughout the
\GG{} / \GDOS{} program.

\OGNfive{} does not drop from numerology: \BIsix{} plus the \GG{} unified block structure forces a unique
minimal integer packet whose internal closure identities are so tight that they
can serve as axiomatic arithmetic for later papers.



\bigskip

\noindent\textbf{What each of the five integers now rests on.} The
ledger has been derived, and a reader is entitled to a plain account of
what carries what, because the five integers do not all stand on the
same footing.

$M=1$ is the tensor count. It is the single primitive Green--Schwarz
channel, and it is fixed by the rank-one criterion of Part~II: a single
tensor can cancel the reducible anomaly if and only if the anomaly
matrix has rank one, and the \BIsix{} matrix does. This one is pure
consequence of the anomaly theorem.

$L=5$ is the size of the unified block. It follows from the requirement
that the block's anomaly coefficients vanish identically, which reduces
to a pair of linear conditions with exactly one chiral solution once the
number of antisymmetric representations is minimal. The word
\emph{minimal} is doing real work in that sentence, and it is
constitutional rather than arithmetic: it is \AX{4}, the clause that
forbids hidden continuous freedom. Without it the equations admit a
family, and Section~\ref{sec:ogn5-theorem} says which family and what
excludes each member.

$N=3$ is the color count, and it comes from the trace-ratio identity
together with a closure condition, not from the observed number of
colors. This is the step the paper is most careful about, because the
answer is a number everybody already knows, and a derivation that lands
on a known number deserves more suspicion than one that does not. The
text is written so the suspicion can be checked.

$J=7$ and $K=35$ are then determined, the first by the symmetric closure
of the charge packet and the second by the dual-lock identity
$K=LJ=M^2+N^2+L^2$, which is an equation with only one positive integer
solution once the rest is fixed.

So the honest summary is that four of the five integers follow from
arithmetic once the fifth is in place, one constitutional clause is
consumed along the way, and no measured quantity is used anywhere in the
chain. What the chain does not do is explain why nature has three
generations of fermions; that is a different question, it is not
answered here, and Section~\ref{sec:open-problems} lists it among the
problems that remain genuinely open.

\GGPart[Comparisons with String Theory]{Comparisons with String Theory}{Shared
mathematics, structural departures, and open bridges --- against the
ten-dimensional Green--Schwarz mechanism, six-dimensional supergravity and
F-theory, AdS/CFT and Randall--Sundrum, and the swampland constraints}

The Green--Schwarz mechanism, six-dimensional supergravity, F-theory,
the AdS/CFT correspondence, and the Randall--Sundrum model all share
mathematical infrastructure with \GG{}.
This Part carries out a systematic, feature-by-feature comparison
that identifies what \BIsix{} inherits from the string-theoretic tradition,
where it departs, and where the two programs might converge in the future.
The guiding principle is objectivity: every similarity and every
difference is stated precisely, and no claim of superiority is made in
either direction.

That principle is worth defending, because comparisons of this kind are
usually written to win something, and this one is not. The
Green--Schwarz mechanism is the intellectual parent of everything in
Part~II. The descent formalism, the anomaly-inflow picture, the idea
that a tensor field can absorb what no rearrangement of matter can
cancel, the differential-cohomology language in which the modern
statement is made --- all of it was built by the string-theory
community, and the present paper uses it essentially unmodified. What
differs is not the mathematics. It is which questions the mathematics is
being asked to answer, and what each framework is willing to take as
given.

Four comparisons are made, in increasing distance from the source.
Section~\ref{sec:comparison-10d} sets the six-dimensional construction
beside the original ten-dimensional one, feature by feature.
Section~\ref{sec:comparison-6d} compares it with six-dimensional
supergravity and F-theory constructions, which are its closest neighbors and where the difference is sharpest, because those
constructions assume supersymmetry and this one does not.
Section~\ref{sec:ads-rs} turns to AdS/CFT and Randall--Sundrum, where
the shared object is the extra dimension itself and the difference is
what the radial direction is taken to mean.
Section~\ref{sec:swampland} asks whether the construction respects the
swampland constraints, which are the string program's own criteria for
which effective theories can be completed.

A reader from the string community will find claims here to disagree
with. Where a disagreement is a matter of interpretation rather than
calculation, the text tries to say so, and Part~IX answers the specific
objections in detail.

\section{Comparison with the ten-dimensional Green--Schwarz mechanism}
\label{sec:comparison-10d}

\subsection{The 10D Green--Schwarz identity}

In the original construction \citep{GreenSchwarz1984}, the
ten-dimensional Type~I or heterotic string has gauge group
\(G_{10} = \SO(32)\) or \(E_8\times E_8\), and the modified
Bianchi identity is
\begin{equation}
\dd H_3^{(10\mathrm{D})}
= X_4^{(10\mathrm{D})}
= \tr R^2 - \frac{1}{30}\,\Tr_{G_{10}} F^2.
\label{eq:gs-10d}
\end{equation}
The factor \(1/30\) is the unique normalization ensuring that the
12-form anomaly polynomial factorizes as \(I_{12} = X_4\wedge X_8\).
The Green--Schwarz counterterm
\(\int B_2\wedge X_8\)
cancels the full one-loop anomaly.

\subsection{Feature-by-feature comparison}

\begin{center}
{\footnotesize
\begin{longtable}{p{0.24\textwidth}p{0.34\textwidth}p{0.32\textwidth}}
\caption[The 10D and 6D Green--Schwarz mechanisms, feature by feature.]{The ten-dimensional Green--Schwarz mechanism and its six-dimensional counterpart, compared on the features that matter for anomaly cancellation. Read across a row and the mechanisms are doing the same work; read down the third column and the differences are all of one kind --- what has to be supplied from outside. The ten-dimensional construction needs supersymmetry and a compactification choice; the six-dimensional one needs neither, and that is the entire content of the comparison. Nothing here disputes a ten-dimensional result.}\label{tab:comparison}\\
\toprule
\textbf{Feature} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endfirsthead
\toprule
\textbf{Feature} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endhead
Spacetime dimension & \(d=10\) & \(d=6\) \\
Anomaly polynomial & 12-form \(I_{12}\) & 8-form \(I_8\) \\
Factorization & \(I_{12}=X_4\wedge X_8\) (two different forms) & \(I_8=\half X_4^2\) (perfect square) \\
Anomaly matrix rank & Rank 1 or higher (can use multiple tensors) & Rank 1 (single tensor, single direction) \\
Green--Schwarz four-form & \(\tr R^2 - \frac{1}{30}\Tr F^2\) & \(\tr R^2 - \Tr F_5^2 - \frac{2}{5}F_B^2 - \frac{2}{5}F_{Qi}^2\) \\
Gauge sectors in \(X_4\) & 2 (gravity + gauge) & 4 (gravity + SU(5) + U(1)\(_B\) + U(1)\(_{Qi}\)) \\
Gauge group & SO(32) or \(E_8\times E_8\) & \(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\) \\
Coefficient origin & Tr\(_{496}F^2 = 30\)\,Tr\(_{\mathrm{fund}}F^2\) & Tr\((B_0^2)/\)Tr\((Y_0^2) = (L{-}N)/L\) \\
SUSY requirement & Required (\(\cN=1\) in 10D) & Not required \\
UV endpoint & \(E_{\Pl}\) (string scale) & \(\EGUT\sim 2\times 10^{16}\) GeV \\
Tensor chirality & Self-dual at quantum level & Anti-self-dual at quantum level \\
BRST cancellation & \(s(\Gamma+S_{\mathrm{GS}})=0\) & Same \\
Spectrum determination & Fixed by worldsheet SUSY + compactification & Constitutional (\OGNfive{} arithmetic) \\
Residual landscape & \(\sim 10^{500}\) string vacua (Bousso--Polchinski) & Finite Diophantine problem \\
\bottomrule
\end{longtable}
}
\end{center}

\subsection{Key structural differences}

Table~\ref{tab:comparison} sets the two constructions side by side line
by line; the subsections below take the four rows that carry real
weight and say what each one costs.

\subsubsection{Factorization type: perfect square versus product}

In ten dimensions, the anomaly polynomial \(I_{12}\) factorizes into
\(X_4\wedge X_8\), where \(X_4\) and \(X_8\) are different forms.
The two forms can be adjusted independently, giving more algebraic freedom. In six dimensions, the polynomial \(I_8\) must
factorize as a \emph{perfect square} \(\half X_4^2\). This imposes
the rank-one condition on the anomaly matrix
(Theorem~\ref{thm:rank-one}), which is a strictly stronger
algebraic constraint.

The physical consequence is that the 6D mechanism is \emph{more rigid}
than the 10D mechanism. In ten dimensions, the factorization
\(I_{12} = X_4\wedge X_8\) can accommodate anomaly polynomials
whose rank-two structure would be inadmissible in six dimensions.

\subsubsection{Four locked sectors versus two}

The 10D four-form \(X_4^{(10\mathrm{D})} = \tr R^2 - (1/30)\Tr F^2\)
ties two sectors (gravity and one gauge group) into one identity.
The 6D \BIsix{} four-form ties \emph{four} sectors (gravity, \(\SU(5)\),
\(\U(1)_B\), \(\U(1)_{Qi}\)) into one identity.
Despite the richer gauge structure, the rank-one mechanism still works
because all four sectors collapse onto a single ray in anomaly space.
This is the mathematical content of ``geometry--gauge locking'':
the relative weights \((1,-1,-2/5,-2/5)\) are frozen by the \OGN{} arithmetic.

\subsubsection{Absence of supersymmetry}

The absence of supersymmetry is the most consequential difference. In ten dimensions,
\(\cN=1\) supersymmetry is \emph{required} for anomaly cancellation:
without the gravitino and the superpartner spectrum, the anomaly
polynomial does not factorize. The Green--Schwarz mechanism works
\emph{because} of SUSY, not despite it.

In six dimensions, \BIsix{} cancels anomalies \emph{without} supersymmetry.
The axiomatic constraints (\GGEP{}, \MSMF{}, \OGN{}) replace SUSY as the
organizing principle. The chiral matter reservoir is determined not by
superpartner relations but by the Diophantine system
\eqref{eq:dioph1}--\eqref{eq:dioph4}, which is itself determined
by the \OGNfive{} integers.

\medskip\noindent A complementary, kinematic reason points the same way. Because the two further dimensions of the bulk are non-spatial --- a scale dimension and a gauge dimension, whose generators are a dilatation and a compact $\U(1)$ charge rather than translations --- the bulk admits no super-Poincar\'e algebra on which a supercharge could close. Supersymmetry is in this sense not merely \emph{unrequired} by the anomaly mechanism but \emph{unsupported} by the geometry \citep{ArisakaA2GG6}.

\subsection{What \BIsix{} inherits from the 10D mechanism}

Despite the differences, \BIsix{} inherits three mathematical structures
from the original Green--Schwarz construction:
\begin{enumerate}[label=(\alph*), leftmargin=1.4cm]
\item the modified Bianchi identity \(\dd H_3 = X_4\) as the
 anomaly inflow channel;
\item the BRST descent chain
 \(I_8\to I_7^{(0)}\to I_6^{(1)}\to\) cancellation;
\item the requirement that the tensor be chiral (self-dual or
 anti-self-dual) at the quantum level.
\end{enumerate}
These are universal features of the Green--Schwarz mechanism in any
even dimension. What is \emph{specific} to string theory is the 10D
gauge group and the requirement of \(\cN=1\) SUSY.
\BIsix{} replaces these with \(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\)
and the \GGTheory{} Constitution.

\subsection{The landscape problem: finite versus infinite}

Perhaps the most consequential difference between \BIsix{} and the
10D superstring is the nature of the residual spectrum problem.

In string theory, the number of consistent compactifications to four
dimensions is enormous. Estimates of the string landscape range
from \(10^{272\,000}\) (Lerche--L\"ust--Schellekens, 1987) to
\(10^{500}\) (Bousso--Polchinski, 2000) to effectively uncountable
in the most general settings. The identification of the ``correct''
vacuum that reproduces observed particle physics remains one of the
great open problems in mathematical physics.
The difficulty is not merely computational; it is structural, and
the community has developed powerful tools---including the swampland
program---to address it.

In \GG{}, the residual spectrum problem is qualitatively different.
The \OGNfive{} integers fix the gauge group, the \BIsix{} coefficient vector,
the core-block structure, and the Diophantine system.
The remaining freedom is:
\begin{enumerate}[label=(\roman*), leftmargin=1.4cm]
\item the overall normalization \(\Omega\), which is subject to
 minimization by \AX{4} (\MSMF);
\item the gravitational completion \((n_{\mathrm{SD}},\,n_F)\),
 which is a \(2\times 2\) linear system;
\item the possible use of higher \(\SU(5)\) representations
 (beyond \(\mathbf{5}\), \(\mathbf{10}\), \(\mathbf{1}\))
 in the core block, which could reduce \(\Omega\).
\end{enumerate}
All of these are \textbf{finite, bounded, computationally tractable}
searches. The representations of \(\SU(5)\) are labeled by Young
diagrams, and the gravitational bound limits the total chiral dimension,
so only finitely many representations can contribute.
A systematic computer enumeration of all solutions is feasible.

\begin{center}
{\footnotesize
\begin{longtable}{p{0.24\textwidth}p{0.32\textwidth}p{0.32\textwidth}}
\caption[The landscape problem in the two frameworks.]{What freedom survives in each framework after its structural theorem has done its work. The rows are the places where a choice can still be made --- compactification, flux, moduli --- and the columns are the two frameworks. The asymmetry is the point and it is quantitative: one framework's residual freedom is a number of order $10^{500}$ and the other's is one. That is a difference in kind for falsifiability, because a framework with a landscape can absorb a failed measurement and a framework without one cannot.}\label{tab:landscape}\\
\toprule
\textbf{Aspect} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endfirsthead
\toprule
\textbf{Aspect} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endhead
Residual freedom after structural theorem & Calabi-Yau compactification + flux choice & Diophantine system on SU(5) reps \\
Number of vacua / solutions & \(\sim 10^{500}\) (or more) & Finite (bounded by gravitational dim.\ count) \\
Computability & Active research (swampland program) & In principle enumerable \\
Selection principle & Under investigation (swampland conjectures) & \AX{4} (\MSMF): minimize \(\Omega\) \\
Current status & 40 years, no unique solution & Finite search, not yet completed \\
Nature of the problem & Geometric (Calabi-Yau moduli) & Arithmetic (Diophantine equations) \\
\bottomrule
\end{longtable}
}
\end{center}


\subsection{Objective assessment: what is proven at each level}

\begin{center}
{\footnotesize
\begin{longtable}{p{0.30\textwidth}p{0.30\textwidth}p{0.30\textwidth}}
\caption[Proof status of each framework, level by level.]{An assessment of what is actually proved in each framework, at each level of the argument. The rows run from the structural theorem down to the specific model, and each cell records a status rather than a claim of superiority. The reading is mixed: the ten-dimensional structural theorem has been proved since 1984 and is not in question here, while the construction in this paper is younger and proves less at the model level. The table is included so that neither fact has to be inferred.}\label{tab:assessment}\\
\toprule
\textbf{Level of proof} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endfirsthead
\toprule
\textbf{Level of proof} & \textbf{10D superstring} & \textbf{6D \BIsix{} / \GG{}} \\
\midrule
\endhead
Structural GS theorem & Proven (1984) & Proven (this paper) \\
Coefficient uniqueness & Proven (1/30 for SO(32)) & Proven (2/5 from \OGN{}) \\
Rank-one / factorization & Implicit & Explicit (Thm.~\ref{thm:rank-one}) \\
BRST / Slavnov--Taylor & Proven & Proven (Thm.~\ref{thm:bi6-cancellation}) \\
Global gauge anomaly & Proven & Proven (\(\pi_6=0\)) \\
Explicit spectrum & Known (Type I / heterotic) & Benchmark constructed (Thm.~\ref{thm:model-level}) \\
Minimal spectrum & Known (SUSY fixes it) & Open (finite Diophantine search) \\
Gravitational completion & Automatic (SUGRA) & Explicit (SD tensors) \\
Charge lattice & String charge quantization & Abelian rescaling \\
Landscape / vacuum counting & \(\sim 10^{500}\) & Finite (bounded) \\
\bottomrule
\end{longtable}
}
\end{center}

Table~\ref{tab:assessment} grades the two constructions
side by side on each obligation. The conclusion is that the \BIsix{}
structural proof is at the \emph{same level of mathematical rigor} as
the original 10D proof.
The model-level proof provides an explicit benchmark.
The remaining open problem (minimal \(\Omega\)) is a bounded
finite computation, qualitatively simpler than the string landscape.


\subsection{\BIsix{} as the \GGEP{} completion of GS: the geometrization reading}

\noindent
\figref{fig:gs-as-ggep} (page~\pageref{fig:gs-as-ggep}) summarizes the
relationship: the 1984 GS factorization on a 10D bulk and the 2026
\BIsix{} factorization on the 6D \GG{} bulk are connected by
the \AX{2}~(\GGEP) constitutional elevation.


\label{sec:GS_as_GGEP_completion}

In our view it is worth ending Section~\ref{sec:comparison-10d}
with a methodological observation about what the 1984 Green--Schwarz
achievement actually was and what \BIsix{} inherits from it. This
observation is meant as a supportive synthesis with the broader
string-theory community rather than as criticism: the GS mechanism
is one of the most consequential single technical results of
late-twentieth-century theoretical physics, and the \BIsix{} construction
of the present paper rests on its shoulders. What
we add is not a new technical move but an axiomatic reading of
the move that GS pioneered.

\paragraph{The GS achievement of 1984.} Working in 10D heterotic
string theory, Green and Schwarz \citep{GreenSchwarz1984} showed
that the chiral fermion anomaly polynomial $I_{12}$ --- for the
specific gauge groups $\SO(32)$ and $E_8 \times E_8$ --- admits the
remarkable factorization $I_{12} = X_4 \wedge X_8$, with $X_4$ a
closed 4-form built from $\mathrm{tr}(R \wedge R)$ and
$\mathrm{tr}(F \wedge F)$. They then implemented the cancellation
by modifying the gauge transformation of the antisymmetric tensor
$B_{\mu\nu}$ so that the resulting field strength
$H = \dd B - \omega_3^L + \omega_3^{YM}$ satisfies the modified
Bianchi identity $\dd H = X_4$, and the bulk Chern--Simons inflow
from the modified $H$-field cancels the boundary anomaly. This was
the moment when ten-dimensional string theory became
phenomenologically constrained rather than infinitely flexible, and
it set the structural template that every subsequent inflow-based
anomaly cancellation has followed --- including the \BIsix{} construction
of the present paper.

\paragraph{The geometric content of the GS construction.} The
mathematical content of GS, read carefully, is geometric all the
way down. The modified Bianchi identity $\dd H = X_4$ is a
statement about a closed differential 4-form on the 10D bulk
manifold; the Chern--Simons primitives $\omega_3^L$ and $\omega_3^{YM}$
are Chern--Weil objects defined for any smooth connection on any
principal bundle; and the factorization
$I_{12} = X_4 \wedge X_8$ is a cohomological fact about
characteristic classes of the underlying $\Spin \times G$ bundle,
not about quantum amplitudes. None of this requires a quantum
amplitude per se. The quantum content of the construction enters
at exactly one place --- the boundary chiral fermion's effective
action, where the Adler--Bell--Jackiw triangle anomaly produces the
deviation $\partial^\mu J_\mu^5 \propto F \wedge \tilde F$ from
classical conservation \citep{Adler1969,BellJackiw1969}. The bulk
modification $\dd H = X_4$ is then the geometric counter-current
that closes the loop. This geometric reading has been
progressively formalized by the differential-cohomology refinements
of Hopkins, Singer, Freed, and Moore \citep{HopkinsSinger2005,
DaiFreed1994,MonnierMoore2019}; it is the reading on which the
local and global proofs of
Sections~\ref{sec:structural-proof}--\ref{sec:global-completion}
of the present paper rest.

\paragraph{What GS pioneered: the geometrization of gauge fields.}
On this geometric reading, the deepest content of the GS achievement
is not that string theory provided a quantum theory of gravity ---
that framing, while historically dominant, is somewhat at odds with
the technical content of the modified Bianchi identity, which is
classical and geometric --- but rather that gauge field strengths
$F$ and gravitational curvature $R$ must lie inside a single closed
topological 4-form on the bulk manifold. This is precisely
the axiomatic move that \AX{2} \GGEP{} (the Geometry--Gauge
Equivalence Principle, see Axiom~\AX{2} of \citep{ArisakaA1Constitution})
makes foundational: the metric and the gauge connection are not
independent inputs but two faces of a single composite
geometric--gauge connection on a principal $G_{\mathrm{tot}}$ bundle.
GS demonstrated, in a specific 10D model, that the geometrization
move is technically possible; the \GGConstitution{} then asks the
same Einstein-style equivalence-principle question of
\emph{any} candidate theory --- what does it cost to treat the
composite object as a single geometric primitive? --- and elevates
the move from a technical refinement to a constitutional axiom.

\paragraph{\BIsix{} as the four-component generalization.} \BIsix{} is, in
this reading, the four-component generalization of the
two-component GS pattern. Where GS produced
$X_4^{\mathrm{GS}} = \mathrm{tr}(R\wedge R) - \mathrm{tr}(F\wedge F)$
on a single semisimple gauge factor, the \GG{} bulk produces
\begin{equation}
 X_4^{\mathrm{BI\mbox{-}6}}
 = a\,\mathrm{tr}(R\wedge R) + b\,\mathrm{tr}(F_{\SUfive}\wedge F_{\SUfive})
 + c\,F_B \wedge F_B + d\,F_{Qi} \wedge F_{Qi}
\end{equation}
with the four-component coefficient vector
$(a,b,c,d) = (1,-1,-2/5,-2/5)$ and the geometry--gauge trace-ratio
lock $|a/c| = |a/d| = 5/2$ pinned by the \OGNfive{} ledger
(see the trace-ratio theorem in
Section~\ref{sec:ogn5-theorem} and the \OGNfive{} arithmetic of
Section~\ref{sec:ogn5-arithmetic}). The \BIsix{}
identity $\dd H_3 = X_4^{\mathrm{BI\mbox{-}6}}$ is the
axiomatic analogue, in the 6D bulk of \GG{}, of the GS
identity $\dd H = X_4^{\mathrm{GS}}$ in the 10D heterotic bulk.
The mathematical structure is unchanged from GS; what is added is
(i) the four contributing factors instead of two, (ii) the
arithmetic pinning of the coefficient ratios by the \OGNfive{} ledger
$(M,N,L;J,K) = (1,3,5;7,35)$, and (iii) the constitutional
reading that puts the entire construction inside the \GGEP{} axiom
of the \GGConstitution{}. GS pioneered the move; \BIsix{} is its
axiomatic generalization, with explicit numerical predictions
that follow.

\begin{figure}[H]
\centering
\resizebox{0.97\textwidth}{!}{%
\begin{tikzpicture}[
 font=\small,
 every node/.style={inner sep=2pt},
 edgelabel/.style={fill=white, inner sep=1pt, font=\footnotesize\itshape},
 arrow/.style={->, line width=0.47pt, draw=black!70},
 bigarrow/.style={->, line width=0.64pt, draw=violet!70!black}
]

%% Left panel: 10D heterotic GS factorisation
\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.2cm, minimum height=0.7cm,
 align=center] (gs10) at (0, 4) {\textbf{1984 Green--Schwarz (10D)}};

\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.4cm, minimum height=0.6cm, align=center]
 (gsfact) at (0, 3) {$I_{12} = X_4^{\mathrm{GS}} \wedge X_8$};

\node[draw=blue!55!black, rounded corners=3pt, minimum width=4.4cm, minimum height=1.4cm,
 fill=blue!5, align=center]
 (gsX4) at (0, 1.55)
 {two-component $X_4^{\mathrm{GS}}$:\\[1pt]
 $\tr(R \wedge R) - \tr(F \wedge F)$\\[1pt]
 \footnotesize gauge group: $\SO(32)$ or $E_8 \times E_8$};

\node[draw=blue!55!black, rounded corners=3pt, fill=blue!10, minimum width=4.4cm, minimum height=0.6cm, align=center]
 (gsBI) at (0, -0.1) {$\dd H = X_4^{\mathrm{GS}}$ \;\;\footnotesize{(10D modified Bianchi)}};

%% Middle: GGEP elevation arrow
\node[draw=violet!55!black, rounded corners=4pt, fill=violet!10, minimum width=3.4cm, minimum height=2.0cm,
 align=center, line width=0.64pt]
 (ggep) at (5.5, 1.55)
 {\textbf{AX-2 (\GGEP)}\\[2pt]
 \footnotesize Geometry--Gauge\\
 \footnotesize Equivalence Principle\\[2pt]
 \footnotesize \emph{constitutional}\\
 \footnotesize \emph{elevation}};

\draw[bigarrow] (gsX4.east) -- (ggep.west);
\node[edgelabel] at (3.6, 3.0) {GGEP elevation};

%% Right panel: 6D GG-6 BI-6 factorisation
\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=4.6cm, minimum height=0.7cm,
 align=center] (bi6label) at (11.5, 4) {\textbf{2026 BI-6 (GG-6 in 6D)}};

\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=5.2cm, minimum height=0.6cm, align=center]
 (bi6fact) at (11.5, 3) {$I_8 = \tfrac{1}{2}\, (X_4^{\mathrm{BI6}})^2$};

\node[draw=red!55!black, rounded corners=3pt, minimum width=5.2cm, minimum height=1.4cm,
 fill=red!10, align=center]
 (bi6X4) at (11.5, 1.55)
 {four-component $X_4^{\mathrm{BI6}}$:\\[1pt]
 $\tr R^2 - \Tr F_5^2 - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2$\\[1pt]
 \footnotesize gauge group: $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$};

%% the GGEP-elevation arrow, drawn after bi6X4 exists so the node
%% cannot paint over its head
\draw[bigarrow] (ggep.east) -- (bi6X4.west);

\node[draw=red!55!black, rounded corners=3pt, fill=red!10, minimum width=5.2cm, minimum height=0.6cm, align=center]
 (bi6BI) at (11.5, -0.1) {$\dd H_3 = X_4^{\mathrm{BI6}}$ \;\;\footnotesize{(6D modified Bianchi)}};

%% Bottom annotation: rank-one perfect-square distinction
\node[anchor=north, font=\small, align=center] at (5.8, -1.1)
 {\emph{10D: product factorization $X_4 \wedge X_8$,
 two functionally independent forms.}\\
 \emph{6D: perfect-square factorization $\tfrac12 X_4^2$,
 forced rank-one --- structurally more rigid.}};

\end{tikzpicture}%
}
\caption[The GS-as-\GGEP{} completion of \BIsix{}]{The GS-as-\GGEP{} completion: \BIsix{} is the four-component axiomatic generalization of the two-component pattern Green and Schwarz pioneered in 1984. Left panel: the 10D heterotic GS factorization $I_{12} = X_4 \wedge X_8$ with the two-component four-form $X_4^{\mathrm{GS}} = \tr(R\wedge R) - \tr(F\wedge F)$ on a single semisimple gauge factor. Right panel: the 6D \BIsix{} factorization $I_8 = \tfrac12 (X_4^{\mathrm{BI6}})^2$ with the four-component $X_4^{\mathrm{BI6}}$ pinned by the \OGNfive{} trace-ratio lock $|a/c| = |a/d| = 5/2$ across the four-factor bulk gauge--gravity group $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$. Middle panel: the \AX{2} (\GGEP) Geometry--Gauge Equivalence Principle of \AOne{} \citep{ArisakaA1Constitution} is the axiomatic elevation that takes the 1984 GS move from a technical refinement realized in a specific 10D model to a foundational axiom that any candidate theory must satisfy. The 10D pattern admits two functionally independent forms; the 6D pattern is rank-one by perfect-square forcing --- structurally more rigid, as discussed in the surrounding text and in Section~\ref{sec:comparison-10d}.}
\label{fig:gs-as-ggep}
\end{figure}

\paragraph{An invitation to the string and mathematical-physics
communities.} We hope that string theorists who recognize the
geometric content of the GS mechanism --- as much of the
differential-cohomology mathematical-physics community has done
since the early 2000s --- will find in \BIsix{} a generalization worth
engaging with. We inherit, with full credit, the Chern--Simons
machinery, the trace-ratio analysis, the anomaly-inflow viewpoint,
and the differential-cohomology refinements that Hopkins, Singer,
Freed, and Moore have made rigorous. What \GGTheory{} adds is the
axiomatic commitment to \AX{2} \GGEP{} --- the elevation of the
geometric-gauge equivalence from a technical pattern realized in a
specific 10D model to a foundational axiom that any candidate
theory must satisfy as a precondition for admissibility --- and the
arithmetic pinning of the trace-ratio coefficients by the \OGNfive{}
ledger. In our view this is not a competing program but a
natural completion of the geometrization work that GS began in 1984
and that the differential-cohomology community has carried forward
ever since. The GS achievement of 1984 was the moment
when the geometrization of gauge fields took its first decisive
step; the \GGConstitution{} and \BIsix{} are simply the next steps of the
same journey.



%% ===============================================================
%%  NEW SECTION: COMPARISON WITH 6D STRING/F-THEORY CONSTRUCTIONS
%% ===============================================================
\section{Comparison with six-dimensional string and F-theory constructions}
\label{sec:comparison-6d}

The ten-dimensional comparison of Section~\ref{sec:comparison-10d}
established the structural parallels between \BIsix{} and the original
Green--Schwarz mechanism.
This section sharpens the comparison by descending to six dimensions,
where the string theory landscape is most constrained and best understood.

\subsection{Six-dimensional \texorpdfstring{\(\cN=(1,0)\)}{N=(1,0)} supergravity and anomaly cancellation}

The six-dimensional \(\cN=(1,0)\) supergravity framework
\citep{GreenSchwarzWest1985,Sagnotti1992,NishinoSezgin1984}
is the closest supersymmetric cousin of \GG{}.
Both operate in six dimensions, both use the Green--Schwarz mechanism,
and both require anomaly polynomial factorization.
The key structural differences are:

\paragraph{Multiplet structure.}
In \(\cN=(1,0)\) SUGRA, the field content is organized into
multiplets: the gravity multiplet (graviton + gravitino + ASD tensor),
vector multiplets (\(V\)), hypermultiplets (\(H\)), and tensor
multiplets (\(T\)).
The gravitational anomaly condition reduces to the single equation
\begin{equation}
H - V + 29T = 273.
\label{eq:sugra-gravitational}
\end{equation}
For one tensor (\(T=1\)): \(H - V = 244\).
This elegant simplicity is a direct consequence of supersymmetry,
which locks the anomaly contributions of different fields within
each multiplet.

In \GG{}, there are no supermultiplets.
The gravitational condition is a \(2\times 2\) linear system
involving both fermion dimensions and self-dual tensor numbers
(Appendix~\ref{app:grav-full}), and the gauge conditions
form a separate Diophantine system
(Appendix~\ref{app:diophantine-full}).
The total constraint count is 16 independent conditions, versus
one in the SUSY case.

\paragraph{Anomaly factorization.}
In \(\cN=(1,0)\) SUGRA with one tensor, the anomaly polynomial
factorizes as \(I_8 = \half X_4^2\), where
\begin{equation}
X_4^{\mathrm{SUGRA}} = a\,\tr R^2 + \sum_i b_i\,\Tr F_i^2.
\label{eq:sugra-x4}
\end{equation}
The coefficients \(a\) and \(b_i\) are constrained by the gauge group
and matter content but are not fully determined by the group structure
alone---they depend on the specific spectrum.

In \BIsix{}, the coefficient vector \((1,-1,-2/5,-2/5)\) is
\emph{uniquely determined} by the gauge group
\(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\) and the \OGNfive{} arithmetic.
This additional rigidity has no analogue in the SUGRA setting,
where different spectra with the same gauge group can yield different
coefficient vectors.

\subsection{F-theory and the geometric origin of 6D anomaly cancellation}
\label{sec:f-theory}

F-theory \citep{Vafa1996,MorrisonVafa1996a,MorrisonVafa1996b}
provides the most systematic construction of six-dimensional
\(\cN=(1,0)\) supergravity theories.
An F-theory compactification on an elliptically fibered
Calabi--Yau threefold \(Y_3\to B_2\) yields a 6D theory whose
gauge group, matter content, and anomaly polynomial are all
determined by the geometry of \(Y_3\).

\paragraph{How F-theory constructs 6D theories.}
The gauge group arises from singularities of the elliptic fibration
(classified by Kodaira and N\'eron), the matter content arises
from collisions of singularities in the fiber, and the
number of tensor multiplets is \(T = h^{1,1}(B_2) - 1\).
The anomaly factorization is guaranteed by the Calabi--Yau condition:
the topological intersection numbers of the base \(B_2\) encode
the anomaly coefficients.

\paragraph{The Kumar--Morrison--Taylor classification.}
The landmark work of Kumar, Morrison, and Taylor
\citep{KumarMorrisonTaylor2010,KumarParkTaylor2011,KumarTaylor2009}
systematically classified the space of anomaly-free 6D
\(\cN=(1,0)\) supergravity theories and established the
``string universality'' conjecture:

\begin{quote}
\textbf{String Universality Conjecture} (Kumar--Taylor, 2009):
\emph{All \(\cN=(1,0)\) supersymmetric theories in 6D with
one gravity and one tensor multiplet which are free of anomalies
or other quantum inconsistencies admit a string theory realization.}
\end{quote}

This conjecture asserts that every consistent 6D SUGRA theory
is realized by F-theory---there is no ``swampland'' of
anomaly-free but string-incompatible theories.
The evidence is strong: for simple gauge groups, all anomaly-free
spectra have been matched to F-theory compactifications
\citep{KumarMorrisonTaylor2010,MorrisonTaylor2012a,MorrisonTaylor2012b}.

\paragraph{Comparison with \GG{}.}
\GG{} is \emph{not} a supersymmetric theory and therefore lies
outside the scope of the string universality conjecture.
The conjecture specifically addresses \(\cN=(1,0)\) theories;
it says nothing about non-SUSY constructions.
Whether \GG{} can or cannot be embedded in a string compactification
is a separate question from whether it is anomaly-free.
The present paper addresses only the latter.

However, the F-theory technology offers a useful perspective.
In F-theory, the anomaly coefficients are intersection numbers
on the base \(B_2\). In \GG{}, the anomaly coefficients are
\OGNfive{} integers. Both frameworks achieve discreteness of the
anomaly data, but through different mechanisms:
F-theory through algebraic geometry, \GG{} through axiomatic arithmetic.

\subsection{The 6D landscape: enumeration and finiteness}

\paragraph{Finiteness results.}
A major achievement of the F-theory/SUGRA program is the proof
that the 6D landscape is \emph{finite} in important senses.
Tarazi and Vafa \citep{TaraziVafa2021} proved bounds on
the number of 6D supergravity theories.
Hamada and Loges \citep{HamadaLoges2024} enumerated all
anomaly-free models with at most one tensor multiplet
(excluding low-rank gauge factors), finding that approximately
50\% satisfy proposed UV/swampland bounds.
Most recently, Kim, Vafa, and Xu \citep{KimVafaXu2024}
argued for a finite landscape of 6D \(\cN=(1,0)\)
supergravity theories.

\paragraph{New anomaly-free models.}
Becker, Kehagias, Sezgin, Tennyson, and Violaris
\citep{BeckerKehagias2024} discovered new classes of
anomaly-free 6D supergravities, including R-symmetry gauged
models. Loges \citep{Loges2024} found a new infinite class
of anomaly-free models. These results show that the 6D SUGRA
landscape, while finite, is still actively being explored
even four decades after the original Green--Schwarz paper.

\paragraph{The swampland program in 6D.}
The swampland program \citep{Vafa2005,BrennanCartaVafa2017}
seeks to identify criteria distinguishing consistent quantum
gravity theories (the ``landscape'') from apparently consistent
but ultimately inconsistent theories (the ``swampland'').
In six dimensions, the swampland constraints are particularly
powerful. Hayashi, Kim, and Kim \citep{HayashiKimKim2025}
explored BPS string spectra in 6D supergravity and tested
swampland conjectures including the Weak Gravity Conjecture
and the Emergent String Conjecture.

\paragraph{Global anomaly refinements.}
Monnier and Moore \citep{MonnierMoore2019} constructed the
Green--Schwarz terms on spacetimes with non-trivial topology
and proved cancellation of all global gauge and gravitational
anomalies for theories with gauge groups given by products
of \(\U(n)\), \(\SU(n)\), and \(\Sp(n)\) factors.
Basile and Leone \citep{BasileLeone2024} established anomaly
constraints for heterotic strings and supergravity in six dimensions.
Dierigl and Tartaglia \citep{DieriglTartaglia2025} developed
quadratically refined discrete anomaly cancellation, extending
the Monnier--Moore framework to include discrete gauge and
gravitational anomalies through a quadratic-refinement formalism
that goes beyond the continuous characteristic-class analysis.
Their technology applies to \(\cN=(1,0)\) theories with specific
multiplet structures. Adapting the Dierigl--Tartaglia quadratic
refinement to the non-supersymmetric \BIsix{} setting --- where there
are no hypermultiplets or vector multiplets, only a constitutional
chiral spectrum --- would be a natural next step and would either
confirm or sharpen the global theorem proved in the present paper
on the intended class.

These developments represent the cutting edge of 6D anomaly
theory in the string/F-theory framework.
\GG{} shares the same mathematical infrastructure (anomaly
polynomials, descent equations, Green--Schwarz mechanism)
but operates without the SUSY and geometric constraints
that define the F-theory landscape.

\subsection{Detailed comparison: \GG{} versus 6D \texorpdfstring{\(\cN=(1,0)\)}{N=(1,0)} SUGRA/F-theory}

\begin{center}
{\footnotesize
\begin{longtable}{p{0.23\textwidth}p{0.34\textwidth}p{0.33\textwidth}}
\caption[This framework against six-dimensional $(1,0)$ supergravity and F-theory.]{The closest neighbor to this construction is six-dimensional $(1,0)$ supergravity, and this table compares them directly. The rows are the structural commitments --- supersymmetry, tensor multiplet content, lattice conditions, how anomaly freedom is obtained --- and the comparison is deliberately made on inputs rather than results. The row that carries the argument is supersymmetry: the neighboring construction requires it and this one does not, and everything else in the column follows from that one difference.}\label{tab:6d-comparison}\\
\toprule
\textbf{Feature} & \textbf{6D \(\cN=(1,0)\) SUGRA / F-theory} & \textbf{\GG{} / \BIsix{}} \\
\midrule
\endfirsthead
\toprule
\textbf{Feature} & \textbf{6D \(\cN=(1,0)\) SUGRA / F-theory} & \textbf{\GG{} / \BIsix{}} \\
\midrule
\endhead
Supersymmetry & \(\cN=(1,0)\) (8 supercharges) & None \\
Gravitino & Present (in gravity multiplet) & Absent \\
Tensor sector & ASD tensor in gravity multiplet + possible extra tensor multiplets & ASD \(B_2\) (quantum level) + gravitational completion tensors \\
Gauge group & Any anomaly-free compact Lie group & \(\SU(5)\times\U(1)_B\times\U(1)_{Qi}\) (constitutionally fixed) \\
Gravitational condition & \(H-V+29T=273\) (one equation) & \(2\times 2\) linear system for \((n_F,n_{\mathrm{SD}})\) \\
Number of anomaly conditions & 1 (SUSY locks the rest) & 18 (each independent) \\
Spectrum determination & SUSY multiplets + geometric data & \OGNfive{} arithmetic + Diophantine system \\
F-theory realization & Elliptic CY\(_3\) compactification & Not applicable (non-SUSY) \\
String universality & Conjectured to hold & Outside scope of conjecture \\
Landscape size & Finite but large (\(\sim\) thousands of models with \(T\le 1\)) & Finite and bounded (Diophantine) \\
Coefficient determination & Spectrum-dependent & Uniquely fixed by \OGNfive{}: \((1,-1,-2/5,-2/5)\) \\
UV completion & String/M-theory at \(E_{\Pl}\) & Constitutional at \(\EGUT\) \\
Proton stability & Model-dependent & Exact (gauged \(\U(1)_B\)) \\
Information sector & Absent & \(\U(1)_{Qi}\) (constitutionally mandated) \\
\bottomrule
\end{longtable}
}
\end{center}

\subsection{Where \GG{} and string theory could converge}

Table~\ref{tab:6d-comparison} states the comparison on inputs rather
than on results, which is the only fair way to make it. Despite the
structural differences it records, there are potential points of
convergence that deserve investigation.

\paragraph{(i) Non-SUSY string constructions.}
Recent work on non-supersymmetric heterotic constructions
\citep{KaidiTachikawaYonekura2025} and non-geometric
compactifications \citep{BaykataHamadaTarazi2024} shows that
string theory is not limited to supersymmetric vacua.
If a non-SUSY string construction reproduces the \GG{} gauge
group and anomaly structure, it would establish a connection
between the two frameworks without requiring SUSY.

\paragraph{(ii) The 6D--4D boundary.}
\GG{} uses an interval compactification \(M_6 = M_4\times I_r\times S^1_\theta\)
with orbifold boundary conditions. This is structurally similar
to heterotic orbifold compactifications \citep{DixonHarveyVafaWitten1985}.
The boundary anomaly inflow mechanism is identical in both cases.

\paragraph{(iii) The tensor sector.}
Both F-theory and \GG{} use anti-self-dual tensors for anomaly
cancellation. The global consistency conditions (charge lattice,
Wu structure, eta invariants) are the same mathematical objects
\citep{MonnierMoore2019,HopkinsSinger2005}.
A string theorist studying global anomalies in \GG{} would
use exactly the same tools as in F-theory.

\paragraph{(iv) The finiteness of the landscape.}
Both programs have finite landscapes, but for different reasons:
F-theory is finite because the number of elliptic CY threefolds
is bounded \citep{TaraziVafa2021,KimVafaXu2024};
\GG{} is finite because the \OGNfive{} arithmetic bounds the
Diophantine search space.
Understanding whether these two finiteness results are related
is an open question.





%% ===============================================================
%%  NEW SECTION: AdS/CFT, RANDALL-SUNDRUM, AND THE OGGEP DEPARTURE
%% ===============================================================
\section{From AdS/CFT and Randall--Sundrum to \OGGEP{}: shared geometry, transformed physics}
\label{sec:ads-rs}

The six-dimensional \GG{} bulk with its scale-profile metric, logarithmic
radial coordinate, and bulk-to-boundary projection is
structurally indebted to two of the most influential ideas in modern
string theory: the anti-de~Sitter/conformal field theory (AdS/CFT)
correspondence \citep{Maldacena1998,Gubser1998,Witten1998AdS}
and the Randall--Sundrum (RS) model of warped extra dimensions
\citep{RandallSundrum1999}.
\GG{} retains the core mathematics of both but replaces the
underlying physics with \OGGEP{}---the Operational/Open Geometry--Gauge
Equivalence Principle.
This section explains what is borrowed, what is replaced, and why
the replacement constitutes a departure rather than a minor variation.

\subsection{The AdS/CFT correspondence: a brief review}

The AdS/CFT correspondence, proposed by Maldacena in 1997
\citep{Maldacena1998} and made precise by Gubser, Klebanov,
and Polyakov \citep{Gubser1998} and by Witten \citep{Witten1998AdS},
establishes an exact duality between:
\begin{enumerate}[label=(\roman*), leftmargin=1.4cm]
\item a gravitational theory (Type~IIB string theory or supergravity)
 on \(\mathrm{AdS}_5\times S^5\), and
\item a four-dimensional conformal field theory (\(\cN=4\)
 super-Yang--Mills with gauge group \(\SU(N_c)\)) living on the
 boundary of AdS.
\end{enumerate}
The radial coordinate \(z\) of AdS encodes the energy scale of the
boundary CFT: the boundary at \(z\to 0\) corresponds to the UV,
and the interior at \(z\to\infty\) corresponds to the IR.
Bulk fields act as sources for boundary operators, and the on-shell
bulk action generates the boundary generating functional:
\begin{equation}
Z_{\mathrm{CFT}}[\phi_0]
= \exp\!\paren{-S_{\mathrm{bulk}}[\phi\big|_{z\to 0}=\phi_0]}.
\label{eq:ads-cft-dictionary}
\end{equation}
Holographic renormalization \citep{deHaroSkenderis2001}
makes this correspondence precise by regulating the on-shell action
near the boundary and identifying boundary counterterms with
renormalization-group counterterms.

Three key features of AdS/CFT are relevant for comparison:
\begin{enumerate}[label=(\alph*), leftmargin=1.4cm]
\item The radial direction is dual to the RG scale.
\item An on-shell bulk functional generates boundary correlators.
\item The duality is \emph{exact} (in principle) but requires
 supersymmetry and a large-\(N_c\) limit for computational control.
\end{enumerate}

\subsection{The Randall--Sundrum model: a brief review}

The Randall--Sundrum model \citep{RandallSundrum1999}
introduces a slice of \(\mathrm{AdS}_5\) bounded by two branes:
\begin{equation}
\dd s^2 = e^{-2k|y|}\,\eta_{\mu\nu}\,\dd x^\mu\dd x^\nu + \dd y^2,
\qquad
y\in[-\pi R_c,\,\pi R_c],
\label{eq:rs-metric}
\end{equation}
where \(k\) is the AdS curvature scale and \(R_c\) is the
compactification radius.
The UV brane sits at \(y=0\) and the IR brane at \(y=\pi R_c\).
The exponential warp factor generates a large hierarchy between
the Planck scale (set by the UV brane) and the TeV scale (set by
the IR brane):
\begin{equation}
\frac{m_{\mathrm{IR}}}{m_{\mathrm{UV}}} = e^{-k\pi R_c}.
\label{eq:rs-hierarchy}
\end{equation}
The Randall--Sundrum model was the first concrete realization
of the idea that an extra dimension with warp factor can solve
the gauge hierarchy problem without supersymmetry.

Three features are relevant for comparison:
\begin{enumerate}[label=(\alph*), leftmargin=1.4cm]
\item The fifth dimension is an interval (not a circle), with
 different physics at the two boundaries.
\item The warp factor generates a mass hierarchy geometrically.
\item The model does not require supersymmetry (though it is
 compatible with it).
\end{enumerate}

\subsection{What \GG{} borrows: the shared mathematics}

\GG{} inherits three mathematical structures from these constructions.

\paragraph{(i) The logarithmic radial coordinate.}
The \GG{} radial coordinate
\begin{equation}
r = \ln\!\paren{\frac{\EGUT}{\mu}}
\end{equation}
is the direct analogue of the AdS/CFT radial coordinate and
the Randall--Sundrum fifth dimension.
In all three frameworks, the extra dimension geometrizes
the renormalization-group scale.
The mathematical content is the same: the RG semigroup
\(\cG_{\lambda_1}\circ\cG_{\lambda_2}=\cG_{\lambda_1\lambda_2}\)
is additively represented by a logarithmic coordinate.

\paragraph{(ii) The warped metric.}
The \GG{} metric
\begin{equation}
\dd s_6^2 = e^{-2A(r)}\,g_{\mu\nu}(x)\,\dd x^\mu\dd x^\nu
+ \dd r^2 + R_\theta^2\,\dd\theta^2
\end{equation}
with \(A(r) = kr\) is structurally identical to the RS metric
\eqref{eq:rs-metric}, with the replacement \(|y|\to r\) and
the addition of the compact circle \(S^1_\theta\).
The warp factor \(e^{-2kr}\) generates the mass hierarchy
between the GUT scale and the infrared boundary.

\paragraph{(iii) The bulk-to-boundary functional.}
In AdS/CFT, the on-shell bulk action generates the boundary
generating functional \eqref{eq:ads-cft-dictionary}.
In \GG{}, the bulk functional generates the boundary effective
action through the \OGGEP{} channel:
\begin{equation}
\cD_{\OGGEP}:\quad
\rho_{\mathrm{Bulk}} \;\longmapsto\;
(\rho_{\partial},\;\Phi_{t,s},\;\{\cI_\alpha\}).
\end{equation}
Both are ``holographic'' in the sense that lower-dimensional
boundary physics is encoded in and generated by higher-dimensional
bulk physics.

\subsection{What \GG{} replaces: the transformed physics}

Despite sharing the mathematical infrastructure, \GG{} makes
three fundamental conceptual departures from AdS/CFT and
Randall--Sundrum.

\subsubsection{Departure 1: from exact duality to operational projection}

\paragraph{In AdS/CFT,} the bulk and boundary theories are
\emph{exactly dual}: they are different descriptions of the same
physics. The boundary CFT is a complete, self-contained theory;
the bulk gravity is another complete, self-contained theory.
Neither is ``more fundamental'' than the other.
The duality is \emph{ontologically symmetric}.

\paragraph{In \GG{}/\GDOS{},} the relationship is \emph{asymmetric}.
The six-dimensional bulk is the constitutionally complete theory.
The four-dimensional boundary is an \emph{operational projection}
obtained by tracing over the inaccessible bulk degrees of freedom.
The boundary theory is derived from the bulk, not dual to it.
The mathematical implementation is not a duality but a
completely positive trace-preserving (CPTP) channel:
\begin{equation}
\rho_{\partial} = \Tr_{\mathrm{env}}\!\paren{\rho_{\mathrm{Bulk}}},
\end{equation}
followed by the exact Nakajima--Zwanzig memory-kernel equation
for the reduced boundary dynamics.
The boundary observer does not ``see'' an equivalent dual theory;
the boundary observer sees a \emph{reduced} theory with
environmental decoherence, memory effects, and measurement-induced
state updates.

The difference is fundamental. AdS/CFT is a statement
about two complete theories. \OGGEP{} is a statement about how an
incomplete observer extracts lawful physics from a more complete bulk.

\subsubsection{Departure 2: from conformal symmetry to open-system dynamics}

\paragraph{In AdS/CFT,} the boundary theory is a conformal field
theory with exact conformal symmetry (at least in the
large-\(N_c\), strong-coupling limit).
The boundary dynamics is unitary, reversible, and exactly solvable
in many cases.

\paragraph{In \GG{}/\GDOS{},} the boundary dynamics is generically
\emph{non-unitary}, \emph{irreversible}, and \emph{open}.
The Nakajima--Zwanzig equation has a memory kernel and an
inhomogeneous term; the Markovian limit yields the GKLS
(Gorini--Kossakowski--Lindblad--Sudarshan) master equation with
a dissipator.
The mesoscopic coarse-graining yields the \GDOS{} continuity equation
with gradient, solenoidal, and diffusive current components.

This replacement is physically motivated: actual scientific
observations are performed by finite apparatuses interacting with
environmental degrees of freedom.
No real experiment is a conformal field theory.
The \OGGEP{} channel produces a boundary theory that is
\emph{operationally realistic} rather than conformally symmetric.

\subsubsection{Departure 3: from string-theoretic completeness to
 axiomatic measurement theory}

\paragraph{In AdS/CFT,} the bulk-to-boundary map is a tool for
computing correlators, partition functions, and entanglement
entropies.
It does not, by itself, address the measurement problem of
quantum mechanics.
The question ``what is a measurement?'' is external to the formalism.

\paragraph{In \GG{}/\GDOS{},} the bulk-to-boundary map \emph{is}
the measurement theory.
\AX{3} (\OGGEP{}) constitutionally requires that the theory provide
its own operational dictionary. The boundary instruments
\begin{equation}
\cI_\alpha(\rho_{\partial})
= \Tr_A\!\left[
(\mathbf{1}\otimes\Pi_\alpha)\,
U_{SA}(\rho_{\partial}\otimes\sigma_{\mathrm{app}})
U_{SA}^\dagger\,
(\mathbf{1}\otimes\Pi_\alpha)
\right]
\end{equation}
are not interpretational additions; they are derived from the
6D\(\to\)4D projection.
The Born rule, the instrument formalism, and the Hilbert-space
structure of four-dimensional quantum mechanics all emerge as
consequences of the operational channel.

No other aspect of \GG{} departs as sharply from the string-theoretic
tradition as this one. In string theory, quantum mechanics is a prerequisite
for the theory: one quantizes the worldsheet, computes amplitudes,
and takes the classical limit.
In \GG{}/\GDOS{}, quantum mechanics is a \emph{consequence} of the
theory: the boundary observer sees quantum behavior because the
6D\(\to\)4D channel is a CPTP map that produces density operators,
instruments, and Born-rule statistics.

\subsection{Comparison table: AdS/CFT, Randall--Sundrum, and \GG{}/\GDOS{}}

\begin{center}
{\footnotesize
\begin{longtable}{p{0.219\textwidth}p{0.229\textwidth}p{0.229\textwidth}p{0.219\textwidth}}
\caption[Three frameworks with warped extra dimensions, compared.]{Three constructions that share a warped extra dimension, set side by side so the shared machinery and the different commitments are both visible. All three use the same mathematics of a warped metric; they differ in dimension, in what fixes the warp factor, and in whether the extra direction is spatial. The third of those is the one to attend to, because it is where this framework departs from both neighbors: the radial direction here carries scale rather than length, and that reading is what makes the warp factor computable rather than chosen.}\label{tab:ads-rs-gg6}\\
\toprule
\textbf{Feature} & \textbf{AdS/CFT} & \textbf{Randall--Sundrum} & \textbf{\GG{} / \GDOS{}} \\
\midrule
\endfirsthead
\toprule
\textbf{Feature} & \textbf{AdS/CFT} & \textbf{Randall--Sundrum} & \textbf{\GG{} / \GDOS{}} \\
\midrule
\endhead
Spacetime dim. & 5 (AdS\(_5\)) & 5 & 6 (\(M_4\!\times\!I_r\!\times\!S^1_\theta\)) \\
Radial coord.\ meaning & RG scale / holog.\ depth & Hierarchy direction & \(r=\ln(\EGUT/\mu)\) \\
Warp factor & \(e^{-2z/L}\) & \(e^{-2k|y|}\) & \(e^{-2kr}\) \\
Boundary & Conformal bdry of AdS & UV and IR branes & \(M_4\) at \(r=0\) \\
Bulk--boundary relation & Exact duality & Localization of gravity & \OGGEP{} operational projection \\
SUSY & Required (\(\cN=4\) SYM) & Not required & Not required \\
Boundary dynamics & Unitary CFT & Standard QFT on brane & Open-system (GKLS/\GDOS{}) \\
Measurement theory & External to formalism & External to formalism & Derived from 6D\(\to\)4D channel \\
Gauge group & \(\SU(N_c)\) (any \(N_c\)) & Model-dependent & \(\SU(5)\!\times\!\U(1)_B\!\times\!\U(1)_{Qi}\) \\
UV completion & String theory at \(E_{\Pl}\) & Not specified & Constitutional at \(\EGUT\) \\
Anomaly cancellation & Via SUSY & Not addressed & \BIsix{} (non-SUSY GS mechanism) \\
Hierarchy solution & AdS/CFT duality & Warp factor & Warp factor + \OGNfive{} \\
Landscape & Large (many AdS vacua) & No landscape issue & Finite (Diophantine) \\
Born rule & Assumed & Assumed & Derived \\
\bottomrule
\end{longtable}
}
\end{center}

\subsection{What string theorists contributed and what \GG{} adds}

The intellectual debt to the string theory community is substantial
and should be stated without ambiguity.

\paragraph{What \GG{} inherits from string theory:}
\begin{enumerate}[label=(\roman*), leftmargin=1.4cm]
\item The Green--Schwarz mechanism for anomaly cancellation
 (Green and Schwarz, 1984).
\item The idea that an extra radial dimension can geometrize
 the renormalization-group scale
 (holographic RG, Maldacena--Gubser--Witten, 1997--1998).
\item The warped metric as a hierarchy-generating mechanism
 (Randall and Sundrum, 1999).
\item The technology of anomaly polynomials, descent equations,
 and charge lattices developed over four decades of string/F-theory
 research.
\end{enumerate}
Without these contributions, \GG{} could not exist in its present form.

\paragraph{What \GG{} adds beyond string theory:}
\begin{enumerate}[label=(\roman*), leftmargin=1.4cm]
\item A constitutional axiom system (\AX{1}--\AX{4}, \POS{1}--\POS{7}) that
 derives the bulk geometry, gauge group, and anomaly structure
 from admissibility requirements alone, without postulating strings,
 branes, or compactification manifolds. The fundamental excitations remain point-like local field quanta ($\AX{1}$ locality and causality), not extended objects; because the two further dimensions are non-spatial --- a scale dimension and a gauge dimension --- \GG{} remains a point-particle framework even as it geometrizes the gauge sector.
\item The \OGGEP{} channel: a lawful operational map from bulk to
 boundary that turns quantum measurement from an external postulate
 into a derived consequence of the 6D\(\to\)4D projection.
\item The \OGNfive{} arithmetic: a discrete integer ledger that freezes
 all dimensionless normalization freedom, replacing the continuous
 moduli spaces of string compactifications with a finite discrete
 structure.
\item The \BIsix{} identity: a non-supersymmetric six-dimensional
 Green--Schwarz mechanism with a constitutionally determined
 coefficient vector, achieving anomaly cancellation without
 SUSY, strings, or a Planck-scale UV completion.
\item The \GDOS{} boundary dynamics: an open-system evolution theory
 (Nakajima--Zwanzig \(\to\) GKLS \(\to\) \GDOS{}) that unifies
 quantum mechanics, measurement, and the Born rule as boundary
 projections of one bulk variational engine.
\end{enumerate}

\subsection{The philosophical shift: from ontology to operational law}

The most far-reaching difference between \GG{}/\GDOS{} and the AdS/CFT tradition
is not mathematical but philosophical.

In AdS/CFT, the bulk and boundary are two \emph{equally real}
descriptions of the same ontological reality.
The holographic principle is a statement about the nature of
spacetime itself: the number of degrees of freedom scales with
the boundary area, not the bulk volume.

In \GG{}/\GDOS{}, the bulk is the \emph{constitutionally complete}
description. The boundary is what a \emph{finite observer} can
register. The distinction between bulk and boundary is not
ontological but \emph{operational}: it is the distinction between
the full lawful dynamics and the reduced dynamics accessible to
an agent who cannot retain all degrees of freedom.

This shift has a concrete consequence for the measurement problem.
In AdS/CFT, the measurement problem is no easier than in ordinary
quantum mechanics, because the boundary theory is itself a unitary
quantum theory.
In \GDOS{}, the measurement problem is dissolved: boundary observations
are CPTP channels with instruments, and the Born rule is a
consequence of the channel structure, not an additional postulate.

Whether this philosophical shift is ultimately correct is an
empirical question.
But the conceptual clarity of this shift is testable:
if quantum mechanics is the operational shadow of a
higher-dimensional constitutional law, then the apparent
``strangeness'' of quantum measurement is not strange at all---it
is the inevitable consequence of projecting a closed bulk onto
an open boundary.


%% --------------------------------------------------------------
%%  NEW SUBSECTION (v7.2): Constitutional classification of AdS/CFT,
%%  Randall-Sundrum, and BI-6 as Stratum 1 (GGEP-closed) theorems.
%% --------------------------------------------------------------
\subsection{AdS/CFT, Randall--Sundrum, and \BIsix{} as Stratum~2 specializations of the GGEP-closed axiom layer}
\label{ssec:bi6-stratum1}

The comparisons developed in
Sections~\ref{sec:comparison-10d}--\ref{sec:ads-rs} can be sharpened
by adopting the constitutional-classification vocabulary established
in \AFive{} (\AFive{} Part~VIII, Figure~5;
\citealp{ArisakaA5GDOS}). We do so explicitly here, because the
classification places the present anomaly theorem in a sharply
defined axiomatic position relative to the broader twentieth-century
high-energy edifice, and the position matters both for the scope of
what the theorem accomplishes and for the scope of what remains to
be done by the companion papers.

\paragraph{The GGEP-closed axiom layer (Stratum~2).} By the
constitutional classification of \AFive{} Part~VIII, the
\emph{GGEP-closed axiom layer} is the subset
$\{\AX{1},\AX{2}\}$ of the four-axiom \GGConstitution{}. It
activates local causality (\AX{1}, \LC) and the geometry--gauge
equivalence principle (\AX{2}, \GGEP) but neither the
operational/open extension (\AX{3}, \OGGEP) nor the maximum-symmetry
/ minimum-freedom selection (\AX{4}, \MSMF). Stratum~2 of the
constitutional tower is the framework class admitted by exactly
this axiom subset.

\paragraph{Frameworks placed on Stratum~2.} The frameworks that
sit on Stratum~2 of the constitutional tower include, in
chronological order:
\begin{itemize}[leftmargin=2em,itemsep=0.25em]
\item the Bianchi identity in its classical form
 \citep{Bianchi1902};
\item general relativity \citep{Einstein1915}, whose composite
 geometry--gauge content is the Einstein tensor itself
 $G_{\mu\nu}$ with the contracted Bianchi identity $\nabla^\mu
 G_{\mu\nu} = 0$ as the local conservation law;
\item Yang--Mills gauge theory \citep{YangMills1954}, which is
 the prototype geometrization of internal symmetry;
\item the Green--Schwarz anomaly-cancellation mechanism
 \citep{GreenSchwarz1984}, which is the moment higher-dimensional
 gauge theory recognized that gauge curvature $F$ and gravitational
 curvature $R$ must lie inside a single closed four-form on the
 bulk manifold (Section~\ref{sec:GS_as_GGEP_completion} of the
 present paper);
\item superstring theory at the GGEP-closed level (perturbative
 Type~I, IIA, IIB, heterotic; \citealp{Polchinski1998a,
 Polchinski1998b});
\item the AdS/CFT correspondence
 \citep{Maldacena1998,Gubser1998,Witten1998AdS} and its
 Gubser--Klebanov--Polyakov--Witten dictionary
 (Section~\ref{sec:ads-rs} above);
\item the Randall--Sundrum warped construction
 \citep{RandallSundrum1999} and the family of holographic
 dualities that descend from it;
\item the present \BIsix{} anomaly theorem, which is the
 six-dimensional four-component generalization of the Green--Schwarz
 factorization, and the entire descent + global-completion
 proof chain of Sections~\ref{sec:structural-proof}
 through~\ref{sec:global-completion}.
\end{itemize}
All eight items in this list are theorems of the same axiom subset
$\{\AX{1},\AX{2}\}$; they differ from each other only in which
specific bulk geometry they take as input, not in which
constitutional clauses they invoke.

\paragraph{What Stratum~2 can derive.} The GGEP-closed axiom layer
suffices to derive:
\begin{enumerate}[leftmargin=2em,itemsep=0.25em]
\item the existence of a composite geometry--gauge connection on a
 principal $G_{\mathrm{tot}}$ bundle over an admissible bulk
 manifold (\AX{2});
\item the closed gauge-invariant four-form
 $X_4 = \tr(R\wedge R) - \Tr(F \wedge F) - \ldots$ that lives on
 this composite bundle (Chern--Weil theory applied to the
 composite connection);
\item the modified Bianchi identity $\dd H_3 = X_4$ for the tensor
 sector, in any number of even spacetime dimensions where the
 anomaly polynomial admits a Green--Schwarz factorization
 $I_{2n+2} = \tfrac12 X_4 \wedge X_{2n-2}$ or, in six dimensions
 with rank-one factorization, $I_8 = \tfrac12 X_4 \wedge X_4$
 (Theorem~\ref{thm:bi6-cancellation} of the present paper);
\item one-loop and global anomaly cancellation by inflow, given the
 Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore
 differential-cohomology framework on the chosen bulk class
 (Theorem~\ref{thm:bi6-global-complete} of the present paper);
\item the existence of a bulk-to-boundary projection,
 in the closed-system limit, on any bulk geometry with a conformal
 boundary (AdS/CFT as the canonical example).
\end{enumerate}
This is the theorem class admitted by Stratum~2. It is large enough
to contain the entire mathematical content of the GS mechanism, of
AdS/CFT, of holographic dualities, of Randall--Sundrum, and of the
present \BIsix{} construction.

\paragraph{What Stratum~2 cannot derive without \AX{3}.} The
following structural facts are \emph{not} theorems of the GGEP-closed
axiom layer alone; they require activation of \AX{3} (\OGGEP):
\begin{enumerate}[leftmargin=2em,itemsep=0.25em]
\item an operational dictionary $\cD_{\OGGEP}:
 (\cH_\Bulk,\rho_\Bulk,U_\Bulk) \mapsto
 (\cH_\Bdy,\rho_\Bdy,\Phi_{t,s},\{\cI_\alpha\})$ from inaccessible
 bulk data to registered boundary observables, with the
 bulk-to-boundary projection a completely positive trace-preserving
 (CPTP) channel rather than a unitary equivalence;
\item the seven textbook postulates of quantum mechanics --- Hilbert
 space, the Born rule, the L\"uders projection, decoherence,
 einselection, the violation of Bell inequalities --- recovered as
 theorems of the constitutional projection rather than as separate
 postulates (\AFive{} Part~III \citep{ArisakaA5GDOS});
\item the second law of thermodynamics as the data-processing
 inequality of the boundary CPTP channel, with the Spohn quantum
 $H$-theorem as the operator-level realization;
\item the dissolution of the black-hole information paradox by
 identifying Hawking radiation as the output of the OGGEP channel
 at the event horizon, with the Page curve as a bipartite-pure-state
 bookkeeping argument and the 2020 islands formula as the bulk-side
 geometric reconstruction.
\end{enumerate}
The absence of \AX{3} from Stratum~2 is the constitutional source
of the QM-foundations gap that has divided the string-theory and
QM-foundations communities for fifty years.

\paragraph{What Stratum~2 cannot derive without \AX{4}.} The
following structural facts are not theorems of GGEP-closed alone;
they require activation of \AX{4} (\MSMF):
\begin{enumerate}[leftmargin=2em,itemsep=0.25em]
\item a unique vacuum: GGEP-closed admits any bulk geometry whose
 Bianchi identity factorizes through a single tensor sector,
 including the perturbative string landscape of $\sim 10^{500}$
 inequivalent compactifications \citep{Susskind2003,Douglas2003}.
 \AX{4} selects the unique bulk that minimizes continuous parameter
 freedom subject to the maximum-symmetry constraint, eliminating
 the landscape by construction. In the present paper, the
 six-dimensional bulk
 $\cM_6 = \cM_4 \times I_r \times S^1_\theta$ of \citet{ArisakaA2GG6}
 is the \AX{4}-selected representative of GGEP-closed-admissible
 bulks;
\item the \OGNfive{} integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$ as
 the unique Diophantine solution of the global consistency of
 \BIsix{} (Section~\ref{sec:ogn5-theorem} of the present paper).
 Without \AX{4}, the integers would be a chosen ledger;
 \emph{with} \AX{4}, they are the unique ledger compatible with
 the rest of the constitutional clauses.
\end{enumerate}
On this classification, what the cohort program adds at Stratum~2 is
\AX{4}, and it is \AX{4} that removes the continuous freedom a landscape
of vacua represents. That is a statement about what the two frameworks
assume, not a claim that one has failed: a framework without \AX{4} is
not required to have a unique vacuum, and the string program has never
claimed that it must.

\paragraph{The Schr\"odinger--GKLS analogy at the holographic layer.}
The relationship between AdS/CFT (a GGEP-closed framework on
Stratum~2) and the cohort 6D-to-4D projection (a Full-Constitution
framework on Stratum~4 that uses all four axioms) is precisely the
holographic-layer analogue of the relationship between the
Schr\"odinger equation (closed-system dynamics) and the
Gorini--Kossakowski--Lindblad--Sudarshan master equation (open-system
dynamics). Stated as a parallel pair:
\[
\text{GKLS (open)} \quad
\xrightarrow[\gamma_\alpha \to 0]{} \quad
\text{Schr\"odinger (closed)},
\]
\[
\text{Cohort 6D--4D projection (open)} \quad
\xrightarrow[\AX{3}\text{ off}]{} \quad
\text{AdS/CFT duality (closed)}.
\]
The structural content of both limits is the same: switching off the
open-system axiom (the GKLS dissipators $\gamma_\alpha$ at the
dynamical layer, the operational coarse-graining \AX{3} at the
holographic layer) collapses an open CPTP evolution to a closed
unitary evolution. AdS/CFT is the closed-bulk limit of the cohort
6D-to-4D projection; this is why AdS/CFT realizes holography as
information-preserving \emph{exact duality} (the boundary CFT and
the bulk gravitational theory share the same Hilbert space, with the
on-shell bulk action generating boundary correlators exactly),
whereas the cohort projection realizes holography as
information-coarsening CPTP map (the boundary state is mixed even
when the bulk state is pure, with the Born rule a Gleason-type
theorem on the boundary algebra and the second law a Lindblad-type
monotonicity statement on the boundary entropy). Both readings are
correct on their respective axiom subsets; neither is the other's
competitor. The present paper's anomaly theorem closes the
GGEP-closed prerequisite for either reading. \AFive{} \S69.4
\citep{ArisakaA5GDOS}, where the second law appears as a
data-processing inequality, develops the analogy in full.

\paragraph{Where this places \BIsix{}.} The constitutional
classification therefore places the central theorem of the present
paper precisely: \BIsix{} anomaly freedom is a Stratum~2 theorem
of the constitutional tower. It is the strongest Stratum~2 result
the framework needs. It does \emph{not} yet establish the
operational \OGGEP{} channel that turns the bulk theory into a
boundary observational program --- that is the job of the
companion paper \AFive{} \citep{ArisakaA5GDOS}. It does \emph{not}
yet select the unique \GG{} bulk uniquely among GGEP-closed-admissible
bulks --- that is the job of \AX{4} as deployed in
\ATwo{} \citep{ArisakaA2GG6} the dimensional-minimality theorem
(dimensional minimality), Theorem on metric uniqueness, and the
\OGNfive{} closure of the present paper's Section~\ref{sec:ogn5-theorem}.
But it does, at this stratum, deliver the global anomaly consistency
that is the gating prerequisite for both the \AX{3} and the \AX{4}
applications. In the language of the constitutional flowchart of
\AFive{} Figure~5: \BIsix{} closes the lock on the bottom-right cell
of Stratum~2.

\paragraph{Historical analogy: Riemann to Einstein, Green--Schwarz to the cohort.}
The reader who finds this constitutional reading unfamiliar may find
the following historical parallel useful. Riemannian geometry was
developed by \citet{Riemann1854} as pure mathematics, sixty years
before Einstein \citep{Einstein1915} identified the physical setting
(gravitation as curvature of spacetime) in which Riemannian geometry
became operative. The mathematical content of Riemann's
\emph{Habilitationsschrift} was correct, complete, and unchanged by
Einstein's later constitutional identification; what changed was the
\emph{recognition} of which physical question the mathematics was
the right tool for. The relationship between Green--Schwarz / AdS/CFT
/ Randall--Sundrum and the cohort constitutional setting is structurally
analogous, with the temporal direction reversed: the geometric
machinery was developed inside string theory between 1984 and 2020,
and the constitutional setting in which that machinery is recognized
as the GGEP-closed content of the \GGConstitution{} is identified by
the cohort program of 2026. None of the string-theoretic
mathematical content is altered or contradicted; what is added is
the constitutional setting in which it is recognized as the
geometrization of gauge in higher dimensions. The 1984 GS mechanism
becomes, in this reading, the moment string theory glimpsed
\AX{2}~(\GGEP) at the technical level; the 1997 AdS/CFT
correspondence becomes the moment string theory glimpsed the
GGEP-closed bulk-to-boundary projection in the
closed-system limit; the 2020 islands formula becomes the moment
string theory glimpsed the bulk-side geometric reconstruction of
what the operational \AX{3} channel does on the boundary side. Each
move was correct as far as it went, and each was limited only by
the axiomatic clauses (\AX{3} and \AX{4}) that the cohort framework
now makes explicit.

After all, the test of a constitutional framework is not whether it
makes its predecessors wrong but whether it makes their contributions
intelligible inside a wider picture. By that test, the constitutional
classification places the entire string-theoretic edifice cleanly
into the GGEP-closed layer of a constitutional tower whose three
remaining layers (\AX{1}~only at Stratum~1, plus \AX{3} at Stratum~3,
plus \AX{4} at Stratum~4) are the contributions of the cohort
program.



\subsection{Scope and limits: what the cohort framework does and does not claim about string theory}
\label{ssec:scope-and-limits}

A senior reader trained on string theory will reasonably ask whether
the cohort framework's claim that ``string theory is the
geometrization of gauge, not the quantization of geometry'' is the
right summary of four decades of string-theoretic results. We owe
that reader a precise answer.

\paragraph{What the cohort claim DOES include.}
The GGEP-closed sector of string theory --- the descent formalism of
\citet{WessZumino1971} and \citet{Becchi1976}, the modified Bianchi
identity of \citet{GreenSchwarz1984}, the characteristic-class
content of anomaly cancellation, the differential-cohomology
classification of \citet{HopkinsSinger2005} and
\citet{MonnierMoore2019}, and the AdS bulk-to-boundary projection
\emph{in the closed-system limit} of the
Gubser--Klebanov--Polyakov--Witten dictionary --- is captured by
\AX{2}~(\GGEP) of the \GGConstitution{}. This sector is
classical-topological in content: the mathematical statements
($\dd H_3 = X_4$, Stora--Zumino descent, Hopkins--Singer
differential cohomology) make no reference to quantum loops in
their formulation. The cohort framework recognizes this sector as
the \AX{2} content of any candidate theory and elevates the
recognition to a constitutional axiom.

\paragraph{What the cohort claim DOES NOT include.}
The quantum-mechanical content of string theory is a separate body
of work that the cohort framework does not subsume. The
closed-string vertex operator that yields the graviton mode is a
quantum-mechanical operator on the worldsheet conformal field
theory; the $\alpha'$ and $1/N$ expansions in AdS/CFT are quantum
corrections to the bulk supergravity description; the islands
formula of Almheiri--Penington--Stanford--Yang and
Penington--Shenker--Stanford--Yang of 2020
\citep{Almheiri2020, Penington2020} computing the Page curve of
Hawking radiation is a quantum-gravitational result with no
classical-topological analogue. These results are not in the
GGEP-closed sector; they belong to the quantum-dynamical sector
that runs on top of the GGEP-closed backbone. The cohort
framework neither replaces nor reproduces this content in the
present paper.

\paragraph{Precise scope of the geometrization-of-gauge claim.}
When we say in this paper that \emph{string theory is the
geometrization of gauge in higher dimensions, not the quantization
of geometry}, the precise claim is this: the structural backbone
that the Green--Schwarz mechanism, AdS/CFT, holographic dualities,
the modified Bianchi identity, and the islands formula all share
--- the classical-topological content of differential characters
on a higher-dimensional bulk manifold --- is the
geometrization-of-gauge content forced by \AX{2}. The
string-theoretic quantum dynamics on top of this backbone is a
separate sector that the cohort framework neither replaces nor
reproduces.

\paragraph{Precise scope of the AdS/CFT analogy.}
The AdS/CFT-as-\AX{3}-switched-off-limit framing of
Section~\ref{sec:ads-rs} (and developed in detail in
\AFive{} \citep{ArisakaA5GDOS}) should be read in the same
scope-restricted sense: it identifies the structural backbone
shared by AdS/CFT and the cohort 6D-to-4D projection at the
\GGEP-closed layer. It does \emph{not} claim a precise
GKP/Witten-level dictionary
\citep{Gubser1998, Witten1998AdS} from the 6D \GG{} bulk
action to the IIB SUGRA action on $\mathrm{AdS}_5 \times S^5$ in
the \AX{3}-off limit; that explicit dictionary, including the
matching of the holographic correlator generating functional, of
the $a$- and $c$-coefficients between bulk SUGRA and boundary
CFT, of Wilson loops via minimal surfaces, and of the
Ryu--Takayanagi entanglement entropy formula, is recorded as
Open Problem~O12 of Section~\ref{sec:open-problems}.

\paragraph{Why we make this qualification explicit.}
The cohort framework's contribution at the GGEP-closed layer is
intellectually serious: it elevates the geometric content of forty
years of string-theoretic results to an axiomatic recognition.
That contribution does not require, and does not benefit from,
the suggestion that the quantum content of string theory has been
displaced or subsumed. The honest summary is: the cohort framework
adds a constitutional setting in which the classical-topological
backbone is recognized as \AX{2}; the quantum sector of string
theory remains the body of work that the string community has
built over four decades, and the cohort framework neither replaces
nor disputes it.

\section{Swampland compatibility of \GG{} and \BIsix{}}
\label{sec:swampland}

The swampland program of \citet{Vafa2005},
\citet{BrennanCartaVafa2017}, and \citet{TaraziVafa2021} proposes that
not every internally consistent effective field theory of gravity is
the low-energy limit of a UV-complete quantum theory of gravity:
those that are not lie in the ``swampland''. The program has
proposed a family of conjectures (no global symmetries, completeness
hypothesis, weak-gravity conjecture, distance conjecture, de~Sitter
conjecture, cobordism conjecture) that any candidate quantum-gravity
EFT must satisfy. The conjectures were derived from string-theoretic
intuition, and their universality outside string theory is itself a
matter of active debate.

A reader trained on the swampland program will reasonably ask
whether the non-stringy cohort framework \GG{}\,$+$\,\BIsix{}
satisfies these constraints. We check the framework against the
five most widely accepted conjectures and state the result
honestly.

\paragraph{Conjecture 1: No global symmetries.}
Every continuous global symmetry of a consistent quantum-gravity
EFT must be either gauged or broken at high energy. In the cohort
framework, the bulk gauge group is
$G_\Bulk = \Spin(1,5) \times \SU(5) \times \UoneB \times \UoneQi$
with all four factors gauged by \AX{2}~(\GGEP), and the
constitutional admissibility theorem (Theorem~\ref{thm:UoneQi-admissibility})
confirms that the boundary $\UoneB$ baryon-number symmetry and the
optional $\UoneQi$ information-current symmetry both descend from
bulk gauge structure. The cohort framework therefore has no
ungauged continuous \emph{structural} symmetries. The
gravitational-completion sector introduces the accidental flavor
symmetry of its gauge-singlet spectators (as any large singlet
sector does); its fate --- lifted by bulk masses or broken by the
boundary projection --- is part of the completion-sector physics
carried by the spectrum papers. At the structural level:
\textbf{satisfied}.

\paragraph{Conjecture 2: Completeness hypothesis.}
Every charge in the gauge lattice must be populated by some physical
state. The cohort framework realizes this in the integer
\OGNfive{} ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$, which fixes
the bulk charge spectrum to integer multiples of the primitive
lattice with $J = 7$ giving the symmetric closure cardinality
$\{0, \pm 1, \pm 2, \pm 3\}$. Whether every lattice site in this
range is populated by a physical state is the subject of the
benchmark spectrum construction of Section~\ref{sec:model-level}
and Theorem~\ref{thm:model-level}; the spectrum is constructed
explicitly and the lattice is saturated by the cohort fermion
reservoir for every $\Omega \in 25\mathbb{Z}$.
\textbf{Satisfied at the existence level.}

\paragraph{Conjecture 3: Weak-gravity conjecture (WGC).}
For every $\U(1)$ gauge symmetry in a quantum-gravity EFT, there
must exist a particle with charge $q$ and mass $m$ satisfying
$g q M_\Pl / m \geq 1$ (in appropriate units), so that
extremal black holes can decay. For the cohort bulk, the WGC
constraint applies to $\UoneB$ and $\UoneQi$. The cohort spectrum
delivers particles with charges $\pm 1, \pm 2, \pm 3$ under $\UoneB$
and similarly under $\UoneQi$; whether the lightest charged state
in the descended 4D spectrum saturates the WGC inequality is a
calculation that requires the boundary spectrum derivation of
\AFive{}, \ASeven{}, and \ANine{}. At the bulk level, the
existence of charged states populating the lattice is established;
the precise mass-to-charge ratio remains to be computed.
\textbf{Open: structurally compatible, quantitative check
deferred} to the boundary spectrum derivation.

\paragraph{Conjecture 4: Distance conjecture (swampland distance).}
Every infinite-distance limit in moduli space is accompanied by a
tower of light states. The cohort framework has \emph{no infinite-distance
moduli limits}: the scale interval is $r \in [0, K]$ with
$K = 35$ frozen by the \OGNfive{} ledger, the compact direction is
$\theta \in [0, 2\pi)$ with no continuous radius parameter, and
\AX{4} eliminates the moduli space of bulk geometries to a single
point (the cohort canonical bulk). There is no infinite-distance
limit at which the conjecture could be tested.
\textbf{Vacuously satisfied} (no moduli space to traverse).

\paragraph{Conjecture 5: Cobordism conjecture.}
In a consistent quantum-gravity EFT, the bordism classes of the
theory's backgrounds must be trivializable in the full theory: any
non-trivial class defines a conserved global charge, which must be
broken by defects or vanish \citep{BrennanCartaVafa2017}. For the cohort \GG{} bulk
with structure group $\Spin(1,5) \times \SU(5) \times \UoneB
\times \UoneQi$, the relevant bordism groups in low dimensions are
standard \citep{FreedHopkins2016}:
$\Omega^{\Spin}_n(\mathrm{pt}) = \mathbb{Z},\ \mathbb{Z}_2,\
\mathbb{Z}_2,\ 0,\ \mathbb{Z},\ 0,\ 0,\ 0$ for $n = 0, \ldots, 7$;
gauge-twisted bordism with the cohort $G_\Bulk$ adds finite
contributions that do not introduce new infinite-rank obstructions
(see~\citealp{HamadaLoges2024}). The structural verification of
hypothesis H3 in Theorem~\ref{thm:bi6-global-complete} confirms
bordism-trivializability of the anomaly cocycle on the cohort
background class. \textbf{Satisfied at the structural level}; the
exhaustive enumeration of all twisted bordism contributions for
the four-component $G_\Bulk$ is Open Problem~O10.

\paragraph{Summary of the swampland audit.}
Of the five widely accepted swampland conjectures, the cohort
\GG{}\,$+$\,\BIsix{} framework satisfies four (no-global-symmetries,
completeness, distance, cobordism) at the structural level, and the
fifth (weak-gravity) is structurally compatible with the precise
quantitative check deferred to the boundary-spectrum derivation in
\AFive{}, \ASeven{}, and \ANine{}. The framework is therefore
swampland-compatible by the most widely accepted criteria. We
omit the de~Sitter conjecture from the audit because its status as
a universal swampland constraint is itself disputed in the recent
literature \citep{TaraziVafa2021, KimVafaXu2024} and the cohort
cosmological inventory $\Omega_\Lambda = 1 - 1/\pi$ of
\ATen{} \citep{ArisakaA10Cosmos} is the subject of a dedicated
treatment there.

After all, the test of a constitutional framework against the
swampland constraints is not whether the framework reproduces the
specific string-theoretic derivations of those constraints but
whether it satisfies the universal content the constraints
encode. By that test, the cohort framework appears to be in the
landscape rather than the swampland, with the caveat that
quantitative confirmation of WGC and the explicit cobordism
enumeration remain follow-up calculations.



\bigskip
\noindent The comparisons developed in Sections~\ref{sec:comparison-10d}--\ref{sec:ads-rs} can be summarized in one sentence: \GG{} inherits the mathematical infrastructure of the Green--Schwarz mechanism, six-dimensional anomaly-polynomial technology, holographic radial coordinates, and warped extra dimensions, while replacing supersymmetry, fundamental strings, Planck-scale completion, exact duality, and the landscape of string vacua with the \GGTheory{} Constitution, the \OGNfive{} arithmetic, and the \OGGEP{} operational projection.

\bigskip

\noindent\textbf{What the comparison found.} Five differences carry the
weight, and they can be stated in a sentence each.

Supersymmetry: the neighboring six-dimensional constructions require
it, and this one does not. Almost every other entry in the comparison
follows from that one difference, which is why the tables are built on
inputs rather than on results.

Fundamental strings: the ten-dimensional mechanism is a statement about
a theory of extended objects, and the six-dimensional one is a statement
about a field theory on a bulk. The cancellation calculus is identical;
what it is a calculus \emph{of} is not.

Completion: the string program takes the Planck scale as the place where
the description is completed, and the present framework does not offer a
completion at all. That is a limitation and it is recorded as an open
problem, not argued away.

Duality: AdS/CFT reads the radial direction as an exact correspondence
between a gravity theory in the bulk and a field theory on the boundary.
Here the radial direction is read as a literal, irreversible interval of
scale. These are different claims about the same geometry, and only one
of them can be right about any given system.

Freedom: a construction with a landscape of vacua can absorb a failed
measurement by moving within the landscape; a construction with no
continuous parameters cannot. This is a difference in how the two
frameworks can be tested, and it cuts both ways --- the rigid framework
is easier to kill, which is an advantage only if it survives.

None of this establishes that one program is better than the other,
and the paper does not claim it does. What the comparison establishes is
narrower and more useful: the six-dimensional construction is not a
rival derivation of the same thing, and it is not a special case of the
ten-dimensional one either. It borrows a technique and applies it under
a different set of assumptions, and the honest way to evaluate it is on
whether those assumptions produce numbers that turn out to be right.
Part~V takes up that question.

\GGPart[Discussions]{Discussions}{What the anomaly theorem signifies,
what interlocks to make it unique, and what is new here; the predictions
that can falsify it; the map of what this paper gives the cohort and
takes from it; and the honest triage of what remains open}

The proofs are now complete. Parts~II and~III established local and
global anomaly cancellation and derived the \OGNfive{} integers from
\BIsix{}; Part~IV placed these results against the major frameworks of
modern high-energy theory. This Part steps back to ask the broader questions in the canonical
cohort order: what the theorem signifies, what interlocks to make it
unique and what is new here, where it can be killed, what the paper
gives the rest of the cohort and takes from it, and what remains open.
The Conclusions follow in Part~VI, written in plain language.

A discussion section is where a paper is most tempted to overstate, so
it is worth fixing the standard in advance. Three gradings are used
throughout and they are used strictly. A result is \emph{derived} if
there is a chain of theorems from the constitutional clauses to the
statement, with every link written down. It is \emph{conditional} if
such a chain exists but rests on a hypothesis that has not itself been
established --- in which case the hypothesis is named in the same
sentence. It is \emph{open} if no theorem in this paper or in any
current companion carries the answer, and saying so is not a failure of
nerve but the only way the grading means anything.

Section~\ref{sec:discussion} argues the significance and states what
interlocks with what, then lists the ten things that are structurally
new. Section~\ref{sec:predictions-falsification} gives the falsifiers:
the specific measurements that would end the framework, with the
precision each would need. Section~\ref{sec:coherence-map} makes the
cohort dependencies explicit in both directions.
Section~\ref{sec:open-problems} triages twelve open problems into
resolved, partially resolved, and genuinely open, and names the two that
are genuinely open without softening either.

The last of those is the one a skeptical reader should turn to first.

\section{Significance, uniqueness, and what is new}
\label{sec:discussion}

Four things make the anomaly theorem of this paper significant beyond
its technical statement.

\emph{First, it is the cohort's license to exist.} Every downstream
paper of the \GGTheory{} program --- the boundary law of \AFive{},
the pillars of \ASix{} and \ASeven{}, the Standard-Model codebook of
\ANine{}, the cosmological inventory of \ATen{}, the capstone of
\AEleven{} --- builds on the \GG{} bulk. A six-dimensional chiral
theory that failed the anomaly test would not be inaccurate; it would
be inconsistent, and everything built on it would be void. The theorem
proved here is therefore not one result among many: it is the gate
through which the rest of the program walks. That is also why the
paper takes the global half of the theorem so seriously: a bulk theory
that is only infinitesimally consistent could still hand the
\OGGEP{} boundary channel a topological phase ambiguity, and the
lawfulness of the entire boundary program rests on its absence.

\emph{Second, it converts arithmetic into physics.} The same
factorization that closes the anomaly forces the trace-ratio lock
$|a/c| = |a/d| = 5/2$ and, through it, the integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$. Those five integers are not bookkeeping:
they reappear as $\alpha^{-1}(0) = 2^J + N^2 = 137$ at leading order,
as the kaxion twin-peak spacing $\Delta\nu/\nu = 1/K$, and as the
throat depth that fixes $\Mlock \approx 3.85$~TeV. The anomaly test
--- the same test that selected $\SO(32)$ and $E_8 \times E_8$ in ten
dimensions --- here selects numbers that experiments can check.

\emph{Third, it demonstrates that the Green--Schwarz mechanism does
not need supersymmetry.} Forty years of six-dimensional anomaly
freedom have been written almost entirely inside $\cN = (1,0)$
supergravity. The present paper closes local and global anomaly
freedom for a non-supersymmetric chiral theory, with the
constitutional clauses doing the selection work that supersymmetry
would have done --- a proof of concept that the anomaly-inflow
tradition extends beyond the supersymmetric island where it was born.

\emph{Fourth, it explains why the ledger cannot run.} The
couplings of physics run with scale; the anomaly coefficients do
not, and the cohort's recent campaign (\textsf{GG-W1-Symmetry};
\citealp{ArisakaP5CPT}) has made the reason exact: \emph{holonomy
requires closed loops, and an interval has none.} The two
non-spatial fibers of the \GG{} bulk are of opposite topological
kind. The compact circle $S^1_\theta$ carries all of the bulk's
holonomy --- quantized, gauge-invariant, protected; the radial
interval $I_r$ carries none --- only the one-way RG semigroup of
\POS{5}. Anomaly inflow is circle-side bookkeeping: every
coefficient matched in this paper is topological pairing data on
the fiber that remembers, which is why the ledger's entries are
integers, exact at every scale, rather than couplings with
anomalous dimensions --- the scale-dependence of physics is the
\emph{other} fiber's business, and the interval has no holonomy to
renormalize. That is why one and the same ledger
$(M,N,L;J,K) = (1,3,5;7,35)$, fixed once by the factorization at
the top of the descent, can surface unchanged at
$\alpha^{-1}(0) = 2^J + N^2$, in the kaxion spacing $1/K$, and at
the TeV lock --- across the whole of cosmic history, the $e^{140}$
of boundary time chronicled in the capstone
\citep{ArisakaA11Origin} --- with no radiative erosion in between.
Exactness here is not an approximation that improves; it is the
topology of the fiber on which the ledger is written. The anomaly
theorem is thereby the constitution's guarantee that its
arithmetic is permanent: \emph{the circle remembers, and the
ledger is written on the circle.}

\subsection{Uniqueness: what interlocks with what}
\label{sec:uniqueness-nobility}

\paragraph{Uniqueness.} The construction is rigid at every joint where
a conventional model would have a dial. The coefficient vector
$(1,-1,-2/5,-2/5)$ is the unique rank-one factorization of the
microscopically computed anomaly matrix (Theorem~\ref{thm:rank-one}),
unique up to the overall sign that the inflow convention fixes. The
Abelian fraction $2/5$ is the trace ratio of the Standard-Model
embedding and is invariant on the one-point moduli space of
normalizations that preserve the integer charges
(Section~\ref{sec:ogn5-theorem}). The integer family $(5k, 3k)$
collapses to $(5,3)$ under the \MSMF{} minimality clause
(Theorem~\ref{thm:lN-minimality}); the closure pair $(J,K)$ is then
overdetermined --- two independent identities, $K = LJ$ and
$K = M^2 + N^2 + L^2$, meet at the same integer $35$. And the
gravitational completion is a $2\times 2$ linear system with the
unique solution $n_{\mathrm{SD}} = 4\Omega$, $n_F = 32\Omega$ (exactly
eight singlets per tensor). Nothing in this chain can be adjusted
without breaking something else in it: that interlocking is the
uniqueness claim, stated at the calibrated register of
Section~\ref{sec:proved-suggested-open}.



\begingroup\emergencystretch=3em\sloppy
\subsection{What is new: the ten structural innovations}
\label{sec:ten-innovations}

The contribution of \AThree{} is to prove that the \GG{} stage defines a
consistent quantum theory: a complete local-and-global anomaly theorem for
\BIsix{}, together with the integer ledger that keeps the sector finite. We
collect here the ten structural innovations that proof delivers, each carrying
its cohort-global identifier (\S\ref{sec:naming-convention}).

\paragraph{Innovation 1 (I-BI6-1): A complete local-and-global anomaly theorem
in six dimensions.} \AThree{} proves anomaly freedom for \BIsix{} on the
physical intended \GG{} class --- not a symbolic restatement of Green--Schwarz
but a concrete 6D result (T-BI6-6.3 local, T-BI6-7.2 global) bringing the
explicit factorization, the descent proof, the differential-cohomology
completion, and the automatic-admissibility theorem under one roof.

\paragraph{Innovation 2 (I-BI6-2): The explicit \BIsix{} four-form and its
single anomaly polynomial.} The four-form
$X_4=\tr R^2-\Tr F_5^2-\tfrac25 F_B^2-\tfrac25 F_{Qi}^2$ with
$I_8=\tfrac12 X_4^2$ is written explicitly, so the entire anomaly is carried by
one composite invariant rather than an open-ended set of counterterms.

\paragraph{Innovation 3 (I-BI6-3): Single-tensor Green--Schwarz cancellation.}
One two-form tensor $B_2$ cancels all gauge, gravitational, and mixed anomalies
through a rank-one inflow (T-BI6-3.2, T-BI6-6.1), in contrast to the multiple
tensor fields of generic 6D supergravity.

\paragraph{Innovation 4 (I-BI6-4): Global anomaly freedom by differential
cohomology.} The Wu--Chern--Simons / Hopkins--Singer completion of the tensor
counterterm establishes global (not merely local) consistency on the intended
class (T-BI6-7.2), closing the topological phase ambiguity a purely local proof
would leave open.

\paragraph{Innovation 5 (I-BI6-5): The trace-ratio lock $|a/c|=|a/d|=5/2$.} The
\BIsix{} coefficient vector $(1,-1,-\tfrac25,-\tfrac25)$ is forced, not fitted:
the Abelian trace ratio is determined by the \OGN{} integers (T-BI6-9.1), fixing
the anomaly polynomial with no free coefficient.

\paragraph{Innovation 6 (I-BI6-6): The \OGNfive{} ledger as a theorem.} The
integer packet $(M,N,L;J,K)=(1,3,5;7,35)$ is stated and used at theorem level
(T-BI6-10.2), with the Diophantine minimality of $(L,N)=(5,3)$ proved
(T-BI6-10.3) --- the arithmetic lock that keeps the \BIsix{} sector finite and
sharply constrained, rather than an external input.

\paragraph{Innovation 7 (I-BI6-7): Anomaly freedom as the prerequisite for a
lawful boundary law.} Global consistency is shown to be a precondition for the
\OGGEP{}/\GDOS{} bulk-to-boundary map: a globally multi-valued parent would hand
the boundary functional an unphysical topological phase, so the theorem opens
the door to the entire boundary program rather than merely repairing a bulk
gauge theory.

\paragraph{Innovation 8 (I-BI6-8): Constitutional admissibility of both bulks.}
\AThree{} proves that the constitution admits both the three-component
$\Spin(1,5)\times\SUfive\times\UoneB$ and the four-component bulk with $\UoneQi$
(T-BI6-7.5): the informational sector is an admissible extension adopted on
empirical grounds, not an axiomatic forcing --- an honesty distinction the
cohort carries downstream.

\paragraph{Innovation 9 (I-BI6-9): Anomaly cancellation as geometric smoothness.}
The Green--Schwarz mechanism is read as the technical completion of AX-2
(\GGEP{}): canceling an anomaly is closing a snag in the composite
gauge--gravity fabric, a geometric-smoothness condition rather than a
quantization prescription.

\paragraph{Innovation 10 (I-BI6-10): \BIsix{} as the Stratum-2 anomaly theorem
of the cohort.} The result is logically prior to every downstream application:
\[
\begin{aligned}
&\text{Constitution}\Rightarrow\text{\GG{} stage}+\BIsix{}\\
&\quad\Rightarrow\text{local/global anomaly freedom}\xRightarrow{\OGGEP{}}\text{lawful \GDOS{} channel},
\end{aligned}
\]
so Standard-Model closure, cosmology, and the life-science branches all rest on
the anomaly theorem established here.
\endgroup

\section{Predictions for falsification}
\label{sec:predictions-falsification}

A theoretical framework earns its place in physics not by being internally consistent alone --- many consistent theories describe no aspect of the real world --- but by making predictions that can be tested and, in principle, refuted. The \BIsix{}/\GDOS{} framework makes several such predictions, each carrying a global falsifier identifier F-BI6-$k$ and tied directly to the \OGNfive{} arithmetic or the \GG{} gauge structure:
\begin{enumerate}[label=(\roman*), leftmargin=1.4cm]
\item \textbf{(F-BI6-1) Proton stability.}
 Gauged \(\U(1)_B\) protects the proton: the observable falsifier is
 any confirmed \(\Delta B = 1\) or \(\Delta B = 2\) process (proton
 decay or \(n\)--\(\bar n\) oscillation); only the electroweak-anomaly
 channel \(\Delta B \in 3\mathbb{Z}\), which cannot deplete a single
 nucleon, is exempt. The protection is conditional on one named
 step --- the holonomy computation that the descended boundary
 charge is baryon number on the split spectrum, stated and fenced
 in \citep{ArisakaA2GG6} and carried as the boundary-side theorem
 of \citep{ArisakaA9SM}.
\item \textbf{(F-BI6-2) The 28 Standard Model parameters.}
 The companion paper \citep{ArisakaA9SM} carries the 28 continuous
 Standard-Model parameters (6~quark masses, 3~lepton masses,
 3~CKM angles + 1~phase, 3~gauge couplings, the Higgs mass and VEV,
 the QCD \(\theta\) angle, and the neutrino sector) from the
 \OGNfive{} arithmetic and the \GDOS{} boundary projection, with an
 explicit per-parameter provenance triage (derived,
 derived-conditional, or asserted, each with a stated error budget).
 The falsifier is executable parameter by parameter: a measurement
 outside the budget \ANine{} states for a derived entry falsifies
 the framework.
\item \textbf{(F-BI6-3) KK lock cluster at 3.85~TeV.}
 The Kaluza--Klein tower of the \GG{} throat is anchored at
 \(\Mlock \approx 3.85\)~TeV, fixed by the \OGNfive{} throat depth
 \(K=35\) and the scale profile with no continuous parameter. Per
 \citep{ArisakaP1KK}, the HL-LHC falsifier is carried by the
 colored members of the cluster (the color octet), while the
 leptophobic \(Z'_B\) is collider-invisible as a separate signal
 with its coupling confined to a discrete menu; the mass anchor is
 theorem-level, the cluster morphology model-level.
\item \textbf{(F-BI6-4) Kaxion dark matter near \(8.1~\mu\)eV.}
 The Green--Schwarz tensor sector produces an axion-like particle
 (the ``Kaxion'') with benchmark mass \(8.1~\mu\)eV
 (\(\approx 1.95\)~GHz) and twin-peak signature
 \(\Delta\nu/\nu = 1/35\). The canonical falsifier of
 \citep{ArisakaC1Kaxion} is the fenced discrete window
 \(\nu_K \in [1.59, 2.13]\)~GHz (cap index
 \(\Xi_{\mathrm{cap}} \in \{35, \ldots, 47\}\), the named
 mass-uniqueness conjecture): a null across the closed window at
 the stated coupling depth kills the framework. The band is being
 swept by IBS-CAPP and ADMX-EFR.
\item \textbf{(F-BI6-5) GUT-adjacent slow smooth inflation.}
 The inflationary phase is a GUT-adjacent plateau one e-fold below
 \(\EGUT\), with \(V_{\mathrm{inf}}^{1/4} = 8.3\times 10^{15}\)~GeV,
 predicting the spectral tilt \(n_s = 0.9620\) and the two-sided
 tensor-to-scalar ratio \(r = 0.0043\)
 \citep{ArisakaA10Cosmos}. A measured \(n_s\) or \(r\) firmly
 outside these values would falsify the descent.
\item \textbf{(F-BI6-6) Dark matter abundance.}
 The Kaxion must account for the full geometric inventory
 \(\Omega_{\mathrm{DM}} = 1/\pi - 1/(2\pi^2) \approx 0.27\) of
 \citep{ArisakaA10Cosmos}; the abundance mechanism is the
 consistency requirement carried in \citep{ArisakaC1Kaxion}, and a
 demonstrated failure of the Kaxion to saturate the inventory
 falsifies the dark-sector closure.
\item \textbf{(F-BI6-7) The \(\U(1)_{Qi}\) sector.}
 The informational gauge boson coupling to the \(\U(1)_{Qi}\)
 current is a unique structural prediction of \GG{} with no
 analogue in the Standard Model or in any string compactification;
 its operational observables are carried by the boundary program
 of \citep{ArisakaA7Qi4} (the \QiFour{} pillars and the \(\QB\)
 calibration), so it is listed here as a structural discriminator
 rather than a dated laboratory falsifier.
\end{enumerate}

The scope of these predictions---from the proton lifetime to
collider resonances to dark matter to inflation to the complete
Standard Model parameter set---is the framework's full exposure
surface, and each item carries its provenance and its conditioning
in the companion paper that owns it. The companion papers
(\citealp{ArisakaA9SM}; \citealp{ArisakaA10Cosmos};
\citealp{ArisakaP1KK}; \citealp{ArisakaC1Kaxion}) develop the
quantitative detail.


\section{Cross-Paper Coherence Map}
\label{sec:coherence-map}

\AThree{} does not stand alone. The local and global anomaly-freedom theorems of Sections~\ref{sec:structural-proof}--\ref{sec:global-completion}, the constitutional-admissibility theorem of Section~\ref{sec:UoneQi-constitutional-status}, the model-level benchmark spectrum of Section~\ref{sec:model-level}, the trace-ratio and \OGNfive{} theorems of Section~\ref{sec:ogn5-theorem}, and the GS-as-\GGEP{} completion of Section~\ref{sec:GS_as_GGEP_completion} together feed into several companion papers of the \GGTheory\ program; conversely, the constitutional axioms and postulates that drive those theorems live in the companion paper \AOne{}, and the 6D bulk plus the \BIsix{} coefficient vector that the present paper takes as input live in \ATwo{}. The present paper sits at the \emph{anomaly heart} of the program: \AOne{} supplies the constitutional clauses, \ATwo{} supplies the geometric stage and the \BIsix{} four-form, \AThree{} closes the local and global quantum-consistency layer, \AFive{} carries the bulk-to-boundary projection that depends on \AThree{}'s global theorem for its lawfulness, and \AFour{}, \ASix{}, \ASeven{}, \ANine{}, \ATen{} build the boundary-side phenomenology in their respective domains. This section states that structure in prose. The two supporting tables --- which \AThree{} theorems are used by which companion papers, and which axiomatic inputs \AThree{} imports from \AOne{} and \ATwo{} --- are given with the dependency ledger in Appendix~\ref{app:checklist}, together with the audit trail that ties each pointer to a specific section or theorem of the cited companion.

\section{Problems Solved, Open Questions, and Future Directions}
\label{sec:open-problems}

\GG\ closes the bulk-side anomaly question, but the present paper does not close the \GGTheory\ program. Several of the questions raised by this paper are answered fully within the present text or by named companion papers; several are settled in their core but await a specific technical sub-task; and a small number remain genuinely open even within the broader program. Honesty requires that we mark each one with its status. We follow the cohort's triage convention: every entry $O_k$ is labeled \textbf{Resolved} (the question has a complete answer in the present paper or in a named companion paper, with explicit citation), \textbf{Partially resolved} (the central claim is settled but a specific sub-task is still pending), or \textbf{Genuinely open} (no theorem, in this paper or any current companion paper, currently carries the answer). We address twelve open problems below.

\subsection*{O1 (PS-BI6-1). Local anomaly cancellation of $\BIsix$ in six dimensions}

\textbf{Resolved.} The factorization $I_8 = \tfrac12 X_4^2$ with $X_4 = \tr R^2 - \Tr F_5^2 - (2/5) F_B^2 - (2/5) F_{Qi}^2$ is established in Theorem~\ref{thm:rank-one} (rank-one criterion) and Theorem~\ref{thm:bi6-cancellation} (\BIsix{} cancellation) of the present paper. The single-tensor Green--Schwarz counterterm $S_{\mathrm{GS}} = \kappa\int B_2 \wedge X_4$ cancels the full perturbative one-loop anomaly via standard descent, and the BRST consistency of the resulting quantum gauge theory follows by Slavnov--Taylor identity. This is the local content of the central theorem of the paper.

\subsection*{O2 (PS-BI6-2). Global anomaly cancellation of $\BIsix$ on the intended \GG{} background class}

\textbf{Resolved.} Theorem~\ref{thm:bi6-global-complete} of the present paper proves that, on the eight-clause intended \GG{} background class, the combined phase of the chiral determinant and the exponentiated \BIsix{} counterterm is globally well-defined and gauge-invariant, using the modern Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore differential-cohomology framework. The four hypotheses (H1)--(H4) of the global theorem are verified automatically on the intended class: the product / collar mapping tori bound spin eight-manifolds with extending data (H1), $\pi_6$ vanishes for $\SUfive\times\UoneB\times\UoneQi$ (H2), the \BIsix{} lattice lift obeys the Wu-shifted integrality law (H3), and $X_4$ lifts to a global degree-four differential cocycle (H4). Hence $\BIsix$ is completely anomaly-free at both the local and global levels on the physical intended class.

\subsection*{O3 (PS-BI6-3). Constitutional status of $\UoneQi$: forced or admissible extension?}

\textbf{Resolved.} Theorem~\ref{thm:UoneQi-admissibility} of the present paper (Section~\ref{sec:UoneQi-constitutional-status}) settles this question. The legacy reading that ``\AX{3} + \POS{4} + \POS{1} forces $\UoneQi$'' is shown to overstate the axiomatic content: CPTP is a structural property of the GKLS channel (Lindblad 1976; Gorini--Kossakowski--Sudarshan 1976), not a covariance demand under a fresh $U(1)$. The constitutional clauses admit both the three-component bulk $G_{\mathrm{Bulk}}^{(3)} = \Spin(1,5) \times \SUfive \times \UoneB$ and the four-component bulk $G_{\mathrm{Bulk}}^{(4)} = \Spin(1,5) \times \SUfive \times \UoneB \times \UoneQi$. The present paper adopts the four-component version on empirical grounds: the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ requires $\UoneQi$ on the boundary with bioenergetic conversion factor $\QB \approx 0.1555$~eV per qubit matching ATP hydrolysis at biological temperatures; the \OGNfive{} dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$ closes only for the four-component case under the equal-Abelian-treatment convention; and the boundary $Qi$-4 pillars of \ASeven{} require $\UoneQi$ as host symmetry. The companion paper Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6} cross-cites this resolution at its Theorem on Six Forces, at the \BIsix{} coefficient theorem, and at Objection~4 of its Common Objections section.

\subsection*{O4 (PS-BI6-4). Trace-ratio lock $|a/c| = |a/d| = 5/2$ from the \OGNfive{} ledger}

{\sloppy \textbf{Resolved.} The trace-ratio identity that fixes the \BIsix{} coefficient vector $(a, b, c, d) = (1, -1, -2/5, -2/5)$ is established as the trace-ratio theorem in Section~\ref{sec:ogn5-theorem} of the present paper, and then used to derive the \OGNfive{} integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ as a Diophantine cascade. The integer $5$ in the ratio $|a/c| = |a/d| = 5/2$ is the $L = 5$ size of the $\SUfive$ fundamental representation; the integer $2$ is the rank of the equally-treated Abelian sector $\UoneB \times \UoneQi$. No continuous rescaling freedom remains; the ratio is rigid.\par}

\subsection*{O5 (PS-BI6-5). Existence of a finite chiral spectrum saturating the $\BIsix$ factorization}

\textbf{Resolved.} Theorem~\ref{thm:model-level} of Section~\ref{sec:model-level} constructs an explicit benchmark spectrum --- a modular core block plus an orbit family plus gravitational completion --- that realizes the factorization $I_8 = \tfrac\Omega2 X_4^2$ with all ten coefficient-matching conditions and all eight vanishing conditions satisfied. The benchmark proves that the structural theorem is satisfiable; minimality of the integer normalization $\Omega$ is the next sub-question (see O8 below).

\subsection*{O6 (PS-BI6-6). The 1984 Green--Schwarz mechanism as the technical pioneering of \AX{2} \GGEP{}}

\textbf{Resolved.} Section~\ref{sec:GS_as_GGEP_completion} of the present paper carries the methodological reading that frames the 1984 GS achievement as the constitutional-precursor instance of the \AX{2} (\GGEP) Geometry--Gauge Equivalence Principle: GS demonstrated, in a specific 10D heterotic string model, that the metric and the gauge connection can sit together inside a single closed differential 4-form on the bulk manifold; \AX{2} elevates that move from a technical refinement to a constitutional axiom. $\BIsix$ is the four-component generalization of the two-component GS pattern, with the Lorentz, $\SUfive$, $\UoneB$, and $\UoneQi$ factors all entering on equal logical footing inside one closed gauge-invariant 4-form $X_4$. Section~\ref{sec:GS_as_GGEP_completion} is the home of this reading; \ATwo{} \citep{ArisakaA2GG6} cross-cites it at its $\BIsix$ coefficient theorem.

\subsection*{O7 (PS-BI6-7). Bulk-to-boundary projection through $\OGGEP$, with anomaly freedom as prerequisite}

\textbf{Resolved.} The $\OGGEP$ operational projection from the 6D bulk to the 4D boundary --- whose lawfulness depends on the global anomaly freedom proved in the present paper --- is the subject of \AFive{} \citep{ArisakaA5GDOS}. The anomaly proof of Sections~\ref{sec:structural-proof}--\ref{sec:global-completion} of the present paper supplies the prerequisite (Lemma on phase propagation through the $\OGGEP$ channel of Section~\ref{sec:global-completion}); \AFive{} then constructs the \GDOS{} runtime as the boundary CPTP-channel image of the bulk and derives 4D quantum mechanics, 4D general relativity, and the low-energy Standard Model as boundary projections. The handshake between the two papers is clean: \BIsix\ anomaly freedom in / \GDOS{} runtime out.

\subsection*{O8 (PS-BI6-8). Minimal integer normalization $\Omega$ for the $\BIsix$ spectrum}

\textbf{Partially resolved.} Theorem~\ref{thm:model-level} establishes \emph{existence} of an integer $\Omega$ for which the \BIsix{} factorization $I_8 = \tfrac\Omega2 X_4^2$ admits a finite chiral spectrum satisfying all eighteen coefficient-matching and vanishing conditions. The minimality sub-question --- the smallest such $\Omega$ --- is a bounded finite Diophantine search whose answer is not provided by the present paper. The companion paper \ATen{} \citep{ArisakaA10Cosmos} drives $\Omega \to 1$ dynamically as a cosmological attractor under the GUT-adjacent slow smooth inflationary descent, but the static minimal-$\Omega$ lower bound for the bulk Diophantine system itself is the open sub-task. The search is qualitatively simpler than the string landscape (it is bounded and finite), but it has not yet been carried out.

\subsection*{O9 (PS-BI6-9). Explicit Monnier--Moore mapping-torus partition functions for the non-supersymmetric \BIsix{} setting}

\textbf{Resolved (constructive).} Subsection~\ref{ssec:sd-partition} now closes this problem at the level at which it was open: Theorem~\ref{thm:refinement-unique} proves the Wu--Chern--Simons refinement of the \BIsix{} tensor sector is unique (no discrete choice exists on a unimodular lattice), Lemma~\ref{lem:twist-exclusion} excludes every deformation both structurally and constitutionally, the supersymmetry audit shows the Monnier--Moore construction consumes no supersymmetry, and Theorem~\ref{thm:sd-partition} writes the partition function down --- the canonical theta block of Eq.~\eqref{eq:Z-tensor}, single-valued by Theorem~\ref{thm:unrestricted-global}. The earlier status of this entry stands as history: the structural global theorem verifies the four hypotheses (H1)--(H4) of the framework automatically on the intended \GG{} class, so the global anomaly phase vanishes by hypothesis-checking rather than by explicit partition-function computation. An explicit numerical evaluation of the anomaly field theory's partition function on each individual mapping torus --- of the kind that Monnier and Moore carry out for $\cN = (1,0)$ supergravity --- would serve as an independent confirmation of the structural argument; the same computation should treat the benchmark's full chiral-tensor sector --- the \(X_4\)-coupled mediator together with its \(4\Omega - 1\) gauge-singlet spectators --- as a single anomaly field theory. On the intended \GG{} class the phase is now computed exactly (Lemma~\ref{lem:computed-phase}: an APS index per species, and a vanishing filling signature for the tensor sector), and Theorem~\ref{thm:unrestricted-global} now extends the structural computation to the winding and mixed-clutching classes as well, on signature-normalized fillings. With Subsection~\ref{ssec:sd-partition} in place, the explicit numerical evaluation guards no assumption at all: the refinement is unique, the characteristic is canonical, and the phase is one, each by theorem. The evaluation retains the standing of re-running a verified computation --- an error-catching cross-check the 6D-anomaly community is welcome to perform, no longer a load-bearing open problem of this paper.

\subsection*{O10 (PS-BI6-10). Extension of the global theorem beyond the intended \GG{} background class}

\textbf{Resolved for the \GG{} gauge content; open only as a general classification program.} Theorem~\ref{thm:unrestricted-global} (Subsection~\ref{ssec:unrestricted-global}) extends the global theorem beyond the intended class: the reduced spin bordism groups \(\widetilde\Omega^{\mathrm{spin}}_5(BG)\) and \(\widetilde\Omega^{\mathrm{spin}}_7(BG)\) vanish for \(G = \SUfive\times\UoneB\times\UoneQi\), every closed spin seven-datum bounds, fillings can be normalized to signature zero, and the phase is one --- with no restriction on the base topology or on the homotopy class of the transformation, within the differential-cohomology accounting of Part II (whose defining refinement is now proved unique and whose partition function is constructed, Subsection~\ref{ssec:sd-partition}). What remains open under this heading is not a gap in the \GG{} chain but the community-wide question: the same analysis for \emph{arbitrary} gauge packets and tensor lattices, where the bordism groups need not vanish and genuinely obstructed cases exist.

\subsection*{O11 (PS-BI6-11). Non-perturbative completion of the bulk theory}

\textbf{Genuinely open.} Theorems~\ref{thm:bi6-cancellation} and \ref{thm:bi6-global-complete} of the present paper establish, respectively, perturbative local and global anomaly freedom of $\BIsix$ on the intended class. But a fully non-perturbative quantum theory of the 6D bulk --- including the lattice (or other) regularization that defines the path integral, the global topology of the principal $G_{\Bulk}$-bundle on $M_6$, and the rigorous existence of the corresponding quantum field theory in the Wightman / Haag--Kastler axiomatic sense --- is genuinely open. The relevant mathematical tools (constructive QFT, lattice gauge theory in dimension six, Osterwalder--Schrader reflection positivity for scale-profile backgrounds) are all in active development outside the \GGTheory\ program, and connecting them to the bulk theory is a long-term task.

\subsection*{O12 (PS-BI6-12). Generation count: why exactly three generations of fermions?}

\textbf{Genuinely open.} The \OGNfive{} ledger of Section~\ref{sec:ogn5-theorem} supplies the integer $N = 3$ as the triplicity index, and the Quaternionic Goldilocks reading of \ATwo{} \citep{ArisakaA2GG6} suggests that three generations are the algebraically forced count for the $\mathrm{SL}(2, \mathbb{H})$ rung of the magic Lorentzian-dimension ladder. But neither the present paper nor any current companion paper derives the three-generation structure of the boundary fermion content from the constitutional clauses as a theorem. \ATwo{} lists this as O11 of its own Open Problems section and identifies three candidate mechanisms (topological zero-mode counting, magic-ladder algebraic forcing, Hosotani Wilson-line breaking on $S^1_\theta$); none has been carried to a theorem. This is, in our view and in the view recorded by \ATwo{}, the single most important open theoretical problem of the \GGTheory\ program.

\paragraph{Summary of the triage.} Of the twelve open problems above, eight are \textbf{Resolved} (O1--O7 and O9), two are \textbf{Partially resolved} (O8 and O10), and two are \textbf{Genuinely open} (O11--O12). The eight Resolved problems are answered either by theorems of the present paper (O1--O6 and O9) or by a named companion paper that builds on this paper's anomaly proof (O7). The two Partially resolved problems each name the specific sub-task that is still pending --- the minimal-$\Omega$ Diophantine search (O8), and the extension of global consistency from the \GG{} gauge content to a general classification (O10). The two Genuinely open problems --- non-perturbative completion (O11) and generation count (O12) --- carry the framework's theoretical research agenda forward. The test of an honest framework is not that it answers every question on the first pass, but that it knows which questions it has answered, which it has half-answered, and which it has not yet touched.

\paragraph{Future directions.} The triage above sets the near-term
research program in order of leverage. (i) \emph{Global phases beyond
the intended class}: the winding and mixed-clutching transformations of
Open Problem O10 are the one place where a genuinely new Dai--Freed
evaluation is required; the intended-class phases are now computed
exactly (Lemma~\ref{lem:computed-phase}). (ii) \emph{The completion
sector's spectrum}: the gauge-singlet fermions admit Dirac pairings
that can lift most of the sector, while the spectator tensors, all of
one chirality, cannot pair --- their boundary projection and
cosmological exposure are the sector's real open physics. (iii)
\emph{The holonomy-descent theorem}: the single fenced step behind
F-BI6-1, that the descended boundary charge is baryon number on the
split spectrum --- the subject of the dedicated proton-stability
treatment. (iv) \emph{One variational principle for bulk and
boundary}: an action whose boundary term delivers the localized inflow
of Section~\ref{subsec:boundary-mixed-4d} while the bulk variation
reproduces the sealed descent of Section~\ref{sec:structural-proof}.
(v) \emph{The minimal-\(\Omega\) search} (O8) and, above all,
(vi) \emph{the generation count} (O12) --- still, in our view, the
single most important open problem of the program.

\paragraph{Scope and limitations.} Before we close, let us state the scope and the limits of the present paper as a self-contained block. \emph{Scope.} This paper proves complete local and global anomaly freedom of the modified Bianchi identity $\BIsix$ on the physical intended \GG{} background class, using the modern Wu--Chern--Simons / Hopkins--Singer / Monnier--Moore differential-cohomology framework. The local theorem (Theorem~\ref{thm:bi6-cancellation}) is the standard Green--Schwarz descent argument, applied to the four-component $X_4 = \tr R^2 - \Tr F_5^2 - (2/5) F_B^2 - (2/5) F_{Qi}^2$. The global theorem (Theorem~\ref{thm:bi6-global-complete}) verifies the four hypotheses (H1)--(H4) of the differential-cohomology framework automatically on the intended class. The constitutional-admissibility theorem (Theorem~\ref{thm:UoneQi-admissibility}) shows that the \GGTheory{} Constitution admits both the three-component and four-component bulks; the four-component version is preferred on empirical grounds. The model-level benchmark spectrum theorem (Theorem~\ref{thm:model-level}) constructs an explicit finite chiral spectrum saturating the \BIsix{} factorization. The \OGNfive{} theorem of Section~\ref{sec:ogn5-theorem} derives the integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ as a Diophantine cascade. The GS-as-\GGEP{} completion subsection (Section~\ref{sec:GS_as_GGEP_completion}) frames the 1984 Green--Schwarz mechanism as the technical pioneering of \AX{2} (\GGEP). \emph{Limits.} The paper does \emph{not} carry out the bulk-to-boundary projection that produces 4D quantum mechanics, 4D general relativity, and the low-energy Standard Model --- this is the subject of \AFive{} \citep{ArisakaA5GDOS}. The paper does \emph{not} reconstruct the universal constants $\{c, G_N, \hbar; k_B, Q_B\}$ from the bulk geometry --- this is the subject of \AFour{}. The paper does \emph{not} carry out the empirical calibration of the boundary phenomenology (CKM, neutrino masses, fine-structure constant, bioenergetic $\QB$, cosmological inventory) --- these are the subjects of \ASeven{} \citep{ArisakaA7Qi4}, \ANine{} \citep{ArisakaA9SM}, \ATen{} \citep{ArisakaA10Cosmos}. The paper does \emph{not} compute explicit Monnier--Moore mapping-torus partition functions for the \BIsix{} setting (Open Problem~O9), nor does it determine the minimal integer normalization $\Omega$ of the \BIsix{} spectrum (Open Problem~O8), nor does it extend the global theorem beyond the intended class (Open Problem~O10), nor does it provide a non-perturbative completion of the bulk theory (Open Problem~O11). \emph{Dependencies.} The validity of every theorem in the present paper is conditional on the correctness of the constitutional clauses \AX{1}--\AX{4} and \POS{1}, \POS{3}, \POS{4}, \POS{5}, \POS{6} of \AOne{} \citep{ArisakaA1Constitution}. What the present paper takes from \ATwo{} \citep{ArisakaA2GG6} is the \emph{object} and not the theorems: the six-dimensional stage, the four-factor gauge packet, the rank-one tensor channel and the \BIsix{} four-form enter here as clauses (C1)--(C4) of the admissible class of Definition~\ref{def:admissible-main}, which is background data quantified over, not a premise imported from a proof. Every proof below is carried out inside that class and consumes nothing further from any companion. The two papers are therefore parallel consumers of the constitution rather than a chain: \ATwo{} establishes that the constitution \emph{selects} this bulk, the present paper that the bulk so specified is \emph{consistent}, and neither derivation runs through the other. Should the constitutional clauses be revised, the chain of theorems must be re-walked; should the specification of the object change, the admissible class must be restated and the proofs re-run inside it. The full inward map of dependencies is recorded in
Section~\ref{ssec:inward-map}.

\bigskip

\noindent\textbf{Where the discussion leaves things.} Three summaries
are worth carrying into the Conclusions.

The first is about what kind of result this is. The anomaly theorem is
not a prediction; it is a license. It says the six-dimensional stage is
a consistent place to do physics, and it says so without consuming a
single measured quantity. Everything the cohort builds on that stage ---
the boundary projection, the Standard-Model codebook, the cosmological
inventory --- would be void if this Part had come out the other way, and
none of it is made true by this Part coming out the way it did. A
license is a necessary condition, not a sufficient one, and the paper is
careful not to trade on the difference.

The second is about where the framework can be killed. Seven falsifiers
are listed, and each names a measurement, a value, and a precision.
Three of them will be tested in the present decade: the resonance
cluster near $3.85$~TeV, the kaxion line near $8.1$~\textmu eV with its
companion one part in thirty-five away, and the inflationary parameters.
The others are longer-term. What the framework does not have is a way to
survive any of them failing --- there is no continuous parameter to
adjust and no landscape to move within. That is a genuine risk and it is
taken deliberately.

The third is about what remains undone, and it is the part a reader
should weigh most heavily. Twelve open problems are triaged. Two are
genuinely open: a non-perturbative completion of the bulk theory, and
the generation count. The second of these is the one that matters most
for how this framework should be judged, because the number three
appears in the ledger as a color count and appears in nature as a
generation count, and the paper does \emph{not} derive the second from
the first. It says so in four places, and it says so here again, because
the temptation to blur the two is exactly what an honest framework has
to resist. Anyone who can close that gap --- or show it cannot be closed
--- would be doing the most useful work available on this program, and
the author would welcome the collaboration.

\GGPart[Conclusions]{Conclusions}{What has been proved, what it means,
and why it matters --- in plain language first, then in the precise
register of the theorem chain, and last the place of the result in the
series}

This Part says what has been established, and it says it twice: once
in ordinary language, for a reader who has not followed the
calculations, and once in the register of the theorem chain, for a
reader who has.

The order is deliberate. A result that cannot be stated plainly is
usually a result that is not yet understood, and the discipline of
saying it without symbols is a check on the author as much as a service
to the reader. Section~\ref{sec:conclusions} is that statement. It
uses no notation and assumes no background beyond a first course, and a
reader who reads only that section will come away with an accurate
picture of what this paper does and does not claim.

Section~\ref{sec:established-precisely} then restates the same content
precisely: what was proved, what it rests on, which claims are
conditional and on what, which remain open, and where the result sits in
the twenty-two-paper cohort. Section~\ref{sec:big-picture} closes with
the wider reading --- what it would mean, if the argument holds, for the
relationship between geometry and the forces --- and marks clearly where
that reading stops being a theorem and starts being a proposal.

%% ===============================================================
%% SECTION 10: CONCLUSIONS
%% ===============================================================
\section{The result in plain language}
\label{sec:conclusions}

Here is what this paper proves, said as plainly as we can say it.

Every force we know is described by a gauge theory, and every gauge
theory built from chiral matter carries a hidden risk: the quantum
theory can betray the very symmetry that defines it. When that
happens the theory does not become approximate --- it dies. Probability
leaks, ghosts fail to decouple, and no measurement it predicts can be
trusted. Physicists call this failure an \emph{anomaly}, and passing
the anomaly test is the entrance examination that every candidate
theory of nature must sit.

\GGTheory{} proposes that the world we see is the four-dimensional
face of a six-dimensional geometry. This paper sat that geometry down
for the examination and graded every question. The result: it passes
--- completely. The one-loop failure is canceled exactly by a single
tensor field flowing in from the bulk; the subtler global failures, the ones that only show up when the
theory is carried around non-contractible loops in the space of its own
backgrounds, cancel too --- first on the class of backgrounds the
theory actually uses, and then, once the topological fine print is
removed, on every shape of space the question can be asked about; and the passing grade is not generic. The theory passes because five
small integers --- $1, 3, 5, 7, 35$ --- take exactly the values they
take, and it would fail for almost any other choice.

That is the deep physical intuition worth carrying away. In this
framework the anomaly is not a quantum disease to be cured; it is a
classical fact about the geometry --- a statement, descended from
Bianchi's 1902 identity, that curvature is stitched together
smoothly --- and the quantum loop is merely the instrument on the
boundary that measures it. Canceling the anomaly is not surgery; it
is the recognition that gravity and the gauge forces were cut from one
cloth in the bulk, so precisely that the seams --- the coefficients
$(1, -1, -2/5, -2/5)$ --- cannot be re-sewn in any other pattern.
The examination is passed not by tuning but by rigidity.

And rigidity is testable. The same integers that pass the anomaly test
put numbers on the sky and in the laboratory: an inverse
fine-structure constant $\alpha^{-1} = 137$ at leading order --- the
number that says the electric force between two electrons is about
$137$ times weaker than the crude quantum unit --- a dark-matter note near
$1.95$~GHz with a companion line one part in thirty-five away, a
resonance cluster near $3.85$~TeV, a proton that does not decay. If
any of those numbers comes out otherwise, the geometry that passed
this examination is the wrong geometry --- and the paper you are
holding tells you exactly which page fails. Few theories offer their
critics that courtesy; we consider it the point.

\section{What has been established, what it rests on, and what remains open}
\label{sec:established-precisely}

\subsection{The four ingredients under one roof}

The central achievement of this paper is a complete local-and-global anomaly theorem for \BIsix{} on the physical intended \GG{} background class. The theorem does more than restate the Green--Schwarz idea in new symbols. It is a concrete six-dimensional result with four distinct ingredients brought under one roof: the explicit \BIsix{} factorization
\[
X_4=\tr R^2-\Tr F_5^2-\frac25F_B^2-\frac25F_{Qi}^2,
\qquad
I_8=\frac12 X_4^2,
\]
the descent proof of the local anomaly cancellation, the Wu--Chern--Simons / differential-cohomology completion of the tensor counterterm, and the automatic admissibility theorem for the intended \GG{} category.

A second achievement is that the constitutive integer packet
\[
(M,N,L;J,K)=(1,3,5;7,35)
\]
is now stated and used at theorem level. In particular, the paper makes clear that \(N=3\) is fixed first as the bulk color triplicity index by the \BIsix{} trace ratio, that \(L=5\) is the defining unified block size of \(\SUfive\), that \(J=7\) is the minimal symmetric closure cardinality of the charge packet \(\{0,\pm1,\pm2,\pm3\}\), and that \(K=35\) is both the rectangular lock index and the quadratic witness. This is the arithmetic lock that keeps the \BIsix{} sector finite and sharply constrained.

A third achievement is conceptual. Global anomaly freedom is shown to be a prerequisite for a lawful \OGGEP{}/\GDOS{} bulk-to-boundary map. If the six-dimensional parent theory were only locally consistent but globally multi-valued, then the projected boundary functional would inherit an unphysical topological phase ambiguity. The present theorem therefore does more than repair a bulk gauge theory. It opens the door to the entire boundary program.

With this summary in hand, we can now place the paper in the broader context of the series.

\subsection{The place of this paper in the series}

This paper should be read as the direct mathematical continuation of \AOne{} \citep{ArisakaA1Constitution} and \ATwo{} \citep{ArisakaA2GG6}. \AOne{} proposed the \GGTheory{} Constitution as a compact admissibility system. \ATwo{} derived the six-dimensional \GG{} stage and wrote down \BIsix{} as the structurally distinguished modified Bianchi identity. The present paper asks the next unavoidable question: does that stage define a consistent quantum theory? The answer given here is yes, on the physical intended \GG{} class, and the proof is explicit.

In that sense the derivation chain now reads
\begin{equation}
\boxed{
\begin{aligned}
&\text{\GGTheory{} Constitution}
\;\Longrightarrow\;
\text{\GG{} stage + \BIsix{}}\\
&\quad\Longrightarrow\;
\text{complete local/global anomaly freedom}
\;\xRightarrow{\;\OGGEP\;}
\text{lawful \GDOS{} boundary channel}.
\end{aligned}
}
\end{equation}
The later applications to Standard-Model closure, cosmology, chemistry, life, and mind depend on this chain, but the anomaly theorem established here is logically prior to all of them.

\subsection{What is proved, what is suggested, and what remains open}
\label{sec:proved-suggested-open}

Theorem-level statements in this paper are category-precise. \BIsix{} is completely anomaly-free on the physical intended \GG{} class. \OGNfive{} is theorem-level inside that same class. The explicit off-diagonal cancellations, the affine orbit family, and the chirality reinterpretation of the apparent negative multiplicity are all closed. The constitutional admissibility of $\UoneQi$ as an optional fourth Abelian factor of the bulk gauge group is settled in Section~\ref{sec:UoneQi-constitutional-status} (Theorem~\ref{thm:UoneQi-admissibility}). The methodological reading of the 1984 Green--Schwarz mechanism as the technical pioneering of \AX{2} (\GGEP) is settled in Section~\ref{sec:GS_as_GGEP_completion}.

Two kinds of statements are intentionally separated from that theorem core. First, the stronger claim for every arbitrarily enlarged non-supersymmetric six-dimensional category is not proved here and should not be advertised as a corollary. Second, the late-time operational normalization \(\Omegaeff\to1\) is given here as a condensed \GDOS{} attractor sketch rather than as the main anomaly theorem itself. The purpose of that sketch is self-contained logical clarity, not inflation of the anomaly claim. These distinctions are not weaknesses. They are part of the paper's design principle: to keep theorem, corollary, and downstream dynamical application separated cleanly enough that each can be judged on its own terms.

The explicit triage of every open problem raised by the present paper --- with each entry $O_k$ labeled \textbf{Resolved} (in the present paper or in a named companion paper), \textbf{Partially resolved} (with a specific sub-task pending), or \textbf{Genuinely open} (no theorem in any current paper carries the answer) --- is given in Section~\ref{sec:open-problems}, the closing section of Part~V, immediately preceding these Conclusions.

\section{Big picture and closing reflection}
\label{sec:big-picture}

A higher-dimensional theory becomes physically meaningful only when three layers close simultaneously: geometry, gauge law, and global quantum consistency. \GG{} supplies the six-dimensional geometric stage. \BIsix{} supplies the locked gauge--gravity tensor identity. The present theorem supplies the global quantum consistency. Only after all three layers are closed is it legitimate to project the theory operationally to the four-dimensional world.

This is why anomaly freedom matters so much in the axiomatic program. It is not a technical footnote appended to a geometrical construction. It is the sharpest filter that nature imposes on any candidate quantum theory in higher dimensions. The Green--Schwarz mechanism taught the string community this lesson in ten dimensions in 1984; the six-dimensional anomaly-polynomial program deepened it through the work of Sagnotti, Schwarz, Kumar, Morrison, Taylor, Monnier, Moore, and many others. The present paper applies the same lesson --- with the same mathematical tools --- to a framework that departs from the string tradition in its physical assumptions while respecting its mathematical standards.

There is a broader point worth making. The history of theoretical physics suggests that the most productive ideas are not always those that are most popular at the time of their introduction. The original Green--Schwarz anomaly cancellation result of 1984 was initially met with skepticism by parts of the particle physics community; it took years for its full implications to become clear. The six-dimensional anomaly program itself was a relatively quiet corner of string theory until the modern differential-cohomology and bordism-theoretic tools matured. The \GGTheory{} axiomatic approach is new, unfamiliar, and comes from outside the string-theory tradition. It should be judged not by its provenance but by its mathematical content, its internal consistency, and its falsifiable predictions.

The five integers $(1,3,5;7,35)$ encode all dimensionless normalization freedom in the \BIsix{} identity. They are not chosen to fit data; they are derived from the gauge architecture of the six-dimensional bulk. The same integers later reappear in the derivation of Standard Model parameters, in the KK mass scale, in the cosmological hierarchy, and throughout the \GDOS{} boundary program. Whether this arithmetic compression reflects a deep truth about nature is, ultimately, an empirical question. But the fact that it passes the anomaly-cancellation test --- the same test that selected $\SO(32)$ and $E_8 \times E_8$ in ten dimensions --- is a non-trivial piece of evidence.

If the theorem established in this paper survives further scrutiny, then \GG{} will stand as more than an attractive geometric picture: as a finitely locked and quantum-mechanically admissible parent theory. The observed four-dimensional world, with its three generations of fermions, its specific gauge couplings, its particular Higgs vacuum, and its peculiar pattern of masses and mixings, would then be not an accident to be fitted but a necessary consequence of a constitutional law operating in six dimensions. Whether that vision is correct is for experiment and further mathematical analysis to decide. The present paper has taken one step --- proving that the six-dimensional parent theory is consistent --- and the road ahead is open.


The anomaly
theorem proved in the present paper is, in this constitutional
reading, the strongest result the framework requires at Stratum~2.
The further constitutional content of \AX{3} and \AX{4} --- the
operational bulk-to-boundary CPTP channel, the QM postulates as
theorems, the dissolution of the measurement problem, the unique
\OGNfive{} ledger, the elimination of the string landscape --- is the
subject of the companion papers \AFive{}, \ASix{}, \ASeven{}, \AEight{},
\ANine{}, \ATen{}, and \AEleven{}. Each of those companion contributions rests on the Stratum~2 anomaly
consistency proved here. In that sense, the present paper closes the
gating prerequisite for the entire cohort program: the seven cohort
papers downstream of \AThree{} are each, in their different ways,
applications of \AX{3} and \AX{4} on a six-dimensional bulk whose
Stratum~2 consistency is the subject of the present \BIsix{} theorem.

\bigskip

\noindent\textbf{A last word on how to read this.} The strongest claim
in this paper is a consistency theorem, and consistency theorems are
modest things. They do not say a theory is true. They say it is not
disqualified --- that one particular way of being wrong has been ruled
out. What makes this one worth writing down is that the way of being
wrong is severe, the test admits no adjustable parameters, and the same
structure that passes the test hands over five integers that turn into
numbers an experiment can check.

Whether those numbers are right is not something this paper can settle,
and it should not be read as trying to. The proper response to a
framework of this kind is neither belief nor dismissal but attention to
the short list of things it forbids. If a resonance cluster near
$3.85$~TeV is excluded, if the kaxion line is not where the ledger puts
it, if the proton decays, then the geometry that passed this examination
is the wrong geometry, and the paper has said in advance which page
fails. That is the whole of the offer.

What remains open is stated plainly and is not small. The generation
count is not derived. The bulk theory has no non-perturbative
completion. Several of the sharper questions raised here --- the minimal
spectrum problem, the extension of the global theorem beyond the
intended class, the relation between the integer ledger and the F-theory
lattice conditions --- are well-posed mathematical problems that this
paper does not solve. They are stated in the hope that someone will,
and the author would be glad of the company.

\GGAcknowledgments

\GGPart[For the Curious Reader]{For the Curious Reader}{Twenty-five
questions a thoughtful reader without a physics degree actually carries
into a paper like this one --- answered plainly, and in the order they
usually arrive}

The Conclusions have said what the paper proves. This Part answers the
questions a reader is entitled to ask next, and it assumes no physics
beyond a first course. It is placed here, and not at the back, because
a reader who wants these answers should not have to walk through sixty
pages of bordism theory to reach them. Nothing in the Part is needed by
the technical argument; nothing in the technical argument is needed to
read it. The specialist objections --- the ones a particle theorist or a
differential-cohomology specialist raises --- are a separate matter and
are answered in their own Part, further on.

The twenty-five questions were chosen the way a teacher chooses them:
they are the ones that actually get asked. Several are hostile, and they
are printed hostile. One of them --- why a demand for internal
consistency should be entitled to forbid anything at all in a world
under no obligation to be mathematically comfortable --- is the hardest
question anyone has put to this framework, and the answer to it concedes
more than it settles.

Two habits are followed throughout. Where a claim rests on a step that
is not closed, the answer says so and names the step, rather than
rounding it up to a result. And where the honest answer is that the
framework does not reach the question, the answer says that too. A
reader who finishes this Part should be able to state not only what the
paper claims but where it stops, which is the more useful of the two.

Nothing here is a summary of the proof. The proof is Part~II, and it is
not summarizable in this register without becoming false. What these
pages give instead is the shape of the claim: what an anomaly is, why
anything has to cancel, where a handful of whole numbers could come
from, and what would have to be measured for the whole construction to
be abandoned.

\section{The big questions, plainly answered}
\label{sec:curious-reader}
\label{sec:qa-curious}

The twenty-five questions that follow are not a specialist's. They are the
ones a thoughtful reader without a physics degree actually carries
into a paper like this one: what an anomaly is and why anything should have to cancel,
where a handful of whole numbers could possibly come from if nobody
measured them, why a demand for internal consistency is allowed to
predict the weight of a star, and whether an integer that cannot be
adjusted is a strength or simply a hostage. Several of these have been settled in
popular science in directions this paper does not take, and a reader
is owed plain answers before the specialist objections begin. They
are met first, and plainly.

The hardest of the twenty-five is the fifth --- why a consistency requirement --- a statement about mathematics
--- should be entitled to forbid anything at all in a world that was
under no obligation to be mathematically comfortable. The response concedes the
difficulty rather than dissolving it, and says which part of it the
paper can carry and which part it cannot. Where a claim rests on a
step that is not closed, the response says so and names the step;
where the honest answer is that the framework does not reach the
question, it says that too.


\bigskip
\noindent\textbf{Question 1 (Q-BI6-1) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is an `anomaly', in plain terms, and why does anything have to cancel?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} An anomaly is a symmetry that survives the classical description of
a system and dies when the quantum description is written down. That is
the whole of it, and the word is badly chosen --- nothing anomalous in
the everyday sense is happening. \\ Here is why it matters. Some
symmetries in physics are decorative: if they fail, a quantity you
thought was conserved turns out not to be, and you adjust. Others are
structural --- they are what makes the description consistent at all,
because the theory is written in terms of quantities that are only
defined up to a choice, and the symmetry is what guarantees the choice
does not matter. If one of those dies quantum-mechanically, the theory
does not become slightly wrong. It becomes meaningless: probabilities
stop summing to one, and the same calculation gives different answers
depending on an arbitrary convention. \\ So ``must cancel'' is not an
aesthetic preference. It is the requirement that the description be a
description. What makes this powerful is that the cancellation condition
is an equation relating the contents of the theory --- what kinds of
matter there are, what charges they carry --- and equations of that kind
have solutions rather than accommodating anything you like. \\ The
historical demonstration is worth knowing. In 1984 Green and Schwarz
showed that in ten dimensions the anomaly cancels for exactly two gauge
groups out of the enormous space of candidates. That single calculation
redirected a field. The argument in this paper is of the same type, run
in six dimensions.
\end{objqabody}

\bigskip
\noindent\textbf{Question 2 (Q-BI6-2) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should nature care about a number like 35? It seems completely arbitrary.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would be arbitrary if it had been chosen, and the reason to
think it was not is that it arrives with four other numbers, all
determined together, none of them adjustable. \\ The way to see this is
to notice what would happen if 35 were wrong. It is not a coefficient
sitting in front of a term where a different value would give a slightly
different prediction. It participates in a closure condition alongside
the others, and the condition either closes or it does not. Change the
35 and nothing in the framework closes --- there is no repaired version
with a different number, because the numbers are a solution rather than
a parameterization. \\ It also helps to see where such numbers come
from generally, because they are less exotic than they look. Integers
appear in physics wherever something has to fit: the energy levels of an
atom, the number of ways a lattice can be arranged, the winding of a
loop around a hole. In none of those cases did anyone choose the number.
It is what a counting problem returned. The ledger here is the same kind
of object --- the answer to a counting problem posed by the arena's
structure. \\ The fair form of your suspicion is not that 35 looks odd.
It is that a framework producing a specific integer should be asked what
else that integer commits it to, so that the number is exposed rather
than merely stated. Here it fixes the strength of the electric force at low energy
(the number $137$), the frequency of a dark-matter line near
$1.95$~gigahertz, and the mass of a cluster of new particles near
$3.85$~TeV. All three are measurable, and instruments now running are
measuring them. What the ledger does \emph{not} fix is the number of
families of matter --- the paper is explicit that the three colors it
derives and the three families nature shows are not the same three, and
that connecting them is an open problem it does not solve.
\end{objqabody}

\bigskip
\noindent\textbf{Question 3 (Q-BI6-3) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Where do these whole numbers come from if nobody measured them?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} From a counting problem that the arena poses and that has one
answer. \\ It is worth being concrete about the shape of such an
argument, because ``derived from geometry'' is the kind of phrase that
can mean anything. The structure of the arena determines what kinds of
twisting are possible on it. The consistency requirement demands that
certain contributions from those twistings sum to zero. That is a
system of equations in whole numbers --- how many of this kind, how many
of that --- and such systems are famously rigid: most have no solutions
at all, and those that do usually have exactly one. Here there is one.
\\ Nothing in that chain involves a measurement, and nothing in it
involves a choice. It also means the numbers are not free to move if a
measurement disagrees. That is the whole point, and it is the property
being bought at the price of no longer having anywhere to retreat to.
\\ The honest limit is that the argument is conditional on the arena.
Six dimensions, a specific symmetry group, and the constitutional
requirements are all inputs, established elsewhere in the cohort and not
derived here. Given those, the ledger follows and its uniqueness is
proved. Deny those, and the whole calculation is about nothing. A
skeptical reader is better served attacking the arena than attacking the
integers, because the integers are the easy part.
\end{objqabody}

\bigskip
\noindent\textbf{Question 4 (Q-BI6-4) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What actually breaks if the ledger is wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nearly everything downstream, and it is worth naming the pieces so
the answer is not just an alarming adjective. \\ The value of the electric force at low energy follows from the
ledger --- the number $137$ is $2^7 + 3^2$, and the $7$ and the $3$ are
two of the five integers --- so if the ledger is wrong that agreement is
a coincidence and the framework's most-cited result is gone. The
dark-matter line near $1.95$~gigahertz and the particle cluster near
$3.85$~TeV are computed directly from the same integers; a different
ledger gives different numbers and there is no version of the framework
that gives no number. The relationships among charges --- which is why charge comes in
the multiples it does --- follow from the same source. And the
constitutional claim that the framework has no free parameters depends
entirely on the ledger being forced rather than fitted, so if it were
adjustable the central methodological boast would be false. \\ This is
not a comfortable answer and it should not be softened. A framework in
which every piece is independently replaceable is a framework in which
nothing carries weight. The ledger carries a great deal, which is why
the paper spends its length proving uniqueness rather than asserting
it, and why a reader wanting to find a fatal error should look there
first. \\ What does \emph{not} break is worth noting too. The arena
itself, its dimension and its symmetry group are upstream and survive
--- they would simply no longer force the physics they are claimed to
force, which would leave a mathematically respectable structure with
nothing to say about the world.
\end{objqabody}

\bigskip
\noindent\textbf{Question 5 (Q-BI6-5) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should a demand for internal consistency be allowed to predict anything about the world?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the hardest question here and it does not have a clean
answer. Saying so is better than managing it. \\ The worry is exact.
Consistency is cheap. One can write down any number of internally
impeccable structures and almost all of them are about nothing. From
the fact that a framework hangs together, nothing follows about whether
nature implements it. Any argument of the form ``it must be so, or the
mathematics would not close'' is, taken by itself, an argument about
mathematics. \\ What can be claimed is narrower. Consistency is not
being used here to establish that the world is a certain way; it is
being used to \emph{eliminate}. Given an arena that is assumed rather
than derived, the closure requirement admits one ledger instead of a
family of them. That is a statement about a search space. Whether nature
is in the search space at all is settled by measurement and by nothing
else, and the framework's entire empirical exposure lives in that gap.
\\ There is a historical reason to take the method seriously without
mistaking it for proof. Anomaly cancellation --- exactly a demand for
internal consistency --- has repeatedly narrowed the field of admissible
theories in particle physics, and the narrowing has survived contact
with data. The Green--Schwarz calculation is the famous case. That is an
encouraging record and it is not an argument. The accurate summary is
that consistency is a filter of unusual reach and no kind of evidence at
all, and a reader who keeps those two facts apart will read this paper
correctly.
\end{objqabody}

\bigskip
\noindent\textbf{Question 6 (Q-BI6-6) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``You say a handful of whole numbers fixes what happens in a particle detector. Why should a detector care about six-dimensional algebra?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should not care, and the framing hides what is actually being claimed. The detector is not responding to the algebra. It is responding to
a small extra term in the equation that governs how a dense object
balances its own weight against its internal pressure --- and the size
of that term is what the algebra fixes. \\ The chain is short enough to
state. The consistency condition forces a specific depth for the extra dimension, and a depth is a mass. It is not a mass anyone chose: it is the ledger integer $35$, used once. The mass comes out near $3.85$~TeV. The value is not adjustable, because it is fixed by the same whole numbers that had to be whole for the theory to exist at all. \\ What makes this a good prediction rather than an
impressive-sounding one is the absence of freedom. There is no coupling
to tune, no scale to choose. The framework says the mass is a specific value, with a specific pattern of companion states around it. \\ And it is
currently measurable. The High-Luminosity LHC reaches that energy, and
a second prediction --- a dark-matter line between $1.59$ and
$2.13$~gigahertz --- is being swept by haloscopes now. A null
result at better than the predicted precision is not a setback to be
absorbed; it ends the ledger, and with it the paper.
\end{objqabody}

\bigskip
\noindent\textbf{Question 7 (Q-BI6-7) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't anomaly cancellation just bookkeeping about infinities? That sounds like a technicality, not a discovery.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The bookkeeping reputation is real and it is a misreading of what
the calculation does. \\ The reputation comes from renormalization,
where infinities genuinely are absorbed by a procedure that many
physicists found distasteful for decades and that Dirac never accepted.
Anomaly cancellation is a different operation and the difference is
worth seeing. Renormalization takes a quantity that came out infinite
and relates it to a measured one. Anomaly cancellation does not absorb
anything: it is a condition that either holds or fails, given the
contents of the theory, and when it fails there is no procedure that
rescues the theory. \\ That is why it constrains. A procedure that can
be applied to anything tells you nothing about what exists. A condition
that most candidate theories fail tells you a great deal --- and the
overwhelming majority of ways one might imagine assembling matter and
forces do fail it. The Standard Model passes, and the passing is
delicate: the charges of the quarks and leptons enter a sum that cancels
only because the values are what they are. This is standard physics and
not a claim of this framework. \\ What this paper adds is running the
same requirement one level up, on the arena rather than within a fixed
four-dimensional theory, and finding that the condition again admits
almost nothing. Whether that step is legitimate is a technical question
with a technical answer, and it is the right place for a skeptical
reader to spend effort.
\end{objqabody}

\bigskip
\noindent\textbf{Question 8 (Q-BI6-8) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the ledger is unique, why did it take until now to find it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the question it answers was not being asked in a form that
had an answer, and the reason for that is more interesting than the
delay. \\ For most of the twentieth century the project was to write
down a four-dimensional theory matching the measurements, and the
parameters that theory needed were treated as things to be measured
rather than derived. That is not a failure of imagination; it is the
correct strategy when you have data and no organizing structure, and it
produced the Standard Model, which is the most accurate theory anyone
has ever had. Nobody asks where twenty-eight numbers come from while
they are still being measured. \\ Asking the question in a form with an
answer requires committing to a specific arena first, and that commitment
was not available. The algebraic argument that bounds the dimension is
old mathematics, but connecting it to the constitutional requirements
used here is not, and without both halves the counting problem cannot
even be posed. \\ It should also be said plainly that lateness is weak
evidence in either direction. Long-unnoticed results do exist and so do
long-unnoticed errors, and the fact that something was not found before
tells you almost nothing about whether it is correct now. The reason to take the ledger seriously is that it is exposed --- a coupling strength, a particle mass, a charge pattern --- and not that it is new.
\end{objqabody}

\bigskip
\noindent\textbf{Question 9 (Q-BI6-9) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why five numbers rather than one? A theory of everything ought to need one.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It ought not, and the intuition that it should is worth examining
because it has misled people. \\ There is no principle saying a
fundamental description must be specified by a single quantity, and
nothing in the history of physics supports the expectation. What matters
is not how many numbers appear but whether they were chosen. Five
integers that are the unique solution of a closure condition constitute
one fact, not five. Five continuous parameters fitted to data constitute
five decisions. The count of symbols is a poor guide to the count of
assumptions. \\ There is also a structural reason the number here is not
one. The ledger describes how several different kinds of structure fit
together on the arena --- how many of each kind of twisting, at what
relative weights --- and a single number could not carry that
information any more than a single number could specify the shape of a
crystal. What is being reported is a pattern, and patterns have several
entries by nature. \\ The version of your question that does bite is
this: could a different arena have produced a different five-number
solution, so that the uniqueness is only uniqueness-given-the-arena?
Yes, and the paper says so. The arena is an input. What is proved here
is that given it, there is no second ledger --- which is a real result
and a conditional one, and conflating the two would be the error.
\end{objqabody}

\bigskip
\noindent\textbf{Question 10 (Q-BI6-10) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Charge comes in whole multiples of a smallest unit. Is that the same phenomenon?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is, and noticing that is the fastest route into why any of this
should be believed. \\ The quantization of charge is one of the oldest
unexplained facts in physics. Every electron carries exactly the same
charge as every other, to the limits of measurement, and every charge
ever measured is a whole multiple of a common unit. Nothing in the
Standard Model explains this. The values are put in by hand, and their
exact equality --- the reason atoms are neutral to better than one part
in $10^{21}$ --- is an unexplained coincidence of the deepest kind.
\\ Geometry supplies a natural mechanism, and it is old: if charge
counts how many times something wraps around a closed internal
direction, then it is a whole number for the same reason that the number
of times a rope winds around a post is a whole number. There is no
continuous path from three windings to four. Dirac saw a version of this
in 1931 and it has been part of the subject ever since. \\ What this
framework contributes is not the mechanism but the specifics: which
windings the arena permits, and therefore which charges can exist. That
converts a qualitative explanation into a list, and a list can be wrong.
If a particle turns up with a charge outside the permitted set, the
account fails --- which is a considerably better position than an
explanation that predicts only that charges are quantized without saying
which.
\end{objqabody}

\bigskip
\noindent\textbf{Question 11 (Q-BI6-11) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is a `Bianchi identity' and why does yours need modifying?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} An identity in this sense is a statement that is true automatically
--- not a law imposed on the world but a consequence of how the
quantities were defined, in the way that the statement that the
boundary of a boundary is empty is true of any shape whatever. The
standard Bianchi identity says that a certain combination of field
quantities vanishes identically, and it is the reason magnetic charge
does not appear in ordinary electromagnetism. \\ Modifying an identity
sounds like a contradiction and is not. What is modified is not the
mathematics but which quantity the identity applies to. When the arena
carries additional structure, the natural field quantity is not the one
you first write down; there is an extra piece, and the identity holds
for the corrected object rather than the naive one. The corrected
statement is again automatic once the correct object is identified.
\\ Why this matters is that the extra piece is not free. Its form is
fixed by the requirement that the anomaly cancel, and its coefficients
are the ledger. So the modification is where the geometry enters the
physics: the identity is the equation, the correction term is the
unknown, and the closure condition is what solves it. \\ It is worth
adding that this kind of modification is not exotic. It is exactly the
structure Green and Schwarz used in ten dimensions in 1984, and the
mechanism has been standard since. What is claimed here is its
consequence in six, where the counting comes out differently.
\end{objqabody}

\bigskip
\noindent\textbf{Question 12 (Q-BI6-12) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is a shift of a fraction of a percent in stellar weight a real prediction, or is it lost in the noise?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is at the edge of what is currently measurable, which is the
uncomfortable and honest answer. \\ The dark-matter line sits in a window between $1.59$ and $2.13$~gigahertz, and the framework names the coupling depth at which a null result counts. No single scan settles it; the window has to be swept. That is
a real limitation and no amount of framing removes it. \\ What makes the
prediction testable anyway is that the shift is \emph{coherent}: it acts
in the same direction, by the same fractional amount, for every compact
star. Random measurement error does not do that. A population of stars
measured independently can therefore reveal a systematic offset far
smaller than any individual uncertainty, in the same way that averaging
many noisy measurements of a fixed quantity converges. Instruments now
operating are accumulating exactly such a population, and the precision
of the ensemble is approaching the required level. \\ The risk this
creates should be stated. If the population study returns a null result
at better than the predicted precision, the ledger is wrong, and there
is no adjustment available --- the size of the shift is a ratio of
integers. A framework whose most accessible prediction sits just at the
edge of current capability is a framework about to find out, which is
the position a theory should want to be in and an uncomfortable one to
occupy.
\end{objqabody}

\bigskip
\noindent\textbf{Question 13 (Q-BI6-13) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why do integers show up in physics at all? Nature isn't obviously counting anything.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is counting rather often, and once you start noticing it is hard
to stop. \\ The electron in an atom does not occupy arbitrary energies
but a discrete ladder, and the reason is that a wave has to fit around a
closed path a whole number of times. A crystal admits a small finite
number of symmetry types and no others, which is why there is no
five-fold periodic lattice. The number of ways a loop can wind around a
hole is an integer because there is no continuous path from one winding
number to another. In none of these cases is anything counting in the
sense of keeping a tally. The integer is the answer to a question about
fitting or about topology. \\ What all of them share is a structure with
a constraint that admits only certain configurations, and this is why
integers are a signature of geometry rather than of arithmetic. Where
you find one in physics, there is usually a shape underneath. \\ The
claim in this paper is of that type: the ledger integers are the answer
to a fitting question posed by the arena, and their whole-numberedness
is not a coincidence but a consequence of the question having that form.
Which is also why they are the framework's most exposed feature. An
integer cannot be nudged, and a framework that has been forced onto
integers has given up the ability to survive a disagreement by a small
adjustment.
\end{objqabody}

\bigskip
\noindent\textbf{Question 14 (Q-BI6-14) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is the difference between `forced' and `fitted' if both end with the same number?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The difference is what happens next, and it is the only difference
that matters. \\ A fitted number is answerable to the measurement it was
fitted to and to nothing else. If a second measurement disagrees, the
number moves. It will always be possible to make it agree with whatever
is in front of it, which is why a fitted number, taken alone, is not
evidence of anything --- it is a record of what was measured, written in
the theory's notation. \\ A forced number is answerable to everything it
touches. It was produced by a condition that knew nothing about the
measurement, and it now determines other quantities that were not used
to obtain it. If any one of those disagrees, there is no move available.
\\ So the test of whether a number in a framework is forced or fitted is
not to examine how it was obtained --- accounts of derivation can be
constructed after the fact, and have been. The test is to ask what else
it commits the framework to, and whether those commitments were made
before the checking. Here the ledger fixes a coupling strength, a charge pattern and two particle masses, none of which was used in deriving it,
and all of which are on the record in advance. That is the whole of the
defense, and it is checkable rather than rhetorical.
\end{objqabody}

\bigskip
\noindent\textbf{Question 15 (Q-BI6-15) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Could a different arena give a different unique ledger? Then what have you proved?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes, and stating that plainly is more useful than the theorem is.
\\ What is proved here is conditional: given this arena --- six
dimensions, this symmetry group, these constitutional requirements ---
the closure condition admits exactly one ledger. Change the arena and
the counting problem changes, and one would expect a different answer or
none. The uniqueness is uniqueness \emph{within} a specified setting,
not across all possible settings, and a reader who took it for the
latter would be taking more than is on offer. \\ What that conditional
buys is still worth something. It means the framework has no internal
freedom: having accepted the arena, nobody can adjust anything to
accommodate a measurement. All the freedom is concentrated in one place,
at the top, in the choice of arena --- which is declared, is a small
number of statements, and is argued for on separate grounds in the
companion that establishes it. \\ It also tells a critic exactly where
to attack. There is no point looking for slack in the ledger; there is
none. The argument to break is the one that fixes the arena, and if it
breaks, everything here becomes a correct calculation about a structure
nature does not use. That is a clean place for a disagreement to live,
and clean is the most one can ask of a framework's exposure.
\end{objqabody}

\bigskip
\noindent\textbf{Question 16 (Q-BI6-16) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What would a failed anomaly cancellation actually look like, physically?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It would not look like a small discrepancy. It would look like the
theory failing to be a theory, and that is worth making concrete
because ``inconsistent'' is an abstract-sounding verdict. \\ In a theory
with an uncanceled anomaly of this kind, the probabilities computed for
a set of mutually exclusive outcomes do not sum to one. That is not a
prediction that is slightly off; it is the absence of a prediction,
because the numbers being computed have stopped being probabilities.
Related symptoms follow: the same physical process gives different
answers depending on a choice that was supposed to be arbitrary, and
quantities that must be positive come out negative. \\ This is why the
condition is treated as a filter rather than as a hypothesis. One does
not test whether a theory with an uncanceled anomaly describes the
world; such a theory does not describe anything. It is discarded before
any measurement is considered. \\ The physically interesting statement
is the contrapositive, and it is the one this framework uses: since the
world manifestly does have consistent probabilities, whatever describes
it must satisfy the condition --- and the condition is restrictive enough
that satisfying it is a real constraint on what can exist. That is the step from an abstract requirement to a definite spectrum, and it is the paper's central move.
\end{objqabody}

\bigskip
\noindent\textbf{Question 17 (Q-BI6-17) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one of countless others, and we happen to be in the one where the numbers work?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It might be. What this paper offers is a reason the question may
not need the answer. \\ The multiverse addresses a specific problem: the
constants look arranged, and if they could have been otherwise the
arrangement demands explanation. An ensemble supplies one. That argument
is coherent and is made by careful people. \\ Its premise is that the
constants could have been otherwise. Here they could not --- not because
of a philosophical objection to variation, but because the closure
condition has one solution and there is no second ledger to be drawn.
If that is right, the observation that motivated the ensemble is no
longer surprising and the ensemble has nothing to explain. \\ It is
worth adding the asymmetry that makes this checkable rather than a
matter of preference. The ensemble picture makes no prediction about the strength of the forces, or about which charges can exist, or about where a new particle must be. This one does. Whichever is right, only one of them is in a
position to be embarrassed by a measurement, and that is the more useful
kind of theory to have even if it turns out to be the wrong one.
\end{objqabody}

\bigskip
\noindent\textbf{Question 18 (Q-BI6-18) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``The universe looks fine-tuned for life. Doesn't that point to a designer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It points to something requiring explanation, and the move from
there to a designer needs a premise that this framework denies. \\ The
observation is genuine: small changes in certain quantities and there is
no chemistry, no structure, nothing that lasts. Feeling the pull of
design when confronted with that is not naive. \\ The inference requires
that the quantities could have been set differently. A dial that was
turned implies a hand on it. If the values are the unique solution of a
closure condition, there was no dial, and asking who chose them is like
asking who chose the ratio between a circle's circumference and its
diameter. \\ None of this argues against a designer, and it should not
be recruited to. Someone who believes on other grounds is untouched by
this paper, and so is someone who does not. What changes is one
particular argument, which has carried a great deal of popular weight
and which loses its premise if the ledger is forced. Whether it is forced is settled by a haloscope scan and a collider search, which is a strange place for a theological argument to be decided and is nonetheless where this one now sits.
\end{objqabody}

\bigskip
\noindent\textbf{Question 19 (Q-BI6-19) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the integers are fixed, is my future already written? What happens to free will?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing here touches your future, and the inference does not hold
even though it feels tight. \\ What is fixed is the rule book: what charges are permitted, how
strong the forces are, what new particles must exist and where. What is not fixed is any particular history within it. The
distinction is between the rules of chess and the game played last
Tuesday, and a rule book admitting no alternative says nothing about
which of an enormous number of permitted games occurred. \\ There is a
sharper point underneath. Even a fully deterministic rule book delivers
no written future in any sense that could trouble anyone, because the
only thing capable of computing what the rules imply for a system as
complicated as a person is a system at least that complicated, running
at least that long. There is no shortcut, no vantage point, and no book.
A universe can be lawful all the way down and still be, in the only
sense available to anything inside it, unwritten until it happens.
\\ And the limit: this paper is about a cancellation condition and its
solution. What agency is, and whether the concept survives contact with
physics at all, is not a question it reaches.
\end{objqabody}

\bigskip
\noindent\textbf{Question 20 (Q-BI6-20) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This framework does not touch it, and pretending otherwise would be
a sales pitch. \\ Everything here is conditional --- given an arena with
certain properties, the closure condition has this solution. The
question of why there is an arena sits outside the construction
entirely, and no rearrangement of the argument reaches it. That is not a
gap further work will close; it is a different kind of question. \\ What
can be contributed is narrower and is not nothing. If the framework is
right, the space of possible somethings is far smaller than it appears:
not a landscape of coherent worlds from which one was drawn, but very
nearly a single admissible structure. That converts ``why this, out of
all there might have been?'' into ``why anything?'' --- which is harder
and cleaner. Narrowing a question is a service and is not an answer.
\\ The temptation is to say something about nothingness being unstable
or the question being ill-posed. Both are available, both have serious
defenders, and neither follows from anything in this paper.
\end{objqabody}

\bigskip
\noindent\textbf{Question 21 (Q-BI6-21) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``A handful of integers explaining this much makes me suspicious. Shouldn't it make you suspicious too?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should and it does, and Eddington is the name to keep in mind.
He was among the finest astrophysicists of his generation and spent his
last years deriving the fine-structure constant and the number of
particles in the universe from combinatorial arguments that convinced
almost nobody. His reputation never recovered. Anyone deriving integers
is one bad step from that. \\ What separates a derivation from a
coincidence is not the quality of the agreement. It is whether the
numbers came from machinery that exists for another reason, and whether
they commit you to anything you did not already know. A number
reverse-engineered from a measurement will always match it. \\ Here the
ledger comes out of a cancellation condition on an arena fixed
beforehand by an argument that knows nothing about detectors, and it then fixes a coupling strength, a charge pattern and two particle masses that were not used to obtain it. That is the only defense available and
it is the right one: not ``too good to be chance'' but ``here are three
further consequences, go and look.'' \\ Keep the suspicion. If the
consequences fail, the numbers were a coincidence, and no elegance in
their derivation will save them.
\end{objqabody}

\bigskip
\noindent\textbf{Question 22 (Q-BI6-22) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should I take one person's framework seriously against thousands of physicists working on other approaches?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not because of who wrote it, and not despite it either --- both are
the same error with opposite sign. \\ The practical question is fair,
since attention is finite. The useful filter is exposure, not headcount.
A framework supported by thousands that forbids nothing measurable is a
worse use of your attention than one supported by a single person that
forbids specific things and names the instrument that will decide.
Community size tracks funding history and fashion, and does not track
correctness. \\ There is a genuine cost to working outside a large
community. Errors are found faster where more people are looking, and a
framework developed largely by one person carries a higher chance of
containing a mistake a hundred readers would have caught in a week.
Internal coherence does nothing to reduce that. \\ So do not take it on
trust. Take the compact-star prediction, which needs no six-dimensional
mathematics to evaluate --- a fractional shift of a stated size, in a
stated direction, across a population --- and ask whether it is stated
precisely enough to be wrong. If it is not, ignore the rest. If it is,
that deserves attention regardless of whose framework it came from.
\end{objqabody}

\bigskip
\noindent\textbf{Question 23 (Q-BI6-23) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What would convince you that you are wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Four things, named rather than gestured at. \\ A sweep of the microwave window between $1.59$ and $2.13$~gigahertz
finding no dark-matter line, at the coupling depth the framework states,
would end the ledger. A proton decay would do the same by a different route, since the protection follows from the same gauge structure and there is no parameter to adjust. A
particle carrying a charge outside the permitted pattern would break the
link between the arena's structure and what can exist. And an internal
failure counts fully: if the uniqueness proof has a hole, the claim that
the ledger is forced rather than chosen is gone whatever the
instruments say, and that is the most likely of the four. \\ What would
not convince: a disagreement in a regime the framework does not claim to
cover, or the failure of a result whose hypotheses are already flagged
as open. A falsifier list is only honest if it excludes things, and a
reader should be wary of any proponent unwilling to say which failures
are survivable. \\ The framework's exposure is unusually concentrated.
Almost everything rests on the ledger, so almost everything can be ended
by one measurement. That is a feature and it is also a reason to check
the uniqueness argument before checking anything else.
\end{objqabody}

\bigskip
\noindent\textbf{Question 24 (Q-BI6-24) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should mathematics be able to forbid a universe?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It cannot, and the framing is where the trouble starts. Mathematics
forbids nothing about the world. What it can do is tell you that certain
\emph{descriptions} are unavailable --- and since we have no access to
the world except through descriptions, that turns out to be almost as
useful and is not the same thing. \\ The distinction matters here.
Nothing in this paper says the universe could not have had four families
of matter. It says that no consistent description of the kind under
consideration produces four. If nature has four, then the kind of
description under consideration is not the right kind, and the framework
is wrong rather than nature being illegal. That is the correct direction
of the inference and it is easy to state backwards. \\ Physics has been
in this position before and the outcome has gone both ways. The
requirement that a description be relativistically consistent forbade a
great many candidate theories and the survivors turned out to be right.
The requirement that a description be built on absolute simultaneity
forbade a great many candidate theories and was itself wrong. There is
no way to tell in advance which case one is in. \\ What follows is a
discipline rather than a reassurance: a framework arguing from
consistency owes an unusually clear account of what it forbids, so that
if nature turns out to be outside its reach the fact is discovered
quickly rather than argued about.
\end{objqabody}

\bigskip
\noindent\textbf{Question 25 (Q-BI6-25) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``I'm not a physicist and I've got this far. What is the one thing you want me to take away?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That the interesting question is not whether a framework is true
but whether it has arranged to be caught, and that this one has arranged
it unusually tightly. \\ Almost everything written about fundamental
physics for a general audience --- in both directions, enthusiastic and
debunking --- argues about whether a set of ideas is right. That argument
is rarely settleable from an armchair and it is not what moves science.
What moves it is the duller question of whether a claim has been put in
a form some measurement could contradict, and whether that was said
beforehand. \\ This paper takes a requirement that any consistent
description must satisfy, applies it to a specific arena, and finds that
it admits exactly one set of whole numbers. Those numbers then say which charges can exist, how strong the electric force is at low energy, and where two new particles must be. Instruments now running will settle the last of those within
a few years. \\ So take away a habit rather than a belief. When you next
meet a grand claim about the universe --- this one included --- ask what
it forbids. If the answer is nothing, or keeps changing, you have
learned what you needed without following any mathematics. If the answer is a short list of things somebody is about to measure,
that claim is worth your attention whether or not it survives.
\end{objqabody}

\bigskip

\noindent\textbf{Where to go from here.} A reader who has come this far
without the mathematics has, in fact, the substance of the paper. The
claim is that a consistency requirement in six dimensions, applied
without any freedom to tune, forces five small whole numbers, and that
those numbers turn into quantities that can be measured. Everything
before this Part is the demonstration that the requirement really is a
requirement and that the numbers really are forced.

If you want to go further, the shortest useful route is not through the
proof. It is through Section~\ref{sec:predictions-falsification}, which
lists what would end the framework, and through
Section~\ref{sec:open-problems}, which lists what it has not managed to
do. Between them those two sections say more about how much confidence a
framework deserves than any summary of its successes could.

If you want the argument itself, Part~VI states the conclusions
precisely, and Part~II is the proof. If a specific technical objection
occurred to you while reading these answers, Part~IX may already have it
--- the fifty-three there are the ones specialists raise, written in
their language, and several of them are sharper versions of questions
answered plainly above.

\GGPart[Technical Appendices]{Technical Appendices}{A graduate-level course in six-dimensional anomaly cancellation, read side by side with string theory --- from-scratch derivations of the descent formalism, the ten-dimensional Green--Schwarz mechanism and its six-dimensional dictionary, the rank-one criterion, the \OGNfive{} arithmetic, the benchmark spectrum, the gravitational anomaly polynomials, and the global (Dai--Freed) completion --- preceded by the reference dictionary and followed by the theorem checklist}

The body of this paper has emphasized the theorem chain. The ten
appendices that follow supply the detailed technical content behind
each step, and taken together they are intended to be read as \emph{a
graduate-level course in anomaly cancellation seen through \GGTheory{}
and through string theory side by side}, addressed to an incoming
graduate student in particle physics. We organize them into two groups. The \emph{reference} appendices
(\ref{app:abbrev} and \ref{app:checklist}) provide the abbreviation
dictionary and the theorem checklist matrix, and serve as the reader's
quick lookup. The \emph{constructive} appendices are the course
proper: the complete descent proof, sector by sector
(\ref{app:descent-full}); the ten-dimensional Green--Schwarz mechanism
worked from scratch, with the dictionary that translates every one of
its moves into the six-dimensional setting of this paper
(\ref{app:gs10d}); the rank-one criterion and all eighteen
coefficient-matching and vanishing conditions
(\ref{app:matrix-and-coefficients}); the \OGNfive{} integers derived
from \BIsix{} (\ref{app:ogn5-complete}); the benchmark spectrum ---
core block, Diophantine orbit system, and gravitational completion
(\ref{app:core-block-full}--\ref{app:grav-full}), the last with the
gravitational anomaly polynomials themselves derived from the index
theorem; and the global anomaly proof on the admissible class
(\ref{app:global-proof-dossier}), opened by a self-contained primer on
the eta invariant and the Dai--Freed formalism. Where the material
exists in the string-theory tradition, each appendix treats it in
\emph{three steps}: first the standard machinery, from scratch, in the
form a graduate student meets it in the textbooks; then a frank
statement of how string theory deploys that machinery and what its
deployment assumes; and finally the \GG{} derivation, in which the
same machinery runs on the six-dimensional bulk with the constitutional
clauses doing the work that supersymmetry or worldsheet consistency
would have done.

Throughout these appendices we adhere to two stylistic commitments,
following the cohort convention. First, every derivation is presented
step by step with no skipped intermediate manipulations: a graduate
student should be able to reproduce each calculation with paper and
pencil. Second, the formal steps are accompanied by \emph{physics
intuition} paragraphs that translate the algebra into a mental
picture. The combination is designed to make the technical material
usable as a course while preserving the rigor of the main text.

\paragraph{Prerequisites and recommended textbooks.}
The course assumes the standard first-year graduate sequence in quantum
field theory and is meant to be read \emph{alongside} the canonical
texts, not in place of them. The recommended path through each
appendix is: read the matching textbook treatment first, return here
for the from-scratch recap, and then follow the \GG{} derivation and
the string-side comparison. For self-study we particularly recommend
the following, each cited again in the appendix to which it is best
matched:
\begin{itemize}
\item \textbf{Anomalies, first pass.} \citet{PeskinSchroeder1995}
 (Ch.~19) and \citet{WeinbergQFT1996} (Vol.~II, Ch.~22) --- the
 triangle anomaly, the Fujikawa measure, and anomaly matching as a
 graduate student first meets them.
\item \textbf{Anomalies, in depth.} \citet{Bilal2008} and
 \citet{Harvey2005} --- the two standard lecture-note treatments of
 the descent formalism, anomaly polynomials, and inflow; and
 \citet{Bertlmann1996} and \citet{FujikawaSuzuki2004} --- the two
 book-length treatments, the first cohomological, the second
 path-integral.
\item \textbf{Geometry and topology background.}
 \citet{Nakahara2003} --- characteristic classes, index theorems, and
 the eta invariant at the level this paper uses them;
 \citet{MilnorStasheff1974} for the underlying characteristic-class
 theory.
\item \textbf{The string side.} \citet{GSW1987} (Vol.~II, Ch.~13) ---
 the original textbook treatment of the ten-dimensional Green--Schwarz
 mechanism; \citet{Polchinski1998b} (Ch.~12) and
 \citet{BeckerBeckerSchwarz2007} (Ch.~5) --- the modern treatments;
 \citet{IbanezUranga2012} --- the phenomenological bridge; and
 \citet{Taylor2011} --- the six-dimensional supergravity landscape
 this paper's Part~IV compares against.
\item \textbf{Group theory for the spectrum computations.}
 \citet{Slansky1981} and \citet{Georgi1999} --- indices, Casimirs, and
 branching rules for the $\SU(5)$ representation data of
 Appendix~\ref{app:core-block-full}.
\item \textbf{The global (nonperturbative) side.} \citet{WittenRMP2016}
 --- fermion path integrals, the eta invariant, and anomaly inflow in
 the modern reflection-positivity language; and
 \citet{GarciaEtxebarriaMontero2019} --- Dai--Freed anomalies deployed
 on particle-physics gauge groups, the closest published analogue of
 the checks run in Appendix~\ref{app:global-proof-dossier}.
\end{itemize}
\noindent Throughout, \AOne{}--\AEleven{}, \PaperPOne{}--\PaperPSix{},
and \PaperCOne{}--\PaperCThree{} refer to the cohort papers, to which the \GG{}-side
steps repeatedly point.

\appendix

\section{Summary of Abbreviations}
\label{app:abbrev-full}
\label{app:abbrev}

This appendix is the full dictionary: every abbreviation used in this paper, together with the program-defined names of the companion papers it cites and of the acronym plate below, together with the companion paper that introduces each one. The handful of names that appear before they are defined are glossed in the Notation block of the front matter; this is where the rest live.
All program-defined names are rendered in \emph{sans-serif type}
(\GGTheory{}, \GGConstitution{}, \GG{}, \GDOS{}, \BIsix{}, \OGNfive{},
\QGL{}, \GiFour{}, \QiFour{}, \GRIP{}, \BFFT{}, etc.) so the reader
can see at a glance which terms belong to the \GGTheory{} program
proper. Standard physics abbreviations imported from the wider
literature (KK, RS, QFT, TOV, etc.) are rendered in plain uppercase
Roman, following the convention of the major physics journals.
Mathematical objects are in roman or italic in the usual way.

{\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.153\linewidth} >{\raggedright\arraybackslash}p{0.538\linewidth} >{\raggedright\arraybackslash}p{0.230\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Every abbreviation used in this paper, together with the program-defined names of the companion papers it cites and of the acronym plate below, each with its meaning and the place it is introduced in full. Typography carries information here: program-defined names are set in sans serif, standard physics abbreviations in plain uppercase Roman, and mathematical objects in roman or italic as usual, so a reader can tell at a glance whether a name belongs to this framework or to the general literature. The third column links each entry to its primary source --- the present paper for names defined here, the companion paper otherwise. A reader who meets an unfamiliar name anywhere in the volume should come here first.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

\multicolumn{3}{c}{\itshape Table~\ref{tab:appA-abbrev} continued from previous page}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endhead

\midrule
\multicolumn{3}{r}{\itshape continued on next page}\\
\endfoot

\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape Framework name and core program objects} \\
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry-Gauge Equivalence Theory) & \AOne{}, Preface \\
\GGConstitution{} & The constitutional charter of the \GGTheory{} program: four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) & \AOne{}, \S 4--\S 5 \\
\GG{} & Geometry-Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 12 (\TX{1}{GG6}) \\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law implied by \OGGEP{} & \AFive{} \\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D (the principal object of the present paper) & \AThree{}, \S\ref{sec:structural-proof} \\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1, 3, 5; 7, 35)$ & \AThree{}, \S\ref{sec:ogn5-arithmetic}--\S\ref{sec:ogn5-theorem} \\
\QGL{} & Quadruple Gauge Locking: the four-factor bulk gauge group $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$ & \ATwo{}, \S 19 (\TX{8}{GG6}) \\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & \ASix{} \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{} \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms (\AX{1}--\AX{4})} \\
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S 1.5 \\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S 4.2 \\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}, \S 4.3 \\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle (boundary observation dictionary) & \AOne{}, \S 4.4 \\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}, \S 4.5 \\
\midrule

\multicolumn{3}{l}{\itshape Seven postulates (\POS{1}--\POS{7})} \\
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}, \S 1.5 \\
\GI{} & \POS{1} Gauge invariance across scales (\LGI{} / \DGI{} / \GGI{}) & \AOne{}, \S 5.3 \\
\UEP{} & \POS{2} Unified Extremal Principle (PLA $\Leftrightarrow$ PLT / \MEPP{} / FEP) & \AOne{}, \S 5.4 \\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow (the constitutional clause central to the present paper) & \AOne{}, \S 5.5 \\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}, \S 5.6 \\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}, \S 5.7 \\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization; \emph{formerly} MRHN) & \AOne{}, \S 5.8 \\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}, \S 5.9 \\
\LGI{} & Local Gauge Invariance (microscopic / closed-system layer of \GI{}) & \AOne{}, \S 5.3 \\
\GGI{} & Global Gauge Invariance (macroscopic / ledger layer of \GI{}) & \AOne{}, \S 5.3 \\
\midrule

\multicolumn{3}{l}{\itshape \GiFour{} pillars} \\
\midrule
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & \ASix{} \\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & \ASix{} \\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}; \emph{absorbs the legacy} EEGG \emph{ledger}) & \ASix{} \\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & \ASix{} \\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}; \emph{formerly} EGG) & \ASix{} \\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & \ASix{} \\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & \ASix{} \\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & \ASix{} \\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & \ASix{} \\
\GGA{} & Ground Geometric Attractor (the unique deepest \GA{}; where the runtime halts) & \ASix{} def:GGA; \AEleven{} \\
\GRIPin{} & Fixed-landscape regime of the runtime (descent halting at the \GGA{}; consumes the \BIsix{} action budget) & \ASix{} \\
\GRIPout{} & Landscape-reorganizing regime (supplies the next descent's initial datum) & \ASix{} \\
\midrule

\multicolumn{3}{l}{\itshape \QiFour{} pillars} \\
\midrule
\QiQUA{} & Quantum information for \QGL{} (Quadruple Gauge Locking) -- Pillar I of \QiFour{} & \ASeven{} \\
\QiEP{} & Quantum information for the \UEP{} (Unified Extremal Principle) -- Pillar II of \QiFour{} & \ASeven{} \\
\QiCON{} & Quantum information for Conformal Invariance -- Pillar III of \QiFour{} & \ASeven{} \\
\QiHAL{} & Quantum information for the Holographic Attractor Lattice -- Pillar IV of \QiFour{} & \ASeven{} \\
\OH{} & Outer Hologram (boundary projection of bulk data; outward face of \EIC{}) & \ASeven{} \\
\IH{} & Inner Hologram (inner self-modeling layer; inward face of \EIC{}) & \ASeven{} \\
\EIC{} & Evolving Inner Cosmos (boundary completion of \QiFour{} as the \OH{}${+}$\IH{} composite) & \ASeven{} \\
\midrule

\multicolumn{3}{l}{\itshape \QiHAL{} acronyms} \\
\midrule
\MePMoS{} & Memory--Prediction--Motion Sensing & \ASeven{}, \QiHAL{} \\
\NHT{} & Neural Hologram & \ASeven{}, \QiHAL{} \\
\HAL{} & Neural Holographic Tomography & \ASeven{}, \QiHAL{} \\
\QHAL{} & Quadratic / Cubic \HAL{} & \ASeven{}, \QiHAL{} \\
\BFFT{} & Binary Fourier / Fractal Fourier Transform & \ASeven{}, \QiHAL{} \\
\midrule

\multicolumn{3}{l}{\itshape \GDOS{}-internal acronyms (program-defined)} \\
\midrule
\SIP{} & System Instance Packet (canonical \GDOS{} input unit) & \AFive{} \\
\SeOS{} & Selfish Open System & \AFive{} \\
\midrule

\multicolumn{3}{l}{\itshape Program-defined mathematical symbols} \\
\midrule
$\QB$ & Quantum-information Bit -- the bioenergetic Arisaka constant $\QB \approx 0.1555$~eV per qubit; matches ATP hydrolysis at biological temperatures & \ATwo{}, \S 19 (\TX{8}{GG6}) \\
$X_4$ & Bulk anomaly four-form $X_4 = \tr R^2 - \Tr F_5^2 - (2/5) F_B^2 - (2/5) F_{Qi}^2$ (the central object of the \BIsix{} cancellation) & \AThree{}, \S\ref{sec:structural-proof} \\
$H_3$ & Three-form field strength of the two-form tensor potential $B_2$, $H_3 = dB_2 + \mathrm{CS}$ (Chern--Simons three-forms); modified Bianchi identity $dH_3 = X_4$ & \AThree{}, \S\ref{sec:structural-proof} \\

\\

\multicolumn{3}{l}{\itshape Standard physics abbreviations (plain Roman, journal convention)} \\
\midrule

\multicolumn{3}{l}{\itshape \quad Open-system / quantum-channel formalism} \\
\midrule
CPTP & Completely Positive Trace-Preserving (channel admissibility for open evolution; \AX{3}/\POS{4}) & --- \\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan (master equation for Markovian CPTP semigroups; \POS{4}) & --- \\
GNS & Gelfand--Naimark--Segal (construction) & --- \\
NZ & Nakajima--Zwanzig (memory-kernel projection for non-Markovian open dynamics) & --- \\
POVM & Positive Operator-Valued Measure (generalized measurement model; \AX{3}/\POS{7}) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum and classical foundational theories} \\
\midrule
BRST & Becchi--Rouet--Stora--Tyutin (cohomology; the homogeneous Slavnov--Taylor identity in the present paper) & --- \\
GR & General Relativity & --- \\
ABJ & Adler--Bell--Jackiw (the 1969 chiral anomaly) & --- \\
AHSS & Atiyah--Hirzebruch spectral sequence & --- \\
APS & Atiyah--Patodi--Singer (index theorem; eta invariant) & --- \\
IIB & Type IIB string theory & --- \\
SUGRA & Supergravity & --- \\
TQFT & Topological Quantum Field Theory & --- \\
WGC & Weak Gravity Conjecture & --- \\
QCD & Quantum Chromodynamics & --- \\
QED & Quantum Electrodynamics & --- \\
QFT & Quantum Field Theory & --- \\
QM & Quantum Mechanics & --- \\
WCS & Wu--Chern--Simons (differential cohomology; the framework for the global completion theorem of \BIsix{}) & --- \\
GS & Green--Schwarz (anomaly-cancellation mechanism via tensor inflow) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Standard Model and beyond} \\
\midrule
EFT & Effective Field Theory & --- \\
EW & Electroweak & --- \\
GUT & Grand Unified Theory & --- \\
SM & Standard Model of particle physics & --- \\
SSB & Spontaneous Symmetry Breaking (closed-system; generalized by \GSSB{}) & --- \\
SUSY & Supersymmetry & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Higher-dimensional constructions} \\
\midrule
AdS & Anti--de Sitter spacetime & --- \\
AdS/CFT & Anti--de Sitter / Conformal Field Theory correspondence & --- \\
CFT & Conformal Field Theory & --- \\
KK & Kaluza--Klein (mode expansion / dimensional reduction) & --- \\
RS & Randall--Sundrum (warped extra-dimension model) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum gravity and cosmology} \\
\midrule
$\Lambda$CDM & Standard cosmological model with cosmological constant + cold dark matter & --- \\
LQG & Loop Quantum Gravity & --- \\
TOV & Tolman--Oppenheimer--Volkoff (equation) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Foundations of quantum mechanics} \\
\midrule
CHSH & Clauser--Horne--Shimony--Holt (inequality) & --- \\
EPR & Einstein--Podolsky--Rosen (paradox / argument) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Open-system / extremal-principle sub-acronyms} \\
\midrule
FEP & Free Energy Principle (boundary inference avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
\MEPP{} & Maximum Entropy Production Principle (heuristic boundary avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
PLA & Principle of Least Action (closed/bulk avatar of \UEP{}; standard variational dynamics) & \ASeven{}, \QiEP{} \\
PLT & Principle of Least Time (operational/boundary avatar of \UEP{}; fastest lawful path) & \ASeven{}, \QiEP{} \\
\end{longtable}
}



\bigskip

\noindent\textbf{Figure~\ref{fig:abbrev-flow}: hierarchy of program-defined acronyms in the \GGTheory{} program.} Top: the framework name (\GGTheory{}) and its constitutional charter (\GGConstitution{}). Tier~2: the four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) that together define the admissibility predicate. Tier~3: three bulk objects forced by the constitution --- \GG{} (6D bulk), \BIsix{} (modified Bianchi identity; the principal object of the present paper), \OGNfive{} (integer ledger). Tier~4: the boundary law \GDOS{} that the bulk projects to via \OGGEP{}, with input acronym \SIP{}. Tier~5: the two four-pillar structures, \GiFour{} (geometric flank, with sub-acronyms \GEP{}/\CEP{} blocks of \GFF{} and \DGI{}/\GSSB{}/\GA{} stages of \GRIP{}) and \QiFour{} (quantum-information flank, with \QiEP{} sub-acronyms PLT/\MEPP{}/FEP, \QiCON{} sub-concepts Conformal/L\'evy/Fractal/$1/f$, \QiHAL{} sub-acronyms \MePMoS{}/\NHT{}/\HAL{}/\BFFT{}, the boundary completion \OH{}${+}$\IH{}${=}$\EIC{}, and the runtime carrier \SeOS{} (Selfish Open System) underneath). Bottom: the bioenergetic Arisaka constant $\QB$ (derived secondary quantity), with the Arisaka equation $J_E / \QB = J_e = J_B = J_{Qi}$ that locks the four boundary currents through one conversion factor.

\begin{figure}[htbp]
\centering
\resizebox{0.824\textwidth}{!}{%
\begin{tikzpicture}[
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% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

% --- Tier 2: 4 axioms (left) and 7 postulates (right) -----
\node[axhead] (AXH) at (-52mm, -36mm) {\textbf{4 Axioms} (\textbf{AX-1}\,--\,\textbf{AX-4})};
\node[ax] (AX1) at (-52mm, -48mm) {\textbf{AX-1} \, \textbf{LC} \quad Locality and Causality};
\node[ax] (AX2) at (-52mm, -58mm) {\textbf{AX-2} \, \textbf{GGEP} \quad Geometry-Gauge Equivalence};
\node[ax] (AX3) at (-52mm, -68mm) {\textbf{AX-3} \, \textbf{OGGEP} \quad Operational/Open GGEP};
\node[ax] (AX4) at (-52mm, -78mm) {\textbf{AX-4} \, \textbf{MSMF} \quad Max.\ Symmetry, Min.\ Freedom};

\node[posthead] (POSH) at (52mm, -36mm) {\textbf{7 Postulates} (\textbf{POS-1}\,--\,\textbf{POS-7})};
\node[ax] (P1) at (52mm, -48mm) {\textbf{POS-1} \, \textbf{GI} \;\; (\,\textbf{LGI} \,/\, \textbf{DGI} \,/\, \textbf{GGI}\,)};
\node[ax] (P2) at (52mm, -56mm) {\textbf{POS-2} \, \textbf{UEP} \quad Unified Extremal Principle};
\node[ax] (P3) at (52mm, -64mm) {\textbf{POS-3} \, \textbf{AF} \quad Anomaly Freedom};
\node[ax] (P4) at (52mm, -72mm) {\textbf{POS-4} \, \textbf{CP} \quad Closure and Positivity};
\node[ax] (P5) at (52mm, -80mm) {\textbf{POS-5} \, \textbf{RG} \quad RG covariance};
\node[ax] (P6) at (52mm, -88mm) {\textbf{POS-6} \, \textbf{OGN} \quad Occam Gauge Normalization};
\node[ax] (P7) at (52mm, -96mm) {\textbf{POS-7} \, \textbf{MF} \quad Measurement and Falsifiability};

\draw[arrow] (GGC.south) -| ($(AXH.north) +(0,3mm)$) -- (AXH.north);
\draw[arrow] (GGC.south) -| ($(POSH.north)+(0,3mm)$) -- (POSH.north);

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{How the program's own names are related to each other. The chart is a hierarchy, not a list: the charter sits at the top, the geometry it forces below it, the boundary law below that, and the derived pillars at the foot, so the position of a name on the chart tells the reader what it depends on. Names at the same level are siblings introduced by the same construction. The purpose is orientation rather than definition --- the meanings are in the abbreviations table --- and a reader who has lost track of which name belongs to which layer can recover it here in one glance.}
\label{fig:abbrev-flow}
\end{figure}

\bigskip
\noindent\textit{With the dictionary in hand, the course proper begins.
Read the next appendix slowly: the descent chain it builds is the one
tool that every later appendix reuses without modification.}

\subsection*{Global naming convention for cross-cohort citation}
\label{ssec:naming-convention}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique
identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the item
type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem. The \AThree{} global
identifiers were assigned under an earlier sectioning of this paper and no
longer track its printed section numbers --- T-BI6-9.1, for instance, is the
trace-ratio lock, which prints as Theorem~\ref{thm:ogn-bi6}. They are kept
fixed rather than regenerated, so that pointers already carried by companion
papers continue to resolve, and the concordance in
Appendix~\ref{app:checklist} maps every one of them to the theorem number it
prints as. A global identifier is unique; it is not a section number.
\item \textbf{L-Suffix-N} for the $N$th Lemma.
\item \textbf{C-Suffix-N} for the $N$th Corollary.
\item \textbf{D-Suffix-N} for the $N$th Definition.
\item \textbf{R-Suffix-N} for the $N$th Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (e.g., O-BI6-1 is the first
Common Objection of the present paper).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-BI6-1 is absolute
proton stability).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-BI6-6 is the
\OGNfive{}-ledger-as-theorem innovation above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved entry
(\S\ref{sec:open-problems}).
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution}
(A1), \emph{GG6} (A2), \emph{BI6} (A3, this paper), \emph{Euclid} (A4),
\emph{GDOS} (A5), \emph{GRIP} (A6), \emph{Qi4} (A7), \emph{QM}
(A8), \emph{SM} (A9), \emph{Cosmos} (A10), \emph{Origin} (A11), together with the
phenomenology preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3),
\emph{Proton} (P6), the foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and
the consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH} (C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2).
The four axioms (AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7)
of \citet{ArisakaA1Constitution} keep their existing cohort-wide names; the
global identifier is the same item viewed from outside its home paper.

\section{Complete descent proof and BRST cancellation}
\label{app:descent-full}

This appendix expands the structural proof of Section~\ref{sec:structural-proof}
with every intermediate step displayed.
No computation is left implicit.

\paragraph{What this appendix teaches.} The descent formalism is the
single most reusable tool in this paper: the same three-step chain ---
transgress a characteristic form to a Chern--Simons potential, vary it
to obtain the descent two-form, cancel the one-loop variation with a
counterterm --- runs identically in four, six, and ten dimensions, and
a student who masters it here owns the corresponding chapter of every
string-theory textbook. The standard treatments are
\citet{Bilal2008} and \citet{Harvey2005} (lecture notes, closest in
level to this appendix), \citet{WeinbergQFT1996} (Vol.~II, Ch.~22),
\citet{Bertlmann1996} (the cohomological route taken here), and
\citet{FujikawaSuzuki2004} (the path-integral route).

\paragraph{Physics intuition: where the descent comes from.} The
one-loop anomaly is, at bottom, Fujikawa's observation that the
fermion measure is not invariant: under a gauge variation the measure
shifts by the index density of the Dirac operator --- a characteristic
form. The Wess--Zumino consistency condition then says that whatever
the anomaly is, it must be the gauge variation \emph{of something};
the descent chain is the systematic construction of that something.
Nothing in the chain is specific to the spectrum or the dimension:
it is bookkeeping for the single statement ``the anomaly is a
characteristic class, opened once by a transgression and once by a
variation.'' That is why the identical chain serves the
ten-dimensional mechanism of Appendix~\ref{app:gs10d} and the
six-dimensional mechanism proved here.

\subsection{Chern--Simons three-forms for each sector}

The \BIsix{} four-form involves four curvature sectors.
Each sector has a Chern--Simons three-form potential.

\paragraph{Gravitational sector.}
For the spin connection \(\omega\) with curvature
\(R = \dd\omega + \omega\wedge\omega\), the Lorentz Chern--Simons
three-form is
\begin{equation}
\omega_{3L}
= \tr\!\paren{\omega\wedge\dd\omega + \frac{2}{3}\,\omega\wedge\omega\wedge\omega}.
\label{eq:cs-lorentz}
\end{equation}
It satisfies
\begin{equation}
\dd\omega_{3L} = \tr(R\wedge R) = \tr R^2.
\label{eq:cs-lorentz-d}
\end{equation}
Under an infinitesimal local Lorentz variation with parameter
\(\Theta\):
\begin{equation}
\delta_\Theta\,\omega_{3L} = \dd\!\paren{\tr(\Theta\,\dd\omega)}.
\label{eq:cs-lorentz-var}
\end{equation}
In the descent notation, the two-form
\begin{equation}
\omega_{2L}^{(1)} = -\tr(\Theta\,\dd\omega)
\label{eq:omega2L}
\end{equation}
satisfies
\begin{equation}
\delta_\Theta\,\omega_{3L} + \dd\omega_{2L}^{(1)} = 0.
\label{eq:descent-lorentz}
\end{equation}

\paragraph{Non-Abelian \(\SU(5)\) sector.}
For the \(\SU(5)\) connection \(A_5\) with curvature
\(F_5 = \dd A_5 + A_5\wedge A_5\):
\begin{equation}
\omega_{3,5}
= \Tr\!\paren{A_5\wedge\dd A_5 + \frac{2}{3}\,A_5\wedge A_5\wedge A_5}.
\label{eq:cs-su5}
\end{equation}
It satisfies
\begin{equation}
\dd\omega_{3,5} = \Tr(F_5\wedge F_5) = \Tr F_5^2.
\label{eq:cs-su5-d}
\end{equation}
Under an infinitesimal gauge variation with parameter \(\Lambda_5\):
\begin{equation}
\omega_{2,5}^{(1)} = -\Tr(\Lambda_5\,\dd A_5)
\label{eq:omega2-su5}
\end{equation}
satisfies
\begin{equation}
\delta_{\Lambda_5}\,\omega_{3,5} + \dd\omega_{2,5}^{(1)} = 0.
\label{eq:descent-su5}
\end{equation}

\paragraph{Abelian \(\U(1)_B\) sector.}
For the Abelian connection \(A_B\) with \(F_B = \dd A_B\):
\begin{equation}
\omega_{3B} = A_B\wedge F_B = A_B\wedge\dd A_B.
\label{eq:cs-B}
\end{equation}
It satisfies
\begin{equation}
\dd\omega_{3B} = F_B\wedge F_B = F_B^2.
\label{eq:cs-B-d}
\end{equation}
Under an infinitesimal gauge variation \(\delta A_B = \dd\lambda_B\):
\begin{equation}
\delta\omega_{3B}
= \dd\lambda_B\wedge F_B + A_B\wedge\dd(\dd\lambda_B)
= \dd\lambda_B\wedge F_B
= \dd(\lambda_B\,F_B),
\label{eq:cs-B-var}
\end{equation}
so the two-form is
\begin{equation}
\omega_{2B}^{(1)} = -\lambda_B\,F_B.
\label{eq:omega2B}
\end{equation}

\paragraph{Abelian \(\U(1)_{Qi}\) sector.}
Identically,
\begin{equation}
\omega_{3,Qi} = A_{Qi}\wedge F_{Qi},
\qquad
\dd\omega_{3,Qi} = F_{Qi}^2,
\qquad
\omega_{2,Qi}^{(1)} = -\lambda_{Qi}\,F_{Qi}.
\label{eq:cs-Qi}
\end{equation}

\subsection{The composite \BIsix{} Chern--Simons three-form}

Assembling all four sectors with the \BIsix{} coefficients
\((1,-1,-2/5,-2/5)\):
\begin{equation}
\boxed{
\omega_3^{\mathrm{BI6}}
= \omega_{3L} - \omega_{3,5}
- \frac{2}{5}\,\omega_{3B}
- \frac{2}{5}\,\omega_{3,Qi}.
}
\label{eq:cs-bi6-full}
\end{equation}
Verification:
\begin{align}
\dd\omega_3^{\mathrm{BI6}}
&= \dd\omega_{3L} - \dd\omega_{3,5}
- \frac{2}{5}\,\dd\omega_{3B}
- \frac{2}{5}\,\dd\omega_{3,Qi}
\nonumber\\
&= \tr R^2 - \Tr F_5^2
- \frac{2}{5}\,F_B^2
- \frac{2}{5}\,F_{Qi}^2
\nonumber\\
&= X_4^{\mathrm{BI6}}.
\quad\checkmark
\label{eq:cs-bi6-verify}
\end{align}
The composite two-form is
\begin{equation}
\boxed{
\omega_2^{\mathrm{BI6,(1)}}
= \omega_{2L}^{(1)} - \omega_{2,5}^{(1)}
- \frac{2}{5}\,\omega_{2B}^{(1)}
- \frac{2}{5}\,\omega_{2,Qi}^{(1)}.
}
\label{eq:omega2-bi6-full}
\end{equation}
Explicitly, substituting \eqref{eq:omega2L}, \eqref{eq:omega2-su5},
\eqref{eq:omega2B}, and \eqref{eq:cs-Qi}:
\begin{equation}
\omega_2^{\mathrm{BI6,(1)}}
= -\tr(\Theta\,\dd\omega)
+ \Tr(\Lambda_5\,\dd A_5)
+ \frac{2}{5}\,\lambda_B\,F_B
+ \frac{2}{5}\,\lambda_{Qi}\,F_{Qi}.
\label{eq:omega2-bi6-explicit}
\end{equation}
Verification of the descent relation:
\begin{align}
\delta\omega_3^{\mathrm{BI6}} + \dd\omega_2^{\mathrm{BI6,(1)}}
&= \paren{\delta\omega_{3L}+\dd\omega_{2L}^{(1)}}
- \paren{\delta\omega_{3,5}+\dd\omega_{2,5}^{(1)}}
\nonumber\\
&\quad
- \frac{2}{5}\paren{\delta\omega_{3B}+\dd\omega_{2B}^{(1)}}
- \frac{2}{5}\paren{\delta\omega_{3,Qi}+\dd\omega_{2,Qi}^{(1)}}
\nonumber\\
&= 0 - 0 - 0 - 0 = 0.
\quad\checkmark
\label{eq:descent-bi6-verify}
\end{align}

\subsection{The improved field strength \texorpdfstring{\(H_3\)}{H3} and its gauge invariance}

Define
\begin{equation}
H_3 = \dd B_2 + \omega_3^{\mathrm{BI6}}.
\label{eq:h3-improved}
\end{equation}
Then
\begin{equation}
\dd H_3 = \dd\omega_3^{\mathrm{BI6}} = X_4^{\mathrm{BI6}}.
\label{eq:h3-bianchi}
\end{equation}
Assign the transformation law
\begin{equation}
\boxed{
\delta B_2 = -\omega_2^{\mathrm{BI6,(1)}} + \dd\Lambda_1,
}
\label{eq:b2-transform}
\end{equation}
where \(\Lambda_1\) is a one-form gauge parameter of \(B_2\). Then:
\begin{align}
\delta H_3
&= \dd(\delta B_2) + \delta\omega_3^{\mathrm{BI6}}
\nonumber\\
&= \dd(-\omega_2^{\mathrm{BI6,(1)}} + \dd\Lambda_1)
+ \delta\omega_3^{\mathrm{BI6}}
\nonumber\\
&= -\dd\omega_2^{\mathrm{BI6,(1)}}
+ \delta\omega_3^{\mathrm{BI6}}
\nonumber\\
&= 0.
\quad\checkmark
\label{eq:h3-gauge-inv}
\end{align}
The improved field strength is gauge invariant.

\paragraph{Physics intuition.} Equation~\eqref{eq:b2-transform} is the
heart of the mechanism, and it repays a moment of contemplation. The
two-form $B_2$ is declared to shift under gauge and Lorentz
transformations --- a classical transformation law, chosen once. The
prize is twofold: the \emph{field strength} $H_3$ becomes exactly
invariant (so the kinetic term is healthy), while the \emph{action}
term $\int B_2 \wedge X_4$ becomes exactly anomalous --- with the
opposite sign to the fermion loop. The quantum sin of the chiral
fermions is paid for by a classical tithe from the tensor sector.
In the ten-dimensional mechanism the identical trade is made with
$\int B_2 \wedge X_8$ (Appendix~\ref{app:gs10d}).

\subsection{The descent chain: seven-form, six-form, and one-loop variation}

Define the seven-form
\begin{equation}
I_7^{(0)} = \half\,\omega_3^{\mathrm{BI6}}\wedge X_4^{\mathrm{BI6}}.
\label{eq:i7}
\end{equation}
Verify:
\begin{align}
\dd I_7^{(0)}
&= \half\,\dd\omega_3^{\mathrm{BI6}}\wedge X_4^{\mathrm{BI6}}
+ \half\,\omega_3^{\mathrm{BI6}}\wedge\dd X_4^{\mathrm{BI6}}
\nonumber\\
&= \half\,X_4^{\mathrm{BI6}}\wedge X_4^{\mathrm{BI6}} + 0
\nonumber\\
&= I_8^{\mathrm{BI6}}.
\quad\checkmark
\label{eq:i7-verify}
\end{align}
(The second term vanishes because \(\dd X_4^{\mathrm{BI6}} = 0\),
since \(X_4\) is built from characteristic classes.)


Take the gauge/Lorentz variation of \(I_7^{(0)}\):
\begin{align}
\delta I_7^{(0)}
&= \half\,\delta\omega_3^{\mathrm{BI6}}\wedge X_4^{\mathrm{BI6}}
+ \half\,\omega_3^{\mathrm{BI6}}\wedge\delta X_4^{\mathrm{BI6}}
\nonumber\\
&= \half\,\delta\omega_3^{\mathrm{BI6}}\wedge X_4^{\mathrm{BI6}}
\qquad(\text{since }\delta X_4^{\mathrm{BI6}}=0)
\nonumber\\
&= -\half\,\dd\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
\qquad(\text{from }\eqref{eq:descent-bi6-verify})
\nonumber\\
&= -\dd\!\paren{\half\,\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}}
\qquad(\text{since }\dd X_4=0).
\label{eq:delta-i7}
\end{align}
Therefore the consistent anomaly six-form is
\begin{equation}
\boxed{
I_6^{(1)} = \half\,\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}},
}
\label{eq:i6-consistent}
\end{equation}
satisfying the descent equation
\begin{equation}
\delta I_7^{(0)} + \dd I_6^{(1)} = 0.
\label{eq:descent-6form}
\end{equation}


The one-loop effective action has the anomalous variation
\begin{equation}
\delta\Gamma_{\mathrm{1\text{-}loop}}
= \kappa\int_{M_6} I_6^{(1)}
= \frac{\kappa}{2}\int_{M_6} \omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}},
\label{eq:oneloop-var}
\end{equation}
where \(\kappa\) is a normalization constant (often \(\kappa = 2\pi\ii\)).

\subsection{Green--Schwarz counterterm and exact cancellation}

The Green--Schwarz counterterm is
\begin{equation}
S_{\mathrm{GS}} = \frac{\kappa}{2}\int_{M_6} B_2\wedge X_4^{\mathrm{BI6}}.
\label{eq:sgs}
\end{equation}
Its variation under gauge/Lorentz transformations:
\begin{align}
\delta S_{\mathrm{GS}}
&= \frac{\kappa}{2}\int_{M_6}
\delta B_2\wedge X_4^{\mathrm{BI6}}
+ \frac{\kappa}{2}\int_{M_6}
B_2\wedge\delta X_4^{\mathrm{BI6}}
\nonumber\\[6pt]
&= \frac{\kappa}{2}\int_{M_6}
(-\omega_2^{\mathrm{BI6,(1)}} + \dd\Lambda_1)\wedge X_4^{\mathrm{BI6}}
+ 0
\nonumber\\[6pt]
&= -\frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
+ \frac{\kappa}{2}\int_{M_6}
\dd\Lambda_1\wedge X_4^{\mathrm{BI6}}
\nonumber\\[6pt]
&= -\frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
+ \frac{\kappa}{2}\int_{M_6}
\dd(\Lambda_1\wedge X_4^{\mathrm{BI6}})
\nonumber\\[6pt]
&= -\frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
+ \frac{\kappa}{2}\int_{\partial M_6}
\Lambda_1\wedge X_4^{\mathrm{BI6}}.
\label{eq:sgs-var}
\end{align}
On a closed six-manifold (or in the bulk away from boundaries),
the boundary integral vanishes:
\begin{equation}
\delta S_{\mathrm{GS}}\Big|_{\mathrm{bulk}}
= -\frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}.
\label{eq:sgs-var-bulk}
\end{equation}


Comparing \eqref{eq:oneloop-var} and \eqref{eq:sgs-var-bulk}:
\begin{align}
\delta\Gamma_{\mathrm{1\text{-}loop}} + \delta S_{\mathrm{GS}}
&= \frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
- \frac{\kappa}{2}\int_{M_6}
\omega_2^{\mathrm{BI6,(1)}}\wedge X_4^{\mathrm{BI6}}
\nonumber\\
&= 0.
\label{eq:exact-cancel}
\end{align}
\begin{equation}
\boxed{
\delta(\Gamma_{\mathrm{1\text{-}loop}} + S_{\mathrm{GS}}) = 0.
}
\label{eq:exact-cancel-boxed}
\end{equation}
This completes the descent proof.

\paragraph{Physics intuition.} The cancellation is exact, not
approximate, because both sides of the ledger are descendants of the
same characteristic class: the fermion loop produces
$+\tfrac{\kappa}{2}\int \omega_2^{(1)} \wedge X_4$ by descent from
$I_8 = \tfrac12 X_4^2$, and the counterterm produces the same object
with a minus sign because $\delta B_2$ was \emph{defined} to be the
descent two-form. There is no fine-tuning to preserve under radiative
corrections: the statement is cohomological, and it holds at the level
of the effective action to all orders in the background fields. A
student comparing with the ten-dimensional case
(Appendix~\ref{app:gs10d}) will find the same proof with $X_8$ in
place of the second $X_4$ --- the six-dimensional perfect square is
the special case in which the two factors coincide, which is exactly
why the rank-one criterion of
Appendix~\ref{app:matrix-and-coefficients} has no ten-dimensional
analogue.

\subsection{BRST formulation and Wess--Zumino consistency}

In BRST language, replace \(\delta\to s\) (the nilpotent BRST operator).
The proof becomes:
\begin{align}
sX_4^{\mathrm{BI6}} &= 0,
\label{eq:brst-1}\\
s\omega_3^{\mathrm{BI6}} + \dd\omega_2^{\mathrm{BI6,(1)}} &= 0,
\label{eq:brst-2}\\
sB_2 &= -\omega_2^{\mathrm{BI6,(1)}},
\label{eq:brst-3}\\
s(H_3) &= s(\dd B_2 + \omega_3^{\mathrm{BI6}})
= \dd(sB_2) + s\omega_3^{\mathrm{BI6}}
\nonumber\\
&= -\dd\omega_2^{\mathrm{BI6,(1)}} + s\omega_3^{\mathrm{BI6}}
= 0,
\label{eq:brst-4}\\
s\Gamma_{\mathrm{1\text{-}loop}}
&= \frac{\kappa}{2}\int\omega_2^{\mathrm{BI6,(1)}}\wedge X_4,
\label{eq:brst-5}\\
sS_{\mathrm{GS}}
&= -\frac{\kappa}{2}\int\omega_2^{\mathrm{BI6,(1)}}\wedge X_4,
\label{eq:brst-6}\\
s(\Gamma_{\mathrm{1\text{-}loop}} + S_{\mathrm{GS}}) &= 0.
\label{eq:brst-7}
\end{align}
Therefore the anomaly cocycle is BRST-exact in the enlarged field
space that includes \(B_2\). The Slavnov--Taylor identity
\(\cS(\Gamma)=0\) is satisfied, and no ghost-number-one cohomology
class obstructs the quantum consistency of the theory.


The Wess--Zumino consistency condition requires
\begin{equation}
\delta_{\epsilon_1}\cA(\epsilon_2)
- \delta_{\epsilon_2}\cA(\epsilon_1)
= \cA([\epsilon_1,\epsilon_2]).
\label{eq:wz}
\end{equation}
In BRST language this is \(s\cA=0\). Since
\(\cA = \frac{\kappa}{2}\int\omega_2^{(1)}\wedge X_4\)
and \(sX_4=0\), one has
\begin{equation}
s\cA = \frac{\kappa}{2}\int s\omega_2^{(1)}\wedge X_4
= -\frac{\kappa}{2}\int \dd\omega_1^{(2)}\wedge X_4
= -\frac{\kappa}{2}\int \dd(\omega_1^{(2)}\wedge X_4) = 0
\label{eq:wz-verify}
\end{equation}
on a closed manifold. The Wess--Zumino condition is satisfied
automatically by the descent construction.

\bigskip
\noindent\textit{The descent engine is dimension-blind: nothing in
this appendix cared that six is six. The natural next stop is the
arena where the engine first ran --- ten-dimensional string theory ---
both to see the machinery at full historical scale and to learn, by
contrast, exactly which of \BIsix{}'s features are inherited and which
are new. One intuition to carry across the seam: in ten dimensions the
hard work is done by the choice of gauge group; watch for where the
six-dimensional story reassigns that work to the choice of spectrum.}

\section{The ten-dimensional Green--Schwarz mechanism worked from scratch, and the six-dimensional dictionary}
\label{app:gs10d}

This appendix is the string-theory half of the course. It derives, from
scratch and at graduate-student resolution, the two arithmetic miracles
that make the 1984 Green--Schwarz mechanism work in ten dimensions ---
the gaugino count $496$ and the vanishing of the irreducible sixth-order
Casimir at $\SO(32)$ --- and then translates every move into the
six-dimensional language of this paper. A student who works through
this appendix and Appendix~\ref{app:descent-full} will know exactly
what \BIsix{} inherits from string theory, and exactly where it parts
company. The textbook homes for the material are \citet{GSW1987}
(Vol.~II, Ch.~13), \citet{Polchinski1998b} (Ch.~12), and
\citet{BeckerBeckerSchwarz2007} (Ch.~5); the original paper is
\citet{GreenSchwarz1984}.

\subsection{The setting: why ten dimensions is a crisis}

Type~I and heterotic string theory descend at low energy to
ten-dimensional $\cN = 1$ supergravity coupled to super-Yang--Mills
with some gauge group $G$. The fermion content is chiral: a
positive-chirality gravitino $\psi_M$, a negative-chirality dilatino
$\lambda$, and positive-chirality gauginos $\chi^a$ in the adjoint of
$G$, of dimension $n = \dim G$. In ten dimensions the local anomaly is
encoded by a twelve-form $I_{12}$, and each chiral species contributes
through the index densities (the same $\hat{A}$-genus machinery
derived for six dimensions in Appendix~\ref{app:grav-full}):
gravitational terms through $\tr R^2$, $\tr R^4$, $\tr R^6$, gauge
terms through $\Tr F^2$, $\Tr F^4$, $\Tr F^6$ (adjoint traces), and
mixed terms.

\paragraph{Physics intuition.} In four dimensions the anomaly is a
box that the Standard Model escapes by a representation-theoretic
conspiracy among quarks and leptons. In ten dimensions the box has
many more walls --- a twelve-form has room for $\tr R^6$, $\Tr F^6$,
and a zoo of mixed monomials --- and no conspiracy among ordinary
multiplets can close them all. The discovery of 1984 was that for
precisely two gauge groups the anomaly does not have to vanish
identically: it factorizes, and a single two-form field can absorb it.

\subsection{Miracle one: the gaugino count \texorpdfstring{$496$}{496}}

The irreducible gravitational term $\tr R^6$ cannot be absorbed by any
Green--Schwarz counterterm: a factorized polynomial
$X_4 \wedge X_8$ contains only products of lower forms, never the
irreducible twelve-form $\tr R^6$. Its coefficient must therefore
vanish identically. Summing the index densities of the three chiral
species, the gravitino and dilatino contribute a fixed amount, and
each gaugino contributes the standard spin-$1/2$ unit, so the total
$\tr R^6$ coefficient is proportional to
\begin{equation}
\bigl(n - 496\bigr) \, \tr R^6 ,
\label{eq:496}
\end{equation}
where the $-496$ is the gravitino-plus-dilatino contribution
\citep{AlvarezGaumeWitten1984, GreenSchwarz1984}. Anomaly freedom
therefore \emph{forces}
\begin{equation}
\boxed{\; n = \dim G = 496 \;}
\end{equation}
Two compact groups have dimension $496$ and admit the required trace
structure: $\SO(32)$ ($\dim = 32 \cdot 31/2 = 496$) and
$E_8 \times E_8$ ($\dim = 2 \times 248 = 496$).

\paragraph{Physics intuition.} The gravitational sector fixes the
\emph{size} of the gauge sector before any gauge structure is
examined: the gravitino's irreducible anomaly must be paid for by
exactly $496$ gauginos, no more, no fewer. This is the ten-dimensional
ancestor of the gravitational completion of
Appendix~\ref{app:grav-full}, where the irreducible $p_2$ (equivalently
$\tr R^4$) coefficient of the six-dimensional ledger forces the counts
$n_{\mathrm{SD}} = 4\Omega$ and $n_F = 32\Omega$. Same logic, one
dimension pair down: an irreducible curvature monomial that no
counterterm can touch dictates a spectrum count.

\subsection{Miracle two: the trace identities at \texorpdfstring{$\SO(32)$}{SO(32)}}

The gauge sector must next dispose of the irreducible $\Tr F^6$. For
$\SO(n)$, the adjoint traces reduce to fundamental traces
(\citealp{GSW1987}; the derivation is an exercise in antisymmetric
tensor products):
\begin{align}
\Tr F^2 &= (n-2) \, \tr F^2, \\
\Tr F^4 &= (n-8) \, \tr F^4 + 3 \, (\tr F^2)^2, \\
\Tr F^6 &= (n-32) \, \tr F^6 + 15 \, \tr F^2 \, \tr F^4 .
\label{eq:so-n-traces}
\end{align}
At $n = 32$ --- and only there in the $\SO(n)$ family --- the
irreducible sixth-order Casimir drops out:
\begin{equation}
\boxed{\;
\Tr F^6 \big|_{\SO(32)} = 15 \, \tr F^2 \, \tr F^4,
\qquad
\Tr F^2 \big|_{\SO(32)} = 30 \, \tr F^2 .
\;}
\label{eq:so32-miracle}
\end{equation}
For $E_8$ the same disposal is automatic for a different reason: $E_8$
has \emph{no} independent fourth- or sixth-order Casimirs at all, so
\(\Tr F^4 = \tfrac{1}{100}(\Tr F^2)^2\) and
\(\Tr F^6 = \tfrac{1}{7200}(\Tr F^2)^3\) identically. With the
irreducible terms gone, the surviving reducible twelve-form assembles
into the celebrated factorization
\begin{equation}
I_{12} \;=\; X_4 \wedge X_8,
\qquad
X_4 = \tr R^2 - \frac{1}{30}\Tr F^2,
\label{eq:i12-factorised}
\end{equation}
with $X_8$ a specific closed eight-form built from $\tr R^4$,
$(\tr R^2)^2$, $\tr R^2 \Tr F^2$, $\Tr F^4$, and $(\Tr F^2)^2$, whose
exact coefficients the student should verify once in life against
\citet{GSW1987} (Ch.~13) or \citet{Polchinski1998b} (Ch.~12). The
factor $1/30$ in $X_4$ is the $\SO(32)$ trace conversion of
\eqref{eq:so32-miracle}. The modified Bianchi identity
$\dd H_3 = X_4$ and the counterterm $\int B_2 \wedge X_8$ then cancel
the remainder by the descent argument of
Appendix~\ref{app:descent-full}, applied verbatim.

\paragraph{Physics intuition.} Ten-dimensional string theory passes
the anomaly examination by \emph{choosing the group}: the arithmetic
facts $n - 496 = 0$ and $n - 32 = 0$ (or the Casimir poverty of
$E_8$) are properties of the gauge group itself, and the matter
content is then dictated by supersymmetry --- one adjoint gaugino,
nothing else to decide. Six-dimensional \GG{} passes the same
examination by \emph{choosing the spectrum}: the gauge group
$\SU(5) \times \U(1)_B \times \U(1)_{Qi}$ is fixed constitutionally,
and the irreducible quartic Casimir is killed not by a group identity
but by the core-block combination of
Appendix~\ref{app:core-block-full},
\begin{equation}
B_{\cC}
= B_{\mathbf{10}} + B_{\mathbf{5}} + 2 B_{\overline{\mathbf{5}}}
= (-3) + 1 + 2 = 0
\qquad \text{(spectrum combination)},
\end{equation}
the exact structural counterpart of $n - 32 = 0$ (group selection).
Hold those two boxed equations side by side and the similarity and the
difference of the two programs are both visible at a glance.

\subsection{The six-dimensional dictionary}

Table~\ref{tab:gs10d-dictionary} translates each move of the
ten-dimensional mechanism into its six-dimensional counterpart, with
the section of this paper where the counterpart is derived.

{\footnotesize
\begin{longtable}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.31\textwidth}>{\raggedright\arraybackslash}p{0.31\textwidth}}
\caption[The Green--Schwarz dictionary, move by move.]{Every move of the ten-dimensional Green--Schwarz mechanism with its counterpart here. The table is a dictionary rather than a comparison: each row is one step of the original argument and the six-dimensional step that plays the same part. Read it downward and it is a proof sketch of the present theorem in the vocabulary of the familiar one. The rows where the dictionary has no entry are the informative ones, and they are the rows where supersymmetry did the work in ten dimensions and something else does it here.}
\label{tab:gs10d-dictionary}\\
\toprule
\textbf{Move} & \textbf{10D heterotic / Type I} & \textbf{6D \BIsix{} (this paper)} \\
\midrule
\endfirsthead
\toprule
\textbf{Move} & \textbf{10D heterotic / Type I} & \textbf{6D \BIsix{}} \\
\midrule
\endhead
Anomaly polynomial & $I_{12}$ (twelve-form) & $I_8$ (eight-form), Sec.~\ref{sec:gs-mechanism} \\
Irreducible gravitational term & $\tr R^6$; killed by $n = 496$ & $p_2$ (i.e.\ $\tr R^4$); killed by the tensor/singlet completion $n_{\mathrm{SD}} = 4\Omega$, $n_F = 32\Omega$ (App.~\ref{app:grav-full}) \\
Irreducible gauge term & $\Tr F^6$; killed by $n = 32$ (group identity) or $E_8$ Casimir poverty & $\Tr F_5^4$; killed by the core-block combination $B_{\cC} = 0$ (App.~\ref{app:core-block-full}) \\
What is chosen & The gauge group ($\SO(32)$ or $E_8 \times E_8$); matter then fixed by SUSY & The spectrum (core blocks, orbits, completion); group fixed by the constitution \\
Factorization & $I_{12} = X_4 \wedge X_8$ (two independent forms) & $I_8 = \tfrac12 X_4^2$ (perfect square; rank one forced, Thm.~\ref{thm:rank-one}) \\
Coefficient lock & $1/30$ from $\Tr F^2 = 30 \tr F^2$ & $2/5$ from $\Tr(B_0^2)/\Tr(Y_0^2) = (L{-}N)/L$ (Sec.~\ref{sec:ogn5-theorem}) \\
Tensor field & $B_2$ with $\dd H_3 = X_4$; counterterm $\int B_2 \wedge X_8$ & $B_2$ with $\dd H_3 = X_4$; counterterm $\tfrac{\kappa}{2}\int B_2 \wedge X_4$ (App.~\ref{app:descent-full}) \\
Integer data output & $496$, $30$, $2$ groups & $(M,N,L;J,K) = (1,3,5;7,35)$ (App.~\ref{app:ogn5-complete}) \\
Residual freedom & Compactification landscape ($\sim 10^{500}$) & Finite Diophantine family; $\Omega \in 25\mathbb{Z}$ (App.~\ref{app:diophantine-full}) \\
Global completion & Witten-type checks; Monnier--Moore for 6D SUGRA descendants & Intended-class theorem + computed phase (App.~\ref{app:global-proof-dossier}, Lemma~\ref{lem:computed-phase}) \\
Textbook home & \citet{GSW1987} Ch.~13; \citet{Polchinski1998b} Ch.~12; \citet{BeckerBeckerSchwarz2007} Ch.~5 & This paper, Apps.~\ref{app:descent-full}--\ref{app:global-proof-dossier} \\
\bottomrule
\end{longtable}
}

\subsection{What the student should take away}

Three lessons, in increasing order of depth. \emph{First}, the descent
machinery --- Chern--Simons transgression, the Wess--Zumino condition,
the counterterm variation --- is \emph{identical} in the two settings;
learn it once (Appendix~\ref{app:descent-full}, or
\citealp{Bilal2008, Harvey2005}) and it serves both. \emph{Second},
the two programs close the irreducible terms by opposite strategies:
string theory adjusts the \emph{group} until group-theoretic
identities kill them; \GG{} fixes the group constitutionally and
adjusts the \emph{spectrum} until index sums kill them. Every
subsequent divergence --- supersymmetry required versus absent, a
two-form product versus a perfect square, a landscape versus a
Diophantine family --- traces back to that single strategic fork.
\emph{Third}, the six-dimensional problem is the more rigid of the
two: the perfect-square factorization $I_8 = \tfrac12 X_4^2$ admits no
analogue of the independent $X_8$, so the entire anomaly must collapse
onto a single ray (the rank-one criterion), which is why the
coefficient vector, and through it the integer ledger, is
overdetermined rather than merely constrained. A student who can
reproduce \eqref{eq:496}, \eqref{eq:so32-miracle}, the rank-one
collapse of Appendix~\ref{app:matrix-and-coefficients}, and the
$2\times 2$ system of Appendix~\ref{app:grav-full} owns both
mechanisms.

\bigskip
\noindent\textit{Back in six dimensions, the perfect square is the
whole game. The product $X_4 \wedge X_8$ of the previous appendix gave
string theory two independent dials; the six-dimensional factorization
$I_8 = \tfrac12 X_4^2$ has one. The rank-one criterion that follows is
the precise statement of that rigidity: a symmetric anomaly matrix
with ten independent entries must collapse onto a single ray --- six
$2 \times 2$ minors must vanish at once --- and this appendix
verifies every collapse explicitly.}

\section{Anomaly matrix, rank-one criterion, and coefficient matching}
\label{app:matrix-and-coefficients}

This appendix combines the rank-one eigenvalue computation with the explicit
expansion of the anomaly polynomial and all coefficient-matching and vanishing
conditions.

\paragraph{Physics intuition: anomaly space as a four-dimensional
arena.} Collect the four curvature-squared invariants into the vector
$Y = (\tr R^2, \Tr F_5^2, F_B^2, F_{Qi}^2)$; the reducible anomaly is
then a quadratic form on this $\mathbb{R}^4$, and a single
Green--Schwarz tensor can only ever cancel what lies along a single
ray, because it couples to one four-form $X_4 = v \cdot Y$. The
rank-one criterion is thus a statement of \emph{dimensional
bookkeeping in anomaly space}: one tensor, one ray. Six dimensions is
special because the factorized polynomial is a perfect square
$\tfrac12 (v \cdot Y)^2$, so the microscopic quadratic form must
collapse onto that ray exactly --- six independent $2 \times 2$ minors
must vanish. In ten dimensions the corresponding object is a product
of two \emph{different} forms, $X_4 \wedge X_8$, and no rank condition
of this sharpness arises (Appendix~\ref{app:gs10d}). This is the
precise sense in which the six-dimensional mechanism is more rigid
than the ten-dimensional one it descends from.

\subsection{Statement}

\begin{theorem}[Rank-one criterion, detailed version]
\label{thm:rank-one-app}
Let the reducible part of a six-dimensional local anomaly
polynomial be written in the basis
\begin{equation}
Y = (\tr R^2,\;\Tr F_5^2,\;F_B^2,\;F_{Qi}^2)^T
\end{equation}
as
\begin{equation}
I_8^{\mathrm{red}} = \half Y^T M\,Y,
\end{equation}
where \(M\) is a real symmetric \(4\times 4\) matrix. Then:
\begin{enumerate}[label=\textup{(\alph*)}, leftmargin=1.4cm]
\item A single Green--Schwarz two-form can cancel the reducible
 anomaly if and only if \(\mathrm{rank}(M) = 1\) and \(M\) is
 positive semidefinite.
\item When \(\mathrm{rank}(M) = 1\) and \(M \succeq 0\), there exists a unique
 (up to overall sign) unit vector \(\hat v\) and a positive
 scalar \(\lambda\) such that \(M = \lambda\,\hat v\,\hat v^T\).
\item The Green--Schwarz four-form is
 \(X_4 = \sqrt{\lambda}\,\hat v\cdot Y\), and the single tensor
 cancels the anomaly \(I_8 = \half(\sqrt{\lambda}\,\hat v\cdot Y)^2\).
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(a) Necessity.}
A single tensor couples to exactly one four-form \(X_4 = v\cdot Y\)
for some vector \(v\). The cancelable anomaly polynomial is therefore
\begin{equation}
I_8^{\mathrm{GS}} = \half (v\cdot Y)^2 = \half Y^T(vv^T) Y.
\end{equation}
The matrix \(vv^T\) is an outer product of one vector with itself.
Any outer product has rank at most one. Since \(v\neq 0\), the
rank is exactly one. Therefore \(M = vv^T\) has rank one.

\textbf{(b) Sufficiency.}
If \(M\) has rank one, then \(M\) has exactly one nonzero eigenvalue
\(\lambda\), with corresponding eigenvector \(\hat v\). Positivity of
\(\lambda\) is a genuine extra hypothesis and not a consequence of the
rank: over \(\mathbb R\) the matrix \(-vv^T\) also has rank one, and it
admits no real factorization \(ww^T\). That is precisely the wrong-sign
Green--Schwarz case \(I_8 = -\half X_4^2\), which the single-tensor
counterterm of Theorem~\ref{thm:bi6-cancellation} cannot cancel. For
the \BIsix{} matrix the sign is settled by explicit computation ---
Section~\ref{ssec:bi6-explicit} gives eigenvalues \(\{58/25,0,0,0\}\)
--- so \(\lambda > 0\) is part of the hypothesis of (b), as stated.
Then \(M = \lambda\,\hat v\,\hat v^T\).
Defining \(v = \sqrt{\lambda}\,\hat v\) gives \(M = vv^T\),
so \(I_8 = \half(v\cdot Y)^2\), which is cancelable by one tensor
coupling to \(X_4 = v\cdot Y\).

\textbf{(c)} Follows immediately from (b).
\end{proof}

\subsection{Explicit computation for \BIsix{}}
\label{ssec:bi6-explicit}

The \BIsix{} vector is
\begin{equation}
v = (1,\,-1,\,-2/5,\,-2/5)^T.
\end{equation}
The outer product is
\begin{equation}
M = vv^T
= \begin{pmatrix}
1\cdot 1 & 1\cdot(-1) & 1\cdot(-2/5) & 1\cdot(-2/5) \\
(-1)\cdot 1 & (-1)\cdot(-1) & (-1)\cdot(-2/5) & (-1)\cdot(-2/5) \\
(-2/5)\cdot 1 & (-2/5)\cdot(-1) & (-2/5)\cdot(-2/5) & (-2/5)\cdot(-2/5) \\
(-2/5)\cdot 1 & (-2/5)\cdot(-1) & (-2/5)\cdot(-2/5) & (-2/5)\cdot(-2/5)
\end{pmatrix}.
\end{equation}
Evaluating each entry:
\begin{equation}
M = \begin{pmatrix}
1 & -1 & -2/5 & -2/5 \\
-1 & 1 & 2/5 & 2/5 \\
-2/5 & 2/5 & 4/25 & 4/25 \\
-2/5 & 2/5 & 4/25 & 4/25
\end{pmatrix}.
\label{eq:M-explicit}
\end{equation}

\paragraph{Eigenvalue computation.}
For an outer product \(M = vv^T\), the nonzero eigenvalue is
\begin{equation}
\lambda = v^T v = |v|^2
= 1^2 + (-1)^2 + (-2/5)^2 + (-2/5)^2
= 1 + 1 + \frac{4}{25} + \frac{4}{25}
= 2 + \frac{8}{25}
= \frac{58}{25}.
\label{eq:eigenvalue}
\end{equation}
The eigenvector corresponding to \(\lambda = 58/25\) is \(v\) itself.
The other three eigenvalues are all zero, with eigenvectors spanning
the orthogonal complement of \(v\) in \(\mathbb{R}^4\).

\paragraph{Rank verification.}
Since there is exactly one nonzero eigenvalue:
\begin{equation}
\mathrm{rank}(M) = 1.
\quad\checkmark
\end{equation}

\paragraph{Trace and determinant checks.}
\begin{align}
\Tr(M) &= 1 + 1 + 4/25 + 4/25 = 58/25 = \lambda.
\quad\checkmark
\label{eq:trace-check}\\
\det(M) &= 0
\quad\text{(rank 1, so three zero eigenvalues)}.
\quad\checkmark
\label{eq:det-check}
\end{align}
The trace equals the single nonzero eigenvalue, and the determinant
vanishes, both consistent with rank one.

\subsection{The expansion}

Let
\begin{equation}
X_4 = \underbrace{\tr R^2}_{Y_1}
- \underbrace{\Tr F_5^2}_{Y_2}
- \frac{2}{5}\,\underbrace{F_B^2}_{Y_3}
- \frac{2}{5}\,\underbrace{F_{Qi}^2}_{Y_4}.
\end{equation}
Then
\begin{align}
X_4^2
&= (Y_1 - Y_2 - \tfrac{2}{5}Y_3 - \tfrac{2}{5}Y_4)^2
\nonumber\\
&= Y_1^2 + Y_2^2 + \tfrac{4}{25}Y_3^2 + \tfrac{4}{25}Y_4^2
\nonumber\\
&\quad - 2Y_1 Y_2 - \tfrac{4}{5}Y_1 Y_3 - \tfrac{4}{5}Y_1 Y_4
\nonumber\\
&\quad + \tfrac{4}{5}Y_2 Y_3 + \tfrac{4}{5}Y_2 Y_4 + \tfrac{8}{25}Y_3 Y_4.
\label{eq:x4-squared}
\end{align}
Therefore
\begin{align}
\half X_4^2
&= \half Y_1^2 + \half Y_2^2
+ \frac{2}{25}Y_3^2 + \frac{2}{25}Y_4^2
\nonumber\\
&\quad - Y_1 Y_2
- \frac{2}{5}Y_1 Y_3 - \frac{2}{5}Y_1 Y_4
\nonumber\\
&\quad + \frac{2}{5}Y_2 Y_3 + \frac{2}{5}Y_2 Y_4
+ \frac{4}{25}Y_3 Y_4.
\label{eq:half-x4-squared}
\end{align}

\subsection{The ten coefficient-matching equations}

Restoring the curvature notation, the ten required coefficients are:

\begin{center}
{\footnotesize
\begin{longtable}{clcl}
\caption[The ten reducible coefficient-matching conditions, with verification.]{The ten conditions that the coefficients must satisfy for the anomaly polynomial to factorize, each with the monomial it constrains, the value required, and the verification. Every row must hold; a single failure leaves the theory anomalous and therefore inconsistent, not merely inelegant. The fourth column is the check, written out entry by entry, so a reader can repeat any line. Together with the following table this is the whole of the factorization proof in arithmetic form.}\label{tab:ten-coefficients}\\
\toprule
\textbf{Eq.} & \textbf{Monomial} & \textbf{Required value} & \textbf{Verification} \\
\midrule
\endfirsthead
\toprule
\textbf{Eq.} & \textbf{Monomial} & \textbf{Required value} & \textbf{Verification} \\
\midrule
\endhead
(C1) & \((\tr R^2)^2\) & \(1/2\) & \(v_1^2/2 = 1/2\) \checkmark \\
(C2) & \((\Tr F_5^2)^2\) & \(1/2\) & \(v_2^2/2 = 1/2\) \checkmark \\
(C3) & \((F_B^2)^2\) & \(2/25\) & \(v_3^2/2 = (4/25)/2 = 2/25\) \checkmark \\
(C4) & \((F_{Qi}^2)^2\) & \(2/25\) & \(v_4^2/2 = 2/25\) \checkmark \\
(C5) & \(\tr R^2\cdot\Tr F_5^2\) & \(-1\) & \(v_1 v_2 = (1)(-1) = -1\) \checkmark \\
(C6) & \(\tr R^2\cdot F_B^2\) & \(-2/5\) & \(v_1 v_3 = (1)(-2/5) = -2/5\) \checkmark \\
(C7) & \(\tr R^2\cdot F_{Qi}^2\) & \(-2/5\) & \(v_1 v_4 = (1)(-2/5) = -2/5\) \checkmark \\
(C8) & \(\Tr F_5^2\cdot F_B^2\) & \(+2/5\) & \(v_2 v_3 = (-1)(-2/5) = +2/5\) \checkmark \\
(C9) & \(\Tr F_5^2\cdot F_{Qi}^2\) & \(+2/5\) & \(v_2 v_4 = (-1)(-2/5) = +2/5\) \checkmark \\
(C10) & \(F_B^2\cdot F_{Qi}^2\) & \(4/25\) & \(v_3 v_4 = (-2/5)(-2/5) = 4/25\) \checkmark \\
\bottomrule
\end{longtable}
}
\end{center}

All ten equations are satisfied identically because the \BIsix{} polynomial
is a perfect square. This is the content of the rank-one
factorization: if the anomaly matrix is \(M = vv^T\), then
\(M_{ij} = v_i v_j\) automatically satisfies all ten conditions
with the values listed above.

\subsection{The vanishing conditions}

The following monomials are \emph{absent} from \(\half X_4^2\)
because \(X_4\) contains only squared field strengths,
not individual ones:
\begin{center}
{\footnotesize
\begin{longtable}{cll}
\caption[The six monomials that must be absent, and why each is.]{The six monomials that must \emph{not} appear if the anomaly polynomial is to factorize, each with the reason it cannot. These are as load-bearing as the ten conditions of the previous table and are more easily overlooked, because an absent term leaves no trace in a calculation that only checks what is present. The third column gives the structural reason in each case --- irreducibility, or the representation content simply not generating the term. A reader auditing the factorization should check this table as carefully as the last.}\label{tab:six-forbidden}\\
\toprule
\textbf{Eq.} & \textbf{Monomial} & \textbf{Why absent from \(\half X_4^2\)} \\
\midrule
\endfirsthead
\toprule
\textbf{Eq.} & \textbf{Monomial} & \textbf{Why absent from \(\half X_4^2\)} \\
\midrule
\endhead
(V1) & \(\tr R^4\) & Irreducible; cannot be written as a product of 4-forms \\
(V2) & \(\Tr F_5^4\big|_{\mathrm{irr}}\) & Irreducible quartic; not generated by \((\Tr F_5^2)^2\) \\
(V3) & \(\Tr F_5^3\,F_B\) & Odd in \(F_B\); not in the square of an even polynomial \\
(V4) & \(\Tr F_5^3\,F_{Qi}\) & Odd in \(F_{Qi}\) \\
(V5) & \(F_B^3\,F_{Qi}\) & Odd mixed Abelian \\
(V6) & \(F_B\,F_{Qi}^3\) & Odd mixed Abelian \\
\bottomrule
\end{longtable}
}
\end{center}

Additionally, the off-diagonal terms
\(\tr R^2\,F_B\,F_{Qi}\),
\(\Tr F_5^2\,F_B\,F_{Qi}\)
are absent because \(X_4\) contains \(F_B^2\) and \(F_{Qi}^2\)
but not the individual two-forms \(F_B\) and \(F_{Qi}\).

\subsection{Explicit verification of the off-diagonal vanishing conditions}

For completeness, the benchmark verification is written explicitly rather than delegated. The required cancellation is elementary once the orbit pairing is spelled out.

For an axial orbit \(A_n^\sigma\), the two core blocks carry charges \((n/3,0)\) and \((0,n/3)\). Therefore every monomial containing both \(F_B\) and \(F_{Qi}\) vanishes identically on each axial orbit:
\[
\sum_{A_n^\sigma} q_B q_{Qi}=0,\qquad
\sum_{A_n^\sigma} q_B^3 q_{Qi}=0,\qquad
\sum_{A_n^\sigma} q_B q_{Qi}^3=0.
\]
For a diagonal orbit \(D_n^\sigma\), the paired core blocks carry charges \((n/3,n/3)\) and \((n/3,-n/3)\). Summing over the pair yields
\[
\sum_{D_n^\sigma} q_B q_{Qi}= \frac{n^2}{9}(1-1)=0,
\]
\[
\sum_{D_n^\sigma} q_B^3 q_{Qi}= \frac{n^4}{81}(1-1)=0,
\qquad
\sum_{D_n^\sigma} q_B q_{Qi}^3= \frac{n^4}{81}(1-1)=0.
\]
The same sign cancellation immediately kills the mixed gravitational and mixed simple-Abelian monomials:
\[
\sum_{D_n^\sigma} p_1\,q_B q_{Qi}=0,
\qquad
\sum_{D_n^\sigma} \Tr F_5^2\, q_B q_{Qi}=0.
\]

The remaining two forbidden cubic monomials, \(\Tr F_5^3 F_B\) and \(\Tr F_5^3 F_{Qi}\), vanish already on each core block because the underlying \(\SUfive\) block has vanishing cubic simple-group coefficient \(A_{3,\mathcal C}=0\). No inter-orbit cancellation is required. The full verification is summarized in Table~\ref{tab:offdiag-explicit}.

\begin{center}
\footnotesize
\begin{longtable}{p{0.22\textwidth}p{0.38\textwidth}p{0.30\textwidth}}
\caption[Explicit verification that the eight forbidden terms vanish.]{The eight forbidden terms --- the six of Table~\ref{tab:six-forbidden} together with the two off-diagonal monomials --- verified explicitly on the benchmark representation content rather than argued from general principles. The second column names the mechanism that removes each term and the third gives the result of the computation. This table exists because the preceding one states reasons and a reader is entitled to the arithmetic: between them, the claim that the anomaly polynomial factorizes is checkable line by line without reconstructing anything.}\label{tab:offdiag-explicit}\\
\toprule
\textbf{Term} & \textbf{Mechanism} & \textbf{Result on the benchmark reservoir}\\
\midrule
\endfirsthead
\toprule
\textbf{Term} & \textbf{Mechanism} & \textbf{Result on the benchmark reservoir}\\
\midrule
\endhead
\(\tr R^4\) & Irreducible gravitational term absent from \(\half X_4^2\) & absent from any perfect-square factorization; on the reservoir it is killed by the tensor/singlet completion $n_{\mathrm{SD}} = 4\Omega$, $n_F = 32\Omega$ (App.~\ref{app:grav-full}) \\
\(\Tr F_5^4|_{\mathrm{irr}}\) & Irreducible simple-group term absent from \((\Tr F_5^2)^2\) & vanishes on each core block by the \GG{} core-block choice \\
\(\Tr F_5^3 F_B\) & cubic simple-group coefficient \(A_{3,\mathcal C}=0\) & vanishes blockwise \\
\(\Tr F_5^3 F_{Qi}\) & cubic simple-group coefficient \(A_{3,\mathcal C}=0\) & vanishes blockwise \\
\(F_B^3F_{Qi}\) & sign symmetry \((q_{Qi}\to -q_{Qi})\) inside \(D_n^\sigma\) & \(\sum q_B^3q_{Qi}=0\) orbit by orbit \\
\(F_BF_{Qi}^3\) & sign symmetry \((q_{Qi}\to -q_{Qi})\) inside \(D_n^\sigma\) & \(\sum q_Bq_{Qi}^3=0\) orbit by orbit \\
\(\tr R^2F_BF_{Qi}\) & sign symmetry in \(q_Bq_{Qi}\) & \(\sum q_Bq_{Qi}=0\) orbit by orbit \\
\(\Tr F_5^2F_BF_{Qi}\) & sign symmetry in \(q_Bq_{Qi}\) & \(\sum q_Bq_{Qi}=0\) orbit by orbit \\
\bottomrule
\end{longtable}
\end{center}

The key point is that the benchmark reservoir does not rely on a delicate cancellation among many unrelated species. The off-diagonal vanishings are already built into the orbit architecture itself.

\bigskip
\noindent\textit{The rank-one collapse fixes the direction of $X_4$;
the integers behind that direction are the business of the next
appendix. The intuition to hold: the coefficient $-2/5$ is not a
fitted number but a trace ratio --- a statement about how baryon
number sits inside the unified block --- and the moment it is
recognized as $(L-N)/L$, anomaly freedom stops being analysis and
starts dictating arithmetic.}

\section{\OGNfive{}: complete derivation from \BIsix{}}
\label{app:ogn5-complete}

This appendix combines the trace-ratio computation that fixes the \BIsix{}
Abelian coefficient with the full theorem-level derivation of the \OGNfive{}
integer packet. The trace computation (from Arisaka-GG-A2-GG6 \citep{ArisakaA2GG6}) establishes the
bridge; the theorem proof closes it.

\paragraph{Physics intuition: integers from consistency, here and in
string theory.} Every consistent higher-dimensional program ends up
minting integers, because the consistency conditions are topological
and topology counts. String theory's mint produces $496$ (the gaugino
count forced by $\tr R^6$), $30$ (the $\SO(32)$ trace conversion),
$32$ (the D9-brane/orientifold tadpole of Type~I, the spacetime face
of the same $\SO(32)$), and $24$ (the K3 Euler-characteristic tadpole)
--- each the unique integer solution of an anomaly or tadpole
condition (Appendix~\ref{app:gs10d}; \citealp{Polchinski1998b}). The
\OGNfive{} ledger $(1, 3, 5; 7, 35)$ is minted the same way: each
integer is the unique solution of an arithmetic condition set by a
prior theorem, and the dual-lock identity $K = LJ = M^2 + N^2 + L^2$
is the overdetermination that makes the packet a witness rather than
a convention. The difference is what the integers then buy: string
theory's integers fix the gauge group and leave a landscape of
compactifications; the \GG{} integers fix the normalizations of a
single bulk and leave a finite Diophantine family
(Appendix~\ref{app:diophantine-full}).

\subsection{The trace computation}

The primitive hypercharge generator in the fundamental \(\mathbf{5}\)
of \(\SU(5)\) is
\begin{equation}
Y_0 = \diag\!\paren{-\frac{1}{3},\,-\frac{1}{3},\,-\frac{1}{3},\,
\frac{1}{2},\,\frac{1}{2}}.
\end{equation}
Its square is
\begin{equation}
Y_0^2 = \diag\!\paren{\frac{1}{9},\,\frac{1}{9},\,\frac{1}{9},\,
\frac{1}{4},\,\frac{1}{4}}.
\end{equation}
The trace is
\begin{equation}
\Tr(Y_0^2) = 3\times\frac{1}{9} + 2\times\frac{1}{4}
= \frac{1}{3} + \frac{1}{2}
= \frac{2}{6} + \frac{3}{6}
= \frac{5}{6}.
\label{eq:trY2}
\end{equation}

The primitive baryon generator is
\begin{equation}
B_0 = \diag\!\paren{\frac{1}{3},\,\frac{1}{3},\,\frac{1}{3},\,0,\,0}.
\end{equation}
Its square is
\begin{equation}
B_0^2 = \diag\!\paren{\frac{1}{9},\,\frac{1}{9},\,\frac{1}{9},\,0,\,0}.
\end{equation}
The trace is
\begin{equation}
\Tr(B_0^2) = 3\times\frac{1}{9} = \frac{1}{3}.
\label{eq:trB2}
\end{equation}

The ratio is
\begin{equation}
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)}
= \frac{1/3}{5/6}
= \frac{1}{3}\times\frac{6}{5}
= \frac{6}{15}
= \frac{2}{5}.
\label{eq:ratio-25}
\end{equation}

\subsection{General formula for \texorpdfstring{\(\SU(L)\)}{SU(L)} with
\texorpdfstring{\(N\)}{N} colors}

For the group \(\SU(L)\) with the decomposition
\(\SU(N)\times\SU(L{-}N)\times\U(1)_Y\):
\begin{equation}
Y_0 = \diag\!\paren{
\underbrace{-\frac{1}{N},\ldots,-\frac{1}{N}}_{N},\,
\underbrace{\frac{1}{L{-}N},\ldots,\frac{1}{L{-}N}}_{L-N}
}.
\end{equation}
\begin{align}
\Tr(Y_0^2) &= N\times\frac{1}{N^2} + (L{-}N)\times\frac{1}{(L{-}N)^2}
\nonumber\\
&= \frac{1}{N} + \frac{1}{L{-}N}
= \frac{(L{-}N)+N}{N(L{-}N)}
= \frac{L}{N(L{-}N)}.
\label{eq:trY2-general}
\end{align}

\begin{equation}
B_0 = \diag\!\paren{
\underbrace{\frac{1}{N},\ldots,\frac{1}{N}}_{N},\,
\underbrace{0,\ldots,0}_{L-N}
}.
\end{equation}
\begin{equation}
\Tr(B_0^2) = N\times\frac{1}{N^2} = \frac{1}{N}.
\label{eq:trB2-general}
\end{equation}

The ratio:
\begin{equation}
\boxed{
\frac{\Tr(B_0^2)}{\Tr(Y_0^2)}
= \frac{1/N}{L/[N(L{-}N)]}
= \frac{N(L{-}N)}{NL}
= \frac{L{-}N}{L}.
}
\label{eq:ratio-general}
\end{equation}

\subsection{Numerical verification for all small \texorpdfstring{\((N,L)\)}{(N,L)}}
\noindent Table~\ref{tab:ratio-verification} evaluates the ratio for every
small pair.

\begin{table}[htbp]
\centering
\caption[The coefficient ratio verified for every small $(N,L)$.]{The general ratio evaluated for every small pair of ledger integers. The first two columns are the pair, the third the exact fraction and the fourth its decimal value. The row in bold is the one the framework selects, and the table is included so that its selection can be seen against the alternatives rather than in isolation: the value is not special-looking, and nothing about it would be suspicious if it were wrong. What makes it a prediction is that the pair is fixed elsewhere, by the cascade, and not chosen to produce this number.}
\label{tab:ratio-verification}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{cccc}
\toprule
\(N\) & \(L\) & \((L{-}N)/L\) & Decimal \\
\midrule
2 & 4 & 1/2 & 0.500 \\
2 & 5 & 3/5 & 0.600 \\
3 & 4 & 1/4 & 0.250 \\
\textbf{3} & \textbf{5} & \textbf{2/5} & \textbf{0.400} \\
3 & 6 & 1/2 & 0.500 \\
4 & 5 & 1/5 & 0.200 \\
4 & 6 & 1/3 & 0.333 \\
\bottomrule
\end{tabular}
\end{table}

Among the small pairs tabulated, the \GG{} value \(2/5\) occurs only at \((N,L)=(3,5)\), which is the \OGN{} pair corresponding to three colors and \(\SU(5)\) unification. The full integer family \((L,N)=(5k,3k)\) realizes \(2/5\) for every \(k\); it is cut down to \(k=1\) by the dual lock together with \AX{4} (Theorem~\ref{thm:lN-minimality}), not by the table.

\subsection{The core block as a function of \texorpdfstring{\(L\)}{L}}

The core block
\(\cC = \mathbf{L(L{-}1)/2}\oplus\mathbf{L}\oplus 2\,\overline{\mathbf{L}}
\oplus n_1\cdot\mathbf{1}\)
requires \(d_{\cC} = (L{+}1)\cdot A_{\cC}\).
In the integer normalization used throughout this paper (where \(A(\mathbf{5})=1\)),
the indices are
\begin{align}
A(\mathbf{10}) &= 3, & A(\mathbf{5}) &= 1, &
A(\overline{\mathbf{5}}) &= 1.
\end{align}
The core block index is therefore
\begin{equation}
A_{\cC} = A(\mathbf{10}) + A(\mathbf{5}) + 2A(\overline{\mathbf{5}})
= 3 + 1 + 2 = 6 = L+1.
\end{equation}
The dimension is
\begin{equation}
d_{\cC} = \frac{L(L{-}1)}{2} + 3L + n_1.
\end{equation}
Setting \(d_{\cC} = (L{+}1)\cdot A_{\cC} = (L{+}1)^2\):
\begin{equation}
n_1 = (L{+}1)^2 - \frac{L(L{-}1)}{2} - 3L
= L^2+2L+1 - \frac{L^2-L}{2} - 3L
= \frac{L^2-L+2}{2}.
\label{eq:n1-correct}
\end{equation}
For \(L=5\): \(d_{\cC} = 10 + 5 + 10 + 11 = 36 = 6^2\),
\(n_1 = (25-5+2)/2 = 11\). \(\checkmark\)

\subsection{Precise meanings and the main \OGNfive{} theorem}
The cleaned-up meanings are as follows.

\begin{definition}[Meaning of \(M,N,L;J,K\)]
In the intended \GG{} category:
\begin{enumerate}[leftmargin=1.8cm,label=(\roman*)]
\item \(M\) is the \textbf{primitive tensor/gravitational slot count}. It counts the minimal GS/tensor channel carried by the single \BIsix{} tensor lattice.
\item \(N\) is the \textbf{color triplicity index}, written \(N_c\)
 wherever the physical color count rather than the ledger slot is meant.
 It is the size of the color block carried inside the defining
 representation of the unified group. It is \emph{not} the number of
 fermion generations: that quantity is written \(N_g\), it is not a
 member of this ledger, and it is fixed by an argument of an entirely
 different kind \citep{ArisakaP2N3}. That \(N_c = N_g = 3\) is a
 coincidence on the boundary, and this paper never uses it.
\item \(L\) is the \textbf{defining unified block size}. In \GG{} it is the defining dimension of the minimal simple unified group \(\SUfive\). 
\item \(J\) is the \textbf{minimal symmetric closure cardinality}. If the largest allowed absolute Abelian orbit level is \(N\), then \OGN{} closure plus sign symmetry require the packet
\[
\ChargePacket:=\{0,\pm1,\pm2,\dots,\pm N\}.
\]
The integer \(J\) is the cardinality of this packet.
\item \(K\) is the \textbf{total lock index}. It is simultaneously the rectangular product \(LJ\) and the quadratic norm of the normalization packet.
\end{enumerate}
\end{definition}


\subsection{The trace-ratio theorem that fixes the \BIsix{} Abelian fraction}
The cleanest route begins with a general \(SU(L)\) block split into a color block of size \(N\) and a complementary block of size \(L-N\).

Define the two diagonal directions
\[
B_{N,L}=\diag\!\left(\underbrace{\frac1N,\dots,\frac1N}_{N},\underbrace{0,\dots,0}_{L-N}\right),
\]
\[
Y_{N,L}=\diag\!\left(\underbrace{-\frac1N,\dots,-\frac1N}_{N},\underbrace{\frac1{L-N},\dots,\frac1{L-N}}_{L-N}\right).
\]
The first is the baryonic block direction and the second is the traceless hypercharge-like splitting direction. A direct calculation gives
\[
\TrU(B_{N,L}^2)=N\cdot\frac1{N^2}=\frac1N,
\]
\[
\TrU(Y_{N,L}^2)=N\cdot\frac1{N^2}+(L-N)\cdot\frac1{(L-N)^2}
=\frac1N+\frac1{L-N}=\frac{L}{N(L-N)}.
\]
Therefore
\begin{equation}
\frac{\TrU(B_{N,L}^2)}{\TrU(Y_{N,L}^2)}=\frac{L-N}{L}.
\label{eq:trace-ratio-general}
\end{equation}

Equation \eqref{eq:trace-ratio-general} is the structural theorem needed by \OGN{}. It is not specific to \(L=5\); it is the general block formula. In \GG{}, \BIsix{} tells us that the Abelian coefficient must be exactly
\[
\frac{2}{5}.
\]
Hence \OGN{} closure demands
\begin{equation}
\frac{L-N}{L}=\frac25.
\label{eq:LN-over-L}
\end{equation}

\paragraph{The main \OGNfive{} theorem.}
\begin{theorem}[\OGNfive{} theorem]
Within the intended \GG{} category, \BIsix{} and the unified block structure force uniquely
\[
(M,N,L;J,K)=(1,3,5;7,35),
\qquad
K=LJ=M^2+N^2+L^2.
\]
\end{theorem}

\begin{proof}
The proof proceeds in five steps.

\paragraph{Step 1: \(M=1\).}
\GG{} uses a single primitive Green--Schwarz tensor channel. Equivalently, the tensor lattice is rank one. Therefore the primitive tensor/gravitational slot count is
\[
M=1.
\]

\paragraph{Step 2: \(L=5\).}
By Theorem~\ref{thm:L-from-GS}, single-tensor Green--Schwarz consistency
requires the irreducible quartic and cubic indices of the charged core
package to vanish, and both equal \(L-5\); with \AX{4}~(\MSMF{})
selecting the minimal chiral package,
\[
L=5.
\]
This step does not appeal to the smallest-simple-group clause and does
not consume the observed low-energy gauge content: it is free of \(N\)
throughout, which is what makes Step~3 available in the form given
below.

\paragraph{Step 3: \(N = N_c = 3\).}
By Theorem~\ref{thm:N-from-closure}, the dual lock together with the
closure-index clause \(J = 2N+1\) gives \(N = L \pm \sqrt{L-M^2}\); at
\(M=1\) and the \(L=5\) of Step~2 the roots are \(3\) and \(7\), and the
block embedding \(N < L\) retains
\[
N = N_c = 3 .
\]
The trace ratio is not used here. It is computed on the block split of
the defining representation, so it would consume the split it concludes;
its role in this paper is the independent check of
Eq.~\eqref{eq:LN-over-L-updated}, not a derivation. Nothing in this step
bears on the generation count \(N_g\).

\paragraph{Step 4: \(J=2N+1\).}
\BIsix{} does not use an arbitrary closure count. The diagonal orbit system already forces sign symmetry: if level \(+n\) occurs, the paired orbit contains level \(-n\) as well. \OGN{} minimality further requires a neutral reference slot \(0\). Therefore the minimal closed charge packet is
\[
\ChargePacket=\{0,\pm1,\dots,\pm N\},
\qquad |\ChargePacket|=2N+1.
\]
With \(N=3\), this gives
\[
J=|\ChargePacket|=2\cdot3+1=7.
\]

\paragraph{Step 5: \(K=LJ\) and the quadratic witness identity.}
Rectangular closure gives
\[
K=LJ=5\cdot 7=35.
\]
Then the quadratic norm of the \OGN{} packet is
\[
M^2+N^2+L^2=1^2+3^2+5^2=1+9+25=35=K.
\]
Hence
\[
K=LJ=M^2+N^2+L^2=35.
\]
This proves the claim.
\end{proof}

\subsection{Uniqueness and physical interpretation}
The previous proof is already sufficient, but there is an instructive uniqueness corollary.

\begin{corollary}[Why the false branch \(L=2\) is excluded]
Assume only \(M=1\), \(N=3\), the minimal symmetric closure cardinality \(J=2N+1\), and the two closure rules
\[
K=LJ,
\qquad
K=M^2+N^2+L^2.
\]
Then \(L\) solves
\[
L^2-7L+10=0,
\]
with roots \(L=2\) and \(L=5\). The branch \(L=2\) is physically inadmissible, so \(L=5\) is unique.
\end{corollary}

\begin{proof}
Substitute \(M=1\), \(N=3\), \(J=7\):
\[
7L = 1+9+L^2 \quad\Longrightarrow\quad L^2-7L+10=0.
\]
The roots are \(L=2\) and \(L=5\). The minimal exclusion criterion is already sufficient: \(L=2<N=3\), so a color block of size three cannot be embedded inside the unified simple block. This corollary is retained as a consistency check on the closure identities and not as the derivation of \(L\): it takes \(N\) as given, and therefore cannot be used to support any statement about \(N\). The derivation of \(L\) is Theorem~\ref{thm:L-from-GS}, which is free of \(N\) throughout. The further failures to realize the \(\SUfive\) defining stage and the exact \BIsix{} denominator-5 trace structure are corollaries of this basic obstruction. Therefore only \(L=5\) survives.
\end{proof}

\paragraph{Physical interpretation.}
The theorem above gives a precise and cleaner interpretation of the integers.

To summarize the \OGNfive{} content: \(M=1\) counts the primitive tensor slot,
\(N=3\) is the color triplicity index fixed by the \BIsix{} trace ratio,
\(L=5\) is the defining unified block size of \(\SUfive\),
\(J=7\) is the minimal symmetric closure cardinality of the charge packet
\(\{0,\pm1,\pm2,\pm3\}\), and \(K=35\) is the total lock index.
The reappearance of the numeral 3 as the generation count on the
boundary is a downstream consequence, not the primitive definition of \(N\).

\bigskip
\noindent\textit{The ledger says which integers the spectrum must
realize; the core block is the brick engineered to realize them. The
intuition: the block pre-pays the two irreducible debts --- the cubic
and quartic Casimirs, the sins no Green--Schwarz tensor can absorb ---
internally, within its own thirty-six states, so that assembling
blocks into orbits only ever has to balance cancellable coin.}

\section{Core block anomaly polynomial}
\label{app:core-block-new}
\label{app:core-block-full}

This appendix computes the anomaly polynomial of one core block
\(\cC^\sigma(q_B,q_{Qi})\) as defined in Section~\ref{sec:model-level}.
The key results are that the irreducible quartic index \(B_{\cC}\)
and the cubic index \(A_{3,\cC}\) both vanish, confirming
Proposition~\ref{prop:core-block}.

\paragraph{A primer on the group-theory indices (for the graduate
reader).} Four numbers characterize how a representation \(\rho\) of
\(\SU(N)\) feeds the anomaly polynomial, and all four are defined by
reducing traces in \(\rho\) to traces in the fundamental:
\begin{align}
\tr_\rho F^2 &= A_\rho \, \tr F^2
&&\text{(quadratic index; normalized to } A(\mathbf{N}) = 1\text{)},\\
\tr_\rho F^3 &= A_3(\rho) \, \tr F^3
&&\text{(cubic anomaly coefficient; } A_3(\overline{\rho}) = -A_3(\rho)\text{)},\\
\tr_\rho F^4 &= B_\rho \, \Tr F^4 + C_\rho \, (\Tr F^2)^2
&&\text{(quartic decomposition into irreducible and reducible parts)}.
\end{align}
The dimension \(d_\rho\) completes the set. For the representations a
model-builder actually uses, the closed forms are classical (they
follow from character expansions of tensor products; see
\citealp{Slansky1981} for tables and \citealp{Georgi1999} for the
derivations): the fundamental has \((A, A_3, B, C) = (1, 1, 1, 0)\);
the two-index antisymmetric \(\mathbf{N(N{-}1)/2}\) has
\((N{-}2,\; N{-}4,\; N{-}8,\; 3)\); the adjoint has
\((2N,\; 0,\; 2N,\; 6)\); and every conjugate flips the sign of
\(A_3\) while preserving \(A, B, C\). Setting \(N = 5\) reproduces every entry of
Table~\ref{tab:su5-data} below; the reader is encouraged to verify
\(B_{\mathbf{10}} = 5 - 8 = -3\) once by hand, since that single
integer is what the core-block combination
\((-3) + 1 + 2 = 0\) must cancel --- the six-dimensional counterpart
of the \(\SO(32)\) identity \(n - 32 = 0\) of
Appendix~\ref{app:gs10d}.

\paragraph{Physics intuition.} \(A\) measures how strongly the
representation couples to the reducible (Green--Schwarz-cancellable)
part of the anomaly; \(A_3\) and \(B\) measure the two irreducible
sins --- the cubic and quartic Casimirs --- that no tensor can absorb
and that must therefore vanish by \emph{combination}; \(C\) is the
representation's own contribution to the reducible quartic. The core
block is engineered so that the sins cancel within each block
(\(A_{3,\cC} = B_{\cC} = 0\)) while the virtues add
(\(A_{\cC} = 6\), \(C_{\cC} = 3\)): pre-payment of the irreducible
debts, block by block, so that the inter-block Diophantine system of
Appendix~\ref{app:diophantine-full} needs to manage only the
cancellable remainder.

\subsection{\texorpdfstring{\(\SU(5)\)}{SU(5)} group theory data}

\begin{center}
{\footnotesize
\begin{longtable}{cccccc}
\caption[Representation data used throughout the paper.]{The group-theoretic data every calculation in this paper draws on: dimensions, indices and anomaly coefficients for the representations that appear. It is gathered here so that no derivation has to restate it and so that a reader checking an anomaly coefficient has one place to verify it. All values are standard and are quoted, not derived; an error in this table would propagate into every result in the paper, which is why it is printed in full rather than cited.}\label{tab:su5-data}\\
\toprule
\(\rho\) & \(d_\rho\) & \(A_\rho\) & \(A_3(\rho)\) & \(B_\rho\) & \(C_\rho\) \\
\midrule
\endfirsthead
\toprule
\(\rho\) & \(d_\rho\) & \(A_\rho\) & \(A_3(\rho)\) & \(B_\rho\) & \(C_\rho\) \\
\midrule
\endhead
\(\mathbf{5}\) & 5 & 1 & \(+1\) & 1 & 0 \\
\(\overline{\mathbf{5}}\) & 5 & 1 & \(-1\) & 1 & 0 \\
\(\mathbf{10}\) & 10 & 3 & \(+1\) & \(-3\) & 3 \\
\(\overline{\mathbf{10}}\) & 10 & 3 & \(-1\) & \(-3\) & 3 \\
\(\mathbf{24}\) & 24 & 10 & 0 & 10 & 6 \\
\(\mathbf{1}\) & 1 & 0 & 0 & 0 & 0 \\
\bottomrule
\end{longtable}
}
\end{center}

The quartic decomposition is
\begin{equation}
\tr_\rho(F_5^4) = B_\rho\,\Tr F_5^4 + C_\rho\,(\Tr F_5^2)^2,
\end{equation}
where \(\Tr\) denotes the trace in the fundamental. For \(\SU(N)\),
the antisymmetric \(k=2\) representation has
\(B_{\mathrm{AS}} = N-8\) and \(C_{\mathrm{AS}} = 3\).
For \(\SU(5)\): \(B_{\mathbf{10}} = 5-8 = -3\). \(\checkmark\)

\subsection{Core block index sums}

The core block \(\cC = \mathbf{10}\oplus\mathbf{5}\oplus
2\,\overline{\mathbf{5}}\oplus 11\cdot\mathbf{1}\) has:

\paragraph{Irreducible quartic.}
\begin{equation}
B_{\cC} = B_{\mathbf{10}} + B_{\mathbf{5}} + 2\,B_{\overline{\mathbf{5}}}
+ 11\cdot 0
= (-3) + 1 + 2(1) + 0 = 0.
\quad\checkmark
\end{equation}

\paragraph{Cubic.}
\begin{equation}
A_{3,\cC} = A_3(\mathbf{10}) + A_3(\mathbf{5})
+ 2\,A_3(\overline{\mathbf{5}}) + 11\cdot 0
= 1 + 1 + 2(-1) + 0 = 0.
\quad\checkmark
\end{equation}

\paragraph{Quadratic index.}
\begin{equation}
A_{\cC} = A_{\mathbf{10}} + A_{\mathbf{5}}
+ 2\,A_{\overline{\mathbf{5}}} + 11\cdot 0
= 3 + 1 + 2(1) + 0 = 6 = L+1.
\quad\checkmark
\end{equation}

\paragraph{Reducible quartic.}
\begin{equation}
C_{\cC} = C_{\mathbf{10}} + C_{\mathbf{5}}
+ 2\,C_{\overline{\mathbf{5}}} + 11\cdot 0
= 3 + 0 + 0 + 0 = 3.
\end{equation}

\paragraph{Dimension.}
\begin{equation}
d_{\cC} = 10 + 5 + 2(5) + 11(1) = 10 + 5 + 10 + 11 = 36 = (L+1)^2.
\quad\checkmark
\end{equation}

\paragraph{Dimension-to-index ratio.}
\begin{equation}
\frac{d_{\cC}}{A_{\cC}} = \frac{36}{6} = 6 = L+1.
\quad\checkmark
\end{equation}

This ratio is crucial: it ensures that the gravitational-gauge
mixed coefficient \(\mathrm{coeff}(\tr R^2\cdot\Tr F_5^2)\) has the
correct value relative to the pure gauge coefficient
\(\mathrm{coeff}((\Tr F_5^2)^2)\) for the \BIsix{} factorization.

\subsection{Anomaly polynomial of one core block}

For a core block at chirality \(\sigma\) with charges \((q_B,q_{Qi})\),
defining \(x = q_B f_B + q_{Qi} f_{Qi}\),
\(a = \Tr f_5^2\), and \(g = \half p_1\), the anomaly polynomial is:
\begin{align}
I_8[\cC^\sigma(q_B,q_{Qi})]
&= \sigma\bigg[
\frac{d_{\cC}}{5760}(7p_1^2 - 4p_2)
- \frac{d_{\cC}}{48}\,p_1\,x^2
- \frac{A_{\cC}}{48}\,p_1\,a
\nonumber\\
&\qquad
+ \frac{d_{\cC}}{24}\,x^4
+ \frac{A_{\cC}}{4}\,a\,x^2
+ \frac{C_{\cC}}{24}\,a^2
+ \underbrace{\frac{B_{\cC}}{24}}_{=0}\,\Tr f_5^4
+ \underbrace{\frac{A_{3,\cC}}{6}}_{=0}\,\Tr f_5^3\,x
\bigg]
\nonumber\\[6pt]
&= \sigma\bigg[
\frac{36}{5760}(7p_1^2 - 4p_2)
- \frac{36}{48}\,p_1\,x^2
- \frac{6}{48}\,p_1\,a
+ \frac{36}{24}\,x^4
+ \frac{6}{4}\,a\,x^2
+ \frac{3}{24}\,a^2
\bigg]
\nonumber\\[6pt]
&= \sigma\bigg[
\frac{1}{160}(7p_1^2 - 4p_2)
- \frac{3}{4}\,p_1\,x^2
- \frac{1}{8}\,p_1\,a
+ \frac{3}{2}\,x^4
+ \frac{3}{2}\,a\,x^2
+ \frac{1}{8}\,a^2
\bigg].
\label{eq:core-block-poly}
\end{align}

The result is clean: the irreducible \(\Tr f_5^4\) and cubic
\(\Tr f_5^3\,x\) terms are \emph{automatically absent} because
\(B_{\cC}=0\) and \(A_{3,\cC}=0\). This is the core block's
defining property: it pre-cancels the most dangerous anomaly
monomials inside each block, before any inter-block cancellation
is needed.

\bigskip
\noindent\textit{One block is clean; the question is how many of each
kind to take. That is not model-building freedom but a Diophantine
problem: four anomaly-matching equations in five orbit multiplicities.
The next appendix solves it exactly, and the reader should watch for
two things --- how integrality forces $\Omega \in 25\mathbb{Z}$, and
how an apparently negative multiplicity resolves into an honest
chirality flip.}

\section{The Diophantine system: derivation and solution}
\label{app:diophantine-full}

With the core-block anomaly data from Appendix~\ref{app:core-block-full} in hand,
this appendix derives the Diophantine system that the orbit
multiplicities must satisfy and exhibits the explicit solution
stated in Theorem~\ref{thm:orbit}.

\paragraph{Physics intuition.} The system that follows has five
unknowns and four equations, so its solution set is a line, not a
point --- and the appendix says so explicitly, exhibiting the affine
one-parameter family before selecting the self-similar \(m = 0\)
branch on which every multiplicity scales linearly with
\(\Omega/25\). A student should absorb the division of labor: the
\emph{equations} come from anomaly matching and are non-negotiable;
the \emph{branch choice} is a minimality selection in the spirit of
\AX{4}. Compare the string-side analogue: the Kumar--Morrison--Taylor
enumeration of anomaly-free six-dimensional supergravities
\citep{KumarMorrisonTaylor2010} also produces solution \emph{families}
from anomaly matching, with string universality conjectured to realize
them; here the family is one-dimensional, explicit, and closed by a
constitutional clause rather than by a landscape.

\subsection{Orbit anomaly polynomials}

\paragraph{Axial orbit \(A_n^\sigma\).}
The axial orbit consists of two core blocks with charges
\((n/3,0)\) and \((0,n/3)\):
\begin{equation}
A_n^\sigma = \cC^\sigma\!\paren{\frac{n}{3},0}
\oplus \cC^\sigma\!\paren{0,\frac{n}{3}}.
\end{equation}
For the first block, \(x = (n/3)f_B\), so \(x^2 = (n/3)^2 f_B^2\)
and \(x^4 = (n/3)^4 f_B^4\).
For the second block, \(x = (n/3)f_{Qi}\), giving the same with
\(B\leftrightarrow Qi\). The sum produces:
\begin{align}
I_8[A_n^\sigma]
&= \sigma\bigg[
\frac{72}{5760}(7p_1^2 - 4p_2)
- \frac{1}{4}\,p_1\,a
+ \frac{1}{4}\,a^2
\nonumber\\
&\qquad
- \frac{n^2}{12}\,p_1\,(f_B^2+f_{Qi}^2)
+ \frac{n^2}{6}\,a\,(f_B^2+f_{Qi}^2)
+ \frac{n^4}{54}\,(f_B^4+f_{Qi}^4)
\bigg].
\label{eq:axial-poly-full}
\end{align}
Note that \emph{no mixed \(f_B^2 f_{Qi}^2\) or \(f_B f_{Qi}\) terms appear},
because the two blocks have charges in orthogonal Abelian directions.

\paragraph{Diagonal orbit \(D_n^\sigma\).}
The diagonal orbit consists of two core blocks with charges
\((n/3,n/3)\) and \((n/3,-n/3)\):
\begin{equation}
D_n^\sigma = \cC^\sigma\!\paren{\frac{n}{3},\frac{n}{3}}
\oplus \cC^\sigma\!\paren{\frac{n}{3},-\frac{n}{3}}.
\end{equation}
For the first block, \(x = (n/3)(f_B+f_{Qi})\), so
\begin{align}
x^2 &= (n/3)^2(f_B^2 + 2f_B f_{Qi} + f_{Qi}^2),
\\
x^4 &= (n/3)^4(f_B^4 + 4f_B^3 f_{Qi} + 6f_B^2 f_{Qi}^2
+ 4f_B f_{Qi}^3 + f_{Qi}^4).
\end{align}
For the second block, \(x = (n/3)(f_B-f_{Qi})\), giving
\begin{align}
x^2 &= (n/3)^2(f_B^2 - 2f_B f_{Qi} + f_{Qi}^2),
\\
x^4 &= (n/3)^4(f_B^4 - 4f_B^3 f_{Qi} + 6f_B^2 f_{Qi}^2
- 4f_B f_{Qi}^3 + f_{Qi}^4).
\end{align}
When the two blocks are summed:
\begin{enumerate}[label=\textbullet, leftmargin=1.4cm]
\item The \(f_B f_{Qi}\) terms in \(x^2\) cancel: \(+2-2=0\).
\item The \(f_B^3 f_{Qi}\) terms in \(x^4\) cancel: \(+4-4=0\).
\item The \(f_B f_{Qi}^3\) terms in \(x^4\) cancel: \(+4-4=0\).
\item The diagonal terms \(f_B^2\) and \(f_{Qi}^2\) in \(x^2\) add: \(1+1=2\).
\item The \(f_B^2 f_{Qi}^2\) terms in \(x^4\) add: \(6+6=12\).
\item The diagonal quartics \(f_B^4\) and \(f_{Qi}^4\) add: \(1+1=2\).
\end{enumerate}
Therefore:
\begin{align}
I_8[D_n^\sigma]
&= \sigma\bigg[
\frac{72}{5760}(7p_1^2 - 4p_2)
- \frac{1}{4}\,p_1\,a
+ \frac{1}{4}\,a^2
\nonumber\\
&\qquad
- \frac{n^2}{6}\,p_1\,(f_B^2+f_{Qi}^2)
+ \frac{n^2}{3}\,a\,(f_B^2+f_{Qi}^2)
\nonumber\\
&\qquad
+ \frac{n^4}{27}\,(f_B^4+f_{Qi}^4)
+ \frac{2n^4}{9}\,f_B^2 f_{Qi}^2
\bigg].
\label{eq:diagonal-poly-full}
\end{align}
All off-diagonal terms have canceled by the
\(q_{Qi}\to -q_{Qi}\) symmetry.

\subsection{Matching to the \BIsix{} target}

Using the shorthand \(b=f_B^2\), \(c=f_{Qi}^2\), \(g=\half p_1\),
the target (excluding the pure gravitational \(g^2\) term) is:
\begin{equation}
\frac{\Omega}{2}\bigg[
-2ga + a^2 - \frac{4}{5}\,g(b+c)
+ \frac{4}{5}\,a(b+c)
+ \frac{4}{25}\,(b^2+c^2)
+ \frac{8}{25}\,bc
\bigg].
\label{eq:target-gauge-mixed}
\end{equation}
From the reservoir
\(\cM = \bigoplus_n a_n A_n^+ \oplus \bigoplus_n d_n D_n^+\),
the coefficients are:
\begin{align}
\mathrm{coeff}(ga) &= -\half\!\paren{\sum_n a_n + \sum_n d_n}
= -\Omega,
\label{eq:match-ga}\\
\mathrm{coeff}(a^2) &= \frac{1}{4}\paren{\sum_n a_n + \sum_n d_n}
= \frac{\Omega}{2},
\label{eq:match-a2}\\
\mathrm{coeff}(g(b{+}c)) &= -\frac{1}{6}\sum_n n^2 a_n
- \frac{1}{3}\sum_n n^2 d_n
= -\frac{2\Omega}{5},
\label{eq:match-gbc}\\
\mathrm{coeff}(a(b{+}c)) &= \frac{1}{6}\sum_n n^2 a_n
+ \frac{1}{3}\sum_n n^2 d_n
= \frac{2\Omega}{5},
\label{eq:match-abc}\\
\mathrm{coeff}(b^2{+}c^2) &= \frac{1}{54}\sum_n n^4 a_n
+ \frac{1}{27}\sum_n n^4 d_n
= \frac{2\Omega}{25},
\label{eq:match-b2c2}\\
\mathrm{coeff}(bc) &= \frac{2}{9}\sum_n n^4 d_n
= \frac{4\Omega}{25}.
\label{eq:match-bc}
\end{align}
Equations \eqref{eq:match-ga} and \eqref{eq:match-a2} both require
\begin{equation}
\sum_n a_n + \sum_n d_n = 2\Omega.
\label{eq:system-1}
\end{equation}
Equations \eqref{eq:match-gbc} and \eqref{eq:match-abc} both require
\begin{equation}
\sum_n n^2 a_n + 2\sum_n n^2 d_n = \frac{12\Omega}{5}.
\label{eq:system-2}
\end{equation}
Equation \eqref{eq:match-bc} gives
\begin{equation}
\sum_n n^4 d_n = \frac{18\Omega}{25}.
\label{eq:system-4}
\end{equation}
Equation \eqref{eq:match-b2c2} combined with \eqref{eq:system-4}:
\begin{equation}
\frac{1}{54}\sum_n n^4 a_n + \frac{1}{27}\cdot\frac{18\Omega}{25}
= \frac{2\Omega}{25},
\end{equation}
so
\begin{equation}
\frac{1}{54}\sum_n n^4 a_n = \frac{2\Omega}{25} - \frac{18\Omega}{675}
= \frac{54\Omega-18\Omega}{675}
= \frac{36\Omega}{675}
= \frac{4\Omega}{75},
\end{equation}
hence
\begin{equation}
\sum_n n^4 a_n = \frac{4\Omega\times 54}{75} = \frac{216\Omega}{75}
= \frac{72\Omega}{25}.
\label{eq:system-3}
\end{equation}


\subsection{The solution}

The linear system \eqref{eq:system-1}--\eqref{eq:system-4} has five unknowns
\((a_1,a_2,d_1,d_2,d_3)\) and four equations, so one should expect an affine
one-parameter family rather than a unique point. Solving the system exactly gives
\begin{align}
a_1 &= -\frac{64}{3}d_3 + \frac{152\Omega}{75},\label{eq:affine-a1}\\
a_2 &= \phantom{-}\frac{4}{3}d_3 + \frac{4\Omega}{75},\label{eq:affine-a2}\\
d_1 &= \phantom{-}\frac{77}{3}d_3 - \frac{2\Omega}{15},\label{eq:affine-d1}\\
d_2 &= -\frac{20}{3}d_3 + \frac{4\Omega}{75}.\label{eq:affine-d2}
\end{align}
To preserve integrality when \(\Omega\in25\mathbb Z\), write
\[
\Omega=25k,\qquad d_3=2k+3m,\qquad k,m\in\mathbb Z.
\]
Then the affine family becomes
\begin{align}
a_1 &= 8k-64m,\label{eq:affine-int-a1}\\
a_2 &= 4k+4m,\label{eq:affine-int-a2}\\
d_1 &= 48k+77m,\label{eq:affine-int-d1}\\
d_2 &= -12k-20m,\label{eq:affine-int-d2}\\
d_3 &= 2k+3m.\label{eq:affine-int-d3}
\end{align}

\paragraph{Why the benchmark branch is chosen as \(m=0\).}
The benchmark used in the paper is the self-similar branch
\[
m=0,
\qquad
d_3=\frac{2\Omega}{25},
\]
for which all coefficients scale linearly with \(\Omega/25\):
\begin{equation}
\boxed{
a_1 = \frac{8\Omega}{25},\quad
a_2 = \frac{4\Omega}{25},\quad
d_1 = \frac{48\Omega}{25},\quad
d_2 = -\frac{12\Omega}{25},\quad
d_3 = \frac{2\Omega}{25}.
}
\label{eq:benchmark-branch}
\end{equation}
This is not the unique affine solution; it is the simplest scaling branch,
and it is best to state this explicitly rather than leave it implicit.

\paragraph{Row-reduction in full.}
Starting from
\begin{align}
a_1 + 16a_2 &= \frac{72\Omega}{25},\label{eq:rr1}\\
d_1 + 16d_2 + 81d_3 &= \frac{18\Omega}{25},\label{eq:rr2}\\
a_1 + 4a_2 + 2d_1 + 8d_2 + 18d_3 &= \frac{60\Omega}{25},\label{eq:rr3}\\
a_1 + a_2 + d_1 + d_2 + d_3 &= \frac{50\Omega}{25},\label{eq:rr4}
\end{align}
eliminate \(a_1\) using \eqref{eq:rr1}:
\[
a_1=\frac{72\Omega}{25}-16a_2.
\]
Substituting into \eqref{eq:rr3} and \eqref{eq:rr4} gives
\begin{align}
-12a_2 + 2d_1 + 8d_2 + 18d_3 &= -\frac{12\Omega}{25},\label{eq:rr5}\\
-15a_2 + d_1 + d_2 + d_3 &= -\frac{22\Omega}{25}.\label{eq:rr6}
\end{align}
Double \eqref{eq:rr6} and subtract from \eqref{eq:rr5}:
\[
18a_2 + 6d_2 + 16d_3 = \frac{32\Omega}{25},
\]
hence
\begin{equation}
3a_2 + d_2 + \frac{8}{3}d_3 = \frac{16\Omega}{75}.
\label{eq:rr7}
\end{equation}
Eliminate \(d_1\) from \eqref{eq:rr2}:
\[
d_1=\frac{18\Omega}{25}-16d_2-81d_3.
\]
Substitute this into \eqref{eq:rr6}:
\[
-15a_2 -15d_2 -80d_3 = -\frac{40\Omega}{25},
\]
or
\begin{equation}
a_2+d_2+\frac{16}{3}d_3=\frac{8\Omega}{75}.
\label{eq:rr8}
\end{equation}
Subtracting \eqref{eq:rr8} from \eqref{eq:rr7} yields
\[
2a_2-\frac{8}{3}d_3=\frac{8\Omega}{75},
\]
so
\[
a_2=\frac{4}{3}d_3+\frac{4\Omega}{75}.
\]
Then \eqref{eq:rr8} gives
\[
d_2=-\frac{20}{3}d_3+\frac{4\Omega}{75},
\]
and back-substitution into \eqref{eq:rr2} and \eqref{eq:rr1} produces
\eqref{eq:affine-d1} and \eqref{eq:affine-a1}. This is the complete row-reduction.

\paragraph{Negative multiplicity is really opposite chirality.}
A potential confusion is that \(d_2<0\) requires interpretation. The
right interpretation is not negative particle number. Because chirality flips
the sign of the six-dimensional anomaly polynomial,
\[
I_8[D_n^-]=-I_8[D_n^+].
\]
Therefore a negative coefficient multiplying \(D_n^+\) is equivalent to a
positive coefficient multiplying \(D_n^-\):
\[
d_2\,D_2^+ = |d_2|\,D_2^- \qquad\text{when }d_2<0.
\]
On the benchmark branch \eqref{eq:benchmark-branch}, the physical reservoir can
therefore be rewritten with nonnegative multiplicities as
\begin{equation}
\boxed{
\cM_{\mathrm{gauge}}^{\mathrm{phys}}(\Omega)
=
\frac{8\Omega}{25}A_1^{+}
\oplus
\frac{4\Omega}{25}A_2^{+}
\oplus
\frac{48\Omega}{25}D_1^{+}
\oplus
\frac{12\Omega}{25}D_2^{-}
\oplus
\frac{2\Omega}{25}D_3^{+}.
}
\label{eq:physical-benchmark-branch}
\end{equation}
This resolves the physical concern cleanly: the reservoir contains no
negative multiplicity, only a definite chirality pattern.

\paragraph{Minimum \(\Omegaorb\).}
The benchmark branch has coefficients proportional to \(\Omega/25\), so
integrality requires
\[
\Omegaorb\in25\mathbb Z.
\]
At the first static benchmark \(\Omegaorb=25\),
\[
(a_1,a_2,d_1,d_2,d_3)=(8,4,48,-12,2),
\]
equivalently
\[
(8,4,48,12,2)\quad\text{with the fourth orbit of negative chirality.}
\]

\bigskip
\noindent\textit{The gauge and mixed books now balance; gravity's
remain. Pure gravitational anomalies cannot be traded against gauge
choices --- they see only the count and the chirality of what
propagates --- so what follows is the most rigid accounting in the
paper: two curvature invariants, two species of field, one
$2 \times 2$ linear system, and no freedom at all.}

\section{Gravitational completion: detailed computation}
\label{app:grav-full}

The gauge-matter reservoir constructed in Appendix~\ref{app:diophantine-full} satisfies all
non-Abelian and mixed anomaly conditions. The purely gravitational
sector remains: the \(p_1^2\) and \(p_2\) coefficients must also
match. This appendix solves the \(2\times 2\) system for the
number of gauge-singlet anti-self-dual tensors \(n_{\mathrm{SD}}\)
and singlet fermions \(n_F\).

\subsection{From the index theorem to the anomaly polynomials: a
from-scratch derivation}

The two per-field polynomials used below are quoted everywhere in the
6D literature but rarely derived in one place at graduate resolution;
we derive them here. The textbook homes are
\citet{AlvarezGaumeWitten1984} (the original),
\citet{Nakahara2003} (Ch.~12, index theorems), and
\citet{Bilal2008} (\S5); the string-side deployment is
\citet{GSW1987} (Ch.~13).

\paragraph{Step 1: the two genera.} The gravitational anomaly of a
chiral field is governed by an index density: the \(\hat{A}\)-genus for spin-\(1/2\) fermions and the Hirzebruch
\(L\)-genus for (anti-)self-dual tensors. Both are characteristic
classes built from the curvature: a genus assigns to a manifold a
polynomial in its Pontryagin classes whose integral computes the index
of a differential operator on it. The \(\hat{A}\)-genus does this for
the Dirac operator and the \(L\)-genus for the signature operator, and
that is the whole of what is used below. Their expansions in Pontryagin classes are
\begin{align}
\hat{A} &= 1 \;-\; \frac{p_1}{24} \;+\; \frac{7 p_1^2 - 4 p_2}{5760}
\;+\; \cdots,
\label{eq:ahat-expansion}\\
L &= 1 \;+\; \frac{p_1}{3} \;+\; \frac{7 p_2 - p_1^2}{45}
\;+\; \cdots.
\label{eq:l-expansion}
\end{align}
Both follow from the defining products
\(\hat{A} = \prod_i \frac{x_i/2}{\sinh(x_i/2)}\) and
\(L = \prod_i \frac{x_i}{\tanh x_i}\)
by expanding to fourth order in the Chern roots \(x_i\) and
re-expressing the symmetric polynomials through
\(p_1 = \sum x_i^2\) and \(p_2 = \sum_{i<j} x_i^2 x_j^2\); the
\(5760 = 2^7 \cdot 3^2 \cdot 5\) is the honest denominator of the
fourth-order \(\sinh\) expansion, and the student should carry it
through once by hand.

\paragraph{Step 2: the fermion eight-form.} For a positive-chirality
Weyl fermion in six dimensions, the anomaly polynomial is the
degree-eight term of \(\hat{A}\) (times the gauge character, set to
one for a singlet):
\begin{equation}
I_8^{F}
= \bigl[\hat{A}\bigr]_8
= \frac{7 p_1^2 - 4 p_2}{5760}.
\label{eq:fermion-from-ahat}
\end{equation}

\paragraph{Step 3: the tensor eight-form.} For one chiral two-form,
the contribution is \(-\tfrac{1}{8}\) of the degree-eight term of the
\(L\)-genus, the normalization fixed by the signature theorem for the
(anti-)self-dual field strength \citep{AlvarezGaumeWitten1984}:
\begin{equation}
I_8^{\mathrm{T}}
= -\frac{1}{8}\bigl[L\bigr]_8
= -\frac{1}{8}\cdot\frac{7 p_2 - p_1^2}{45}
= \frac{p_1^2 - 7 p_2}{360}
= \frac{16 p_1^2 - 112 p_2}{5760}.
\label{eq:tensor-from-l}
\end{equation}

\paragraph{Physics intuition.} Why does one tensor ``count as
twenty-eight fermions''? Compare the \(p_2\) coefficients of
\eqref{eq:fermion-from-ahat} and \eqref{eq:tensor-from-l}:
\(-112/5760\) against \(-4/5760\), a ratio of exactly \(28\). Add the
one Weyl fermion that accompanies the tensor in a supergravity tensor
multiplet and one obtains the famous \(29\) of the
\(n_H - n_V + 29\,n_T = 273\) constraint \eqref{eq:sugra-273} --- which
the student can now regard as an exercise rather than an incantation.
The deeper point is the \emph{ratio mismatch}: the fermion produces
\(p_2\) and \(p_1^2\) in the fixed ratio \(-4 : 7\), the tensor in the
ratio \(-7 : 1\); because the target polynomial
\((\Omega/2) X_4^2\) contains no irreducible \(p_2\) at all, no
quantity of fermions alone can land on it, and the two-species
\(2 \times 2\) system solved below is forced. The same mismatch, one
dimension pair up, is what forces the gravitino's \(496\) in the
ten-dimensional mechanism (Appendix~\ref{app:gs10d}): an irreducible
curvature monomial always dictates a spectrum count.

\subsection{Anomaly polynomials for the gravitational sector}

\paragraph{Weyl fermion.}
For one positive-chirality gauge-singlet Weyl fermion, the
gravitational anomaly polynomial is (from the \(\hat A\)-genus):
\begin{equation}
I_8^{F} = \frac{7p_1^2 - 4p_2}{5760}.
\label{eq:fermion-grav}
\end{equation}

\paragraph{Chiral tensor.}
For one real chiral two-form tensor, the gravitational anomaly comes
from the Hirzebruch \(L\)-genus \citep{AlvarezGaumeWitten1984}. For the
tensor chirality aligned with the positive-chirality fermion convention
above, the contribution is
\begin{equation}
I_8^{\mathrm{T}} = \frac{16p_1^2 - 112p_2}{5760}
= \frac{p_1^2 - 7p_2}{360},
\label{eq:sd-grav}
\end{equation}
and the opposite (anti-self-dual) chirality contributes
\(-I_8^{\mathrm{T}}\). The ratio of the tensor's \(p_2\) coefficient to
the fermion's is \(-112/-4 = 28\); together with the one Weyl fermion of
a tensor multiplet, this is the origin of the familiar factor \(29\) in
the supergravity constraint \eqref{eq:sugra-273}. The tensor's
\(p_2/p_1^2\) ratio \(-7\) differs from the fermion's \(-4/7\); this
difference is what breaks the universal ratio and allows the
gravitational system to be solved.

\subsection{The gravitational deficit}

The gauge-matter reservoir at normalization \(\Omega\) contributes
\(144\Omega\) effective fermion dimensions.
Its gravitational polynomial is:
\begin{equation}
I_8^{\mathrm{gauge,grav}}
= \frac{144\Omega}{5760}(7p_1^2 - 4p_2)
= \frac{\Omega}{40}(7p_1^2 - 4p_2).
\label{eq:gauge-grav}
\end{equation}
The target is
\begin{equation}
I_8^{\mathrm{target,grav}} = \frac{\Omega}{2}\cdot\frac{p_1^2}{4}
= \frac{\Omega\,p_1^2}{8},
\label{eq:target-grav}
\end{equation}
since the gravitational slot of \(X_4\) is \(p_1/2\) in the
characteristic-class normalization fixed by the mixed-sector matching of
Appendix~\ref{app:diophantine-full} and by the integral-lift lemma of
Appendix~\ref{app:global-proof-dossier}.
The deficit is
\begin{equation}
\Delta I_8 = \frac{\Omega\,p_1^2}{8}
- \frac{\Omega}{40}(7p_1^2 - 4p_2)
= \Omega\paren{\frac{p_1^2}{8} - \frac{7p_1^2}{40} + \frac{p_2}{10}}
= \Omega\paren{-\frac{p_1^2}{20} + \frac{p_2}{10}}.
\label{eq:grav-deficit}
\end{equation}

\subsection{The \texorpdfstring{\(2\times 2\)}{2x2} system}

We need
\begin{equation}
n_F\,I_8^{F,-} + n_{\mathrm{SD}}\,I_8^{\mathrm{ASD}} = \Delta I_8,
\end{equation}
where both completion species are gauge singlets:
\(n_{\mathrm{SD}}\) anti-self-dual tensors, each contributing
\(I_8^{\mathrm{ASD}} = -\tfrac{1}{5760}(16p_1^2 - 112p_2)\)
(the chirality opposite to \eqref{eq:sd-grav}), and \(n_F\) Weyl
fermions of chirality opposite to the gauge reservoir, each
contributing \(I_8^{F,-} = -\tfrac{1}{5760}(7p_1^2 - 4p_2)\).
Matching the \(p_1^2\) and \(p_2\) coefficients of the deficit
\eqref{eq:grav-deficit} in units of \(1/5760\):
\begin{align}
-7\,n_F - 16\,n_{\mathrm{SD}} &= -288\,\Omega,
\label{eq:grav-eq1}\\
+4\,n_F + 112\,n_{\mathrm{SD}} &= 576\,\Omega.
\label{eq:grav-eq2}
\end{align}
Equation~\eqref{eq:grav-eq2} simplifies to
\(n_F + 28\,n_{\mathrm{SD}} = 144\,\Omega\), hence
\(n_F = 144\,\Omega - 28\,n_{\mathrm{SD}}\). Substituting into
\eqref{eq:grav-eq1}:
\begin{equation}
-7\,(144\,\Omega - 28\,n_{\mathrm{SD}}) - 16\,n_{\mathrm{SD}}
= -288\,\Omega
\quad\Longrightarrow\quad
180\,n_{\mathrm{SD}} = 720\,\Omega,
\end{equation}
so that
\begin{equation}
\boxed{\;
n_{\mathrm{SD}} = 4\,\Omega,
\qquad
n_F = 144\,\Omega - 112\,\Omega = 32\,\Omega.
\;}
\label{eq:grav-solution}
\end{equation}
Both counts are positive integers for every \(\Omega\in 25\mathbb Z\),
linear in \(\Omega\), and they satisfy the exact structural ratio
\begin{equation}
\frac{n_F}{n_{\mathrm{SD}}} = 8:
\end{equation}
exactly eight opposite-chirality singlet Weyl fermions accompany each
anti-self-dual tensor, independently of \(\Omega\). The chirality flip
of the completion sector relative to the gauge reservoir is the same
bookkeeping as the \(D_2^-\) orbit of the Diophantine solution
(Appendix~\ref{app:diophantine-full}): the reservoir over-produces the
\(p_1^2\) coefficient relative to the target
(\(7\Omega/40 > \Omega/8\)), so the completion must enter with the
opposite sign of both gravitational coefficients.

\paragraph{Accounting for the Green--Schwarz tensor itself.}
The count \(n_{\mathrm{SD}} = 4\,\Omega\) is the \emph{total}
chiral-tensor content required by the gravitational ledger, and it
includes the Green--Schwarz mediator \(B_2\) itself, which is
anti-self-dual at the quantum level (Proposition~\ref{prop:asd}) and
carries the same gravitational anomaly as each spectator. The
\(X_4\)-coupled channel remains unique and primitive (the tensor
clause of Definition~\ref{def:admissible-main}); the remaining
\(4\Omega - 1\) chiral tensors are gauge-singlet spectators carrying
gravitational anomaly only. At the minimal branch \(\Omega = 25\):
one mediator plus \(99\) spectators.

\paragraph{Verification.} In units of \(1/5760\), at general
\(\Omega\): the \(p_1^2\) coefficient is
\(-7\,(32\,\Omega) - 16\,(4\,\Omega) = -288\,\Omega
= 5760\times(-\Omega/20)\)~\(\checkmark\); the \(p_2\) coefficient is
\(+4\,(32\,\Omega) + 112\,(4\,\Omega) = 576\,\Omega
= 5760\times(\Omega/10)\)~\(\checkmark\).

For the minimal benchmark \(\Omega = 25\):
\begin{equation}
n_{\mathrm{SD}} = 100,
\qquad
n_F = 800,
\end{equation}
and for \(\Omega = 50\): \(n_{\mathrm{SD}} = 200\), \(n_F = 1600\).
At the minimal branch \(\Omega = L^2 = 25\) the tensor count is the
perfect square \(n_{\mathrm{SD}} = 4L^2 = (2L)^2 = 100\); whether this
reflects a deeper \OGN{} structure is an open question. Because
\(n_{\mathrm{SD}} = 4\,\Omega\), the tensor count is divisible by \(8\)
exactly when \(\Omega\) is even: the \(\Omega = 50\) branch is
manifestly signature-friendly
(Appendix~\ref{app:global-proof-dossier}), while on the
\(\Omega = 25\) branch the tensor-sector phase vanishes identically on
the intended-class fillings, where \(\sigma(X_6 \times D^2) = 0\)
(Lemma~\ref{lem:computed-phase}).

\bigskip
\noindent\textit{Everything so far is local: the ledger balances point
by point on the bulk. But an anomaly can hide where no local
computation looks --- in the holonomy the theory picks up when carried
around a loop of backgrounds, an eigenvalue slipping through zero far
from any Feynman diagram. The final constructive appendix chases that
last hiding place with the eta invariant, and closes it on the
intended class.}

\section{Global anomaly proof and automatic admissibility}
\label{app:global-proof-dossier}

The preceding appendices have established the local anomaly cancellation:
the descent proof (\ref{app:descent-full}), the rank-one factorization and coefficient
matching (\ref{app:matrix-and-coefficients}), the \OGNfive{} derivation
(\ref{app:ogn5-complete}), the core block and Diophantine spectrum
(\ref{app:core-block-full}--\ref{app:diophantine-full}), and the
gravitational completion (\ref{app:grav-full}).
Together they discharge proof obligations (PO1)--(PO6) of
Section~\ref{sec:eight-parts}.

The present appendix addresses the remaining obligations
(PO7)--(PO8): global gauge anomalies and the well-definedness of
the exponentiated Green--Schwarz term on the charge lattice.
The proof upgrades the local \BIsix{} cancellation to a full
global theorem on the admissible \GG{} background class.


\subsection{A primer on the eta invariant and the Dai--Freed
formalism (for the graduate reader)}

The global analysis below uses machinery that most graduate curricula
postpone; this primer supplies the working minimum. The modern
references are \citet{WittenRMP2016} (the physicist's treatment),
\citet{GarciaEtxebarriaMontero2019} (the same machinery deployed on
particle-physics gauge groups), \citet{DaiFreed1994} (the original
determinant-line construction), and \citet{Nakahara2003} (Ch.~12) for
the index-theory background.

\paragraph{The eta invariant.} For a self-adjoint Dirac operator
\(D\) on a closed odd-dimensional manifold, define
\begin{equation}
\eta(s) = \sum_{\lambda \neq 0} \mathrm{sign}(\lambda)\,
|\lambda|^{-s},
\qquad
\eta \equiv \eta(0)_{\mathrm{reg}},
\end{equation}
the regularized spectral asymmetry: it counts how lopsided the
spectrum is between positive and negative eigenvalues. The
Atiyah--Patodi--Singer theorem ties it to an index on a bounding
manifold: for \(\partial W = U\),
\begin{equation}
\mathrm{index}\, D_W
= \int_W \hat{A} \wedge \mathrm{ch}(F)
\;-\; \frac{\eta(U) + h(U)}{2},
\label{eq:aps-primer}
\end{equation}
with \(h\) the number of zero modes. Rearranged, this is exactly the
statement used by Lemma~\ref{lem:computed-phase}: the combination
\(\int_W I - \xi(U)\) (with \(\xi = (\eta + h)/2\)) is an
\emph{integer}, so its exponential is trivial.

\paragraph{A one-line worked example.} Take the free fermion on a
circle with Wilson-line twist \(\theta\): eigenvalues
\(\lambda_n = n + \theta/2\pi\), \(n \in \mathbb{Z}\). A short
computation gives \(\eta = 1 - \theta/\pi\) (mod even integers) for
\(0 < \theta < 2\pi\): as the twist winds once, one eigenvalue flows
through zero and \(\eta\) jumps by \(2\). This \emph{spectral flow} in
miniature is the entire content of the global anomaly: transporting
the theory around a loop in background space can slide eigenvalues
through zero, and each crossing flips the sign of the fermion
determinant. The mapping torus construction below is this example
writ large --- the twist \(\theta\) becomes a large gauge or
diffeomorphism transformation, the circle becomes the loop in the
space of backgrounds, and \(\exp(2\pi i \xi)\) is the accumulated
phase.

\paragraph{The Dai--Freed prescription.} On manifolds with boundary
the chiral partition function is not a number but an element of a
determinant \emph{line}; \citet{DaiFreed1994} showed that
\(\exp(2\pi i \xi(U))\) of a bounding \(U\) supplies the canonical
trivialization, making the combined bulk-boundary object well defined
whenever the APS combination \eqref{eq:aps-primer} is used. The
practical recipe for the physicist, and the one Appendix sections
below implement: to test a large transformation, build its mapping
torus \(U_7\), fill it with any \(W_8\) carrying the extended data,
and evaluate \(\exp[2\pi i(\int_{W_8} I_8 - \xi(U_7))]\); consistency
requires the answer to be filling-independent (the Wu-shift lemma)
and equal to one (the cancellation lemmas).

\paragraph{Physics intuition.} The local anomaly is a statement about
the \emph{curvature} of the anomaly line over the space of
backgrounds; the global anomaly is a statement about its
\emph{holonomy}. Green--Schwarz cancellation flattens the line ---
that is the local theorem. A flat line can still have holonomy around
non-contractible loops --- that is the residual global question, and
it is answered on the intended class by
Lemma~\ref{lem:computed-phase} (every allowed loop is contractible,
so the flat line has no holonomy to show) and routed to Open
Problem~O10 off it.

\subsection{What exactly has to be proved?}

The phrase \emph{complete anomaly freedom} means more than the statement that the descent polynomial vanishes locally. For a six-dimensional chiral theory one must distinguish three layers.

\begin{enumerate}[label=\textbf{Layer \arabic*.},leftmargin=1.8cm]
\item \textbf{Infinitesimal / local anomaly.} Under small gauge or local Lorentz transformations the one-loop effective action varies by the six-form descent term \(I_6^{(1)}\). \BIsix{} cancels this part exactly if the anomaly polynomial factorizes as \(I_8=\frac12 X_4^2\).
\item \textbf{Large gauge / diffeomorphism anomaly.} Around a non-contractible loop in the space of backgrounds, the partition function can return with a nontrivial phase even when the local descent variation vanishes.
\item \textbf{Self-dual / anti-self-dual tensor global phase.} In six dimensions a chiral two-form is not described globally by the naive local action \(\int B_2\wedge X_4\); it needs a quadratic refinement / Wu-shifted differential-cocycle definition.
\end{enumerate}

In other words, the target theorem is:
\begin{quote}
\emph{for every admissible large transformation of every admissible \GG{} background, the total exponentiated path integral returns with phase one.}
\end{quote}

That is stronger than the local theorem, and the distinction is essential.

\paragraph{Dimensional ladder.}
The correct dimensional hierarchy for the global anomaly is:
\[
\text{6D theory on } M_6 
\quad\Longrightarrow\quad
\text{7D mapping torus } U_7
\quad\Longrightarrow\quad
\text{8D filling } W_8.
\]

Large gauge or diffeomorphism transformations of the 6D theory are encoded by the closed mapping torus \(U_7\). To evaluate the anomaly phase one chooses an 8-manifold \(W_8\) with \(\partial W_8 = U_7\) and extends the spin/gauge/tensor data to \(W_8\). That is the correct modern APS/Dai--Freed setting.

In short: the global anomaly is a seven-dimensional holonomy problem and an eight-dimensional filling problem, not a direct six-dimensional descent problem.

\noindent\textit{The local descent proof is given in full in Appendix~\ref{app:descent-full} above.}

\subsection{Admissible background class and hypotheses H1--H4}

To close the global proof cleanly, we must state the precise hypotheses. We label them \(H1\)--\(H4\) and sharpen each one.

\begin{definition}[Admissible \GG{} background class]
An admissible background consists of:
\begin{enumerate}[leftmargin=1.6cm,label=(\alph*)]
\item a compact Euclideanized six-manifold \(M_6\) obtained either by doubling the interval geometry \(M_4\times I_r\times S^1_\theta\) or by imposing APS-compatible boundary conditions at the ends of \(I_r\);
\item a spin structure on \(M_6\);
\item principal bundles for \(\SUfive\), \(\UoneB\), and \(\UoneQi\);
\item a tensor-sector differential cocycle \(\checkB\) whose modified field strength has curvature representative \(X_4\).
\end{enumerate}
\end{definition}

\noindent This abstract definition gives the minimal data needed to formulate the conditional global theorem below. The concrete physical realization is the eight-clause class \(\AdmissibleClass\) defined in Definition~\ref{def:admissible-main} of the main text; the automatic admissibility lemmas in Section~\ref{app:auto-admissibility} below prove that every background in \(\AdmissibleClass\) satisfies hypotheses (H1)--(H4) automatically.

\begin{enumerate}[label=(H\arabic*),leftmargin=1.8cm]
\item \textbf{Bordism / extension hypothesis.} For every admissible large gauge or diffeomorphism transformation of an admissible background, the associated mapping torus \(U_7\) represents the trivial class in the relevant spin-gauge bordism group, so there exists a spin eight-manifold \(W_8\) with \(\partial W_8=U_7\) to which the gauge and tensor data extend.

\item \textbf{No independent Witten-type gauge obstruction.} The gauge group is exactly \(\SUfive\times\UoneB\times\UoneQi\), and
\[
\pi_6(\SUfive)=0,
\qquad
\pi_6(\UoneB)=0,
\qquad
\pi_6(\UoneQi)=0.
\]
Hence there is no separate six-dimensional homotopy anomaly of the Witten type.

\item \textbf{Tensor-lattice / quadratic-refinement hypothesis.} The lattice-valued lift of the \BIsix{} four-form obeys the correct shifted integrality law, equivalently the exponentiated Wu--Chern--Simons functional is single-valued. On every closed eight-manifold \(Z_8\),
\begin{equation}
\frac12 \int_{Z_8} \bigl(X_\Lambda\cup X_\Lambda - X_\Lambda\cup \lambda \bigr) \in \mathbb Z,
\label{eq:quadratic-integrality}
\end{equation}
where \(X_\Lambda\) is the lattice-valued lift of the \BIsix{} cocycle and \(\lambda\) is the appropriate Wu / characteristic lift.

\item \textbf{Differential-cocycle hypothesis.} The factorizing four-form \(X_4\) lifts to a globally defined degree-4 differential cocycle \(\checkX\) on all admissible mapping tori and their fillings. This is the global version of the local identity \(dH_3=X_4\).
\end{enumerate}

\begin{remark}
(H1) and (H4) are the global geometric hypotheses. (H2) removes an independent gauge-homotopy obstruction. (H3) is the crucial quantum statement: it is what makes the chiral two-form / Green--Schwarz functional globally well-defined instead of merely locally meaningful.
\end{remark}

\subsection{The anomaly line and the mapping-torus formulation}

Let \(\mathcal B\) be the space of admissible \GG{} background fields and \(\mathcal G\) the group of gauge / local Lorentz / diffeomorphism transformations. The chiral matter partition function is not a function on \(\mathcal B/\mathcal G\); it is a section of an anomaly line bundle
\[
\cL_{\an} \longrightarrow \mathcal B/\mathcal G.
\]
Its curvature is the local anomaly polynomial, while its holonomy around a loop is the global anomaly phase.

A large transformation defines a loop \(\gamma\subset \mathcal B/\mathcal G\). The corresponding global anomaly is
\begin{equation}
\hol_{\cL_{\an}}(\gamma) = \cA_\chi(U_7),
\end{equation}
where \(U_7\) is the mapping torus of the 6D background under that transformation.

The APS / Dai--Freed form of the anomaly phase can be written schematically as
\begin{equation}
\cA_\chi(U_7)
=
\exp\Bigl[2\pi i\Bigl(\int_{W_8} I_8 - \xi_\chi(U_7)\Bigr)\Bigr].
\label{eq:DaiFreed-schematic}
\end{equation}
Here \(W_8\) is any filling of \(U_7\), and \(\xi_\chi(U_7)\) denotes the reduced eta-invariant / torsion correction carried by the seven-dimensional boundary operator. The exact normalization of \(\xi_\chi\) is not the important issue here. The important issue is that the global anomaly contains \emph{two} kinds of terms:
\begin{enumerate}[leftmargin=1.6cm]
\item an integral \(\int_{W_8} I_8\), which is the extension of the local descent data;
\item a torsion / eta piece \(\xi_\chi\), which is invisible to the naive local formula \(\int B_2\wedge X_4\).
\end{enumerate}

This is exactly why the global Green--Schwarz term must be written as a differential-cohomology / Wu--Chern--Simons object.

\subsection{The globally defined \BIsix{} counterterm}

\paragraph{Why the local expression is not enough.}
The local expression
\[
\frac12\int_{M_6} B_2\wedge X_4
\]
is perfectly adequate on a topologically trivial patch. But under large gauge transformations or on a manifold with nontrivial cohomology it is not, by itself, a globally defined number. The chiral two-form is a higher gauge field, not an ordinary globally defined 2-form everywhere.

The correct global object is a \emph{Wu--Chern--Simons} / differential-cohomology functional on the mapping torus.

\paragraph{Definition of the global phase.}
Let \(\checkX\) be the degree-4 differential cocycle lifting \(X_4\). On a closed mapping torus \(U_7\) bounding \(W_8\), define
\begin{equation}
S_{\WCS}(U_7;\checkX)
:=
\frac12\int_{W_8}
\bigl(\checkX\cup\checkX - \checkX\cup\checklambda \bigr)
\quad \in \mathbb R/\mathbb Z.
\label{eq:WCS-def}
\end{equation}
The exponentiated \BIsix{} counterphase is then
\begin{equation}
\cA_{\GS}(U_7)
:=
\exp\bigl[-2\pi i\,S_{\WCS}(U_7;\checkX)\bigr].
\label{eq:AGS-def}
\end{equation}

Equation \eqref{eq:WCS-def} is the global replacement for the local \BIsix{} term. The cup-product notation is deliberate: global anomaly cancellation is a statement in differential cohomology, not only in differential forms.

\begin{remark}
On topologically simple backgrounds, or on an admissible class where the Wu shift trivializes, \eqref{eq:WCS-def} reduces to the familiar form
\[
\cA_{\GS}(U_7) \sim \exp\Bigl[-2\pi i\,\frac12\int_{W_8} X_4^2\Bigr].
\]
This is the fully global formula. The full \(\checklambda\)-shift is what makes the expression globally meaningful.
\end{remark}

\subsection{Detailed proof of global anomaly cancellation}

We now prove the theorem rigorously on the admissible class.

\begin{theorem}[Complete \BIsix{} anomaly freedom on the admissible class]
\label{thm:main}
Assume the admissible \GG{} background class satisfies (H1)--(H4). Then for every admissible large gauge transformation, local Lorentz transformation, or diffeomorphism, the combined holonomy of the chiral determinant and the exponentiated \BIsix{} counterterm is trivial:
\begin{equation}
\cA_{\tot}(U_7)
:=
\cA_\chi(U_7)\,\cA_{\GS}(U_7)
=
1.
\end{equation}
Hence \BIsix{} is completely anomaly-free on the admissible \GG{} background class.
\end{theorem}

The proof will be split into four lemmas.

\paragraph{Lemma 1: the \BIsix{} global counterterm is well-defined}

\begin{lemma}
Under (H1), (H3), and (H4), the phase \(\cA_{\GS}(U_7)\) defined in \eqref{eq:AGS-def} is independent of the chosen filling \(W_8\).
\end{lemma}

\begin{proof}
Suppose \(W_8\) and \(W_8'\) are two fillings of the same mapping torus \(U_7\), both carrying the extended cocycle \(\checkX\). Glue them along the common boundary with reversed orientation to obtain a closed eight-manifold
\[
Z_8 = W_8 \cup_{U_7} (-W_8').
\]
The ratio of the two exponentiated actions is
\begin{align}
\frac{\exp[-2\pi i S_{\WCS}(U_7;\checkX)]_{W_8}}
 {\exp[-2\pi i S_{\WCS}(U_7;\checkX)]_{W_8'}}
&=
\exp\Bigl[-2\pi i\,\frac12\int_{Z_8}
\bigl(X_\Lambda\cup X_\Lambda - X_\Lambda\cup\lambda\bigr)\Bigr].
\end{align}
By hypothesis (H3), the exponent is an integer. Therefore the ratio is one. So the phase is independent of the filling and is globally well-defined.
\end{proof}

\paragraph{Lemma 2: the non-torsion integral part cancels exactly}

\begin{lemma}
Under \(I_8=\frac12 X_4^2\), the extension-dependent integral part of the chiral anomaly phase is canceled exactly by the extension-dependent integral part of the \BIsix{} counterterm.
\end{lemma}

\begin{proof}
The chiral determinant contributes the factor
\[
\exp\Bigl[2\pi i\int_{W_8} I_8\Bigr]
=
\exp\Bigl[2\pi i\int_{W_8} \frac12 X_4^2\Bigr].
\]
The global \BIsix{} counterterm contributes
\[
\exp\Bigl[-2\pi i\int_{W_8} \frac12 X_4^2\Bigr]
\times
\exp\bigl[\text{Wu/torsion correction}\bigr].
\]
Multiplying the first factors gives one. Thus the entire non-torsion integral contribution cancels identically. This is the global lift of the local factorization theorem.
\end{proof}

\paragraph{Lemma 3: there is no independent Witten-type gauge obstruction}

\begin{lemma}
Under (H2), there is no independent six-dimensional Witten-type global gauge anomaly for the \GG{} gauge group.
\end{lemma}

\begin{proof}
For a six-dimensional chiral gauge theory, an independent Witten-type global gauge anomaly is classified by a homotopy class in \(\pi_6(G)\). In \GG{},
\[
G = \SUfive\times\UoneB\times\UoneQi.
\]
Now Bott periodicity gives \(\pi_6(\SU(n))=0\) for \(n\ge 4\), hence
\[
\pi_6(\SUfive)=0.
\]
For \(\mathrm U(1)\), one has \(\pi_k(\mathrm U(1))=0\) for all \(k>1\), so
\[
\pi_6(\UoneB)=0,
\qquad
\pi_6(\UoneQi)=0.
\]
Because the homotopy group of a direct product is the direct sum of the homotopy groups of the factors, \(\pi_6(G)=0\). Hence there is no independent Witten-type gauge phase beyond the anomaly-field-theory phase already encoded by the mapping torus formalism.
\end{proof}

\paragraph{Lemma 4: the eta-invariant / tensor-sector phase is exactly the quadratic-refinement phase}

\begin{lemma}
Under (H3) and (H4), the residual eta-invariant / torsion phase of the chiral determinant is exactly the phase canceled by the Wu--Chern--Simons / self-dual-tensor quadratic refinement.
\end{lemma}

\begin{proof}
The local form \(\frac12\int B_2\wedge X_4\) knows only about the de Rham class of \(X_4\). The missing data are the torsion and half-integral shifts that arise for a chiral two-form. These are encoded mathematically by the quadratic refinement of the cup product. In modern language, the anomaly field theory of the self-dual tensor is a Wu--Chern--Simons theory; its partition function on \(U_7\) carries precisely the quadratic-refinement phase that trivializes the Dai--Freed eta phase of the chiral matter sector.

Concretely, hypothesis (H3) says that the lattice-valued cocycle \(X_\Lambda\) is chosen so that
\[
\frac12\int_{Z_8} (X_\Lambda\cup X_\Lambda - X_\Lambda\cup\lambda)\in\mathbb Z
\]
on all closed eight-manifolds. This is exactly the integrality condition that makes the quadratic refinement single-valued. Therefore the residual phase of the self-dual / anti-self-dual tensor sector is not an extra uncanceled anomaly; it is the missing piece required to cancel the eta-invariant contribution of the chiral determinant. Hence the residual torsion phase cancels.
\end{proof}

\begin{remark}[Methodological scope of Lemma~4]
The argument uses the general Hopkins--Singer / Monnier--Moore framework \citep{HopkinsSinger2005,MonnierMoore2019} at the structural level: the quadratic-refinement machinery guarantees cancellation whenever \((H3)\) and \((H4)\) hold. What is not done in this paper is an explicit computation of the anomaly field theory's partition function on every relevant mapping torus of the non-supersymmetric \BIsix{} spectrum---the kind of case-by-case verification that Monnier and Moore carry out for \(\cN=(1,0)\) SUGRA.
For the intended \GG{} class, this additional verification is not strictly required, because the admissible mapping tori are topologically simple enough (product structure, null-homotopic clutching, simply connected base) that the structural argument is sufficient. A fully explicit non-SUSY anomaly-field-theory computation, parallel to the Monnier--Moore treatment but without a multiplet structure, would constitute a valuable independent confirmation and a productive direction for collaboration with the six-dimensional anomaly community.
\end{remark}

\begin{lemma}[Computed phase on the intended class]
\label{lem:computed-phase}
On \(\AdmissibleClass\), the combined global phase of every allowed
large transformation equals \(1\) exactly.
\end{lemma}

\begin{proof}
Two independent routes. \emph{Route A (flat holonomy).} The combined
anomaly line \(\cL_{\an} \otimes \cL_{\mathrm{GS}}\) over the space of
backgrounds is globally defined (Lemma 1) and \emph{flat}: its
curvature is the local anomaly two-form, which vanishes identically by
the exact Green--Schwarz cancellation of
Theorem~\ref{thm:bi6-cancellation}. Every allowed transformation is
isotopic to the identity (clause (C6)), so the loop it defines in the
space of backgrounds is contractible, the isotopy providing the
contracting disk over which the background data extend
(product/collar clauses). The holonomy of a flat line around a
contractible loop is \(1\). \emph{Route B (APS index on the product
filling).} The doubled mapping torus factorizes as \(\Useven \cong
X_6 \times S^1\) with bounding structure and bounds \(\Weight = X_6
\times D^2\) with all data extending (Lemma for H1). For each chiral
fermion species, the Atiyah--Patodi--Singer index theorem gives
\(\int_{\Weight} I_8^{(s)} - \xi_s(\Useven) = \mathrm{index}\,
D^{(s)}_{\Weight} \in \mathbb{Z}\), so its Dai--Freed factor is
\(\exp(2\pi i \cdot \mathbb{Z}) = 1\). For the Green--Schwarz sector,
the Wu--Chern--Simons functional is filling-independent (Lemma 1) and
its cocycle on \(\Weight\) is a pullback from \(X_6\), so no torsion
pairing survives. For the chiral-tensor sector,
\(\sigma(X_6 \times D^2) = 0\): every class in
\(H^4(\Weight;\mathbb{R})\) is a pullback \(\pi^*\alpha\) from
\(X_6\), and \(\pi^*\alpha \cup \pi^*\beta = \pi^*(\alpha \cup
\beta)\) with \(\alpha \cup \beta \in H^8(X_6) = 0\); the relative
intersection form therefore vanishes identically, and the
signature-type phase \(\exp(2\pi i\, n_{\mathrm{SD}}\,
\sigma(\Weight)/8)\) equals \(1\) at \emph{every}
\(\Omega \in 25\mathbb{Z}\) --- including the \AX{4}-minimal branch
\(\Omega = 25\), where the mod-8 shortcut is unavailable. The two
routes agree, and no residual phase remains.
\end{proof}

\begin{remark}
Lemma~\ref{lem:computed-phase} upgrades Lemmas~2--4 from structural to
computed on the intended class: the eta content of the Dai--Freed
formula is evaluated (it is an integer index for every allowed loop),
and the tensor phase is evaluated (the filling signature vanishes).
What remains genuinely open is exactly the complement --- the winding,
mixed-clutching, and non-isotopic classes excluded by clause (C6) ---
which is the content of Open Problem O10.
\end{remark}

\paragraph{Completion of the proof of Theorem \ref{thm:main}}

\begin{proof}[Proof of Theorem \ref{thm:main}]
Take any admissible large gauge/diffeomorphism transformation and form its mapping torus \(U_7\).

\begin{enumerate}[leftmargin=1.6cm]
\item By Lemma 1, the exponentiated \BIsix{} counterterm is globally well-defined on \(U_7\).
\item By Lemma 2, the extension-dependent integral part of the chiral determinant cancels exactly against the extension-dependent integral part of the \BIsix{} counterterm because \(I_8=\frac12 X_4^2\).
\item By Lemma 3, there is no independent Witten-type six-dimensional gauge obstruction because \(\pi_6(G)=0\).
\item By Lemma 4, the residual eta-invariant / torsion phase is precisely the quadratic-refinement phase canceled by the Wu--Chern--Simons definition of the tensor sector.
\end{enumerate}

Therefore no residual phase remains. Hence
\[
\cA_{\tot}(U_7)=1
\]
for every admissible mapping torus. The anomaly line has trivial holonomy on the admissible class, so the combined exponentiated path integral is globally well-defined and gauge invariant. This is complete anomaly freedom on the admissible \GG{} background class.
\end{proof}

The four lemmas exhaust all known sources of global anomaly in six dimensions: the extension-dependent integral (Lemma 2), independent Witten-type homotopy (Lemma 3), and the residual eta/torsion phase (Lemma 4), with Lemma 1 ensuring that the counterterm is well-defined in the first place.

\subsection{Specialization, scope, and reference data}

\paragraph{The \GG{} gauge group and the vanishing of \texorpdfstring{$\pi_6$}{pi6}}
The \GG{} gauge group is constitutionally fixed to
\[
\SUfive\times\UoneB\times\UoneQi.
\]
This is unusually favorable for the global-anomaly proof, because the dangerous six-dimensional homotopy groups vanish exactly as required in Lemma 3.

\paragraph{Why the \BIsix{} coefficients are compatible with integrality.}
In integral characteristic-class normalization, define
\begin{equation}
\frac{X_4}{8\pi^2}
=
\frac12 p_1(TM_6) - c_2(E_5) - \frac{2}{5}c_1(L_B)^2 - \frac{2}{5}c_1(L_{Qi})^2.
\label{eq:X4-integral-norm}
\end{equation}
Because \(M_6\) is spin, \(\lambda:=p_1/2\in H^4(M_6,\mathbb Z)\). Therefore
\begin{equation}
5\frac{X_4}{8\pi^2}
=
5\lambda - 5c_2(E_5) - 2c_1(L_B)^2 - 2c_1(L_{Qi})^2
\in H^4(M_6,\mathbb Z).
\label{eq:5X4-integral}
\end{equation}
So the \BIsix{} form is not arbitrarily fractional. Its only denominator is exactly the denominator that is killed by multiplying by five.

This is why the present paper naturally arrives at the arithmetic condition
\[
\Omega \in 25\mathbb Z.
\]
The factor of \(25\) is the square of the denominator that must be absorbed when one exponentiates the quadratic GS/WCS action.

\paragraph{The particularly clean benchmark \texorpdfstring{$\Omega=50$}{Omega=50}}
The present paper identifies a distinguished benchmark with
\[
\Omega = 50,
\qquad
n_{SD}=200,
\qquad
n_F=1600.
\]
For this benchmark the model-level anomaly polynomial is
\[
I_8^{\tot} = \frac{50}{2}X_4^2 = 25X_4^2.
\]
Since \(5X_4\) is integral by \eqref{eq:5X4-integral}, the non-torsion part is especially transparent:
\[
25X_4^2 = (5X_4)^2.
\]
So the form-level integrality is manifest.

The same benchmark also makes the pure signature part of the self-dual completion especially clean. A single self-dual tensor carries a signature-type phase of the schematic form
\[
\exp\Bigl(2\pi i\,\frac{\sigma(W_8)}{8}\Bigr).
\]
For \(200\) chiral tensors this becomes
\[
\exp\Bigl(2\pi i\,200\,\frac{\sigma(W_8)}{8}\Bigr)
=
\exp\bigl(2\pi i\,25\,\sigma(W_8)\bigr)
=
1.
\]
So the benchmark chosen in the present paper is not only locally consistent; it is also globally friendly.

The benchmark values \(\Omega=50\) and \(n_{\mathrm{SD}}=200\) are a concrete explicit branch on which the global anomaly phase closes cleanly; whether they are the unique or minimal solution is a separate question.

\paragraph{What has now really been proved.}

This appendix proves a complete cancellation theorem \emph{on the admissible background class}. That is already a real theorem and not merely a heuristic. But it is equally important to state what remains open.

Three items remain outside the scope of the present theorem.
First, the paper does not prove that every non-supersymmetric \GG{} background (beyond the intended class) defines a mapping torus in the trivial spin-gauge bordism class required by (H1).
Second, the full differential-cocycle / Wu-shift formalism is applied at the structural level; a self-contained exposition within the paper would strengthen the presentation.
Third, uniqueness of the tensor lattice and Wu shift used in the benchmark is not established.
The cancellation argument on the intended class is complete; extending it to the full non-supersymmetric \GG{} category is the main open problem.

\paragraph{Self-contained theorem formulation.}

Stated in a form independent of the surrounding construction, the theorem reads as follows.

\begin{theorem}[Self-contained \BIsix{} global-anomaly theorem]
For the \GG{} bulk with gauge group \(\SUfive\times\UoneB\times\UoneQi\) and \BIsix{} four-form \(X_4\) satisfying \(I_8=\frac12 X_4^2\), the perturbative local anomaly is canceled exactly by the Green--Schwarz mechanism. Moreover, on the admissible class of spin mapping-torus backgrounds for which the gauge/tensor data extend to eight-manifold fillings and the \BIsix{} cocycle satisfies the required shifted integrality / quadratic-refinement condition, the combined phase of the chiral determinant and the exponentiated \BIsix{} counterterm is globally well-defined and gauge invariant. Hence \BIsix{} is completely anomaly-free on that admissible class.
\end{theorem}

That statement is mathematically precise and physically clear.

\paragraph{Homotopy groups used in the proof.}

The proof of Lemma 3 used only three standard facts:
\[
\pi_6(\SU(n))=0\quad (n\ge 4),
\qquad
\pi_k(\mathrm U(1))=0\quad (k>1),
\qquad
\pi_6(G_1\times G_2)=\pi_6(G_1)\oplus\pi_6(G_2).
\]
Thus the \GG{} gauge group has no independent six-dimensional Witten anomaly.

\paragraph{Local versus global theorem.}
Table~\ref{tab:local-vs-global} separates what each settles.

\begin{center}
{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.32\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}}
\caption[Local and global anomaly theorems, question by question.]{What the local and the global anomaly theorems each do and do not settle. The rows are the questions one must answer to claim a theory is anomaly-free; the columns are the two theorems. They are commonly conflated, and the conflation is dangerous in both directions --- a theory can pass the local test and fail the global one, and the global test is silent about the perturbative question the local one answers. The table separates them so that the claim made in this Part can be read as the specific claim it is.}\label{tab:local-vs-global}\\

\toprule
\textbf{Question} & \textbf{Local theorem} & \textbf{Global theorem}\\
\midrule
\endfirsthead
\toprule
\textbf{Question} & \textbf{Local theorem} & \textbf{Global theorem}\\
\midrule
\endhead
Object & descent polynomial \(I_8\to I_6^{(1)}\) & mapping-torus holonomy on \(U_7\)\\
Counterterm & \(\frac12\int B_2\wedge X_4\) & Wu--Chern--Simons / differential cocycle\\
Main condition & \(I_8=\frac12 X_4^2\) & \(I_8=\frac12 X_4^2\) plus (H1)--(H4)\\
Potential obstruction & none once descent closes & eta invariant, torsion phase, lattice quantization, bordism\\
Status in the present paper & exact & exact on the admissible class ((H1)--(H4) discharged in Section~\ref{app:auto-admissibility})\\
\bottomrule
\end{longtable}}
\end{center}

\subsection{Automatic admissibility on the intended \GG{} class}
\label{app:auto-admissibility}

The following appendix integrates the two later closure theorems that are most
important for the present paper: first, the proof that the admissibility
hypotheses are automatic on the intended \GG{} background category; second, the
proof that the \OGNfive{} integers are theorem-level consequences of \BIsix{} rather than
external numerological inputs.



\paragraph{The main missing implication.}
The complete global theorem is naturally stated on an \emph{admissible \GG{} background class}. The proof requires four hypotheses:
\begin{enumerate}[label=(H\arabic*),leftmargin=1.6cm]
\item every relevant seven-dimensional mapping torus bounds an eight-manifold with extending spin/gauge/tensor data;
\item no independent Witten-type six-dimensional gauge obstruction survives;
\item the tensor counterterm is correctly quantized by the required quadratic refinement / Wu shift;
\item the \BIsix{} four-form has a differential-cocycle lift.
\end{enumerate}
The remaining task is not to re-prove cancellation. It is to prove that those four hypotheses are automatic once one says carefully what counts as an intended \GG{} background.


\paragraph{Definition of the intended \GG{} category.}
The needed definition is more precise than the informal phrase ``non-supersymmetric \GG{} background.''

\begin{definition}[Physical intended \GG{} background category]
The intended non-supersymmetric \GG{} background category \(\AdmissibleClass\) consists of data
\[
(\Msix,\mathcal P_5,L_B,L_{Qi},\check B_2)
\]
satisfying the following clauses.
\begin{enumerate}[label=(C\arabic*),leftmargin=1.8cm]
\item \textbf{Stage clause.} The Lorentzian bulk has the \GG{} product stage
\[
\Msix = \Mfour\times I_r\times S^1_\theta,
\]
with \(\Mfour\) spin and the two structural directions \(r\) and \(\theta\) distinguished. For Euclidean anomaly problems one either uses APS-compatible collars at the ends of \(I_r\) or doubles the interval to obtain the closed structural torus
\[
\Ttwo = S^1_r\times S^1_\theta.
\]
\item \textbf{Gauge clause.} The exact gauge group is
\[
\SUfive\times\UoneB\times\UoneQi.
\]
No extra gauge factors are present in the intended category.
\item \textbf{Tensor clause.} There is a single primitive \(X_4\)-coupled tensor channel, equivalently a rank-one unimodular tensor lattice \(\Lambda\cong \mathbb Z\) for the Green--Schwarz sector, with characteristic vector \(a=1\). (The gauge-singlet completion tensors are \(X_4\)-decoupled spectators; they enter only through their signature-type phase, Appendix~\ref{app:grav-full}.)
\item \textbf{\BIsix{} clause.} The four-form entering the modified Bianchi identity is
\[
X_4 = \trU R^2 - \TrU F_5^2 - \gamma\bigl(F_B^2 + F_{Qi}^2\bigr),
\qquad \gamma = \frac{s}{t}\in\mathbb Q_{>0},\quad \gcd(s,t)=1 .
\]
The primitive integrality condition is
\[
\widehat X := \frac{t}{8\pi^2}X_4 \in H^4(\Msix,\mathbb Z).
\]
\BIsix{} is the point \((s,t)=(2,5)\).
\item \textbf{Structural-holonomy clause.} The structural torus \(\Ttwo\) is fixed as part of the background data. Allowed transformations preserve the product/collar structure and do not introduce modular monodromy of the \((r,\theta)\) torus.
\item \textbf{Allowed large transformations.} Large diffeomorphisms are required to be identity on the structural collars (or isotopic to identity after doubling) and product-preserving on \(\Ttwo\). Gauge clutching on the structural torus is therefore null-homotopic in the intended category; the same clause excludes mixed clutching (gauge classes in \(H^2(\Mfour;\mathbb{Z}) \otimes H^1(\Ttwo;\mathbb{Z})\)) and Abelian winding transformations on the structural circles. Transformations outside this category are covered by the unrestricted theorem of Subsection~\ref{ssec:unrestricted-global}; the clause is retained as a description of the physical intended backgrounds.
\item \textbf{Physical base clause.} The Euclideanized compact four-manifolds used in the intended \GG{} category satisfy
\[
H^1(\Mfour;\mathbb Z)=0,
\qquad
H^3(\Mfour;\mathbb Z)=0.
\]
Equivalently, the connected physical backgrounds have no independent \(U(1)\) winding sector and no independent \(\SUfive\) gauge-transformation class on the base. For noncompact Minkowski backgrounds, the allowed large transformations are taken with compact support so that the same null-homotopy conclusion holds after compactification.
\item \textbf{Pullback-bundle clause.} The principal \(\SUfive\), \(U(1)_B\), and \(U(1)_{Qi}\) bundles are pullbacks from the base and are topologically trivial on the structural torus. The \BIsix{} differential cocycle is likewise a pullback/refinement of this base data.
\end{enumerate}
\end{definition}

This definition is more precise than an informal phrase, but it is not ad hoc. It simply makes explicit the class of backgrounds actually used throughout \Ptwo{}, \Pfive{}, and the \GDOS{} bridge papers: the \GG{} product stage, fixed structural \(r\)- and \(\theta\)-directions, exact \(\SUfive\times\UoneB\times\UoneQi\) gauge content, a single primitive tensor, and physically relevant four-dimensional bases without extra topological gauge sectors.

\paragraph{Statement of the automatic admissibility theorem.}

\begin{theorem}[Automatic admissibility theorem on the physical intended class]
Every background in the physical intended \GG{} category \(\AdmissibleClass\) automatically satisfies the four hypotheses used in the \BIsix{} global anomaly proof. Consequently, the complete \BIsix{} anomaly-freedom theorem is unconditional on \(\AdmissibleClass\). The theorem is not claimed for arbitrary enlarged non-supersymmetric six-dimensional categories that violate clauses \((C7)\) or \((C8)\).
\end{theorem}

The proof is split into four lemmas corresponding to \((H1)\)--\((H4)\).

\paragraph{Lemma for H2: absence of independent Witten-type gauge obstructions}
\begin{lemma}[Vanishing of the relevant \(\pi_6\) groups]
For the intended \GG{} gauge group,
\[
\pi_6(\SUfive)=0,
\qquad
\pi_6(\UoneB)=0,
\qquad
\pi_6(\UoneQi)=0.
\]
Therefore there is no independent Witten-type gauge anomaly in six dimensions for the allowed large gauge transformations of the intended category.
\end{lemma}

\begin{proof}
For \(\SU(n)\) with \(n\ge 4\), Bott periodicity places \(\pi_6(\SU(n))\) in the stable range and gives
\[
\pi_6(\SU(n))=0.
\]
In particular \(\pi_6(\SUfive)=0\). For \(U(1)\cong S^1\), all higher homotopy groups above degree one vanish, so
\[
\pi_6(U(1))=0.
\]
The only remaining caveat would be a transformation acting nontrivially on the structural torus rather than in the gauge bundle. But by definition of the intended category, modular monodromies of the structural torus are excluded; allowed transformations are product-preserving and collar-trivial. Therefore there is no extra six-dimensional gauge obstruction in the intended category.
\end{proof}

\paragraph{Lemma for H4: the differential-cocycle lift is automatic}
\begin{lemma}[Integral lift of the \BIsix{} four-form]
\label{lem:integral-lift}
Let \(\gamma=s/t\) in lowest terms and
\[
\widehat X := \frac{t}{8\pi^2}X_4.
\]
For every intended \GG{} background,
\[
\widehat X = t\lambda - t\,c_2(E_5) - s\,c_1(L_B)^2 - s\,c_1(L_{Qi})^2
\in H^4(\Msix,\mathbb Z),
\]
where \(\lambda=p_1/2\), and the class is primitive because \(\gcd(s,t)=1\). Consequently
\(X_4\) possesses a canonical differential-cocycle refinement \(\check X\). At the \BIsix{}
point \((s,t)=(2,5)\) this reads
\(\widehat X = 5\lambda - 5c_2(E_5) - 2c_1(L_B)^2 - 2c_1(L_{Qi})^2\).
\end{lemma}

\begin{proof}
The intended category includes a spin structure, so
\[
\frac{1}{8\pi^2}\trU R^2 = \lambda = \frac{p_1}{2}\in H^4(\Msix,\mathbb Z).
\]
Likewise, with the standard characteristic-class normalizations,
\[
\frac{1}{8\pi^2}\TrU F_5^2 = c_2(E_5),
\qquad
\frac{1}{8\pi^2}F_B^2 = c_1(L_B)^2,
\qquad
\frac{1}{8\pi^2}F_{Qi}^2 = c_1(L_{Qi})^2.
\]
Multiplying the \BIsix{} four-form by \(t\) therefore clears the Abelian denominator and gives
\[
\widehat X=t\lambda-t\,c_2(E_5)-s\,c_1(L_B)^2-s\,c_1(L_{Qi})^2,
\]
which is integral because \(s\) and \(t\) are; and it is primitive because no integer
greater than one divides all of \(t\), \(t\), \(s\), \(s\) when \(\gcd(s,t)=1\). Note that
the only property of the Abelian coefficient used here is that it is rational: the
denominator is cleared whatever it is. Any integral characteristic class equipped with a connection refines canonically to a Cheeger--Simons differential character; equivalently, there exists a differential cocycle \(\check X\) with curvature \(X_4\) and characteristic class \([\widehat X]\). Hence \((H4)\) is automatic.
\end{proof}

\paragraph{Lemma for H3: quadratic integrality of the tensor counterterm}
\begin{lemma}[Automatic Wu--Chern--Simons quantization]
Let \(Z_8\) be any closed spin eight-manifold carrying an extension of the intended \GG{} data, and let \(x=[\widehat X]\in H^4(Z_8,\mathbb Z)\). Then
\[
\frac12\int_{Z_8}\bigl(x^2-x\lambda\bigr)\in \mathbb Z,
\qquad \lambda=\frac{p_1}{2}.
\]
Therefore the exponentiated \BIsix{} tensor counterterm is single-valued and correctly Wu-shifted on the intended category.
\end{lemma}

\begin{proof}
For spin eight-manifolds, \(\lambda=p_1/2\) is a characteristic element of the middle-dimensional intersection form. Equivalently, for every integral class \(x\in H^4(Z_8,\mathbb Z)\),
\[
\int_{Z_8}x^2 \equiv \int_{Z_8}x\lambda \pmod 2.
\]
This standard fact is the eight-dimensional incarnation of the Wu-form / characteristic-vector property. Rearranging yields
\[
\int_{Z_8}(x^2-x\lambda)\in 2\mathbb Z,
\]
and therefore
\[
\frac12\int_{Z_8}(x^2-x\lambda)\in \mathbb Z.
\]
Because \(x=[\widehat X]\) is integral by the previous lemma, the needed \WCS{} functional is automatically well defined. No extra parity assumption on \(x\) is needed beyond spin plus integrality.
\end{proof}


\paragraph{Lemma for H1: every mapping torus in the intended category bounds}
\begin{lemma}[Bounding theorem for physical intended \GG{} mapping tori]
Let \(f\) be an allowed large transformation of a background in \(\AdmissibleClass\). Then the corresponding seven-dimensional mapping torus \(\Useven\) bounds a spin eight-manifold \(\Weight\) carrying extensions of the intended spin/gauge/tensor data.
\end{lemma}

\begin{proof}
Because allowed transformations preserve the product/collar structure and act trivially on the structural torus up to isotopy, the mapping torus factorizes after Euclidean doubling as
\[
\Useven \cong T_{f_4}(\Mfour)\times \Ttwo,
\]
where \(T_{f_4}(\Mfour)\) is the five-dimensional mapping torus of the induced transformation on the four-dimensional base.

The diffeomorphism part is controlled by clause \((C6)\): on the intended class the allowed base diffeomorphisms are isotopic to the identity after doubling. Therefore
\[
T_{f_4}(\Mfour)\cong \Mfour\times S^1,
\]
and hence
\[
W_{\rm geom}:=\Mfour\times D^2\times \Ttwo
\]
is a spin eight-manifold with
\[
\partial W_{\rm geom}\cong \Mfour\times S^1\times \Ttwo\cong \Useven
\]
at the purely geometric level.

It remains to extend the gauge and tensor data. By clause \((C7)\), gauge transformations on the base are null-homotopic. For the Abelian factors,
\[
[\Mfour,U(1)]\cong H^1(\Mfour;\mathbb Z)=0.
\]
For \(\SUfive\), the Postnikov \(4\)-type is \(K(\mathbb Z,3)\) because \(\pi_1=\pi_2=\pi_4=0\) and \(\pi_3(\SUfive)=\mathbb Z\) \citep{Bott1959}. Hence for a four-dimensional CW complex,
\[
[\Mfour,\SUfive]\cong H^3(\Mfour;\mathbb Z)=0.
\]
Therefore every allowed gauge transformation on the base extends across \(\Mfour\times D^2\). Since the gauge bundles are pullbacks from \(\Mfour\) and trivial on the structural torus by clause \((C8)\), their extensions over \(W_{\rm geom}\) are obtained by pullback from the extending bundles on \(\Mfour\times D^2\). The integral class \([\widehat X]\) and its differential-cocycle lift \(\check X\) extend with them.

Thus the seven-dimensional mapping torus of every allowed transformation bounds a spin eight-manifold with extending gauge and tensor data. This proves \((H1)\) on \(\AdmissibleClass\).
\end{proof}

\begin{remark}[Status of this lemma after the unrestricted theorem]
The construction above is the \emph{constructive} route: on the intended class the filling is exhibited explicitly as a product, at the price of clauses (C6)--(C8). Lemma~\ref{lem:bordism-vanishing} of Subsection~\ref{ssec:unrestricted-global} supersedes it in generality --- every closed spin seven-datum with \GG{} gauge bundle bounds, by bordism vanishing rather than by null-homotopy --- while this lemma retains its value as the explicit, inspectable instance of the general fact.
\end{remark}

\paragraph{Completion of the automatic admissibility theorem.}

\begin{proof}[Proof of the automatic admissibility theorem]
The preceding four lemmas establish \((H2)\), \((H4)\), \((H3)\), and \((H1)\), respectively, for every background in the intended \GG{} category. The conditional Theorem~\ref{thm:main} (which assumes H1--H4) therefore applies to every such background without additional assumptions. This completes the proof of the unconditional Theorem~\ref{thm:bi6-global-complete} stated in the main text: \BIsix{} is completely anomaly-free, both locally and globally, on \(\AdmissibleClass\).
\end{proof}

With the automatic admissibility established, the conditional phrasing of
Theorem~\ref{thm:main} can be removed for the physical intended \GG{} category.
The unconditional statement is: \BIsix{} is completely anomaly-free on
\(\AdmissibleClass\), and \(\AdmissibleClass\) satisfies the admissibility
hypotheses H1--H4 automatically.
The broader claim for arbitrary non-supersymmetric six-dimensional
backgrounds remains open.

\bigskip
\noindent\textit{The proof is complete; what remains is bookkeeping
honesty. The checklist matrix that follows states, row by row, what
each theorem consumes, what it delivers, and its present status ---
the reader's map for auditing the whole chain, and the course's final
exam: a student who can defend every row owns the paper.}

\section{Theorem checklist matrix and dependency ledger}
\label{app:checklist}


To make the logical architecture of the present paper maximally transparent,
this appendix summarizes the main theorem chain, its premises, its mathematical
machinery, its present status, and its downstream physical consequence. The goal
is not merely convenience. A theorem program becomes much easier to criticize and
much easier to trust when the dependency structure is written out explicitly.

\renewcommand{\arraystretch}{1.18}
\footnotesize
\begin{longtable}{>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.243\textwidth}>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.268\textwidth}}
\caption[The theorem checklist matrix: one row per load-bearing claim, with its status.]{The theorem checklist matrix: one row per load-bearing claim of this paper. The columns give the target statement, the primary input it consumes, its present status, and the downstream consequence it delivers, so the dependency structure of the whole chain is written out rather than left implicit --- a theorem program becomes much easier to criticize, and much easier to trust, when it is. This is the reader's map for auditing the chain end to end, and the course's final exam: a student who can defend every row owns the paper.}\label{tab:checklist-matrix}\\
\toprule
\textbf{Target statement} & \textbf{Primary input} & \textbf{Status} & \textbf{Downstream consequence}\\
\midrule
\endfirsthead
\toprule
\textbf{Target statement} & \textbf{Primary input} & \textbf{Status} & \textbf{Downstream consequence}\\
\midrule
\endhead
\AX{1} through \AX{4} plus \POS{1} through \POS{7} define an internally coherent admissibility constitution & \AOne{} \citep{ArisakaA1Constitution} and the axiomatic analysis developed in this paper & Adopted axiomatic base & Fixes the type of explanation pursued throughout the series: admissibility before model-building.\\

\GG{} bulk stage is \(M_4\times I_r\times S^1_\theta\) with exact gauge packet \(SU(5)\times U(1)_B\times U(1)_{Qi}\) & \ATwo{} \citep{ArisakaA2GG6} construction & Adopted \GG{} base & Provides the geometric stage, the precise gauge factors, and the UV endpoint at \(E_{\mathrm{GUT}}\).\\

\BIsix{} four-form is \(X_4=\tr R^2-\Tr F_5^2-(2/5)F_B^2-(2/5)F_{Qi}^2\) & \ATwo{} \citep{ArisakaA2GG6} plus Appendix-P style trace logic & Proved in structural form & Supplies the exact coefficient vector that every later theorem inherits.\\

Local anomaly polynomial factorizes as \(I_8=\half X_4^2\) & Section~\ref{sec:structural-proof} and Appendix~\ref{app:descent-full} & Proved & Makes a one-tensor Green--Schwarz cancellation possible.\\

One Green--Schwarz tensor cancels the full infinitesimal anomaly & Descent formalism, BRST formulation, factorized polynomial & Proved & Establishes local six-dimensional gauge/Lorentz consistency.\\

Anti-self-dual tensor sector is necessary & Gravitational coefficient comparison and L-genus structure & Proved & Explains why a purely fermionic spectrum cannot reproduce the target polynomial.\\

No independent Witten-type obstruction survives for the intended gauge group & Bott periodicity, vanishing of \(\pi_6\) for \(SU(5)\) and \(U(1)\) factors & Proved on intended class & Removes the simplest possible source of uncanceled global gauge anomaly.\\

The \BIsix{} class admits an integral differential-cocycle lift & Characteristic-class normalization, spin condition, primitive integrality of \(\widehat X\) & Proved on intended class & Allows the counterterm to be written globally rather than only locally.\\

The tensor counterterm is correctly Wu-shifted and single-valued; the underlying refinement is unique and the self-dual partition function is constructed & Characteristic-vector property of \(\lambda=p_1/2\); coset uniqueness on unimodular lattices (T-BI6-7.7); canonical theta block (T-BI6-7.8) & Proved, canonical & Closes the self-dual tensor phase problem with no residual discrete choice.\\

Every closed spin seven-datum with \GG{} gauge bundle bounds a compatible eight-manifold & Vanishing of \(\widetilde\Omega^{\mathrm{spin}}_{5,7}(BG)\) (AHSS + Wu formula), signature surgery & Proved, unrestricted & Supplies the filling manifolds needed by the APS / Dai--Freed evaluation, with no topological fine print.\\

Complete local and global anomaly freedom of \BIsix{} under \GG{} & Combination of the previous four rows with the local descent theorem; unrestricted extension by T-BI6-7.6 & Proved; unrestricted in the gauge sector & Upgrades \BIsix{} from a local consistency check to a full quantum-consistency theorem with no topological fine print.\\

Model-level finite benchmark spectrum exists & Core block construction, orbit families, gravitational completion & Proved as existence & Shows that the structural polynomial is not empty: an explicit finite realization exists.\\

\(U(1)_B\) is structurally essential & Proton protection, split multiplets, baryonic trace ratio & Proved / clarified & Explains why baryon symmetry is not a cosmetic add-on.\\

\(U(1)_{Qi}\) is anomaly-compatible and optional at the narrow theorem level & Abelian \(\pi_6\) vanishing, quadratic \BIsix{} slot structure & Proved / clarified & Separates narrow anomaly logic from later \GDOS{}/\QiFour{} development.\\

The trace-ratio theorem gives \((L-N)/L=2/5\) & Unified \(SU(L)\) block split into an \(N\)-block and an \((L-N)\)-block & Proved & Bridges \BIsix{} coefficients to the \OGNfive{} integers.\\

\OGNfive{} integers are \((M,N,L;J,K)=(1,3,5;7,35)\) & Primitive tensor slot, unified block size, trace-ratio theorem, closure identities & Proved & Converts \OGNfive{} from mnemonic to theorem and underwrites later arithmetic formulae.\\

Comparison with the 1984 GS mechanism is substantive but non-identical & Shared factorization plus different ontology and dimensional setting & Proved / clarified & Positions \BIsix{} relative to the original benchmark of anomaly cancellation.\\

Comparison with 6D SUGRA / F-theory is mathematically close but physically distinct & Shared global-anomaly technology plus sharply different admissible theory space & Proved / clarified & Justifies dialogue with the 6D anomaly literature without collapsing \GG{} into the landscape program.\\

Landscape-style proliferation is strongly suppressed in the intended \GG{} category & Fixed gauge packet, fixed tensor content, fixed \BIsix{} vector, \OGNfive{} lock arithmetic & Clarified & Supports the claim that \GG{} is a highly restrictive, constitutionally organized framework.\\
\bottomrule
\end{longtable}
\normalsize

Table~\ref{tab:checklist-matrix} is the summary of the whole paper:
one row per load-bearing claim, with its status, its primary input, and
where it is established. A reader who wants to know how much of this
paper is theorem and how much is reading should start there.

\subsection{Theorems of \AThree{} used by companion papers (outward map)}
\label{ssec:outward-map}

\AThree{}'s principal theorems are the local rank-one criterion (Theorem~\ref{thm:rank-one}), the local \BIsix{} cancellation theorem (Theorem~\ref{thm:bi6-cancellation}), the complete local + global anomaly freedom theorem (Theorem~\ref{thm:bi6-global-complete}), the constitutional-admissibility theorem for $\UoneQi$ (Theorem~\ref{thm:UoneQi-admissibility}), the model-level benchmark spectrum theorem (Theorem~\ref{thm:model-level}), the trace-ratio theorem of Section~\ref{sec:ogn5-theorem}, the \OGNfive{} theorem of Section~\ref{sec:ogn5-theorem}, and the GS-as-\GGEP{} completion subsection (Section~\ref{sec:GS_as_GGEP_completion}). Table~\ref{tab:outward-map-A3} records, for each, which companion paper(s) cite it as input and the role the theorem plays in the companion's derivation. Rows point to the companion papers under their canonical cohort keys; where a pointer is section-level rather than a numbered theorem, the structural role has been verified and the pointer will be sharpened as the companions' theorem numbering stabilizes.

{\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.152\textwidth}>{\raggedright\arraybackslash}p{0.152\textwidth}>{\raggedright\arraybackslash}p{0.248\textwidth}>{\raggedright\arraybackslash}p{0.343\textwidth}@{}}
\caption[Outward map: this paper's theorems and who consumes them.]{Where each result of this paper is used downstream. The columns give the theorem or section, the companion paper that consumes it, the location there, and the role it plays in that derivation. The table is the paper's contract with the cohort: anything listed here is a load-bearing export, and a failure in this paper propagates along exactly these rows and no others. Entries marked pointer-only name a companion section that does not yet carry a numbered theorem label --- the structural role is verified, the section number is approximate, and the distinction is made here rather than glossed.}
\label{tab:outward-map-A3}\\
\toprule
\textbf{\AThree{} theorem / section} & \textbf{Companion paper} & \textbf{Companion location} & \textbf{Role in companion derivation} \\
\midrule
\endfirsthead
\multicolumn{4}{c}{\itshape Table~\ref{tab:outward-map-A3} continued from previous page} \\
\toprule
\textbf{\AThree{} theorem / section} & \textbf{Companion paper} & \textbf{Companion location} & \textbf{Role in companion derivation} \\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
Theorem~\ref{thm:rank-one} (rank-one criterion) & \ATwo{} & \TX{4}{GG6} (\BIsix{} coefficient vector) and \TX{5}{GG6} (anomaly factorization) & Single-tensor cancellation requires the $4\times4$ anomaly matrix to have rank one; \ATwo{}'s coefficient vector $(1,-1,-2/5,-2/5)$ saturates this condition. \\
Theorem~\ref{thm:bi6-cancellation} (\BIsix{} local cancellation) & \ATwo{} & \TX{5}{GG6} (anomaly factorization), \TX{7}{GG6} (BRST + Slavnov--Taylor) & The local $I_8 = \tfrac12 X_4^2$ factorization closes \TX{5}{GG6}; the BRST cohomology argument feeds \TX{7}{GG6}'s homogeneous Slavnov--Taylor identity. \\
Theorem~\ref{thm:bi6-global-complete} (global anomaly freedom) & \AFive{} & Lawfulness lemma for the $\OGGEP$ channel & Without global anomaly freedom, the bulk-to-boundary CPTP map fails to descend to gauge-equivalence classes; \AThree{}'s global theorem is the prerequisite. \\
Theorem~\ref{thm:bi6-global-complete} & \ATwo{} & \TX{7}{GG6} (forward pointer to global proof) & \ATwo{} establishes BRST consistency at the perturbative level; the global completion is supplied by \AThree{}. \\
Theorem~\ref{thm:UoneQi-admissibility} (constitutional admissibility of both bulks) & \ATwo{} & \TX{3}{GG6} Step (b/c) (six forces); Common Objection~4; Open Problem O8 & The legacy reading ``\AX{3} + \POS{4} + \POS{1} forces $\UoneQi$'' is corrected: the constitution admits both the three- and four-component bulks; \ATwo{}'s empirical adoption of the four-component version is now a derived choice, not an axiomatic forcing. \\
Theorem~\ref{thm:UoneQi-admissibility} & \ASix{}, \ASeven{} & Boundary \GiFour{} / \QiFour{} pillar derivations & The four-component bulk admissibility is the structural prerequisite for the \GiFour{} and \QiFour{} pillars (their host symmetry is $\UoneQi$). \\
Theorem~\ref{thm:model-level} (model-level benchmark spectrum) & \ATwo{} & \TX{3}{GG6} Step~(b/c), \TX{5}{GG6} (\POS{3} anomaly-free admissibility input) & Existence of a finite chiral spectrum saturating the \BIsix{} factorization closes the spectrum-existence step of \ATwo{}'s six-forces theorem. \\
Trace-ratio theorem (Sec.~\ref{sec:ogn5-theorem}) & \ATwo{} & \TX{4}{GG6} (\BIsix{} coefficients lock $|a/c|=|a/d|=5/2$) & The $(L{-}N)/L = 2/5$ identity is the structural bridge from \BIsix{} to \OGNfive{}; \ATwo{}'s coefficient vector lock cites this directly. \\
Trace-ratio theorem & \AFour{} & Universal-constants reconstruction & The $-2/5$ Abelian fraction enters \AFour{}'s parameter-free reconstruction of $\{c, G_N, \hbar; k_B, Q_B\}$. \\
\OGNfive{} theorem (Sec.~\ref{sec:ogn5-theorem}) & \AFour{} & \OGNfive{} historical reconstruction & The integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$ supplies \AFour{}'s universal-constants pentad. \\
\OGNfive{} theorem & \ANine{} & Fine-structure-constant prediction & $\alpha^{-1}(0) = 2^J + N^2 = 2^7 + 9 = 137$ uses $J = 7$, $N = 3$ from \AThree{}'s \OGNfive{}. \\
\OGNfive{} theorem & \ATen{} & Density inventory and Kaxion frequency & $K = 35$ sets the Kaxion twin-peak haloscope signature $\Delta\nu/\nu = 1/35$; the $\pi$-locked density inventory $\Omega_m = 1/\pi$, etc., depends on $K$. \\
GS-as-\GGEP{} completion (Sec.~\ref{sec:GS_as_GGEP_completion}) & \ATwo{} & The echo paragraph after \TX{4}{GG6} & The 1984 Green--Schwarz mechanism is reframed as the technical pioneering of \AX{2} (\GGEP); \BIsix{} is the four-component generalization. \ATwo{} cross-cites this section directly. \\
\end{longtable}
}

\subsection{Constitutional inputs to \AThree{} (inward map)}
\label{ssec:inward-map}

Conversely, \AThree{} imports its primitive inputs from the axiomatic layer of \AOne{} and the geometric layer of \ATwo{}. Every theorem of the present paper depends on at least one constitutional clause and at least one \ATwo{} input. Table~\ref{tab:inward-map-A3} records the constitutional clauses that \AThree{} invokes and the role each plays in the principal theorems.

{\footnotesize
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.171\textwidth}>{\raggedright\arraybackslash}p{0.171\textwidth}>{\raggedright\arraybackslash}p{0.171\textwidth}>{\raggedright\arraybackslash}p{0.40\textwidth}@{}}
\caption[Inward map: what this paper imports, and from where.]{Every constitutional clause and structural input this paper takes from elsewhere. The first column names the input, the second its source paper, the third the theorem here that consumes it, and the fourth the role it plays. This is the audit trail for the claim that the paper derives rather than assumes: nothing may be used that is not listed, and every row can be followed to its source. Read in the other direction it is a dependency list --- it says exactly what would fail here if a given import failed there.}
\label{tab:inward-map-A3}\\
\toprule
\textbf{Constitutional / structural input} & \textbf{Source paper} & \textbf{Used by \AThree{} theorem / section} & \textbf{Role in \AThree{}'s derivation} \\
\midrule
\endfirsthead
\multicolumn{4}{c}{\itshape Table~\ref{tab:inward-map-A3} continued from previous page} \\
\toprule
\textbf{Constitutional / structural input} & \textbf{Source paper} & \textbf{Used by \AThree{} theorem / section} & \textbf{Role in \AThree{}'s derivation} \\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\itshape continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
\AX{1} (\LC) & \AOne{} & Theorem~\ref{thm:bi6-cancellation}, Theorem~\ref{thm:bi6-global-complete} & Lorentzian $(1,5)$ signature on the 6D bulk; defines the spinor structure for the chiral-fermion anomaly polynomial. \\
\AX{2} (\GGEP) & \AOne{} & Sec.~\ref{sec:GS_as_GGEP_completion}; Theorem~\ref{thm:rank-one} & Geometry--gauge equivalence is the axiomatic content that \BIsix{} modifies; the GS-as-\GGEP{} completion subsection is the home of the methodological reading. \\
\AX{3} (\OGGEP) & \AOne{} & Sec.~\ref{sec:global-completion} (phase-propagation lemma) & The $\OGGEP$ bulk--boundary correspondence forces the bulk-to-boundary handshake whose lawfulness depends on global anomaly freedom. \\
\AX{4} (\MSMF) & \AOne{} & Theorem~\ref{thm:rank-one}; Sec.~\ref{sec:ogn5-theorem}; Theorem~\ref{thm:UoneQi-admissibility} & Single-tensor minimality forces rank-one factorization; equal-Abelian treatment locks the coefficient $-2/5$ symmetrically across $\UoneB$ and $\UoneQi$; minimum-Diophantine closure produces \OGNfive{}. \\
\POS{1} (\GI) & \AOne{} & Theorem~\ref{thm:bi6-cancellation}; Theorem~\ref{thm:UoneQi-admissibility} & Layered gauge invariance for the BRST cohomology; gauge invariance under the existing bulk gauge content is what makes the three-component bulk admissible. \\
\POS{3} (\AF) & \AOne{} & All anomaly theorems of Sec.~\ref{sec:structural-proof}--\ref{sec:global-completion} & Anomaly-free admissibility by inflow; the central postulate of the entire paper. \\
\POS{4} (\CP) & \AOne{} & Theorem~\ref{thm:UoneQi-admissibility} & Closure and positivity; the GKLS-channel CPTP property that \POS{4} imposes on the bulk-to-boundary registration map (a structural property, not a covariance demand under a fresh $U(1)$). \\
\POS{5} (\RG) & \AOne{} & Sec.~\ref{sec:intro-gg6} (6D bulk discussion) & RG compatibility forces the radial coordinate $r = \ln(\EGUT/\mu)$. \\
\POS{6} (\OGN) & \AOne{} & Sec.~\ref{sec:ogn5-theorem}; Theorem~\ref{thm:rank-one} (Sec.~\ref{sec:intro-gg6}) & Diophantine closure of all dimensionless freedom; the constitutional source of the integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$. \\
\ATwo{} bulk gauge group $\Spin(1,5) \times \SUfive \times \UoneB \times \UoneQi$ & \ATwo{} (\TX{3}{GG6}) & All theorems of Sec.~\ref{sec:structural-proof}--\ref{sec:global-completion} & The four-factor gauge--gravity group is the input on which the \BIsix{} four-form $X_4$ is built. \\
\ATwo{} \BIsix{} coefficient vector $(1, -1, -2/5, -2/5)$ & \ATwo{} (\TX{4}{GG6}) & Theorem~\ref{thm:rank-one}; Theorem~\ref{thm:bi6-cancellation} & The microscopic anomaly matrix factorizes as $vv^T$ with $v = (1, -1, -2/5, -2/5)^T$ exactly when this vector is the input. \\
\ATwo{} \OGNfive{} ledger $(1, 3, 5; 7, 35)$ & \ATwo{} (\TX{6}{GG6}) & Sec.~\ref{sec:ogn5-theorem} & The same five integers are reached here by a route internal to the \BIsix{} logic. The agreement of the two routes is a result and not an import: the present paper's cascade consumes no \ATwo{} theorem, and Section~\ref{sec:ogn5-theorem} states which of its steps are forced and which inherit the trace-ratio identity. \\
\end{longtable}
}

\paragraph{Note on companion versions.} Each companion paper in this map is on its own revision track, and the rows above point to the current deployed version of each companion under its canonical cohort key. When a companion undergoes a substantive revision that introduces, removes, or renumbers theorems, the corresponding rows of Tables~\ref{tab:outward-map-A3} and~\ref{tab:inward-map-A3} should be re-walked: structural roles are stable, and only the precise theorem-number pointers (where currently marked ``pointer-only'' or given by a section reference rather than a theorem label) may need sharpening.

\paragraph{Audit trail.} For a reader re-walking the map, we record below the actual internal theorem and section structure of each cited companion paper.

\begin{itemize}[leftmargin=2.5em]
\item \textbf{\AOne{}.} Carries no Euclidean-style numbered theorems in the body; the axiomatic content is organized as 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). \AThree{} cites \AX{1}, \AX{2}, \AX{3}, \AX{4}, \POS{1}, \POS{3}, \POS{4}, \POS{5}, and \POS{6} as listed in Table~\ref{tab:inward-map-A3}.
\item \textbf{\ATwo{}.} Carries nine principal theorems \TX{0}{GG6}--\TX{8}{GG6} in Part~II: \TX{1}{GG6} dimensional minimality; \TX{2}{GG6} canonical metric; \TX{3}{GG6} six forces from the constitution; \TX{4}{GG6} \BIsix{} coefficients; \TX{5}{GG6} anomaly factorization; \TX{6}{GG6} \OGNfive{} cascade; \TX{7}{GG6} BRST + homogeneous Slavnov--Taylor identity; \TX{8}{GG6} the Arisaka equation. \AThree{} and \ATwo{} describe the same object and prove different things about it. The stage, the four-factor gauge packet and the \BIsix{} four-form are specified in both, and enter the present paper as clauses (C1)--(C4) of Definition~\ref{def:admissible-main}; no theorem of \ATwo{} is consumed by any proof here. In the other direction the present paper supplies the global anomaly proof, the rank-one theorem, the constitutional-admissibility theorem, the model-level existence witness for the benchmark spectrum, and the GS-as-\GGEP{} completion.
\item \textbf{\AFour{}.} Reconstructs the universal constants $\{c, G_N, \hbar; k_B, Q_B\}$ from the bulk geometry of \GG; carries an \OGNfive{} derivation theorem and a fine-structure-constant theorem ($\alpha^{-1}(0) = 2^J + N^2 = 137$). \AThree{} supplies the trace-ratio theorem and the \OGNfive{} theorem to \AFour{} as inputs.
\item \textbf{\AFive{}.} Carries \TX{1}{\GDOS{}}--\TX{11}{\GDOS{}} of \GDOS{} (the bulk-to-boundary projection of \GG\ to the open-system runtime). \AThree{}'s global anomaly freedom theorem is the lawfulness prerequisite for the $\OGGEP$ channel; specifically, the phase-propagation lemma in Section~\ref{sec:global-completion} of \AThree{} establishes that a non-trivial bulk global anomaly phase would prevent the channel from descending to gauge-equivalence classes.
\item \textbf{\ASix{}.} Carries the four-pillar \GiFour{} structure: \TGG{} (Total Geometric--Gauge object), \CCI{} (Covariant Conservation Identity), \GFF{} (Geometric Free-Energy Functional), \GRIP{} (Gauge-Runtime Invariance Pipeline, the cascade $\DGI \to \GSSB \to \GA{}/\GGA{}$ in the two regimes \GRIPin{}/\GRIPout{}). \AThree{}'s constitutional-admissibility theorem (Theorem~\ref{thm:UoneQi-admissibility}) is the structural prerequisite for the four-component bulk on which the \GiFour{} pillars are built. One resource flows directly: the \BIsix{} instanton-action budget $S_{\rm BI}^{\max} = 3\pi$ --- quoted here and delivered by the capacity theorem T-N3-6 of \PaperPTwo{} (its Appendix~C records the two viable formulations), conditional on its coupling clause and the integrality theorem L-INT (residue R-INT); it is not derived in the present paper --- is precisely what the runtime's \GRIPin{} descent \emph{consumes} --- the Bogomolnyi tolls of the \GSSB{} gates accumulate against it, and its saturation is the closure that forces $N \le 3$ (\ASix{} Theorem T4.1).
\item \textbf{\ASeven{}.} Carries the four-pillar \QiFour{} structure: \QiQUA, \QiEP, \QiCON, \QiHAL. Empirical-calibration section calibrates $\QB \approx 0.1555$~eV/qubit against ATP hydrolysis at biological temperatures. \AThree{}'s constitutional-admissibility theorem is the structural prerequisite for $\UoneQi$ as host symmetry of the \QiFour{} pillars.
\item \textbf{\ANine{}.} Carries the SM-side phenomenology: Standard-Model parameters as boundary \GRIP{} attractor coordinates; gauge couplings as Wilson--Hosotani branch attractors; CKM/PMNS as projector misalignments; strong-CP angle $|\bar\theta| \lesssim 4.7 \times 10^{-18}$. \AThree{}'s \OGNfive{} theorem feeds the fine-structure prediction $\alpha^{-1}(0) = 2^J + N^2 = 137$.
\item \textbf{\ATen{}.} Carries the cosmological recast: $\pi$-locked density inventory ($\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$, $\Omega_{\mathrm{DM}} = 1/\pi - 1/(2\pi^2)$); Kaxion twin-peak haloscope signature $\Delta\nu/\nu = 1/35$. \AThree{}'s \OGNfive{} theorem supplies the integer $K = 35$ that sets the Kaxion frequency separation.
\end{itemize}

The audit trail records what the present paper actually consumes. When any companion paper undergoes a substantive revision that introduces, removes, or renumbers theorems, this audit trail should be updated to reflect the new internal numbering of that companion paper, and the corresponding rows of Tables~\ref{tab:outward-map-A3} and~\ref{tab:inward-map-A3} should be re-walked.

\paragraph{What the map sustains.} After all, a program of twenty-two coordinated papers stands or falls on the consistency of its cross-references. The present map sustains that consistency in two complementary directions: outward (\AThree{}'s principal theorems are inputs to seven companion papers, each named with the role it plays in the companion's derivation) and inward (\AThree{}'s principal theorems consume nine constitutional clauses from \AOne{} plus three structural inputs from \ATwo{}). The audit trail above ties each pointer to a specific section or theorem of the cited companion paper, so a reader can re-walk the entire program without re-reading every paper. This is the cross-referencing discipline the \GGTheory{} program demands of itself, and it is what \AThree{} delivers.


\subsection{Concordance: global identifiers and printed theorem numbers}
\label{ssec:tbi6-concordance}

The global identifiers \textbf{T-BI6-$x.y$} used by this paper, and cited by
its companions, were assigned under an earlier sectioning and do not track the
printed section numbers. They are kept fixed deliberately: regenerating them
would silently break every pointer a companion paper already carries. The
table below is therefore the authoritative map, and it is built from live
cross-references, so it cannot go stale.

\begin{center}
\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.19\textwidth}>{\raggedright\arraybackslash}p{0.21\textwidth}>{\raggedright\arraybackslash}p{0.54\textwidth}@{}}
\caption[The numbered results of this paper under their cohort identifiers.]{Every numbered result of this paper under its cohort-global identifier, the number it prints as in this document, and what it states. The middle column is generated from live cross-references rather than typed, so renumbering the document cannot make this map wrong: a companion paper citing T-BI6-6.1 will land on the rank-one criterion whatever section it ends up in.}\label{tab:a3-identifier-map}\\
\toprule
\textbf{Global identifier} & \textbf{Prints as} & \textbf{Result} \\
\midrule
\endfirsthead
\toprule
\textbf{Global identifier} & \textbf{Prints as} & \textbf{Result} \\
\midrule
\endhead
T-BI6-3.2  & Theorem~\ref{thm:gs-general}              & Green--Schwarz mechanism in general even dimensions \\
T-BI6-6.1  & Theorem~\ref{thm:rank-one}                & Rank-one criterion \\
T-BI6-6.2  & Theorem~\ref{thm:single-tensor-necessity} & Necessity of the single-tensor channel \\
T-BI6-6.3  & Theorem~\ref{thm:bi6-cancellation}        & Local \BIsix{} anomaly cancellation \\
T-BI6-7.2  & Theorem~\ref{thm:bi6-global-complete}     & Global anomaly freedom on the intended class \\
T-BI6-7.5  & Theorem~\ref{thm:UoneQi-admissibility}    & Constitutional admissibility of $\U(1)_{Qi}$ \\
T-BI6-7.6  & Theorem~\ref{thm:unrestricted-global}     & Unrestricted global anomaly freedom \\
T-BI6-7.7  & Theorem~\ref{thm:refinement-unique}       & Uniqueness of the Wu--Chern--Simons refinement \\
T-BI6-7.8  & Theorem~\ref{thm:sd-partition}            & The chiral-tensor partition function \\
T-BI6-8.8  & Theorem~\ref{thm:orbit}                   & Solution of the orbit Diophantine system \\
T-BI6-8.9  & Theorem~\ref{thm:model-level}             & Model-level benchmark spectrum \\
T-BI6-9.0  & Theorem~\ref{thm:L-from-GS}               & $L=5$ from the Green--Schwarz conditions \\
T-BI6-9.1  & Theorem~\ref{thm:ogn-bi6}                 & The trace-ratio lock $(L-N)/L = 2/5$ \\
T-BI6-9.2  & Theorem~\ref{thm:ogn-core}                & \OGNfive{} on the core block \\
T-BI6-9.3  & Theorem~\ref{thm:ogn-omega}               & Minimal normalization $\Omega \in L^2\mathbb{Z}$ \\
T-BI6-10.1 & Theorem~\ref{thm:N-from-closure}          & $N=3$ from the dual-lock closure \\
T-BI6-10.2 & Theorem~\ref{thm:ogn5-theorem-updated}    & The \OGNfive{} ledger theorem \\
T-BI6-10.3 & Theorem~\ref{thm:lN-minimality}           & Diophantine minimality: $k=1$ \\
\bottomrule
\end{longtable}
\end{center}

\noindent Two entries deserve a word. \textbf{T-BI6-10.1} was, until this
revision, carried as a suffixed variant of \textbf{T-BI6-10.2}; an identifier
that has to be suffixed to distinguish two theorems is not doing its job, and
the collision is now resolved in favor of the ledger theorem, which is the
one companion papers cite. \textbf{T-BI6-9.0} sorts before \textbf{T-BI6-9.1}
but prints after it --- an artifact of the same historical assignment, left as
it stands for the same reason the rest are.

\subsection{How to read the matrix}

The matrix is intentionally conservative. A row is marked ``proved'' only when
the present paper now contains the mathematical chain needed to justify the
statement within the intended \GG{} category. A row is marked ``clarified'' when
the present paper now makes the scope of the statement sharp, even if the
statement is more interpretive than theorem-like. A row is marked ``existence''
when the preprint demonstrates that at least one explicit realization exists but
does not yet settle uniqueness or absolute minimality.

This conservative reading is important for publication. The strength of the present paper is not that it declares every
attractive idea proved; it is that it separates theorem, interpretation,
existence result, and future direction in a way that can be checked line
by line.

\bigskip

\noindent\textbf{What the appendices were for.} Ten appendices is a
great deal of machinery, and it is worth saying what the reader was
meant to get from them. Two things.

The first is auditability. Every step of the argument in Part~II is
carried out here at a level of detail where it can be checked with paper
and pencil, including the steps that the literature usually quotes
rather than derives --- the descent chain, the two arithmetic facts
behind the ten-dimensional mechanism, the genus expansions that produce
the gravitational anomaly polynomials, the spectral-sequence computation
behind the vanishing bordism groups. Nothing in the main text rests on a
result stated without a derivation somewhere in this Part.

The second is that the machinery is not ours. The descent formalism, the
index theorems, the differential-cohomology framework and the
Dai--Freed construction were built by other people for other purposes,
and they work here unmodified. A reader who finishes these appendices
should be able to run the same calculation on a different bulk with a
different gauge group and get a different answer, which is the only
sense in which a method is worth learning.

The checklist matrix of Appendix~\ref{app:checklist} is the summary: one
row per load-bearing claim, with its status, its primary input, and
where it is established. A reader who wants to know how much of this
paper is theorem and how much is reading should start there.


%% ===============================================================
%%  ACKNOWLEDGMENTS + BIBLIOGRAPHY
%% ===============================================================


\GGPart[Common Objections]{Common Objections and Responses}{The
fifty-three recurring objections a particle theorist, string theorist,
or differential-cohomology specialist is most likely to raise on first
reading --- answered in ten clusters, with pointers into the paper}
\label{sec:common-objections}


A reader arriving at the present paper from a particle-theoretic,
string-theoretic, or differential-cohomology background will raise one
of a small number of recurring objections. Fifty-three of them are
answered below, grouped into ten clusters, each response pointing back
into the paper at the place the objection is really about. The plain-language questions a non-specialist asks are a different set
and were answered in Part~\ref{sec:curious-reader}.

The objections are grouped into ten clusters, running from the anomaly
proof itself outward: the trace ratio and the optional Abelian factor;
global anomaly freedom without supersymmetry; minimality and the
parsimony axiom; the convergences and the empirical claims; the internals
of the local descent; the global completion and the eta invariant;
whether the integer ledger is a theorem or a target; the exact relation
to string theory; the constitutional method itself; and the honest
status of the empirical exposure.

Three conventions are followed throughout. The objection is stated in
the form a specialist would actually put it, not in a weakened form that
is easier to answer. Where the objection is right, the response says so
and says what part of the paper has to be read down accordingly ---
several of the sharpest concessions in this document are in this Part
rather than in the main text. And every response points back to the
specific section, theorem, or appendix where the matter is settled, so
that a reader can check the answer rather than take it.

A reader who is deciding how much of this framework to credit could
reasonably read this Part before any other.



\bigskip

%% =====================================================================
%% Cluster 0 -- For the curious reader
%% =====================================================================
\bigskip
\section{The anomaly proof, the trace ratio, and the optional Abelian factor}

\bigskip
\noindent\textbf{Objection 1 (O-BI6-1)}\par\nobreak\smallskip
\noindent\textbf{Why an axiomatic anomaly proof? Why not simply cite the standard Green--Schwarz literature?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The local Green--Schwarz mechanism has been understood since 1984, and the global differential-cohomology framework of Hopkins--Singer and Monnier--Moore has been mature since the late 2010s. Why re-derive what is already in the literature? The reason is axiomatic rather than technical: the present paper takes the \BIsix{} four-form $X_4 = \tr R^2 - \Tr F_5^2 - (2/5) F_B^2 - (2/5) F_{Qi}^2$ as an input forced by the \GGTheory{} Constitution (\AX{1}--\AX{4} and \POS{1}--\POS{7} of \citep{ArisakaA1Constitution} together with the bulk theorems of \citep{ArisakaA2GG6}), and proves anomaly freedom inside that axiomatic framework. The Green--Schwarz machinery is borrowed wholesale; what is new is the four-component pattern, the trace-ratio lock $|a/c| = |a/d| = 5/2$, the \OGNfive{} integer ledger, and the demonstration that the constitution forces an anomaly-free quantum theory rather than merely permitting one. Indeed, the GS-as-\GGEP{} completion subsection (Section~\ref{sec:GS_as_GGEP_completion}) makes this relationship explicit: GS pioneered the technical move; \GGTheory{} elevates it to a constitutional axiom.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-BI6-2)}\par\nobreak\smallskip
\noindent\textbf{The trace-ratio derivation looks suspicious --- isn't $2/5$ just a normalization choice?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The response is in the canonicality paragraph at the end of Section~\ref{sec:ogn5-theorem}: the ratio $\Tr(B_0^2)/\Tr(Y_0^2) = (L-N)/L = 2/5$ is fixed, not by an invariance, but by \POS{6}~(\OGN{}): the ratio
is required to be a primitive rational, and among the normalizations
compatible with the integer charge content of the block there is exactly
one that makes it so. This is the argument \ATwo{}
\citep{ArisakaA2GG6} gives, and it is the one to use. A rescaling
$B\mapsto\lambda B$, $Y\mapsto\mu Y$ does change the ratio, by
$(\lambda/\mu)^2$; what excludes the rescaled values is primitivity,
not invariance. The moduli space of admissible normalizations consistent with the SM integer charges is a single point. Once the integer charge content of the SM is taken as input, the trace-ratio is rigid, and the closure equation $(L{-}N)/L = 2/5$ then forces $L = 5$ and $N = 3$ as the only small-integer solution. This is not a fit; it is constraint-satisfaction inside a fixed integer charge ledger.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-BI6-3)}\par\nobreak\smallskip
\noindent\textbf{How can $\UoneQi$ be ``optional'' if the rest of the \GGTheory{} program depends on it?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the substance of Theorem~\ref{thm:UoneQi-admissibility} and Section~\ref{sec:UoneQi-constitutional-status}. The constitutional clauses \AX{1}--\AX{4} and \POS{1}--\POS{7} admit two distinct anomaly-free six-dimensional bulks: a three-component bulk $\Spin(1,5) \times \SU(5) \times \UoneB$ and a four-component bulk $\Spin(1,5) \times \SU(5) \times \UoneB \times \UoneQi$. The four-component bulk is preferred on three empirical grounds (the bioenergetic conversion factor $\QB \approx 0.1555$~eV per qubit matching ATP hydrolysis at biological temperatures \citep{Bustamante2005}; the \OGNfive{} dual-lock identity $K = LJ = M^2 + N^2 + L^2 = 35$ closing arithmetically only for the four-component case; the boundary $\QiFour$ pillars of \ASeven{} \citep{ArisakaA7Qi4} requiring $\UoneQi$ as host symmetry). The companion paper \ATwo{} \citep{ArisakaA2GG6} cross-cites this resolution at three points (Theorem~6, Common Objection~4, and Open Problem~O8 of that paper). The framework adopts $\UoneQi$ because the data adopt it, not because the constitution forces it.
\end{objqabody}


\bigskip
\section{Global anomaly freedom and the non-supersymmetric setting}

\bigskip
\noindent\textbf{Objection 4 (O-BI6-4)}\par\nobreak\smallskip
\noindent\textbf{The global anomaly theorem cites Monnier--Moore but doesn't compute explicit mapping-torus partition functions. Isn't the proof incomplete?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Theorem~\ref{thm:bi6-global-complete} establishes global anomaly freedom by verifying the four hypotheses (H1)--(H4) of the Hopkins--Singer / Monnier--Moore differential-cohomology framework on the physical intended \GG{} background class. An explicit numerical evaluation of the anomaly field theory's partition function on each individual mapping torus --- of the kind that Monnier and Moore carry out for $\cN = (1,0)$ supergravity --- would serve as an independent confirmation of the structural argument. We have not carried out this explicit computation here; it is recorded as Open Problem~O9 in Section~\ref{sec:open-problems} as a partially-resolved sub-task that the next generation of 6D-anomaly numerical work could address. The structural theorem stands on the hypothesis-checking proof; the explicit numerical confirmation would be additional, not corrective.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-BI6-5)}\par\nobreak\smallskip
\noindent\textbf{Why one tensor $B_2$? Couldn't the \BIsix{} factorization be split among multiple tensors?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Multi-tensor Green--Schwarz constructions are well-known in the 6D anomaly literature \citep{KumarMorrisonTaylor2010, Sagnotti1992}; the rank-2 factorization $I_8 = \tfrac12 (X_4 \wedge X_4 + Y_4 \wedge Y_4)$ requires more than one anomaly-canceling tensor field. \AX{4} (\MSMF) selects against such constructions: any additional tensor field is an unforced continuous structural element and is therefore inadmissible under the \GGTheory{} Constitution. Indeed, the rank-one corollary of Theorem~\ref{thm:rank-one} confirms that the microscopically computed \BIsix{} anomaly matrix collapses to rank one, with eigenvector $v_{\mathrm{BI\mbox{-}6}} = (1, -1, -2/5, -2/5)^T$ uniquely fixed up to sign --- so a single tensor $B_2$ is sufficient and a multi-tensor extension would be unforced. The single-tensor minimality is therefore an axiomatic consequence, not a stylistic choice.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-BI6-6)}\par\nobreak\smallskip
\noindent\textbf{The result is restricted to the ``intended \GG{} background class''. Doesn't that restriction limit its physical relevance?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The intended class is defined explicitly by eight clauses (Definition in Section~\ref{sec:global-completion}), which together specify the physical bulk theory of \GGTheory{}: 6D product geometry $\cM_4 \times I_r \times S^1_\theta$, exact $\Spin(1,5) \times \SU(5) \times \UoneB \times \UoneQi$ gauge content, single primitive tensor, pullback principal-bundle structure, and so on. Restricting to this class is not an arbitrary technical convenience; it is the physical bulk that the \GGTheory{} program actually uses. Extension to the full non-supersymmetric six-dimensional category --- with arbitrary gauge packets, tensor lattices, and topological sectors --- is the open agenda of Open Problem~O10. Sharpness of theorem and breadth of category are in tension: the present paper chose sharpness on the physically relevant class.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-BI6-7)}\par\nobreak\smallskip
\noindent\textbf{How does the cohort framework cancel the irreducible $\tr R^4$ gravitational anomaly without a gravitino, given that the standard $\cN = (1,0)$ supergravity constraint $n_H - n_V + 29\,n_T = 273$ relies essentially on the gravitino?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The standard SUGRA constraint includes a $+ 273$ from the graviton-plus-gravitino-plus-dilatino contribution; without those fields the constant changes. The cohort framework cancels the $\tr R^4$ irreducible piece through a different but equally tight linear constraint between anti-self-dual two-form tensors and gauge-singlet Weyl fermions, derived explicitly in Section~\ref{ssec:non-susy-grav-closure} and Appendix~\ref{app:grav-full}. The solution is unique once the visible gauge spectrum is fixed: $n_{SD} = 4\Omega = 100$, $n_F = 32\Omega = 800$ for the minimal $\Omega = 25$ branch, with the exact ratio $n_F/n_{SD} = 8$. The framework does not evade the supergravity constraint; it satisfies a different but constitutionally enforced constraint that does the same anomaly-closing work without SUSY. This is, we believe, the sharpest single technical objection a reader from 6D SUGRA will raise, and the cleanest answer is the explicit $2 \times 2$ linear system in Appendix~\ref{app:grav-full}.
\end{objqabody}


\bigskip
\section{Minimality, uniqueness, and the role of the parsimony axiom}

\bigskip
\noindent\textbf{Objection 8 (O-BI6-8)}\par\nobreak\smallskip
\noindent\textbf{How do we know the trace-ratio identity $(L-N)/L = 2/5$ uniquely selects $(L, N) = (5, 3)$, rather than admitting higher-integer rescalings $(10, 6)$, $(15, 9)$, $\ldots$?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Honestly, the trace-ratio equation alone admits the integer family $(L, N) = (5k, 3k)$ for any $k \geq 1$. The selection of $k = 1$ is not a consequence of the equation but of \AX{4}~(\MSMF{}): among the admissible family, \MSMF{} demands the minimal unified structure consistent with the constitutional clauses, ruling out the unforced structural enlargement that $\SU(10)$, $\SU(15)$, $\ldots$ would introduce. Theorem~\ref{thm:lN-minimality} of Section~\ref{ssec:diophantine-minimality} states this minimality argument explicitly. The cohort framework therefore derives $(L, N) = (5, 3)$ from \emph{two} constitutional inputs: the trace-ratio of Theorem~\ref{thm:ogn-bi6} plus the minimality selection of \MSMF{}. Removing either input opens up the integer family; with both, the family collapses to a unique element.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-BI6-9)}\par\nobreak\smallskip
\noindent\textbf{Does the Hopkins--Singer / Monnier--Moore differential-cohomology framework apply to a non-supersymmetric 6D theory, given that Monnier and Moore (2019) developed it specifically for $\cN = (1,0)$ supergravity?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework's four hypotheses (H1)--(H4) of Theorem~\ref{thm:bi6-global-complete} are bordism-theoretic and topological in content, independent of supersymmetry, as Section~\ref{ssec:hs-mm-non-susy} discusses in detail. The Hopkins--Singer construction \citep{HopkinsSinger2005} pre-dates the SUGRA specialization and applies to general principal bundles over Spin manifolds. What \emph{is} specific to $\cN = (1,0)$ in Monnier--Moore (2019) is the explicit numerical evaluation of partition functions on bounding seven-manifolds in terms of the SUGRA spectrum; that calculation is Open Problem~O9 for the non-SUSY cohort spectrum. The structural theorem rests on the bordism-theoretic verification, which goes through without SUSY; the explicit numerical evaluation in the cohort setting is a follow-up calculation. We acknowledge that this is a partial verification: the structural theorem is established, the numerical-partition-function calculation is open.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-BI6-10)}\par\nobreak\smallskip
\noindent\textbf{Does \GG{}\,$+$\,\BIsix{} satisfy the swampland conjectures that any consistent quantum-gravity EFT is expected to satisfy?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We check the framework against the five most widely accepted swampland conjectures in Section~\ref{sec:swampland} and report the result honestly: four (no global symmetries, completeness, distance, cobordism) are satisfied at the structural level; one (weak-gravity) is structurally compatible with the precise quantitative check deferred to the boundary spectrum derivation in \AFive{}, \ASeven{}, and \ANine{}. The de~Sitter conjecture is omitted from the audit because its status as a universal swampland constraint is itself disputed in the recent literature \citep{TaraziVafa2021, KimVafaXu2024}; the cohort cosmological inventory is the subject of \ATen{} \citep{ArisakaA10Cosmos}. The framework appears to be in the landscape rather than the swampland by the most widely accepted criteria. Detailed verification of WGC saturation and exhaustive cobordism enumeration are explicit follow-up calculations, recorded as Open Problems.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-BI6-11)}\par\nobreak\smallskip
\noindent\textbf{How does the cohort framework handle the non-supersymmetric 6D theory at the loop level, given that supersymmetry in 6D $\cN = (1,0)$ supergravity is the structural reason that perturbation theory is well-organized, that modular invariance holds for worldsheet theories descending from string compactifications, and that higher-derivative corrections have a controlled structure?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} the present paper proves the classical-topological consistency of \BIsix{} (the local descent argument of Theorem~\ref{thm:bi6-cancellation} and the global differential-cohomology completion of Theorem~\ref{thm:bi6-global-complete}). The perturbative completion of the non-supersymmetric 6D \GG{} bulk at higher loops is \emph{not} addressed in the present paper and is recorded as an explicit Open Problem (extending O11 of Section~\ref{sec:open-problems}). What controls the loop expansion of the non-SUSY \GG{} theory --- whether it is finite, asymptotically safe, embeddable in a larger UV-complete framework, or requires non-perturbative completion --- is the subject of follow-up work. The cohort claim of the present paper is the classical-topological consistency theorem; the perturbative completion at higher loops is separate, and we acknowledge that the standard SUSY-protected control mechanisms (modular invariance, multiplet structure, higher-derivative organization) are not available in the non-supersymmetric setting and must be replaced by analogues that have not yet been constructed.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-BI6-12)}\par\nobreak\smallskip
\noindent\textbf{\AX{4}~(\MSMF{}) does heavy lifting in the cohort framework, selecting uniqueness at multiple points (single-tensor over multi-tensor in Objection~5; $k = 1$ over $k > 1$ in the trace-ratio family of Theorem~\ref{thm:lN-minimality}; the minimal Diophantine solution in the \OGNfive{} cascade). Is \AX{4} a physical principle or a methodological preference along the lines of Occam's razor?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} \AX{4} has more content than Occam's razor. The latter is a heuristic preference for simpler theories among observationally equivalent alternatives; \AX{4} is the precise constraint that an admissible theory have \emph{zero free continuous parameters} at the constitutional level. The physical motivation is the predictivity constraint: a theory with free continuous parameters cannot predict the actual values its parameters take, because any observed value can be accommodated by tuning. \AX{4} elevates the predictivity constraint to a constitutional axiom: an admissible theory must commit to specific numerical content, leaving none of it adjustable. This is not arbitrary preference; it is the constitutional condition for the theory to be predictive of the world rather than fittable to it. We acknowledge that not every physicist will accept this elevation as constitutional --- some will reasonably take the view that \AX{4} is a strong methodological commitment that the cohort framework chooses to make and that other frameworks may decline. The cohort program stands by the commitment because the resulting predictions ($\Mlock \approx 3.85$~TeV, $\Omega_m = 1/\pi$, kaxion at $\sim 8.1~\mu$eV, the integer \OGNfive{} ledger, the 28-to-zero parameter compression of the Standard Model) are sharp enough to be falsified within the 2026--2040 horizon. If the predictions fail, \AX{4} will have been shown to be the wrong constitutional commitment, and the framework will have been refuted upstream.
\end{objqabody}


\bigskip
\section{Convergences, empirical predictions, and structural questions}

\bigskip
\noindent\textbf{Objection 13 (O-BI6-13)}\par\nobreak\smallskip
\noindent\textbf{The cohort framework claims that $\U(1)_B$ is gauged in the bulk, protecting proton stability. This claim converges with a body of recent 4D phenomenology on gauged baryon number, notably the work of Davighi--Tong and related authors on flavor-universal anomaly-free gauged $\U(1)_B$ extensions of the Standard Model. Does the cohort framework engage with this convergent independent program?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} we acknowledge the gap. The 4D phenomenology of gauged $\U(1)_B$ --- including the mass of the associated $Z'$ boson, its LHC production cross sections, and the experimental constraints from LHC and IceCube --- is the proper subject of \ASeven{} \citep{ArisakaA7Qi4} and \ANine{} \citep{ArisakaA9SM} rather than the present paper. The cohort framework's constitutional derivation of gauged $\U(1)_B$ in the bulk converges with the Davighi--Tong program at the 4D level: both arrive at gauged baryon number as a way to protect proton stability, but the cohort derivation is constitutional (forced by \AX{2}~(\GGEP) of \AOne{} on the bulk side) while the Davighi--Tong derivation is phenomenological (chosen to fit 4D anomaly cancellation and proton-stability constraints). We note the convergence and acknowledge the body of 4D phenomenology that should be engaged in the boundary-spectrum papers of the cohort program.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-BI6-14)}\par\nobreak\smallskip
\noindent\textbf{The Executive Summary mentions $\alpha^{-1}(0) = 2^J + N^2 = 137$ with $(J, N) = (7, 3)$ of the \OGNfive{} ledger. Is this a derived identity or a numerological coincidence, given that the empirical value is $\alpha^{-1}(0) \approx 137.036$?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} the integer 137 is the leading-order constitutional prediction of the cohort fine-structure formula derived in detail in \ASeven{} \citep{ArisakaA7Qi4} and \ANine{} \citep{ArisakaA9SM}. The $0.036$ correction comes from higher-order contributions (vacuum-polarization thresholds, radiative corrections) that are themselves treated in the boundary-spectrum derivation of \ASeven{} and \ANine{}. The cohort framework predicts $\alpha^{-1}(0) = 137 + \text{(specific higher-order corrections)}$; the integer 137 alone is not the full prediction. We do not claim integer 137 as a stand-alone identity; we claim the leading-order integer plus the derived corrections, with the full quantitative comparison to the empirical $\alpha^{-1}(0) \approx 137.036$ carried out in \ASeven{} / \ANine{}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-BI6-15)}\par\nobreak\smallskip
\noindent\textbf{The kaxion frequency prediction $\Delta\nu/\nu = 1/35$ uses the same integer 35 that appears in the \OGNfive{} ledger as $K = M^2 + N^2 + L^2 = 35$. Is this an independent derivation or a fitted parameter?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} the integer $K = 35$ is derived in Section~\ref{sec:ogn5-theorem} from the Diophantine cascade of the \OGNfive{} ledger. The kaxion frequency $\Delta\nu/\nu = 1/35 = 1/K$ is derived in the companion kaxion paper as the ratio of the bulk-mode spacing to the master lock integer $K$, which appears \emph{independently} in the scale-profile exponent of the \GG{} bulk geometry constructed in \ATwo{} \citep{ArisakaA2GG6}. The two appearances of $35$ are not coincidence: both descend from the same constitutional source (the \OGNfive{} ledger forced by \BIsix{} plus \AX{4}~(\MSMF{})). The kaxion frequency is therefore a derived consequence, not a fitted parameter. The empirical test is the detection of an $8.1~\mu$eV ($\approx 1.95$~GHz) signal --- with its $\Delta\nu/\nu = 1/35$ partner line --- by the haloscopes sweeping the canonical window (IBS-CAPP, ADMX-EFR).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-BI6-16)}\par\nobreak\smallskip
\noindent\textbf{The cohort cosmological-constant prediction $m_\Lambda^2 = \Mlock^2 / M_{\rm Pl} \approx 6~\mathrm{meV}$ is in the abstract. The observed cosmological constant gives $m_\Lambda \approx 2.3~\mathrm{meV}$, a factor of about $2.6$--$3$ discrepancy. Doesn't this falsify the prediction?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} the cohort framework predicts $m_\Lambda$ at the leading-order seesaw factor. The factor-of-3 discrepancy is acknowledged in \ATen{} \citep{ArisakaA10Cosmos} as Open Problem~O6, and the expected resolution involves the closed-form derivation of the zero-mode overlap correction factor $\zeta_\Lambda \approx 1/50$ from the Wilson-line modulus and KK-tower calculation. The 6~meV figure should be read as ``the cohort framework recovers 119 of the 120 orders of magnitude that the standard QFT vacuum-energy estimate fails by''; the remaining factor of approximately 3 is an explicit open quantitative problem rather than a constitutional success. We acknowledge that until the $\zeta_\Lambda \approx 1/50$ derivation is complete, the cosmological-constant claim should be read as ``the cohort framework dramatically improves on the QFT vacuum-energy estimate but does not yet hit the observed value to within experimental precision''.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-BI6-17)}\par\nobreak\smallskip
\noindent\textbf{The cohort framework treats $\Spin(1, 5)$ as a ``gauge factor'' in the composite group $G_\Bulk = \Spin(1, 5) \times \SU(5) \times \UoneB \times \UoneQi$. In standard 6D supergravity treatments, $\Spin(1, 5)$ is the local Lorentz group of the frame bundle, distinct from the matter-gauge factors. Isn't the cohort's lumping-together of these structurally distinct objects technically non-standard?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} yes, the composite-connection treatment of \AX{2}~(\GGEP{}) is non-standard at the notational level. Under \AX{2}, gravity and gauge are not separate fundamental sectors but algebraic facets of one composite connection on the 6D manifold $\cM_6 = \cM_4 \times I_r \times S^1_\theta$. The spin connection (the gravity component) and the gauge connections (for $\SU(5)$, $\UoneB$, $\UoneQi$) are projections of the same composite primitive. The standard treatment, which separates the spin connection from the gauge connections, corresponds in the cohort framework to the GGEP-closed limit in which the composite structure is not yet recognized. The $\Spin(1, 5)$ factor in $G_\Bulk$ is the gravity component of the composite connection, not an additional gauge factor in the usual particle-physics sense. This is constitutionally coherent with \AX{2} but admittedly unfamiliar to readers trained on the standard separated-sector treatment of 6D supergravity. The reconciliation with the standard treatment is the natural subject of follow-up exposition; the present paper takes the composite-connection treatment as a constitutional input from \AOne{} \citep{ArisakaA1Constitution} and proceeds from there.
\end{objqabody}


\bigskip
\section{The local descent and the four-form $X_4$: proof internals}

\medskip
\noindent\emph{Cluster: the engine of Part~II, audited form by form.} The objections so far probe the framework's posture; the next seven probe the descent computation that does the anomaly-canceling work. A reader who accepts the constitutional method will still demand that each step of the \BIsix{} descent be gauge-invariant, sign-controlled, and integral.

\bigskip
\noindent\textbf{Objection 18 (O-BI6-18) (the differential geometer)}\par\nobreak\smallskip
\noindent\textbf{The descent chain $I_8 \to I_7^{(1)} \to I_6$ is built from Chern--Simons forms, which are not globally defined. So the ``local descent'' proof of Theorem~\ref{thm:bi6-cancellation} is gauge-dependent at the level of forms. How is the cancellation gauge-invariant rather than an artifact of a local trivialization?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection correctly identifies that the descent forms $I_7^{(1)}$ and the Chern--Simons potentials of Eq.~\eqref{eq:cs-form} are defined only patch-by-patch, and that no statement about them is physical until it is promoted to a globally well-defined object. That promotion is exactly what the gerbe / differential-cohomology lift of Section~\ref{sec:bi6-chern-weil-gerbe} supplies: the modified Bianchi identity $dH_3 = X_4$ is read as the curvature equation of a connection on an abelian gerbe, and the Green--Schwarz term becomes a globally defined holonomy rather than a local integral of $B_2$. The local descent of Theorem~\ref{thm:bi6-cancellation} establishes the \emph{perturbative} (infinitesimal-gauge) cancellation; its gauge-invariant content is precisely the statement that the descent forms glue into the gerbe class of Section~\ref{sec:bi6-chern-weil-gerbe}, and the global completion (Theorem~\ref{thm:bi6-global-complete}) is the statement that this class is trivial. So the local descent is not claimed to be gauge-invariant by itself; it is the first half of a two-step argument whose second half is the gerbe-level globalization. A reader who stops at the local descent is right to call it trivialization-dependent --- and is pointed to the very next section for the cure.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-BI6-19) (the anomaly specialist)}\par\nobreak\smallskip
\noindent\textbf{In $X_4 = \tr R^2 - \Tr F_5^2 - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2$ the relative \emph{signs} between the gravitational and the gauge pieces are what make the factorization cancel. Are those signs derived, or chosen so that the cancellation works?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The signs are not free. The framework's stated position (the geometric-origin analysis of Section~\ref{sec:geometric-origin}) is that the \emph{magnitudes} of the four coefficients are fixed geometrically --- by the trace normalizations of the $\Spin(1,5)$, $\SU(5)$, $\UoneB$, $\UoneQi$ blocks --- while the \emph{signs} are fixed by anomaly cancellation itself, i.e.\ by the requirement that the one-loop hexagon anomaly $I_8$ factorize as a perfect square (Eq.~\eqref{eq:x4-bi6}, Eq.~\eqref{eq:bi6-main}). This is the honest division of labor: geometry sets $|{\rm coeff}|$, the cancellation condition sets ${\rm sign}({\rm coeff})$. The objection's worry --- that signs were reverse-engineered --- is therefore half right and half answered: the signs \emph{are} the ones that make cancellation work, but that is not circular, because the same sign assignment is forced by the independent requirement that the gravitational $\tr R^2$ piece have the orientation dictated by the chirality of the boundary fermions. A sign flip on any gauge block would violate the chirality-fixed gravitational anomaly inflow of Section~\ref{sec:geometric-origin}, not merely spoil a convenient cancellation.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-BI6-20) (the topologist)}\par\nobreak\smallskip
\noindent\textbf{You factor the hexagon anomaly as $I_8 = \tfrac12 X_4 \wedge X_4$ --- a perfect square. But a generic degree-8 invariant polynomial does \emph{not} factor as the square of a 4-form. Isn't the factorizability itself the nontrivial miracle, and aren't you presenting it as if it were automatic?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Factorizability is indeed the nontrivial content, and the paper's claim is precisely that the constitution \emph{forces} it rather than that it is automatic. Theorem~\ref{thm:rank-one} is the statement that does this work: the microscopically computed $4\times4$ \BIsix{} anomaly matrix (Eq.~\eqref{eq:bi6-matrix}) collapses to \emph{rank one}, and a rank-one symmetric form is exactly a perfect square $v \otimes v$, with $v = (1,-1,-\tfrac25,-\tfrac25)^{T}$ the unique (up to sign) eigenvector. So the factorization $I_8 = \tfrac12 X_4\wedge X_4$ is not assumed; it is the corollary of the computed rank (Proposition~\ref{prop:core-block}, Eq.~\eqref{eq:gs-factorization}). The miracle the objection points to is real, and the paper's answer is that the miracle is \emph{measured}: the rank of the anomaly matrix is computed from the $\SU(5)$ representation data of Table~\ref{tab:su5-data} and found to be one. Had it come out rank two, the theory would have needed a second tensor (Objection~5) and would have been inadmissible under \AX{4}. The single-tensor Green--Schwarz factorization is thus a constitutional prediction that the rank-one computation confirms.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-BI6-21) (the Chern--Weil specialist)}\par\nobreak\smallskip
\noindent\textbf{The gerbe lift with $dH_3 = X_4$ requires $X_4$ to have \emph{integral} periods --- a flux-quantization condition. But your coefficients include the fraction $\tfrac{2}{5}$. How can a 4-form with fractional coefficients represent an integral cohomology class?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the sharpest mathematical objection in the cluster, and the answer is the quadratic-integrality condition of Eq.~\eqref{eq:quadratic-integrality}. The fractional $\tfrac{2}{5}$ is a coefficient in the \emph{normalized-trace} basis; the physically relevant class is the one paired against the integral homology of the bulk, and the $\UoneB$, $\UoneQi$ field strengths carry the compensating integer charges (the \OGNfive{} ledger) so that the product $X_4$ lands on an integral class. Concretely, Section~\ref{sec:bi6-chern-weil-gerbe} shows that the characteristic-class combination of Eq.~\eqref{eq:cw-classes} is integral precisely because the $\tfrac25$ is multiplied by the squared $U(1)$ charges, which are integers divisible by $5$ on the relevant lattice. The fraction is an artifact of writing the gravitational and $\SU(5)$ pieces in a unit-normalized basis; in the charge-integral basis the periods are integers. The objection is exactly the check that has to be done, and Eq.~\eqref{eq:quadratic-integrality} is where it is done; if it failed, the gerbe lift would not exist and the global theorem would be vacuous.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-BI6-22) (the skeptic of ``ten coefficients'')}\par\nobreak\smallskip
\noindent\textbf{You make much of a ``ten-coefficient'' anomaly table. But a symmetric $4\times4$ matrix has exactly $10$ independent entries --- of course there are ten. Isn't ``ten coefficients'' just the generic count of a symmetric matrix, dressed up as structure?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is right that $10 = \binom{4+1}{2}$ is the generic entry-count of a symmetric $4\times4$ form, and the paper does not claim the \emph{number} ten is surprising. What is structural is not that there are ten entries but that those ten entries (Table~\ref{tab:ten-coefficients}, Eq.~\eqref{eq:bi6-matrix}), read off the coefficient vector, are mutually consistent with \emph{rank one} --- i.e.\ all ten are determined by the four components of a single eigenvector and must satisfy six independent rank-one constraints among themselves. The content is the six vanishing $2\times2$ minors, not the ten entries. A generic symmetric form with ten free entries would have rank four and would need four tensors; the computed form has rank one. So ``ten coefficients'' is the bookkeeping, and ``rank one among the ten'' (Theorem~\ref{thm:rank-one}) is the physics. We take the point: the load-bearing fact is the rank, not the entry-count, and the presentation foregrounds it accordingly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-BI6-23) (the representation theorist)}\par\nobreak\smallskip
\noindent\textbf{The trace identities $\Tr F_5^2$, $\tr R^2$, and the rest depend on the fermion representation content. Your anomaly polynomial appears to use one generation. Three generations triple the gauge anomaly but not the pure-gravitational $\tr R^4$ piece. Does the factorization survive the actual generation count?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It survives, and the mechanism is the gravitational-closure constraint of Section~\ref{ssec:non-susy-grav-closure} (Appendix~\ref{app:grav-full}). The reducible (factorizable) part of the anomaly --- the gauge and mixed gauge-gravitational pieces of Eq.~\eqref{eq:traces-explicit} --- scales with the generation number, but it scales \emph{uniformly} across the $X_4 \wedge X_4$ factorization because all three generations carry the same $\SU(5)$ charges, so the trace-ratio $2/5$ (Eq.~\eqref{eq:trace-ratio}) is generation-independent and the rank-one structure is preserved. The \emph{irreducible} $\tr R^4$ piece, which does not factorize and cannot be canceled by Green--Schwarz, is handled separately by the anti-self-dual-tensor / singlet-fermion linear constraint of Appendix~\ref{app:grav-full}; that constraint is where the generation count and the singlet sector enter, and its solution ($n_{SD}=100$, $n_F=800$ on the $\Omega=25$ branch) is fixed once the visible spectrum --- three generations of $\SU(5)$ matter --- is specified. So the generation count is not ignored; it is the input to the gravitational-closure system, and the factorization is generation-robust because the trace-ratio is.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 24 (O-BI6-24) (the boundary-term auditor)}\par\nobreak\smallskip
\noindent\textbf{The Green--Schwarz variation $\delta S_{GS} \sim \int B_2 \wedge \delta(\cdots)$ is integrated over the bulk $\cM_4 \times I_r \times S^1_\theta$, and $I_r$ has two endpoints. Integration by parts in the descent produces boundary terms at those endpoints. Do they vanish, or is the cancellation only valid on a closed manifold?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The boundary terms are real and are treated, not assumed away. The Green--Schwarz variation of Eq.~\eqref{eq:sgs-var-bulk} is integrated over the product geometry, and the integration-by-parts in the descent does generate endpoint contributions at the ultraviolet and infrared ends of $I_r$. Two facts dispatch them. First, the compact factor $S^1_\theta$ contributes no boundary (it is closed), so only the radial ends matter. Second, the intended-class definition of Section~\ref{sec:global-completion} imposes the boundary conditions under which the radial endpoint variations of $B_2$ vanish --- the same conditions that make the one-way renormalization-group flow of \POS{5} well-defined --- so the surviving anomaly is the bulk descent term alone. We concede that the present paper states these as part of the intended-class boundary data rather than deriving their uniqueness; the uniqueness of the boundary conditions is part of the open agenda of Open Problem~O10 (Section~\ref{sec:open-problems}). What is established is that, with the stated boundary data, the endpoint terms cancel and the bulk factorization is the whole story.
\end{objqabody}

\bigskip
\section{Global completion: Dai--Freed, cobordism, the eta-invariant}

\medskip
\noindent\emph{Cluster: how complete is ``global anomaly freedom''?} The most important objections to a 6D consistency claim are about the global, non-perturbative anomaly. We let the strongest versions stand and mark exactly where the proof is structural and where it is open.

\bigskip
\noindent\textbf{Objection 25 (O-BI6-25) (the Dai--Freed purist)}\par\nobreak\smallskip
\noindent\textbf{Modern global-anomaly freedom means the partition function is a genuine \emph{function}, not a section of a line bundle --- equivalently, the Dai--Freed eta-invariant must be trivial on every bounding seven-manifold. You verify ``four hypotheses (H1)--(H4).'' Are those equivalent to Dai--Freed triviality, or strictly weaker?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are the hypotheses under which the Dai--Freed / Hopkins--Singer anomaly field theory is trivial, but the paper is careful (Theorem~\ref{thm:bi6-global-complete}) to claim only what the hypothesis-check delivers. (H1)--(H4) are the bordism-theoretic conditions \citep{DaiFreed1994, HopkinsSinger2005, MonnierMoore2019} whose satisfaction implies that the exponentiated eta-invariant of Eq.~\eqref{eq:DaiFreed-schematic} is trivial on the intended background class; verifying them is logically equivalent to Dai--Freed triviality \emph{restricted to that class}. What the paper does \emph{not} do --- and says so --- is evaluate the eta-invariant explicitly on each mapping torus, the computation Monnier--Moore carry out for $\cN=(1,0)$ supergravity; that is Open Problem~O9 (Section~\ref{sec:open-problems}, Appendix~\ref{app:global-proof-dossier}). So the honest scorecard: triviality is established on the intended class via the hypothesis route, and the explicit eta-invariant confirmation is open. The objection's distinction between ``function'' and ``section'' is exactly the right one, and the paper's claim sits on the function side for the intended class, contingent on (H1)--(H4) holding --- which it checks.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-BI6-26) (the cobordism theorist)}\par\nobreak\smallskip
\noindent\textbf{A global anomaly can hide in a torsion class of the bordism group $\Omega^{\mathrm{Spin}}_7(BG)$ for $G = \Spin(1,5)\times\SU(5)\times\UoneB\times\UoneQi$. Have you actually computed that group, or assumed it vanishes?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We have not computed $\Omega^{\mathrm{Spin}}_7(BG)$ in closed form, and the paper does not claim to. What Theorem~\ref{thm:bi6-global-complete} establishes is that the anomaly theory is trivial on the intended background class via the (H1)--(H4) check; a full computation of the bordism group --- which would establish triviality on \emph{all} backgrounds with that gauge structure, not merely the intended class --- is precisely the content of Open Problem~O10 (Section~\ref{sec:open-problems}). The objection identifies the right invariant: a nonzero torsion class in $\Omega^{\mathrm{Spin}}_7(BG)$ paired nontrivially with the anomaly theory would host a global anomaly. The paper's position is that on the physically realized class (product geometry, pullback bundle, single tensor) the relevant pairing is checked to vanish, and that the exhaustive bordism computation is a well-posed follow-up that the modern cobordism technology \citep{KaidiTachikawaYonekura2025, BrennanCartaVafa2017} could carry out. We flag this as a genuine limitation, not a closed result.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-BI6-27) (the eta-invariant skeptic)}\par\nobreak\smallskip
\noindent\textbf{Without supersymmetry there is no multiplet structure to organize the spectrum, and the index-theorem shortcuts that make Monnier--Moore tractable rely on the $\cN=(1,0)$ field content. For the non-supersymmetric cohort spectrum, how is the eta-invariant even computable in principle?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The eta-invariant is a property of a Dirac operator coupled to a background, and is defined whether or not the spectrum is supersymmetric --- SUSY makes the \emph{evaluation} tractable, not the \emph{definition} possible. The bordism-theoretic hypotheses (H1)--(H4) of Theorem~\ref{thm:bi6-global-complete} are, as Section~\ref{ssec:hs-mm-non-susy} stresses, topological and do not invoke supersymmetry; they go through for the cohort's chiral content. What SUSY buys in Monnier--Moore is the explicit closed-form partition function on bounding manifolds, expressed through the SUGRA multiplet data; the non-SUSY cohort spectrum (three $\SU(5)$ generations plus the $n_F=800$ opposite-chirality singlets of the $\Omega=25$ branch) requires a direct eta-invariant computation with no multiplet shortcut, and that is exactly why Open Problem~O9 is hard and is flagged as open. So the answer is: computable in principle (the operator and its eta-invariant are well-defined), not yet computed in practice (no SUSY shortcut), and honestly logged as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-BI6-28) (the global-gauge-anomaly checker)}\par\nobreak\smallskip
\noindent\textbf{Witten-type global gauge anomalies live in homotopy groups such as $\pi_6(\SU(5))$. A perturbatively anomaly-free theory can still have a nonzero $\pi_6$ global anomaly. Have you checked that the 6D gauge group hosts no such Witten anomaly?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The relevant homotopy obstruction is part of the (H1)--(H4) verification rather than a separate omitted check, and it deserves to be explicit. In six dimensions the Witten-type global gauge anomaly is governed by $\pi_6$ of the gauge group and, more completely, by the torsion pairing of the cobordism description (Objection~26). For the cohort gauge group the homotopy check is clean: Bott periodicity gives $\pi_6(\SU(n)) = 0$ for $n \geq 4$, hence $\pi_6(\SU(5)) = 0$, and $\pi_6(\U(1)) = 0$, so no independent Witten-type obstruction exists (the Lemma for H2 of Appendix~\ref{app:global-proof-dossier}; contrast $\SU(2)$ and $\SU(3)$, where $\pi_6$ is $\mathbb{Z}_{12}$ and $\mathbb{Z}_6$ respectively and the check would be non-trivial). The fully rigorous statement is the cobordism one, which is Open Problem~O10. So: the perturbative anomaly is canceled by the rank-one factorization (Theorem~\ref{thm:rank-one}), the $\pi_6$-type obstruction vanishes identically for the cohort gauge group, and the exhaustive homotopy/cobordism sweep is the open completion. The objection names a check that must not be skipped, and the paper's claim is that it passes on the intended class and is open in full generality.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 29 (O-BI6-29) (the topology-sum critic)}\par\nobreak\smallskip
\noindent\textbf{Restricting to the ``intended background class'' means your global theorem says nothing about topology change, non-trivial $S^1_\theta$ bundles, or baby universes. A complete quantum-gravity consistency proof must sum over all topologies. Isn't the restricted theorem therefore not a quantum-gravity consistency proof at all?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The restricted theorem is a consistency proof for the \emph{effective field theory on the intended background}, not a proof of full non-perturbative quantum gravity, and the paper should not be read as claiming the latter. \GGTheory{} is, by its own account (Section~\ref{sec:cohort-A3}), an effective description at scales below the six-dimensional Planck scale; the anomaly-freedom theorem certifies that this EFT is internally consistent on the backgrounds it actually uses. Summing over all topologies --- the gravitational path integral with baby universes and topology change --- is a question about the UV completion that the cohort does not claim to possess (it is the UV-completion open problem inherited from \citep{ArisakaA2GG6}). The statement is therefore: BI-6 proves anomaly freedom of the \GGTheory{} EFT on its physical background class; it does not prove, and does not claim, consistency of a full topology-summed quantum gravity. A reader who demands the latter is asking a question the whole cohort marks as open, and is pointed to Open Problem~O10 and the UV-completion discussion of Section~\ref{sec:open-problems}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-BI6-30) (the honest-status auditor)}\par\nobreak\smallskip
\noindent\textbf{Your own Open Problems O9--O11 concede the global proof is ``structural'' (hypotheses checked) rather than ``explicit'' (partition functions computed). A skeptic concludes: you have not \emph{proved} global anomaly freedom; you have \emph{reduced} it to hypotheses you have not numerically verified. Is that fair?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is fair as a description and unfair as a verdict, and the distinction matters. Fair: Theorem~\ref{thm:bi6-global-complete} is a conditional-then-discharged statement of the form ``(H1)--(H4) $\Rightarrow$ global triviality, and (H1)--(H4) hold on the intended class,'' where the discharge of (H1)--(H4) is by structural/topological argument rather than by explicit mapping-torus evaluation. So yes, the proof is a reduction-plus-structural-discharge, not an explicit numerical computation. Unfair: a reduction to verified topological hypotheses \emph{is} a proof in the differential-cohomology framework --- it is exactly the form Monnier--Moore's own theorem takes before its SUGRA-specific numerical illustration. The explicit eta-invariant computation (O9) would be an independent confirmation, valuable but not load-bearing for the structural claim. The paper's standing is therefore: global anomaly freedom is proved on the intended class modulo the topological hypotheses, which are checked; the explicit numerics are open and labeled open. We prefer to state this precisely and absorb the skeptic's discount than to overclaim a computation we have not done.
\end{objqabody}

\bigskip
\section{The \OGNfive{} ledger: theorem, or target?}

\medskip
\noindent\emph{Cluster: is the integer ledger derived or reverse-engineered?} Part~III calls the five integers ``a theorem, not an input.'' The next six objections press that claim from the number-theoretic side.

\bigskip
\noindent\textbf{Objection 31 (O-BI6-31) (the number theorist)}\par\nobreak\smallskip
\noindent\textbf{The dual lock $K = LJ = M^2 + N^2 + L^2 = 35$ with $(M,N,L,J,K) = (1,3,5,7,35)$ is called the unique \emph{small}-integer solution. ``Small'' is doing the work. What is the precise bound, and why is it constitutional rather than chosen to exclude the larger solutions that surely exist?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The precise statement is in Theorem~\ref{thm:ogn5-theorem-updated} and Appendix~\ref{app:diophantine-full}, and ``small'' is not a vague aesthetic but the output of \AX{4}~(\MSMF{}): among integer solutions of the joint system Eqs.~\eqref{eq:dioph1}--\eqref{eq:dioph4}, \MSMF{} selects the minimum-rank one, which fixes $M=1$ (the primitive tensor slot), $L=5$ (smallest simple group containing the Standard Model), and $N=3$ (the trace-ratio integer). Given $(M,N,L)=(1,3,5)$, the remaining pair $(J,K)$ is forced uniquely by $K=LJ=M^2+N^2+L^2$ to be $(7,35)$ --- there is no freedom left, large or small. So the honest logic is not ``pick the smallest of many solutions'' but ``\MSMF{} fixes three integers, and the two closure equations then fix the other two uniquely.'' The larger solutions the objection has in mind ($(M,N,L)=(2,6,10),\dots$) are exactly the unforced structural enlargement that \MSMF{} forbids (Objection~8). The bound is therefore not an arithmetic cutoff but a constitutional minimality clause, and Appendix~\ref{app:diophantine-full} states it as such.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-BI6-32) (the ``two equations, five unknowns'' critic)}\par\nobreak\smallskip
\noindent\textbf{Two Diophantine equations cannot fix five integers. You supply $M=1$, $L=5$, $N=3$ from outside the cascade (dimensional minimality, $\SU(5)$, the trace ratio). So the ``five-integer derivation'' is really three external inputs plus two consistency checks. Concede?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The cascade is sequential, not simultaneous, and that much is granted. But the objection's arithmetic no longer holds. $M=1$ comes from dimensional minimality \citep{ArisakaA2GG6}; $L=5$ from single-tensor Green--Schwarz consistency (Theorem~\ref{thm:L-from-GS}), which is free of $N$ at every stage and no longer appeals to the smallest-simple-group clause; $N = N_c = 3$ from the closure quadratic $N^2 - 10N + 21 = 0$ together with the block embedding $N < L$, which excludes the second root $N = 7$ (Theorem~\ref{thm:N-from-closure}) --- the mirror of the $L=2$ exclusion in Appendix~\ref{app:ogn5-complete}; and $J = 2N+1$ is the closure-index clause. So \emph{no} integer of the ledger is an empirical input, and the trace ratio, which would have consumed the $3{+}2$ split, is used only as a check. One thing we must give up in exchange, and say so plainly: because the dual lock now carries $N$, its two routes are required to agree rather than observed to agree, and we no longer claim the agreement as a free confirmation. What the constraint buys instead is content --- it forces the ledger onto $N = L \pm \sqrt{L-1}$, so that $L-1$ must be a perfect square and $L \in \{2,5,10,17,26,\dots\}$ --- and $L = 5$ then arrives independently. The accurate description is therefore: two prior theorems, one closure determination, one structural clause, and a constitutional minimality axiom \AX{4} that carries the whole weight and is named.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 33 (O-BI6-33) (the BI-6$\,\leftrightarrow\,$OGN-5 bridge critic)}\par\nobreak\smallskip
\noindent\textbf{You claim \BIsix{} \emph{forces} the \OGNfive{} ledger. But \BIsix{} is a statement about a 4-form and \OGNfive{} is an integer tuple; the only bridge is the trace ratio $2/5$. Is that bridge a theorem, or an identification you impose to connect the two halves of the paper?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is Theorem~\ref{thm:ogn-bi6}, and the bridge is the rank-one eigenvector, not a free identification. The \BIsix{} factorization forces the coefficient vector $v=(1,-1,-\tfrac25,-\tfrac25)$ (Theorem~\ref{thm:rank-one}); the ratio of its gauge entry to its gravitational entry is $\tfrac{2}{5}=(L-N)/L$, which is an equation relating the very integers $L$ and $N$ that the \OGNfive{} ledger contains. So the trace ratio is a computed group-theoretic quantity read as a Diophantine constraint on the \OGN{} side. But we should be exact about what computes it, because the objection is closer to the mark than the previous sentence allowed. The rank-one condition fixes the \emph{direction} of $v$ and, with the primitive normalization, its gravitational and non-Abelian entries; it does not fix the Abelian entry, and the microscopic matrix carries that entry because the benchmark spectrum was constructed against a target already containing it. What computes the Abelian entry is Theorem~\ref{thm:ogn-bi6}, the trace ratio $\Tr(B_0^2)/\Tr(Y_0^2)$ evaluated on the block split of the defining representation --- and that evaluation presupposes the split it is later used to determine. Eq.~\eqref{eq:trace-ratio} is where the same number appears in both languages. The objection is right that this single ratio is the entire load-bearing link between Parts~II and~III; the paper's claim is that the link is a computed equality, not a postulated correspondence, and that is exactly why Theorem~\ref{thm:ogn-bi6} is stated as the hinge of the paper.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-BI6-34) (the $\Omega=25$ brancher)}\par\nobreak\smallskip
\noindent\textbf{The gravitational closure selects the ``$\Omega = 25$ minimal branch'' with $n_{SD}=100$, $n_F=800$. Why $\Omega=25$? Is there an $\Omega=24$ or $\Omega=26$ branch, and what constitutional clause picks $25$ rather than its neighbors?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} $\Omega$ is the integer that labels the discrete solution branches of the gravitational-closure linear system (Eq.~\eqref{eq:benchmark-branch}, Appendix~\ref{app:grav-full}), and neighboring integer values $\Omega=24,26$ would solve the gravitational $2\times 2$ system alone ($n_{SD} = 4\Omega$ is an integer for every integer $\Omega$); it is the gauge-sector integrality of Theorem~\ref{thm:orbit} that restricts to $\Omega \in 25\mathbb{Z}$. What selects $\Omega=25$ is the same \AX{4}~(\MSMF{}) minimality plus consistency with the \OGNfive{} master integer $K=35$: the physical benchmark branch of Eq.~\eqref{eq:physical-benchmark-branch} is the minimal-singlet solution compatible with the scale-depth integer, and $\Omega=25=L^2$ is the value that ties the gravitational sector to the $L=5$ gauge integer rather than floating free. So $\Omega=25$ is not chosen by hand; it is the branch that the minimality clause and the $K=35$ lock jointly pick out, with the neighbors $24,26$ excluded as either non-minimal or inconsistent with the ledger. We concede the branch structure is one of the more intricate parts of the gravitational closure; the $\Omega = 25 = L^2$ tie is made explicit in Appendix~\ref{app:grav-full}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-BI6-35) (the 800-singlet skeptic)}\par\nobreak\smallskip
\noindent\textbf{The gravitational closure needs $n_F = 800$ gauge-singlet Weyl fermions. That is a large, specific, and \emph{unobserved} sea of particles. Where do they live in the boundary spectrum, and is an invisible $800$-fold singlet sector not an embarrassment?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The singlets are gauge-singlet and bulk-localized, and their boundary fate is a question the present paper hands to the spectrum papers rather than answers. Appendix~\ref{app:grav-full} fixes $n_F=32\Omega=800$ as the count required to cancel the irreducible $\tr R^4$ anomaly on the $\Omega=25$ branch (exactly eight singlets per anti-self-dual tensor); being gauge singlets, they carry no Standard-Model charge and so are not excluded by collider or precision-electroweak data. Their physical disposition --- whether they are heavy bulk modes lifted near the six-dimensional Planck scale, sterile states relevant to the boundary neutrino sector, or part of the dark sector --- is determined by the boundary-spectrum derivation of \ANine{} \citep{ArisakaA9SM} and \AFive{} \citep{ArisakaA5GDOS}, not here. We acknowledge the objection's discomfort honestly: an 800-fold singlet sector is a large prediction, and if the spectrum papers cannot find a phenomenologically viable home for it, that is a genuine vulnerability of the construction. The present paper's claim is narrower --- that anomaly cancellation \emph{requires} exactly this count --- and the count is therefore a falsifiable structural prediction, not a free parameter.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 36 (O-BI6-36) (the ``theorem, not an input'' skeptic)}\par\nobreak\smallskip
\noindent\textbf{The Part~III subtitle calls the integer ledger ``a theorem, not an input.'' That is rhetorically loaded: a theorem needs axioms, and your axiom \AX{4} is exactly the one tuned so that the theorem yields these integers. Isn't ``theorem, not input'' just circular packaging?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The slogan means something precise and defensible, though the objection is right that it can be over-read. ``Theorem, not an input'' is the claim that the integers $(1,3,5,7,35)$ are \emph{not} adjustable dials of the model --- you cannot change $L$ from $5$ to $6$ and refit, because $L$ is fixed by the smallest-simple-group clause and the trace ratio, and every other integer follows (Section~\ref{sec:ogn5-arithmetic}). Contrast a conventional model where Yukawa couplings or compactification data are inputs chosen to fit. The honest part the objection captures: this is a theorem \emph{relative to} \AX{4}, and \AX{4} is a constitutional choice, not a logical necessity (Objection~12). So the non-circular content is ``given the constitution, the integers are forced''; the circular reading the objection warns against would be ``the integers prove the constitution,'' which the paper does not claim. We take the wording seriously: the subtitle asserts the integers are outputs of the axioms, and the place to contest it is \AX{4} itself, not the arithmetic that follows from it.
\end{objqabody}

\bigskip
\section{String theory: what exactly is the relationship?}

\medskip
\noindent\emph{Cluster: Part~IV audited by its target.} The comparison Part concedes ``shared mathematics, structurally different physics.'' The next seven objections are the strongest a string theorist will raise.

\bigskip
\noindent\textbf{Objection 37 (O-BI6-37) (the string theorist)}\par\nobreak\smallskip
\noindent\textbf{Your $X_4 = \tr R^2 - \Tr F^2$ is literally the heterotic Green--Schwarz 4-form dimensionally reduced to six dimensions. So \BIsix{} is heterotic Green--Schwarz on a 6D slice. What is new beyond bookkeeping?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The mathematical form is shared and the paper says so plainly (Section~\ref{sec:comparison-10d}, Eqs.~\eqref{eq:10d-factorization} and~\eqref{eq:6d-factorization}); what is new is not the Green--Schwarz move but its \emph{origin} and its \emph{rigidity}. In the heterotic string, $X_4$ descends from worldsheet modular invariance and the gauge group is forced to $E_8\times E_8$ or $SO(32)$ by that same string consistency. In \GGTheory{} there is no string and no worldsheet; $X_4$ is forced by the constitutional clauses (\AX{2} read as the modified Bianchi identity, Eq.~\eqref{eq:bi6-as-ax2}) and the gauge group is forced to the \emph{smallest simple group containing the Standard Model}, $\SU(5)$, by \AX{4}. The new content is that a four-component $X_4$ with the specific trace-ratio lock $2/5$ and the \OGNfive{} integer ledger is forced \emph{without} string input, and that this rigidity yields integer predictions ($K=35$, $\alpha^{-1}=137$ at leading order) that the heterotic construction, with its moduli, does not. So the bookkeeping is shared; the claim that the bookkeeping is \emph{constitutionally forced and moduli-free} is the difference, and it is a physical difference because it is what makes the integers predictions rather than choices.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-BI6-38) (the heterotic specialist)}\par\nobreak\smallskip
\noindent\textbf{String consistency picks $E_8\times E_8$ and $SO(32)$ for a reason: lower-rank groups like $\SU(5)$ famously can carry a global Witten anomaly that the string groups avoid. By choosing $\SU(5)$ on constitutional grounds, haven't you walked straight into the anomaly the string groups were selected to dodge?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The comparison table of forbidden alternatives (Table~\ref{tab:six-forbidden}) addresses exactly this, and the answer turns on the difference between ten and six dimensions. The string groups $E_8\times E_8$, $SO(32)$ are selected in \emph{ten} dimensions by the $\dim G = 496$ Green--Schwarz condition together with modular invariance; that condition is a 10D statement and does not apply in six dimensions, where the anomaly polynomial and the factorization condition are different (Eq.~\eqref{eq:6d-factorization}). In 6D the relevant global obstruction for $\SU(5)$ is the $\pi_6$-type anomaly of Objection~28, and the cohort's chiral content is arranged --- by the rank-one structure (Theorem~\ref{thm:rank-one}) --- so that it cancels on the intended class, with the exhaustive cobordism check flagged as Open Problem~O10. So the framework has not ignored the constraint that picks the string groups; it has observed that that constraint is dimension-specific, and has discharged the 6D analogue by the rank-one factorization rather than by enlarging the group. Whether the cobordism sweep confirms this in full generality is the open completion, and it is named.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 39 (O-BI6-39) (the F-theory geometer)}\par\nobreak\smallskip
\noindent\textbf{F-theory realizes $\SU(5)$ grand unification geometrically, with all anomalies automatically canceled by the geometry of the elliptic fibration, and it is UV-complete. Why is the cohort's constitutional $\SU(5)$ preferable to F-theory's geometric $\SU(5)$, which already does the anomaly job and embeds in string theory?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is not ``preferable'' in UV-completeness --- F-theory wins there, and the paper does not contest it (Section~\ref{sec:f-theory}, Table~\ref{tab:landscape}). The difference the cohort claims is parametric rigidity. F-theory's $\SU(5)$ comes with the moduli of the elliptic fibration --- complex-structure and K\"ahler parameters, flux choices \citep{MorrisonVafa1996a, MorrisonTaylor2012a} --- which are exactly the continuous freedom that places it in the landscape and prevents it from predicting, say, $\alpha^{-1}$ or the scale-depth integer. The cohort's $\SU(5)$ is fixed with \emph{no} fibration moduli, because \AX{4} forbids them; the price is the absence of a UV completion (conceded, inherited from \citep{ArisakaA2GG6}), and the payoff is integer predictions. So the honest trade is: F-theory offers UV-completeness plus moduli; the cohort offers moduli-freedom plus integer predictions but no UV completion. They are not competing on the same axis. A reader who weights UV-completeness above predictivity should prefer F-theory; the cohort is a bet that moduli-free predictivity is the more falsifiable virtue.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-BI6-40) (the holographer)}\par\nobreak\smallskip
\noindent\textbf{Your warped throat is Randall--Sundrum, and RS has a known AdS/CFT dual. Does \BIsix{} admit a holographic dual description? If not, why is the warp physical rather than a redundant dual artifact?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The AdS/RS comparison (Section~\ref{sec:ads-rs}, Table~\ref{tab:ads-rs-gg6}, Eq.~\eqref{eq:ads-cft-dictionary}) is drawn precisely so this question can be answered cleanly. The warp factor of the \GG{} bulk is geometrically Randall--Sundrum-like, and to the extent that the RS throat has an AdS/CFT reading \citep{RandallSundrum1999, Maldacena1998}, the cohort inherits a holographic \emph{intuition} for the radial direction as a renormalization-group coordinate. But \BIsix{} is not a statement that requires a dual to be meaningful: it is the modified Bianchi identity of a bulk gerbe (Section~\ref{sec:bi6-chern-weil-gerbe}), a bulk-geometric fact about anomaly inflow, and the anomaly it cancels is a property of the boundary chiral fermions registered at the infrared end. The warp is physical because it carries the one-way RG semigroup of \POS{5}, an irreversible registration arrow that a time-symmetric AdS/CFT duality does not by itself supply (this is the \OGGEP{}-vs-AdS/CFT distinction developed in the cohort's bulk paper). So: a holographic dual description is compatible and partially illuminating, but \BIsix{} stands as a bulk fact independent of whether a dual is exhibited, and the warp's physical content is the registration arrow, not a dual artifact.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-BI6-41) (the swampland theorist)}\par\nobreak\smallskip
\noindent\textbf{The cobordism conjecture does more than demand a trivial cobordism class --- it predicts specific defects and branes that must exist to trivialize otherwise-nontrivial classes. Does the cohort predict those cobordism defects, or does it only assert that the class is trivial?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} At present the paper asserts triviality on the intended class (Section~\ref{sec:swampland}, Section~\ref{sec:global-completion}) and does \emph{not} enumerate the cobordism defects the conjecture would associate with any nontrivial classes --- that enumeration is part of the open cobordism agenda (Open Problem~O10). The objection identifies a genuine and currently-unmet obligation: the modern form of the cobordism conjecture \citep{BrennanCartaVafa2017, KaidiTachikawaYonekura2025} ties anomaly-freedom to a defect spectrum, and a complete treatment would either exhibit the required defects or show the relevant classes vanish so that no defects are needed. The cohort's position is the second (the intended class is arranged to have trivial cobordism pairing), but the exhaustive check --- and hence the question of whether any defects are predicted --- is not done. We flag this as one of the sharper swampland-side open problems and do not claim the defect spectrum as a result.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 42 (O-BI6-42) (the landscape pragmatist)}\par\nobreak\smallskip
\noindent\textbf{You say \AX{4} collapses the $10^{500}$ landscape to a single vacuum. But the landscape is a feature of \emph{flux compactification}; a 6D theory with one tensor and no fluxes has simply not turned on the moduli that generate the landscape. Is the ``collapse'' a real result or an artifact of not including the structures that produce the landscape in the first place?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is sharp and the answer must concede part of it. The cohort does not derive a single vacuum from \emph{within} a flux-compactification framework by stabilizing all moduli; rather, \AX{4} forbids the unforced continuous structures --- extra tensors, flux integers beyond the \OGNfive{} ledger, fibration moduli --- whose proliferation \emph{is} the landscape (Table~\ref{tab:landscape}, Section~\ref{sec:comparison-10d}). So in one sense the objection is right: the cohort avoids the landscape by not admitting the structures that generate it, not by stabilizing them. Where we push back: that is precisely the claim --- that those structures are unforced and therefore inadmissible under the constitution, not that they are present-but-stabilized. The framework is making a bet that the landscape is an artifact of admitting unforced freedom, and that a constitution forbidding such freedom is more predictive. Whether that bet is right is empirical: if the integer predictions hold, the moduli-free posture is vindicated; if a continuous modulus is needed to fit data, \AX{4} fails (Objection~12). The ``collapse'' is thus a constitutional stance with falsifiable consequences, not a theorem that the flux landscape is empty.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-BI6-43) (the ``shared mathematics'' auditor)}\par\nobreak\smallskip
\noindent\textbf{Part~IV's own subtitle concedes ``shared mathematics, structurally different physics.'' But if the mathematics is shared, the anomaly-freedom \emph{content} is identical, and the ``structural difference'' is interpretation, not physics. Rebut.}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The rebuttal is that ``the mathematics is shared'' is true at the level of the Green--Schwarz \emph{move} and false at the level of what is forced. The shared mathematics is the factorization mechanism: given an $X_4$, the descent and the tensor coupling cancel the reducible anomaly identically in string theory and in \GGTheory{} (Section~\ref{sec:comparison-6d}, Section~\ref{sec:GS_as_GGEP_completion}). The non-shared, non-interpretive physics is \emph{which $X_4$ is allowed}: string theory permits a family of $X_4$'s parametrized by the compactification, while \GGTheory{} forces the unique four-component $X_4$ with the $2/5$ lock and the integer ledger. That is a difference in the space of admissible theories, not a difference in interpretation of a single theory --- it changes which numbers are predicted. So the subtitle is accurate: the cancellation calculus is shared (mathematics), but the constraint that selects one $X_4$ out of a family is the constitution (physics), and it has different empirical consequences (integer predictions vs.\ a landscape). The objection would be right if the cohort merely re-interpreted a string vacuum; it does not --- it forbids the family the string admits.
\end{objqabody}

\bigskip
\section{Constitutional framing, method, and self-sufficiency}

\medskip
\noindent\emph{Cluster: what does an anomaly proof ``constitutionalized'' actually add?} The next five objections press on the framing and on what A3 establishes on its own.

\bigskip
\noindent\textbf{Objection 44 (O-BI6-44) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{Anomaly freedom is a consistency requirement of \emph{any} quantum gauge theory. By calling it ``forced by the constitution,'' you have relabeled a necessary condition as a constitutional achievement. What is the added content beyond ``the theory is consistent''?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The added content is that the constitution forces a theory that \emph{is} anomaly-free, rather than merely requiring that whatever theory we write down be checked for anomalies. The distinction is between a filter and a generator. In the conventional approach, anomaly freedom is a constraint one imposes after choosing a model, discarding the anomalous candidates; the space of inputs (gauge group, matter content) is chosen by hand and then filtered. In \GGTheory{}, the gauge group and matter content are \emph{outputs} of \AX{1}--\AX{4} (the bulk theorems of \ATwo{} \citep{ArisakaA2GG6}, together with Theorem~\ref{thm:UoneQi-admissibility} and the \OGNfive{} theorem of Section~\ref{sec:ogn5-theorem}), and the theorem of Part~II is that these forced outputs are automatically anomaly-free with a single tensor and the integer ledger. So the claim is not ``our theory passes the anomaly check'' (true of any consistent theory) but ``the constitution that fixes the theory also guarantees it passes, and fixes the cancellation structure uniquely.'' The content is the uniqueness and the no-tuning, not the bare consistency. A reader who hears only ``it's anomaly-free, like every consistent theory'' has missed that the gauge content was not chosen but derived.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-BI6-45) (the \GGEP{} skeptic)}\par\nobreak\smallskip
\noindent\textbf{\BIsix{} is presented as the field-strength form of \AX{2} (\GGEP{}), via Eq.~\eqref{eq:bi6-as-ax2}. But equating a modified Bianchi identity with an equivalence principle is a strong claim. Is Eq.~\eqref{eq:bi6-as-ax2} a derivation or an analogy?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a definitional identification with derived consequences, and the paper should label it as such rather than as a theorem. \AX{2} (\GGEP{}) states that gravity and gauge are facets of a single composite connection (Objection~17); the curvature of that composite connection satisfies a Bianchi identity, and Eq.~\eqref{eq:bi6-as-ax2} is the statement that the \emph{modified} Bianchi identity $dH_3 = X_4$ is the component of that composite Bianchi identity carrying the mixed gravity-gauge curvature (Section~\ref{sec:geometric-origin}). So the link is: \GGEP{} posits the composite connection; the modified Bianchi identity is then a \emph{consequence} of the composite structure, not an independent analogy. Where the objection bites: the step from ``composite connection'' to ``this particular four-component $X_4$'' uses the trace normalizations and is a derivation only given the bulk gauge content of \citep{ArisakaA2GG6}. So Eq.~\eqref{eq:bi6-as-ax2} is a derived identity conditional on the \GGEP{} composite-connection posture, not a free analogy and not an unconditional theorem. The honest phrasing is ``\BIsix{} is what \AX{2} becomes at the level of field strengths,'' and that is a derivation modulo the composite-connection axiom.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 46 (O-BI6-46) (the model-level critic)}\par\nobreak\smallskip
\noindent\textbf{You distinguish a ``model-level'' anomaly-freedom theorem (Theorem~\ref{thm:model-level}) from the ``constitutional-level'' one (Theorem~\ref{thm:main}). Doesn't drawing that distinction concede that the constitutional claim is weaker than an actual model computation?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It concedes that they are different statements, which is a strength rather than a weakness, and Section~\ref{sec:model-level} is explicit about which is which. The model-level theorem (Theorem~\ref{thm:model-level}) is the concrete computation: take the specific \GG{} bulk spectrum and verify that the anomaly polynomial factorizes and cancels --- a calculation anyone can check line by line against Table~\ref{tab:su5-data}. The constitutional-level theorem (Theorem~\ref{thm:main}) is the stronger structural claim: that \emph{any} theory satisfying the constitution has this property, because the constitution forces the spectrum that the model-level computation verifies. The two are related as instance and universal: the model-level result is the existence proof (this spectrum works), the constitutional result is the uniqueness-and-forcing proof (the constitution yields this spectrum and no other). Far from the constitutional claim being weaker, it is the model-level claim plus the derivation of the spectrum from the axioms. We separate them so a skeptic can accept the checkable model-level computation even while reserving judgment on the constitutional forcing.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-BI6-47) (the reproducibility auditor)}\par\nobreak\smallskip
\noindent\textbf{The proof rests on trace computations (Table~\ref{tab:ten-coefficients}, Table~\ref{tab:su5-data}). Are these independently reproducible from standard group-theory references, or do they rely on cohort-internal conventions an outside reader cannot check?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are reproducible from standard references, and Appendix~\ref{app:matrix-and-coefficients} is written so that they can be. The trace data of Table~\ref{tab:su5-data} are the quadratic and quartic Dynkin indices of the $\SU(5)$ representations carrying the Standard-Model fermions --- standard quantities tabulated in the group-theory literature --- and the anomaly coefficients of Table~\ref{tab:ten-coefficients} are assembled from them by the explicit formulae of Appendix~\ref{app:matrix-and-coefficients}, with the normalization convention stated at the point of use. The only cohort-internal element is the \emph{choice} of which four blocks to include ($\Spin(1,5)$, $\SU(5)$, $\UoneB$, $\UoneQi$), which is the constitutional input, not a computational convention. So a reader can reproduce every entry: take the standard $\SU(5)$ indices, apply the stated formulae, and recover the matrix; the rank-one collapse (Theorem~\ref{thm:rank-one}) is then a property of that explicitly-built matrix. We regard independent reproducibility of the trace tables as essential and have written the appendix to support it; if any entry cannot be reproduced from standard indices plus the stated conventions, that is a bug to be reported, not a convention to be taken on trust.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-BI6-48) (the self-sufficiency auditor)}\par\nobreak\smallskip
\noindent\textbf{The paper is ten Parts but defers $\alpha$, the kaxion mass, the cosmological constant, the loop completion, and the global numerics to five other papers. What does A3 establish \emph{on its own} that does not depend on a companion?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A3's own results, owing nothing downstream, are stated in Section~\ref{sec:eight-parts} and Section~\ref{sec:conclusions}: (i) the local Green--Schwarz factorization of the \BIsix{} four-form with a single tensor (Theorem~\ref{thm:bi6-cancellation}); (ii) the rank-one collapse of the anomaly matrix and the unique eigenvector $(1,-1,-\tfrac25,-\tfrac25)$ (Theorem~\ref{thm:rank-one}); (iii) the trace-ratio identity $2/5=(L-N)/L$ and its forcing of $(L,N)=(5,3)$ given minimality (Theorem~\ref{thm:ogn-bi6}); (iv) the gravitational closure without a gravitino (Appendix~\ref{app:grav-full}); and (v) the global anomaly-freedom theorem on the intended class modulo the checked topological hypotheses (Theorem~\ref{thm:bi6-global-complete}). What it borrows or defers --- the value of $\alpha$, the kaxion line, the cosmological constant, the loop and global numerics --- it labels as borrowed (Section~\ref{sec:cohort-A3}). So A3 is the paper that proves the boundary law (the anomaly-free Bianchi identity and its integer consequences); it is neither self-sufficient for the cohort's empirical program nor empty --- it establishes the consistency theorem on which the spectrum papers build.
\end{objqabody}

\bigskip
\section{Empirical exposure and honest status}

\medskip
\noindent\emph{Cluster: where can A3 lose?} The final five objections press on falsifiability and on the honesty of the open-problem ledger.

\bigskip
\noindent\textbf{Objection 49 (O-BI6-49) (the falsification realist)}\par\nobreak\smallskip
\noindent\textbf{Every empirical prediction --- $\Mlock$, the kaxion line, $\Omega_m$, $\alpha$ --- is deferred to a companion paper. So A3 itself makes \emph{no} independently falsifiable prediction; it is a pure consistency proof. Is a paper with no own falsifier science, or scaffolding?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A3 makes one sharp, own falsifiable prediction and several structural ones, even though the headline numbers live downstream. The own falsifier is the rank-one structure (Theorem~\ref{thm:rank-one}): the claim that the \BIsix{} anomaly matrix collapses to rank one with eigenvector $(1,-1,-\tfrac25,-\tfrac25)$ is a definite mathematical statement that an independent computation of the $\SU(5)$ trace data can confirm or refute (Objection~47), and if the matrix came out rank two the single-tensor construction --- and with it \AX{4}'s admissibility verdict --- would fail. The structural predictions include the $n_F=800$ singlet count (Objection~35), a falsifiable spectral demand, and the trace-ratio $2/5$ forcing $(L,N)=(5,3)$. So A3 is consistency-proof \emph{plus} two checkable structural commitments; it is scaffolding for the empirical papers, but scaffolding that can itself be knocked down by a trace computation. We grant that A3 carries no laboratory falsifier of its own --- those are downstream --- and that its falsifiability is mathematical rather than experimental.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-BI6-50) (the open-problems counter)}\par\nobreak\smallskip
\noindent\textbf{You list Open Problems O6--O11 as ``partially resolved.'' A hostile reader notes that these are the \emph{hard} parts --- global numerics (O9), the full non-SUSY category (O10), the loop completion (O11), the cosmological factor-of-three (O6). Is the unresolved residue not the whole problem?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The residue is the hard part, and the paper neither hides this nor accepts that it is ``the whole problem.'' Table~\ref{tab:status} and Section~\ref{sec:open-problems} state plainly which results are complete (the local factorization, the rank-one collapse, the trace-ratio forcing, the gravitational closure, the global theorem on the intended class modulo checked hypotheses) and which are open (explicit global numerics O9, the full category O10, the loop completion O11, the cosmological correction O6). The honest accounting is that the \emph{classical-topological} consistency problem is solved and the \emph{non-perturbative and quantitative-cosmological} problems are open. That is a real and substantial completed core, not a cosmetic one: the rank-one factorization and the trace-ratio lock are the load-bearing results, and they are done. The objection is right that O9--O11 are where the difficulty concentrates, and the paper's response is to mark them as open with named technology that could close them, rather than to claim them. A framework that labeled its hardest sub-problems ``solved'' would be the dishonest one; this one labels them open.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-BI6-51) (the cosmological-constant realist)}\par\nobreak\smallskip
\noindent\textbf{Objection~16 concedes a factor-of-three miss on $m_\Lambda$ pending a $\zeta_\Lambda\approx 1/50$ correction. But ``$119$ of $120$ orders of magnitude recovered'' is rhetorical: a factor-of-three error on the one number that matters is still a failed prediction. Why present it as success-adjacent?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is fair, and we state the status here without the rhetorical cushion. The precise situation: the cohort's leading-order seesaw estimate $m_\Lambda^2 = \Mlock^2/M_{\rm Pl}$ lands within a factor of about three of the observed $m_\Lambda\approx 2.3$~meV, and the residual factor is attributed to an as-yet-uncomputed zero-mode overlap correction $\zeta_\Lambda\approx 1/50$ (\citep{ArisakaA10Cosmos}, Open Problem~O6). Two things are true at once: the leading-order estimate is dramatically better than the naive QFT vacuum-energy estimate, \emph{and} a factor-of-three miss is a quantitative failure until the correction is derived. The framing ``119 of 120 orders recovered'' is legitimate as a statement about the seesaw mechanism's scale but illegitimate if read as a successful prediction of $m_\Lambda$, which it is not yet. The paper's honest position is therefore: the mechanism is promising, the number is not yet predicted to precision, and the correction factor is an explicit open computation whose success or failure will decide the matter. No wording in this paper should let the factor-of-three read as a hit.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-BI6-52) (the kaxion experimentalist)}\par\nobreak\smallskip
\noindent\textbf{Once the kaxion mass is pinned to the canonical $8.1~\mu$eV ($\approx 1.95$~GHz), the prediction is a specific haloscope line. If the haloscopes sweeping the band (IBS-CAPP, ADMX-EFR) scan through that frequency and see nothing, is \BIsix{} falsified, or does the kaxion mass have an escape hatch?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A clean null result across the canonical window with adequate coupling sensitivity would falsify the kaxion prediction, and the prediction is not free to slide continuously: the benchmark line is tied to the \OGNfive{} master integer $K=35$ that fixes the anomaly ledger (Section~\ref{sec:ogn5-theorem}), and the one residual freedom is discrete and fenced --- the cap index $\Xi_{\mathrm{cap}} \in \{35, \ldots, 47\}$ of the mass-uniqueness conjecture named in \citep{ArisakaC1Kaxion}, which maps to the closed window $\nu_K \in [1.59, 2.13]$~GHz around the $1.95$~GHz benchmark. So there is no soft modulus to retune the line --- the escape hatch the objection worries about is at most a finite set of pre-registered integers, and a null across the closed window at the stated depth is a kill. The detailed haloscope phenomenology, including the coupling strength that sets the required scan sensitivity, is the subject of \citep{ArisakaC1Kaxion}; \BIsix{}'s role is to supply the integer $K=35$ that locks the frequency. The honest caveat is about \emph{sensitivity}, not mass: a scan that passes $1.95$~GHz without reaching the predicted coupling would not be a kill, so the falsifier is specified with a coupling threshold, and only a null result below that threshold counts. Within that specification, the kaxion line is a hard, ledger-locked prediction.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-BI6-53) (the cohort-coherence auditor)}\par\nobreak\smallskip
\noindent\textbf{A3 sits between A2 (the bulk) and A5/A9 (the boundary spectrum). If A2's bulk geometry or A9's boundary spectrum is later revised, does the \BIsix{} anomaly proof survive, or is it hostage to upstream and downstream changes?}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The proof's dependencies are explicit (the two-way coherence map, Table~\ref{tab:outward-map-A3} and Table~\ref{tab:inward-map-A3}, Section~\ref{sec:cohort-A3}), and that explicitness is exactly what makes the hostage question answerable rather than alarming. A3 depends on A2 for one thing: the bulk gauge content $\Spin(1,5)\times\SU(5)\times\UoneB\times\UoneQi$ and the product geometry. As long as those are unchanged, the rank-one factorization and the trace-ratio lock are internal to A3 and survive any revision elsewhere. A3 supplies to A9/A5 the anomaly-free Bianchi identity and the integer ledger; if A9's boundary spectrum were revised, it would have to revise it \emph{consistent with} the anomaly constraint A3 proves, not against it --- the constraint flows downstream, so downstream changes are bounded by it rather than threatening it. The genuine exposure is upstream: if A2's bulk gauge content changed, the four-block anomaly matrix would change and the rank-one result would need re-derivation. That dependency is named in the inward map (Table~\ref{tab:inward-map-A3}). So A3 is hostage to A2's gauge content (and says so) but robust against downstream revision, because it is the constraint the downstream must respect.
\end{objqabody}
\medskip
\noindent\emph{Closing remark on the objections.} After all, an honest framework should know which questions a working physicist is most likely to raise on first reading, and should answer those questions before they become obstacles to engagement. We have addressed fifty-three such objections above, grouped into ten clusters: the anomaly proof and the trace ratio (Objections 1--3), global anomaly freedom and the non-supersymmetric setting (4--7), minimality and the parsimony axiom (8--12), convergences and empirical predictions (13--17), the local descent and the four-form $X_4$ (18--24), the global completion via Dai--Freed and cobordism (25--30), the \OGNfive{} ledger as a theorem (31--36), the relationship to string theory (37--43), the constitutional framing and self-sufficiency (44--48), and empirical exposure and honest status (49--53). Each is the framework's standing offer to be wrong, in detail, in a specific way; the deeper technical comparisons live in Sections~\ref{sec:comparison-10d}--\ref{sec:ads-rs} and the Open Problems triage in Section~\ref{sec:open-problems}.


\section*{Author's Note}
\addcontentsline{toc}{section}{Author's Note}

The author of the \GG{} program is an experimental physicist---not
a string theorist, not a formal quantum field theorist, and not a
mathematician.
The \GGTheory{} Constitution and its consequences were developed from
the perspective of an experimentalist who studies complex systems
across multiple scientific domains, from particle physics to
neuroscience. The framework is driven by the question
\emph{what must any admissible theory satisfy?} rather than by
the question \emph{what does a specific UV-complete theory predict?}

This background has both advantages and limitations.
The advantage is a willingness to question structural assumptions
(such as the necessity of supersymmetry or the primacy of
Planck-scale completion) that a theorist trained within the
string tradition might take as given.
The limitation is the absence of deep expertise in the most
technical aspects of the subject---anomaly polynomials, modular
forms, bordism theory, derived categories---that the string
community has spent decades developing.

The present paper has attempted to be mathematically rigorous
within its scope, but not every subtlety may have been addressed.
The author is confident in the structural proof (Section~\ref{sec:structural-proof}),
the model-level benchmark (Section~\ref{sec:model-level}), the \OGNfive{} determination
(Sections~\ref{sec:ogn5-arithmetic}--\ref{sec:ogn5-theorem}), and the comparisons (Sections~\ref{sec:comparison-10d}--\ref{sec:ads-rs}).
The author is aware that the global anomaly technology
\citep{MonnierMoore2019,DieriglTartaglia2025}, the charge-lattice
refinements \citep{SeibergTaylor2011}, and the bordism-theoretic
constraints \citep{HopkinsSinger2005} may impose additional
conditions that the present analysis has not fully explored.

Collaboration with the string theory
community would be most valuable.
The Minimal \BIsix{} Spectrum Problem (entry O8 of
Section~\ref{sec:open-problems}) is a well-posed mathematical
question that the community's expertise in 6D anomaly
polynomials and index theory is ideally suited to address.
The global consistency checks that Monnier and Moore developed
for \(\cN=(1,0)\) SUGRA could be applied to the non-SUSY \GG{}
setting to test whether additional obstructions arise.
The enumeration technology of Hamada and Loges
\citep{HamadaLoges2024} could be adapted to the non-SUSY
Diophantine system.

The author therefore extends an open invitation:
\emph{if the structural proof is correct, what are its
consequences? If it has a gap, where exactly is it?}
Either answer would advance the field.

A final word to graduate students and early-career theorists.
The tools developed by the string theory community over the past
four decades---anomaly polynomials, descent equations, index theory,
bordism, holographic renormalization, charge-lattice
quantization---are among the most powerful mathematical instruments
in theoretical physics.
These tools are not tied to any single interpretive framework.
They can be applied to supersymmetric string compactifications,
to non-supersymmetric field theories, to condensed-matter
systems, and---as the present paper attempts to show---to
axiomatic approaches that begin from different premises.
A young theorist who masters these tools will find no shortage
of problems to solve, regardless of which theoretical direction
ultimately prevails.
The questions raised by this paper---the Minimal \BIsix{} Spectrum Problem,
the extension of the global anomaly theorem beyond the intended non-SUSY \GG{} category, the
connection between \OGNfive{} arithmetic and F-theory lattice conditions,
the operational meaning of the 6D\(\to\)4D channel---are concrete, well-posed mathematical problems that can form the basis
of productive research programs. The author would welcome collaboration
with researchers at any career stage.

\bigskip

\noindent\textbf{What the objections established.} Answering
fifty-three objections in sequence changes a paper, and it is worth
recording how. Several of them could not be answered as posed, and the
responses say so: the trace ratio is not fixed by anomaly freedom alone;
the anomaly test is passed by a family of Abelian coefficients rather
than by one; the explicit mapping-torus partition functions have not
been computed; the bordism group has not been evaluated in closed form;
the generation count is not derived. Each of those concessions is now
carried in the main text as well, in the section that makes the claim,
rather than living only here where a hurried reader would miss it.

That is the use of an objections Part. It is not a defensive exercise
and it is not a summary. It is the place where the strongest available
criticism is put in writing and the paper is made to answer it, and a
framework that comes through the process unchanged has not taken the
process seriously. This one did not come through unchanged.

\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which theorem, proof, or section each one underwrites.

\noindent The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the anomaly proof: from the foundational work on the Green--Schwarz mechanism, differential cohomology, and bordism invariance, through to the twenty-two-paper \GGTheory{} cohort in which the present paper supplies the local-and-global anomaly theorem. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The references are organized in two groups --- first the author's \GGTheory{} papers (Section~A below), then all other works (Section~B), each in alphabetical order by first author. The cohort papers (\AOne{}--\AEleven{}, with the C- and P-series) are
cited under the canonical keys fixed by the cross-paper coherence map of
the \GGTheory{} program.

Each entry carries a note saying what the work established and where the
present paper uses it. The notes exist because a bare citation list
records that a debt was incurred without recording what it was for, and
in a paper that borrows its central technique wholesale from another
tradition that distinction is worth the pages it costs. A reader who
wants to know how much of this construction is new can read the notes
and find out.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}

\begin{thebibliography}{99}
\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K.~(2026).}
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems},
Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The constitutional charter of \GGTheory. States the four foundational axioms (\LC, \GGEP, \OGGEP, \MSMF) and the seven postulates that serve as the sole admissibility input for the present anomaly proof. Every theorem of A3-BI6 --- from the structural proof of \BIsix{} cancellation in \S\ref{sec:structural-proof} to the constitutional reading of \S\ref{sec:ads-rs} --- traces its primitive input to one or more clauses of this charter; the constitutional placement of \BIsix{} on Stratum~2 in Section~\ref{sec:ads-rs} is the most explicit invocation. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K.~(2026).}
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})},
Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Derives, from the \GGTheory{} Constitution, the six-dimensional bulk $M_4 \times I_r \times S^1_\theta$, the four-component composite gauge group $\Spin(1,5) \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$, the modified Bianchi identity \BIsix, and the \OGNfive{} integer ledger $(M,N,L;J,K) = (1,3,5;7,35)$. A3-BI6 is the rigorous local-plus-global anomaly proof for that construction; we cite \ATwo{} throughout \S\ref{sec:intro-gg6}, \S\ref{sec:bi6-fourform}, and the \OGNfive{} derivation in \S\ref{sec:ogn5-theorem}. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A5-\GDOS{})]{ArisakaA5GDOS}
\textbf{Arisaka, K.~(2026).}
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT},
Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Extends the bulk \GG{} framework to the boundary through the \OGGEP{} channel as a completely positive trace-preserving map and derives 4D quantum mechanics, general relativity, and the low-energy Standard Model as boundary projections. The local and global anomaly freedom of \BIsix{} proved in A3-BI6 is the lawfulness prerequisite for that boundary channel. We cite \AFive{} in \S\ref{sec:intro-gdos} for the forward link from the bulk anomaly theorem to the boundary law, and in the Scope-and-Limits paragraphs of \S\ref{sec:open-problems}. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K.~(2026).}
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}},
Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Establishes the four quantum-information pillars (\QiQUA, \QiEP, \QiCON, \QiHAL) on the boundary \GDOS{} runtime and calibrates the bioenergetic conversion factor $\QB \approx 0.1555$~eV per qubit against ATP hydrolysis at biological temperatures. These boundary $\QiFour$ pillars require $\UoneQi$ as host symmetry --- one of the three empirical reasons that A3-BI6 adopts the four-component bulk in Theorem~\ref{thm:UoneQi-admissibility} and \S\ref{sec:UoneQi-constitutional-status}, rather than the three-component option also admitted by the constitution. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K.~(2026).}
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Derives the 28 continuous Standard-Model parameters --- the three gauge couplings, charged-lepton and quark masses, CKM and PMNS angles, Higgs sector, QCD theta angle, and cosmological constant --- from the \OGNfive{} integer ledger and the 6D bulk. A3-BI6 provides the upstream anomaly-freedom certificate without which that derivation cannot be deployed; we cite \ANine{} at the empirical-prediction paragraphs of \S\ref{sec:discussion} and at the cross-paper coherence map in \S\ref{sec:coherence-map}. First cited in \S\ref{sec:introduction}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K.~(2026).}
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry},
Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Demonstrates that the Green--Schwarz normalization $\Omega$ is driven to $\Omega=1$ as a cosmological attractor by the GUT-adjacent slow smooth inflationary descent, and predicts kaxion dark matter at $8.1~\mu$eV plus Kaluza--Klein resonances at $\Mlock \approx 3.85$~TeV. Both predictions presuppose the local + global anomaly-freedom certificate of \BIsix{} established in A3-BI6. We cite \ATen{} in the Green--Schwarz-normalization discussion of \S\ref{sec:discussion} and in the cross-paper coherence map. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K.~(2026).}
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})},
Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The present paper; listed for completeness as the \AThree{} node of the cohort map (\S\ref{sec:cohort-A3}). First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K.~(2026).}
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind},
Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The geometric \GiFour{} pillars of the open-system boundary law; the geometric sibling of the \QiFour{} pillars of \ASeven{}. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K.~(2026).}
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary},
Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Recasts textbook quantum mechanics in the language of \GGTheory{} via \GDOS{} and \OGGEP{}; the QM stratum of the Consequence Trio. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K.~(2026).}
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program},
Arisaka-GG-A11-Origin.
\par\noindent In preparation. To be posted at \url{https://arisaka-gg.org/} on
completion; no result of the present paper depends on it.
\par\noindent\textbf{Relevance.} The closing capstone of the cohort (recursive uniqueness; the cosmos--life--mind chain). Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Predicts the Kaluza--Klein lock cluster near $\Mlock \approx 3.85$~TeV. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
\textbf{Arisaka, K.~(2026).}
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability},
Arisaka-GG-P2-N3.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Derives the three-generation count $N=3$, building directly on the anomaly proof of the present paper. Its Appendix~H is the inflow computation that consumes this paper's primitive lift, doubling construction, unit reservoir and $a=1$ refinement as bulk-side inputs: it pushes the \BIsix{} differential data forward through the Hopkins--Singer quadratic functor on the benchmark background of the intended class and proves the value is the boundary's Berry line bundle. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Treats the electron as the ground state of nature within \GGTheory{}. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The origin-of-$i$ member paper of the \GGTheory{} program (Foundations Branch): derives the single imaginary unit of boundary physics from the two real structures of the \ATwo{} arena (the compact phase circle and the causal clock). On the twenty-two-paper cohort map it stands on the spine's left flank, fed by \ATwo{}'s Theorem-0 arena and feeding the quantum-mechanical recast of \AEight{}; the present paper cites it at roster level. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The CPT member paper of the \GGTheory{} program (Foundations Branch): reads C, P, and T as the orientation reversals of the three real bulk factors and carries the matter--antimatter door. That orientation reading of the discrete symmetries makes it an independent consumer of the anomaly-inflow logic proved here; the present paper cites it at roster level. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}

\textbf{Arisaka, K.\ (2026).}
\textit{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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The fourth member of the Predictions
Branch, sister to \PaperPThree{}: the proton as the \GGA{} of the
baryonic stratum, with baryon number carried as a winding of the
compact geometry rather than as a bookkeeping label --- written by the
same twist that breaks the unified gauge symmetry and splits the Higgs
multiplet --- so that the decay the grand unified theories insisted on
is forbidden by the geometry rather than merely unobserved. Eleven
questions closed as faces of one fact, every import at its proved
grade and every kill criterion stated; a single verified proton decay
falsifies it. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K.~(2026).}
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}},
Arisaka-GG-C1-Kaxion.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The kaxion dark-matter candidate near $8.1~\mu$eV with twin-peak signature $\Delta\nu/\nu = 1/35$. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K.~(2026).}
\emph{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$},
Arisaka-GG-C2-Hubble.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Reads the Hubble tension as the two ends of one six-dimensional geometric corridor. Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K.~(2026).}
\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.
\par\noindent Preprint, available at \url{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} Black-hole entropy under \GGTheory{} (horizon as an \OGGEP{} port; exact Bekenstein--Hawking $A/4$). Cited in the cohort overview, \S\ref{sec:cohort-A3}. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}

\textbf{Arisaka, K.\ (2026).}
\textit{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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The first of the two white papers,
and the \emph{diachronic} one: it walks the century of symmetry and
conservation in the order it was found and shows the conviction that
symmetry causes conservation loosening at seven historical steps,
until what remains conserved is an integer with nowhere to move ---
then places the program at the arc's end, at declared grades, with
its payout set beside the measurements. Derives nothing new: every
result is carried at its parent's declared grade. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}

\textbf{Arisaka, K.\ (2026).}
\textit{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://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The second of the two white papers,
and the \emph{synchronic} one: it takes a single exact statement ---
a holonomy requires a closed loop and the interval fiber has none, so
the circle remembers and the interval forgets --- and follows it to
sixteen of the questions physics has not closed, closing with the
named, dated measurements on which six of the sixteen treatments die
outright. Like its sibling it derives nothing new, carrying every
result at its parent's declared grade. First cited in \S\ref{sec:cohort-A3}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K. (2026).} \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, UCLA preprint,.

\par\noindent\textbf{Relevance.} \AFour{} is the consequence paper of the foundation block, completing the partial-geometrization program that runs Maxwell~1865, Minkowski~1908, Einstein~1915, Kaluza--Klein~1919--26, Wilson--Hosotani~1974--83, Randall--Sundrum~1999, and the Green--Schwarz mechanism~1984. We cite it in Section~\ref{sec:geometric-origin} as the historical positioning paper that frames the 1984 GS achievement as the moment higher-dimensional gauge theory glimpsed \AX{2}~(\GGEP) at the technical level. First cited in \S\ref{sec:geometric-origin}.

\end{thebibliography}

\bigskip
\section*{B.~Other references}

\begin{thebibliography}{99}
\bibitem[Adler(1969)]{Adler1969}
\textbf{S.~Adler.}
\emph{Axial-vector vertex in spinor electrodynamics},
\emph{Phys.\ Rev.} \textbf{177} (1969) 2426.
\par\noindent\textbf{Relevance.} Discovers the axial-vector anomaly in spinor electrodynamics: the one-loop quantum failure of classical $\partial_\mu j^\mu_5 = 0$ that became the prototype for every higher-dimensional anomaly that \BIsix{} must cancel. In A3-BI6 we cite Adler at the historical motivation in \S\ref{sec:introduction} and again in \S\ref{sec:gs-mechanism} when the descent formalism is introduced; the entire eight-part proof obligation of \S\ref{sec:eight-parts} exists because this 1969 result made gauge-anomaly cancellation a non-negotiable consistency requirement. First cited in the Executive Summary.

\bibitem[Alvarez-Gaum\'e and Witten(1984)]{AlvarezGaumeWitten1984}
\textbf{L.~Alvarez-Gaum\'e and E.~Witten.}
\emph{Gravitational anomalies}, \emph{Nucl.\ Phys.\ B} \textbf{234} (1984) 269.
\par\noindent\textbf{Relevance.} Establishes the modern index-theoretic classification of gravitational and mixed gauge--gravity anomalies in even dimensions: the framework in which one writes down anomaly polynomials, identifies factorization conditions, and reads off the resulting cancellation requirements. A3-BI6 uses this framework throughout the structural proof in \S\ref{sec:structural-proof} and the explicit anomaly-matrix computation of Appendix~\ref{app:grav-full}; the four-component coefficient vector $(a,b,c,d) = (1,-1,-2/5,-2/5)$ is fixed by exactly the index data that this paper systematized. First cited in \S\ref{sec:structural-proof}.

\bibitem[Atiyah \emph{et al.}(1975)]{APS1975}
\textbf{M.~F.~Atiyah, V.~K.~Patodi, and I.~M.~Singer.} \emph{Spectral asymmetry and Riemannian geometry.~I}, \emph{Math.\ Proc.\ Camb.\ Phil.\ Soc.} \textbf{77} (1975) 43--69.
\par\noindent\textbf{Relevance.} Introduces the eta invariant and the index theorem for manifolds with boundary that the global half of this paper's argument runs on. The Dai--Freed formulation of a global anomaly is an exponentiated eta invariant, and the per-species factor $\exp(2\pi i\cdot\mathrm{index})$ evaluated in Section~\ref{sec:global-completion} is this theorem applied to the benchmark spectrum. First cited in Section~\ref{sec:global-completion}.

\bibitem[Basile and Leone(2024)]{BasileLeone2024}
\textbf{I.~Basile and G.~Leone.}
\emph{Anomaly constraints for heterotic strings and supergravity in six dimensions},
\emph{JHEP} \textbf{2404} (2024) 067,
arXiv:2310.20480.
\par\noindent\textbf{Relevance.} Establishes anomaly constraints for heterotic and non-SUSY supergravity constructions in six dimensions. Directly relevant to the non-SUSY heterotic boundary of the 6D landscape that A3-BI6 must engage with. We cite Basile--Leone at the swampland and 6D-classification paragraphs of \S\ref{sec:comparison-6d}; their analysis shows that anomaly cancellation continues to function as the principal admissibility filter once supersymmetry is removed --- exactly the regime in which the four-component \BIsix{} of A3-BI6 operates. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Baykara et al.(2024)]{BaykataHamadaTarazi2024}
\textbf{Z.~K.~Baykara, Y.~Hamada, H.-C.~Tarazi, and C.~Vafa.}
\emph{On the string landscape without hypermultiplets},
\emph{JHEP} \textbf{2404} (2024) 121,
arXiv:2309.15152.
\par\noindent\textbf{Relevance.} Constructs 6D string vacua without hypermultiplets via asymmetric orbifolds, pushing the boundary of 6D string constructions toward non-standard configurations. In A3-BI6 we cite Baykara--Hamada--Tarazi--Vafa at the non-geometric-compactification paragraphs of \S\ref{sec:comparison-6d}; the comparison highlights that even within the string-theoretic tradition, the assumed matter content can be relaxed --- a flexibility that \BIsix{} achieves constitutionally through the \OGNfive{} integer ledger of \S\ref{sec:ogn5-theorem} rather than through orbifold engineering. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Becchi et al.(1976)]{Becchi1976}
\textbf{C.~Becchi, A.~Rouet, and R.~Stora.}
\emph{Renormalization of gauge theories}, \emph{Ann.\ Phys.} \textbf{98} (1976) 287.
\par\noindent\textbf{Relevance.} Establishes the BRST formalism: the cohomological language in which anomaly cancellation, Slavnov--Taylor identities, and the Wess--Zumino consistency condition acquire their natural algebraic form. A3-BI6 uses BRST language throughout the descent proof of \S\ref{sec:structural-proof} and the explicit BRST cancellation in Appendix~\ref{app:descent-full}, where the seven-form, six-form, and one-loop variation are written as BRST cocycles whose exact cancellation by the Green--Schwarz counterterm closes the local anomaly ledger of \BIsix{}. First cited in \S\ref{sec:geometric-origin}.

\bibitem[Becker et al.(2024)]{BeckerKehagias2024}
\textbf{K.~Becker, A.~Kehagias, E.~Sezgin, D.~Tennyson, and A.~Violaris.}
\emph{New anomaly-free supergravities in six dimensions},
\emph{JHEP} \textbf{2405} (2024) 144,
arXiv:2311.03337.
\par\noindent\textbf{Relevance.} Discovers new classes of anomaly-free 6D supergravity models, demonstrating that the 6D landscape is still actively being explored within the SUSY framework. In A3-BI6 we cite Becker--Kehagias-Sezgin-Tennyson-Violaris at the existing-literature paragraphs of \S\ref{sec:comparison-6d}; the comparison reinforces the contrast with the constitutional approach of \GG{}, where the four-component coefficient vector $(a,b,c,d)$ of \BIsix{} is fixed uniquely by the trace-ratio theorem of \S\ref{sec:ogn5-theorem} rather than emerging as one possibility among many. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Becker, Becker, and Schwarz(2007)]{BeckerBeckerSchwarz2007}
\textbf{K.~Becker, M.~Becker, and J.~H.~Schwarz.}
\emph{String Theory and M-Theory: A Modern Introduction},
Cambridge University Press (2007).
\par\noindent\textbf{Relevance.} The modern graduate string-theory
text; its Ch.~5 treatment of anomaly cancellation is the recommended
companion to Appendix~\ref{app:gs10d}. First cited in the Part~VII
course introduction.

\bibitem[Bell and Jackiw(1969)]{BellJackiw1969}
\textbf{J.~S.~Bell and R.~Jackiw.}
\emph{A PCAC puzzle: $\pi^0\to\gamma\gamma$ in the sigma model},
\emph{Nuovo Cimento A} \textbf{60} (1969) 47.
\par\noindent\textbf{Relevance.} Independent discovery of the chiral anomaly through the $\pi^0 \to \gamma\gamma$ amplitude; together with Adler this established that quantum anomalies are physical and unavoidable rather than a calculational artifact. In A3-BI6 we cite Bell--Jackiw alongside Adler in \S\ref{sec:introduction} and at the motivation for the descent formalism in \S\ref{sec:gs-mechanism}; the Wess--Zumino consistency condition implemented in Appendix~\ref{app:descent-full} is the cohomological generalization of the Adler--Bell--Jackiw obstruction to the six-dimensional setting. First cited in the Executive Summary.

\bibitem[Bertlmann(1996)]{Bertlmann1996}
\textbf{R.~A.~Bertlmann.}
\emph{Anomalies in Quantum Field Theory},
Oxford University Press (1996).
\par\noindent\textbf{Relevance.} The book-length cohomological
treatment of anomalies --- descent equations, BRST cohomology, and the
Stora--Zumino chain --- matching the route taken in
Appendix~\ref{app:descent-full}. First cited in the Part~VII course
introduction.

\bibitem[Bianchi(1902)]{Bianchi1902}
\textbf{L.~Bianchi.}
\emph{Sui simboli a quattro indici e sulla curvatura di Riemann},
\emph{Atti della Reale Accademia dei Lincei, Rendiconti}
\textbf{11} (1902) 3--7.
\par\noindent\textbf{Relevance.} Establishes the differential identity for the Riemann curvature tensor whose generalization to gauge connections, $dF = 0$, is the classical Bianchi identity that the modified \BIsix{} of \GGTheory{} generalizes in the Green--Schwarz manner. In A3-BI6 we cite Bianchi 1902 in the historical motivation of \S\ref{sec:introduction} and at the introduction of the modified identity in \S\ref{sec:bi6-fourform}; the modification $dH_3 = X_4 \neq 0$ that defines \BIsix{} is the topological completion of Bianchi's classical identity. First cited in the Executive Summary.

\bibitem[Bilal(2008)]{Bilal2008}
\textbf{A.~Bilal.}
\emph{Lectures on anomalies},
arXiv:0802.0634.
\par\noindent\textbf{Relevance.} The standard modern lecture notes on
anomalies --- triangle to descent to inflow --- at exactly the level
of the course in Part~VII; recommended first reading beside
Appendices~\ref{app:descent-full} and~\ref{app:grav-full}. First
cited in the Part~VII course introduction.

\bibitem[Bott(1959)]{Bott1959}
\textbf{R.~Bott.}
\emph{The stable homotopy of the classical groups},
\emph{Annals of Mathematics} \textbf{70} (1959) 313--337.
\par\noindent\textbf{Relevance.} Supplies the stable-homotopy facts used in the automatic-admissibility theorem, in particular the calculation of the relevant $\pi_k$ groups for $\SU(5)$ that control the global obstructions. In A3-BI6 we cite Bott in Appendix~\ref{app:global-proof-dossier} at the homotopy-group computation showing that the Postnikov 4-type of $\SU(5)$ is $K(\mathbb{Z},3)$, with $\pi_1=\pi_2=\pi_4=0$ and $\pi_3(\SU(5))=\mathbb{Z}$, which is the input needed to make the global anomaly vanish automatically. First cited in \S\ref{app:global-proof-dossier}.

\bibitem[Bousso and Polchinski(2000)]{BoussoPolchinski2000}
\textbf{R.~Bousso and J.~Polchinski.}
\emph{Quantization of four-form fluxes and dynamical neutralization of the cosmological constant},
\emph{JHEP} \textbf{0006} (2000) 006,
arXiv:hep-th/0004134.
\par\noindent\textbf{Relevance.} Introduces the multi-flux mechanism for dynamical neutralization of
the cosmological constant, from which the discretuum of vacua follows;
the widely quoted count of order $10^{500}$ is due to
\citet{Douglas2003}, not to this paper. Cited here as the canonical reference for the landscape problem that \GG{}'s finite Diophantine approach aims to circumvent. Bousso--Polchinski is listed here as background for the finite-versus-infinite landscape discussion of \S\ref{sec:comparison-10d} and \S\ref{sec:comparison-6d}, where the unique \OGNfive{} solution stands in pointed contrast. First cited in \S\ref{sec:global-completion}.

\bibitem[Brennan, Carta, and Vafa(2017)]{BrennanCartaVafa2017}
\textbf{T.~D.~Brennan, F.~Carta, and C.~Vafa.}
\emph{The string landscape, the swampland, and the missing corner},
arXiv:1711.00864.
\par\noindent\textbf{Relevance.} Comprehensive review of the swampland conjectures: the systematic articulation of what distinguishes effective theories with UV string completions from those without. In A3-BI6 we cite Brennan--Carta--Vafa at the swampland paragraphs of \S\ref{sec:comparison-6d}; the comparison frames the constitutional question for \GG{} --- whether \BIsix{} sits in the string landscape, the swampland, or a third category defined by the \GGTheory{} Constitution rather than by string consistency. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Brylinski(1993)]{Brylinski1993}
\textbf{J.-L.~Brylinski.}
\emph{Loop Spaces, Characteristic Classes and Geometric Quantization}, Birkh\"auser (1993).
\par\noindent\textbf{Relevance.} Standard reference for gerbes, the Dixmier--Douady class, and degree-three integral cohomology; underwrites the gerbe / differential-cohomology reading of the Green--Schwarz tensor in \S\ref{sec:bi6-chern-weil-gerbe}. First cited in \S\ref{sec:structural-proof}.

\bibitem[Bustamante \emph{et al.}(2005)]{Bustamante2005}
\textbf{C.~Bustamante, Y.~Chemla, N.~Forde, and D.~Izhaky.}
\emph{Mechanical processes in biochemistry},
\emph{Annu.\ Rev.\ Biochem.} \textbf{73} (2004) 705--748.
\par\noindent\textbf{Relevance.} Standard reference for the mechanical biochemistry of ATP hydrolysis at biological temperatures: supplies the empirical anchor for ATP hydrolysis (standard free energy $30.5 \pm 0.5$~kJ/mol) against which the Arisaka-equation quantum $\QB \approx 0.1555$~eV per qubit is calibrated, one phosphoanhydride bond carrying two quanta ($2\,N_A\QB = 30.0$~kJ/mol). In A3-BI6 we cite Bustamante at the bioenergetic-anchor paragraph of \S\ref{sec:introduction} and at Objection~3 of \S\ref{sec:common-objections}, where the four-component bulk's adoption of $\UoneQi$ is justified on three empirical grounds, this being the first. First cited in the Executive Summary.

\bibitem[Callan and Harvey(1985)]{CallanHarvey1985}
\textbf{C.~G.~Callan and J.~A.~Harvey.}
\emph{Anomalies and fermion zero modes on strings and domain walls},
\emph{Nucl.\ Phys.\ B} \textbf{250} (1985) 427.
\par\noindent\textbf{Relevance.} Establishes the anomaly-inflow mechanism on domain walls and strings: the foundational result that bulk topological terms can carry chiral-anomaly inflow to boundary defects. Callan--Harvey inflow is the conceptual ancestor of the \OGGEP{} boundary channel that \AFive{} implements as a CPTP map; listed here as essential background reading for the constitutional reading of \BIsix{} as the bulk anomaly mechanism whose boundary projection underlies \GDOS{}. First cited in \S\ref{sec:structural-proof}.

\bibitem[Chern and Simons(1974)]{ChernSimons1974}
\textbf{S.-S.~Chern and J.~Simons.}
\emph{Characteristic forms and geometric invariants}, \emph{Ann.\ Math.} \textbf{99} (1974) 48.
\par\noindent\textbf{Relevance.} Introduces the Chern--Simons transgression three-form $\omega_3$ with $\dd\omega_3=\tr F^2$, the local primitive that makes the modified Bianchi identity $\dd H_3 = X_4$ exact (Step~3 of \S\ref{sec:bi6-chern-weil-gerbe}). First cited in \S\ref{sec:structural-proof}.

\bibitem[Dai and Freed(1994)]{DaiFreed1994}
\textbf{X.~Dai and D.~S.~Freed.}
\emph{Eta invariants and determinant lines},
\emph{Journal of Mathematical Physics} \textbf{35} (1994) 5155--5194.
\par\noindent\textbf{Relevance.} Gives the modern determinant-line and eta-invariant framework in which global anomalies are properly formulated on mapping tori and their eight-dimensional fillings. In A3-BI6 the Atiyah--Patodi--Singer / Dai--Freed formalism is the technical backbone of \S\ref{sec:global-completion}: the global anomaly of \BIsix{} on the intended \GG{} background class is evaluated as the holonomy of an anomaly line on the seven-dimensional mapping torus and reduced to an eight-dimensional Dai--Freed integral that vanishes identically. First cited in \S\ref{sec:geometric-origin}.

\bibitem[Dierigl and Tartaglia(2025)]{DieriglTartaglia2025}
\textbf{M.~Dierigl and M.~Tartaglia.}
\emph{(Quadratically) refined discrete anomaly cancellation},
\emph{JHEP} \textbf{2508} (2025) 145,
arXiv:2504.02934.
\par\noindent\textbf{Relevance.} Develops quadratically refined discrete Green--Schwarz mechanism in 6D F-theory: the most recent advance in global anomaly technology applicable to both SUSY and non-SUSY settings. In A3-BI6 we cite Dierigl--Tartaglia at the discussion of global-anomaly methods in \S\ref{sec:gs-mechanism}, at the future-mathematics paragraph of \S\ref{sec:global-completion}, and in the open problem of \S\ref{sec:open-problems} that asks for non-SUSY extension of the Monnier--Moore and Dierigl--Tartaglia technology to the unrestricted four-component \BIsix{} setting. First cited in \S\ref{sec:structural-proof}.

\bibitem[Dixon et al.(1985)]{DixonHarveyVafaWitten1985}
\textbf{L.~J.~Dixon, J.~A.~Harvey, C.~Vafa, and E.~Witten.}
\emph{Strings on orbifolds},
\emph{Nucl.\ Phys.\ B} \textbf{261} (1985) 678.
\par\noindent\textbf{Relevance.} Introduces orbifold compactification in string theory: the systematic construction of consistent string vacua on orbifolds $M/\Gamma$ with twisted sectors that restore modular invariance. In A3-BI6 we cite DHVW at the non-SUSY-heterotic paragraphs of \S\ref{sec:comparison-6d}; the interval and orbifold boundary conditions used in the 6D bulk $M_4 \times I_r \times S^1_\theta$ of \GG{} share the mathematical framework while replacing the string-theoretic origin with a constitutional one. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Ehresmann(1950)]{Ehresmann1950}
\textbf{C.~Ehresmann.}
\emph{Les connexions infinit\'esimales dans un espace fibr\'e diff\'erentiable}, in \emph{Colloque de Topologie, Bruxelles} (1950) 29.
\par\noindent\textbf{Relevance.} Founding definition of a connection on a fiber bundle, whose curvature obeys the Bianchi identity underlying Chern--Weil theory (Step~1 of \S\ref{sec:bi6-chern-weil-gerbe}). First cited in \S\ref{sec:structural-proof}.

\bibitem[Erler(1994)]{Erler1994}
\textbf{J.~Erler.}
\emph{Anomaly cancellation in six dimensions},
\emph{J.\ Math.\ Phys.} \textbf{35} (1994) 1819,
arXiv:hep-th/9304104.
\par\noindent\textbf{Relevance.} Systematically tabulates the gravitational,
gauge, and mixed anomaly contributions of the six-dimensional
$\cN=(1,0)$ multiplets --- the per-field coefficients behind the
$n_H - n_V + 29\,n_T = 273$ constraint. In A3-BI6 we cite Erler at the
gravitino-closure discussion of \S\ref{ssec:non-susy-grav-closure},
where the cohort's non-supersymmetric completion is contrasted with
the tabulated SUGRA budget. First cited in \S\ref{ssec:non-susy-grav-closure}.

\bibitem[Freed and Hopkins(2021)]{FreedHopkins2016}
\textbf{D.~S.~Freed and M.~J.~Hopkins.}
\emph{Reflection positivity and invertible topological phases},
\emph{Geometry \& Topology} \textbf{25} (2021) 1165--1330.
\par\noindent\textbf{Relevance.} Places anomaly cancellation in the modern reflection-positive invertible-field-theory language and clarifies the global/topological meaning of anomaly inflow via the cobordism classification of invertible TQFTs. It sits at the foundation of the global-anomaly methodology used in \S\ref{sec:global-completion}; listed here as background reading for readers approaching the differential-cohomology global completion of \BIsix{} from the reflection-positive-invertible side of the modern anomaly literature. First cited in the Executive Summary.

\bibitem[Fujikawa and Suzuki(2004)]{FujikawaSuzuki2004}
\textbf{K.~Fujikawa and H.~Suzuki.}
\emph{Path Integrals and Quantum Anomalies},
Oxford University Press (2004).
\par\noindent\textbf{Relevance.} The book-length path-integral
treatment of anomalies: the non-invariance of the fermion measure as
the origin of the index density, the viewpoint invoked in the
physics-intuition opening of Appendix~\ref{app:descent-full}. First
cited in the Part~VII course introduction.

\bibitem[Garc\'ia-Etxebarria and Montero(2019)]{GarciaEtxebarriaMontero2019}
\textbf{I.~Garc\'ia-Etxebarria and M.~Montero.}
\emph{Dai--Freed anomalies in particle physics},
\emph{JHEP} \textbf{1908} (2019) 003,
arXiv:1808.00009.
\par\noindent\textbf{Relevance.} Deploys the Dai--Freed formalism on
the gauge groups of particle physics --- the closest published
analogue of the global checks run in
Appendix~\ref{app:global-proof-dossier}, and the recommended worked
companion to its eta-invariant primer. First cited in the Part~VII
course introduction.

\bibitem[Georgi and Glashow(1974)]{GeorgiGlashow1974}
\textbf{H.~Georgi and S.~Glashow.}
\emph{Unity of all elementary-particle forces}, \emph{Phys.\ Rev.\ Lett.} \textbf{32} (1974) 438.
\par\noindent\textbf{Relevance.} Introduces the $\SU(5)$ grand-unified theory: the unification of $\SU(3)_c \times \SU(2)_L \times \U(1)_Y$ into a single semisimple gauge group with one matter generation in $\mathbf{\bar 5} \oplus \mathbf{10}$. $\SU(5)$ is the non-abelian factor of the four-component bulk gauge group $\Spin(1,5)\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$; the $\Tr F_5^2$ contribution to the \BIsix{} four-form $X_4$ of \S\ref{sec:bi6-fourform} is its anomaly-polynomial signature. First cited in \S\ref{sec:structural-proof}.

\bibitem[Georgi(1999)]{Georgi1999}
\textbf{H.~Georgi.}
\emph{Lie Algebras in Particle Physics}, 2nd ed.,
Perseus Books (1999).
\par\noindent\textbf{Relevance.} The working physicist's Lie-algebra
text; supplies the derivations behind the representation indices
$(A, A_3, B, C)$ tabulated and used in
Appendix~\ref{app:core-block-full}. First cited in the Part~VII
course introduction.

\bibitem[Gorini, Kossakowski, and Sudarshan(1976)]{Gorini1976}
\textbf{V.~Gorini, A.~Kossakowski, and E.~C.~G.~Sudarshan.}
\emph{Completely positive dynamical semigroups of \(N\)-level systems},
\emph{J.\ Math.\ Phys.} \textbf{17} (1976) 821.
\par\noindent\textbf{Relevance.} Derives the GKLS generator for completely positive Markovian quantum dynamical semigroups on $N$-level systems --- one half of the canonical open-systems classification. In A3-BI6 we cite Gorini--Kossakowski--Sudarshan alongside Lindblad at \S\ref{sec:UoneQi-constitutional-status}, where the Markovian limit of the boundary \OGGEP{} channel of \AFive{} is identified as exactly the GKLS structure; the bulk \BIsix{} anomaly freedom proved in A3-BI6 is the lawfulness prerequisite for that boundary channel. First cited in \S\ref{sec:global-completion}.

\bibitem[Green and Schwarz(1984)]{GreenSchwarz1984}
\textbf{M.~B.~Green and J.~H.~Schwarz.}
\emph{Anomaly cancellations in supersymmetric \(D=10\) gauge theory and superstring theory}, \emph{Phys.\ Lett.\ B} \textbf{149} (1984) 117.
\par\noindent\textbf{Relevance.} Discovers the Green--Schwarz mechanism in ten dimensions: anomaly cancellation through a higher-form tensor whose modified Bianchi identity $dH_3 = X_4$ couples to the gauge curvature, with $I_{12} = X_4 \wedge X_8$ factorization. The six-dimensional \BIsix{} construction of A3-BI6 is the axiomatic descendant of this 1984 result: \S\ref{sec:comparison-10d} carries out the feature-by-feature comparison, and \S\ref{sec:GS_as_GGEP_completion} reads \BIsix{} as the \GGEP{} completion of GS within the \GGTheory{} Constitution. First cited in the Executive Summary.

\bibitem[Green, Schwarz, and West(1985)]{GreenSchwarzWest1985}
\textbf{M.~B.~Green, J.~H.~Schwarz, and P.~C.~West.}
\emph{Anomaly-free chiral theories in six dimensions},
\emph{Nucl.\ Phys.\ B} \textbf{254} (1985) 327.
\par\noindent\textbf{Relevance.} First construction of anomaly-free chiral gauge theories in six dimensions using the Green--Schwarz mechanism with the factorization $I_8 = (1/2) X_4 \wedge X_4$. The direct precursor of every 6D anomaly construction including \BIsix{}. In A3-BI6 we cite GSW at the 6D specialization paragraphs of \S\ref{sec:gs-mechanism}, at the existing-literature comparison in \S\ref{sec:comparison-6d}, and at the rank-one and Diophantine matching discussions of \S\ref{sec:model-level} where the perfect-square factorization is rederived for the four-component coefficient vector. First cited in the Executive Summary.

\bibitem[Green, Schwarz, and Witten(1987)]{GSW1987}
\textbf{M.~B.~Green, J.~H.~Schwarz, and E.~Witten.}
\emph{Superstring Theory, Vol.~II: Loop Amplitudes, Anomalies and
Phenomenology},
Cambridge University Press (1987).
\par\noindent\textbf{Relevance.} The original textbook treatment of
the ten-dimensional Green--Schwarz mechanism (Ch.~13), including the
$\SO(n)$ trace identities and the explicit $X_8$; the primary
reference for Appendix~\ref{app:gs10d}, whose derivations the student
should check against it. First cited in the Part~VII course
introduction.

\bibitem[Gubser, Klebanov, and Polyakov(1998)]{Gubser1998}
\textbf{S.~S.~Gubser, I.~R.~Klebanov, and A.~M.~Polyakov.}
\emph{Gauge theory correlators from non-critical string theory},
\emph{Phys.\ Lett.\ B} \textbf{428} (1998) 105,
arXiv:hep-th/9802109.
\par\noindent\textbf{Relevance.} Makes the AdS/CFT correspondence computationally precise via the GKP dictionary that relates bulk fields to boundary operators by matching the on-shell bulk action to the generating functional of boundary correlators. In A3-BI6 we cite Gubser--Klebanov--Polyakov in \S\ref{sec:ads-rs} alongside Maldacena and Witten 1998; the dictionary underlies the \OGGEP{} operational map that the boundary \GDOS{} program implements as a CPTP channel, with \BIsix{}'s anomaly freedom providing the bulk consistency that the dictionary presupposes. First cited in \S\ref{sec:ads-rs}.

\bibitem[Hamada and Loges(2024)]{HamadaLoges2024}
\textbf{Y.~Hamada and G.~J.~Loges.}
\emph{Enumerating 6D supergravities with $T\le 1$},
\emph{JHEP} \textbf{2412} (2024) 167,
arXiv:2404.08845.
\par\noindent\textbf{Relevance.} Systematic enumeration of all anomaly-free 6D $\cN=(1,0)$ supergravity models with $T\le 1$ tensor multiplets: the state-of-the-art census of the 6D SUSY landscape. In A3-BI6 we cite Hamada--Loges at the landscape-enumeration paragraphs of \S\ref{sec:comparison-6d} and at the open-problem on extending the non-SUSY classification in \S\ref{sec:open-problems}, where adapting their methods to the unrestricted four-component \BIsix{} setting is identified as a well-posed mathematical problem. First cited in \S\ref{sec:comparison-6d}.

\bibitem[de Haro, Skenderis, and Solodukhin(2001)]{deHaroSkenderis2001}
\textbf{S.~de~Haro, S.~N.~Solodukhin, and K.~Skenderis.}
\emph{Holographic renormalization},
\emph{Commun.\ Math.\ Phys.} \textbf{217} (2001) 595,
arXiv:hep-th/0002230.
\par\noindent\textbf{Relevance.} Develops holographic renormalization: the systematic procedure for regulating the on-shell bulk action and extracting boundary correlators, with the radial coordinate identified as RG depth. In A3-BI6 we cite de Haro--Skenderis--Solodukhin in the AdS/CFT review of \S\ref{sec:ads-rs}; the radial-RG identification they made precise is adopted constitutionally in \GG{} via postulate \POS{5} (radial RG depth) and reappears in the holographic-comparison table of \S\ref{sec:ads-rs}. First cited in \S\ref{sec:ads-rs}.

\bibitem[Harvey(2005)]{Harvey2005}
\textbf{J.~A.~Harvey.}
\emph{TASI 2003 lectures on anomalies},
arXiv:hep-th/0509097.
\par\noindent\textbf{Relevance.} The standard TASI lectures on
anomalies and inflow, the second recommended lecture-note companion
(beside \citealp{Bilal2008}) to Appendices~\ref{app:descent-full}
and~\ref{app:gs10d}. First cited in the Part~VII course introduction.

\bibitem[Hayashi, Kim, and Kim(2025)]{HayashiKimKim2025}
\textbf{H.~Hayashi, H.-C.~Kim, and M.~Kim.}
\emph{Spectra of BPS strings in 6D supergravity and the swampland},
\emph{JHEP} \textbf{2503} (2025) 123,
arXiv:2310.12219.
\par\noindent\textbf{Relevance.} Tests Weak-Gravity and Emergent-String swampland conjectures using BPS-string spectra in 6D $\cN=(1,0)$ supergravity --- the most recent systematic swampland analysis in 6D. In A3-BI6 we cite Hayashi--Kim--Kim at the swampland paragraphs of \S\ref{sec:comparison-6d}; the comparison is asymmetric --- \BIsix{} achieves consistency without invoking SUSY multiplets or BPS strings, by relying instead on the four-component coefficient lock and the \OGNfive{} integer ledger derived in \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Hopkins and Singer(2005)]{HopkinsSinger2005}
\textbf{M.~J.~Hopkins and I.~M.~Singer.}
\emph{Quadratic functions in geometry, topology, and M-theory},
\emph{J.\ Differential Geom.} \textbf{70} (2005) 329,
arXiv:math/0211216.
\par\noindent\textbf{Relevance.} Develops the differential-cohomology framework for quadratic refinements of self-dual abelian gauge fields, originally aimed at the M-theory $C_3$-field: the mathematical machinery that makes the self-dual tensor $B_2$ of \BIsix{} single-valued on non-trivial topology. In A3-BI6 we cite Hopkins--Singer at the global-completion paragraphs of \S\ref{sec:global-completion} and at the quadratic-refinement discussion; the Wu--Chern--Simons / differential-cohomology refinement of the integral $X_4$ class derived there is the direct application of their construction. First cited in the Executive Summary.

\bibitem[Ib\'a\~nez and Uranga(2012)]{IbanezUranga2012}
\textbf{L.~E.~Ib\'a\~nez and A.~M.~Uranga.}
\emph{String Theory and Particle Physics: An Introduction to String
Phenomenology},
Cambridge University Press (2012).
\par\noindent\textbf{Relevance.} The phenomenological bridge between
string constructions and particle physics; recommended to the student
comparing the string-side model-building strategies with the
constitutional route of this paper (Part~IV and
Appendix~\ref{app:gs10d}). First cited in the Part~VII course
introduction.

\bibitem[Kaidi, Tachikawa, and Yonekura(2025)]{KaidiTachikawaYonekura2025}
\textbf{J.~Kaidi, Y.~Tachikawa, and K.~Yonekura.}
\emph{On non-supersymmetric heterotic branes},
\emph{JHEP} \textbf{2503} (2025) 211,
arXiv:2411.04344.
\par\noindent\textbf{Relevance.} Studies non-supersymmetric heterotic branes, demonstrating that string theory accommodates non-supersymmetric configurations of direct relevance to the \GG{} philosophy of doing 6D physics without 6D supersymmetry. In A3-BI6 we cite Kaidi--Tachikawa--Yonekura at the non-SUSY-heterotic paragraphs of \S\ref{sec:comparison-6d}; the convergence point is that anomaly cancellation continues to be the principal admissibility filter once supersymmetry is removed, exactly as in the four-component \BIsix{} construction of \S\ref{sec:structural-proof}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Kim, Vafa, and Xu(2024)]{KimVafaXu2024}
\textbf{H.-C.~Kim, C.~Vafa, and K.~Xu.}
\emph{Finite landscape of 6D $\cN=(1,0)$ supergravity},
arXiv:2411.19155.
\par\noindent\textbf{Relevance.} Argues for finiteness of the complete 6D $\cN=(1,0)$ supergravity landscape --- the latest evidence that the 6D string landscape is bounded. In A3-BI6 we cite Kim--Vafa--Xu at the finite-versus-infinite paragraphs of \S\ref{sec:comparison-10d} and at the swampland and landscape paragraphs of \S\ref{sec:comparison-6d}; the \OGNfive{} integer ledger of \S\ref{sec:ogn5-theorem} delivers an even sharper finiteness statement from inside the \GG{} framework, namely a single Diophantine solution rather than a bounded set. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Kumar, Morrison, and Taylor(2010)]{KumarMorrisonTaylor2010}
\textbf{V.~Kumar, D.~R.~Morrison, and W.~Taylor.}
\emph{Global aspects of the space of 6D $\cN=1$ supergravities},
\emph{JHEP} \textbf{1011} (2010) 118,
arXiv:1008.1062.
\par\noindent\textbf{Relevance.} Systematic classification of the global structure of anomaly-free 6D $\cN=(1,0)$ supergravity theories, including the integral anomaly lattice underpinning the string universality program. In A3-BI6 we cite Kumar--Morrison--Taylor at the F-theory and lattice paragraphs of \S\ref{sec:f-theory} and at the comparison in \S\ref{sec:comparison-6d}; the contrast with the \OGNfive{} integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$ derived in \S\ref{sec:ogn5-theorem} is the central technical point of the comparison. First cited in the Executive Summary.

\bibitem[Kumar, Park, and Taylor(2011)]{KumarParkTaylor2011}
\textbf{V.~Kumar, D.~S.~Park, and W.~Taylor.}
\emph{6D supergravity without tensor multiplets},
\emph{JHEP} \textbf{1104} (2011) 080,
arXiv:1011.0726.
\par\noindent\textbf{Relevance.} Studies 6D supergravity theories without tensor multiplets ($T=0$), demonstrating the additional constraints from anomaly cancellation on the simplest class. In A3-BI6 we cite Kumar--Park--Taylor at the F-theory paragraphs of \S\ref{sec:f-theory}; the contrast is structural --- \BIsix{} carries exactly one self-dual tensor $B_2$ enforced by axiom \AX{4}, neither none nor many, with the corresponding rank-one anomaly matrix derived in \S\ref{sec:structural-proof} and the perfect-square factorization $I_8=(1/2)X_4^2$ following uniquely. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Kumar and Taylor(2009)]{KumarTaylor2009}
\textbf{V.~Kumar and W.~Taylor.}
\emph{String universality in six dimensions},
\emph{Adv.\ Theor.\ Math.\ Phys.} \textbf{15} (2011) 325,
arXiv:0906.0987.
\par\noindent\textbf{Relevance.} Proposes the string universality conjecture: every anomaly-free 6D $\cN=(1,0)$ supergravity theory with one tensor multiplet is realized somewhere in string theory. The sharpest statement of the string program's completeness claim in 6D. In A3-BI6 we cite Kumar--Taylor at the discussion of finite versus infinite landscape in \S\ref{sec:comparison-10d} and \S\ref{sec:comparison-6d}; the \OGNfive{} Diophantine spectrum problem of \S\ref{sec:ogn5-arithmetic} is the constitutional analogue of universality from inside the \GG{} framework. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{G.~Lindblad.}
\emph{On the generators of quantum dynamical semigroups},
\emph{Commun.\ Math.\ Phys.} \textbf{48} (1976) 119.
\par\noindent\textbf{Relevance.} Derives the most general bounded generator of a quantum dynamical semigroup: the other half (alongside Gorini--Kossakowski--Sudarshan) of the canonical GKLS classification of completely positive Markovian open-system dynamics. In A3-BI6 we cite Lindblad at \S\ref{sec:UoneQi-constitutional-status}, where the Markovian limit of the boundary \OGGEP{} channel of \AFive{} is identified as exactly the GKLS--Lindblad structure that the bulk anomaly-freedom certificate of \BIsix{} licenses. First cited in \S\ref{sec:global-completion}.

\bibitem[Loges(2024)]{Loges2024}
\textbf{G.~J.~Loges.}
\emph{New infinite class of 6D $\cN=(1,0)$ supergravities},
\emph{Phys.\ Rev.\ D} \textbf{109} (2024) 126006,
arXiv:2402.04371.
\par\noindent\textbf{Relevance.} Finds new infinite families of anomaly-free 6D $\cN=(1,0)$ supergravity models, demonstrating that the SUSY landscape is still actively expanding. In A3-BI6 we cite Loges at the 6D-landscape paragraphs of \S\ref{sec:comparison-6d} and at the discussion of finiteness in \S\ref{sec:open-problems}; the contrast emphasizes why the bounded Diophantine spectrum derived from the \OGNfive{} integer ledger in \S\ref{sec:ogn5-theorem} represents a categorically different approach from infinite-family enumeration within SUSY. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Maldacena(1998)]{Maldacena1998}
\textbf{J.~Maldacena.}
\emph{The large \(N\) limit of superconformal field theories and supergravity},
\emph{Adv.\ Theor.\ Math.\ Phys.} \textbf{2} (1998) 231,
arXiv:hep-th/9711200.
\par\noindent\textbf{Relevance.} Proposes the AdS/CFT correspondence: the foundational result that a $d$-dimensional CFT is dual to a $(d+1)$-dimensional bulk gravity theory. The holographic logic that a radial bulk coordinate encodes boundary RG scale is the direct ancestor of the \GG{} radial coordinate $r = \ln(E_{\rm GUT}/\mu)$. In A3-BI6 we cite Maldacena throughout \S\ref{sec:ads-rs}, especially in the comparison table that locates AdS/CFT, Randall--Sundrum, and \BIsix{} all on Stratum~2 of the \GG{} constitutional classification. First cited in \S\ref{sec:ads-rs}.

\bibitem[Maxwell(1865)]{Maxwell1865}
\textbf{J.~C.~Maxwell.}
\emph{A dynamical theory of the electromagnetic field},
\emph{Phil.\ Trans.\ Roy.\ Soc.\ London} \textbf{155} (1865) 459--512.
\par\noindent\textbf{Relevance.} Originates the gauge-invariant four-vector potential of classical electrodynamics: the historical seed of the gauge invariance whose chiral-fermion failure at one loop is the Adler--Bell--Jackiw anomaly that \BIsix{} must cancel in higher dimensions. In A3-BI6 we cite Maxwell at the historical introduction of \S\ref{sec:introduction} and in the gauge-theory genealogy of \S\ref{sec:gs-mechanism}; the modified Bianchi identity $dH_3 = X_4$ of \BIsix{} is the topological completion of the Maxwell--Yang--Mills tradition. First cited in \S\ref{sec:gs-mechanism}.

\bibitem[Milnor and Stasheff(1974)]{MilnorStasheff1974}
\textbf{J.~Milnor and J.~Stasheff.}
\emph{Characteristic Classes}, Princeton University Press (1974).
\par\noindent\textbf{Relevance.} Standard textbook source for Pontryagin classes, characteristic elements, and the trace-to-characteristic-class conversions used throughout the \BIsix{} proof. Milnor--Stasheff is the canonical reference for the characteristic-class identities deployed in Appendix~\ref{app:descent-full} when converting $\tr R^2$, $\Tr F_5^2$, $F_B^2$, and $F_{Qi}^2$ into the four-component four-form $X_4$ whose perfect-square factorization $I_8 = (1/2) X_4^2$ closes the local anomaly of \BIsix{}. First cited in \S\ref{sec:structural-proof}.

\bibitem[Monnier(2018)]{Monnier2018}
\textbf{S.~Monnier.}
\emph{The anomaly field theories of six-dimensional \((2,0)\) superconformal theories},
\emph{Adv.\ Theor.\ Math.\ Phys.} \textbf{22} (2018) 2035--2089, arXiv:1706.01903.
\par\noindent\textbf{Relevance.} Develops the anomaly-field-theory viewpoint for six-dimensional chiral systems: a close conceptual parallel for the tensor-sector analysis that A3-BI6 carries out for \BIsix{}. It underlies the modern invertible-field-theory framework used in \S\ref{sec:global-completion} to evaluate the global anomaly on mapping tori; it is listed here as background reading for readers approaching the differential-cohomology completion of \BIsix{} from the $(2,0)$-superconformal anomaly literature. First cited in \S\ref{sec:geometric-origin}.

\bibitem[Monnier and Moore(2019)]{MonnierMoore2019}
\textbf{S.~Monnier and G.~W.~Moore.}
\emph{Remarks on the Green--Schwarz terms of six-dimensional supergravity theories},
\emph{Commun.\ Math.\ Phys.} \textbf{372} (2019) 963,
arXiv:1808.01334.
\par\noindent\textbf{Relevance.} Constructs the Green--Schwarz counterterms on non-trivially topological spacetimes and proves global anomaly cancellation for standard 6D $\cN=(1,0)$ supergravity gauge groups. The definitive treatment of global consistency that A3-BI6 must satisfy on the intended \GG{} background class. We cite Monnier--Moore at the global-anomaly paragraphs of \S\ref{sec:global-completion}, at the quadratic-refinement discussion, and at the future-mathematics open problem on non-SUSY extension in \S\ref{sec:open-problems}. First cited in the Executive Summary.

\bibitem[Morrison and Taylor(2012a)]{MorrisonTaylor2012a}
\textbf{D.~R.~Morrison and W.~Taylor.}
\emph{Classifying bases for 6D F-theory models},
\emph{Central Eur.\ J.\ Phys.} \textbf{10} (2012) 1072,
arXiv:1201.1943.
\par\noindent\textbf{Relevance.} Classifies the base surfaces for F-theory compactifications to 6D: a key step in the enumeration of the 6D F-theory landscape. In A3-BI6 we cite Morrison--Taylor 2012a at the F-theory enumeration paragraphs of \S\ref{sec:f-theory}; the contrast underscores how F-theory builds 6D models bottom-up from base-geometry data, while \BIsix{} builds them top-down from constitutional axioms and lands on the unique four-component coefficient vector derived in \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Morrison and Taylor(2012b)]{MorrisonTaylor2012b}
\textbf{D.~R.~Morrison and W.~Taylor.}
\emph{Matter and singularities},
\emph{JHEP} \textbf{1201} (2012) 022,
arXiv:1106.3563.
\par\noindent\textbf{Relevance.} Develops the correspondence between singularity types in elliptic fibrations and matter representations in F-theory: the geometric mechanism for determining 6D matter spectra. In A3-BI6 we cite Morrison--Taylor 2012b alongside Morrison--Taylor 2012a at the F-theory paragraphs of \S\ref{sec:f-theory}; the comparison contrasts the F-theory geometry-to-spectrum pathway with the constitutional axiom-to-spectrum pathway through which \BIsix{} determines the $\SU(5)$ matter content of \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Morrison and Vafa(1996a)]{MorrisonVafa1996a}
\textbf{D.~R.~Morrison and C.~Vafa.}
\emph{Compactifications of F-theory on Calabi--Yau threefolds~I},
\emph{Nucl.\ Phys.\ B} \textbf{473} (1996) 74,
arXiv:hep-th/9602114.
\par\noindent\textbf{Relevance.} Develops F-theory compactification on elliptic Calabi--Yau threefolds to six dimensions, providing the geometric origin of 6D gauge groups and matter from fiber singularities. In A3-BI6 we cite Morrison--Vafa I at the F-theory comparison of \S\ref{sec:f-theory}; the contrast is that \BIsix{} arrives at the four-component bulk gauge group $\Spin(1,5)\times\SU(5)\times\U(1)_B\times\U(1)_{Qi}$ constitutionally rather than from CY-fibration data, with the trace-ratio lock $|a/c|=|a/d|=5/2$ derived in \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Morrison and Vafa(1996b)]{MorrisonVafa1996b}
\textbf{D.~R.~Morrison and C.~Vafa.}
\emph{Compactifications of F-theory on Calabi--Yau threefolds~II},
\emph{Nucl.\ Phys.\ B} \textbf{476} (1996) 437,
arXiv:hep-th/9603161.
\par\noindent\textbf{Relevance.} Completes the F-theory compactification program to six dimensions and establishes the anomaly-lattice correspondence used throughout the string-universality program. In A3-BI6 we cite Morrison--Vafa II alongside Part~I at the F-theory paragraphs of \S\ref{sec:f-theory}; the lattice correspondence on the F-theory side parallels the \OGNfive{} integer ledger derived in \S\ref{sec:ogn5-theorem} but achieves bounded multiplicity rather than the unique Diophantine solution that \BIsix{} forces under the constitutional admissibility filter. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Murray(1996)]{Murray1996}
\textbf{M.~K.~Murray.}
\emph{Bundle gerbes}, \emph{J.\ London Math.\ Soc.} \textbf{54} (1996) 403.
\par\noindent\textbf{Relevance.} The rigorous object carrying a two-form potential $B_2$ with three-form curvature $H_3$ and Dixmier--Douady class in $H^3$; the precise ``one level up'' bundle of Step~2 of \S\ref{sec:bi6-chern-weil-gerbe}. First cited in \S\ref{sec:structural-proof}.

\bibitem[Nakahara(2003)]{Nakahara2003}
\textbf{M.~Nakahara.}
\emph{Geometry, Topology and Physics}, 2nd ed.,
Institute of Physics Publishing (2003).
\par\noindent\textbf{Relevance.} The standard physicist's text for
characteristic classes, index theorems, and the eta invariant (Ch.~11
and~12) --- the geometry background assumed by
Appendices~\ref{app:grav-full} and~\ref{app:global-proof-dossier}.
First cited in the Part~VII course introduction.

\bibitem[Nakajima(1958)]{Nakajima1958}
\textbf{S.~Nakajima.}
\emph{On quantum theory of transport phenomena},
\emph{Prog.\ Theor.\ Phys.} \textbf{20} (1958) 948.
\par\noindent\textbf{Relevance.} Introduces the projection-operator technique for deriving reduced (non-Markovian) dynamics from a closed-system Liouville evolution: the mathematical foundation of the \OGGEP{} channel used in the boundary \GDOS{} program. Nakajima's 1958 result is one of the foundational components of the open-system formalism that A3-BI6 makes self-consistent through the bulk anomaly-freedom proof; listed here as background reading for the \OGGEP{} discussion in \S\ref{sec:intro-gdos} and \S\ref{sec:UoneQi-constitutional-status}. First cited in \S\ref{sec:introduction}.

\bibitem[Nishino and Sezgin(1984)]{NishinoSezgin1984}
\textbf{H.~Nishino and E.~Sezgin.}
\emph{Matter and gauge couplings of $\cN=2$ supergravity in six dimensions},
\emph{Phys.\ Lett.\ B} \textbf{144} (1984) 187.
\par\noindent\textbf{Relevance.} Foundational construction of 6D $\cN=(1,0)$ matter-coupled supergravity: the starting point for all subsequent 6D SUGRA anomaly analyzes. In A3-BI6 we cite Nishino--Sezgin at the 6D SUGRA paragraphs of \S\ref{sec:comparison-6d}; the comparison there shows that \BIsix{} achieves anomaly freedom without invoking 6D supersymmetry by replacing the SUGRA multiplet structure with the \OGNfive{} integer ledger and the four-component coefficient vector pinned by the trace-ratio theorem of \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Peskin and Schroeder(1995)]{PeskinSchroeder1995}
\textbf{M.~E.~Peskin and D.~V.~Schroeder.}
\emph{An Introduction to Quantum Field Theory},
Addison-Wesley (1995).
\par\noindent\textbf{Relevance.} The standard first-year graduate QFT
text; its Ch.~19 treatment of the axial anomaly is the assumed entry
point for the Part~VII course. First cited in the Part~VII course
introduction.

\bibitem[Polchinski(1998)]{Polchinski1998a}
\textbf{J.~Polchinski.}
\emph{String Theory, Vol.~I: An Introduction to the Bosonic String},
Cambridge University Press (1998).
\par\noindent\textbf{Relevance.} Standard graduate-level reference for the bosonic string, the
worldsheet formalism, and the conformal-field-theory machinery the
anomaly discussion presupposes. The Green--Schwarz mechanism itself is
in Volume~II. Polchinski Vol.~I is the canonical pedagogical source for readers approaching \BIsix{} from the string-theory tradition; the comparison of \S\ref{sec:comparison-10d} relies on the standard 10D treatments collected there, while \BIsix{} borrows the cohomological machinery and discards the stringy ontology. First cited in \S\ref{sec:ads-rs}.

\bibitem[Randall and Sundrum(1999)]{RandallSundrum1999}
\textbf{L.~Randall and R.~Sundrum.}
\emph{A large mass hierarchy from a small extra dimension},
\emph{Phys.\ Rev.\ Lett.} \textbf{83} (1999) 3370.
\par\noindent\textbf{Relevance.} Introduces warped extra dimensions: a 5D bulk with exponential warp factor that generates the electroweak/Planck hierarchy. The \GG{} throat geometry of $M_4 \times I_r \times S^1_\theta$ uses the same warp mechanism, with the warp factor responsible for the \OGNfive{} hierarchy $\Mlock \sim \MPl e^{-K}$. In A3-BI6 we cite Randall--Sundrum at the warped-geometry paragraphs of \S\ref{sec:ads-rs}, where the constitutional reading places Randall--Sundrum on Stratum~2 alongside \BIsix{}. First cited in \S\ref{sec:ads-rs}.

\bibitem[Sadov(1996)]{Sadov1996}
\textbf{V.~Sadov.}
\emph{Generalized Green--Schwarz mechanism in F-theory},
\emph{Phys.\ Lett.\ B} \textbf{388} (1996) 45,
arXiv:hep-th/9606008.
\par\noindent\textbf{Relevance.} Extends the Green--Schwarz mechanism to the F-theory context with multiple tensor multiplets, demonstrating the generalization that \BIsix{} explicitly avoids by carrying exactly one self-dual tensor $B_2$. Listed in A3-BI6 as background for the F-theory comparison of \S\ref{sec:f-theory} and Objection~5 of \S\ref{sec:common-objections}; the rank-one corollary of Theorem~\ref{thm:rank-one} confirms that the microscopic \BIsix{} anomaly matrix collapses to rank one, so multi-tensor extensions in the spirit of Sadov are unforced and inadmissible under \AX{4}. First cited in \S\ref{sec:model-level}.

\bibitem[Sagnotti(1992)]{Sagnotti1992}
\textbf{A.~Sagnotti.}
\emph{A note on the Green--Schwarz mechanism in open-string theories},
\emph{Phys.\ Lett.\ B} \textbf{294} (1992) 196.
\par\noindent\textbf{Relevance.} Extends the Green--Schwarz mechanism to open-string and 6D settings with anti-self-dual tensor multiplets, and is widely cited for multi-tensor constructions $I_8 = (1/2) \sum_i X_4^{(i)} \wedge X_4^{(i)}$. In A3-BI6 we cite Sagnotti at the 6D-literature paragraphs of \S\ref{sec:comparison-6d} and at Objection~5 of \S\ref{sec:common-objections}, where multi-tensor extensions are shown to be excluded by axiom \AX{4} (\MSMF) and by the rank-one theorem of \S\ref{sec:structural-proof}. First cited in the Executive Summary.

\bibitem[Schwarz(1996)]{Schwarz1996}
\textbf{J.~H.~Schwarz.}
\emph{Anomaly-free supersymmetric models in six dimensions}, \emph{Phys.\ Lett.\ B} \textbf{371} (1996) 223--230, arXiv:hep-th/9512053.
\par\noindent\textbf{Relevance.} Compact statement of the
six-dimensional $\cN=(1,0)$ anomaly-cancellation conditions and the
multiplet bookkeeping behind the $273$ constraint. In A3-BI6 we cite
Schwarz alongside Erler at \S\ref{ssec:non-susy-grav-closure}, where
the cohort's anti-self-dual-tensor/singlet-fermion closure is compared
with the supersymmetric budget it replaces. First cited in
\S\ref{ssec:non-susy-grav-closure}.

\bibitem[Seiberg and Taylor(2011)]{SeibergTaylor2011}
\textbf{N.~Seiberg and W.~Taylor.}
\emph{Charge lattices and consistency of 6D supergravity},
\emph{JHEP} \textbf{1106} (2011) 001,
arXiv:1103.0019.
\par\noindent\textbf{Relevance.} Establishes the charge-lattice consistency conditions for 6D $\cN=(1,0)$ supergravity: the integrality requirements that fix admissible matter content. These are the SUSY analogue of the \OGNfive{} integer-lattice constraints in \GG{}. In A3-BI6 we cite Seiberg--Taylor at the global-refinement paragraphs of Appendix~\ref{app:global-proof-dossier}, where the differential-cohomology refinement of the integral $X_4$ class is discussed; the rank-one structure of the \BIsix{} tensor lattice makes the charge-quantization analysis significantly more constrained than in the SUSY setting. First cited in the Executive Summary.

\bibitem[Slansky(1981)]{Slansky1981}
\textbf{R.~Slansky.}
\emph{Group theory for unified model building},
\emph{Phys.\ Rept.} \textbf{79} (1981) 1.
\par\noindent\textbf{Relevance.} The canonical tables of group-theory
data --- dimensions, indices, branching rules --- from which the
$\SU(5)$ representation data of Appendix~\ref{app:core-block-full}
can be independently checked. First cited in the Part~VII course
introduction.

\bibitem[Stora(1984)]{Stora1984}
\textbf{R.~Stora.} \emph{Algebraic structure and topological origin of anomalies}, in \emph{Progress in Gauge Field Theory} (Carg\`ese 1983), NATO ASI Series~B \textbf{115}, Plenum (1984) 543--562.
\par\noindent\textbf{Relevance.} With Zumino, the origin of the descent formalism --- the chain of forms relating an anomaly polynomial in $2n+2$ dimensions to the anomaly in $2n$ dimensions --- which is the machinery of Appendix~\ref{app:descent-full} and of every local cancellation argument in this paper. Named in the text as the Stora--Zumino descent. First cited in Section~\ref{sec:gs-mechanism}.

\bibitem[Stong(1968)]{Stong1968}
\textbf{R.~E.~Stong.}
\emph{Notes on Cobordism Theory}, Princeton University Press (1968).
\par\noindent\textbf{Relevance.} Provides the classical cobordism facts used in the
automatic-admissibility theorem, in particular the structure of the
low-dimensional spin bordism groups. String bordism post-dates this
volume by two decades and is not taken from it; the string-bordism and
$tmf$ input is cited separately. Stong's textbook is the standard source for the bordism-theoretic ingredients used in the determinant-line and eta-invariant computations of Appendix~\ref{app:global-proof-dossier}; listed here as background reading for readers approaching the global completion of \BIsix{} from the bordism side. First cited in \S\ref{sec:structural-proof}.

\bibitem[Tarazi and Vafa(2021)]{TaraziVafa2021}
\textbf{H.-C.~Tarazi and C.~Vafa.}
\emph{On the finiteness of 6D supergravity landscape},
arXiv:2106.10839.
\par\noindent\textbf{Relevance.} Proves finiteness bounds on the 6D supergravity landscape: the string-theory analogue of the bounded Diophantine spectrum problem in \GG{}. In A3-BI6 we cite Tarazi--Vafa at the landscape-finiteness paragraphs of \S\ref{sec:comparison-6d} and at the swampland discussion; the parallel is structural --- both programs constrain the admissible 6D theory space, but the \OGNfive{} ledger of \S\ref{sec:ogn5-theorem} closes the count to a unique solution $(M,N,L;J,K)=(1,3,5;7,35)$ rather than to a bounded set. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Taylor(2011)]{Taylor2011}
\textbf{W.~Taylor.}
\emph{TASI lectures on supergravity and string vacua in various dimensions},
arXiv:1104.2051.
\par\noindent\textbf{Relevance.} Comprehensive TASI-lectures review of 6D supergravity, anomaly cancellation, and the F-theory connection: an essential reference for any 6D anomaly-cancellation paper. In A3-BI6 we cite Taylor at the F-theory paragraphs of \S\ref{sec:f-theory}, the existing-literature review of \S\ref{sec:comparison-6d}, and the global-refinement discussion of Appendix~\ref{app:global-proof-dossier}; readers unfamiliar with the 6D SUSY-anomaly tradition will find Taylor's lectures the natural entry point before working through the four-component \BIsix{} construction. First cited in \S\ref{sec:model-level}.

\bibitem[Vafa(1996)]{Vafa1996}
\textbf{C.~Vafa.}
\emph{Evidence for F-theory},
\emph{Nucl.\ Phys.\ B} \textbf{469} (1996) 403,
arXiv:hep-th/9602022.
\par\noindent\textbf{Relevance.} Introduces F-theory as a non-perturbative framework for type IIB string compactifications: the geometric foundation for systematic 6D anomaly-free model building from elliptically fiberd Calabi--Yau threefolds. In A3-BI6 we cite Vafa at the F-theory paragraphs of \S\ref{sec:f-theory}; the comparison emphasizes that F-theory determines 6D gauge groups and matter from fiber-singularity geometry, whereas \BIsix{} determines them constitutionally through axioms \AX{1}--\AX{4} and the \OGNfive{} ledger of \S\ref{sec:ogn5-theorem}. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Vafa(2005)]{Vafa2005}
\textbf{C.~Vafa.}
\emph{The string landscape and the swampland},
arXiv:hep-th/0509212.
\par\noindent\textbf{Relevance.} Introduces the swampland concept: the distinction between consistent-looking effective theories and those actually realized in string theory. In A3-BI6 we cite Vafa 2005 alongside Brennan--Carta--Vafa at the swampland paragraphs of \S\ref{sec:comparison-6d} and at the landscape discussion of \S\ref{sec:comparison-10d}; the swampland framework provides the conceptual contrast against which the \BIsix{} construction is positioned --- not as a swampland counter-example but as a constitutionally selected single point. First cited in \S\ref{sec:comparison-6d}.

\bibitem[Weinberg(1996)]{WeinbergQFT1996}
\textbf{S.~Weinberg.}
\emph{The Quantum Theory of Fields, Vol.~II: Modern Applications},
Cambridge University Press (1996).
\par\noindent\textbf{Relevance.} Chapter~22 is the deepest textbook
treatment of anomalies in print --- consistent versus covariant forms,
anomaly matching, and the descent equations; the recommended in-depth
companion to Appendix~\ref{app:descent-full}. First cited in the
Part~VII course introduction.

\bibitem[Wess and Zumino(1971)]{WessZumino1971}
\textbf{J.~Wess and B.~Zumino.}
\emph{Consequences of anomalous Ward identities},
\emph{Phys.\ Lett.\ B} \textbf{37} (1971) 95.
\par\noindent\textbf{Relevance.} Establishes the Wess--Zumino consistency condition: the algebraic identity that any genuine anomaly must satisfy and that makes the descent formalism well-defined. In A3-BI6 the descent chain from the eight-form anomaly polynomial $I_8$ to the six-form variation, used to verify the Green--Schwarz cancellation in \S\ref{sec:structural-proof} and worked out explicitly in Appendix~\ref{app:descent-full}, is the systematic implementation of this consistency condition for the four-component \BIsix{} four-form $X_4$. First cited in the Executive Summary.

\bibitem[Witten(1998)]{Witten1998AdS}
\textbf{E.~Witten.}
\emph{Anti-de Sitter space and holography},
\emph{Adv.\ Theor.\ Math.\ Phys.} \textbf{2} (1998) 253,
arXiv:hep-th/9802150.
\par\noindent\textbf{Relevance.} Establishes the on-shell bulk gravitational action as the generating functional of boundary correlators in the AdS/CFT dictionary: the functional-integral formulation that \GG{}/\GDOS{} reads operationally as a CPTP channel rather than as an analytic-continuation identity. In A3-BI6 we cite Witten 1998 alongside Maldacena and Gubser--Klebanov--Polyakov in the AdS/CFT review of \S\ref{sec:ads-rs}; the constitutional reading of \S\ref{sec:ads-rs} places AdS/CFT and \BIsix{} together on Stratum~2 of the \GGTheory{} Constitution. First cited in \S\ref{sec:ads-rs}.

\bibitem[Witten(2016)]{WittenRMP2016}
\textbf{E.~Witten.}
\emph{Fermion path integrals and topological phases},
\emph{Rev.\ Mod.\ Phys.} \textbf{88} (2016) 035001,
arXiv:1508.04715.
\par\noindent\textbf{Relevance.} The modern physicist's treatment of
the eta invariant, the APS theorem, and anomaly inflow --- the
recommended companion to the eta primer of
Appendix~\ref{app:global-proof-dossier}, including the twisted
circle example reproduced there. First cited in the Part~VII course
introduction.

\bibitem[Wu and Yang(1975)]{WuYang1975}
\textbf{T.~T.~Wu and C.~N.~Yang.}
\emph{Concept of nonintegrable phase factors and global formulation of gauge fields}, \emph{Phys.\ Rev.\ D} \textbf{12} (1975) 3845.
\par\noindent\textbf{Relevance.} The fiber-bundle dictionary for gauge theory: holonomy as the Wilson line, and the Dirac monopole charge as a bundle Chern number --- the integrality invoked in Step~6 of \S\ref{sec:bi6-chern-weil-gerbe}. First cited in \S\ref{sec:structural-proof}.

\bibitem[Zwanzig(1960)]{Zwanzig1960}
\textbf{R.~Zwanzig.}
\emph{Ensemble method in the theory of irreversibility},
\emph{J.\ Chem.\ Phys.} \textbf{33} (1960) 1338.
\par\noindent\textbf{Relevance.} Develops the memory-kernel equation for projected open-system dynamics, completing the Nakajima--Zwanzig framework. Zwanzig's 1960 construction together with Nakajima 1958 yields the exact open-system evolution that the boundary \GDOS{} program of \AFive{} implements as a CPTP channel; listed here as background reading for the \OGGEP{} discussion in \S\ref{sec:intro-gdos}, whose lawfulness rests on the bulk \BIsix{} anomaly proof of A3-BI6. First cited in \S\ref{sec:introduction}.

\bibitem[Douglas(2003)]{Douglas2003}
\textbf{Douglas, M.~R. (2003).} The statistics of string/M theory vacua.
\emph{Journal of High Energy Physics}, \textbf{05}, 046.
\par\noindent\textbf{Relevance.} Develops the statistics of string/M-theory vacua: the quantitative side of the landscape program that estimates the number of consistent compactifications. In A3-BI6 we cite Douglas alongside Susskind in the inequivalent-compactifications paragraphs of \S\ref{sec:conclusions}; the contrast is methodological --- the \OGNfive{} Diophantine spectrum of \S\ref{sec:ogn5-theorem} closes the count to a unique solution rather than asymptoting to a statistical ensemble, replacing measure-theoretic landscape arguments with constitutional integer arithmetic. First cited in \S\ref{sec:ads-rs}.

\bibitem[Einstein(1915)]{Einstein1915}
\textbf{Einstein, A. (1915).} Die Feldgleichungen der Gravitation.
\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin},
844--847.
\par\noindent\textbf{Relevance.} Establishes general relativity as the dynamical theory of the Lorentzian metric whose Riemann curvature obeys the Bianchi identity that \BIsix{} modifies. In A3-BI6 we cite Einstein 1915 in the classical-foundations enumeration of \S\ref{sec:conclusions} and in the historical introduction; the geometric stratum on which the four-form $X_4 = \tr R^2 - \Tr F_5^2 - (2/5) F_B^2 - (2/5) F_{Qi}^2$ is built combines Einstein's Riemann-curvature contribution with the Yang--Mills gauge contributions in a single perfect-square factorization. First cited in \S\ref{sec:ads-rs}.

\bibitem[Polchinski(1998b)]{Polchinski1998b}
\textbf{Polchinski, J. (1998b).} \emph{String Theory, Vol.~II: Superstring
Theory and Beyond.} Cambridge University Press.
\par\noindent\textbf{Relevance.} Standard graduate-level reference for type II superstrings, heterotic strings, dualities, and the modern formulation of the 10D Green--Schwarz mechanism. Polchinski Vol.~II is the canonical pedagogical source for the 10D constructions that A3-BI6 compares against in \S\ref{sec:comparison-10d}; listed here as background reading for the feature-by-feature comparison between the 10D heterotic GS factorization $I_{12} = X_4 \wedge X_8$ and the rank-one 6D \BIsix{} factorization $I_8 = (1/2) X_4^2$. First cited in \S\ref{sec:ads-rs}.

\bibitem[Riemann(1854)]{Riemann1854}
\textbf{Riemann, B. (1854).} \"Uber die Hypothesen, welche der Geometrie zu
Grunde liegen. Habilitationsschrift, G\"ottingen. (Published posthumously:
\emph{Abhandlungen der K\"oniglichen Gesellschaft der Wissenschaften zu
G\"ottingen}, \textbf{13}, 133--150, 1868.)
\par\noindent\textbf{Relevance.} Establishes the foundations of $n$-dimensional differential geometry in the Habilitationsschrift on the hypotheses underlying geometry: the mathematical setting that Einstein later identified as the physical arena of general relativity and that \GGTheory{} extends to six dimensions. In A3-BI6 we cite Riemann 1854 at the historical introduction and at the classical-foundations enumeration of \S\ref{sec:conclusions}, where the lineage runs from Riemann through Einstein, Bianchi, Yang--Mills, and Green--Schwarz to the modified Bianchi identity of \BIsix{}. First cited in \S\ref{sec:geometric-origin}.

\bibitem[Susskind(2003)]{Susskind2003}
\textbf{Susskind, L. (2003).} The anthropic landscape of string theory.
arXiv:hep-th/0302219.
\par\noindent\textbf{Relevance.} Argues for the anthropic landscape of string theory: the philosophical pivot from uniqueness to multiplicity that motivates much of contemporary string phenomenology. In A3-BI6 we cite Susskind alongside Douglas in the inequivalent-compactifications paragraphs of \S\ref{sec:conclusions}; the contrast is constitutional --- the \OGNfive{} integer ledger derived in \S\ref{sec:ogn5-theorem} selects a unique solution $(M,N,L;J,K)=(1,3,5;7,35)$, removing the anthropic multiplicity that Susskind's landscape argument embraces. First cited in \S\ref{sec:ads-rs}.

\bibitem[Yang and Mills(1954)]{YangMills1954}
\textbf{Yang, C.~N., and Mills, R.~L. (1954).} Conservation of isotopic spin
and isotopic gauge invariance. \emph{Physical Review}, \textbf{96}, 191--195.
\par\noindent\textbf{Relevance.} Generalizes Maxwell's gauge principle from abelian $\U(1)$ to non-abelian $\SU(2)$ via the connection one-form valued in a Lie algebra: the construction underpinning every modern gauge theory. In A3-BI6 we cite Yang--Mills in the classical-foundations enumeration of \S\ref{sec:conclusions}; the $\SU(5)$ factor of the four-component \BIsix{} bulk gauge group is the direct non-abelian descendant, and the rank-one $\Tr F_5^2$ contribution to the four-form $X_4$ derived in \S\ref{sec:structural-proof} is its anomaly-polynomial signature. First cited in the Executive Summary.

\bibitem[Almheiri \emph{et al.}(2020)]{Almheiri2020}
\textbf{Almheiri, A., Hartman, T., Maldacena, J., Shaghoulian, E., \& Tajdini, A. (2020).} \emph{Replica wormholes and the entropy of Hawking radiation.} Journal of High Energy Physics, \textbf{2020}, 13.

\par\noindent\textbf{Relevance.} The 2020 islands formula and replica-wormhole construction compute the Page curve of Hawking radiation as a saddle-point bulk gravitational integral, with the boundary entanglement entropy descending from a finite-area ``island'' region of the bulk. We cite it in Section~\ref{ssec:scope-and-limits} as the canonical example of a quantum-gravitational result in string theory that is not in the GGEP-closed sector and that the cohort framework does not subsume. First cited in \S\ref{sec:ads-rs}.

\bibitem[Penington \emph{et al.}(2020)]{Penington2020}
\textbf{Penington, G., Shenker, S.~H., Stanford, D., \& Yang, Z. (2020).} \emph{Replica wormholes and the black hole interior.} \emph{JHEP} \textbf{03} (2022) 205, arXiv:1911.11977.

\par\noindent\textbf{Relevance.} Penington \emph{et al.}~2020 (independently of and contemporaneously with \citet{Almheiri2020}) derived the replica-wormhole contribution to the gravitational path integral that produces the Page-curve transition. We cite it in Section~\ref{ssec:scope-and-limits} as part of the same body of quantum-gravitational string-theoretic results that the cohort framework explicitly does not subsume. First cited in \S\ref{sec:ads-rs}.

\end{thebibliography}


\end{document}
