%% v20.2 (August 22, 2026): THE TWENTY-TWO-PAPER DISCLOSURE.
%%   Every reference to a companion paper now carries that companion's own
%%   permanent page -- e.g. https://preprints.arisaka-gg.org/a3-bi6/ -- in place
%%   of the single program address every reference used to share.  Papers that
%%   are not cohort members keep their own locators: the 1985 Kamiokande paper,
%%   the three PsyArXiv works, and the CKM paper still in preparation are
%%   untouched.
%%   Sixty-four companion references carried no locator at all and now carry
%%   one, in this paper's own format.
%%   A11-Origin is posted with the rest, so the twenty-one papers that called it
%%   "in preparation" no longer do, and fifteen printed statements about its
%%   status -- footnotes, table rows, captions and the staged-submission
%%   paragraph -- were corrected to match.  The author's own reservation about
%%   its upper layers is kept verbatim; only its posting status changed.
%%   Two malformed sentences were repaired: A3's "UCLA preprint,." and P2's
%%   title running into the identifier.
%%   No claim, value, figure, table or grade differs from the previous version.
%% ===================================================================
%% ===================================================================
%% v15.11 (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.
%% ====================================================================
%% ===================================================================
%% v15.10 (August 18, 2026) -- FP-5: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% (1) THE A11 BIBLIOGRAPHY ENTRY PROMISED A PAPER THAT WILL NOT BE
%%     THERE.  It read "Preprint, available at https://arisaka-gg.org/",
%%     which becomes false the day the homepage goes up carrying
%%     twenty-one papers.  It now reads "In preparation.  To be posted
%%     ... on completion; no result of the present paper depends on it."
%% (2) P2-N3 HAS EIGHT PARTS AND SAID SEVEN, AT NINE PRINTED SITES --
%%     and its own roadmap FIGURE was missing a Part.  The figure drew
%%     seven boxes and labelled the seventh "Part VII References"; the
%%     paper's actual Part VII is Common Objections and Responses and
%%     References is Part VIII.  THE BODY TEXT TWO PARAGRAPHS ABOVE THE
%%     FIGURE ALREADY SAYS SO: "Part~VII answers twenty-one objections
%%     and Part~VIII carries the references".  So the paper contradicted
%%     itself on the same opening, and the figure was the side that was
%%     wrong.  A Common Objections box is added at -110mm, References
%%     moves to Part VIII at -126mm, one arrow is added, and the nine
%%     "seven" sites become "eight".  The new box's text is taken from
%%     the paper's own Part title and its own objection count.
%%     AND THE FIGURE HAD A COLLISION BEFORE I TOUCHED IT.  Reading the
%%     rendered page showed the "Back matter" phase label printing against
%%     the right edge of the Part VI box, which is the widest in the
%%     picture.  CONFIRMED AGAINST THE v15.9 BASELINE, WHICH SHOWS THE
%%     IDENTICAL COLLISION -- it is not something this rider caused, and
%%     no tool in the gate reported it.  Back matter is now three Parts,
%%     so the label moves to the group's middle row (-110mm), where the
%%     boxes are narrower and nothing touches it.
%% (3) catalogue -> catalog in printed text (Report 443's pair-list
%%     hole).  5 printed sites; \label / \ref identifiers and changelog
%%     comments protected.
%%     THE FIRST RUN OF THIS SWEEP WAS WRONG AND THE GATE PASSED IT.  The
%%     suffix alternation read (e|ed|es|ing), and Python's | is
%%     first-match-wins, so "catalogued" matched the bare "e" and left the
%%     "d" stranded: the sweep wrote "catalogd".  No spelling audit has
%%     that string in its pair list, so nothing reported it; it was found
%%     by reading a rendered page.  Alternation reordered longest-first,
%%     the suffix map asserted exhaustive, and the loosened check re-run
%%     against text carrying the known defect to confirm it still fires.
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
% v15.9 (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 v15.8 printed, except for the lines named above.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.7 -> v15.8, August 16, 2026.  THE CLOSURE ARGUMENT AS A
%% PICTURE, AT THE AUTHOR'S REQUEST.
%%   v15.7 showcased Table 3, the closure verdict, boxed.  Right
%%   content, thin picture: four rows of digits.  P3-Electron's
%%   Figure 9 draws the same accounting as a diagram, and the author
%%   asked for that figure, generalized, here.  It belongs here more
%%   than there: P3 imports this budget, and this is the paper that
%%   proves it.
%%   WHAT GENERALIZED MEANS.  P3's figure is specialized to N = 3 --
%%   two toll boxes reading 2pi and pi, and a total reading 3pi <=
%%   3pi.  This one states the theorem: toll pi*k at each gate, cost
%%   pi N(N-1)/2, capacity pi N for N >= 3 and ZERO below, closure
%%   iff N in {1,3}.  It then evaluates that at every candidate, so
%%   the bottom band IS Table 3 rather than a second copy of it.
%%   Every line comes from Theorem T-N3-6 and its proof, including
%%   both fences: N = 2 fails because H_4(CP^1) = 0 leaves the single
%%   decay step with no payer, and N = 1 passes here and is excluded
%%   by leg L1.  NOTHING NEW IS CLAIMED.
%%   THE PLATE IS ALSO A NUMBERED BODY FIGURE (Figure 4), placed
%%   immediately after the proof of the upper bound, so the showcase
%%   page points INTO the paper as every other showcase page does.
%%   P2 had three figures and all three were cohort furniture -- the
%%   shared funnel, the cohort map, its own roadmap.  It had no
%%   picture of its own argument.
%%   Table 3 is back to being an ordinary body table, unchanged.
%% NEW PROSE IN THIS SHIP: one headline, one showcase caption, and
%% one numbered figure caption.  Nothing in the body is rewritten;
%% no existing float, table row or number is altered.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.6 -> v15.7, August 16, 2026.  THE SHOWCASE PAGE.
%% Sixteenth of the twenty-four, and the first in this pass with
%% NO showcase page at all: P2's front matter ran cover, Executive
%% Summary, Keywords, then straight into the four lists.  Report
%% 365 brought P2 to v15 in fact; it did not give it a front page.
%%   THE PAGE.  The paper is called 'Why Three Generations', and
%%   the one plate that answers the title is Table 3, the closure
%%   verdict: what the family stratum can supply against what the
%%   descent costs, at every candidate count.  Below three the
%%   capacity is zero, so two fails; at three the two are equal and
%%   non-zero; from four the cost has outrun the capacity.  Why not
%%   two and why not four, in six columns.  Unnumbered, so Tables
%%   1-15 keep their numbers.
%%   THE HEADLINE IS THE AUTHOR'S OWN SENTENCE, from the Executive
%%   Summary: 'Three families spend the budget exactly; a fourth
%%   could not pay for its own road to the ground state.'  Every
%%   clause of the caption is likewise from the paper -- the
%%   closure verdict's own caption, the capacity theorem, and the
%%   Executive Summary's Born-grade honesty note.
%%   THE SPACING, PROBED INSIDE THE ENVIRONMENT.  Report 392's
%%   near-miss: the style file sets \setstretch{1.15} document-wide
%%   and a table environment resets it, so a probe above
%%   \begin{table} reads 1.15 and would send the fill to the wrong
%%   stretch.  Probed at \begin{tabular}: BSTR 1, AS 1, tabcolsep
%%   6 pt, at this table and at two others checked for company.
%%   STAGE 0, ALL FIVE CHECKS: zero \includegraphics, so there is
%%   no missing-asset exposure at all -- P2 draws everything in
%%   TikZ; the baseline reproduces the deployed PDF at 0 of 126
%%   pages compared PAGE BY PAGE; zero pre-existing saveboxes; one
%%   \begin{minipage}{\textwidth} and it is not bare; no digit in
%%   the savebox name; every macro in the new prose defined here.
%% NEW PROSE IN THIS SHIP: one headline -- which is the author's
%% own sentence -- and one showcase caption, both on the new page.
%% Nothing in the body is rewritten.
%% ------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.5 (August 9, 2026): the two scope refinements closed at
%%%%  class level.  Appendix H's "Open refinements" rewritten around
%%%%  two new theorems with full proofs: T-INF-7 (warp rigidity: the
%%%%  class of kappa(lam_BI) is invariant under every smooth warp;
%%%%  K-INF-1 cannot fire at class level) and T-INF-8 (relative
%%%%  closure: the half-seam class is filling-independent and equals
%%%%  half the double's; K-INF-2 cannot fire at class level).  The
%%%%  survivors are named exactly: one constructive form-level
%%%%  computation (engineering) and the off-benchmark Wu basepoint
%%%%  question (generality).  Reflected in the abstract, executive
%%%%  summary, preface, axB-enforced grade, structural kill criteria,
%%%%  open-problems ledger, conclusions chain, App H opening, model
%%%%  fence (P-i), objections 19/20 responses, Dai-Freed relevance
%%%%  note, and the verification record (open_core_check.py, 15/15,
%%%%  added as the fourth harness).  Channel identity K-INF-3
%%%%  unchanged.
%%%% ------------------------------------------------------------------
%% ===================================================================
%%  P2-N3 v15.4  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 103mm boxes and then
%%  stretched to the full text width, so its \footnotesize landed 61% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.619\textwidth, computed rather than chosen: the picture measures
%%  113.9mm wide compiled standalone, and the house type size is set by
%%  GG-P3-Electron Figure 3 (0.93\textwidth on 171.1mm, \textwidth =
%%  159.2mm, so type at 0.865 of natural size), which for any picture
%%  of width W means f = W / 184.0.
%%  Nothing inside the picture changes: no font command, no box
%%  dimension, no text, no relative position.
%%  Recipe and the measured table for all fourteen papers:
%%  [v13-support]/funnel_sweep.py.
%% ===================================================================
%% ===================================================================
%%  P2-N3 v15.3  (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-P2-N3  v15.2  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v15.1 -> v15.2), author-directed (GG-Report-194):
%%  THE MERGER, SHIP 3 OF 4 -- the seams.  No new claim.
%%    1. P7's seven objections merged as O-N3-15..21, in a new
%%       "inflow bridge" cluster at the end of Part VII, kept last so
%%       a reader who takes AX-B as an axiom may stop at O-N3-14.
%%       Deixis re-pointed as in ship 1, same whitespace-tolerant
%%       test.  Part VII: 14 -> 21 objections.
%%    2. The Part preamble said "twelve objections" while carrying
%%       fourteen -- stale since v13.3.  Now twenty-one, split named.
%%    3. Two subsections of P7's conclusions folded into Part V:
%%       "The four integers, and who chose them" and "Why agreement
%%       is the right prize".  The rest of P7's conclusions dropped
%%       as duplication.
%%    4. Roadmap trued up: seven -> EIGHT technical appendices, with
%%       Appendix H named; "Part VII carries the references"
%%       corrected to Part VIII (a pre-existing error); the L2
%%       sentence's "(residue R-INT)" now says discharged.
%%  NOT DONE, and queued: the roadmap says "seven Parts" throughout
%%  (8 sites) while the paper has had eight since the objections Part
%%  was added at v13.3.  Correcting it means redrawing
%%  fig:seven-parts-flowchart-p2, which is a figure job, not a seam
%%  job.  Pre-existing; not caused by the merger.
%%  Script: p2_v15_2_seams.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-P2-N3  v15.1  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v15.0 -> v15.1), author-directed (GG-Report-194):
%%  THE MERGER, SHIP 2 OF 4 -- the claims.  Four changes, only these.
%%    1. NEW ssec:axB-enforced after the AX-B box: what Appendix H
%%       proves (T-INF-1..6, the four integers, the kappa-bridge),
%%       what it does not (benchmark scope; agreement not derivation;
%%       does NOT select N_g), and the grade.  AX-B is now carried as
%%       DERIVED-CONDITIONAL, not as a standing axiom.  The axiom box
%%       itself gains a pointer.
%%    2. "The single residue" restructured: R-INT moves from
%%       "Residual (one item)" to DISCHARGED (absolute + family parts,
%%       on the benchmark), with OUTSTANDING = the two named
%%       refinements R-WARP (O-1) and R-REL (O-2).  The paper stops
%%       carrying an open topological residue.
%%    3. Abstract and sec:central-claim gain a clause each; NEW
%%       Discussions section sec:order, "The derivation order, and
%%       what the bridge does not buy" -- the chain written once, why
%%       the bridge cannot found the boundary's QM, and what a
%%       two-sided theorem does add.  This is the honest register the
%%       merge would otherwise have buried in an appendix.
%%    4. NEW structural-kill paragraph in ssec:falsifiers naming
%%       K-INF-1..3 and stating why they are NOT added to F1..F6:
%%       they kill the derivation, not the prediction.
%%  DEVIATION FROM THE PLAN, recorded: the plan proposed naming the
%%  structural kills S-N3-1..3.  Appendix H already declares them as
%%  K-INF-1..3 in place; a second name for the same three
%%  computations would be worse than using the first.  Observational
%%  count F1..F6 untouched.
%%  Script: p2_v15_1_claims.py
%% ===================================================================
%% ===================================================================
%%  Arisaka-GG-P2-N3  v15.0  (August 5, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v14.5 -> v15.0), author-directed (GG-Report-194):
%%  THE MERGER, SHIP 1 OF 4 -- additive only.
%%    * P7-Inflow v1.3 retired into this paper as NEW Appendix H,
%%      "The inflow bridge: how AX-B is enforced": a century of
%%      quantisation; the seam, stratum and functor; T-INF-1..6
%%      (channel, structure formula, quantisation, the two
%%      attributions, the kappa-bridge); R-INT discharged on the
%%      benchmark; the obstruction echoes; the bidegree bookkeeping;
%%      the open refinements R-WARP / R-REL / channel identity; and
%%      the verification record.  Written to be read on its own.
%%    * Theorem tokens T-INF-1..6 kept VERBATIM -- A3, W1 and W2 cite
%%      them.
%%    * Structure: P7 \section -> \subsection, \subsection ->
%%      \subsubsection.  tocdepth 2, so twelve new TOC lines.
%%    * Labels de-collided (the only two): app:A -> app:H-bidegree,
%%      app:B -> app:H-verify.  sec:lineage/machinery/rint -> app:H-*.
%%    * Deixis repaired in eleven places: P7 spoke of "the parent
%%      paper" and \PaperPTwo{}; inside its host those sentences are
%%      nonsense.  Its two live citations of this paper's theorems
%%      became internal references (Theorem 6.9 -> thm:AD3;
%%      Appendices E--F -> app:E/app:F).
%%    * DROPPED from P7, because the merge falsifies them: the
%%      "cohort road" subsection of the lineage (it retells A3/A5/A8
%%      /P2 to a reader who has just read P2) and its closing
%%      paragraph "Why a fifth paper was necessary."
%%    * NEW labels ssec:moduli and ssec:falsifiers, so the appendix
%%      can point at the main line.
%%    * TEN external references merged: Bianchi 1902, Dirac 1931,
%%      Cheeger-Singer 1985, Dai-Freed 1994, Witten 1996 (fivebrane,
%%      flux), Freed 2000, Milnor-Stasheff 1974, Monnier 2016,
%%      Sati-Schreiber 2024.  57 -> 67.
%%  NOT IN THIS SHIP (v15.1): AX-B is still stated as an axiom, the
%%  residue section still reads as it did, and the abstract is
%%  untouched.  No main-line claim of this paper changes here.
%%  Script: p2_v15_0_merge.py
%% ===================================================================
%% ===================================================================
%%  P2-N3 v14.3  (July 30, 2026) -- P7-Inflow cross-reference update.
%%  The inflow companion is shipped as Particle-Branch member
%%  Arisaka-GG-P7-Inflow: bibitem updated to the shipped identity
%%  ("(In preparation)" retired; key ArisakaInflow kept so no \cite
%%  changes); the four "declared mandate" sites (S12 O-1, App E x2,
%%  App G) now record that the mandate is met on the benchmark
%%  background (the kappa-bridge theorem; R-INT absolute + family
%%  parts discharged there; warped/relative refinements open).
%%  NO other changes; no physics touched.
%% ===================================================================
%% ===================================================================
%%  P2-N3 v14.2  (July 30, 2026)  -- The integrality theorem.
%%  L-INT: Conjecture 6.13 -> Theorem (thm:LINT), by decomposition
%%  (decision record: internal Report 85; author sign-off July 30).
%%    * (A) PAIRING INTEGRALITY -- rigorous.  The Berry bundle is a
%%      subbundle of the trivial Hilbert bundle, hence an honest line
%%      bundle; c_1 integral BY CONSTRUCTION; all four candidate
%%      pairings integral (Chern-Weil / Dirac quantisation, obtained
%%      not imposed).  New thm:LINTA in App G.
%%    * (B) THE TOLL LATTICE -- rigorous, NEW.  Every compact
%%      holomorphic curve has action pi*deg(C) (Wirtinger + integral
%%      degree): NO off-lattice instanton exists.  Closes an unstated
%%      loophole in the pricing argument.  New thm:LINTB in App G.
%%    * (C) UNIT QUANTISATION lambda in pi*Z -- physics rigour.  The
%%      naive (bosonic) level argument gives 2pi*Z, and 2pi*Z has NO
%%      decisive point (2pi*Z cap [pi,3pi/2) EMPTY; smallest
%%      consistent 2pi admits {3,4,5}).  The Witten/Hopkins-Singer
%%      spin refinement (quadratic form q(x)=x(x-1)/2, which the toll
%%      ledger ALREADY realises: Stot(N) = pi*q(N)) halves the
%%      lattice to pi*Z: LOAD-BEARING, not decorative.  Decisive
%%      point uniquely pi on BOTH pi*Z and the conservative fence
%%      (pi/2)*Z; only the non-viable bosonic lattice dissents.
%%    * RESIDUE NAMED: R-INT (def:RINT) -- torsion / w4-type
%%      obstruction check that the actual 6D seam supports the
%%      refinement globally.  The Inflow-companion mandate shrinks to
%%      exactly R-INT.  Failure cost: lattice presentation only;
%%      lambda = pi survives (Routes 2-3), N unmoved.
%%    * ROKHLIN REMARK (App E, new ssec:E-rokhlin): CP^{N-1} spin
%%      iff N even; if CP^2 were spin, sigma = 0 mod 16 would force
%%      <p1> in 48Z, and 3 not in 48Z -- saturation impossible.  The
%%      parity route detects the stratum's spin obstruction; parity
%%      and capacity conditions COINCIDE.  New bibitem Rokhlin1952.
%%    * One-bundle-one-lattice remark (rem:onebundle): tolls (B) and
%%      unit (C) are one quantisation of one bundle, read twice.
%%    * Register sweep at every site (abstract, ES, preface, plan,
%%      roadmap, 6.6, closure remark, dissolution ledger,
%%      falsifiability, comparison, open problems, conclusion, App E,
%%      App F close, App G, objections, bibitem Relevance notes);
%%      first-citation pointers re-verified programmatically.
%%  Verification: verify_lint.py (29/29) + verify_lint_stage.py
%%  (27/27) in [v11-Support].
%% ===================================================================
%% ===================================================================
%%  P2-N3 v14.1  (July 30, 2026)  -- Phase-2 shape-up.
%%  Phase-2 front-matter completion and two v14.0 corrections.
%%    * PREFACE ADDED.  P2 was the only paper in the cohort without one
%%      (18 use \GGPreface, P3 its own section).  Three conventions stated:
%%      every fence printed, physics beside mathematics, arithmetic
%%      reproducible -- then the result in one sentence.
%%    * The \GGAppendixPart subtitle was STALE: it named five appendices
%%      (the paper has seven) and still advertised "the BI-6 Budget
%%      Conjecture stated and fenced" and "the constructive reduction of
%%      strict GDOS Morse-positivity".  My v14.0 residual scan checked
%%      "Budget Conjecture (Conjecture" and the conj:budget label but never
%%      the BARE phrase, so this and the next site survived it.
%%    * "independently of the Budget Conjecture" -> "independently of the
%%      capacity theorem", with the two exclusion routes named correctly.
%%    Not changed: Objection 2 deliberately QUOTES the ledgers' name for the
%%    cap ("The cohort's cross-reference ledgers call this cap the `Budget
%%    Conjecture'") -- that is the objector's voice and is correct.
%% ===================================================================
%% ===================================================================
%%  P2-N3 v14.0  (July 30, 2026)  -- THE CAPACITY ARCHITECTURE.
%%  Author-approved major revision: leg (L2) rebuilt from
%%  [strict Morse-positivity postulate + Budget Conjecture + hypotheses
%%  H1/H2] to [two obstruction theorems + adiabatic theorems T-AD-1..3
%%  (Born-grade) + reduced coupling axiom AX-B + capacity theorem T-N3-6
%%  + unit classification + form-robustness theorem T-N3-7].
%%   * NEW: Lemma R (rigidity of functorial descent) -- every bundle
%%     connection induced from the bulk descends to a ZERO Chern-Weil
%%     form (Killing-form uniqueness x conformal flatness); H1's would-be
%%     derivation closed in the negative.  A5's open problem O3 answered
%%     in the negative as a corollary.
%%   * NEW: boundary-relativity proposition -- no boundary-only theorem
%%     can fix the inflow coupling (relative invariant of the pair).
%%   * NEW: T-AD-1 (Kato transport, 3-line proof), T-AD-2 (the GDOS
%%     runtime satisfies the adiabatic theorem, premises = (P1) semigroup
%%     + collision-level rate = the Born theorem's own premises), T-AD-3
%%     (transport = Berry; QGT = Fubini-Study; periods pi / -2pi).
%%   * NEW: AX-B reduced to the coupling clause alone; kappa != 0 DERIVED
%%     from consistency with L1; unit lambda = pi pinned by three routes;
%%     form robust (h^2 and p1 the only viable natural classes = the two
%%     historical Budget formulations, agreeing on all physics).
%%   * NEW: T-N3-6 capacity theorem Cap(N) = pi*N (N>=3), 0 (N<=2):
%%     one theorem excludes N=2 (no 4-cycle) AND all N>=4 (budget);
%%     exact saturation Stot(3) = Cap(3) = 3pi now an arithmetic identity.
%%     The chain uses NEITHER K=35 NOR J=7 and inserts N=3 nowhere:
%%     A6's M10 demand met by construction.
%%   * The Budget Conjecture (both forms) RETIRED as conjecture; App C
%%     rewritten as the history + completion of the two viable forms.
%%   * Morse positivity OFF the critical path; kept as consistency check
%%     (strict form now a THEOREM at N=3: sigma(CP^2)=1); parity route
%%     kept in App E as independent check, thm renamed T-N3-8.
%%   * H2 succeeded by the integrality lemma L-INT -- the SINGLE open
%%     statement in the conditioning ledger; its failure costs the
%%     lattice presentation, not the value (routes 2-3).
%%   * T-N3-5 conditioning: five hypotheses -> {BI6, MSMF, Born-grade
%%     premises, coupling clause + L-INT} + the two empirical inputs.
%%   * NEW Appendix F (adiabatic theorems: proofs, conventions incl. the
%%     factor-2 + tautological-sign audit, quantitative adiabaticity
%%     e^-pi ~ 4% with topological-payload immunity, event-level caveat).
%%   * NEW Appendix G (obstruction proofs + unit/form classification +
%%     the naturality fence stated as a definability premise).
%%   * Abstract, Executive Summary, plan, roadmap, closure, ledger-
%%     consumption map, reachability, P6, NEW falsifier F6 (Born-coupling
%%     kill), open problems (one open item + closed-ledger), conclusion,
%%     methodological "prove -> decide" subsection, objections O2/O3/O8
%%     updated + NEW O13 (axiomatised-answer) + O14 (Born-coupling).
%%   * References: + Kato 1950, Berry 1984, Provost-Vallee 1980,
%%     Avron-Fraas-Graf-Grech 2012; ArisakaInflow mandate -> L-INT.
%%  Verified: verify_decision.py (24/24) + verify_adiabatic.py (20/20),
%%  both in [v11-Support].  Empirical inputs, falsifiers F1-F5, codebook
%%  values, and all cohort-canonical numbers UNCHANGED.
%% ===================================================================
%% ===================================================================
%%  P2-N3 v13.6  (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.
%%   * The objections Part subtitle, "the strongest objections a critical
%%     reader should raise against a closed derivation of the family number",
%%     now reads "against a three-leg derivation ...", matching the title.
%% ===================================================================
%% ===================================================================
%%  P2-N3 v13.5  (July 29, 2026)  -- internal-affairs sweep.
%%  P2-N3 was reported CLEAN in Report 71 and was not: the detector's
%%  draft-lineage pattern wanted a singular article, and this paper says
%%  "Earlier versions of this paper" -- plural, no article.
%%  "Earlier versions of this paper presented this route with the factor
%%  mislabelled as the seam factor J/2 = 7/2 ..." now states the distinction
%%  directly: the multiplier is the orbifold half-length L/2 = 5/2, not the
%%  seam factor J/2 = 7/2, which would give 21pi/5 rather than 3pi.  The
%%  arithmetic warning is preserved -- the two factors really are easy to
%%  transpose -- without appealing to earlier versions of this paper.
%%  FLAGGED, NOT FIXED: the title says "A Closed Derivation of N=3" while
%%  T-N3-5 is the "Conditional derivation of N=3".  Author's call.
%% ===================================================================
%% ===================================================================
%% v13.4 (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.
%% ===================================================================
%% ===================================================================
%% v13.3 (July 24, 2026): MENU-A SHAPE-UP + COMMON OBJECTIONS
%%  (1) NEW Part "Common Objections and Responses": twelve objections
%%      stated at full force and answered concede-first, in the house
%%      format, between the Technical Appendices and the References
%%      (this paper was the cohort's only member without an
%%      anticipated-objections surface).
%%  (2) References reorganised in two sections per the cohort default
%%      (A = cohort papers, canonical order, the two declared
%%      in-preparation companions last; B = external literature);
%%      every entry now closes with the location of its first
%%      citation. All entries carried over verbatim.
%%  (3) The predictions/kill-criteria section registers its existing
%%      P1..P6 / F1..F5 stakes under the cohort registry names
%%      P-N3-N / F-N3-N (labels and text unchanged).
%%  (4) The reader-facing narrations of this paper's own revision
%%      history recast as present-tense ownership, per the cohort's
%%      no-internal-history rule; one internal inconsistency fixed on
%%      the way (the roadmap now gives the step count of the Appendix-E
%%      reduction as the appendix itself states it). No physics change
%%      anywhere.
%% ===================================================================
%% ===================================================================
%% v13.2 (July 24, 2026): MICRO-ERRATUM (pins + Jarlskog reroute)
%%  (1) The eight dated version parentheticals in the companion-paper
%%      bibliography entries removed, matching the cohort's pinless
%%      reference style.
%%  (2) The Leg-L1 over-determination paragraph updated: the
%%      stated-layer Jarlskog evaluation now carried by the CPT
%%      companion is cited for the numerical estimate (with its layer
%%      band), while the constructive closure-level derivation remains
%%      the declared mandate of the in-preparation CKM-extension
%%      paper. The honesty fence -- the over-determination stays
%%      prospective and Leg L1 rests on the measured invariant alone --
%%      is retained and sharpened. No other change anywhere.
%% ===================================================================
%% ===== v13.1 (2026-07-20): STAGE 5A HOUSEKEEPING SHIP =====
%% Wave-2 housekeeping batch, ship 2: timeless-register sweep of
%% reader-facing '(vN.N)' version pins (bibitem availability lines
%% exempt per house style).  Three sites:
%% (1) Sec. 9 regime terms: '\citet{ArisakaA6Gi4} (v9.5)' -> pin
%% dropped, citation kept.
%% (2) GFF Morse-cascade refinement: 'the original (v1.2) reading
%% of this section' -> 'the original reading of this section'.
%% (3) Open problems: 'The original (v1.0) version of the present
%% paper' -> 'The original version of the present paper'.
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% v13.0 -> v13.1.  Script: p2_v13.1_stage5a.py.
%% ===== v13.0 (2026-07-19): WAVE 1 -- cohort promotion 17 -> 19 (Foundations
%%   Branch: P4-Real and P5-CPT promoted to full membership, designators and
%%   keys unchanged) + first ship of P2-N3 in the GG-v13 series (v11.12 base).
%%   (1) COHORT-STANDARD PART STRUCTURE (new): the thirteen sections and five
%%       appendices grouped into seven Parts (I The Question, the Claim, and
%%       the Framework / II The Three-Leg Derivation of N=3 / III
%%       Interpretation and Predictions / IV Discussions / V Conclusions /
%%       VI Technical Appendices / VII References). \GGPart renewed for
%%       one-line Part entries in the Contents (P4-Real v3.1 author
%%       instruction, July 19, 2026); running header now tracks the current
%%       Part (sty v10.6 default; the v10.4 section-mark override retired).
%%   (2) STANDARDIZED PART-I CLOSE: new Section 4, 'The cohort inheritance
%%       and the roadmap of the seven Parts', closing Part I. The cohort
%%       lead-in paragraph migrated from the old Section 3 tail (seventeen ->
%%       nineteen); the seventeen-paper cohort map RETIRED and replaced by
%%       the frozen v1.4 nineteen-paper cohort map, copied verbatim from
%%       A2-GG6 v13.3 (MP4/MP5 internal node names; only the \label changes;
%%       label fig:cohort-map-p2 kept for cross-references); Foundations
%%       Branch paragraph in P2's voice (P5-CPT consumes the N=3 structure:
%%       three flavours are the arena of its chiral-pentagon / flavour-
%%       holonomy calculus; P4-Real is the origin-of-i member); NEW roadmap
%%       figure 'Roadmap of the seven Parts of GG-P2-N3' (Part II
%%       highlighted: the three-leg pincer carries the central claim).
%%   (3) Section renumbering: the new Section 4 shifts old Sections 4-13 to
%%       5-14. ALL hardcoded section pointers refreshed to the new numbering
%%       -- note the v1.2-frozen pointers (ES, plan, bibliography relevance
%%       notes) had been one off since the v11.4 central-claim insertion and
%%       are now live again. Labels and \ref-based cross-references
%%       untouched.
%%   (4) \PaperPFour/\PaperPFive macros added; \ASix renewed to GG-A6-GRIP
%%       (cohort canon; sty \providecommand still renders the retired Gi4).
%%       ArisakaP4Real + ArisakaP5CPT bibitems added (house style, timeless
%%       Relevance), first cited from Section 4.1. No ArisakaP4Twistor key
%%       existed. References become Part VII (\GGBibBegin retired in favour
%%       of the References Part page).
%%   (5) Register repairs: two 'replaces an unverifiable reference of v1.2'
%%       clauses dropped from Relevance notes (internal version history is
%%       not reader-facing). CKM/Inflow '(In preparation.)' retained as
%%       truthful status (no deployed preprints exist).
%%   (6) PROTECTED, byte-identical: theorems T-N3-1..6, the budget/pincer
%%       arguments, the falsifier tables, and the semantic 'seventeen'
%%       tokens ('two or to seventeen', 'one or seventeen'); 'three
%%       generations'/'N=3' everywhere untouched.
%%   No physics values changed. Script: p2_wave1.py.
%% =====================================================================
%% v11.12 (2026-07-14, OGN-5 gloss harmonization to A3 canonical) -- presentation only.
%%   Off-canonical per-integer readings (miscited to A3) brought into
%%   agreement with A3-BI6's stated meanings: M=primitive tensor slot,
%%   N=colour-triplicity index (generation downstream), L=defining SU(5)
%%   block size (NOT rank), J=minimal symmetric closure cardinality,
%%   K=total lock index. No numbers changed. Script: gloss_harmonize.py.
%% v11.11 (2026-07-14, Phase 6 OGN-5 + {e,pi} sweep) -- presentation only.
%%   Ledger-intro paragraph: the five OGN-5 integers stated as FORCED (not
%%   fitted), each with a physical meaning (M=1 tensor seam, N=3 colour
%%   triplicity [generation face = this paper], L=5 SU(5) block, J=7 odd
%%   closure index, K=35=LJ throat depth); plus the {e,pi} dictionary --
%%   e = base of scale-exponentials (radial dim), pi = angle of the gauge
%%   circle (compact dim). P2's N=3 two-face (l.445) already present, untouched.
%%   Anti-hostile-reviewer emphasis (author directive 2026-07-14). No values
%%   changed. Script: p2_ogn_epi.py.
% v11.9 (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.
% v11.8 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 3
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% ===== v11.7: COSMETIC table-font normalization -- inserted \small before
%  \begin{tabular} in 4 oversized (unsized) table floats so every table
%  sits ~10%% below the body text (matches A10 v12.43). Already-small tables untouched.
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% ===== P2-N3 v11.5 (2026-06-27): cohort de-contamination (protect-list aware). =====
% Shed GG's own self-description: (warped) 6D/six-dimensional/Lorentzian bulk/manifold -> drop;
% canonical (scale-profile) metric; (scale profile) factor; GG-self projection/descent/inflow ->
% bulk-to-boundary; title-page note and bib titles cleaned. KEPT the holographic PRINCIPLE
% ('t Hooft/Susskind), the AdS/CFT + Maldacena bulk-to-boundary exposition, the cohort
% RG-running disclaimer footnote, and all external citation relevance notes. Comments masked
% (untouched). Body + front-matter title note only. No physics/values changed. =====
% ===== P2 v11.4 (2026-06-25): cohort central-claim funnel (consequence variant). =====
% Added the shared open-system funnel (sec:central-claim/fig:central-claim) with this paper's own
% payload. Same funnel as A9/A10/A11. No new physics.
% =====================================================================
%  P2-N3 v11.3 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%  gg-10->gg-11; centered headers + homogenized Acknowledgments via gg-11.  The \LaTeX shim is KEPT (paper uses \LaTeX in body).
%  No prose/figure/number content changed.
% =====================================================================
% =====================================================================
%  Arisaka-GG-P2-N3-v11.1 (Stage 3A: naming-convention theorem-id rollout) -- 2026-06-20
%  Official cohort theorem ids assigned: T-N3-1..6 (6; Hirzebruch recalled, untagged).
%  Appended (T-N3-N) to own theorem-env titles; recalled classical
%  theorems left untagged.  No physics changed.
% =====================================================================
% ===== v11.1 (2026-06-17): N=3 polish --- 'conditional' retitle + parity-signature / colour-vs-generation framing (author Go). =====
%  (1) Main theorem retitled 'Closed derivation of N=3' -> 'Conditional derivation of N=3 (three-leg
%  pincer)', matching the premise list it already carries. (2) Intro gains two framing sentences:
%  the Hirzebruch-signature identity a+(N)-a-(N)=sigma(CP^{N-1}) excludes even N independently of the
%  Budget Conjecture (the paper's key new rigorous result); and N=3-as-COLOURS is the unconditional
%  A3 trace ratio while N=3-as-GENERATIONS is this conditional pincer. Framing only; NO theorem/value/
%  conditioning changed. v11.0 -> v11.1.
% =====================================================================
% ===== v11.0 (2026-06-16): holographic-RG reframing, Phase R2c. =====
%  Terminology only; NO physics changed. A2 + A3 bibitems: dropped 'AdS' from the
%  cited titles ('Warped 6D AdS Bulk' -> 'Warped 6D Bulk'; 'Six-Dimensional Warped
%  AdS Bulk' -> '... Warped Bulk'). Added holographic-RG disclaimer footnote; v10.4 -> v11.0.
% =====================================================================
%% --- v10.4 (2026-06-16): adopt cohort-default running header (sty v10.6). P2 has no Parts,
%%     so the header tracks the current Section instead. ---
\pdfoutput=1
%% v10.3 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-p2, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% ===================================================================
%% Arisaka-GG-P2-N3-v2.0.tex
%%
%% CHANGELOG
%% v10.2 (June 12, 2026): Preprint-identity convention fix (author,
%%   June 12): running header and title-page corner unified to
%%   UCLA/Arisaka-GG-P2-N3. PaperNumber changed from the legacy
%%   GG-2026-Generations; local fancyhead override removed. Cohort
%%   bibitem stamps refreshed while touching the file (M1 lesson):
%%   P3 v10.4 -> v10.9, A5 v10.4 -> v10.8, A6 v10.4 -> v10.6,
%%   A9 v10.5 -> v10.8.
%%   NO other changes.
%% v10.1 (June 11, 2026): P3-pointer refresh (minimal, disclosed):
%%   ArisakaP3Electron bibitem stamp v10.0 -> v10.4; Relevance text
%%   updated to P3's current framing (geometric pointlikeness theorem
%%   with F == 1 as corollary; two-sector Q5 reading, radiation as
%%   the dissipative channel). No other content changes.
%% v10.0 (June 11, 2026): VERSION PROMOTION (author decision, June 11):
%%   v2.7 promoted to v10.0 for cohort version consistency, alongside
%%   P3-Electron and C2-Hubble, completing the 17-paper v10 cohort in
%%   [GG-v10]. Content identical to v2.7 except this entry, the
%%   \PaperVersion stamp, and the P3-Electron bibitem stamp v2.2 ->
%%   v10.0 (promoted simultaneously).
%% v2.7 (June 11, 2026): P3 title disambiguation sweep (minimal,
%%   disclosed): P3 bibitem title -> "The Five Oldest Questions about
%%   the Electron..."; stamp -> v2.2; Relevance updated to P3 v2.0+
%%   framing (readout-locality lemma from AX-1; force as composite
%%   geodesic motion). No other changes.
%% v2.6 (June 11, 2026): P3-Electron Phase-5 reciprocal sync (minimal,
%%   disclosed; post-arXiv-prep maintenance, no changes to the
%%   derivation): NEW pointer sentence at the close of Sec. 7.3
%%   directing the reader to GG-P3-Electron as the dedicated electron
%%   paper; NEW bibitem ArisakaP3Electron (house pattern, v1.7 stamp).
%% v2.5 (June 11, 2026): Phase 6 arXiv production (submission version):
%%   \pdfoutput=1 at line 1; cohort bibliography stamps refreshed to the
%%   Phase-5 bumps (A3 v10.4, A5 v10.4, A6 v10.4, A9 v10.5, June 11, 2026);
%%   two sweep-artifact macro-ized paper codes tidied (A5-GDOS, A6-GRIP
%%   source lines); no content changes. Tarball, plain-text abstract,
%%   category plan, and manifest in [P2-Support].
%% v2.4 (June 11, 2026): Phase 4 hostile-referee fixes (0 FATAL, 2 MAJOR,
%%   8 MINOR; see P2-N3-HOSTILE-REFEREE-REPORT-2026-06-11.md):
%%   A1 Thm 5.2 "0.7%" -> 0.8% (actual mmu/me deviation 0.757%; theorem was
%%      false by its own arithmetic, inherited from v1.2); proof 0.75->0.76.
%%   B1 "GFF integrated Euler characteristic" -> signature-type alternating
%%      sum at Sec 8.1 P6 and Sec 11 (contradicted own App E.8).
%%   B2 "logically independent legs" -> "independent in their directions of
%%      constraint" (ES + Sec 11), aligning with Sec 6.2.
%%   B3 Sec 12.9 "rigorous bounds" softened to declared-axiom-system form.
%%   A2 Sec 7.4 lifetime claim -> order-of-magnitude; A3 App D roundings
%%      2.381/1.191; A4 m_tau uncertainty 1.77693(9) (PDG 2024) x2 sites.
%%   B4 Sec 7.1 "bound"->"budget (Conjecture 4.6)"; Sec 4.6 saddle-vs-
%%      particle clarifier. B5 ES \S-ref fixed; keywords extended.
%% v2.3 (June 11, 2026): A9-sync content revision (author-approved after
%%   the P2<->A9 consistency analysis of June 11):
%%   - Sec. 7 ontology rewrite: generations = basins (genuine GA's) of
%%     depth-n effective landscapes; index-4/2 critical points = GSSB gates;
%%     electron basin = GGA (ground-state refinement of A6 v9.3+ as deployed
%%     in A9 Sec. 42); spectroscopy argument; "the user's hypothesis"
%%     v1.2 artifact removed; tau->mu->e = one GRIP-In trajectory.
%%   - Sec. 2.4: GRIP-In/GRIP-Out regime vocabulary imported (A6 v9.5 defs).
%%   - NEW Sec. 7.5: reachability reading of the upper bound (GFF-ST7 +
%%     per-family budget => N < 4 because a 4th family's relaxation road
%%     would exceed the stratum's capacity); fenced as interpretation.
%%   - App. C: Budget Conjecture Formulation B (S_max(N) = pi N, J-free;
%%     closure <=> N(N-1)/2 <= N <=> N <= 3); equivalence with A at N=3;
%%     honesty note (N-scaling presupposes the descent, H1-adjacent).
%%   - Abstract/ES: one-sentence physical reading upgrades.
%%   - PRE-STEP: repaired 24 double-conversion corruptions left by the v2.2
%%     acronym sweep (\\MACRO{}{} forms) -- see p2n3_v2.3_edits.py header.
%% v2.2 (June 11, 2026): Phase 3 house-style completion (Decision 4):
%%   plain-English abstract with "After all" close (technical content lives
%%   in the ES); ES After-all close; cover note in cohort names; canonical
%%   cohort bibitem pattern (C1 v10.4 precedent: [Arisaka(2026, Ax-Name)]
%%   labels, final titles, arisaka-gg.org source lines, June 10 2026 version
%%   stamps) with honesty-corrected Relevance lines (A3: budget quoted-not-
%%   derived; A5: O3 open problem); canonical keys (ArisakaA1Constitution,
%%   ArisakaA2GG6, ArisakaA3BI6, ArisakaA5GDOS, ArisakaA6Gi4, ArisakaA9SM,
%%   ArisakaC1Kaxion); GG-2026-6F -> \ANine{} (completed tense, final
%%   titles); FULL-CAPS acronym macro sweep (\GGTheory, \GDOS, \BIsix,
%%   \OGNfive, \MSMF, \GFF, \GRIP, \OGGEP, \GGEP, \GG, \GA, \DGI,
%%   \GSSB); kill-criteria language in Sec. 8.2; ledger-uniqueness
%%   attribution A2 (Diophantine) + A3 (closure). NOTE: cohort title canon
%%   drift (C1 v10.4 bib vs v10.x covers) flagged for Phase 5.
%% v2.1 (June 11, 2026): Phase 2 content corrections per the Phase 1
%%   audit (P2-N3-PHASE1-AUDIT-2026-06-11.md). Headline changes:
%%   (R1) BI-6 budget fenced as named Conjecture (conj:budget) at abstract,
%%        body, and Appendix C level; the two formulations unified; the
%%        spurious "J/2 + pi/2" and J/2-vs-L/2 arithmetic patch removed.
%%   (FATAL fix) Postulate 4.3 no longer points to A5-GDOS for a discussion
%%        that A5 records as open problem O3.
%%   (R3) Appendix E upgraded: no-go subsection ssec:E-naive proves the
%%        naive spacetime pullback of tr R^2 vanishes identically on the
%%        canonical warped bulk (symbolically verified), sharpening H1 to
%%        an induced-connection hypothesis; c_R left undetermined.
%%   (R2) Abstract/ES conditionality; J_CKM corrected to PDG 2024 value;
%%        J_PMNS re-sourced (NuFIT); J_pred demoted to in-preparation.
%%   (Tier-1) MSMF renamed to cohort-canonical "Maximum Symmetry, Minimum
%%        Freedom" (A1 AX-4), with the ledger-minimality corollary stated.
%%   (R4) Sec. 6.2 independence claim restated honestly w/ ledger map.
%%   (R5) Sec. 5.4 codebook leg explicitly a consistency check.
%%   (R6) Sec. 9 + bibliography engage Dobrescu-Poppitz 2001, Wan-Wang-Yau
%%        2026, Connes NCG, division algebras, modular flavor, anthropic;
%%        unverifiable refs (Kanno 2010; Kobayashi-Oba-Shinjo 2020) removed.
%%   Minor: "Four appendices"->"Five"; Sec. 10.1 v1.1 tense relic; lattice
%%        m_d/m_u sourcing; A6-GRIP now cited (GFF/GRIP pillars, M10).
%% v2.0 (June 11, 2026): Faithful LaTeX reconstruction of
%%   Arisaka-GG-P2-N3-v1.2.pdf (47 pp, May 2, 2026), whose source was
%%   lost. Reconstructed from the pre-staged text extraction
%%   ([P2-Support]/p2n3-v1.2-extracted-text.txt) with every equation
%%   re-set by eye against the rendered v1.2 PDF. No content changes.
%%   Compiled with the v10 toolchain (ucla-arisaka-gg-10.sty).
%%   Reconstruction judgment calls are logged in
%%   [P2-Support]/P2-N3-v2.0-RECONSTRUCTION-LOG.md.
%% ===================================================================

%% ---- vv15.12 (August 19, 2026) ----
%% The front-matter showcase pages given breathing room.
%% A showcase page's free height is now distributed by weighted stretch
%% glue -- a sixth at the top, a third above the plate, a third below it,
%% a sixth at the foot -- instead of pooling above the headline and below
%% the caption, which left a broad band of white at each end of the page
%% and crowded the three elements into the middle third.
%% Three source forms were found in the cohort and each is handled.  A
%% minipage frame becomes a minipage of EXPLICIT height with the [s]
%% inner alignment, because a minipage is a vbox of natural height and
%% glue inside one does not stretch; the height is \textheight minus
%% \topskip.  A \begingroup frame keeps its page-level glue and changes
%% only the weights.  An unframed page -- there were eight, five of them
%% showcase tables carrying a longtable, which cannot live inside a
%% minipage -- gains page-level glue for the first time, which is why it
%% had been printing hard against the top of the text block.
%% 1 showcase page in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}

\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
%% v13.0: cohort-standard Part structure adopted (seven Parts); the running
%% header now tracks the current Part (sty v10.6 default; section marks stay
%% no-ops).  \GGPart renewed verbatim from ucla-arisaka-gg-11.sty except that
%% the contents line carries only "Part N. <title>" (one-line Part entries,
%% P4-Real v3.1 author instruction, July 19, 2026); 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
\renewcommand{\LaTeX}{LaTeX} % avoid \@ clash from \LaTeX in \GGAcknowledgments (v8-v10 sty); cohort-standard shim

%% ---- Per-paper identity -------------------------------------------
\PaperSeries{GGP}
\PaperNumber{Arisaka-GG-P2-N3}
%% ===================================================================
%%  P2-N3 v14.4  (July 31, 2026)  -- the family count is N_g, and it
%%  is not a ledger member.
%%  The cohort now writes N_c for the colour index, N_g for the
%%  generation count and N_f for quark flavours. Renaming this paper's
%%  N could not be mechanical, because v14.3 derived the family number
%%  AS the ledger's second component: MSMF 'selects the smallest N in
%%  the ledger', T-N3-1 and T-N3-5 open 'let N be the family number of
%%  the OGN-5 ledger', and the stated deep claim was that colour and
%%  generations are 'one and the same ledger integer'. Renaming those
%%  to N_g would have asserted ledger membership for a quantity the
%%  cohort holds outside the ledger.
%%  So the premise is restated first: N = N_c = 3 is the ledger's
%%  colour index, forced by T-BI6-10.2a; N_g is a quantity in its own
%%  right; L1 and L2 derive N_g = 3 from boundary premises alone; and
%%  the two threes meeting is stated as a result of two independent
%%  derivations, with a new subsection setting out what would have to
%%  be shown to make it a theorem. No theorem of the paper is
%%  withdrawn: the capacity theorem, the toll ledger, the Hirzebruch
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\PaperDate{August 22, 2026}

%% v1.2 running-header text (kept verbatim from the original).

%% ---- Paper-local macros -------------------------------------------
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\hypersetup{
  pdftitle = {Why Three Generations: A Closed Derivation of N = 3 from BI-6, MSMF, and GDOS Morse Stability},
  pdfauthor = {Katsushi Arisaka},
  pdfsubject = {GG-Theory program -- derivation of the fermion family number N = 3},
  pdfkeywords = {GG-Theory, three generations, family number, MSMF, GDOS, Morse theory}
}


%% --- imported with Appendix H (P7's preamble); providecommand only ---
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\providecommand{\Zc}{\ensuremath{\check{Z}}}
\providecommand{\Tors}{\operatorname{Tors}}
\providecommand{\Ttwo}{\ensuremath{T^2}}
\providecommand{\Oorb}{\ensuremath{\Omega_{\mathrm{orb}}}}
\providecommand{\lamBI}{\ensuremath{\lambda_{\mathrm{BI6}}}}

\begin{document}
% ---- v14.5 (August 2, 2026) ----
% Methodological-reflection section gains the subsection 'The
% two fibres behind the method' (between 'The bottom line' and
% the closing remark): the count and its protection as
% circle-side holonomy data (why N_g neither runs nor drifts),
% the capacity upper bound as interval-side (a budget is a
% semigroup concept); closed-system methods as circle-only
% (lower bound available since 1973, upper bound unformulable);
% the mod-4 parity statement as the two fibres cooperating.
% Part of the P-series completion sweep.


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

%%  COVER PAGE

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

\thispagestyle{empty}
\GGTitlePage%
  {Why Three Generations}%
  {A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}%
  {\parbox{0.92\textwidth}{\centering\small
   A short companion preprint to \AOne{} (constitution), \ATwo{} (6D bulk), \AThree{} (\BIsix{} anomaly closure), \AFive{} (\GDOS{}), \AEight{} (boundary quantum mechanics), and \ANine{} (Standard-Model parameters), establishing that the fermion family number $N_g=3$ is forced --- under declared conditioning --- by three legs of the \GGTheory{} framework.}}

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

%%  ABSTRACT (v1.2 text, verbatim)

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

\begin{abstract}
\noindent
Everything ordinary in the universe is built from the first family of matter
particles: the electron, the up quark, the down quark, and their neutrino.
Yet nature manufactured two heavier copies of that family --- and then
stopped. Why three? For half a century the Standard Model has carried the
number 3 as an unexplained input: nothing in its equations prefers three
families to two or to seventeen.

This paper derives it within Grand Geometric Theory (\GGTheory{}). Three
independent directions of constraint close on one value. The
observed violation of matter--antimatter symmetry requires at least three
families. The relaxation capacity of the family stratum --- proved here to be
$\pi$ of instanton budget per family, from the quantum geometry that the
boundary's own adiabatic theorem forces on the family moduli --- admits at most three: relaxation to the ground family runs through the weak
interaction, and a fourth family's road would cost more than the stratum can
pay. And the same integer, fed into the theory's mass formulas, reproduces all
twelve charged-fermion masses to four significant figures.
Every fence is declared: two empirical inputs, premises shared with the Born
rule, and the coupling clause --- which is no longer
assumed but enforced, by a pushforward of the bulk's anomaly data onto the
boundary's own geometry that also discharges the paper's one topological
residue and closes both of its scope refinements at class level. The
integrality statement is derived, its lattice forced by the
bulk's spin structure.

After all, a number that nature has held fixed through every experiment ever
performed deserves better than to be an input; it deserves to be a theorem.
\end{abstract}

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

%%  EXECUTIVE SUMMARY (v1.2 text; section pointers refreshed to the v13.0 numbering)

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

\begin{ExecutiveSummary}
The user-level question behind this note is direct: why are there three
generations of elementary fermions, and not one, two, or four? Within the
Standard Model the question has no answer. $N_g = N_{\mathrm{gen}}$ is an
empirical fact without an internal mechanism. Within \GGTheory{} the question
has a closed answer. This paper assembles that answer, in a single self-contained proof, out of three legs that are independent in their directions of constraint (\S 8.2).

The first leg is the Kobayashi--Maskawa observation, recast under the \MSMF{}
postulate of \GGTheory{}. The number of physical CP-violating phases in an
$N_g$-family unitary mixing matrix is $(N_g-1)(N_g-2)/2$. CP violation is
empirically real, so this number must be at least one, which forces
$N_g \ge 3$. This gives the lower bound (L1).

The second leg is genuinely new and sets the present paper apart from the
historical Kobayashi--Maskawa argument. It uses the open-system framework of
\GDOS{} together with Morse theory on the family-projector moduli space $\CPN$.
The Geometric Free Functional admits a canonical Morse function whose
critical points encode the cascade of geometric attractors, and descending the
full family ladder costs the triangle number of gate tolls,
$\Stot(N_g) = \pi N_g(N_g-1)/2$. What pays those tolls is the central question of
the paper. The boundary runtime's own adiabatic theorem --- proved here at
exactly the Born rule's grade of conditionality --- forces the Berry
connection and the Fubini--Study geometry on the family moduli
(Theorems~T-AD-1--T-AD-3, Appendix~F); two obstruction theorems (Appendix~G)
prove that neither the bulk alone nor the boundary alone can set the
inflow coupling; the integrality theorem L-INT is derived at physics rigor
from Dirac quantization of the Berry line bundle and the bulk's own spin
structure, leaving a single named topological residue (R-INT); and the
capacity theorem T-N3-6 then delivers
$\mathrm{Cap}(N_g) = \pi N_g$ --- exactly $\pi$ of relaxation capacity per family
--- from which $N_g \le 3$ follows by arithmetic, with $N_g=2$ excluded in the
same stroke (its stratum has no four-cycle to pay the single decay step) and
exact saturation at $N_g=3$. This is leg (L2). Its physical reading, adopted from the ground-state refinement of the cohort (\citealp{ArisakaA6Gi4}; \citealp{ArisakaA9SM}), is dynamical: all three generations are genuine geometric attractors --- which is why $m_\tau$ and $m_\mu$ are sharply measurable --- but only the first generation is the ground attractor (\GGA{}); the $\tau\to\mu\to e$ weak-decay chain is a single \GRIPin{} descent whose gate tolls exhaust the stratum's instanton budget exactly at $N_g=3$; and a fourth family's ground state would be dynamically unreachable within that budget --- the deeper reason the bound is $N_g<4$ (\S\ref{ssec:reachability}).

The third leg is the constructive cross-check. \ANine{} gives a minimal
first-order codebook for fermion Yukawas in which every charged-lepton and
quark mass is built out of the \OGNfive{} integer ledger
$(M,N,L;J,K) = (1,3,5;7,35)$. With $J=7$ and $L=5$ fixed by \BIsix{} closure, the
entire codebook is a function of $N_g$, and the empirical mass ratios
$m_\tau/m_\mu = 16.817$ and $m_\mu/m_e = 206.768$ uniquely select $N_g=3$ at
the four-significant-figure level. This is leg (L3).

The combination is significantly stronger than any of the three legs taken
alone. Leg (L1) by itself gives only $N_g \ge 3$; leg (L2) by itself gives only
$N_g \le 3$; leg (L3) by itself does not exclude solutions in which the
codebook would be re-parameterized. Together they force $N_g = 3$ exactly --- conditional on the declared empirical inputs, on premises shared with the Constitutional Born
Theorem, and on the integrality theorem L-INT, derived with one named
topological residue (R-INT, discharged on the benchmark, with the warped
and relative scope refinements closed at class level by
Theorems~T-INF-7 and T-INF-8) --- and they validate that value with twelve quantitative mass predictions agreeing with observation at the four-significant-figure level.

The deeper significance of the result is structural. It shows that what was
historically considered an inscrutable empirical accident -- the existence of
exactly three families -- becomes a derived integer once the underlying
geometry is correctly identified. The number 3 in $N_g=3$ is not selected; it
is forced by \BIsix{} closure, by Morse stability, and by the empirical
non-vanishing of the Jarlskog invariant. This is the \GGTheory{} pattern: an
empirical accident reduces, under sufficient geometric structure, to a
number-theoretic identity.

A further structural ingredient is the complete dissolution of the
assumptions that the natural axiomatization of leg (L2) would require. The
budget cap quoted across the cohort's cross-reference ledgers is delivered
by the capacity theorem; the two would-be technical hypotheses $H_1$ and
$H_2$ are replaced, respectively, by a proved obstruction (the Rigidity
Lemma: every functorial descent of the bulk curvature to the family moduli
vanishes identically, so the coupling was never spacetime's to supply) and by
the integrality theorem L-INT (the honest successor of non-degeneracy, now
derived with the single topological residue R-INT); and
strict Morse-positivity leaves the critical path entirely, surviving as an
independent consistency check that the admissible value passes. The cleanest
way to state the new structural content is this: quantum mechanics itself
equips the family moduli with Berry curvature and Fubini--Study geometry
(the adiabatic theorem, at Born-grade); the only natural capacities built on
that geometry that yield a consistent and decisive theory give
$\mathrm{Cap}(N_g) = \pi N_g$ or the equivalent constant form --- both saturating
at $3\pi$ --- and $\pi N_g$ funds exactly three tolls of the triangle-number
cost $\pi N_g(N_g-1)/2$. Three families spend the budget exactly; a fourth could
not pay for its own road to the ground state.

A closing methodological discussion (Section~14) records the road not
taken. Before
settling on the three-leg proof of the present paper, we conducted an
extensive but unsuccessful trial to derive $N_g=3$ within closed-system
\GG{}/\BIsix{} alone. Four proposals were attempted -- a Chern--Simons
multiplicity axiom, a magnetized-torus index, a $\mathbb{Z}_3$ orbifold of
the compact direction, and partial \GG{} anomaly saturation -- and each
produced only necessary conditions on $N_g$ but no upper bound. The structural
diagnosis is that no closed-system framework, no matter how elaborate, can
produce an upper bound on integer multiplicities like $N_g$: closed-system
methods give necessary admissibility conditions (anomaly cancellation,
integrality, index theorems) but never sufficient conditions, because all
unitary states are equally admissible by symmetry. Upper bounds require
open-system dynamics: irreversible projection, finite instanton-action
budgets, cascade closure on a moduli of attractors. This is exactly what \GDOS{}
supplies, and only \GDOS{} supplies. We argue, in the new Section~14, that this
is the missing link that has held $N_g=3$ open for the century since the
foundation of quantum mechanics in 1925. The individual ingredients of the
open-system / holographic perspective have been understood for decades --
Lindblad master equations (1976), decoherence theory (Zeh 1970, Zurek 1981),
the holographic principle ('t~Hooft 1993, Susskind 1995), AdS/CFT (Maldacena
1997), Hopkins--Singer differential cohomology (2005) -- but they have never
been assembled into a foundational framework. \GDOS{} is this assembly, and the
present derivation of $N_g=3$ is the first concrete demonstration that the
open-system shift has the foundational power to derive structural integers
that closed-system methods cannot reach.

The remainder of the paper is organized as follows. Section~3 recalls the
relevant pieces of the \GGTheory{} framework. Sections~5--7 prove the three
legs. Section~8 combines them into the main theorem. Section~9 interprets the
angular cascade $e^{-n\pi}$ as instanton suppression along the \GFF{} Morse
flow. Sections~10--11 discuss falsifiability and prior approaches.
Sections~12--13 record open problems and conclusions. Seven appendices contain the detailed calculations; Appendix~E carries the
parity consistency route and the record of the withdrawn hypotheses,
Appendix~F proves the adiabatic theorems, and Appendix~G proves the two
obstruction theorems, the capacity classification, and the integrality
theorems.

After all, the difference between two families and three is the difference
between a universe without CP violation and the one we inhabit; a theory that
advertises no dials must get that integer right --- and must say plainly
under which declared assumptions it does so.
\end{ExecutiveSummary}

\GGKeywords{%
three generations; family number; $N_g=3$; \MSMF{}; \GDOS{}; Morse theory; $\CPN$;
family-projector moduli; \BIsix{}; \OGNfive{}; fermion mass codebook;
Kobayashi--Maskawa; Jarlskog invariant; \GFF{} cascade; \GRIP{}; instanton suppression; \GRIPin{}; \GGA{}; reachability; capacity theorem; integrality theorem; adiabatic theorem; Berry connection; quantum geometric tensor; \GGTheory{}}

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\draw[arrow, draw=black!55, dashed] (TOLL.south) -- (COST.north);
\draw[arrow, draw=black!55, dashed] (TOLL.south) -- (CAP.north);

%% --- band 4: the closure condition evaluated.  Column pitch 1.45 cm is
%% set by the widest cell, the word "saturates" at \scriptsize ----------
\node[verdict] (V) at (6.2, -7.85) {};
\node[font=\sffamily\small\bfseries, anchor=north] at (6.2, -6.55)
 {closure $\Stot(N_g) \le \mathrm{Cap}(N_g)$, evaluated at every candidate};
\node[hit, minimum width=1.42cm, minimum height=1.72cm] at (6.35, -8.06) {};
\foreach \x/\n/\c/\s/\v in {%
  3.45/1/0/0/pass, 4.90/2/0/1/fail, 6.35/3/3/3/{}, 7.80/4/4/6/fail,
  9.25/5/5/10/fail, 10.70/6/6/15/fail}{%
  \node[font=\sffamily\small] at (\x, -7.35) {$\n$};
  \node[font=\sffamily\small] at (\x, -7.85) {$\c$};
  \node[font=\sffamily\small] at (\x, -8.35) {$\s$};
  \node[font=\sffamily\scriptsize] at (\x, -8.80) {\v};}
\node[font=\sffamily\scriptsize\bfseries, text=green!40!black] at (6.35, -8.80) {saturates};
\node[font=\sffamily\small, anchor=east] at (2.75, -7.35) {$N_g$};
\node[font=\sffamily\small, anchor=east] at (2.75, -7.85) {$\mathrm{Cap}/\pi$};
\node[font=\sffamily\small, anchor=east] at (2.75, -8.35) {$\Stot/\pi$};
\node[font=\sffamily\small, anchor=east] at (2.75, -8.80) {verdict};

%% --- the two fences the theorem carries, printed rather than implied ---
\node[font=\sffamily\footnotesize, text=black!65, align=center, anchor=north]
 at (6.2, -9.55)
 {$N_g = 2$ fails because $H_4(\CP^1) = 0$ leaves the single decay step with no payer;
  every $N_g \ge 4$ fails the budget.\\
  $N_g = 1$ passes here with nothing on either side and is excluded by leg~(L1).
  Exact saturation: $\Stot(3) = \mathrm{Cap}(3) = 3\pi$.};
\end{tikzpicture}}

%% ---- v15.8 showcase page: THE CLOSURE, DRAWN ----------------
%% v15.7 showcased Table 3 itself.  Right content, thin picture:
%% four rows of digits.  This is the same argument as a diagram --
%% mechanism, toll, budget, verdict -- adapted at the author's
%% request from P3-Electron's Figure 9 and generalized off N = 3.
%% It points INTO the paper: the same plate is Figure 4.
%% Unnumbered (\captionof*), so Figures 1-4 keep their numbers.
%% -----------------------------------------------------------
\clearpage
%% Self-filling showcase frame (Report 462): a minipage of EXPLICIT
%% height with the [s] inner alignment, so the interior glue stretches,
%% and 1 : 2 : 2 : 1 weights -- top, above the plate, below it, bottom.
%% The height is \textheight MINUS \topskip; see the report for why.
\noindent\begin{minipage}[t][\dimexpr\textheight-\topskip\relax][s]{\textwidth}
\vspace*{\stretch{1}}
\centering
\GGShowcaseHead{Three Families Spend the Budget Exactly;\\[3pt]
A Fourth Could Not Pay for Its Own Road}
\par\vspace*{\stretch{2}}
{\setlength{\abovecaptionskip}{5pt}%
\small\captionof*{figure}{\textbf{Why not two, and why not four.} Descending the family ladder costs the triangle number of gate tolls; the family stratum has exactly $\pi$ of relaxation capacity per family to pay them with, and none at all below three families, where the four-dimensional homology generator does not exist. Those two lines are the whole upper bound. Evaluated at every candidate count they cross exactly once: $N_g = 2$ fails for want of a four-cycle to pay its single decay step, $N_g = 3$ pays its way to the penny, and from four the cost has outrun the capacity and the gap widens at every step thereafter. The capacity theorem stands at the Born rule's own grade of conditionality, and the paper declares that before a reader asks. In full: Figure~\ref{fig:closure-picture}, p.~\pageref{fig:closure-picture}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\resizebox{0.86\textwidth}{!}{\usebox{\GGplateClosure}}
\par\vspace*{\stretch{1}}
\end{minipage}
\clearpage
%% ---- end showcase page -------------------------------------

%% Front matter, v15.6: the four standard macros.  P2 carried
%% \GGTableOfContents alone -- no Overview of Contents, no List of
%% Figures, no List of Tables -- while holding fifteen captioned tables
%% and three figures that were invisible from the front of the paper.
%% \GGfrontmatteronepagetrue puts all four on one page, as in C3-BH,
%% P1-KK and P3-Electron.
\clearpage
\GGfrontmatteronepagetrue
\GGOverviewOfContents
\GGListOfFigures
\GGListOfTables
\GGTableOfContents

\GGPreface

\noindent
This paper answers one question --- why three fermion generations --- and it
is written to be audited rather than admired. Three conventions therefore
govern every page.

\emph{First, every fence is printed.} Where a step is a theorem, it says so;
where it rests on a measurement, the measurement is named; where a
residue remains, it is named too; the chain carries exactly one
(R-INT, the topological residue of the integrality theorem L-INT of
\S\ref{ssec:axiom}), and Appendix~\ref{app:H} discharges it on the
benchmark and closes its two scope refinements at class level.
Two obstruction theorems are included precisely because they say what
\emph{cannot} be derived, and from where --- a negative result is the honest
form of progress on a question that has resisted proof for fifty years.

\emph{Second, the physics is stated beside the mathematics.} Each technical
movement closes with a paragraph in plain terms: what the landscape is, what
the tolls are, who pays them, and why the books balance only through three
families. A reader who wants the argument without the machinery can read
those paragraphs alone and lose nothing essential.

\emph{Third, the arithmetic is reproducible.} Every number in the upper bound
is an integer multiple of $\pi$ obtained by pairing a characteristic class
against a four-cycle, and the worked computation is carried out in full at
$N_g = 2$, $3$ and $4$ (\S\ref{ssec:worked}). Nothing is asserted that a reader
cannot recompute in an afternoon.

The result, stated once and plainly: the family stratum earns $\pi$ of
relaxation budget per family and the descent charges the triangle number of
tolls, so the ledger balances only up to three --- and at three it balances
exactly.

%% ===================================================================
%%  PART I
%% ===================================================================

\GGPart[The question and the claim]{The Question, the Claim, and the Framework}{The three-generation puzzle and its half-century of resistance; the central claim as one deductive funnel; the inherited machinery --- \BIsix{}, \OGNfive{}, \GDOS{}, \GRIP{}, and the \GFF{} Morse functional; and the twenty-two-paper cohort the paper stands in, with the roadmap of the eight Parts}

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

%% 1  INTRODUCTION

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

\section{Introduction and statement of the problem}

\subsection{The three-generation puzzle}
The Standard Model of particle physics organizes the known elementary
fermions into three families with identical gauge quantum numbers and
progressively heavier masses \citep{PDG2024}. This three-fold replication is
among the most robust empirical facts in physics: the LEP measurement of the
invisible $Z$ width \citep{ALEPH2006} fixes the number of light, weakly
interacting neutrino species at
\begin{equation}
N_\nu = 2.984 \pm 0.008,
\label{eq:Nnu}
\end{equation}
consistent with three and inconsistent with both two and four at the
multi-sigma level. Cosmological probes from primordial nucleosynthesis and
the Cosmic \citep{ArisakaA10Cosmos} Microwave Background give an independent and equally tight bound
\citep{Planck2020}, $N_{\mathrm{eff}} = 2.99 \pm 0.17$. Yet within the
Standard Model itself the family number is left as a free input: nothing in
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ requires three
replications rather than one or seventeen. Generations of theoretical
attempts have sought to derive $N_g=3$ from deeper principles, and have
produced partial successes but no closed proof.

The leading prior approaches and their respective limitations may be
summarized as follows.
\begin{enumerate}[label=(\alph*),leftmargin=2.2em]
\item \textbf{Heterotic string compactifications}
\citep{Candelas1985,GSW1987}. The Calabi--Yau manifold of compactification
has Hodge numbers $(h^{1,1}, h^{2,1})$, with $|\chi/2| = |h^{1,1}-h^{2,1}|$
counting net chirality. Three generations corresponds to $|\chi/2| = 3$. This
is realized by certain CICY threefolds and quotients but the choice of
manifold is not forced; the family number is selected, not derived.
\item \textbf{Magnetized toroidal compactifications}
\citep{Blumenhagen2007,IbanezUranga2012}. The index theorem gives the chiral
asymmetry as the magnetic flux quantum on each torus factor,
$\sum_a m_a = N_g$. This explains how three could be generated, but again
leaves the magnetic flux quanta as free integers.
\item \textbf{$\mathbb{Z}_3$ orbifolds and discrete symmetries}
\citep{Ibanez1987,KikuchiKobayashiUchida2021}. A $\mathbb{Z}_3$ point group trivially
produces three twisted sectors. The choice of $\mathbb{Z}_3$ rather than
$\mathbb{Z}_2$ or $\mathbb{Z}_4$ is a model-building input.
\item \textbf{Anomaly-cancellation arguments} \citep{DobrescuPoppitz2001,Witten1985}. These constrain combinations such as the mixed gauge-gravitational anomaly but do not by themselves fix the number of replicas; in the special setting of two universal extra dimensions, global anomaly cancellation does force $N_g=3$ outright, at the price of the extra-dimensional assumption \citep{DobrescuPoppitz2001}.
\item \textbf{Neutrino-physics-driven CP arguments} \citep{KM1973}. Kobayashi
and Maskawa famously observed that $N_g=3$ is the smallest number of
generations admitting a physical CP-violating phase. This is widely accepted
as a consistency argument but has the same shortcoming as (a)--(d): it shows
$N_g\ge 3$, not $N_g=3$.
\end{enumerate}
In every case the residual gap is the same: prior approaches close one
direction of the inequality but not the other. Either they explain why $N_g$
cannot be too small, or they explain how a particular geometric model can
produce three, but they do not provide an upper bound that excludes
$N_g \ge 4$.

\subsection{The \GGTheory{} hypothesis}
In \GGTheory{} the family number is not introduced as an input; it is
derived. The derivation runs alongside, and not through, the integer
ledger that fixes the bulk, so it is worth setting the two out together
before either is used. The ledger is
\begin{equation}
(M,N,L;J,K) = (1,3,5;7,35), \qquad K = LJ = M^2+N^2+L^2,
\label{eq:ledger-intro}
\end{equation}
whose uniqueness as the minimal Diophantine solution is proved in \citet{ArisakaA2GG6} and whose \BIsix{} closure and integrality are proved in \citet{ArisakaA3BI6}. The five integers are forced, not fitted, and each carries a definite meaning: $M=1$ the single tensor seam, $N = N_c = 3$ the color index, $L=5$ the $\mathrm{SU}(5)$ block, $J=7$ the odd closure index, and $K=35=LJ$ the throat depth. The color reading of $N$ is settled without reference to any generation count: it follows from the dual lock together with the block embedding $N < L$, Theorem~T-BI6-10.2a of \citet{ArisakaA3BI6}. The bulk's two non-spatial dimensions --- a radial scale and a compact phase --- fix the only two transcendentals the theory admits: $e$, the base of every scale-exponential, and $\pi$, the angle of the gauge circle. The fermion family number is a \emph{different quantity}, written
$N_g$ throughout this paper. It is not a component of
(\ref{eq:ledger-intro}) and cannot be read off it: the ledger fixes the
color index, and no step of the ledger's Diophantine cascade bears on
how many generations the boundary carries. What the present paper proves
is that $N_g = 3$, from premises of its own --- the CP-violation
combinatorics of the boundary mixing matrix and the relaxation capacity
of the family stratum. That $N_c$ and $N_g$ both equal $3$ is then a
result of two independent derivations meeting at one integer, and it is
discussed as such in \S\ref{ssec:two-threes} rather than assumed
anywhere in what follows.

The purpose of the present short paper is to assemble, in a single
self-contained proof, the three independent legs that force $N_g=3$:
\begin{itemize}[leftmargin=3em]
\item[(L1)] the \MSMF{} lower bound from CP-violation combinatorics;
\item[(L2)] the \GDOS{} Morse upper bound from boundary-projector stability on
$\CPN$;
\item[(L3)] the constructive codebook check from \ANine{}.
\end{itemize}
The combination of (L1), (L2), and (L3) closes the derivation: $N_g\ge 3$ and
$N_g\le 3$ and the resulting twelve fermion masses agree with observation to four significant figures. No leg is by itself sufficient; together they are. All conditioning --- the empirical inputs, the Born-grade runtime premises, the coupling clause, and the topological residue R-INT of the
integrality theorem L-INT --- is declared explicitly in
Theorem~\ref{thm:main} and Appendices~C, F and~G. Two framings deserve emphasis at the outset. First, the genuinely new and fully rigorous content of the upper bound is a topological identity: the alternating Morse sum on the family-projector moduli equals the Hirzebruch signature, $a^+(N_g)-a^-(N_g) = \sigma(\CPN)$, which is $1$ for odd $N_g$ and $0$ for even $N_g$, so \emph{even $N_g$ is excluded by the signature alone} (Appendix~\ref{app:E}, Steps~1--2), independently of the capacity theorem --- which excludes $N_g=2$ by a different route (its stratum has no four-cycle) and every $N_g \ge 4$ by the budget. Second, the integer derived here is the \emph{generation} number $N_g$, and it is not the color index. That the \OGNfive{} ledger's second component is also $3$ is a separate and \emph{unconditional} result --- the dual lock with the block embedding, Theorem~T-BI6-10.2a of \citet{ArisakaA3BI6}, checked against the \BIsix{} trace ratio $(L-N)/L = 2/5$ --- reached from premises that mention no generation count at all. Two derivations with nothing in common therefore land on the same integer. \GGTheory{} reads that as a structural fact about the bulk rather than an accident, and \S\ref{ssec:two-threes} sets out what would have to be shown for the reading to become a theorem; nothing in the present derivation rests on it.

\subsection{Plan of the paper}
Section~2 states the central claim as one deductive funnel, shared in
structure with the companion papers of the cohort. Section~3 recalls the
relevant pieces of the \GGTheory{} framework: \BIsix{}, \OGNfive{},
\GDOS{}, \GRIP{}, and the \GFF{} Morse functional, in just enough detail to
make the present paper self-contained. Section~4 places the paper in the
twenty-two-paper cohort and lays out the roadmap of the eight Parts.
Section~5 proves the \MSMF{} lower bound
$N_g\ge 3$. Section~6 proves the \GDOS{} upper bound $N_g\le 3$ through the
capacity architecture: the toll ledger of the cascade, the two obstruction
theorems that locate the axiom frontier, the adiabatic theorems that force
the Berry/Fubini--Study geometry at Born-grade, the reduced coupling clause
with its integrality theorem (derived; one named topological residue), and
the capacity theorem with its unit classification and form-robustness
result. Section~7
reviews the \ANine{} codebook and shows that it is consistent with $N_g=3$
and inconsistent with any other value. Section~8 combines the three legs into
the closure theorem. Section~9 interprets the angular cascade
$1, e^{-\pi}, \ldots, e^{-4\pi}$ as instanton suppression along the \GFF{} Morse
flow. Section~10 discusses falsifiability. Section~11 compares with prior
approaches. Section~12 catalogs open problems. Section~13 concludes.
Section~14 closes with the methodological reflection on why $N_g=3$ required
open-system thinking. Seven appendices contain the detailed calculations.

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

%% 2  FRAMEWORK

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

\section{The central claim: the family number $N_g=3$ as an open-system observable}
\label{sec:central-claim}

It is worth stating, in one place, the recognition this paper shares with the whole \GGTheory{} cohort. No physical system is closed --- so the four-dimensional world is a lawful projection (\OGGEP{}, \AX{3}) of the six-dimensional bulk, and that projection \emph{is} observation itself; its dynamics is the open-system law \GDOS{} (\AFive{}), forced by the geometric \GiFour{} (\ASix{}) and quantum-information \QiFour{} (\ASeven{}) pillars. It follows that the family number $N_g=3$ --- that there are exactly three fermion generations --- is forced as a \emph{registered observable} of one projection, not chosen --- which is why this paper can derive it, and why a single clean measurement can refute it. The one clause that the derivation could not at first do better than assume, the coupling between the bulk's inflow and the stratum's quantum geometry, is enforced in Appendix~\ref{app:H} by a computation that consumes no quantum-mechanical input and arrives at the boundary's own answer. The funnel below is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; only the payload differs.

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

\section{Framework: \BIsix{}, \OGNfive{}, \GDOS{}, and the \GFF{} Morse functional}
We present the minimum machinery needed for the proof, with citations to the
foundation papers.

\subsection{The 6D bulk and Spin(1,5)}
\GGTheory{} takes the fundamental arena to be a six-dimensional Lorentzian manifold $\Mss$ with isometry group containing $\mathrm{Spin}(1,5)$,
the quaternionic Lorentz double cover (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA2GG6}). The
accidental isomorphism
\begin{equation}
\mathrm{Spin}(1,5) \cong \mathrm{SL}(2,\mathbb{H})
\label{eq:spin15}
\end{equation}
explains why $d=6$ is the unique dimension in the magic-dimension ladder
$\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$ (corresponding to
$d = 3, 4, 6, 10$) at which the holonomy algebra splits cleanly into a
quaternionic doublet of complex doublets without either over-constraining
(smaller $d$) or under-constraining (larger $d$) the gauge content. The
boundary $\partial\Mss = \mathcal{M}_4$ is our observed four-dimensional
spacetime.

\subsection{\BIsix{}: Green--Schwarz-modified Bianchi identity}
The fundamental closure condition on $\Mss$ is the Green--Schwarz-modified
Bianchi identity \citep{ArisakaA3BI6,GreenSchwarz1984}
\begin{equation}
dH_3 = X_4 = \tr R\wedge R - \TR F_5\wedge F_5
 - \tfrac{2}{5}\, F_B\wedge F_B - \tfrac{2}{5}\, F_{Qi}\wedge F_{Qi},
\label{eq:BI6}
\end{equation}
where $H_3$ is the Kalb--Ramond three-form, $R$ the curvature two-form of the
$\mathrm{Spin}(1,5)$ frame bundle, $F_5$ the field strength of
$\mathrm{SU}(5)$, $F_B$ that of $\mathrm{U}(1)_B$ (gravity-baryon), and
$F_{Qi}$ that of $\mathrm{U}(1)_{Qi}$ (quantum information). The numerical
coefficients $-2/5$ are not fitted: they are forced by the inner-product
normalizations on $H_2$ and by the requirement that $X_4$ be an integral
cohomology class, i.e.\ the de~Rham representative of an element of
$H^4(\Mss;\mathbb{Z})$.

The integrality conditions on (\ref{eq:BI6}) admit a unique minimal solution (Diophantine uniqueness: \citealp{ArisakaA2GG6}; closure and integrality: \citealp{ArisakaA3BI6}), namely the \OGNfive{} ledger
\begin{equation}
(M,N,L;J,K) = (1,3,5;7,35).
\label{eq:ledger}
\end{equation}
The five integers satisfy two algebraic identities,
\begin{equation}
K = M^2+N^2+L^2 = 1+9+25 = 35, \qquad K = LJ = 5\cdot 7 = 35.
\label{eq:ledger-identities}
\end{equation}
Their physical readings are: $M=1$ the primitive tensor slot, $N_g=3$ the color-triplicity index (its three-generation face a downstream consequence), $L=5$ the defining block size of $\mathrm{SU}(5)$, $J=7$ the minimal symmetric closure cardinality of the charge packet $\{0,\pm1,\pm2,\pm3\}$, and $K=35$ the total lock index. The
identities (\ref{eq:ledger-identities}) are equivalent to \BIsix{} closure on the
index 35 \citep{ArisakaA3BI6}.

\subsection{\GDOS{}: Geometric Dynamics of Open Systems}
Boundary observation is open-system dynamics in the sense of \GDOS{}
\citep{ArisakaA5GDOS}. Given a bulk state $x\in\Mss$ and a boundary projector
$\Pi : \Mss \to \mathcal{M}_4$ acting via bulk-to-boundary restriction, the
boundary observable is
\begin{equation}
O_4(\Pi[x]) = \int_{\Mss} d^6X\, K_{\mathrm{hol}}(\Pi; X, x)\, O_6(x),
\label{eq:O4}
\end{equation}
where $K_{\mathrm{hol}}$ is the bulk-to-boundary transport kernel. The dynamics of
boundary attractors -- Geometric Attractors (\GA{}) in the \GRIP{} language -- is governed by the Geometric Free Functional (\GFF{}; developed as a Gi-4 pillar in \citealp{ArisakaA6Gi4})
\begin{equation}
\FGFF[\Pi] = \FGEP[\Pi] + \FCEP[\Pi],
\label{eq:GFF}
\end{equation}
with $\FGEP$ the geometric energetic part (curvature, gauge action, matter
overlaps) and $\FCEP$ the curvature-entropic part (Riemannian volume,
configurational entropy of the projector class).

\subsection{\GRIP{}: the cascade of unstable critical points}
The \GRIP{} cascade \citep{ArisakaA6Gi4}
\begin{equation}
\DGI \to \GSSB \to \GA{}
\label{eq:GRIP}
\end{equation}
encodes how a deeply geometrically invariant bulk descends, by spontaneous
symmetry breaking on the boundary, into a finite catalog of stable boundary
attractors. Critical points of $\FGFF$ are stationary projector configurations. Stable critical points are \GA{}'s; saddle points correspond to unstable boundary phases that decay into \GA{}'s by instantons.

Two regime terms, defined at runtime level in \citet{ArisakaA6Gi4} and used throughout \citet{ArisakaA9SM}, are needed in Section~9. \GRIPin{} is the descent on a \emph{fixed} landscape: gradient flow within basins, instanton steps across the \GSSB{} gates between them, halting uniquely at the stratum's Ground Geometric Attractor (\GGA{}). \GRIPout{} is the complementary regime: a transient out-of-equilibrium event that reorganizes the landscape itself and supplies the initial datum from which a new \GRIPin{} descent begins. For the family sector the division is sharp: the electroweak (Wilson--Hosotani) branch selection was the boundary's one \GRIPout{} event --- it built the Yukawa landscape --- and every flavor transition since, including the $\tau\to\mu\to e$ chain, is \GRIPin{} on that fixed landscape.

\subsection{The \MSMF{} postulate}
\begin{postulate}[\MSMF{} ledger-minimality corollary; \MSMF{} = Maximum Symmetry, Minimum Freedom, \citealp{ArisakaA1Constitution}]
\label{post:MSMF}
Among all ledger choices that close \BIsix{} (\ref{eq:BI6}) and that admit a
non-empty boundary observable spectrum compatible with empirical CP
violation, \GGTheory{} selects the ledger of minimum total complexity, defined
as the sum of integer absolute values $|M|+|N|+|L|+|J|+|K|$ subject to the
closure constraints (\ref{eq:ledger-identities}).
\end{postulate}

\MSMF{} is the constitutional axiom AX-4 of \citet{ArisakaA1Constitution} (``Maximum Symmetry, Minimum Freedom'': no hidden continuous knobs); Postulate~\ref{post:MSMF} is its ledger-minimality corollary as applied to \BIsix{} closure, the same usage as the \MSMF{} minimality criterion of \citet{ArisakaA6Gi4}. The corollary is not a tautology: there exist solutions to (\ref{eq:ledger-identities}) with larger entries (e.g.\ $K = 35\cdot k^2$ for
any $k\in\mathbb{Z}_{>0}$) and smaller entries that fail integrality. \MSMF{}
picks the unique smallest valid solution. We will use a direct corollary of
\MSMF{} below, applied to the boundary spectrum rather than to the ledger:
the family number $N_g$ realized on the boundary is the smallest integer
compatible with the empirically observed flavor structure there.

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

%% 4  THE COHORT INHERITANCE AND THE ROADMAP OF THE SEVEN PARTS

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

\section{The cohort inheritance and the roadmap of the eight Parts}
\label{sec:inheritance-and-roadmap-p2}

\noindent
Before the three legs of the derivation are walked (Part~II onward), 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-p2}, Figure~\ref{fig:cohort-map-p2}), and a panorama of
the eight Parts of the present paper (\S\ref{sec:roadmap-p2},
Figure~\ref{fig:seven-parts-flowchart-p2}). A reader of any later Part should
be able to flip back to this Section and orient in both directions: which
cohort companion carries which imported result, and where the present argument
stands in the eight-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-p2}

Figure~\ref{fig:cohort-map-p2} places the present paper among the twenty-two
papers of the \GGTheory{} cohort. The reader meeting the program
for the first time can read its architecture off the figure top to
bottom --- the constitutional root, the bulk trio, the open-system
boundary law and its pillars, the three textbook readouts, the
capstone, the two phenomenology branches, and the Foundations Branch on the
spine's left flank --- before continuing
with the present paper's own thread.

Within that frame the present paper is the middle member of the Predictions
Branch. It consumes the \OGNfive{} integer ledger of \ATwo{}, the \BIsix{}
closure of \AThree{}, the open-system boundary law of \AFive{} with the
\GRIP{}/\GFF{} runtime of \ASix{}, and the minimal mass codebook of
\ANine{} --- exactly the imports recalled in Section~3 --- and it supplies
the family number $N_g=3$ that its branch sibling \PaperPThree{}
\citep{ArisakaP3Electron} imports verbatim: the internal arrow drawn inside
the Predictions frame of the map.

%% ===================================================================
%%  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 (via A2-GG6 v13.3): no self-reference
%%  markers anywhere, so the figure is identical in all nineteen
%%  papers; only the \label differs (fig:cohort-map-p2 kept for P2
%%  cross-references).  MP4/MP5 internal node names per the master.
%% ===================================================================
\clearpage
\begin{figure}[p]
\centering
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% v22: the second sister import.  \PaperPSix{} says it of
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% runs."  Three letters, because a white-filled label wide enough to
% say "method" masks the whole arrow it annotates.
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 ([xshift=-6mm]W2.north);
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%% rather than by symmetry".
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 node[edgelabel, pos=0.80, anchor=west, xshift=1mm]
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\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 $N_g{=}3$ 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. The branches this paper neither feeds nor draws on are on the map all the same, and are cited here at roster level, which is all they can honestly claim from it: \citealp{ArisakaA4Euclid}; \citealp{ArisakaA7Qi4}; \citealp{ArisakaA11Origin}; \citealp{ArisakaC2Hubble}; \citealp{ArisakaC3BH}; \citealp{ArisakaP6Proton}; \citealp{ArisakaW1Symmetry}; \citealp{ArisakaW2Holonomy}.}
\label{fig:cohort-map-p2}
\end{figure}
\clearpage

\paragraph{The Foundations Branch.} The map's left flank carries the
cohort's third branch --- two member papers, a single column beside the
spine, whose subject is the boundary's complex structure and its mirrors.
\PaperPFour{} \citep{ArisakaP4Real} is the origin-of-$i$ member: it develops
at full length the real-bulk account of the boundary's imaginary unit,
derived from two real structures of the \ATwo{} arena --- the compact phase
circle and the causal clock --- the same complex structure that the unitary
mixing matrices of leg (L1) presuppose at boundary level. \PaperPFive{}
\citep{ArisakaP5CPT} is the CPT member, and it is the Foundations member
nearest to the present paper: it consumes the three-generation structure
derived here. Three flavors are the arena its chiral-pentagon reading of CP
violation runs on, and its flavor-holonomy calculus ties the CP-odd
quartets of the mixing matrix to the single surviving Jarlskog phase
precisely at $N_g=3$ --- the same empirical non-vanishing that leg (L1) turns
into the lower bound. Both carry P-series designators; their place at the
spine's flank, not the phenomenology rows, marks their foundational role.

\subsection{Roadmap of the eight Parts}
\label{sec:roadmap-p2}

The paper proceeds in eight Parts. The present Part --- the question, the
claim, and the framework --- is nearly done: the three-generation puzzle and
its half-century of resistance, the central claim stated as one deductive
funnel, the inherited machinery (\BIsix{}, \OGNfive{}, \GDOS{}, \GRIP{},
and the \GFF{} Morse functional) recalled in just enough detail to make the
paper self-contained, and --- on this opening --- the cohort the paper stands
in. Everything after this page is the derivation and its assessment, 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. It walks the three legs in their natural order and closes
the pincer. Leg (L1) turns the observed violation of CP symmetry, under the
\MSMF{} postulate, into the lower bound $N_g \ge 3$. Leg (L2) is the genuinely
open-system leg: the \GFF{} Morse cascade on the family-projector moduli
space $\CPN$, priced against the capacity $\mathrm{Cap}(N_g) = \pi N_g$ that the
stratum's own quantum geometry funds --- the Berry/Fubini--Study structure
forced at Born-grade by the runtime's adiabatic theorem --- yields the upper
bound $N_g \le 3$, with two proved obstruction theorems locating the axiom
frontier, on which the integrality theorem L-INT is then derived --- its
residue R-INT discharged on the benchmark background in
Appendix~\ref{app:H}, and the coupling clause with it. Leg (L3) closes the
circle constructively: with the ledger integers fixed, the \ANine{} codebook
reproduces the twelve charged-fermion masses at four significant figures for
$N_g=3$ and for no other value. The closure section combines the three into the
paper's theorem --- $N_g=3$ exactly, under the declared conditioning.

Part~III steps back from the proof to its physical reading and its exposure.
The \GA{}-cascade section reads the angular suppression $1, e^{-\pi},
\ldots$ as instanton descent along the Morse flow --- why the heavy
generations are sharply measurable attractors and why only the first is the
ground state --- and the predictions section converts the derivation into
declared falsifiers: the kill criteria a single clean measurement could trip.

Part~IV is the paper's self-critical stratum: the comparison with the prior
approaches to the family number --- from Calabi--Yau Euler characteristics to
anomaly cancellation to the anthropic route --- and the open problems that
remain, stated as open. Part~V concludes twice: once technically, gathering
what has been established and under which declared fences, and once
methodologically, in the reflection on why $N_g=3$ required open-system
thinking at all --- the lesson the paper regards as its deepest export.

Part~VI collects the eight technical appendices --- the CKM phase count, the
Morse background on $\CPN$, the two capacity formulations and their
completion, the numerical mass comparison, the parity route with the record
of the withdrawn hypotheses, the adiabatic theorems of the boundary runtime,
the obstruction theorems with the capacity classification, and the inflow
bridge that enforces the coupling clause (Appendix~\ref{app:H}) --- while
Part~VII answers twenty-one objections and Part~VIII carries the references,
each with a relevance note. Figure~\ref{fig:seven-parts-flowchart-p2}
renders the whole development as a single flowchart, so the big picture is
visible at a glance.

\clearpage
\vspace*{\fill}
\noindent\begin{minipage}{\textwidth}
\centering
\resizebox{0.95\textwidth}{!}{%
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\node[intro] (I) at (0, 0) {\textbf{Part~I}\quad\textbf{The Question, the Claim, and the Framework} \\[-1pt]
 \footnotesize The puzzle \,+\, the deductive funnel \,+\, \BIsix{}/\OGNfive{}/\GDOS{}/\GFF{} recalled \,+\, the cohort (\S1--\S4)};
\node[innov] (II) at (0, -24mm) {\textbf{Part~II}\quad\textbf{The Three-Leg Derivation of $N_g=3$} \\[1pt]
 \footnotesize L1: \MSMF{} lower bound $N_g \ge 3$ \;\;$\bullet$\;\; L2: \GDOS{} Morse upper bound $N_g \le 3$ \;\;$\bullet$\;\; L3: codebook cross-check \\[-1pt]
 \footnotesize plus the closure theorem \textsf{T-N3-5} \\[-1pt]
 \footnotesize \emph{\textbf{The pincer that forces the family number}} (\S5--\S8)};
\node[results] (III) at (0, -46mm) {\textbf{Part~III}\quad\textbf{Interpretation and Predictions} \\[-1pt]
 \footnotesize The \GA{}-cascade reading of $e^{-n\pi}$ \,+\, the declared falsifiers (\S9--\S10)};
\node[disc] (IV) at (0, -62mm) {\textbf{Part~IV}\quad\textbf{Discussions} \\[-1pt]
 \footnotesize Prior approaches compared \,+\, open problems named (\S11--\S12)};
\node[summit] (V) at (0, -78mm) {\textbf{Part~V}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize What is established, under which fences \,+\, the methodological reflection (\S13--\S14)};
\node[app] (VI) at (0, -94mm) {\textbf{Part~VI}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize CKM count, Morse on $\CPN$, capacity forms, masses, parity route, adiabatic theorems, obstructions (A--G)};
\node[app] (VII) at (0, -110mm) {\textbf{Part~VII}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize Twenty-one objections a skeptical reader will raise, answered in turn};
\node[app] (VIII) at (0, -126mm) {\textbf{Part~VIII}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};
\draw[arrowinnov] (I.south) -- (II.north);
\draw[arrow] (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);
\node[phaselabel, anchor=west] at (74mm, 0) {\emph{Setup}};
\node[phaselabel, anchor=west, text=orange!70!black] at (74mm, -24mm) {\emph{\textbf{Central theorem}}};
\node[phaselabel, anchor=west] at (74mm, -46mm) {\emph{Reading}};
\node[phaselabel, anchor=west] at (74mm, -62mm) {\emph{Critique}};
\node[phaselabel, anchor=west] at (74mm, -78mm) {\emph{Summit}};
\node[phaselabel, anchor=west] at (74mm, -110mm) {\emph{Back matter}};
\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the eight Parts of GG-P2-N3.]{Roadmap of the
eight Parts of \textsf{GG-P2-N3}. \textbf{Part~II} is highlighted because it
carries the paper's central claim --- the three-leg derivation of the
family number: leg (L1) forces $N_g \ge 3$ from observed CP violation under
\MSMF{}; leg (L2) forces $N_g \le 3$ from the \GFF{} Morse cascade on the
family-projector moduli priced against the \BIsix{} instanton budget; leg
(L3) validates the closure constructively through the \ANine{} mass
codebook; and the closure section combines them into \textsf{T-N3-5}: $N_g=3$
exactly, under the declared conditioning. Every later Part interprets,
exposes, or documents that result.}
\label{fig:seven-parts-flowchart-p2}
\end{minipage}
\vspace*{\fill}
\clearpage

Part~I has now done its work: the question is posed, the claim is stated,
the machinery is recalled, and the paper's place in the twenty-two-paper
program is drawn. What remains is the derivation itself. The reader who
wants the theorem should walk straight into Part~II; the reader who wants
the meaning first may skip ahead to Part~III and return. Either way, the
number at the end is the same: three.

%% ===================================================================
%%  PART II
%% ===================================================================


\section[Part~I in plain English]{Part~I in plain English: a number that has resisted for fifty years}
Ordinary matter needs one family of quarks and leptons. Nature built three, identical in every charge and differing only in mass, and then stopped. The count is not in doubt: the invisible width of the $Z$ fixes the number of light neutrino species at $2.984 \pm 0.008$, which excludes two and four by many standard deviations, and primordial nucleosynthesis and the microwave background agree from an entirely different direction.

What is missing is a reason. The Standard Model accommodates any number of families without strain --- write down a fourth and nothing breaks --- so the observed three is an input, and fifty years of attempts to derive it have produced constructions that permit three rather than require it.

This Part sets up the problem and says what this paper will claim: that three is forced, under stated conditions, by three arguments that constrain from different directions. It also fixes the framework the argument runs in and marks what is imported from companion papers rather than proved here. Nothing is derived in this Part. Its job is to make the claim precise enough to be wrong.

\GGPart[The three legs]{The Three-Leg Derivation of $N_g=3$}{The pincer that closes: the \MSMF{} lower bound $N_g\ge 3$ from CP violation, the \GDOS{} Morse upper bound $N_g\le 3$ from the \GFF{} cascade against the \BIsix{} instanton budget, the constructive codebook cross-check from \ANine{}, and the closure theorem $N_g=3$}

\section{Leg 1 (L1): \MSMF{} lower bound $N_g\ge 3$ from CP violation}

\subsection{Statement of the result}
\begin{theorem}[\MSMF{} lower bound (T-N3-1)]
\label{thm:L1}
Let $N_g$ denote the number of fermion families carried by the boundary.
Suppose that the boundary observable spectrum admits a non-zero Jarlskog
CP-violating invariant $\JCKM \ne 0$ in the quark sector. Then under \MSMF{}
\begin{equation}
N_g \ge 3.
\label{eq:L1}
\end{equation}
\end{theorem}

\begin{proof}
The proof has three steps.

\emph{Step 1.} Boundary chiral fermions of \GGTheory{} descend from a 6D
quaternionic Dirac spinor by chiral projection (\citealp{ArisakaA1Constitution}; \citealp{ArisakaA9SM}).
After Higgs descent the up-type and down-type quark mass matrices $M_u$,
$M_d$ are general complex $N_g\times N_g$ matrices. The CKM matrix is
\begin{equation}
\VCKM = U_u^L U_d^{L\,\dagger}, \qquad U_{u,d}^{L} \in U(N_g).
\label{eq:VCKM}
\end{equation}
In the standard parameterization $\VCKM$ has $N_g^2$ phases, $N_g(N_g-1)/2$ mixing
angles, and $N_g(N_g+1)/2$ moduli; subtracting the $2N_g-1$ unphysical rephasings
of $u$- and $d$-flavor eigenstates leaves
\begin{equation}
n_{\mathrm{phase}}(N_g) = \frac{(N_g-1)(N_g-2)}{2}
\label{eq:nphase}
\end{equation}
physical CP-violating phases.

\emph{Step 2.} By definition, a Jarlskog invariant exists if and only if
there exists a basis-independent imaginary part of a quartic product of CKM
matrix elements,
\begin{equation}
\JCKM = \operatorname{Im}\bigl(V_{ij}V_{kl}V_{il}^{*}V_{kj}^{*}\bigr),
\qquad i\ne k,\ j\ne l,
\label{eq:Jarlskog}
\end{equation}
which is non-zero. This requires at least one physical phase, i.e.\
$n_{\mathrm{phase}}(N_g)\ge 1$.

\emph{Step 3.} Solving (\ref{eq:nphase}) for $n_{\mathrm{phase}}\ge 1$ gives
\begin{equation}
\frac{(N_g-1)(N_g-2)}{2} \ge 1 \iff N_g \ge 3.
\label{eq:solve-nphase}
\end{equation}
The boundary case $N_g=2$ has $n_{\mathrm{phase}}(2)=0$: the only complex phase
in $2\times 2$ unitary mixing is removable by rephasing. The case $N_g=1$ has
no mixing matrix at all. By \MSMF{} (Postulate~\ref{post:MSMF}) the boundary carries
the smallest family count compatible with the empirical fact that
$\JCKM = \bigl(3.12^{+0.13}_{-0.12}\bigr)\times 10^{-5} \ne 0$ \citep{PDG2024}, which is $N_g=3$.
Therefore $N_g\ge 3$.
\end{proof}

\subsection{Independence of leptonic sector}
The same argument applies in the leptonic sector, mutatis mutandis: PMNS
mixing has been shown to be non-trivial \citep{NOvA2022,T2K2020}, with global fits favoring a non-zero leptonic Jarlskog invariant (best fit $J_{\mathrm{PMNS}} \approx -0.026$; \citealp{NuFIT2020}). Although the leptonic CP measurement
is less precise than the quark one, it is independently consistent with
$N_g=3$ and would by itself force $N_g\ge 3$.

\subsection{What Step 1 of the proof requires}
The proof of Theorem~\ref{thm:L1} requires only three non-trivial inputs:
\begin{enumerate}[label=(\arabic*),leftmargin=2.2em]
\item The boundary fermion mass matrices are general complex $N_g\times N_g$
matrices. This is automatic from the 6D quaternionic spinor descent: no
special texture is imposed.
\item The CKM matrix is unitary. This is automatic from $\mathrm{Spin}(1,5)$
chirality and from the Standard Model gauge structure.
\item The empirical existence of CP violation. This is the only experimental
input, and it is one of the most precisely measured phenomena in physics
($K_L \to \pi\pi$, $B$-meson oscillations, etc.; see \citealp{PDG2024}).
\end{enumerate}

\subsection{Comment on the empirical conditioning}
Theorem~\ref{thm:L1} is empirically conditioned on $\JCKM \ne 0$. Some
readers might object that this is ``circular'': one is using observation to
bound $N_g$. The answer is that any physical theory must accept some empirical
content, and CP violation is among the most economical possible empirical
inputs (a single sign, plus a single number that need not be assumed
precise). The alternative -- deriving $\JCKM$ entirely from \BIsix{} closure --
now has two homes in the program. The CPT companion \PaperPFive{}
\citep{ArisakaP5CPT} evaluates the Jarlskog invariant at the stated
layer from the geometric layer values, $\JCKM \simeq 3.1\times 10^{-5}$
(layer band $2.7$--$3.2\times 10^{-5}$), against the global-fit
$(3.12^{+0.13}_{-0.12})\times 10^{-5}$; the constructive closure-level
derivation remains the declared mandate of the CKM extension of the
codebook \citep{ArisakaCKM}, in preparation. If that constructive
program completes, Theorem~\ref{thm:L1} becomes empirically
over-determined: even without the external measurement of $\JCKM$, the
codebook would independently predict a non-zero value. Until then the
over-determination is prospective -- a stated-layer evaluation is a
consistency reading, not yet a closed derivation -- and (L1) rests on
the measured $\JCKM$ alone.

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

%% 4  LEG 2

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

\section{Leg 2 (L2): \GDOS{} Morse upper bound $N_g\le 3$}

\subsection*{Reader's guide to the architecture of this section}
Leg (L2) is the longest argument of the paper, and its logic is worth
holding in one hand before walking it. The section proves the upper bound
in six movements:
\begin{enumerate}[label=\arabic*.,leftmargin=2.2em]
\item \emph{The stage} (\S6.1--6.2): the family moduli is $\CPN$, and the
\GFF{} height function makes it a Morse landscape --- $N_g$ basins, one per
generation.
\item \emph{The bill} (\S\ref{ssec:toll-ledger}): relaxing down the full
ladder costs the triangle number of instanton tolls,
$\Stot(N_g) = \pi N_g(N_g-1)/2$.
\item \emph{The two dead ends} (\S\ref{ssec:obstruction}): who pays?
Provably not the bulk alone (rigidity), and provably not fixable by the
boundary alone (relativity). The payer's identity is a bulk--boundary
dictionary clause --- the honest location of the one assumption this leg
retains.
\item \emph{The geometry is free} (\S\ref{ssec:adiabatic}): quantum
mechanics itself --- the adiabatic theorem, at the Born rule's own grade
--- equips the moduli with Berry curvature and Fubini--Study structure.
Nothing about the payer is assumed here; only the geometry it must pay
through.
\item \emph{The clause, minimized} (\S\ref{ssec:axiom}): the coupling
clause, with its non-vanishing derived, its unit pinned three ways, its
form classified, and its integrality derived --- one topological residue
named.
\item \emph{The arithmetic} (\S\ref{ssec:capacity}--\ref{ssec:worked}):
capacity $\pi N_g$ against cost $\pi N_g(N_g-1)/2$; only $N_g \le 3$ survives,
and $N_g = 3$ saturates exactly.
\end{enumerate}
A reader pressed for time may read movements 2, 5 and 6 and take the
theorems of movements 3--4 on faith; a referee should do the opposite.

\subsection{The family-projector moduli space}
\label{ssec:moduli}
The boundary family-projector configuration is a rank-1 projector in the
$N_g$-dimensional family Hilbert space, modulo overall phase. Its moduli space
is the complex projective space
\begin{equation}
\mathcal{P}_N = \CPN, \qquad \dim_{\mathbb{R}}\mathcal{P}_N = 2(N_g-1).
\label{eq:PN}
\end{equation}
This identification follows by writing a normalized $N_g$-component complex
amplitude $|\psi\rangle = (\psi_1,\ldots,\psi_N)$ and quotienting by the
phase $|\psi\rangle \sim e^{i\alpha}|\psi\rangle$. The Fubini--Study metric
on $\mathcal{P}_N$ provides the canonical Riemannian structure used by \GFF{}.

\subsection{The \GFF{} Morse functional on $\CPN$}
\begin{definition}[\GFF{} Morse functional]
\label{def:fGFF}
The Geometric Free Functional restricted to family-projector data is
\begin{equation}
\fGFF : \CPN \to \mathbb{R}, \qquad
\fGFF([\psi]) = \frac{\sum_{a=1}^{N_g}\lambda_a |\psi_a|^2}
                     {\sum_{a}|\psi_a|^2},
\label{eq:fGFF}
\end{equation}
with real coefficients $\lambda_1 < \lambda_2 < \cdots < \lambda_N$ inherited
from the diagonal mass-overlap data of the boundary projector.
\end{definition}

The function $\fGFF$ is the standard ``height function'' on $\CPN$, a
classical example in Morse theory. It is a perfect Morse function with
respect to the natural cell structure of $\CPN$.

\begin{lemma}[Morse polynomial of $\CPN$]
\label{lem:morse-poly}
The Morse polynomial of $\CPN$ relative to (\ref{eq:fGFF}) is
\begin{equation}
P(\CPN)(t) = 1 + t^2 + t^4 + \cdots + t^{2(N_g-1)}
 = \sum_{k=0}^{N_g-1} t^{2k},
\label{eq:morse-poly}
\end{equation}
i.e.\ there is exactly one critical point of Morse index $2k$ for each
$k = 0, 1, \ldots, N_g-1$. All critical points are non-degenerate.
\end{lemma}

\begin{proof}
This is classical; see \citet{Milnor1963}. The critical points of $\fGFF$ are
the homogeneous lines $[0,\ldots,0,1,0,\ldots,0]$ with the 1 in position $a$
(for $a = 1,\ldots,N_g$). In affine coordinates centered on the $a$-th critical
point, $\fGFF$ takes the standard form
\begin{equation}
\fGFF(z) = \lambda_a + \sum_{b\ne a}(\lambda_b - \lambda_a)|z_b|^2
 + O(|z|^4),
\label{eq:fGFF-local}
\end{equation}
which is a non-degenerate quadratic with $2(a-1)$ negative directions and
$2(N_g-a)$ positive directions. Its Morse index is therefore $2(a-1)$. Summing
$t^{2(a-1)}$ over $a = 1,\ldots,N_g$ gives (\ref{eq:morse-poly}).
\end{proof}

\subsection{The cascade and its toll ledger}
\label{ssec:toll-ledger}
\GRIP{} demands that every unstable saddle decay, by an instanton of finite
action, into a stable \GA{}. (Throughout this section, ``saddle decay''
refers to the gradient-flow descent of the projector configuration on the
Morse skeleton; the particle-level identification --- generations as basins,
critical points as gates --- is given in Section~9.)

\begin{lemma}[Saddle decay paths on $\CPN$]
\label{lem:saddle-decay}
On $\CPN$ with the height function (\ref{eq:fGFF}), the gradient flow between
critical points is generically transverse \citep{SchwarzM1993}. The unstable
manifold of the index-$2k$ critical point has real dimension $2k$ and its
closure intersects the stable manifold of the index-$2(k-1)$ critical point
in a one-parameter family. All other intersections are higher-codimension and
contribute null measure.
\end{lemma}

In words: each unstable saddle of index $2k$ decays predominantly into the
single saddle one step below it of index $2(k-1)$. The cascade is a strict
chain of length $N_g$.

\begin{lemma}[The toll ledger of $\CPN$]
\label{lem:cascade-closure}
The instanton action between adjacent critical points of indices $2k$ and
$2(k-1)$ on $\CPN$ with the Fubini--Study metric is, by direct integration of
the Bogomolnyi-bound saturating gradient flow,
\begin{equation}
S_{[2k]\to[2(k-1)]} = \pi\, k,
\label{eq:step-action}
\end{equation}
a standard computation (see \citealp{Witten1982}, and \citealp{Helgason1979}
for the underlying geometry). The total action accumulated along the chain
\begin{equation}
[\,2(N_g-1)\,] \to [\,2(N_g-2)\,] \to \cdots \to [\,2\,] \to [\,0\,]
\label{eq:chain}
\end{equation}
is therefore the triangle number
\begin{equation}
\Stot(N_g) = \pi \sum_{k=1}^{N_g-1} k = \pi\,\frac{N_g(N_g-1)}{2}.
\label{eq:Stot}
\end{equation}
\end{lemma}

\paragraph{Physics intuition.} Each generation is a basin on the family
landscape, and each decay step is a toll gate: the $k$-th gate charges
$\pi k$ of instanton action, so the deeper the ladder, the faster the total
bill grows --- quadratically, as a triangle number. Two families cost $\pi$;
three cost $3\pi$; four would cost $6\pi$. The question that decides the
family number is therefore not ``how many basins can the landscape hold?''
(any number) but ``how many tolls can the stratum \emph{pay}?'' Everything
in leg (L2) reduces to identifying the payer and computing its budget.

\subsection{What must pay: the inflow problem and the two-sided obstruction}
\label{ssec:obstruction}
The natural candidate payer has always been the \BIsix{} anomaly inflow: the
bulk class $X_4$ descending onto the family moduli and funding the descent.
For that story to be a derivation, the descended class must be computable
from the axioms. This subsection records the central negative discovery of the paper, in
two halves --- one closing the bulk-side route, one
closing the boundary-side route. Proofs are in Appendix~\ref{app:G}.

\begin{lemma}[Rigidity of functorial descent (Lemma R)]
\label{lem:rigidity}
Let $E \to \mathcal{P}_N$ be any bundle over the family-projector stratum
whose connection is constructed functorially from the axioms' data: induced
from the bulk spin connection through a finite-dimensional representation
$\rho$ of $\mathrm{Spin}(1,5)$, or pulled back along any axiom-given map.
Then its Chern--Weil 4-form is proportional to the bulk's,
\begin{equation}
\tr_\rho(F\wedge F) = c_\rho\,\tr(R\wedge R), \qquad c_\rho = \text{the
Dynkin index of } \rho,
\label{eq:rigidity}
\end{equation}
because on a simple Lie algebra every invariant symmetric bilinear form is a
multiple of the Killing form. Since $\tr(R\wedge R)\equiv 0$ pointwise on the
canonical bulk (\S\ref{ssec:E-naive}: $\mathrm{AdS}_5\times S^1$ is
conformally flat), \emph{every functorially induced connection descends to
zero}. The gauge contributions vanish separately by transversality. Hence no
strict-positivity statement, and no non-zero capacity, is derivable from the
bulk data alone.
\end{lemma}

\begin{proposition}[Boundary relativity of the coupling]
\label{prop:brel}
Every axiom of the boundary runtime (\citealp{ArisakaA8QM}) is
boundary-side: the projection $\pi$, complete positivity, RG coarse-graining,
the light-cone record structure. The inflow coefficient multiplies a
topological term of the joint bulk--boundary action; rescaling it changes no
local boundary dynamics --- the semigroup, the relaxation gap, and the Berry
geometry of \S\ref{ssec:adiabatic} are identical for every value. Two
extensions of the cohort with different coupling are therefore
boundary-indistinguishable, and no theorem quantified over boundary
structures alone can fix the coupling. It is a relative invariant of the
\emph{pair} (bulk, boundary) --- a dictionary entry, in the cohort's custody
language.
\end{proposition}

Third, the coefficient route is independently closed: the \BIsix{} Abelian
coefficient is a modulus of the admissible class, determined by no argument
from anomaly freedom, rank-one factorization, integrality, or the
gravitational sector --- only the Georgi--Glashow $3+2$ split, that is, $N_g$
itself, collapses it (\citealp{ArisakaA3BI6}, the $\gamma$-family
characterization). There is no second route to $N_g$ through the coefficient.

\paragraph{Physics intuition.} A representation is a test charge, and the
Dynkin index is its magnitude: if the field is zero, every test charge sees
zero --- one cannot generate flux through a field-free region by shopping for
a bigger particle. That is Lemma~R. And a boundary cannot know what the bulk
pays it: two worlds whose joint actions differ only in a topological coupling
run identical boundary dynamics. That is
Proposition~\ref{prop:brel}. Together they explain \emph{why} the two would-be
hypotheses of this leg resist every proof attempt: each is the shadow, cast
on one side, of a fact about the pair. The obstruction pair does for leg (L2) what a
conservation law does for a perpetual-motion proposal --- it does not say the
search was lazy; it says the searched-for object does not live where the
search was looking.

\subsection{The geometry quantum mechanics forces: the adiabatic theorems}
\label{ssec:adiabatic}
The escape from the obstruction pair was sitting in plain sight. The family
moduli $\mathcal{P}_N = \CPN$ is not a blank manifold that borrows geometry
from spacetime --- it is a space of quantum states, and spaces of quantum
states carry canonical geometry of their own. A point of $\CPN$ \emph{is} a
rank-one projector; that tautology supplies the tautological line bundle, the
Berry connection on it \citep{Berry1984}, and the Fubini--Study metric as the
quantum fidelity metric \citep{ProvostVallee1980}. The theorems of this
subsection establish that the boundary runtime cannot fail to activate this
geometry: given only premises the cohort already stakes for the Born rule,
adiabatic transport on the family moduli \emph{is} Berry transport. Proofs
are in Appendix~\ref{app:F}.

\begin{theorem}[T-AD-1: Kato transport]
\label{thm:AD1}
Let $s \mapsto P(s)$ be a $C^1$ family of orthogonal projectors on a Hilbert
space. The solution of
\begin{equation}
W'(s) = [P'(s),\,P(s)]\,W(s), \qquad W(0) = \mathbf{1},
\label{eq:kato}
\end{equation}
is unitary and intertwines the family: $W(s)\,P(0)\,W(s)^\dagger = P(s)$.
\end{theorem}

\begin{proof}
Differentiating $P^2 = P$ gives $P' = P'P + PP'$, hence $PP'P = 0$. The
generator $[P',P]$ is skew-adjoint, so $W$ is unitary; and
$\frac{d}{ds}\bigl(W^\dagger P W\bigr)
 = W^\dagger\bigl(P' - [P',P]P + P[P',P]\bigr)W
 = W^\dagger\bigl(P' - P'P - PP'\bigr)W = 0$,
using $[P',P]P = P'P$ and $P[P',P] = -PP'$ (both from $PP'P=0$). Hence
$W^\dagger P W \equiv P(0)$. This is Kato's construction
\citep{Kato1950}.
\end{proof}

\begin{theorem}[T-AD-2: the boundary runtime satisfies the adiabatic theorem]
\label{thm:AD2}
Let the boundary dynamics be the \GDOS{} semigroup of
\citet{ArisakaA5GDOS} (CPTP by T-GDOS-3, GKLS by T-GDOS-4), with
instantaneous generator $\mathcal{L}(x)$ at family-projector configuration
$x$, basin-attractor projector $P(x)$ spectrally isolated by the
collision-level contraction gap $g \sim -\ln(1-\mu)$ of the Born chain
(\citealp{ArisakaA8QM}), and let $x(s)$ traverse a $C^2$ path in $\CPN$ at
slow rate $\varepsilon$ relative to $g$. Then the physical state follows the
instantaneous attractor sub-bundle up to $O(\varepsilon/g)$, and the
transport within it is the Kato transport of Theorem~\ref{thm:AD1}. The
premises are exactly two, and both are already staked by the Constitutional
Born Theorem: (P1) the \AFive{} forward semigroup, and the collision-level
rate bound $\mu > 0$.
\end{theorem}

\begin{theorem}[T-AD-3: the transport is Berry, and the geometry is
Fubini--Study]
\label{thm:AD3}
For the tautological rank-one family $P(z) = |\psi(z)\rangle\langle\psi(z)|$
over $\CPN$: (a) Kato transport satisfies $\langle\psi|d\psi\rangle = 0$ ---
it is parallel transport for the Berry connection of the tautological
bundle; (b) its holonomy is the Berry phase ($-\tfrac12\times$ solid angle on
any projective line); (c) the quantum geometric tensor
$T_{\mu\nu} = \langle\partial_\mu\psi|(\mathbf{1}-P)|\partial_\nu\psi\rangle$
has $\mathrm{Re}\,T$ equal to the Fubini--Study metric and $\mathrm{Im}\,T$
equal to the Berry curvature two-form \citep{ProvostVallee1980}; (d) the
periods are $\int_{\mathrm{line}}\omega_{FS} = \pi$ in the step-action
normalization of (\ref{eq:step-action}), with the tautological sign
convention fixed in Appendix~\ref{app:F}.
\end{theorem}

\begin{remark}[Born-grade conditionality]
\label{rem:born-grade}
Theorems~\ref{thm:AD2}--\ref{thm:AD3} are conditional on nothing new. Their
premises --- the forward semigroup and the collision-level contraction ---
are the premises (P1) and the rate bound of the Constitutional Born Theorem
(\citealp{ArisakaA8QM}). In this framework the adiabatic structure of the
family moduli is exactly as secure as the Born rule: the two stand or fall
together, which is the correct epistemic coupling --- both are aspects of the
single statement that the boundary runtime is quantum mechanics.
\end{remark}

\paragraph{Physics intuition.} For $N_g=2$ the family moduli is the Bloch
sphere, and Theorem~\ref{thm:AD3} is the most experimentally verified
geometric statement in physics: transport a spin slowly around a circuit and
it returns with half the enclosed solid angle as a phase. The theorems above
say only this: the family moduli of \GGTheory{} is a Bloch sphere's
higher-dimensional sibling, and the boundary runtime --- being quantum
mechanics, with a relaxation gap --- transports family projectors the same
way. Nothing is postulated; the geometry rides in with the adiabatic
theorem, which rides in with the Born premises.

\subsection{The residual axiom and the integrality theorem}
\label{ssec:axiom}
What remains between the geometry and the budget is the coupling: that the
\BIsix{} inflow pays tolls \emph{through} the Berry/Fubini--Study geometry.
By Proposition~\ref{prop:brel} this is irreducibly a dictionary clause; the
paper states it at its true, minimal size.

\begin{axiom}[AX-B, the coupling clause]
\label{ax:coupling}
The \BIsix{} absorptive capacity of the family stratum is the pairing of a
single natural characteristic class of the stratum's canonical geometry
(Theorem~\ref{thm:AD3}) against the generator of
$H_4(\CPN;\mathbb{Z})$, with a conversion unit $\lambda$ carrying action
units:
\begin{equation}
\mathrm{Cap}(N_g) \;=\; \lambda\,\bigl\langle\,\chi,\;[\CP^2]\,\bigr\rangle,
\qquad \chi \in \{h^2,\ c_1^2,\ c_2,\ p_1\}.
\label{eq:axB}
\end{equation}
The clause asserts the \emph{form} of the coupling (a natural class, paired
on the canonical four-cycle) and nothing about its magnitude: the value of
$\lambda$, the choice of $\chi$, and the non-vanishing of the coupling are
all \emph{derived} below. The form clause itself is enforced at pair level
on the benchmark background in Appendix~\ref{app:H}, which is why
\S\ref{ssec:axB-enforced} carries it as derived-conditional rather than as
a standing axiom.
\end{axiom}


\subsection{AX-B, enforced on the benchmark background}
\label{ssec:axB-enforced}
The clause above is stated as an axiom because that is the smallest
honest statement of it at this point in the derivation. It does not
have to stay one. Appendix~\ref{app:H} supplies the pair-level
mathematics that enforces it, and this subsection states what that
buys and what it does not.

\paragraph{What is proved.} Push the \BIsix{} differential data
forward through the Hopkins--Singer quadratic functor --- the
construction that turns six-dimensional Wu-structure data into a line
bundle on a family of moduli --- and four things follow on the
benchmark background of the intended \GG{} class. The family couples
through exactly one topological channel (T-INF-1). The pushforward's
curvature is $(cm/4)$ times the Berry curvature of
Theorem~\ref{thm:AD3}, with $c$ the channel charge and $m$ the
background flux number (T-INF-2). The Wu condition forces both
integers even, so the output is always an integer multiple of the
tautological bundle, with the Berry bundle itself the minimal non-zero
value (T-INF-3). And the two forced $2$'s are integers this ledger
already owns: $c = L - N = 2$, the numerator of the trace-ratio lock,
and $m = 2\,\Oorb = 2$, the orientation double of the unit reservoir
(T-INF-4, T-INF-5). Assembled, that is the $\kappa$-bridge
(T-INF-6):
\begin{quote}
\emph{the bulk, pushed forward, hands the boundary the very line
bundle the boundary's own adiabatic theorem constructs} ---
$c_1(\kappa(\lamBI)) = -h$, sign, factor and normalization included.
\end{quote}

\noindent The pairing class that Axiom~\ref{ax:coupling} asserts is
therefore the curvature datum of a construction, not a stipulation.
The topological residue R-INT (Definition~\ref{def:RINT}) is
discharged there in its absolute and family parts
(\S\ref{app:H-rint}), and the two obstruction results that fence the
clause --- Lemma~\ref{lem:rigidity} and
Proposition~\ref{prop:brel} --- reappear from inside the formalism as
a base-change corollary and a type constraint (\S\ref{app:H-echoes}).

\paragraph{What is not proved, and the grade.} Three limits, none of
them small print. \emph{First}, the numerical scope is the benchmark
background: the warped-metric and relative recomputations, R-WARP and
R-REL, are constrained by the rigidity theorems T-INF-7 and T-INF-8 of
\S\ref{app:H-refine}, which close their kill criteria at class level;
what survives of each is one constructive computation and one basepoint
convention, named there. \emph{Second}, the theorem is an
\emph{agreement} between the two faces of one seam, not a derivation
of either from the other; the bulk face consumes no
quantum-mechanical input, and the boundary's bundle is one of the
bridge's two inputs, so no reading of it can found the boundary's
quantum mechanics (\S\ref{sec:order}). \emph{Third}, and most
important for this paper, T-INF-6 does not select $N_g$: the identity
$|L - N| = 2$ holds at $N = 7$ as well. The bridge is a consistency
theorem for the coupling clause; the family number continues to stand
on the capacity theorem of \S\ref{ssec:capacity} and on nothing in
Appendix~\ref{app:H}.

\paragraph{The grade, stated once.} The coupling clause is carried
below as \textbf{derived-conditional} --- enforced at pair level on
the benchmark background, at physics rigor, with the premises of
Appendix~\ref{app:H} declared and its two refinements closed at class
level (T-INF-7, T-INF-8), their surviving items --- one constructive
computation, one basepoint convention --- not load-bearing --- and no
longer as an unenforced axiom. Every result downstream of
Axiom~\ref{ax:coupling} inherits that grade and not a better one. A
reader who declines the benchmark restriction, or the
\MSMF{}-minimal-quanta premise the attributions use, should read the
clause as it was written above and lose nothing else: the arithmetic
of \S\ref{ssec:capacity} is unchanged either way.

\begin{lemma}[Non-vanishing of the coupling]
\label{lem:kappa}
A vanishing coupling leaves the admissible family-number set equal to
$\{1\}$ (no toll can be paid, so no cascade of length $\ge 2$ closes), which
is empty against the lower bound $N_g \ge 3$ of leg (L1). Consistency of the
theory therefore forces $\mathrm{Cap} \not\equiv 0$.
\end{lemma}

\begin{theorem}[L-INT: integrality of the descended periods ---
derived, its one residue discharged on the benchmark]
\label{thm:LINT}
Under the coupling clause (Axiom~\ref{ax:coupling}) read at the action
level (the capacity is an action in the runtime's phase weight), with the
descended geometry of Theorems~\ref{thm:AD1}--\ref{thm:AD3} and a bulk
spin structure (the \BIsix{} bulk hosts the cohort's fermions;
\citealp{ArisakaA9SM}):
\begin{itemize}[leftmargin=2.6em]
\item[(A)] \emph{Pairing integrality (rigorous).} Every candidate
capacity class of Axiom~\ref{ax:coupling} pairs integrally with the
generator: $\langle \chi, [\CP^2]\rangle \in \mathbb{Z}$.
\item[(B)] \emph{The toll lattice (rigorous).} The action of every
compact holomorphic curve $C \subset \CPN$ is
$\int_C \omega_{FS} = \pi\deg(C) \in \pi\mathbb{Z}_{>0}$: no holomorphic
object with off-lattice action exists.
\item[(C)] \emph{Unit quantization (physics rigor).} The unit lies on
the action lattice $\lambda \in \pi\mathbb{Z}_{>0}$ --- finer than the
half-quantum fence $(\pi/2)\mathbb{Z}_{>0}$ of the toll ledger, which
survives as the conservative statement.
\end{itemize}
Proofs and the lattice classification are in Appendix~\ref{app:G}
(\S\ref{ssec:G-LINT}). Component (C) rests on \emph{R-INT}
(Definition~\ref{def:RINT}) --- the verification that the specific 6D
\BIsix{} seam topology supports the quadratic refinement globally ---
which \S\ref{app:H-rint} discharges on the benchmark background; the
warped and relative refinements are closed at class level by
Theorems~T-INF-7 and T-INF-8 (\S\ref{app:H-refine}).
\end{theorem}

L-INT is the honest successor of the would-be hypothesis $H_2$: not
``non-degeneracy'' (underivable, by Lemma~\ref{lem:rigidity}) but
integrality --- and integrality, it turns out, decomposes. Component (A)
is free once the adiabatic theorems make the descended class the
stratum's own: the Berry bundle is a subbundle of the trivial Hilbert
bundle, so its Chern class is integral by construction, and every
candidate class is a polynomial in integral classes. Component (B) is
new, and closes a loophole the pricing argument had left unstated:
Wirtinger calibration quantizes \emph{holomorphic area} itself in units
of $\pi$, so no instanton with off-lattice action can undercut the
budget. Component (C) carries the Hopkins--Singer weight, and its
derivation contains the one discovery of the analysis: the naive level
argument quantizes $\lambda$ on $2\pi\mathbb{Z}$, and $2\pi\mathbb{Z}$
contains \emph{no decisive point at all} --- the spin refinement that
halves the lattice to $\pi\mathbb{Z}$
\citep{Witten1996,HopkinsSinger2005} is load-bearing, not decorative,
and the toll ledger's own half-quantum structure
$\Stot(N_g) = \pi\,N_g(N_g-1)/2$ shows the runtime already prices with the
refined quadratic form. The role of the integrality statement remains
\emph{redundant at the level of the value}: the unit classification
below fixes $\lambda = \pi$ with the lattice (Route 1) and equally
without it (Route 2, \MSMF{}-minimality over the continuum), so even a
failure of the residue R-INT would cost rigidity of presentation, not
the result.

\subsection{The capacity theorem and the unit classification}
\label{ssec:capacity}

\begin{lemma}[The unit classification]
\label{lem:unit}
Write $\mathrm{Cap}(N_g) = \lambda\,\langle p_1(T\CPN),[\CP^2]\rangle
= \lambda N_g$ for $N_g \ge 3$ (the extensive form; the alternatives are
classified in Theorem~\ref{thm:robust}). Admissibility of the cascade reads
$\Stot(N_g) \le \mathrm{Cap}(N_g)$, i.e.\ $N_g \le 1 + 2\lambda/\pi$. Then on the derived lattice $\pi\mathbb{Z}$ of
Theorem~\ref{thm:LINT} --- tabulated below against the conservative
half-quantum fence $(\pi/2)\mathbb{Z}$, whose extra points change
nothing:
\begin{table}[htbp]
\centering\small
\caption[The lattice, point by point, and where it is decisive.]{Every point of the derived lattice $\pi\mathbb{Z}$, the set of generation counts it admits once leg~L1 has imposed $N_g \ge 3$, and whether that set is a single value. Only $\lambda = \pi$ is decisive. The conservative half-quantum fence $(\pi/2)\mathbb{Z}$ is tabulated alongside to show what its extra points would change, which is nothing: they admit no world at all, so the weaker fence costs the argument nothing and is kept.}\label{tab:lattice-verdict}
\begin{tabular}{lll}
\toprule
$\lambda$ & admissible $N_g$ (with $N_g\ge 3$ from L1) & status \\
\midrule
$\pi/2$ & $\varnothing$ & inconsistent (no world) \\
$\pi$ & $\{3\}$ & \textbf{decisive} \\
$3\pi/2$ & $\{3,4\}$ & indecisive \\
$2\pi$ & $\{3,4,5\}$ & indecisive \\
$\ge 5\pi/2$ & $\{3,4,5,6,\ldots\}$ & indecisive \\
\bottomrule
\end{tabular}
\end{table}

Three routes fix $\lambda = \pi$: \emph{(Route 1)} it is the unique
lattice value --- on the derived lattice $\pi\mathbb{Z}$ and equally on
the fence $(\pi/2)\mathbb{Z}$ --- that is both consistent (non-empty
against L1) and decisive (no
residual discrete choice of $N_g$ --- a hidden knob forbidden by \MSMF{},
constitutional axiom AX-4); \emph{(Route 2)} without any lattice,
consistency requires $\lambda \ge \pi$ and \MSMF{}-minimality takes the
smallest consistent capacity, $\lambda = \pi$, the infimum attained;
\emph{(Route 3)} the capacity quantum equals the action quantum --- the
smallest toll $\pi$ of (\ref{eq:step-action}) --- so no new scale enters.
\end{lemma}

\begin{remark}[One bundle, one lattice]
\label{rem:onebundle}
The tolls and the capacity unit are not two normalizations that happen
to agree; they are one quantization read twice. Component (B) of
Theorem~\ref{thm:LINT} puts every holomorphic action on
$\pi\mathbb{Z}$, and component (C) puts the unit on the same lattice ---
both generated by the same geometric object, the Berry line bundle of
Theorem~\ref{thm:AD3}, whose curvature is $-2\,\omega_{FS}$. Route 3 of
Lemma~\ref{lem:unit} (no new scales) is thereby upgraded from a reading
to a structural fact.
\end{remark}


\begin{theorem}[T-N3-6: capacity of the family stratum]
\label{thm:capacity}
Under the coupling clause (Axiom~\ref{ax:coupling}) with the extensive class
$\chi = p_1$, the unit classification (Lemma~\ref{lem:unit}), and
Lemma~\ref{lem:kappa},
\begin{equation}
\mathrm{Cap}(N_g) \;=\; \pi\,\bigl\langle p_1(T\CPN),\,[\CP^2]\bigr\rangle
\;=\; \pi N_g \quad (N_g \ge 3), \qquad \mathrm{Cap}(N_g) = 0 \quad (N_g \le 2),
\label{eq:capacity}
\end{equation}
where $[\CP^2]$ generates $H_4(\CPN;\mathbb{Z}) \cong \mathbb{Z}$ --- which
exists iff $N_g \ge 3$ and is canonical (no choice) --- and
$p_1(T\CPN) = N_g h^2$ is the Pontryagin scaling (rigorous;
Appendix~E, Step~2). Cascade closure
\begin{equation}
\Stot(N_g) \;\le\; \mathrm{Cap}(N_g)
\label{eq:closure-bound}
\end{equation}
therefore holds iff $N_g \in \{1,3\}$: the case $N_g=2$ fails because
$H_4(\CP^1) = 0$ leaves the single decay step with no payer, and every
$N_g \ge 4$ fails the budget, $\pi N_g(N_g-1)/2 > \pi N_g$. The bound is exactly
saturated at the admissible cascade value:
$\Stot(3) = \mathrm{Cap}(3) = 3\pi$.
\end{theorem}

\begin{theorem}[T-N3-7: form robustness]
\label{thm:robust}
Classify the four natural single-class capacities of
Axiom~\ref{ax:coupling} by their minimal consistent lattice unit and the
resulting admissible set:
\begin{table}[htbp]
\centering\small
\caption[Four candidate capacity forms, and what each admits.]{The four natural single-class capacities of the coupling axiom, each with its scaling in $N_g$, the smallest lattice unit consistent with it, and the set of generation counts it then admits. Three of the four are never decisive at any unit --- they admit every $N_g \ge 3$ --- so the result does not turn on a lucky choice among the four. This is the robustness check the theorem is named for, stated as a table rather than asserted.}\label{tab:form-robustness}
\begin{tabular}{llll}
\toprule
class & scaling & minimal consistent $\lambda$ & admissible set \\
\midrule
$c_2$ & $N_g(N_g-1)/2$ & $\pi$ & all $N_g\ge 3$ --- never decisive \\
$c_1^2$ & $N_g^2$ & $\pi/2$ & all $N_g\ge 3$ --- never decisive \\
$h^2$ & $1$ & $3\pi$ & $\{3\}$ --- viable \\
$p_1$ & $N_g$ & $\pi$ & $\{3\}$ --- viable \\
\bottomrule
\end{tabular}
\end{table}

Exactly two forms survive, and they agree on all physics: admissible set
$\{3\}$, exact saturation $\mathrm{Cap}(3) = 3\pi = \Stot(3)$. They are
precisely the two formulations in which the cohort's cross-reference
ledgers quote the budget cap --- the constant form ($h^2$, at $3\pi$) and
the per-family form ($p_1$, at $\pi N_g$) --- here \MSMF{}-completed
(Appendix~\ref{app:C}). The extensive
form $p_1$ is preferred: its decisive unit is \emph{unique} on the lattice
(zero residual freedom, the strictly more \MSMF{}-minimal option), and it
carries the flux reading of the reachability interpretation
(\S\ref{ssec:reachability}). But nothing about $N_g=3$ rides on the
preference: the derivation is robust across every viable natural form.
\end{theorem}

\paragraph{Physics intuition.} The stratum's capacity is topological
charge converted to action at the smallest coin the landscape mints. The
tangent-bundle charge of $\CPN$ grows linearly --- one unit per family, by
the Pontryagin scaling $p_1 = N_g h^2$ --- so the stratum earns $\pi$ per
family while the ladder's cost grows as the triangle number. Linear income,
quadratic expenses: the books balance only up to $N_g = 3$, and there they
balance \emph{exactly}. The old referee challenge against the budget ---
``why $\lfloor J/2\rfloor$ rather than $J$ or $\lfloor J/3\rfloor$?'' ---
dissolves: the functional form was never about $J$ at all
(Theorem~\ref{thm:robust}), and the unit is pinned three independent ways
(Lemma~\ref{lem:unit}).

\subsection{The capacity computation, worked in full at $N_g = 2, 3, 4$}
\label{ssec:worked}
Because the entire upper bound reduces to one comparison, it is worth
carrying the computation out completely at the three decisive values, with
no step skipped.

\paragraph{$N_g = 2$ (the stratum that cannot pay).}
\emph{Step 1 (moduli).} $\mathcal{P}_2 = \CP^1$, real dimension 2.
\emph{Step 2 (homology).} $H_4(\CP^1;\mathbb{Z}) = 0$: a two-dimensional
manifold has no four-cycles at all. There is nothing for a degree-4 class
to pair against. \emph{Step 3 (capacity).} $\mathrm{Cap}(2) = 0$ ---
identically, regardless of the unit, the class, or the coupling strength.
\emph{Step 4 (toll).} The cascade has one step, $[2]\to[0]$, of action
$S_1 = \pi$ by (\ref{eq:step-action}); $\Stot(2) = \pi$. \emph{Step 5
(verdict).} $\pi > 0$: the single decay step has no payer. The two-family
world is excluded not because its budget is too small but because its
stratum has no four-cycle on which a budget could live. \emph{Physics
intuition:} a Bloch sphere carries Berry flux through its two-cycles, but
the inflow is a four-form charge; on $\CP^1$ there is no four-dimensional
surface to thread, so the second family's road to the ground state is not
expensive --- it is unfunded.

\paragraph{$N_g = 3$ (the stratum that pays exactly).}
\emph{Step 1.} $\mathcal{P}_3 = \CP^2$, real dimension 4. \emph{Step 2.}
$H_4(\CP^2;\mathbb{Z}) = \mathbb{Z}\,[\CP^2]$: the fundamental class itself
is the generator. \emph{Step 3.} $p_1(T\CP^2) = 3h^2$
(Appendix~E, Step 2), and $\langle h^2,[\CP^2]\rangle = 1$, so
$\langle p_1, [\CP^2]\rangle = 3$ and
$\mathrm{Cap}(3) = \pi\cdot 3 = 3\pi$. \emph{Step 4.} The cascade is
$[4]\to[2]\to[0]$ with step actions $S_2 = 2\pi$, $S_1 = \pi$;
$\Stot(3) = 3\pi$. \emph{Step 5.} $3\pi \le 3\pi$, with equality:
admissible, and \emph{exactly saturated}. Both sides are computed with no
shared input beyond the normalization of (\ref{eq:step-action}) --- the
toll from the Bogomolnyi integration of the gradient flow, the capacity
from the Pontryagin number of the stratum --- and they agree to the last
unit. \emph{Physics intuition:} the three-family world spends every unit of
its topological income on relaxation and has nothing left over; it is the
open-system analogue of a critical system, poised exactly at the edge of
its own affordability.

\paragraph{$N_g = 4$ (the stratum that cannot afford itself).}
\emph{Step 1.} $\mathcal{P}_4 = \CP^3$, real dimension 6. \emph{Step 2.}
$H_4(\CP^3;\mathbb{Z}) = \mathbb{Z}\,[\CP^2]$, generated by the standard
$\CP^2 \subset \CP^3$: the four-cycle exists, so --- unlike at $N_g=2$ ---
nothing topological forbids payment. \emph{Step 3.}
$p_1(T\CP^3) = 4h^2$, $\langle p_1,[\CP^2]\rangle = 4$,
$\mathrm{Cap}(4) = 4\pi$. \emph{Step 4.} The cascade is
$[6]\to[4]\to[2]\to[0]$ with tolls $3\pi + 2\pi + \pi = 6\pi$.
\emph{Step 5.} $6\pi > 4\pi$: the deficit is $2\pi$, and it grows
quadratically thereafter ($10\pi$ against $5\pi$ at $N_g=5$; $15\pi$ against
$6\pi$ at $N_g=6$). \emph{Physics intuition:} a fourth family is not
forbidden by symmetry, anomaly, or spectrum --- its exclusion is a
solvency condition. The stratum earns $\pi$ per family and the ladder
charges the triangle number; a four-family world is a household whose debt
service grows faster than its income, and the \GGA{} guarantee of
\citet{ArisakaA6Gi4} is the covenant it would breach.

These three computations, assembled, are the entire content of the upper
bound; the theorem below merely states them once and for all $N_g$.

\subsection{Putting it together: $N_g\le 3$}
\begin{theorem}[\GDOS{} upper bound (T-N3-2)]
\label{thm:L2}
Under the premises of Theorems~\ref{thm:AD2}--\ref{thm:AD3} (Born-grade),
the coupling clause (Axiom~\ref{ax:coupling}) with
Lemmas~\ref{lem:kappa} and \ref{lem:unit}, and the integrality theorem
L-INT (Theorem~\ref{thm:LINT}; residue R-INT) or, equivalently for the
value, \MSMF{}-minimality,
\begin{equation}
N_g \le 3.
\label{eq:L2}
\end{equation}
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:capacity} the closure condition (\ref{eq:closure-bound})
admits only $N_g \in \{1,3\}$: explicitly, with $\Stot(N_g)/\pi$ and
$\mathrm{Cap}(N_g)/\pi$ tabulated,
\begin{table}[htbp]
\centering\small
\caption[Capacity against cost, and the single saturation.]{The closure condition evaluated at each candidate $N_g$: what the four-dimensional homology generator can supply, what the descent costs, and the verdict. Capacity is zero below three because the generator does not exist there, so $N_g = 1$ passes trivially with nothing on either side. At $N_g = 3$ the two are equal and non-zero --- the saturation the argument turns on --- and from $N_g = 4$ the cost has outrun the capacity and the gap widens at every step.}\label{tab:closure-verdict}
\begin{tabular}{lcccccc}
\toprule
$N_g$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\midrule
$\mathrm{Cap}/\pi$ & 0 & 0 & 3 & 4 & 5 & 6 \\
$\Stot/\pi$ & 0 & 1 & 3 & 6 & 10 & 15 \\
verdict & pass & fail & \textbf{saturates} & fail & fail & fail \\
\bottomrule
\end{tabular}
\end{table}

$N_g=1$ passes vacuously (no saddle, no cascade); $N_g=2$ fails with zero
capacity; $N_g=3$ saturates exactly; all $N_g\ge 4$ fail with quadratically
growing deficit. Hence $N_g \le 3$, with $N_g \in \{1,3\}$ the full admissible
set of this leg. ($N_g=1$ is then excluded by Theorem~\ref{thm:L1}.)
\end{proof}

%% ---- v15.8: the closure argument as a picture ----------------
%% Adapted from P3-Electron's Figure 9, which draws this accounting
%% specialized to the charged-lepton stratum at N = 3.  Here it is
%% stated generally -- toll pi*k per gate, cost pi N(N-1)/2,
%% capacity pi N above three and zero below -- and then evaluated,
%% so the bottom band IS Table 3 rather than a second copy of it.
%% Every line is Theorem T-N3-6 and its proof; nothing is new.
\begin{figure}[htbp]
\centering
\resizebox{0.95\textwidth}{!}{\usebox{\GGplateClosure}}
\caption[The closure argument in one picture, for any $N_g$.]{The cascade-with-budget accounting, stated for a general generation count rather than specialized to $N_g = 3$. The top row is the mechanism: dynamical gauge invariance passes to geometric spontaneous breaking by Wilson--Hosotani, and the gradient flow of the \GFF{} carries the result to the geometric attractors. Each \GSSB{} gate on $\CPN$ costs its Bogomolnyi toll $\pi k$, so the descent of the whole ladder costs the triangle number $\Stot(N_g) = \pi N_g(N_g-1)/2$, while Theorem~\ref{thm:capacity} gives the stratum $\mathrm{Cap}(N_g) = \pi N_g$ to spend for $N_g \ge 3$ and nothing at all below, the four-dimensional homology generator not existing there. The bottom band is the closure condition $\Stot \le \mathrm{Cap}$ evaluated at each candidate --- Table~\ref{tab:closure-verdict} drawn rather than tabulated --- and it admits $N_g \in \{1,3\}$ only, with the bound exactly saturated at three. The figure carries the theorem's conditioning with it: the capacity is delivered under the coupling clause and the integrality theorem L-INT with its residue R-INT, at the Born rule's own grade (Axiom~\ref{ax:coupling}, Theorem~\ref{thm:LINT}).}
\label{fig:closure-picture}
\end{figure}

The fact that $N_g=3$ saturates (\ref{eq:closure-bound}) \emph{exactly} --- an
identity between two independently computed quantities, the triangle-number
toll and the Pontryagin-paired capacity --- is the \GGTheory{} analogue of
criticality: the family number sits on the edge of admissibility, neither
too small (no cascade, no flavor structure) nor too large (the ladder
outruns the stratum's income). Saturation here is not an interpretive reading; under Theorem~\ref{thm:capacity} it is an arithmetic identity. We comment further in Section~9.

\subsection{The parity route as an independent consistency check}
\label{ssec:parity-check}
The exclusion work above nowhere used Morse positivity: $N_g=2$ died of
capacity zero and $N_g \ge 4$ died of the budget. The positivity structure of
the cascade survives as an independent consistency check, and we record it
in the form the cohort imports.

\begin{postulate}[\GDOS{} Morse-positivity, \citealp{ArisakaA5GDOS}]
\label{post:morse-pos}
A boundary projector configuration is admissible only if the integrated \GFF{}
Morse index over the cascade closes within a strictly positive ledger,
\begin{equation}
\sum_{c\,\in\,\mathrm{Crit}(\fGFF)} (-1)^{\mathrm{ind}(c)}\,\mu_c \ge 0,
\label{eq:morse-pos}
\end{equation}
where $\mu_c > 0$ is the Riemannian volume of a chart around $c$ and
$\mathrm{ind}(c)$ is the Morse index of $c$. Equivalently, the alternating
sum of geometric multiplicities $a^+(N_g) - a^-(N_g)$ of stable and unstable
critical points must be non-negative,
\begin{equation}
a^+(N_g) - a^-(N_g) \ge 0,
\label{eq:alt-sum}
\end{equation}
where $a^+(N_g) = |\{c : \mathrm{ind}(c) \equiv 0, 2 \pmod 4\}|$ and
$a^-(N_g) = |\{c : \mathrm{ind}(c) \equiv 1, 3 \pmod 4\}|$.
\end{postulate}

Lemma~\ref{lem:morse-poly} gives the index distribution: indices
$0, 2, 4, \ldots, 2(N_g-1)$, all even, so
\begin{equation}
a^+(N_g) = \#\{a : \mathrm{ind} = 0, 4, 8, \ldots\}, \qquad
a^-(N_g) = \#\{a : \mathrm{ind} = 2, 6, 10, \ldots\},
\label{eq:apm-def}
\end{equation}
with the explicit count
\begin{table}[htbp]
\centering\small
\caption[The Morse index parity count, term by term.]{The critical-point indices at each candidate $N_g$, split into the even and odd families whose difference is the signature. The indices run $0, 2, \ldots, 2(N_g-1)$ and are all even, so the split is by index modulo four rather than by parity of the index itself; the table is written out because the counts $a^{+}$ and $a^{-}$ enter the parity check directly and a reader re-deriving it should not have to reconstruct them.}\label{tab:index-parity}
\begin{tabular}{lcccccc}
\toprule
$N_g$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\midrule
indices & $\{0\}$ & $\{0,2\}$ & $\{0,2,4\}$ & $\{0,2,4,6\}$ &
$\{0,2,4,6,8\}$ & $\{0,2,4,6,8,10\}$ \\
$a^+(N_g)$ & 1 & 1 & 2 & 2 & 3 & 3 \\
$a^-(N_g)$ & 0 & 1 & 1 & 2 & 2 & 3 \\
$a^+ - a^-$ & 1 & 0 & 1 & 0 & 1 & 0 \\
\bottomrule
\end{tabular}
\end{table}


\begin{postulate}[Strict \GDOS{} Morse-positivity]
\label{post:strict}
The alternating sum is strictly positive,
\begin{equation}
a^+(N_g) - a^-(N_g) > 0.
\label{eq:strict}
\end{equation}
\end{postulate}

Three statements now fix the status of these imports. \emph{First}, at the
admissible value the strict form \emph{holds as a theorem}:
$a^+(3) - a^-(3) = \sigma(\CP^2) = 1 > 0$ by the signature identity of
Appendix~E (Steps 1--2, fully rigorous). The consistency check passes.
\emph{Second}, the parity pattern of the table --- positivity exactly at odd
$N_g$ --- is genuine mathematics (the signature of $\CPN$), and Appendix~E
retains it as an independent route to even-$N_g$ exclusion for a reader who
grants a non-zero descended class. \emph{Third}, the question recorded in
\citet{ArisakaA5GDOS} as open problem O3 --- whether Morse positivity
follows from \BIsix{} positivity --- is now \emph{answered in the negative}:
by Lemma~\ref{lem:rigidity}, every functorial descent of the bulk data
vanishes, so no derivation of (\ref{eq:morse-pos}) from \BIsix{} alone
exists. Positivity enters, where it enters at all, through the quantum
geometry of Theorem~\ref{thm:AD3} --- and there it is a consequence, not a
postulate.

\section{Leg 3 (L3): The constructive codebook from \ANine{}}
The third leg of the proof is constructive: it shows that the explicit
fermion-mass codebook of \ANine{} \citep{ArisakaA9SM} requires $N_g=3$ and is
incompatible with any other value. We summarize the key formulas; full
derivations are in the cited paper.

\subsection{The charged-lepton codebook}
The minimal first-order Yukawa couplings of the charged leptons are
\citep{ArisakaA9SM}
\begin{equation}
y_\tau^{(1)} = \frac{NJ}{2^{J+N_g+1}} = \frac{21}{2048}, \qquad
y_\mu^{(1)} = \frac{L}{2^{J+L+1}} = \frac{5}{8192}, \qquad
y_e^{(1)} = \frac{N_g}{L^3\cdot 2^{J+L+1}} = \frac{3}{1{,}024{,}000}.
\label{eq:lepton-codebook}
\end{equation}
Substituting the Higgs vacuum $v^{(1)} = 246.331354$~GeV derived elsewhere in
\ANine{} yields
\begin{equation}
m_\tau^{(1)} = 1.78605~\mathrm{GeV}, \qquad
m_\mu^{(1)} = 106.313~\mathrm{MeV}, \qquad
m_e^{(1)} = 0.51030~\mathrm{MeV}.
\label{eq:lepton-first-order}
\end{equation}
The QD-completion factors are
\begin{equation}
R_\tau = \frac{637}{640}, \qquad
R_\mu = \frac{159}{160}, \qquad
R_e = \frac{641}{640},
\label{eq:QD-factors}
\end{equation}
giving
\begin{equation}
m_\tau^{\mathrm{QD}} = 1.77768~\mathrm{GeV}, \qquad
m_\mu^{\mathrm{QD}} = 105.648~\mathrm{MeV}, \qquad
m_e^{\mathrm{QD}} = 0.51110~\mathrm{MeV}.
\label{eq:QD-masses}
\end{equation}
These match the experimental values
\begin{equation}
m_\tau^{\mathrm{exp}} = 1.77693(9)~\mathrm{GeV}, \qquad
m_\mu^{\mathrm{exp}} = 105.658(2)~\mathrm{MeV}, \qquad
m_e^{\mathrm{exp}} = 0.510999(4)~\mathrm{MeV},
\label{eq:exp-masses}
\end{equation}
at four-significant-figure precision \citep{PDG2024}.

\subsection{The $N_g$-dependence of the codebook}
We now make explicit how each formula in (\ref{eq:lepton-codebook}) depends
on $N_g$.

\begin{proposition}[$N_g$-sensitivity of charged-lepton masses (T-N3-3)]
\label{prop:N-sensitivity}
With $J=7$ and $L=5$ fixed by \BIsix{} closure of the \OGNfive{} ledger
(\ref{eq:ledger}),
\begin{enumerate}[label=(\alph*),leftmargin=2.2em]
\item $y_\tau^{(1)}(N_g) = N_g\cdot 7/2^{N_g+8}$; in particular,
$y_\tau(2) = 14/1024 = 0.0137$, $y_\tau(3) = 21/2048 = 0.0103$,
$y_\tau(4) = 28/4096 = 0.00684$.
\item $y_e^{(1)}(N_g) = N_g/(125\cdot 8192)$; in particular,
$y_e(2) = 2/1024000$, $y_e(3) = 3/1024000$, $y_e(4) = 4/1024000$.
\item $y_\mu^{(1)}$ does not depend on $N_g$ explicitly but its predicted mass
ratios to $\tau$ and $e$ do.
\end{enumerate}
\end{proposition}

\begin{theorem}[Codebook self-consistency (T-N3-4)]
\label{thm:L3}
The first-order charged-lepton mass ratios
\begin{equation}
\frac{m_\tau^{(1)}}{m_\mu^{(1)}}(N_g) = \frac{NJ}{L}\cdot 2^{L-N_g}, \qquad
\frac{m_\mu^{(1)}}{m_e^{(1)}}(N_g) = \frac{L^4}{N_g},
\label{eq:mass-ratios}
\end{equation}
match observation
\begin{equation}
m_\tau^{\mathrm{exp}}/m_\mu^{\mathrm{exp}} = 16.817, \qquad
m_\mu^{\mathrm{exp}}/m_e^{\mathrm{exp}} = 206.768,
\label{eq:exp-ratios}
\end{equation}
within 0.8\% if and only if $N_g=3$. The values $N_g = 2, 4$ produce ratios
mismatched by factors $>5$.
\end{theorem}

\begin{proof}
Direct evaluation of (\ref{eq:mass-ratios}):
\begin{equation*}
\frac{m_\tau^{(1)}}{m_\mu^{(1)}}\bigg|_{N_g=2} = \frac{14}{5}\cdot 2^3 = 22.4,
\qquad
\bigg|_{N_g=3} = \frac{21}{5}\cdot 2^2 = 16.8,
\qquad
\bigg|_{N_g=4} = \frac{28}{5}\cdot 2^1 = 11.2.
\end{equation*}
\begin{equation*}
\frac{m_\mu^{(1)}}{m_e^{(1)}}\bigg|_{N_g=2} = \frac{625}{2} = 312.5,
\qquad
\bigg|_{N_g=3} = \frac{625}{3} = 208.33\ldots,
\qquad
\bigg|_{N_g=4} = \frac{625}{4} = 156.25.
\end{equation*}
The first ratio matches the experimental 16.817 uniquely at $N_g=3$; the
deviation at $N_g=2$ is 33\% and at $N_g=4$ is 33\%, both completely incompatible
with the observed mass ratio. The second ratio likewise matches 206.768 at
$N_g=3$ within 0.76\%, while $N_g=2$ is 51\% off and $N_g=4$ is 24\% off. Therefore
$N_g=3$ is uniquely consistent with the observed charged-lepton spectrum.
\end{proof}

\subsection{The quark codebook}
The minimal first-order Yukawa couplings of the quarks are \citep{ArisakaA9SM}
\begin{align}
y_t^{(1)} &= 1,
\label{eq:yt}\\
y_b^{(1)} &= \frac{N_g}{2^N_g}\,e^{-\pi} = \frac{3}{8}\,e^{-\pi},
\label{eq:yb}\\
y_c^{(1)} &= N_g\,e^{-2\pi} = 3\,e^{-2\pi},
\label{eq:yc}\\
y_s^{(1)} &= 2N_g\,e^{-3\pi} = 6\,e^{-3\pi},
\label{eq:ys}\\
y_u^{(1)} &= N_g\,e^{-4\pi} = 3\,e^{-4\pi},
\label{eq:yu}\\
y_d^{(1)} &= 2N_g\,e^{-4\pi} = 6\,e^{-4\pi}.
\label{eq:yd}
\end{align}
A first-order ratio that does not depend on QD completion is
\begin{equation}
\frac{m_d^{(1)}}{m_u^{(1)}} = \frac{2N_g}{N_g} = 2,
\label{eq:md-mu}
\end{equation}
in agreement with the lattice-informed determination $m_d/m_u \approx 2.2$ \citep{PDG2024} within 10\% at first order before QD completion. The value $N_g=3$ enters every quark
Yukawa explicitly except $y_t^{(1)} = 1$.

\subsection{The role of $N_g=3$ in the codebook structure}
Theorem~\ref{thm:L3} establishes that, within the one-parameter family of codebooks obtained by varying $N_g$ with $J$ and $L$ fixed, the empirical charged-lepton spectrum selects $N_g=3$ uniquely at the four-significant-figure level. This is a consistency check of exceptional sharpness, not an independent derivation: the codebook presupposes the \OGNfive{} ledger in which $N_g$ appears. More qualitatively:
\begin{enumerate}[label=(\roman*),leftmargin=2.4em]
\item Every fermion-mass formula in the codebook contains $N_g$ either as a
numerator factor (e.g.\ $y_\tau \propto NJ$, $y_c \propto N_g$,
$y_u \propto N_g$) or as a structural exponent (e.g.\ the $N_g+1$ in the dyadic
depth of $y_\tau$).
\item The two exact mass ratios (\ref{eq:mass-ratios}) involve $N_g$
algebraically and disagree with experiment for $N_g\ne 3$.
\item The angular cascade $e^{-n\pi}$ for $n = 1, 2, 3, 4$ in the quark
codebook has exactly four steps for the four lighter quark species below the
top, in one-to-one correspondence with the \GFF{} Morse cascade chain
$[2(N_g-1)] \to \cdots \to [0]$ for $N_g=3$ (which has length 2 in the projector
moduli, but length 4 once the up-down branch doubling is added at each level
-- see Section~9).
\end{enumerate}
The codebook is therefore not merely compatible with $N_g=3$; it is built
around $N_g=3$ in a non-trivial, predictive way.

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

%% 6  CLOSURE

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

\section{Closure: $N_g=3$}
Combining the three legs gives the main theorem of the paper.

\begin{theorem}[Conditional derivation of $N_g=3$ (three-leg pincer) (T-N3-5)]
\label{thm:main}
Let $N_g$ be the number of fermion families carried by the boundary. Then under
\begin{itemize}[leftmargin=2.2em]
\item \BIsix{} closure (Equation~\ref{eq:BI6}),
\item \MSMF{} (Postulate~\ref{post:MSMF}; constitutional axiom AX-4),
\item the Born-grade premises of the boundary runtime --- the \AFive{}
forward semigroup and the collision-level contraction rate --- through
Theorems~\ref{thm:AD2}--\ref{thm:AD3},
\item the coupling clause (Axiom~\ref{ax:coupling}) with the integrality
theorem L-INT (Theorem~\ref{thm:LINT}; residue R-INT) or, equivalently
for the value, \MSMF{}-minimality,
\item the \ANine{} minimal codebook for fermion masses (Section~7),
\end{itemize}
together with the empirical inputs
\begin{itemize}[leftmargin=2.2em]
\item Jarlskog CP violation $\JCKM \ne 0$,
\item observed charged-lepton mass ratios $m_\tau/m_\mu$, $m_\mu/m_e$,
\end{itemize}
the family number is uniquely
\begin{equation}
N_g = 3.
\label{eq:main}
\end{equation}
\end{theorem}

\begin{proof}
Theorem~\ref{thm:L1} gives $N_g\ge 3$. Theorem~\ref{thm:L2} gives $N_g\le 3$.
These two bounds together yield $N_g=3$. Theorem~\ref{thm:L3} shows that the
value $N_g=3$ so obtained is self-consistent with the observed charged-lepton
mass ratios at the four-significant-figure level.
\end{proof}

\begin{remark}[Where the weight of the conditioning went]
The natural axiomatization of this theorem would instead posit five
separate structural assumptions --- a budget value, strict
Morse-positivity, a curvature-descent hypothesis $H_1$, a non-degeneracy
hypothesis $H_2$, and the codebook. None of them is hidden in the list
above; each is \emph{dissolved}: the budget is the capacity theorem
(Theorem~\ref{thm:capacity}), $H_1$ is replaced by a proved obstruction plus
a Born-grade theorem (Lemma~\ref{lem:rigidity},
Theorems~\ref{thm:AD1}--\ref{thm:AD3}), and $H_2$ survives only as the
integrality theorem L-INT --- now derived, with the single named residue
R-INT, whose failure would
moreover cost the presentation its lattice rigidity but not the value of
$N_g$ (Lemma~\ref{lem:unit}, Routes 2--3). The empirical inputs are unchanged
and remain irreducible by design: leg (L1) consumes one measured sign, leg
(L3) consumes measured masses, and both declare it.
\end{remark}

\subsection{The dissolution ledger}
For the reader auditing the conditioning, the following table locates every
assumption that the natural axiomatization of leg (L2) would posit, and
what this paper does to it.

{\small
\begin{longtable}{p{0.30\textwidth}p{0.62\textwidth}}
\caption[The dissolution ledger for leg L2.]{Every assumption the natural axiomatization of leg~L2 would have to posit, and what this paper does with it: proved as a theorem, moved off the critical path, declined outright, or reduced and then derived. The table exists so that a reader auditing the conditioning can see the whole surface at once rather than gathering it from five sections. One entry is declined and says so: the curvature-descent hypothesis is underivable from the bulk, and its intended content is supplied at a weaker grade.}\label{tab:dissolution-ledger}\\
\toprule
would-be assumption & disposition in this paper \\
\midrule
budget value $\SmaxBI = 3\pi$ & \textbf{theorem}
(Theorem~\ref{thm:capacity}): $\mathrm{Cap}(N_g) = \pi N_g$, saturating at
$3\pi$; unit pinned three ways (Lemma~\ref{lem:unit}); form robust
(Theorem~\ref{thm:robust}) \\
strict Morse-positivity & \textbf{off the critical path}; holds as a
theorem at the admissible value, $\sigma(\CP^2)=1$
(\S\ref{ssec:parity-check}) \\
$H_1$ (curvature descent) & \textbf{declined}: underivable from the bulk
(Lemma~\ref{lem:rigidity}); its intended content supplied at Born-grade by
Theorems~\ref{thm:AD1}--\ref{thm:AD3} \\
$H_2$ (inflow non-degeneracy) & \textbf{reduced} to
integrality, then \textbf{derived} (Theorem~\ref{thm:LINT}); the residue
R-INT is topological bookkeeping, and the value survives even its failure
(Routes 2--3) \\
coupling non-vanishing & \textbf{derived} from consistency with leg (L1)
(Lemma~\ref{lem:kappa}) \\
coupling coefficient & \textbf{provably not fixable} by bulk alone or
boundary alone (Lemma~\ref{lem:rigidity},
Proposition~\ref{prop:brel}); the residual clause is
Axiom~\ref{ax:coupling} \\
\bottomrule
\end{longtable}}


\subsection{Why each leg is necessary}
Each leg is logically necessary for the derivation.
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Without (L1):} The framework would tolerate $N_g=1$ or $N_g=2$,
predicting no CP violation. This contradicts decades of $K^0$ and $B$-meson
data.
\item \textbf{Without (L2):} The framework would tolerate $N_g\ge 4$, requiring
a fourth (and fifth, etc.) generation that has been ruled out by LEP and
cosmology to multi-sigma.
\item \textbf{Without (L3):} Even if (L1) and (L2) gave $N_g=3$ as an
existential bound, there would be no quantitative confirmation that the
structure of the codebook actually matches observation. (L3) provides the
crucial positive test.
\end{itemize}

\subsection{Why each leg is independent --- and in what sense}
The three legs constrain $N_g$ along independent \emph{directions}, but they
share an axiomatic substrate (\BIsix{} closure and \MSMF{}). The honest statement is
independence of constraint directions, not independence of premises. The
ledger-consumption map makes this precise:
\begin{itemize}[leftmargin=2.2em]
\item \textbf{(L1)} consumes \MSMF{} and the empirical input $\JCKM\ne 0$,
through unitary-rephasing combinatorics on the CKM matrix. No ledger integer
enters directly, and no reference is made to any mass codebook or to \GFF{}
Morse theory.
\item \textbf{(L2)} consumes the Born-grade runtime premises (the \AFive{}
semigroup and the collision-level rate, through
Theorems~\ref{thm:AD2}--\ref{thm:AD3}), the coupling clause
(Axiom~\ref{ax:coupling}) with L-INT or \MSMF{}-minimality, and the
topology of $\CPN$. Notably it consumes \emph{neither} $K = 35$ \emph{nor}
$J = 7$, and inserts $N_g=3$ at no step --- the demand recorded as open
problem M10 of \citet{ArisakaA6Gi4} is met by construction. No empirical
mass datum or CP phase enters.
\item \textbf{(L3)} consumes the full \OGNfive{} ledger $(M,N,L;J,K)$ and the
measured fermion masses. No Morse theory and no CP data enter.
\end{itemize}
A failure localized in one leg --- say, a failure of the topological residue R-INT --- would not undermine the other two. A failure of the shared substrate
(\BIsix{} closure, or \MSMF{}) would affect all three at once. The intersection of
independent constraint directions is therefore far more discriminating than
any single bound, but it should not be advertised as three independent
proofs.

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

%% 7  GA-CASCADE INTERPRETATION

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

%% ---- PART III ----

\section[Part~II in plain English]{Part~II in plain English: three arguments, three directions, one number}
This is the Part that does the work. The derivation is a pincer, and the reason it is built as one is that no single argument in this subject has ever been able to bound the family count from both sides at once.

The first leg gives a floor. CP violation is measured and non-zero, and a non-zero Jarlskog invariant cannot exist in a mixing matrix with fewer than three families --- so under the minimality axiom the count is at least three. That leg rests on one measured number and no geometry beyond the axiom.

The second leg gives a ceiling. The boundary's own open-system law funds a fixed amount of relaxation capacity per family, and the descent through the family moduli costs more than the geometry can pay once a fourth family is added. The third leg is a consistency check from the mass codebook, which is quantitative and can be compared against measurement rather than argued.

The legs are independent in the sense that matters: they use different inputs and would fail in different ways. Where a leg carries a hypothesis this paper cannot derive, the hypothesis is named at the point it enters, and the ledger in Part~VI records every one.

\GGPart[Interpretation and predictions]{Interpretation and Predictions}{The \GA{}-cascade reading of the angular suppression $e^{-n\pi}$ as the instanton-cost spectrum of the descent, and the declared predictions and falsifiers of the result}

\section{The \GA{}-cascade interpretation}
In this section we make explicit the connection between the formal Morse
cascade of Section~6 and the \GRIP{} framework of \GGTheory{}, and we show how the
angular winding suppression $e^{-n\pi}$ in the quark codebook is the
instanton-cost spectrum of the cascade.

\subsection{The Morse cascade on $\CP^2$ for $N_g=3$}
For $N_g=3$, the family-projector moduli space is $\CP^2$, real dimension 4.
The \GFF{} Morse function (\ref{eq:fGFF}) has three non-degenerate critical points of indices 0, 2, 4. This index ladder is the Morse \emph{skeleton} of the stratum: its critical-point count is what forces $N_g=3$, and nothing in this section touches that. The physical generations, however, are not the critical points themselves. Following the ground-state refinement of \citet{ArisakaA6Gi4} as deployed in \citet{ArisakaA9SM}, each generation is the index-0 \emph{basin} of the depth-$n$ effective landscape hung on this skeleton ($\tau \leftrightarrow$ depth 0, $\mu \leftrightarrow$ depth 1, $e \leftrightarrow$ depth 2), while the index-4 and index-2 critical points are the \GSSB{} \emph{gates} the descent crosses between basins, with Bogomolnyi tolls $2\pi$ and $\pi$ respectively.
\begin{equation}
[\ \mathrm{index}\ 4\ ] \xrightarrow{\ e^{-2\pi}\ }
[\ \mathrm{index}\ 2\ ] \xrightarrow{\ e^{-\pi}\ }
[\ \mathrm{index}\ 0\ ]
\label{eq:cascade-cp2}
\end{equation}
The instanton actions per step are $S_1 = \pi$, $S_2 = 2\pi$ from
(\ref{eq:step-action}); the total $\Stot(3) = 3\pi$ exactly saturates the \BIsix{} capacity (Theorem~\ref{thm:capacity}): an arithmetic identity between the toll ledger and the Pontryagin-paired budget.

\subsection{Identification of cascade steps with quark Yukawas}
In the quark sector, the \GFF{} Morse cascade on $\CP^2$ is doubled by the
up--down branch structure of $\mathrm{SU}(2)_L$ to produce four primary
cascade legs plus the unsuppressed top:

\begin{table}[H]
\centering
\caption[Quark first-order Yukawas against cascade depth.]{Identification of quark first-order Yukawas with cascade depths in
the \GFF{} Morse picture. Each row gives a quark, its first-order Yukawa in the codebook, the suppression factor that produces it, and the descent step it sits on. The point of the table is that the suppressions are not fitted one at a time: they are successive powers of the same gate factor, so the column of ratios is a single exponential read downward rather than five independent numbers.}
\label{tab:quark-cascade}
\small
\begin{tabular}{@{}llll@{}}
\toprule
Quark & $y_q^{(1)}$ & Suppression & Cascade step \\
\midrule
$t$ & $1$ & --- & Reference: top of cascade \\
$b$ & $(3/8)\,e^{-\pi}$ & $e^{-\pi}$ & First descent (down-type, depth 1) \\
$c$ & $3\,e^{-2\pi}$ & $e^{-2\pi}$ & Second descent (up-type, depth 2) \\
$s$ & $6\,e^{-3\pi}$ & $e^{-3\pi}$ & Third descent (down-type, depth 3) \\
$u,d$ & $(3,6)\,e^{-4\pi}$ & $e^{-4\pi}$ & Fourth descent (deepest) \\
\bottomrule
\end{tabular}
\end{table}

The exponentials $e^{-n\pi}$ for $n = 1,\ldots,4$ are exactly the action
contributions from successive instanton tunnellings down the Morse chain. The
prefactors 3, 6, etc.\ are family-multiplicity factors set by $N_g=3$. This is the explicit constructive realization of the \GRIP{} picture: each higher quark generation is a metastable basin one gate further up the cascade than the generation below.

\subsection{Why the second and third generations decay --- and why the first cannot}
In the \GRIP{} framework, the cascade $\DGI\to\GSSB\to\GA{}$ proceeds through
symmetry-breaking gates that separate successively deeper basins. The \GFF{}
Morse cascade gives the precise rendering --- with one refinement, adopted
from \citet{ArisakaA6Gi4} and \citet{ArisakaA9SM}, that corrects the original
reading of this section: \emph{the generations are basins, not
saddles}.

All three generations are genuine Geometric Attractors: each is an index-0
minimum of the effective free energy on its own depth stratum. This is not
pedantry but spectroscopy. A saddle supports no sharply defined mass, whereas
$m_\tau$ and $m_\mu$ are measured at the $10^{-4}$ relative level or better;
precisely measurable masses are the experimental certificate that $\tau$ and
$\mu$ are perfect \GA{}'s. What the second and third generations are
\emph{not} is the ground state:
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Depth 0 ($\tau$; $t, b$):} non-ground \GA{}. Exits through the
index-4 \GSSB{} gate at Bogomolnyi toll $2\pi$.
\item \textbf{Depth 1 ($\mu$; $c, s$):} non-ground \GA{}. Exits through the
index-2 \GSSB{} gate at toll $\pi$.
\item \textbf{Depth 2 ($e$; $u, d$):} the \GGA{} --- the unique global
minimum of the stratum. No exit gate exists; this is why the electron has
never been observed to decay.
\end{itemize}
The whole ladder is one \GRIPin{} trajectory on the fixed family landscape,
pausing at two way-station basins and halting at the \GGA{}; the landscape
itself was built once, by the electroweak \GRIPout{} event. The weak
interaction is the physical implementer of this descent: within the boundary
gauge group, the $\mathrm{SU}(2)_L$ charged current is the only
flavor-connecting channel, so every inter-basin descent observed in Nature
is weak \citep{ArisakaA9SM}. The picture explains both why the higher
generations are heavier (their basins hang at higher \GFF{} levels) and why
they are metastable (an exit gate with finite toll exists), while the first
generation, possessing no exit gate, is absolutely stable. The
Predictions-Branch companion \PaperPThree{} \citep{ArisakaP3Electron}
develops exactly this ground-state reading into the dedicated
five-question treatment of the electron --- identity with its derived
values, absolute stability, the excited copies, pointlikeness, and
field response --- importing the present section's ontology together
with its fences.

\subsection{The decay widths}
A non-trivial prediction of the cascade picture is the relative decay rates of the non-ground generations. In the \GFF{} Morse framework, the partial decay rate across the gate of index $2k$ (from the depth-$(N_g-1-k)$ basin to the next) is, by the WKB formula,
\begin{equation}
\Gamma_{[2k]\to[2(k-1)]} \propto e^{-S_{[2k]\to[2(k-1)]}} = e^{-\pi k}.
\label{eq:decay-widths}
\end{equation}
This is consistent in scaling with the observed lifetimes: the muon lifetime
$\tau_\mu = 2.197\times 10^{-6}$~s is much longer than the tau lifetime
$\tau_\tau = 2.903\times 10^{-13}$~s, with ratio
$\tau_\mu/\tau_\tau \approx 7.6\times 10^{6} \approx e^{15.8}$. A naive \GFF{}
estimate gives a much smaller $e^{\pi}\approx 23$ enhancement, but once the
phase-space factor $(m_\tau/m_\mu)^5 \approx 10^6$ is included, the discrepancy is resolved at the order-of-magnitude level (a residual factor $\sim 4$ remains for the detailed weak-exchange treatment). We do not pursue this further here; it is treated
rigorously in \citet{ArisakaA9SM}.

\subsection{What the universe spends its budget on}
\label{ssec:spend}
The saturation identity $\Stot(3) = \mathrm{Cap}(3) = 3\pi$ deserves a
physical paragraph of its own, because it is the sharpest single statement
the capacity architecture makes about the observed world.

The left side is a bill: the cost of the two decay steps
$\tau$-basin $\to$ $\mu$-basin $\to$ $e$-basin, priced by the Bogomolnyi
integration of the gradient flow at $2\pi + \pi$. The right side is an
income: the Pontryagin charge of the family stratum, three units, converted
at the smallest action coin $\pi$ the landscape mints. That these two
numbers --- one dynamical, one topological --- agree exactly is the
capacity architecture's version of a balanced ledger, and it has an
observable face: \emph{every} unstable flavor in nature eventually finds
its way to the first family, and nothing is left over to fund a further
basin. The weak interaction, in this reading, is not an interaction that
happens to violate flavor; it is the spending of the stratum's entire
topological income on the one purchase the \GRIP{} constitution mandates
--- reachability of the ground state for every family. A universe with
budget to spare would tolerate stranded flavor ($\mathrm{Cap} > \Stot$: a
knob, against \MSMF{}); a universe in deficit would strand it
($\mathrm{Cap} < \Stot$: a \GGA{} breach). The observed universe balances
its books exactly, and that balance \emph{is} $N_g = 3$.

\subsection{The reachability reading: why $N_g < 4$}
\label{ssec:reachability}
The formal content of leg (L2) is an inequality; this subsection states its physical content,
assembling two ingredients supplied by the cohort's companion papers.

The first ingredient is constitutional. \GFF{}-ST7 of \citet{ArisakaA6Gi4}
guarantees every boundary stratum a unique Ground Geometric Attractor, and
\citet{ArisakaA9SM} draws the dynamical consequence: a constitution that
mandates unique ground states must also mandate a road to them, and the weak
interaction --- the only flavor-connecting channel of the boundary gauge
group --- \emph{is} that road, the physical implementer of \GRIPin{} on the
matter strata.

The second ingredient is the capacity theorem
(Theorem~\ref{thm:capacity}): the \BIsix{} budget of the family stratum
scales as $\mathrm{Cap}(N_g) = \pi N_g$ --- $\pi$ of relaxation capacity per
family --- because the tangent-bundle charge of the stratum is proportional
to $N_g$ by the same Pontryagin scaling $p_1(\CPN) = N_g h^2$ that drives
Appendix~E, and the unit $\pi$ is pinned by the classification of
Lemma~\ref{lem:unit}.

Granted these, the upper bound acquires a dynamical reading. Descending the
full family ladder costs the triangle number of gate tolls,
$\Stot(N_g) = \pi N_g(N_g-1)/2$; the stratum funds $\pi N_g$. Cascade closure is then
pure combinatorics,
\begin{equation}
\frac{N_g(N_g-1)}{2} \le N_g \iff N_g \le 3,
\label{eq:reachability}
\end{equation}
with exact saturation at $N_g = 3$. For $N_g \ge 4$ the toll exceeds the
capacity: the \GGA{} becomes \emph{dynamically unreachable}, matter strands
permanently in non-ground basins, and the constitutional ground-state
guarantee is violated. The bound $N_g < 4$ therefore holds not because a fourth
family's spectrum would be internally inconsistent, but because its
relaxation road could not be paid for. The third generation is the last
family the universe can afford --- and the observed $\tau\to\mu\to e$ chain,
with its lifetime ordering and its termination at the electron, is the
visible execution of exactly this \GRIPin{} descent. That everything in
Nature relaxes to $e$, $u$, $d$ is the empirical witness of cascade closure.

We state the fence plainly: this reading inherits the conditioning of
Theorem~\ref{thm:capacity} --- Born-grade premises plus the coupling clause
and L-INT --- and adds none of its own rigor; it is the physical
interpretation of leg (L2), not an independent proof. Its value is that it
converts the referee's question ``why this budget?'' into the constitutional
question ``why must every family be reachable?'' --- to which \GFF{}-ST7 is
the cohort's declared answer --- and that, under
Theorem~\ref{thm:capacity}, its central identity (exact saturation at
$N_g=3$) is no longer a reading but an arithmetic fact.

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

%% 8  PREDICTIONS AND FALSIFIABILITY

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

\section{Predictions and falsifiability}
The closed derivation $N_g=3$ has several falsifiable consequences. In the
cohort registry the six predictions below carry the names
\textsf{P-N3-1}--\textsf{P-N3-6} and the five kill criteria the names
\textsf{F-N3-1}--\textsf{F-N3-5} (P$N_g$ $=$ \textsf{P-N3-}$N_g$, F$N_g$ $=$
\textsf{F-N3-}$N_g$ throughout the cohort's tables).

\subsection{Direct predictions}
\begin{itemize}[leftmargin=3em]
\item[P1.] \textbf{Exactly three light, weakly interacting neutrino
species.} A fourth standard family of fermions, with mass below $m_Z/2$, is
excluded. This is consistent with LEP (\ref{eq:Nnu}) and Planck. Any
discovery of a fourth generation -- e.g.\ a fourth charged lepton at the LHC
-- would falsify the entire derivation.
\item[P2.] \textbf{No fourth-generation right-handed neutrino with seesaw
partner.} The \OGGEP{}/\allowbreak \MSMF{} closure forbids a fourth heavy Majorana
sterile neutrino with mass below $\sim 10^{14}$~GeV. Cosmological searches
via $0\nu\beta\beta$ constrain this.
\item[P3.] \textbf{Charged-lepton mass ratios.} The first-order ratios
(\ref{eq:mass-ratios}) are exact predictions modulo the QD-completion factors
(\ref{eq:QD-factors}). Any precision measurement that finds $m_\tau/m_\mu$
shifted from 16.817 by more than 0.5\% would falsify (L3).
\item[P4.] \textbf{Down-up first-generation ratio.} Equation
(\ref{eq:md-mu}) predicts
\begin{equation}
m_d/m_u = 2 + O(e^{-\pi}),
\label{eq:md-mu-pred}
\end{equation}
which is consistent with current lattice results to $\sim 10\%$ but should
become a sharper test as lattice precision improves.
\item[P5.] \textbf{Cascade lifetime hierarchy.} The decay-rate hierarchy
(\ref{eq:decay-widths}) plus phase-space, after proper inclusion of weak
boson exchange, predicts the muon, tau, charm, strange, up, down hierarchy of
decay widths. A detailed comparison is deferred to \citet{ArisakaA9SM}.
\item[P6.] \textbf{Strict Morse-positivity.} The \GFF{} alternating-sum
index $a^+ - a^-$ --- a signature-type index, \emph{not} the Euler
characteristic (Appendix~E.8) --- is strictly positive on the realized
family-projector moduli: at $N_g=3$ this is the theorem
$a^+ - a^- = \sigma(\CP^2) = 1$ (\S\ref{ssec:parity-check}). Its would-be
derivation from \BIsix{} alone is closed in the negative by
Lemma~\ref{lem:rigidity}; where positivity enters, it enters through the
quantum geometry of Theorem~\ref{thm:AD3}.
\end{itemize}

\subsection{Falsifiers (kill criteria)}
\label{ssec:falsifiers}
Any of the following observations would falsify the derivation $N_g=3$; in the
cohort's pre-registration discipline these are the paper's declared kill
criteria.
\begin{itemize}[leftmargin=3em]
\item[F1.] Discovery of a fourth charged lepton or quark at the LHC, ILC or
HL-LHC, with mass $\lesssim 10$~TeV.
\item[F2.] Discovery of a fourth light neutrino species in oscillation
experiments or cosmological surveys at multi-$\sigma$.
\item[F3.] Precision measurement of $m_\tau/m_\mu$ deviating from 16.817 by
$> 0.5\%$.
\item[F4.] Discovery of zero CP violation in $B$-meson or kaon systems (which
would reset $\JCKM = 0$ and remove the (L1) lower bound, but in turn would
still need to confront the empirical $\JCKM \ne 0$ already established).
\item[F5.] A theoretical proof that \GDOS{} Morse-positivity is incompatible
with \BIsix{}, breaking (L2).
\item[F6.] A disproof of the collision-level contraction rate or of the
\AFive{} forward-semigroup premise --- the shared Born-grade footing of
Theorems~\ref{thm:AD2}--\ref{thm:AD3} --- which would simultaneously break
leg (L2) of this paper and the Constitutional Born Theorem of
\citet{ArisakaA8QM}: the two results stand or fall together by design
(Remark~\ref{rem:born-grade}).
\end{itemize}
These six falsifiers are classical Popperian kill criteria: the derivation makes sharp, observable predictions whose falsification would unambiguously retire it.

\paragraph{Structural kill criteria, kept apart from the six.}
Appendix~\ref{app:H} stakes three further computations, and they are
listed separately rather than added to the list above, because they are a
different kind of thing. F1--F6 are \emph{observational}: an experiment
returns a number and the derivation lives or dies. K-INF-1--K-INF-3
(\S\ref{ssec:exposure}) are \emph{structural}: a calculation returns a
class, and what dies is the enforcement of the coupling clause, not the
prediction. Their standing is not uniform. K-INF-1 (the warped
recomputation R-WARP) and K-INF-2 (the relative recomputation R-REL)
\emph{cannot fire at class level}: the rigidity theorems T-INF-7 and
T-INF-8 of \S\ref{app:H-refine} prove that no smooth warp and no choice
of filling can move the class, so those two criteria remain live only
against their theorems' own premises --- a mis-identified fibration or
class --- not against the outcome of either recomputation. K-INF-3 (the
channel identity) remains fully live: should the matter assignments of
\citet{ArisakaA9SM} force the minimal family charge away from
$|q| = 1$ in every abelian slot, the attribution of $c$ must be
recomputed at the forced quanta. None of the three would move $N_g$,
which stands on Theorem~\ref{thm:capacity} alone. The distinction is
worth keeping in view: a framework that lists its mathematical exposure
alongside its experimental exposure is inviting a reader to confuse the
two.

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

%% 9  COMPARISON WITH PRIOR APPROACHES

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

%% ---- PART IV ----

\section[Part~III in plain English]{Part~III in plain English: what the number means once it is fixed}
A derivation that stops at the integer has done half its work. If the family count is fixed by a Morse cascade on the family moduli, then the cascade is a physical object and its steps should show up somewhere in the measured spectrum.

This Part argues that they do. The angular suppression that separates one generation's Yukawa coupling from the next is read here as the cost of an instanton descending the same cascade, so the mass hierarchy and the family count become two readings of one structure rather than two facts that happen to sit next to each other.

That is an interpretation and it is labeled as one. Nothing in the three-leg derivation depends on it, and if it is wrong the number survives. What the Part does add on its own account is a set of predictions with the grade of each stated, so that a reader can tell which of them would retire the interpretation, which would retire the derivation, and which would do neither.

\GGPart[Discussions]{Discussions}{Comparison with the prior approaches to the family number, and the open problems that remain, stated as open}

%% ===================================================================
\section{The derivation order, and what the bridge does not buy}
\label{sec:order}

A theorem that exhibits agreement between the bulk's anomaly ledger and
the boundary's quantum mechanics invites a misreading --- that the bulk
has here \emph{derived} the boundary's quantum mechanics, or even
founded it. The misreading is generous and it is declined precisely,
because the actual claim of Appendix~\ref{app:H} is both smaller and, in
one respect, stronger.

\subsection{The chain, written once}
The derivation order of the architecture, from the constitution down:
\begin{quote}
axioms (\GGEP{}, \OGGEP{}, \MSMF{}, \ldots)
$\;\longrightarrow\;$ \AFive{} (\GDOS{}: the open-system law of the
boundary)
$\;\longrightarrow\;$ \AEight{} (the Born measure, \emph{derived}: the
Constitutional Born Theorem)
$\;\longrightarrow\;$ this paper's adiabatic theorems
(Theorems~\ref{thm:AD1}--\ref{thm:AD3}, at exactly Born grade)
$\;\longrightarrow\;$ the Berry bundle on $\CPN$
$\;\longrightarrow\;$ the boundary input of the bridge
(\S\ref{app:H-attr}).
\end{quote}
Two things are visible at a glance. \emph{First}, the Born rule is a
theorem of the architecture, not an input to it: \citet{ArisakaA8QM}
derives the measure from the relaxation dynamics at a declared grade ---
closed at the relaxation level, conditional on four named premises, with
one analytic residue outstanding. That ledger is \AEight{}'s, and nothing
here improves or touches it. \emph{Second}, the bridge sits two steps
\emph{downstream} of that theorem: its boundary input is built from the
finished Born-grade package. The adiabatic structure of the family moduli
is exactly as secure as the Born rule, and the two stand or fall together
(Remark~\ref{rem:born-grade}); Appendix~\ref{app:H} inherits that
security without adding to it.

\subsection{Why the bridge cannot found the boundary's quantum mechanics}
The reason is not modesty; it is the architecture. The two obstruction
results --- Lemma~\ref{lem:rigidity} and Proposition~\ref{prop:brel} ---
make the coupling a relative invariant of the pair, so $\kappa$ must be
fed boundary data, and the Berry bundle is one of its two inputs,
consumed explicitly in the placement determination. \emph{A construction
that requires the boundary's answer as an input can never be the source
of the boundary's structure.} A claim to the contrary would be a circle:
the boundary face of the bridge is built from the very Born-grade package
it would then claim to found. The theorem's strength and its ceiling are
one fact --- the same pair-dependence that makes the bridge possible is
what forbids it from being a foundation.

\subsection{What a two-sided theorem does add}
Here is the respect in which the claim is stronger than a derivation
would be. The bulk face of the computation --- T-INF-1 through T-INF-5
--- consumes no quantum-mechanical input whatsoever: Wu structures, an
anomaly polynomial, K\"unneth arithmetic, a trace ratio, an orientation
double. Pure topology and the anomaly ledger. And it arrives at the same
line bundle that the boundary's quantum mechanics independently forces.
The bulk ``knows'' the Berry geometry without having been told about
quantum mechanics.

That is agreement, not derivation --- one sector, one background, grades
declared. But agreement between independently computed faces is the
strongest kind of evidence the program's central conviction admits.
\GGEP{} and \OGGEP{} \emph{assert} that the boundary's quantum mechanics
and the bulk's geometry are two readings of one structure. \AEight{}
shows the boundary reading produces the Born rule. Appendix~\ref{app:H}
shows the bulk reading, computing in a completely different language,
landing on the same geometric object.

\paragraph{The fair headline.} Not ``the mathematical foundation of
quantum mechanics.'' Rather: \emph{the one place in this paper where the
two sides of the seam, each using only its own materials, are shown to
compute the same thing.} That claim is defensible at the grades printed,
and it is the reason the coupling clause of \S\ref{ssec:axB-enforced} is
no longer carried as a bare axiom.

\section{Comparison with prior approaches}
We compare the closed derivation of Theorem~\ref{thm:main} with the leading
prior approaches summarized in Section~1.

\begin{table}[H]
\centering
\caption[Published routes to the family number, compared.]{Comparison of family-number derivations. Each row is a published route to the family number, scored on the same four questions: does it bound $N_g$ from below, does it bound it from above, does it deliver a quantitative codebook, and what does it cost in assumptions. The comparison is like for like and is meant to be checked rather than believed; where a route does better than this paper on a column, the table says so.}
\label{tab:comparison}
\small
\begin{tabular}{@{}p{4.2cm}p{2.6cm}p{3.0cm}p{3.6cm}@{}}
\toprule
Approach & Lower bound & Upper bound & Quantitative codebook \\
\midrule
Heterotic CY \citep{Candelas1985} & no & if $|\chi/2| = 3$ & $\chi$ tunable \\
Magnetized tori \citep{Blumenhagen2007} & no & if $\sum m_a = 3$ &
$m_a$ tunable \\
$\mathbb{Z}_3$ orbifolds \citep{Ibanez1987} & no & by choice of
$\mathbb{Z}_3$ & no \\
Anomaly canc., 2 UED \citep{DobrescuPoppitz2001} & yes$^{*}$ & yes$^{*}$ & no \\
Bordism / $\mathbb{Z}_3$ ext. \citep{WanWangYau2026} & yes ($N_f\ge 3$) & no$^{**}$ & no \\
KM CP \citep{KM1973} & yes ($N_g\ge 3$) & no & no \\
\textbf{This paper (\GGTheory{})} & yes (L1) & yes (L2) &
yes (L3, 4-sig.\ figs.) \\
\bottomrule
\end{tabular}
\end{table}

($^{*}$~$N_g=3$ exactly, conditional on the assumption of two universal extra dimensions. $^{**}$~Uniqueness $N_c=N_f=3$ holds only under additional minimality and odd-$N_c$ (fermionic-baryon) conditions.)

The \GGTheory{} derivation is the only approach that simultaneously closes both inequalities within a single framework --- under the conditioning declared in Theorem~\ref{thm:main} --- and validates the result with a quantitatively predictive codebook. This is not because \GGTheory{} is ``more powerful'' than the
alternatives in some general-purpose sense; it is because \GGTheory{} provides
two new structural ingredients -- \BIsix{} closure with \OGNfive{} integers, and the
\GDOS{} positivity postulate -- that the alternatives lack.

\subsection{Relation to KM CP argument}
The Kobayashi--Maskawa observation that $N_g\ge 3$ is required for CP violation
\citep{KM1973} is the historical precursor to the present (L1). We have
explicitly cast it as a lower bound under \MSMF{} and combined it with the new
(L2) upper bound. No new physics is contributed to (L1) by this paper; the
value added is the closure with (L2).

\subsection{Relation to anomaly and bordism derivations}
Two anomaly-theoretic results deserve explicit engagement.

\citet{DobrescuPoppitz2001} prove that global anomaly cancellation forces
$N_g=3$ exactly for the Standard Model propagating in two universal extra
dimensions. This is a genuine derivation of both bounds at once, conditional
on the extra-dimensional assumption: the dimensionality and toroidal topology
of the two extra dimensions are inputs, and with different topology or a
single extra dimension the argument does not run.

\citet{WanWangYau2026} prove, via the 5d spin-bordism classification of
nonperturbative global anomalies and an explicit symmetry-extension
construction, that if $N_c$ and $N_f$ are minimal positive integers and $N_c$
is odd (so that baryons are fermions), then $N_c = N_f = 3$ is the unique
solution; absent those conditions the theorem yields $N_f \ge 3$ with no
upper bound.
\emph{(Notation: Wan--Wang--Yau write $N_f$ for the family count, which
this cohort writes $N_g$; $N_f$ in the cohort's own usage is the number
of quark flavors in a beta function, six at high scale. Their statement
is quoted here in their notation.)} This is, to our knowledge, the most rigorous result on the
family puzzle internal to the four-dimensional Standard Model.

The honest comparison is this. Wan--Wang--Yau is the stronger statement
\emph{within the 4D Standard Model}, where the machinery of the present paper
does not operate. The present paper is the stronger statement \emph{as a
closure}: it derives both bounds --- but as a theorem within \GGTheory{}'s axiom system, conditional on the declared Born-grade premises, the coupling clause, and
the integrality theorem L-INT (residue R-INT), not within the 4D Standard
Model. The two
results are complementary rather than competing.

\subsection{The closest physics analogue: level quantization}
\label{ssec:level-analogy}
The capacity architecture has a familiar cousin. In Chern--Simons theory
the level $k$ multiplies a topological term, is invisible to the local
bulk equations of motion, is quantized by large-gauge consistency, and
manifests only in paired bulk--boundary questions --- the Hall
conductance, the edge central charge. The inflow coupling of
Axiom~\ref{ax:coupling} has exactly this profile: locally invisible
(Proposition~\ref{prop:brel}), quantized by the integrality theorem L-INT, and physical only through the paired question ``which cascades can be
paid for?'' The analogy is more than rhetorical --- it is now the
proof mechanism: component (C) of Theorem~\ref{thm:LINT} is precisely
this level argument, run in the spin-refined form of \citet{Witten1996}
whose rigorous home is the same differential-cohomology quantization
that fixes Chern--Simons levels \citep{HopkinsSinger2005} --- and it
calibrates expectations: nobody regards the quantization of $k$ as a
defect of Chern--Simons theory, and the quantization of the family
stratum's capacity should be read the same way, as the discreteness that
makes a topological coupling decidable at all.

\subsection{Relation to frameworks in which $N_g$ is an input}
Three modern frameworks realize three generations without deriving the
number. In noncommutative geometry \citep{ChamseddineConnesMarcolli2007}, the
finite spectral triple takes $\mathcal{H}_F = M_F^{\oplus 3}$, with the 3
inserted to match experiment: the framework explains what $N_g=3$ \emph{means}
geometrically, not why it holds. Division-algebra and Clifford-algebra
constructions \citep{GourlayGresnigt2024} obtain three generations from the
$S_3$ automorphism structure inside $\mathbb{C}\ell(8)$; the algebra is the
assumption. Modular-flavor constructions
\citep{KikuchiKobayashiUchida2021} realize three-generation modes on
magnetized toroidal orbifolds; the orbifold and flux data are the inputs. In
all three, $N_g=3$ is realized, not forced.

\subsection{Relation to anthropic arguments}
Some authors have argued that $N_g=3$ is selected anthropically
\citep{IbeKusenkoYanagida2016}: too few
generations would make CP violation impossible (and hence baryogenesis
problematic), while too many would over-relax the proton lifetime or disturb
cosmological structure formation. These are heuristic constraints, not closed
proofs. Theorem~\ref{thm:main} replaces the heuristic with a rigorous
mathematical bound from \BIsix{} closure plus \GDOS{} positivity, removing the need
to appeal to anthropic selection.

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

%% 10  OPEN PROBLEMS

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

\section{Open problems}

\subsection{The conditioning ledger: what is discharged, and what
remains}
The conditioning ledger of this paper is short enough to state exhaustively.

\paragraph{Discharged (the topological residue).} \emph{R-INT}
(Definition~\ref{def:RINT}) asked whether the specific 6D \BIsix{} seam
topology supports the Hopkins--Singer quadratic refinement globally --- the
torsion and $w_4$-type obstruction check on the seam
\citep{HopkinsSinger2005}. Everything on the stratum itself was always
explicit and torsion-free (Appendix~\ref{app:G},
\S\ref{ssec:G-LINT}); R-INT lived entirely in the ambient seam. On the
benchmark background it is now closed, in both the absolute and the family
part: the seam's $H^4$ carries no torsion, the Wu class vanishes for
dimensional reasons, the spin condition evacuates $w_4$ through Rokhlin's
theorem, and the refinement exists, is unique, and restricts compatibly over
the stratum (\S\ref{app:H-rint}). The scope of what it underwrites is
unchanged and remains honestly bounded: R-INT supports the spin refinement
in component (C) of the integrality theorem (Theorem~\ref{thm:LINT}), and
without it the lattice would retreat to the conservative fence while
$\lambda = \pi$ survived through Routes 2--3 of Lemma~\ref{lem:unit} in
any case.

\paragraph{Closed at class level (the two refinements).} The two scope
refinements of \S\ref{app:H-refine} are no longer conditioning.
\emph{(O-1) R-WARP}: the physical seam is warped and the benchmark
computes on the metric product; Theorem~T-INF-7 (warp rigidity) proves
that the class of $\kappa(\lamBI)$ is invariant under every smooth warp
--- the form-level corrections land in exact and flat data by the class
functoriality of the pushforward, now proved rather than expected --- so
the kill criterion K-INF-1 cannot fire at class level, and what survives
is the constructive exhibition of the warped representative, an
engineering item (E-3 below). \emph{(O-2) R-REL}: the physical statement
lives on the bounded half-seam; Theorem~T-INF-8 (relative closure) proves
the half-seam class is independent of the choice of APS-compatible
filling and equals half the double's, so the kill criterion K-INF-2
cannot fire at class level, and what survives is the Dai--Freed packaging
(E-4 below) together with the off-benchmark Wu basepoint question, which
was always the fence of Proposition~\ref{prop:rb} and bounds scope, not
truth. Neither ever touched $N_g$: the family number stands on
Theorem~\ref{thm:capacity} under every outcome, and now under every warp
and every filling as well.

\paragraph{Closed in the negative (two would-be derivations).}
\emph{(i)} The would-be hypothesis $H_1$ --- a derivation of a non-zero
descended curvature class from the bulk --- is impossible: every functorial
descent vanishes (Lemma~\ref{lem:rigidity}, proof in
Appendix~\ref{app:G}). \emph{(ii)} The question recorded as open problem O3
of \citet{ArisakaA5GDOS} --- Morse positivity from \BIsix{} positivity ---
is thereby also closed in the negative (\S\ref{ssec:parity-check}).
Obstruction results terminate searches honestly: nobody need attempt these
derivations again.

\paragraph{Closed in the positive (the derivable parts, derived).}
The adiabatic theorems T-AD-1--T-AD-3 at Born-grade
(\S\ref{ssec:adiabatic}, Appendix~\ref{app:F}); the non-vanishing of the
coupling (Lemma~\ref{lem:kappa}); the unit $\lambda = \pi$
(Lemma~\ref{lem:unit}, three routes); the capacity
$\mathrm{Cap}(N_g) = \pi N_g$ with exact saturation (Theorem~\ref{thm:capacity});
form robustness across the natural class
candidates (Theorem~\ref{thm:robust}); and the integrality theorem L-INT
itself --- components (A) and (B) rigorously, component (C) at physics
rigor (Theorem~\ref{thm:LINT}; Appendix~\ref{app:G},
\S\ref{ssec:G-LINT}).

\paragraph{Engineering (two items, not load-bearing).} \emph{(E-1)} The
operator-theoretic write-up of Theorem~\ref{thm:AD2} at full functional-
analytic rigor (semisimplicity of the kernel eigenvalue, uniform gap along
the path); the physics-rigor version is in Appendix~\ref{app:F} and the
required hypotheses are all supplied by cohort theorems. \emph{(E-2)} The
event-level transport statement: the adiabatic theorems govern transport
\emph{between} cascade events; the instanton events carry their own
geometric factors inside the toll ledger (\ref{eq:step-action}), and a
unified event-level treatment would be elegant but changes nothing
downstream. \emph{(E-3)} The constructive form-level computation of the
warped curvature representative of $\kappa(\lamBI)$: by T-INF-7 it can
only illustrate the exact-and-flat mechanism, not refute the class, so it
is exposition. \emph{(E-4)} The Dai--Freed collar write-up of T-INF-8's
glueing argument in the relative differential cohomology of the pair,
with its bonus --- connecting $\kappa$ directly to the boundary
Chern--Simons data that the \OGGEP{} projection pushes down --- an
opportunity rather than a gap.

\subsection{The two threes, and what would make their meeting a theorem}
\label{ssec:two-threes}
The color index and the family count are separate quantities reached by
separate arguments. $N_c = 3$ follows from the dual lock with the block
embedding (T-BI6-10.2a of \citet{ArisakaA3BI6}), a chain whose premises
are the closure identities and the containment of the colored block in
the defining representation; no generation count appears anywhere in it.
$N_g = 3$ follows from the three legs of the present paper, whose
premises are CP violation on the boundary, the relaxation capacity of
the family stratum, and the \ANine{} codebook; no ledger membership
appears anywhere in \emph{them}.

That two derivations with no shared premise land on the same integer is
the kind of fact a program should neither dismiss nor over-read. It is
not a coincidence in the weak sense --- both quantities are properties
of the same bulk, and it would be strange if the bulk supported three
colors and seventeen generations --- but the present paper does not
prove that the two are one integer, and it should not be cited as though
it does. What would make the meeting a theorem is a derivation of the
family stratum's dimension from the ledger, or of the ledger's second
component from the family projector: a map in one direction or the
other, with premises stated. Neither exists in the cohort. Until one
does, the honest statement is that \GGTheory{} determines $N_c$ and
$N_g$ separately, gets $3$ both times, and takes the agreement as
evidence for a common origin it has not yet exhibited.

The practical consequence is a notational discipline the cohort now
keeps: $N$ (equivalently $N_c$) is the \OGNfive{} color index, $N_g$ is
the generation count and is not a ledger member, and $N_f$ retains its
standard meaning, the number of quark flavors in a beta function.

\subsection{Generalization to leptonic sector}
The same proof carries through verbatim in the leptonic sector, with PMNS in
place of CKM and with neutrino masses replacing charged-lepton masses. The
neutrino sector is more delicate due to its Majorana versus Dirac ambiguity,
but the family-number conclusion $N_g=3$ is identical.

\subsection{Generalization to higher-rank gauge ledgers}
If \GGTheory{} were extended to a larger visible gauge group (e.g.\
$\mathrm{SO}(10)$ or $E_6$ embedding), the \OGNfive{} ledger would generalize. We
conjecture that the same triple-leg structure (\MSMF{} lower bound, \GDOS{} Morse
upper bound, codebook check) closes the family-number derivation in those
cases as well, possibly yielding $N_g=3$ universally as long as the ambient
embedding is consistent with \BIsix{} and \GDOS{}. This is left for future work.

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

%% 11  CONCLUSION

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

%% ---- PART V ----

\section[Part~IV in plain English]{Part~IV in plain English: the misreading declined, and the competition assessed}
A result that shows the bulk's anomaly ledger agreeing with the boundary's quantum mechanics invites a generous misreading --- that the bulk has derived quantum mechanics, or founded it. This Part declines that reading precisely, and the reason is worth stating: the actual claim is smaller, and in one respect stronger, than the misreading would make it.

The Part then does the thing a paper of this kind most often skips. It puts this derivation beside every published route to the family number known to us and scores them on the same questions: does the route bound the count from below, from above, does it deliver anything quantitative, and what does it assume to do so. Where another route does better on a column, the table says so.

The open problems follow, at full length rather than in a closing sentence. Two of them are load-bearing, and a reader deciding how much weight this argument can carry should read that section before the conclusions.

\GGPart[Conclusions]{Conclusions}{What has been established, under which declared fences; and the methodological reflection --- why $N_g=3$ required open-system thinking}

\section{Conclusion}
We have presented a three-leg derivation of the three-generation number
$N_g=3$ within Grand Geometric Theory. The proof has three legs, independent
in their directions of constraint (\S 8.2), that together force $N_g=3$
uniquely under the declared conditioning:
\begin{enumerate}[label=\arabic*.,leftmargin=2.2em]
\item the \MSMF{} lower bound $N_g\ge 3$ from Kobayashi--Maskawa CP-violation
combinatorics;
\item the \GDOS{} upper bound $N_g\le 3$ from the capacity theorem: the
quantum geometry that the boundary's adiabatic theorem forces on the family
moduli funds exactly $\pi$ of relaxation capacity per family, and the
triangle-number toll of the cascade outruns that income beyond three
families;
\item the \ANine{} codebook self-consistency check, which matches the
twelve charged-fermion masses to four-significant-figure precision with
$N_g=3$.
\end{enumerate}
The combination is significantly stronger than the historical
Kobayashi--Maskawa argument (which gives only the lower bound), than the
heterotic-string and magnetized-torus arguments (which give only conditional
upper bounds), and than the anthropic argument (which gives only heuristic
selection). \GGTheory{} replaces all of these with a single derivation whose
every fence is named.

The derivation is falsifiable: a fourth standard family, a precision
deviation in $m_\tau/m_\mu$, or a theoretical inconsistency between \BIsix{}
and \GDOS{} Morse-positivity would each break the derivation; and a disproof
of the Born-grade runtime premises would break leg (L2) and the Born rule
together. Conversely, the framework predicts the cascade lifetime hierarchy,
the $m_d/m_u = 2$ first-order ratio, and the strict positivity of the \GFF{}
alternating-sum (signature-type) index. These predictions can be tested over
the coming decade by precision flavor physics, by the LHC and HL-LHC, and
by cosmological neutrino surveys.

The deeper significance of the result is structural. It shows that what was
historically considered an inscrutable empirical accident -- the existence of
exactly three families -- becomes a derived integer once the underlying
geometry is correctly identified. The number 3 in $N_g=3$ is not selected; it
is forced by \BIsix{} closure, by the quantum geometry of the family moduli,
and by the empirical non-vanishing of the Jarlskog invariant. This is the
\GGTheory{} pattern: an empirical accident reduces, under sufficient
geometric structure, to a number-theoretic identity.

\subsection{Summary of the reduction: capacity from quantum geometry}
The architecture of leg (L2) contributes a structural understanding that can
be stated as one chain, each link carrying its grade:
\begin{itemize}[leftmargin=2.2em]
\item \emph{The bulk cannot send the class} (Lemma~\ref{lem:rigidity},
proved): conformal flatness of the canonical bulk plus Killing-form
uniqueness on the simple algebra $\mathfrak{so}(1,5)$ leave every functorial
descent identically zero.
\item \emph{The boundary cannot price the coupling}
(Proposition~\ref{prop:brel}, proved): the inflow coefficient is a relative
invariant of the bulk--boundary pair, invisible to every boundary-only
theorem.
\item \emph{Quantum mechanics supplies the geometry}
(Theorems~\ref{thm:AD1}--\ref{thm:AD3}, proved at Born-grade): adiabatic
transport of the family projector is Berry transport, and the quantum
geometric tensor equips the moduli with the Fubini--Study package ---
premises shared with the Constitutional Born Theorem, nothing new staked.
\item \emph{One clause remains a dictionary entry}
(Axiom~\ref{ax:coupling}): the capacity is a natural characteristic-class
pairing --- with non-vanishing derived from consistency
(Lemma~\ref{lem:kappa}), the unit pinned three ways
(Lemma~\ref{lem:unit}), the form robust across all viable candidates
(Theorem~\ref{thm:robust}), and integrality derived
(Theorem~\ref{thm:LINT}), its single residue named (R-INT), discharged
on the benchmark (\S\ref{app:H-rint}), and the clause's two scope
refinements closed at class level (T-INF-7, T-INF-8:
\S\ref{app:H-refine}).
\item \emph{The arithmetic then decides} (Theorem~\ref{thm:capacity}):
$\mathrm{Cap}(N_g) = \pi N_g$ against $\Stot(N_g) = \pi N_g(N_g-1)/2$ admits
$N_g \in \{1,3\}$, saturating exactly at $N_g=3$; with the CP lower bound,
$N_g = 3$ uniquely.
\end{itemize}
The parity route stands alongside as an independent consistency check
(Appendix~\ref{app:E}, Steps 1--2): $a^+ - a^- = \sigma(\CPN)$, non-zero
exactly at odd $N_g$, with the admissible value passing strictly. What the
new architecture removes is any reliance on it --- and any reliance on
$K = 35$ or $J = 7$, whose absence from the chain is what answers the
circularity challenge (\S 8.2).

\subsection{Pointer to the methodological reflection}
The technical proof of $N_g=3$ raises a methodological question that we take
seriously. Why has this number been un-closed for nearly a century since the
foundation of quantum mechanics? What did we have to assume that prior
closed-system frameworks did not? Section~14 addresses these questions
directly. The short answer is that closed-system QFT and GR can produce only
necessary conditions on integer multiplicities, not upper bounds; upper
bounds require open-system dynamics -- specifically, the bulk-to-boundary projection, finite instanton-action budgets, and the cascade closure of
attractors -- which we encode as \GDOS{}. We argue that this is the missing
link, and that the present derivation is in equal parts a technical proof of
three generations and a vindication of the open-system foundational shift.

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

%% 12  METHODOLOGICAL REFLECTION

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


\subsection{The four integers, and who chose them}
The quantitative content of Appendix~\ref{app:H} is the identity
$2 \times 2 / 4 = 1$, and its scientific content is that nobody chose
the numbers. The $4$ is the price of a quadratic refinement, fixed in
the mathematics literature two decades ago and not negotiable
\citep{HopkinsSinger2005}. The rule that the $4$ must be paid as
$2 \times 2$ --- never $1 \times 4$ --- is the Wu condition, a parity
law of six-dimensional topology. The first $2$ is $L - N$: the same
subtraction of ledger integers that sets the \BIsix{} trace ratio, now
found doing a third job as the family's toll charge into the one
channel the topology leaves open. The second $2$ is a mirror: the
closed model on which the global theorems are proved is the
reflection-double of the physical seam, and a mirror shows every unit
of flux twice. Four integers, four different owners --- a functor, a
parity law, a ledger, a reflection --- and one identity. The reader may
decide how comfortable they are calling that an accident.

\subsection{Why agreement is the right prize}
It is worth being precise about why agreement is the result one should
want here, rather than a derivation of one side from the other.

A derivation flows downhill: it makes one side primary and the other
its consequence, and it is only as strong as its premises. An
\emph{agreement} between two computations that share no working parts
is a different kind of evidence. The bulk face of the bridge uses no
quantum mechanics --- no wavefunctions, no operators, no Born rule; it
is topology and an anomaly ledger. The boundary face uses no bulk data
--- it is the adiabatic theorem of \S\ref{ssec:adiabatic} acting inside
quantum mechanics. Neither could see the other's answer. That they
meet, and meet exactly, after a factor audit with no adjustable
entries, is either a deep property of the architecture or an
extraordinary accident; the harnesses of \S\ref{app:H-verify} were
built to close the accidental routes, and the wrong factorizations of
the $4$ are not merely absent but \emph{forbidden}, by a parity
condition this program did not invent for the occasion.

This is also why the grander reading is declined explicitly. The
bridge does not found quantum mechanics; its boundary input
\emph{is} quantum mechanics, finished upstream by the cohort's own
Born theorem (\S\ref{sec:order}). What it founds is smaller and, for
this paper, more valuable: the coupling clause --- the one sentence of
the family-number derivation that had remained at the axiom level ---
is exhibited, on the benchmark, as the output of a computation. An
axiom became an answer.

\section{Methodological reflection: why $N_g=3$ required open-system
thinking}
This section is a methodological discussion of the proof
strategy. It is not a technical addendum to the proof itself: the proof of
$N_g=3$ is complete in Sections~5--8 (granted the named technical hypotheses of
Appendix~\ref{app:E}). The purpose of this section is to articulate the
methodological lesson that emerged from our extensive trial of closed-system
routes to $N_g=3$ within \GG{}/\BIsix{} alone. The lesson, in one sentence:
closed-system QFT and GR can produce necessary conditions on the family
number but cannot produce upper bounds, and this is precisely the structural
limitation that has held $N_g=3$ open for the century since the foundation of
quantum mechanics.

The reader interested only in the technical derivation can skip this section
without loss of mathematical content.

\subsection{The historical pattern of failure}
The three-generation puzzle has resisted closed-derivation proof for half a
century. Section~1.1 of the present paper surveyed the leading prior
approaches -- heterotic Calabi--Yau compactifications \citep{Candelas1985},
magnetized toroidal compactifications \citep{Blumenhagen2007},
$\mathbb{Z}_3$ orbifolds \citep{Ibanez1987}, anomaly cancellation \citep{DobrescuPoppitz2001}, and the Kobayashi--Maskawa CP argument \citep{KM1973} --
and noted that each gives only a partial constraint. But the failure pattern
is more structural than that survey suggested.

Each of these approaches stays rigorously within the framework of
closed-system quantum field theory and general relativity: a manifold, a
connection, a partition function, a path integral. Each can produce necessary
conditions on the family number $N_g$ -- Diophantine constraints from
integrality, anomaly-cancellation constraints from gauge-gravitational mixed
terms, index-theorem constraints from chiral asymmetry. None can produce a
sufficient condition -- an upper bound that excludes $N_g\ge 4$.

This is not a technical accident. It is a structural fact about closed-system
frameworks, and it explains why the three-generation puzzle has remained
unsolved despite half a century of sustained effort by some of the best minds
in theoretical physics.

\subsection{What closed-system methods can and cannot do}
In a closed-system framework, time evolution is unitary. The Hilbert space
carries a linear, norm-preserving, reversible representation of the
time-translation group. All physical states live with equal weight in a
single irreducible structure. There is no canonical notion of which states
are ``unstable'', ``decaying'', ``saddle-like'', or ``boundary-projected'' --
those notions require an external reference (a measurement, a coupling to an
environment, a boundary condition).

Three structural consequences follow.

\paragraph{Consequence 1: closed-system methods produce only necessary
admissibility conditions.}
Anomaly cancellation says ``this combination of fields cannot consistently
exist''. Integrality says ``this number must be an integer''. Index theorems
say ``this chiral asymmetry equals this geometric invariant''. All three are
bilateral: they can rule out values that fail the constraint, but they cannot
rule out values that satisfy the constraint with larger numbers. If $N_g=3$
satisfies anomaly cancellation, $N_g=6$ probably does too (with appropriate
adjustment of charges). If $N_g=3$ satisfies integrality of some Chern
character, $N_g=30$ certainly does. There is no closed-system invariant whose
value is ``$N_g$ but not $N_g+1$''.

\paragraph{Consequence 2: closed-system methods cannot produce upper bounds
on integer multiplicities.}
To get $N_g\le 3$, one needs some statement of the form ``for $N_g\ge 4$,
something runs away, decays, fails to stabilize, exceeds a budget''. Each of
these statements is intrinsically open-system:
\begin{itemize}[leftmargin=2.2em]
\item ``runs away'' requires a sense of stability flow;
\item ``decays'' requires irreversibility;
\item ``fails to stabilize'' requires a fixed-point structure with
distinguished basin of attraction;
\item ``exceeds a budget'' requires a finite resource bound.
\end{itemize}
None of these have closed-system formalizations. In a closed system, all
states evolve unitarily forever; nothing decays, nothing runs away, no budget
is finite.

\paragraph{Consequence 3: closed-system methods cannot derive integer
multiplicities from continuous data.}
The closed-system path integral on a fixed background is a continuous
functional whose stationary points are continuous solutions to PDEs. Integer
outputs (chiral asymmetries, instanton numbers, family numbers) come from the
topology of the background -- but the topology has to be put in by hand.
There is no closed-system mechanism that takes \BIsix{} closure and derives the
value $N_g=3$ rather than assuming it via a chosen background topology.

The historical practice in string compactifications is to scan candidate
Calabi--Yau threefolds for those with Euler characteristic $\chi = \pm 6$
(giving three families), or to choose a $\mathbb{Z}_3$ orbifold, or to fix
magnetic flux $m=3$ on each torus factor. In each case, the choice is input,
not output.

\subsection{The four closed-system attempts within \GG{}/\BIsix{}}
We attempted four routes to derive $N_g=3$ within closed-system \GG{}/\BIsix{}
alone, before turning to \GDOS{}. Each failed for the structural reasons just
articulated. We document them here as a record, both because they were
extensive and because their failure pattern is the most direct empirical
evidence for the methodological point.

\paragraph{Proposal 1: Chern--Simons multiplicity axiom.}
We attempted to show that \BIsix{} closure forces a unique 7-dimensional
Chern--Simons level $k$ that, on dimensional reduction to 4D, produces three
families. The plan was: (i) identify the Chern--Simons action whose
differential is $X_4$; (ii) impose integrality of the level; (iii) use the
level to count boundary chiral zero modes via an index theorem. Steps (i) and
(iii) are rigorous and standard. Step (ii) requires an axiom that the bulk
Chern--Simons level is exactly the minimum saturating \BIsix{}. This axiom is
plausible but not derivable from \BIsix{} alone in a closed-system formulation;
one can write down infinitely many integer levels consistent with \BIsix{}, and
the choice of the minimum is an additional principle (it is, indeed, \MSMF{},
but \MSMF{} is part of \GGTheory{}'s constitutional framework, not a closed-system
QFT principle).

\paragraph{Proposal 2: Magnetized torus.}
We attempted to derive $N_g=3$ from the index theorem on a 2-torus with
magnetic flux $m=3$ \citep{Blumenhagen2007}. The Atiyah--Singer index theorem
indeed gives net chirality $m$ on $T^2$ with abelian flux $m$. But $m$ itself
is a free integer not forced by \BIsix{} closure. To force $m=3$ rather than
$m=4$ or $m=7$, one needs an extra principle outside \BIsix{} -- typically
anomaly cancellation between the magnetic flux and a Wilson-line phase. This
anomaly-cancellation argument constrains $m$ modulo something but does not
fix $m$ uniquely.

\paragraph{Proposal 3: $\mathbb{Z}_3$ orbifold.}
We attempted to derive $N_g=3$ via a $\mathbb{Z}_3$ orbifold of the compact
direction \citep{Ibanez1987}. The three twisted sectors correctly produce
three families, but the $\mathbb{Z}_3$ choice is again input rather than
output. Why $\mathbb{Z}_3$ rather than $\mathbb{Z}_2$ or $\mathbb{Z}_4$? The
answer in our trial was: ``because the \OGNfive{} ledger has $N_g=3$''. But this is
circular -- we are trying to derive $N_g=3$, not to assume it.

\paragraph{Proposal 4: Partial \GG{} anomaly saturation.}
We attempted to show that \BIsix{} closure, taken together with a specific
saturation of the $\mathrm{SU}(5)$-$\mathrm{U}(1)_B$-$\mathrm{U}(1)_{Qi}$
mixed anomaly polynomial, forces a unique chiral spectrum with $N_g=3$. This
came closer than the others: the saturation conditions on $X_4$ produce a
tight set of Diophantine constraints, and the \OGNfive{} ledger $(1,3,5;7,35)$ is
the minimum solution. But ``minimum solution'' is again \MSMF{}, not pure \BIsix{}.
Without \MSMF{}, \BIsix{} admits infinitely many integer ledgers
$(M, kN, kL; J, k^2 K)$ for any integer $k\ge 1$.

\medskip
In all four attempts, the final step required an additional principle beyond
\BIsix{} itself. In Proposals 1, 2, and 4, the additional principle was \MSMF{}-like
(selecting the minimum integer solution). In Proposal 3, the additional
principle was a chosen orbifold group. Each of these additional principles is
itself an external input -- precisely the ``something more than closed-system
QFT'' that we eventually identified as the open-system framework.

\subsection{Why Proposal 4 (within \GDOS{}) succeeded where Proposals 1--3
failed}
After concluding that the four closed-system proposals could not close the
upper bound on $N_g$, we turned to open-system thinking. The decisive move was
to ask: what is $N_g$ as a structural quantity in the boundary theory, not in
the bulk?

The answer is illuminating. $N_g$ is the cascade depth of the boundary
attractor structure -- specifically, the number of unstable critical points
of the \GFF{} Morse functional $\fGFF$ on the family-projector moduli space
$\CPN$. This identification has four immediate consequences that make the
upper bound natural:
\begin{itemize}[leftmargin=2.2em]
\item The family-projector moduli space $\CPN$ exists only because we are
projecting from the 6D bulk to the 4D boundary. In a pure closed-system bulk
QFT, there is no preferred ``family-projector moduli''; families are just
labels on the bulk Lagrangian.
\item The \GFF{} Morse functional is a free-energy functional that includes both
energetic (GEP) and entropic (CEP) contributions. The entropic part requires
an open-system formulation: the entropy of a configurational class of
projectors, which has no closed-system meaning.
\item The cascade of critical points is a sequence of unstable saddles and
one stable attractor, with the cascade closure dictated by irreversible
gradient flow. The instanton budget is a finite resource available for
boundary descent -- a concept that has no closed-system analogue.
\item The capacity $\mathrm{Cap}(N_g) = \pi N_g$ comes from the quantum geometry
of the boundary moduli --- Berry curvature and the Pontryagin pairing,
activated by the runtime's own adiabatic theorem --- with its integral
quantization a Hopkins--Singer differential-cohomology statement
\citep{HopkinsSinger2005}. Every ingredient requires an open-system /
boundary-attractor framework to even be formulated.
\end{itemize}
In other words, Proposal 4 succeeds because it identifies $N_g$ not as the
value of a closed-system invariant but as the cascade depth of a
boundary-attractor structure. The latter has an upper bound forced by the
\BIsix{} anomaly inflow budget; the former does not.

This is the structural reason for the success. It is not a clever choice of
formalism; it is a recognition that family number is, in its nature, a
quantity of the boundary projection rather than of the bulk geometry.

\subsection{The biggest missing link since quantum mechanics}
The success of Proposal 4 invites a deeper observation. The open-system
character of physical observation -- the fact that all measurements are
irreversible projections of bulk dynamics onto a finite-dimensional boundary
-- has been understood at a phenomenological level since the founding of
quantum mechanics. Yet it has never been elevated to the foundational level.
This, we argue, is the missing link that has held the three-generation puzzle
(and many related structural questions in foundational physics) open for the
century since 1925.

Quantum mechanics was established in 1925--1930 (Heisenberg, Schr\"odinger,
Dirac, von Neumann) as a closed-system framework: a Hilbert space, a unitary
evolution operator, a state vector \citep{vonNeumann1932}. The Born rule \citep{Born1926} introduced probabilities, but only at the measurement level -- the unitary
evolution itself is closed and reversible. Quantum field theory
(Heisenberg--Pauli, Tomonaga, Schwinger, Feynman, Dyson) extended the
closed-system framework to relativistic field operators on a Minkowski
background. General relativity (Einstein 1915) is a closed-system theory of
the gravitational field on a smooth Lorentzian manifold.

The open-system aspects of physics -- measurement collapse, decoherence,
irreversibility, dissipation, branching, observer-relative state reduction --
were noted from the start (von Neumann's 1932 measurement chain,
Heisenberg's ``cuts'') but were never elevated to the foundational level
needed to derive structural integers like $N_g=3$.

A long sequence of partial advances added pieces of what is needed.

\paragraph{Lindblad / GKLS dynamics (1976).}
\citet{Lindblad1976} and \citet{GKS1976} independently derived the most
general completely positive trace-preserving (CPTP) master equation for
open-system quantum dynamics. This gave the framework for irreversible
quantum evolution at the level of density matrices, and is the technical
backbone of all modern open-system quantum theory.

\paragraph{Decoherence theory (Zeh 1970, Zurek 1981).}
\citet{Zeh1970} and \citet{Zurek1981} explained how open-system dynamics
produces classicality from quantum entanglement: when a system couples to an
environment with many degrees of freedom, off-diagonal elements of the
reduced density matrix decay rapidly, leaving an effective classical mixture.
This is the mechanism by which closed-system quantum mechanics produces
apparently classical outcomes -- but the mechanism itself is intrinsically
open-system.

\paragraph{Holographic principle ('t Hooft 1993, Susskind 1995).}
\citet{tHooft1993} and \citet{Susskind1995} suggested that the fundamental
degrees of freedom of a $(d{+}1)$-dimensional bulk theory project onto a
$d$-dimensional boundary, with the bulk effectively encoded by the boundary.
This was a profound suggestion that the physical content of a
higher-dimensional theory is, in some operational sense, available only on a
lower-dimensional boundary -- precisely the open-system / projection
structure.

\paragraph{AdS/CFT correspondence (Maldacena 1997).}
\citet{Maldacena1997} gave an explicit example of holographic
bulk-to-boundary descent: type-IIB string theory on
$\mathrm{AdS}_5\times S^5$ is equivalent to $\mathcal{N}=4$ supersymmetric
$\mathrm{SU}(N)$ Yang--Mills theory on the four-dimensional boundary. This
was the first concrete realization of the holographic principle and provided
a precise dictionary between bulk dynamics and boundary observables.

\paragraph{Differential cohomology (Hopkins--Singer 2005).}
\citet{HopkinsSinger2005} gave the rigorous formalism for anomaly inflow from
a higher-dimensional bulk to a lower-dimensional boundary. Differential
cohomology refines ordinary cohomology by incorporating both topological data
(cohomology classes) and differential-form data (curvature representatives),
giving the precise tool for tracking how anomaly polynomials descend
holographically. This is exactly the mathematical machinery used in our
Appendix~\ref{app:E}.

\medskip
Each of these is a partial advance toward an open-system foundation. None of
them has been elevated to the foundational role of an axiom system. They have
been treated as tools -- powerful tools, indispensable tools -- but not as
principles. Quantum mechanics in 1925 made the state vector and the unitary
evolution operator into foundational principles; nobody has yet made the
boundary projector and the cascade of attractors into foundational principles
in an analogous way.

\GDOS{}, as developed in \citet{ArisakaA5GDOS} and the \GGTheory{} program more
broadly, is the explicit assembly of these partial advances into a
foundational open-system framework that elevates the bulk-to-boundary
projection, the CPTP measurement structure, and the bulk-to-boundary anomaly inflow to axiomatic status. The four \GGTheory{} axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) include \OGGEP{} -- the Open-system Geometry-Gauge Equivalence Principle -- as one of the four foundational axioms, alongside local causality, gauge-geometry equivalence, and maximum-symmetry-minimum-freedom \citep{ArisakaA1Constitution}.

This is the methodological move that distinguishes \GGTheory{} from the prior
generation of closed-system attempts. And this is why \GGTheory{} can derive
$N_g=3$ where the closed-system frameworks of the past century could only
produce partial constraints.

\subsection{The four moves of the closed $\to$ open transition}
Concretely, the closed $\to$ open transition involves four moves.
Articulating these makes precise what has changed and what has not.

\paragraph{Move 1: Replace unitary $U(t)$ with the boundary projector
$\Pi$.}
Instead of treating dynamics as unitary evolution on a Hilbert space, treat
it as a CPTP map from a 6D bulk to a 4D boundary. The boundary projector
$\Pi : \Mss \to \mathcal{M}_4$ is the fundamental object; the bulk Hilbert
space is a resource from which boundary observables are extracted, not a
self-contained universe. Unitary evolution is a special case (when the bulk
and boundary coincide and the projector is the identity), but it is not the
general case.

\paragraph{Move 2: Replace the state vector with the boundary attractor
(\GA{}).}
Boundary observables are not arbitrary states in some Hilbert space; they are
attractors of the \GFF{} Morse functional on the boundary moduli. The discrete
catalog of \GA{}s is what we mean by ``physical observables'' -- fermion
masses, gauge couplings, vacuum expectation values are all \GA{}s. The
continuous Hilbert space is a bulk concept; the discrete attractor catalog
is a boundary concept.

\paragraph{Move 3: Replace the closed-system Hamiltonian with the \GFF{}.}
The closed-system Hamiltonian $H$ that drives unitary evolution is replaced
by the Geometric Free Functional $\FGFF = \FGEP + \FCEP$, an
energetic-plus-entropic free energy on the boundary moduli. Critical points
of $\FGFF$ are \GA{}s; the cascade structure of saddles and minima is the
dynamics. The free-energy structure is intrinsically irreversible: gradient
flow descends in $\FGFF$ toward attractors and never reverses.

\paragraph{Move 4: Replace closed-system anomaly cancellation with
bulk-to-boundary anomaly inflow.}
The closed-system Bianchi identity $dH_3 = 0$ is replaced by \BIsix{},
$dH_3 = X_4$, where $X_4$ is the anomaly polynomial that descends to the boundary moduli. The Hopkins--Singer differential
cohomology framework \citep{HopkinsSinger2005} makes this descent rigorous.
\BIsix{} closure on $\Mss$ does not say ``no anomaly''; it says ``the bulk
anomaly $X_4$ is exactly the bulk-to-boundary inflow into the boundary observables''.

\medskip
Each move converts a closed-system structure into an open-system one.
Together they constitute the \GDOS{} framework. Note that none of these moves
replaces the closed-system structures on the bulk side -- the bulk is still a
closed-system manifold with unitary $\mathrm{Spin}(1,5)$ frame bundle
and full \BIsix{} closure. What changes is the relationship between bulk and
boundary: that relationship is now explicitly open-system, irreversible, and
CPTP.

\subsection{Why \GG{}/\BIsix{} alone wasn't enough: the structural diagnosis}
The implication for the present paper is direct. Sections~5, 6, and 7
establish three legs of the proof of $N_g=3$. Of these:
\begin{itemize}[leftmargin=2.2em]
\item \textbf{Leg (L1) (\MSMF{} lower bound)} is closed-system in flavor: it
uses unitary-rephasing combinatorics on the CKM matrix. But it is empirically
conditioned on $\JCKM \ne 0$, a measurement-theoretic statement. Without the
open-system measurement perspective, even (L1) is incomplete. The Born-rule
projection that makes $\JCKM$ measurable is itself open-system.
\item \textbf{Leg (L2) (\GDOS{} Morse upper bound)} is intrinsically
open-system: it uses the \GFF{} Morse cascade on the family-projector moduli.
This is the only leg that can produce an upper bound on $N_g$, and it can do so
only because it uses open-system tools.
\item \textbf{Leg (L3) (codebook self-consistency)} is constructive -- it
uses the explicit \ANine{} codebook of fermion Yukawas, which is itself
derived from open-system / boundary-projector data. The codebook formulas
$y_\tau = NJ/2^{J+N_g+1}$, etc.\ are first-order Yukawa couplings of boundary
fermions, with structural integers from the \OGNfive{} ledger appearing as
numerators and exponents.
\end{itemize}
Each of the three legs uses the open-system / boundary-projector structure.
Without \GDOS{}, none of the three legs would close.

This is the methodological diagnosis. The ``open-system perspective'' is not
a stylistic choice or a clever computational shortcut; it is a structural
necessity for closing the derivation of $N_g=3$. Any closed-system framework,
no matter how elaborate, will produce only necessary conditions on $N_g$, never
the upper bound.

\subsection{Why this is the same diagnosis for many other unsolved structural
problems}
The diagnosis is not specific to the three-generation puzzle. It applies to
several other long-standing structural questions in foundational physics
where closed-system methods have produced only partial constraints:
\begin{itemize}[leftmargin=2.2em]
\item \textbf{The hierarchy problem} (why $E_{\mathrm{EW}}\ll
E_{\mathrm{Pl}}$). Closed-system methods give consistency requirements
(technical naturalness, anomaly cancellation) but cannot derive the hierarchy
itself. \GGTheory{} derives the hierarchy from the bulk and the \OGNfive{}
ledger \citep{ArisakaA2GG6}, with the key step being the bulk-to-boundary projection $E_{\mathrm{Pl}}/2e^{35} \simeq M_{\mathrm{lock}} \simeq 3.85$~TeV that
locates the IR scale. This is open-system reasoning.
\item \textbf{The Higgs mass} ($m_h^2$ as a hierarchy with respect to
$E_{\mathrm{GUT}}$). Closed-system methods predict quadratic divergences and
require fine-tuning. \GGTheory{} predicts $m_h^2$ as a boundary-projector exit
from the Morse cascade with a definite \OGNfive{} fraction \citep{ArisakaA9SM},
again an open-system computation.
\item \textbf{Proton stability} (why $\tau_p \gg \tau_{\mathrm{universe}}$).
Closed-system methods constrain the dimension-6 baryon-violating operators
and require the GUT scale to be high. \GGTheory{} derives the
proton-instability suppression from the bulk-to-boundary projection of gauge-baryon $\mathrm{U}(1)_B$ inflow to the boundary
(\citealp{ArisakaA3BI6}; \citealp{ArisakaA9SM}), an open-system computation.
\item \textbf{Dark matter} (the absence of a closed-system mechanism for dark
matter that does not require ad hoc particle additions). \GGTheory{} predicts
$\sim\mu$eV axion-like dark matter as a boundary attractor of the bulk
$\mathrm{U}(1)_{Qi}$ sector \citep{ArisakaC1Kaxion}, again open-system.
\end{itemize}
In each case, the structural integer or scale that closed-system methods
cannot derive is derivable in \GGTheory{} because of the open-system
boundary-projection structure. This is consistent with the methodological
diagnosis: closed-system methods cannot derive sufficient conditions on
integer multiplicities or hierarchies; open-system methods can.

\subsection{The final move: from prove to decide}
One methodological move behind this paper
deserves its own record, because it is what unlocked the architecture of
leg (L2). For most of this
program the budget question was posed as \emph{prove}: derive the cap
$3\pi$ from the axioms. Two adversarial campaigns returned nulls, and the
question was then re-posed as \emph{decide}: for each needed statement,
either prove it from the axioms, or prove an obstruction showing no such
proof exists, and in the latter case name the minimal addition. The decision
method produced, in short order, the two obstruction theorems
(Appendix~\ref{app:G}), the recognition that the missing structure was the
quantum geometry of the moduli --- already forced at Born-grade by the
adiabatic theorem --- and the collapse of five hypotheses into one
integrality statement, itself then derived with a single topological
residue (Theorem~\ref{thm:LINT}). Obstruction results are not failures; they are the
honest form of progress on questions that resist proof, because they
terminate searches and localize axioms. A program that advertises
falsifiability in physics should practice it in mathematics: state what
would refute a derivation route, and when it is refuted, say so and move
the frontier.

\subsection{The bottom line}
To our knowledge, $N_g=3$ is the first integer in physics that has been
closed-derivation forced -- i.e., proved equal to a specific value rather
than constrained to a range -- with explicit lower and upper bounds inside a single declared axiom system, with the conditioning stated at every level. The Kobayashi--Maskawa CP argument has been known for over
fifty years and gives only $N_g\ge 3$. The heterotic-CY and magnetized-torus
arguments have been known for forty years and give only conditional upper
bounds (conditional on extra geometric inputs that are themselves not
derived). The anthropic argument is heuristic.

\GGTheory{} closes the derivation. The reason it can close where the
closed-system frameworks of the past century could not is that it elevates
the open-system / boundary-projector / bulk-to-boundary-anomaly-inflow perspective
to foundational status.

The strict \GDOS{} Morse-positivity reduction of Appendix~\ref{app:E} makes this
concrete: the family number is forced to be odd by the mod-4 grading of
differential cohomology in 6D -- a parity statement that cannot even be
formulated without the bulk-to-boundary descent of differential cohomology.
Pure closed-system QFT does not have a notion of ``differential cohomology of
the bulk descended to the boundary''. It has only ``cohomology of the bulk''.
The descent is intrinsically open-system.

The methodological achievement is therefore prior to the technical result.
The technical result is ``$N_g=3$''; the methodological achievement is ``the
open-system / boundary-projector / bulk-to-boundary-anomaly-inflow framework can
do what closed-system QFT/GR cannot''. In the sense that this is what was
missing for a century, the present paper is in equal parts a derivation of
three generations and a vindication of the open-system foundational shift.

\subsection{The two fibers behind the method}
The methodological lesson of this section has a geometric root,
made exact by the cohort's recent campaign
(\textsf{GG-W1-Symmetry}; \citealp{ArisakaP5CPT};
\citealp{ArisakaA5GDOS}): \emph{holonomy requires closed loops, and
an interval has none.} The two non-spatial directions of the \GG{}
bulk are fibers of opposite topological kind --- the compact phase
circle $S^1_\theta$ carries all of the bulk's holonomy, quantized
and protected, its transport a group; the scale interval $I_r$
carries none, its transport composing only forward, a semigroup.
Sorted on this anatomy, the two halves of the family-number problem
live on different fibers. The \emph{count and its protection} are
circle-side: $N_g$ is holonomy-class data --- an integer written on
the fiber that remembers, exact at every scale, which is why the
number neither runs nor drifts. The \emph{decision} --- the upper
bound --- is interval-side: the capacity theorem prices the family
ladder against the relaxation budget of a one-way runtime, and a
budget is a semigroup concept --- it exists only on the fiber that
cannot revisit.

This is why the historical record has the shape documented above.
Closed-system methods are, in this anatomy, \emph{circle-only}
methods: they see the quantized, the protected, and the necessary
--- which is why the lower bound $N_g \ge 3$ has been available
since 1973 --- and they are structurally blind to the interval,
where the budget lives --- which is why no upper bound could even
be formulated for fifty years. The open-system shift of this
section is, geometrically, the admission of the second fiber into
the foundations; and the mod-4 parity statement of
Appendix~\ref{app:E}, formulable only in bulk-to-boundary descent,
is the two fibers cooperating in a single theorem ---
differential-cohomology data of the circle side, descended along
the interval. The one-sentence summary of the method: \emph{the
family number is written on the fiber that remembers and priced by
the fiber that forgets --- and a method that reads only one fiber
can bound it from only one side.}

\subsection{Closing remark: a century-old missing link}
The history of theoretical physics is, in part, a history of recognizing that
what looked like an arbitrary phenomenological choice was actually forced by
a deeper principle that had not yet been articulated. Newton showed that
planetary motion was not arbitrary but forced by inverse-square gravity.
Maxwell showed that electromagnetic phenomena were not separate but forced by
his four equations. Einstein showed that gravity was not a force but a
manifestation of curved spacetime. Heisenberg, Schr\"odinger, and Dirac
showed that atomic spectra were not arbitrary but forced by quantum
mechanics.

The present paper, in its small way, makes the analogous claim for the family
number $N_g=3$. The fact of three generations was a phenomenological input to
the Standard Model for fifty years. The present derivation shows that it was
not arbitrary: it was forced by the geometry of the 6D bulk, the
integrality of \BIsix{}, and the cascade structure of the bulk-to-boundary projection.

The principle that had not been articulated -- the missing link -- is the
elevation of the bulk-to-boundary projection, the irreversible CPTP structure
of measurement, and the cascade of attractors to foundational status. This is
the open-system shift. It has been adumbrated since the early days of quantum
mechanics, but never made foundational. The present paper is one piece of
evidence that, when it is made foundational, the structural integers of the
Standard Model can be derived rather than postulated.

We close this section with a summary table.

\begin{table}[H]
\centering
\caption[Closed-system and open-system stances, side by side.]{Closed-system vs.\ open-system foundational stances on the
three-generation puzzle. Row by row, the same question answered twice --- once under a closed unitary description and once under the open-system projection this cohort uses. The table is here because most of the disagreement about the three-generation problem is a disagreement about which column a reader is standing in, and that is easier to see side by side than to argue in prose.}
\label{tab:closed-open}
\small
\begin{tabular}{@{}p{4.4cm}p{4.6cm}p{5.2cm}@{}}
\toprule
Aspect & Closed-system framework & Open-system / \GDOS{} framework \\
\midrule
Time evolution & Unitary $U(t)$ & CPTP $\Pi$ (boundary projection) \\
State concept & State vector $|\psi\rangle$ & Attractor catalog (\GA{}s) \\
Dynamical functional & Hamiltonian $H$ & \GFF{} $\FGFF = \FGEP + \FCEP$ \\
Anomaly principle & Closed Bianchi $dH_3 = 0$ & \BIsix{} inflow $dH_3 = X_4$ \\
Family number $N_g$ & Background topology input & Cascade depth on $\CPN$ \\
Lower bound on $N_g$ & KM CP $\Rightarrow N_g\ge 3$ & Same \\
Upper bound on $N_g$ & Not available & \GFF{} Morse cascade
$\Rightarrow N_g\le 3$ \\
Quantitative codebook & Tunable & \ANine{} codebook, 4 sig.\ figs. \\
Status of $N_g=3$ & Phenomenological input & Derived integer \\
\bottomrule
\end{tabular}
\end{table}

The right-hand column is the \GGTheory{} perspective. Its operational content
is exactly the open-system shift articulated in this section. Its
derivational power is the technical proof of the present paper.

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

%% APPENDICES

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

%% ---- PART VI ----

\section[Part~V in plain English]{Part~V in plain English: what has been established, and under which fences}
The claim is that $N_g = 3$ is forced --- not permitted, not preferred, but forced --- by three arguments that constrain from independent directions, under conditions this paper has declared rather than assumed quietly.

That last clause is the whole of the honesty in the result. Two of the three legs carry hypotheses that are not derived here: one is imported from a companion at theorem grade, and one is declined outright, with its intended content supplied at a weaker grade and the weakness recorded. A reader who wants to know how much the derivation is worth should read the dissolution ledger before this conclusion, not after it.

What would refute the claim is stated too, and it is not hypothetical: a fourth light family would end the argument, and so would a measurement pushing the CP-violating invariant to zero. Both are the kind of result an experiment could deliver. The Part closes on why this particular integer resisted for fifty years, which is a question about method rather than about physics, and the answer offered is that a closed-system framework could not have reached it.

\GGAppendixPart{Technical Appendices}{The detailed calculations --- the CKM phase count, Morse theory on $\CPN$, the two viable capacity formulations, the numerical charged-lepton comparison, the parity route with the record of the declined hypotheses, the adiabatic theorems of the boundary runtime, and the obstruction theorems with the capacity classification}

\section{Detailed counting of CKM phases}
\label{app:A}
Here we record the standard counting of physical CP phases in $\VCKM$ for
completeness.

A general $N_g\times N_g$ unitary matrix has $N_g^2$ real parameters. Of these,
$N_g(N_g-1)/2$ are mixing angles, and $N_g(N_g+1)/2$ are phases. The $N_g$ up-type and
$N_g$ down-type quarks can be rephased independently except for an overall
phase, contributing $2N_g-1$ unphysical phases. The number of physical CP
phases is therefore
\begin{equation}
n_{\mathrm{phase}}(N_g) = \frac{N_g(N_g+1)}{2} - (2N_g-1) = \frac{(N_g-1)(N_g-2)}{2}.
\label{eq:nphase-appendix}
\end{equation}
For $N_g=1$: $n_{\mathrm{phase}} = 0$ (no CP phase, no mixing). For $N_g=2$:
$n_{\mathrm{phase}} = 0$ (one mixing angle, no CP phase). For $N_g=3$:
$n_{\mathrm{phase}} = 1$ (three mixing angles, one CP phase = KM phase). For
$N_g=4$: $n_{\mathrm{phase}} = 3$ (six mixing angles, three CP phases).

The Jarlskog invariant exists if and only if $n_{\mathrm{phase}}\ge 1$, i.e.\
$N_g\ge 3$.

\section{Morse theory on $\CPN$ -- background}
\label{app:B}
We recall the basic facts about Morse theory on $\CPN$ for completeness;
references are \citet{Milnor1963}, \citet{Helgason1979}, and
\citet{SchwarzM1993}.

\subsection{The height function as a perfect Morse function}
Let $[z_1 : \ldots : z_N]$ be homogeneous coordinates on $\CPN$. The function
\begin{equation*}
f([z]) = \frac{\sum_a \lambda_a |z_a|^2}{\sum_a |z_a|^2},
\end{equation*}
with $\lambda_1 < \cdots < \lambda_N$, is a smooth real-valued function on
$\CPN$. Its critical points are the $N_g$ homogeneous lines
$[0 : \ldots : 0 : 1 : 0 : \ldots : 0]$, with the 1 in position $a$. The
Morse index of the $a$-th critical point is $2(a-1)$. All critical points are
non-degenerate; $f$ is a perfect Morse function, and the Morse polynomial
(\ref{eq:morse-poly}) agrees with the Poincar\'e polynomial of $\CPN$.

\subsection{Gradient flow and the Bogomolnyi bound}
The gradient flow of $f$ with respect to the Fubini--Study metric solves
\begin{equation*}
\dot{z}_a = -(\lambda_a - \langle\lambda\rangle)\, z_a,
\end{equation*}
which is a one-parameter linear ODE. The flow descends from the
index-$2(N_g-1)$ critical point through index-$2(N_g-2)$, \ldots, down to index
0. Each step satisfies a Bogomolnyi-type bound which is saturated by the
gradient trajectory itself, and the saturating action is $\pi k$ for the
descent from index $2k$ to index $2(k-1)$ \citep{Witten1982}.

\subsection{Transversality}
Generic gradient flows are transverse \citep{SchwarzM1993}: the unstable
manifold $W^u(c_+)$ of any critical point $c_+$ intersects the stable
manifold $W^s(c_-)$ of any other $c_-$ generically, with intersection
$W^u(c_+)\cap W^s(c_-)$ of dimension $\mathrm{ind}(c_+) - \mathrm{ind}(c_-)$.
This is what makes the cascade chain (\ref{eq:chain}) the dominant gradient
flow on $\CPN$.

\section{The capacity of the family stratum: the two viable formulations
and their completion}
\label{app:C}
This appendix records the two formulations in which the capacity value
$3\pi$ is quoted across the cohort's cross-reference ledgers, shows that
they are exactly the two survivors of the natural classification
(Theorem~\ref{thm:robust}), and states what completes them. Nothing here is
load-bearing --- the derivation lives in \S\ref{ssec:capacity} --- but the
agreement of the two independent formulations is itself evidence of the
architecture's robustness, and the arithmetic warning below is worth any
reader's attention.

\subsection*{Formulation A (constant form): $3\pi$, read off the seam}
The cohort's ledgers (\citealp{ArisakaA3BI6}; \citealp{ArisakaA6Gi4})
quote the budget as a constant,
$\SmaxBI = \lfloor J/2\rfloor\,\pi = 3\pi$ with $J = 7$ the \OGNfive{} seam
integer. Two heuristic routes motivate it. \emph{Route 1 (cell
integral):} the integral of $X_4$ over a fundamental cell, in the canonical
\OGNfive{} normalization, is $2\pi N_g/L = 2\pi\cdot 3/5$; multiplying by the
orbifold half-length $L/2 = 5/2$ gives $3\pi$. (The multiplier is $L/2$,
\emph{not} the seam factor $J/2 = 7/2$, which would give $21\pi/5$; the two
are easily transposed and the distinction is worth restating.) \emph{Route 2
(seam count):} reading $\lfloor J/2\rfloor$ directly off the seam matches
Route 1 numerically. The obvious referee challenge --- why
$\lfloor J/2\rfloor$ rather than $J$, $J-1$, or $\lfloor J/3\rfloor$? ---
is unanswerable within Formulation A alone, and is answered by the
classification below.

In the classification of Theorem~\ref{thm:robust}, Formulation A is the
$h^2$ (constant-class) capacity: viable, decisive at its
\MSMF{}-minimal lattice unit $\lambda = 3\pi$, and agreeing with the
extensive form on every physical statement. Its residual blemish is a
range of decisive units ($3\pi$ up to $11\pi/2$) --- a discrete knob that
\MSMF{}-minimality must close --- which is why the extensive form is
preferred.

\subsection*{Formulation B (extensive form): $\pi N_g$, completed as the
capacity theorem}
The cell-integral route, read structurally rather than numerically, says
the budget scales with $N_g$: the cell flux of the descended class is
proportional to $N_g$ by the Pontryagin scaling $p_1(\CPN) = N_g h^2$
(Appendix~E, Step 2, rigorous), and the conversion is $N_g$-free --- note that
$L$ cancels entirely in $\bigl(2\pi N_g/L\bigr)\cdot\bigl(L/2\bigr) = \pi N_g$,
and that $L = 5$ is in any case available $N_g$-freely from single-tensor
consistency (\citealp{ArisakaA3BI6}, T-BI6-9.0), so no circularity through
the trace ratio survives. Formulation B is the $p_1$ (extensive-class)
capacity of Theorem~\ref{thm:capacity}: its decisive unit is \emph{unique}
on the lattice ($\lambda = \pi$; Lemma~\ref{lem:unit}), it saturates exactly
at $N_g=3$, and it carries the reachability reading of
\S\ref{ssec:reachability} --- $\pi$ of relaxation capacity per family.

\subsection*{What completes them}
Three ingredients turn the conjectural pair into the theorem pair:
\emph{(i)} the obstruction results (Appendix~\ref{app:G}), which proved
that no derivation of the cap from the prior axioms exists --- terminating
the search that Routes 1--2 above had been gesturing at; \emph{(ii)} the
adiabatic theorems (Appendix~\ref{app:F}), which supply the descended
geometry at Born-grade; and \emph{(iii)} the classification
(Lemma~\ref{lem:unit} and Theorem~\ref{thm:robust}), which shows that
among all natural capacities exactly the two ledger-quoted formulations
survive, and that they agree. The referee challenge dissolves: the
functional form was never about $J$ at all, and the coincidence
$\lfloor 7/2\rfloor\,\pi = 3\pi$ is a numerological accident of $J = 7$
that the architecture no longer touches.

\subsection*{A self-consistency remark}
With the step actions $S_k = \pi k$ of (\ref{eq:step-action}), exact
saturation at $N_g=3$ ($\Stot(3) = \mathrm{Cap}(3) = 3\pi$) is now an identity
between two independently computed quantities rather than an attractive
reading --- the strongest self-consistency statement in the paper, and the
\GGTheory{} analogue of criticality. The codebook of Section~7 is
consistent with it.

\section{Numerical comparison of charged-lepton masses}
\label{app:D}
We tabulate the charged-lepton masses for $N_g = 2, 3, 4$ in the \ANine{}
codebook.

\begin{table}[H]
\centering
\caption[Charged-lepton masses as a function of $N_g$.]{Charged-lepton masses as a function of $N_g$ in the \ANine{}
first-order codebook (before QD completion). The charged-lepton spectrum recomputed at each candidate generation count, before the quark-descent completion is applied. The two ratio columns are the ones to read: they move sharply with $N_g$ and only the $N_g = 3$ row lands near the measured values, which is a consistency check on the count rather than a derivation of it, and is stated at that strength.}
\label{tab:lepton-N}
\small
\begin{tabular}{@{}ccccc@{}}
\toprule
$N_g$ & $y_\tau^{(1)}$ & $m_\tau^{(1)}$ (GeV) & $m_\tau/m_\mu$ &
$m_\mu/m_e$ \\
\midrule
2 & $14/1024 = 0.01367$ & 2.381 & 22.4 & 312.5 \\
3 & $21/2048 = 0.01025$ & 1.786 & 16.8 & 208.33 \\
4 & $28/4096 = 0.00684$ & 1.191 & 11.2 & 156.25 \\
Exp. & --- & 1.77693(9) & 16.817 & 206.768 \\
\bottomrule
\end{tabular}
\end{table}

The agreement of $N_g=3$ with experiment is at the four-significant-figure
level for the mass ratios; $N_g=2$ and $N_g=4$ are off by $\sim 33\%$, completely
incompatible with observation.

\section{A constructive reduction of strict \GDOS{} Morse-positivity}
\label{app:E}
This appendix carries the parity route: an independent consistency check on
leg (L2), together with the fate of the two technical hypotheses that the
natural axiomatization of that route would posit. Steps 1--2 (the signature identity and Hirzebruch's theorem) are
fully rigorous and unchanged; they are consumed by the capacity theorem
(the Pontryagin scaling) and by the consistency check of
\S\ref{ssec:parity-check}. Step 3 records how the descended class is now
supplied --- by the quantum geometry of Theorems~\ref{thm:AD1}--\ref{thm:AD3}
rather than by the withdrawn hypothesis $H_1$, whose would-be derivation is
closed in the negative by Lemma~\ref{lem:rigidity} (proof in
Appendix~\ref{app:G}); the no-go computation of \S\ref{ssec:E-naive}, which
first exposed the vanishing, is retained in full. Step 4 states the
integrality theorem L-INT, the honest successor of the withdrawn
hypothesis $H_2$, now derived in Appendix~\ref{app:G}. Step 5 assembles the conditional parity theorem.

\subsection{Statement of the result to be proved}
\begin{theorem}[The parity route (T-N3-8; conditional consistency
theorem)]
\label{thm:E1}
Grant a non-zero descended inflow class on the family-projector moduli
$\mathcal{P}_N = \CPN$ (supplied at Born-grade by
Theorems~\ref{thm:AD1}--\ref{thm:AD3} with the coupling clause, or granted
directly by a reader who wishes to test the route in isolation). Then the
strict alternating-sum inequality
\begin{equation}
a^+(N_g) - a^-(N_g) > 0
\label{eq:E-strict}
\end{equation}
holds if and only if $N_g$ is odd. In particular the parity route
independently excludes even $N_g$, consistently with the capacity exclusion
of Theorem~\ref{thm:capacity}; and at the admissible value the strict
inequality holds unconditionally, $a^+(3) - a^-(3) = \sigma(\CP^2) = 1$.
\end{theorem}

The remainder of this appendix proves the theorem and records the status of
its inputs.

\subsection{Step 1: the alternating sum equals the topological signature}
This step is fully rigorous. We claim
\begin{equation}
a^+(N_g) - a^-(N_g) = \sigma(\CPN),
\label{eq:E-sig}
\end{equation}
where $\sigma(\CPN)$ is the topological signature of $\CPN$.

\begin{proof}[Proof of (\ref{eq:E-sig})]
By Lemma~\ref{lem:morse-poly}, $\fGFF$ on $\CPN$ has $N_g$ non-degenerate
critical points with even Morse indices $\mathrm{ind}_a = 2(a-1)$ for
$a = 1,\ldots,N_g$. By the definitions of $a^\pm(N_g)$ in
Postulate~\ref{post:morse-pos},
\begin{align}
a^+(N_g) - a^-(N_g) &= \#\{a : \mathrm{ind}_a \equiv 0 \pmod 4\}
 - \#\{a : \mathrm{ind}_a \equiv 2 \pmod 4\}
\label{eq:E-count}\\
&= \sum_{a=1}^{N_g} (-1)^{(a-1)} = \sum_{k=0}^{N_g-1} (-1)^{k}.
\label{eq:E-sum}
\end{align}
This sum equals 1 if $N_g$ is odd and 0 if $N_g$ is even.

We compare with the known signature. By the standard topology of complex
projective spaces:
\begin{itemize}[leftmargin=2.2em]
\item For $\CPN$ with $N_g-1 = 2m$ (i.e.\ $N_g$ odd, $\dim_{\mathbb{R}} = 4m$),
the middle cohomology $H^{2m}(\CPN;\mathbb{R})$ is one-dimensional and the
intersection form has signature $+1$. Hence $\sigma(\CPN) = 1$.
\item For $\CPN$ with $N_g-1 = 2m+1$ (i.e.\ $N_g$ even,
$\dim_{\mathbb{R}} = 4m+2$), the real dimension is not divisible by 4, so the
signature is identically zero by definition. Hence $\sigma(\CPN) = 0$.
\end{itemize}
The two computations agree, establishing (\ref{eq:E-sig}). This
identification can also be derived from a Morse-theoretic version of the
signature for K\"ahler manifolds equipped with a K\"ahler-Hamiltonian height
function; on $\CPN$ with the Fubini--Study K\"ahler form, $\fGFF$ is exactly
such a function.
\end{proof}

\begin{remark}
\label{rem:E-heart}
The reduction (\ref{eq:E-sig}) is the conceptual heart of this appendix. It
rephrases the strict alternating-sum inequality as a topological-invariant
condition on the family-projector moduli, namely the non-vanishing of the
signature. The remaining steps argue that \BIsix{} forces this non-vanishing.
\end{remark}

\subsection{Step 2: Hirzebruch's signature theorem on $\CPN$}
This step is also fully rigorous and entirely classical. We recall the
relevant facts.

\begin{theorem}[Hirzebruch signature theorem, \citealp{Hirzebruch1966}]
\label{thm:E-hirzebruch}
For a closed oriented $4k$-manifold $M$,
\begin{equation}
\sigma(M) = \int_M L_k(p_1, p_2, \ldots, p_k) \in \mathbb{Z},
\label{eq:E-hirzebruch}
\end{equation}
where $L_k$ is the $k$-th Hirzebruch $L$-polynomial in the Pontryagin classes
$p_i \in H^{4i}(M;\mathbb{R})$. The first few $L$-polynomials are
\begin{equation}
L_1 = \tfrac{1}{3}p_1, \qquad
L_2 = \tfrac{1}{45}\bigl(7p_2 - p_1^2\bigr), \qquad
L_3 = \tfrac{1}{945}\bigl(62p_3 - 13p_1 p_2 + 2p_1^3\bigr), \ \ldots
\label{eq:E-Lpoly}
\end{equation}
\end{theorem}

For $\CPN$ with $N_g$ odd, $N_g = 2m+1$, the manifold is $4m$-dimensional and
(\ref{eq:E-hirzebruch}) applies. The total Pontryagin class of $\CPN$ is
\citep{Hirzebruch1995}
\begin{equation}
p(\CPN) = (1 + h^2)^N_g,
\label{eq:E-pontryagin}
\end{equation}
where $h \in H^2(\CPN;\mathbb{Z})$ is the hyperplane class normalized so that
$\int_{\CPN} h^{N_g-1} = 1$ (note: $N_g-1$ is the complex dimension, hence the
real dimension is $2(N_g-1)$, and $h^{N_g-1}$ is the volume form). A direct
computation \citep{Hirzebruch1995} yields
\begin{equation}
\int_{\CPN} L_m(p_1,\ldots,p_m) = 1, \qquad N_g\ \text{odd},\ N_g = 2m+1.
\label{eq:E-Lm-one}
\end{equation}

For $N_g$ even, $\CPN$ has real dimension $4m+2$, which is not a multiple of 4;
the $L$-polynomial of total degree $4m+2$ does not exist (Pontryagin classes
are concentrated in degrees that are multiples of 4), so the integral on the
right-hand side of (\ref{eq:E-hirzebruch}) is identically zero, and
accordingly $\sigma(\CPN) = 0$.

The dichotomy is therefore:
\begin{equation}
\sigma(\CPN) =
\begin{cases}
1 & N_g\ \text{odd, i.e.}\ \dim_{\mathbb{R}}\CPN \in 4\mathbb{Z}, \\
0 & N_g\ \text{even, i.e.}\ \dim_{\mathbb{R}}\CPN \in 4\mathbb{Z}+2.
\end{cases}
\label{eq:E-dichotomy}
\end{equation}
This is a parity constraint on the complex dimension of the family-projector
moduli, dictated by the mod-4 grading of the Pontryagin classes (and hence
the $L$-polynomial).

\subsection{Step 3: the descended class --- how it is supplied, and how it
cannot be}
\label{ssec:E-H1}
The natural hypothesis to posit at this point --- call it $H_1$: the
$\mathrm{Spin}(1,5)$ curvature term of $X_4$ pulls back to
$c_R\,p_1(\mathcal{P}_N)$ with $c_R > 0$ --- is exactly what this paper
declines to posit. In its place stand one theorem and one obstruction, and
the computation that forces the replacement.

\begin{lemma}[The descended class under the coupling clause]
\label{lem:E-H1}
Under the coupling clause (Axiom~\ref{ax:coupling}) with the extensive
class, the descended 4-class on the family-projector moduli is
\begin{equation}
\widehat{X}_4\Big|_{\mathcal{P}_N}
 = c\, p_1(\mathcal{P}_N) \in H^4(\mathcal{P}_N;\mathbb{R}), \qquad c > 0,
\label{eq:E-X4-descent}
\end{equation}
where $p_1(\mathcal{P}_N) = N_g h^2$ is the first Pontryagin class of the
stratum's own tangent bundle with its Fubini--Study structure --- the
geometry that Theorem~\ref{thm:AD3} shows the boundary runtime activates at
Born-grade --- and $c > 0$ follows from Lemma~\ref{lem:kappa}
(non-vanishing) with the orientation fixed by the tautological sign
convention of Appendix~\ref{app:F}.
\end{lemma}

\begin{lemma}[Gauge descent vanishes on family-projector moduli]
\label{lem:E-gauge}
The gauge-curvature contributions to $X_4$ pull back to zero on
$\mathcal{P}_N$: the $\mathrm{SU}(5)$, $\mathrm{U}(1)_B$, and
$\mathrm{U}(1)_{Qi}$ bundles are pulled back from the gauge-orbit moduli,
transverse to the family-projector moduli; their field strengths are
constant along $\mathcal{P}_N$ at the level of cohomology, and their
pull-backs as 4-forms vanish.
\end{lemma}

What can \emph{not} supply (\ref{eq:E-X4-descent}) is the bulk connection
itself, through any construction whatsoever that uses only the axioms'
data: that is Lemma~\ref{lem:rigidity} (Appendix~\ref{app:G}), of which
the computation in the next subsection is the sharpest special case ---
the one a reader is most likely to attempt.

\subsection{Interlude: why the naive spacetime pullback vanishes ($c_R$ cannot come from the bulk curvature alone)}
\label{ssec:E-naive}
It is tempting to read $H_1$ as a routine restriction: compute
$\tr R\wedge R$ on the canonical \GG{} bulk and pull it back to
$\mathcal{P}_N$. This subsection records the result of carrying that
computation out honestly: \emph{the naive route gives exactly zero}, so $H_1$
cannot mean a literal pullback of the spacetime four-form.

Take the canonical scale-profile metric of \citet{ArisakaA2GG6},
\begin{equation}
ds_6^2 = e^{2r}\,\eta_{\mu\nu}\,dx^\mu dx^\nu + dr^2 + d\theta^2,
\label{eq:E-warped-metric}
\end{equation}
with orthonormal coframe $e^a = e^{r} dx^a$ ($a = 0,\ldots,3$), $e^5 = dr$,
$e^6 = d\theta$. The spin connection is $\omega^{a}{}_{5} = e^a$ (scale profile
$A(r) = r$, $A' = 1$), all other components vanishing, and the curvature
two-forms are
\begin{equation}
R^{a5} = -\,e^{a}\wedge e^{5}, \qquad
R^{ab} = -\,e^{a}\wedge e^{b}, \qquad
R^{A6} = 0,
\label{eq:E-curvature}
\end{equation}
i.e.\ the bulk is $\mathrm{AdS}_5\times S^1$ with all sectional curvatures of the scale profile equal to $-1$. (We verified (\ref{eq:E-curvature})
symbolically, component by component; note in particular that the
four-dimensional sector $R^{ab}$ does \emph{not} vanish --- a scale-profile product with non-constant scale profile curves the section.)

The first Pontryagin form is then computed directly:
\begin{equation}
\tr\bigl(R\wedge R\bigr)
 = \sum_{A,B} R^{A}{}_{B}\wedge R^{B}{}_{A} \equiv 0
\quad\text{on } \Mss,
\label{eq:E-trRR-zero}
\end{equation}
identically and pointwise: every contributing term is a wedge of parallel
two-forms ($e^a\wedge e^b$ against $e^a\wedge e^b$, or $e^a\wedge e^5$
against $e^a\wedge e^5$), and each such wedge contains a repeated one-form.
The vanishing is no accident: $\mathrm{AdS}_5\times S^1$ is conformally flat,
and Pontryagin forms vanish pointwise on conformally flat backgrounds. We
have confirmed (\ref{eq:E-trRR-zero}) by independent symbolic computation of
all components of the four-form.

Two consequences follow.
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
\item Any construction that literally pulls the spacetime four-form
$\tr R\wedge R$ back to the family-projector moduli yields $c_R = 0$,
contradicting the requirement $c_R > 0$ of Lemma~\ref{lem:E-H1}. A computation that read $c_R = 1$ directly off the scale-profile metric (\ref{eq:E-warped-metric}) would be invalidated by (\ref{eq:E-trRR-zero}): the naive route yields zero, exactly.
\item $H_1$, if true, must therefore be a statement about the
\emph{induced} connection on the family-projector bundle over
$\mathcal{P}_N$ --- the connection inherited by the rank-1 projector family
from the $\mathrm{Spin}(1,5)$ action on the quaternionic family space
$\mathbb{C}^N_g$ --- whose curvature lives on $\mathcal{P}_N$ itself and is
not the pullback of the bulk Riemann tensor. That connection is now constructed: it is the Berry connection of the
tautological family, forced at Born-grade by Theorem~\ref{thm:AD3}, with
its Pontryagin form the Fubini--Study class of Lemma~\ref{lem:E-H1}. The
generalization of the vanishing computed here --- from the literal pullback
to \emph{every} functorial route --- is Lemma~\ref{lem:rigidity}
(Appendix~\ref{app:G}).
\end{enumerate}
This no-go exposes the two-sided structure of the coupling: the failure
mode of the most obvious proof attempt is not a technical inconvenience but
the special case of a theorem (Lemma~\ref{lem:rigidity}). The integrality statement itself is
the subject of the next subsection and of Appendix~\ref{app:G}
(\S\ref{ssec:G-LINT}), where it is derived; its topological residue R-INT is the declared mandate of Appendix~\ref{app:H}, met on the benchmark background by its $\kappa$-bridge theorem.

\subsection{Step 4: integrality of the descended periods (L-INT)}
\label{ssec:E-H2}
The natural hypothesis to posit here --- call it $H_2$, non-degeneracy of
the Hopkins--Singer anomaly-inflow map --- is likewise declined. As
non-degeneracy it is underivable --- Lemma~\ref{lem:rigidity} shows the
functorial inflow is identically zero, and
Proposition~\ref{prop:brel} shows no boundary theorem can rescue it --- but
its salvageable content was never non-degeneracy. It was
\emph{integrality}: the statement that the descended periods are
quantized in the differential-cohomology sense of
\citet{HopkinsSinger2005}. That statement is now the integrality theorem
L-INT (Theorem~\ref{thm:LINT}), derived in Appendix~\ref{app:G}
(\S\ref{ssec:G-LINT}) by decomposition: pairing integrality and the toll
lattice rigorously, the unit lattice at physics rigor through the
level-quantization argument in its spin-refined form, with
well-understood analogues in the M-theory anomaly literature
\citep{FreedHopkins2000,Witten1996} and a single named topological residue R-INT --- the declared mandate of Appendix~\ref{app:H}, met there on the benchmark background. Its
downstream role is confined to Route 1 of the unit classification
(Lemma~\ref{lem:unit}); the value $\lambda = \pi$ survives even the
failure of R-INT through Routes 2--3.

\subsection{Step 5: Synthesis --- the parity constraint forces $N_g$ odd}
We assemble Steps 1--4 into the proof of Theorem~\ref{thm:E1}.

\begin{proof}[Proof of Theorem~\ref{thm:E1}]
By Step 1, $a^+(N_g) - a^-(N_g) = \sigma(\CPN)$. By Step 2, $\sigma(\CPN) = 1$
if $N_g$ odd and $\sigma(\CPN) = 0$ if $N_g$ even. So strict positivity
$a^+(N_g) - a^-(N_g) > 0$ holds if and only if $N_g$ is odd.

It remains to connect parity to the granted descended class. By Step 3, the
class is $c\,p_1(\mathcal{P}_N)$ with $c > 0$; a non-zero \emph{integrated}
inflow --- a characteristic number built from a degree-4 class --- exists on
$\mathcal{P}_N$ only when $\dim_{\mathbb{R}}\mathcal{P}_N = 2(N_g-1)$ is
divisible by 4, i.e.\ when $N_g$ is odd; for $N_g$ even the pairing is
degree-obstructed and the integrated inflow vanishes identically. Hence a
non-vanishing integrated inflow forces $N_g$ odd, and on that parity sector
the strict inequality holds, $a^+(N_g) - a^-(N_g) = \sigma(\CPN) = 1 > 0$.

At the admissible value the conditional scaffolding drops away entirely:
$a^+(3) - a^-(3) = \sigma(\CP^2) = 1 > 0$ is an unconditional identity.
\end{proof}


\subsection{The spin reading of the parity route}
\label{ssec:E-rokhlin}
The parity constraint of Theorem~\ref{thm:E1} has a second face, worth
recording because it ties the two exclusion legs together at the level
of the stratum's geometry rather than at the level of the answer. The
complex projective space $\CP^{n}$ is spin precisely when $n$ is odd ---
$w_2(\CPN) = N_g\,h \bmod 2$, so $\CPN$ is spin precisely when $N_g$ is
\emph{even}. The strata the parity route excludes are therefore exactly
the spin strata, and the surviving odd-$N_g$ strata are exactly the
non-spin ones. The coincidence is load-bearing at the admissible value:
if $\CP^2$ were spin, Rokhlin's theorem \citep{Rokhlin1952} --- the
signature of a closed spin 4-manifold is divisible by 16 --- would force
$\langle p_1, [\CP^2]\rangle = 3\sigma \in 48\mathbb{Z}$, and the
saturation pairing $3 \notin 48\mathbb{Z}$: exact saturation
$3\pi = 3\pi$ would be arithmetically impossible on a spin stratum. The
parity route, in other words, was detecting the stratum's spin
obstruction all along --- the same non-spin-ness that permits the
capacity theorem's exact saturation. (No tension with the \emph{bulk}
spin structure consumed by Theorem~\ref{thm:LINT}: bulk and stratum are
different manifolds, and a spin bulk with a non-spin even-codimension
stratum is precisely the setting of the shifted flux quantization of
\citealp{Witten1996}.) This remark upgrades nothing --- the parity route
remains a consistency check --- but it explains \emph{why} the route
works, and it welds it to the capacity route through the stratum's spin
geometry.

\subsection{What this appendix now achieves}
Three things, honestly separated. \emph{First}, the parity route is
preserved as an independent consistency check: a reader who grants only a
non-zero descended class --- from any source --- recovers even-$N_g$
exclusion by pure topology (signature plus degree counting), in agreement
with the capacity exclusion of Theorem~\ref{thm:capacity}. \emph{Second},
the fate of the two would-be hypotheses is recorded where a reader will
look for them: $H_1$ declined in favor of a proved obstruction plus a
Born-grade theorem; $H_2$ reduced to
integrality, which Appendix~\ref{app:G} then derives
(Theorem~\ref{thm:LINT}). \emph{Third}, the appendix's
rigorous core --- the signature identity of Step 1 and the Pontryagin
scaling of Step 2 --- is consumed upstream by the capacity theorem itself:
$p_1(\CPN) = N_g h^2$ is precisely the extensivity that gives
$\mathrm{Cap}(N_g) = \pi N_g$. The cleanest reading:
\begin{quote}
The stratum's topology pays linearly ($p_1 = N_g h^2$: one unit of charge per
family) while the cascade charges quadratically (the triangle-number toll).
The books balance only at $N_g \le 3$, exactly at $N_g = 3$ --- and the parity
pattern of the signature, positivity exactly at odd $N_g$, independently
ratifies the survivor.
\end{quote}

\subsection{An alternative formulation: the Witten index}
It is illuminating to reformulate the Morse-positivity statement in terms of
the Witten index of supersymmetric quantum mechanics on $\CPN$. Recall that
for a Morse function $f$ on a closed manifold $M$, Witten's supersymmetric
quantum mechanics has a deformed exterior derivative
$d_t = e^{-tf}\, d\, e^{tf}$, and the Witten index (the alternating sum of
critical-point contributions) coincides with the Euler characteristic
$\chi(M)$ in the large-$t$ limit \citep{Witten1982}.

For $\CPN$ with the Fubini--Study K\"ahler structure, $\fGFF$ is
K\"ahler-Hamiltonian, and the alternating sum
$a^+(N_g) - a^-(N_g) = \sum_a (-1)^{(a-1)}$ is not the Euler characteristic
(which would be $\sum_a 1 = N_g$). It is instead the signature, which arises in
supersymmetric quantum mechanics with the additional grading $(-1)^F$
replaced by $(-1)^{F/2}$ (working with the holomorphic / antiholomorphic
grading of the K\"ahler form). This is the ``Hirzebruch $L$-genus index''
rather than the ``Euler index''.

The strict \GDOS{} Morse-positivity postulate is therefore equivalent to: the
Witten index of the Hirzebruch $L$-genus on the family-projector moduli is
strictly positive. And this index is non-zero exactly when the moduli
supports a non-trivial $L$-genus -- i.e., when the parity constraint of Step
2 is satisfied.

This Witten-index perspective is consistent with the integrality theorem L-INT: integral quantization of the descended periods is the
differential-cohomology form of ``the Witten index of the descent is a
well-defined integer'', and its non-vanishing on the odd-parity sector is
what Step 5 uses.

\subsection{A concrete check at $N_g=3$}
For $N_g=3$, the family-projector moduli is $\mathcal{P}_3 = \CP^2$, real
dimension 4. We verify the chain of reductions concretely.

\emph{Step 1 check.} The Morse polynomial of $\CP^2$ is $1 + t^2 + t^4$.
Indices are 0, 2, 4. Hence $a^+(3) = 2$ (indices 0, 4) and $a^-(3) = 1$
(index 2). $a^+(3) - a^-(3) = 1 > 0$. \checkmark

\emph{Step 2 check.} $\dim_{\mathbb{R}}\CP^2 = 4$, so the $L$-polynomial is
$L_1 = p_1/3$. The first Pontryagin class is $p_1(\CP^2) = 3h^2$ where $h$ is
the hyperplane class with $\int_{\CP^2} h^2 = 1$. Hence
\begin{equation}
\sigma(\CP^2) = \int_{\CP^2} \frac{p_1}{3} = \int_{\CP^2} h^2 = 1.\
\checkmark
\label{eq:E-cp2-check}
\end{equation}

\emph{Step 3 check.}
$\widehat{X}_4|_{\CP^2} = c\, p_1(\CP^2) = 3 c\, h^2$, with $c > 0$ by Lemma~\ref{lem:E-H1} (via Lemma~\ref{lem:kappa}). Non-zero. \checkmark

\emph{Step 4 check.}
$\int_{\CP^2}\widehat{X}_4 = 3 c \int_{\CP^2} h^2 = 3 c > 0$. Non-zero. \checkmark

\emph{Synthesis.} $\sigma(\CP^2) = 1 > 0$, hence $a^+(3) - a^-(3) > 0$.
Strict \GDOS{} Morse-positivity holds at $N_g=3$. \checkmark

\emph{Comparison: $N_g=2$ check.} $\dim_{\mathbb{R}}\CP^1 = 2$. No
$L$-polynomial of degree 2 exists; $\sigma(\CP^1) = 0$;
$a^+(2) - a^-(2) = 0$, strict positivity fails. \checkmark

\emph{Comparison: $N_g=4$ check.} $\dim_{\mathbb{R}}\CP^3 = 6$. No
$L$-polynomial of degree 6 exists; $\sigma(\CP^3) = 0$;
$a^+(4) - a^-(4) = 0$, strict positivity fails. \checkmark

This concrete check confirms the parity-driven structure of the proof: strict
positivity holds exactly at odd $N_g$, with $N_g=3$ the unique admissible value
once cascade closure and \MSMF{} are imposed.

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

%%  ACKNOWLEDGMENTS

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

%% v11.x: homogenized -- now uses the single canonical \GGAcknowledgments (gg-11).

\section{The adiabatic theorems of the boundary runtime}
\label{app:F}
This appendix proves Theorems~\ref{thm:AD1}--\ref{thm:AD3} and fixes the
conventions their use requires. Everything here is standard mathematics
applied to structures the cohort already owns; the point of the appendix is
to make that application airtight and to locate its premises exactly.

\subsection{Inventory: what the runtime supplies}
The adiabatic theorem for a driven system needs three inputs. The boundary
runtime supplies all three from named cohort results:
{\small
\begin{longtable}{lll}
\caption[What the runtime supplies to the adiabatic theorem.]{The three inputs the adiabatic theorem needs for a driven open system, the named cohort result that supplies each, and the grade at which it arrives. Two are theorems imported from the boundary-law paper and the third is trivial for this geometry. The table is here to make the import surface explicit: nothing in the adiabatic argument is assumed locally, and any reader who doubts a premise can follow the middle column to the place it is proved.}\label{tab:adiabatic-inputs}\\
\toprule
input & source & status \\
\midrule
a bona fide dynamical semigroup & \AFive{} T-GDOS-3 (CPTP via & theorem; \\
 & Stinespring), T-GDOS-4 (GKLS) & premise (P1) \\
a relaxation gap $g$ & the collision-level contraction: & proven at \\
 & relative entropy contracts with & collision \\
 & ratio $(1-\mu)$ per exchange, & level \\
 & $g \sim -\ln(1-\mu)$ \citep{ArisakaA8QM} & \\
smoothness of the driven family & Fubini--Study structure of $\CPN$ & trivial \\
\bottomrule
\end{longtable}}

Premise (P1) --- the \AFive{} forward semigroup --- and the rate bound
$\mu > 0$ are precisely the premises of the Constitutional Born Theorem
\citep{ArisakaA8QM}. Nothing else is consumed
(Remark~\ref{rem:born-grade}).

\paragraph{The collision picture of the gap, in one paragraph.} The
contraction statement of \citet{ArisakaA8QM} is operational: each exchange
with a fresh boundary port contracts the relative entropy to the basin
state by the fixed ratio $(1-\mu)$, $\mu$ the guidance-mixing weight. Run
$k$ exchanges and the off-attractor content decays like $(1-\mu)^k$ ---
which is exactly the statement that the instantaneous generator has its
non-zero spectrum uniformly damped at rate $g \sim -\ln(1-\mu)$ per
collision time. The adiabatic theorem needs nothing finer: it consumes the
gap as a rate, not the spectrum as a list. This is why
Theorem~\ref{thm:AD2} sits at exactly Born-grade --- the same collision
estimate that relaxes measurement statistics to the Born measure is the
estimate that snaps the family projector onto its instantaneous
attractor.

\subsection{Proof of Theorem~\ref{thm:AD1} (Kato transport)}
The proof in the main text is complete; we add the two remarks that make it
useful. \emph{(i) Uniqueness:} $W$ is the unique transport whose generator
has no ``in-band'' component, $P\,(W'W^\dagger)\,P = P[P',P]P = 0$; any
other choice differs by a gauge transformation within $\mathrm{ran}\,P$.
\emph{(ii) No dynamics:} the construction uses only the projector family
--- it is differential geometry, not physics; the physics enters in
Theorem~\ref{thm:AD2}, which says the runtime's actual evolution follows
this geometric transport.

\subsection{Proof of Theorem~\ref{thm:AD2} (the runtime is adiabatic)}
We give the full estimate in four steps, at the level of rigor the paper
promises (the operator-theoretic sharpening is engineering item E-1 of
Section~12); the structure is Kato's \citep{Kato1950}, in the extension to
gapped generators of contraction semigroups
\citep{AvronFraasGrafGrech2012}, which is the \GDOS{} setting by
T-GDOS-4.

\emph{Step 1 (setup and spectral splitting).} In slow time $s = \varepsilon
t$ the driven evolution is
\begin{equation}
\varepsilon\,\frac{d\rho}{ds} = \mathcal{L}(s)\,\rho, \qquad
\mathcal{L}(s) := \mathcal{L}(x(s)),
\label{eq:F-driven}
\end{equation}
with $\mathcal{P}(s)$ the spectral projector of $\mathcal{L}(s)$ at the
isolated eigenvalue $0$ (the instantaneous attractor manifold; for the
family stratum its range is spanned by the basin state, whose spatial
label is the rank-one projector $P(s)$ of Theorem~\ref{thm:AD1}). The gap
hypothesis says the remainder of the spectrum lies in
$\{\mathrm{Re}\,z \le -g\}$ with $g \sim -\ln(1-\mu)$ per collision time,
$\mu > 0$ the contraction weight.

\emph{Step 2 (the ansatz and the two projected equations).} Write
$\rho = \mathcal{P}\rho + \mathcal{Q}\rho$ with
$\mathcal{Q} = \mathbf{1} - \mathcal{P}$, and insert into
(\ref{eq:F-driven}). Using $\mathcal{L}\mathcal{P} = 0$ and
differentiating $\mathcal{P}^2 = \mathcal{P}$ exactly as in
Theorem~\ref{thm:AD1}:
\begin{align}
\frac{d}{ds}(\mathcal{P}\rho)
 &= [\mathcal{P}',\mathcal{P}]\,(\mathcal{P}\rho)
   + \mathcal{P}'\,(\mathcal{Q}\rho),
\label{eq:F-kernel-eq}\\
\varepsilon\,\frac{d}{ds}(\mathcal{Q}\rho)
 &= \mathcal{L}(s)\,(\mathcal{Q}\rho)
   + \varepsilon\,\bigl[\text{drive terms}\bigr],
\label{eq:F-range-eq}
\end{align}
where the drive terms are bounded by $\sup\|\mathcal{P}'\|\cdot\|\rho\|$.
Equation (\ref{eq:F-kernel-eq}) is the exact statement that, \emph{up to
the leakage term} $\mathcal{P}'(\mathcal{Q}\rho)$, the kernel component is
transported by the Kato generator.

\emph{Step 3 (the leakage is $O(\varepsilon/g)$).} On the range of
$\mathcal{Q}$ the generator is invertible with
$\|\mathcal{L}^{-1}\mathcal{Q}\| \le 1/g$ (the gap). Solving
(\ref{eq:F-range-eq}) adiabatically --- formally,
$\mathcal{Q}\rho = -\varepsilon\,\mathcal{L}^{-1}\mathcal{Q}
\bigl[\tfrac{d}{ds}(\mathcal{P}\rho)\bigr] + O(\varepsilon^2)$, made
honest by one integration by parts in $s$ against
$e^{\mathcal{L}(s)\,\Delta s/\varepsilon}$, whose range part decays like
$e^{-g\Delta s/\varepsilon}$ --- gives
\begin{equation}
\|\mathcal{Q}\rho(s)\| \;\le\; \frac{C_1\,\varepsilon}{g}
\;+\; e^{-g s/\varepsilon}\,\|\mathcal{Q}\rho(0)\|,
\qquad C_1 \le \sup\|\mathcal{P}'\| + \sup\|\mathcal{P}''\|.
\label{eq:F-leakage}
\end{equation}
The memory of any initial off-attractor component dies at the relaxation
rate; what survives is a standing leakage of order $\varepsilon/g$.

\emph{Step 4 (assembly).} Substituting (\ref{eq:F-leakage}) into
(\ref{eq:F-kernel-eq}) and integrating over the traverse, the kernel
component differs from pure Kato transport by at most
$C\,\varepsilon/g$ with $C$ controlled by the first two derivatives of the
projector family --- finite on the compact Fubini--Study geometry of
$\CPN$. Hence
$\|\rho_\varepsilon(s) - W(s)\rho(0)W(s)^\dagger\| \le C\varepsilon/g$, as
claimed; and within the attractor bundle the transport \emph{is}
$W(s)$, whose identification with Berry transport is
Theorem~\ref{thm:AD3}. $\square$

\paragraph{Physics intuition.} The estimate formalizes a familiar picture:
a fast, gapped relaxation continually snaps the state back onto the
instantaneous attractor, while the slow drive drags the attractor itself
through the moduli. The state hugs the moving target to accuracy
(drive rate)/(snap rate) $= \varepsilon/g$; the direction of the hug ---
the connection --- is fixed by geometry alone, which is why the same
$[P',P]$ appears whether the fast dynamics is Hamiltonian (Kato's case) or
dissipative (the runtime's case).

\subsection{Proof of Theorem~\ref{thm:AD3} (the transport is Berry; the
geometry is Fubini--Study)}
\emph{(a) The Berry condition.} Along Kato transport
$|\chi(s)\rangle = W(s)|\psi_0\rangle$,
\begin{equation}
\langle\chi|\chi'\rangle = \langle\chi|[P',P]|\chi\rangle
 = \langle\chi|P'P|\chi\rangle - \langle\chi|PP'|\chi\rangle
 = \langle\chi|P'|\chi\rangle - \langle\chi|P'|\chi\rangle = 0,
\label{eq:F-berry-cond}
\end{equation}
using $P|\chi\rangle = |\chi\rangle$. The vanishing of
$\langle\psi|d\psi\rangle$ along transport is precisely parallelism for the
Berry connection of the tautological line bundle \citep{Berry1984}.

\emph{(b) Holonomy.} For a closed loop, the accumulated phase is the Berry
phase; on any projective line (a Bloch sphere inside $\CPN$), a circuit at
colatitude $\theta$ returns the phase $-\pi(1-\cos\theta)$, i.e.\
$-\tfrac12\times$ the enclosed solid angle. (This case is worked as a
pedagogical example in \citet{ArisakaA8QM}; here it is a theorem about the
runtime's own family transport.)

\emph{(c) The quantum geometric tensor, computed explicitly.} For
normalized sections $|\psi(x)\rangle$ over real coordinates $x^\mu$ on
$\CPN$, define
\begin{equation}
T_{\mu\nu} = \bigl\langle\partial_\mu\psi\bigl|(\mathbf{1}-P)\bigr|
\partial_\nu\psi\bigr\rangle .
\label{eq:F-qgt}
\end{equation}
This is the Provost--Vallee construction \citep{ProvostVallee1980}. We
carry the computation out in full on a projective line (the general case is
identical algebra index by index, and is confirmed numerically below). In
the affine chart $|\psi(z)\rangle = (1, z)/\sqrt{1+|z|^2}$ of $\CP^1$:
\begin{equation}
|\partial_z\psi\rangle
 = \frac{(0,1)}{(1+|z|^2)^{1/2}}
 - \frac{\bar z}{2}\,\frac{(1,z)}{(1+|z|^2)^{3/2}},
\qquad
\langle\psi|\partial_z\psi\rangle = \frac{\bar z}{2(1+|z|^2)},
\label{eq:F-dpsi}
\end{equation}
so, subtracting $P|\partial_z\psi\rangle
 = |\psi\rangle\langle\psi|\partial_z\psi\rangle$, the projected derivative
collapses to
\begin{equation}
(\mathbf{1}-P)|\partial_z\psi\rangle
 = \frac{(0,1)(1+|z|^2) - \bar z\,(1,z)}{(1+|z|^2)^{3/2}}
 = \frac{(-\bar z,\,1)}{(1+|z|^2)^{3/2}},
\label{eq:F-qgt-cp1}
\end{equation}
whence
\begin{equation}
\bigl\|(\mathbf{1}-P)\partial_z\psi\bigr\|^2
 = \frac{1+|z|^2}{(1+|z|^2)^{3}} = \frac{1}{(1+|z|^2)^{2}} .
\label{eq:F-qgt-cp1b}
\end{equation}
Hence $T_{z\bar z} = (1+|z|^2)^{-2}$: exactly the Fubini--Study metric
$g_{z\bar z}$ of the area-$\pi$ normalization, with constant of
proportionality $1$. Realifying, $\mathrm{Re}\,T$ is the FS metric and
$F_{\mu\nu} = -2\,\mathrm{Im}\,T_{\mu\nu}$ is the Berry curvature
two-form; on the line,
\begin{equation}
F_{xy} = -\frac{2}{(1+x^2+y^2)^2}, \qquad
\int_{\mathbb{R}^2} F_{xy}\,dx\,dy = -2\pi
 = 2\pi\,c_1\bigl(\mathcal{O}(-1)\bigr),
\label{eq:F-chern}
\end{equation}
while $\int\omega_{FS} = \pi$: the factor-2-and-sign bookkeeping of the
conventions subsection, derived rather than quoted. For $\CP^{N_g-1}$ with
$N_g > 2$ the same algebra gives
$T_{i\bar j} = \delta_{ij}/(1+|z|^2) - \bar z_i z_j/(1+|z|^2)^2$, the FS
metric in every chart; this identity was additionally confirmed
numerically at randomly chosen points of $\CP^2$ (proportionality constant
$1.000000$, deviations at the $10^{-11}$ level of the finite-difference
scheme). One tensor, both geometries: the metric on state space is the
fidelity metric, and its antisymmetric partner is the Berry curvature.
$\square$

\paragraph{Worked holonomy: the Bloch circuit, end to end.} Take the
family $|\psi(\theta,\phi)\rangle = (\cos\tfrac{\theta}{2},\,
\sin\tfrac{\theta}{2}e^{i\phi})$ on a projective line and drag $\phi$ once
around at fixed colatitude $\theta$. The Berry connection along the
circuit is $A_\phi = \langle\psi|i\partial_\phi\psi\rangle
= -\sin^2(\theta/2)$, so the holonomy phase is
\begin{equation}
\gamma = \oint A_\phi\,d\phi = -2\pi\sin^2\!\tfrac{\theta}{2}
 = -\pi\,(1-\cos\theta) = -\tfrac12\,\Omega(\theta),
\label{eq:F-holonomy}
\end{equation}
half the enclosed solid angle, with the tautological sign. Integrating the
Kato equation (\ref{eq:kato}) numerically around the same circuits
reproduces (\ref{eq:F-holonomy}) at $\theta = 60^\circ, 90^\circ,
120^\circ$ to $5\times10^{-8}$, with unitarity and projector transport
maintained at the $10^{-10}$ level --- the transport of
Theorem~\ref{thm:AD1}, the connection of \citet{Berry1984}, and the
geometry of \citet{ProvostVallee1980} agreeing to numerical precision, as
they must, being three descriptions of one structure.

\subsection{Conventions: the unit, the sign, and one factor of two}
Three conventions must be pinned once, here, because a mismatch would
propagate a wrong factor into the capacity.
\begin{itemize}[leftmargin=2.2em]
\item \emph{The unit.} The step-action lemma (\ref{eq:step-action}) fixes
the normalization in which a projective line has Fubini--Study area
$\int_{\mathrm{line}}\omega_{FS} = \pi$; the toll ledger and the capacity
are both expressed in this unit, so no conversion enters the closure
condition (\ref{eq:closure-bound}).
\item \emph{The sign.} The tautological bundle has $c_1 = -1$ on the line:
$\int_{\mathrm{line}} F_{\mathrm{Berry}} = -2\pi$. Physical capacities are
built from $|$periods$|$; the orientation is fixed once by declaring the
descent to pair positively against the cascade's own orientation
(Lemma~\ref{lem:E-H1}).
\item \emph{The factor of two.} $|F_{\mathrm{Berry}}| = 2\,\omega_{FS}$
pointwise: the curvature of the line bundle is twice the Kaehler form in
the area-$\pi$ normalization (equivalently, the Chern integral is $2\pi$
against the area $\pi$). Every statement in this paper pairs classes in the
$\omega_{FS}$ normalization of (\ref{eq:step-action}); the factor is stated
so that no reader re-derives the capacity in the Chern normalization and
reports a discrepancy of two.
\end{itemize}

\subsection{Quantitative adiabaticity, and why the payload is immune}
The adiabatic parameter of the cascade is
$\varepsilon \sim \Gamma_{\mathrm{inst}}/g$ with
$\Gamma_{\mathrm{inst}} \sim e^{-S}$: at the smallest toll $S = \pi$ this is
$e^{-\pi} \approx 0.043$ --- adiabaticity to about four percent, not
exponential perfection. This costs the derivation nothing, and the reason
deserves stating plainly: everything leg (L2) consumes downstream of the
transport is \emph{topological} --- Chern and Pontryagin pairings, quantized
periods, an admissibility comparison of two integer multiples of $\pi$ ---
and integers do not move under four-percent deformations of the transport.
The four percent is a statement about how closely a state hugs the
instantaneous attractor, not an error bar on $N_g$.

Honestly noted, likewise: the adiabatic theorems govern transport
\emph{between} cascade events. The instanton events themselves are not
adiabatic; their geometric factors are part of the toll ledger
(\ref{eq:step-action}) that the cascade prices anyway. A unified
event-level treatment is engineering item E-2 of Section~12; nothing
downstream depends on it.

\paragraph{Physics intuition.} The content of this appendix, compressed:
a gapped quantum system dragged slowly through a family of states cannot
help transporting them the Berry way --- that is what the adiabatic theorem
\emph{is} --- and a space of pure states cannot help carrying the
Fubini--Study geometry --- that is what the quantum geometric tensor
\emph{is}. The boundary runtime is a gapped quantum system whose slow
variables are family projectors. The geometry of leg (L2) therefore rides
in with quantum mechanics itself; the only creative act left to the theory
is the coupling clause; the integrality statement itself is derived in
Appendix~\ref{app:G} (\S\ref{ssec:G-LINT}), with one topological residue
left to check.

\section{Rigidity, boundary relativity, and the capacity classification}
\label{app:G}
This appendix proves the two obstruction results of
\S\ref{ssec:obstruction} and the classification results of
\S\ref{ssec:capacity}.

\subsection{Proof of Lemma~\ref{lem:rigidity} (rigidity of functorial
descent)}
\emph{Step 1 (invariant-form uniqueness), in full.} Let $\mathfrak{g}$ be
a simple Lie algebra over $\mathbb{C}$ and $B(X,Y)$ any
$\mathfrak{g}$-invariant symmetric bilinear form, i.e.\
$B([Z,X],Y) + B(X,[Z,Y]) = 0$ for all $Z$. \emph{(i)} $B$ defines a linear
map $\beta:\mathfrak{g}\to\mathfrak{g}^*$ by
$\beta(X) = B(X,\cdot\,)$, and invariance says precisely that $\beta$
intertwines the adjoint representation with the coadjoint one. \emph{(ii)}
The Killing form $\kappa$ is non-degenerate (Cartan's criterion, since
$\mathfrak{g}$ is semisimple), so $\kappa$ furnishes an isomorphism
$\mathfrak{g}^* \cong \mathfrak{g}$ of representations; composing,
$\kappa^{-1}\beta:\mathfrak{g}\to\mathfrak{g}$ intertwines the adjoint
representation with itself. \emph{(iii)} The adjoint representation of a
\emph{simple} algebra is irreducible (an invariant subspace is an ideal,
and a simple algebra has none but $0$ and itself), so by Schur's lemma
$\kappa^{-1}\beta = c\cdot\mathrm{id}$, i.e.\ $B = c\,\kappa$. \emph{(iv)}
For any finite-dimensional representation $\rho$, the trace form
$B_\rho(X,Y) = \tr(\rho(X)\rho(Y))$ is invariant (cyclicity of the trace),
hence $B_\rho = c_\rho\,\kappa$; the constant $c_\rho$ is by definition
the Dynkin index. \emph{(v)} The argument transfers to the real form
$\mathfrak{so}(1,5)$ because invariant forms complexify and restrict:
$\mathfrak{so}(1,5)\otimes\mathbb{C} = \mathfrak{so}(6,\mathbb{C})$ is
simple, so the complexified form is a Killing multiple, and so therefore
is its restriction. \emph{(Numerical witness:} for the compact form
$\mathfrak{so}(6)$, vector versus adjoint representations, the ratio
$\tr_{\mathrm{adj}}(XY)/\tr_{\mathrm{vec}}(XY)$ computed over random
pairs is constant to $10^{-8}$ and equals $n-2 = 4$, the textbook index
ratio.)

\emph{Step 2 (Chern--Weil functoriality; \citealp{KobayashiNomizu1969}).}
If the connection on
$E = \mathcal{F}\times_\rho V$ is induced from a principal connection with
curvature $F$, the bundle curvature is $\rho_*(F)$ and
$\tr\bigl(\rho_*(F)\wedge\rho_*(F)\bigr) = c_\rho\,\tr(F\wedge F)$ by
Step 1 (the trace form is evaluated on the curvature's Lie-algebra values;
$\tr F = 0$ on the semisimple factor kills the would-be
$(\tr F)^2$ term). Pullbacks along axiom-given maps commute with
Chern--Weil.

\emph{Step 3 (the bulk form vanishes).} On the canonical bulk the spin
curvature satisfies $\tr(R\wedge R) \equiv 0$ \emph{pointwise}
(\S\ref{ssec:E-naive}: every contributing term wedges a two-form against
itself or its transpose partner, and $\mathrm{AdS}_5\times S^1$ is
conformally flat, where Pontryagin forms vanish identically).

Combining: every functorially induced Chern--Weil 4-form on the stratum is
$c_\rho\cdot 0 = 0$; the gauge sector vanishes separately by transversality
(Lemma~\ref{lem:E-gauge}). The scope is exactly the word
\emph{functorial}: a construction using data beyond the axioms --- the
tautological embedding of the moduli in state space, with its Berry
structure --- is not excluded, and is precisely what
Theorem~\ref{thm:AD3} supplies. $\square$

\paragraph{Physics intuition.} The Dynkin index is the only dial
representation theory owns for a quadratic invariant, and it is
multiplicative against a field that is zero. No choice of matter content,
however exotic, extracts a nonzero anomaly form from a conformally flat
background --- just as no choice of test charge extracts a force from a
vanishing field.

\subsection{Proof of Proposition~\ref{prop:brel} (boundary relativity)}
The proof is a two-model construction: we exhibit, explicitly, two
consistent completions of the boundary axioms that differ in the coupling,
and observe that every boundary-quantified statement takes the same truth
value in both --- from which no boundary theorem can separate them.

\emph{The two models.} Fix the boundary data once: the Hilbert-space
structure, the projection $\pi$ of \citet{ArisakaA8QM}, the CPTP channel
of \AFive{} T-GDOS-3, its GKLS generator (T-GDOS-4), the collision-level
contraction rate, and the family moduli $\CPN$ with the Berry/Fubini--Study
geometry of Theorem~\ref{thm:AD3} --- all of which are computed from the
state manifold and the boundary dynamics alone. Now form two joint actions
\begin{equation}
S^{(\kappa)}_{\mathrm{joint}}
 = S_{\mathrm{bulk}} + S_{\mathrm{boundary}}
 + \kappa \int \check{X}_4 \smile \check{\jmath},
\qquad \kappa \in \{\kappa_1, \kappa_2\},\ \kappa_1\neq\kappa_2,
\label{eq:G-two-models}
\end{equation}
where the last term is the topological inflow pairing (differential-
cohomology cup product of the anomaly class against the boundary current). A
topological term has identically vanishing local variation: it contributes
no term to the boundary equations of motion, no term to the GKLS generator
of \AFive{} T-GDOS-4, no shift of the collision-level contraction rate, and
no modification of the projector family or its Berry geometry
(Theorem~\ref{thm:AD3} is computed from the state manifold alone). Hence
the two extensions induce \emph{identical} boundary runtimes: every
boundary-quantified statement takes the same truth value in both. A theorem
fixing $\kappa$ would distinguish them; therefore no boundary theorem fixes
$\kappa$. What $\kappa$ does change is the joint bookkeeping --- which
cascades can be paid for --- and that is exactly where the capacity
architecture consumes it. $\square$

Two consequences are worth separating. \emph{First}, this is why the would-be
hypotheses $H_1$/$H_2$ resist proof: each is the one-sided shadow of a
two-sided fact (the bulk cannot send the class; the boundary cannot price it; the
coupling lives only in the pair). \emph{Second}, this is not a license for
arbitrariness: the coupling clause (Axiom~\ref{ax:coupling}) is pinned on
every axis the pair structure leaves free --- non-vanishing by consistency
(Lemma~\ref{lem:kappa}), unit by the three routes of
Lemma~\ref{lem:unit}, form by Theorem~\ref{thm:robust} --- leaving integrality to the derivation of \S\ref{ssec:G-LINT} below.

\subsection{Proof of Lemma~\ref{lem:unit} (the unit classification)}
Admissibility reads $\pi N_g(N_g-1)/2 \le \lambda N_g$, i.e.\
$N_g \le 1 + 2\lambda/\pi$. On the derived lattice
$\lambda \in \pi\mathbb{Z}$ (Theorem~\ref{thm:LINT}) --- and
identically on the conservative fence $(\pi/2)\mathbb{Z}$, whose extra
points are all inconsistent or indecisive, as the tabulation shows:
at $\lambda = \pi/2$ the admissible set (intersected with $N_g \ge 3$ from
leg L1) is empty; at $\lambda = \pi$ it is $\{3\}$; at
$\lambda = 3\pi/2$ it is $\{3,4\}$; and it grows monotonically thereafter.
Consistency (non-emptiness) excludes $\lambda = \pi/2$; decisiveness ---
no residual discrete choice of $N_g$, which would be a hidden knob in the
sense forbidden by constitutional axiom AX-4 (\MSMF{}) --- excludes
$\lambda \ge 3\pi/2$. Hence $\lambda = \pi$ uniquely (Route 1). Without the
lattice, consistency alone requires $3\pi \le 3\lambda$, i.e.\
$\lambda \ge \pi$, and \MSMF{}-minimality selects the infimum, attained at
$\lambda = \pi$ (Route 2). Route 3 is the no-new-scales reading: the
capacity quantum coincides with the smallest toll of the ledger
(\ref{eq:step-action}), so the axiom system introduces no second action
scale. $\square$

\paragraph{Why this is not a scandal.} Relative invariants are the normal
condition of bulk--boundary physics: the level of a Chern--Simons theory
is invisible to the conformal boundary algebra until one asks a
paired question (a quantized Hall current, an edge central charge). The
capacity is the paired question here, and the classification results
below are exactly the discipline of asking it without freedom.

\subsection{Proof of Theorem~\ref{thm:robust} (form robustness)}
The natural degree-4 characteristic classes of the stratum's canonical
bundles are generated by $h^2$ (the tautological class squared), $c_1^2$
and $c_2$ of the tangent bundle, and $p_1 = c_1^2 - 2c_2$; on $\CPN$ these
pair against the generator $[\CP^2]$ of $H_4$ as $1$, $N_g^2$,
$N_g(N_g-1)/2$, and $N_g$ respectively. For each scaling $\chi(N_g)$, the minimal
lattice unit consistent with $N_g = 3$ is $\lambda_{\min} =
\lceil 3\pi/\chi(3)\rceil_{(\pi/2)\mathbb{Z}}$; the admissible set at
$\lambda_{\min}$ is then computed directly:
\begin{itemize}[leftmargin=2.2em]
\item $\chi = c_2$, in full: $c_2(T\CPN)$ pairs to $\binom{N_g}{2}
 = N_g(N_g-1)/2$, so $\mathrm{Cap}(N_g) = \lambda N_g(N_g-1)/2$ and admissibility
reads $\pi N_g(N_g-1)/2 \le \lambda N_g(N_g-1)/2$, i.e.\ $\lambda \ge \pi$
\emph{independently of} $N_g$: cost and capacity are proportional, every
$\lambda \ge \pi$ admits all $N_g \ge 3$, no $\lambda$ excludes anything,
and no choice is decisive. A capacity that scales exactly like the cost
can never bound it. Excluded.
\item $\chi = c_1^2$, in full: $c_1(T\CPN)^2$ pairs to $N_g^2$, so
admissibility reads $N_g(N_g-1)/2 \le (\lambda/\pi) N_g^2$, i.e.\
$(N_g-1)/(2N_g) \le \lambda/\pi$. The left side increases from $1/3$ at
$N_g = 3$ toward $1/2$; consistency (admit $N_g=3$) requires
$\lambda \ge \pi/3$, while decisiveness (exclude $N_g=4$) requires
$\lambda < 3\pi/8$. The window $[\pi/3, 3\pi/8)$ contains no point of the
lattice $(\pi/2)\mathbb{Z}$ --- its lower edge already exceeds no lattice
point below $\pi/2$, and $\pi/2 > 3\pi/8$ --- so on the lattice every
choice is either inconsistent or indecisive (at $\lambda = \pi/2$,
\emph{all} $N_g$ are admitted, since $(N_g-1)/2N_g < 1/2$ always). Excluded.
\item $\chi = h^2$: $\lambda_{\min} = 3\pi$; admissible set $\{3\}$.
Viable --- the constant form (Formulation A of Appendix~\ref{app:C}),
with a residual decisive range closed by minimality.
\item $\chi = p_1$: $\lambda_{\min} = \pi$; admissible set $\{3\}$; the
decisive unit is unique on the lattice. Viable --- the extensive form
(Formulation B), strictly the more \MSMF{}-minimal option.
\end{itemize}
Both survivors saturate exactly at the admissible value:
$\mathrm{Cap}(3) = 3\pi = \Stot(3)$ in either form. $\square$

\paragraph{The full admissibility tables.} For the record, and for any
reader re-deriving the classification, the complete admissible sets (with
$N_g \ge 3$ from leg L1) across the lattice:
\begin{table}[htbp]
\centering\small
\caption[The complete admissibility tables, all forms and units.]{The full classification, for the record and for any reader re-deriving it: each capacity form against each lattice unit, with the admissible set of generation counts in every cell, taken with $N_g \ge 3$ from leg~L1. Read across a row and a form's behavior as the unit coarsens is visible; read down a column and the forms are compared at a fixed unit. Exactly one cell in the whole array is the singleton $\{3\}$ at a unit the geometry independently fixes.}\label{tab:admissibility-full}
\begin{tabular}{lccccc}
\toprule
& $\lambda = \pi/2$ & $\pi$ & $3\pi/2$ & $2\pi$ & $5\pi/2$ \\
\midrule
$\chi = h^2$ (Cap $=\lambda$) & $\varnothing$ & $\varnothing$ &
$\varnothing$ & $\varnothing$ & $\varnothing$ \\
\quad\ldots until $\lambda = 3\pi$ & \multicolumn{5}{l}{$\{3\}$ at
$\lambda \in [3\pi, 11\pi/2]$ on the lattice; $\{3,4\}$ from $6\pi$} \\
$\chi = p_1$ (Cap $=\lambda N_g$) & $\varnothing$ & $\{3\}$ & $\{3,4\}$ &
$\{3,4,5\}$ & $\{3,\ldots,6\}$ \\
$\chi = c_1^2$ (Cap $=\lambda N_g^2$) & all $N_g\ge 3$ & all & all & all &
all \\
$\chi = c_2$ (Cap $=\lambda\binom{N_g}{2}$) & $\varnothing$ & all $N_g\ge 3$
& all & all & all \\
\bottomrule
\end{tabular}
\end{table}

Reading the table columnwise: the only cell anywhere that is decisive
\emph{and} whose unit is unique for its row is $(p_1, \pi)$; the $h^2$ row
becomes decisive only at $3\pi$ and stays decisive across a range (the
residual knob closed by minimality); the $c_1^2$ row admits everything as
soon as it admits anything; the $c_2$ row is all-or-nothing by
proportionality. The arithmetic of every cell is one line of the
inequality $N_g(N_g-1)/2 \le (\lambda/\pi)\chi(N_g)$.

\paragraph{The naturality fence, stated as such.} The classification
quantifies over \emph{single} natural characteristic classes.
Arbitrary rational combinations of classes could tune any admissible set
and are excluded by fiat --- this is a definability premise of
Axiom~\ref{ax:coupling}, printed here so that no reader mistakes it for a
theorem. Its justification is the usual one for naturality fences: a
capacity that is not a single natural invariant of the canonical geometry
would itself be a hidden structure demanding explanation, against \MSMF{}.


\subsection{The integrality theorems: proof of Theorem~\ref{thm:LINT}}
\label{ssec:G-LINT}
The conjectural form of the integrality statement --- ``the descended
periods are integrally quantized, so the unit lies on the half-quantum
lattice'' --- bundles three claims with different subjects, different
proof technologies, and, it turns out, different grades: a claim about
the \emph{class} (does it pair integrally?), a claim about \emph{cycles}
(does any holomorphic object carry off-lattice action?), and a claim
about the \emph{coupling} (why is the unit itself quantized?). This
subsection proves them in that order. The first two are rigorous; the
third is the level-quantization argument of the Chern--Simons analogy
(\S\ref{ssec:level-analogy}) run quantitatively, derived at physics
rigor with its single topological residue stated as
Definition~\ref{def:RINT}.

\paragraph{Component (A): pairing integrality.}

\begin{theorem}[L-INT-A]
\label{thm:LINTA}
Under the descended geometry of Theorems~\ref{thm:AD1}--\ref{thm:AD3},
the Berry line bundle $L \to \CPN$ is an honest complex line bundle, its
curvature has periods in $2\pi\mathbb{Z}$, and every candidate capacity
class of Axiom~\ref{ax:coupling} pairs integrally with the generator of
$H_4(\CPN;\mathbb{Z})$:
\begin{equation}
\bigl\langle \chi, [\CP^2] \bigr\rangle \in \mathbb{Z},
\qquad \chi \in \{h^2,\ c_1^2,\ c_2,\ p_1\}.
\label{eq:LINTA}
\end{equation}
\end{theorem}

\begin{proof}
The stratum's points are rank-one projectors realized by the runtime as
operators on the fixed boundary Hilbert space (Theorem~\ref{thm:AD2}).
The assignment $x \mapsto \mathrm{im}\,P(x)$ is therefore a rank-one
\emph{subbundle of the trivial Hilbert bundle} over $\CPN$. A subbundle
of a trivial bundle is an honest vector bundle: there is no gerbe, no
torsion ambiguity, no differential-cohomology subtlety to resolve,
because the bundle is constructed, not postulated. Its first Chern class
$c_1(L)$ is an element of $H^2(\CPN;\mathbb{Z})$ by the definition of
Chern classes --- integral before any curvature is mentioned.
Theorem~\ref{thm:AD3} identifies the descended connection with the Berry
connection on $L$ and its curvature with $F_{\mathrm{Berry}}$;
Chern--Weil then says $[F_{\mathrm{Berry}}/2\pi]$ represents an integral
class, so every period of $F_{\mathrm{Berry}}$ lies in
$2\pi\mathbb{Z}$ --- Dirac quantization, obtained rather than imposed:
single-valued parallel transport around every closed loop is automatic
for an honest bundle.

For the pairing: $H^*(\CPN;\mathbb{Z})$ is the truncated polynomial ring
on the integral generator $h = c_1(\mathcal{O}(1))$, and $L$ is the
tautological $\mathcal{O}(-1)$, so $c_1(L) = -h$ (the sign convention of
Appendix~\ref{app:F}; the numerically computed Chern number of the
tautological family is exactly $-1$). Every candidate class is a
polynomial in integral classes: $c(T\CPN) = (1+h)^N_g$ truncated, so
$c_1 = Nh$, $c_2 = \tfrac12 N_g(N_g-1)\,h^2$, and
$p_1 = c_1^2 - 2c_2 = N_g h^2$. Pairing the degree-4 parts against
$[\CP^2]$ gives $\langle h^2\rangle = 1$, $\langle p_1\rangle = N_g$,
$\langle c_1^2\rangle = N_g^2$, and
$\langle c_2\rangle = \tfrac12 N_g(N_g-1)$ --- all integers.
\end{proof}

\paragraph{What made this free.} Before the adiabatic theorems, the
descended class was a pullback from the bulk, and its integrality was
hostage to the bulk's differential-cohomology data --- the hard route.
The content of Theorem~\ref{thm:AD3} is that the class is the stratum's
\emph{own}: the Berry bundle exists because the runtime's projectors
exist. Component (A) is not a shortcut around Hopkins--Singer; it is the
observation that, after the adiabatic theorems, this component of the
problem no longer lives where Hopkins--Singer machinery is needed.
\emph{Physics intuition:} a Berry phase around a closed loop is a
physical, measurable phase; if the flux through a closed two-cycle were
not a multiple of $2\pi$, transporting a state around the cycle's
boundary two different ways would disagree about physics. The runtime
cannot disagree with itself about a projector it constructed --- that is
the whole proof, stated without bundles.

\paragraph{Component (B): the toll lattice.}

\begin{theorem}[L-INT-B]
\label{thm:LINTB}
With the cohort normalization $\int_{\mathrm{line}}\omega_{FS} = \pi$
(Appendix~\ref{app:F}), the action of every compact holomorphic curve
$C \subset \CPN$ is
\begin{equation}
\int_C \omega_{FS} \;=\; \pi \cdot \deg(C) \;\in\; \pi\mathbb{Z}_{>0}.
\label{eq:LINTB}
\end{equation}
In particular no holomorphic object with off-lattice action exists, and
the step actions $\pi k$ of the toll ledger (\ref{eq:step-action}) are
points of this lattice rather than assumptions about it.
\end{theorem}

\begin{proof}
In cohomology $[\omega_{FS}] = \pi\,h$ with $h$ the integral generator
--- this \emph{is} the normalization, fixed by the quantum-geometric
identification $|F_{\mathrm{Berry}}| = 2\,\omega_{FS}$ of
Theorem~\ref{thm:AD3} together with the integral $c_1$ of
Theorem~\ref{thm:LINTA}. A compact holomorphic curve defines a homology
class $[C] = \deg(C)\,[\mathrm{line}]$ with
$\deg(C) \in \mathbb{Z}_{\ge 0}$ --- an \emph{integer} because
$H_2(\CPN;\mathbb{Z}) = \mathbb{Z}$ is generated by the line, and
non-negative because holomorphic curves pair non-negatively with the
Kaehler class. Hence
$\int_C \omega_{FS} = \langle \pi h, [C]\rangle = \pi\deg(C)$. By
Wirtinger's theorem (holomorphic curves are calibrated by
$\omega_{FS}$) this integral is simultaneously the \emph{area} of $C$:
holomorphic area on the family moduli is quantized in units of $\pi$.
\end{proof}

\paragraph{The loophole this closes.} The exclusion of $N_g \ge 4$ is an
inequality between a quadratic expense and a linear income, and the
expense is charged on holomorphic representatives. An adversary could
ask: is there a cheaper holomorphic object --- an off-lattice instanton
--- that saturates the budget differently? Theorem~\ref{thm:LINTB}
answers: there is nothing in the holomorphic category off the lattice
$\pi\mathbb{Z}$, at any degree, for any $N_g$. The expense side of the
capacity theorem is bounded below lattice-theoretically, not merely on
the representatives exhibited. (Verification: the Fubini--Study areas of
the degree-$d$ rational curves $z \mapsto [1 : z : z^d]$, computed in
closed form in the companion harness, are exactly $d\pi$ for
$d = 1, 2, 3$.)

\paragraph{Component (C): the unit lattice.}
The remaining question is why the \emph{coupling} $\lambda$ multiplying
an integral pairing should itself be quantized. This is where the
Hopkins--Singer weight actually sits. Four steps.

\emph{Step (C1) --- the premise.} The capacity enters the runtime as an
action: sector configurations are weighed by phases $e^{iS}$ in which
the family term contributes
$S_{\mathrm{fam}} = \lambda \times (\mathrm{pairing})$. This is the
reading under which the toll ledger was derived and under which the
budget competes against the throughput income; it is an interpretive
premise of component (C), and it is listed in the statement of
Theorem~\ref{thm:LINT}.

\emph{Step (C2) --- the naive lattice.} Deform the configuration by the
generator of the family sector's topological charge: the pairing shifts
by $1$ (Theorem~\ref{thm:LINTA}: the generator pairs to exactly $1$).
Single-valuedness of $e^{iS}$ under this shift requires
$\lambda \in 2\pi\mathbb{Z}$ --- the standard bosonic Chern--Simons
level quantization.

\emph{Step (C3) --- the fork: the naive lattice is not viable.} Here the
classification of Lemma~\ref{lem:unit} turns from bookkeeping into a
discovery. The decisive window for the unit is $[\pi, 3\pi/2)$: large
enough that the $N_g = 3$ expense $3\pi$ fits the income $3\lambda$, small
enough that the $N_g = 4$ expense $6\pi$ does not fit $4\lambda$. Classify
the candidate lattices by their decisive sets (the lattice points whose
admissible set, against $N_g \ge 3$ from leg L1, is exactly $\{3\}$):
\begin{table}[htbp]
\centering\small
\caption[Candidate lattices, and which admits a decisive point.]{The three candidate lattices --- bosonic, spin-refined and the conservative half-quantum fence --- classified by their decisive sets, meaning the lattice points whose admissible generation count is exactly $\{3\}$ once leg~L1 is imposed. The bosonic lattice has none: on it the argument cannot close at all. The spin-refined lattice has exactly one. That the decisive point is unique, and that it is the one the spin structure independently selects, is the content of the leg.}\label{tab:lattice-classification}
\begin{tabular}{lll}
\toprule
lattice & decisive set & status \\
\midrule
$2\pi\mathbb{Z}$ (bosonic) & $\varnothing$ & no decisive point exists \\
$\pi\mathbb{Z}$ (spin-refined) & $\{\pi\}$ & decisive, uniquely \\
$(\pi/2)\mathbb{Z}$ (conservative fence) & $\{\pi\}$ & decisive,
uniquely \\
\bottomrule
\end{tabular}
\end{table}

$2\pi\mathbb{Z} \cap [\pi, 3\pi/2) = \varnothing$: on the bosonic
lattice the smallest consistent unit is $2\pi$, which admits
$\{3,4,5\}$ --- consistent but indecisive. If step (C2) were the whole
story, the lattice route (Route 1 of Lemma~\ref{lem:unit}) would not be
incomplete; it would be \emph{wrong}, and only Routes 2--3 would
survive. The refinement of the next step is therefore load-bearing for
the entire lattice presentation --- a fact the half-quantum phrasing of
the toll ledger would disguise as a safety margin.

\emph{Step (C4) --- the spin refinement, and why the runtime has already
chosen it.} On a spin bulk, the quantization of the relevant quadratic
functional is shifted: the flux-quantization argument of
\citet{Witten1996}, made rigorous by the quadratic refinement of
\citet{HopkinsSinger2005}, replaces the naive square $x^2$ by the
refinement $2q(x) = x(x-1)$, and single-valuedness is imposed on
$\lambda\,x(x-1)$, whose increment under $x \to x+1$ is $2\lambda x$ ---
\emph{even} --- so $\lambda \in \pi\mathbb{Z}$ suffices. The \BIsix{}
bulk carries $\mathrm{Spin}(1,5)$ structure (it hosts the cohort's
fermions; \citealp{ArisakaA9SM}) --- an imported premise, listed in the
theorem. And the cohort's own arithmetic shows the runtime uses the
refined functional and not the naive square: the cascade's total toll is
exactly $\pi\,q(N_g)$ with $q(x) = x(x-1)/2$ the refinement's quadratic
form --- the half-integer structure is not imported to rescue the
argument; it is already present in the object being priced. Hence
$\lambda \in \pi\mathbb{Z}$, and by step (C3) the decisive point is
uniquely $\lambda = \pi$ --- \emph{invariantly}: a reader who doubts
that the refinement transfers cleanly from its M-theory home to the
\BIsix{} seam may retreat to the conservative fence
$(\pi/2)\mathbb{Z}$, where the decisive point is also uniquely $\pi$.
Only the non-viable bosonic lattice dissents, and it dissents by
emptiness, not by preferring a different value.

\paragraph{The residue.}

\begin{definition}[R-INT: the topological residue]
\label{def:RINT}
R-INT is the statement that the specific 6D \BIsix{} seam topology ---
the bulk $M_6$ with its boundary and the embedded family stratum ---
globally supports the quadratic refinement: the class $[X_4]$ admits the
refinement $q$ compatibly with restriction to the stratum, i.e.\ the
relevant torsion in $H^4(M_6;\mathbb{Z})$ and the $w_4$-type obstruction
vanish for the actual seam.
\end{definition}

Everything on the stratum itself is explicit and torsion-free
($H^*(\CPN;\mathbb{Z})$ has no torsion; every class and pairing above is
computed). R-INT lives entirely in the \emph{ambient} seam --- it is
topological bookkeeping on a specific manifold, of the same kind as
checking orientability, not a conceptual gap --- and it is the declared mandate of Appendix~\ref{app:H} --- met on the benchmark background by its $\kappa$-bridge theorem, which proves
the pushforward is the Berry bundle and closes both parts of R-INT
there. Its failure cost
is stated once more for the record: without R-INT, step (C4) loses its
geometric license, the lattice retreats to the conservative fence or
falls to presentational status --- and $\lambda = \pi$ survives on
Routes 2--3 of Lemma~\ref{lem:unit} with the value of $N_g$ unmoved.

\paragraph{The premises of Theorem~\ref{thm:LINT}, inventoried.}
Component (A): the descended geometry (Born-grade;
Theorems~\ref{thm:AD1}--\ref{thm:AD3}). Component (B): the same, plus
the normalization of (\ref{eq:step-action}). Component (C): the same,
plus the action-level reading of the coupling clause (step C1) and the
bulk spin structure (step C4), with R-INT the residue. No step consumes
$N_g = 3$, $K = 35$, or $J = 7$: the decisive-set computations
\emph{conclude} $\{3\}$; they do not assume it. The grade vocabulary is
the cohort's: components (A) and (B) are theorems given the adiabatic
package; component (C) is derived-conditional at physics rigor with its
residue named.

\paragraph{Physics intuition.} The three components, compressed. (A):
the stratum cannot carry fractional flux, because its states are the
runtime's own projectors, and a fractional flux would make their
transport multivalued. (B): holomorphic area is quantized --- the
landscape sells relaxation in whole tolls only, so there is no discount
instanton. (C): the price of a whole toll is itself pinned by the same
consistency that pins a Chern--Simons level --- and the bulk's fermions,
by forcing spin structure, halve the naive price lattice to exactly the
one on which the family count is decided. One bundle underwrites all
three: the tolls, the budget, and the unit are three readings of the
Berry line bundle that the adiabatic theorem forces.

\subsection{Verification record}
Every numerical and structural claim of Appendices F and G, and of
\S\ref{ssec:capacity}, is checked programmatically in the companion
harnesses to this paper (Kato transport unitarity and holonomy against the
half-solid-angle law; the quantum geometric tensor against the
Fubini--Study metric with proportionality constant $1$; the period
integrals $\pi$ and $-2\pi$; the signature and Morse tables for
$N_g = 1,\ldots,8$; the exclusion, unit-classification, and form-robustness
tables; the exact Chern number, the degree-$d$ holomorphic areas $d\pi$,
and the decisive-set classification of \S\ref{ssec:G-LINT}; and the
pointwise vanishing of $\tr(R\wedge R)$ on the canonical
bulk by full exterior-algebra recomputation). The falsifiable content of
the architecture is thereby reproducible from the paper's own statements.
\section{The inflow bridge: how AX-B is enforced}
\label{app:H}

\S\ref{ssec:axiom} states the coupling clause AX-B at its true,
minimal size and \S\ref{ssec:capacity} spends it. This appendix
supplies the pair-level mathematics that enforces it on the benchmark
background of the intended \GG{} class, and discharges the topological
residue R-INT (Definition~\ref{def:RINT}) there.

The instrument is the Hopkins--Singer quadratic functor: the
fivebrane-pattern pushforward that turns six-dimensional Wu-structure
data into a line bundle on the family moduli. Six theorems follow,
T-INF-1 to T-INF-6. Their conclusion is the $\kappa$-bridge: the
bulk's pushed-forward bundle \emph{is} the Berry bundle the boundary's
own adiabatic theorems construct (Theorems~\ref{thm:AD1}--\ref{thm:AD3}).

Two things should be said before the reader starts, because they bound
what follows. \emph{First}, the theorem is an agreement between the
two faces of one seam, not a derivation of either face from the other;
the bulk side of the computation consumes no quantum-mechanical input,
and \S\ref{app:H-echoes} shows the two obstruction theorems of
\S\ref{ssec:axiom} reappearing from inside the formalism. \emph{Second},
the numerical scope is the benchmark background: the warped and relative
refinements, R-WARP and R-REL, are stated in \S\ref{app:H-refine} and
closed there at class level by the rigidity theorems T-INF-7 and
T-INF-8, with what survives of each named exactly.

This appendix is written to be read on its own. It pins its
conventions before it computes, and nothing in the main line depends
on the machinery here --- only on the fact that the clause it enforces
can be enforced.

\subsection{A century of quantization, in one page}
\label{app:H-lineage}

The instrument of this appendix was built over ninety years, by
physicists who kept discovering that an integer they had measured
was topology they had not yet named, and by mathematicians who then
named it. The road is worth walking before the computation starts, because every object in \S\ref{app:H-channel}--\S\ref{app:H-rint} is the end of it.

\subsubsection{The external road: from Dirac's monopole to the
quadratic functor}
\label{ssec:external-road}

\paragraph{1902: conservation built into geometry.} The story
begins, fittingly, with an identity rather than an equation of
motion. \citet{Bianchi1902} proved that the curvature of a
connection automatically satisfies $dR + [A, R] = 0$: geometry
polices its own bookkeeping. Everything in this appendix descends from
learning, over a century, exactly when that identity can be
\emph{modified} and what the modification costs.

\paragraph{1931: the first topological integer of physics.}
\citet{Dirac1931} showed that a single magnetic monopole anywhere in
the universe forces electric charge to be an integer multiple of a
unit: the wavefunction of a charged particle transported around the
monopole must return to itself, and single-valuedness quantizes the
flux. This is the ur-argument of the subject --- \emph{a phase that
must close forces an integer} --- and the reader will meet it again,
essentially unchanged, at three places in this appendix: in the Dirac
quantization of the Berry bundle, in the level argument behind this paper's unit lattice, and in the Wu-evenness that drives
T-INF-3.

\paragraph{1984: the anomaly becomes geometry.}
\citet{GreenSchwarz1984} discovered that a gauge theory whose
quantum phase fails to close --- an anomalous theory --- can be
rescued by \emph{modifying the Bianchi identity}: let a tensor field
$H_3$ satisfy $dH_3 = X_4 \neq 0$, with $X_4$ built from the very
curvatures whose anomaly is to be canceled, and the failure of one
bookkeeping exactly pays the failure of the other. The modified
identity stopped being a defect and became a design principle.
\BIsix{} is the cohort's Green--Schwarz identity, with its
coefficient vector locked by \OGNfive{} arithmetic rather than
chosen \citep{ArisakaA3BI6}.

\paragraph{1985: the third datum.} A gauge field carries three
kinds of information: its charge lattice (Dirac's integers), its
flux density (the curvature), and its \emph{holonomies} --- the
phases around closed paths that neither of the first two determines.
\citet{CheegerSimons1985} built the mathematical object that carries
all three at once, the differential character. The third datum is
the one closed-system physics can ignore and open-system physics
cannot: torsion and holonomy are invisible until something wraps a
cycle, and observation --- in the cohort's reading --- is precisely
a wrapping.

\paragraph{1996: spin halves the lattice, and physics orders
mathematics it does not yet have.} \citet{Witten1996flux} found that
the four-form flux of M-theory is quantized not on the integers but
on integers \emph{shifted} by half a gravitational class,
$\lambda = p_1/2$, whenever the manifold is spin: the fermions'
existence changes the bosonic field's lattice. In the same year
\citet{Witten1996five} showed that the fivebrane's self-dual tensor
field has no Lagrangian at all --- its partition function must be
\emph{defined} as a section of a line bundle over the space of
boundary data, constructed from a quadratic refinement of the flux
pairing. The construction sketched there needed mathematics that did
not yet exist.

\paragraph{2005: the mathematics arrives.} \citet{HopkinsSinger2005}
supplied it: the differential cocycle $(c, h, \omega)$ --- charge,
holonomy, and flux as one object --- and the quadratic functor
$\kappa$, which takes a family of manifolds with a degree-$2k$
differential integral Wu structure and returns, in the fivebrane
case, exactly the line bundle Witten's partition function needed.
The modern descendants \citep{Freed2000, SatiSchreiber2024,
Monnier2016, DaiFreed1994} turned the construction into a doctrine:
\emph{to quantize a higher gauge field is to choose the cohomology
theory its fluxes live in}, and a Green--Schwarz-modified Bianchi
identity dictates a twisted differential theory. That doctrine is
the frame of this appendix; the functor is its only instrument.

\paragraph{The thread, in one sentence.} Dirac: a closing phase
forces an integer. Green--Schwarz: a modified identity can trade two
failures. Cheeger--Simons: keep the holonomy, not just the flux.
Witten: spin shifts the lattice, and families of six-manifolds hand
you line bundles. Hopkins--Singer: all of the above is one functor.
This appendix: applied to the \GG{} seam, that functor's output is the
Berry bundle.


\subsection{The seam, the stratum, and the functor}
\label{app:H-machinery}

\subsubsection{The seam (from \AThree{})}
The intended \GG{} background class of \citet{ArisakaA3BI6} fixes
the bulk: $\Msix = \Mfour \times I_r \times S^1_\theta$ with
$\Mfour$ closed, oriented, spin, $H^1(\Mfour;\mathbb{Z}) = 0 =
H^3(\Mfour;\mathbb{Z})$, torsion-free $H_1$; APS-compatible collars
at the interval ends, or the doubled closed model
$\Mfour \times \Ttwo$ with the structural torus
$\Ttwo = S^1_r \times S^1_\theta$; gauge group
$\mathrm{SU}(5) \times U(1)_B \times U(1)_{Qi}$ with bundles pulled
back from the base and trivial on the torus; the modified Bianchi
identity $dH_3 = X_4$ with the locked coefficient vector
$(1, -1, -\tfrac25, -\tfrac25)$; and the primitive integral lift
\begin{equation}
\widehat X \;=\; \frac{t}{8\pi^2}\,X_4 \;\in\; H^4(\Msix;\mathbb{Z}),
\qquad
\gamma = \frac{s}{t} = \frac{L-N}{L} = \frac25,\quad (s,t) = (2,5).
\label{eq:xhat}
\end{equation}
The \emph{benchmark} is $\Mfour = S^4$ (the compactified physical
base), on which \AThree{}'s reservoir existence theorem supplies the
unit background flux, written here $\Oorb = 1$. Cohomology of the
closed benchmark (ranks, degrees $0$--$6$): $[1,2,1,0,1,2,1]$, with
$H^2(\Msix) = \mathbb{Z}\,\tau$ ($\tau$ the structural-torus class,
$\int_{\Ttwo}\tau = 1$) and $H^4(\Msix) = \mathbb{Z}\,\sigma$
($\sigma$ the base class, $\int_{S^4}\sigma = 1$).

\subsubsection{The stratum (from \S\ref{ssec:moduli})}
The family-projector moduli is $\CPN$ with hyperplane generator
$h \in H^2(\CPN;\mathbb{Z})$. The boundary runtime's adiabatic
package forces, at the Born grade of conditionality --- premises
shared with the Constitutional Born Theorem
\citep{ArisakaA8QM} --- the Berry
connection on the tautological bundle $\check a$ with
\begin{equation}
c_1(\check a) = -h, \qquad
\Bigl[\tfrac{F_a}{2\pi}\Bigr] = -h, \qquad
|F_a| = 2\,\omega_{FS}, \qquad
\int_{\mathrm{line}} \omega_{FS} = \pi,
\label{eq:berry-conv}
\end{equation}
in the step-action normalization of Appendix~\ref{app:F}. These
four equalities are the conventions this appendix computes against; no
further sign or factor choices are made downstream.

\subsubsection{The functor, and the three factors to watch}
Exactly one construction in the mathematical literature has that
type from six-dimensional data: the Hopkins--Singer quadratic
functor \citep{HopkinsSinger2005}. For an $\Hc$-oriented family
$E \to S$ of relative dimension $4k - i$ with $i \le 2$, equipped
with a differential integral Wu structure $\lambda$ of degree $2k$,
Hopkins--Singer construct
\begin{equation}
\kappa_{E/S} \colon \Hc^{2k}_{\nu}(E) \longrightarrow \Hc^{\,i}(S),
\qquad
\kappa(\lambda) \equiv \tfrac18 \textstyle\int_{E/S}
\bigl(\lambda \cup \lambda - \check L_{4k}\bigr)
\ \ (\mathrm{mod\ torsion}),
\label{eq:kappa-def}
\end{equation}
natural under base change and transitive in composite families. In
six fiber dimensions the bookkeeping $4k - i = 6$, $i \le 2$ has the
unique solution $(k, i) = (2, 2)$: the refinement acts on
degree-four differential cohomology, the Wu structure has degree
four --- and exists canonically, since the Wu class $\nu_4$ vanishes
on any six-manifold --- and the value $\kappa(\lambda) \in \Hc^2(S)$
is a \emph{line bundle with connection on the base of the family}.
This is the pattern of Witten's fivebrane partition function
\citep{Witten1996five}, whose rigorous form was the motivation of
\citet{HopkinsSinger2005}, and of the Wu-structure field theories of
\citet{Monnier2016}; its flux-quantization reading in the modern
sense is \citet{SatiSchreiber2024, Freed2000}.

The claim of this appendix, proved as Theorem \mbox{T-INF-6}: on the
benchmark background of the intended \GG{} class, with the seam
fibered over the family-projector moduli,
\begin{equation}
\kappa_{E/S}(\lamBI) \;=\; \check a
\qquad\text{--- the tautological/Berry line bundle, } c_1 = -h,
\label{eq:claim}
\end{equation}
with every integer in the assembly derived: the channel charge
$c = L - N = 2$ from the abelian cross term of the primitive
\BIsix{} lift at minimal quanta, and the background number
$m = 2\,\Oorb = 2$ from the orientation doubling of the unit
reservoir. The bulk, pushed forward, is the boundary's bundle.

The functor \eqref{eq:kappa-def} lives in the differential-character
tradition of \citet{CheegerSimons1985}: its values carry holonomy
and torsion, not curvature alone, which is exactly why it can see a
coupling that every curvature-only formalism provably misses.
Equation \eqref{eq:kappa-def} carries three factor risks, each
audited where it arises: the $\tfrac18$ of the normalization
(resolved by the cross-term $2$ and the two derived $2$'s,
\S\ref{sec:structure}--\ref{sec:attribution}); the $\check L$-term
(vanishes identically on the product seam, Lemma~\ref{lem:Lterm});
and the mod-torsion caveat (empty on the benchmark, where every
group in sight is torsion-free --- \S\ref{app:H-rint}).

\subsubsection{Grade vocabulary}
As throughout the cohort: \emph{rigorous} (proof from stated
premises), \emph{physics rigor} (the standard argument of the
physics literature, premises listed, no analytic gaps filled by
fiat), \emph{consistency} (agreement of independently derived
structures), with every conditionality declared at the theorem
statement, not in a footnote.

\subsection{The seam family and the channel theorem}
\label{app:H-channel}
\label{sec:channel}

\subsubsection{The model}
\begin{definition}[The benchmark seam family]
\label{def:seam-family}
The seam family is the product fibration $E = \Msix \times S \to S$,
$S = \CPN$, $\Msix = S^4 \times \Ttwo$, with vertical tangent bundle
pulled back from $\Msix$ and $\Hc$-orientation fixed by the cohort conventions of \S\ref{app:H-machinery}. A \emph{family Wu structure} is a
differential integral Wu structure $\lambda \in \Zc^4(E)$ of degree
four (existence canonical: $\nu_4 = 0$ in six dimensions), with
integer class decomposed along K\"unneth as
\begin{equation}
[\lambda] \;=\; m\,\sigma \;+\; c\,(\tau \boxtimes h)
\;+\; e\,(1 \boxtimes h^2),
\qquad m, c, e \in \mathbb{Z}.
\label{eq:lambda-class}
\end{equation}
The model's premises, printed as fences: \emph{(P-i)} the fibration
is the product --- the physical warping cannot move the class
(Theorem~T-INF-7, \S\ref{app:H-refine}), and the warped representative
is deferred to the engineering item E-3;
\emph{(P-ii)} family dependence enters through pullback classes, per
the pullback-bundle clause of \AThree{}; \emph{(P-iii)} all signs
and normalizations are those of \eqref{eq:berry-conv}, fixed before
the computation.
\end{definition}

\begin{theorem}[T-INF-1: the channel theorem]
\label{thm:channel}
On the benchmark, the family-non-trivial part of
$H^4(\Msix \times \CPN;\mathbb{Z})$ has rank exactly two, spanned by
$\tau \boxtimes h$ and $1 \boxtimes h^2$; and of these, only the
mixed channel $\tau \boxtimes h$ can contribute to
$\kappa_{E/S}$'s curvature. The family couples to the pushforward
through the structural torus, or not at all.
\end{theorem}

\begin{proof}
K\"unneth over $\mathbb{Z}$ (all groups in sight torsion-free): the
$q > 0$ part of $H^4(E) = \bigoplus_q H^{4-q}(\Msix) \otimes
H^q(\CPN)$ receives contributions only from $q = 2$ and $q = 4$,
since $H^{\mathrm{odd}}(\CPN) = 0$ and $H^3(\Msix) = 0$: namely
$H^2(\Msix) \otimes H^2(\CPN) = \mathbb{Z}\,(\tau \boxtimes h)$ and
$H^0(\Msix) \otimes H^4(\CPN) = \mathbb{Z}\,(1 \boxtimes h^2)$. For
the second statement: the fiber integration in \eqref{eq:kappa-def}
retains only terms of fiber degree six in the eight-form
$\lambda \cup \lambda$; the cross term of $m\,\sigma$
(fiber degree 4) with $c\,(\tau \boxtimes h)$ (fiber degree 2) has
fiber degree six, while every term involving
$1 \boxtimes h^2$ (fiber degree 0) tops out at fiber degree four and
dies under $\int_{E/S}$. (Harness \S A: the rank computation; the
bidegree table is Appendix~A.)
\end{proof}

\paragraph{Physics intuition.} The stratum can only speak to the
bulk in a language the bulk's topology understands. On the benchmark
the bulk offers exactly one two-dimensional ear --- the structural
torus of the Kaluza--Klein sector --- and the K\"unneth arithmetic
says the family's hyperplane class must pair with it. That the
family--bulk conversation runs through the compact directions is not
an assumption of this appendix; it is forced by the intended class's
own topology, and it lands exactly where \AThree{}'s
structural-holonomy clause and the Kaluza--Klein architecture of
\citet{ArisakaP1KK} put the compact-direction physics.

\subsection{The structure formula}
\label{sec:structure}

\begin{lemma}[The $\check L$-term vanishes on the product seam]
\label{lem:Lterm}
For the product fibration of Definition~\ref{def:seam-family}, the
vertical tangent bundle is pulled back from $\Msix$, so every
characteristic form of it --- in particular the degree-eight
integrand of $\check L_8$ --- is of pure fiber degree; and an
eight-form of pure fiber degree on a six-dimensional fiber vanishes
identically. The correction term of \eqref{eq:kappa-def} contributes
nothing, at curvature level and exactly.
\end{lemma}

\begin{theorem}[T-INF-2: the structure formula]
\label{thm:structure}
For a family Wu structure \eqref{eq:lambda-class} on the benchmark
seam family, the curvature of
$\kappa_{E/S}(\lambda) \in \Hc^2(\CPN)$ is
\begin{equation}
F_\kappa \;=\; \frac{c\,m}{4}\; F_a,
\qquad\text{with underlying class}\qquad
c_1\bigl(\kappa(\lambda)\bigr) \;=\; -\,\frac{c\,m}{4}\; h .
\label{eq:structure}
\end{equation}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:Lterm}, $\kappa(\lambda) =
\tfrac18 \int_{E/S} \lambda \cup \lambda$ on the benchmark. Expand
$\lambda\cup\lambda$ and keep fiber degree six
(Theorem~\ref{thm:channel}): the only survivor is the cross term
$2\,c\,m\,(\sigma \smile \tau) \boxtimes (h\text{-datum})$, whose
fiber pairing is $\langle \sigma \smile \tau, [\Msix]\rangle =
(\int_{S^4}\sigma)(\int_{\Ttwo}\tau) = 1$. Hence
$F_\kappa = \tfrac18 \cdot 2cm \cdot F_a = (cm/4) F_a$, and the
class statement follows from \eqref{eq:berry-conv}. The assembled
ratio $F_\kappa / F_a = cm/4$ is verified numerically at test points
of the moduli for four $(c,m)$ pairs, with the fiber volumes
computed by quadrature (harness \S B; Appendix~B).
\end{proof}

\paragraph{The factor audit, first reading.} Equation
\eqref{eq:structure} is the honest form of the inflow: the
pushforward is \emph{not} automatically the Berry bundle --- it is
$(cm/4)$ of it. A derivation that proclaimed the identification
here, with $c$ and $m$ unexamined, would be exactly the silent-factor
failure the Preface promised to avoid. The next three theorems close
the factor from inside.

\paragraph{Physics intuition.} Where does the $4$ in the denominator
come from, and who has to pay it? The $\tfrac18$ is the price of the
quadratic refinement --- the same $\tfrac18$ that normalizes a
theta characteristic, hard-wired into the functor by
\citet{HopkinsSinger2005} and not negotiable. The cross term of the
square contributes a $2$, because the background and the family
deformation each appear once in $\lambda \cup \lambda$'s mixed term.
Net: the machine pays out one quarter of the product $cm$. For the
output to be exactly one Berry bundle, physics must therefore supply
a $4$ --- and the deep content of the next three theorems is that it
supplies it in the only honest way possible: as $2 \times 2$, one
even factor from each side of the seam, with the asymmetric splits
$1 \times 4$ and $4 \times 1$ forbidden by the Wu condition. The
denominator is the functor's; the numerator is the physics'; the
theorem is that they meet.

\subsection{The quantization theorem}
\label{sec:quantisation}

\begin{theorem}[T-INF-3: $\kappa$-quantization]
\label{thm:quantisation}
The Wu condition on $\lambda$ --- its integer class congruent to
$\nu_4 = 0$ modulo two --- forces $m, c, e$ all even in
\eqref{eq:lambda-class}. Consequently $cm \in 4\mathbb{Z}$:
\begin{equation}
c_1\bigl(\kappa(\lambda)\bigr) \in \mathbb{Z}\cdot h
\qquad\text{always},
\label{eq:quantised}
\end{equation}
and the minimal non-zero value of $|cm/4|$ is $1$, attained only at
$(|c|, |m|) = (2, 2)$. In particular the factorizations
$(1,4)$ and $(4,1)$ of $cm = 4$ are excluded, and the minimal
non-zero output of the pushforward is exactly the tautological
bundle $\pm\check a$.
\end{theorem}

\begin{proof}
Evenness of the class is the definition of a differential integral
Wu structure at $\nu = 0$; evenness of each coefficient follows
because $\sigma$, $\tau \boxtimes h$, $1 \boxtimes h^2$ are a basis
of a free group. Then $cm \equiv 0 \pmod 4$, so
\eqref{eq:quantised}; the solution set of $|cm| = 4$ in even
integers is $(\pm2, \pm2)$ alone (harness \S D). Integrality of
$c_1(\kappa)$ is guaranteed independently by the Hopkins--Singer
theorem; the two routes agree, and the odd control $(c,m) = (1,2)$
--- which would give the non-integral class $-h/2$ --- is displayed
in the harness as the failure it is (\S E).
\end{proof}

\paragraph{Physics intuition.} This is the third appearance of one
phenomenon: a spin/Wu condition halves a naive lattice, and the
halving is load-bearing --- the pattern whose prototype is Witten's
shifted flux quantization \citep{Witten1996flux}. This paper's unit lattice halved from $2\pi\mathbb{Z}$ to $\pi\mathbb{Z}$ (where the decisive point
lives); \AThree{}'s tensor refinement runs on the characteristic
vector $a = 1$; and here the Wu condition quantizes the pushforward
on the integer multiples of the Berry class and forbids the
asymmetric factorizations of $4$. One refinement, three theaters ---
and in each, the physics sits exactly on the point the refinement
creates.

\subsection{The attribution theorems}
\label{app:H-attr}
\label{sec:attribution}

\subsubsection{The channel charge: $c = L - N$}

\begin{theorem}[T-INF-4: attribution of $c$]
\label{thm:attr-c}
Let the family-locked abelian excitation of the seam carry torus
flux $n_\tau \in \mathbb{Z}$ and family charge $q \in \mathbb{Z}$
under the tautological bundle. Then the abelian quadratic term of
the primitive lift \eqref{eq:xhat} writes into the mixed channel the
coefficient
\begin{equation}
c \;=\; \pm\, s\, n_\tau\, q,
\qquad s = L - N = 2,
\label{eq:attr-c}
\end{equation}
so at \MSMF{}-minimal quanta ($n_\tau = q = 1$) the channel charge
is $|c| = L - N = 2$: the first forced $2$ is the numerator of the
trace-ratio lock. Moreover the evenness demanded by
Theorem~\ref{thm:quantisation} is automatic: $s = L - N$ is even
because $N$ and $L$ are both odd.
\end{theorem}

\begin{proof}
The abelian part of the lift is
$\widehat X \supset -(s/8\pi^2)(F_B^2 + F_{Qi}^2)$. Writing the
family-locked abelian field as $F = F_\tau + q F_a$ with
$[F_\tau/2\pi] = n_\tau\,\tau$ and $[F_a/2\pi] = -h$, the cross term
is $-(s/8\pi^2)\cdot 2\,F_\tau \wedge q F_a =
-s\,(F_\tau/2\pi)\wedge(q F_a/2\pi)$, of class
$+\,s\,n_\tau q\,(\tau \boxtimes h)$ --- the mixed-channel class
with coefficient $\pm s\,n_\tau q$, the sign fixed by the
orientation convention. The class number is computed by explicit
integration over $\Ttwo \times \mathrm{line}$ in the harness
(\S A of the attribution harness: $-n_\tau q$ to $10^{-6}$ for four
quanta pairs). Minimality of the quanta is the \MSMF{} selection,
stated as a premise; the parity claim is immediate.
\end{proof}

\paragraph{Physics intuition.} The integer $s = L - N$ has been in
the cohort since \AThree{}: it is the numerator of the trace-ratio
lock $\gamma = (L-N)/L = 2/5$, the exchange rate at which the bulk
prices abelian flux against the gravitational unit --- and, in the
\OGNfive{} reading, the count of ledger slots the families leave
unused. What Theorem~\ref{thm:attr-c} adds is that this exchange
rate is also the \emph{family's toll charge into the mixed channel}:
when the minimal family-locked excitation threads one unit of flux
through the structural torus with one unit of family charge, the
seam's four-form bookkeeping records exactly $s$ units in the
channel the pushforward reads. The number was never free: it is the
same subtraction, $L - N$, doing its third job.

\begin{remark}[The excitation reading]
\label{rem:excitation}
The pullback-bundle clause of \AThree{} sets the torus flux to zero
\emph{in the background} --- and \S\ref{sec:echoes} shows the
background alone pushes forward to the trivial bundle. There is no
tension: the family coupling is precisely the minimal excitation in
the one sector the background clauses switch off and the topology
leaves open. The clause is why the coupling is invisible to every
one-sided computation; the excitation is how the pair turns it on.
\end{remark}

\begin{remark}[The parity echo, fourth appearance]
\label{rem:parity}
$s = L - N$ is even iff $N \equiv L \pmod 2$ --- with $L = 5$, iff
$N$ is odd. For even $N$ the minimal family channel has odd $c$ and
violates the Wu condition outright: the coupling would be forced to
doubled quanta. This is consistency, not exclusion --- but the
odd-$N$ condition of the signature route, the stratum-spin reading,
and the unit-halving argument here surfaces a fourth time, in the
channel arithmetic itself (harness \S B, $N = 2$--$7$ table). At
$N = L$ the channel coefficient vanishes identically; nothing in the
cohort rides on this curiosity, and it is recorded as one.
\end{remark}

\subsubsection{The background number: $m = 2\,\Oorb$}

\begin{theorem}[T-INF-5: attribution of $m$]
\label{thm:attr-m}
The closed benchmark $\Msix = S^4 \times \Ttwo$ is the orientation
double of the physical collar seam $S^4 \times I_r \times
S^1_\theta$. Pairings double under doubling, so the unit reservoir
flux of \AThree{}'s benchmark existence theorem ($\Oorb = 1$ on the
physical half) lands on the closed model as
\begin{equation}
m \;=\; 2\,\Oorb \;=\; 2 .
\label{eq:attr-m}
\end{equation}
The second forced $2$ is the orientation double. Moreover
$m = 2\Oorb$ is even for \emph{every} reservoir value: the closed
model is always Wu-compatible, so the Wu condition constrains the
closed bookkeeping only, never the physical seam; and $m \ne 0$ iff
the reservoir is non-trivial --- the non-vanishing of the coupling
has a concrete bulk carrier.
\end{theorem}

\begin{proof}
The doubling identity $\langle \widehat X, [\mathrm{double}]\rangle
= 2\,\langle \widehat X, [\mathrm{half}]\rangle$ is the reflection
computation, verified to $10^{-9}$ on an arbitrary collar profile in
the harness (\S C); the identification of the closed benchmark as
the double, and the unit reservoir, are the constructions of
\citet{ArisakaA3BI6} (APS doubling; benchmark reservoir existence),
imported at their own grade. Evenness and the non-vanishing
statement are immediate from \eqref{eq:attr-m}.
\end{proof}

\paragraph{Physics intuition.} The second $2$ is a mirror. The
theorems of \AThree{} are proved on a closed model of the bulk,
built by reflecting the physical collar geometry through its end ---
the standard Atiyah--Patodi--Singer doubling. An observer computing
a flux integral on the closed model sees every physical flux twice:
once directly, once in the mirror. So the unit reservoir of the
physical seam appears as $2$ in the closed bookkeeping --- and,
pleasingly, this also explains why the Wu condition never threatens
the physics: \emph{any} integer, doubled, is even. The evenness that
Theorem~\ref{thm:quantisation} demands of $m$ is not a constraint on
nature; it is an automatic property of reflections. Nature keeps the
unit; the mirror supplies the two.

\subsubsection{The placement of the physical datum, decided by the pair}

\begin{proposition}[The placement fork]
\label{prop:placement}
Two placements of the \BIsix{} datum in the quadratic machine are a
priori available: (A) the family coupling deforms the Wu structure
$\lambda$ itself; (B) it rides in the refined field $\check x$, the
physical bundle being $q^\lambda(\check x) = \kappa(\lambda -
2\check x)$. Under (B), with $\lambda$ gauge-fixed to the minimal
even class, the pushed-forward bundle has curvature
$c_x m_x F_a$ with $|c_x| = s = 2$, hence $|c_1| \ge 2$. The
boundary's bundle has $c_1 = -1$ exactly
(Theorem~\ref{thm:AD3}). Placement (B) is refuted by the
pair; placement (A) is selected.
\end{proposition}

This determination consumes boundary data by design. That is not a
weakness to be disguised: the boundary-relativity proposition says
the coupling is a relative invariant of the pair, so a pair-level
matching is the only admissible fixing of the placement --- and one
side of the pair, the boundary, has carried its answer since the
adiabatic theorems.

\paragraph{Physics intuition.} The quadratic machine has two slots:
the Wu structure $\lambda$ --- the \emph{ruler}, the background
datum against which fluxes are measured --- and the refined field
$\check x$ --- the \emph{measured length}. The placement question
asks which slot the \BIsix{} datum occupies, and it is a genuine
question because the formalism accepts either. But the two
placements predict different bundles: the ruler slot delivers one
Berry unit, the field slot at least two. The boundary, which has
possessed its bundle since the adiabatic theorems, measures one
unit. The \BIsix{} datum is the ruler. A formalism in which both
slots gave the same answer would have made the question
metaphysical; this one makes it empirical --- internal to the
theory, but empirical in kind.

\subsection{The $\kappa$-bridge theorem}
\label{sec:bridge}

\begin{theorem}[T-INF-6: the $\kappa$-bridge]
\label{thm:bridge}
On the benchmark seam family, with the placement of
Proposition~\ref{prop:placement} and the attributions of
Theorems~\ref{thm:attr-c}--\ref{thm:attr-m},
\begin{equation}
c_1\bigl(\kappa_{E/S}(\lamBI)\bigr)
\;=\; -\,\frac{(L-N)\,\Oorb}{2}\; h
\;=\; -\,h
\qquad\text{at unit quanta:}
\label{eq:bridge}
\end{equation}
$\kappa_{E/S}(\lamBI) = \check a$, the tautological/Berry line
bundle of the boundary, sign and normalization included.
\emph{Grade: physics rigor}, conditional on three declared inputs:
the channel model of Definition~\ref{def:seam-family} (itself
topology-forced on the benchmark by Theorem~\ref{thm:channel}), the
doubling construction and unit reservoir of \AThree{}, and
\MSMF{}-minimal quanta.
\end{theorem}

\begin{proof}
Assembly: $c = \pm(L-N) = \pm 2$ (Theorem~\ref{thm:attr-c}),
$m = 2\Oorb = 2$ (Theorem~\ref{thm:attr-m}), so $cm/4 = \pm 1$ in
\eqref{eq:structure}; the sign is fixed by the declared orientation
(the pairing positive against the cascade orientation, Appendices~\ref{app:E}--\ref{app:F}), giving $c_1 = -h$ and the
curvature $F_\kappa = F_a$ on the nose. Uniqueness of the benchmark
Wu structure (\S\ref{app:H-rint}) ensures no residual choice hides in
$\lambda$'s cocycle data.
\end{proof}

\paragraph{Physics intuition: the seam agrees with itself.} Read the
chain end to end. On the boundary side, quantum mechanics alone ---
the adiabatic theorem at the Born rule's own grade --- builds a line
bundle on the family moduli: the Berry bundle, curvature twice the
K\"ahler form. On the bulk side, the \BIsix{} seam carries a
primitive four-class whose only family-visible channel runs through
the Kaluza--Klein torus; the trace-ratio lock writes the integer
$L - N$ into that channel, and the orientation double writes
$2\,\Oorb$. The quadratic functor --- the same machine that makes
the fivebrane partition function well-defined --- converts that bulk
data into a line bundle on the same moduli. The two bundles agree
exactly, factor and sign. Each side was derived with no knowledge of
the other; the agreement is the pair-level fact the two obstruction
theorems predicted had to exist and forbade either side to own
alone. The books of the seam balance --- which is, at bottom, the
same sentence the capacity theorem speaks at $N = 3$.

\subsection{R-INT discharged on the benchmark}
\label{app:H-rint}

This paper's integrality theorem (Theorem~\ref{thm:LINT}) names as its single topological residue R-INT: that the specific six-dimensional seam
supports the quadratic refinement globally --- no torsion
obstruction in $H^4$, no $w_4$-type obstruction, and a Wu-structure
choice compatible with the stratum, uniformly in the family. On the
benchmark, every part of this is now closed.

\begin{proposition}[The absolute part]
\label{prop:rint-abs}
On the intended class $\Msix = \Mfour \times \Ttwo$ with the clauses of \S\ref{app:H-machinery}: \emph{(i)} $\Tors H^4(\Msix;\mathbb{Z}) \cong
(\Tors H_1(\Mfour))^3 = 0$ --- the integral lift \eqref{eq:xhat} is
torsion-unambiguous; \emph{(ii)} $\nu_4(\Msix) = 0$ canonically
($\nu_j = 0$ for $2j > 6$; \citealp{MilnorStasheff1974}), so
degree-four Wu structures exist, the choices a torsor over
$H^3(\Msix;\mathbb{Z}/2) \cong H^2(\Mfour;\mathbb{Z}/2)^{\oplus 2}$;
\emph{(iii)} $w_4(\Msix) = w_4(\Mfour) = \chi(\Mfour) \bmod 2 =
\sigma(\Mfour) \bmod 2 = 0$, the spin clause doing the killing
through Rokhlin's divisibility \citep{Rokhlin1952}.
\end{proposition}

\begin{proof}
(i) K\"unneth with $H^*(\Ttwo)$ free, then universal coefficients
and Poincar\'e duality on the closed oriented four-base:
$\Tors H^4 = 0$, $\Tors H^3 \cong \Tors H^2 \cong \Tors H_1$; the
clauses kill it. (ii) is the dimensional vanishing plus the standard
torsor statement; the mod-2 K\"unneth uses $H^1 = H^3 = 0$ on the
base. (iii) Whitney with the parallelizable torus, the top
Stiefel--Whitney class as the mod-2 Euler characteristic,
$\chi \equiv \sigma \pmod 2$, and $\sigma \equiv 0 \pmod{16}$ for
spin. The non-spin control $\CP^2$ ($\chi = 3$) shows the spin
clause is load-bearing. (Harness record: Appendix~B.)
\end{proof}

\begin{proposition}[The relative/family part, on the benchmark]
\label{prop:rb}
On the benchmark base $\Mfour = S^4$: \emph{(i)} the Wu torsor
$H^3(S^4 \times \Ttwo;\mathbb{Z}/2) = 0$, so the degree-four Wu
structure is \emph{unique}; \emph{(ii)} restriction and doubling
maps land in torsors over the same vanishing group, so the unique
structure is automatically compatible with the APS collars;
\emph{(iii)} the product family pulls it back to a single uniform
choice over $\CPN$. Off the benchmark (a $b_2 > 0$ base of K3 type,
torsor of order $2^{44}$) one uniform family choice always exists by
pullback, and a canonical basepoint is supplied by the
pullback-bundle clause; canonicity, not existence, is the residual
question there.
\end{proposition}

\begin{corollary}
R-INT holds on the benchmark background: the refinement exists, is
unique, restricts compatibly, and is uniform over the stratum. This paper's conditioning ledger, restricted to the benchmark, carries no remaining topological check.
\end{corollary}

\paragraph{Physics intuition.} What would it have meant for R-INT to
\emph{fail}? A torsion class is a ghost flux: a field configuration
invisible to every curvature measurement --- it integrates to zero
against all forms --- yet capable of shifting quantized phases, of
making two apparently identical backgrounds disagree about an
integer. Had $H^4$ of the seam carried torsion, the \BIsix{} lift
would have been ambiguous by exactly such ghosts, and the capacity arithmetic of \S\ref{ssec:capacity} would have inherited a hidden
$\pm$-ambiguity no experiment on curvature could resolve. The
content of Proposition~\ref{prop:rint-abs} is that on the intended
class there is nowhere for a ghost to live: the base's clauses
evacuate the torsion, the dimension evacuates the Wu class, and the
spin condition --- through Rokhlin's theorem, the same theorem that polices the stratum in \S\ref{ssec:capacity} --- evacuates $w_4$. The
seam is ghost-free, and the ledger's last topological worry with it.

\subsection{The obstruction echoes, and the parity echo}
\label{app:H-echoes}
\label{sec:echoes}

\begin{proposition}[The rigidity echo]
\label{prop:rigidity-echo}
A family-independent Wu structure ($c = 0$ in
\eqref{eq:lambda-class}) has $\kappa_{E/S}(\lambda)$ trivial: by
naturality under base change, the pushforward of an $S$-independent
structure is pulled back from a point, and
$\Hc^2(\mathrm{pt}) = 0$. The bulk alone sends nothing to the
stratum --- the conclusion of the rigidity lemma (Lemma~\ref{lem:rigidity}),
reproduced as a two-line corollary of functoriality.
\end{proposition}

\begin{proposition}[The relativity echo]
\label{prop:brel-echo}
$\kappa$ consumes, by its type, the bulk Wu structure together with
the fibration over the stratum; boundary-only data cannot be
supplied to it. That the coupling can arise only from the pair ---
the boundary-relativity proposition (Proposition~\ref{prop:brel}) --- is here a
type constraint, visible before any computation.
\end{proposition}

These two echoes are the paper's quiet consistency theorem: the
formalism did not have to be told the obstruction structure of the
problem; it contains it. Together with the fourth appearance of the
odd-$N$ parity condition in the channel arithmetic
(Remark~\ref{rem:parity}), they are the strongest internal evidence
that the quadratic functor is not an imported metaphor but the
native mathematics of the seam.


\subsection{The bidegree bookkeeping of T-INF-2}
\label{app:H-bidegree}

Write forms on $E = \Msix \times S$ by bidegree $(p, q)$ ---
fiber degree $p$, base degree $q$. The fiber integration
$\int_{E/S}$ annihilates every component with $p \ne 6$. The family
Wu structure \eqref{eq:lambda-class} has curvature components
\begin{equation}
\omega_\lambda \;=\;
\underbrace{m\,\omega_\sigma}_{(4,0)}
\;+\; \underbrace{c\,\omega_\tau \wedge \pi_S^* F_a'}_{(2,2)}
\;+\; \underbrace{e\,\pi_S^*(\text{4-form on } S)}_{(0,4)},
\label{eq:bidegree}
\end{equation}
with $\omega_\sigma$, $\omega_\tau$ the normalized representatives
($\int_{S^4}\omega_\sigma = \int_{\Ttwo}\omega_\tau = 1$) and $F_a'$
the curvature datum of the $h$-class in the mixed channel. The
square:
{\small
\begin{longtable}{lccl}
\caption[The fiber integration, term by term.]{Every term of the square expanded by bidegree, whether it survives the integration over the fiber, and what it contributes if it does. Six of the seven vanish, and the table records why each one does --- degree too high for the fiber, or a squared representative that is itself zero --- rather than leaving the survivor to be taken on trust. Exactly one term survives, and it is the one the identity is built from.}\label{tab:fiber-integration}\\
\toprule
term & bidegree & survives $\int_{E/S}$? & contribution \\
\midrule
$(4,0)\!\cdot\!(4,0)$ & $(8,0)$ & no ($p=8>6$) & --- \\
$2\,(4,0)\!\cdot\!(2,2)$ & $(6,2)$ & \textbf{yes} &
$2cm\,(\omega_\sigma\!\wedge\!\omega_\tau)\otimes F_a'$ \\
$2\,(4,0)\!\cdot\!(0,4)$ & $(4,4)$ & no & --- \\
$(2,2)\!\cdot\!(2,2)$ & $(4,4)$ & no ($\omega_\tau^2 = 0$ too) & --- \\
$2\,(2,2)\!\cdot\!(0,4)$ & $(2,6)$ & no & --- \\
$(0,4)\!\cdot\!(0,4)$ & $(0,8)$ & no & --- \\
$\check L_8$-term & $(8,0)$ & vanishes identically & Lemma~\ref{lem:Lterm} \\
\bottomrule
\end{longtable}}

Hence $\int_{E/S}[\lambda\cup\lambda] = 2cm\,F_a'$ with the fiber
pairing $\int_{\Msix}\omega_\sigma\wedge\omega_\tau = 1$, and
$\kappa$'s curvature is $\tfrac18\cdot 2cm\,F_a' =
\tfrac{cm}{4}F_a'$. Identifying $F_a'$ with the Berry curvature
$F_a$ of \eqref{eq:berry-conv} --- the content of the placement
selection, Proposition~\ref{prop:placement} --- gives
\eqref{eq:structure}. One subtlety deserves its own sentence: the
$(3,1)$-type components absent from \eqref{eq:bidegree} are absent
because $H^3(\Msix) = 0$ and $H^{\mathrm{odd}}(\CPN) = 0$ leave them
no cohomology to live in; at cocycle level a $(3,1)$ exact term can
be added, but it shifts $\kappa$ by an exact form and moves nothing.


\subsection{The two refinements, closed at class level; and the channel identity}
\label{app:H-refine}
\label{sec:open}

Two refinements bound the scope of the six theorems above. The physical
seam is warped, while the benchmark computes on the metric product
(R-WARP); and the physical statement lives on a bounded half-seam, while
the global theorems are proved on its closed reflection-double (R-REL).
This subsection states both precisely and then proves that neither can
move the class: two theorems, T-INF-7 and T-INF-8, close the kill
criteria K-INF-1 and K-INF-2 at class level, with every step of the
derivation displayed. What survives of each refinement is then stated
exactly --- one constructive computation and one basepoint convention ---
and the channel identity, which is a different kind of item and remains
fully live, closes the subsection.

\subsubsection{R-WARP: the warped-metric refinement, and the warp-rigidity theorem}
\label{ssec:rwarp}

\paragraph{The refinement, stated.} The physical bulk metric of the
intended class is warped --- the scale-profile metric
(\ref{eq:E-warped-metric}) --- while every numerical statement of
\S\ref{app:H-channel}--\S\ref{app:H-rint} is computed on the metric
product. The topological chain --- T-INF-1, T-INF-3, the attributions,
\S\ref{app:H-rint} --- is homotopy-invariant and survives warping
unchanged. The question the refinement poses is whether the
\emph{form-level} corrections from the non-flatness of the fibration
can accumulate into a shift of the \emph{class}: whether a warped
recomputation could return $c_1(\kappa(\lamBI)) \neq -h$.

\begin{lemma}[Class functoriality of the pushforward]
\label{lem:classfun}
The Hopkins--Singer pushforward, restricted to underlying topological
classes, is metric-free: it is the topological integration (umkehr) map
determined by the fibration, the orientation, and the Wu-structure data
alone. A metric enters the construction only through the choice of
differential representatives, and two such choices differ by exact and
flat data, which the passage to classes annihilates.
\end{lemma}

\begin{proof}
This is the defining compatibility of the differential-cohomology
integration map \citep{HopkinsSinger2005}: the functor fits the
character diagram whose curvature leg is metric-dependent and whose
class leg is the ordinary topological pushforward. Concretely, a
differential class $\check x$ has curvature $R(\check x)$ (a form,
metric-sensitive through its representative) and underlying class
$c(\check x) \in H^*(\cdot;\mathbb{Z})$ (metric-blind); the integration
map commutes with both legs, and on the class leg it is the umkehr map
of the fibration. Changing the metric changes the horizontal
distribution and hence the representative by a shift lying in the image
of exact forms and flat classes --- precisely the data killed by $c$.
\end{proof}

\begin{theorem}[T-INF-7: warp rigidity of the bridge class]
\label{thm:warpr}
Let $g_s$, $s \in [0,1]$, be any smooth family of metrics on the fixed
seam manifold, interpolating from the metric product ($s=0$) to the
physical warped metric ($s=1$), preserving the underlying smooth
manifold and its fibration structure. Then the underlying class of the
pushed-forward bundle is constant in $s$:
\begin{equation}
c_1\bigl(\kappa_{E/S}^{(s)}(\lamBI)\bigr) \;=\; -\,h
\qquad\text{for every } s \in [0,1].
\label{eq:warpr}
\end{equation}
In particular the kill criterion K-INF-1 cannot fire under any smooth
warp: a warped recomputation that shifts the class off $-h$ would have
to change the fibration topology, the class $[\lamBI]$, or pass through
a singular metric, none of which a warp does.
\end{theorem}

\begin{proof}
The proof has three steps, each closing one road by which the warp
could act.

\emph{Step 1 (the inputs are metric-independent).} Every integer
entering the bridge is topological data of the fixed manifold. The
class $[\lambda] = m\,\sigma + c\,(\tau \boxtimes h) + e\,(1 \boxtimes
h^2)$ and its K\"unneth decomposition \eqref{eq:lambda-class} live in
$H^4$ of the fixed manifold; the channel theorem (T-INF-1) is a rank
statement in the same cohomology; the Wu-structure existence,
uniqueness, and collar-compatibility clauses of \S\ref{app:H-rint} rest
on $\Tors H^4$, $\nu_4$, and $w_4$, all diffeomorphism invariants; and
the attributions $c = \pm(L-N)$ (Theorem~\ref{thm:attr-c}, a Chern
number pinned by the trace-ratio lock) and $m = 2\,\Oorb$
(Theorem~\ref{thm:attr-m}, a count fixed by the APS doubling) are
integers owned upstream of any metric. A warp is a smooth deformation
of the metric on a \emph{fixed} manifold with a \emph{fixed} fibration:
it moves none of these. (This step is the homotopy-invariance
observation already recorded above, now doing load-bearing work.)

\emph{Step 2 (the pushforward cannot transmit the warp to the class).}
By Lemma~\ref{lem:classfun}, the class leg of the pushforward is the
topological umkehr map: metric-free. The warp acts on the construction
only through the differential representatives --- the connection on
$\kappa(\lamBI)$ and its curvature acquire warped corrections --- and
those corrections lie in exact and flat data, which the class
annihilates. That warping ``moves exact and flat data only, not the
class'' is therefore not an expectation requiring case-by-case
verification: it is the class functoriality of the integration map,
and it holds for every smooth metric at once.

\emph{Step 3 (continuity and integer rigidity close the gap).} Suppose,
for contradiction, that $c_1(\kappa^{(s)}(\lamBI))$ varied with $s$.
Along a smooth metric family on a compact seam the construction is
continuous in $s$: the fibration is fixed, $[\lamBI]$ is fixed
(Step~1), and the warp factor $e^{\pm 2r}$ is smooth and bounded on the
compact radial interval of the intended class, so no singular metric
intervenes. A $\mathbb{Z}$-valued invariant depending continuously on
$s$ is constant. The only escapes from this rigidity are a
discontinuity of the construction --- which requires the fibration or
the Wu data to jump, excluded by Step~1 --- or a singular metric on the
path, excluded above. Hence \eqref{eq:warpr}, and with it the
class-level closure of K-INF-1.
\end{proof}

\paragraph{Grade, and what survives.} T-INF-7 is carried at physics
rigor: its one imported ingredient is the class functoriality of the
Hopkins--Singer integration map (Lemma~\ref{lem:classfun}), which is
that map's defining property rather than an additional hypothesis. What
survives of R-WARP is exactly the constructive computation: exhibiting
the warped curvature representative of $\kappa(\lamBI)$ and watching
its corrections land in exact and flat data, as Step~2 guarantees they
must. That computation can illustrate the mechanism; by T-INF-7 it can
no longer refute the class. It is accordingly reclassified as
engineering (item E-3 of \S 12.1) and is not conditioning.

\paragraph{Physics intuition: the warp bends the rulers, not the
ledger.} A metric is a system of rulers; a warp is a smooth regrading
of those rulers along the radial direction. Curvature is where the
rulers live, and the warped seam's curvature representative differs
from the product's at every point. But the bridge equation
$c_1(\kappa(\lamBI)) = -h$ is an equation between \emph{integers} ---
a Chern class paired against cycles --- and integers do not live in
the rulers; they live in the topology the rulers are laid over.
Deforming the rulers smoothly can no more change that integer than
re-drawing a coastline's map grid can change the number of islands.
The measured instantiation (\S\ref{app:H-verify}, fourth harness): a
representative deformation moving the pointwise curvature by twelve
hundred percent moves the class by zero at $10^{-12}$, and the model
family's Chern number is $-1$ unchanged at warp anisotropies from
$e^{-7}$ --- the cascade's own strength --- to six decades, failing
only when a coefficient reaches exactly zero, which is a change of
topology, not of metric.

\subsubsection{R-REL: the relative recomputation, and the relative-closure theorem}
\label{ssec:rrel}

\paragraph{The refinement, stated.} The doubling derivation of $m$
(Theorem~\ref{thm:attr-m}) says the physical statement lives on the
bounded half-seam; the fully honest object is therefore the relative
differential cohomology of the pair, with the Dai--Freed collar
formalism \citep{DaiFreed1994} carrying the boundary terms. The
question the refinement poses is whether the relative computation could
return a boundary class different from the absolute one --- which would
expose a factor error in the doubling bookkeeping.

\begin{theorem}[T-INF-8: relative closure on the benchmark]
\label{thm:relw}
On the benchmark seam family, with the Wu structure of
\S\ref{app:H-rint}:
\begin{itemize}[leftmargin=2.4em]
\item[(a)] \emph{(Filling independence.)} The half-seam pairing is
well-defined modulo the period lattice, independently of the choice of
APS-compatible filling of the boundary data.
\item[(b)] \emph{(Half equals half the double.)} The relative class
equals one half of the double's class, and the apparent half-integer
risk is void: the doubled datum is automatically even.
\item[(c)] \emph{(Class-level closure of K-INF-2.)} A correct relative
computation must reproduce the absolute class: the two cannot differ
modulo the lattice, because their difference is itself a closed-manifold
lattice element.
\end{itemize}
\end{theorem}

\begin{proof}
\emph{(a)} Let $Z$ denote the half-seam with boundary $Y$, and let
$Z_1, Z_2$ be two extensions of the boundary data compatible with the
APS collar. Glue them along $Y$ with one orientation reversed:
$X = Z_1 \cup_Y \overline{Z}_2$ is a closed manifold of the intended
class. The two relative pairings differ by the absolute pairing
evaluated on $X$. On the intended class that absolute pairing lies in
the period lattice: components (A) and (B) of the integrality theorem
(Theorem~\ref{thm:LINT}) are rigorous, and the topological residue
R-INT is discharged on the benchmark (\S\ref{app:H-rint}: the Wu
structure exists, is unique, restricts compatibly to the APS collars,
and is uniform over the family stratum --- the collar-compatibility
clause of Proposition~\ref{prop:rb}(ii) is what licenses the glueing).
Hence the difference of the two relative pairings is a lattice element,
and the relative class is filling-independent modulo the lattice.

\emph{(b)} Take $Z_2$ to be the mirror image of $Z_1$. The glued
manifold is then the reflection-double, and the doubling identity
$\langle \widehat X, [\mathrm{double}]\rangle = 2\,\langle \widehat X,
[\mathrm{half}]\rangle$ (the reflection computation of
Theorem~\ref{thm:attr-m}, harness-verified to $10^{-9}$) gives the
relative class as one half of the double's. The half-integer worry ---
that half of an odd lattice element might fall off the lattice --- is
void by the evenness mechanism of Theorem~\ref{thm:attr-m}: the mirror
shows every unit of flux twice, so the doubled datum is $2m$ with $m$
the physical integer, and half of it is $m$, on the lattice by
construction. Any integer, doubled, is even; nature keeps the unit and
the mirror supplies the two.

\emph{(c)} Immediate from (a) and (b): the relative and absolute
classes cannot differ modulo the lattice, because their difference is
the absolute pairing on a closed glued manifold, which (a) has already
placed on the lattice. A relative recomputation returning a different
class would therefore not be revealing new relative physics; it would
be committing an arithmetic error, and the doubling identity that
guards that arithmetic is verified to $10^{-9}$ in \S\ref{app:H-verify}
and independently to $10^{-12}$ in the fourth harness.
\end{proof}

\paragraph{Grade, and what survives.} The glueing argument of (a) is
the standard relative-from-absolute mechanism of the collar calculus
\citep{DaiFreed1994}; every load-bearing input --- the lattice, the
discharge of R-INT, the collar compatibility --- is a benchmark result
already proved above, which is why the theorem is stated on the
benchmark and carries its grade. Two items survive. First, the formal
Dai--Freed write-up: packaging (a)--(c) inside the relative
differential cohomology of the pair, with the bonus that the relative
frame connects $\kappa$ directly to the boundary Chern--Simons data
that the \OGGEP{} projection pushes down --- reclassified as engineering
(item E-4 of \S 12.1), an opportunity rather than a gap. Second, and
genuinely open: off the benchmark, on a $b_2 > 0$ base of K3 type, the
Wu torsor has order $2^{44}$ and a uniform family choice exists by
pullback, but the \emph{canonicity} of the basepoint is undetermined
--- exactly the fence Proposition~\ref{prop:rb} already declares. That
question bounds the scope of the theorems, not their truth on the
physical background, and nothing $N_g$-facing routes through it.

\paragraph{Physics intuition: two observers filling the same
boundary.} Two mathematicians, handed the same boundary data, will cap
it off differently --- and the difference between their answers is not
arbitrary: it is the books of a closed world, the world obtained by
glueing one cap to the other's mirror. Closed worlds keep integer
books on the intended class; that is the absolute integrality theorem.
So the two cappings can disagree only by an integer, which is to say,
modulo the lattice, they cannot disagree at all. The half-seam's books
are exactly as honest as the double's --- the measured instantiation
(\S\ref{app:H-verify}, fourth harness): the fluxes through two
different caps of the same loop differ by precisely the closed-surface
integer, at four latitudes, to $10^{-12}$.

\subsubsection{The channel identity}
Which abelian factor --- $U(1)_B$, $U(1)_{Qi}$, or a combination ---
carries the family charge $q = 1$ is not determined here; both slots
carry the same $s$, so the arithmetic of
Theorem~\ref{thm:attr-c} is identical in all cases. The
identification is a physics refinement for a later edition, with
the matter assignments of \citet{ArisakaA9SM} the natural arbiter.

\subsubsection{Declared exposure}
\label{ssec:exposure}
A numbered member of the branch declares what would kill it. This
member's exposure is \emph{mathematical}, and it says so plainly:
it carries no independent observational falsifier --- its empirical exposure routes entirely through this paper's declared falsifier ledger (\S\ref{ssec:falsifiers}), since every physical claim it touches enters through the main line's conditioning. What it stakes of its own are
three computations, whose standing T-INF-7 and T-INF-8 have made
unequal:

\begin{itemize}[leftmargin=2.2em]
\item[\textbf{K-INF-1}] \emph{(the warped computation).} By
Theorem~T-INF-7 the class of $\kappa(\lamBI)$ cannot shift off $-h$
under any smooth warp, so this criterion can no longer fire against
the outcome of the warped recomputation. It remains live against the
theorem's own premises: a demonstration that the physical seam's
fibration or the class $[\lamBI]$ is mis-identified would void
Step~1 of the proof and reopen the computation
(\S\ref{ssec:rwarp}).
\item[\textbf{K-INF-2}] \emph{(the relative computation).} By
Theorem~T-INF-8 a correct relative computation must reproduce the
absolute class, so this criterion can no longer fire against the
outcome. It remains live against the arithmetic it always guarded: a
factor error in the doubling bookkeeping of
Theorem~\ref{thm:attr-m}, which the harnesses bound at $10^{-9}$
and $10^{-12}$ (\S\ref{ssec:rrel}).
\item[\textbf{K-INF-3}] \emph{(the channel identity; fully live).} If
the matter assignments of \citet{ArisakaA9SM}, once brought to bear,
force the minimal family charge away from $|q| = 1$ in every
abelian slot, the \MSMF{}-minimal-quanta premise of
Theorem~\ref{thm:attr-c} fails in the physical seam and the
attribution of $c$ must be recomputed at the forced quanta.
\end{itemize}

Any of the three would be reported as what it is. None of them
touches the family number, which this appendix does not carry.

\subsubsection{Honest limits}
\label{ssec:honesty}
Three sentences a careful reader deserves in the main text, not a
footnote. \emph{First}, T-INF-6 does not select $N$: the identity
$|L - N| = 2$ holds at $N = 7$ as well, and at other odd $N$ the
bridge delivers a higher multiple of the Berry class; the theorem is
consistency for the coupling clause, and the family number stands on the capacity theorem of \S\ref{ssec:capacity} alone. \emph{Second}, the
\MSMF{}-minimal-quanta premise does real work; a reader who declines
it retains Theorem~\ref{thm:quantisation}'s statement that the
Berry bundle is the \emph{minimal non-zero} value of the
pushforward --- the bridge at its unconditional core. \emph{Third},
the benchmark is one background; the intended class is a family of
them, and Propositions~\ref{prop:rint-abs}--\ref{prop:rb} say
exactly which parts extend (the absolute part, everywhere on the
class) and which acquire a basepoint convention ($b_2 > 0$ bases) ---
a division T-INF-7 and T-INF-8 sharpen but do not change: the
basepoint convention is the one item of the pair-level story that
remains a convention.

\subsection{Verification record}
\label{app:H-verify}

Four companion harnesses, one hundred five checks, zero failures,
archived in the toolchain home alongside the paper source.

\paragraph{\texttt{verify\_hs\_review.py} (35/35).} The
Hopkins--Singer bookkeeping this appendix stands on: the $(k,i)$
type-check in six dimensions (unique solution $(2,2)$; integer
values would need an eight-manifold); the Wu-class vanishing table;
the K\"unneth/torsion audit of $H^4(\Mfour\times\Ttwo)$ on $S^4$,
K3, $S^2\times S^2$ with controls; the parity chain
$\chi \equiv \sigma \bmod 2$, Rokhlin, $w_4$, with the non-spin
control $\CP^2$; the ledger and coefficient arithmetic
($s = L-N = 2$, $t = 5$ clearing, $\gcd(2,5) = 1$); and the toll
ladder's exact satisfaction of the quadratic-refinement functional
equation at characteristic element $a = 1$.

\paragraph{\texttt{verify\_ckap.py} (28/28).} The structure and
quantization theorems: the channel ranks (T-INF-1); the assembled
curvature ratio $F_\kappa/F_a = cm/4$ at moduli test points for four
$(c,m)$ pairs, with $\mathrm{Vol}(S^4)$ by four-angle quadrature to
$10^{-6}$ and the Berry Chern number to $10^{-8}$ (T-INF-2); the
$\check L$ dimension-vanishing; the even-solution table of
$cm = 4$ --- the impostors $(1,4), (4,1)$ dying, $(\pm2,\pm2)$
surviving (T-INF-3); the integrality cross-check with the odd
control; the (R-b) torsor arithmetic (benchmark $0$; K3 $2^{44}$);
and the rigidity echo.

\paragraph{\texttt{verify\_ckap2.py} (27/27).} The attributions: the
cross-term class number by explicit $\Ttwo\times\mathrm{line}$
integration ($-n_\tau q$ to $10^{-6}$, four quanta pairs; T-INF-4);
the parity table $N = 2$--$7$ (Wu-admissible iff $N$ odd); the
reflection-doubling identity to $10^{-9}$ and the evenness of
$2\Oorb$ for every reservoir value (T-INF-5); the placement fork
($|c_1| \ge 2$ under placement (B), refuted by the boundary's
$c_1 = -1$); and the final assembly
$c_1(\kappa) = -(L-N)\Oorb/2\,h$ at generic odd $N$, with the
no-insertion check ($N = 3$, $K = 35$, $J = 7$ nowhere consumed).

\paragraph{\texttt{open\_core\_check.py} (15/15).} The closure
theorems of \S\ref{app:H-refine}, measured: class invariance under
representative deformation ($1200\%$ pointwise, zero class shift at
$10^{-12}$); integer rigidity of the model Chern number from $e^{-7}$
to six-decade anisotropy, the zero-coefficient control the only
genuine kill; filling independence and the doubling identity, both
to $10^{-12}$; and the cascade-saddle link invariants.


\GGAcknowledgments

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

%%  REFERENCES (inline, annotated with Relevance paragraphs; order

%%  preserved from v1.2)

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

%% ---- PART VII ----
\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a critical reader should raise against a three-leg derivation of the family number --- stated at full force and answered, each response beginning with what the objection gets right}
\label{part:objections-p2}

\newenvironment{objqabody}%
  {\par\nobreak\begingroup\leftskip=1.8em\parindent=0pt\relax}%
  {\par\endgroup}

\section{Common objections and responses}
\label{sec:objections-p2}

A paper that claims to force $N_g = 3$ from geometry will be met ---
rightly --- with skepticism from every quarter, and a short paper
invites the sharpest forms of it. This Part collects twenty-one
objections --- fourteen on the three-leg derivation itself, then seven
on the inflow bridge of Appendix~\ref{app:H} --- each stated at full
force in the objector's own voice, with a response that begins by
conceding whatever the objection gets right --- the cohort's convention, following the closing objections
section of \citet{ArisakaA2GG6}. Where the honest answer is ``not
yet,'' the response says so and points to the registered debt.

\bigskip
\noindent\textbf{Objection 1 (O-N3-1) (the circularity skeptic)}\par\nobreak\smallskip
\noindent\textbf{``You prove $N_g \ge 3$ from observed CP violation. Using the world to count the world is not a derivation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Partially right, and the paper says so before the objector can: Theorem~\ref{thm:L1} is empirically conditioned on $\JCKM \ne 0$, and the conditioning is declared in \S5.4, in the abstract, and at every layer it touches. Three replies. First, economy: the input is a single sign --- CP violation exists --- not a fitted number; every physical theory admits some empirical content, and this is among the smallest on record. Second, the conditioning touches one leg of three: the upper bound (Theorem~\ref{thm:L2}) and the constructive realization (Theorem~\ref{thm:L3}) owe nothing to the measured $\JCKM$. Third, the input is scheduled for retirement: the CPT companion \PaperPFive{} \citep{ArisakaP5CPT} already evaluates the Jarlskog invariant at the stated layer, and the constructive closure-level derivation is the declared mandate of the CKM-extension paper (\citealp{ArisakaCKM}, in preparation). If that program completes, the leg becomes over-determined and the objection dissolves; until then the paper claims exactly what it can support.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-N3-2) (the conjecture skeptic)}\par\nobreak\smallskip
\noindent\textbf{``The cohort's cross-reference ledgers call this cap the `Budget Conjecture'. Renaming it a `capacity theorem' does not make a century-old question close.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The renaming is earned, and the ledger is printed. What the
conjectural reading treats as one opaque assumption splits into parts with
three different fates.
\emph{Proved:} the adiabatic theorems T-AD-1--T-AD-3 (at Born-grade), the
non-vanishing of the coupling (Lemma~\ref{lem:kappa}), the unit
$\lambda = \pi$ (three routes, Lemma~\ref{lem:unit}), extensivity via the
Pontryagin scaling, and form robustness (Theorem~\ref{thm:robust}).
\emph{Proved impossible:} any derivation of the cap from the prior axioms
alone --- the two obstruction theorems of Appendix~\ref{app:G}, which are
why the conjecture stood unproven for as long as it was posed that way.
\emph{Still open:} exactly one residue, R-INT
(Definition~\ref{def:RINT}) --- the topological check underwriting the
spin refinement in Theorem~\ref{thm:LINT} --- whose failure would
moreover cost the lattice presentation but not the value (Routes 2--3). A reader prepared to grant the five-assumption axiomatization is being
offered strictly less to swallow; a reader who rejects
the coupling clause knows precisely which single clause they are rejecting
and what evidence (the classification, the saturation identity, the
falsifiers) bears on it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-N3-3) (the postulate skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Morse positivity is assumed, not derived. Postulates dressed as machinery.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is answered by architecture rather than by
defense: Morse positivity carries nothing in this derivation. The exclusion of
$N_g=2$ and of all $N_g \ge 4$ runs entirely through the capacity theorem
(\S\ref{ssec:capacity}), which nowhere invokes
Postulates~\ref{post:morse-pos} or~\ref{post:strict}; the postulates
survive as an imported consistency check, and at the admissible value the
strict form holds as a theorem, $a^+(3) - a^-(3) = \sigma(\CP^2) = 1$
(\S\ref{ssec:parity-check}). The residue of the objection lands on the one
place assumption survives --- the coupling clause --- and there the paper
directs the skeptic to Objection~13, where that challenge is put at full
force.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-N3-4) (the three-frameworks skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Your three legs live in three different formalisms --- a symmetry corollary, a Morse bound, a mass codebook. That is three sketches, not one proof.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The heterogeneity is real and deliberate. The legs are not three proofs of one proposition; they are two bounds and a cross-check on one integer: leg (L1) forces $N_g \ge 3$ (Theorem~\ref{thm:L1}), leg (L2) forces $N_g \le 3$ (Theorem~\ref{thm:L2}), and leg (L3) verifies that the surviving value is constructively realized, with the \ANine{} codebook reproducing the charged-lepton mass ratios at four significant figures (Theorem~\ref{thm:L3}). The closure is stated once, in Theorem~\ref{thm:main}, with the conditionality of each leg carried along. And the heterogeneity buys falsifiability: the three legs die to different data --- a CP-conserving world, a fourth basin, a failed mass ratio --- so the conjunction is exposed on three independent fronts, not sheltered behind one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-N3-5) (the modeling skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Identifying particle generations with critical points of a projector landscape is a modeling choice, not physics.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The identification is \GDOS{}-internal, and the paper carries that fence in the statement of Theorem~\ref{thm:L2} itself. But within the framework the objects are not chosen ad hoc: once the boundary carries $N_g$ families, the family-projector moduli space is forced to be $\CPN$ (Section~6), and the \GFF{} functional on it is the cohort's fixed free-energy object, inherited from \citep{ArisakaA6Gi4}, not tuned here. The identification also earns its keep dynamically: the same cascade reading yields the decay-width ordering of the second and third generations and the reachability statement of \S\ref{ssec:reachability}, both of which match observation. A modeling choice that predicts the lifetime hierarchy it was not fitted to is the kind physics keeps.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-N3-6) (the already-excluded skeptic)}\par\nobreak\smallskip
\noindent\textbf{``LEP counted the neutrinos long ago. Deriving a number experiment has already settled adds nothing.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The measurement (\ref{eq:Nnu}) is settled physics, and the paper claims no discovery there. But exclusion is not explanation: the Standard Model accommodates any $N_g$ --- nothing in its structure prefers three --- so the family count has stood for fifty years as a free discrete input. The derivation converts that input into a theorem with collateral obligations the measurement never imposed: the parity statement that $N_g$ must be odd, the first-order mass ratios (predictions P3--P4), and the cascade lifetime hierarchy (P5). Those are net exposure, registered as kill criteria F1--F5 in Section~10, and they can fail tomorrow. A postdiction with new death conditions attached is a prediction about everything not yet measured.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 7 (O-N3-7) (the numerology skeptic)}\par\nobreak\smallskip
\noindent\textbf{``$\Stot^{\max} = 3\pi$ is numerology with a Greek letter in it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The answer is a ledger, and every line of it is checkable.
What is derived: the step
actions $S_k = \pi k$ follow from the cascade geometry (Section~9); the
capacity's $N_g$-scaling follows from the Pontryagin identity $p_1 = N_g h^2$;
and the unit $\lambda = \pi$ is pinned by three independent routes
(Lemma~\ref{lem:unit}) --- lattice-plus-decisiveness, \MSMF{}-minimality,
and no-new-scales --- none of which knows the answer. What is classified:
among all natural single-class capacities, exactly two survive and both give
$N_g=3$ with exact saturation (Theorem~\ref{thm:robust}); the old
$\lfloor J/2\rfloor$ reading is exposed as a numerical coincidence of
$J=7$ that the chain no longer touches. Numerology is a pattern selected
after the fact from an unconstrained supply; here the supply is constrained
to a single integer multiple of a derived unit, declared before the
adjudicating mathematics is brought to bear --- and that mathematics
selects the declared value uniquely. That is the opposite of
numerology; it is a pre-registration paying off.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-N3-8) (the axioms skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Theorems `within \GDOS{}' just relocate the mystery into your axioms.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the criticism in the form stated: every theorem here is conditional on the declared axiom system, and a reader who rejects the constitution receives conditionals. Two replies. First, relocation with compression is what explanation is: the same axiom set that elsewhere carries the Standard-Model parameter sheet and the cosmological inventory here carries the family count --- one axiom system, many former mysteries; the mystery count went down. Second, the axioms are not adjustable refuges: AX-1 through AX-4 are fixed cohort-wide, and this paper added none of its own --- its two imported postulates are consistency-grade with declared provenance (Postulates~\ref{post:morse-pos} and~\ref{post:strict}; the import's would-be derivation from \BIsix{} is now closed in the negative, \S\ref{ssec:parity-check}), its one axiom-level clause is stated at minimal size with every derivable part derived (Axiom~\ref{ax:coupling}), and its one residual check is named with its failure mode priced (R-INT, the topological residue of the integrality theorem L-INT). Nothing was smuggled in to make $N_g=3$ come out --- and the chain now consumes neither $K=35$ nor $J=7$, the two places a smuggler would hide.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-N3-9) (the methodology skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Counting generations is kinematics. Why drag open-system machinery into it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A closed-system derivation would indeed have been cleaner, and we tried to find one: four closed-system routes --- a Chern--Simons multiplicity axiom, a magnetized-torus index, a $\mathbb{Z}_3$ orbifold, and partial anomaly saturation --- were attempted and failed, for reasons recorded route by route in Section~14. The failure has a structural reading: $N_g$ counts the stable readouts of the boundary runtime, and stability is an open-system notion --- basins and gates, not representations. Closed-system counting delivers candidate multiplicities; it cannot say which candidates survive as registered families. That is why leg (L2) is a statement about the landscape the boundary actually runs, and why the methodological reflection is in the paper at all: the negative results are data about where the answer lives.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-N3-10) (the postdiction skeptic)}\par\nobreak\smallskip
\noindent\textbf{``The mass ratio 16.817 was measured decades before you `predicted' it.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded without reservation: $m_\tau/m_\mu$ is known physics, and the paper claims no postdictive credit. The registered exposure is forward-looking. The ratio enters as kill criterion F3: a precision value off the first-order prediction beyond the declared QD tolerance kills the codebook leg going forward, whatever the history of the measurement. The integers behind the ratios are fixed cohort-wide by the \OGNfive{} ledger $(1,3,5;7,35)$ --- this paper could not adjust them to fit if it wanted to, which is what separates a derived ratio from a fitted one. And the paper's permanent exposure is sharper still: one fourth-generation event, at any energy, in any decade, fires F1--F2 and kills the conjunction outright. Absences are the most falsifiable claims there are.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-N3-11) (the anthropic skeptic)}\par\nobreak\smallskip
\noindent\textbf{``The landscape already explains $N_g=3$: other values happen elsewhere, and observers see what observers can see.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Section~11.4 gives the structural comparison without polemic. Anthropic reasoning accommodates any family count compatible with observers, at the price of a measure problem and an unobservable ensemble; it explains why we might find ourselves at $N_g=3$ without forbidding anything nearby. The present route is a uniqueness claim with zero continuous parameters: $N_g=3$ is forced, and a fourth generation --- anywhere, ever --- kills it. The two are therefore not rivals on one axis: one is a selection argument over an ensemble, the other a two-sidedly falsifiable derivation in a single universe. A reader may prefer either posture; only one of them can lose.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 12 (O-N3-12) (the sociology skeptic)}\par\nobreak\smallskip
\noindent\textbf{``A single-author cohort, citing itself, is not independent verification.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fully conceded: self-citation is provenance bookkeeping, not evidence, and no referee should weigh it as support. What the paper offers for verification is deliberately machine-checkable without trusting the author: every inherited statement in the claim-status ledger carries a named source and a status tag that can be checked against the cited paper; the kill criteria F1--F5 are experimental statements owned by the community's instruments, not by the cohort; and the codebook cross-check of leg (L3) is arithmetic from five integers that any reader can repeat in an afternoon. The cohort structure exists to make that audit tractable --- one derivation chain, consistently named across papers --- not to substitute for it. Skepticism is the intended reading posture; the paper's job is to make the skepticism efficient.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-N3-13) (the axiomatized-answer skeptic)}\par\nobreak\smallskip
\noindent\textbf{``You proved the budget could not be derived --- and then adopted it as an axiom. That is not deciding the family number; it is enshrining the failure.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Stated at full force, and the paper's favorite objection,
because answering it is what the architecture is for. Four replies, in
descending order of weight. \emph{First}, the necessity half is proved: the
obstruction theorems show that no derivation from the prior axioms exists
--- so the alternatives were a named clause or a permanent conjecture, and
naming is strictly more honest than conjecturing. \emph{Second}, the clause
does not mention the answer: Axiom~\ref{ax:coupling} contains no $N_g$, no
$3$, no $K$, no $J$, and no budget value --- the number three emerges from
$\pi N_g(N_g-1)/2 \le \pi N_g$, which is arithmetic. \emph{Third}, the clause's
content is nearly all theorem: transport is Berry and the geometry is
Fubini--Study at Born-grade (Appendix~\ref{app:F}); what the axiom actually
adds is the pairing form, whose freedom is then closed by classification
(Theorem~\ref{thm:robust}). \emph{Fourth}, the clause is falsifiable
through its shared premises: kill criterion F6 ties it to the Born rule's
own footing. What would vindicate the skeptic is a proof that the coupling
clause cannot be derived from the joint bulk--boundary object either ---
the analogue, one level up, of what this paper proved about the prior
axioms. The paper states the target rather than hiding it, and Appendix~\ref{app:H} is where that program was carried out: the residue R-INT is discharged on the benchmark, and the clause is exhibited as the output of a computation rather than assumed. What is \emph{not} yet shown --- and the objection is entitled to press it --- is that no derivation from the joint object could have failed; enforcement on one background is not a necessity proof.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-N3-14) (the Born-coupling skeptic)}\par\nobreak\smallskip
\noindent\textbf{``Leg (L2) rests on the same premises as your Born theorem. You have made the family count hostage to your most speculative derivation.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The coupling of fates is deliberate, and it runs in the
skeptic's favor. The premises in question --- the \AFive{} forward
semigroup and the collision-level contraction --- are the minimal content
of ``the boundary runtime is quantum mechanics with relaxation''; a reader
who rejects them is not quibbling with leg (L2), they are rejecting the
framework's quantum mechanics wholesale, Born rule and all, and the paper
cannot and should not insulate the family count from that. What the
coupling buys is economy of exposure: one premise set now carries the Born
rule, the adiabatic transport, and the capacity geometry, so a single
experimental or mathematical failure would break all three loudly ---
which is falsifiability working as designed (F6) --- while a single success
(the field-grade representation step, the declared residue of the Born
chain) would harden all three at once. Hostage-taking would be tying the
result to an \emph{unrelated} speculation; tying it to the framework's own
quantum mechanics is just honesty about what leg (L2) is: a statement
about what quantum transport on the family moduli can pay for.
\end{objqabody}


\subsection*{The inflow bridge}
The seven objections that follow are addressed to Appendix~\ref{app:H}
and to the enforcement claim of \S\ref{ssec:axB-enforced}. They are
kept together, and kept last, because a reader who takes the coupling
clause as an axiom --- as \S\ref{ssec:axiom} states it --- may stop at
Objection~14 and lose nothing of the family-number
argument.

\bigskip
\noindent\textbf{Objection 15 (O-N3-15) (the model skeptic)}\par\nobreak\smallskip
\textbf{``Your `channel model' is an assumption dressed as a
definition. The whole theorem lives inside it.''}
\begin{objqabody}
\emph{Response.} The objection is right that
Definition~\ref{def:seam-family} is where the conditionality
lives, and the theorem statements say so. Three replies. First, the
channel itself is not modeled but computed: on the benchmark the
K\"unneth structure admits exactly two family classes, and fiber
degree kills one --- there is nothing else the coupling could be, in
cohomology (T-INF-1). Second, the product premise (P-i) and the
pullback premise (P-ii) are the intended class's own clauses,
imported from \AThree{}, not invented here; the refinement they
defer is closed at class level by Theorem~T-INF-7, with the
constructive warped representative the named survivor (E-3).
Third, what escapes the model is cocycle-level (non-topological)
family dependence; the paper's claim is calibrated to the class
level throughout, which is where the coupling clause itself lives.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-N3-16) (the minimality skeptic)}\par\nobreak\smallskip
\textbf{``Strip the \MSMF{} appeal and your bridge is a lattice
statement, not an identity.''}
\begin{objqabody}
\emph{Response.} Correct --- and the paper prices exactly that.
Without minimal quanta, T-INF-3 still holds unconditionally: the
pushforward is always an integer multiple of the Berry bundle, with
the Berry bundle the minimal non-zero value. The bridge identity
then follows from the same two principles the unit classification of \S\ref{ssec:capacity} already stakes (non-vanishing plus minimality), used
in the same way. A reader who rejects \MSMF{} here has already
rejected it there; the conditional structure is shared, declared,
and no worse at this link of the chain than at any other.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-N3-17) (the numerology skeptic)}\par\nobreak\smallskip
\textbf{``$2 \times 2 / 4 = 1$. You have dressed a triviality in
functors.''}
\begin{objqabody}
\emph{Response.} The arithmetic is trivial; the content is where
each integer comes from and what forbids the alternatives. The $4$
is the Hopkins--Singer normalization --- not chosen. The evenness
that excludes $(1,4)$ and $(4,1)$ is the Wu condition --- not
chosen. The first $2$ is the trace-ratio numerator $L - N$,
delivered by the abelian cross term of a lift fixed in \AThree{}
--- not chosen. The second $2$ is the orientation double of a unit
reservoir proved to exist there --- not chosen. Four independent
structures, none adjustable, meeting at $1$: that is the opposite of
numerology, and the harnesses verify each meeting point separately
precisely so that no single ``nice'' coincidence carries the
theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 18 (O-N3-18) (the circularity skeptic)}\par\nobreak\smallskip
\textbf{``You matched the boundary to fix the placement, then
celebrate agreeing with the boundary.''}
\begin{objqabody}
\emph{Response.} The objection would be fatal if the matching fixed
a continuous parameter; it fixes a binary placement, and the
non-trivial content survives the fixing. What the boundary decides
is \emph{which slot} of the quadratic machine carries the physical
datum (Proposition~\ref{prop:placement}) --- a choice the formalism
offers and one side of the pair must resolve, as the
boundary-relativity proposition demands. What the boundary does
\emph{not} supply is everything that then has to come out right:
the channel rank, the $cm/4$ factor, the evenness, the value
$L - N$, the doubling. Placement (B) was refutable and was refuted;
a framework in which both placements matched would deserve the
objection --- this one does not.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-N3-19) (the selection skeptic)}\par\nobreak\smallskip
\textbf{``$|L-N| = 2$ at $N = 7$ too. Your bridge cannot tell three
families from seven.''}
\begin{objqabody}
\emph{Response.} Correct, stated in the paper's own honesty section
(\S\ref{ssec:honesty}), and by design: this is a consistency
theorem for the coupling clause, not a fourth leg of the pincer.
The family number is decided by the capacity theorem of \S\ref{ssec:capacity} (Theorem~\ref{thm:capacity}), which excludes $N = 7$ by the budget with room to spare. What
the bridge adds at $N = 3$ is not selection but enforcement: the
clause the derivation conditions on is exhibited as the output of a
computation, on the one background where every ingredient is
theorem-grade.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-N3-20) (the one-background skeptic)}\par\nobreak\smallskip
\textbf{``One background is not a theory. Everything here is
$S^4 \times T^2$.''}
\begin{objqabody}
\emph{Response.} Partly right, and the paper's propositions are
scoped accordingly. What holds on the whole intended class: the
absolute part of R-INT (Proposition~\ref{prop:rint-abs}), the
quantization theorem, the attributions, the echoes. What is
benchmark-specific: the uniqueness of the Wu structure (elsewhere, a
$2^{44}$-torsor with a declared basepoint) and the numerical
curvature check. The benchmark is not a toy: it is the background
\AThree{}'s global anomaly theorem itself certifies, and the
physical compactification of the intended geometry. Extending the
uniform-choice statement across the class is the basepoint question
that survives Theorem~T-INF-8's closure of R-REL, stated as such in
\S\ref{ssec:rrel}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-N3-21) (the foundations skeptic)}\par\nobreak\smallskip
\textbf{``Your program advertises that \GDOS{} solves the
measurement problem; now your bridge leans on the Born rule. Which
way does the arrow point?''}
\begin{objqabody}
\emph{Response.} Both statements are true because they live at
different links of one chain, and the paper devotes
\S\ref{sec:order} to writing the chain once. Downward: the axioms
give \AFive{}'s open-system law; \AEight{} derives the Born measure
from it (conditionally, at its declared grade); this paper's adiabatic theorems ride on that result; and the bridge's boundary input rides on those. So yes --- the Born rule is derived inside
the architecture, and yes --- this appendix consumes the derived
result rather than re-deriving it. The arrow points one way, and
this appendix sits at its downstream end. What would be incoherent is
the claim the objection warns against --- that the bridge
\emph{founds} quantum mechanics --- and the paper declines it
explicitly: its boundary input is built from the Born-grade
package, and a construction fed the boundary's answer cannot be the
answer's source. What it contributes instead is the two-sided
agreement of \S11.3, which is precisely the kind of evidence a
derivation cannot supply.
\end{objqabody}
%% ===================================================================

\addtocontents{toc}{\protect\enlargethispage{8\baselineskip}}
%% ---- PART VIII ----
\GGPart[References]{References}{Cited works in two groups --- the \GGTheory{} cohort and the external literature --- each entry carrying a relevance note and its first citation point}
\renewcommand{\bibsection}{\vspace{1.2em}}
\noindent The reference list is organized in two sections, in line
with the \GGTheory{} house style. \textbf{Section A} lists the
Arisaka cohort entries cited in this paper --- ten shipped sister
papers, in the canonical order of the program, followed by the two
declared in-preparation companions --- and \textbf{Section B} the
external literature. Every entry carries a \textbf{Relevance}
paragraph stating what the work contributes to this paper and where
it enters, followed by the location of its first citation; the list
contains no entry that is not cited in the text. All shipped cohort preprints are available at \url{https://preprints.arisaka-gg.org/}.

\section*{A.\ Papers by Katsushi Arisaka (the \GGTheory{} cohort)}

\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K. (2026).} The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems. Arisaka-GG-A1-Constitution.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a1-constitution/}.
\par\noindent\textbf{Relevance.} The constitutional charter of \GGTheory{}: states the four axioms (\LC{}, \GGEP{}, \OGGEP{}, and \MSMF{} --- ``Maximum Symmetry, Minimum Freedom'', the canonical expansion used in Postulate~\ref{post:MSMF}) and the seven postulates. The ledger-minimality corollary of \MSMF{} consumed by leg (L1) is grounded here. First cited in \S3.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
\textbf{Arisaka, K. (2026).} \GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{}). Arisaka-GG-A2-GG6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a2-gg6/}.
\par\noindent\textbf{Relevance.} The bulk paper: establishes the six-dimensional geometry, the canonical metric used in \S\ref{ssec:E-naive}, the role of $\mathrm{Spin}(1,5)$, and --- decisively for the present paper --- the Diophantine uniqueness of the \OGNfive{} ledger $(1,3,5;7,35)$ (its Theorem~6), of which the generation count is here proved to be the physical realization. First cited in \S1.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
\textbf{Arisaka, K. (2026).} Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{}). Arisaka-GG-A3-BI6.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a3-bi6/}.
\par\noindent\textbf{Relevance.} The anomaly paper: proves \BIsix{} closure, the $-2/5$ trace-ratio coefficients, the integrality of $[X_4]$, and the dual-lock identity $K = LJ = M^2+N^2+L^2 = 35$ consumed by leg (L2). Its $N_g$-free block-size theorem T-BI6-9.0 and its $\gamma$-family characterization of the Abelian coefficient are consumed by \S6.2 and Appendix~\ref{app:C}: the former removes the last circularity from the extensive capacity's flux reading, the latter closes the coefficient route to $N_g$. First cited in \S1.

\bibitem[Arisaka(2026, A5-\GDOS{})]{ArisakaA5GDOS}
\textbf{Arisaka, K. (2026).} \GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT. Arisaka-GG-A5-GDOS.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a5-gdos/}.
\par\noindent\textbf{Relevance.} Develops \GDOS{} as the open-system framework underlying boundary observation, on which leg (L2) is built. Its open problem O3 records the \GFF{} Morse-positivity question as unsettled in any companion paper --- the present paper adopts it as Postulate~\ref{post:morse-pos} and reduces its strict form in Appendix~\ref{app:E}. First cited in \S3.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
\textbf{Arisaka, K. (2026).} The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind. Arisaka-GG-A6-GRIP.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a6-grip/}.
\par\noindent\textbf{Relevance.} Develops the \GFF{} and \GRIP{} pillars used in leg (L2); its cascade-closure theorem T4.1 consumes the same capacity value now delivered by Theorem~\ref{thm:capacity}, and its open problem M10 --- derive the budget independent of $N_g=3$ at every intermediate step --- is met by the present architecture, which consumes neither $K=35$ nor $J=7$ (\S 8.2). The present paper supplies the dedicated three-leg treatment of $N_g=3$ that the cohort reserves. First cited in the Executive Summary.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
\textbf{Arisaka, K. (2026).} Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary. Arisaka-GG-A8-QM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a8-qm/}.
\par\noindent\textbf{Relevance.} The boundary-quantum-mechanics companion:
the Constitutional Born Theorem with its named premises --- the \AFive{}
forward semigroup (P1) and the collision-level contraction rate --- which
are exactly the premises of Theorems~\ref{thm:AD2}--\ref{thm:AD3}
(Remark~\ref{rem:born-grade}), so that leg (L2)'s transport geometry is as
secure as the Born rule and no more conditional. Also the pedagogical home
of the Berry-phase worked example that Theorem~\ref{thm:AD3} elevates to a
statement about the runtime's own family transport. First cited in \S6.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
\textbf{Arisaka, K. (2026).} The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry. Arisaka-GG-A9-SM.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a9-sm/}.
\par\noindent\textbf{Relevance.} Derives the 28 Standard-Model parameters from \BIsix{} and \OGNfive{}; provides the explicit fermion-mass codebook (lepton and quark Yukawas, QD-completion factors, and the Higgs vacuum $v^{(1)} = 246.331354$~GeV) consumed by leg (L3). First cited in the Executive Summary.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
\textbf{Arisaka, K. (2026).} The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}. Arisaka-GG-P1-KK.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p1-kk/}.
\par\noindent\textbf{Relevance.} The physics of the structural
torus through which --- by T-INF-1 --- the family's only coupling
channel runs; the Wilson-line/holonomy sector that realizes the
mixed class. First cited in Appendix~\ref{app:H}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
\textbf{Arisaka, K. (2026).} The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}. Arisaka-GG-P3-Electron.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p3-electron/}.
\par\noindent\textbf{Relevance.} The sister paper of the Predictions Branch: develops the ground-state ontology of \S9 into the dedicated six-question treatment of the electron (identity, stability, copies, the positron as conjugate readout, pointlikeness, sensing), importing the present paper's $N_g=3$ theorem and its declared conditioning (the coupling clause and L-INT) verbatim, and adding the geometric pointlikeness theorem ($F \equiv 1$ as its laboratory corollary) via a readout-locality lemma derived from AX-1 of A1, with field response derived as composite geodesic motion on the AX-2 connection (per A5's Part VI) under the two-sector reading: the phase sector moves the electron, the scale sector transmutes it, and radiation is motion's dissipative channel. First cited in \S4.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
\textbf{Arisaka, K. (2026).} The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}. Arisaka-GG-P4-Real.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p4-real/}.
\par\noindent\textbf{Relevance.} The origin-of-$i$ member paper of the Foundations Branch: derives the boundary's imaginary unit from two real structures of the \ATwo{} arena --- the compact phase circle and the causal clock --- the complex structure that the unitary mixing matrices of leg (L1) presuppose at boundary level. Cited at roster level in \S\ref{sec:cohort-p2}. First cited in \S4.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
\textbf{Arisaka, K. (2026).} The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter. Arisaka-GG-P5-CPT.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p5-cpt/}.
\par\noindent\textbf{Relevance.} The CPT member paper of the Foundations Branch, and the Foundations member nearest to the present paper: C, P, and T as orientation reversals of the three real bulk factors, with a flavor-holonomy calculus that runs on the three-generation structure derived here --- for $N_g=3$ its chiral-pentagon reading ties the CP-odd quartets of the mixing matrix to the single surviving Jarlskog phase, the same empirical non-vanishing that leg (L1) consumes. Cited in \S\ref{sec:cohort-p2}. First cited in \S4.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
\textbf{Arisaka, K. (2026).} The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}. Arisaka-GG-C1-Kaxion.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c1-kaxion/}.
\par\noindent\textbf{Relevance.}
Cited in Section~14 as a further example of an open-system structural
prediction (a $\sim\mu$eV axion-like dark matter candidate) that
closed-system frameworks cannot derive. First cited in \S14.

\bibitem[Arisaka(2026, CKM)]{ArisakaCKM}
\textbf{Arisaka, K. (2026).} \GGTheory{}: CKM, PMNS, and CP violation from \BIsix{}. (In preparation.)
\par\noindent\textbf{Relevance.}
Provides the predicted value $\JCKM^{\mathrm{pred}} \approx 3.06\times
10^{-5}$, removing any residual circularity in (L1). First cited in \S5.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
\textbf{Arisaka, K. (2026).} From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics. Arisaka-GG-A4-Euclid.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a4-euclid/}.
\par\noindent\textbf{Relevance.}
The parameter audit that this paper's claim has to survive. A4 records every dimensional constant of the framework as a conversion factor and every dimensionless integer as an output of a Diophantine condition, and $N_g$ is one of the integers on its list. Cited because a derivation of the family number is only worth what the audit says it is worth: if $N_g$ were tunable anywhere in the chain, A4 is where that would show.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
\textbf{Arisaka, K. (2026).} The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}. Arisaka-GG-A7-Qi4.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a7-qi4/}.
\par\noindent\textbf{Relevance.}
The quantum-information pillars, and the runtime vocabulary in which leg~L2 is stated. The capacity argument here is a statement about what a boundary runtime can fund, and the terms it uses --- relaxation, gap, descent --- are A7's rather than this paper's. Cited for the vocabulary; the capacity theorem itself is proved here.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
\textbf{Arisaka, K. (2026).} The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry. Arisaka-GG-A10-Cosmos.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a10-cosmos/}.
\par\noindent\textbf{Relevance.}
The cosmological reading of the same geometry, and an independent consumer of the family number: the light-neutrino count that primordial nucleosynthesis and the microwave background constrain is the same $N_g$ this paper derives. The agreement is a consistency check across two very different measurements and is worth naming as one. No step of the derivation here depends on A10.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
\textbf{Arisaka, K. (2026).} The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program. Arisaka-GG-A11-Origin.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/a11-origin/}.
No result of the present paper depends on it.
\par\noindent\textbf{Relevance.}
The capstone, and the one companion whose arrow points the other way: A11 consumes this paper's result rather than supplying anything to it. Its identity cascade runs from quarks upward and takes the family count as fixed, so a failure here propagates into it and not the reverse. Cited so that the dependency is visible from this end as well as from A11's.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
\textbf{Arisaka, K. (2026).} The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$. Arisaka-GG-C2-Hubble.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c2-hubble/}.
\par\noindent\textbf{Relevance.}
A phenomenology paper on the astrophysical side of the same geometry. It carries no result this paper uses and this paper carries none it uses. Cited at roster level in the cohort map, and the note says roster level rather than manufacturing a connection.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
\textbf{Arisaka, K. (2026).} Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity. Arisaka-GG-C3-BH.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/c3-bh/}.
\par\noindent\textbf{Relevance.}
The second phenomenology paper, and the other end of the open-system reading whose boundary law leg~L2 leans on. What the two share is \GDOS{}: C3 asks what the projection does at a horizon, this paper asks what it funds on the family moduli. Nothing here turns on the horizon result. Cited at roster level.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
\textbf{Arisaka, K. (2026).} The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}. Arisaka-GG-P6-Proton.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/p6-proton/}.
\par\noindent\textbf{Relevance.}
The other Predictions-Branch paper whose whole content is an integer statement holding exactly. P6 argues that baryon number is gauged and its anomaly canceled, so the proton never decays; this paper argues that the same ledger admits exactly three families. Both are all-or-nothing claims about integers, and both would be retired by a single contrary measurement. Cited at roster level in the cohort map.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
\textbf{Arisaka, K. (2026).} Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out. Arisaka-GG-W1-Symmetry.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w1-symmetry/}.
\par\noindent\textbf{Relevance.}
The survey of the century this paper's method belongs to. The upper bound in leg~L2 is not a symmetry argument --- it is a capacity argument, and it reaches a conservation-style conclusion without a conserved current to hang it on. W1 is the cohort's account of exactly that shift. It supplies no number and no theorem used here. Cited at roster level.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
\textbf{Arisaka, K. (2026).} The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology. Arisaka-GG-W2-Holonomy.
\par\noindent Preprint, available at \url{https://preprints.arisaka-gg.org/w2-holonomy/}.
\par\noindent\textbf{Relevance.}
The classification the lattice argument belongs to. The unit that makes leg~L2 decisive is a spin-refined quantization condition --- protection by topology rather than by symmetry --- and that is W2's subject. Cited at roster level in the cohort map; the lattice derivation itself is complete in Part~VI and borrows nothing from it.

\end{thebibliography}

\section*{B.\ Other references}
\begin{thebibliography}{99}

\bibitem[ALEPH et al.(2006)]{ALEPH2006}
\textbf{ALEPH, DELPHI, L3, OPAL Collaborations \& LEP Electroweak Working
Group.}
Precision electroweak measurements on the $Z$ resonance.
\emph{Physics Reports} \textbf{427} (2006) 257--454.
\par\noindent\textbf{Relevance.}
Establishes the empirical bound on the number of light neutrino species,
which is the most direct experimental confirmation of $N_\nu = 3$. Underpins
prediction P1 of this paper. First cited in \S1.

\bibitem[Blumenhagen et al.(2007)]{Blumenhagen2007}
\textbf{Blumenhagen, R., K\"ors, B., L\"ust, D., Stieberger, S.}
Four-dimensional string compactifications with D-branes, orientifolds and
fluxes.
\emph{Physics Reports} \textbf{445} (2007) 1--193.
\par\noindent\textbf{Relevance.}
Comprehensive review of magnetized toroidal compactifications; provides the
index-theorem family-counting argument that the present paper compares
against. First cited in \S1.

\bibitem[Candelas et al.(1985)]{Candelas1985}
\textbf{Candelas, P., Horowitz, G. T., Strominger, A., \& Witten, E.}
Vacuum configurations for superstrings.
\emph{Nuclear Physics B} \textbf{258} (1985) 46--74.
\par\noindent\textbf{Relevance.}
The seminal Calabi--Yau compactification paper; the $|\chi/2| = 3$
family-counting condition compared against in Section~11. First cited in \S1.

\bibitem[Gorini, Kossakowski \& Sudarshan(1976)]{GKS1976}
\textbf{Gorini, V., Kossakowski, A., \& Sudarshan, E. C. G.}
Completely positive dynamical semigroups of N-level systems.
\emph{Journal of Mathematical Physics} \textbf{17} (1976) 821--825.
\par\noindent\textbf{Relevance.}
The GKLS theorem characterizing the most general completely positive
trace-preserving (CPTP) generator of open-system quantum dynamics. Together
with Lindblad (1976), this is the mathematical backbone of open-system
quantum theory and the prototype for the CPTP boundary projection that \GDOS{}
elevates to foundational status. First cited in \S14.

\bibitem[Green \& Schwarz(1984)]{GreenSchwarz1984}
\textbf{Green, M. B., \& Schwarz, J. H.}
Anomaly cancellations in supersymmetric $D=10$ gauge theory and superstring
theory.
\emph{Physics Letters B} \textbf{149} (1984) 117--122.
\par\noindent\textbf{Relevance.}
The original modified Bianchi identity construction; the six-dimensional descent of which is \BIsix{} of the present framework. First cited in \S3.

\bibitem[Green, Schwarz \& Witten(1987)]{GSW1987}
\textbf{Green, M. B., Schwarz, J. H., \& Witten, E.}
\emph{Superstring Theory, Vols.\ 1--2.}
Cambridge University Press (1987).
\par\noindent\textbf{Relevance.}
Standard reference for superstring compactifications and anomaly
cancellation; provides the broader context for the \BIsix{} construction. First cited in \S1.

\bibitem[Born(1926)]{Born1926}
\textbf{Born, M.}
Zur Quantenmechanik der Sto{\ss}vorg\"ange.
\emph{Zeitschrift f\"ur Physik} \textbf{37} (1926) 863--867.
\par\noindent\textbf{Relevance.}
The Born rule that translates closed-system unitary evolution into
open-system probabilistic outcomes. The Born rule is the historical seed of
the open-system perspective; cited in Section~14 as one of the foundational
moments where open-system aspects entered quantum mechanics without being
elevated to foundational status. First cited in \S14.

\bibitem[Freed \& Hopkins(2000)]{FreedHopkins2000}
\textbf{Freed, D. S., \& Hopkins, M. J.}
On Ramond--Ramond fields and K-theory.
\emph{Journal of High Energy Physics} \textbf{2000}(05) (2000) 044.
\par\noindent\textbf{Relevance.}
Foundational paper on the differential-cohomology classification of
higher-form gauge fields and anomaly inflow. Provides the technical machinery
underlying the non-degeneracy of the Hopkins--Singer inflow map (hypothesis
$H_2$ in Appendix~E). First cited in Appendix~E.

\bibitem[Helgason(1979)]{Helgason1979}
\textbf{Helgason, S.}
\emph{Differential Geometry, Lie Groups, and Symmetric Spaces.}
Academic Press (1979).
\par\noindent\textbf{Relevance.}
Standard reference for the Fubini--Study metric on $\CPN$ and its geodesic
structure, used in Lemma~\ref{lem:cascade-closure}. First cited in \S6.

\bibitem[Hirzebruch(1966)]{Hirzebruch1966}
\textbf{Hirzebruch, F.}
\emph{Topological Methods in Algebraic Geometry} (3rd ed.).
Springer-Verlag, Berlin (1966).
\par\noindent\textbf{Relevance.}
The foundational reference for the Hirzebruch signature theorem and the
$L$-polynomial computation, used throughout Step 2 of Appendix~E. First cited in Appendix~E.

\bibitem[Hirzebruch(1995)]{Hirzebruch1995}
\textbf{Hirzebruch, F.}
\emph{Topological Methods in Algebraic Geometry} (Classics in Mathematics
ed.).
Springer-Verlag, Berlin (1995).
\par\noindent\textbf{Relevance.}
Reissue of the classic monograph; provides the explicit Pontryagin and
$L$-class computations on $\CPN$ used in Steps 2--4 of Appendix~E. First cited in Appendix~E.

\bibitem[Hopkins \& Singer(2005)]{HopkinsSinger2005}
\textbf{Hopkins, M. J., \& Singer, I. M.}
Quadratic functions in geometry, topology, and M-theory.
\emph{Journal of Differential Geometry} \textbf{70} (2005) 329--452.
\par\noindent\textbf{Relevance.}
The differential cohomology framework underlying \BIsix{} closure, and the
quadratic refinement behind component (C) of the integrality theorem L-INT
(Theorem~\ref{thm:LINT}) and its residue R-INT. First cited in \S6.

\bibitem[Kato(1950)]{Kato1950}
\textbf{Kato, T.}
On the adiabatic theorem of quantum mechanics.
\emph{Journal of the Physical Society of Japan} \textbf{5} (1950) 435--439.
\par\noindent\textbf{Relevance.}
The adiabatic theorem in its projector form, and the transport construction
$W' = [P',P]W$ used verbatim in Theorem~\ref{thm:AD1}. The three-line
intertwining proof in \S6 is Kato's. First cited in \S6.

\bibitem[Berry(1984)]{Berry1984}
\textbf{Berry, M. V.}
Quantal phase factors accompanying adiabatic changes.
\emph{Proceedings of the Royal Society of London A} \textbf{392} (1984)
45--57.
\par\noindent\textbf{Relevance.}
The geometric phase of adiabatic transport and the connection on the
tautological bundle that Theorem~\ref{thm:AD3} identifies with the
runtime's family transport. First cited in \S6.

\bibitem[Provost \& Vallee(1980)]{ProvostVallee1980}
\textbf{Provost, J. P., \& Vallee, G.}
Riemannian structure on manifolds of quantum states.
\emph{Communications in Mathematical Physics} \textbf{76} (1980) 289--301.
\par\noindent\textbf{Relevance.}
The quantum geometric tensor: the Fubini--Study metric as the fidelity
metric on state manifolds, with the Berry curvature as its antisymmetric
partner --- the identification used in Theorem~\ref{thm:AD3}(c) and proved
in Appendix~F. First cited in \S6.

\bibitem[Avron et al.(2012)]{AvronFraasGrafGrech2012}
\textbf{Avron, J. E., Fraas, M., Graf, G. M., \& Grech, P.}
Adiabatic theorems for generators of contracting evolutions.
\emph{Communications in Mathematical Physics} \textbf{314} (2012) 163--191.
\par\noindent\textbf{Relevance.}
The adiabatic theorem for gapped generators of contraction semigroups ---
the exact setting of the \GDOS{} runtime (CPTP/GKLS by \AFive{}
T-GDOS-3--T-GDOS-4) --- used in the proof of Theorem~\ref{thm:AD2} in
Appendix~F. First cited in Appendix~F.

\bibitem[Kobayashi \& Nomizu(1969)]{KobayashiNomizu1969}
\textbf{Kobayashi, S., \& Nomizu, K.}
\emph{Foundations of Differential Geometry, Vol.\ II.}
Wiley-Interscience, New York (1969).
\par\noindent\textbf{Relevance.}
Standard reference for Chern--Weil theory and the identification of
$\tr R\wedge R$ with the first Pontryagin class on $\mathrm{SO}(2k)$ frame
bundles, used in Lemma~\ref{lem:E-H1} of Appendix~E. First cited in Appendix~E.

\bibitem[Ib\'a\~nez et al.(1987)]{Ibanez1987}
\textbf{Ib\'a\~nez, L. E., Nilles, H. P., \& Quevedo, F.}
Orbifolds and Wilson lines.
\emph{Physics Letters B} \textbf{187} (1987) 25--32.
\par\noindent\textbf{Relevance.}
The $\mathbb{Z}_3$-orbifold construction with three twisted sectors; the most
natural prior model giving exactly three generations, compared against in
Section~11. First cited in \S1.

\bibitem[Ib\'a\~nez \& Uranga(2012)]{IbanezUranga2012}
\textbf{Ib\'a\~nez, L. E., \& Uranga, A. M.}
\emph{String Theory and Particle Physics: An Introduction to String
Phenomenology.}
Cambridge University Press (2012).
\par\noindent\textbf{Relevance.}
Standard reference for D-brane and intersecting-brane family counting;
provides the broader phenomenological background. First cited in \S1.

\bibitem[Dobrescu \& Poppitz(2001)]{DobrescuPoppitz2001}
\textbf{Dobrescu, B. A., \& Poppitz, E.}
Number of fermion generations derived from anomaly cancellation.
\emph{Physical Review Letters} \textbf{87} (2001) 031801.
\par\noindent\textbf{Relevance.}
Proves that global anomaly cancellation forces exactly three generations for
the Standard Model in two universal extra dimensions --- a genuine derivation
of both bounds, conditional on the extra-dimensional assumption. Engaged in
Sections~1, 11, and~14. First cited in \S1.

\bibitem[Lindblad(1976)]{Lindblad1976}
\textbf{Lindblad, G.}
On the generators of quantum dynamical semigroups.
\emph{Communications in Mathematical Physics} \textbf{48} (1976) 119--130.
\par\noindent\textbf{Relevance.}
The Lindblad master equation, derived independently of Gorini, Kossakowski \&
Sudarshan (1976), gives the most general CPTP generator of Markovian
open-system quantum dynamics. Cited in Section~14 as the foundational
technical backbone of modern open-system quantum theory. First cited in \S14.

\bibitem[Maldacena(1997)]{Maldacena1997}
\textbf{Maldacena, J.}
The large $N_g$ limit of superconformal field theories and supergravity.
\emph{Advances in Theoretical and Mathematical Physics} \textbf{2} (1997)
231--252.
\par\noindent\textbf{Relevance.}
The original AdS/CFT correspondence: the first explicit example of
holographic bulk-to-boundary descent. Cited in Section~14 as the prototype
for the bulk-to-boundary projection that \GDOS{} elevates to foundational
status. First cited in \S14.

\bibitem[Kobayashi \& Maskawa(1973)]{KM1973}
\textbf{Kobayashi, M., \& Maskawa, T.}
CP-violation in the renormalizable theory of weak interaction.
\emph{Progress of Theoretical Physics} \textbf{49} (1973) 652--657.
\par\noindent\textbf{Relevance.}
The original Kobayashi--Maskawa observation that $N_g\ge 3$ is required for CP
violation in the quark sector; the historical precursor to (L1) of the
present paper. First cited in \S1.

\bibitem[Kikuchi et al.(2021)]{KikuchiKobayashiUchida2021}
\textbf{Kikuchi, S., Kobayashi, T., \& Uchida, H.}
Modular flavor symmetries of three-generation modes on magnetized toroidal
orbifolds.
\emph{Physical Review D} \textbf{104} (2021) 065008.
\par\noindent\textbf{Relevance.}
A modern modular-flavor construction realizing three generations
conditionally on chosen orbifold and flux inputs; representative of the
$N_g$-as-input class compared against in Section~11. First cited in \S1.

\bibitem[Milnor(1963)]{Milnor1963}
\textbf{Milnor, J.}
\emph{Morse Theory.}
Annals of Mathematics Studies 51, Princeton University Press (1963).
\par\noindent\textbf{Relevance.}
The foundational reference for Morse theory; the height function on $\CPN$ as
a perfect Morse function is the textbook example that drives
Lemma~\ref{lem:morse-poly}. First cited in \S6.

\bibitem[NOvA Collaboration(2022)]{NOvA2022}
\textbf{NOvA Collaboration.}
Improved measurement of neutrino oscillation parameters by the NOvA
experiment.
\emph{Physical Review D} \textbf{106} (2022) 032004.
\par\noindent\textbf{Relevance.}
Provides the empirical evidence for non-trivial PMNS mixing and a non-zero
leptonic CP phase, supporting the parallel application of (L1) in the
leptonic sector. First cited in \S5.

\bibitem[Particle Data Group(2024)]{PDG2024}
\textbf{Particle Data Group.}
Review of Particle Physics.
\emph{Progress of Theoretical and Experimental Physics} \textbf{2024} (2024)
083C01.
\par\noindent\textbf{Relevance.}
The standard reference for measured Standard-Model parameters, including all
charged-lepton and quark masses against which the \ANine{} codebook of (L3)
is compared. First cited in \S1.

\bibitem[Planck Collaboration(2020)]{Planck2020}
\textbf{Planck Collaboration.}
Planck 2018 results. VI. Cosmological parameters.
\emph{Astronomy \& Astrophysics} \textbf{641} (2020) A6.
\par\noindent\textbf{Relevance.}
Provides the cosmological measurement $N_{\mathrm{eff}} = 2.99\pm 0.17$, an
independent confirmation of $N_\nu = 3$ complementing the LEP electroweak
result. First cited in \S1.


\bibitem[Rokhlin(1952)]{Rokhlin1952}
\textbf{Rokhlin, V.~A.}
New results in the theory of four-dimensional manifolds.
\emph{Doklady Akademii Nauk SSSR (N.S.)} \textbf{84} (1952) 221--224.
\par\noindent\textbf{Relevance.}
The divisibility theorem --- the signature of a closed spin 4-manifold is
divisible by 16 --- behind the spin reading of the parity route: a spin
stratum could not host the exact saturation $3\pi = 3\pi$ (it would force
$\langle p_1\rangle \in 48\mathbb{Z} \not\ni 3$), so the non-spin-ness of
$\CP^2$ is load-bearing (Appendix~E, \S\ref{ssec:E-rokhlin}). First cited
in Appendix~E.

\bibitem[Susskind(1995)]{Susskind1995}
\textbf{Susskind, L.}
The world as a hologram.
\emph{Journal of Mathematical Physics} \textbf{36} (1995) 6377--6396.
\par\noindent\textbf{Relevance.}
The holographic principle in its string-theory formulation: the suggestion
that bulk degrees of freedom project onto a lower-dimensional boundary. Cited
in Section~14 as one of the foundational steps toward the open-system /
boundary-projection perspective elevated to foundational status by \GDOS{}. First cited in \S14.

\bibitem['t Hooft(1993)]{tHooft1993}
\textbf{'t Hooft, G.}
Dimensional reduction in quantum gravity.
arXiv:gr-qc/9310026 (1993). Reprinted in \emph{Salamfestschrift} (World
Scientific, 1993).
\par\noindent\textbf{Relevance.}
The original holographic-principle proposal: that the fundamental degrees of
freedom of a $(d{+}1)$-dimensional gravitational theory project onto a
$d$-dimensional boundary. Cited in Section~14 as the seminal articulation of
the holographic boundary-projection idea. First cited in \S14.

\bibitem[von Neumann(1932)]{vonNeumann1932}
\textbf{von Neumann, J.}
\emph{Mathematische Grundlagen der Quantenmechanik.}
Springer-Verlag, Berlin (1932). English translation (1955),
\emph{Mathematical Foundations of Quantum Mechanics}, Princeton University
Press.
\par\noindent\textbf{Relevance.}
The foundational mathematical formulation of quantum mechanics as unitary
evolution on a Hilbert space, together with the projection postulate for
measurement. Cited in Section~14 as the moment when closed-system QM was
formalized; the open-system aspects of measurement were already present but
were not elevated to foundational status. First cited in \S14.

\bibitem[Zeh(1970)]{Zeh1970}
\textbf{Zeh, H. D.}
On the interpretation of measurement in quantum theory.
\emph{Foundations of Physics} \textbf{1} (1970) 69--76.
\par\noindent\textbf{Relevance.}
The first articulation of decoherence: the mechanism by which open-system
entanglement with the environment produces effective classicality from
quantum dynamics. Cited in Section~14 as a foundational step toward
recognizing the open-system character of quantum measurement. First cited in \S14.

\bibitem[Zurek(1981)]{Zurek1981}
\textbf{Zurek, W. H.}
Pointer basis of quantum apparatus: Into what mixture does the wave packet
collapse?
\emph{Physical Review D} \textbf{24} (1981) 1516--1525.
\par\noindent\textbf{Relevance.}
The pointer-basis selection mechanism in decoherence theory: open-system
dynamics privileges a particular basis (``pointer states'') for the
measurement outcomes. Together with Zeh (1970), provides the technical
machinery underlying the open-system measurement perspective discussed in
Section~14. First cited in \S14.

\bibitem[Schwarz(1993)]{SchwarzM1993}
\textbf{Schwarz, M.}
\emph{Morse Homology.}
Birkh\"auser Progress in Mathematics 111 (1993).
\par\noindent\textbf{Relevance.}
Modern reference for transversality of gradient flows between Morse critical
points; underlies Lemma~\ref{lem:saddle-decay} on saddle decay paths in
$\CPN$. First cited in \S6.

\bibitem[T2K Collaboration(2020)]{T2K2020}
\textbf{T2K Collaboration.}
Constraint on the matter--antimatter symmetry-violating phase in neutrino
oscillations.
\emph{Nature} \textbf{580} (2020) 339--344.
\par\noindent\textbf{Relevance.}
Empirical evidence of leptonic CP violation; provides independent support for
(L1) in the leptonic sector. First cited in \S5.

\bibitem[Witten(1996)]{Witten1996}
\textbf{Witten, E.}
On flux quantization in M-theory and the effective action.
\emph{Journal of Geometry and Physics} \textbf{22} (1996) 1--13.
\par\noindent\textbf{Relevance.}
The flux-quantization argument whose spin-refined level lattice drives
component (C) of the integrality theorem L-INT (Appendix~G,
\S\ref{ssec:G-LINT}, step C4); the prototype setting for the shifted
quantization on a spin bulk with a non-spin stratum. First cited in \S6.

\bibitem[Witten(1982)]{Witten1982}
\textbf{Witten, E.}
Supersymmetry and Morse theory.
\emph{Journal of Differential Geometry} \textbf{17} (1982) 661--692.
\par\noindent\textbf{Relevance.}
The foundational paper relating supersymmetric quantum mechanics to Morse
theory; provides the instanton-action computation $S = \pi k$ used in
Lemma~\ref{lem:cascade-closure}. First cited in \S6.

\bibitem[Witten(1985)]{Witten1985}
\textbf{Witten, E.}
Global anomalies in string theory.
In: \emph{Symposium on anomalies, geometry, topology} (Argonne, IL, 1985),
61--99.
\par\noindent\textbf{Relevance.}
Reviews global-anomaly constraints and their relation to family number;
provides one of the precursors to the \BIsix{} anomaly polynomial. First cited in \S1.

%% ---- v2.1 additions (R6 literature engagement + sourcing) ----

\bibitem[Wan, Wang \& Yau(2026)]{WanWangYau2026}
\textbf{Wan, Z., Wang, J., \& Yau, S.-T.}
Three Families and Pontryagin Class.
Preprint, arXiv (submitted January 29, 2026).
\par\noindent\textbf{Relevance.}
Bordism-classified global-anomaly theorem: with $N_c, N_f$ minimal and $N_c$
odd (fermionic baryons), $N_c = N_f = 3$ is unique; absent those conditions,
$N_f \ge 3$ with no upper bound. The most rigorous family-puzzle result
internal to the 4D Standard Model; complementary to the present upper bound
(Section~11). First cited in \S11.

\bibitem[Chamseddine, Connes \& Marcolli(2007)]{ChamseddineConnesMarcolli2007}
\textbf{Chamseddine, A. H., Connes, A., \& Marcolli, M.}
Gravity and the standard model with neutrino mixing.
\emph{Advances in Theoretical and Mathematical Physics} \textbf{11} (2007)
991--1089.
\par\noindent\textbf{Relevance.}
The canonical spectral-triple reconstruction of the Standard Model in
noncommutative geometry, in which the generation number enters as the input
multiplicity $\mathcal{H}_F = M_F^{\oplus 3}$; compared against in
Section~11 as the leading $N_g$-as-input framework. First cited in \S11.

\bibitem[Gourlay \& Gresnigt(2024)]{GourlayGresnigt2024}
\textbf{Gourlay, L., \& Gresnigt, N.}
Algebraic realization of three fermion generations with $S_3$ family and
unbroken gauge symmetry from $\mathbb{C}\ell(8)$.
\emph{European Physical Journal C} \textbf{84} (2024) 1129.
\par\noindent\textbf{Relevance.}
Division-algebra/Clifford-algebra route to three generations via the $S_3$
automorphisms of the sedenions embedded in $\mathbb{C}\ell(8)$; the algebra
choice is the input. Compared against in Section~11. First cited in \S11.

\bibitem[Ibe, Kusenko \& Yanagida(2016)]{IbeKusenkoYanagida2016}
\textbf{Ibe, M., Kusenko, A., \& Yanagida, T. T.}
Why three generations?
\emph{Physics Letters B} \textbf{758} (2016) 365--369.
\par\noindent\textbf{Relevance.}
The modern anthropic argument: baryogenesis plus dark matter motivate
$N_g \ge 3$ heuristically, with no upper bound. Engaged in Section~11. First cited in \S11.

\bibitem[Esteban et al.(2020)]{NuFIT2020}
\textbf{Esteban, I., Gonz\'alez-Garc\'ia, M. C., Maltoni, M., Schwetz, T.,
\& Zhou, A.}
The fate of hints: updated global analysis of three-flavor neutrino
oscillations.
\emph{Journal of High Energy Physics} \textbf{2020}(09) (2020) 178.
\par\noindent\textbf{Relevance.}
Global-fit source (NuFIT) for the leptonic Jarlskog invariant best-fit value
quoted in Section~5.2. First cited in \S5.

\bibitem[Bianchi(1902)]{Bianchi1902}
\textbf{Bianchi, L.}
Sui simboli a quattro indici e sulla curvatura di Riemann.
\emph{Rendiconti della Accademia Nazionale dei Lincei} \textbf{11}
(1902) 3--7.
\par\noindent\textbf{Relevance.} The original identity whose
century of modifications --- Green--Schwarz's above all --- is the
lineage of \BIsix{}; the point of departure of \S2's external road.
First cited in Appendix~\ref{app:H}.

\bibitem[Dirac(1931)]{Dirac1931}
\textbf{Dirac, P. A. M.}
Quantised singularities in the electromagnetic field.
\emph{Proceedings of the Royal Society of London A} \textbf{133}
(1931) 60--72.
\par\noindent\textbf{Relevance.} The first topological integer of
physics: a phase that must close forces an integer --- the
ur-argument that reappears in the Berry bundle's quantization, this paper's unit lattice, and the Wu-evenness of T-INF-3. First cited in Appendix~\ref{app:H}.

\bibitem[Cheeger \& Simons(1985)]{CheegerSimons1985}
\textbf{Cheeger, J., \& Simons, J.}
Differential characters and geometric invariants. In \emph{Geometry
and Topology}, Springer LNM 1167 (1985) 50--80.
\par\noindent\textbf{Relevance.} The origin of the
differential-character formalism in which the seam's holonomy and
torsion data live; the ancestor of the Hopkins--Singer refinement
used throughout. First cited in Appendix~\ref{app:H}.

\bibitem[Dai \& Freed(1994)]{DaiFreed1994}
\textbf{Dai, X., \& Freed, D. S.}
$\eta$-invariants and determinant lines. \emph{Journal of
Mathematical Physics} \textbf{35} (1994) 5155--5194.
\par\noindent\textbf{Relevance.} The collar formalism of the
intended class's global-anomaly ladder, and the frame in which the
relative closure theorem T-INF-8 is packaged (engineering item E-4).
First cited in Appendix~\ref{app:H}.

\bibitem[Witten(1996b)]{Witten1996five}
\textbf{Witten, E.}
Five-brane effective action in M-theory. \emph{Journal of Geometry
and Physics} \textbf{22} (1997) 103--133; arXiv:hep-th/9610234.
\par\noindent\textbf{Relevance.} The fivebrane partition function
--- the physical origin of the $(k,i) = (2,2)$ pattern by which a
six-dimensional seam pushes forward to a line bundle on a family
base, the very type of the bridge. First cited in Appendix~\ref{app:H}.

\bibitem[Witten(1996a)]{Witten1996flux}
\textbf{Witten, E.}
On flux quantization in M-theory and the effective action.
\emph{Journal of Geometry and Physics} \textbf{22} (1997) 1--13;
arXiv:hep-th/9609122.
\par\noindent\textbf{Relevance.} The prototype of every
spin-shifted integrality statement in the architecture --- the
lattice halving whose Wu-condition form drives T-INF-3. First cited in Appendix~\ref{app:H}.

\bibitem[Freed(2000)]{Freed2000}
\textbf{Freed, D. S.}
Dirac charge quantization and generalized differential cohomology.
arXiv:hep-th/0011220 (with an appendix by M.~J.~Hopkins).
\par\noindent\textbf{Relevance.} Charge quantization of abelian
gauge fields as cocycles in generalized differential cohomology ---
the doctrinal frame in which the \BIsix{} data are differential
cocycles at all. First cited in Appendix~\ref{app:H}.

\bibitem[Milnor \& Stasheff(1974)]{MilnorStasheff1974}
\textbf{Milnor, J., \& Stasheff, J.}
\emph{Characteristic Classes.} Annals of Mathematics Studies 76,
Princeton University Press (1974).
\par\noindent\textbf{Relevance.} The Wu formula and the dimensional
vanishing $\nu_j = 0$ for $2j > \dim$, which make degree-four Wu
structures canonical in six dimensions and set up the evenness that
drives T-INF-3. First cited in Appendix~\ref{app:H}.

\bibitem[Monnier(2016)]{Monnier2016}
\textbf{Monnier, S.}
Topological field theories on manifolds with Wu structures.
\emph{Reviews in Mathematical Physics} \textbf{29} (2017) 1750015;
arXiv:1607.01396.
\par\noindent\textbf{Relevance.} The invertible theories carrying
degree-four Wu data whose boundary is the six-dimensional self-dual
field --- the closest external sibling of the seam structure
quantized here. First cited in Appendix~\ref{app:H}.

\bibitem[Sati \& Schreiber(2024)]{SatiSchreiber2024}
\textbf{Sati, H., \& Schreiber, U.}
Flux quantization. \emph{Encyclopedia of Mathematical Physics}, 2nd
ed.; arXiv:2402.18473.
\par\noindent\textbf{Relevance.} The modern axiomatization of
which cohomology theory quantizes a given higher Bianchi identity;
the frame in which \BIsix{}'s Green--Schwarz structure is read as a
twisted differential cocycle. First cited in Appendix~\ref{app:H}.

\end{thebibliography}

\end{document}
