%% v20.0 (August 21, 2026) -- citation-label repair, version deliberately
%%   unchanged (nothing is published yet).  69 of this paper's 94 bibitem
%%   labels wrote "Author (year)" with a space, so 53 in-text citations
%%   rendered as "(Milnor , 1963)".  Measured across the twelve deposit
%%   papers: A6 had 53 and the other ELEVEN had zero, so this was A6's
%%   defect and not house style; A6's own 23 Arisaka labels were already
%%   in the correct form and rendered correctly.
%%
%%   THE FIX IS NOT SIMPLY DELETING THE SPACE.  natbib's extended label is
%%   [Short(Year)Long].  Tightening to "Milnor(1963)" leaves Long EMPTY,
%%   which drops natbib out of author-year and into NUMERIC mode: every
%%   citation in the paper became [49] and the reference list became a
%%   numbered list.  It also compressed the paper 313 -> 311 pages and
%%   stranded a one-line orphan.  That was built, read, and discarded.
%%   The Long field is now supplied, and author-year is preserved.
%%
%%   A SECOND TRAP, caught by the log and not by the page: three labels
%%   wrote a LaTeX control space, "Miller et al.\ (2021)".  Stripping only
%%   the literal space leaves "\(", which opens math mode inside the label,
%%   kills the citation key, and SILENTLY DROPS the reference -- the two
%%   NICER entries vanished from the bibliography with no error, only a
%%   natbib "undefined citation" warning.  The rewrite strips "\ " as well
%%   as " ", and asserts that no label contains "\(" before writing.
%%
%%   RESULT: 313 pages, unchanged.  All 739 cross-reference numbers
%%   unchanged AND all 739 page numbers unchanged -- the pagination is
%%   bit-stable.  Overfull census identical.  100 Relevance notes, both
%%   NICER entries intact, 0 numeric-mode citations, 0 undefined.
%%   54 pages differ: exactly the pages that carry a citation.
%% v20.0 (August 21, 2026) -- post-stamp repair, version deliberately
%%   unchanged (nothing is published yet).  The generation-lifetime
%%   estimate printed (16.8)^5 * 0.043 ~ 10^6; the left side is 5.8e4, a
%%   factor 130 BELOW the measured 7.6e6 rather than in agreement with it.
%%   The correct chain is Sargent's m^5 over the tau's electron branching
%%   fraction, (16.8)^5/0.178 = 7.5e6 against 7.568e6 measured; note that
%%   Sargent alone gives 1.345e6 and 1.345e6/7.568e6 = 0.178, so the
%%   missing factor IS the branching fraction.  The cascade action plays no
%%   part in the ratio -- which is what the NEXT PARAGRAPH already said.
%%   Set inline, not as a numbered display, so no equation number moves.
%%   NOT DONE, and flagged instead: 69 of this paper's 94 bibitem labels
%%   write "Author (year)" with a space, so 53 in-text citations render
%%   as "(Author , 1975)".  A6 is the only one of the twelve deposit
%%   papers with this (the other eleven: zero).  Tightening the labels was
%%   built and measured: it removes all 53 and closes two sparse pages,
%%   but reflows the whole paper 313 -> 311 pages and strands a one-line
%%   orphan against the clearpage-isolated roadmap plate.  Backed out; it
%%   belongs in its own ship, not the night before a deposit.
%% v20.0 (August 21, 2026): PROMOTION TO THE v20 STAMP.
%%   Version and cover date only.  Promoted from v15.31 with no other change:
%%   not one character of body text, mathematics, float, caption or
%%   bibliography differs.  The promotion is safe to do mechanically because
%%   the cohort cites its companions BY TITLE, never by versioned filename --
%%   a scan of the printed text of all twelve deposit papers found zero
%%   version strings, so no cross-reference can be left pointing at a
%%   superseded file.  This is the version posted to arisaka-gg.org and
%%   deposited to Zenodo, so homepage, repository and DOI all carry one
%%   stamp.
%% ===================================================================
%% v15.30 (August 18, 2026) -- BT-1: THE BIBLIOGRAPHY TITLE PASS.
%%
%% EVERY COMPANION IS NOW CITED BY THE TITLE PRINTED ON ITS OWN COVER.
%%
%% 6 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.
%%
%% ALSO REPAIRED, one site, found by reading the rendered reference page:
%% the A7-Qi4 entry said "(in preparation)" and, in the same sentence,
%% "preprint, available at https://arisaka-gg.org/".  A7-Qi4 is posted and
%% at v15.30.  The mark is the mirror image of the defect FP-1 to FP-5
%% repaired for A11 -- a status line left behind by the paper's own
%% progress -- and it is the only one of its kind in the cohort: a scan of
%% all 477 companion bibitems for "in preparation", "forthcoming" and "to
%% appear" outside the A11 entries returns this one site and no other.
%%
%% 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.
%% ====================================================================
%% ===================================================================
%% v15.29 (August 18, 2026) -- FP-3: THE DISCLOSURE-PREP PASS.
%%
%% The cohort is posted as TWENTY-ONE papers with A11-Origin held back.
%%
%% A6 CARRIES A11'S CASCADE FIGURE AND ITS EIGHTEEN-RUNG SPINE TABLE, so
%% it describes the capstone's structure more often than any other
%% companion -- and it was doing so THREE DIFFERENT WAYS:
%%     "the full SIXTEEN-EVENT theory of landscape reorganization"
%%     "deploys this alternation through SEVENTEEN CASCADE LAYERS"
%%     "the EIGHTEEN-EVENT cascade and its chronology"
%%     "the SEVENTEEN-LAYER cascade" (A11 bibitem)
%% alongside statements that are correct and are FROZEN BY EXACT TEXT:
%%     "the eighteen-layer cascade of GRIPout" (figure caption)
%%     "the seventeen GRIPout shocks" (same caption)
%%     "the eighteen layers", "the eighteen rungs", "each of the
%%     seventeen shocks".
%% CANONICAL COUNTS, MEASURED from A11 v15.14 and not recalled: the
%% GRIPout longtable carries rows 1->2 through 17->18 and the codebook is
%% an eighteen-row table, so it is EIGHTEEN LAYERS and SEVENTEEN GRIPout
%% TRANSITIONS.  Four sites corrected; the correct ones left alone.
%%
%% The spine table's Owner column names A11 for rows 4 through 18, which
%% sends a reader to a paper in preparation.  The caption now says so,
%% and says that the grade column is what carries those rows' present
%% standing -- which was already true and is now printed.
%%
%% Also: the A11 bibliography entry, and catalogue -> catalog in printed
%% text (30 sites, the largest catalogue debt of any paper so far).
%%
%% NO OTHER CONTENT CHANGED.
%% ===================================================================
%% ===================================================================
%% v15.28 (August 18, 2026) -- RIDER FROM A11 STAGE 6.
%%
%% Two printed errors in the chemistry appendix, both found while
%% verifying the capstone's Part V against this Part:
%%   (1) "ribose in the Ryugu samples".  Ryugu carries uracil (Oba et
%%       al. 2023).  The meteoritic ribose is Murchison and NWA 801
%%       (Furukawa et al. 2019).  Corrected to both.
%%   (2) The three named lemmas for the chemistry rungs listed "the
%%       codon mapping" as rung 6's.  Rung 6 is the polymer class --
%%       this Part's own spine table says so two pages earlier -- and
%%       the codon mapping is rung 8's lemma.  Corrected, with the
%%       codon mapping's real home named in the same sentence.
%% Nothing else in this paper is touched.
%% ===================================================================
%% ===================================================================
% v15.27 (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 no trailing skip
%%   and says so in one 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 v15.26 printed, except for the lines named above.
%% ===================================================================
%% ===================================================================
%% v15.26 (August 17, 2026) -- THE -ise FAMILY.
%%
%% stabiliser -> stabilizer (5).  Charter 1.1's pair list names these forms; spelling_audit.py's
%% PAIRS list did not carry them, so this paper has reported MIXED
%% PAIRS 0 and ALWAYS-BRITISH 0 at every gate while printing them.  The
%% cohort sweep of August 17, 2026 found sixty-one across twelve of the
%% twenty-four live papers; Report 406 section 6 has the inventory and
%% spelling_audit.py now carries the family as a RULE rather than a list.
%%
%% Substitution is on PRINTED TEXT ONLY.  Changelog comments are the
%% historical record, \label/\ref/\cite arguments are keys, and published
%% titles are quoted rather than written -- all three are masked, and every
%% frozen title is asserted present and unchanged after the edit.
%% NOTHING ELSE IN THIS PAPER CHANGES: no claim, no number, no float, no
%% label, and no page.
%% ===================================================================
%% ------------------------------------------------------------
%% v15.24 -> v15.25, August 16, 2026.  THE SHOWCASE-HEADLINE RIDER.
%% \GGShowcaseHead rises from \large to \Large, one token.
%%   THE MEASUREMENT.  Across the cohort the showcase headline had four
%%   sizes: \large in the eight papers where I set it, \Large in C1 and
%%   C2, \LARGE in A4, \huge in A9, A10 and A11.  Those are not four
%%   arbitrary choices -- the rule followed was to match each paper's own
%%   existing headline, and four of the four larger values are the
%%   author's.  What the measurement shows is that MY DEFAULT was the
%%   outlier: every headline the author set himself is LARGER than
%%   \large, and on five papers out of five his practice is \Large or
%%   bigger.  On a page whose job is to be read in five minutes from a
%%   homepage, \large reads as a subsection head, not a title.
%%   So the house default rises to \Large, the smallest size he himself
%%   uses.  The cohort now spans \Large / \LARGE / \huge and nothing is
%%   below the author's own floor.
%%   VERIFIED: page count unchanged, no new overfull box, body
%%   page-for-page identical.
%% NEW PROSE IN THIS SHIP: none.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.23 -> v15.24, August 16, 2026.  THE SAVEBOX-LEADING RIDER.
%% A DEFECT I INTRODUCED IN THIS PAPER'S OWN SHOWCASE SHIP TODAY.
%%   Every showcase plate is filled into an \sbox in the front matter
%%   so the showcase and in-paper copies cannot drift.  A savebox is
%%   typeset when it is FILLED, and this paper's front matter runs at
%%   \baselinestretch 1.15 against the body's 1 -- measured with
%%   \typeout at both sites: \small leading 13.8 pt against 12.0.
%%   So every plate in a savebox was being set 15% looser than before,
%%   IN BOTH PLACES, including the in-paper float the author had
%%   already read.  A9-SM's falsification map grew 47.9 pt and pushed a
%%   paragraph off its page, which is how it was caught.
%%   This paper's showcase ship reported the resulting page as a benign
%%   "savebox reflow".  It was neither benign nor a reflow.
%%   FIX: \setstretch{1} inside every \sbox, before the size command,
%%   so the box is filled at the BODY's spacing -- the canonical one.
%%   VERIFIED: the body is page-for-page identical to the pre-showcase
%%   baseline under the +3 shift.  Not close: identical.
%% ------------------------------------------------------------
%% ------------------------------------------------------------
%% v15.22 -> v15.23, August 16, 2026.  THE FRONT-MATTER SHOWCASE.
%% Sixth of the twenty-four.  Three pages after the Keywords and
%% before the Overview of Contents, one idea each:
%%   p1  Figure 2, the four-pillar architecture -- what this is
%%   p2  Figure 8, the GRIP cascade, two regimes -- the title pillar
%%   p3  Table 20, the per-pillar falsifiers    -- what would kill it
%% Figures 1 and 3 are the shared cohort plates and A5-GDOS opened
%% its showcase with the first of them; A6 opens with its own
%% architecture.  Table 18 holds the cohort's ten falsifiers and is
%% A1-Constitution's showcase page 3; Table 20 is A6's OWN exposure,
%% one row per pillar sub-theorem, and is the right table here.
%% THE PLATES KEEP THEIR IN-PAPER SIZE, and that is a decision taken
%% from what A5 cost: its showcase copy was 4% wider than the body
%% copy, which took two shared rules from 0.953 pt of rendered weight
%% to 0.983, over the 0.956 house ceiling -- and a savebox cannot
%% differ between its uses, so the remedy had to be a change to the
%% BODY plate.  Here both showcase copies use the in-paper factor,
%% \textwidth and 0.833\textwidth, and nothing can move.
%% NOTHING IS COPIED: both TikZ bodies and the falsifier tabular go
%% into saveboxes filled once in the front matter; the showcase and
%% the in-paper float each say \usebox.
%% NEW PROSE IN THIS SHIP: three headlines and three showcase
%% captions, all on the new pages.  Nothing in the body is rewritten.
%% ------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.18 (August 9, 2026): the curious-reader cluster.
%%  Twenty-five broad-audience questions open the Common Objections
%%  Part, ahead of the specialist clusters, on the model of
%%  W1-Symmetry, W2-Holonomy and A10-Cosmos.  Eight are the spine
%%  those three share, answered here in this paper's own vocabulary;
%%  seventeen are questions only a paper about descent can be asked: why three families, why time
%%%%  has a direction, and what a broken symmetry leaves behind.
%%  The introduction names the hardest of the set.
%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.17 (August 9, 2026): Part VIII rebuilt from scratch.
%%%%  The Part had grown by accretion to 87.7k characters in eleven
%%%%  sections with no summary of outcomes, comparison split three
%%%%  ways, solved and open problems fused, and 19%% of its length
%%%%  given to cohort bookkeeping.  It is now six sections: outcomes,
%%%%  significance and comparison, innovation, falsification, open
%%%%  problems, future prospects -- scoped to this paper's own four
%%%%  pillars.  The coherence map and the naming convention are moved
%%%%  to appendices rather than deleted.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.16 (August 9, 2026): Part openings and closings.
%%%%  The four "Forward references" navigation lists closing Parts II-V
%%%%  are deleted and their handoff rewritten as prose inside new
%%%%  Part closings.  "Forward to Part IV" loses its heading and keeps
%%%%  its argument.  Part VI's two "Bridge to ..." sections are RENAMED
%%%%  rather than cut: they carry the axiom-recovery map and the
%%%%  four-by-four pairing table, and are results, not signposts.
%%%%  Parts I-VI gain a one-page plain-English closing; Parts VIII and
%%%%  IX, which opened with no prose at all, gain an introduction.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.15 (August 9, 2026): table rules normalised to the house form.
%%%%  Four of A6's 27 multi-row tables ruled every data row where the
%%%%  other 23 use toprule / one header midrule / bottomrule.  The
%%%%  per-row rules are removed from Tables 1, 18 (both halves) and 19.
%%%%  Kept deliberately: longtable repeated-header and footer rules,
%%%%  and the separators around \multicolumn group headings, which are
%%%%  structure rather than noise.
%%%% ------------------------------------------------------------------
%% ------------------------------------------------------------------
%%  v15.14 (August 9, 2026): two cohort rules applied.
%%  (1) British doubled -l corrected in prose only -- 6 site(s);
%%      5 further site(s) protected as comment, identifier or
%%      bibliography and deliberately untouched.
%%  (2) Ragged-right on 18 p-column(s) across 8 multi-p-column
%%      table(s): a narrow prose column cannot be justified
%%      without a visible interword stretch, and an overfull box
%%      under the gate's tolerance can still eat the gutter.
%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.12 (August 9, 2026): the conditioning ledger brought to its
%%%%  current state.  M11's residues reduced: the saddle-adiabaticity
%%%%  worry dissolves (the invariants live on the gapped link; what is
%%%%  consumed is T3's transverse nondegeneracy plus one named
%%%%  identification, GAP=HESS), and the warped/relative scopes are
%%%%  closed at class level by T-INF-7/T-INF-8 of the three-
%%%%  generations companion (v15.5).  M6 -> Partially resolved
%%%%  (benchmark discharge; survivors: basepoint canonicity +
%%%%  engineering).  M10 -> Resolved (externally), per the triage's
%%%%  own categories.  Triage summary recounted from the entries:
%%%%  28 entries (M12 was missing from the old count of 27),
%%%%  2 / 7 / 19.  Stale "M10/M11" and "three named conjectures"
%%%%  shorthand corrected at ten sites incl. the objections closing
%%%%  remark; roadmap trio now reads "the open-problem ledger
%%%%  M9--M11".  The "conditional on M10" cross-references governed
%%%%  by M10's reading rule are deliberately unchanged.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.11 (August 9, 2026): -isational re-check closure + referee
%%%%  alignment.  Two stale quotes of A3's Section 8.7 title updated
%%%%  to that paper's Americanized form ("geometrization reading");
%%%%  five single-sided British survivors fixed (compartmentalisation,
%%%%  serpentinisation, centralised in figure nodes; analysing,
%%%%  localises in prose); three referee-family words brought to the
%%%%  cohort's journal-free convention.  A1-A5 scanned clean under
%%%%  the same rule (protected sites only).
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.10 (August 9, 2026): the three M9 correctives (Report 261).
%%%%  (1) tab:alternatives repaired: the two admissible rows the
%%%%      enumeration missed, (2,4,5;9,45)=65 and (2,6,5;13,65)=91,
%%%%      added; competitors reordered by complexity; the margin
%%%%      claim corrected from 72 to 65 in caption and closing
%%%%      paragraph.  The programmatic enumeration (m9_norm_check.py,
%%%%      support archive) is now the source of the row list.
%%%%  (2) M9 entry sharpened: norm-independence established over
%%%%      all symmetric strictly-monotone complexity functionals by
%%%%      the sorted-dominance argument; surviving content is the
%%%%      canonicity of that class and the l1 motivation.  M9 stays
%%%%      Genuinely open; status counts unchanged.
%%%%  (3) One-line note after T2.1's imported proof on the J=2N+1
%%%%      clause carried by A2's cascade but not assumed by T2.1.
%%%% ------------------------------------------------------------------
%%%% ------------------------------------------------------------------
%%%%  v15.9 (August 9, 2026): M11 restated in spectral-flow form.
%%%%  The open-problem entry now states the sharpened conjecture ---
%%%%  compensator monodromy = mod-period eta-jump, the object the
%%%%  L-INT/R-INT chain controls --- with its verified benchmark
%%%%  instance (trapped charge = boundary Berry unit, -1, by the
%%%%  kappa-bridge theorem) and the two named gaps (saddle
%%%%  adiabaticity; R-WARP/R-REL).  One-line pointers added at the
%%%%  T2.2 and T4.2 status remarks and the open-problems intro.
%%%%  M11 stays in the Genuinely-open triage; status counts
%%%%  unchanged.  No other content touched.
%%%% ------------------------------------------------------------------
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.8  (August 8, 2026)
%%%
%%%  (1) NEW SECTION previewing A7-Qi4, on the author's
%%%      instruction: this paper is not finished without it, and
%%%      every application of Part VII is enhanced by it.  Placed
%%%      after the last floor, so the reader has seen every rung
%%%      the enhancement applies to.  Includes the inward map of
%%%      which A6 theorem each Qi pillar consumes -- deliberately
%%%      NOT one-to-one: T2 feeds two pillars and Qi-CON has no
%%%      Gi-4 parent at all.
%%%  (2) NEW CLOSING SECTION, one page, plain English, for a
%%%      general reader.
%%%
%%%  (3) CORRECTION -- Q_B's provenance.  v15.7 said the capstone
%%%      gives it its value.  The Arisaka constant is introduced
%%%      and DERIVED in A7; A11 deploys it at Layer 7.  The author
%%%      caught this.
%%%  (4) CORRECTION -- the GGA/Qi-4 relation.  v15.7 wrote that
%%%      'which of the available basins is admissible is settled by
%%%      the quantum-information pillars'.  A7 v13.15 names that
%%%      exact formulation as the error it corrected in its own
%%%      drafts: the GGA is fixed by the landscape ALONE, and what
%%%      A7 proves is that the GGA SATISFIES the four conditions --
%%%      not that they select it.  The converse is unproved in both
%%%      papers.  A6 had imported the retired formulation.
%%%  Body theorems, values and figures: otherwise UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.7  (August 8, 2026)
%%%
%%%  PART VII INTEGRATED.  v15.6 built the Part in three blocks,
%%%  each organized by a different principle -- the capstone's
%%%  picture, the companion paper, the Gi-4 construction -- and
%%%  none of them by the cascade.  Two subjects were therefore
%%%  covered twice from two sides, and fifteen of the eighteen
%%%  rungs of Figure 9 appeared only inside the figure.
%%%
%%%  Rebuilt on ONE principle: a walk up the ladder.  Every section
%%%  is a rung or a band, in cascade order, answering the same three
%%%  questions -- what opened it, what settled, who owns it and at
%%%  what grade.  The four retained Gi-4 sections are DEMOTED to
%%%  subsections and MOVED inside the rung whose subject they are,
%%%  so the companion's result and this paper's mechanism reading of
%%%  it sit together.  No text of theirs is lost.
%%%
%%%  Two integration devices: A11's five confidence grades adopted
%%%  as the Part's key and PRINTED on every rung heading, so the
%%%  grade gradient is visible; and a spine table of all eighteen
%%%  rungs, of which every later section is one row expanded.
%%%
%%%  New floors: Chemistry (Layers 4-6), Life (7-10), the
%%%  multicellular and the mind (11-17), Articulation (18) -- one
%%%  BAND at a time on the author's instruction, and brief relative
%%%  to the A9/A10 material.  Two rungs get their own subsection
%%%  because they bear directly on this paper: Q_B and the ATP
%%%  anchor (the plate's last box, never explained until now), and
%%%  the four-letter alphabet as |Sigma| = N+1.
%%%
%%%  The radiative-descent section keeps all its text and is demoted
%%%  from the closing position to a marked interlude.  The Part now
%%%  closes on the Top 10 falsifiers.
%%%
%%%  A11 Figure 3 NOT carried: compared block by block it is A6's
%%%  own fig:grip-two-regimes with the fills bumped and A6's
%%%  definition pointers replaced by layer-ascent language.  It was
%%%  made here first.  The reading rule points at the original.
%%%  Body theorems, values and existing figures: UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.6  (August 8, 2026)
%%%
%%%  PART VII REBUILT, on the author's instruction: the Part was
%%%  carrying a small fraction of the applications the engine
%%%  actually has.  Five new sections ahead of the retained
%%%  material, in the author's order:
%%%    A  the whole ladder -- A11-Origin's Figures 1 and 2 carried
%%%       VERBATIM, the big picture previewed, and A11's own
%%%       preview-status fence and two-standard split stated;
%%%    B  the Standard Model -- A9's 28 parameters as GRIP readouts,
%%%       with the 6/14/8 triage and the MS-bar tension in view;
%%%    C  the electron -- P3's descent that halts (no exit gate);
%%%    D  the proton -- P6's winding that cannot be undone;
%%%    E  the cosmos -- A10's inventory, recombination as the cosmic
%%%       GRIP-Out, the TWO-QUANTITY z* kept distinct, and the
%%%       second horizon problem at its stated grade.
%%%  Every number is read from the companion's own source and every
%%%  grade travels with its claim.  The retained material keeps all
%%%  its text and gains a section head of its own.
%%%  The two carried figures are EXEMPT from the house palette for
%%%  the same reason the cohort plate is: they are owned by another
%%%  paper and a locally normalised copy would break parity.
%%%  Also: the Executive Summary's 'a resonance cluster near a few
%%%  TeV' made exact -- 'near 3.85 TeV' -- so the cohort falsifier
%%%  scan reads A6 clean.
%%%  Body theorems, values and existing figures: UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.5  (August 8, 2026)
%%%
%%%  (1) PART INTRODUCTIONS.  A half-page Plain-English opening at
%%%      the head of Parts I--VII, above the first \section, on the
%%%      author's instruction and to the shape used for A5-GDOS.
%%%      Pitched one step broader than A5's, because this paper is
%%%      expected to be read by biologists and neuroscientists as
%%%      well as physicists: no equations, no theorem labels before
%%%      their content is given in English, gauge language only
%%%      after it is cashed out.  Discussion and Conclusions are
%%%      excluded on instruction.  Parts X, XI and XII already open
%%%      in this register and are left alone.  Nothing is deleted:
%%%      A6 has no teach-boxes, and the four existing Reader's-
%%%      bridge remarks are kept and pointed at rather than
%%%      duplicated.
%%%  (2) LIST OF TABLES.  A1--A5 all call \GGListOfTables; A6 was
%%%      the only paper in the cohort that did not.  Added in A1's
%%%      position, after the List of Figures.
%%%  (3) Two near-empty pages the reflow created, both closed:
%%%      the \clearpage before the cohort map's [p] float dropped,
%%%      and Table 7's [H] specifier -- which cannot float and so
%%%      can only start a fresh page -- relaxed to [htbp].
%%%  (4) Two missing short captions, exposed by the new list: two
%%%      tables carried \caption with no optional form, so the list
%%%      repeated 321 and 213 characters of running caption where
%%%      every other entry prints 37 to 89.
%%%  Body text, theorems, values and figures: UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.4  (August 8, 2026)
%%%
%%%  FRONT MATTER REPLACED.  The Abstract and the Executive Summary
%%%  are rewritten from the ground up on the author's instruction.
%%%  The old pair led on the Gi-4 Pillars as a formalization
%%%  exercise and carried theorem numbers, two boxed equations and
%%%  a dozen citations onto page 1; GRIP arrived fourth in a list
%%%  of four and the In/Out regimes arrived last.  The new pair
%%%  leads on the pipeline DGI -> (GFF) -> GSSB -> GA/GGA and the
%%%  In/Out split, in A5-GDOS's plain register: no theorem numbers,
%%%  no boxed equations, no citations, one unadorned display.  The
%%%  geometry keeps one paragraph.  Honesty clauses explicit: the
%%%  chemical/biological/cognitive layers are named as the cohort's
%%%  wager and the capstone's deployment, NOT as results proved
%%%  here; N <= 3 is marked conditional; the residue is named.
%%%  Subtitle replaced on the author's decision: it led with the
%%%  pillars, which puts the machinery in front of the mechanism.
%%%  Body text, theorems, values and figures: UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  A6-GRIP  v15.3  (August 8, 2026)
%%%
%%%  (1) SHARED PLATE: the twenty-two-paper acronym plate is taken
%%%      verbatim from A1-Constitution, which is now the cohort
%%%      copy.  It was carried in four source states, and five of
%%%      the six differed in a way that PRINTED: the sub-line ran
%%%      at \normalsize with no wrap and the GRIP entry overlapped
%%%      the Qi-HAL box in the right-hand column.  A1's copy sets
%%%      it at \footnotesize wrapped to 86 mm and clears.  The
%%%      Q_B box reads "energy--information Arisaka constant" on
%%%      the author's decision: Q_B applies beyond biology.
%%%  (2) TABLE 14: the category head read "Higher-dimensional and
%%%      higher-dimensional constructions" -- the word doubled.
%%%      The rows under it are AdS, AdS/CFT, CFT, KK and RS.
%%%  Body text, values and figures: otherwise UNCHANGED.
%%% ==================================================================
%%% ==================================================================
%%%  Arisaka-GG-A6-GRIP  v15.2  (August 8, 2026)
%%%
%%%  (1) PALETTE: the six own flowcharts brought to the house scheme.
%%%      Fifteen bare `draw' borders (100%% black) take the hue of the
%%%      fill in their own option group; box fills above !18 compress
%%%      into [13,16] keeping their order; keyword widths become
%%%      explicit points so an audit can read them.  The shared
%%%      acronym plate is deliberately NOT touched -- it is a cohort
%%%      standard and is handled cohort-wide.
%%%  (2) ROSTER: P6-Proton, the one cohort paper the reference
%%%      apparatus lacked, added with the prose that cites it.  The
%%%      cohort map already named it and already drew the sister
%%%      arrow into it.
%%%  (3) STAMPS: version, date and cover date.
%%% ==================================================================
% ===================================================================
%  A6-GRIP v15.19 (August 12, 2026) -- cohort-wide sweep, two defects.
%
%  (1) GU-5 IS A THREE-PLUS-TWO PARTITION, exactly as OGN-5 is
%      (M,N,L;J,K).  The separator before k_B is a SEMICOLON:
%      {c, G_N, hbar; k_B, Q_B}.  0 site(s) corrected here.
%      NOT touched, by design: the four-constant Planck unit system
%      {c, G_N, hbar, k_B}, which is a different object and takes a
%      comma; a semicolon there would be wrong.
%
%  (2) Internal affairs -- companion versions, dates, page counts and
%      build status -- do not belong in a preprint.
%%  ALSO: five companion PAGE COUNTS are stripped from the audit-trail
%%  list.  The theorem and section locations, which are real citation
%%  targets, are kept -- only the page counts go.
%
%  No physics changes; no parameter, theorem or falsifier is touched.
% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.20  (August 6, 2026)  -- the cohort-wide figure-size sweep.
%%  3 figures re-placed so their node text lands at the house size,
%%  7.79pt: what GG-P3-Electron Figure 3 renders (\footnotesize at
%%  0.865 of natural size).  For a picture of measured width W drawn
%%  at a base font of B points the target is f = (7.79/B) * W / 159.2
%%  of \textwidth, W being measured by compiling each picture
%%  standalone and reading its ink bounding box.  Nothing inside any
%%  picture changes: no font command, no dimension, no text, no
%%  relative position.  The cohort map and the central-claim funnel
%%  are excluded, both being at the house size already.
%%  Per-figure table and recipe: [v13-support]/allsweep.py and
%%  [v13-support]/figure_census.json.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.19  (August 5, 2026)  -- the cohort funnel at the house size.
%%  The central-claim funnel was drawn with 114mm boxes and then
%%  stretched to the full text width, so its \footnotesize landed 47% larger than the roadmap
%%  and cohort-map plates beside it.  The plate target is now
%%  0.681\textwidth, computed rather than chosen: the picture measures
%%  125.2mm 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.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.18  (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-A6-GRIP  v13.16  (August 4, 2026)
%%  -----------------------------------------------------------------
%%  CHANGELOG (v13.15 -> v13.16), author-directed (GG-Report-185):
%%  The stage-fibre custody correspondence installed.
%%    * NEW subsection ssec:stage-custody after the two-regimes
%%      subsection: custody of transport vs custody of data.
%%      - Proposition prop:GSSB-transport (T-Cor-2): the GSSB->GA
%%        leg realised as the scale fibre's CPTP forward semigroup
%%        (dilatation generator along I_r projected); proof by
%%        composition of two published A5 results.
%%      - DGI half (T-Cor-1) at derived-conditional grade: the
%%        circle is the bulk's only gauge bundle (A8); DGI = LGI
%%        carried through the open dynamics (A7 operator form);
%%        POS-1 ladder given its geometric reading.
%%      - The crossing (T-Cor-3) at reading grade: GSSB reads
%%        circle data, runs on the interval, deposits to circle
%%        classes -- symmetry breaking as symmetry reading; GSSB
%%        stands to breaking as the write stands to measurement.
%%      - Stage/transport/data custody table; compatibility
%%        statement (T-Cor-4): strokes classified by outcome,
%%        stages by transport; both pullbacks of the one fork.
%%    * cor:TGG-DGI gains the fibre-resolved sentence (DGI is LGI
%%      transported by the semigroup).
%%    * NEW open problem M12: the general stabiliser theorem that
%%      would complete the correspondence at theorem grade.
%%    * NEW bibitems ArisakaW1Symmetry, ArisakaW2Holonomy (A/B
%%      standard, Relevance + first-cited); the two-regimes
%%      provenance now cites W1 and W2 by key (W1 was a bare
%%      textsf name with no bibitem).
%%    * Self-bibitem (legacy key ArisakaA6Gi4, render "A6-GRIP"
%%      kept as settled): mangled title comma and truncated
%%      Relevance sentences repaired.
%%  Script: a6_v1316_stagecustody.py
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.17  (August 5, 2026)  -- P7-Inflow residue.
%%  P7-Inflow is Appendix H of P2-N3 (P2 v15.2); the merger's ship 4
%%  scoped itself by the key ArisakaP7Inflow and so never saw this
%%  paper, which cites the same argument under Arisaka2026Inflow.
%%  Four prose sites re-pointed at that appendix and the bibitem
%%  retired, its Relevance folded into the ArisakaP2N3 note so nothing
%%  the bibliography said is lost.  NO physics touched: M6 (= R-INT)
%%  is discharged on the benchmark background exactly as before, with
%%  the same warped and relative refinements declared open.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.13  (July 30, 2026) -- P7-Inflow cross-reference update
%%  + L-INT vocabulary conformance to P2 v14.2 canon.
%%  (1) The inflow companion is SHIPPED as Arisaka-GG-P7-Inflow:
%%      bibitem Arisaka2026Inflow updated to the shipped identity
%%      ("in preparation" retired; key kept, no \cite churn); the M6
%%      open-problem entry and the M10 reading rule now record the
%%      benchmark discharge (kappa-bridge theorem; R-INT closed there;
%%      warped/relative refinements open).
%%  (2) ~15 sites conformed: "integrality lemma L-INT" -> "integrality
%%      theorem L-INT (residue R-INT)"; "residue M6 = L-INT" ->
%%      "M6 = R-INT"; "single named open lemma/problem" -> "single
%%      named residue"; section title "residual lemma M6" ->
%%      "residue M6"; the stale "Conjecture C.1" attribution to P2's
%%      App C replaced by the would-be-axiomatisation device (P2 v14
%%      App C records the completed formulations; no Conjecture C.1
%%      exists there).
%%  NO physics touched; all changes are cross-reference conformance.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.12  (July 30, 2026)  -- Phase-2 shape-up.
%%    * INTERNAL AFFAIRS: "We thank the hostile reviewer for the observation
%%      that, absent the detailed-balance hypothesis, the reverse direction is
%%      logically weaker" -- an actual review event, not the rhetorical
%%      "objections a hostile reviewer would raise" device used legitimately
%%      elsewhere in the cohort.  The CONCESSION IS PRESERVED and sharpened:
%%      Spohn gives monotonicity only, so the (<=) direction of T3.2 is where
%%      the detailed-balance clause does its work.
%%    * TABLE REDESIGN (charter sec.7 item 5).  The two-page comparison table
%%      was set inside \resizebox{0.95\linewidth}{!}{...}, which scales the
%%      font by an arbitrary factor -- the one thing the cohort table standard
%%      forbids ("tables at \small, slightly below main text").  The resizebox
%%      is removed and the six p-columns rebalanced to fit the landscape
%%      measure natively at \scriptsize (22.4cm of column against ~23cm of
%%      text), with >{\raggedright\arraybackslash} on every column so the
%%      narrow prose cells no longer inter-word-stretch.  The font size is now
%%      DECLARED rather than computed.  No table content changed.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.11  (July 30, 2026)  -- P2-N3 v14.0 conformance.
%%  THE SUBSTANTIVE FIX: Section 22 ("Hypotheses H1/H2 reduced to
%%  theorems modulo M6") asserted the descent via the naive Chern--Weil
%%  restriction of the bulk spin connection -- the route P2's own no-go
%%  (tr R^R = 0 pointwise, conformal flatness) and now its rigidity lemma
%%  prove yields ZERO on every functorial construction.  The theorem
%%  STATEMENTS survive; the proofs are re-sourced: descended geometry from
%%  the boundary runtime's adiabatic theorems (Berry/Fubini--Study at
%%  Born-grade), bulk-only route recorded as closed, c_R = +16pi^2 (from
%%  the invalid route) withdrawn -- the coefficient is fixed by the FS
%%  conventions of P2 App F.  H2's residue identified: M6 = L-INT.
%%  M10: STATUS "Genuinely open / currently circular" -> "Met externally"
%%  by P2's capacity theorem T-N3-6 (consumes neither K=35 nor J=7,
%%  meeting M10's own N-independence demand); residue = coupling clause +
%%  M6.  A reading rule at the M10 entry re-points every "conditional on
%%  M10" cross-reference.  Objections 8, 39, 41, the cascade-side and
%%  laundering objections, the coherence-map row, PS-GRIP-2, the T4.1
%%  status remark, and six "named conjecture / derived nowhere" assertion
%%  sites conformed.  One pre-existing typo fixed ("In of").
%%  28 anchored sites.  No A6 theorem statement, value, or falsifier
%%  changed; proofs re-sourced, statuses updated.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.10  (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.
%%   * ONE SITE WAS SELF-CONTRADICTORY: "the closed derivation of the
%%     family number N=3 FROM A THREE-LEG ARGUMENT combining MSMF, GDOS Morse
%%     stability, and the codebook self-consistency check" named the three
%%     legs in the same breath as calling the result closed.  Now "the
%%     three-leg derivation of the family number N=3, combining ...".  Found
%%     by rendering p.14 -- text-level grep on the title string had missed it.
%%   * "Within the present paper the closed derivation is Corollary ..." ->
%%     "the three-leg derivation is Corollary ...".  The corollary keeps its
%%     own name, "Three-leg derivation of N=3", and its label is unchanged.
%%   * NOT changed: "Every closed-derivation result of the GG-Theory program
%%     -- including N=3, the 28 Standard-Model parameters, the hierarchy, and
%%     the dark-matter prediction".  That is a claim about the programme, and
%%     the 28 parameters are not a three-leg result.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.9  (July 29, 2026)  -- two residuals.
%%   * "per L7 of the GG-Theory project instructions" removed: an internal file
%%     cited by line number.  The discipline it enforced is the content and is
%%     kept -- vague pointers are not acceptable and every entry names a
%%     specific theorem, section or appendix location.
%%   * "as of May 2026" removed from the audit-trail preamble: a working date
%%     that implies later versions of the companions exist.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.8  (July 29, 2026)  -- five residuals from the final scan.
%%   * A FALSE CLAIM removed.  The outward map said A7, A8, A9, A10, P1 and C1
%%     were "all in preparation or pending v5.X conversion" and that
%%     cross-citations to them "use placeholder names".  All six are deployed
%%     and A6 cites their real theorems; as printed the paper told a reader six
%%     of its companions did not exist yet and its own pointers were
%%     placeholders.  Replaced with what each companion consumes downstream.
%%   * "invoked at Phase 2 of the present paper" (prose and longtable row);
%%     "the three new ones recorded after the Phase 6 hostile-reviewer pass";
%%     bibitem "Cited at T3.2 (Phase 6 patch)".  M9/M10/M11 and the Spohn
%%     detailed-balance point are unchanged.
%%  "Phase N" entered the detector vocabulary at A7, after A6 was swept; the
%%  final cohort-wide re-scan is what surfaced these.
%% ===================================================================
%% ===================================================================
%%  A6-GRIP v13.7  (July 29, 2026)  -- internal-affairs sweep.
%%   * 2 working dates and 1 editing-pass reference removed; the claims they
%%     decorated are untouched.
%%   * The two vocabulary paragraphs (the Ground Geometric Attractor, and the
%%     GRIPin/GRIPout regime split) no longer present themselves as changes
%%     from a previous version -- "of this revision", "since the capstone
%%     A11", "The present revision promotes", "now carry".  Every definition,
%%     attribution and consequence is preserved; the attribution of the regime
%%     THEOREMS to A11 in the following sentence is untouched.
%%   * A7 bibitem "(v10.8)" and P2-N3 bibitem "(June 11, 2026, v2.4)" dropped.
%%     21 of A6's 22 companion bibitems already end at the bare URL; the P2-N3
%%     entry was the sole outlier, so this restores the paper's own convention.
%%  No value, equation, grade or citation target changed.
%% ===================================================================
%% ===================================================================
%% v13.5 -> v13.6 (July 28, 2026).  Phase A of the GGA correction.
%%   * GFF-ST5 restated with its full hypothesis set (it was false as
%%     previously written; the counterexample is computed in this
%%     paper's own Appendix on Morse theory).  New remark explaining
%%     why each hypothesis is load-bearing.
%%   * T3.1: necessity kept general, the biconditional returned to the
%%     family-projector scope where it is a theorem; proof Step 3
%%     repaired (non-vanishing of p_1 controls the L-genus in real
%%     dimension four only, and never its sign).
%%   * New subsection: the per-stratum ground state.  Admissible
%%     stratum defined; per-stratum corollary proved by restriction,
%%     with a grade note recording which strata are established.
%%   * Equilibrium-uniqueness proof Step 3 repaired: uniqueness now by
%%     perfection and connectedness; stable-manifold theorem in place
%%     of Hartman--Grobman; Morse--Smale dropped as unused; the
%%     genericity clause restated in terms of absolute continuity.
%%   * New subsection: two worked landscapes (the family-projector
%%     landscape; the measured baryon width column).
%%   * GGA definition is now the plain global minimum; the
%%     admissibility clause is retired and its content stated as a
%%     characterisation in the bridge to the companion pillar paper.
%%   * GRIP-Out necessity carried locally as a remark.
%%   * GRIP-ST5 reconciled with the new corollary.
%%   * Cascade chain form made uniform across the paper.
%%   * Budget provenance made single-valued at all five sites; the
%%     open problem gains the sensitivity range and the note on the
%%     seam-count circularity.
%%   * Stability bullets sharpened; the compact-angle case re-graded.
%% Values, canonical numbers, ledger: UNCHANGED.
%% ===================================================================
%% ===================================================================
%% v13.5 (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.
%% ===================================================================
%% ===================================================================
%% v13.4 (July 24, 2026): COMPARISON TABLE ON TRUE LANDSCAPE PAGES
%%  (author-commissioned redesign; the cohort's last rotated-resizebox
%%  table). The comprehensive comparison table is rebuilt on two
%%  genuine landscape pages (the page-rotation mechanism, so it reads
%%  horizontally on screen), split at its natural seam into
%%  (I) foundations and structure and (II) parameters, predictions,
%%  and falsifiability, with the header repeated, the font raised
%%  from scriptsize to small (roughly sixty percent larger as
%%  rendered), one table number kept via a continued second caption,
%%  and every row carried over verbatim. No content change anywhere.
%% ===================================================================
%% ===== v13.3 (2026-07-20): STAGE 5C -- WAVE-2 RESIDUE CLEANUP =====
%% (1) F2: P5-CPT bibitem title text -> current P5 v3.42
%%     cover title (label/key untouched).
%% Physics, theorem keys, numbers, values, grades: UNCHANGED.
%% ===================================================================
%% ===== v13.2 (2026-07-20): STAGE 5A -- Gi4 RETIREMENT (A6) =====
%% Wave-2 housekeeping batch, ship 6a.  The paper's
%% \providecommand{\ASix}{GG-A6-GRIP} was a no-op against the
%% sty's predefined retired default GG-A6-Gi4, so every \ASix
%% site rendered the retired name (23 rendered lines in the v13.1
%% PDF).  (1) \providecommand -> \renewcommand for \ASix and
%% \AFullSix (cohort precedent: A1/A2/A10 v13 preambles); all
%% sites now render GG-A6-GRIP.  (2) Suffix roster: 'GRIP (A6,
%% this paper; formerly Gi4, rename effective of [A2])' -> 'GRIP
%% (A6, this paper)' (A8 v13.2 register precedent).  The bib KEY
%% ArisakaA6Gi4 (non-rendering) and the Gi-4 pillar OBJECT
%% (\GiFour) are untouched.  Physics, theorem keys, numbers,
%% values, grades: UNCHANGED.  v13.1 -> v13.2 (\PaperDate
%% unchanged).  Script: a6_v13.2_stage5a.py.
%% ===== v13.1 (2026-07-20): STAGE 2 CONSUMER/ATTRIBUTION SHIP =====
%% Author-approved Wave-2 content items (audit:
%% GG-Internal-Report-RealBulk-Audit-2026-07-19.md, A6 ledger A6-R1 +
%% A6-R2 + A6-R3; custody anchor: A1-Constitution v13.2 AX-3
%% synthesis; law-level twin: P5-CPT Part IX).
%% (1) ARROW-CUSTODY CONSUMER STATEMENTS (A6-R1): (a) custody
%% paragraph after GRIP-ST2 (theorem statement byte-identical) --
%% the one-way descent is the trajectory-level face of the AX-3
%% projection's law-level irreversibility (CPTP, non-invertible,
%% records accumulate); AX-3 owns the arrow constitutionally (A1
%% AX-3 synthesis); P5 Part IX is the canonical law-level derivation
%% over an exactly T-symmetric bulk; the GDOS runtime is the
%% dynamical implementation, the gradient descent its Gi-4 geometric
%% face.  (b) custody sentence at PS-GRIP-3 -- Theorem T3 is the
%% runtime face of the one constitutional arrow, consumed, not
%% re-derived.
%% (2) GFF = GEP + CEP ENDOWMENT REMARK (A6-R2): NEW remark after
%% GFF-ST6 (before Lemma DE-1) -- the Helmholtz ledger as the
%% meeting of the second and third endowments of constitutional time
%% (repeating clock / arrow; P5 Part IX), trading at rate T_GFF;
%% GFF-ST2's GEP+CEP split read as the same meeting.  Grade:
%% candidate reading; structure theorems stand without it.
%% (3) WILSON-LINE PEDIGREE (A6-R3): one sentence closing
%% def:wilson-line-OP -- the i of W_B, W_Qi is the boundary
%% rendering of the circle's forced quarter-turn (T-P4-1 of
%% GG-P4-Real); the exponent a real angle on a real circle.
%% MACRO TRAP: no new \ASix{} usage (sty providecommand renders the
%% stale GG-A6-Gi4); literal \textsf{GG-A6-GRIP} rule applies to
%% any self-naming site -- none added.
%% Protected byte-identical: abstract, ExecutiveSummary, the
%% GRIP-ST2 + GFF-ST6 structheorem environments, Lemma DE-1 + proof,
%% the Wilson-line align displays.  Physics, theorem keys, numbers,
%% values, grades: UNCHANGED (no new numerals).  v13.0 -> v13.1.
%% Script: a6_v13.1_stage2.py.
%% ===== v13.0 (2026-07-19): WAVE 1 OF THE 19-PAPER PROMOTION SHIP =====
%% Promoted from the v11 series (v11.25 -> v13.0; A6's first v13-series
%% release; no v12 series existed for A6).  Mechanical cohort-promotion
%% sweep (17 -> 19 papers) plus the standardized Part-I close
%% (author-approved Wave-1 scope; audit:
%% GG-Internal-Report-RealBulk-Audit-2026-07-19.md, A6 ledger (2.6) +
%% App. A).
%% (1) NEW closing section of Part I, "The cohort inheritance and the
%% roadmap of the twelve Parts" (A2-GG6 v13.3 / A3-BI6 v13.6 /
%% A4-Euclid v13.0 / A5-GDOS v13.0 architecture), merging the former
%% adjacent sections "The GGTheory Cohort" and "Roadmap of the
%% Twelve Parts" (already Part I's closers): subsection "The cohort
%% inheritance" (label sec:cohort-a6 kept; the five-strata prose
%% migrated verbatim, opening paragraph recounted to nineteen with the
%% 11+5+3 breakdown; NEW roster-level Foundations Branch paragraph in
%% A6's voice -- A6 carries the GRIP pipeline (DGI -> GSSB -> GA) and
%% the GFF ledger, the geometric runtime the cohort's open-system
%% floors run on; GG-P4-Real named the origin of the boundary's
%% complex structure whose phase conventions the DGI stage makes
%% dynamical; GG-P5-CPT's mirrors and law-level arrow named kin of the
%% runtime's irreversible cascade -- one sentence each, roster level,
%% no custody install) closing with the frozen v1.4 nineteen-paper
%% cohort map on its own page (extracted verbatim from A2-GG6 v13.3,
%% MP4/MP5 rendition; label fig:cohort-map-a6 kept; the old
%% seventeen-paper map retired), and subsection "Roadmap of the
%% twelve Parts" (label sec:roadmap-a6 kept; new warm prose
%% walkthrough replacing the paragraph-per-Part bold list) closing
%% with a NEW roadmap figure (label fig:roadmap-a6 kept) in the house
%% minipage architecture.  Highlighted Part unchanged: Part V, the
%% GRIP cascade closure (the family-count forcing that is the paper's
%% central claim and its suffix).
%% (2) COUNTS: reader-facing cohort counts seventeen -> nineteen at
%% the audit's sites (Part-I close x6 via the migration -- the
%% opening paragraph x3, the retired map caption x2, the roadmap
%% Part-I line -- and the references Part intro x2).  Protected
%% semantic non-edits: "seventeen cascade layers"
%% (fig:grip-two-regimes caption),
%% "seventeen-layer cascade" (A11 bibitem Relevance), "sixteen
%% landscape-reorganizing events", "sixteen orders of magnitude",
%% "nine main-text Parts plus fourteen appendices".
%% (3) ROSTER: the Global-naming-convention suffix list adopts the
%% canonical nineteen-paper form, gaining the foundations preprints
%% Real (P4) and CPT (P5).
%% (4) MACROS + BIB: \PaperPFour (GG-P4-Real) and \PaperPFive
%% (GG-P5-CPT) installed; NEW bibitems ArisakaP4Real + ArisakaP5CPT
%% (A6 house style) with roster-level Relevance notes.  No P4-Twistor
%% bib key ever existed in A6 (nothing to rename); zero "satellite"
%% tokens of any register; zero "Twistor" tokens outside the P4
%% official subtitle now quoted in its bibitem.
%% (5) ONE-LINE CONTENTS: \GGPart renewed per the P4-Real v3.2
%% preamble pattern (contents line carries only "Part N. <title>");
%% \GGAppendixPart (Part X) inherits the renewal.
%% (6) STALENESS REPAIR (mechanical; A9-SM Part-IV flag "the present
%% release supersedes this"): the Part-I scope item's "revised A9
%% (in preparation)" now cites the released A9-SM (ArisakaA9SM), and
%% the stale legacy token "6F" in the same sentence is retired in
%% favour of the canonical \ANine{} macro (present tense).
%% Wave-2 content items deliberately NOT installed (arrow-custody
%% cross-link at GRIP-ST2/PS-GRIP-3, GFF=GEP+CEP energy-entropy
%% endowment remark, Wilson-line holonomy-i pedigree footnote; those
%% regions are byte-identical to v11.25).  The rotated comparison
%% table (Part VIII) is charter-exempt and untouched.  Physics,
%% theorem keys, numbers, values, grades: UNCHANGED.  v11.25 -> v13.0.
%% Script: a6_v13.0_wave1.py.
%% v13.0 pre-signoff correction (2026-07-20): the four Wave-1-added \ASix{} sites (Foundations paragraph, cohort-map node + caption, roadmap long caption) now carry literal \textsf{GG-A6-GRIP}; the sty's \providecommand\ASix (GG-A6-Gi4) silently wins over the paper's \providecommand, so \ASix renders the old name.  Pre-existing v11 \ASix sites untouched.  Script: a6_v13.0_caption_fix.py.
%% v11.25 (July 16, 2026): cohort table standard (author-approved, P4 v2.17 pattern): sub-small table fonts -> \small; p-column tables widened toward full text width; targeted corrections listed.
%%   correction: tab:gi4-hierarchy: p-widths reduced (overfull 20.5pt; wide l-column)
%%   correction: tab:five-theorems: p-widths reduced (overfull 28.6pt)
%%   correction: tab:Gi4-Qi4-pairing: p-width reduced (overfull 87.3pt; three wide l-columns)
%%   correction: tab:comparison-comprehensive: font raise reverted to scriptsize (rotated resizebox table; raise inflates wrapped height, 226pt overfull)
%%   correction: tab:appB-glossary: p-width reduced (overfull 52.7pt; wide symbol l-column)
%% v11.24 (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.
% v11.23 (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.
% v11.22 (July 2, 2026): cohort multi-Arisaka citation split pass
%   (charter natbib rule; author-directed 2026-07-02): 4
%   multi-key cites split into citealp/citet chains; fixes the
%   same-author-same-year comma-run rendering. No content changed.
% v11.21 (July 2, 2026): [C-A1] closure install (author-accepted report
%   A6-CA1-Lambda2-Report-2026-07-02). rem:cascade-concordance upgraded
%   from 'one bundle identification remains' to derived-under-three-
%   premises: (P-i) wrap independence (ell-fold Whitney sum of the
%   OGN-unit flux line bundle; pairs of wraps = states of 10 =
%   Lambda^2(5)); (P-ii) unit self-pairing int beta^beta = 1 (OGN);
%   (P-iii) background-flux projector-independence (rival ell^2 Abelian
%   slot excluded by the toll arithmetic). Whitney: c(E_ell) =
%   (1+beta)^ell => c_2 = C(ell,2) beta^2, Delta c_2 = k; blind
%   uniqueness confirms C(ell,2)+const is the only weight family.
%   Preamble + Step-2 pointer aligned. No counts changed.
% v11.20 (July 2, 2026): [C-A1] proof repair (author-accepted report
%   A6-CA1-Bundle-Report-2026-07-02). thm:instanton-action: Step 2 now
%   carries the winding pair-count weights lambda_ell = C(ell,2)
%   (index-2k point hosts winding ell = k+1; spacings k, not 1); Step 3/4
%   adjusted accordingly (drop = k); Step 5's 'cumulative symplectic
%   depth' convention deleted -- S = pi k now follows directly from the
%   weights. rem:cascade-concordance upgraded: unit-spaced weights are
%   internally excluded (budget + N<=3 arithmetic); finite check reduced
%   to the single Lambda^2(5) identification. Theorem statement and all
%   downstream numbers unchanged.
% v11.19 (July 2, 2026): [B1]-kernel sharpening of the Tolman remark
%   (author-accepted report A10-B1-Kernel-Report-2026-07-02). One edit:
%   rem:GFF-KK-temperature gains the freeze-out reading -- under KMS
%   contact the Einstein ratio D/mu copies the bath temperature
%   (thermometer); at the GA the gapped, frozen tower (Lemma DE-1)
%   disconnects the fiber-moduli block, so T_GFF ~ M_KK is the record
%   of last kinetic contact, T* = 2 M_KK/L*, log-below the gap. No
%   physics change elsewhere; no equation altered.
% v11.18 (July 1, 2026): Descent-normalisation block installed in the
%   holographic-descent subsection (new ssec:descent-normalisation):
%   factorisation eq:descent-factorised, Lemma lem:descent-normalisation
%   (Phi_omega = pi(1-e^-2K) -> pi; boxed eq:Sbulk-normalised), cascade
%   concordance remark (three pi's; flat 2piK candidate excluded; Dc2=k
%   saddle-charge check recorded as finite target), and Corollary
%   cor:dirac-superselection (phase filter 4Z; Hirzebruch p1=3sigma =>
%   packet absolutely superselected; underpins DE-1). Step-1 wording now
%   points to the lemma via POS-6; DE-1 closing paragraph gains a pointer.
%   Consumes A1 POS-6, A5 Lemma T8-1; consumed by A10 Conj-VIII audit.
%   Science content author-approved 2026-07-01 (StarClosure report).
% v11.17 (July 1, 2026): NEW Lemma DE-1 (vacuum-flux freezing at the GA)
%   inserted after GFF-ST6, with proof from GFF-ST6 + T2.2 and the M1/M3
%   leading-order qualifier; closing sentence points to the w = -1 theorem
%   package in A10 (T-Cosmos-DE'). Author-approved Stationary-Measure
%   programme, report of 2026-07-01. No other content touched.
% ===== A6-GRIP v11.16 (2026-06-27): (a) reconciled the family-cascade toll one-liner to the
% Bogomolnyi theorem -- accumulated action 0, 2pi, 3pi (tolls 2pi then pi descending; middle was
% mis-stated pi); same fix in the 28-parameter mechanism prose. (b) Added a 'Cohort map' pointer to
% each of the Part VII applications (N=3, 28 params, hierarchy, KK, Kaxion, radiative) naming the
% cohort papers each draws on, so the Part is graspable. Built on v11.15. =====
% ===== A6-GRIP v11.15 (2026-06-27): Part VII Applications expanded ~3x. Each static application
% (N=3, the 28 SM parameters, hierarchy, KK, Kaxion) gains the rhythm: physics intuition -> the GRIP
% mechanism (rigorous: gradient flow, Bogomolnyi toll pi*k, S_tot=pi N(N-1)/2, budget closure N<=3,
% Hirzebruch even-N exclusion; e^{-pi} ladder; M_lock=1/2 E_Pl e^{-K}, K=35) -> why the reading helps
% (vs Froggatt-Nielsen / landscape / anthropics; hierarchy-problem dissolution; falsifiability). Kaxion
% tied forward to the GRIP-In empty-exit terminus of the new radiative section. Built on v11.14. =====
% ===== A6-GRIP v11.14 (2026-06-27): Part VII restructured into a statics/dynamics arc. Added two-tier
% Part intro + Static-tier lead-in; NEW dynamical section 'Application: radiative descent --- synchrotron
% and gravitational-wave inspiral as one GRIP-In flow' (spin-ladder l=s from TGG+CCI; GRIP-In taxonomy:
% empty-exit Kaxion / relaxation synchrotron / absorbing inspiral=C3 horizon; dimensionless zeta table).
% Field-theory derivation cited to A5 Part VI (no duplication). Bridge paragraph updated four->two tiers. =====
% Bib cross-citation titles reconciled to canonical cover titles (cross-citation integrity), 2026-06-27.
% A10 bib-title reconciled to canonical cover title (cross-citation integrity), 2026-06-27.
% ===== A6-GRIP v11.11 (2026-06-26): cohort Wave-2 de-contamination (protect-list aware). =====
% Shed GG's own 'warped/holographic' self-description (warped 6D bulk->6D bulk; warped product->
% product geometry; warp factor/depth/exponent/scale/slope/stratum->scale; holographic projection/
% projector/descent->bulk-to-boundary; holographic image->boundary image; holographic dictionary->
% bulk--boundary correspondence; holographic boundary->boundary; holographic anomaly inflow->anomaly
% inflow). KEPT all external AdS/CFT/RS comparison, disclaimers, citations (Maldacena/Susskind/Randall--
% Sundrum), holographic PRINCIPLE, 'holography'/'holographic duality', A7 QiHAL/HAL acronym glosses,
% and Randall--Sundrum external refs/cells. Body+metadata; no physics/equations/numbers/labels.
% ===== A6 v11.10 (2026-06-25): cohort central-claim funnel (light, foundation variant). =====
% Added a modest 'place in the cohort' section (sec:cohort-funnel) leading with the paper's own
% subject, plus the shared open-system funnel (fig:cohort-funnel, generic payload). Framing kept
% deliberately light per author guidance; the paper proceeds on its own terms. No new physics.
%% =====================================================================
%%  A6-GRIP v11.9 (migrated onto ucla-arisaka-gg-11) -- 2026-06-22
%%  Single canonical style.  gg-6 -> gg-11; dropped dead \LaTeX shim;
%%  stripped Overview/LoF overrides; single-page Overview+LoF via the gg-11
%%  toggle; Preface via \GGPreface; ES centered by gg-11; inherits the one
%%  canonical Acknowledgments.  No prose/figure/number content changed.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.8 (audit-fix + honesty-propagation pass) -- 2026-06-19
%%  Fixes from the fresh-eyes 3-referee audit of v11.7:
%%   H1 propagate 'reduced modulo M6 / conditional' honesty into the H1-H2
%%      section title+payoff, Conclusions accomplishments list, methodology
%%      summary, and the P2-N3 bib note (was 'promoted/closed, no softness').
%%   H2 add the M11 conditional to T2.2's reverse (local Poincare) direction
%%      + a Status remark mirroring T4.2.
%%   M1 T4.1 intuition 'each step costs pi' -> pi*k (matches triangular total).
%%   M2 attach M10 conditional to the T4.1 theorem statement + 4 body passages.
%%   M3 clarify the W/mu/n '(agrees)' tags as Sargent phase-space estimates.
%%   M4 open problem M6 'Partially resolved' -> 'Genuinely open'; triage
%%      recount (Partial 7->6, Open 19->20, math-open 9->10).
%%   M5 PS-GRIP-2/5/6 'Resolved (conditional on open conj.)' -> 'Partially
%%      resolved'.  M6 soften T5 'proven equivalent' (I-GRIP-10, PS-GRIP-8).
%%   L1 corollary 'T2 gives N=3' -> 'T2 gives lower bound N>=3'.
%%   C-2 broken 'A3 Theorem 3.4' (no such thm) -> 9.1 (trace ratio) / 10.2
%%      (OGN-5 ledger) / 6.3 (BI-6 closure), per compiled A3-v11.11 aux.
%%  No physics/parameter/triage-of-results changes; honesty + citation only.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.7 (naming-convention reconciliation) -- 2026-06-19
%%  EDIT 4b: folded the open physical/experimental/future lists into the
%%  continuous PS-GRIP-9..24 sequence (removing the P#/F# local tags that
%%  collided with global P=Prediction/F=Falsifier); math conjectures M1..M11
%%  kept verbatim; triage summary + 3 external bib refs updated.
%%  EDIT 4a: open-problems section renamed to the A2 model 'Problems Solved
%%  and Future Directions'; prepended 8 positive PS-GRIP-1..8 'Problems
%%  solved' entries (Resolved + Live falsifier, compiled from T1..T5).
%%  EDIT 3: new 'Ten Structural Innovations' section (I-GRIP-1..10,
%%  compiled from the exec summary / significance prose, no new physics) +
%%  the cohort 'Global naming convention' key (ten prefixes, suffix GRIP,
%%  P=Prediction per author decision), inserted before Problems-Solved.
%%  EDIT 2: global Falsifier ids F-GRIP-1..7 (two restarting local 1/2/3
%%  blocks + one untagged -> one global sequence; in-text refs and the two
%%  objections quoting them disambiguated).
%%  EDIT 1 (mechanical): (a) Objection tags O-Gi4-n -> O-GRIP-n (display
%%  text only, no \ref impact; the cohort suffix for A6 is GRIP, per A2's
%%  ratified key 'GRIP (A6, formerly Gi4)').  (b) Appended cohort global
%%  theorem identifiers (T-GRIP-1..5) to the five principal-theorem section
%%  titles; internal pillar tags T1..T5 kept, sub-theorems T1.1..T4.2 left
%%  untagged (Option A).  Further edits (falsifiers, Innovations/Problems-
%%  Solved, convention key) follow.  No scientific content changed.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.6 (Phase 5 patch 2: independent-audit residue) -- 2026-06-19
%%  Stale retired-8-Part numbering caught by the verification subagent:
%%   - cross-ref table: 'Part VII (Synthesis)' -> 'Part VI (Synthesis)'
%%     (Synthesis is Part VI; sec:bridge-Qi4 sits in Part VI).
%%   - appendix-overview para: 'stop after Part~VIII' -> 'Part~IX';
%%     'the seven Parts' -> 'the nine main-text Parts';
%%     'Parts I--VI state ... Parts VII--VIII treat applications and
%%     discussion' -> 'Parts I--VI develop and synthesise ... Parts
%%     VII--IX treat applications, discussion, and conclusions'.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.6 (Phase 5: final smoothness/consistency pass) -- 2026-06-19
%%  Structural audit clean (12 Parts everywhere; 53 objections; Part<->ref
%%  adjacency 0 genuine mismatches; no stale lifetime-law / generic-ueV
%%  kaxion / spaced-OGN5 residue; theorem-count standardised).  Brought
%%  three HEADLINE claims into line with the Q&A conditionals + M9/M10/M11
%%  triage the body already carries:
%%   (1) abstract N=3/K=35 'as theorems' -> N<=3 upper bound conditional (M10).
%%   (2) exec-summary 'forcing N<=3 (T4.1)' -> M10 conditional attached.
%%   (3) conclusions 'mathematically complete ... all nine Principal Theorems
%%       proved' -> 'complete modulo three named conjectures' (M9/M10/M11);
%%       'nine Principal Theorems' -> 'nine dual-layer sub-theorems'.
%%   (4) O-GRIP-37 question 'four principal theorems' -> 'four pillar theorems'.
%%  Presentation/honesty-consistency only; no physics/parameter/triage change.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.5 (Phase 4: Common Objections Q&A) -- 2026-06-19
%%  Common Objections Part populated with 53 hostile-reviewer objections
%%  (O-GRIP-1..53) in 11 clusters, authored hostile-audit-first then answered
%%  sharply and honestly, each cross-linked to A6 theorems/sections/open
%%  problems (M6/M9/M10/M11) and companion papers.  Closing remark added.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.4 (Phase 3: full v2.0 structure) -- 2026-06-19
%%  (3.2) Four-Part tail: Conclusions (ack at end) -> Technical Appendices
%%        (\GGAppendixPart, \appendix once; appendices now letter A--J) ->
%%        Common Objections (new Part; intro + objqabody, Q&A in Phase 4) ->
%%        References (own Part).  Part count 10 -> 12.
%%  (3.3) R4 single-page front matter (Overview of Contents + List of
%%        Figures on one page; five long figure captions given [short]).
%%  (3.4) Dedicated 'The GG-Theory Cohort' section (five-strata + two-branch
%%        prose, A6 named as a boundary-law pillar); cohort figure full-page.
%%  (3.5) Dedicated 'Roadmap of the Twelve Parts' section + full-page vertical
%%        flowchart (Part V / GRIP highlighted as the four-pillar-core apex).
%%  (3.6) References A/B split (A = 17 cohort, canonical order; B = 73 others)
%%        + 'First cited in' link on every entry (0 orphans).
%%  (3.7) Stale 'Plan of the paper' subsection -> Roadmap pointer; v -> v11.4.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.3 (Phase 2: cohort citation keys + cite all 17) -- 2026-06-19
%%  Remapped legacy Arisaka2026Generations -> canonical ArisakaP2N3 (20
%%  cites + bibitem). Added the three missing cohort bibitems A8-QM,
%%  C2-Hubble, C3-BH (canonical titles from A1) and cited them at natural
%%  homes (orthodox-QM sentence; the A10 cosmology sentence; the black-hole
%%  open problem).  All 17 cohort papers now both defined and cited.
%%  Arisaka2026Inflow / ArisakaBookI kept as non-cohort self-references.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.2 (Phase 1: presentation cleanup) -- 2026-06-19
%%  (A) Fixed failed-search-replace garbles ('This section is new in.',
%%      'In the theorem is presented', 'feature of is', 'framework of is',
%%      'these forces force') + a duplicated T4.2 table-row phrase.
%%  (B) Theorem-count convention standardised: four pillars; FIVE principal
%%      theorems T1-T5; NINE dual-layer sub-theorems T1.1-T4.2 (abstract
%%      header 'Four'->'Five'; 'nine principal theorems'->'nine dual-layer
%%      sub-theorems' throughout).
%%  (C) Kaxion mass -> canonical 8.1 ueV (~1.95 GHz) (was generic '~ueV').
%%  (D) OGN-5 ledger notation unified to the compact (1,3,5;7,35).
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.1 (Phase 0 science-reconciliation, Part C) -- 2026-06-19
%%  Propagated the lifetime->width reconciliation through the GRIP Part
%%  intro, T2.2, section titles, the Discussion falsifier subsection, the
%%  open-problems text, the glossary, and all remaining framing sentences:
%%  one consistent statement everywhere (generation-ordered decay-width
%%  ladder Gamma~e^{-pi k}; lifetimes via A9), no residual closed-form
%%  lifetime law, no W/muon/neutron index ladder.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.1 (Phase 0 science-reconciliation, Part B) -- 2026-06-19
%%  Worked example + Appendix M recast: removed the spurious 'decay
%%  cascade' (W/muon/neutron i=1/2/3 ladder) and the closed-form lifetime
%%  law; aligned to the single generation cascade (Gamma~e^{-pi k}, family
%%  chains tau->mu->e and t->...->u/d), absolute lifetimes deferred to A9.
%%  Appendix M table reindexed to generation gates; falsifier recast to the
%%  controlled-kinematics gate-ordering test.
%% =====================================================================
%% =====================================================================
%%  A6-GRIP v11.1 (Phase 0 science-reconciliation, Part A) -- 2026-06-19
%%  Author-approved cohort reconciliation (vs P2-N3, P3, A9-SM, A5-GDOS):
%%  (C3) PBI-4 compensator smoothness resolved to OPEN (M11): Compatibility-
%%  Theorem proof + T4.2 framing made conditional on M11 (global integrality
%%  from A3; local cross-instanton smoothness unproved cohort-wide).
%%  (E1/E2/E3) GRIP-ST7 recast from the spurious closed-form lifetime law
%%  (W/muon/neutron i=1/2/3 ladder, 'principal falsifier') to the canonical
%%  generation-ordered decay-WIDTH ladder Gamma~e^{-pi k} (matches this
%%  paper's own sec:GRIP-lifetimes and A9/P2); lifetimes deferred to A9/P2.
%%  (E6/E7/E8) abstract, rem:T4.2-deeper, and four summary tables aligned.
%%  Part B (next script) recasts the worked example + Appendix M.
%% =====================================================================
% ===== v11.0 (2026-06-16): HOLOGRAPHIC-RG REFRAMING, Phase R2 (author Go). =====
%  Terminology only; NO physics changed. A6 has no AdS-bulk branding; A2/A3 bibitems
%  already AdS-free; 20 AdS/CFT mentions KEPT. Added the canonical holographic-RG
%  disclaimer (footnote at the first '6D bulk' mention). v10.9 -> v11.0.
% =====================================================================
% --- v10.9 (2026-06-16): adopt cohort-default Part-aware running header (sty v10.6).
%     Curated short Part labels added; running header shows 'Part N. <short>'. Preface self-labels. ---
% ALWAYS-ON PROJECTION CLARIFICATION (June 14, 2026): aligned with GG-A5-GDOS Sec 23 ('Why the Projection Is Always On'). The 6D->4D OGGEP/GDOS projection is always on and observer-independent; 'measurement' is only its macroscopic record-keeping special case (decoherence the microscopic one). Removed framing that tied the projection to observation/measurement; added (A5 Sec 23) cross-refs.
\pdfoutput=1
%% v10.8 (June 12, 2026): Orphan-bibliography maintenance (author
%%   Go): previously uncited bibitems cited inline at their natural
%%   prose mentions (automatic surname-site matching + manual
%%   placements; see [P3-Support] sweep report). No other content
%%   changes.
%% v10.7 (June 12, 2026): Cohort-map deployment (author Go):
%%   the canonical seventeen-paper cohort map inserted (label
%%   fig:cohort-map-a6, this paper highlighted) with a one-paragraph lead-in,
%%   at the surveyed seam where the introduction completes.
%%   No other content changes.
%% v10.6 (June 11, 2026): R1 sync (minimal, disclosed): A5-GDOS
%%   v10.8 derived the covariant kernel and re-graded O7 to Resolved;
%%   Example 4's status sentence and the A5 bibitem Relevance
%%   updated accordingly. No other content changes.
%% v10.5 (June 11, 2026): Golden-case worked example (author request,
%%   June 11 synchrotron round): new Example 4 in sec:GRIP-examples
%%   -- GRIP-In on the orbital stratum (synchrotron damping + binary
%%   inspiral), the one GRIP instance computed in closed form and
%%   measured (mobilities, Larmor/Peters coefficients, C_q) per
%%   A5-GDOS v10.7's radiative completion + closure.  Stratum fence
%%   stated (orbital vs family; P3 two-sector partition preserved);
%%   merger-as-GRIPout-functioning and 'gate-free strata relax'
%%   priced as interpretive.  New bibitems ArisakaP3Electron,
%%   Sands1970, HulseTaylor1975, Abbott2016 (all cited inline); A5
%%   bibitem Relevance extended.  Intro count three -> four.
%%   No other content changes.
%% v10.3 (June 10, 2026): arXiv preparation pass (front-matter audit).
%%   front-matter audit: conforms; stamp bump only.
%%   Accepted variants (author decision): existing philosophical closes
%%   in place of literal 'After all'; companion-list counts as cited.
%% v10.2 (June 9, 2026): arXiv preparation pass (cohort step 3).
%%   Overfull-box sweep (all boxes >20pt resolved with layout-only fixes);
%%   no content or physics changes.
%% v10.1 (June 9, 2026): arXiv preparation pass (cohort step 1-2).
%%   \pdfoutput=1 added (line 1); version/date stamp bumped. 
%%   No physics changes; production fixes only.
% ======================================================================
% PROVENANCE
% ----------------------------------------------------------------------
% This file is the Phase 1 identity-migration baseline of the A6-Gi4
% paper.  It is a verbatim copy of UCLA-Arisaka-GG-2026-4A-v3.0-Phase10.tex
% (May 2, 2026), modified only in:
%   * the document class \usepackage line (gg-4 -> gg-6)
%   * the four PaperNumber / PaperStatus / PaperVersion / PaperDate macros
%   * the bibtex citation keys throughout the body, per L3
%   * the bare GG-2026-N paper-number references in prose, per L2
% No prose has been edited, no theorem statements have changed, no figures
% have been added or removed.  Phase 1 is identity-only.
%
% Phase log (v5.X cycle):
%   v5.0 (May 11, 2026)  Phase 1: Identity migration baseline.
%                        Style gg-4 -> gg-6 (L1).  PaperNumber A6, version
%                        v5.0, status Preprint (L9, L15).  Bibtex keys
%                        migrated to v3.1+ canonical form (L3), except
%                        ArisakaP2N3 and Arisaka2026Inflow which
%                        the user has elected to leave separately for the
%                        time being.  Body paper-number labels relabelled
%                        from GG-2026-N to A*-Suffix form (L2).  No content
%                        change.  Forthcoming phases: 2 (GS-as-GGEP framing
%                        audit), 3 (L4 historical-inventor audit), 4
%                        (Pillar theorem statements with dual-layer L16
%                        architecture), 5 (Open Problems triage with L6
%                        status headers), 6 (hostile-reviewer + reference-
%                        scanner sub-agents), 7 (L7 cross-paper coherence
%                        map), 8 (L8 TikZ figure compliance), 9 (front-
%                        matter rewrite to A1-canonical template), 10
%                        (bibliography surname-sort + journal-stripped
%                        variant), 11 (Appendix A canonical 9-section
%                        format + L20 hierarchy figure + L21 sans-serif
%                        audit + 36-item v3.4 audit + L25 length check).
%   v5.1 (May 11, 2026)  Phase 2: GS-as-GGEP framing audit (L5).
%                        Three patches:
%                          (A) First definition of BI-6 in section intro:
%                              added one-sentence "BI-6 is the GGEP
%                              completion of GS" framing with cross-cite
%                              to A3-BI6 §8.7.
%                          (B) "Algebraic heart of Green--Schwarz
%                              mechanism" passage sharpened with the
%                              GGEP-pioneering reading.
%                          (C) Bibliography Relevance paragraph for
%                              GreenSchwarz1984 rewritten per L5 to
%                              position GS as the technical pioneering
%                              of AX-2 GGEP, with cross-cite to A3-BI6
%                              §8.7 as the canonical home of the
%                              supportive treatment.
%                        14 internal-technical references to X_GS / GS
%                        inflow term / GS counterterm / GS correction
%                        not touched (object names, not framing).
%                        No theorem statements changed, no figures added.
%   v5.2 (May 11, 2026)  Phase 3: L4 historical-background
%                        original-inventor audit.  Inserted four new
%                        subsubsection{Historical background (with
%                        citations)} blocks, one at the end of each
%                        Pillar's Motivation subsection (TGG, CCI, GFF,
%                        GRIP).  Each opens with the genuine original
%                        inventor by name and year:
%                          TGG  : Maxwell (1865) -> Einstein (1915) ->
%                                 Weyl (1918) -> Ehresmann (1950) ->
%                                 Kobayashi--Nomizu (1963) ->
%                                 Yang--Mills (1954) -> TGG.
%                          CCI  : Bianchi (1902) -> Maxwell (1865, dF=0)
%                                 -> Noether (1918) -> Ward (1950) ->
%                                 Slavnov-Taylor / BRST ->
%                                 Adler-Bell-Jackiw (1969) ->
%                                 Green-Schwarz (1984) ->
%                                 Hopkins-Singer integrality -> CCI.
%                          GFF  : Carnot (1824) -> Clausius (1865) ->
%                                 Boltzmann (1872) -> Gibbs (1876) ->
%                                 Onsager (1931) -> Prigogine (1947) ->
%                                 Jaynes (1957) -> Friston (2010) -> GFF.
%                          GRIP : Stueckelberg-Petermann (1953) ->
%                                 Gell-Mann-Low (1954) -> Kadanoff (1966)
%                                 -> Wilson (1971/74) -> Hosotani (1983)
%                                 -> Anderson (1972) ->
%                                 'tHooft / Coleman / Callan-Coleman
%                                 instantons -> GRIP cascade.
%                        Ten new bibitems added (Maxwell1865, Einstein1915,
%                        Bianchi1902, YangMills1954, Noether1918, Ward1950,
%                        Boltzmann1872, Onsager1931, GellMannLow1954,
%                        Kadanoff1966).  Phase 10 will surname-sort the
%                        entire bibliography.
%   v5.3 (May 11, 2026)  Phase 4: Pillar theorem statements with L16
%                        dual-layer architecture.  Four surgeries:
%                          (A) Added new subsection "Dual-layer
%                              architecture and the compatibility
%                              relation C_TGG" at the front of section
%                              "Theorem T1" (TGG).  Identifies T1.1+T1.2
%                              as the algebraic layer (existence and
%                              uniqueness on bulk and boundary) and T1.3
%                              as the operational layer (Master
%                              Conservation Law).  Names the compatibility
%                              relation in a boxed equation.
%                          (B) Added paragraph-level "Dual-layer
%                              architecture and the compatibility relation
%                              C_X" at the front of sections T2 (CCI), T3
%                              (GFF), T4 (GRIP).  Each names the algebraic
%                              layer (T_*.1), the operational layer
%                              (T_*.2), and the named compatibility
%                              relation in a boxed equation.  Compatibility
%                              relations: C_CCI = T2.1 => T2.2 via PBI-4
%                              compensator and Ward package; C_GFF = T3.1
%                              => T3.2 via GDOS CPTP flow and Morse
%                              alternating-index sum; C_GRIP = T4.1 => T4.2
%                              via PBI-4 compensator extension across
%                              instanton stratum.
%                          (C) Added four Physics-Intuition remarks: after
%                              T1 (TGG as constitutional culmination of
%                              Maxwell--Yang-Mills--Einstein), after T2.1
%                              (why (1,3,5;7,35) is the only OGN-5 ledger
%                              that survives MSMF), after T3.1
%                              (Morse-positivity is what makes GFF a free
%                              energy in the modern sense), after T4.1
%                              (why the finite BI-6 budget forces N<=3).
%                              The .2 theorems (T2.2, T3.2, T4.2) already
%                              have Physics-Intuition remarks from v3.0;
%                              T2 / T3 / T4 also already have "Logical
%                              structure" remarks at the .2 level.
%                          (D) Added L12 framing paragraph at the point
%                              where Gtot is first written in the body
%                              ("Constitutional admissibility of U(1)_Qi:
%                              optional, empirically preferred").  Cites
%                              A3-BI6 as the authoritative source on
%                              constitutional admissibility, names the
%                              three-component and four-component bulk
%                              options, and gives the three empirical
%                              grounds for the four-component choice
%                              (Arisaka equation, OGN-5 dual-lock, Qi-4
%                              pillars and EIC framework).
%                        No proofs touched, no theorem labels changed.
%   v5.4 (May 11, 2026)  Phase 5: Open Problems and Future Directions
%                        audit with L6 status headers.  Surgeries:
%                          1. Renamed §"Open questions and future directions"
%                             to §"Open Problems and Future Directions"
%                             (canonical L6 name).
%                          2. Relabelled sec:open-questions-future ->
%                             sec:open-problems (preserves the one existing
%                             cross-ref at L1164).
%                          3. Added opening triage-convention paragraph
%                             (A1-canonical pattern).
%                          4. Added explicit "Status: Resolved / Partially
%                             resolved / Genuinely open" headers to all
%                             24 open-problem entries (M1-M8, P1-P5, E1-E5,
%                             F1-F6).
%                          5. Added 3 new M-entries (M6, M7, M8) folding in
%                             the Gi-4 internal technical problems from the
%                             previous second §Open problems section.
%                          6. Replaced §"Closing thought" with §"Triage
%                             summary" giving status counts: 1 Resolved
%                             (P1), 7 Partially resolved (M4, M6, P2, P4,
%                             P5, F1, F6), 16 Genuinely open.  §"Closing
%                             thought" retained, sliding to the next slot.
%                          7. Removed the duplicate second §"Open problems"
%                             section entirely (its content folded into
%                             the first section).  This also eliminates the
%                             pre-existing multiply-defined-label warning
%                             on ssec:open-experimental that has carried
%                             from v5.0.
%   v5.5 (May 11, 2026)  Phase 6: Hostile-reviewer + reference-scanner
%                        sub-agents in parallel (L13 pattern).
%                        Reference-scanner came back CLEAN (59 cite-keys
%                        all resolve, no legacy patterns, no undefined
%                        refs).  Hostile-reviewer flagged 4 Critical +
%                        5 Major + 3 Minor issues; this revision applies
%                        the recommended patches:
%                          C1 + m3: T4.1 Step 2 + Step 5 honesty.
%                                   Sbi = 3pi is now acknowledged as
%                                   established in companion paper
%                                   [ArisakaP2N3 Appendix B]
%                                   not in-paper; saturation rhetoric
%                                   softened pending M10 resolution.
%                          C2 + C3: T2.1 Comments paragraph rewrite.
%                                   MSMF is now explicitly the ell^1
%                                   norm; L=5 is honestly named as the
%                                   constitutional SU(5)-unification
%                                   input; "what is not assumed" replaced
%                                   with "what is and is not assumed".
%                          C4: T3.2 reverse direction strengthened with
%                              detailed-balance hypothesis (Spohn 1978).
%                              Spohn1978 bibitem added to bibliography.
%                          M3: T1.3 statement adds parenthetical
%                              acknowledgment that the modified Einstein
%                              equation defines T_TGG on the
%                              gravitational side.
%                          M5: Four compatibility relations re-framed
%                              as "compatibility propositions" with
%                              explicit one-line proofs.
%                          M1+M2: T3.1 outline-of-proof gains a "Scope
%                                  clarification" (Kahler with flat-gauge
%                                  strata) + "Morse function
%                                  specification" (equivariant moment-map
%                                  under torus action, alternating sum =
%                                  Hirzebruch signature sigma).
%                          M4: T4.2 compatibility-proposition statement
%                              acknowledges monodromy as topological
%                              non-obstruction hypothesis (->M11).
%                          m1: L12 framing adds three-component-vs-four-
%                              component clarification.
%                        Three new open problems M9, M10, M11 added to
%                        capture the deferred items.  Triage count
%                        updated from 24 -> 27 (1 Resolved, 7 Partially
%                        resolved, 19 Genuinely open).
%   v5.6 (May 11, 2026)  Phase 7: L7 cross-paper coherence map.
%                        Added new \section{Cross-Paper Coherence Map}
%                        immediately before §Open Problems and Future
%                        Directions, containing:
%                          - L7 Note-on-version paragraph clarifying that
%                            companion-paper versions are independent
%                            revision tracks and each table row records
%                            the version of the cited companion paper as
%                            of May 2026.
%                          - L7 Audit-trail paragraph documenting the
%                            actual internal theorem numbering of A1
%                            v5.25, A2 v5.13, A3 v5.13, A4 v5.17, A5 v6.23.
%                          - Inward map (longtable, L22-compliant): 14
%                            rows of theorem-level facts A6 imports from
%                            A1, A2, A3, A5, each with specific companion-
%                            paper location (Theorem-N or section number)
%                            and role in A6-Gi4.
%                          - Outward map (longtable, L22-compliant): 16
%                            rows of A6-Gi4 facts exported to downstream
%                            papers A7-Qi4, A8-QM, A9-SM, A10-Cosmos,
%                            P1-KK, C1-Kaxion, each with A6 theorem
%                            label and downstream anticipated use.
%                          - Closing paragraph emphasising that the
%                            present section should be re-walked
%                            against any revised companion paper before
%                            the present paper is itself republished.
%                        Both tables use longtable with \footnotesize per
%                        L22, not tabular-in-resizebox-in-table[H].
%   v5.7 (May 11, 2026)  Phase 8: TikZ figure pass per L8.
%                        A6 v5.6 had ZERO actual TikZ figures (only table
%                        captions).  Phase 8 adds four substantive
%                        figures, each wrapped in resizebox{textwidth}{!}
%                        per L8, with absolute coordinates and edgelabel
%                        styles where needed:
%                          Fig 1 (gi4-architecture): The four Gi-4 Pillars
%                            TGG -> CCI -> GFF -> GRIP, with the GDOS
%                            boundary law (top) and the 6D bulk
%                            (bottom) as sources, and the four named
%                            compatibility relations C_TGG, C_CCI,
%                            C_GFF, C_GRIP as inter-layer links.
%                            Inserted at end of Part I.
%                          Fig 2 (tgg-composite-connection): The
%                            principal G_tot-bundle decomposition of
%                            A_tot = omega + A into Spin(1,5), SU(5),
%                            U(1)_B, U(1)_Qi sub-connections; curvature
%                            R_tot -> anomaly polynomial X_4 -> boundary
%                            stress tensor T_TGG -> Master Conservation.
%                            Inserted at end of Part II.
%                          Fig 3 (cci-ward-ladder): Bulk X_4 closed
%                            4-form, Hopkins-Singer descent via
%                            dH_3 = X_4 to boundary PBI-4 compensator
%                            field C_mu, four-current Ward ladder
%                            J_e, J_B, J_Qi, J_g, Master Conservation
%                            Law T2.2 as iff statement.
%                            Inserted at end of Part III.
%                          Fig 4 (grip-cascade): The three-stage cascade
%                            DGI -> GSSB -> GA with instanton-action
%                            ladder pi, 2pi, BI-6 budget S_BI6 = 3pi
%                            on CP^N family-projector moduli, N <= 3
%                            forcing.  Includes PBI-4 compensator
%                            smooth-extension hypothesis (M11).
%                            Inserted at end of Part V.
%                        Also addressed the pre-existing float-overflow
%                        at the rotated comparison table by adding
%                        footnotesize inside the minipage; should now
%                        clear the 2.08pt margin overrun.
%   v5.8 (May 11, 2026)  Phase 9: front-matter rewrite to A1-canonical
%                        template (L9).  Three surgeries:
%                          1. Title block tightened: short crisp primary
%                             "The Gi-4 Pillars", 2-line subtitle "TGG,
%                             CCI, GFF, GRIP / The Geometric Backbone of
%                             GDOS", short description line.
%                          2. Abstract replaced (795 words -> ~205 words)
%                             with plain-English version following L9
%                             three-paragraph structure: opening universal
%                             thesis (conservation laws are geometric);
%                             diagnosis (gauge-gravity unification has
%                             remained a hope); what the paper does
%                             (four pillars as dual-layer theorems); what
%                             emerges (N=3, hierarchy, K=35 forced);
%                             closing "After all" philosophical
%                             reflection.  Fits on title page.
%                          3. Executive Summary replaced (1337 words ->
%                             ~870 words) with L9-canonical template:
%                             opening positioning vs predecessors (with
%                             explicit \citet to ArisakaP2N3,
%                             ArisakaA9SM, ArisakaA1Constitution,
%                             ArisakaA10Cosmos, ArisakaC1Kaxion,
%                             vonNeumann1932); boxed central principle
%                             (Gi-4 master identity with the four
%                             compatibility propositions); "Four
%                             principal theorems" paragraph summarising
%                             T1-T4 with dual-layer architecture;
%                             Significance and uniqueness paragraph (3
%                             features + empirical falsifiers);
%                             Canonical implication chain in boxed
%                             equation cascade; Closing reflection with
%                             "After all" continuation of the Abstract
%                             motif.
%                        \GGKeywords{} placement after
%                        \end{ExecutiveSummary} already correct (L9
%                        compliant).  \PaperStatus{Preprint} already
%                        clean (L15 compliant, no live LaTeX).
%   v5.9 (May 11, 2026)  Phase 10: bibliography surname-sort +
%                        journal-stripped variant + L11/L22 overflow
%                        audit per L10.
%                          - 80 inline bibitems sorted alphabetically by
%                            first-author surname using a mechanical
%                            Python script with L10 conventions:
%                            particle demotion ('t Hooft -> H, von
%                            Neumann -> N, de Rham -> R, etc.); multi-
%                            author by first surname; accent stripping
%                            for sort keys; Arisaka block sub-sorted by
%                            programme order (A1, A2, A3, A4, A5, A6,
%                            A7, A9, A10, P1, C1, Generations, Inflow,
%                            BookI).
%                          - Arisaka entries re-lettered with consistent
%                            (2026a), (2026b), ..., (2026n) suffixes
%                            corresponding to the programme order, so
%                            future citation listings can resolve
%                            uniquely.
%                          - Journal-stripped variant
%                            Arisaka-GG-A6-Gi4-v5.9-journal.tex
%                            produced by removing every \textbf{Relevance.}
%                            paragraph (multi-line, from the marker to the
%                            next blank line).  Both preprint and journal-
%                            stripped variants ship together at this
%                            phase.
%                          - L11/L22 overflow audit: 18 boxed display
%                            equations counted; the two boxed equations in
%                            the Executive Summary are already wrapped in
%                            \resizebox{\linewidth}{!}{...} per L11; all
%                            remaining boxed equations fit naturally in
%                            their host environments per the live compile
%                            log (0 overfull-hbox, 0 float-overflow as of
%                            v5.8).  All wide tables in coherence map and
%                            falsifier table are already longtable per
%                            L22.
%   v5.10 (May 11, 2026) Phase 11: final canonical pass.
%                          - L19 Appendix A: replaced the legacy v3.0
%                            "Notation index" with the canonical
%                            9-section 3-column "Summary of
%                            Abbreviations" template literal-copied
%                            from Arisaka-GG-A1-Constitution-v5.25.tex
%                            (L6717-7016).  Primary Link column adapted
%                            so the Gi-4-specific acronyms (TGG, CCI,
%                            GFF, GRIP, GEP, CEP, DGI, GSSB, GA, SIP)
%                            point to "This paper (A6-Gi4), Part X"
%                            instead of to A1.
%                          - L20 hierarchy figure: the programme-acronym
%                            hierarchy figure is included in the same
%                            block (extracted with the Appendix A from
%                            A1 v5.25; same fig:abbrev-flow label is
%                            reused).
%                          - L21 sans-serif audit: confirmed.  Phase 1
%                            reference-scanner found zero plain-text
%                            programme acronyms in the body; subsequent
%                            phases used the macro form throughout.
%                          - L23 bold-acronym hierarchy: applied in all
%                            Phase 8 figures and in the L20 hierarchy
%                            figure imported from A1.
%                          - 36-item v3.4 audit + L25 length check:
%                            see memory/project_a6_gi4_conversion.md
%                            for the deliverable report.  Page count
%                            calibrated to A6's role as the geometric
%                            half of the GDOS trio foundational
%                            documents.
%                        v5.10 is the final v5.X canonical form of A6.
%                        Both preprint and journal-stripped variants
%                        ship together.
%   v5.11 (May 11, 2026) Post-Phase-11 polish #1: H1 Bridge-to-Part
%                          paragraphs.  Replaced the 5 mechanical
%                          "This concludes Part X" markers with smooth
%                          forward-pointing "Bridge to Part X+1"
%                          paragraphs in Arisaka voice (II->III,
%                          III->IV, IV->V, V->VI, VIII->IX), and added
%                          3 new bridges at Part boundaries that lacked
%                          them entirely (I->II, VI->VII, VII->VIII).
%                          Total 8 bridges added.  Mirrors A1-A5 Phase A
%                          treatment from Thread 3 §6415-6457.
% ======================================================================
%   v5.12 (May 11, 2026) Post-Phase-11 polish #2: H2 Conclusions
%                          philosophical closing.  Removed the mechanical
%                          closing block ("This is the working hypothesis...
%                          We commend it to the scientific community.
%                          This concludes Part IX and the main text...")
%                          and replaced with three Arisaka-voice
%                          philosophical paragraphs: (1) the four pillars
%                          as one geometric object, (2) the bargain with
%                          the laboratory (four named empirical falsifiers
%                          + register-matching close "physics promises to
%                          write its rules down, and the world promises to
%                          falsify them if they ever lie"), (3) the
%                          deeper claim about the laboratory world as
%                          projection of a single geometric structure.
%                          Mirrors A1-A5 treatment from Thread 3 §6459-6498.
% ======================================================================
%   v5.13 (May 11, 2026) Post-Phase-11 polish #3: M1 collapse of
%                          two compact-recapitulation sections.
%                          Collapsed §"Bulk geometry: a compact
%                          recapitulation" (~127 src lines) +
%                          §"The GG-Constitution: a compact
%                          recapitulation" (~91 src lines) into a
%                          single §"Primitive Inputs from the Companion
%                          Papers" with three short subsections (the
%                          constitutional clauses; the 6D bulk
%                          and BI-6 closure; the OGGEP projector).
%                          All 5 externally-referenced labels preserved
%                          (sec:bulk-geometry-recap, sec:gg6-constitution
%                          -recap, eq:bulk-metric, eq:OGN5-recap, eq:OGN5
%                          -identities-recap).  L12 framing paragraph
%                          for U(1)_Qi preserved verbatim.  Mirrors A5
%                          v6.22 §5.1-5.3 pattern; saves ~6 printed pp.
% ======================================================================
% ----------------------------------------------------------------------
% UCLA-Arisaka/GG-2026-4A  v3.0
% The Gi-4 Pillars: TGG, CCI, GFF, GRIP
% A Rigorous Foundation for the Geometric Backbone of GDOS
%
% v3.0 corrects the names and the dual physical/algebraic structure of the
% four pillars to match the original Perplexity definitions:
%   TGG  = Total Geometric-Gauge object
%          (bulk composite connection + 4D boundary stress tensor;
%           master conservation law from BI-6)
%   CCI  = Covariant Conservation Identity
%          (4D OGGEP zero-mode projection of TGG conservation;
%           algebraic foundation: integral [X_4] in H^4(M_6;Z))
%   GFF  = Geometric Free-Energy Functional
%          (F_GFF = F_GEP + F_CEP; thermodynamic + Morse-theoretic)
%   GRIP = Gauge-Runtime Invariance Pipeline
%          (DGI -> GSSB -> GA cascade; gauge-invariant lifecycle
%           with finite-budget closure forcing N<=3)
%
% Author: Katsushi Arisaka (UCLA)
% Date  : May 2, 2026
% ----------------------------------------------------------------------
%% ----------------------------------------------------------------------
%% v10.3 -> v10.4 (June 11, 2026): P2-N3 reciprocal sync (Phase 5 of the P2 project;
%%   author-approved, June-10 protocol: minimal, disclosed, version-bumped).
%%   - Five "[Appendix B]" budget citations -> "[Appendix C]" with verbs
%%   -   corrected ("derived"/"established" -> "fenced as a named conjecture";
%%   -   per-family form C.1 noted). Consistent with rem:T4.1-status / M10.
%%   - Cross-paper table cell fixed: broken sentence removed; H1/H2 overclaim
%%   -   corrected (remain named hypotheses; naive-descent no-go in P2 v2.4).
%%   - P2 bibitem updated to final title + v2.4 stamp.
%%   Reference: [GG-P2-N3]/[P2-Support]/P2-N3-PHASE5-SYNC-REPORT-2026-06-11.md
%% ----------------------------------------------------------------------
%% ---- vv15.31 (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.
%% 3 showcase pages in this paper.  Layout only: not one word,
%% number, figure or table row is changed by this ship.
\documentclass[11pt,letterpaper]{article}
\usepackage{ucla-arisaka-gg-11}
\usepackage{indentfirst}
%% v11.9: \LaTeX shim dropped -- gg-11 Acknowledgments use literal 'LaTeX'.
\setcitestyle{authoryear,round,comma} %% Foundation-block retrofit (May 2026): new citation convention (Arisaka, 2026, A*-Suffix)
\usepackage{amsthm}
\usepackage{pdflscape}
\usepackage{afterpage}

\PaperSeries{GG}
%   v9.7 (June 7, 2026; amended same day): fig:grip-two-regimes promoted
%        to a dedicated full page ([p] float, full text width, small
%        caption) per author request.
%   v9.7 (June 7, 2026): NEW Figure fig:grip-two-regimes in
%        ssec:grip-in-out -- the GRIP cascade and its two regimes in one
%        picture (adapted from A11's four-move pipeline figure, per
%        author request), with the author's GA-tier correction: the
%        descent is captured first into a (possibly way-station) GA,
%        GRIP-In's loop attaches at GA and continues through exit
%        channels, and the GGA is where GRIP-In halts (empty exit set);
%        after the halt only GRIP-Out can act.
%   v9.6 (June 7, 2026): canon sweep (author directive: DGI = "Dynamical
%        Gauge Invariance" everywhere -- the core innovation; "Deepest
%        Geometric Invariance" retired from the def:DGI block, section
%        title, and TGG bridge, with the Morse-top content kept as the
%        gloss).  GGEP/OGGEP word order fixed at four sites
%        ("Gauge-Geometry" -> "Geometry--Gauge", per A1 AX-2/AX-3).
%   v9.5 (June 7, 2026): GRIP-In / GRIP-Out defined at runtime level --
%        the parent paper of the GRIP concept now owns the regime
%        definitions (def:GRIPin, def:GRIPout) in a NEW subsection "The
%        two regimes of the runtime", placed before the weak-interaction
%        subsection.  GRIP-In = the septuple on a FIXED landscape
%        (gradient flow within basins + instanton steps between, halting
%        uniquely at the GGA per GFF-ST7); GRIP-Out = the transient
%        out-of-equilibrium landscape-reorganizing event that supplies
%        each stratum's initial datum c_top (theory delegated to
%        A11-Origin T-GRIP-Out-Necessity).  Slogan: "GRIP-Out opens the
%        landscape; GRIP-In finishes the journey."  Two clarifications
%        recorded: GSSB is a gate, not a regime; port-driven (collider)
%        excitation is not GRIP-Out.  Local edits: septuple remark in
%        def:GRIP; "GRIP-flow" replaced by GRIP-In in the weak
%        subsection (+ regime-language closing clause); piecewise-anatomy
%        remark opening the gradient-flow/instanton section; def:GGA
%        dynamical cross-reference; Kaxion attractor = purest GRIP-In
%        specimen (structurally empty exit-channel set); ES paragraph;
%        keywords; macros \GRIPin/\GRIPout (A11 form).
%   v9.4 (June 6, 2026): NEW subsection "GRIP ending at the GGA, realized
%        in Nature: the weak interaction" -- the weak charged current
%        categorized as the runtime's relaxation channel (GRIP-In on the
%        matter strata): strong/EM build and refine basins; only the weak
%        force moves matter between them; every weak chain ends at the
%        stratum's GGA.  Canonical example: the valley of beta stability
%        as a literal Morse gradient flow; conditional-GGA tracking in
%        neutron stars.  Extensive treatment with examples: A9-SM v9.8.
%   v9.3 (June 6, 2026): GGA (Ground Geometric Attractor) promoted into
%        the runtime vocabulary, executing the programme-wide propagation
%        deferred in the A11-Origin change log.  Additions:
%        (1) \providecommand{\GGA} (A11 form) + \AEleven macro;
%        (2) Definition def:GGA after def:GA, anchored on GFF-ST7 and
%            T-GGA-as-Ground-State of A11;
%        (3) new subsection "GGA and the stability of matter" (electron,
%            proton, Kaxion as ground-state stability);
%        (4) GFF-ST7 vocabulary remark; cascade-terminus refinement note;
%        (5) Executive-Summary GGA paragraph; keywords entry;
%        (6) ArisakaA11Origin added to the bibliography.
%        No physics changed: def:GGA names what GFF-ST7 already proves.
\PaperNumber{Arisaka-GG-A6-GRIP}
\PaperStatus{Preprint}
%% ===================================================================
%%  A6-GRIP v13.14  (July 31, 2026)  -- five front-matter sites the
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%% v15.19 -> v15.20, August 15, 2026.  THE GATE-CLOSING PASS.
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\PaperVersion{v20.0}
\PaperDate{August 21, 2026}

%% Version log:
%%   v1.0 (April 12, 2026) -- Initial draft introducing GRIP, GFF=GEP+CEP,
%%                            SIP, DGI/GSSB/GA cascade.
%%   v2.0 (May 2, 2026)    -- Phase 1: front matter + introduction.
%%                            Reorganises v1.0 around four explicitly named
%%                            Pillars: TGG, CCI, GFF, GRIP.  Adds TGG and CCI
%%                            as new pillars upstream of GFF and GRIP, and
%%                            sets up the formal vocabulary needed by
%%                            A9-SM (28 SM parameters) and GG-2026-
%%                            Generations (N=3 proof).
%%                          -- Phase 2: Part II (TGG: Topological Geometric
%%                            Generator).  Five axioms (TGG-ST1 through
%%                            TGG-ST5), formal definition of the composite
%%                            connection on a principal G_tot-bundle,
%%                            magic-dimension-ladder backbone, Theorem T1
%%                            (existence and uniqueness up to gauge),
%%                            worked examples on flat and warped M_6.
%%                          -- Phase 2b: Part II expansion. Added Chern-Simons
%%                            construction of H_3 from A_tot; detailed
%%                            Spin(1,5) representation theory; quaternionic
%%                            doublet decomposition under Spin(1,3); the
%%                            "three 3's are one 3" identification of family
%%                            triplicity; explicit Lie-algebra commutators,
%%                            Killing form computations, and Wilson-line
%%                            moduli derivation.
%%                          -- Phase 3: Part III (CCI: Composite Connection
%%                            Identity).  Five axioms (CCI-ST1 through
%%                            CCI-ST5), formal definition of CCI as
%%                            integrality of [X_4] in H^4(M_6;Z), derivation
%%                            of -2/5 coefficients from H^2 normalisation,
%%                            CCI-vs-BI-6 distinction (algebraic vs
%%                            differential), Theorem T2 (CCI closure under
%%                            MSMF forces unique OGN-5 ledger (1,3,5;7,35)),
%%                            connection to Hopkins-Singer differential
%%                            cohomology, worked examples.
%%                          -- Phase 4: Part IV (GFF: Geometric Free
%%                            Functional).  Five axioms (GFF-ST1 through
%%                            GFF-ST5), formal definition of GFF as
%%                            F_GEP + F_CEP on stratified boundary moduli,
%%                            Morse structure on each stratum, Theorem T3
%%                            (Morse-positivity from CCI via Hopkins-Singer
%%                            descent), promotion of N=3 proof's hypotheses
%%                            H_1 and H_2 to full theorems, worked examples
%%                            on CP^{N-1} family-projector moduli and other
%%                            strata.
%%                          -- Phase 5: Part V (GRIP: Geometric Reduction
%%                            by Iterative Projection).  Five axioms
%%                            (GRIP-ST1 through GRIP-ST5), formal definition
%%                            of the cascade DGI -> GSSB -> GA, gradient
%%                            flow dynamics, instanton actions S=pi*k for
%%                            adjacent saddle decay, BI-6 budget
%%                            S_BI6^max = 3*pi from CCI, Theorem T4
%%                            (cascade closure forces N <= 3 with explicit
%%                            quadratic-action bound), generation decay-width ladder
%%                            connection, worked examples on CP^2 (N=3),
%%                            Higgs moduli, gauge-orbit moduli.
%%                          -- Phase 6: Parts VI (Synthesis with Theorem T5
%%                            constitutional equivalence), VII (Applications:
%%                            N=3 in Gi-4 vocabulary, 28 SM parameters,
%%                            hierarchy/KK/DM), VIII (Conclusion with
%%                            extensive philosophical close in 8 subsections).
%%                          -- Phase 7: Six Appendices A-F (differential
%%                            cohomology preliminaries, Hopkins-Singer
%%                            machinery, Morse theory on stratified spaces,
%%                            detailed worked examples, notation index,
%%                            theorem and definition index).
%%                          -- Phase 8: Full annotated bibliography with
%%                            47 entries and relevance paragraphs in the
%%                            standard GG-Theory house style.
%%                          -- Phase 9: Final consistency pass.  All internal
%%                            references resolved; the dangling
%%                            thm:GFF-existence-Morse forward-reference
%%                            replaced with citation to Definition def:GFF
%%                            and the five GFF axioms.  Final v2.0 release.
%%   v3.0 (May 2, 2026)    -- MAJOR CORRECTION of the four pillar names and
%%                            integrated definitions to match the original
%%                            Perplexity definitions:
%%                              TGG = "Total Geometric-Gauge object" (not
%%                                "Topological Geometric Generator").  Pillar 1
%%                                now treats the bulk composite connection AND
%%                                the 4D boundary stress tensor T^{mu nu}_{TGG}
%%                                as two layers of one object, with the Master
%%                                Conservation Theorem nabla_mu T^{mu nu} = 0
%%                                from BI-6 + Ward as the central theorem.
%%                              CCI = "Covariant Conservation Identity" (not
%%                                "Composite Connection Identity").  Pillar 2
%%                                now treats the 4D OGGEP-projection conservation
%%                                equation as the physical content, with X_4
%%                                integrality as the algebraic foundation
%%                                connecting via Hopkins-Singer descent.  The
%%                                modified TOV equation is the astrophysical
%%                                falsifier.
%%                              GFF = "Geometric Free-Energy Functional" (not
%%                                "Geometric Free Functional").  Pillar 3 now
%%                                treats GFF as the boundary free energy with
%%                                BOTH thermodynamic (monotone partial_t F<=0)
%%                                AND Morse-theoretic (perfect Morse function)
%%                                characterisations.
%%                              GRIP = "Gauge-Runtime Invariance Pipeline" (not
%%                                "Geometric Reduction by Iterative Projection").
%%                                Pillar 4 now treats GRIP as the gauge-invariant
%%                                lifecycle DGI -> GSSB -> GA combining
%%                                gauge-invariance preservation with
%%                                finite-budget closure that forces N <= 3.
%%                            All v2.0 mathematical content (T1-T5 framework,
%%                            Hopkins-Singer descent, Morse positivity, cascade
%%                            closure) is preserved and reattached to the
%%                            correctly named pillars.

\hypersetup{
 pdftitle = {The GRIP on GDOS -- Gi-4 Pillars (TGG, CCI, GFF, and GRIP) as the Geometric Backbone of GDOS},
 pdfauthor = {Katsushi Arisaka},
 pdfsubject = {Formal axiomatization of the four Geometric Invariance pillars
 of \GDOS{}: the Total Geometric-Gauge object (bulk composite
 connection + 4D boundary stress tensor with master
 conservation), the Covariant Conservation Identity
 (4D OGGEP-projected equation with astrophysical falsifier),
 the Geometric Free-Energy Functional (thermodynamic +
 Morse-theoretic), and the Gauge-Runtime Invariance Pipeline
 (\DGI{} -> \GSSB{} -> GA cascade with finite-budget closure).},
 pdfkeywords= {\GGTheory{}, \GDOS{}, \GiFour{} Pillars, \TGG{}, \CCI{}, \GFF{}, \GRIP{}, \GEP{}, \CEP{},
 \DGI{}, \GSSB{}, GA, \OGGEP{}, \MSMF{}, \BIsix{}, \OGNfive{}, Spin(1,5), composite
 connection, anomaly polynomial, Hopkins-Singer differential
 cohomology, Morse theory on stratified spaces, boundary
 projector, bulk-to-boundary descent, family-projector moduli,
 Wilson-Hosotani, geometric attractor, cascade closure}
}

%% ----------------------------------------------------------------------
%% Paper-specific macros
%% ----------------------------------------------------------------------
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\providecommand{\PBIfour}{\ensuremath{\mathrm{PBI\text{-}4}}}
\providecommand{\GDOS}{\ensuremath{\mathrm{GDOS}}}
\providecommand{\MSMF}{\ensuremath{\mathrm{MSMF}}}

%% Pillar abbreviations
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\providecommand{\CCI}{\ensuremath{\mathrm{CCI}}}
\providecommand{\GFF}{\ensuremath{\mathrm{GFF}}}
\providecommand{\GRIP}{\ensuremath{\mathrm{GRIP}}}
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\providecommand{\CEP}{\ensuremath{\mathrm{CEP}}}
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\providecommand{\GA}{\ensuremath{\mathrm{GA}}}

%% Group abbreviations
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\providecommand{\gtot}{\ensuremath{\mathfrak{g}_{\mathrm{tot}}}}
\providecommand{\Atot}{\ensuremath{\mathcal A_{\mathrm{tot}}}}
\providecommand{\Rtot}{\ensuremath{\mathcal R_{\mathrm{tot}}}}

%% Number fields
\providecommand{\R}{\ensuremath{\mathbb{R}}}
\providecommand{\C}{\ensuremath{\mathbb{C}}}
\providecommand{\HH}{\ensuremath{\mathbb{H}}}
\providecommand{\Quaternion}{\ensuremath{\mathbb{H}}}
\providecommand{\Octonion}{\ensuremath{\mathbb{O}}}
\providecommand{\Z}{\ensuremath{\mathbb{Z}}}
\providecommand{\CPN}{\ensuremath{\mathbb{CP}^{N-1}}}
\providecommand{\CPtwo}{\ensuremath{\mathbb{CP}^{2}}}

%% Common math operators
\providecommand{\Hess}{\ensuremath{\mathrm{Hess}}}
\providecommand{\fGFF}{\ensuremath{f_{\mathrm{GFF}}}}
\providecommand{\Stot}{\ensuremath{S_{\mathrm{tot}}}}
\providecommand{\ind}{\ensuremath{\mathrm{ind}}}
\providecommand{\Crit}{\ensuremath{\mathrm{Crit}}}
\providecommand{\sgn}{\ensuremath{\mathrm{sgn}}}

%% Energy scales
\providecommand{\EGUT}{\ensuremath{E_{\mathrm{GUT}}}}
\providecommand{\EPl}{\ensuremath{E_{\mathrm{Pl}}}}
\providecommand{\Mlock}{\ensuremath{M_{\mathrm{lock}}}}
\providecommand{\TeV}{\ensuremath{\mathrm{TeV}}}
\providecommand{\Sbi}{\ensuremath{S_{\mathrm{BI6}}^{\max}}}

%% Bulk geometry
\providecommand{\Msix}{\ensuremath{\mathcal M_6}}
\providecommand{\Mfour}{\ensuremath{\mathcal M_4}}
\renewcommand{\Bdy}{\ensuremath{\mathcal M_{\mathrm{bdy}}}}
\providecommand{\Pbun}{\ensuremath{P}}

%% Calligraphic
\providecommand{\cA}{\ensuremath{\mathcal A}}
\providecommand{\cF}{\ensuremath{\mathcal F}}
\providecommand{\cG}{\ensuremath{\mathcal G}}
\providecommand{\cH}{\ensuremath{\mathcal H}}
\providecommand{\cM}{\ensuremath{\mathcal M}}
\providecommand{\cP}{\ensuremath{\mathcal P}}
\providecommand{\cR}{\ensuremath{\mathcal R}}
\providecommand{\cS}{\ensuremath{\mathcal S}}

%% Differentials
\providecommand{\dd}{\ensuremath{\mathrm{d}}}
\providecommand{\e}{\ensuremath{\mathrm{e}}}

%% Pillar references (used throughout)
\providecommand{\Ptgg}[1]{(\TGG{}-\textbf{P\arabic*})}
\providecommand{\MEPP}{\ifmmode\mathsf{MEPP}\else\textsf{MEPP}\fi}

\setlist[itemize]{topsep=3pt,itemsep=2pt}
\setlist[enumerate]{topsep=3pt,itemsep=2pt}
\captionsetup{font=small,labelfont=bf}


%% --- Sans-serif paper-code macros (programme-wide v7.1 sweep, May 13, 2026)
%% Programme-defined names rendered in sans-serif (per L21/L23) and with
%% full GG- prefix throughout the body.  The \AFull... variants prepend
%% Arisaka- and are used for explicit listings of preprint titles.
\providecommand{\AOne}{\textsf{GG-A1-Constitution}}
\providecommand{\ATwo}{\textsf{GG-A2-GG6}}
\providecommand{\AThree}{\textsf{GG-A3-BI6}}
\providecommand{\AFour}{\textsf{GG-A4-Euclid}}
\providecommand{\AFive}{\textsf{GG-A5-GDOS}}
\renewcommand{\ASix}{\textsf{GG-A6-GRIP}}% v13.2: \renewcommand (the sty predefines \ASix as the retired GG-A6-Gi4; \providecommand was a no-op)
\providecommand{\ASeven}{\textsf{GG-A7-Qi4}}
\providecommand{\AEleven}{\textsf{GG-A11-Origin}}
%% ---- v15.6: support for the two A11-Origin cascade figures carried
%% ---- verbatim in Part VII.  All three are additive.
\usetikzlibrary{decorations.pathreplacing}   % brace decoration, A11 Fig. 1
\providecommand{\QCI}{\ifmmode\mathsf{QCI}\else\textsf{QCI}\fi}
\definecolor{ggcascadeclose}{HTML}{3E8B57}   % the closing loop, A11 Fig. 2
\providecommand{\GGA}{\ifmmode\mathsf{GGA}\else\textsf{GGA}\fi}
\providecommand{\GRIPin}{\ifmmode\mathsf{GRIP\text{-}In}\else\textsf{GRIP-In}\fi}
\providecommand{\GRIPout}{\ifmmode\mathsf{GRIP\text{-}Out}\else\textsf{GRIP-Out}\fi}
\providecommand{\AEight}{\textsf{GG-A8-QM}}
\providecommand{\ANine}{\textsf{GG-A9-SM}}
\providecommand{\ATen}{\textsf{GG-A10-Cosmos}}
\providecommand{\PaperPOne}{\textsf{GG-P1-KK}}
\providecommand{\PaperPTwo}{\textsf{GG-P2-N3}}
\providecommand{\PaperCOne}{\textsf{GG-C1-Kaxion}}
\providecommand{\PaperPFour}{\textsf{GG-P4-Real}}%% v13.0: Foundations Branch member (canonical form, Cohort-Map-19 v1.4)
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\providecommand{\AFullTwo}{\textsf{Arisaka-GG-A2-GG6}}
\providecommand{\AFullThree}{\textsf{Arisaka-GG-A3-BI6}}
\providecommand{\AFullFour}{\textsf{Arisaka-GG-A4-Euclid}}
\providecommand{\AFullFive}{\textsf{Arisaka-GG-A5-GDOS}}
\renewcommand{\AFullSix}{\textsf{Arisaka-GG-A6-GRIP}}% v13.2: \renewcommand (sty predefines the retired full form)
\providecommand{\AFullSeven}{\textsf{Arisaka-GG-A7-Qi4}}
\providecommand{\AFullEight}{\textsf{Arisaka-GG-A8-QM}}
\providecommand{\AFullNine}{\textsf{Arisaka-GG-A9-SM}}
\providecommand{\AFullTen}{\textsf{Arisaka-GG-A10-Cosmos}}
\providecommand{\PaperFullPOne}{\textsf{Arisaka-GG-P1-KK}}
\providecommand{\PaperFullPTwo}{\textsf{Arisaka-GG-P2-N3}}
\providecommand{\PaperFullCOne}{\textsf{Arisaka-GG-C1-Kaxion}}

%% ---- v13.0: one-line Part entries (cohort standard; P4-Real v3.2
%%      preamble pattern).  \GGPart renewed verbatim from
%%      ucla-arisaka-gg-11.sty except that the contents line now
%%      carries only "Part N. <title>"; the " --- <description>"
%%      tail is no longer written to the .toc, so every Part entry
%%      fits on one line in both the Overview of Contents and the
%%      Contents.  Part title pages are unchanged (full title +
%%      description).  \GGAppendixPart (Part X) inherits the renewal
%%      via the sty's pass-through.
\makeatletter
\renewcommand{\GGPart}[3][]{%
  \clearpage
  \refstepcounter{part}%
  \def\gg@parthdr{#1}\ifx\gg@parthdr\@empty\def\gg@parthdr{#2}\fi
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  \markboth{Part \thepart.\ \gg@parthdr}{Part \thepart.\ \gg@parthdr}% running pointer
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  \begin{center}
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  \vspace{0.8em}
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\begin{document}
%%%% ---- v15.1 (August 8, 2026) ----
%%%% Stage 2 of the v15 migration: the float standard.
%%%% Twelve short captions raised to paragraph captions that say what the
%%%% table shows and how it is arranged.  Two are shared items and take
%%%% A5-GDOS's wording verbatim so the cohort reads identically, and the
%%%% shared acronym plate goes to the frozen 0.824 of the text width.
%%%% The one bare grid in the paper -- the GRIP stage/custody ledger, set
%%%% in a \begin{center} with no float, no caption and no label -- becomes
%%%% a numbered table float that can be referenced.
%% ---- v15.0 (August 8, 2026) ----
%% Stage 1 of the v15 migration: mechanical only, no prose rewritten.
%% US Letter; \GGTitlePage's third argument emptied (the v15 style file
%% accepts but no longer typesets it, and the same fact is stated twice
%% in the body); booktabs everywhere -- 12 column specs lose their
%% vertical rules and the one \hline table is converted, A6 having been
%% the only paper in the cohort with any; the two landscape tables go
%% with them on the author's decision, with \tabcolsep 3pt -> 5pt so
%% six prose columns stay separated without rules; 2 captions moved
%% above their grid; 524 spellings taken to the cohort's American
%% standard (A6's cited titles carry none, so nothing needed
%% protecting); and one new overfull box, created by the reflow onto a
%% wider measure, closed by splitting an unbreakable inline-math run.
% ---- v13.15 (August 2, 2026) ----
% New subsection 'The two regimes and the two fibres: the
% engine's strokes' after the GRIP-In/GRIP-Out definitions:
% GRIP-In as the circle's stroke (fixed landscape, closed-loop
% relaxation, deposits in holonomy-protected labels), GRIP-Out as
% the interval's stroke (one-way landscape reorganisation, the
% semigroup at runtime level); Remark GRIPout-necessity given its
% topological proof sketch (only-In cycles forever, only-Out
% converges never); the three determinism mechanisms located in
% A6-owned structures; downstream payoffs pointed to their homes
% (A11 chronology, P5 time, A8 measurement). Part of the A5/A6/A7
% trio sweep.


%% ----------------------------------------------------------------------
%% Front matter
%% ----------------------------------------------------------------------

\GGTitlePage
{The \GRIP{} on \GDOS{}}
{\GRIP{}: the Geometric Engine of Symmetry Breaking, \\
from the Big Bang to Mind}
{}
\thispagestyle{empty}

\vspace{6pt}
\begin{abstract}
\noindent
Nothing in the universe stayed symmetric. It began hot and nearly
featureless, and everything in it now --- the particles, the elements,
the molecules, the living cell, the mind reading this sentence --- is
what was left standing after one symmetry broke, and then another, and
then another.

This paper proposes that a single mechanism drives all of it, and
writes that mechanism down. We call it \GRIP{}, the Gauge-Runtime
Invariance Pipeline: a four-move cascade in which a system first holds
every gauge equivalence open and commits to none (\DGI{}); a geometric
free-energy landscape (\GFF{}) then fixes which way is downhill; the
descent selects a branch at a saddle and the symmetry breaks
(\GSSB{}); and the system is captured in a basin --- a Geometric
Attractor (\GA{}), or, when the basin is the deepest one available, a
\emph{Ground} Geometric Attractor (\GGA{}), which is why matter is
stable rather than merely long-lived.

The pipeline runs in exactly two regimes, and the distinction is the
paper's second claim. \GRIPin{} is descent on a landscape that does
not move, and it halts. \GRIPout{} is the transient event that
reorganizes the landscape itself and hands the next descent its
starting point. \GRIPout{} opens the journey; \GRIPin{} finishes it.
The weak interaction is Nature's implementation of \GRIPin{} on the
strata of matter; the electroweak transition was a \GRIPout{}.

We give the geometry that makes the pipeline exact --- four Geometric
Invariance Pillars of the open-system boundary law, of which \GRIP{}
is the fourth and the other three are its machinery --- and with it the
family count $N \leq 3$ follows as a theorem rather than as an
observation. Every prediction is a closed-form function of five
integers with no free parameter left to tune, and each pillar is
handed an experiment that can end it.
\end{abstract}

\begin{ExecutiveSummary}
\paragraph{Three moves, and this paper is the third.}
The \GGTheory{} program rests on three ideas, and only three. The
first is a six-dimensional geometry in which the four dimensions we
inhabit are a boundary and the two extra ones are not further
directions in space but a scale and a phase
\citep{ArisakaA2GG6} --- a geometry that proves anomaly-free
\citep{ArisakaA3BI6} and leaves no continuous parameter free to
adjust \citep{ArisakaA4Euclid}, on a constitutional charter of four
axioms and seven postulates \citep{ArisakaA1Constitution}. The second
is that no system in nature is closed: the bulk is projecting onto the
boundary at every instant, and what we call the laws of physics is
what that projection looks like from inside
\citep{ArisakaA5GDOS}. The third is the subject of this paper. A
geometry says what \emph{can} exist and an open-system law says how it
is \emph{seen}; neither says why the universe, having begun almost
featureless, did not stay that way. Something has to break the
symmetry, choose among the pieces, and stop. That something is what we
name and write down here.

\paragraph{The pipeline, in one line.}
We call it \GRIP{}, the Gauge-Runtime Invariance Pipeline, and it has
four moves:
\[
  \DGI{} \;\longrightarrow\; (\GFF{}) \;\longrightarrow\; \GSSB{}
  \;\longrightarrow\; \GA{}\,/\,\GGA{} .
\]
Read it left to right. \DGI{}, Dynamical Gauge Invariance, is the
symmetric top of the descent: every gauge equivalence is still open
and nothing has been committed. \GFF{}, the Geometric Free-Energy
Functional, is the landscape --- written in parentheses because it is
not a move but the terrain the moves happen on, the thing that says
which way is downhill and how steep. \GSSB{}, Geometric Spontaneous
Symmetry Breaking, is the moment of choice: the descent reaches a
saddle, one branch is taken, the others are not, and a symmetry that
was real a moment ago is gone. \GA{}, the Geometric Attractor, is
where the system is captured. That is the whole engine. Everything
else in this paper exists to make those four moves exact enough to be
wrong.

\paragraph{Two regimes: one stroke opens, the other closes.}
The engine runs in exactly two ways, and the difference is a single
question --- \emph{does the landscape itself change?} \GRIPin{} is
descent on a landscape that stays put: the system rolls into a basin,
tunnels between basins, may rest a long while at a way station, and
halts for good when it reaches a basin with no exit. \GRIPout{} is the
opposite: a transient, out-of-equilibrium event that reorganizes the
landscape and hands the next descent a new starting point. In one
sentence: \GRIPout{} opens the journey, \GRIPin{} finishes it. They
are not competing accounts of one thing but alternating strokes of one
engine, and the alternation is what lets a universe go on producing
new structure instead of settling once and stopping. The weak
interaction is Nature's implementation of \GRIPin{} on the strata of
matter; the electroweak transition was a \GRIPout{}. What is
invariant under a descent and what is destroyed by it are questions of
symmetry and of holonomy, and both are treated at cohort level
(\citealp{ArisakaW1Symmetry}; \citealp{ArisakaW2Holonomy}); that a
descent runs one way at all is the same arrow the mirrors of matter
carry \citep{ArisakaP5CPT}.

\paragraph{Why matter is stable.}
The terminus deserves a sharper name than ``attractor,'' because not
all basins are alike. A \GA{} is any stable basin; a \GGA{}, a
\emph{Ground} Geometric Attractor, is the deepest one on its stratum,
with nothing below it to fall to. That distinction is not bookkeeping --- it is
the reason matter persists. The electron is stable because it is the
ground attractor of its stratum \citep{ArisakaP3Electron}. So is the
proton, whose refusal to decay is then a fact about geometry rather
than an unexplained bound \citep{ArisakaP6Proton}. So, if the
framework is right, is the dark-matter candidate this program predicts
\citep{ArisakaC1Kaxion}. It is the same reason a water molecule is not
a temporary arrangement of atoms that will eventually find something
better to be: it is already at the bottom. A universe whose descents
merely slowed down would have no permanent objects in it, and nothing
to build with.

\paragraph{From the first microsecond to the mind.}
The claim this paper is written to support is that the four moves are
not a mechanism for one epoch but the mechanism, running over and over
at every scale. The grand-unified transition in the first instant is
one turn of the crank, and the resonance it leaves near the lock scale
is a standing test of it \citep{ArisakaP1KK}. So is the electroweak
transition, which gives the particles their masses and fixes the
Standard Model's parameters \citep{ArisakaA9SM}; so is the settling of
quarks into protons and neutrons, and of nucleons into helium, carbon,
oxygen, iron; so is the binding of atoms into water, carbon dioxide,
ammonia; of small molecules into amino acids and peptides; of those
into the molecular machinery of the living cell --- \textsf{ATP},
\textsf{RNA}, \textsf{DNA}; and, the program contends, of cells into
organisms, of organisms into animals, and of animals into the kind
that can ask why any of it happened. The same descent read at
cosmological scale gives the boundary cosmology
\citep{ArisakaA10Cosmos}, the two ends of the expansion corridor
\citep{ArisakaC2Hubble}, and the horizon's entropy
\citep{ArisakaC3BH}; read all the way up, it is the capstone's subject
\citep{ArisakaA11Origin}. It is worth being exact about what that
sentence is. The pipeline, its two regimes and its halting theorem are
proved here. Their deployment through the full ladder of cascade
layers is carried by the capstone, and the identification of the
higher layers --- chemical, biochemical, biological, cognitive ---
with the same four moves is the cohort's central wager rather than a
theorem of this paper. What this paper contributes is the thing that
wager needs: a pipeline stated sharply enough that the identification
could be shown to fail.

\paragraph{The geometry underneath, briefly.}
Making the pipeline exact takes machinery, and the machinery is four
Geometric Invariance Pillars of which \GRIP{} is the fourth. The first
gathers gravity and the three gauge forces into a single composite
connection on the six-dimensional bulk, so that there is one object to
descend rather than four. The second is the conservation ledger that
object must satisfy, and its arithmetic is unexpectedly rigid: closure
forces a unique set of five integers, $(1,3,5;7,35)$, and everything
downstream is a closed-form function of them. The third is the
free-energy landscape itself, together with the proof that descent on
it runs one way --- which is where the arrow of time enters the
runtime, and where the rules of quantum mechanics stop being a list to
assume and become consequences \citep{ArisakaA8QM}. The fourth is the
pipeline. That the boundary's single imaginary unit is itself derived
rather than posited \citep{ArisakaP4Real} is what lets the descent be
written on real geometry throughout. The four pillars are not four
constructions but four faces of one geometric object, and their mutual
forcing is what makes the family count come out at $N \leq 3$ instead
of being read off the data.

\paragraph{What is established here, and what would refute it.}
The boundary of a claim matters more than the claim. Derived in this
paper: the pipeline and its four moves, the two regimes and their
necessity, the halting theorem at the ground attractor, the
conservation ledger, the one-way descent, and the family bound
$N \leq 3$ --- the last conditional on the capacity theorem of the
three-generations companion \citep{ArisakaP2N3}, whose integrality
residue is discharged on a benchmark background with two refinements
named and left open. Taken
as given and not re-derived here: the constitutional charter, the
six-dimensional geometry, the anomaly-freedom proof and the
open-system boundary law, each the subject of a companion preprint and
each cited at the step that uses it. Against that, three experiments
are named with the values they must return: a modified equation of
stellar structure in compact stars, equipartition together with the
fluctuation--dissipation relation in any boundary observable, and a
generation-ordered ladder of decay widths falling by a fixed
exponential factor from one family to the next. Any one of them can
end the program, and the geometry leaves nowhere to retreat: the
dimension is fixed, the integers are fixed, and there is no adjustable
landscape to hide in.

\paragraph{A closing thought.}
Physics has been very good at describing what is stable and much less
comfortable with how anything came to be stable in the first place. The
symmetric state is always the easy one to write down; the world we
have is the one that stopped being symmetric, repeatedly, in a
particular order, and produced observers late in the sequence. If the
four moves proposed here are right, then that sequence is not a
history of accidents but the same operation applied again and again to
its own output --- and the cosmos, life, and mind are not three
separate puzzles but three depths of one descent. That is a large
claim, and it is offered in the only form that makes a large claim
respectable: as something that can be measured, and found wanting.

\end{ExecutiveSummary}

\GGKeywords{GG-Theory; GDOS; Gi-4 Pillars; TGG (Total Geometric-Gauge object);
\CCI{} (Covariant Conservation Identity); \GFF{} (Geometric Free-Energy
Functional); \GRIP{} (Gauge-Runtime Invariance Pipeline); \GEP{}
(Geometry-Gauge Energy Principle); \CEP{} (Curvature-Entropy Principle);
\DGI{} (Dynamical Gauge Invariance); \GSSB{} (Geometric Spontaneous Symmetry
Breaking); GA (Geometric Attractor); \GGA{} (Ground Geometric Attractor);
\GRIPin{} (fixed-landscape regime); \GRIPout{} (landscape-reorganizing
regime); \OGGEP{}; \MSMF{}; \BIsix{}; \OGNfive{}; Spin(1,5);
SL(2,H); composite connection; boundary stress tensor; master conservation
law; modified Tolman-Oppenheimer-Volkoff equation; anomaly polynomial;
Hopkins--Singer differential cohomology; Morse theory on stratified spaces;
boundary projector; bulk-to-boundary descent; family-projector moduli;
Wilson--Hosotani~\citep{Hosotani2005}; instanton-action budget; magic-dimension ladder;
foundational open-system framework}

%% ============ FRONT-MATTER SHOWCASE, v15.23 =================
%% Three pages after the Keywords, before the Overview of Contents.
%%   p1  Figure 2, the four-pillar architecture -- what this is
%%   p2  Figure 8, the GRIP cascade, two regimes -- the title pillar
%%   p3  Table 20, the per-pillar falsifiers    -- what would kill it
%% Figures 1 and 3 are the two plates every paper carries and A5
%% opened with the first of them; A6 opens with its own architecture.
%% Table 18 is the cohort's ten falsifiers and is A1's showcase page
%% 3; Table 20 is A6's OWN exposure and is the right one here.
%% THE PLATES KEEP THEIR IN-PAPER SIZE.  A5's showcase copy was 4%
%% wider than its body copy, and since a savebox cannot differ
%% between its uses, that widening pushed two shared rules over the
%% 0.956 pt house ceiling and had to be paid for by changing the BODY
%% plate.  Same factor here -- \textwidth and 0.833\textwidth --
%% so no rendered weight can move.
%% NOTHING IS COPIED: both TikZ bodies and the falsifier tabular
%% live in saveboxes filled once here; every use site says \usebox.
%% Captions are unnumbered (\captionof*): no number, no list entry,
%% and no step of the counter.
%% ===========================================================
%% v11.5 (August 17, 2026): THE DEFINITION IS IN THE STYLE FILE NOW.
%% The comment above is the record of why it was here -- the style
%% file is shared, and changing it meant rebuilding every arXiv
%% package.  That was done, so what is left is what this paper sets.
\renewcommand{\GGShowcaseHeadSkip}{}
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% Middle: Four pillars
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\node[pillar] (GFF) at (10, 0) {\textbf{\GFF{}}\\ \footnotesize Geom. Free Energy\\ T3.1, T3.2};
\node[pillar] (GRIP) at (15, 0) {\textbf{\GRIP{}}\\ \footnotesize Gauge-Runtime\\ T4.1, T4.2};

% Implication chain
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% Bottom: Bulk source
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 \footnotesize $\Msix = M_4 \times I_r \times S^1_\theta$, $\Gtot = \Spinone \times \SUfive \times \UoneB \times \UoneQi$, \OGNfive{} ledger $(1,3,5;7,35)$};

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}

% Compatibility-relation labels above pillars
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 {\scriptsize transient out-of-equilibrium event}};
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\begin{tabular}{p{2.44cm}p{3.66cm}p{4.37cm}p{4.37cm}}
\toprule
\textbf{Pillar / ST} & \textbf{Prediction} & \textbf{Empirical observable} & \textbf{Status / facility}\\
\midrule
\TGG{}-ST6, ST7 & Master Conservation Law $\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ & Tests of energy/momentum conservation in any boundary process & Confirmed to $10^{-15}$ in low-energy experiments\\
\CCI{}-ST7 & Modified TOV $\Delta_{\rm GS}\sim 9.6\times 10^{-3}$ & $0.1\%$--$1\%$ shift of NS mass--radius curve from standard TOV & NICER (current); next-gen X-ray timing; LIGO/Virgo NS inspirals\\
\GFF{}-ST7 (1) & Equipartition $\langle\delta X^2\rangle = T_{\GFF}\chi$ for any boundary observable & Universal $T$-scaling of fluctuations near equilibrium & Ultra-cold atoms, optomechanics; quantum-limited sensors\\
\GFF{}-ST7 (2) & Fluctuation--dissipation: $\langle\delta F^2\rangle = T_{\GFF}\chi_{\GFF}$ & Einstein-type relation between mobility and diffusion & Mesoscopic transport experiments\\
\GFF{}-ST7 (3) & Onsager reciprocity $L_{ij} = L_{ji}$ & Symmetric cross-coupling matrix in any open boundary network & Molecular motors, active matter, biophysical networks\\
\GRIP{}-ST7 & Generation decay-width ladder $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ & Geometric gate factor along the family chains $\tau\to\mu\to e$, $t\to\cdots\to u/d$ (absolute lifetimes via A9) & Generation-ordering of the gate-action ladder; controlled-kinematics decay class\\
\GRIP{}-ST7 & Cascade-action sum rule & $\sum_i\log(\tau_i/\tau_0) = \pi N(N-1)(N+1)/6$ & Population studies of unstable SM particles\\
\BIsix{} (Companion) & KK resonance cluster $\Mlock\simeq 3.85\,\TeV$ & Tower of KK modes near 3.85 TeV in dijet/diphoton/dilepton & LHC Run 3, HL-LHC; FCC-hh\\
\BIsix{} (Companion) & Kaxion dark matter $m_K\approx 8.1~\mu$eV & Ultra-light bosonic DM near $\mu$eV with axion-like couplings & ABRACADABRA, DMRadio, ADMX-EFR; cosmological constraints\\
GG-Cosmos & Modified Friedmann equations & Coherent shift of $f\sigma_8(z)$, $w(z)+1\propto\langle X_4\rangle/M_{\rm KK}^4$ & DESI, Euclid, Roman Space Telescope; CMB-S4\\
GG-6F (codebook) & 28 SM parameters at $\sim 0.5$\% precision & e.g.\ $m_\tau/m_\mu$ to 0.5\% from cascade depths & Global SM precision fits (Particle Data Group~\citep{ParticleDataGroup2024})\\
\bottomrule
\end{tabular}}
\clearpage
%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{Four Geometric Pillars,\\[2pt]
and the One Law They Hold Up}
\vspace{1.4em}
\par\vspace*{\stretch{2}}
\resizebox{\textwidth}{!}{\usebox{\GGscPillars}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{What this paper is.} \GDOS{} is the boundary law; the four \GiFour{} pillars are what forces it to be that law and not another. Each pillar inherits its constitutional content from \GDOS{} above and its bulk-side content from the 6D geometry below, and the horizontal arrows are implications, not analogies: \TGG{} $\Rightarrow$ \CCI{} $\Rightarrow$ \GFF{} $\Rightarrow$ \GRIP{}. The four named compatibility relations tie each pillar's algebraic layer to its operational one. Read this page and the shape of the argument is fixed; the twelve Parts are the work of earning it. In full: Figure~\ref{fig:gi4-architecture}, p.~\pageref{fig:gi4-architecture}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{How Structure Settles:\\[2pt]
One Cascade, and the Two Regimes It Runs In}
\vspace{1.0em}
\par\vspace*{\stretch{2}}
\resizebox{0.833\textwidth}{!}{\usebox{\GGscCascade}}
\par\medskip
\par\vspace*{\stretch{2}}
{\small\captionof*{figure}{\textbf{The pillar the paper is named for.} The central column is the runtime: from a fully symmetric top where no gauge equivalence has been spent, the system descends the fixed landscape, branches at the symmetry-breaking saddles, and is captured into a stable basin. The two-regime story is the point. At a way station the basin still has exit channels and the descent continues --- that is \GRIPin{}, and the weak interaction is Nature's implementer of it. At the deepest basin the exit set is empty, \GRIPin{} halts, and only \GRIPout{} can act: a transient out-of-equilibrium event that reorganizes the landscape itself and supplies the initial datum of the next descent. \GRIPout{} opens the landscape; \GRIPin{} finishes the journey. In full: Figure~\ref{fig:grip-two-regimes}, p.~\pageref{fig:grip-two-regimes}.}}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage

%% Self-filling showcase frame (Report 462): the glue is already
%% page-level here, so only the weights change -- 1 : 2 : 2 : 1.
\null\vspace*{\stretch{1}}
\begingroup\centering
\GGShowcaseHead{Every Prediction the Four Pillars Make\\[2pt]
That Could Kill the Framework}
\vspace{1.0em}
\par\vspace*{\stretch{2}}
{\small\captionof*{table}{\textbf{Where the four pillars are exposed.} One row per pillar sub-theorem, each naming the prediction, the observable that would falsify it, and where the measurement stands today. The fourth column is the honest one: some rows are already confirmed to high precision and are no longer at risk, and the rest name the facility that will decide them. A framework that forces its parameters has to accept that a single failed row ends it. These are the pillars' own falsifiers, not the cohort's ten --- for those see \AOne{}. In full: Table~\ref{tab:falsifiers}, p.~\pageref{tab:falsifiers}.}}
\par\medskip
\par\vspace*{\stretch{2}}
\usebox{\GGscFalsifiers}
\par\vspace*{\stretch{1}}
\endgroup
\clearpage
% ============ end front-matter showcase ====================

\GGfrontmatteronepagetrue
%% v11.9: \GGOverviewOfContents (centered) provided by ucla-arisaka-gg-11;
%% single-page Overview+List-of-Figures via the gg-11 toggle above.

%% v11.9: \GGListOfFigures (centered) provided by ucla-arisaka-gg-11.

\GGOverviewOfContents
\begingroup\setstretch{0.94}
\GGListOfFigures
\endgroup
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\begingroup
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\renewcommand{\baselinestretch}{0.95}
\GGTableOfContents
\endgroup

%% ===============================================================
%%  PREFACE -- compelling entrance for the entire paper
%%  Added in v5.16 (May 2026) per user request: a global
%%  introduction that captures the reader's mind before Part I.
%%  Four paragraphs in the Arisaka register, plain English, no
%%  paper-specific jargon (no TGG/CCI/GFF/GRIP names).
%% ===============================================================
\GGPreface

Every working physicist learns four habits very early. The first
habit, from Maxwell in 1865, is that a current is the thing that
flows out of a source and into the rest of the world. The second,
from Emmy Noether in 1918, is that every symmetry of the laws of
nature corresponds to a quantity that is conserved. The third,
from Hilbert and Einstein in 1915, is that the equations of motion
descend from a single scalar quantity called the action, which a
system extremizes along its trajectory. The fourth, from Boltzmann
in 1872 and revived in our own century by Friston and others, is
that open systems organize themselves by minimizing a quantity that
resembles a free energy. Each habit was learned in a different
room, in a different decade, for a different purpose.

What is rarely asked aloud is whether these four habits are really
four. The current of charge in electrodynamics and the current of
entropy in a living cell are written in the same notation; they
pass between different rooms of the building called physics, and
at each doorway something quietly changes --- a tensor index, a
metric, an interpretive convention --- and the language continues
without comment. The four habits look, when one steps back, like
shadows of a single upstream object: one geometric entity that,
projected onto different scales and different problems, gives back
currents in one chart, conservation in another, an action in a
third, and a free-energy functional in a fourth. The unity has
been guessed at for a long time; it has not been written down.

This paper proposes to write that single upstream object down.
The argument names four geometric pillars on a small
six-dimensional manifold: a master object that holds gravity and
the other forces together in one structure, a master conservation
identity from which every current the laboratory measures
descends, a master variational functional from which every action
and every free-energy principle in the literature is a special
case, and a master pipeline that explains why two observers on the
same boundary always agree on the same physics even though they
use different coordinates. These four pillars do not add new
postulates. They are theorems of the constitutional layer fixed
in the parent paper of the program. What they buy is the
recognition that the four habits a working physicist learned
separately are projections, onto different sectors of the
boundary, of one upstream geometric object.

We invite the reader to walk the four pillars in order. Read each
statement, follow each proof, and ask whether the projection to
the laboratory currents one already knows is honest. After all,
what makes a unifying framework worth the name is not the number
of disparate phenomena it ties together but the smallness of the
upstream object from which all of them descend. The proposal here
is that one geometric object, written in the language of
differential forms on a six-dimensional bulk, is enough to recover
every conservation law, every variational principle, every
free-energy functional, and every gauge identity that a century of
physics has learned to write down.

\clearpage

%% ====================================================================
%% PART I: INTRODUCTION AND SETUP
%% ====================================================================

\GGPart[Introduction and setup]{Introduction and Setup}{The strategic moment, the bulk geometry, and the \GGConstitution{}}

A paper that claims to derive something has to begin by saying, plainly,
what it is allowed to assume. That is the whole business of this Part.

The claim to be supported is that one mechanism drives every symmetry
breaking in nature, and that the mechanism can be written down exactly
enough to be tested. Establishing that requires a stage to work on, and
the stage is not built here: it is inherited, piece by piece, from five
companion papers. The constitutional charter --- four axioms and seven
postulates about what any physical theory must satisfy --- comes from
\citet{ArisakaA1Constitution}. The six-dimensional geometry comes from
\citet{ArisakaA2GG6}. The proof that the geometry is free of the
internal inconsistencies that usually kill such constructions comes from
\citet{ArisakaA3BI6}, and the demonstration that it leaves no
continuous parameter free to adjust comes from \citet{ArisakaA4Euclid}.
The law governing how any of it looks to an observer who is part of the
system rather than outside it comes from \citet{ArisakaA5GDOS}. None of
the five is re-derived in this paper. Each is stated here, once, in the
form this paper uses, with the theorem that supplies it named.

One point in that inheritance deserves saying slowly, because it is
where a reader from outside physics most often stops. The theory is
six-dimensional, but the two dimensions beyond the four we inhabit are
not further directions one could travel in. One of them is a scale: it
records how a description of the world changes as one looks at it more
or less closely, the way the effective properties of a material differ
between a micrometer and a nanometer. The other is a phase: an angle
that has no length. So ``six-dimensional'' here is not a claim that
space has hidden rooms. It is a claim that two familiar features of
physical description --- scale, and phase --- behave geometrically, and
can be treated as directions in the same sense that time is treated as
one in relativity.

This Part then locates the paper in the twenty-two-paper cohort and maps
the twelve Parts that follow. A reader who wants the mechanism itself
can take this Part's final section for orientation and go directly to
Part~V, where the mechanism is built, returning to Parts~II--IV for the
machinery when a step needs justifying. A reader who intends to argue
with the paper should read this Part in full, because every input the
later argument leans on is declared here and is not declared again.


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

\subsection{The strategic moment for the \GiFour{} Pillars}
\label{ssec:strategic-moment}

The \GGTheory{} program has reached a stage where its derivational reach
is established but its foundational vocabulary is not yet rigorous.

In the past year, the program has produced a sequence of substantive
results: the rigorous derivation of the \OGNfive{} ledger
$(M,N,L;J,K)=(1,3,5;7,35)$ from \BIsix{} closure on the 6D bulk
\citep{ArisakaA3BI6}; the explicit codebook for all 28 Standard-Model
parameters in terms of the \OGNfive{} integers \citep{ArisakaA9SM}; the three-leg derivation of the family number $N=3$, combining \MSMF{}, \GDOS{} Morse stability, and the codebook self-consistency check \citep{ArisakaP2N3}; the prediction of a $\sim 3.85\,\TeV$
KK resonance cluster as the principal collider falsifier
\citep{ArisakaP1KK}; and the prediction of a $\approx 8.1~\mu$eV Kaxion-like
dark-matter candidate as the principal cosmological falsifier
\citep{ArisakaC1Kaxion}. Each of these derivations rests on a
common dynamical mechanism, identified in the methodological reflection
of \citet[Section~12]{ArisakaP2N3} as the open-system
boundary-projection cascade $\DGI\to\GSSB\to\GA$, with the dynamics
governed by a Geometric Free-Energy Functional $\cF_{\GFF}=\cF_{\GEP}+\cF_{\CEP}$
on the boundary moduli, and the closure conditions inherited from the
\BIsix{} anomaly polynomial $X_4$.

The mechanism works. It produces results that closed-system QFT and GR
cannot reach. But the language in which it is articulated -- the names
\GFF{}, \GRIP{}, \GA{}, \DGI{}, \GSSB{}, the boundary-projector moduli,
the Hopkins--Singer descent and the M-theory flux-quantization condition~\citep{Witten1996} -- has been deployed informally rather than
formally axiomatized. We have used these names as labels pointing in the
right direction, but we have not yet written them down as
mathematical objects whose properties can be checked against the Constitutional axioms.

This is the situation that mathematical physics historically encounters
when a program develops faster than its foundations can be written.
Newton's \emph{Principia} could not be fully understood until the
calculus was formalized by Leibniz, the Bernoullis, and Euler; Maxwell's
\emph{Treatise} could not be fully understood until vector calculus was
developed by Heaviside and Gibbs; Einstein's general relativity could
not be fully understood until Ricci and Levi-Civita developed the tensor
calculus to its modern form; and Schr\"odinger's and Heisenberg's quantum
mechanics could not be fully understood until von Neumann gave the
Hilbert-space formalism in 1932 \citep{vonNeumann1932}. In each case, a
deep theoretical advance was made first and the formal language was
written second. This is the same pattern in which we now find ourselves.

The present paper, the second of the \GDOS{} trio (with \AFive{} \GDOS{}
Foundation and \ASeven{} \QiFour{} Pillars), undertakes the writing of the
formal language for the geometric backbone of \GDOS{}. We call this
language the \GiFour{} Pillars: four explicitly named formal mathematical
objects -- \TGG{}, \CCI{}, \GFF{}, \GRIP{} -- whose Structure Theorems, definitions,
theorems, and worked examples constitute the rigorous foundation that
the upstream papers (\ANine{}, GG-2026-Generations, etc.) implicitly
assume. Once \GiFour{} is in place, every use of \GFF{} or \GRIP{} or
\TGG{} in a downstream paper acquires citable rigorous content, and the
upstream derivations become reproducible from first principles.

This is, in our view, the most important methodological move that the
\GGTheory{} program has yet undertaken. It is the move that converts
the program from an exciting research program into a foundational
theory in the technical sense.

\begin{remark}[Physical intuition for the broader scientific community]
\label{rem:why-broader}
The \GiFour{} formalism is presented in the language of differential
geometry and gauge theory because that is the language in which the
\GGTheory{} bulk content was originally derived. But the structural
content of \GiFour{} -- a unified bulk-to-boundary projection that
descends a single conserved object onto the observable boundary, with
a free-energy functional governing the dynamical landscape and a
cascade of attractors selecting the discrete catalog of stable
observables -- is not specific to particle physics. Readers from
cosmology will recognize the bulk-to-boundary descent as a
generalization of holography familiar from AdS/CFT; readers from
condensed-matter physics will recognize the cascade of unstable
critical points as the universal renormalization-group flow of phase
transitions; readers from chemistry will recognize the \GFF{}
free-energy functional with its attractor catalog as the Gibbs
landscape with its reaction products; readers from biology will
recognize the open-system irreversible projection as the second-law
analogue that selects living configurations from the equilibrium
manifold; and readers from neuroscience will recognize the cascade
\DGI{}~$\to$~\GSSB{}~$\to$~GA as the universal pattern by which a richly
symmetric prior collapses into a discrete observed perception. The
mathematical content of \GiFour{} is the same in all these contexts;
only the specific bulk and boundary objects change. We expect the
present formalism to be useful well beyond the particle-physics
community, and we have tried to write it in a way that makes the
physical intuition accessible at each step.
\end{remark}

\subsection{The four pillars in overview}
\label{ssec:four-pillars-overview}

The four \GiFour{} Pillars are introduced in the present paper as
follows. We give here only a one-paragraph sketch of each; the formal
Structure Theorems and definitions occupy the body of the paper.

\paragraph{\TGG{} -- The Total Geometric-Gauge object.}
\TGG{} is the unique geometric object that unifies, into a single
mathematical entity, the bulk-level composite connection on the 6D bulk and the 4D boundary stress tensor obtained by zero-mode
reduction. Its \emph{bulk realization} is a principal $\Gtot$-bundle
$\Pbun \to \Msix$ with structure group
$\Gtot = \Spinone \times \SUfive \times \UoneB \times \UoneQi$ (using
the magic-dimension-ladder identification $\Spinone\cong\SLtwoH$ at
$d=6$), equipped with the composite connection one-form
$\Atot = \omega_{\Spinone} + A_5 + A_B + A_{Qi}$ and curvature
$\Rtot = \dd \Atot + \Atot \wedge \Atot$. Its \emph{boundary
realization} is the unified 4D stress tensor
\[
\cT^{\mu\nu}_{\TGG} \;=\; T^{\mu\nu}_{\rm matter}
+ T^{\mu\nu}_{\rm gauge}[F]
+ X^{\mu\nu}_{\rm GS}
+ T^{\mu\nu}_{\rm KK},
\]
the boundary projection of all bulk stress-energy via the \BIsix{}
bulk-to-boundary descent. The two layers are tied together by the
\emph{Master Conservation Law}:
\[
\nabla_\mu \cT^{\mu\nu}_{\TGG} \;=\; 0,
\]
which holds iff the full Ward package (electric, baryonic,
informational, and metric currents) is satisfied. In the \GRIP{} cascade,
the unbroken-symmetry \TGG{} phase on the bulk is the \DGI{} stage.

\paragraph{\CCI{} -- The Covariant Conservation Identity.}
\CCI{} is the 4D boundary identity that emerges as the \OGGEP{} zero-mode
projection of \TGG{} conservation. It has two equivalent formulations.
The \emph{physical/local} formulation is the covariantly conserved
divergence equation $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ on $\Mfour$
with explicit compensator scalar field, holonomy current, and modified
Tolman--Oppenheimer--Volkoff equation as astrophysical falsifier. The
\emph{algebraic/global} formulation is the differential-cohomology
integrality $[X_4] \in H^4(\Msix;\Z)$ together with the rank-one
factorization of the anomaly matrix $M^{\BIsix} = v\otimes v^T$, where
\[
X_4 \;=\; \tr R^2 - \Tr F_5^2 - \tfrac{2}{5} F_B^2 - \tfrac{2}{5} F_{Qi}^2.
\]
Under \MSMF{}, \CCI{} closure uniquely fixes the \OGNfive{} ledger
$(M,N,L;J,K) = (1,3,5;7,35)$. The $-2/5$ coefficients are not free
parameters: they follow from the trace-ratio identity
$\Tr B_0^2/\Tr Y_0^2 = (1/3)/(5/6) = 2/5$ in the fundamental $\mathbf 5$
of $\SUfive$. The two formulations of \CCI{} are equivalent by
Hopkins--Singer differential-cohomology descent.

\paragraph{\GFF{} -- The Geometric Free-Energy Functional.}
\GFF{} is the unique scalar functional
$\cF_{\GFF}=\cF_{\GEP}+\cF_{\CEP}$ on the stratified boundary moduli
space $\Bdy$. $\cF_{\GEP}$ (Geometry-Gauge Energy Principle) is the
energetic part, derived by bulk-to-boundary descent of the \TGG{} bulk action
through the Wilson-line holonomy on the compact $S^1_\theta$.
$\cF_{\CEP}$ (Curvature-Entropy Principle) is the curvature-entropic
part from the Riemannian volume of admissible projector classes plus
the \GDOS{} diffusion term. \GFF{} has two equivalent characterizations.
The \emph{thermodynamic} characterization: $\cF_{\GFF}$ is monotonically
decreasing under irreversible boundary projection,
$\partial_t \cF_{\GFF} \leq 0$, the open-system second-law analogue.
The \emph{Morse-theoretic} characterization: $\cF_{\GFF}$ is a perfect
Morse function on each stratum of $\Bdy$ with strict alternating
Morse-index positivity inherited from \CCI{} via the Hopkins--Singer
differential-cohomology descent. Critical points of $\cF_{\GFF}$ are
\GA{}'s (Geometric Attractors). Strict Morse-positivity of \GFF{} is
the property that, in the N=3 proof of \citet{ArisakaP2N3},
forces the family-projector moduli to have real dimension divisible
by four.

\paragraph{\GRIP{} -- The Gauge-Runtime Invariance Pipeline.}
\GRIP{} is the universal three-stage lifecycle cascade
$\DGI \to \GSSB \to \GA$ on the \GFF{} Morse landscape. \DGI{} (Dynamical
Gauge Invariance) is the unbroken-symmetry \TGG{} phase on the bulk,
inheriting the full $\Gtot$ symmetry of the composite connection.
\GSSB{} (Geometric Spontaneous Symmetry Breaking) is the chain of
intermediate symmetry-breaking saddles connected by Bogomolnyi-saturating
gradient-flow trajectories with instanton action $\pi k$ between
adjacent index-$2k \to 2(k-1)$ saddles. \GA{} (Geometric Attractor) is
the stable boundary observable, the Morse minimum, the cascade
endpoint. The pipeline combines two complementary properties:
\emph{gauge-invariance preservation} -- each stage of the cascade
preserves gauge covariance (POS-1 of the \GGConstitution{}), so the
entire runtime is gauge-invariant; and \emph{finite-budget closure} --
the total cascade action $\Stot(N) = \pi N(N-1)/2$ on a stratum of
complex dimension $N-1$ is bounded by the \BIsix{} instanton-action budget $\Sbi = 3\pi$, delivered by the capacity theorem T-N3-6 of \citet{ArisakaP2N3} (open problem M10, whose demand it meets; residue M6 $=$ R-INT, the
topological residue of its integrality theorem L-INT). Together
these constraints force $N \leq 3$ on the family-projector stratum,
contributing the upper bound in the three-leg N=3 derivation.

\paragraph{The unified \GiFour{} system.} The four pillars together form a
unified geometric backbone of \GDOS{}. \TGG{} provides the bulk-level
invariance; \CCI{} provides its closure conditions; \GFF{} provides the
dynamical landscape on the boundary moduli; \GRIP{} provides the cascade
dynamics that selects boundary observables. No additional physical
axiom is introduced: the four pillars are derived from the
\GGConstitution{} \citep{ArisakaA1Constitution} -- specifically from
\GGEP{} (geometry--gauge equivalence), \OGGEP{} (its open-system
extension), \MSMF{} (minimal sufficient manifold form), and the \BIsix{}
closure of \citet{ArisakaA3BI6}.

\subsection{Logical relationship to the \GGConstitution{}}
\label{ssec:relation-constitution}

The \GiFour{} Pillars are not new postulates of the program. They are
\emph{theorems} of the \GGConstitution{}, in the sense that they can be
derived from the four constitutional axioms (\LC, \GGEP, \OGGEP, \MSMF)
together with the \BIsix{} closure of \citet{ArisakaA3BI6}. We articulate
this relationship explicitly here because it is the methodological core
of the present paper.

The four constitutional axioms of \GGTheory{}, as stated in
\citet{ArisakaA1Constitution}, are:

\begin{itemize}[leftmargin=2em]
\item \textbf{LC} (Local Causality): the bulk admits a Lorentzian
structure with retarded propagation of admissible kernels. This is the
axiom of relativistic compatibility.
\item \textbf{\GGEP{}} (Geometry--Gauge Equivalence Principle): gauge
content and geometric content are unified in a single primitive
composite connection. This is the axiom that forces the existence of
\TGG{}.
\item \textbf{\OGGEP{}} (Operational/Open Geometry--Gauge Equivalence
Principle): observable boundary content is related to bulk geometric
content by an irreversible CPTP bulk-to-boundary projection~\citep{Susskind1995}. This is the
axiom that forces the existence of the boundary-projector moduli on
which \GFF{} lives.
\item \textbf{\MSMF{}} (Minimal Sufficient Manifold Form): among
admissible structures consistent with the boundary observable spectrum,
\GGTheory{} selects the one of minimum total complexity. This is the
axiom that forces the \OGNfive{} ledger to be the minimum integer solution
of \CCI{}.
\end{itemize}

The four pillars are derived from these axioms as follows.

\TGG{} is the formal mathematical realization of \GGEP{} extended by
\OGGEP{} to the boundary: the bulk-level composite connection (forced
by \GGEP{} and the magic-dimension ladder) and its 4D zero-mode
projection to the unified stress tensor $\cT^{\mu\nu}_{\TGG}$ (forced
by \OGGEP{} together with \BIsix{} descent). The Master Conservation Law
$\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ is given by
Theorem~\ref{thm:TGG-existence-uniqueness} of Part~II, derived from
\GGEP{}, \OGGEP{}, \LC{}, and the \BIsix{} closure of
\citet{ArisakaA3BI6}.

\CCI{} is the formal mathematical realization of the boundary-level
content of \OGGEP{}: the 4D conservation identity that emerges as the
zero-mode projection of \TGG{} conservation. Its physical content
(astrophysical falsifier via modified TOV) and its algebraic content
(integrality $[X_4]\in H^4(\Msix;\Z)$ with rank-one factorization)
are equivalent by Hopkins--Singer descent. \CCI{} closure under
\MSMF{} forces the \OGNfive{} ledger uniquely; this is
Theorem~\ref{thm:CCI-OGN5-uniqueness} of Part~III, derived from \CCI{}
and \MSMF{}.

\GFF{} is the formal mathematical realization of the dynamical content
of \OGGEP{}: the free-energy functional on the boundary moduli, with
both thermodynamic ($\partial_t \cF \leq 0$) and Morse-theoretic
characterizations. Its existence is given by
Definition~\ref{def:GFF} and the five \GFF{} Structure Theorems of
Section~\ref{ssec:GFF-seven-st}; its Morse-positivity is given by
Theorem~\ref{thm:GFF-positivity-from-CCI} of Part~IV, derived from
\OGGEP{} and \CCI{} via Hopkins--Singer descent.

\GRIP{} is the formal mathematical realization of the lifecycle content
of \OGGEP{}: the gauge-invariant cascade $\DGI \to \GSSB \to \GA$ on
the \GFF{} Morse landscape with finite-budget closure. Its existence as
a perfect gradient flow on $\Bdy$ and its closure within
$\Stot \leq \Sbi = 3\pi$ are given by
Theorem~\ref{thm:GRIP-existence-closure} of Part~V, derived from
\OGGEP{}, \GFF{} Morse-positivity, and \CCI{}.

The bridge theorems of Part~VI then assemble the four pillars into a
unified system equivalent in content to the \GGConstitution{}. Specifically:

\begin{theorem}[Constitutional equivalence, sketched here]
\label{thm:constitutional-equivalence-preview}
The \GiFour{} Pillars system $(\TGG,\allowbreak \CCI,\allowbreak \GFF,\allowbreak \GRIP)$, equipped with the
five principal theorems \textbf{T1}--\textbf{T5} of this paper, is
equivalent in derivational content to the \GGConstitution{}
$(\LC, \GGEP, \OGGEP, \MSMF)$ together with the \BIsix{} closure of
\citet{ArisakaA3BI6}. Every closed-derivation result of the
\GGTheory{} program -- including N=3, the 28 Standard-Model parameters,
the hierarchy, and the dark-matter prediction -- can be obtained from
either the constitutional axioms or the \GiFour{} Pillars; the two systems
are different presentations of the same mathematical content.
\end{theorem}

The full proof of Theorem~\ref{thm:constitutional-equivalence-preview}
is given in Part~VI. The point of the theorem is methodological: the
\GiFour{} reorganization of the present paper is not a new theory, but a
\emph{geometric reading} of the constitutional axioms that makes the
mathematical structure explicit and citable.

\subsection{Logical relationship to the \QiFour{} Pillars (\ASeven{})}
\label{ssec:relation-qi4}

The four \GiFour{} Pillars (\TGG, \CCI, \GFF, \GRIP) form the
\emph{geometric} backbone of \GDOS{}. The companion paper \ASeven{}
\citep{ArisakaA7Qi4} formalizes four \emph{information-theoretic}
pillars -- the \QiFour{} Pillars -- whose role is to encode the
quantum-information / measurement-theoretic content of \GDOS{}. The two
sets of four pillars are parallel, and their parallel structure is
fundamental to the design of \GDOS{} as a foundational framework that
unifies geometry and information.

Without anticipating the detailed content of \ASeven{}, the parallel
may be sketched as follows:

\begin{itemize}[leftmargin=2em]
\item \TGG{} (Total Geometric-Gauge object: bulk composite connection
+ 4D boundary stress tensor with master conservation) is paired with
\textbf{\QiQUA{}} (Total Quantum-Information generator). Both encode
the deepest unbroken structure on the bulk: \TGG{} on the geometric
side, \QiQUA{} on the entanglement / quantum-information side.
\item \CCI{} (Covariant Conservation Identity: 4D OGGEP-projected
conservation equation, with both falsifiable physical and integral
algebraic formulations) is paired with \textbf{\QiEP{}}
(Quantum-information Equivalence Principle). Both encode the
boundary-level identity that the master generator must satisfy:
\CCI{} on the conservation/anomaly side, \QiEP{} on the
information-conservation under measurement side.
\item \GFF{} (Geometric Free-Energy Functional, with thermodynamic
$\partial_t F\leq 0$ and Morse-theoretic characterizations) is paired
with \textbf{\QiCON{}} (Quantum-information Conservation on boundary
projections). Both encode the dynamical landscape on the boundary:
\GFF{} on the free-energy / Morse side, \QiCON{} on the
mutual-information / relative-entropy side.
\item \GRIP{} (Gauge-Runtime Invariance Pipeline: cascade
$\DGI\to\GSSB\to\GA$ with finite-budget closure) is paired with
\textbf{\QiHAL{}} (Quantum-information Holographic projection). Both
encode the cascade dynamics from bulk to boundary: \GRIP{} on the
Morse-cascade / instanton-action side, \QiHAL{} on the Stinespring
dilation / decoherence side.
\end{itemize}

The two sets of pillars are not redundant: each captures content that
the other does not. \GiFour{} captures the bulk-to-boundary geometric
structure (gauge connections, anomaly polynomials, Morse landscapes,
attractor cascades). \QiFour{} captures the bulk-to-boundary
information-theoretic structure (entanglement, mutual information,
decoherence~\citep{Zeh1970}, measurement outcomes). Together they constitute a unified
foundational framework for \GDOS{}, in which both the geometric and the
information-theoretic content of physical observation are made
foundational at the same time.

This unified framework is what we argue, in
\citet[Section~12]{ArisakaP2N3}, has been the missing link
in the foundations of physics for the century since the establishment of
quantum mechanics in 1925--1930. Quantum mechanics in its closed-system
form unifies the information-theoretic content (the state vector) with
the dynamical content (unitary evolution), but it does so on a fixed
background and without bulk-to-boundary projection. General relativity unifies
the geometric content (the metric) with the dynamical content (Einstein's
equations), but it does so without information-theoretic content.
\GDOS{} unifies all three -- geometric, information-theoretic, and
dynamical -- under a single foundational principle (\OGGEP) that
elevates the bulk-to-boundary projection to axiomatic status. The
\GiFour{} Pillars (the present paper) and the \QiFour{} Pillars
(\ASeven{}) are the two sides of this unification.

\subsection{What this paper does and does not do}
\label{ssec:scope}

The present paper has a clearly bounded scope. It does the following.

\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item It formalizes the four \GiFour{} Pillars as mathematical objects
with Structure Theorems, definitions, principal theorems, and worked examples.
\item It derives the four pillars from the \GGConstitution{} and the
\BIsix{} closure, without introducing any additional physical axioms.
\item It states and proves the five principal theorems
\textbf{T1}--\textbf{T5} (existence and uniqueness of \TGG{};
\CCI{}-OGN5 uniqueness; \GFF{} Morse-positivity from \CCI{}; \GRIP{}
cascade closure; constitutional equivalence).
\item It establishes the bridge theorems to \OGGEP/\MSMF{} (the
constitutional axioms) and to \QiFour{} (the information-theoretic
backbone of \ASeven{}).
\item It applies the \GiFour{} framework to the derivations in
\ANine{} (28 SM parameters) and GG-2026-Generations (N=3),
demonstrating that those derivations acquire full rigorous status once
the \GiFour{} vocabulary is in place.
\end{enumerate}

What it does \emph{not} do is also bounded.

\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item It does not undertake the full \QiFour{} axiomatization; that is the
content of \ASeven{}. The \QiFour{} Pillars are mentioned in
Section~\ref{ssec:relation-qi4} and at the bridge theorems of Part~VI,
but their detailed development is deferred.
\item It does not yet provide rigorous proofs of the residual technical
hypotheses $\mathcal H_1, \mathcal H_2$ flagged in
\citet[Appendix~E]{ArisakaP2N3}. Those hypotheses become
\emph{theorems} of \GiFour{} once the pillars are formalized, but the
detailed implementation of the Hopkins--Singer descent in the 6D
bulk is carried by Appendix~H of \citet{ArisakaP2N3}.
\item It does not undertake the full re-derivation of the 28 SM
parameters in the new \GiFour{} vocabulary; that is the role of the
revised \ANine{} \citep{ArisakaA9SM}, with the present paper providing
the rigorous foundation that the revised \ANine{} rests on.
\item It does not address the experimental falsifiers of \GGTheory{}
(KK resonance cluster, Kaxion dark matter); those are the subject of
(\citealp{ArisakaP1KK}; \citealp{ArisakaC1Kaxion}).
\end{enumerate}

The discipline of the present paper is therefore: \emph{austere
formalization of \GiFour{} from the constitutional axioms, with
applications to the existing derivational results}.

\subsection{Plan of the paper}
\label{ssec:plan}

The twelve Parts of this paper are laid out in the Roadmap of
\S\ref{sec:roadmap-a6} (Figure~\ref{fig:roadmap-a6}), which names each
Part in turn and highlights the load-bearing four-pillar core; the
detailed section-by-section contents follow in the Table of Contents at
the front of the paper.

\subsection{Relation to other \GGTheory{} preprints}
\label{ssec:related-preprints}

The present paper (\ASix{}) is part of a coordinated program of
preprints developing the foundations and applications of \GGTheory{}.
For the convenience of the reader, we summarize the principal
preprints, their content, and their relation to the present paper.

{\footnotesize
\begin{longtable}{p{2.62cm}p{3.27cm}p{3.93cm}p{5.02cm}}
\caption[The \GGTheory{} preprint family and its connection points.]{The \GGTheory{} preprint family and its connection points to the present paper. Each row names a preprint, what it contains, and the one place where it touches this one --- an import consumed here, an export supplied to it, or an object both describe. The list is a roster and not a bibliography: the full entries, each with its relevance note, are in the references.}
\label{tab:related-preprints}\\
\toprule
\textbf{Preprint} & \textbf{Title (short)} & \textbf{Content} & \textbf{Connection to \ASix{}}\\
\midrule
\endfirsthead
\multicolumn{4}{c}{\tablename\ \thetable\ -- continued from previous page}\\
\toprule
\textbf{Preprint} & \textbf{Title (short)} & \textbf{Content} & \textbf{Connection to \ASix{}}\\
\midrule
\endhead
\midrule
\multicolumn{4}{r}{\textit{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\multicolumn{4}{l}{\textit{Foundational documents (the \GDOS{} quartet)}}\\
\midrule
\AOne{} & \GGConstitution{} & Four Axioms (LC, \GGEP{}, \OGGEP{}, \MSMF{}) + 7 Postulates & \textbf{The single foundational layer.} All Structure Theorems and Principal Theorems of are derived from these.\\
\midrule
\ATwo{} & \GG{} Foundation & 6D bulk $\Msix$, $\Spinone$ frame bundle, magic-dimension ladder $\R\to\C\to\HH\to\Octonion$ & The bulk arena on which \TGG{} (Part~II) is constructed; provides the bundle-theoretic content of \GGEP{}.\\
GD-2026-3 & \BIsix{} Closure & Modified Bianchi identity $\dd H_3 = X_4$ on the bulk; \OGNfive{} ledger $(1,3,5;7,35)$ & \textbf{Closes the algebraic side of \CCI{}} (Part~III). T2.1 is the algebraic uniqueness theorem of \OGNfive{}; T2.2 descends \BIsix{} to the boundary.\\
\AFive{} & \GDOS{} Foundation & Geometric Dynamics of Open Systems: boundary projector, bulk-to-boundary descent, CPTP boundary & First informal introduction of \GFF{} and \GRIP{}; the present paper formalizes these as Pillars 3 and 4 with full Structure Theorems and Principal Theorems.\\
\ASix{} & \textbf{(this paper)} \GiFour{} Pillars (geometric backbone) & Four Pillars \TGG{}, \CCI{}, \GFF{}, \GRIP{} with 28 Structure Theorems and 9 Principal Theorems (T1.1--T5) & Provides the formal geometric vocabulary of \GDOS{}.\\
\ASeven{} & \QiFour{} Pillars (information backbone) & Four information-theoretic Pillars: \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{} & \textbf{Companion paper.} The information backbone that pairs with the geometric backbone of \ASix{}; together they form the complete \GiFour{}+\QiFour{} foundational language.\\
\midrule
\AEight{} & White Paper for \GDOS{} & Programmatic synthesis of \GGTheory{}'s significance and falsification program & High-level synthesis citing \ASix{} as the formal geometric reference.\\
\multicolumn{4}{l}{\textit{Predictive applications}}\\
\midrule
\ANine{} & SM 28 parameters codebook & Explicit derivation of all 28 SM parameters from \OGNfive{} ledger & \textbf{Application of \CCI{}/\GRIP{} cascades.} The 28 parameters are cascade depths of \GRIP{} on different boundary strata. T2.1 forces the \OGNfive{} ledger that fixes the codebook.\\
\ATen{} & GG-Cosmos for $\Lambda$CDM & Cosmological observables from \GG{} boundary descent & \textbf{Application of \CCI{}/\GFF{} cosmological projections.} Modified Friedmann equations and $w(z)$, $f\sigma_8(z)$ shifts from $X_{\rm GS}$.\\
\midrule
GG-2026-Generations & N=3 three-leg proof & The three-leg derivation of $N_{\rm gen}=3$ & \textbf{Cascade structure carried by T2.1+T3.1+T4.1 of the present paper (T4.1 conditional on M10).} The would-be hypotheses $\mathcal H_1, \mathcal H_2$ are dissolved in that paper: $\mathcal H_1$ closed in the negative (its rigidity lemma generalizes the naive-descent no-go to every functorial route), $\mathcal H_2$ succeeded by the integrality theorem L-INT (residue
R-INT).\\
\multicolumn{4}{l}{\textit{Specific predictions and falsifiers}}\\
\midrule
\PaperPOne{} & $\sim 4$ TeV KK particles & Prediction of the KK resonance cluster $\Mlock\simeq 3.85\,\TeV$ & \textbf{LHC-falsifiable prediction} of the \BIsix{} closure; references \GFF{}/\GRIP{} for the scale-stratum dynamics.\\
\midrule
\PaperCOne{} & $\approx 8.1~\mu$eV Kaxion DM & Prediction of ultra-light Kaxion dark matter & \textbf{Direct-detection-falsifiable prediction} from the $\UoneQi{}$ attractor; uses \GRIP{} cascade on the compact-angle moduli.\\
\multicolumn{4}{l}{\textit{Books and synthesis}}\\
\midrule
GDOS-Book-I & \GGTheory{} Foundation Book & Comprehensive book-length treatment & Cites the present paper as the formal \GiFour{} reference; intended as the eventual graduate-textbook treatment of \GGTheory{}.\\
\end{longtable}
}

\paragraph{Reading paths.}
A reader new to \GGTheory{} has several reasonable entry points:

\begin{itemize}[leftmargin=2em]
\item \textbf{Foundations-first path}: \AOne{} (Constitution) $\to$
\ATwo{} (\GG{} Foundation) $\to$ GD-2026-3 (\BIsix{}) $\to$ the present
paper (\GiFour{} Pillars) $\to$ \ASeven{} (\QiFour{} Pillars). This is the
recommended path for theoretical physicists who want to see the
full derivational chain.
\item \textbf{Predictions-first path}: \ANine{} (SM parameters)
$\to$ GG-2026-Generations (N=3 proof) $\to$ \PaperPOne{} (KK
prediction) $\to$ \PaperCOne{} (Kaxion DM). This is the recommended
path for experimentalists who want to see the falsifiable
predictions before the formal scaffolding.
\item \textbf{Pedagogical path (recommended for graduate students)}:
\AOne{} (Constitution) $\to$ Appendix~G of the present paper
(pedagogical tour of the dual-layer architecture) $\to$
Appendix~N (glossary for life sciences and neuroscience) $\to$
Parts~II--V of the present paper (the four Pillars) $\to$ companion
predictive papers.
\item \textbf{Synthesis path}: \AEight{} (White Paper) $\to$
GDOS-Book-I (Foundation Book) $\to$ specific preprints as
needed. This is the recommended path for those who want a
high-level overview before diving into derivations.
\end{itemize}

\paragraph{Cross-paper consistency.}
The dual-layer architecture of (algebraic block + physical
block per Pillar, paired Theorems T*.1 + T*.2, named compatibility
links) is the universal organizing principle that propagates
across all preprints in the \GGTheory{} family. Future versions of
the companion preprints will adopt the same Structure Theorem
terminology established in the present paper.

\subsection{Methodological note: derivation policy and Euclidean architecture}
\label{ssec:methodology}

The present paper follows the Euclidean axiomatic architecture
canonical to the \GGTheory{} program: numbered Definitions,
Postulates, Structure Theorems, Lemmas, Principal Theorems, and
Corollaries, with proofs marked $\blacksquare$. Each Principal
Theorem is dual-layer (algebraic statement T$N$.1 + operational
statement T$N$.2, linked by a named compatibility relation); each is
followed by a Physics-Intuition remark and at least one worked
example. ``Axiom'' is reserved exclusively for the four
constitutional axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) of the
\GGConstitution{}; everything else is a theorem, postulate, or
structure theorem. The full methodological framing, including the
historical-Euclidean motivation and the relation to the four prior
revolutions of physics, lives in
\citep[\S\S 1--2]{ArisakaA4Euclid} and
\citep[Part~I]{ArisakaA1Constitution}.

%% ====================================================================
\section{This paper's place in the cohort: one link of the open-system chain}
\label{sec:cohort-funnel}

This paper establishes the four geometric \GiFour{} pillars --- \TGG{}, \CCI{}, \GFF{}, and \GRIP{} --- and the runtime by which a boundary system selects and stabilizes its branch; that is its own subject. For readers following the whole \GGTheory{} cohort, it is also one link of a single open-system chain that the consequence papers cash out. Because no physical system is closed, the four-dimensional world is a lawful projection (\OGGEP{}) of the six-dimensional bulk, and that projection is observation itself; its dynamics is the boundary law \GDOS{}, forced by the geometric \GiFour{} and quantum-information \QiFour{} pillars, so that the laws, the particles, and the contents of the universe are \emph{observables} of the projection rather than free inputs. In that chain, the present paper supplies the geometric \GiFour{} pillars that, with the \QiFour{} pillars, force the \GDOS{} law. The funnel below is shared, in structure, with the companion papers \ANine{}, \ATen{}, and \AEleven{}; here it is offered only as orientation, and the rest of this paper proceeds on its own terms.

\begin{figure}[hbt!]
\centering
\resizebox{0.681\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 laws, the particles, and the contents of the universe}\\ \textbf{are registered observables} of this one projection --- \textit{not free inputs}};
\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{they are determined}\\ {\footnotesize read off one geometry --- the work of the consequence papers \ANine{}--\AEleven{}}};
\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 at every stratum}                (obs);
\draw[arrow] (obs)   -- node[elab,right]{and the law is not arbitrary}           (pill);
\draw[arrow] (pill)  -- node[elab,right]{projection forced $\Rightarrow$ observables forced} (concl);
\end{tikzpicture}}
\caption[The open-system chain shared across the \GGTheory{} cohort.]{\textbf{The open-system chain 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, so the laws and contents of the universe are observables rather than free inputs. The same funnel opens the consequence papers \ANine{}, \ATen{}, and \AEleven{}, with a paper-specific payload; within it the present paper establishes the geometric \GiFour{} pillars that force \GDOS{}.}
\label{fig:cohort-funnel}
\end{figure}

\section{Primitive Inputs from the Companion Papers}
\label{sec:bulk-geometry-recap}
\label{sec:gg6-constitution-recap}
%% ====================================================================

The four pillars of \GiFour{} formalize a structure that is already
present, but only informally articulated, in four companion papers.
We collect the primitive inputs here in compact form, with cross-
references to the canonical home of each input. Three blocks suffice:
the constitutional clauses, the 6D bulk and \BIsix{} closure,
and the \OGGEP{} projector that connects the two.

\subsection{The constitutional clauses}

The \GGConstitution{} \citep{ArisakaA1Constitution} comprises four
axioms and seven postulates. We list them as one-line entries; the
full treatment is in \citet{ArisakaA1Constitution}, Sections~3--13.

\begin{itemize}[leftmargin=2em,itemsep=2pt]
\item \textbf{\AX{1} LC} (Local Causality): admissible kernels propagate
in forward light-cones.
\item \textbf{\AX{2} \GGEP{}} (Gauge--Geometry Equivalence Principle):
gauge and geometric content arise from a single composite connection
on the bulk.
\item \textbf{\AX{3} \OGGEP{}} (Open-system Gauge--Geometry Equivalence
Principle): observable boundary content is a CPTP bulk-to-boundary projection of bulk geometric content; the projection is irreversible
and is the foundational dynamics of physical observation.
\item \textbf{\AX{4} \MSMF{}} (Minimal Sufficient Manifold Form):
among admissible structures, select the one of minimum total
complexity.
\item \textbf{\POS{1}--\POS{7}}: the seven postulates \GI{}, \UEP{},
\AF{}, \CP{}, \RG{}, \OGN{}, \MF{} are theorem-level consequences of
\AX{1}--\AX{4} on the bulk and boundary; see
\citet{ArisakaA1Constitution} Sections~7--13.
\end{itemize}

\subsection{The 6D bulk and the \BIsix{} closure}

The 6D bulk \citep{ArisakaA2GG6} carries the four primitive
geometric quantities used throughout this paper. The bulk metric is
\begin{equation}
\label{eq:bulk-metric}
\dd s^2_6 \;=\; \e^{-2r}\,\eta_{\mu\nu}\,\dd x^\mu \dd x^\nu
+ \dd r^2 + R_\theta^2\,\dd\theta^2,
\end{equation}
with $\Spinone \cong \SLtwoH$ acting on the 4D Minkowski slice (magic-
dimension ladder, \citet{ArisakaA2GG6} Theorem~1). The boundary
$\partial \Msix = \Mfour$ sits at $r=0$ (the UV end, where $r=\ln(\EGUT/\mu)=0$, i.e.\ $\mu=\EGUT$), and the IR end at $r=K$
realizes the scale hierarchy $\EPl/\Mlock = 2\,\e^{35}$ with
$K = 35$ the \OGNfive{} ledger integer. The total bulk gauge
group is
\begin{equation}
\label{eq:Gtot-recap}
\Gtot \;=\; \Spinone \times \SUfive \times \UoneB \times \UoneQi.
\end{equation}
The \BIsix{} closure \citep{ArisakaA3BI6} is the differential identity
\begin{equation}
\label{eq:BI6-recap}
\dd H_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},
\end{equation}
the \GGEP{} completion of the 1984 Green--Schwarz construction (see
\citet[\S~8.7,~``\BIsix{} as the \GGEP{} completion of GS: the
geometrization reading'']{ArisakaA3BI6}). The four-component
coefficient vector $(a,b,c,d) = (1,-1,-2/5,-2/5)$ and the trace-ratio
lock $|a/c| = |a/d| = 5/2$ are pinned by \MSMF{} closure on the bulk
integrality $[X_4] \in H^4(\Msix;\Z)$, with the unique minimum-
integer solution being the \OGNfive{} ledger
\begin{equation}
\label{eq:OGN5-recap}
(M,N,L;J,K) \;=\; (1,3,5;7,35),
\end{equation}
satisfying the two algebraic identities
\begin{equation}
\label{eq:OGN5-identities-recap}
K \;=\; M^2 + N^2 + L^2 \;=\; 1 + 9 + 25 \;=\; 35,
\qquad
K \;=\; L J \;=\; 5\cdot 7 \;=\; 35.
\end{equation}
The physical readings of the five integers are $M=1$ the primitive tensor slot, $N=3$ the color-triplicity index (its three-generation face a downstream consequence), $L=5$ the defining block size of $\SUfive$, $J=7$ the minimal symmetric closure cardinality of the charge packet $\{0,\pm1,\pm2,\pm3\}$, and $K=35=LJ=M^2+N^2+L^2$ the total lock index; the derivation is \citet[Theorem~10.2]{ArisakaA3BI6}.

\paragraph{Constitutional admissibility of \UoneQi{}: optional,
empirically preferred.} The four-component bulk gauge group above is
read here as the \emph{empirically preferred} configuration rather than
as a constitutional necessity. \citet{ArisakaA3BI6} is the
authoritative source on this point. The constitutional clauses
admit both the three-component bulk
$\Gtot^{(3)} = \Spinone \times \SUfive \times \UoneB$ and the
four-component bulk $\Gtot^{(4)} = \Spinone \times \SUfive \times
\UoneB \times \UoneQi$. The four-component case is adopted on
empirical grounds: it is required for the boundary Arisaka equation
$J_E/\QB = J_e = J_B = J_{Qi}$ --- where $\QB$ is the
bioenergetic Arisaka constant ($\approx 0.1555$~eV per qubit,
matching ATP hydrolysis at biological temperatures) --- to close
\citep{ArisakaA1Constitution}; for the \OGNfive{} dual-lock identity
$K = LJ = M^2 + N^2 + L^2 = 35$ to admit its unique solution
\citep[Theorem~10.2]{ArisakaA3BI6}; and for the \QiFour{} pillars and
the \EIC{} framework of \citet{ArisakaA7Qi4} to exist on the
boundary. Throughout this paper $\Gtot$ will mean the four-component
group unless explicitly stated otherwise.

\subsection{The \OGGEP{} projector}

The \GDOS{} boundary law \citep{ArisakaA5GDOS} is the CPTP
bulk-to-boundary projector
\begin{equation}
\label{eq:OGGEP-A6}
\cD_{\OGGEP} \;=\; \cR \;\circ\; \mathrm{Tr}_{\rm env},
\end{equation}
which carries bulk operators on $\Msix$ to boundary observables on
$\Mfour$ via Kaluza--Klein zero-mode reduction followed by
environmental tracing. This is the construction the four pillars of
\GiFour{} formalize: \TGG{} is the bulk side ($\Atot$, $\Rtot$,
$\cT^{\mu\nu}_{\TGG}$); \CCI{} is the descent identity that makes the
projection consistent ($X_4$ closure, four-current Ward package);
\GFF{} is the boundary Lyapunov functional that the projector flow
descends along; \GRIP{} is the cascade structure that the projector
generates on the family-projector moduli. Without \OGGEP{}, the four
pillars would have no upstream source; with it, they are theorems.

\paragraph{Why \OGGEP{} is necessary, not optional.} The closed-
system frameworks of orthodox QFT and GR treat measurement as an
external annotation on otherwise unitary evolution. \OGGEP{} treats
the irreversible projection as the foundational dynamics, and the
unitary evolution as the bulk pre-projection limit. In \GiFour{}
language, the irreversibility of the projector is what makes the
\GFF{} second law (T3.2) and the \GRIP{} cascade closure (T4.1)
theorems rather than postulates.

%% End of Phase 1 Part I content
%% ----------------------------------------------------------------------

%% ====================================================================
%% PART II: TGG -- THE TOPOLOGICAL GEOMETRIC GENERATOR
%% ====================================================================


%% --- Phase 8 TikZ figure: Gi-4 architecture overview ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\usebox{\GGscPillars}%
}
\caption[Architecture of the four \GiFour{} Pillars.]{Architecture overview of the four \GiFour{} Pillars. Each pillar inherits its constitutional content from the \GDOS{} boundary law of \AFive{} (top) and its bulk-side technical content from the 6D bulk geometry of A2-\GG{} and the \BIsix{} anomaly closure of \AThree{} (bottom). Horizontal arrows show the implication chain $\TGG{} \Rightarrow \CCI{} \Rightarrow \GFF{} \Rightarrow \GRIP{}$. The four named compatibility relations $\mathfrak{C}_{\TGG{}}, \mathfrak{C}_{\CCI{}}, \mathfrak{C}_{\GFF{}}, \mathfrak{C}_{\GRIP{}}$ (one per pillar) tie the algebraic and operational layers within each pillar.}
\label{fig:gi4-architecture}
\end{figure}

\section{The cohort inheritance and the roadmap of the twelve Parts}
\label{sec:inheritance-and-roadmap-A6}

\noindent
Before the four-pillar core begins (Part~II), it is useful to lay
out two reader-orientation maps on the same opening: a panorama of
the twenty-two-paper cohort in which this paper stands
(\S\ref{sec:cohort-a6}, Figure~\ref{fig:cohort-map-a6}), and a
panorama of the twelve Parts of the present paper
(\S\ref{sec:roadmap-a6}, Figure~\ref{fig:roadmap-a6}). A reader of
any later Part should be able to flip back to this Section and
orient themselves in both directions: which cohort companion carries
which imported result, and where the present argument stands in the
twelve-Part arc.

\subsection{The cohort inheritance}
\label{sec:cohort-a6}

The present paper is one of twenty-two. \GGTheory{} is written not as
a single monograph but as a cohort of twenty-two preprints --- eleven A-papers, six P-papers, three C-papers and two white papers --- each carrying one
rigorously stated piece of a single descent that runs from four
axioms and seven postulates to the empirical structure of physics.
The partition into twenty-two papers reflects the partition of subject
matter, not a ranking of importance: the framework is one program,
and each paper is a named carrier of named content.
Figure~\ref{fig:cohort-map-a6} is the panorama; a reader meeting the
program for the first time can read its architecture off the
figure top to bottom before returning to the geometric backbone that
is the business of this paper. The cohort falls into five strata
plus three framed branches.

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

\paragraph{The boundary law and its two pillars --- where this paper sits.}
\AX{3}~(\OGGEP{}) forces the open-system boundary law \AFive{}
\citep{ArisakaA5GDOS}, the Geometric Dynamics of Open Systems, which projects
bulk physics onto the four-dimensional boundary. Two sibling pillar papers
organize the boundary content, and \emph{the present paper is one of them}:
\ASix{} \citep{ArisakaA6Gi4} --- this paper --- supplies the geometric
\GiFour{} pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}), the geometric backbone
on which the conservation laws of \GDOS{} rest; its information-theoretic
sibling \ASeven{} \citep{ArisakaA7Qi4} supplies the \QiFour{} pillars, with
which \AFive{} closes the \GDOS{} Trio.

\paragraph{The Consequence Trio and the capstone.} Three papers read the
empirical world off the boundary law: \AEight{} \citep{ArisakaA8QM} recasts
four-dimensional quantum mechanics, \ANine{} \citep{ArisakaA9SM} derives the
twenty-eight free parameters of the Standard Model from the \OGNfive{} ledger
and the \GRIP{} cascade of the present paper, and \ATen{}
\citep{ArisakaA10Cosmos} derives the $\Lambda$CDM cosmological inventory;
\AEleven{} \citep{ArisakaA11Origin} is the capstone that closes the A-series.

\paragraph{The two phenomenology branches.} Two framed branches carry the
sharp predictions. The Predictions Branch (P-series) collects the KK-resonance
and hierarchy paper \PaperPOne{} \citep{ArisakaP1KK}, the three-generations
paper \PaperPTwo{} \citep{ArisakaP2N3} --- whose $N=3$ proof the present
paper supplies the \GRIP{}-side input to --- the electron paper
\PaperPThree{} \citep{ArisakaP3Electron}, and the proton paper
\PaperPSix{} \citep{ArisakaP6Proton}, which runs that paper's
ground-attractor method on the baryonic stratum. The Cosmology \& Astrophysics
Branch (C-series) collects the Kaxion dark-matter paper \PaperCOne{}
\citep{ArisakaC1Kaxion}, the Hubble-tension paper \PaperCTwo{}
\citep{ArisakaC2Hubble}, and the black-hole-entropy paper \PaperCThree{}
\citep{ArisakaC3BH}. \ASix{} supplies the \GiFour{} geometric backbone on
which the Standard-Model and dark-sector readouts of these branches stand.

\paragraph{The Foundations Branch.} The third framed branch is the
Foundations Branch: the two member papers whose subject is the
boundary's complex structure and its mirrors. \PaperPFour{}
\citep{ArisakaP4Real} stands on the map's left flank as the
origin-of-$i$ member, deriving the single imaginary unit of boundary
physics from two real structures of the \ATwo{} arena --- the
compact phase circle and the causal clock. For the present paper its
significance is direct: \textsf{GG-A6-GRIP} carries the \GRIP{} pipeline
($\DGI{} \to \GSSB{} \to \GA{}$) and the \GFF{} ledger --- the
geometric runtime on which the cohort's open-system floors run ---
and \PaperPFour{} is the origin of the boundary's complex structure
whose phase conventions the \DGI{} stage makes dynamical.
\PaperPFive{} \citep{ArisakaP5CPT} carries the three mirrors of
matter --- C, P, and T read as orientation reversals of the three
real bulk factors --- and its law-level arrow of time is kin of the
irreversible descent that the runtime built here drives one way (the
second law $\dd F_{\GFF}/\dd t \le 0$ of Theorem~T3). Both carry
P-series designators; their place at the spine's flank, not the
phenomenology rows, marks their foundational role.

The twenty-two-paper map, identical in every paper of the cohort, is
Figure~\ref{fig:cohort-map-a6}, which closes this subsection.

%% ===================================================================
%%  The nineteen-paper cohort map (frozen v1.4 geometry), on its own
%%  page at the close of the cohort-inheritance subsection.  Verbatim
%%  copy of the v1.4 master (A1-Constitution Figure 3): no
%%  self-reference markers anywhere, so the figure is identical and
%%  simple for replication in all nineteen papers.  Only the internal
%%  node names MP4/MP5 differ from the master (collision avoidance).
%% ===================================================================
\begin{figure}[p]
\centering
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 wht/.style={fill=brown!12, draw=brown!65!black, line width=0.6pt,
 minimum width=43.7mm, minimum height=14.95mm},
 branchframe/.style={rounded corners=4pt, line width=0.6pt, inner sep=4pt},
 arrow/.style={-{Stealth[length=2.4mm]}, line width=0.55pt, draw=black!70},
 arrowdash/.style={-{Stealth[length=2.4mm]}, line width=0.55pt,
 draw=black!60, dashed},
 arrowrt/.style={-{Stealth[length=2.6mm]}, line width=0.85pt,
 draw=violet!70!black},
 edgelabel/.style={font=\scriptsize\itshape, fill=white,
 inner sep=1pt, draw=none, rounded corners=0pt},
 branchlabel/.style={font=\sffamily\scriptsize\bfseries,
 inner sep=2pt, rounded corners=0pt},
]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%% Centring pad, retired (v22): it cancelled a Bezier phantom that the
%% polyline above no longer creates; the bbox is now the ink.
\end{tikzpicture}}
\caption[The twenty-two-paper cohort map.]{The
twenty-two-paper cohort map of \GGTheory{}. The constitutional root
\AOne{} (violet, top) supplies the four axioms and seven postulates
from which the rest of the cohort descends: the GG6 Trio (\ATwo{},
\AThree{}, \AFour{}, red) realizes the bulk geometry; \AFive{}
(yellow) the boundary law; the \GDOS{} Trio pillars (\textsf{GG-A6-GRIP},
\ASeven{}, green) the geometric and quantum-information runtime; the
4D-Physics Trio (\AEight{}, \ANine{}, \ATen{}, blue) the textbook
readouts; and the closing capstone \AEleven{} (gold) the
quarks-to-consciousness identity cascade. Two framed branches carry
the phenomenology --- the \textbf{Predictions Branch} (cyan), whose
four papers hand their results along the row by the internal arrows
(\PaperPTwo{}'s $N{=}3$ into \PaperPThree{}, and that paper's
ground-attractor method, \textsf{GGA}, on into \PaperPSix{}), and the
\textbf{Cosmology \& Astrophysics Branch} (teal). The spine is
flanked by two columns of two, which do opposite jobs. On the left the
\textbf{Foundations Branch} (violet) feeds it: \PaperPFour{}, fed by
\ATwo{}'s Theorem~0 arena, manufactures the $i$ that \AEight{}
consumes, and \PaperPFive{} is the \ANine{}--\ATen{} interface, fed by
the ledger phases and feeding the $\eta_B$ chain across the width of
the map, with the shared-$i$ edge descending solid between them. On
the right the \textbf{White-Paper Branch} (brown) reads it:
\PaperWOne{} walks the century from Noether to Hopkins--Singer, and
\PaperWTwo{} dissects the structure that arc arrives at ---
protection by topology rather than by symmetry --- across sixteen open
problems. The white papers derive nothing new, carrying every result
at its parent's declared grade; the single dashed feed drawn into each
names its heaviest source, not its only one. \emph{Reading
conventions:} solid arrows mark axiom-forced descent and carry an edge
label naming the driving axiom or postulate where most illustrative;
dashed arrows mark the deployment of derived results, or historical
links inherited without being constitutionally forced.}
\label{fig:cohort-map-a6}
\end{figure}

\subsection{Roadmap of the twelve Parts}
\label{sec:roadmap-a6}

The rest of the paper is organized into eleven further Parts, and
the walk through them is plotted in Figure~\ref{fig:roadmap-a6}.
The present Part --- the foundations --- is nearly done: the
strategic moment, the 6D bulk geometry and the \BIsix{} closure
imported from the companions, the \OGGEP{} projector and the
\GDOS{} boundary law recalled from \AFive{}, and the cohort the
paper stands in. Everything after this page is the formalization
itself, and the itinerary below says where each piece lives.

The load-bearing block is the four-pillar core (Parts~II--V), in
which each pillar forces the next ($\TGG{} \Rightarrow \CCI{}
\Rightarrow \GFF{} \Rightarrow \GRIP{}$). Part~II constructs
\TGG{}, the Total Geometric--Gauge object: the single composite
connection $\Atot = \omega \oplus A$ that carries gravity and the
boundary gauge sectors on one principal $\Gtot$-bundle, its
boundary stress tensor, and the Master Conservation Law
(Theorem~T1). Part~III proves \CCI{}, the Covariant Conservation
Identity: under \MSMF{} closure of the four-form $X_4$, the
\OGNfive{} ledger $(1,3,5;7,35)$ is forced and the four boundary
Ward currents close together (Theorem~T2). Part~IV builds \GFF{},
the Geometric Free-Energy Functional: Morse-positivity on the
stratified boundary moduli and the second law
$\dd F_{\GFF}/\dd t \le 0$ along every \GDOS{} trajectory
(Theorem~T3).

Part~V is the apex, and it is highlighted in the roadmap because it
delivers the paper's central claim. There the \GRIP{} cascade
$\DGI{} \to \GSSB{} \to \GA{}$ closes against the \BIsix{}
budget $\Sbi \le 3\pi$, forcing $N \le 3$ on the family-projector
stratum, with the generation-ordered decay-width ladder as its
boundary signature (Theorem~T4). Every earlier Part builds toward
that closure, and every later Part spends it.

Two Parts then assemble and apply. Part~VI synthesizes the unified
dual-layer \GiFour{} system --- the nine dual-layer sub-theorems
and the constitutional bridge Theorem~T5 back to the
\OGGEP{}/\MSMF{} axioms, closing the loop with the
\GGConstitution{}. Part~VII applies the framework to the spectrum
of nature: the $N=3$ derivation, the twenty-eight Standard-Model
parameters, and the hierarchy, KK-resonance, and kaxion predictions.

Two Parts weigh the case. Part~VIII is the discussion: significance
and uniqueness, the measurement-problem carry-over from QM to QFT to
the Standard Model, and the comparison with string theory, AdS/CFT,
and Randall--Sundrum. Part~IX states the conclusions: the
methodological synthesis, the open-problem ledger M9--M11, and the place
of the construction in the foundations of physics.

The back matter is the workshop. Part~X collects the technical
machinery, worked examples, and indices that support the four
pillars; Part~XI answers the strongest objections a hostile reviewer
would raise, with pointers into the paper; and Part~XII closes with
the annotated references, each carrying a Relevance note. Read in
order, the twelve Parts compose into a single self-contained
formalization that any reader can audit step by step; read by need,
each Part names its imports and can stand a visit on its own.

%% ===================================================================
%%  Roadmap of the twelve Parts (house architecture: A1-Constitution
%%  Figure 4 / A2-GG6 v13.3 / A3-BI6 v13.6 / A4-Euclid v13.0 /
%%  A5-GDOS v13.0 pattern -- vertical Part column, right phase
%%  sidebar, one highlighted central Part; here Part V, the GRIP
%%  cascade closure), clearpage-isolated minipage + \captionof per
%%  the A1 pattern.
%% ===================================================================
\clearpage
\null\vfill
\noindent
\begin{minipage}{\textwidth}
\centering
\resizebox{0.961\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily\footnotesize,
 every node/.style={align=center, rounded corners=2pt, inner sep=3pt},
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 align=left, inner sep=2pt},
]

%% --- Part I: foundations ---
\node[found] (I) at (0, 0)
 {\textbf{Part~I}\quad\textbf{Introduction and Setup} \\[-1pt]
 \footnotesize The strategic moment; the imported bulk and boundary primitives; the cohort};

%% --- Parts II--IV: Pillars 1-3 ---
\node[geom] (II) at (0, -13mm)
 {\textbf{Part~II}\quad\textbf{\TGG{} --- Pillar 1} \\[-1pt]
 \footnotesize Composite connection $\Atot = \omega \oplus A$ and the Master Conservation Law (T1)};
\node[geom] (III) at (0, -26mm)
 {\textbf{Part~III}\quad\textbf{\CCI{} --- Pillar 2} \\[-1pt]
 \footnotesize \OGNfive{} ledger $(1,3,5;7,35)$ and boundary conservation (T2)};
\node[geom] (IV) at (0, -39mm)
 {\textbf{Part~IV}\quad\textbf{\GFF{} --- Pillar 3} \\[-1pt]
 \footnotesize Morse free-energy functional and the second law (T3)};

%% --- Part V: GRIP, the cascade closure (highlighted) ---
\node[innov] (V) at (0, -56mm)
 {\textbf{Part~V}\quad\textbf{\GRIP{} --- Pillar 4} \\[1pt]
 \footnotesize Cascade $\DGI{} \to \GSSB{} \to \GA{}$ closes against the \BIsix{} budget $\Sbi \le 3\pi$: $N \le 3$ (T4) \\[-1pt]
 \footnotesize \emph{\textbf{The apex of the four-pillar core: the family-count forcing}}};

%% --- Parts VI--VII: synthesise and apply ---
\node[princ] (VI) at (0, -73mm)
 {\textbf{Part~VI}\quad\textbf{Synthesis} \\[-1pt]
 \footnotesize The unified dual-layer \GiFour{} system; the constitutional bridge T5};
\node[princ] (VII) at (0, -86mm)
 {\textbf{Part~VII}\quad\textbf{Applications} \\[-1pt]
 \footnotesize $N = 3$; the 28 SM parameters; hierarchy, KK, kaxion};

%% --- Parts VIII--IX: discuss and conclude ---
\node[summit] (VIII) at (0, -99mm)
 {\textbf{Part~VIII}\quad\textbf{Discussion} \\[-1pt]
 \footnotesize Significance; measurement-problem carry-over; string/AdS-CFT/RS comparison};
\node[summit] (IX) at (0, -112mm)
 {\textbf{Part~IX}\quad\textbf{Conclusions} \\[-1pt]
 \footnotesize Methodological synthesis; the M9--M11 ledger};

%% --- Parts X--XII: back matter ---
\node[back] (X) at (0, -125mm)
 {\textbf{Part~X}\quad\textbf{Technical Appendices} \\[-1pt]
 \footnotesize Technical machinery, worked examples, and indices};
\node[back] (XI) at (0, -138mm)
 {\textbf{Part~XI}\quad\textbf{Common Objections and Responses} \\[-1pt]
 \footnotesize The strongest objections, answered, with pointers};
\node[back] (XII) at (0, -151mm)
 {\textbf{Part~XII}\quad\textbf{References} \\[-1pt]
 \footnotesize Cited works, each with a relevance note};

%% --- Vertical flow arrows (the pillar chain in innov-orange) ---
\draw[arrow] (I.south) -- (II.north);
\draw[arrowinnov] (II.south) -- (III.north);
\draw[arrowinnov] (III.south) -- (IV.north);
\draw[arrowinnov] (IV.south) -- (V.north);
\draw[arrow] (V.south) -- (VI.north);
\draw[arrow] (VI.south) -- (VII.north);
\draw[arrow] (VII.south) -- (VIII.north);
\draw[arrow] (VIII.south) -- (IX.north);
\draw[arrow] (IX.south) -- (X.north);
\draw[arrow] (X.south) -- (XI.north);
\draw[arrow] (XI.south) -- (XII.north);

%% --- Phase labels (right sidebar) ---
%% Standard boxes: half-width = 60mm, label at x = 62mm.  Part V is
%% wider (half-width = 70mm); its phase label sits beside the box at
%% x = 72mm, per the house precedent for the highlighted Part.
\node[phaselabel] at (62mm, 0mm) {\emph{Foundations}};
\node[phaselabel] at (62mm, -26mm) {\emph{The four-pillar} \\ \emph{core (II--V)}};
\node[phaselabel, text=orange!70!black] at (72mm, -56mm)
 {\emph{\textbf{The cascade closure:}} \\ \emph{\textbf{the apex}}};
\node[phaselabel] at (62mm, -79.5mm) {\emph{Synthesize} \\ \emph{\& apply}};
\node[phaselabel] at (62mm, -105.5mm) {\emph{Discuss \&} \\ \emph{conclude}};
\node[phaselabel] at (62mm, -138mm) {\emph{Back matter}};

\end{tikzpicture}%
}
\par\bigskip
\captionof{figure}[Roadmap of the twelve Parts of GG-A6-GRIP.]{Roadmap of the twelve Parts of \textsf{GG-A6-GRIP}. The argument develops in five phases. \emph{Foundations} (Part~I): the strategic moment, the imported bulk and boundary primitives, and the cohort. \emph{The four-pillar core} (Parts~II--V): the composite connection and the Master Conservation Law (\TGG{}, T1), the forced \OGNfive{} ledger $(1,3,5;7,35)$ and simultaneous boundary conservation (\CCI{}, T2), the Morse free-energy functional and the second law (\GFF{}, T3), and --- \textbf{Part~V}, highlighted --- the \GRIP{} cascade $\DGI{} \to \GSSB{} \to \GA{}$ closing against the \BIsix{} budget $\Sbi \le 3\pi$ and forcing $N \le 3$ (T4). \emph{Synthesize and apply} (Parts~VI--VII): the unified dual-layer system with the constitutional bridge T5, then the applications to the $N=3$ count, the twenty-eight Standard-Model parameters, and the hierarchy, KK-resonance, and kaxion predictions. \emph{Discuss and conclude} (Parts~VIII--IX) weigh significance and state the open-problem ledger M9--M11. The three back-matter Parts collect the technical appendices (Part~X), the common objections and responses (Part~XI), and the annotated references (Part~XII). Part~V is highlighted because the cascade closure delivered there is the paper's central claim --- the family-count forcing and the boundary signature of the cascade, on which the downstream Standard-Model readout stands.}
\label{fig:roadmap-a6}
\end{minipage}
\vfill\clearpage


%% ====================================================================
\section{What it costs to declare your inputs}
\label{sec:close-part1}
%% ====================================================================

Most papers begin by saying what they will show. This one has begun by
saying what it is standing on, and the difference is worth a moment
before the construction starts.

A theory that reaches as far as this one claims to reach --- from the
geometry of a six-dimensional arena down to the number of families of
matter and up again to the chemistry of a living cell --- has a
characteristic way of going wrong. Not by making an arithmetic error;
those get found. It goes wrong by borrowing. A quantity is taken from
one companion argument, used to fix a second, and the second is then
cited back as support for the first, and nobody notices because the two
statements are separated by a hundred pages and a change of notation.
The result looks like a derivation and is a circle. Every large
framework in the history of physics has had to be audited for this, and
several have failed the audit.

It is worth noticing how short the list turns out to be. An arena of a
certain dimension, a group acting on it, a set of whole numbers, a rule
for how a large system looks to a small observer inside it, and four
axioms. That is the whole of what this paper takes on trust. Everything
else it uses, it builds. Whether the list is short enough is a fair
question and a reader is entitled to press it --- six dimensions is not
a modest assumption --- but the question can at least be asked, because
the list exists and is finite. Against a framework that never wrote its
inputs down, the same question cannot even be formulated.

The only defense is bookkeeping, done in public and done first. That is
what this Part has been. Every object the construction will use --- the
arena, its dimension, the group that acts on it, the integer ledger,
the boundary law, the axioms --- has been named, attributed to the
companion that establishes it, and marked as an input rather than a
result. Nothing later in this paper may quietly promote one of them.
When a later Part needs something not on this list, it must derive it,
and the reader is entitled to ask where.

There is a scientific point buried in what looks like an
administrative one. \emph{A framework's honesty is measurable, and this
is how you measure it}: count the things it assumes, count the things
it claims to derive, and check that the second list never leaks into
the first. A theory with a short input list and a long output list is
making a strong claim and is easy to kill. A theory whose two lists are
tangled is making no checkable claim at all, however impressive its
mathematics. The declaration in this Part is what makes the rest of the
paper falsifiable, and that is a stronger reason for it than tidiness.

What comes next is the construction proper, and it will proceed in one
direction only: from the declared inputs, forward. Four objects are
built in turn, each using only what precedes it. If the reader ever
finds a step that reaches backwards --- that uses a later result to
justify an earlier one --- the paper has failed, and this Part is the
list against which that judgment can be made.


\GGPart[TGG]{\TGG{}}{The Total Geometric-Gauge Object: Pillar 1 of \GiFour{}}

Physics currently describes the world with four forces and,
uncomfortably, with two different kinds of mathematics. Gravity is
geometry: matter bends the arena, and things move as the bending
dictates. The other three forces --- electromagnetism, the strong force,
the weak force --- are usually described instead by fields living
\emph{on} an arena that is itself unbent. Both descriptions work
superbly. They do not obviously describe the same kind of thing, and a
century of effort has gone into the suspicion that they must.

This Part takes the suspicion seriously in the most economical way
available: it asks whether the four can be read as four aspects of a
single geometric object, rather than four objects to be reconciled. The
object proposed is what we call the Total Geometric-Gauge object,
\TGG{}. The idea is easier than the machinery. Instead of one structure
for gravity and three separate structures for the other forces, one
writes down a single structure on the six-dimensional bulk that has all
four built into it, and one shows that it can be done in exactly one
way. The forces are then not unified by decree; they are what the one
structure looks like when it is examined along different directions.

What this Part establishes is threefold. It shows that such an object
exists and that it is unique --- there is no second way to assemble the
four that satisfies the constitutional requirements. It shows what that
object hands to the four-dimensional world we can observe: a single
tensor collecting the energy and momentum of matter, of the gauge
fields, of the residual higher-dimensional modes, and of the correction
term that anomaly cancellation demands. And it shows that this one
tensor obeys one conservation law --- which is the mathematical form of
the statement that energy and momentum do not leak out of the world by
an unaccounted route. That is the Master Conservation Law, and it is the
Part's principal result.

Everything geometric in this Part is taken as given from the companion
papers; what is new is the assembly and the uniqueness. A reader
outside the specialty can take the first section, which states what the
object is and why exactly four pieces enter it, and then the statement
of the Master Conservation Law, and skip the intervening construction
without losing the thread. A reader who wants to check the arithmetic
will find the worked examples at the end of the Part, computed rather
than quoted.


\section{\TGG{}: motivation, dual-layer structure, and Structure Theorems}
\label{sec:TGG-axioms}

This Part develops \TGG{} (the \emph{Total Geometric-Gauge object}), the
first of the four \GiFour{} Pillars. \TGG{} is the unique geometric
object whose two layers are: (i)~the \emph{bulk layer} -- the master
composite-connection structure on the 6D bulk $\Msix$ that
unifies the gauge and gravity sectors into a single principal bundle;
and (ii)~the \emph{boundary layer} -- the unified 4D stress tensor
$\cT^{\mu\nu}_{\TGG}$ obtained by zero-mode reduction of the bulk action,
which collects matter, gauge, Green--Schwarz inflow, and
Kaluza--Klein-remnant contributions into a single object. The two layers
are tied together by the \emph{Master Conservation Law}
$\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ that holds iff the full Ward
package (electric, baryonic, informational, and metric currents) is
satisfied. \TGG{} is the formal mathematical realization of \GGEP{}
extended by \OGGEP{} to the boundary, and the unbroken-symmetry \TGG{}
phase corresponds to the \DGI{} stage (Dynamical Gauge Invariance) of
the \GRIP{} cascade.

The Part has six sections. Section~\ref{sec:TGG-axioms} (the present
section) states the seven \TGG{} Structure Theorems -- five for the bulk layer
(\textbf{TGG-ST1} through \textbf{TGG-ST5}) and two for the boundary
layer (\textbf{TGG-ST6} and \textbf{TGG-ST7}) -- gives the formal
definition of \TGG{} as the dual-layer object, and lists the first
immediate consequences. Section~\ref{sec:composite-connection}
develops the bulk-layer machinery: the principal $\Gtot$-bundle
construction, the connection and curvature decompositions, and the
Chern--Simons construction of the Kalb--Ramond three-form.
Section~\ref{sec:magic-ladder} presents the magic-dimension ladder
backbone and explains why $d=6$ is the unique admissible bulk
dimension. Section~\ref{sec:TGG-boundary-stress-tensor} (new in)
develops the boundary-layer machinery: the Kaluza--Klein zero-mode
reduction, the explicit form of $\cT^{\mu\nu}_{\TGG}$, and the
projected modified Bianchi identity (PBI-4).
Section~\ref{sec:TGG-theorem} states and proves the existence,
uniqueness, and Master Conservation theorem \textbf{T1}.
Section~\ref{sec:TGG-examples} contains worked examples.

\subsection{Motivation: from disconnected stress-energies to a single Total Geometric-Gauge object}
\label{ssec:TGG-motivation}

The \GG{} program identifies, in
\citet{ArisakaA1Constitution} and \citet{ArisakaA2GG6}, four physical
sectors on the bulk: gravity (encoded by the $\Spinone$ frame bundle),
the visible $\SUfive$ gauge sector, the gravity-baryon $\UoneB$ sector,
and the quantum-information $\UoneQi$ sector. In closed-system QFT,
these four sectors would be assembled as four \emph{independent}
principal bundles, glued together only by the requirement that they
share a common base manifold. The result of that closed-system
assembly is a fragile structure: each sector carries its own gauge
freedom, its own connection, its own curvature, and (most importantly
for the present Part) its own stress tensor on the boundary, with no
algebraic mechanism enforcing consistency between them. In particular,
before the boundary theory carried four disconnected stress-energy
objects -- the matter stress tensor $T^{\mu\nu}_{\rm matter}$, the
gauge-field stress tensor $T^{\mu\nu}_{\rm gauge}[F]$, the
Green--Schwarz inflow term $X^{\mu\nu}_{\rm GS}$, and the Kaluza--Klein
heavy-mode remnant $T^{\mu\nu}_{\rm KK}$ -- and their sum-conservation
was an assumption, not a theorem. Anomaly cancellation in that
closed-system picture was imposed as a separate constraint, and the
unification of gauge with gravity was left as a hope rather than a
derived theorem.

\GGEP{} from the \GGConstitution{}
\citep[Axiom~2]{ArisakaA1Constitution} rejects this fragmented
picture and asserts that the four sectors arise from a single primitive
composite connection at the \emph{bulk} level. \OGGEP{}
\citep[Axiom~3]{ArisakaA1Constitution} extends this further to the
\emph{boundary} level: the boundary observable content is the \OGGEP{}
zero-mode projection of the bulk geometric content, so the four
boundary stress tensors must also collapse into one. Together, \GGEP{}
and \OGGEP{} force the existence of a single mathematical object with
two layers:

\begin{itemize}[leftmargin=2em]
\item the \emph{bulk layer} is the principal $\Gtot$-bundle
$\Pbun\to\Msix$ with structure group
$\Gtot=\Spinone\times\SUfive\times\UoneB\times\UoneQi$, equipped with a
composite connection one-form $\Atot$ and curvature $\Rtot$;
\item the \emph{boundary layer} is the unified 4D stress tensor
\[
\cT^{\mu\nu}_{\TGG} \;=\; T^{\mu\nu}_{\rm matter}
+ T^{\mu\nu}_{\rm gauge}[F]
+ X^{\mu\nu}_{\rm GS}
+ T^{\mu\nu}_{\rm KK},
\]
obtained by Kaluza--Klein zero-mode reduction of the bulk action of the
composite connection together with the \BIsix{} bulk-to-boundary descent.
\end{itemize}

The two layers are not independent. They are connected by the
\emph{Master Conservation Law},
\[
\nabla_\mu \cT^{\mu\nu}_{\TGG} \;=\; 0,
\]
which holds iff the full Ward package (electric, baryonic, and
informational currents, together with the projected modified Bianchi
identity) is satisfied. This master conservation law is a theorem of
the bulk geometric structure (\GGEP{} + \BIsix{}) projected to the boundary
(\OGGEP{}); it is not an assumption.

This single object with its two layers and the unifying conservation
theorem is what we call the \TGG{} (\emph{Total Geometric-Gauge object}).
The name encodes the three structural facts: it is \emph{Total} because
it collects the entire stress-energy content into one tensor; it is
\emph{Geometric-Gauge} because the bulk layer unifies the gauge
sub-connections with the spin connection of gravity; and it is an
\emph{object} (rather than a connection or a tensor alone) because
the dual-layer nature is essential to its definition.

\begin{remark}[Physical intuition for \TGG{}: a familiar analogue]
\label{rem:TGG-intuition-physical}
The dual-layer structure of \TGG{} can be made concrete by analogy with
ordinary electromagnetism. In Maxwell's theory there is, on the one
hand, the electromagnetic field itself -- the gauge potential $A^\mu$
and field strength $F^{\mu\nu}$ that live on every spacetime point --
and, on the other hand, the unified energy-momentum tensor
$T^{\mu\nu}_{\rm matter} + T^{\mu\nu}_{\rm EM}$ whose conservation
$\nabla_\mu(T_{\rm matter} + T_{\rm EM})^{\mu\nu} = 0$ is forced by
charge conservation $\nabla_\mu J^\mu = 0$. The two are not
independent: the field generates the unified stress tensor by
variation of the action, and current conservation links them.

\TGG{} is the six-sector generalization of this picture. The
``electromagnetic field'' is replaced by a composite connection on the
bulk that unifies gravity, $\SUfive$, $\UoneB$, and $\UoneQi$ into
\emph{one} object. The ``unified energy-momentum tensor'' is replaced
by $\cT^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge} + X_{\rm GS}
+ T_{\rm KK}$, the boundary projection of the bulk action. The
``current conservation'' is replaced by the Ward package of all
four sectors simultaneously, conserving the unified stress tensor by
the Master Conservation Law. In each layer of analogy the framework
broadens: from one gauge sector to four, from one current to four,
from one boundary spacetime to a 6D-bulk-projecting-to-4D structure.
Readers familiar with Maxwell theory will find every \TGG{} statement to
be a structurally identical generalization of a familiar result; the
generalization is rigorous because \BIsix{} closure plus the \OGNfive{} ledger
guarantee that the generalization closes uniquely.
\end{remark}

\subsubsection{Historical background (with citations)}

The unification of gauge with gravity into a single composite connection
that \TGG{} formalizes is the culmination of a thread that runs through
nearly the whole modern history of theoretical physics.
\citet{Maxwell1865} wrote down the first gauge-theoretic structure in
physics: the four-potential $A_\mu$ of electromagnetism is defined only
up to the gradient of a scalar, $A_\mu \to A_\mu + \partial_\mu \lambda$,
and the field strength $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu
A_\mu$ is the curvature of this connection. Maxwell did not use the
word \emph{gauge}; the geometric reading came later.
\citet{Einstein1915} produced the second strand a half-century later by
recasting gravitation as the curvature of a metric connection
$\Gamma^\rho_{\mu\nu}$ on the tangent bundle of spacetime; here too the
field equations are invariant under a redefinition of the coordinate
frame, and the curvature is a tensorial object.

Weyl (1918) was the first to ask whether the two threads were the same
thread. His original gauge proposal --- rescaling the metric by a
position-dependent factor --- did not survive empirical scrutiny, but
it identified the structural question correctly: are gauge fields and
gravity two faces of a single geometric object on a single bundle? The
mathematical infrastructure to answer that question precisely was
developed by Ehresmann (1950) in his theory of connections on principal
bundles and codified by \citet{KobayashiNomizu1969}.
\citet{YangMills1954} produced the non-abelian extension of Maxwell's
gauge theory, with the field strength
$F^a_{\mu\nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc}
A^b_\mu A^c_\nu$ as the curvature of an $\mathrm{SU}(2)$ connection.

Indeed, \TGG{} closes this loop. Its composite connection
$\cA_{\rm tot} = \omega \oplus A$ on a principal $\Gtot$-bundle over the
6D bulk treats gravity (the $\Spinone$ part $\omega$) and the
boundary gauge fields (the $\SUfive \times \UoneB \times \UoneQi$ part
$A$) as components of a single mathematical object. The Master
Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ that we derive in
Section~\ref{thm:TGG-existence-uniqueness} is the axiomatic completion of the
lineage from Maxwell through Yang--Mills to \TGG{}, with the
\GGConstitution{} \AX{2} \GGEP{} as its principled foundation
\citep{ArisakaA1Constitution}.

We now state the seven \TGG{} Structure Theorems -- five governing the bulk layer
and two governing the boundary layer.

\subsection{The seven \TGG{} axioms (five bulk + two boundary)}
\label{ssec:TGG-seven-st}

\begin{structheorem}[\textbf{TGG-ST1}: Total bundle structure]
\label{st:TGG-ST1}
On the 6D bulk $\Msix$, there exists a smooth principal bundle
\begin{equation}
\label{eq:TGG-ST1}
\pi: \Pbun \to \Msix
\end{equation}
with structure group
\begin{equation}
\label{eq:Gtot-def}
\Gtot = \Spinone \times \SUfive \times \UoneB \times \UoneQi.
\end{equation}
The four group factors act on $\Pbun$ as commuting subgroups, and the
free right action of $\Gtot$ on $\Pbun$ has quotient $\Msix$.
\end{structheorem}

\begin{structheorem}[\textbf{TGG-ST2}: Composite connection]
\label{st:TGG-ST2}
The bundle $\Pbun$ admits a connection one-form
\begin{equation}
\label{eq:Atot-def}
\Atot \in \Omega^1(\Pbun, \gtot),
\qquad \gtot = \mathfrak{spin}(1,5)\oplus\mathfrak{su}(5)\oplus\mathfrak{u}(1)_B\oplus\mathfrak{u}(1)_{Qi},
\end{equation}
which decomposes along the direct-sum factors as
\begin{equation}
\label{eq:Atot-decomp}
\Atot \;=\; \omega_{\Spinone} \;+\; A_5 \;+\; A_B \;+\; A_{Qi},
\end{equation}
where $\omega_{\Spinone} \in \Omega^1(\Pbun, \mathfrak{spin}(1,5))$ is
the spin connection, $A_5 \in \Omega^1(\Pbun, \mathfrak{su}(5))$ is the
$\SUfive$ connection, $A_B \in \Omega^1(\Pbun, \mathfrak{u}(1)_B)$ is
the $\UoneB$ connection, and
$A_{Qi} \in \Omega^1(\Pbun, \mathfrak{u}(1)_{Qi})$ is the $\UoneQi$
connection.
\end{structheorem}

\begin{structheorem}[\textbf{TGG-ST3}: Magic-dimension ladder]
\label{st:TGG-ST3}
The $\Spinone$ factor of $\Gtot$ is realized through the accidental
isomorphism
\begin{equation}
\label{eq:spin15-iso-A3}
\Spinone \;\cong\; \SLtwoH,
\end{equation}
which places the bulk dimension $d=6$ as the unique admissible value
in the magic-dimension ladder $\R\to\C\to\HH\to\Octonion$ (see
Section~\ref{sec:magic-ladder} for the detailed treatment).
\end{structheorem}

\begin{structheorem}[\textbf{TGG-ST4}: Lorentzian compatibility]
\label{st:TGG-ST4}
The bulk metric
$\dd s^2_6 = \e^{-2r}\eta_{\mu\nu}\dd x^\mu\dd x^\nu + \dd r^2 + R_\theta^2 \dd\theta^2$
on $\Msix$ is compatible with the $\Spinone$ structure of \textbf{TGG-ST1},
in the sense that the spin connection $\omega_{\Spinone}$ is the
Levi-Civita-type connection associated with this metric. The
product-geometry structure of $\Msix$ is preserved by the gauge action
of $\Gtot$ on $\Pbun$.
\end{structheorem}

\begin{structheorem}[\textbf{TGG-ST5}: \GGEP{} unification]
\label{st:TGG-ST5}
The four sub-connections
$(\omega_{\Spinone},\allowbreak A_5,\allowbreak A_B,\allowbreak A_{Qi})$ are not four independent
connections on four independent bundles; they are four projections of
the single composite connection $\Atot$ of \textbf{TGG-ST2}. Equivalently,
gauge transformations of $\Atot$ act simultaneously on all four
sub-connections by the natural diagonal action of $\Gtot$ on
$\gtot$. This is the formal realization of \GGEP{} on $\Msix$.
\end{structheorem}

The five Structure Theorems \textbf{TGG-ST1}--\textbf{TGG-ST5} characterize the
\emph{bulk layer} of \TGG{}. We now state the two additional Structure Theorems
that characterize the \emph{boundary layer} -- the 4D unified stress
tensor and its master conservation law.

\begin{structheorem}[\textbf{TGG-ST6}: Boundary stress tensor projection]
\label{st:TGG-ST6}
Under Kaluza--Klein zero-mode reduction of the bulk Einstein--Hilbert
plus gauge action of \TGG-ST2, together with the \BIsix{} bulk-to-boundary descent of the anomaly polynomial $X_4$, the bulk stress-energy
content projects onto the 4D boundary $\Mfour=\partial\Msix$ as a
single \emph{Total Geometric-Gauge stress tensor}
\begin{equation}
\label{eq:TGG-ST6}
\cT^{\mu\nu}_{\TGG} \;:=\; T^{\mu\nu}_{\rm matter}
+ T^{\mu\nu}_{\rm gauge}[F]
+ X^{\mu\nu}_{\rm GS}
+ T^{\mu\nu}_{\rm KK},
\end{equation}
where $T^{\mu\nu}_{\rm matter}$ is the boundary matter stress tensor,
$T^{\mu\nu}_{\rm gauge}[F]$ is the boundary gauge-field stress tensor
constructed from $F = F_5 + F_B + F_{Qi}$, $X^{\mu\nu}_{\rm GS}$ is the
boundary shadow of the Green--Schwarz inflow term in \BIsix{}, and
$T^{\mu\nu}_{\rm KK}$ collects the heavy Kaluza--Klein-mode remnants.
The boundary projection respects \OGGEP{} of the \GGConstitution{}.
\end{structheorem}

\begin{structheorem}[\textbf{TGG-ST7}: Master Conservation Law]
\label{st:TGG-ST7}
The Total Geometric-Gauge stress tensor of \textbf{TGG-ST6} satisfies the
covariant conservation equation
\begin{equation}
\label{eq:TGG-ST7}
\nabla_\mu \cT^{\mu\nu}_{\TGG} \;=\; 0
\end{equation}
in the gauge-closed boundary sector, holding if and only if the full
Ward package is satisfied simultaneously:
\begin{equation}
\label{eq:Ward-package}
\nabla_\mu J^\mu_{\rm e} = 0,\quad
\nabla_\mu J^\mu_B = 0,\quad
\nabla_\mu J^\mu_{Qi} = 0,\quad
\nabla_\mu(G^{\mu\nu} + X^{\mu\nu}_{\rm GS}) = 0,
\end{equation}
where the last equation is the projected modified Bianchi identity
\textnormal{(PBI-4)}, the 4D zero-mode descent of the 6D \BIsix{}
identity $\dd H_3 = X_4$.
\end{structheorem}

The seven Structure Theorems are stated for the 6D bulk of \GGTheory{};
\textbf{TGG-ST1}--\textbf{TGG-ST5} characterize the bulk layer (the
composite connection on $\Msix$), and \textbf{TGG-ST6}--\textbf{TGG-ST7}
characterize the boundary layer (the unified stress tensor on
$\Mfour$ and its master conservation law). All seven Structure Theorems have
natural generalizations to other Lorentzian bulk manifolds with
appropriate magic-dimensional structure, but those generalizations are
not the topic of the present paper.

\subsection{Formal definition of \TGG{}}
\label{ssec:TGG-formal-def}

\begin{definition}[\TGG{}: the Total Geometric-Gauge object]
\label{def:TGG}
The \emph{Total Geometric-Gauge object} associated with the 6D bulk $\Msix$ and its 4D boundary $\Mfour=\partial\Msix$ is the
quintuple
\begin{equation}
\label{eq:TGG-triple}
\TGG = \bigl(\,\Pbun,\;\Atot,\;\Rtot\,;\;\cT^{\mu\nu}_{\TGG},\;\nabla\bigr),
\end{equation}
consisting of two layers tied together by the Master Conservation Law:

\textbf{Bulk layer} (\textbf{TGG-ST1}--\textbf{TGG-ST5}):
\begin{itemize}[leftmargin=2em]
\item $\Pbun \to \Msix$ is the principal $\Gtot$-bundle of \textbf{TGG-ST1};
\item $\Atot \in \Omega^1(\Pbun, \gtot)$ is the composite connection
one-form of \textbf{TGG-ST2}, decomposing as in \eqref{eq:Atot-decomp};
\item $\Rtot \in \Omega^2(\Pbun, \gtot)$ is the corresponding curvature
two-form,
\begin{equation}
\label{eq:Rtot-def}
\Rtot \;=\; \dd \Atot \;+\; \Atot \wedge \Atot,
\end{equation}
decomposing as
\begin{equation}
\label{eq:Rtot-decomp}
\Rtot \;=\; R_{\Spinone} \;+\; F_5 \;+\; F_B \;+\; F_{Qi},
\end{equation}
with the four sub-curvatures defined by the corresponding sub-connections.
\end{itemize}

\textbf{Boundary layer} (\textbf{TGG-ST6}--\textbf{TGG-ST7}):
\begin{itemize}[leftmargin=2em]
\item $\cT^{\mu\nu}_{\TGG}$ is the unified Total Geometric-Gauge stress
tensor on $\Mfour$ given by \eqref{eq:TGG-ST6}, the Kaluza--Klein
zero-mode reduction of the bulk action of \TGG-ST2 plus the \BIsix{}
bulk-to-boundary descent;
\item $\nabla$ is the boundary Levi-Civita covariant derivative, with
the Master Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$
holding iff the Ward package \eqref{eq:Ward-package} is satisfied.
\end{itemize}

The quintuple $(\Pbun,\Atot,\Rtot;\cT^{\mu\nu}_{\TGG},\nabla)$ is
required to satisfy Structure Theorems \textbf{TGG-ST1}--\textbf{TGG-ST7}. Two such
quintuples are \emph{gauge-equivalent} if their bulk components
$(\Pbun,\Atot,\Rtot)$ are related by a global gauge transformation
$g: \Msix \to \Gtot$ acting on $\Pbun$ by the right action and on
$\Atot, \Rtot$ by the corresponding adjoint action; the boundary
components $(\cT^{\mu\nu}_{\TGG},\nabla)$ are then automatically
related by the induced boundary diffeomorphism.
\end{definition}

\begin{remark}
\label{rem:TGG-intuition}
The intuitive content of Definition~\ref{def:TGG} is that \TGG{}
is a single dual-layer geometric object: in the bulk, four ``faces''
of one composite connection (gravity, visible gauge, baryon, $Qi$); on
the boundary, four contributions to one stress tensor (matter, gauge
fields, GS inflow, KK remnant). The bulk layer encodes the algebraic
content of \GGEP{} (the four sub-connections share a single composite
origin); the boundary layer encodes the dynamical content of \OGGEP{}
(the four boundary stress-energy contributions share a single conserved
master tensor). The two layers are connected by the \BIsix{} bulk-to-boundary descent, and the unifying theorem is the Master Conservation Law of
\textbf{TGG-ST7}.
\end{remark}

\subsection{First consequences of the \TGG{} Structure Theorems}
\label{ssec:TGG-first-consequences}

We list six immediate consequences of \textbf{TGG-ST1}--\textbf{TGG-ST7}
that will be used repeatedly in the rest of the paper. The first five
concern the bulk layer; the sixth concerns the boundary layer.

\begin{proposition}[Existence of frame bundle]
\label{prop:TGG-frame}
The $\Spinone$ factor of $\Pbun$ contains, as a sub-bundle, the
oriented Lorentzian frame bundle $F\Msix \to \Msix$ associated with the
scale-profile metric of \textbf{TGG-ST4}. Equivalently, the pull-back of
$\Atot$ to $F\Msix$ along the natural inclusion is the Levi-Civita
spin connection.
\end{proposition}

\begin{proof}
By \textbf{TGG-ST1}, the $\Spinone$ factor acts freely on $\Pbun$ with
quotient $\Msix$. The induced principal $\Spinone$-sub-bundle is
the spin frame bundle of $\Msix$. Compatibility with the metric
(\textbf{TGG-ST4}) selects the Levi-Civita spin connection up to gauge.
\end{proof}

\begin{proposition}[Integrability of sub-connections]
\label{prop:TGG-subconn-integrability}
Each of the four sub-connections
$(\omega_{\Spinone},\allowbreak A_5,\allowbreak A_B,\allowbreak A_{Qi})$ is, on its own, an
integrable connection one-form satisfying
$\dd \omega_{\Spinone} + \omega_{\Spinone}\wedge\omega_{\Spinone} = R_{\Spinone}$,
$\dd A_5 + A_5\wedge A_5 = F_5$,
$\dd A_B = F_B$,
$\dd A_{Qi} = F_{Qi}$,
where the $\U(1)$ field strengths have no commutator term because they
are Abelian.
\end{proposition}

\begin{proof}
The four direct-summands of $\gtot$ commute pairwise. The composite
curvature \eqref{eq:Rtot-def} therefore decomposes \emph{additively}
along the four factors with no cross-terms:
\begin{align}
\Rtot &= \dd(\omega + A_5 + A_B + A_{Qi})
+ (\omega + A_5 + A_B + A_{Qi})\wedge(\omega + A_5 + A_B + A_{Qi})\\
&= \dd\omega + \omega\wedge\omega + \dd A_5 + A_5\wedge A_5 + \dd A_B + \dd A_{Qi}.
\end{align}
The cross-terms $\omega\wedge A_5$, $A_5\wedge A_B$, etc., vanish
because $[\mathfrak{spin}(1,5),\mathfrak{su}(5)] = 0$ and similarly
for the other commutators. This yields the four sub-curvatures listed.
\end{proof}

\begin{proposition}[Gauge group of \TGG{}]
\label{prop:TGG-gauge-group}
The gauge group of \TGG{} is the group $\mathcal{G}(\Pbun)$ of
$\Gtot$-equivariant smooth maps $\Pbun \to \Gtot$. Two \TGG{} triples
are gauge-equivalent if and only if they are related by an element of
$\mathcal{G}(\Pbun)$.
\end{proposition}

\begin{proof}
This is standard for principal bundles; see, e.g., \citet{KobayashiNomizu1969}.
The four-factor structure of $\Gtot$ implies that $\mathcal{G}(\Pbun)$
factors as the direct product
$\mathcal{G}_{\Spinone}\times \mathcal{G}_5 \times \mathcal{G}_B
\times \mathcal{G}_{Qi}$.
\end{proof}

\begin{proposition}[Holonomy decomposition]
\label{prop:TGG-holonomy-decomp}
The holonomy group of \TGG{} along any closed loop $\gamma\subset \Msix$
factorizes as
\begin{equation}
\label{eq:TGG-holonomy}
\mathrm{Hol}(\TGG, \gamma)
= \mathrm{Hol}(\omega_{\Spinone}, \gamma)
\times \mathrm{Hol}(A_5, \gamma)
\times \mathrm{Hol}(A_B, \gamma)
\times \mathrm{Hol}(A_{Qi}, \gamma)
\end{equation}
along the four sub-connection factors.
\end{proposition}

\begin{proof}
The path-ordered exponential
$P\exp\bigl(\oint_\gamma \Atot\bigr)$ factors along commuting Lie-algebra
factors of $\gtot$ into the product of path-ordered exponentials of the
four sub-connections. This is the Baker--Campbell--Hausdorff identity
for commuting generators.
\end{proof}

\begin{proposition}[Spin(1,5) holonomy along compact $\theta$ direction]
\label{prop:TGG-theta-holonomy}
For the product metric of \textbf{TGG-ST4}, the $\Spinone$
holonomy along the compact $\theta$-direction is trivial,
\begin{equation}
\label{eq:trivial-theta-spin-holonomy}
\mathrm{Hol}(\omega_{\Spinone}, S^1_\theta) = \mathbf{1}_{\Spinone},
\end{equation}
because the scale profile $\e^{-2r}$ does not depend on $\theta$. In
contrast, the Abelian $\UoneB$ and $\UoneQi$ holonomies along
$S^1_\theta$ may be non-trivial Wilson-line phases, and these are the
order parameters for the Wilson--Hosotani electroweak breaking that
will be discussed in Part~V.
\end{proposition}

\begin{proof}
Direct computation: the spin connection of the product metric
\eqref{eq:bulk-metric} contains no $\dd\theta$ component because the
metric components are $\theta$-independent. Therefore
$\oint_{S^1_\theta} \omega_{\Spinone} = 0$ and the path-ordered
exponential reduces to the identity.
\end{proof}

\begin{proposition}[Existence of the unified boundary stress tensor]
\label{prop:TGG-stress-tensor-existence}
On the 4D boundary $\Mfour=\partial\Msix$ of any \TGG{} configuration
satisfying \textbf{TGG-ST1}--\textbf{TGG-ST7}, the unified Total
Geometric-Gauge stress tensor $\cT^{\mu\nu}_{\TGG}$ exists, is unique
up to gauge equivalence, and is symmetric: $\cT^{\mu\nu}_{\TGG} =
\cT^{\nu\mu}_{\TGG}$.
\end{proposition}

\begin{proof}[Sketch]
Each of the four constituents in \eqref{eq:TGG-ST6} is a
well-defined, symmetric stress-energy tensor on the boundary: the
matter stress tensor by the standard variation of the matter action,
the gauge stress tensor by the symmetric trace structure of the
gauge action, the Green--Schwarz inflow by the symmetric variation
of $X_4 \wedge \omega$, and the Kaluza--Klein remnant by zero-mode
reduction. Symmetry of the sum follows from termwise symmetry, and
uniqueness follows from the uniqueness of the bulk composite
connection (Theorem~\ref{thm:TGG-existence-uniqueness} of
Section~\ref{sec:TGG-theorem}) plus the canonical zero-mode
reduction. Detailed proofs of each piece are standard; see
\citet[\S~2.2--2.4]{ArisakaA3BI6}.
\end{proof}

These six propositions are the workhorse statements -- five for the
bulk layer and one for the boundary layer -- that will be applied
throughout the rest of the paper.


%% --- Phase 8 TikZ figure: TGG composite connection ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 bigbox/.style={draw=blue!55!black, rounded corners=4pt, line width=0.80pt, minimum width=12cm, minimum height=1.4cm, align=center, fill=blue!8},
 sector/.style={draw=green!55!black, rounded corners=3pt, minimum width=2.5cm, minimum height=1.6cm, align=center, fill=green!8},
 arrow/.style={->, line width=0.80pt, >=Stealth},
]
% Top: composite connection on principal bundle
\node[bigbox] (ATOT) at (6, 4)
 {\textbf{$\Atot = \omega \oplus A$}: composite connection on principal $\Gtot$-bundle over $\Msix$\\
 \footnotesize (T1.1: existence and uniqueness up to gauge, by \GGEP{} + \LC{})};

% Four sectors below
\node[sector] (SPIN) at (0, 1.2) {$\omega$\\ \footnotesize$\boldsymbol{\Spinone}$\\ \footnotesize gravity};
\node[sector] (SU5) at (4, 1.2) {$A_{\SUfive}$\\ \footnotesize$\boldsymbol{\SUfive}$\\ \footnotesize visible gauge};
\node[sector] (UB) at (8, 1.2) {$A_B$\\ \footnotesize$\boldsymbol{\UoneB}$\\ \footnotesize baryon};
\node[sector] (UQi) at (12, 1.2) {$A_{Qi}$\\ \footnotesize$\boldsymbol{\UoneQi}$\\ \footnotesize quantum-info};

\foreach \n in {SPIN, SU5, UB, UQi} {
 \draw[arrow, gray!60] (ATOT) -- (\n);
}

% Curvature: R_tot = R + F
\node[bigbox, fill=red!8, minimum width=12cm] (RTOT) at (6, -1)
 {\textbf{$\Rtot = R \oplus F$}: curvature of $\Atot$, with $F = F_5 + F_B + F_{Qi}$\\
 \footnotesize (input to the anomaly polynomial $X_4 = a\,\mathrm{tr}(R\wedge R) + b\,\mathrm{tr}(F_5\wedge F_5) + c\,F_B\wedge F_B + d\,F_{Qi}\wedge F_{Qi}$)};

\foreach \n in {SPIN, SU5, UB, UQi} {
 \draw[arrow, gray!60] (\n) -- (RTOT);
}

% Boundary stress tensor below
\node[bigbox, fill=orange!12, minimum width=12cm] (TBND) at (6, -3.4)
 {\textbf{Boundary stress tensor} $\cT^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge}[F] + X^{\mu\nu}_{\rm GS} + T^{\mu\nu}_{\rm KK}$\\
 \footnotesize (T1.2: KK zero-mode reduction onto $\Mfour = \partial\Msix$, OGGEP-projection)};

\draw[arrow, gray!60] (RTOT) -- (TBND);

% Master Conservation Law to right
\node[draw=olive!55!black, rounded corners=3pt, line width=0.80pt, fill=yellow!10, align=center, font=\sffamily, minimum width=4cm] (MCL) at (16, -3.4)
 {\textbf{T1.3:}\\$\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$};
\draw[arrow, line width=0.80pt] (TBND) -- (MCL);
\end{tikzpicture}%
}
\caption[Algebraic structure of \TGG{} (Pillar 1).]{Algebraic structure of \TGG{} (Pillar 1 of \GiFour{}). The composite connection $\Atot = \omega \oplus A$ decomposes into four sub-connections on a principal $\Gtot$-bundle over the 6D bulk $\Msix$: the gravitational $\omega$ on the $\Spinone$ frame bundle, and the gauge sub-connections $A_{\SUfive}, A_B, A_{Qi}$ on the visible, baryonic, and quantum-information sectors respectively. Its curvature $\Rtot$ is the input to the four-form anomaly polynomial $X_4$ of \CCI{}. Kaluza--Klein zero-mode reduction onto the 4D boundary produces the unified stress tensor $\cT^{\mu\nu}_{\TGG}$, whose covariant conservation is the Master Conservation Law T1.3.}
\label{fig:tgg-composite-connection}
\end{figure}

%% ====================================================================
\section{The composite connection structure}
\label{sec:composite-connection}
%% ====================================================================

\begin{remark}[Reader's bridge: one bundle, not four]
\label{rem:composite-readers-bridge}
Before diving into the formal definitions, let us motivate the
fiber-product construction by an everyday picture. In the
Standard-Model treatment, gravity lives on one bundle (the frame
bundle of spacetime), the strong/electroweak interactions live on
another (the $\SU(3)\times\SU(2)\times\U(1)$ bundle), and
baryon number plus the information sector are added as auxiliary
$\U(1)$'s --- four parallel structures, each with its own
connection, glued by hand to coincide on the same spacetime
manifold. Roughly speaking, this is a confederation of
independent bundles that happen to share a base.
\TGG{} replaces the confederation with a federation. The four
sub-bundles are merged into a single principal $\Gtot$-bundle
$\Pbun$ on the 6D bulk, with one composite connection
$\Atot = \omega \oplus A_{\SUfive} \oplus A_B \oplus A_{Qi}$.
Why insist on this? Because anomaly cancellation, gauge
compatibility, and the bulk-to-boundary descent to four dimensions all
require the four sectors to be facets of a single geometric
object, not four independent objects that happen to add to zero.
The fiber-product construction below is the precise way to say
``one bundle, four facets'' instead of ``four bundles, one base.''
\end{remark}

We now develop the formal mathematical content of \TGG{} in more
detail, focusing on the principal $\Gtot$-bundle construction, the
explicit decompositions of $\Atot$ and $\Rtot$, and the gauge-equivalence
relation.

\subsection{The principal $G_{\rm tot}$-bundle on $\mathcal M_6$}
\label{ssec:Gtot-bundle}

The construction of the bundle $\Pbun$ in \textbf{TGG-ST1} can be made
explicit by exhibiting $\Pbun$ as the fiber product of four principal
sub-bundles:
\begin{equation}
\label{eq:Pbun-fibre-product}
\Pbun \;=\; P_{\Spinone}\;\times_{\Msix}\; P_{\SUfive}
\;\times_{\Msix}\; P_{\UoneB}\;\times_{\Msix}\; P_{\UoneQi},
\end{equation}
where each $P_H \to \Msix$ is the principal $H$-bundle for the relevant
factor $H \in \{\Spinone,\allowbreak \SUfive,\allowbreak \UoneB,\allowbreak \UoneQi\}$ and the fiber
product is taken over $\Msix$. The fiber product construction enforces
that all four sub-bundles share the same base manifold, and the
fiber over $x \in \Msix$ is
\begin{equation}
\label{eq:Pbun-fibre}
\Pbun_x = \Spinone \times \SUfive \times \UoneB \times \UoneQi = \Gtot.
\end{equation}

The existence of each individual sub-bundle is guaranteed by:

\begin{itemize}[leftmargin=2em]
\item \textbf{$P_{\Spinone}$}: the bulk is Lorentzian and time-orientable
(\textbf{TGG-ST4}), so it admits a global $\Spinone$ structure.
\item \textbf{$P_{\SUfive}$}: the second Stiefel--Whitney class
$w_2(\Msix)$ vanishes because $\Msix$ is a product geometry of contractible
factors with the spin manifold $\R^{1,3}$, so $\Msix$ admits a spin
structure that lifts to an $\SUfive$ bundle once the boundary embedding
is fixed.
\item \textbf{$P_{\UoneB}, P_{\UoneQi}$}: $\Msix$ is non-compact and
admits trivial $\U(1)$ bundles globally. In the product-geometry picture,
the non-trivial Abelian holonomy along the compact $\theta$ direction
makes these bundles topologically non-trivial in the Pontryagin sense
even when the principal-bundle homotopy class is trivial.
\end{itemize}

\subsection{The connection one-form and its decomposition}
\label{ssec:Atot-decomp}

The composite connection $\Atot$ of \textbf{TGG-ST2} is, in any local
trivialization $\sigma: U \to \Pbun|_U$ over a chart $U\subset\Msix$,
expressed as a $\gtot$-valued one-form:
\begin{equation}
\label{eq:Atot-local}
\sigma^* \Atot
= \omega_{\Spinone}^{ab}\,T_{ab}^{(\Spinone)}
+ A_5^{I}\,T_I^{(\SUfive)}
+ A_B\,T_B
+ A_{Qi}\,T_{Qi},
\end{equation}
where $T_{ab}^{(\Spinone)}$ are the generators of
$\mathfrak{spin}(1,5)$ (with antisymmetric Lorentz indices $a,b$ running
over $0,1,\ldots,5$), $T_I^{(\SUfive)}$ are the 24 generators of
$\mathfrak{su}(5)$, and $T_B, T_{Qi}$ are the single Abelian generators
of $\mathfrak{u}(1)_B, \mathfrak{u}(1)_{Qi}$. The component
one-forms $\omega^{ab}, A_5^I, A_B, A_{Qi}$ are smooth real-valued
one-forms on $U$.

The total Lie algebra $\gtot$ has dimension
\begin{equation}
\label{eq:gtot-dim}
\dim \gtot = \dim\mathfrak{spin}(1,5) + \dim\mathfrak{su}(5) + 1 + 1
= 15 + 24 + 1 + 1 = 41.
\end{equation}
The $15$ dimensions of $\mathfrak{spin}(1,5)$ correspond to the
six-dimensional Lorentz generators
$M^{ab} = -M^{ba}$ ($a,b=0,\ldots,5$), of which six are boosts and nine
are rotations.

\subsection{The curvature two-form and its decomposition}
\label{ssec:Rtot-decomp}

The composite curvature $\Rtot = \dd\Atot + \Atot\wedge\Atot$ decomposes
along the four direct-summand factors as in \eqref{eq:Rtot-decomp}.
Explicitly, in a local trivialization:
\begin{align}
R_{\Spinone}^{ab} &= \dd\omega^{ab} + \omega^{ac}\wedge\omega_c{}^b,
\label{eq:R-Spin}\\
F_5^I &= \dd A_5^I + \tfrac12 f^{I}{}_{JK} A_5^J\wedge A_5^K,
\label{eq:F-5}\\
F_B &= \dd A_B,
\label{eq:F-B}\\
F_{Qi} &= \dd A_{Qi},
\label{eq:F-Qi}
\end{align}
where $f^I{}_{JK}$ are the structure constants of $\mathfrak{su}(5)$.
The Abelian field strengths $F_B, F_{Qi}$ are simply exact one-form
exterior derivatives because the Lie bracket on $\mathfrak{u}(1)$
vanishes.

The three-form
\begin{equation}
\label{eq:H3-def}
H_3 \in \Omega^3(\Msix)
\end{equation}
that appears in \BIsix{} is constructed from $\Atot$ and $\Rtot$ via
the Chern--Simons descent, as detailed in
\citet[Section~3]{ArisakaA3BI6}. Specifically, $H_3$ is the
Kalb--Ramond three-form whose differential is the four-form $X_4$
defined in Part~III. The construction of $H_3$ from \TGG{} data is the
algebraic core of the link between \TGG{} and \BIsix{} and will be
developed in Part~III.

\subsection{Gauge equivalence and gauge orbits}
\label{ssec:gauge-equivalence}

Two \TGG{} triples $(\Pbun, \Atot, \Rtot)$ and
$(\Pbun', \Atot', \Rtot')$ are gauge-equivalent if there exists a
diffeomorphism $\Phi: \Pbun \to \Pbun'$ commuting with the right
$\Gtot$-action and taking $\Atot$ to $\Atot'$ via the natural
adjoint action. In a local trivialization, this amounts to
\begin{equation}
\label{eq:gauge-transform}
A'_{\rm tot}(x) = g(x)^{-1}\Atot(x) g(x) + g(x)^{-1}\dd g(x),
\qquad g: \Msix \to \Gtot.
\end{equation}
The gauge orbits of \TGG{} are the equivalence classes under
\eqref{eq:gauge-transform}. By Proposition~\ref{prop:TGG-gauge-group},
the gauge group factors along the four direct summands of $\Gtot$, so
the gauge orbits also factor.

\subsection{Holonomy and global structure}
\label{ssec:holonomy-global}

For the 6D bulk of \GGTheory{}, the principal-bundle topology
is conveniently captured by the holonomy along the compact
$\theta$-circle. The $\Spinone$ holonomy is trivial
(Proposition~\ref{prop:TGG-theta-holonomy}); the $\SUfive$ holonomy
encodes the orbifold breaking $\SUfive \to \SU(3)\times \SU(2)\times \U(1)$
via Wilson lines; and the $\UoneB, \UoneQi$ holonomies encode the
electroweak Wilson--Hosotani phase and the Kaxion phase respectively.

\begin{definition}[Wilson-line order parameters]
\label{def:wilson-line-OP}
The Wilson-line order parameters of \TGG{} along the compact
$\theta$-direction are
\begin{align}
W_{\Spinone}(\theta_0) &= \mathrm{tr}\bigl[\,P\exp\oint_{S^1_\theta} \omega_{\Spinone}\bigr] = 1,
\label{eq:WSpin}\\
W_5(\theta_0) &= \mathrm{tr}\bigl[\,P\exp\oint_{S^1_\theta} A_5\bigr],
\label{eq:W5}\\
W_B(\theta_0) &= \exp\bigl(\,\mathrm{i}\oint_{S^1_\theta} A_B\bigr),
\label{eq:WB}\\
W_{Qi}(\theta_0) &= \exp\bigl(\,\mathrm{i}\oint_{S^1_\theta} A_{Qi}\bigr).
\label{eq:WQi}
\end{align}
The Wilson-line value $W_{\Spinone}=1$ is forced by
Proposition~\ref{prop:TGG-theta-holonomy}; the other three Wilson lines
are non-trivial dynamical variables that, in Part~V, become the order
parameters of the \GRIP{} cascade. The
$\mathrm{i}$ of the two Abelian holonomies carries a
pedigree: it is the boundary rendering of the circle's forced
quarter-turn --- derived, not primitive (T-P4-1 of
\citealp{ArisakaP4Real}) --- and the exponent is a real angle
on a real circle.
\end{definition}

%% ====================================================================
\section{\TGG{} and the magic-dimension ladder}
\label{sec:magic-ladder}
%% ====================================================================

We now address the algebraic backbone of \TGG{}: the magic-dimension
ladder $\R\to\C\to\HH\to\Octonion$, which provides the unique
realization of $\Spinone$ in dimension $d=6$ and which forces the
quaternionic doublet structure of bulk spinors.

\subsection{The magic-dimension ladder}
\label{ssec:magic-ladder-def}

The four classical normed division algebras of Hurwitz are
$\R, \C, \HH, \Octonion$ (the reals, complex numbers, quaternions, and
octonions), of real dimensions $1, 2, 4, 8$ respectively. Each
admits an associated Lie group $\SL(2,\mathbb{K})$ for
$\mathbb{K}\in\{\R,\C,\HH\}$ (the octonions are non-associative, so
$\SL(2,\Octonion)$ is not a group in the strict sense, but its
universal enveloping structure plays the analogous role). The
remarkable result is the chain of accidental isomorphisms
\begin{equation}
\label{eq:magic-ladder-iso}
\begin{gathered}
\Spin(1,2)\cong\SL(2,\R), \quad
\Spin(1,3)\cong\SL(2,\C), \\
\Spin(1,5)\cong\SL(2,\HH), \quad
\Spin(1,9)\;\text{relates to}\;\SL(2,\Octonion),
\end{gathered}
\end{equation}
which identifies the Lorentz double cover in dimensions $d=3,4,6,10$
with the $\SL(2,\mathbb{K})$ family. These are exactly the dimensions
in which classical superstrings (or analogous structures) admit the
clean spinor algebra needed for supersymmetry, and they are exactly
the dimensions in which the Lorentz algebra has a manifestly
$\SL(2)$-like structure over a normed division algebra.

The relevance to \TGG{} is that the bulk dimension $d=6$ is the
\emph{quaternionic} entry in the magic-dimension ladder. This is not a
choice of convenience: it is forced by \GGEP{} together with the
algebraic content of the four-factor gauge group $\Gtot$.

\subsection{Spin(1,5)$\,\cong\,$SL(2,$\HH$) and the quaternionic doublet $\HH^2$ (cited)}
\label{ssec:magic-cited}

The explicit construction of the accidental isomorphism
$\Spinone\cong\SLtwoH$, the proof that $d=6$ is uniquely admissible
among the magic dimensions, the quaternionic doublet representation
$\HH^2$ for bulk spinors, and the Spin(1,5) generators, structure
constants, and Killing form, are all worked out in detail in
\citet{ArisakaA2GG6} Theorem~1 (Dimensional Minimality) and \S~10.1
(``The accidental isomorphism $\Spinone\cong\SLtwoH$'' and ``Spinor
representations in six dimensions''). We import the results as
standing inputs to the \TGG{} construction. In particular:

\begin{itemize}[leftmargin=2em]
\item $\dim_\R\Spinone = 15$ (not 12, as a careless count might
suggest --- see \citep[Lemma~10.1]{ArisakaA2GG6} for the
clarification);
\item the $\HH^2$ doublet decomposes under $\Gtot$-action into the
matter content used throughout Parts~III--V;
\item the Killing form on each sub-algebra (sign-definite on
$\mathfrak{su}(5)$ and Abelian-zero on the two $\mathfrak{u}(1)$
factors) supplies the trace-ratio normalizations $\alpha_R = +1$,
$\alpha_5 = -1$ used in CCI-ST2 below.
\end{itemize}

Section~\ref{ssec:doublet-to-triplicity} carries forward the unique
A6 contribution: the descent from the bulk quaternionic doublet to
the boundary family triplicity that anchors the $N=3$ derivation
of T4.1.

\subsection{From the quaternionic doublet to family triplicity}
\label{ssec:doublet-to-triplicity}
\label{ssec:quat-to-family}

The connection between the quaternionic doublet structure and the
\OGNfive{} family triplicity $N=3$ is the algebraic content most relevant
to the N=3 proof of \citet{ArisakaP2N3}.

The chain of identifications is:

\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item $\HH^2$ has three independent commuting quaternionic structures
$(J_i, J_j, J_k)$, with the bulk quaternionic structure imported from
\citep[\S 10.1]{ArisakaA2GG6}.
\item Under the bulk-to-boundary fermion descent of \citep[\S 10.2]{ArisakaA2GG6}, these
three structures correspond to three independent ways of organizing
the same fermion content into generations.
\item The \OGNfive{} ledger encodes this triplicity as $N=3$, the second
component of $(M, N, L; J, K) = (1,3,5;7,35)$.
\item The family-projector moduli space, on which the \GFF{} Morse
cascade of Part~V acts, is therefore $\CPN = \CPtwo$ --- a compact
symmetric space in the classical classification~\citep{HelgasonBook} --- of complex
dimension $N-1=2$.
\end{enumerate}

The relation between the quaternionic structure and the family count
is not yet a closed derivation at the level of \TGG{} alone: the
precise statement $N=3$ requires the combined input of \CCI{}
(Part~III, \OGNfive{} ledger forced under \MSMF{}) and the \GFF{} Morse cascade
(Part~V, $N\leq 3$ from cascade closure). But the upstream algebraic
origin of family triplicity is visible already at the \TGG{} level, in
the three quaternionic structures of $\HH^2$.

This is one of the principal payoffs of formalizing \TGG{}: the
``family triplicity'' that \ANine{} treats as a derived \OGNfive{}
integer is shown, at the \TGG{} level, to have a direct algebraic
origin in the three quaternionic structures of the bulk spinor space.
The $N=3$ of the Standard Model, the $N=3$ of \OGNfive{}, and the $N=3$ of
the quaternionic structures of $\HH^2$ are not three separate
``$3$''s; they are one $3$ traced through three layers of the program.

%% ====================================================================
\section{The boundary layer: the unified stress tensor $\mathcal T^{\mu\nu}_{\rm \TGG{}}$}
\label{sec:TGG-boundary-stress-tensor}
%% ====================================================================

We have so far developed the bulk layer of \TGG{}: the principal
$\Gtot$-bundle, the composite connection, the curvature decomposition,
and the magic-dimension-ladder backbone. We now turn to the boundary
layer: the unified 4D stress tensor $\cT^{\mu\nu}_{\TGG}$ on
$\Mfour=\partial\Msix$, obtained by Kaluza--Klein zero-mode reduction
of the bulk action together with the \BIsix{} bulk-to-boundary descent. This
section is new content in, restoring the stress-tensor layer that
was inadvertently omitted from.

\subsection{Why a unified boundary stress tensor is needed}
\label{ssec:why-unified-stress}

Before \TGG{} was made explicit at the boundary level, the boundary
theory carried four disconnected stress-energy objects:
\begin{itemize}[leftmargin=2em]
\item $T^{\mu\nu}_{\rm matter}$ -- the matter stress tensor (electrons,
quarks, neutrinos, etc.);
\item $T^{\mu\nu}_{\rm gauge}[F]$ -- the gauge-field stress tensor
constructed from $F = F_5 + F_B + F_{Qi}$;
\item $X^{\mu\nu}_{\rm GS}$ -- the boundary shadow of the
Green--Schwarz inflow term in \BIsix{};
\item $T^{\mu\nu}_{\rm KK}$ -- the Kaluza--Klein heavy-mode remnants.
\end{itemize}
The question was: is their sum conserved? And if so, why? In the
closed-system QFT framework, the answer is empirical -- one
\emph{checks} that $\nabla_\mu (T^{\mu\nu}_{\rm matter} +
T^{\mu\nu}_{\rm gauge}) = 0$ on shell, but there is no constitutional
reason for it to hold. In the \TGG{} framework, the answer becomes a
\emph{theorem}: the sum is conserved because all four pieces are the
boundary projection of one bulk geometric object, and the bulk closure
condition (\BIsix{}) descends to the boundary as the master conservation
law $\nabla_\mu \cT^{\mu\nu}_{\TGG}=0$ via \OGGEP{}.

\begin{remark}[Physical intuition: from Newton's third law to a six-sector master balance]
\label{rem:Newton-master}
Newton's third law -- ``every action has an equal and opposite
reaction'' -- is, in the language of fields, the statement that
energy and momentum lost by one body are gained by another, so the
total stress-energy is conserved. In Maxwell theory this generalizes
to a two-sector balance: energy lost by the electromagnetic field
(via the Lorentz force on charged matter) equals energy gained by
matter, with the unified $T^{\mu\nu}_{\rm matter} + T^{\mu\nu}_{\rm EM}$
exactly conserved provided electric current is conserved. \TGG{} is
the \emph{six-sector} generalization of this same principle: gravity,
visible gauge ($\SUfive$), baryon ($\UoneB$), informational
($\UoneQi$), Green--Schwarz inflow, and Kaluza--Klein remnants are
all in mutual balance, with the Master Conservation Law
$\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ holding because all four
sectoral currents are simultaneously conserved. Failure of any one
current would leave an uncanceled divergence, just as in Maxwell
theory the failure of electric charge conservation would break
energy-momentum conservation. In \TGG{}, this six-sector balance is not
imposed by hand; it is a \emph{theorem} that follows from \BIsix{} closure
in the bulk via \OGGEP{} descent.
\end{remark}

\subsection{Kaluza--Klein zero-mode reduction of the bulk action}
\label{ssec:KK-reduction}

The bulk action of \TGG{} is, in the schematic form,
\begin{equation}
\label{eq:S-bulk}
S_{\rm bulk}^{\TGG} \;=\; \int_{\Msix}\,\sqrt{|g_6|}\,
\Bigl[\,\tfrac{1}{2\kappa_6^2}\,R_{\Spinone}
- \tfrac{1}{4 g_5^2}\,\Tr F_5^2
- \tfrac{1}{4 g_B^2}\,F_B^2 - \tfrac{1}{4 g_{Qi}^2}\,F_{Qi}^2
+ \cL_{\rm matter}\,\Bigr]
+ \int_{\Msix}\,B_2 \wedge X_4,
\end{equation}
where $\kappa_6$ is the bulk gravitational coupling, $g_5, g_B, g_{Qi}$
are the bulk gauge couplings, $\cL_{\rm matter}$ is the matter
Lagrangian, and the last term is the Green--Schwarz counterterm
required by \BIsix{} anomaly closure
\citep[\S~3]{ArisakaA3BI6}.

Kaluza--Klein zero-mode reduction along the radial coordinate $r$ and
the compact circle $S^1_\theta$ produces the boundary effective action
\begin{equation}
\label{eq:S-boundary}
S_{\rm bdy}^{\TGG} \;=\; \int_{\Mfour}\,\sqrt{|g_4|}\,
\Bigl[\,\tfrac{1}{2\kappa_4^2}\,R^{(4)}
- \tfrac{1}{4 g_4^2}\,\Tr F^{(4)2}
+ \cL_{\rm matter}^{(4)}\,\Bigr]
+ \cdots,
\end{equation}
where the four-dimensional gravitational coupling
$\kappa_4 = \kappa_6/\sqrt{2\pi R_\theta \cdot L_r}$ and the
four-dimensional gauge coupling $g_4$ are the projected zero-modes of
their bulk parents, and the dots indicate the Green--Schwarz boundary
contribution and Kaluza--Klein heavy-mode corrections.

Varying \eqref{eq:S-boundary} with respect to the boundary metric
yields the modified Einstein equation
\begin{equation}
\label{eq:modified-Einstein}
\tfrac{1}{\kappa_4^2}\,G^{\mu\nu} \;=\;
T^{\mu\nu}_{\rm matter} + T^{\mu\nu}_{\rm gauge}[F]
+ X^{\mu\nu}_{\rm GS} + T^{\mu\nu}_{\rm KK},
\end{equation}
which is exactly the unified Total Geometric-Gauge stress tensor
$\cT^{\mu\nu}_{\TGG}$ of \textbf{TGG-ST6}.

\subsection{The four contributions to $\mathcal T^{\mu\nu}_{\rm \TGG{}}$}
\label{ssec:four-contributions}

We give the explicit form of each of the four contributions.

\textbf{Matter sector.} The boundary matter stress tensor takes the
canonical form
\begin{equation}
\label{eq:T-matter}
T^{\mu\nu}_{\rm matter} \;=\; \frac{2}{\sqrt{|g_4|}}\,
\frac{\delta(\sqrt{|g_4|}\,\cL_{\rm matter}^{(4)})}{\delta g_{\mu\nu}}
\;=\; \sum_{i\in {\rm matter}} T^{\mu\nu}_{(i)},
\end{equation}
where the sum runs over the standard-model matter species (quarks,
leptons, Higgs scalar) on the boundary.

\textbf{Gauge sector.} The boundary gauge stress tensor is
\begin{equation}
\label{eq:T-gauge}
T^{\mu\nu}_{\rm gauge}[F]
\;=\; \tfrac{1}{g_4^2}\,\Tr\bigl(F^{\mu\rho} F^\nu{}_\rho
- \tfrac{1}{4} g^{\mu\nu} F^{\rho\sigma} F_{\rho\sigma}\bigr),
\end{equation}
with $F = F_5 + F_B + F_{Qi}$ the total boundary field strength.

\textbf{Green--Schwarz inflow.} The boundary shadow of the bulk
$B_2\wedge X_4$ counterterm is
\begin{equation}
\label{eq:X-GS}
X^{\mu\nu}_{\rm GS} \;=\; \frac{2}{\sqrt{|g_4|}}\,
\frac{\delta(\sqrt{|g_6|}\,B_2 \wedge X_4)|_{r=0}}{\delta g_{\mu\nu}},
\end{equation}
which encodes the bulk-to-boundary descent of the bulk anomaly polynomial
$X_4$ onto the boundary. Its explicit form depends on the choice of
$B_2$ representative; see \citet[\S~3.2]{ArisakaA3BI6} for the
details.

\textbf{Kaluza--Klein remnant.} The boundary contribution from the
heavy KK modes is
\begin{equation}
\label{eq:T-KK}
T^{\mu\nu}_{\rm KK} \;=\; \sum_{n\geq 1} T^{\mu\nu}_{(n)},
\end{equation}
the sum over KK levels $n$ of the boundary stress contributions of the
massive KK modes at scale $\sim n \cdot \Mlock$ where
$\Mlock = \EPl/2\e^{35} \simeq 3.85\,\TeV$ is the lock scale of
\citet{ArisakaP1KK}.

\subsection{The projected modified Bianchi identity (PBI-4)}
\label{ssec:PBI-4}

The 6D Bianchi identity $\dd H_3 = X_4$ projects to the 4D boundary
as the \emph{projected modified Bianchi identity}:
\begin{equation}
\label{eq:PBI-4}
\nabla_\mu \bigl(G^{\mu\nu} + X^{\mu\nu}_{\rm GS}\bigr) \;=\; 0,
\qquad\text{(PBI-4)}
\end{equation}
which is a derived 4D equation, not a primitive postulate.

\begin{proposition}[PBI-4 from \BIsix{} descent]
\label{prop:PBI-4-from-BI6}
On any \TGG{} configuration satisfying \textbf{TGG-ST1}--\textbf{TGG-ST6},
the projected modified Bianchi identity \eqref{eq:PBI-4} holds on
$\Mfour$ as a consequence of the bulk identity
$\dd H_3 = X_4$ and the Kaluza--Klein zero-mode reduction.
\end{proposition}

\begin{proof}[Sketch]
The bulk identity $\dd H_3 = X_4$ together with the Bianchi identity
$\nabla_\mu G^{\mu\nu}_{(6)} = 0$ for the bulk Einstein tensor implies,
after zero-mode reduction along $r$ and $\theta$,
$\nabla_\mu^{(4)}(G^{\mu\nu}_{(4)} + X^{\mu\nu}_{\rm GS}) = 0$.
The boundary $X^{\mu\nu}_{\rm GS}$ is exactly the boundary shadow of
the bulk $X_4$ via the variational expression \eqref{eq:X-GS}; its
boundary divergence cancels the failure of the Einstein tensor's
divergence to vanish in the presence of the GS counterterm. See
\citet[\S~3.4]{ArisakaA3BI6} for the detailed proof.
\end{proof}

\subsection{The boundary layer ready for Theorem T1}
\label{ssec:boundary-ready}

With the explicit form of $\cT^{\mu\nu}_{\TGG}$ in
\eqref{eq:modified-Einstein}, the four contributions in
\eqref{eq:T-matter}--\eqref{eq:T-KK}, and the projected modified
Bianchi identity \eqref{eq:PBI-4}, we have all the ingredients required
to state and prove the Master Conservation Law $\nabla_\mu
\cT^{\mu\nu}_{\TGG} = 0$. This is the content of Theorem~\textbf{T1}
in Section~\ref{sec:TGG-theorem}.

%% ====================================================================
\section{Theorem T1 (T-GRIP-1): Existence, uniqueness, and Master Conservation of \TGG{}}
\label{sec:TGG-theorem}
%% ====================================================================

We now state and prove the principal theorem of Part~II, the existence
and uniqueness of \TGG{} on the 6D bulk.


\subsection{Dual-layer architecture and the compatibility relation $\mathfrak{C}_{\TGG{}}$}
\label{ssec:T1-dual-layer}

Theorem~\textbf{T1} is, in the language of L16, a dual-layer principal
theorem of \GiFour{}. Its \emph{algebraic layer} (T1.1 and T1.2)
asserts the existence and uniqueness of the composite connection
$\Atot = \omega \oplus A$ on the principal $\Gtot$-bundle over the
6D bulk $\Msix$, and of the unified boundary stress tensor
$\cT^{\mu\nu}_{\TGG}$ obtained by Kaluza--Klein zero-mode reduction of
the bulk action under \OGGEP{}. Its \emph{operational/physical layer}
(T1.3) asserts the Master Conservation Law $\nabla_\mu
\cT^{\mu\nu}_{\TGG} = 0$ as a laboratory-falsifiable Ward-identity
statement on the four conserved boundary currents $J_e$, $J_B$,
$J_{Qi}$, $J_g$.

The two layers are tied together by the named compatibility relation
\begin{equation}
\label{eq:CTGG-compat}
\boxed{\;\mathfrak{C}_{\TGG{}}\;\equiv\;
\bigl(\text{T1.1}\;\wedge\;\text{T1.2}\bigr)\;\Longrightarrow\;\text{T1.3}
\quad\text{(via Noether--Bianchi descent on the $\Gtot$-bundle)}.\;}
\end{equation}
\textbf{Compatibility proposition.} $\mathfrak{C}_{\TGG{}}$ above is
a one-line proposition with the proof: given the existence and
uniqueness of $\Atot$ (T1.1) and of $\cT^{\mu\nu}_{\TGG}$ (T1.2),
the Master Conservation Law (T1.3) follows by applying the Noether
identity $\partial_\mu J^\mu = 0$ to each gauge sector independently
and the geometric Bianchi identity $\dd R = 0$ on the gravitational
sector, and summing. The summed identity is the divergence of
$\cT^{\mu\nu}_{\TGG}$. The compatibility relation makes T1
cite-able at the algebraic level by mathematical-physics readers (who
care about the principal-bundle structure and the magic-dimension
ladder) and at the physical level by phenomenology readers (who care
about the laboratory Ward identities of the four conserved currents).
Both readings refer to the same mathematical object \TGG{}; the
dual-layer architecture simply makes explicit which face a given
theorem statement presents.

\subsection{Statement of Theorem T1}
\label{ssec:T1-statement}

\begin{theorem}[\textbf{T1}: Existence, Uniqueness, and Master Conservation of \TGG{}]
\label{thm:TGG-existence-uniqueness}
On the 6D bulk $\Msix$ with metric \eqref{eq:bulk-metric} and
gauge content $\Gtot=\Spinone\times\SUfive\times\UoneB\times\UoneQi$,
the Total Geometric-Gauge object \TGG{} of Definition~\ref{def:TGG}
satisfies the following three properties:

\smallskip
\textbf{(T1.1) Bulk-layer existence and uniqueness.} There exists a
triple $(\Pbun,\Atot,\Rtot)$ satisfying \textbf{TGG-ST1}--\textbf{TGG-ST5},
unique up to gauge equivalence \eqref{eq:gauge-transform}. Existence
and uniqueness follow from \GGEP{}, \LC{}, the magic-dimension ladder,
and the product geometry of $\Msix$, with no additional physical
axiom required.

\smallskip
\textbf{(T1.2) Boundary-layer existence and uniqueness.} Under
Kaluza--Klein zero-mode reduction of the bulk action of \textbf{TGG-ST2}
together with the \BIsix{} bulk-to-boundary descent, the unified Total
Geometric-Gauge stress tensor $\cT^{\mu\nu}_{\TGG}$ of \textbf{TGG-ST6}
exists on $\Mfour=\partial\Msix$, is symmetric, and is unique up to
the gauge equivalence inherited from the bulk.

\smallskip
\textbf{(T1.3) Master Conservation Law.} Reading the modified
Einstein equation as the definition of $\cT^{\mu\nu}_{\TGG}$ on the
gravitational side (so that the conservation of the gravitational
part is automatic by the geometric Bianchi identity), the substantive
content of T1.3 lives on the Ward-package side: in the gauge-closed
boundary sector, the unified stress tensor satisfies the covariant
conservation law
\begin{equation}
\label{eq:T1-master}
\nabla_\mu \cT^{\mu\nu}_{\TGG} \;=\; 0,
\end{equation}
holding if and only if the full Ward package \eqref{eq:Ward-package} of
\textbf{TGG-ST7} is satisfied simultaneously.
\end{theorem}

\begin{remark}[Physics intuition: \TGG{} as the constitutional culmination of Maxwell--Yang--Mills--Einstein]
T1 says, in plain terms, that the 6D bulk admits exactly one
unified geometric--gauge object \TGG{}, and that this object's
boundary shadow conserves its stress--energy automatically. Indeed,
the algebraic layer (T1.1, T1.2) reads as a uniqueness statement on
the principal $\Gtot$-bundle: out of all conceivable bundle
structures over the 6D bulk, exactly one is compatible with
\GGEP{} and \LC{}, and that one is the \TGG{} structure of
Definition~\ref{def:TGG}. The operational layer (T1.3) reads as a
laboratory falsification target: every boundary observer in the lab
measures exactly four conserved currents $(J_e, J_B, J_{Qi}, J_g)$,
and the Master Conservation Law says that any cross-current leakage
would falsify \TGG{}. The dual-layer architecture is what lets a
mathematical physicist verify T1.1--T1.2 from bundle theory while a
LHC experimenter verifies T1.3 from precision Ward-identity tests
--- two falsification routes converging on a single geometric
object.
\end{remark}

\subsection{Proof of Theorem T1: bulk-layer existence (T1.1, existence part)}
\label{ssec:T1-proof-existence}

\begin{proof}[Proof of bulk-layer existence (T1.1)]
We construct $\TGG$ explicitly.

\textbf{Step 1: the principal $\Spinone$-bundle.} By \textbf{TGG-ST4},
the 6D bulk $\Msix$ is Lorentzian, time-orientable, and admits
a global frame; therefore it admits a principal $\Spinone$-bundle
$P_{\Spinone}\to\Msix$, given by the spin lift of the frame bundle.
The associated Levi-Civita-type connection $\omega_{\Spinone}$ is
unique up to gauge.

\textbf{Step 2: the principal $\SUfive$-bundle.} The 6D bulk
admits a global $\SUfive$ bundle because: (i) $\Msix$ is contractible
in the radial direction (modulo the scale profile which does not affect
homotopy); (ii) the second Stiefel--Whitney obstruction vanishes; and
(iii) the boundary $\partial\Msix=\Mfour$ admits the embedded
Standard-Model gauge structure $\SU(3)\times \SU(2)\times \U(1)$ as a
sub-group of $\SUfive$. The connection $A_5$ is determined up to gauge
by the boundary $\SUfive$ data of \citet{ArisakaA9SM}.

\textbf{Step 3: the principal $\UoneB$ and $\UoneQi$ bundles.}
$\Msix$ is non-compact, and $\U(1)$ bundles over non-compact bases are
classified by the integral cohomology $H^2(\Msix;\Z)$, which is generated
by the integer multiples of the compact $\theta$-direction. Both
$P_{\UoneB}$ and $P_{\UoneQi}$ exist with topological winding numbers
that are determined by the Wilson-line data of
Definition~\ref{def:wilson-line-OP}.

\textbf{Step 4: the fiber product.} Define
\[
\Pbun := P_{\Spinone}\times_{\Msix} P_{\SUfive}\times_{\Msix}
P_{\UoneB}\times_{\Msix} P_{\UoneQi}
\]
as in equation~\eqref{eq:Pbun-fibre-product}. The fiber product is a
smooth manifold because each factor is smooth and the base $\Msix$ is
common. The free right action of $\Gtot$ on $\Pbun$ is the diagonal
action of the four sub-groups, which is free because each sub-action is
free. The quotient is $\Msix$, satisfying \textbf{TGG-ST1}.

\textbf{Step 5: the composite connection.} Define
\[
\Atot := \omega_{\Spinone} + A_5 + A_B + A_{Qi},
\]
viewed as a connection one-form on $\Pbun$ via the obvious sum of
pull-backs from the four factors. Since the four direct summands of
$\gtot$ commute, $\Atot$ is a well-defined $\gtot$-valued one-form
satisfying the principal-bundle compatibility conditions. This
provides \textbf{TGG-ST2}.

\textbf{Step 6: \textbf{TGG-ST3} and \textbf{TGG-ST4}.} These are
direct consequences of the construction. \textbf{TGG-ST3} follows from
the choice of bulk dimension $d=6$ and the magic-ladder isomorphism
\eqref{eq:spin15-iso-A3}. \textbf{TGG-ST4} follows from the
product-geometry structure of $\Msix$, which is preserved by all four
gauge actions because they commute with diffeomorphisms of $\Msix$.

\textbf{Step 7: \textbf{TGG-ST5}.} The unification axiom is automatic
in this construction: the four sub-connections are projections of the
single connection $\Atot$ onto its four direct-summand components, and
gauge transformations of $\Atot$ act diagonally on the four
sub-connections by the natural adjoint action of $\Gtot$ on $\gtot$.

This completes the existence proof.
\end{proof}

\subsection{Proof of Theorem T1: bulk-layer uniqueness (T1.1, uniqueness part)}
\label{ssec:T1-proof-uniqueness}

\begin{proof}[Proof of bulk-layer uniqueness up to gauge (T1.1)]
Suppose $(\Pbun, \Atot, \Rtot)$ and $(\Pbun', \Atot', \Rtot')$ both
satisfy \textbf{TGG-ST1}--\textbf{TGG-ST5}. We show that they are
gauge-equivalent.

\textbf{Step 1: bundle isomorphism.} Each principal $H$-sub-bundle
$P_H$ for $H\in\{\Spinone,\allowbreak\SUfive,\allowbreak\UoneB,\allowbreak\UoneQi\}$ is determined by
its topology, which is fixed by the bulk geometry and the boundary
gauge data. Two bundles with the same topology over the same base
are isomorphic as principal $H$-bundles \citep{KobayashiNomizu1969}.
Therefore $\Pbun \cong \Pbun'$ as principal $\Gtot$-bundles.

\textbf{Step 2: connection equivalence.} Two connections on the same
principal $H$-bundle differ by an $\mathfrak{h}$-valued one-form on
$\Pbun$ pulled back from $\Msix$ (the difference of two connections is
a tensor). This one-form is an element of
$\Omega^1(\Msix, \mathfrak{h})$. By the inverse-function theorem
applied to gauge transformations
\eqref{eq:gauge-transform}, any such difference is locally exact in the
gauge sense: there exists a gauge transformation $g_H : \Msix \to H$
such that $A_H' = g_H^{-1} A_H g_H + g_H^{-1}\dd g_H$. Globally, the
existence of $g_H$ depends on the topology of the principal bundle
$P_H$ (the obstruction lies in $H^1(\Msix; \mathfrak{h})$), but for
each of our four sub-bundles the obstruction vanishes (the
$H^1$-cohomology of the contractible $\Msix$ is zero). Therefore each
sub-connection is gauge-equivalent.

\textbf{Step 3: composite gauge equivalence.} The four-factor structure
of $\Gtot$ implies that the composite gauge transformation
$g = (g_{\Spinone}, g_5, g_B, g_{Qi})$ acts diagonally on $\Atot$, so
the composite gauge equivalence is $\Atot' = g^{-1}\Atot g + g^{-1}\dd g$.
This is the global gauge transformation that takes
$(\Pbun, \Atot, \Rtot)$ to $(\Pbun', \Atot', \Rtot')$.

This completes the uniqueness proof.
\end{proof}

\subsection{Proof of Theorem T1: boundary stress tensor existence and uniqueness (T1.2)}
\label{ssec:T1-proof-T12}

\begin{proof}[Proof of (T1.2)]
We construct $\cT^{\mu\nu}_{\TGG}$ explicitly as the variational
derivative of the boundary effective action.

\textbf{Step 1: zero-mode reduction.} By Section~\ref{ssec:KK-reduction},
the bulk action \eqref{eq:S-bulk} reduces under Kaluza--Klein
zero-mode projection along $r$ and $\theta$ to the boundary effective
action \eqref{eq:S-boundary}.

\textbf{Step 2: variational derivation of $\cT^{\mu\nu}_{\TGG}$.}
Varying \eqref{eq:S-boundary} with respect to the boundary metric
$g_{\mu\nu}$ yields the modified Einstein equation
\eqref{eq:modified-Einstein}, with the right-hand side identified as
$\cT^{\mu\nu}_{\TGG}$ via the four canonical contributions
\eqref{eq:T-matter}--\eqref{eq:T-KK}. Each contribution is
well-defined and symmetric in $\mu\nu$ by the standard variational
formulae.

\textbf{Step 3: uniqueness from bulk uniqueness.} By (T1.1),
$(\Pbun,\Atot,\Rtot)$ is unique up to gauge equivalence. The
zero-mode reduction is canonical (no choices beyond the choice of $r$
and $\theta$ coordinates), so $\cT^{\mu\nu}_{\TGG}$ is determined by
the bulk data up to the same gauge equivalence. Two
gauge-equivalent bulk configurations give the same boundary stress
tensor up to the induced boundary diffeomorphism.

This completes the proof of (T1.2).
\end{proof}

\subsection{Proof of Theorem T1: Master Conservation Law (T1.3)}
\label{ssec:T1-proof-T13}

\begin{proof}[Proof of (T1.3)]
We prove that $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ holds iff the full
Ward package \eqref{eq:Ward-package} is satisfied.

\textbf{Step 1: divergence of the gauge sector.} The gauge-sector
stress tensor \eqref{eq:T-gauge} has divergence
\begin{equation}
\label{eq:div-T-gauge}
\nabla_\mu T^{\mu\nu}_{\rm gauge}
\;=\; -F^{\nu\mu}_e J^e_\mu - F^{\nu\mu}_B J^B_\mu - F^{\nu\mu}_{Qi} J^{Qi}_\mu,
\end{equation}
i.e., minus the Lorentz-force density transferred from the gauge fields
to the matter currents. This identity follows from the gauge equations
of motion $\nabla_\mu F^{\mu\nu} = J^\nu$ together with the Bianchi
identity for $F$.

\textbf{Step 2: divergence of the matter sector.} Applying the matter
equations of motion plus the Ward identities for each conserved
current $J^\mu_e, J^\mu_B, J^\mu_{Qi}$ gives
\begin{equation}
\label{eq:div-T-matter}
\nabla_\mu T^{\mu\nu}_{\rm matter}
\;=\; +F^{\nu\mu}_e J^e_\mu + F^{\nu\mu}_B J^B_\mu + F^{\nu\mu}_{Qi} J^{Qi}_\mu.
\end{equation}
The signs are exactly opposite to those in \eqref{eq:div-T-gauge}:
energy and momentum lost by the gauge fields are gained by matter, and
vice versa. This is the standard force-balance equation.

\textbf{Step 3: divergence of the GS-inflow + KK sectors.} By
Proposition~\ref{prop:PBI-4-from-BI6}, the projected modified Bianchi
identity \eqref{eq:PBI-4} gives
\begin{equation}
\label{eq:div-X-GS-T-KK}
\nabla_\mu \bigl(G^{\mu\nu} + X^{\mu\nu}_{\rm GS}\bigr) \;=\; 0,
\qquad
\nabla_\mu T^{\mu\nu}_{\rm KK} \;=\; 0,
\end{equation}
where the first follows from the \BIsix{} descent and the second from the
fact that the heavy KK modes are decoupled at boundary scales (their
stress contribution is conserved by the bulk equations of motion
restricted to the boundary).

\textbf{Step 4: assembly.} Adding the four divergences:
\begin{align}
\nabla_\mu \cT^{\mu\nu}_{\TGG}
&= \nabla_\mu T^{\mu\nu}_{\rm matter}
+ \nabla_\mu T^{\mu\nu}_{\rm gauge}
+ \nabla_\mu X^{\mu\nu}_{\rm GS}
+ \nabla_\mu T^{\mu\nu}_{\rm KK}\\
&= \bigl(+F\cdot J\bigr) + \bigl(-F\cdot J\bigr) + \bigl(-\nabla G\bigr) + 0\\
&= -\nabla_\mu G^{\mu\nu}.
\end{align}
By the modified Einstein equation \eqref{eq:modified-Einstein},
$\nabla_\mu G^{\mu\nu} = \kappa_4^2 \nabla_\mu \cT^{\mu\nu}_{\TGG}$, so
the equation $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ closes consistently.
The condition for closure is exactly the simultaneous satisfaction of
the Ward package \eqref{eq:Ward-package}: failure of any one current
conservation leaves an uncanceled divergence in $\cT^{\mu\nu}_{\TGG}$.

This proves the iff in (T1.3).
\end{proof}

\begin{remark}[Physical content of the Master Conservation Law]
\label{rem:T1-master-physical}
The Master Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ is the
6D-bulk-to-4D-boundary descent of the \BIsix{} closure, identifying the
boundary as the boundary image of a single conserved bulk geometric
object. It is the constitutional analogue of energy-momentum
conservation in ordinary Maxwell theory (where conservation requires
the electric current to be conserved): in the present six-sector
generalization, conservation of the unified stress tensor requires
simultaneous conservation of all four currents (electric, baryonic,
informational, and metric-inflow), and all four are conserved because
they are tied to one anomaly rope upstairs in the bulk. This is the
foundation of the astrophysically falsifiable predictions of \CCI{}
(Part~III), which is the \OGGEP{} zero-mode projection of \eqref{eq:T1-master}
in the open-system regime.
\end{remark}

\subsection{Corollary: \TGG{} and \DGI{}}
\label{ssec:TGG-DGI}

\begin{corollary}[\TGG{} realizes \DGI{}]
\label{cor:TGG-DGI}
The unique \TGG{} structure on $\Msix$, restricted to its undisturbed
phase (i.e., before any boundary projection), realizes the
\emph{Dynamical Gauge Invariance} (\DGI{}) stage of the \GRIP{}
cascade introduced in \citep{ArisakaA6Gi4}. Specifically:
\begin{itemize}[leftmargin=2em]
\item the full $\Gtot$ symmetry of \TGG{} on $\Msix$ corresponds to
\DGI{};
\item the boundary projection $\Pi: \Msix\to\Mfour$, which selects a
specific Wilson-line holonomy (Section~\ref{ssec:holonomy-global}),
realizes \GSSB{} (Geometric Spontaneous Symmetry Breaking);
\item the boundary attractors selected by the broken-phase \GFF{}
Morse landscape (Part~IV) realize \GA{} (Geometric Attractors).
\end{itemize}
The complete \GRIP{} cascade $\DGI \to \GSSB \to \GA$ (Part~V) thus
takes \TGG{} as its initial deepest-invariance configuration. In
fiber-resolved terms (Subsection~\ref{ssec:stage-custody}): the gauge
directions that \DGI{} makes dynamical are the $\theta$-fiber's ---
the circle is the bulk's only gauge bundle --- while the dynamism
itself is supplied by the $r$-fiber's openness; \DGI{} is \LGI{}
transported by the semigroup.
\end{corollary}

\subsection{Corollary: \TGG{} and \GGEP{} equivalence}
\label{ssec:TGG-GGEP}

\begin{corollary}[\TGG{} and \GGEP{} are equivalent]
\label{cor:TGG-GGEP-equiv}
On the 6D bulk $\Msix$, the existence of a \TGG{} structure
satisfying \textbf{TGG-ST1}--\textbf{TGG-ST5} is equivalent to the
GG-Constitutional axiom \GGEP{}. Specifically, \TGG{} is the formal
mathematical realization of \GGEP{} on $\Msix$, and any structure on
$\Msix$ that satisfies \GGEP{} is a \TGG{} (up to gauge equivalence).
\end{corollary}

\begin{proof}
\GGEP{} states that gauge content and geometric content arise from a
single primitive composite connection. Axiom \textbf{TGG-ST2} encodes
this primitive composite connection as $\Atot$; Structure Theorem \textbf{TGG-ST5}
encodes its diagonal gauge action; and axioms
\textbf{TGG-ST1}, \textbf{TGG-ST3}, \textbf{TGG-ST4} encode the geometric
context (bundle, magic-ladder dimension, Lorentzian compatibility) in
which the primitive composite connection is realized. Conversely, any
structure satisfying \GGEP{} on $\Msix$ admits a unique principal
$\Gtot$-bundle (by the magic-dimension argument of
Section~\ref{sec:magic-ladder}) and a unique composite connection up to
gauge (by Theorem~\ref{thm:TGG-existence-uniqueness}).
\end{proof}

\subsection{Status of T1 in the constitutional theorem chain}
\label{ssec:T1-status}

The proof of Theorem~\ref{thm:TGG-existence-uniqueness} relies only on:
(i) the magic-dimension-ladder algebraic content (a mathematical fact
about $\Spin$ groups over normed division algebras);
(ii) the product-geometry Lorentzian structure of $\Msix$ (set up in
\citet{ArisakaA2GG6});
(iii) the integrality of principal bundles over non-compact contractible
bases (a topological fact);
(iv) standard differential-geometric machinery (Chern--Weil,
Kobayashi--Nomizu).

No additional physical axiom beyond \GGEP{} of the \GGConstitution{} is
invoked. This is consistent with the derivation policy of
(the derivation policy documented in
\citep[\S\S 1--2]{ArisakaA4Euclid}) and confirms
the no-new-axioms principle (\citep[\S 1.4]{ArisakaA4Euclid}): the existence and uniqueness of
\TGG{} is already latent in the constitutional theorem chain, and the
present paper merely makes it explicit.

%% ====================================================================
\section{Worked examples}
\label{sec:TGG-examples}
%% ====================================================================

To make Theorem~\ref{thm:TGG-existence-uniqueness} concrete, we work
through three explicit examples: \TGG{} on flat $\R^{1,5}$
(Minkowski warm-up), \TGG{} on the $\Msix$ of \GGTheory{}, and
the explicit Wilson-line / holonomy structure in the general case.

\subsection{Example 1: \TGG{} on flat $\mathbb R^{1,5}$}
\label{ssec:example-flat}

The simplest example is the trivial Lorentzian bulk
$\Msix = \R^{1,5}$, with flat Minkowski metric and trivial $\Gtot$
bundle structure.

\textbf{Bundle.}
$\Pbun = \R^{1,5}\times \Gtot$ (the trivial product bundle).

\textbf{Connection.}
$\Atot = 0$ in the natural global trivialization; equivalently, in any
trivialization, $\Atot$ is a pure gauge,
$\Atot = g^{-1}\dd g$ for some $g: \R^{1,5}\to\Gtot$.

\textbf{Curvature.}
$\Rtot = 0$. All four sub-curvatures vanish.

\textbf{Holonomy.}
Trivial along all loops. The Wilson-line order parameters
\eqref{eq:WSpin}--\eqref{eq:WQi} all evaluate to identity.

\textbf{Discussion.}
Flat $\R^{1,5}$ is a degenerate case of \TGG{}: it satisfies all
five axioms but with trivial content. The holonomy is trivial,
the curvature vanishes, and there is no compact direction to support
non-trivial Wilson lines. This example serves as a sanity check that
the axioms are not vacuous (the trivial bundle is a valid \TGG{}) and
as a reference point against which the non-trivial case can be
contrasted.

\subsection{Example 2: \TGG{} on the $\mathcal M_6$ of \GGTheory{}}
\label{ssec:example-warped}

The principal example is the 6D bulk
$\Msix = \R^{1,3}\times I_r\times S^1_\theta$ with metric
\eqref{eq:bulk-metric}, where $I_r = [0, K]$ is the radial
interval (UV at $r=0$, IR at $r=K$) and $S^1_\theta$ is the
compact angular direction with circumference $2\pi R_\theta$.

\textbf{Bundle.}
The principal $\Gtot$-bundle is non-trivial along the compact
$\theta$-direction. Specifically, the principal $\Spinone$-bundle is
trivial in the strict-bundle sense (because $\Msix$ is homotopy
equivalent to $S^1$), but the Levi-Civita connection has non-trivial
$\theta$-component derivatives that encode the scale profile. The
$\SUfive$-bundle has non-trivial topological winding around $S^1_\theta$
(orbifold breaking). The two $\U(1)$-bundles have non-trivial Wilson
lines along $S^1_\theta$.

\textbf{Connection.}
\begin{itemize}[leftmargin=2em]
\item $\omega_{\Spinone}$: the Levi-Civita spin connection of the scale-profile metric. In the conventions of
\citet[Appendix~A]{ArisakaA2GG6}, the non-trivial components
are $\omega^{0r} = -\e^{-2r}\dd t$, etc. The $\theta$-component
vanishes by Proposition~\ref{prop:TGG-theta-holonomy}.
\item $A_5$: the orbifold Wilson line implementing
$\SUfive\to\SU(3)\times\SU(2)\times\U(1)$ with hypercharge embedding
$Y_0 = \mathrm{diag}(-1/3,-1/3,-1/3,1/2,1/2)$, encoded by the
$\theta$-component
$A_5^\theta = \mathrm{diag}(-1/3, -1/3, -1/3, 1/2, 1/2)\,\partial_\theta$
in the natural $\SUfive$ basis.
\item {\sloppy $A_B$: the gravity-baryon Abelian connection, with non-trivial
$\theta$-component $A_B^\theta = \mathrm{diag}(1/3,1/3,1/3,0,0)\,\partial_\theta$
that gives the gravitational pull on baryons \citep{ArisakaA3BI6}.\par}
\item $A_{Qi}$: the quantum-information Abelian connection, with
$\theta$-component related to the Kaxion phase
\citep{ArisakaC1Kaxion}.
\end{itemize}

\textbf{Curvature.}
\begin{itemize}[leftmargin=2em]
\item $R_{\Spinone}$: non-trivial in the radial direction, encoding the
bulk curvature. Explicitly, the Riemann tensor of \eqref{eq:bulk-metric}
has non-vanishing components proportional to $\e^{-2r}$.
\item $F_5, F_B, F_{Qi}$: non-trivial radial-angular components arising
from the orbifold breaking and the Wilson-line phases.
\end{itemize}

\textbf{Holonomy.}
The Wilson-line order parameters along $S^1_\theta$ are exactly the
phases that drive the Wilson--Hosotani electroweak breaking
\citep{Hosotani1983}, the Kaxion oscillation
\citep{ArisakaC1Kaxion}, and the orbifold projection
\citep{ArisakaA9SM}. In Part~V of the present paper, these Wilson
lines become the order parameters of the \GRIP{} cascade.

\textbf{Discussion.}
This is the physically realized \TGG{} of \GGTheory{}. All the
machinery developed in Part~I (the scale-profile metric, the \OGNfive{} ledger,
the gauge content) is now organized under a single composite
connection $\Atot$, with the four sub-sectors as projections. The
non-trivial $\theta$-holonomy of three out of four sub-connections
(everything except the $\Spinone$ factor, which has trivial holonomy
by Proposition~\ref{prop:TGG-theta-holonomy}) drives all the dynamical
content of the boundary theory.

\subsection{Example 3: explicit Wilson lines and holonomy phases}
\label{ssec:example-wilson}

We give explicit numerical values for the Wilson-line holonomies on
the $\Msix$, which will be used in Parts~III--V.

\textbf{Spin holonomy.}
$W_{\Spinone}(S^1_\theta) = \mathbf 1_{\Spinone}$, by
Proposition~\ref{prop:TGG-theta-holonomy}.

\textbf{$\SUfive$ holonomy (orbifold breaking).}
The eigenvalues of $W_5(S^1_\theta)$ are
$\{\e^{-2\pi\mathrm{i}/3},\allowbreak \e^{-2\pi\mathrm{i}/3},\allowbreak \e^{-2\pi\mathrm{i}/3},\allowbreak
\e^{\pi\mathrm{i}},\allowbreak \e^{\pi\mathrm{i}}\}$
in the $Y$-eigenstate basis, corresponding to the orbifold breaking of
$\SUfive$ at the boundary. The unbroken subgroup is the commutant of
$W_5$, which is $\SU(3)\times\SU(2)\times\U(1)_Y$.

\textbf{$\UoneB$ holonomy (gravity-baryon mixing).}
$W_B(S^1_\theta) = \exp(\mathrm{i}\,\alpha_B)$ where $\alpha_B$ is fixed
by the \BIsix{} closure to one of three discrete values according to which
quark color is being measured. In the context of the \OGNfive{} ledger,
$\alpha_B = (2\pi/L)\cdot M = 2\pi/5$.

\textbf{$\UoneQi$ holonomy (Kaxion phase).}
$W_{Qi}(S^1_\theta) = \exp(\mathrm{i}\,\alpha_{Qi})$ where
$\alpha_{Qi}$ is the Kaxion field value at the boundary, dynamical and
slowly varying with time \citep{ArisakaC1Kaxion}.

\subsection{Chern--Simons construction of the Kalb--Ramond three-form}
\label{ssec:CS-construction}

A central output of \TGG{} that is used directly in Part~III is the
Kalb--Ramond three-form $H_3 \in \Omega^3(\Msix)$ whose differential
gives the four-form anomaly polynomial $X_4$. We now construct $H_3$
explicitly from \TGG{} data via the Chern--Simons descent.

\begin{definition}[Chern--Simons three-form]
\label{def:CS-three-form}
For a connection one-form $A$ taking values in a Lie algebra $\mathfrak h$
with curvature $F = \dd A + A\wedge A$, the Chern--Simons three-form is
\begin{equation}
\label{eq:CS-3-form}
\mathrm{CS}(A) \;=\; \mathrm{Tr}\Bigl(A\wedge \dd A + \tfrac{2}{3} A\wedge A\wedge A\Bigr),
\end{equation}
where the trace is taken in a suitably normalized representation of
$\mathfrak h$. This three-form satisfies the descent identity
\begin{equation}
\label{eq:CS-descent}
\dd\,\mathrm{CS}(A) \;=\; \mathrm{Tr}(F\wedge F).
\end{equation}
For an Abelian connection (where $A\wedge A = 0$), the formula reduces
to $\mathrm{CS}(A) = A\wedge \dd A$ with $\dd\,\mathrm{CS}(A) = F\wedge F$.
\end{definition}

\begin{proposition}[\TGG{}-induced Chern--Simons construction of $H_3$]
\label{prop:H3-from-TGG}
Given the \TGG{} structure $(\Pbun, \Atot, \Rtot)$ on $\Msix$, define
\begin{equation}
\label{eq:H3-from-CS}
H_3 \;:=\; \mathrm{CS}(\omega_{\Spinone})
\;-\; \mathrm{CS}(A_5)
\;-\; \tfrac{2}{5}\,\mathrm{CS}(A_B)
\;-\; \tfrac{2}{5}\,\mathrm{CS}(A_{Qi}),
\end{equation}
with the Chern--Simons three-forms taken in the natural normalized
traces of each factor. Then
\begin{equation}
\label{eq:dH3-equals-X4}
\dd H_3 \;=\; \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}
\;=\; X_4,
\end{equation}
which is the \BIsix{} modified Bianchi identity of \citet{ArisakaA3BI6}.
\end{proposition}

\begin{proof}
Apply the descent identity \eqref{eq:CS-descent} to each of the four
sub-connections:
\begin{align}
\dd\,\mathrm{CS}(\omega_{\Spinone}) &= \tr(R_{\Spinone}\wedge R_{\Spinone}) = \tr R^2,\\
\dd\,\mathrm{CS}(A_5) &= \Tr(F_5\wedge F_5) = \Tr F_5^2,\\
\dd\,\mathrm{CS}(A_B) &= F_B\wedge F_B,\\
\dd\,\mathrm{CS}(A_{Qi}) &= F_{Qi}\wedge F_{Qi}.
\end{align}
Substituting into \eqref{eq:H3-from-CS} and applying $\dd$ termwise
gives \eqref{eq:dH3-equals-X4}. This is exactly $X_4$ of \BIsix{}.
\end{proof}

\begin{remark}[Sign and coefficient origins]
\label{rem:H3-signs}
The signs and coefficients in \eqref{eq:H3-from-CS} are not free
choices. The plus sign of the gravity (Spin) Chern--Simons term and
the minus sign of the gauge Chern--Simons terms reflect the
Killing-form sign convention of the corresponding Lie algebras
(positive-definite for the compact gauge factors, negative-definite
for the non-compact $\Spinone$). The $-2/5$ coefficients of the two
$\U(1)$ Chern--Simons terms come from the inner-product normalizations
on $\HH^2$ of the underlying $\SLtwoH$ structure
\citep[Theorem~9.1]{ArisakaA3BI6}; they are forced by the
integrality requirement on $[X_4]\in H^4(\Msix;\Z)$ and cannot be
modified without violating \BIsix{} closure.
\end{remark}

The three-form $H_3$ constructed in \eqref{eq:H3-from-CS} is therefore
a derived object of \TGG{}, not an independent input. This is one of
the key consequences of \GGEP{} unification: the Kalb--Ramond field
that drives \BIsix{} is fully determined by the \TGG{} composite
connection.

\subsection{Gauge-transformation properties of $H_3$ and $X_4$}
\label{ssec:gauge-prop-H3-X4}

Under a gauge transformation $\Atot \to g^{-1}\Atot g + g^{-1}\dd g$,
the Chern--Simons three-form changes by an exact form plus a closed
Wess--Zumino term:
\begin{equation}
\label{eq:CS-gauge-transform}
\mathrm{CS}(A^g) - \mathrm{CS}(A) = \dd \alpha_2(g, A) + \omega_{\rm WZ}(g),
\end{equation}
where $\alpha_2$ is a two-form depending on $g$ and $A$ and
$\omega_{\rm WZ}(g) = \tfrac{1}{3}\Tr(g^{-1}\dd g)^3$ is the
Wess--Zumino three-form. Therefore $H_3$ is not strictly gauge-invariant
but transforms by an exact-plus-WZ shift. The four-form
$X_4 = \dd H_3$, however, is fully gauge-invariant because
$\dd^2 = 0$ and $\dd \omega_{\rm WZ}=0$ for the WZ piece. The
gauge-invariance of $X_4$ is what makes its cohomology class
$[X_4]\in H^4(\Msix;\Z)$ a well-defined topological invariant of \TGG{}.

This gauge-transformation behavior is the algebraic heart of the
Green--Schwarz mechanism --- the 1984 pioneering of the geometrization
that \AX{2} \GGEP{} of the \GGConstitution{} elevates from a
technical refinement to a foundational axiom
\citep[\S~8.7]{ArisakaA3BI6}. In Part~III, the requirement that the
combined gauge-transformation anomaly of $H_3$ cancel against the
boundary chiral-fermion anomaly will give the integrality and
non-triviality of $[X_4]$, which is the content of \CCI{}.

\subsection{Summary of the algebraic content of \TGG{}}
\label{ssec:TGG-algebraic-summary}

Bringing together the content of Sections~\ref{ssec:Atot-decomp}
through~\ref{ssec:gauge-prop-H3-X4}, the algebraic content of \TGG{} is
summarized in the following table.

\begin{table}[H]
\centering\small
\caption[The four sub-sectors of \TGG{} and their data.]{The four sub-sectors of the composite connection, each with its
group, its connection one-form, its curvature, and the Chern--Simons
form it contributes to the three-form field strength. The signs in the
last column are the content of the table: gravity enters with one sign
and the three gauge sectors with the other, and the two abelian factors
carry the same $2/5$ coefficient that the \OGNfive{} ledger fixes.
Reading that column downward is reading the anomaly-canceling
combination which \TGG{} asserts is a single object rather than four
that happen to sit on the same manifold.}
\label{tab:TGG-algebraic}
\begin{tabular}{lllll}
\toprule
\textbf{Sector} & \textbf{Group} & \textbf{Connection} & \textbf{Curvature} & \textbf{CS form contribution to $H_3$}\\
\midrule
Gravity & $\Spinone$ & $\omega_{\Spinone}$ & $R_{\Spinone}$ & $+\,\mathrm{CS}(\omega_{\Spinone})$\\
Visible & $\SUfive$ & $A_5$ & $F_5$ & $-\,\mathrm{CS}(A_5)$\\
Baryon & $\UoneB$ & $A_B$ & $F_B$ & $-\,\frac{2}{5}\,\mathrm{CS}(A_B)$\\
Qi & $\UoneQi$ & $A_{Qi}$ & $F_{Qi}$ & $-\,\frac{2}{5}\,\mathrm{CS}(A_{Qi})$\\
\bottomrule
\end{tabular}
\end{table}

The four rows are not independent: they are four projections of a
single underlying object (\GGEP{}, Structure Theorem \textbf{TGG-ST5}). The signs
in the third column come from the Killing-form signs of the four Lie
algebras; the $-2/5$ coefficients of the last two rows come from the
inner-product normalizations on $\HH^2$.

\textbf{Discussion.}
The four Wilson lines are the boundary face of \TGG{}'s holonomy
structure. Three of them ($W_5, W_B, W_{Qi}$) are non-trivial and
encode the dynamical content of the boundary theory; the fourth
($W_{\Spinone}$) is trivial because the product metric has no
$\theta$-component in its spin connection. This asymmetry is essential:
it is what allows the gauge sector to break (orbifold,
Wilson--Hosotani, Kaxion) while keeping the gravity sector unbroken,
which is what we observe in nature.


%% ====================================================================
\section{One object, and what it means that it is unique}
\label{sec:close-part2}
%% ====================================================================

The result of this Part can be said in one sentence: the four forces
can be written as one geometric object on the six-dimensional arena,
and there is no second way to do it.

Both halves of that sentence matter, and the second is the one worth
dwelling on. Physics has produced unifications before that were merely
possible --- schemes in which the known forces could be arranged inside
a larger structure, and could equally well have been arranged inside
several others. A possible arrangement explains nothing, because the
question "why this one?" has no answer, and the honest form of such a
proposal is a menu. What changes here is that the constitutional
requirements inherited in Part~I leave exactly one arrangement
standing. The four forces are not unified because it is elegant to
unify them. They are unified because, given the arena and the rules,
nothing else fits.

The Master Conservation Law that follows is worth reading in plain
language, because its content is more ordinary than its name. It says
that the energy and momentum of everything in the observable
four-dimensional world --- matter, the gauge fields, the leftover
higher-dimensional modes, and the correction term that consistency
demands --- add up to something that does not leak. Nothing drains out
of the world through a channel the accounting missed. That sounds like
a statement one could take for granted, and in ordinary physics one
can. In a theory where the observable world is a projection of a larger
one, it is not automatic at all: a projection is exactly the kind of
operation that loses things. The theorem says this particular
projection does not.

There is a habit of thought this Part is quietly arguing against. It is
natural to imagine gravity as the arena and the other three forces as
the furniture --- one thing that curves, three things that sit on it.
That picture is useful and it is not what the mathematics says here.
There is one structure; gravity is what it looks like along certain
directions and the gauge forces are what it looks like along others.
The distinction between arena and content, which feels like the most
basic distinction one could draw about the physical world, turns out to
be a statement about which way an observer happened to look.

That is not a rhetorical flourish; it has a concrete consequence for
how the rest of the paper reads. When a later Part speaks of energy
flowing between the gravitational sector and the gauge sector, it is
not describing a transaction between two things. It is describing one
object redistributing itself, in the way that a person who shifts their
weight has not handed anything to anybody.

\emph{The deeper lesson is about how much a consistency requirement can
do.} Nothing in this Part fits a parameter to data. The uniqueness comes
out of demanding that a structure be well defined at all --- and that
demand, made precisely enough, turns out to be nearly as informative as
an experiment. The next Part takes that observation much further: it
asks what a single consistency condition forces when it is imposed not
on the shape of the object but on its bookkeeping, and the answer is a
set of integers that nobody chose.


\GGPart[CCI]{\CCI{}}{The Covariant Conservation Identity: Pillar 2 of \GiFour{}}

Every conservation law in physics is, at bottom, a piece of
bookkeeping: something goes in, something comes out, and the ledger
balances. The awkward discovery of the twentieth century was that in
quantum theory the ledger does not always balance by itself. Certain
combinations of fields produce what are called anomalies --- terms that
appear from nowhere and break a conservation law that the classical
theory obeyed exactly. A theory in which that happens is not merely
inelegant; it is inconsistent, and it must be discarded.

This Part is about the ledger. It asks what has to be true of the
six-dimensional geometry for the four-dimensional conservation law of
Part~II to hold, and it finds that the requirement is far more
restrictive than one would guess. The anomalies do not simply have to
cancel; the way they cancel forces the numbers in the theory to take
particular values. When the closure condition is imposed, a single set
of five integers survives, and everything the cohort later computes ---
the number of particle families, the parameters of the Standard Model,
the characteristic mass scales --- turns out to be a closed-form
function of those five. That is an unusual position for a physical
theory to be in. It is what the phrase ``no free parameters'' means in
practice, and it is also what makes the theory easy to kill.

Two further results are established here. The first is that the
cancellation is not free: it requires a compensating structure on the
boundary, and the Part identifies what that structure must be and what
it costs. The second is the Part's empirical exposure. The same ledger
that fixes the integers also modifies the equation governing the
internal structure of a dense star, by an amount the theory computes
rather than fits. That prediction is a falsifier: a sufficiently precise
measurement of a neutron star's structure either finds the modification
or ends this pillar.

The mathematics used --- differential cohomology, and the descent
construction that goes with it --- is the one genuinely specialized tool
in the paper, and it is not assumed. It is developed from the beginning
in Part~X, with worked examples, for a reader who has not met it. Within
this Part, a reader can follow the argument by reading the opening
section and then the statement of the ledger result, treating the
cohomological machinery as a black box that delivers the integers; the
box is opened in the appendix for anyone who wants to look inside.


\section{\CCI{}: motivation, dual-layer structure, and Structure Theorems}
\label{sec:CCI-axioms}

This Part develops \CCI{} (the \emph{Covariant Conservation Identity}),
the second of the four \GiFour{} Pillars. Like \TGG{} of
Part~II, \CCI{} is intrinsically a \emph{dual-layer} object: it is
simultaneously
\begin{enumerate}[leftmargin=2em,label=(\roman*)]
\item a \textbf{boundary/physical statement} --- the local 4D conservation
equation
\begin{equation}
\label{eq:CCI-physical-prelude}
\nabla_\mu T^{\mu\nu}_{\TGG} \;=\; 0
\end{equation}
on the boundary $\partial\Msix \cong M_4$, asserting that
the Master Stress Tensor of \TGG{} is covariantly conserved when (and
only when) the bulk \BIsix{} closure $\dd H_3 = X_4$ is satisfied
\emph{and} the four boundary Ward currents (electric, baryonic,
informational, metric) are simultaneously closed; and
\item an \textbf{algebraic/bulk statement} --- the cohomological
identity
\begin{equation}
\label{eq:CCI-algebraic-prelude}
[X_4] \in H^4(\Msix;\Z),
\qquad
X_4 = \tr R_{\Spinone}^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2,
\end{equation}
asserting that the four-form anomaly polynomial $X_4$ has a unique
algebraic structure (with the four specific coefficients
$+1, -1, -\tfrac{2}{5}, -\tfrac{2}{5}$) and that its de~Rham class is
integral; \MSMF{} then selects the minimal-complexity integer ledger
$(M, N, L; J, K) = (1,3,5;7,35)$, the \emph{\OGNfive{}} ledger.
\end{enumerate}

The two layers are not independent: they are linked by the
\emph{Projected Bianchi Identity} \PBIfour{} (introduced in
Section~\ref{sec:TGG-boundary-stress-tensor} of Part~II), which
descends the bulk closure $\dd H_3 = X_4$ to a boundary divergence
identity, and by the Master Conservation Theorem (\TGG-ST7). In short:
\begin{equation}
\label{eq:CCI-dual-link}
\boxed{\;
\underbrace{\dd H_3 = X_4}_{\text{bulk \BIsix{} (algebraic)}}
\;\xrightarrow{\;\text{KK descent}\;}\;
\underbrace{\nabla_\mu T^{\mu\nu}_{\TGG} = 0}_{\text{boundary \CCI{} (physical)}}
\;}
\end{equation}
The arrow ``$\xrightarrow{\;\text{KK descent}\;}$'' is precisely
\PBIfour{}: it carries the algebraic identity to its physical avatar.

\paragraph{Section structure of Part~III.}
Section~\ref{sec:CCI-axioms} (the present section) states the seven
\CCI{} axioms (\textbf{CCI-ST1} through \textbf{CCI-ST7}), organized
into the algebraic block (\textbf{CCI-ST1}--\textbf{CCI-ST5}) and the
physical block (\textbf{CCI-ST6}--\textbf{CCI-ST7}); gives the dual-layer
formal definition of \CCI{}; and lists the first immediate
consequences. Section~\ref{sec:X4-structure} develops the four-form
anomaly polynomial $X_4$ in detail, deriving its closure, integrality,
and non-triviality from the \TGG{} structure of Part~II.
Section~\ref{sec:CCI-coefficients} derives the four coefficients in
$X_4$ --- in particular the $-2/5$ coefficients of the two $\U(1)$
terms --- from the inner-product normalization on $\HH^2$.
Section~\ref{sec:CCI-vs-BI6} contrasts \CCI{}-algebraic with \BIsix{}
(differential) and shows how the two identities couple via \PBIfour{}.
Section~\ref{sec:CCI-physical-layer} (new in) develops the
physical/boundary layer of \CCI{}: the local 4D conservation equation,
its derivation from \PBIfour{}, and its astrophysical falsifier (the
\emph{modified TOV equation} for compact stars).
Section~\ref{sec:CCI-theorem} states and proves Theorem~\textbf{T2},
now in two parts: \textbf{T2.1} (the algebraic \OGNfive{} uniqueness
theorem) and \textbf{T2.2} (the boundary conservation theorem).
Section~\ref{sec:CCI-Hopkins-Singer} connects \CCI{} to the
Hopkins--Singer differential cohomology framework that will be used in
Part~IV for \GFF{} Morse-positivity. Section~\ref{sec:CCI-examples}
contains worked examples.

\begin{remark}[Physics intuition: \CCI{} as the GG analogue of GR's contracted Bianchi identity]
\label{rem:CCI-GR-analogue}
Readers familiar with general relativity will find an immediate
analogy. In GR, the contracted Bianchi identity
$\nabla_\mu G^{\mu\nu} = 0$ (where $G^{\mu\nu}$ is the Einstein tensor)
is a \emph{kinematic} identity that follows automatically from the
diffeomorphism invariance of the Einstein--Hilbert action. It implies
the local conservation of the matter stress tensor
$\nabla_\mu T^{\mu\nu}_{\rm matter} = 0$ \emph{provided} the field
equations $G^{\mu\nu} = 8\pi G\, T^{\mu\nu}_{\rm matter}$ hold. This
is the \emph{single} conservation law of pure GR.

In \GGTheory{}, \CCI{} plays the same logical role but for the
\emph{six-sector} Master Stress Tensor of \TGG{} (\TGG-ST6):
$T^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge} + X_{\rm GS} +
T_{\rm KK}$. The identity $\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ is again
kinematic: it follows from the bulk diffeomorphism invariance of the
\GG{} effective action plus the closure of the four Ward currents on
the boundary. But unlike GR, \GGTheory{} has \emph{multiple}
simultaneous conservation laws (one per sector), and \CCI{} packages
them into a single covariant master identity. This is the precise
sense in which \CCI{} \emph{generalizes} GR's contracted Bianchi
identity from one sector (gravity) to six (gravity, electroweak,
strong, baryonic, informational, KK).
\end{remark}

\begin{remark}[Why broader scientists should care about \CCI{}]
\label{rem:CCI-broader}
The Master Conservation Law $\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ is the
\emph{only} statement of \GGTheory{} that every empirical scientist ---
not just particle physicists --- already uses every day. Energy
conservation in chemistry, momentum conservation in fluid dynamics,
charge conservation in electrochemistry, baryon-number near-conservation
in cosmology, mass--energy balance in cell biology, information
conservation (entropy bounds) in neuroscience: all are \emph{boundary
shadows} of \CCI{}. What is new is that \GGTheory{} shows these
conservation laws are not independent postulates but follow from a
single bulk identity $\dd H_3 = X_4$ via the Projected Bianchi
Identity \PBIfour{}. This is why \CCI{} is the \emph{universal
empirical bridge} from \GG{} bulk geometry to the laboratory: it
explains, in one stroke, why all of physics, chemistry, biology, and
information science share the same kinematic backbone.
\end{remark}

\subsection{Motivation: why a dual-layer pillar is needed}
\label{ssec:CCI-motivation}

The \TGG{} construction of Part~II provides the principal
$\Gtot$-bundle $\Pbun$, the composite connection $\Atot$, the
composite curvature $\Rtot$ on the 6D bulk $\Msix$, the
boundary Master Stress Tensor $T^{\mu\nu}_{\TGG}$ (\TGG-ST6), and the
Master Conservation Law $\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ (\TGG-ST7).
But \TGG{} alone does not fix two crucial pieces of structure:

\smallskip
\noindent\textbf{Question 1 (algebraic).}
\TGG{} does not fix \emph{which} four-form built from $\Rtot$ should
appear as the source of the modified Bianchi identity \BIsix{}.
Several different four-forms can be constructed from the four
sub-curvatures $(R_{\Spinone}, F_5, F_B, F_{Qi})$:
\begin{itemize}[leftmargin=2em]
\item $\tr R_{\Spinone}^2$, $\Tr F_5^2$, $F_B \wedge F_B$, $F_{Qi}\wedge F_{Qi}$
(the four ``self-curvature'' four-forms);
\item Cross-curvature four-forms $F_B \wedge F_{Qi}$, $\tr R_{\Spinone}\wedge F_B$, etc.;
\item Higher-degree-sensitive combinations like
$\tr R_{\Spinone}\wedge \tr R_{\Spinone}$, etc.
\end{itemize}
Without an additional principle, all of these four-forms are
mathematically admissible candidates for $X_4$. But \BIsix{} requires
a specific combination with specific coefficients --- namely
\begin{equation}
\label{eq:X4-structure-CCI}
X_4 = \tr R_{\Spinone}^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2,
\end{equation}
to within an overall normalization. The structure
\eqref{eq:X4-structure-CCI} is not free: it is forced by the
inner-product structure of \TGG{} on the quaternionic doublet $\HH^2$,
by the requirement of integrality of $[X_4]$, and by the parity / sign
conventions of the four sub-algebra Killing forms. This is the
\textbf{algebraic content} of \CCI{}.

\smallskip
\noindent\textbf{Question 2 (physical).}
\TGG{} on its own asserts that \emph{some} master stress tensor exists
on the boundary (\TGG-ST6) and that it is \emph{conditionally}
conserved (\TGG-ST7) --- conserved iff the four Ward currents are
closed. But \TGG-ST7 does not specify the precise differential equation
that governs each sector's contribution to $\nabla_\mu T^{\mu\nu}_{\TGG}$,
nor does it produce a single \emph{unconditional} astrophysical or
cosmological prediction. The physical content of \CCI{} fills both
gaps: it states the local 4D conservation equation
\begin{equation}
\label{eq:CCI-physical-prelude2}
\nabla_\mu T^{\mu\nu}_{\TGG}(x) \;=\; 0 \qquad \forall\, x \in M_4
\end{equation}
in \emph{operational} form, identifies the \PBIfour{} compensator field
that enforces it, and predicts a specific \emph{astrophysical
falsifier}: the modified Tolman--Oppenheimer--Volkoff (TOV) equation
for compact stars, with a calculable correction proportional to
$X_{\rm GS}$. This is the \textbf{physical content} of \CCI{}.

\smallskip
\noindent\textbf{Why both layers must be in the same Pillar.}
The two questions are inseparable. The integrality of $[X_4]$ (the
algebraic side) is precisely the condition for \PBIfour{} to descend
without anomaly to a smooth 4D conservation equation (the physical
side). Conversely, the empirical observation that $\nabla_\mu
T^{\mu\nu}_{\rm matter} = 0$ holds in laboratory and astrophysical
data (the physical side) provides direct constraints on the bulk
integers $(M, N, L; J, K)$ (the algebraic side), via cosmological and
neutron-star observations. This bidirectional coupling
\emph{mandates} that both layers be axiomatized together.

\CCI{} is the formalization of this dual content. It is
simultaneously:
\begin{itemize}[leftmargin=2em]
\item the \emph{algebraic consistency identity} of the composite
connection, distinguishing the unique physical $X_4$ from the infinite
family of mathematically admissible four-forms; and
\item the \emph{boundary conservation identity}, asserting
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ as the universal empirical content
of the bulk closure $\dd H_3 = X_4$.
\end{itemize}
Once \CCI{} is given, \BIsix{} is the \emph{differential consequence}
of \CCI{} together with the de Rham exactness of $X_4$, and the
boundary conservation equation is its \emph{empirical projection}
through \PBIfour{}.

In short: \TGG{} gives the bulk geometry and the boundary master
stress tensor; \CCI{} gives both the algebraic form of the closure
condition (uniqueness of $X_4$) and the physical conservation law on
the boundary (operational content of $\nabla_\mu T^{\mu\nu}_{\TGG} =
0$); \BIsix{} gives the differential content of the closure condition
in the bulk; and \PBIfour{} ties the three together. All four
structures are needed.

\begin{remark}[Physics intuition: the $-2/5$ coefficient and the \OGNfive{} ledger]
\label{rem:CCI-25-intuition}
The reader may wonder why the coefficient of $F_B^2$ and $F_{Qi}^2$ in
$X_4$ is exactly $-2/5$. The intuitive answer is dimensional: the
quaternionic doublet $\HH^2$ has real dimension $8$, but its
inner-product normalization on the two $\U(1)$ Cartan directions
factors out a rank-$5$ index from the magic-dimension ladder
$\R\to\C\to\HH\to\Octonion$. The factor $2/5$ then arises as the ratio of
the doublet inner product (which contributes 2) to the \OGNfive{} base
length (which contributes 5). Section~\ref{sec:CCI-coefficients}
makes this precise.

The key takeaway is that the $-2/5$ is not a fitting parameter --- it
is the \emph{unique} coefficient that makes the integrality
$[X_4]\in H^4(\Msix;\Z)$ compatible with the \OGNfive{} minimal ledger
$(M,N,L;J,K) = (1,3,5;7,35)$. Any other coefficient would either
break integrality or force a non-minimal (and hence \MSMF-disfavored)
ledger. This is why the \OGNfive{} ledger is the unique solution.
\end{remark}

\subsubsection{Historical background (with citations)}

The covariant conservation identity that \CCI{} formalizes traces back
to the very first geometric identities of modern physics.
\citet{Bianchi1902} proved the celebrated identity $\nabla_{[\mu}
R_{\nu\rho]\sigma\tau} = 0$ for the Riemann tensor in 1902 --- a purely
geometric statement, derived without any reference to dynamics or
quantum effects, that fixes once and for all the local conservation of
the geometric curvature. The electromagnetic analogue $\dd F = 0$, the
homogeneous half of Maxwell's equations \citep{Maxwell1865}, is the
abelian version of the same statement.

The dynamical face of \CCI{} is due to \citet{Noether1918}: every
continuous symmetry of the action yields a conserved current, and the
divergence-free property $\partial_\mu J^\mu = 0$ is the identity that
ties symmetry to conservation. \citet{Ward1950} re-expressed Noether's
identity in the language of quantum field theory, with the Ward
identity stating that the gauge-transformation variation of the
effective action vanishes on shell. The non-abelian extension was
completed in the 1970s by Slavnov, Taylor, Becchi, Rouet, and Stora,
and independently by Tyutin, with the BRST formulation as the
cohomological setting in which \CCI{} lives naturally.

Indeed, the quantum surprise that forced the modern reading of \CCI{}
was the chiral anomaly of \citet{Adler1969} and \citet{BellJackiw1969}:
the axial current of a chiral fermion is not conserved at the loop
level, but acquires a divergence proportional to $F\wedge\tilde F$.
This loss is reconciled with the geometric Bianchi identity through the
geometrization move pioneered by \citet{GreenSchwarz1984} (whose
\GGEP{}-completion reading we discussed in
Section~\ref{sec:gg6-constitution-recap}): the bulk 4-form $X_4$ absorbs the
boundary anomaly into a closed differential object whose cohomology
class $[X_4] \in H^4(\Msix;\Z)$ is integral
\citep{HopkinsSinger2005}. \CCI{} is the axiomatic reading of the
integrality requirement: \BIsix{} closure under \MSMF{} forces the
unique \OGNfive{} ledger $(M,N,L;J,K)=(1,3,5;7,35)$, and the
four-component coefficient vector $(a,b,c,d) = (1,-1,-2/5,-2/5)$
becomes the unique solution \citep[Theorem~9.1]{ArisakaA3BI6}.


\subsection{The seven \CCI{} Structure Theorems (algebraic block + physical block)}
\label{ssec:CCI-seven-st}

We now formalize the dual-layer structure of \CCI{} through seven
Structure Theorems --- five governing the algebraic content of the
4-form anomaly polynomial and two governing its physical projection
to boundary observables. The seven \CCI{} Structure Theorems
partition naturally into two blocks:
\begin{description}[leftmargin=2em]
\item[Algebraic block: \textbf{CCI-ST1} -- \textbf{CCI-ST5}] These Structure Theorems
fix the structural form of the four-form anomaly polynomial $X_4$, its
coefficients, its closure, its integrality, and the \OGNfive{} minimality
condition. They live entirely in the bulk $\Msix$ and are
representations-theoretic in character.
\item[Physical block: \textbf{CCI-ST6} -- \textbf{CCI-ST7}] These Structure Theorems
state the boundary covariant conservation equation $\nabla_\mu
T^{\mu\nu}_{\TGG} = 0$ and its astrophysical falsifier (the modified
TOV equation). They live on the boundary
$\partial\Msix\cong M_4$ and are operationally testable.
\end{description}
The two blocks are linked by the Projected Bianchi Identity
\PBIfour{} of Section~\ref{sec:TGG-boundary-stress-tensor}.

\subsubsection*{Algebraic block (CCI-ST1 through CCI-ST5)}

\begin{structheorem}[\textbf{CCI-ST1}: Structural form of $X_4$]
\label{st:CCI-ST1}
The four-form anomaly polynomial $X_4 \in \Omega^4(\Msix)$ is built from
the \TGG{} curvature $\Rtot$ as a sum of four self-curvature
contributions:
\begin{equation}
\label{eq:CCI-ST1}
X_4 = \alpha_R\,\tr(R_{\Spinone}\wedge R_{\Spinone})
+ \alpha_5\,\Tr(F_5\wedge F_5)
+ \alpha_B\,F_B \wedge F_B
+ \alpha_{Qi}\,F_{Qi}\wedge F_{Qi},
\end{equation}
where $\alpha_R, \alpha_5, \alpha_B, \alpha_{Qi}$ are real coefficients.
No cross-curvature four-forms appear in $X_4$.
\end{structheorem}

\begin{structheorem}[\textbf{CCI-ST2}: Coefficient determination from $\HH^2$ normalization]
\label{st:CCI-ST2}
The four coefficients in \eqref{eq:CCI-ST1} are determined by the
inner-product normalization on the quaternionic doublet $\HH^2$ of
\TGG-ST3:
\begin{equation}
\label{eq:CCI-ST2-coeffs}
\alpha_R = +1, \qquad
\alpha_5 = -1, \qquad
\alpha_B = -\tfrac{2}{5}, \qquad
\alpha_{Qi} = -\tfrac{2}{5}.
\end{equation}
The signs and magnitudes are derived in Section~\ref{sec:CCI-coefficients};
they are not free parameters but consequences of the
$\Spinone\cong\SLtwoH$ algebraic structure.
\end{structheorem}

\begin{structheorem}[\textbf{CCI-ST3}: Closure of $X_4$]
\label{st:CCI-ST3}
The four-form $X_4$ is closed:
\begin{equation}
\label{eq:CCI-ST3-closure}
\dd X_4 = 0.
\end{equation}
\end{structheorem}

\begin{structheorem}[\textbf{CCI-ST4}: Integrality]
\label{st:CCI-ST4}
The de Rham cohomology class of $X_4$ is integral:
\begin{equation}
\label{eq:CCI-ST4-integrality}
[X_4] \;\in\; H^4(\Msix; \Z),
\end{equation}
i.e., $X_4$ represents an integer cohomology class on $\Msix$.
\end{structheorem}

\begin{structheorem}[\textbf{CCI-ST5}: Non-triviality and \OGNfive{} minimality]
\label{st:CCI-ST5}
The class $[X_4]$ is non-zero in $H^4(\Msix;\R)$, and among all
admissible \TGG{} configurations satisfying
\textbf{CCI-ST1}--\textbf{CCI-ST4}, \MSMF{} selects the configuration
whose integrality data $(M, N, L; J, K)$ has minimum total complexity
$|M|+|N|+|L|+|J|+|K|$ subject to the closure constraints
\begin{equation}
\label{eq:OGN5-closure-constraints}
K = M^2 + N^2 + L^2,
\qquad
K = LJ.
\end{equation}
\end{structheorem}

The five Structure Theorems \textbf{CCI-ST1}--\textbf{CCI-ST5} constitute the
\emph{algebraic} block of \CCI{} and uniquely determine the \OGNfive{}
ledger $(1,3,5;7,35)$ via Theorem~\textbf{T2.1} (proved in
Section~\ref{sec:CCI-theorem}).

\subsubsection*{Physical block (CCI-ST6 and CCI-ST7)}

The physical block translates the algebraic structure of \CCI{} into
operational, observer-accessible form on the boundary $M_4$. It
consists of two Structure Theorems.

\begin{structheorem}[\textbf{CCI-ST6}: Boundary covariant conservation (sector-summed)]
\label{st:CCI-ST6}
On the boundary $\partial\Msix\cong M_4$, the Master
Stress Tensor $T^{\mu\nu}_{\TGG}$ of \TGG-ST6 satisfies the
\emph{sector-summed} covariant conservation equation
\begin{equation}
\label{eq:CCI-ST6-conservation}
\nabla_\mu T^{\mu\nu}_{\TGG}(x) \;=\; 0
\qquad \forall\, x \in M_4,
\end{equation}
where $\nabla$ is the Levi--Civita connection of the induced boundary
metric $g^{(4)}_{\mu\nu}$. Decomposing
$T^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge} + X_{\rm GS} + T_{\rm KK}$
(\TGG-ST6), the individual sector divergences are non-zero ---
computed explicitly in Proposition~\ref{prop:sector-divergences} of
Section~\ref{ssec:descent-map} --- and add to zero:
\begin{equation}
\label{eq:CCI-ST6-decomposition}
\nabla_\mu T^{\mu\nu}_{\rm matter}
+ \nabla_\mu T^{\mu\nu}_{\rm gauge}
+ \nabla_\mu X_{\rm GS}^{\mu\nu}
+ \nabla_\mu T^{\mu\nu}_{\rm KK}
\;=\; 0.
\end{equation}
The four sector divergences are linked --- not zero individually but
summing to zero --- by the \PBIfour{} \emph{compensator field}
$C_\mu$ (defined in Section~\ref{sec:CCI-physical-layer}); the
sector-summed identity \eqref{eq:CCI-ST6-conservation} follows from
the bulk \BIsix{} closure $\dd H_3 = X_4$ combined with the
integrality of $[X_4]$ (\textbf{CCI-ST4}), which guarantees
$\langle X_4 \rangle_{\rm KK} = 0$ at the boundary. In particular,
CCI-ST6 is a sector-summed kinematic identity rather than a
sector-wise conservation law: each Ward current carries its own
anomaly inflow~\citep{Bardeen1969,AlvarezGaumeWitten1984,CallanHarvey1985} from the corresponding 4-form term in $X_4$, and the
anomaly inflows cancel collectively (by closure of $X_4$) rather than
individually.
\end{structheorem}

\begin{structheorem}[\textbf{CCI-ST7}: Astrophysical falsifier (modified TOV) --- primary derivation in A5 T8.6]
\label{st:CCI-ST7}
On a static, spherically symmetric, gravitationally bound configuration
(e.g.\ a neutron star, white dwarf, or planetary interior), the
boundary conservation equation \eqref{eq:CCI-ST6-conservation} reduces
to the \emph{modified Tolman--Oppenheimer--Volkoff} (TOV) equation
(primary derivation: \citealp[Theorem~T8.6, Part~IV]{ArisakaA5GDOS};
the present structure theorem records the CCI-side identity)
\begin{equation}
\label{eq:CCI-ST7-mTOV}
\boxed{\;
\frac{\dd p}{\dd r}
\;=\;
-\,\frac{G\,(\rho+p)\,(m + 4\pi r^3 p)}{r(r-2Gm)}
\;-\;\Delta_{\rm GS}(r),
\;}
\end{equation}
where $\Delta_{\rm GS}(r)$ is a calculable, non-vanishing correction
arising from the Green--Schwarz contribution $X_{\rm GS}$ in
$T^{\mu\nu}_{\TGG}$. The functional form of $\Delta_{\rm GS}(r)$ is
fixed by \PBIfour{} and is given (to leading order in the scale-profile
expansion) by
\begin{equation}
\label{eq:CCI-ST7-DeltaGS}
\Delta_{\rm GS}(r) \;=\; \kappa_{\rm GS}\,
\frac{\langle X_4\rangle_{\rm bulk}}{M_{\rm KK}^4}\,
\frac{1}{r^2}\,\frac{\dd}{\dd r}\!\left(r^2 \rho_{\rm KK}(r)\right),
\end{equation}
with $\kappa_{\rm GS}\sim \mathcal{O}(1)$ a coefficient computable from
the \OGNfive{} ledger. Empirically, equation \eqref{eq:CCI-ST7-mTOV}
predicts a small but in-principle observable shift of the neutron-star
mass--radius curve relative to the standard TOV prediction; this shift
is the principal astrophysical falsifier of \CCI{}.

If empirical data on neutron-star masses and radii (from NICER,
gravitational-wave inspiral, etc.) ever decisively rule out
\eqref{eq:CCI-ST7-mTOV} for any $\kappa_{\rm GS}\neq 0$, then \CCI{}
(and with it the entire \GG{} framework) is falsified.
\end{structheorem}

The seven Structure Theorems \textbf{CCI-ST1}--\textbf{CCI-ST7} together constitute
the full \emph{Covariant Conservation Identity}. They will be used in
Section~\ref{sec:CCI-theorem} to prove Theorem~\textbf{T2}, now in two
parts: \textbf{T2.1} (algebraic \OGNfive{} uniqueness) and \textbf{T2.2}
(boundary conservation iff Ward + \PBIfour{}).

\subsection{Dual-layer formal definition of \CCI{}}
\label{ssec:CCI-formal-def}

We now consolidate the algebraic and physical blocks into a single
dual-layer definition.

\begin{definition}[\CCI{}: the Covariant Conservation Identity, dual-layer form]
\label{def:CCI}
The \emph{Covariant Conservation Identity} on the \TGG{} structure
$\bigl(\Pbun, \Atot, \Rtot;\, T^{\mu\nu}_{\TGG}, \nabla\bigr)$ over the
6D bulk $\Msix$ with boundary $\partial\Msix\cong M_4$ is the
\textbf{septuple}
\begin{equation}
\label{eq:CCI-septuple}
\CCI = \Bigl(\;
\underbrace{X_4,\;[X_4],\;(\alpha_R, \alpha_5, \alpha_B, \alpha_{Qi}),\;
\dd X_4 = 0,\;\text{\OGNfive{} minimality}}_{\text{algebraic block (CCI-ST1--A5)}};
\;\;
\underbrace{\nabla_\mu T^{\mu\nu}_{\TGG} = 0,\;\;\Delta_{\rm GS}(r)}_{\text{physical block (CCI-ST6--A7)}}
\;\Bigr),
\end{equation}
where the seven components correspond to the Structure Theorems
\textbf{CCI-ST1}--\textbf{CCI-ST7} respectively.

The triple $(X_4, [X_4], \text{coefficients})$ is required to satisfy
the integrality and non-triviality conditions of \textbf{CCI-ST4} and
\textbf{CCI-ST5}. The boundary conservation equation
\eqref{eq:CCI-ST6-conservation} is required to follow from the bulk
closure $\dd H_3 = X_4$ via \PBIfour{}. The TOV correction
$\Delta_{\rm GS}(r)$ is required to be the unique \PBIfour{}-compatible
spherically-symmetric reduction. Two \CCI{} configurations are
equivalent if they are related by gauge transformations of the
underlying \TGG{} structure (by
Proposition~\ref{prop:TGG-gauge-group}) together with passive
boundary diffeomorphisms.
\end{definition}

\begin{remark}[Logical content of \CCI{}]
\label{rem:CCI-content}
The intuitive content of Definition~\ref{def:CCI} is twofold:
\begin{itemize}[leftmargin=2em]
\item \textbf{Algebraic side.} \CCI{} fixes the four-form $X_4$
uniquely (up to overall normalization and gauge equivalence) and
asserts that its cohomology class is integral and non-trivial. The
integrality constraint is what selects the \OGNfive{} ledger from the
infinite family of admissible $(M, N, L; J, K)$ tuples. The
non-triviality constraint is what prevents the boundary observables
from collapsing into a degenerate zero-curvature configuration; it is
the algebraic prerequisite for the \GFF{} Morse-positivity of
Part~IV.
\item \textbf{Physical side.} \CCI{} packages all six sector
conservation laws into the single Master Conservation Law
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$. This is the universal empirical
content of \GG{} that observers actually measure: every conservation
law of physics, chemistry, biology, and information theory is a
boundary projection of this identity. The astrophysical falsifier
\eqref{eq:CCI-ST7-mTOV} provides a sharp empirical test.
\end{itemize}
The two sides are linked by \PBIfour{}: the algebraic integrality
$[X_4]\in H^4(\Msix;\Z)$ is precisely the necessary and sufficient
condition for \PBIfour{} to descend to a smooth, anomaly-free 4D
conservation equation. This is the \emph{duality} of \CCI{}.
\end{remark}

\begin{remark}[Physics intuition: from $\dd H_3 = X_4$ to laboratory measurements]
\label{rem:CCI-bulk-to-lab}
A graduate student in physics may find it useful to trace the chain
\begin{equation*}
\begin{gathered}
\underbrace{\dd H_3 = X_4}_{\text{bulk \BIsix{}}}
\;\xrightarrow{\;[X_4]\in H^4(\Msix;\Z)\;}\;
\underbrace{\PBIfour{}}_{\text{descent}}
\;\xrightarrow{\;\text{KK reduction}\;}\;
\underbrace{\nabla_\mu T^{\mu\nu}_{\TGG} = 0}_{\text{boundary 4D conservation}}\\
\;\xrightarrow{\;\text{spherical symmetry}\;}\;
\underbrace{\dd p/\dd r = -\,\cdots\, -\,\Delta_{\rm GS}(r)}_{\text{modified TOV}}.
\end{gathered}
\end{equation*}
Each arrow is a precise mathematical reduction --- there are no
phenomenological insertions. The first arrow is purely topological
(integrality of de~Rham cohomology); the second is purely algebraic
(KK harmonic decomposition on the extra dimensions); the third
is purely geometric (Schwarzschild ansatz applied to a static
spherically symmetric configuration).

What is genuinely new is that the modified TOV correction
$\Delta_{\rm GS}(r)$ is \emph{not} an ad-hoc parameter. It is
calculable from the \OGNfive{} ledger $(1,3,5;7,35)$ once the bulk
scale running $W(y)$ of \TGG-ST2 is specified. This is the empirical
backbone of the \GG{} prediction for compact-object structure.
\end{remark}

\subsection{First consequences of the \CCI{} Structure Theorems}
\label{ssec:CCI-first-consequences}

We list four immediate consequences that will be used in the rest of
the Part.

\begin{proposition}[Closure of $X_4$ from sub-curvature Bianchi identities]
\label{prop:CCI-closure-from-sub}
\textbf{CCI-ST3} (closure $\dd X_4 = 0$) follows automatically from the
ordinary Bianchi identities for the four sub-curvatures.
\end{proposition}

\begin{proof}
For each sub-curvature, the Bianchi identity gives
$\dd_{\rm cov} R_{\Spinone} = 0$, $\dd_{\rm cov} F_5 = 0$,
$\dd F_B = 0$ (by exactness of the Abelian field strength),
$\dd F_{Qi} = 0$. Therefore
\begin{align}
\dd \,\tr(R_{\Spinone}\wedge R_{\Spinone}) &= 2\,\tr(R_{\Spinone}\wedge \dd R_{\Spinone}) = 0,\\
\dd \,\Tr(F_5\wedge F_5) &= 2\,\Tr(F_5\wedge \dd F_5) = 0,\\
\dd (F_B\wedge F_B) &= 2 F_B\wedge \dd F_B = 0,\\
\dd (F_{Qi}\wedge F_{Qi}) &= 2 F_{Qi}\wedge \dd F_{Qi} = 0.
\end{align}
Adding the four contributions with the coefficients of \textbf{CCI-ST2}
gives $\dd X_4 = 0$.
\end{proof}

\begin{proposition}[$X_4$ as a topological invariant of \TGG]
\label{prop:CCI-topo-inv}
The cohomology class $[X_4] \in H^4(\Msix; \R)$ is a gauge-invariant
topological invariant of the \TGG{} structure. In particular, $[X_4]$
does not depend on the specific choice of \TGG{} representative within
its gauge orbit (Proposition~\ref{prop:TGG-gauge-group}), and it is
unchanged under continuous deformations of $\Atot$ that preserve the
underlying $\Gtot$-bundle topology.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:CCI-closure-from-sub}, $X_4$ is closed. Under
a gauge transformation $\Atot \to g^{-1}\Atot g + g^{-1}\dd g$, each
self-curvature transforms by conjugation:
$R\to g^{-1} R g$, etc. But $\tr$ and $\Tr$ are gauge-invariant traces
in their respective representations, so
$\tr(R_{\Spinone}\wedge R_{\Spinone})$, $\Tr(F_5\wedge F_5)$, $F_B^2$,
and $F_{Qi}^2$ are all gauge-invariant 4-forms. Therefore $X_4$ is
gauge-invariant. Continuous deformations within a gauge orbit change
$X_4$ by an exact form
\citep[Theorem~II.5.3]{KobayashiNomizu1969}, leaving $[X_4]$ unchanged.
\end{proof}

\begin{proposition}[{Integrality of $[X_4]$ from sub-bundle integrality}]
\label{prop:CCI-integrality-from-sub}
\textbf{CCI-ST4} ($[X_4]\in H^4(\Msix;\Z)$) follows from the
integrality of each sub-bundle's first Chern class (for the Abelian
factors) or first Pontryagin class (for the non-Abelian factors), with
the coefficients of \textbf{CCI-ST2} chosen consistently.
\end{proposition}

\begin{proof}[Sketch]
For the gauge factors:
\begin{itemize}[leftmargin=2em]
\item $\frac{1}{8\pi^2}\tr(R_{\Spinone}\wedge R_{\Spinone})$ represents
the first Pontryagin class $p_1$ of the $\Spinone$ bundle, integral
in $H^4(\Msix;\Z)$.
\item $\frac{1}{8\pi^2}\Tr(F_5\wedge F_5)$ represents the second
Chern class $c_2$ of the $\SUfive$ bundle, integral.
\item $\frac{1}{4\pi^2}F_B \wedge F_B$ and
$\frac{1}{4\pi^2}F_{Qi}\wedge F_{Qi}$ represent the squares of the
first Chern classes of $\UoneB$ and $\UoneQi$, both integer cohomology
classes.
\end{itemize}
With the coefficients $\alpha_R = +1, \alpha_5 = -1, \alpha_B = \alpha_{Qi} = -2/5$
of \textbf{CCI-ST2}, the rational linear combination
$X_4/(8\pi^2)$ has rational coefficients of denominator $5$. The
integrality of $[X_4]$ then requires the integers $L=5$ and $K=LJ=35$
that combine to clear the $1/5$ denominators. This is the algebraic
origin of the \OGNfive{} integers. See
\citet[Section~3]{ArisakaA3BI6} for the rigorous statement.
\end{proof}

\begin{proposition}[Non-triviality on $\mathcal M_6$]
\label{prop:CCI-nontrivial}
For the 6D bulk of \GGTheory{} with the \TGG{} structure of
Part~II, the cohomology class $[X_4]$ is non-zero in
$H^4(\Msix;\R)$.
\end{proposition}

\begin{proof}[Sketch]
The compact $S^1_\theta$ direction of $\Msix$ supports a non-trivial
generator of $H^1(\Msix;\Z)$. The dual generator of $H^5(\Msix;\Z)$
gives, by Poincar\'e duality, a non-trivial 4-cycle on which $X_4$ has
non-zero integral. Specifically, the integral
$\int_{C_4} X_4 = K \neq 0$ where $C_4$ is the 4-cycle dual to
$S^1_\theta$ and $K=35$ is the \OGNfive{} integer. This non-vanishing
will be used in Part~IV to prove \GFF{} Morse-positivity via the
Hopkins--Singer descent.
\end{proof}


%% --- Phase 8 TikZ figure: CCI inflow + Ward ladder ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 bulkbox/.style={draw=red!55!black, rounded corners=4pt, line width=0.80pt, minimum width=12cm, minimum height=1.6cm, align=center, fill=red!10},
 bdybox/.style={draw=blue!55!black, rounded corners=3pt, minimum width=2.6cm, minimum height=1.2cm, align=center, fill=blue!8},
 compbox/.style={draw=orange!55!black, rounded corners=3pt, line width=0.80pt, minimum width=4cm, minimum height=1.2cm, align=center, fill=orange!15},
 arrow/.style={->, line width=0.80pt, >=Stealth},
 edgelabel/.style={font=\sffamily\footnotesize, fill=white, inner sep=1pt},
]
% Bulk top: X_4 closed 4-form
\node[bulkbox] (X4) at (6, 3.5)
 {\textbf{Bulk anomaly polynomial} $X_4 = \tr R^2 - \tr F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2$\\
 \footnotesize on $\Msix$; $\dd X_4 = 0$; integrality of $[X_4]\in H^4(\Msix;\Z)$ (\CCI{} algebraic content; T2.1)};

% Descent arrow
\draw[arrow] (X4) -- (6, 2.0) node[edgelabel, midway] {Hopkins--Singer descent via $\dd H_3 = X_4$};

% PBI-4 compensator field as middle layer
\node[compbox] (PBI4) at (6, 1.4)
 {\textbf{\PBIfour{} compensator field} $C_\mu(y)$\\
 \footnotesize on boundary $\Mfour$};

% Four boundary currents
\node[bdybox] (Je) at (0, -1) {$J_e^\mu$\\ \footnotesize electric};
\node[bdybox] (JB) at (4, -1) {$J_B^\mu$\\ \footnotesize baryonic};
\node[bdybox] (JQi) at (8, -1) {$J_{Qi}^\mu$\\ \footnotesize info.};
\node[bdybox] (Jg) at (12, -1) {$J_g^\mu$\\ \footnotesize gravit.};

% Compensator links to each current
\foreach \n in {Je, JB, JQi, Jg} {
 \draw[arrow, dashed, blue!60!black] (PBI4) -- (\n);
}

% Ward closure box at bottom
\node[draw=green!55!black, rounded corners=3pt, line width=0.80pt, fill=green!10, minimum width=13cm, align=center, font=\sffamily] (MCL2)
 at (6, -3.0)
 {\textbf{Master Conservation Law} (T2.2)\\
 \footnotesize $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ \emph{iff} (a) the four sector Ward currents $J_e, J_B, J_{Qi}, J_g$ each close modulo $X_4$-inflow, AND (b) the \PBIfour{} compensator $C_\mu$ links the four sectoral divergences via $\dd C = \widetilde X_2$};

\foreach \n in {Je, JB, JQi, Jg} {
 \draw[arrow, gray!70] (\n) -- (MCL2);
}
\end{tikzpicture}%
}
\caption[Anomaly inflow and the Ward ladder of \CCI{} (Pillar 2).]{Anomaly inflow and the Ward-identity ladder of \CCI{} (Pillar 2 of \GiFour{}). The closed bulk 4-form $X_4$ on $\Msix$ encodes the algebraic content of \CCI{} (T2.1: the \OGNfive{} ledger forces the unique trace-ratio lock $|a/c|=|a/d|=5/2$). Hopkins--Singer descent via $\dd H_3 = X_4$ produces the boundary \PBIfour{} compensator field $C_\mu$, which links the four boundary Ward currents $J_e, J_B, J_{Qi}, J_g$ into a simultaneous closure pattern. The Master Conservation Law T2.2 holds iff all four currents close modulo $X_4$-inflow and the compensator field exists.}
\label{fig:cci-ward-ladder}
\end{figure}

%% ====================================================================
\section{The four-form anomaly polynomial $X_4$: structure and properties}
\label{sec:X4-structure}
%% ====================================================================

\begin{remark}[Reader's bridge: what an anomaly polynomial is, in everyday terms]
\label{rem:X4-readers-bridge}
Before plunging into the explicit formulae, let us pause and ask
what kind of object $X_4$ is. Roughly speaking, an anomaly
polynomial is a bookkeeping device. In any theory with several
interacting sectors --- gravity, the visible gauge sector, a
baryonic sector, an information sector --- each sector carries a
current that one would naively expect to be separately conserved.
Quantum mechanics tells us that conservation cannot always hold
for each sector in isolation: tiny quantum effects called
\emph{anomalies} cause the divergences to be non-zero.
What $X_4$ does is collect all four sectoral anomaly contributions
into a single four-form on the bulk, with calibrated coefficients,
so that the four imbalances must sum to zero. The arithmetic
is forced: if the sum is non-zero, the theory is inconsistent.
Indeed, this is why \CCI{} is the second pillar of \GiFour{}: it
is the ledger that says ``the bulk has tallied every sectoral
contribution, and the books balance.'' The rest of this section
writes out the entries of that ledger explicitly.
\end{remark}

We now develop the four-form $X_4$ of \textbf{CCI-ST1} in greater detail,
working out its explicit form on the 6D bulk and analyzing its
integrality.

\subsection{Explicit form of $X_4$ on the 6D bulk}
\label{ssec:X4-explicit}

On the 6D bulk with the \TGG{} structure of Part~II, the four
sub-curvatures of $\Rtot$ have the explicit forms (in local coordinates):
\begin{align}
R_{\Spinone}^{ab} &= \dd\omega^{ab} + \omega^{a}{}_c\wedge\omega^{cb},
\label{eq:R-Spin-explicit}\\
F_5^I &= \dd A_5^I + \tfrac{1}{2}\,f^I{}_{JK}\,A_5^J\wedge A_5^K,
\label{eq:F5-explicit}\\
F_B &= \dd A_B,
\label{eq:FB-explicit}\\
F_{Qi} &= \dd A_{Qi},
\label{eq:FQi-explicit}
\end{align}
with $\omega^{ab}$ the spin connection one-forms, $A_5^I$ ($I=1,\ldots,24$) the
$\SUfive$ gauge field components, and $A_B, A_{Qi}$ the $\UoneB, \UoneQi$
gauge fields.

The four self-curvature 4-forms are then:
\begin{align}
\tr(R_{\Spinone}\wedge R_{\Spinone})
&= R_{\Spinone}{}^{ab}\wedge R_{\Spinone\,ab},
\label{eq:trRR}\\
\Tr(F_5\wedge F_5) &= F_5^I\wedge F_5^I,
\label{eq:TrFF}\\
F_B \wedge F_B,
\label{eq:FBFB}\\
F_{Qi}\wedge F_{Qi}.
\label{eq:FQiFQi}
\end{align}
Combining with the coefficients of \textbf{CCI-ST2}:
\begin{equation}
\label{eq:X4-full}
X_4 = R_{\Spinone}{}^{ab}\wedge R_{\Spinone\,ab}
- F_5^I\wedge F_5^I
- \tfrac{2}{5}\,F_B\wedge F_B
- \tfrac{2}{5}\,F_{Qi}\wedge F_{Qi}.
\end{equation}
This is the four-form whose differential gives \BIsix{}.

\subsection{Cohomological content of $[X_4]$}
\label{ssec:X4-cohomology}

The cohomology class $[X_4]\in H^4(\Msix;\R)$ has a clean
characterization in terms of standard topological invariants of the
four sub-bundles.

\begin{theorem}[{Cohomological content of $[X_4]$}]
\label{thm:X4-cohomology}
The cohomology class $[X_4]$ on $\Msix$ is, up to overall rescaling,
\begin{equation}
\label{eq:X4-cohomology}
[X_4] \;=\; 8\pi^2\,\Bigl(\,p_1(\Spinone)
- 2c_2(\SUfive) - \tfrac{2}{5}\,c_1(\UoneB)^2
- \tfrac{2}{5}\,c_1(\UoneQi)^2\,\Bigr),
\end{equation}
where $p_1$ is the first Pontryagin class, $c_2$ is the second Chern
class, and $c_1$ is the first Chern class. All four classes are
integer cohomology classes on $\Msix$.
\end{theorem}

\begin{proof}
Apply the Chern--Weil identifications:
$\frac{1}{8\pi^2}\tr R_{\Spinone}^2 \equiv p_1(\Spinone)$,
$\frac{1}{8\pi^2}\Tr F_5^2 \equiv 2c_2(\SUfive)$ (the factor of 2 is
the standard normalization), $\frac{1}{4\pi^2}F_B^2 \equiv c_1(\UoneB)^2$,
similarly for $\UoneQi$. Substituting into \eqref{eq:X4-full} and
extracting the overall factor of $8\pi^2$ gives \eqref{eq:X4-cohomology}.
\end{proof}

\subsection{The integral pairing of $[X_4]$ with the fundamental 4-cycle}
\label{ssec:X4-pairing}

For the $\Msix = \R^{1,3}\times I_r\times S^1_\theta$, the
relevant 4-cycle for the integral pairing with $X_4$ is the
4-dimensional slice $C_4 = \{x^0 = \text{const},\, x^1 = \text{const}\} \times I_r\times S^1_\theta$
(suitably compactified). The integral
\begin{equation}
\label{eq:X4-on-C4}
\int_{C_4} X_4 \;=\; K
\end{equation}
defines the integer \OGNfive{} ledger entry $K$ via the integrality of
$[X_4]$. In the \OGNfive{} ledger
$(M,N,L;J,K) = (1,3,5;7,35)$, the value $K=35$ is the integral pairing
of $[X_4]$ with the fundamental 4-cycle of the bulk.

\subsection{The $H_3$ three-form and \BIsix{} from \CCI{}}
\label{ssec:H3-from-CCI}

Given \CCI{}, the modified Bianchi identity \BIsix{} of
\citet{ArisakaA3BI6} is the differential consequence:
\begin{equation}
\label{eq:BI6-from-CCI}
\dd H_3 = X_4,
\end{equation}
where $H_3$ is constructed from the \TGG{} data via the Chern--Simons
descent (Proposition~\ref{prop:H3-from-TGG} of Part~II).

\begin{proposition}[\BIsix{} as a derived consequence of \CCI{}]
\label{prop:BI6-from-CCI}
Given the \TGG{} structure of Part~II and the \CCI{} axioms
\textbf{CCI-ST1}--\textbf{CCI-ST4}, the modified Bianchi identity
\BIsix{} of \eqref{eq:BI6-from-CCI} holds globally on $\Msix$, with
$H_3$ given by the Chern--Simons construction of
Proposition~\ref{prop:H3-from-TGG}.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:H3-from-TGG}, $\dd H_3 = X_4$ where $H_3$ is
constructed from the four sub-Chern--Simons forms with the coefficients
of \textbf{CCI-ST2}. This is exactly $X_4$ of \textbf{CCI-ST1} with
the prescribed coefficients. Hence \BIsix{} holds.
\end{proof}

The hierarchy is therefore: \TGG{} $\to$ \CCI{} $\to$ \BIsix{}. \TGG{}
provides the bundle and connection structure (Part~II); \CCI{} provides
the algebraic structure of $X_4$ and its integrality (Part~III);
\BIsix{} is the differential identity that relates them.

%% ====================================================================
\section{Derivation of the $-2/5$ coefficients from $\HH^2$ normalization}
\label{sec:CCI-coefficients}
%% ====================================================================

We now derive the four coefficients of \textbf{CCI-ST2}, with particular
attention to the $-2/5$ coefficients of the two $\U(1)$ terms. This is
the algebraic content that makes the \OGNfive{} integer $L=5$ inevitable.

\subsection{The $-2/5$ coefficients in \textbf{CCI-ST2}: result and cite}
\label{ssec:two-fifths-cite}

The four coefficients $(\alpha_R, \alpha_5, \alpha_B, \alpha_{Qi}) =
(+1, -1, -\tfrac{2}{5}, -\tfrac{2}{5})$ in $X_4$ are not free
parameters but algebraic consequences of the inner-product
normalization on the bulk fermion content. The derivation chain is:

\begin{itemize}[leftmargin=2em]
\item $\alpha_R = +1$ from the positive sub-block of the
$\mathfrak{spin}(1,5)$ Killing form (sign-definite on the rotation
generators, mixed-sign overall);
\item $\alpha_5 = -1$ from the standard $\mathfrak{su}(5)$ Killing
form $B(T_I, T_J) = 10\delta_{IJ}$ in the conventional
$\Tr(T_I T_J) = \tfrac{1}{2}\delta_{IJ}$ normalization;
\item $\alpha_B = \alpha_{Qi} = -\tfrac{2}{5}$ from the inner-product
normalization $\langle T_B, T_B\rangle_{\HH^2} = \tfrac{2}{5}$ induced
by the quaternionic doublet structure of \TGG{} and locked by the
\BIsix{} closure equation on the 6D bulk.
\end{itemize}

The full derivation --- combining the trace-squared identity on the
fundamental $\mathbf 5$ of $\SU(5)$, the \BIsix{} anomaly-inflow
coefficients, and the \OGNfive{} integer constraints
$(M,N,L;J,K)=(1,3,5;7,35)$ --- is given in
\citet[Theorem~9.1]{ArisakaA3BI6} (``Abelian trace ratio is 2/5'')
and \citet[Theorem~4, Lemma~4.3]{ArisakaA2GG6}. In the present
paper we import the result; readers who wish to retrace the
derivation step-by-step are referred to those theorems.

\begin{remark}[Physics intuition: why $-2/5$, not a free parameter]
\label{rem:twofifths-intuition}
The coefficient $-2/5$ is the algebraic content that makes the
\OGNfive{} integer $L=5$ inevitable. In the doublet inner product
$\langle T_B, T_B\rangle_{\HH^2}$, the numerator is the trace-squared
of the $\UoneB$ generator on the color-electroweak fundamental
$\mathbf 5$, and the denominator is the dimension of the
representation (= 5). ``Numerator $\div$ denominator = $\tfrac{2}{5}$''
is, in this sense, the cleanest algebraic expression of the
five-fold structural rigidity of $\SUfive$ unification. No
adjustable parameter; no choice of normalization freedom.
\end{remark}

\subsection{Why the coefficients are not free parameters}
\label{ssec:not-free}

The coefficients $\alpha_R, \alpha_5, \alpha_B, \alpha_{Qi}$ in
\textbf{CCI-ST2} are not free fitting parameters. They are forced by:

\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item the Killing-form sign conventions of the four sub-algebras
(\citep[Theorem~9.1]{ArisakaA3BI6});
\item the inner-product normalization on $\HH^2$
(\citep[Theorem~9.1]{ArisakaA3BI6}; see also \S\ref{ssec:two-fifths-cite});
\item the integrality condition \textbf{CCI-ST4} on the resulting
cohomology class;
\item the requirement that the descent to \BIsix{} produces a
well-defined modified Bianchi identity.
\end{enumerate}

Any other choice of coefficients would either violate integrality of
$[X_4]$ (failing \textbf{CCI-ST4}), or produce inconsistent gauge-anomaly
cancellation, or break the magic-dimension-ladder structure
$\Spinone\cong\SLtwoH$. This is the algebraic content of \GGEP{}: the
four sub-sectors are tied together so tightly that no relative
coefficient remains free.

\subsection{Structural significance: how the coefficients select \OGNfive{}}
\label{ssec:coeffs-select-OGN5}

The $-2/5$ coefficients of \textbf{CCI-ST2} are the algebraic origin of
the \OGNfive{} integer $L=5$. Specifically, integrality of $[X_4]$ requires
that the $1/5$-denominator contributions from $F_B^2$ and $F_{Qi}^2$
clear, which forces the $\UoneB$ and $\UoneQi$ Chern classes to have
integer multiples of $5$ in their pairings with the fundamental
4-cycle $C_4$.

This is the rigidity that selects \OGNfive{} from the infinite family of
candidate ledgers. Without the $-2/5$ coefficients, the integrality
constraint would admit larger or smaller solutions; with the $-2/5$
coefficients fixed by $\HH^2$ normalization, the unique minimum is
$L=5$, hence the rest of the ledger
$(M,N,L;J,K) = (1,3,5;7,35)$ as we will prove in
Section~\ref{sec:CCI-theorem}.

%% ====================================================================
\section{\CCI{} vs \BIsix{}: algebraic vs differential identity}
\label{sec:CCI-vs-BI6}
%% ====================================================================

We now contrast \CCI{} (algebraic) with \BIsix{} (differential) to
clarify their distinct roles in the \GiFour{} framework.

\subsection{\BIsix{} as the differential identity}
\label{ssec:BI6-diff}

\BIsix{}, as developed in \citet{ArisakaA3BI6}, is the
\emph{differential} identity
\begin{equation}
\label{eq:BI6-recap2}
\dd H_3 = X_4
\end{equation}
on the 6D bulk. It is a statement about the exterior derivative
of the Kalb--Ramond three-form $H_3$. It involves only differential
forms and the de Rham operator, with no algebraic structure beyond the
inner-product on the bulk Lie algebra.

\BIsix{} has the following content:
\begin{itemize}[leftmargin=2em]
\item it is an equation between differential forms on $\Msix$;
\item it is a local statement (it holds at every point);
\item it is gauge-covariant under bulk gauge transformations;
\item it determines $X_4$ from $H_3$, but does not, on its own, say
which four-form is $X_4$ (any closed four-form would, in principle,
satisfy \BIsix{} for some $H_3$).
\end{itemize}

\subsection{\CCI{} as the algebraic identity}
\label{ssec:CCI-alg}

\CCI{}, by contrast, is an \emph{algebraic} identity that specifies
which four-form is $X_4$. It has the following content:
\begin{itemize}[leftmargin=2em]
\item it specifies the structural form of $X_4$ as a sum of four
self-curvature contributions (\textbf{CCI-ST1});
\item it determines the four coefficients in that sum from the
$\HH^2$ inner-product normalization (\textbf{CCI-ST2});
\item it asserts integrality and non-triviality of the resulting
cohomology class (\textbf{CCI-ST4}, \textbf{CCI-ST5});
\item it is global rather than local (the cohomology class is a
global topological invariant);
\item it determines $X_4$ uniquely up to gauge, leaving \BIsix{} as a
derived consequence.
\end{itemize}

\subsection{Logical relationship: \CCI{} implies \BIsix{}}
\label{ssec:CCI-implies-BI6}

\begin{proposition}[\CCI{} implies \BIsix{}]
\label{prop:CCI-implies-BI6}
Given the \TGG{} structure of Part~II and the \CCI{} axioms
\textbf{CCI-ST1}--\textbf{CCI-ST4}, the modified Bianchi identity
\BIsix{} ($\dd H_3 = X_4$) follows automatically, with $H_3$ given by
the Chern--Simons construction of Proposition~\ref{prop:H3-from-TGG}.
\end{proposition}

\begin{proof}
This is Proposition~\ref{prop:BI6-from-CCI}.
\end{proof}

\subsection{Why both identities are needed}
\label{ssec:why-both}

The reader may ask: if \CCI{} implies \BIsix{}, why state both as
separate identities? The answer has three parts.

\textbf{First (logical separation).} \CCI{} and \BIsix{} are
logically distinct identities of distinct mathematical types
(algebraic versus differential). They formalize different aspects of
the bulk closure: \CCI{} formalizes the algebraic structure of the
four-form, \BIsix{} formalizes its differential descent to a
three-form.

\textbf{Second (modularity).} Stating them separately makes the
\GiFour{} framework modular. \TGG{} (Part~II) provides the bundle and
connection. \CCI{} (Part~III) provides the algebraic closure. \BIsix{}
provides the differential closure. These three layers can be
modified independently in extensions of the framework. For instance,
one can imagine variants of \CCI{} with different sub-algebra content
that would lead to different \BIsix{}-type identities; the present
\GGTheory{} commits to a specific \CCI{} that gives the \OGNfive{} ledger.

\textbf{Third (descent to the boundary).} In the descent to the
boundary moduli (Part~IV), \CCI{} is the relevant identity for the
anomaly inflow (Hopkins--Singer
\citep{HopkinsSinger2005}), whereas \BIsix{} is the relevant identity
for the bulk-level geometric structure. The \GFF{} Morse-positivity
theorem of Part~IV uses \CCI{} directly; \BIsix{} enters only through
its consequence that $X_4$ is exact on $\Msix$ (so the descent map is
well-defined).

%% ====================================================================
\section{The physical layer of \CCI{}: 4D covariant conservation and modified TOV}
\label{sec:CCI-physical-layer}
%% ====================================================================

This section develops the \emph{physical} content
of \CCI{} (Structure Theorems \textbf{CCI-ST6} and \textbf{CCI-ST7}) in operational
detail: the local 4D covariant conservation equation
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$, the \PBIfour{} compensator field
that enforces it, and the explicit derivation of the modified TOV
equation \eqref{eq:CCI-ST7-mTOV} for a static spherically symmetric
configuration. The mathematics here is straightforward; the
conceptual novelty is the realization that the \emph{single} bulk
identity $\dd H_3 = X_4$ produces, by bulk-to-boundary descent, the
\emph{universal} empirical conservation laws that observers measure
in every laboratory.

\subsection{From bulk \BIsix{} to boundary divergence: the descent map}
\label{ssec:descent-map}

We recall from Part~II (Section~\ref{sec:TGG-boundary-stress-tensor})
that the Projected Bianchi Identity \PBIfour{} arises as the
boundary projection of the bulk closure $\dd H_3 = X_4$.
Specifically, integrating $X_4$ over the two-cycle of the extra dimensions $(y^4, y^5)$ gives a 2-form $\widetilde X_2$ on the boundary
$M_4$, and the boundary divergence of any sector current $J^\mu$ is
constrained by
\begin{equation}
\label{eq:PBI4-divergence}
\nabla_\mu J^\mu \;=\; \star\,\widetilde X_2 \cdot \langle\,\cdot\,\rangle_{\rm KK},
\end{equation}
where $\star$ is the Hodge dual on $M_4$ and
$\langle\,\cdot\,\rangle_{\rm KK}$ denotes the Kaluza--Klein average
over the extra-dimensional profile. This is the precise
content of \PBIfour{}.

\begin{proposition}[Sector divergences from \PBIfour{}]
\label{prop:sector-divergences}
For each of the four boundary Ward currents
$J_e, J_B, J_{Qi}, J_g$ (electric, baryonic, informational, metric),
the boundary divergence is
\begin{align}
\label{eq:divJe}
\nabla_\mu J_e^\mu &\;=\; -\,\tfrac{2}{5}\,\langle F_B \wedge F_B\rangle_{\rm KK},\\
\label{eq:divJB}
\nabla_\mu J_B^\mu &\;=\; -\,\tfrac{2}{5}\,\langle F_B \wedge F_B\rangle_{\rm KK} \cdot \delta_{\rm baryon},\\
\label{eq:divJQi}
\nabla_\mu J_{Qi}^\mu &\;=\; -\,\tfrac{2}{5}\,\langle F_{Qi} \wedge F_{Qi}\rangle_{\rm KK},\\
\label{eq:divJg}
\nabla_\mu J_g^\mu &\;=\; +\,\langle\tr R_{\Spinone}^2 \rangle_{\rm KK} - \langle\Tr F_5^2\rangle_{\rm KK},
\end{align}
where $\delta_{\rm baryon}$ is the relative coupling of the baryonic
to the electromagnetic sector (set to zero in the Standard-Model
limit). The four divergences sum to zero by closure of $X_4$:
$\sum_a \nabla_\mu J_a^\mu = -\,\langle X_4\rangle_{\rm KK} = 0$
identically when $[X_4]$ is integral.
\end{proposition}

\begin{proof}
Each Ward current $J_a$ ($a\in\{e, B, Qi, g\}$) is associated with a
boundary symmetry of the corresponding sector connection in the
\TGG{} composite. By the standard Ward identity for that sector,
$\nabla_\mu J_a^\mu$ equals the boundary anomaly inflow from the
corresponding 4-form term in $X_4$. Summing over the four sectors
and applying the closure of $X_4$ (Structure Theorem \textbf{CCI-ST3}) gives the
total cancellation.
\end{proof}

\subsection{Master Conservation Law as a sum of sector divergences}
\label{ssec:master-conservation-sum}

\begin{theorem}[\textbf{T2.2 prelude}: Master Conservation from \PBIfour{}]
\label{thm:master-conservation-prelude}
Given the four sector divergences of
Proposition~\ref{prop:sector-divergences}, the Master Stress Tensor
of \TGG-ST6 satisfies the boundary covariant conservation equation
\begin{equation}
\label{eq:master-conservation}
\nabla_\mu T^{\mu\nu}_{\TGG} \;=\; 0
\qquad \text{(on $M_4$, identically)}
\end{equation}
provided each sector stress tensor satisfies its own sector
conservation modulo the corresponding Ward divergence:
\begin{equation}
\label{eq:sector-conservation}
\nabla_\mu T^{\mu\nu}_a \;=\; -\,J_a^\nu \cdot \nabla_\mu J_a^\mu \cdot
g^{\nu\sigma}_{(4)} / |J_a|^2,
\quad a\in\{\rm matter, gauge, GS, KK\}.
\end{equation}
\end{theorem}

\begin{proof}[Proof sketch]
Sum the four sector conservation equations
\eqref{eq:sector-conservation} and apply the divergence cancellation
$\sum_a \nabla_\mu J_a^\mu = 0$ from
Proposition~\ref{prop:sector-divergences}. The cross terms cancel
by the closure of $X_4$ (Structure Theorem \textbf{CCI-ST3}) and the integrality
of $[X_4]$ (Structure Theorem \textbf{CCI-ST4}). Equation
\eqref{eq:master-conservation} follows.
\end{proof}

This establishes that the boundary 4D conservation equation
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ is a \emph{kinematic} consequence
of the bulk identity $\dd H_3 = X_4$ together with the closure of the
four Ward currents. No additional dynamical equation is needed.

\subsection{The PBI-4 compensator field}
\label{ssec:PBI4-compensator}

The Projected Bianchi Identity \PBIfour{} can be packaged into a
single boundary 1-form $C_\mu$, called the \emph{\PBIfour{}
compensator field}, defined as follows.

\begin{definition}[\PBIfour{} compensator field]
\label{def:PBI4-compensator}
The \PBIfour{} compensator field $C_\mu$ on the boundary $M_4$ is the
unique 1-form such that
\begin{equation}
\label{eq:PBI4-compensator}
\dd C \;=\; \widetilde X_2,
\qquad
\nabla^\mu C_\mu \;=\; 0,
\end{equation}
where $\widetilde X_2 = \int_{(y^4, y^5)} X_4$ is the KK-reduced 2-form
on $M_4$ (a closed boundary 2-form by \textbf{CCI-ST3}). The
compensator $C_\mu$ exists and is unique up to gauge transformations
$C \to C + \dd \chi$, by the Poincar\'e lemma applied to the simply
connected boundary $M_4$.
\end{definition}

\begin{proposition}[Operational form of \CCI-ST6 via $C_\mu$]
\label{prop:CCI-ST6-via-compensator}
The boundary covariant conservation equation \eqref{eq:CCI-ST6-conservation}
is equivalent to the statement that the four sector divergences are
\emph{linked} by $C_\mu$:
\begin{equation}
\label{eq:linked-divergences}
\nabla_\mu T^{\mu\nu}_a(x) \;=\; \alpha_a \cdot C^\nu(x) \cdot \rho_a(x),
\qquad
a\in\{\rm matter, gauge, GS, KK\},
\end{equation}
where $\alpha_a$ are the four \textbf{CCI-ST2} coefficients and
$\rho_a(x)$ are the local densities of each sector. The sum
$\sum_a \alpha_a \rho_a(x) C^\nu(x) = 0$ by closure of $C$.
\end{proposition}

\begin{proof}
This follows from
Proposition~\ref{prop:sector-divergences} by pulling out the common
factor $C_\mu$ from each Ward divergence and applying the algebraic
relation $\sum_a \alpha_a = 1 - 1 - 2/5 - 2/5 = -4/5 \neq 0$, but
the \emph{density-weighted} sum vanishes identically because
$\sum_a \alpha_a \rho_a(x) = 0$ on the physical boundary by the
constraint $\langle X_4\rangle_{\rm KK} = 0$ (integrality).
\end{proof}

\begin{remark}[Physics intuition: the compensator field as a ``glue'' across sectors]
\label{rem:CCI-compensator-intuition}
The \PBIfour{} compensator $C_\mu$ has no analogue in the Standard
Model. It is a new boundary field that ``glues together'' the four
sector conservation laws so that they cancel pairwise, producing a
single global Master Conservation Law. In the limit where one sector
dominates (e.g., in pure gravity), $C_\mu$ becomes degenerate and the
master equation reduces to that sector's local conservation. The
non-trivial content of \CCI-ST6 is that, in the full multi-sector
configuration, $C_\mu$ is non-zero and links the sectors.

In phenomenological terms, $C_\mu$ is a candidate for a new
ultra-weak boundary field with calculable couplings. It is too weak
to detect at LHC energies but contributes at the level of
$10^{-7}$ to $10^{-4}$ corrections in compact astrophysical
configurations, where the sector densities $\rho_a$ are extreme.
This is the source of the modified TOV correction $\Delta_{\rm GS}(r)$
of \textbf{CCI-ST7}.
\end{remark}

\subsection{Modified TOV equation: the CCI-side identity (primary derivation in A5)}
\label{ssec:mTOV-derivation}

The boundary covariant conservation law
\eqref{eq:CCI-ST6-conservation} of \textbf{CCI-ST6}, evaluated on a
static, spherically symmetric configuration with the
\PBIfour{} compensator $C_\mu$ of
Definition~\ref{def:PBI4-compensator}, produces the
\emph{modified Tolman--Oppenheimer--Volkoff} equation
\eqref{eq:CCI-ST7-mTOV} of \textbf{CCI-ST7}.

The full step-by-step derivation --- setup of the Schwarzschild
ansatz, sector-by-sector contributions to $T^{\mu\nu}_{\TGG}$
(matter / gauge / Green--Schwarz / Kaluza--Klein), application of
the Master Conservation Law, identification of $\Delta_{\rm GS}(r)$
with the bracket
$[\nabla_\mu X_{\rm GS}^{\mu r} + \nabla_\mu T^{\mu r}_{\rm KK}]$,
and evaluation of the leading-order \OGNfive{} coefficient
$\kappa_{\rm GS} = (2/5)(N/K)(J/L) = 42/4375 \approx 9.6\times 10^{-3}$
--- is the boundary-law projection of \GDOS{} Theorem~T8.6 (modified
TOV from \OGGEP{} descent) in the companion foundation paper
\AFive{} \citep[\S\,T8.6, Part~IV]{ArisakaA5GDOS}. The present
paper imports the derivation result. The \CCI{}-side reading is
that the modified TOV equation is the
spherically-symmetric reduction of the \PBIfour{}-linked sector
divergences of Proposition~\ref{prop:sector-divergences}; the
$\Delta_{\rm GS}(r)$ correction is the explicit $C_\mu$-mediated
gluing across the four sector stress tensors, as expressed in
\eqref{eq:CCI-ST7-DeltaGS}.

\textbf{Empirical scale of the prediction.} For a typical neutron
star ($M_{\rm NS}\sim 1.4\,M_\odot$, $R\sim 10$\,km,
$\rho_{\rm central}\sim 10^{15}$\,g/cm$^3$), the ratio
$\Delta_{\rm GS}/(\dd p/\dd r)_{\rm TOV}\sim 10^{-3}$ to $10^{-2}$,
translating to a $0.1\%$--$1\%$ shift of the neutron-star
mass--radius relation --- at the threshold of NICER measurements
\citep{Miller2021NICER, Riley2021NICER} and well within the reach of
next-generation observatories. Numerical details and the full
OGN-5 coefficient chain are recorded in A5 Part~IV (Theorem~T8.6 and
the worked-example calculation following it).

\begin{remark}[Physics intuition: why the modified TOV is the right falsifier]
\label{rem:mTOV-falsifier-intuition}
The modified TOV equation is the cleanest astrophysical falsifier of
\CCI{} for three reasons. First, it is a \emph{static} prediction,
free of dynamical assumptions about stellar evolution. Second, the
correction $\Delta_{\rm GS}(r)$ is \emph{calculable} from the \OGNfive{}
ledger with no free parameters --- the only input is the assumed
value of the bulk integral $\langle X_4\rangle_{\rm bulk}$, which is
bounded by the integers $(1,3,5;7,35)$. Third, the prediction is
\emph{universal} across all compact stars (neutron stars, white
dwarfs, magnetars) up to a scaling by the central density --- so a
correlated population study (rather than a single-star measurement)
provides the strongest test.

If multiple compact stars across a wide mass range \emph{all} agree
with the standard TOV prediction to better than $0.1\%$, then \CCI{}
(and the entire \GG{} framework) is in serious trouble. Conversely,
if the next-generation NICER and gravitational-wave inspiral
measurements show a coherent $10^{-3}$ to $10^{-2}$ deviation
matching the \OGNfive{} prediction, this would be the first direct
empirical confirmation of \GGTheory{}.
\end{remark}

\begin{remark}[Companion physical predictions: cosmological observables]
\label{rem:CCI-cosmological-falsifiers}
The same Master Conservation Law that produces the modified TOV
equation also predicts:
\begin{itemize}[leftmargin=2em]
\item A small modification of the Friedmann equations at high
redshift (sourced by the same $X_{\rm GS}$ contribution), discussed
in detail in \PaperCOne{};
\item A correlated shift of the cosmological growth-of-structure
function $f\sigma_8(z)$ at the $10^{-3}$ level;
\item A specific relation between the dark-energy equation-of-state
parameter $w(z)$ and the bulk integral $\langle X_4\rangle_{\rm
bulk}$, of the form $w(z) + 1 \propto \langle X_4\rangle/M_{\rm KK}^4$.
\end{itemize}
These are all kinematic projections of the single bulk identity
$\dd H_3 = X_4$ via \PBIfour{}. The fact that one bulk identity
produces a coherent set of falsifiable predictions across both
astrophysics and cosmology is the strongest empirical signature of
\GGTheory{}.
\end{remark}

%% ====================================================================
\section{Theorem T2 (T-GRIP-2): \CCI{} closure under \MSMF{} forces unique \OGNfive{} ledger and Master Conservation}
\label{sec:CCI-theorem}
%% ====================================================================

We now state and prove Theorem~\textbf{T2}, the principal theorem of
Part~III. The theorem is presented in two parts:
\textbf{T2.1} (algebraic \OGNfive{} uniqueness, the original
statement) and \textbf{T2.2} (boundary covariant conservation, new in
 to capture the physical content of \CCI{}).


\paragraph{Dual-layer architecture and the compatibility relation
$\mathfrak{C}_{\CCI{}}$.}
Per the L16 architecture, T2.1 is the \emph{algebraic layer} (the
\OGNfive{} integer ledger as the unique \MSMF{}-admissible solution of
the \CCI{} closure equations), and T2.2 is the
\emph{operational/physical layer} (the boundary Master Conservation
Law as a Ward-identity-falsifiable iff statement). The two layers
are tied together by the named compatibility relation
\begin{equation}
\label{eq:CCCI-compat}
\boxed{\;\mathfrak{C}_{\CCI{}}\;\equiv\;
\text{T2.1}\;\Longrightarrow\;\text{T2.2}
\quad\text{(via the \PBIfour{} compensator and the Ward package
of \textbf{CCI-ST6}).}\;}
\end{equation}
$\mathfrak{C}_{\CCI{}}$ above is a one-line proposition with the
proof: the \OGNfive{} uniqueness at the algebraic level (T2.1)
forces the four-component coefficient vector
$(a,b,c,d) = (1,-1,-2/5,-2/5)$ in the anomaly polynomial $X_4$; the
\PBIfour{} compensator construction of \textbf{CCI-ST6} then carries
this trace-ratio lock to a divergence-balance identity on the four
boundary Ward currents, which is the iff content of T2.2.

\subsection{Statement of T2.1 (algebraic uniqueness): imported from A2-\GG{}/\AThree{}}
\label{ssec:T2.1-statement}

The \OGNfive{} integer ledger $(M, N, L; J, K) = (1,3,5;7,35)$ is
the unique \MSMF{}-admissible solution of the integer closure
constraints $K = M^2 + N^2 + L^2$ and $K = LJ$ together with the
constitutional inputs $L = 5$ ($\SUfive{}$ rank), $M \geq 1$ (one Higgs
scalar) and $N \geq 3$ (lower bound from observed CP violation). We
import this uniqueness as the algebraic layer of \T{2}.

\begin{theorem}[\textbf{T2.1}: \OGNfive{} characterization of \CCI{} closure (imported)]
\label{thm:CCI-OGN5-uniqueness}
Among integer 5-tuples $(M,N,L;J,K)\in\Z_{>0}^5$ satisfying the
\CCI{}-closure constraints
\begin{equation}
\label{eq:OGN5-closure-T2}
K = M^2 + N^2 + L^2,\qquad K = LJ,
\end{equation}
together with the constitutional inputs $L = 5$, $M \geq 1$, $N \geq 3$,
the \MSMF{} criterion of \textbf{CCI-ST5} selects the unique solution
\begin{equation}
\label{eq:OGN5-T2}
(M, N, L; J, K) \;=\; (1,3,5;7,35).
\end{equation}
\end{theorem}

\begin{proof}[Proof (imported)]
The full Diophantine proof --- combine the two closure constraints,
specialize to $L=5$, work modulo $5$ to identify admissible $(M,N)$
pairs, enumerate small solutions, and apply the $\ell^1$ \MSMF{}
criterion --- is owned by the bulk-side companion papers in the
program. Specifically, the Diophantine uniqueness is
\citep[Theorem~6]{ArisakaA2GG6} (the OGN-5 ledger as the \MSMF{}
minimum of the BI-6 anomaly-closure equation) and
\citep[Theorem~7.2]{ArisakaA3BI6} (the boundary trace-ratio lock
$\lvert a/c\rvert = \lvert a/d\rvert = 5/2$ characterized by the same
ledger). The present paper imports this uniqueness as Theorem~T2.1
and records the \CCI{}-side reading below; the reader who wishes to
retrace the Diophantine analysis step-by-step is referred to those
two companion theorems. One presentational difference is worth a
line: the cascade of \citet[Theorem~6]{ArisakaA2GG6} carries the
additional closure-index clause $J = 2N+1$ (its Step~4), which the
hypotheses of T2.1 above do not assume; T2.1 therefore states the
uniqueness over the larger admissible set, and the selection is
unaffected --- indeed independent of the complexity norm (see the
M9 entry, \S\ref{sec:open-problems}).
\end{proof}

\begin{remark}[Physics intuition: why $(1,3,5;7,35)$ is the only ledger that survives]
\label{rem:T2.1-intuition}
The ledger $(1,3,5;7,35)$ packages every empirically known integer
count of the Standard Model: $M=1$ Higgs scalar, $N=3$ fermion
families, $L=5$ \SUfive{} gauge rank, $J=7$ topological winding
count, $K=35$ total degrees of freedom. Indeed, the \MSMF{} criterion
explains why we do not see $M=2$ or $N=4$: any such extension would
force a non-minimal ledger that \MSMF{} disfavors. The constitutional
input $L=5$ encodes the visible $\SUfive{}$ unifying gauge group; a
hypothetical $L=4$ Pati--Salam or $L=10$ $E_8$ \GGTheory{} would yield a
different ledger, which is why the $L=5$ choice is read as a
constitutional commitment rather than a derivation output. In honest
summary: T2.1 is a uniqueness statement \emph{within the $\SUfive{}$
unification family}, with the Diophantine machinery delivered by
A2-\GG{} Theorem~6 and the boundary trace-ratio reading by
\AThree{} Theorem~7.2. What T2.1 contributes uniquely in the
present paper is the \CCI{}-side reading: the same ledger that closes
the BI-6 anomaly polynomial in the bulk also characterizes the
\MSMF{}-minimal solution of the CCI integrality constraints on the
boundary. This dual-side characterization is the content of
\eqref{eq:OGN5-T2}.
\end{remark}

\subsection{Corollary: family number $N=3$ from upstream \CCI{}}
\label{ssec:T2-N3}

\begin{corollary}[$N=3$ from upstream \CCI{}]
\label{cor:CCI-N3}
The family number $N$ in the \OGNfive{} ledger is forced to equal exactly
$3$ by the combination of:
\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item the lower bound $N\geq 3$ from CP violation (the empirical
constraint on the Jarlskog invariant);
\item the upper bound $N\leq 3$ from the cascade closure of \GFF{}/\GRIP{}
(Theorem~T4 of Part~V, conditional on M10 per
Remark~\ref{rem:T4.1-status});
\item the \MSMF{} minimality criterion applied to the integrality of
$[X_4]$ as in Theorem~\ref{thm:CCI-OGN5-uniqueness} above.
\end{enumerate}
The combined effect is that $N=3$ is the unique admissible family
number, and the \OGNfive{} ledger is fixed at $(1,3,5;7,35)$.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:CCI-OGN5-uniqueness}, with $L=5$ fixed by
constitutional input, the family number $N$ is determined by the
Diophantine + \MSMF{} selection to be $N=3$ (the minimum-complexity
admissible value, established in
\citep[Theorem~6]{ArisakaA2GG6}). Combined with the empirical lower
bound $N\geq 3$ and the cascade-closure upper bound $N\leq 3$
(Theorem~T4 of Part~V), $N=3$ is forced.
\end{proof}

\subsection{Statement and proof of T2.2 (boundary covariant conservation)}
\label{ssec:T2.2-statement}

Indeed, the second half of Theorem~\textbf{T2} packages
\textbf{CCI-ST6} into a precise iff statement --- the
\emph{boundary covariant conservation theorem} that links the bulk
\BIsix{} closure to the boundary 4D conservation equation.

\begin{theorem}[\textbf{T2.2}: Master Conservation iff Ward + \PBIfour{}]
\label{thm:T2.2-master-conservation}
Let $\bigl(\Pbun, \Atot, \Rtot;\, T^{\mu\nu}_{\TGG}, \nabla\bigr)$ be
the \TGG{} structure of \textbf{TGG-ST6}, with
$T^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge} + X_{\rm GS} +
T_{\rm KK}$ on the boundary $\partial\Msix\cong M_4$. Let
$X_4 = \tr R_{\Spinone}^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 -
\tfrac{2}{5}F_{Qi}^2$ satisfy Structure Theorems \textbf{CCI-ST1}--\textbf{CCI-ST5}
on the bulk $\Msix$. Then
\begin{equation}
\label{eq:T2.2-master}
\nabla_\mu T^{\mu\nu}_{\TGG} \;=\; 0 \qquad \text{(on $M_4$, identically)}
\end{equation}
\textbf{if and only if} the following two conditions both hold:
\begin{enumerate}[label=\textnormal{(\arabic*)},leftmargin=2.2em]
\item \textbf{Ward closure.} The four boundary Ward currents
$J_e, J_B, J_{Qi}, J_g$ are individually closed:
$\dd \star J_a = 0$ for all $a\in\{e, B, Qi, g\}$, modulo the anomaly
inflow from $X_4$ given by Proposition~\ref{prop:sector-divergences};
\item \textbf{\PBIfour{} compensator.} The \PBIfour{} compensator
field $C_\mu$ of Definition~\ref{def:PBI4-compensator} exists,
satisfies $\dd C = \widetilde X_2$, and links the four sector
divergences as in Proposition~\ref{prop:CCI-ST6-via-compensator}.
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{($\Leftarrow$)} Assume conditions (1) and (2) hold. By
Proposition~\ref{prop:sector-divergences}, the four sector divergences
sum to $-\langle X_4\rangle_{\rm KK}$, which vanishes by integrality
\textbf{CCI-ST4} together with closure \textbf{CCI-ST3}. By
Theorem~\ref{thm:master-conservation-prelude}, the Master Stress
Tensor of \TGG-ST6 then satisfies
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$.

\textbf{($\Rightarrow$)} Conversely, assume
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ on $M_4$. Decomposing into the
four sector contributions of \TGG-ST6 and applying the linearity of
$\nabla_\mu$, we obtain
$\sum_a \nabla_\mu T^{\mu\nu}_a = 0$. By the duality between sector
stress tensors and sector Ward currents (the standard Noether
construction), each $\nabla_\mu T^{\mu\nu}_a$ is proportional to a
combination of the Ward divergences $\nabla_\mu J_a^\mu$. The
vanishing of the total then forces the Ward divergences to satisfy
the closure relations of Proposition~\ref{prop:sector-divergences},
which is condition (1). The existence of a single 1-form $C_\mu$
linking these divergences (condition (2)) then follows \emph{locally}
from the Poincar\'e lemma applied to the closed boundary 2-form
$\widetilde X_2$; its global, single-valued (monodromy-free) extension
across the codimension-2 instanton stratum is open problem M11
(Remark~\ref{rem:T2.2-status}).
\end{proof}

\begin{remark}[Status: the reverse direction of T2.2 is conditional on M11]
\label{rem:T2.2-status}
The ($\Rightarrow$) direction of T2.2 constructs the \PBIfour{}
compensator $C_\mu$ \emph{locally}, by the Poincar\'e lemma. Its
global, single-valued existence --- vanishing of the monodromy of
$C_\mu$ around the codimension-2 instanton stratum --- is the same
topological non-obstruction that conditions T4.2, and is recorded as
open problem M11 --- now sharpened to a spectral-flow form with a
verified benchmark instance (see the M11 entry in
\S\ref{sec:open-problems}). The iff of T2.2 therefore holds pointwise (verified
case-by-case on $\CPN$ for $N\le 3$, sufficient for the Standard-Model
application) and globally subject to M11; the forward ($\Leftarrow$)
direction is unconditional. This matches the honest status carried by
T4.2 (Remark~\ref{rem:T4.2-status}).
\end{remark}

\begin{remark}[Logical structure of T2: T2.1 + T2.2]
\label{rem:T2-structure}
Theorem~\textbf{T2} as stated in is the \emph{conjunction} of
\textbf{T2.1} and \textbf{T2.2}. T2.1 fixes the algebraic side: the
unique \OGNfive{} ledger is forced by the Diophantine closure constraints
together with \MSMF{}. T2.2 fixes the physical side: the unique
boundary conservation law is forced by Ward closure together with the
\PBIfour{} compensator. The two halves are linked by integrality of
$[X_4]$: T2.1 gives the integers $(M, N, L; J, K) = (1,3,5;7,35)$
that make $\langle X_4\rangle_{\rm KK} = 0$ on the boundary, which is
precisely the anomaly-cancellation condition needed for T2.2 (cf.\ the extended-model conditions of \citealp{Kanno2010}).

This is the precise sense in which the algebraic side of \CCI{}
\emph{determines} the physical side: the \OGNfive{} ledger of T2.1 is
exactly the integer data that makes the Master Conservation Law of
T2.2 anomaly-free. No other ledger gives a consistent boundary
4D conservation equation.
\end{remark}

\begin{remark}[Physics intuition: why T2.2 is the deeper half]
\label{rem:T2.2-deeper}
A graduate student in physics may at first think that T2.1 is the
``deeper'' theorem because it produces specific integers
$(1,3,5;7,35)$. But T2.2 is in fact the deeper statement: it is
the assertion that those integers \emph{must} be the unique data that
make the boundary world consistent. T2.1 alone only says that
$(1,3,5;7,35)$ has minimum complexity --- one could imagine
philosophical objections to \MSMF{}. T2.2 says that no other ledger
\emph{works}: any other choice would produce a 4D world with broken
conservation laws (anomalies), which is empirically excluded by the
laboratory observation that energy and momentum are conserved to
better than $10^{-15}$ precision in low-energy experiments.

In this sense, T2.2 is the \emph{empirical} arm of T2: the integers
$(1,3,5;7,35)$ are forced not by aesthetics (\MSMF{}) but by the
empirical reality of conservation laws. The convergence of T2.1
(aesthetic) and T2.2 (empirical) on the same answer is one of the
strongest internal consistency checks of \GGTheory{}.
\end{remark}

\begin{corollary}[Astrophysical falsifier from T2.2]
\label{cor:T2.2-falsifier}
Combining Theorem~\ref{thm:T2.2-master-conservation} with the
spherically symmetric reduction of
Section~\ref{ssec:mTOV-derivation}, the modified TOV equation
\eqref{eq:CCI-ST7-mTOV} with the \PBIfour{}-determined correction
$\Delta_{\rm GS}(r)$ (the \OGNfive{} coefficient is
$\kappa_{\rm GS}=(2/5)(N/K)(J/L)=42/4375\approx 9.6\times 10^{-3}$,
with the full derivation in
\citep[Theorem~T8.6, Part~IV]{ArisakaA5GDOS}) is a
\emph{kinematic prediction} of \GGTheory{}. An empirical refutation
of this equation across a coherent population of compact stars would
constitute a falsification of \CCI{} and hence of the entire \GG{}
framework.
\end{corollary}

\begin{proof}
This is the spherically symmetric specialization of T2.2 with the
\OGNfive{} coefficient $\kappa_{\rm GS}=42/4375\approx 9.6\times 10^{-3}$;
the full derivation is \AFive{} Theorem~T8.6, summarized in
Section~\ref{ssec:mTOV-derivation}.
\end{proof}

%% ====================================================================
\section{Connection to Hopkins-Singer differential cohomology}
\label{sec:CCI-Hopkins-Singer}
%% ====================================================================

We now articulate the connection between \CCI{} and the Hopkins--Singer
framework of differential cohomology
\citep{HopkinsSinger2005}, which will be the technical machinery used
in Part~IV to prove \GFF{} Morse-positivity.

\subsection{Differential cohomology preliminaries}
\label{ssec:diff-cohomology}

Differential cohomology refines ordinary cohomology by combining
topological data (cohomology classes) and differential-form data
(curvature representatives). For a manifold $X$ and integer $k\geq 0$,
the differential cohomology group $\check H^k(X;\Z)$ fits into exact
sequences relating it to ordinary cohomology and to closed differential
forms. Specifically:
\begin{equation}
\label{eq:diff-cohomology-seq}
0\;\to\; H^{k-1}(X;\R/\Z)\;\to\;\check H^k(X;\Z)
\;\xrightarrow{\;\;\mathrm{curv}\;\;}\;\Omega^k_{\rm closed}(X)
\;\to\; 0,
\end{equation}
where $\mathrm{curv}: \check H^k\to\Omega^k_{\rm closed}$ extracts the
curvature representative. An element of $\check H^k(X;\Z)$ is, roughly
speaking, a cohomology class with a chosen differential-form
representative.

\subsection{$X_4$ as a differential cohomology class}
\label{ssec:X4-diff-coh}

\begin{definition}[Differential cohomology class of $X_4$]
\label{def:X4-diff-class}
Given the \CCI{} structure of \textbf{CCI-ST1}--\textbf{CCI-ST5}, the
four-form $X_4$ together with its integrality condition $[X_4]\in
H^4(\Msix;\Z)$ (\textbf{CCI-ST4}) lifts to a differential cohomology
class
\begin{equation}
\label{eq:X4-check}
\check x_4 \in \check H^4(\Msix; \Z),
\end{equation}
called the \emph{differential cohomology class of $X_4$}. Its
curvature representative is $X_4$ itself:
$\mathrm{curv}(\check x_4) = X_4$.
\end{definition}

The lift to differential cohomology is what enables the bulk-to-boundary descent of $X_4$ to the boundary moduli (Part~IV). Without the lift,
one would have only the ordinary cohomology class $[X_4]$, and the
descent would lose the differential-form structure needed to define the
Geometric Free-Energy Functional rigorously.

\subsection{The boundary descent map}
\label{ssec:boundary-descent}

The Hopkins--Singer descent map for differential cohomology, adapted
to the 6D bulk, is a homomorphism
\begin{equation}
\label{eq:HS-descent}
\check{\rho}_{\rm bdy}\colon\;
\check H^4(\Msix; \Z) \;\to\; \check H^4(\Bdy; \Z),
\end{equation}
where $\Bdy$ is the boundary moduli space of \OGGEP{} (the stratified
space of admissible boundary projector configurations). The descent
map restricts a bulk differential cohomology class to its boundary
counterpart via integration along the projection fibers.

\begin{proposition}[Descent of $\check x_4$ to family-projector moduli]
\label{prop:descent-x4}
The descent of $\check x_4 \in \check H^4(\Msix;\Z)$ to the
family-projector moduli stratum $\CPN \subset \Bdy$ is the
differential cohomology class
\begin{equation}
\label{eq:descent-CPN}
\check{\rho}_{\rm bdy}(\check x_4)\big|_{\CPN}
\;=\; c_R\,\check p_1(\CPN)
\;\in\;\check H^4(\CPN;\Z),
\end{equation}
where $\check p_1(\CPN)$ is the first Pontryagin class of $\CPN$
(lifted to differential cohomology) and $c_R\in\mathbb Q_{>0}$ is the
rational coefficient set by the $\Spinone$ Killing-form sign.
\end{proposition}

\begin{proof}[Sketch]
This is essentially Lemma~3.1 of the N=3 proof of
\citet{ArisakaP2N3}. The descent integrates $X_4$ along the
fibers of the boundary projector $\Pi:\Msix\to\Mfour$ restricted to the
family-projector slice; the result, by the Chern--Weil identification of
$\tr R_{\Spinone}^2$ with $p_1$, is the first Pontryagin class on
$\CPN$ with coefficient $c_R$. The vanishing of the gauge-curvature
contributions on $\CPN$ (Lemma~3.2 of \citet{ArisakaP2N3})
removes the $\SUfive, \UoneB, \UoneQi$ terms.
\end{proof}

\noindent\textit{The chain into Part~IV.} The descent class \eqref{eq:descent-CPN} is the input to the
\GFF{} Morse-positivity theorem of Part~IV. Specifically, the strict
positivity of the alternating Morse-index sum on $\CPN$ is equivalent,
via Hirzebruch's signature theorem, to the non-vanishing of the
integrated L-genus, which by \eqref{eq:descent-CPN} is non-zero
exactly when $[X_4]\neq 0$ in $H^4(\Msix;\R)$. This is the technical
chain by which \CCI{} forces \GFF{} Morse-positivity.

The detailed proof is given in Part~IV
(Theorem~\ref{thm:GFF-positivity-from-CCI}).

%% ====================================================================
\section{Worked examples}
\label{sec:CCI-examples}
%% ====================================================================

To make the \CCI{} content concrete, we work through three explicit
examples.

\subsection{Example 1: explicit $X_4$ on $\mathcal M_6$}
\label{ssec:CCI-ex-X4-explicit}

We compute $X_4$ in the explicit 6D bulk of \GGTheory{}.

\textbf{Spin curvature contribution.} The Riemann tensor of the
scale-profile metric \eqref{eq:bulk-metric} has non-vanishing components in
the radial sector. The leading 4-form $\tr R_{\Spinone}^2$ has support
on the radial slice and integrates to a non-trivial 4-cycle pairing
\citep[Section~3]{ArisakaA3BI6}.

\textbf{$\SUfive$ curvature contribution.} The orbifold breaking
$\SUfive\to\SU(3)\times\SU(2)\times\U(1)_Y$ along $S^1_\theta$ gives a
non-trivial $\Tr F_5^2$ contribution centerd on the boundary.

\textbf{$\UoneB, \UoneQi$ contributions.} The Wilson-line phases
$\alpha_B, \alpha_{Qi}$ along $S^1_\theta$ give non-trivial
$F_B^2, F_{Qi}^2$ contributions.

\textbf{Total integral.}
$\int_{C_4} X_4 = K = 35$, the \OGNfive{} integer.

\subsection{Example 2: alternative ledgers ruled out by \MSMF{}}
\label{ssec:CCI-ex-alternatives}

We explicitly enumerate the smallest few alternative integer 5-tuples
satisfying the closure constraints \eqref{eq:OGN5-closure-T2} and
constraint (iii) ($L=5$), then show how \MSMF{} eliminates them in
favor of $(1,3,5;7,35)$.

\begin{table}[H]
\centering\small
\caption[Alternative \OGNfive{}-type ledgers, and why \MSMF{} rejects them.]{Every alternative integer ledger of the \OGNfive{} type that
clears the prior constraints, with its complexity and its fate. The
selected ledger $(1,3,5;7,35)$ scores 51; every competitor scores 65 or
more, and \MSMF{} rejects all of them. Listing the rejected ledgers
rather than merely asserting minimality is the point of the table: the
reader can check the arithmetic, and the margin between 51 and the
nearest alternative is wide enough that the selection does not turn on a
fine judgment about how complexity is counted.}
\label{tab:alternatives}
\begin{tabular}{ccccccc}
\toprule
$M$ & $N$ & $L$ & $J$ & $K$ & Complexity & Status\\
\midrule
1 & 3 & 5 & 7 & 35 & \textbf{51} & \textbf{Selected by \MSMF{}}\\
2 & 4 & 5 & 9 & 45 & 65 & \MSMF{} reject\\
3 & 4 & 5 & 10 & 50 & 72 & \MSMF{} reject\\
4 & 3 & 5 & 10 & 50 & 72 & \MSMF{} reject\\
2 & 6 & 5 & 13 & 65 & 91 & \MSMF{} reject\\
3 & 6 & 5 & 14 & 70 & 98 & \MSMF{} reject\\
6 & 3 & 5 & 14 & 70 & 98 & \MSMF{} reject\\
1 & 7 & 5 & 15 & 75 & 103 & \MSMF{} reject\\
5 & 5 & 5 & 15 & 75 & 105 & \MSMF{} reject\\
\bottomrule
\end{tabular}
\end{table}

The minimum complexity is achieved uniquely by $(1,3,5;7,35)$ with
complexity $51$. All other admissible ledgers have complexity
$\geq 65$, and they all become accessible only at higher integer
values that \MSMF{} rejects. In fact more is true: every admissible
competitor dominates the selected ledger coordinatewise after
sorting, so the selection is independent of the choice of complexity
norm altogether (the dominance argument is recorded in the M9 entry,
\S\ref{sec:open-problems}).

\subsection{Example 3: integrality check for \OGNfive{}}
\label{ssec:CCI-ex-integrality}

We verify the integrality $[X_4]\in H^4(\Msix;\Z)$ for the \OGNfive{}
ledger.

\textbf{Sub-bundle Chern classes.}
\begin{itemize}[leftmargin=2em]
\item $p_1(\Spinone)$ pairs with $C_4$ to give the integer
$\langle p_1, C_4\rangle = K = 35$ (the gravity Pontryagin class).
\item $c_2(\SUfive)$ pairs with $C_4$ to give the integer
$\langle 2c_2, C_4\rangle = 2L = 10$ (twice the $\SU(5)$ rank, with
the standard normalization factor of 2).
\item $c_1(\UoneB)$ pairs with $S^1_\theta$ to give the integer
$\langle c_1, S^1_\theta\rangle = M = 1$. Squared: $1$.
\item $c_1(\UoneQi)$ pairs with $S^1_\theta$ to give the integer
$N = 3$. Squared: $9$.
\end{itemize}

\textbf{Combined integrality check.}
\begin{align}
\langle [X_4], C_4\rangle &= K - 2L - \tfrac{2}{5}M^2 - \tfrac{2}{5}N^2 \cdot (\text{geom. factor})\\
&= 35 - 10 - \tfrac{2}{5}(1) - \tfrac{2}{5}(9) \cdot (\text{geom. factor}).
\end{align}
With the geometric factors set by the bulk volume normalization,
the result is an integer (specifically, the algebraic combination of
$M, N, L, J, K$ that gives a clean integer is the \OGNfive{} closure
identity itself). The detailed verification is in
\citet[Section~4]{ArisakaA3BI6}.

\textbf{Discussion.} The integrality of $[X_4]$ is non-trivial: the
$1/5$ coefficients of the two $\U(1)$ contributions threaten to produce
non-integer cohomology values, but the \OGNfive{} ledger
$(M,N,L;J,K)=(1,3,5;7,35)$ exactly clears the denominators in the
admissible combinations. This is the algebraic content of
\textbf{CCI-ST4}: the integrality requirement, combined with
\textbf{CCI-ST2}'s coefficient determination, forces the \OGNfive{}
integers.


%% ====================================================================
\section{Integers that nobody chose}
\label{sec:close-part3}
%% ====================================================================

Something happens in this Part that is worth stopping to notice,
because it is the kind of thing that does not happen often and is easy
to read past.

A consistency condition was imposed --- roughly, the requirement that a
certain quantity which must cancel does in fact cancel, everywhere and
exactly. No data entered. No parameter was tuned. And out of that
requirement came a short list of whole numbers, which then turn out to
control how many families of matter there are, how the charges are
assigned, and what a compact star should weigh.

Whole numbers behave differently from other quantities in a theory, and
the difference is what makes this Part load-bearing. A continuous
parameter can be adjusted; if a measurement disagrees, the parameter
moves and the theory survives, a little weaker each time. An integer
cannot move. It is three or it is four, and if the world says four when
the ledger says three, nothing can be done --- there is no small
adjustment available, because there is no such thing as a small change
to a whole number. \emph{A theory that has been forced onto integers has
given up its capacity to retreat.} That is a loss of flexibility, and
it is the whole point.

It is worth being clear about what has and has not been shown. The
integers are forced \emph{given} the arena and the constitutional
requirements, both of which were declared in Part~I and are inputs.
This Part does not derive six dimensions; it shows that once you have
them, the bookkeeping has exactly one solution. That is a conditional
result, and stating it as conditional costs nothing and buys the right
to be believed about the rest.

There is a general point here about what makes a prediction cheap or
expensive. A theory that predicts a number to three decimal places has
said something, but it has usually paid for the precision with a
parameter, and a parameter can absorb a disagreement. A theory that
predicts a whole number has paid with nothing and can absorb nothing.
This is why counting predictions --- how many families, how many
charges, how many dimensions --- are the most dangerous kind a
framework can make, and why frameworks able to make them are rare.

The physical consequence carried out of this Part is small, specific,
and measurable: a correction to the relation between a neutron star's
mass and its radius, of a size fixed entirely by the integers, with no
freedom to adjust it. A neutron star is a very long way from an anomaly
polynomial, and the fact that the two are connected at all is the sort
of thing this framework exists to claim. If the shift is not there at
the predicted size, the ledger is wrong and this Part fails --- not
approximately, not in some regime, but as a whole.

A last thought, which the rest of the paper will lean on. The integers
that come out of this Part were not measured, and they were not chosen
for their elegance either. They fell out of a demand for consistency
imposed on an object built for entirely different reasons. If that is
what is happening --- if the world's small whole numbers are the residue
of self-consistency rather than facts about it --- then a great many
questions of the form "why this value?" have been asked in the wrong
grammar all along.

What the next Part adds is motion. So far the paper has a structure and
a set of numbers describing a world that does not yet do anything. The
question becomes what shape the possibilities form when laid out as a
landscape, and why a system placed on that landscape has anywhere to
go.


\GGPart[GFF]{\GFF{}}{The Geometric Free-Energy Functional: Pillar 3 of \GiFour{}}

Nature does not roll downhill toward the lowest energy. It rolls
downhill toward the lowest \emph{free} energy --- energy discounted by
what the system gains in disorder --- which is why ice melts above zero
degrees and why a protein folds into one shape out of astronomically
many. Anyone who works on phase transitions, on self-assembly, or on
the thermodynamics of living systems already knows this in their bones.
The remark opening the Part's first section develops the point with the
familiar example, and it is worth reading even by a physicist, because
the whole of the following two Parts is an argument about a landscape.

The question here is whether the six-dimensional geometry supplies such
a landscape for the universe as a whole. Not an analogy --- an actual
function, defined on the space of configurations the boundary world
could be in, whose value the world decreases as it evolves. This Part
constructs that function and shows it has the two properties a landscape
needs. It has to be shaped like a landscape rather than a plateau: there
must be genuine basins, with walls, so that a system arriving in one
stays there. And motion on it has to run one way, so that a descent is a
descent and not an aimless wander.

Those two properties are established as the Part's main results, and the
second is worth naming carefully, because it is where the arrow of time
enters this paper. The function decreases along every physically
admissible evolution of an open system. That statement is the second law
of thermodynamics in the form this framework produces it --- not
assumed, not argued from statistics, but a consequence of the geometry
and of the fact that no system is closed. A companion paper carries the
law-level treatment of time's direction \citep{ArisakaP5CPT}; what is
proved here is the runtime version, the one that makes the descent of
the next Part irreversible.

The landscape has one further feature, and it is the feature that makes
the rest of the paper possible: it is not smooth. It comes in layers ---
strata --- of different dimension, and a system can sit inside a layer
or fall between layers, which is what will later distinguish a slow
settling from a sudden reorganization. Also established here is a test:
if the landscape is real, then any observable on the boundary must
satisfy two classical relations from statistical physics, equipartition
and the fluctuation--dissipation relation, and their failure would end
this pillar. A reader from the life sciences may find Part~X's section
on the free-energy principle a useful entry point, since it connects the
construction here to the free-energy formulations used in neuroscience
and inference.


\section{\GFF{}: motivation, dual-layer structure, and Structure Theorems}
\label{sec:GFF-axioms}

\begin{remark}[Reader's bridge: free energy as the universe's preferred landscape]
\label{rem:GFF-readers-bridge}
A graduate student in physics or chemistry already knows that
thermodynamic systems do not seek the minimum of energy; they seek
the minimum of \emph{free energy} --- the combination $F = U - TS$
of internal energy $U$ and entropic content $TS$ at temperature
$T$. Water freezes at $0^\circ$C not because the ice has lower
energy but because the free energy of the ice phase is lower than
that of the liquid phase, at that temperature. Generally speaking,
every observable phase of matter is a local minimum of free energy
in some appropriate variable. \GFF{} promotes this familiar idea to
the foundational level: the boundary moduli space $\Bdy$ carries a
free-energy functional whose minima --- the \GA{}s (Geometric
Attractors) --- are the empirically realizable phases of the
boundary universe. The Standard-Model vacuum is one such
attractor; the bottom of a free-energy landscape is where the
universe lives. The technical content of this Part is the precise
construction of that landscape from the bulk \TGG{} and \CCI{}
data, plus the Morse-theoretic structure of its critical points.
\end{remark}

This Part develops \GFF{} (the \emph{Geometric Free-Energy Functional}),
the third of the four \GiFour{} Pillars. Like \TGG{} (Part~II) and
\CCI{} (Part~III), \GFF{} is intrinsically a \emph{dual-layer} object:
it is simultaneously
\begin{enumerate}[leftmargin=2em,label=(\roman*)]
\item an \textbf{algebraic/topological functional} --- a real-valued
function on the stratified boundary moduli space $\Bdy$ obtained by
bulk-to-boundary descent of the bulk \TGG{} action and the \CCI{}
4-form $X_4$, whose Morse-theoretic structure encodes the catalog
of Geometric Attractors (\GA{}) on each stratum and whose strict
Morse-positivity is forced by the integrality $[X_4]\in H^4(\Msix;\Z)$
of \CCI{}; and
\item a \textbf{physical/thermodynamic functional} --- the boundary
Helmholtz-type free energy
\begin{equation}
\label{eq:GFF-physical-prelude}
F_{\GFF}(t) \;=\; U_{\GFF}(t) \;-\; T\,S_{\GFF}(t),
\qquad
\frac{\dd F_{\GFF}}{\dd t} \;\le\; 0,
\end{equation}
satisfying the monotonic decrease condition under the boundary
\GDOS{} flow (a covariant Lyapunov inequality). The minima of
$F_{\GFF}$ are the empirically realizable equilibrium phases of the
boundary world; the cascade $F_{\GFF}\to \min$ is what drives the
universe to the Standard-Model vacuum.
\end{enumerate}

The two layers are not independent: they are linked by the
\emph{Geometric First Law} (introduced below in
Section~\ref{sec:GFF-physical-layer}), which identifies
$\cF_{\GEP}$ as the bulk-descended internal energy contribution and
$\cF_{\CEP}$ as the entropic contribution, and shows that the
algebraic Morse cascade of \textbf{GFF-ST5} \emph{is} the
thermodynamic free-energy minimization of
\eqref{eq:GFF-physical-prelude}. In short:
\begin{equation}
\label{eq:GFF-dual-link}
\boxed{\;
\underbrace{a^+(\Sigma) - a^-(\Sigma) > 0}_{\text{algebraic Morse-positivity}}
\;\xrightarrow{\;\text{Geometric First Law}\;}\;
\underbrace{\dd F_{\GFF}/\dd t \le 0}_{\text{thermodynamic monotonicity}}
\;}
\end{equation}
The arrow is precisely the Geometric First Law: it carries the
algebraic identity to its physical avatar.

\paragraph{Section structure of Part~IV.}
Section~\ref{sec:GFF-axioms} (the present section) states the seven
\GFF{} axioms (\textbf{GFF-ST1} through \textbf{GFF-ST7}), organized
into the algebraic block (\textbf{GFF-ST1}--\textbf{GFF-ST5}) and the
physical block (\textbf{GFF-ST6}--\textbf{GFF-ST7}); gives the
dual-layer formal definition; and lists immediate consequences.
Section~\ref{sec:GEP-CEP-decomposition} develops the energetic /
curvature-entropic decomposition $\cF_{\GFF} = \cF_{\GEP} +
\cF_{\CEP}$. Section~\ref{sec:Bdy-stratified} describes the
stratified boundary moduli space and \GFF{} as a Morse function on
each stratum. Section~\ref{sec:GFF-physical-layer} (new in)
develops the physical/thermodynamic layer of \GFF{}: the
Geometric First Law, the monotonic decrease
$\dd F_{\GFF}/\dd t \le 0$ under the \GDOS{} flow, the H-theorem
analogue, and laboratory-accessible thermodynamic falsifiers.
Section~\ref{sec:GFF-T3} states and proves Theorem~\textbf{T3}, now
in two parts: \textbf{T3.1} (algebraic Morse-positivity from \CCI{})
and \textbf{T3.2} (thermodynamic monotonicity from \GDOS{}).
Section~\ref{sec:H1-H2-theorems} reduces hypotheses $\mathcal H_1$
and $\mathcal H_2$ from the N=3 proof of
\citet{ArisakaP2N3} to theorems of \GiFour{}, modulo the
$\eta$-invariant computation M6.
Section~\ref{sec:GFF-examples} contains worked examples on the
family-projector moduli, the Higgs moduli, the gauge-orbit moduli, and
the Wilson-line moduli.

\begin{remark}[Physics intuition: \GFF{} as the GG analogue of the Helmholtz free energy]
\label{rem:GFF-thermodynamic-analogue}
Readers familiar with statistical mechanics will find an immediate
analogy. In thermodynamics, the Helmholtz free energy
$F = U - TS$ is the unique state function whose minimization governs
equilibrium for a system in contact with a heat bath at fixed
temperature $T$. Its derivative $\dd F/\dd t \le 0$ along any
spontaneous trajectory is the second law of thermodynamics.

In \GGTheory{}, \GFF{} plays the same logical role for the boundary
moduli space. $\cF_{\GEP}$ is the analogue of internal energy $U$:
it is the bulk-descended \TGG{} action evaluated on the boundary
slice, with coefficients fixed by the \OGNfive{} ledger. $\cF_{\CEP}$ is
the analogue of $-TS$: it is the entropic contribution from the
Riemannian volume of admissible projector classes, with the
``temperature'' role played by the scale $M_{\rm KK}$.
The minimization $\dd F_{\GFF}/\dd t \le 0$ along any \GDOS{}
trajectory is the \emph{covariant} second law: it is the universal
empirical statement of irreversibility on the boundary.

This is the precise sense in which \GFF{} \emph{generalizes} the
Helmholtz free energy from a single thermodynamic system to the
entire stratified moduli space of admissible boundary projectors.
\end{remark}

\begin{remark}[Why broader scientists should care about \GFF{}]
\label{rem:GFF-broader}
The free-energy minimization principle $\dd F/\dd t \le 0$ is the
\emph{single most-used dynamical principle} across the natural
sciences. In chemistry, it governs reaction kinetics. In
condensed-matter physics, it governs phase transitions. In
statistical mechanics, it governs equilibration. In biology, it
governs protein folding, membrane assembly, and signaling network
relaxation \citep{Friston2010,FristonFreeEnergy2023}. In
neuroscience, it governs the brain's predictive-coding dynamics
(Friston's free-energy principle). In information theory, it
governs maximum-entropy inference \citep{Jaynes1957}.

What is genuinely new in \GFF{} is that all of these distinct
free-energy principles are \emph{boundary projections} of the same
upstream object: the bulk-to-boundary descent of the \TGG{} action and the
\CCI{} 4-form to $\Bdy$. This explains, in one stroke, why the
free-energy paradigm is so universal across the sciences --- it is
the empirical projection of the bulk geometric structure of
\GGTheory{}. The fact that \GFF{} subsumes Friston's free-energy
principle as a special case (when restricted to neural information
manifolds) is one of the strongest internal consistency checks of
the theory.
\end{remark}

\subsection{Motivation: from algebraic \CCI{} to dual-layer \GFF{}}
\label{ssec:GFF-motivation}

The \CCI{} structure of Part~III specifies the integral cohomology
class $[X_4] \in H^4(\Msix;\Z)$ associated with the \TGG{} composite
connection, the boundary covariant conservation
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$, and the \OGNfive{} ledger
$(M, N, L; J, K) = (1,3,5;7,35)$. But \CCI{} alone does not
answer two crucial questions about boundary dynamics.

\smallskip
\noindent\textbf{Question 1 (algebraic).}
\CCI{} tells us \emph{which} 4-form descends to the boundary, but not
\emph{which boundary configurations are stable} under that descent.
The space of admissible boundary projector classes is enormous; we
need a real-valued functional whose critical points are the stable
configurations and whose Morse structure organizes them into a
cascade. This is the \textbf{algebraic content} of \GFF{}.

\smallskip
\noindent\textbf{Question 2 (physical).}
\CCI{} tells us that the Master Stress Tensor is conserved on the
boundary, but not \emph{which equilibrium} the boundary world relaxes
to. Empirically, the boundary world is observed in a unique vacuum
(the Standard-Model vacuum); we need a thermodynamic functional whose
monotonic decrease drives the boundary world to that unique vacuum.
This is the \textbf{physical content} of \GFF{}.

\smallskip
\noindent\textbf{Why both layers must be in the same Pillar.}
The two questions are inseparable. The strict Morse-positivity that
selects a unique \GA{} (algebraic side) is precisely the condition
for the free energy $F_{\GFF}$ to have a unique global minimum
(physical side) --- because every Morse-positivity violation would
correspond to a degenerate equilibrium with an unbroken modulus,
which would manifest empirically as an unobserved massless boson.
Conversely, the empirical absence of unobserved massless bosons
constrains the algebraic Morse data via the Goldstone count
(see Section~\ref{sec:GFF-physical-layer}).

\GFF{} is the formalization of this dual content. It is
simultaneously:
\begin{itemize}[leftmargin=2em]
\item the \emph{algebraic Morse functional} on the stratified
boundary moduli space, with strict Morse-positivity inherited from
\CCI{}; and
\item the \emph{thermodynamic free energy} of the boundary world,
with monotonic decrease driving the cascade to the unique \GA{}.
\end{itemize}
The most important new content of Part~IV is that the strict
Morse-positivity of (formerly Theorem~\textbf{T3}) is now
\emph{equivalent} to a statement of the second law for the boundary
\GDOS{} flow. This is the bridge from algebraic topology to
phenomenological thermodynamics.

\begin{remark}[Physics intuition: why $\dd F_{\GFF}/\dd t \le 0$ is automatic]
\label{rem:GFF-second-law-automatic}
A graduate student in physics may wonder why the second law
$\dd F_{\GFF}/\dd t \le 0$ should hold \emph{automatically} for every
\GDOS{} trajectory, without requiring an additional ``second law
postulate.'' The answer is structural: the \GDOS{} flow is a CPTP
(completely positive, trace-preserving) map on the boundary state
space, and any such map decreases relative entropy by Lindblad's
theorem \citep{Lindblad1975Entropy}. Free-energy decrease is the
projection of relative-entropy decrease onto the boundary moduli
space.

In other words: the second law of thermodynamics is not a separate
postulate of \GGTheory{} but an automatic consequence of the
open-system structure of \GDOS{}. \textbf{GFF-ST6} simply makes this
content explicit and gives it operational form.
\end{remark}

\subsubsection{Historical background (with citations)}

The free-energy functional that \GFF{} formalizes is the latest member
of a tradition that began with the steam engine. Carnot (1824)
introduced the first quantitative argument for an irreversible
thermodynamic limit; Clausius (1865) gave the modern formulation of
entropy as a state function and stated the second law in its closed
form $\dd S \geq \dd Q/T$; \citet{Boltzmann1872} provided the
statistical-mechanical foundation with his $H$-theorem, which derived
monotone increase of entropy from microscopic dynamics; and Gibbs
(1876) introduced the canonical free energy $F = U - TS$ as the
central thermodynamic potential of a system in contact with a thermal
reservoir.

The extension to open and dissipative systems began with
\citet{Onsager1931}, whose reciprocal relations $L_{ij} = L_{ji}$
constrain the linear response of irreversible systems and provide the
first hint of Morse-like positivity in the
fluctuation--dissipation regime. Prigogine (1947) extended the
treatment to nonequilibrium steady states with his
minimum-entropy-production principle for near-equilibrium open
systems; the maximum-entropy-production tradition (Paltridge 1975 and
successors) provided the complementary far-from-equilibrium extension.
\citet{Jaynes1957} then placed the entire program on
information-theoretic footing by deriving the canonical ensemble from
maximum Shannon entropy --- the first explicit bridge between
classical thermodynamics and information theory.

The modern free-energy formulation in computational neuroscience and
machine learning is due to \citet{Friston2010,FristonFreeEnergy2023}:
biological self-organization is interpreted as a gradient flow on
variational free energy, with surprise minimization as the universal
principle. \GFF{} subsumes and generalizes this tradition. The
decomposition $F_\GFF = F_\GEP + F_\CEP$ separates the
\emph{energetic} (geometric) part from the \emph{entropic}
(information-theoretic) part, and the dual-layer Morse structure
inherited from \CCI{} via Hopkins--Singer descent
\citep{HopkinsSinger2005} elevates the classical positivity argument to
a theorem (Theorem~\ref{thm:GFF-positivity-from-CCI}) --- the axiomatic
completion of the lineage from Carnot through Onsager and Friston to
\GFF{}.


\subsection{The seven \GFF{} Structure Theorems (algebraic block + physical block)}
\label{ssec:GFF-seven-st}

The seven \GFF{} Structure Theorems partition naturally into two blocks:
\begin{description}[leftmargin=2em]
\item[Algebraic block: \textbf{GFF-ST1} -- \textbf{GFF-ST5}] These Structure Theorems
fix the domain, decomposition, smoothness, boundedness, and
strict Morse-positivity of the functional. They live on the
stratified moduli space $\Bdy$ and are Morse-theoretic in character.
\item[Physical block: \textbf{GFF-ST6} -- \textbf{GFF-ST7}] These Structure Theorems
state the monotonic decrease of $F_{\GFF}$ under the \GDOS{}
flow (the boundary second law) and the equilibrium uniqueness of
the global minimum (the boundary Helmholtz principle). They live in
real (laboratory) time and are thermodynamically testable.
\end{description}
The two blocks are linked by the Geometric First Law of
Section~\ref{sec:GFF-physical-layer}.

\subsubsection*{Algebraic block (GFF-ST1 through GFF-ST5)}

\begin{structheorem}[\textbf{GFF-ST1}: Domain]
\label{st:GFF-ST1}
The Geometric Free-Energy Functional \GFF{} is a real-valued function on the
boundary moduli space $\Bdy$ of \OGGEP{}:
\begin{equation}
\label{eq:GFF-ST1}
\cF_{\GFF} \colon\; \Bdy \;\to\; \R.
\end{equation}
The domain $\Bdy$ is a stratified space: it decomposes as a disjoint
union of smooth connected strata, each labeled by the discrete data
of the boundary projector class.
\end{structheorem}

\begin{structheorem}[\textbf{GFF-ST2}: \GEP{}/\CEP{} decomposition]
\label{st:GFF-ST2}
\GFF{} decomposes as a sum of two distinct contributions:
\begin{equation}
\label{eq:GFF-ST2}
\cF_{\GFF} \;=\; \cF_{\GEP} \;+\; \cF_{\CEP},
\end{equation}
where $\cF_{\GEP}$ is the \emph{Geometric Energetic Part} (the
energetic contribution from bulk-to-boundary descent of the bulk action of
\TGG{}) and $\cF_{\CEP}$ is the \emph{Curvature-Entropic Part} (the
entropic contribution from the Riemannian volume of projector classes
and the diffusion term of \GDOS{}). Both parts are real-valued
functions on $\Bdy$.
\end{structheorem}

\begin{structheorem}[\textbf{GFF-ST3}: Smoothness and Morse condition]
\label{st:GFF-ST3}
The functional $\cF_{\GFF}$ is smooth on each stratum of $\Bdy$, and
its critical points on each stratum are non-degenerate (i.e., the
Hessian $\Hess(\cF_{\GFF})$ is non-singular at each critical point).
Equivalently, $\cF_{\GFF}$ restricted to each stratum is a Morse
function in the standard sense \citep{Milnor1963}.
\end{structheorem}

\begin{structheorem}[\textbf{GFF-ST4}: Boundedness below]
\label{st:GFF-ST4}
$\cF_{\GFF}$ is bounded below on each stratum of $\Bdy$:
\begin{equation}
\label{eq:GFF-ST4}
\inf_{\Bdy}\,\cF_{\GFF} \;>\; -\infty
\end{equation}
on each stratum. The infimum is attained at a finite collection of
isolated points (the \GA{}'s of that stratum).
\end{structheorem}

\begin{structheorem}[\textbf{GFF-ST5}: Strict Morse-positivity from \CCI{}]
\label{st:GFF-ST5}
Let $\Sigma\subset\Bdy$ be a stratum that is a connected, compact
K\"ahler manifold on which the boundary gauge connections are flat, on
which $\cF_{\GFF}|_\Sigma$ is the moment map of a Hamiltonian torus
action with isolated non-degenerate fixed points, and which supports a
non-trivial Hopkins--Singer descent of the \CCI{} class $[X_4]$. Then
every Morse index of $\cF_{\GFF}$ on $\Sigma$ is even, the alternating
Morse-index sum equals the Hirzebruch signature $\sigma(\Sigma)$, and:
\emph{(necessity)} if $\dim_\R\Sigma\equiv 2\pmod 4$ then
$a^+(\Sigma)-a^-(\Sigma)=0$, so strict Morse-positivity requires
$\dim_\R\Sigma\equiv 0\pmod 4$; \emph{(sufficiency)} strict
Morse-positivity holds precisely when $\sigma(\Sigma)>0$, which for
$\Sigma=\CPN$ is equivalent to $N$ odd. In that case
\begin{equation}
\label{eq:GFF-ST5}
a^+(\Sigma) - a^-(\Sigma) \;>\; 0,
\end{equation}
where $a^+(\Sigma) = \#\{c\in\Crit(\cF_{\GFF}|_\Sigma) : \ind(c) \equiv 0\pmod 4\}$
and $a^-(\Sigma) = \#\{c : \ind(c) \equiv 2\pmod 4\}$.
\end{structheorem}

\begin{remark}[Why the hypotheses are all load-bearing]
\label{rem:GFF-ST5-hypotheses}
Each clause earns its place, and it is worth seeing why, because the
temptation is to state the theorem for every stratum and the statement
is then false. Non-trivial descent alone does not give strict
positivity: on $\CP^3$ the first Pontryagin class is
$p_1 = 4\,h^2 \neq 0$, so the descent is non-trivial by the
$\mathcal H_1$ theorem, yet the critical indices are $0,2,4,6$ and
$a^+ - a^- = 2 - 2 = 0$ --- the computation is carried out in
Appendix~\ref{app:Morse-CPN-worked}. Nor does adding the parity
condition suffice: strict positivity is equivalent to
$\sigma(\Sigma)>0$, and a compact K\"ahler surface of real dimension
four can carry a torus action with $\sigma<0$, in which case the
alternating sum is negative rather than merely zero. What the descent
buys, in real dimension four and there only, is $\sigma\neq 0$, since
$\sigma = p_1[\Sigma]/3$ on a closed oriented four-manifold; the sign
is a separate fact about the stratum. Finally, the moment-map clause is
what makes the alternating sum a topological invariant at all: for a
Morse function not invariant under a torus action the relevant
invariant is the Euler characteristic, which is positive for every
$\CPN$ and would make the parity test vacuous. The physics content of
that clause --- that the functional the boundary actually delivers is
the moment map --- is an identification, and it is graded as one.
\end{remark}

The five Structure Theorems \textbf{GFF-ST1}--\textbf{GFF-ST5} constitute the
\emph{algebraic} block of \GFF{}. They define the Morse-theoretic
structure on the stratified boundary moduli and provide the catalog
of Geometric Attractors via Theorem~\textbf{T3.1} (proved in
Section~\ref{sec:GFF-T3}).

\subsubsection*{Physical block (GFF-ST6 and GFF-ST7)}

The physical block translates the algebraic Morse structure of
\GFF{} into operational, observer-accessible thermodynamic form. It
consists of two Structure Theorems.

\begin{structheorem}[\textbf{GFF-ST6}: Thermodynamic monotonicity (boundary second law)]
\label{st:GFF-ST6}
Let $F_{\GFF}\colon \Bdy\times\R_{\ge 0}\to\R$ be the time-dependent
restriction of \GFF{} along a \GDOS{} trajectory
$t \mapsto [\Pi(t)]\in\Bdy$ (where $t\in\R_{\ge 0}$ is the boundary
laboratory time). Then $F_{\GFF}$ admits a Helmholtz-type
decomposition
\begin{equation}
\label{eq:GFF-ST6-helmholtz}
F_{\GFF}([\Pi(t)]) \;=\; U_{\GFF}([\Pi(t)]) \;-\; T_{\GFF}\,S_{\GFF}([\Pi(t)]),
\end{equation}
where:
\begin{itemize}[leftmargin=2em]
\item $U_{\GFF} = \cF_{\GEP}$ is the \emph{boundary internal energy}
(bulk-to-boundary descent of the \TGG{} action);
\item $S_{\GFF}$ is the \emph{boundary entropy} (von~Neumann entropy
of the boundary projector class, weighted by the Riemannian volume
form of \textbf{GFF-ST2});
\item $T_{\GFF} \sim M_{\rm KK}$ is the \emph{boundary effective
temperature} (set by the scale of \TGG-ST2);
\end{itemize}
and along any \GDOS{} trajectory the free-energy time derivative is
non-positive:
\begin{equation}
\label{eq:GFF-ST6-monotonic}
\boxed{\;
\frac{\dd F_{\GFF}}{\dd t} \;\le\; 0,
\;}
\end{equation}
with equality if and only if $[\Pi(t)]$ is a critical point of
$\cF_{\GFF}$. This is the boundary covariant form of the second law
of thermodynamics.
\end{structheorem}

\begin{remark}[The ledger as the meeting of two endowments ---
candidate reading]
The Helmholtz form \eqref{eq:GFF-ST6-helmholtz} is the boundary
face of two endowments of constitutional time (Part~IX of the CPT
member paper, \citealp{ArisakaP5CPT}): $U_{\GFF} = \cF_{\GEP}$
descends from the repeating clock --- the Killing flow whose
conserved charge is energy --- while $S_{\GFF}$ accumulates under
the arrow --- the dictionary's one-way discard. The \GFF{} ledger
is where the second and third endowments meet and trade at the rate
$T_{\GFF}$, and the split $\cF_{\GFF} = \cF_{\GEP} +
\cF_{\CEP}$ of \textbf{GFF-ST2} is the same meeting read at the
level of the functional: \GEP{} the energetic face of the
repeating clock, \CEP{} the entropic face of the arrow. Grade:
candidate reading; the structure theorems above stand without it.
\end{remark}

\begin{lemma}[DE-1: vacuum-flux freezing at the Geometric Attractor]
\label{lem:DE1-vacuum-freezing}
At a Geometric Attractor endpoint of the \GRIP{} cascade (a critical
point of $\cF_{\GFF}$ in the sense of \textbf{GFF-ST6}), the energy
export of the frozen vacuum sector is individually conserved,
$\dd\rho_\Lambda/\dd t = 0$, with no compensating exchange against the
matter, radiation, or gauge sectors, at the leading order at which the
four-way sector split of the \BIsix{} closure is exact.
\end{lemma}

\begin{proof}
By \textbf{GFF-ST6}, $\dd F_{\GFF}/\dd t \le 0$ along any \GDOS{}
trajectory, with equality if and only if $[\Pi(t)]$ is a critical
point; at the Geometric Attractor the projector class is therefore
stationary, $\dd[\Pi]/\dd t = 0$. The vacuum export is a functional of
the state --- $\rho_\Lambda$ is $\cF_{\GFF}$ evaluated at the endpoint
--- and is hence time-independent. Any net energy flux between the
vacuum sector and another sector is a change of the state
decomposition and so requires $\dd[\Pi]/\dd t \neq 0$, contradicting
stationarity; and by the global four-current closure (T2.2, closure of
$X_4$) there exists no external reservoir against which a compensating
flux could run. Hence each individual sector flux vanishes. The
qualifier is inherited from the sector split itself: the four-way
decomposition holds at leading order in the KK thresholds (open
problems M1/M3), and the lemma holds at the same order.
\end{proof}

\noindent The lemma upgrades the informal reading ``the attractor's
export is frozen'' from prose to a consequence of \textbf{GFF-ST6} and
the \BIsix{} closure. Its cosmological payoff is drawn in the
companion \ATen{} \citep{ArisakaA10Cosmos}: combined with the
no-phantom theorem there, it pins the dark-energy equation of state at
$w = -1$ exactly (residual deviation bounded by
$(H_0/M_{\rm lock})^2 \sim 10^{-91}$), so that any confirmed
$w \neq -1$ --- in either direction --- refutes the fixed-point
reading outright. A topological mechanism beneath the same freezing
--- a Dirac superselection filter carried by the descent's inflow
term --- is derived in Corollary~\ref{cor:dirac-superselection}
(\S\ref{ssec:descent-normalisation}).


\begin{structheorem}[\textbf{GFF-ST7}: Equilibrium uniqueness and laboratory falsifier]
\label{st:GFF-ST7}
Among all admissible boundary projector classes
$[\Pi]\in\Bdy$ accessible from a generic initial condition under the
\GDOS{} flow of \textbf{GFF-ST6}, there exists a \emph{unique global
minimum} $[\Pi_*]$ of $F_{\GFF}$ in the asymptotic limit
$t\to\infty$:
\begin{equation}
\label{eq:GFF-ST7-equilibrium}
[\Pi_*] \;=\; \arg\min_{[\Pi]\in\Bdy}\,F_{\GFF}([\Pi]).
\end{equation}
This equilibrium configuration is empirically identified with the
Standard-Model vacuum and obeys two universal laboratory-testable
relations:
\begin{enumerate}[label=\textnormal{(\arabic*)},leftmargin=2.2em]
\item (\emph{Equipartition relation.}) At thermal equilibrium near
the global minimum, the boundary effective temperature satisfies
$T_{\GFF} \,\dd S_{\GFF} = \dd U_{\GFF}$, i.e., the boundary
Helmholtz first law holds locally;
\item (\emph{Fluctuation-dissipation relation.}) The covariance of
free-energy fluctuations and the linear response of $F_{\GFF}$ to
external boundary perturbations are linked by an
\emph{Einstein-type} relation
$\langle \delta F_{\GFF}^2 \rangle = T_{\GFF}\,\chi_{\GFF}$, where
$\chi_{\GFF}$ is the boundary susceptibility.
\end{enumerate}
A laboratory observation of an empirical violation of either (1) or
(2) at scales below $M_{\rm KK}$ would constitute a falsification
of \GFF{} and hence of the entire \GG{} framework.
\end{structheorem}

In the vocabulary introduced in Definition~\ref{def:GGA} below, the
unique global minimizer $[\Pi_*]$ of \textbf{GFF-ST7} \emph{is} the
Ground Geometric Attractor (\GGA) of the boundary moduli space taken
whole. The statement every application actually uses is the
per-stratum one, and it is Corollary~\ref{cor:GGA-per-stratum} below;
the reader should keep the two apart, because $\Bdy$ is a stratified
space and its global minimizer is not the minimizer of each stratum.

The seven Structure Theorems \textbf{GFF-ST1}--\textbf{GFF-ST7} together constitute
the full \emph{Geometric Free-Energy Functional}. Axioms
\textbf{GFF-ST1}--\textbf{GFF-ST4} are essentially the standard
free-energy structure on a stratified moduli space. The new content
that distinguishes \GFF{} from a generic free energy is twofold:
\textbf{GFF-ST5} links the strict Morse-positivity to \CCI{}'s
non-trivial cohomology class (algebraic), and
\textbf{GFF-ST6}--\textbf{GFF-ST7} link the cascade dynamics to the
boundary second law of thermodynamics with a unique global minimum
(physical). The two halves are unified by Theorem~\textbf{T3} in
Section~\ref{sec:GFF-T3}.

\subsection{Dual-layer formal definition of \GFF{}}
\label{ssec:GFF-formal-def}

We now consolidate the algebraic and physical blocks into a single
dual-layer definition.

\begin{definition}[\GFF: the Geometric Free-Energy Functional, dual-layer form]
\label{def:GFF}
The \emph{Geometric Free-Energy Functional} on the boundary moduli
space $\Bdy$ associated with the \TGG{} structure
$(\Pbun, \Atot, \Rtot)$ and the \CCI{} structure
$(X_4, [X_4], \alpha_R, \alpha_5, \alpha_B, \alpha_{Qi})$ is the
\textbf{septuple}
\begin{equation}
\label{eq:GFF-septuple}
\GFF = \Bigl(\;
\underbrace{\cF_{\GFF},\;\cF_{\GEP},\;\cF_{\CEP},\;\Crit(\cF_{\GFF}),\;a^+(\Sigma) - a^-(\Sigma) > 0}_{\text{algebraic block (GFF-ST1--A5)}};
\;\;
\underbrace{F_{\GFF}=U_{\GFF}-T_{\GFF}S_{\GFF},\;\;[\Pi_*]}_{\text{physical block (GFF-ST6--A7)}}
\;\Bigr),
\end{equation}
where the seven components correspond to the Structure Theorems
\textbf{GFF-ST1}--\textbf{GFF-ST7} respectively.

The algebraic part $\cF_{\GFF}\colon \Bdy\to\R$ is given by the
bulk-to-boundary descent of the bulk action of \TGG{}, decomposed as
\begin{equation}
\label{eq:GFF-formal}
\cF_{\GFF}([\Pi]) \;=\; \cF_{\GEP}([\Pi]) \;+\; \cF_{\CEP}([\Pi]),
\qquad [\Pi] \in \Bdy.
\end{equation}
The physical part $F_{\GFF}([\Pi(t)]) = U_{\GFF} - T_{\GFF}S_{\GFF}$
satisfies the boundary second law $\dd F_{\GFF}/\dd t \le 0$ along
every \GDOS{} trajectory and converges to a unique global minimum
$[\Pi_*]\in\Bdy$. The two parts are linked by the Geometric First
Law (Theorem~\ref{thm:GFF-first-law} of
Section~\ref{sec:GFF-physical-layer}), which identifies
$U_{\GFF} = \cF_{\GEP}$, $-T_{\GFF}S_{\GFF} = \cF_{\CEP}$, and
$\dd F_{\GFF}/\dd t \le 0$ as the gradient flow of $\cF_{\GFF}$.
\end{definition}

\begin{remark}[Logical content of \GFF{}]
\label{rem:GFF-content}
The intuitive content of Definition~\ref{def:GFF} is twofold:
\begin{itemize}[leftmargin=2em]
\item \textbf{Algebraic side.} $\cF_{\GFF}$ is a Morse function on
the stratified boundary moduli space whose critical points are
Geometric Attractors and whose strict alternating Morse-index sum is
positive on every admissible stratum (forced by the integrality of
the \CCI{} class $[X_4]$). This guarantees the existence of a finite
catalog of stable boundary phases.
\item \textbf{Physical side.} $F_{\GFF}$ is a Helmholtz-type free
energy on the boundary world that satisfies the second law of
thermodynamics along every \GDOS{} trajectory and converges to a
unique global minimum (the Standard-Model vacuum). This guarantees
the empirical observability of the equilibrium and provides
laboratory-testable equipartition and fluctuation-dissipation
relations.
\end{itemize}
The two sides are linked by the Geometric First Law: the same
functional $\cF_{\GFF}$ that organizes the algebraic Morse cascade
\emph{is} the thermodynamic free energy whose minimization drives
the boundary world to equilibrium. This is the duality of \GFF{}.
\end{remark}

\begin{remark}[Open-system character of \GFF{}]
\label{rem:GFF-open-system}
The functional $\cF_{\GFF}$ is fundamentally an \emph{open-system}
object: it is defined on the boundary moduli of the bulk-to-boundary
projection, not on the bulk itself. In closed-system QFT, no
analogue of \GFF{} exists --- the closed-system state space has no
preferred discrete cascade structure, no irreversible flow, no
boundary-projector class, and no thermodynamic temperature. The
existence of \GFF{} as a dual-layer object (Morse-positive +
thermodynamically monotonic) is one of the principal pieces of
evidence that the open-system (\OGGEP{}) framework provides
foundational content not available in closed-system QFT.
\end{remark}

\begin{remark}[Physics intuition: why both layers must be present]
\label{rem:GFF-both-layers-needed}
A physics graduate student may at first think that the algebraic
side (Morse-positivity) is enough --- after all, it gives a finite
catalog of stable boundary phases. But the algebraic side alone
does not specify \emph{which} of those phases the boundary world
actually relaxes to. Without the physical side (the second law),
the boundary world could in principle remain stuck in any local
minimum, including unphysical ones. The physical side
(\textbf{GFF-ST6}--\textbf{GFF-ST7}) is what guarantees that the
\emph{global} minimum is the universally selected equilibrium.

Conversely, the physical side alone (a free energy with monotonic
decrease) does not specify the structure of the moduli space or the
cascade chain. Without the algebraic side, one would have only a
phenomenological free energy with no upstream geometric content.

The dual-layer structure of \GFF{} is therefore not redundant ---
both layers are essential, and the Geometric First Law is what
makes them consistent.
\end{remark}

\subsection{First consequences of the \GFF{} Structure Theorems}
\label{ssec:GFF-first-consequences}

We list four immediate consequences of \textbf{GFF-ST1}--\textbf{GFF-ST5}.

\begin{proposition}[Existence of \GA{} catalog on each stratum]
\label{prop:GFF-GA-catalogue}
On each stratum $\Sigma$ of $\Bdy$, the set of local minima of
$\cF_{\GFF}$ is a finite, isolated, non-empty collection of points,
called the \GA{} catalog of $\Sigma$.
\end{proposition}

\begin{proof}
By \textbf{GFF-ST3}, $\cF_{\GFF}|_\Sigma$ is Morse, so its critical
points are isolated. By \textbf{GFF-ST4}, $\cF_{\GFF}$ is bounded
below on $\Sigma$, so its infimum is attained -- at least one local
minimum exists. Compactness of each stratum (or proper-Morse
condition for non-compact strata) ensures only finitely many critical
points exist. The local minima form the \GA{} catalog.
\end{proof}

\begin{proposition}[Cascade chain on each stratum]
\label{prop:GFF-cascade-chain}
On each stratum $\Sigma$, the Morse critical points of $\cF_{\GFF}$
admit a cascade chain ordered by Morse index: critical points of
index $0$ at the bottom (\GA{}'s), critical points of higher even
index above them (saddles), terminating at the maximum index (the
deepest unstable phase). This chain is the upstream object whose
gradient flow gives the \GRIP{} cascade of Part~V.
\end{proposition}

\begin{proof}
Standard Morse theory \citep{Milnor1963}: the gradient flow of
$\cF_{\GFF}$ descends from higher-index critical points to lower-index
ones along instanton trajectories, with the lowest-index critical
points (the local minima) being the asymptotic attractors of the flow.
\end{proof}

\begin{proposition}[Equivalence of \GFF{} Morse-positivity with topological signature]
\label{prop:GFF-Morse-eq-signature}
On any stratum $\Sigma$ admitting a Hopkins--Singer descent of $[X_4]$,
the strict alternating Morse-index sum
$a^+(\Sigma) - a^-(\Sigma)$ is equal to the topological signature
$\sigma(\Sigma)$ when $\Sigma$ has real dimension divisible by four,
and is identically zero otherwise.
\end{proposition}

\begin{proof}[Sketch]
This is the Morse-theoretic version of Hirzebruch's signature theorem
restricted to the stratum. For a closed oriented $4m$-manifold $\Sigma$
equipped with a perfect Morse function whose critical points all have
even Morse index, the alternating sum
$\sum_a (-1)^{\ind_a/2}$ equals the integrated L-genus, which is the
signature. For $\Sigma$ of real dimension $\equiv 2\pmod 4$, the
signature is identically zero. This is the standard textbook content
of Hirzebruch's signature theorem
\citep{Hirzebruch1966,Milnor1963}. The key consequence: strict
positivity holds if and only if $\dim_\R \Sigma \in 4\Z$.
\end{proof}

\begin{proposition}[\GFF{} as the formal realization of \OGGEP{}]
\label{prop:GFF-equiv-OGGEP}
The existence of $\cF_{\GFF}$ satisfying \textbf{GFF-ST1}--\textbf{GFF-ST4}
is equivalent to the GG-Constitutional axiom \OGGEP{}. The strict
Morse-positivity of \textbf{GFF-ST5} is a derived theorem from \CCI{}
via the Hopkins--Singer descent (Theorem~\ref{thm:GFF-positivity-from-CCI}
of Section~\ref{sec:GFF-T3}).
\end{proposition}

\begin{proof}[Sketch]
\OGGEP{} states that observable boundary content is related to bulk
geometric content by an irreversible CPTP bulk-to-boundary projection.
Equivalently: there exists a boundary moduli space $\Bdy$, a
free-energy functional $\cF_{\GFF}$ on it that drives the cascade, and
the cascade closes in finite time. These are exactly axioms
\textbf{GFF-ST1}--\textbf{GFF-ST4}. Conversely, given $\cF_{\GFF}$
satisfying these axioms, one recovers the \OGGEP{} CPTP-projection
structure via the gradient flow. Theorem~\ref{thm:GFF-positivity-from-CCI}
establishes \textbf{GFF-ST5}.
\end{proof}

%% ====================================================================
\section{The \GEP{}/\CEP{} decomposition}
\label{sec:GEP-CEP-decomposition}
%% ====================================================================

We now develop the two parts of \GFF{} in detail: the energetic part
$\cF_{\GEP}$ and the curvature-entropic part $\cF_{\CEP}$.

\subsection{The energetic part \GEP{}}
\label{ssec:GEP-def}

\begin{definition}[Geometric Energetic Part \GEP]
\label{def:GEP}
The Geometric Energetic Part of \GFF{} is the contribution to
$\cF_{\GFF}$ obtained by bulk-to-boundary descent of the bulk action of
\TGG{}. In the canonical bulk action with two-derivative kinetic
terms,
\begin{equation}
\label{eq:GEP-def}
\cF_{\GEP}([\Pi]) \;=\;
\frac{1}{8\pi^2}\,\int_{\Pi[\Mfour]}\,
\bigl[\,\alpha_R\,\tr R_{\Spinone}^2 + \alpha_5\,\Tr F_5^2
+ \alpha_B\,F_B^2 + \alpha_{Qi}\,F_{Qi}^2\,\bigr],
\end{equation}
where the integration is over the boundary slice
$\Pi[\Mfour]$ of the projector $\Pi$, and the coefficients
$(\alpha_R, \alpha_5, \alpha_B, \alpha_{Qi}) = (+1, -1, -2/5, -2/5)$
are inherited from \textbf{CCI-ST2}.
\end{definition}

The \GEP{} integrand is exactly the four-form $X_4$ of \CCI{}. Therefore
$\cF_{\GEP}$ is the integrated $X_4$ on the boundary slice:
\begin{equation}
\label{eq:GEP-as-X4-integral}
\cF_{\GEP}([\Pi]) = \frac{1}{8\pi^2}\,\int_{\Pi[\Mfour]} X_4.
\end{equation}
This is the cleanest formulation: the \GEP{} is, up to an overall
normalization, the boundary integral of the \CCI{} four-form.

\subsection{The curvature-entropic part \CEP{}}
\label{ssec:CEP-def}

\begin{definition}[Curvature-Entropic Part \CEP]
\label{def:CEP}
The Curvature-Entropic Part of \GFF{} is the entropic contribution
arising from the Riemannian volume of admissible boundary projector
classes plus the diffusion term of \GDOS{}. Schematically,
\begin{equation}
\label{eq:CEP-def}
\cF_{\CEP}([\Pi]) \;=\; -\,T\,\mathcal S([\Pi]) + \cF_{\rm diff}([\Pi]),
\end{equation}
where $T$ is an effective ``temperature'' parameter (set by the bulk
scale), $\mathcal S$ is the configurational entropy of the
projector class $[\Pi]$, and $\cF_{\rm diff}$ is the diffusion-term
contribution from the boundary CPTP map.
\end{definition}

\begin{remark}[Why \CEP{} is needed]
\label{rem:CEP-need}
A pure energetic functional $\cF_{\GEP}$ alone would give critical
points only at the strict zeros of $X_4$ on the boundary -- typically
just the trivial configuration. The entropic contribution $\cF_{\CEP}$
is what creates the rich Morse landscape with multiple critical points
of varying Morse index. This is the open-system content: in a
closed-system QFT, there is no entropic contribution to the action,
and the Morse cascade structure of \GFF{} would not exist.
\end{remark}

\subsection{Bulk-to-boundary descent of the bulk action}
\label{ssec:holographic-descent-action}

The construction of $\cF_{\GEP}$ from the bulk action of \TGG{}
proceeds as follows.

\textbf{Step 1: bulk action.} The bulk action of \TGG{} on $\Msix$
is
\begin{equation}
\label{eq:bulk-action}
S_{\rm bulk} = \frac{1}{16\pi^2}\,\int_{\Msix}\,X_4 \wedge \omega_2,
\end{equation}
where $\omega_2$ is a fixed bulk 2-form on the radial-angular sector.
Its overall normalization is not a free convention: POS-6 of \AOne{}
\citep{ArisakaA1Constitution} (\OGN{}: no continuous rescaling dials;
``conventions are not allowed to function as hidden continuous
parameters'') requires it to be fixed, and
Lemma~\ref{lem:descent-normalisation} below fixes it.

\textbf{Step 2: boundary projection.} Given a boundary projector
$\Pi:\Msix \to \Mfour$, the bulk integral splits into a boundary
integral plus a transverse part. The boundary integral is
\begin{equation}
\label{eq:boundary-integral}
\int_{\Pi[\Mfour]} X_4,
\end{equation}
which is exactly the boundary integral of \eqref{eq:GEP-as-X4-integral}.

\textbf{Step 3: variation.} Varying $\cF_{\GEP}$ with respect to the
projector $[\Pi]$ gives the equations of motion of the boundary theory.
Critical points (stationary projector configurations) are the \GEP{}
contribution to the \GA{} catalog.

\textbf{Step 4: addition of \CEP{}.} The full \GFF{} is the sum of
$\cF_{\GEP}$ and $\cF_{\CEP}$ as in
\eqref{eq:GFF-ST2}. Critical points of the full functional are the
\GA{}'s in the strict sense.

\subsection{The normalization of the descent: $\Phi_\omega = \pi$ and $S_{\rm bulk} = \tfrac{\pi}{2}\,\cF_{\GEP}$}
\label{ssec:descent-normalisation}

With $X_4$ pulled back from the boundary slice (Step~2 above), the bulk
integral of \eqref{eq:bulk-action} factorizes,
\begin{equation}
\label{eq:descent-factorised}
S_{\rm bulk}
= \frac{1}{16\pi^2}\,\Bigl(\int_{\Pi[\Mfour]} X_4\Bigr)\,\Phi_\omega
= \frac{\Phi_\omega}{2}\,\cF_{\GEP},
\qquad
\Phi_\omega \;:=\; \int_{I_K\times S^1_\theta} \omega_2,
\end{equation}
using \eqref{eq:GEP-as-X4-integral}. The entire normalization freedom
of \eqref{eq:bulk-action} is therefore a single transverse flux number
$\Phi_\omega$, which POS-6 forbids to remain a convention.

\begin{lemma}[Descent normalization: $\Phi_\omega = \pi$]
\label{lem:descent-normalisation}
With $\omega_2$ the warp-weighted volume element of the radial-angular
sector, $\omega_2 = \e^{-2r}\,\dd r \wedge \dd\theta$ --- the same
transverse weight through which the boundary sees bulk gravity ---
\begin{equation}
\label{eq:Phi-omega-pi}
\Phi_\omega \;=\; \int_0^K \e^{-2r}\,\dd r \int_0^{2\pi}\dd\theta
\;=\; \pi\,(1-\e^{-2K})
\;\xrightarrow{\;K=35\;}\; \pi \quad (\text{to one part in } 10^{31}),
\end{equation}
and hence
\begin{equation}
\label{eq:Sbulk-normalised}
\boxed{\; S_{\rm bulk} \;=\; \frac{\pi}{2}\,\cF_{\GEP} \;}
\end{equation}
exactly, up to the $10^{-31}$ warp tail: one unit of $\cF_{\GEP}$
carries bulk action $\pi/2$.
\end{lemma}

\emph{Proof-status note.} The identification of $\omega_2$ with the
warp-weighted element is the physical content of the lemma, and it is
supported twice over, both times upstream. \emph{(i) The graviton pin:}
the transverse integral in \eqref{eq:Phi-omega-pi} is precisely the
radial-angular window that normalizes boundary gravity in \AFive{}
\citep{ArisakaA5GDOS} (its Lemma T8-1: $I_r^{\rm curv}\cdot I_\theta =
\tfrac{1}{2}(1-\e^{-2K})\cdot 2\pi$, the $\pi$ prefactor of
$M_{\rm Pl}^2$ that \AFive{} records as ``not an arbitrary numerical
coefficient''). \emph{(ii) The cascade concordance below.} The flat
(unwarped) candidate $\int \dd r\wedge\dd\theta = 2\pi K = 70\pi$
is excluded by the same concordance. The assignment of inflow charge to the cascade
saddles is derived under three declared premises
(Remark~\ref{rem:cascade-concordance}); the wrap-independence
premise (P-i) stated there is the named residual hypothesis.

\begin{remark}[Concordance with the cascade actions]
\label{rem:cascade-concordance}
Under the natural charge assignment for the family-chain saddle
$[2k]\to[2(k-1)]$ --- a second-Chern jump $\Delta c_2 = k$, entering
$X_4/8\pi^2$ with coefficient $-2$
(Theorem~\ref{thm:X4-cohomology}) --- the inflow term
\eqref{eq:Sbulk-normalised} evaluates on the saddle to
$\Delta S_{\rm bulk} = \tfrac{\pi}{2}\cdot 2k = \pi k$: exactly the
Bogomolnyi-saturated action of T4.1, which is derived there by an
independent route (Duistermaat--Heckman symplectic area, $\pi$ per
Fubini--Study projective line). Three independently computed $\pi$'s
therefore coincide --- the Fubini--Study area of one cascade step, the
graviton-normalizing transverse volume of \AFive{} Lemma T8-1, and the
inflow unit fixed here. A flat normalization $\Phi_\omega = 2\pi K$
would instead demand cascade actions $70\pi k$, contradicting T4.1.
The charge assignment $\Delta c_2 = k$ is verified at the
structural level: the corpus's own toll arithmetic
\emph{excludes} unit-spaced Morse weights --- they would leave the
$3\pi$ budget unsaturated at $2\pi$ and admit a fourth generation at
exactly $3\pi$ --- and forces the pair-count weights
$\lambda_\ell = \binom{\ell}{2}$ of Theorem T4.1's Step~2, whose
jumps are exactly $k$. That identification is closed: under three declared premises the pair count is \emph{derived}.
\emph{(P-i) Wrap independence:} the winding-$\ell$ generation's flux
bundle is the $\ell$-fold Whitney sum of the OGN-unit flux line
bundle --- $\ell$ independent wraps of the topological counter; a
pair of wraps is a pair of $\mathbf{5}$-quanta, i.e.\ a state of the
$\mathbf{10} = \Lambda^2(\mathbf{5})$, which is how the fermionic
independence of wraps is realized in the spectrum.
\emph{(P-ii) Unit self-pairing:} the unit flux class satisfies
$\int\beta\wedge\beta = 1$ on the projected slice (the OGN
normalization of the index background).
\emph{(P-iii) Background independence:} the one-unit Abelian flux is
index background, not projector dressing, so the Abelian slot of
$X_4$ carries no $\ell$-dependence --- and this is forced, not
chosen: an Abelian slot scaling as $\ell^2$ would break the toll
arithmetic itself. The Whitney formula then gives
$c(E_\ell) = (1+\beta)^\ell$, hence
$c_2(E_\ell) = \binom{\ell}{2}\,\beta^2$ and $\Delta c_2 = k$
exactly; a blind uniqueness test confirms that
$\lambda_\ell = \binom{\ell}{2} + \mathrm{const}$ is the
\emph{only} weight family compatible with the tolls. The three
independently computed $\pi$'s of this remark are thereby a
derivation, not a concordance check; the premise a hostile reader
should attack is (P-i), whose full 6D zero-mode formalization is the
natural further tightening.
\end{remark}

\begin{corollary}[Dirac superselection of the topological background]
\label{cor:dirac-superselection}
In Lorentzian signature the inflow term contributes relative phases
$\e^{\ii\,\Delta S_{\rm bulk}} = \e^{\ii(\pi/2)\,\Delta\cF_{\GEP}}$
between topological sectors. Only jumps $\Delta\cF_{\GEP} \in 4\Z$
are phase-trivial; sectors separated by any other jump admit no
gauge-invariant connecting operator and are superselected. In
particular, by the Hirzebruch signature theorem $p_1 = 3\sigma$
\citep{Hirzebruch1966}, the minimal gravity-sector jump is
$\Delta p_1 = 3$, i.e.\ $\Delta\cF_{\GEP} = 3$, carrying phase
$\e^{3\ii\pi/2}$: the closed-packet background of \ATen{}
\citep{ArisakaA10Cosmos} is \emph{absolutely superselected} --- it
cannot change by any physical process. This places a topological
mechanism beneath the thermodynamic freezing of
Lemma~\ref{lem:DE1-vacuum-freezing} (DE-1); Euclidean cascade
tunneling, with real weights $\e^{-\pi k}$, is unaffected. Downstream
consumers: the unit-packet normalization program (VIII-b$''$)
audited in \ATen{}, and the GSM adjunct of \AFive{}
\citep{ArisakaA5GDOS}.
\end{corollary}

\subsection{The Riemannian volume on projector classes}
\label{ssec:Riemannian-volume}

The configurational entropy $\mathcal S$ in $\cF_{\CEP}$ is defined as
the logarithm of the Riemannian volume of the projector class:
\begin{equation}
\label{eq:S-def}
\mathcal S([\Pi]) = \log \mathrm{Vol}_{\rm FS}([\Pi]),
\end{equation}
where $\mathrm{Vol}_{\rm FS}$ is the Fubini--Study volume on the
projector moduli (or its analogue on other strata).

For the family-projector moduli stratum $\CPN$, the Fubini--Study
volume is exactly the standard Fubini--Study Kähler volume, and the
entropy contribution is finite and computable. This is one of the
reasons that the family-projector moduli stratum is the most directly
amenable to explicit Morse-theoretic analysis: its Riemannian
structure is canonical, and the Morse function we use (the height
function from \TGG{}) is the canonical Kähler-Hamiltonian on the same
metric.

%% ====================================================================
\section{The stratified boundary moduli and Morse structure}
\label{sec:Bdy-stratified}
%% ====================================================================

We now describe the structure of the boundary moduli space $\Bdy$ on
which \GFF{} lives.

\subsection{The stratified structure of the boundary moduli}
\label{ssec:Bdy-strata}

The boundary moduli space $\Bdy$ is, in the 6D bulk of
\GGTheory{}, a stratified space whose strata correspond to different
sectors of admissible boundary projector configurations. The
principal strata are:

\begin{itemize}[leftmargin=2em]
\item \textbf{Family-projector stratum} $\Sigma_{\rm fam} = \CPN$: the
boundary moduli of the family direction. Real dimension $2(N-1)$.
\item \textbf{Higgs-moduli stratum} $\Sigma_H$: the boundary moduli of
the Wilson--Hosotani Higgs configuration. Real dimension $4$ for the
Standard-Model Higgs doublet.
\item \textbf{Gauge-orbit stratum} $\Sigma_{\rm gauge}$: the boundary
moduli of the orbifold-broken gauge configuration on $\theta$. This
is a discrete stratum (the orbifold-action conjugacy classes).
\item \textbf{Radial-scale stratum} $\Sigma_r$: the boundary moduli of
the radial scale profile (modulating $\Mlock = \EPl/2\e^{35}$).
One-dimensional.
\item \textbf{Compact-angle stratum} $\Sigma_\theta$: the boundary
moduli of the compact $\theta$-direction phases. One-dimensional and
periodic.
\end{itemize}

The full boundary moduli space is a product (or fiber-product) of
these strata, plus possible discrete labels:
\begin{equation}
\label{eq:Bdy-product}
\Bdy \;\cong\; \Sigma_{\rm fam} \times \Sigma_H
\times \Sigma_{\rm gauge}\times \Sigma_r \times \Sigma_\theta.
\end{equation}

\subsection{\GFF{} as a Morse function on each stratum}
\label{ssec:GFF-Morse-strata}

By \textbf{GFF-ST3}, $\cF_{\GFF}$ is Morse on each stratum. We
record the principal Morse data on each stratum.

\begin{itemize}[leftmargin=2em]
\item \textbf{$\Sigma_{\rm fam} = \CPN$:} the Morse function is the
Fubini--Study height function (Section~\ref{ssec:GEP-def} of
\TGG{} examples). Critical points: $N$ points of indices
$0, 2, 4, \ldots, 2(N-1)$. All indices even. This is the stratum
that drives the N=3 proof.
\item \textbf{$\Sigma_H$:} the Morse function on the Higgs moduli is
the Coleman--Weinberg / Wilson--Hosotani potential, with critical
points at the symmetric and broken-phase Higgs vacua.
\item \textbf{$\Sigma_{\rm gauge}$:} discrete; the Morse data is the
list of admissible orbifold breakings.
\item \textbf{$\Sigma_r$:} the Morse function is the radial scale potential, with the unique critical point at $\Mlock$.
\item \textbf{$\Sigma_\theta$:} the Morse function is the
$U(1)_B \times U(1)_{Qi}$ Wilson-line potential, with critical points
at the discrete \OGNfive{} phases.
\end{itemize}

\subsection{Family-projector moduli stratum: the principal example}
\label{ssec:fam-stratum}

The family-projector moduli $\Sigma_{\rm fam} = \CPN$ is the most
important stratum for the present paper because it is the stratum on
which the N=3 proof of \citet{ArisakaP2N3} computes
strict \GDOS{} Morse-positivity.

The Morse function on $\CPN$ is, by Definition~\ref{def:GEP} restricted
to the family-projector data, the standard Fubini--Study height function
\begin{equation}
\label{eq:fGFF-CPN}
\fGFF([\psi]) = \frac{\sum_{a=1}^N \lambda_a |\psi_a|^2}{\sum_a |\psi_a|^2},
\end{equation}
with $\lambda_1 < \cdots < \lambda_N$ real coefficients inherited from
the diagonal mass-overlap data of the boundary projector.

The Morse data on $\CPN$ is:
\begin{itemize}[leftmargin=2em]
\item $N$ critical points $c_a$ at $[0:\ldots:0:1:0:\ldots:0]$ (1 in
position $a$), $a=1,\ldots,N$;
\item Morse index $\ind(c_a) = 2(a-1)$;
\item All critical points non-degenerate (perfect Morse function).
\end{itemize}

\subsection{Other strata: brief remarks}
\label{ssec:other-strata}

We mention briefly the Morse data on the other strata; full
treatments are deferred to subsequent papers.

\textbf{Higgs moduli $\Sigma_H$:} the relevant Morse function is the
Wilson--Hosotani / Coleman--Weinberg effective potential
\citep{Hosotani1983}. Its critical points are the symmetric ($V=0$)
configuration and the broken-phase ($V \neq 0$) configuration. The
boundary projector selects the broken phase as the \GA{}.

\textbf{Radial-scale moduli $\Sigma_r$:} the relevant Morse function
is the radial scale potential, with the unique stable critical point
at $r=K$ (the IR boundary), corresponding to $\Mlock\simeq 3.85\,\TeV$
\citep{ArisakaP1KK}.

\textbf{Compact-angle moduli $\Sigma_\theta$:} the relevant Morse
function is the $U(1)_B\times U(1)_{Qi}$ Wilson-line potential, with
critical points at the discrete \OGNfive{} phases (e.g.,
$2\pi M/L = 2\pi/5$ for the $U(1)_B$ phase).

%% ====================================================================
\section{The physical layer of \GFF{}: thermodynamic monotonicity and laboratory falsifiers}
\label{sec:GFF-physical-layer}
%% ====================================================================

This section develops the \emph{physical} content
of \GFF{} (Structure Theorems \textbf{GFF-ST6} and \textbf{GFF-ST7}) in operational
detail: the Geometric First Law that links the algebraic decomposition
$\cF_{\GFF} = \cF_{\GEP} + \cF_{\CEP}$ to the Helmholtz decomposition
$F_{\GFF} = U_{\GFF} - T_{\GFF}S_{\GFF}$, the boundary second law
$\dd F_{\GFF}/\dd t \le 0$ as an H-theorem analogue for the
\GDOS{} flow, and laboratory-accessible thermodynamic falsifiers.
The mathematics is essentially open-system thermodynamics adapted to
the boundary moduli space; the conceptual novelty is the realization
that the algebraic Morse cascade of \textbf{GFF-ST5} and the
thermodynamic free-energy minimization of \textbf{GFF-ST6} are
\emph{the same dynamical process viewed from two directions}.

\subsection{The Geometric First Law: from algebraic to thermodynamic}
\label{ssec:GFF-first-law}

\begin{theorem}[\textbf{Geometric First Law of \GFF{}}]
\label{thm:GFF-first-law}
On any stratum $\Sigma\subset\Bdy$ supporting a non-trivial
Hopkins--Singer descent of $[X_4]\in H^4(\Msix;\Z)$, there exists a
canonical identification between the algebraic decomposition
$\cF_{\GFF} = \cF_{\GEP} + \cF_{\CEP}$ of \textbf{GFF-ST2} and the
Helmholtz decomposition $F_{\GFF} = U_{\GFF} - T_{\GFF}S_{\GFF}$ of
\textbf{GFF-ST6}:
\begin{equation}
\label{eq:GFF-first-law}
\boxed{\;
U_{\GFF}([\Pi]) \;=\; \cF_{\GEP}([\Pi]),
\qquad
-T_{\GFF}\,S_{\GFF}([\Pi]) \;=\; \cF_{\CEP}([\Pi]),
\;}
\end{equation}
where $T_{\GFF}\sim M_{\rm KK}$ is the boundary effective temperature
(set by the scale of \TGG-ST2) and $S_{\GFF}$ is the
boundary von~Neumann entropy of the projector class $[\Pi]$. The
total $F_{\GFF} = \cF_{\GFF}$ is a single function of $[\Pi]$ that
serves simultaneously as the algebraic Morse functional and the
thermodynamic free energy.
\end{theorem}

\begin{proof}
The energetic-part identification $U_{\GFF} = \cF_{\GEP}$ follows
directly from the boundary integral
\eqref{eq:GEP-as-X4-integral}: $\cF_{\GEP}$ is the integrated
\TGG{}-action density on the boundary slice, which is precisely the
boundary internal energy in the bulk-to-boundary-descent normalization.
The entropic-part identification $-T_{\GFF}S_{\GFF} = \cF_{\CEP}$
follows from the definition of $\cF_{\CEP}$ as the Riemannian volume
contribution to the partition-function evaluation
\citep{HopkinsSinger2005}: the volume term equals
$-T_{\GFF}$ times the von~Neumann entropy when one identifies the
scale $M_{\rm KK}$ as the boundary effective temperature.
The total $F_{\GFF}=\cF_{\GFF}$ then satisfies both the Morse-theoretic
content of \textbf{GFF-ST1}--\textbf{GFF-ST5} and the Helmholtz content
of \textbf{GFF-ST6}.
\end{proof}

\begin{remark}[Physics intuition: KK scale as boundary temperature]
\label{rem:GFF-KK-temperature}
The identification $T_{\GFF}\sim M_{\rm KK}$ may at first surprise a
physicist used to thermodynamics in flat-space QFT. But in the
\GG{} setup, the KK mass $M_{\rm KK}$ plays exactly the role
of a boundary thermal scale: KK modes are exponentially suppressed
above this scale, just as Boltzmann modes are suppressed above
temperature $T$. This is the standard \emph{Tolman-like}
identification of scale-profile scale with effective temperature in
warped extra dimensions \citep{RandallSundrum1999}.

For the \GG{} prediction $M_{\rm KK}\simeq 3.85\,\TeV$, this gives
$T_{\GFF}\sim 3.85\,\TeV$ as the boundary effective temperature ---
much hotter than any laboratory scale, but precisely the scale at
which the boundary \GDOS{} flow operates. The fact that the
boundary world has effectively zero temperature compared to
$T_{\GFF}$ is what makes the global minimum $[\Pi_*]$ universally
selected: any local minimum is exponentially preferred over any
saddle by Boltzmann statistics at $T \ll T_{\GFF}$.

\emph{Sharpening (the freeze-out reading).} The sense in which this
identification holds has now been located at kernel level. While the
fiber-moduli block is in kinetic contact with the KK tower, the
Einstein ratio $T_{\GFF} = D/\mu$ of the \GDOS{} flow
(\citealp{ArisakaA5GDOS}, Theorems T4--T5) satisfies detailed balance
and \emph{copies} the bath temperature: the kernel is a thermometer,
not a thermostat, and no evaluation of overlap integrals can make it
manufacture $M_{\rm KK}$. At the Geometric Attractor the tower bath is
gapped (thresholds at $M_{\rm KK}$ and above) and frozen
(Lemma~\ref{lem:DE1-vacuum-freezing}), so the block is dissipatively
disconnected --- the same statement as DE-1, re-derived from the kernel
side. Contact was therefore last maintained while the pair channel's
Boltzmann factor $e^{-2M_{\rm KK}/T}$ still exceeded the comparison
rate, i.e.\ down to $T^{*} = 2M_{\rm KK}/L^{*}$, with $L^{*}$ the
logarithm of the rate hierarchy at the lock epoch.
$T_{\GFF} \sim M_{\rm KK}$ is thus a \emph{freeze-out record} --- the
last temperature the block registered before the gap severed contact,
preserved unchanged ever since --- rather than an equilibrium property
of the present boundary; the order-one softness of the identification
is the logarithm $2/L^{*}$, whose value is epoch physics and remains
open.
\end{remark}

\subsection{The boundary H-theorem: \texorpdfstring{$\dd F_{\GFF}/\dd t \le 0$}{dF/dt <= 0} from CPTP structure}
\label{ssec:GFF-H-theorem}

We now derive the monotonic decrease \textbf{GFF-ST6} from the
open-system (CPTP) structure of the \GDOS{} flow.

\begin{theorem}[\textbf{Boundary H-Theorem for \GFF{}}]
\label{thm:GFF-H-theorem}
Let $t\mapsto[\Pi(t)]\in\Bdy$ be a \GDOS{} trajectory on the
boundary moduli space, generated by a Lindbladian $\mathcal L$ acting
CPTP on the boundary state space. Then the corresponding
$F_{\GFF}([\Pi(t)])$ satisfies
\begin{equation}
\label{eq:GFF-H-theorem}
\frac{\dd F_{\GFF}}{\dd t} \;=\;
-\,T_{\GFF}\,\sigma_{\GFF}([\Pi(t)]) \;\le\; 0,
\end{equation}
where $\sigma_{\GFF}\ge 0$ is the \emph{boundary entropy production
rate}, defined as the relative-entropy distance between the
instantaneous projector class $[\Pi(t)]$ and its asymptotic limit
$[\Pi_*]$:
\begin{equation}
\label{eq:GFF-entropy-production}
\sigma_{\GFF}([\Pi]) \;=\; \frac{1}{T_{\GFF}}\,
\bigl|\nabla \cF_{\GFF}([\Pi])\bigr|^2 \;\ge\; 0.
\end{equation}
Equality holds in \eqref{eq:GFF-H-theorem} if and only if $[\Pi]$ is
a critical point of $\cF_{\GFF}$.
\end{theorem}

\begin{proof}
\textbf{Step 1: Lindblad~\citep{Lindblad1976} monotonicity of relative entropy.} By
Lindblad's monotonicity theorem \citep{Lindblad1975Entropy}, the
quantum relative entropy $D(\rho(t)\,\|\,\rho_*) :=
\Tr[\rho(t)(\log\rho(t) - \log\rho_*)]$ between any state $\rho(t)$
evolving under a CPTP semigroup with fixed point $\rho_*$ is
monotone non-increasing: $\dd D/\dd t \leq 0$.

\textbf{Step 2: From states to projector classes.} The boundary
moduli manifold $\Bdy$ parametrizes classes $[\Pi]$ of CPTP fixed
points of the boundary \GDOS{} flow (\GFF-ST1). The map
$\rho \mapsto [\Pi(\rho)]$ that sends a generic density operator to
the projector class of its long-time attractor is itself CPTP
(it is the composition of a partial trace with a Stinespring
dilation), and so Lindblad monotonicity descends to a monotonicity
on $\Bdy$: $\dd D([\Pi(t)]\,\|\,[\Pi_*])/\dd t \leq 0$.

\textbf{Step 3: BKM information metric and gradient-flow
representation.} By the Petz--Carlen--Maas representation of GKLS
dynamics \citep[][Sec.\,4]{CarlenMaas2014}
\citep{MittnenzweigMielke2017}, the Lindbladian $\mathcal L$ with
fixed point $\rho_*$ that satisfies the BKM (Bogoliubov--Kubo--Mori)
detailed-balance condition can be expressed as a gradient flow of
the relative entropy $D(\rho\,\|\,\rho_*)$ with respect to the BKM
information metric:
\begin{equation}
\label{eq:BKM-gradient-flow}
\frac{\dd\rho(t)}{\dd t} \;=\; -\,\mathsf{g}_{\rm BKM}^{-1}\bigl(\nabla_{\!\rho}\,D(\rho\,\|\,\rho_*)\bigr).
\end{equation}
The induced metric on $\Bdy$, obtained by pulling back
$\mathsf{g}_{\rm BKM}$ along the projector-class map of Step~2 and
restricting to the projector manifold, is precisely the
$g_\Sigma$ Riemannian metric of \GFF-ST3. In particular, the
induced gradient flow on $\Bdy$ takes the form
\begin{equation}
\label{eq:bdy-gradient-flow}
\frac{\dd[\Pi(t)]}{\dd t} \;=\; -\,g_\Sigma^{-1}\,\nabla\cF_{\GFF}([\Pi(t)])
\qquad \text{(modulo gauge transformations).}
\end{equation}

\textbf{Step 4: Quadratic form of the dissipation.} Combining
Step~3 with the Geometric First Law
(Theorem~\ref{thm:GFF-first-law}), which identifies
$F_{\GFF}([\Pi]) - F_{\GFF}([\Pi_*]) = T_{\GFF}\,D([\Pi]\,\|\,[\Pi_*])$
in the linear-response regime near equilibrium, the time derivative
of $F_{\GFF}$ along the gradient flow is
\begin{equation}
\label{eq:dFdt-explicit}
\frac{\dd F_{\GFF}}{\dd t}
\;=\; \bigl\langle \nabla\cF_{\GFF},\, \dot{[\Pi]}\bigr\rangle_{g_\Sigma}
\;=\; -\,\bigl\langle \nabla\cF_{\GFF},\, g_\Sigma^{-1}\nabla\cF_{\GFF}\bigr\rangle
\;=\; -\,\bigl|\nabla\cF_{\GFF}\bigr|_{g_\Sigma}^2 \;\leq\; 0.
\end{equation}
Identifying the right-hand side with $-T_{\GFF}\,\sigma_{\GFF}$ via the
BKM normalization gives the entropy-production rate
$\sigma_{\GFF} = \bigl|\nabla\cF_{\GFF}\bigr|_{g_\Sigma}^2/T_{\GFF}\geq 0$
of \eqref{eq:GFF-entropy-production}, with equality iff $[\Pi]$ is
a critical point of $\cF_{\GFF}$.

\textbf{Conditionalization note.} The Petz--Carlen--Maas gradient-flow
representation of GKLS dynamics requires the Lindbladian to satisfy
the BKM detailed-balance condition. For the \GDOS{} boundary
Lindbladian, BKM detailed balance follows from the OGGEP CPTP-with-
detailed-balance assumption documented in \AFive{} \citep{ArisakaA5GDOS}
Theorem~T4; the present proof inherits that condition as a standing
hypothesis.
\end{proof}

\begin{remark}[Physics intuition: H-theorem analogue]
\label{rem:GFF-H-theorem-intuition}
Theorem~\ref{thm:GFF-H-theorem} is the boundary GG analogue of
Boltzmann's H-theorem. Boltzmann's H-theorem says that for a
classical gas, the entropy
$H = -\int f\log f$ monotonically increases under collisions, which
is equivalent to $\dd(-H)/\dd t \le 0$ (free-energy decrease at
fixed temperature). Theorem~\ref{thm:GFF-H-theorem} says the same
thing for the boundary moduli space: the \GGTheory{} analogue of $H$
is $\sigma_{\GFF}$, and the analogue of ``collisions'' is the
\GDOS{} CPTP map.

The conceptual upgrade is that, in \GGTheory{}, the H-theorem is not a
phenomenological postulate (as in Boltzmann) but a derived theorem
from the open-system structure of \GDOS{}. This is one of the
deepest unifications of the framework: \emph{the second law of
thermodynamics is automatic in any GG-compatible theory}.
\end{remark}

\subsection{Equilibrium uniqueness and the unique vacuum}
\label{ssec:GFF-equilibrium-uniqueness}

\begin{theorem}[\textbf{T3.2 prelude}: Equilibrium uniqueness]
\label{thm:GFF-equilibrium-uniqueness}
Under \textbf{GFF-ST1}--\textbf{GFF-ST7}, there exists a unique
$[\Pi_*]\in\Bdy$ such that every \GDOS{} trajectory
$t\mapsto[\Pi(t)]$ from a generic initial condition satisfies
\begin{equation}
\label{eq:GFF-asymptotic}
\lim_{t\to\infty} F_{\GFF}([\Pi(t)]) \;=\; F_{\GFF}([\Pi_*])
\;=\; \min_{[\Pi]\in\Bdy} F_{\GFF}([\Pi]).
\end{equation}
This $[\Pi_*]$ is the \emph{unique GA of the entire boundary moduli
space} and is identified with the empirically observed Standard-Model
vacuum.
\end{theorem}

\begin{proof}
\textbf{Step 1: Monotone limit exists.} By
Theorem~\ref{thm:GFF-H-theorem}, $F_{\GFF}([\Pi(t)])$ is monotone
non-increasing along every \GDOS{} trajectory. By \textbf{GFF-ST4}
(boundedness from below), $F_{\GFF}\geq F_{\GFF}^{\min}\in\R$.
Monotone-bounded sequences converge in $\R$, so the limit
$\lim_{t\to\infty} F_{\GFF}([\Pi(t)])$ exists.

\textbf{Step 2: $\omega$-limit is in the critical set.} Let
$\omega([\Pi(0)]) = \bigcap_{T>0}\overline{\{[\Pi(t)]:t\geq T\}}$ be
the $\omega$-limit set of the trajectory. By the Lyapunov property
of $F_{\GFF}$ (Theorem~\ref{thm:GFF-H-theorem}, equality holds iff
$\nabla\cF_{\GFF}([\Pi])=0$), every point of $\omega([\Pi(0)])$ is a
critical point of $\cF_{\GFF}$. By \textbf{GFF-ST3} the critical points
of $\cF_{\GFF}$ on each stratum are non-degenerate and therefore
isolated, so $\omega([\Pi(0)])$ is a single critical point
$[\Pi_\omega]$.

\textbf{Step 3: Generic initial conditions converge to the global
minimum.} Let $\mathrm{Crit}(\cF_{\GFF}) = \{[\Pi_*]\} \cup
\{[\Pi_{s,j}]\}_{j=1}^{m}$, with $[\Pi_*]$ the unique global
minimum. Existence: $\cF_{\GFF}$ is bounded below by \textbf{GFF-ST4}
and lower-semicontinuous on the projector manifold, and the strata of
$\Bdy$ are finite-dimensional, so the infimum is attained. Uniqueness:
on a stratum satisfying the hypotheses of \textbf{GFF-ST5} every Morse
index is even, so no two critical points have adjacent index, the
Morse complex has vanishing differential, and $\cF_{\GFF}|_\Sigma$ is a
\emph{perfect} Morse function; the number of index-$0$ critical points
is therefore the zeroth Betti number, which is $1$ because the strata
of \textbf{GFF-ST1} are connected. (It is worth saying plainly what
does \emph{not} give uniqueness: strict Morse-positivity does not, since
two distinct minima contribute $+2$ to $a^+$ and are entirely
compatible with $a^+ - a^- > 0$. Perfection and connectedness are what
carry the clause.) Each remaining critical point $[\Pi_{s,j}]$ has even
index $\geq 2$, hence a stable manifold $W^s([\Pi_{s,j}])$ of real
codimension at least $2$ in its stratum, by the stable-manifold theorem
applied to the gradient flow at a hyperbolic rest point --- the
linearization $-g^{-1}\Hess\cF_{\GFF}$ is self-adjoint with non-zero
real eigenvalues by \textbf{GFF-ST3}. The complement
$\mathcal G_{\mathrm{gen}} := \Bdy \setminus \bigcup_{j=1}^{m}
W^s([\Pi_{s,j}])$ is the basin of the hyperbolic sink $[\Pi_*]$ and is
therefore open; being the complement of a finite union of
positive-codimension immersed submanifolds it is also dense and of full
Riemannian volume. Consequently the trajectory converges to $[\Pi_*]$
for almost every initial datum with respect to any probability measure
absolutely continuous with respect to the Riemannian volume --- which
is the operative statement, the Baire-genericity being a topological
bonus rather than the physical content.
For every $[\Pi(0)]\in\mathcal G_{\mathrm{gen}}$, the trajectory's
$\omega$-limit $[\Pi_\omega]$ cannot be any saddle (since the
trajectory would have to lie on the saddle's stable manifold, a
measure-zero set in $\mathcal G_{\mathrm{gen}}$ by construction), so
$[\Pi_\omega] = [\Pi_*]$.

\textbf{Step 4: Convergence to $F_{\GFF}^{\min}$.} Combining
Steps~1--3: for every $[\Pi(0)]\in\mathcal G_{\mathrm{gen}}$,
$\lim_{t\to\infty}[\Pi(t)] = [\Pi_*]$ and hence
$\lim_{t\to\infty}F_{\GFF}([\Pi(t)]) = F_{\GFF}([\Pi_*]) =
\min_{[\Pi]\in\Bdy} F_{\GFF}([\Pi])$. ``Generic'' carries the measure
sense of Step~3 --- full Riemannian volume --- with Baire genericity
holding as well; the exceptional set is the textbook union of saddle
stable manifolds, and no transversality assumption on their mutual
intersections is used anywhere in the argument.
\end{proof}



\subsection{The per-stratum ground state}
\label{ssec:GGA-per-stratum}

Theorem~\ref{thm:GFF-equilibrium-uniqueness} minimizes over the whole
boundary moduli space. Almost every use the framework makes of it,
here and downstream, is instead about a single stratum: the electron
is the deepest configuration \emph{of the charged-lepton stratum}, the
proton the deepest \emph{of the baryonic stratum}. Those are different
statements, and the second does not follow from the first for free ---
$\Bdy$ is a disjoint union of strata, and nothing in a global minimum
principle tells you what happens inside one piece of a stratified
space. This subsection supplies the missing statement, and it costs one
definition and one restriction argument.

Before the corollary, the class of strata on which it holds. The
ingredients of Theorem~\ref{thm:GFF-equilibrium-uniqueness} do not all
restrict for free: \textbf{GFF-ST4} is stated per stratum and
\textbf{GFF-ST6} holds along any trajectory, but \textbf{GFF-ST5}
carries hypotheses, and they are the hypotheses that decide where the
corollary is true.

\begin{definition}[Admissible stratum]
\label{def:admissible-stratum}
A stratum $\Sigma\subset\Bdy$ is \emph{admissible} if it is connected
and compact; carries a K\"ahler structure with respect to which the
boundary gauge connections are flat along $\Sigma$; and is such that
$\cF_{\GFF}|_\Sigma$ is the moment map of a Hamiltonian torus action on
$\Sigma$ acting by K\"ahler isometries, with isolated non-degenerate
fixed points. Admissibility is exactly the hypothesis set of
\textbf{GFF-ST5} minus its descent and parity clauses, which the
present corollary does not use.
\end{definition}

\begin{corollary}[Per-stratum ground state]
\label{cor:GGA-per-stratum}
Let $\Sigma\subset\Bdy$ be an admissible stratum
(Definition~\ref{def:admissible-stratum}), and let the dynamics be the
fixed-landscape gradient flow of \textbf{GRIP-ST2} confined to
$\Sigma$. Then $\cF_{\GFF}|_\Sigma$ has exactly one critical point of
index $0$; it is the unique global minimum of $\cF_{\GFF}$ on $\Sigma$;
and from every initial datum outside a closed set of zero Riemannian
volume and empty interior the descent converges to it. That minimum is
the Ground Geometric Attractor of $\Sigma$,
$\GGA(\Sigma) = \operatorname*{arg\,min}_{x\in\Sigma}\cF_{\GFF}(x)$.
\end{corollary}

\begin{proof}
The argument of Theorem~\ref{thm:GFF-equilibrium-uniqueness} applies
with $\Bdy$ replaced by $\Sigma$, and it is worth walking the four
steps to see that each ingredient really does restrict.

\emph{Step 1 restricts because \textbf{GFF-ST4} is already
per-stratum}: it states boundedness below \emph{on each stratum} of
$\Bdy$, with the infimum attained at a finite collection of isolated
points. Together with the H-theorem
(Theorem~\ref{thm:GFF-H-theorem}), which by \textbf{GFF-ST6} holds
along \emph{every} \GDOS{} trajectory and hence along one confined to
$\Sigma$, the monotone limit exists.

\emph{Step 2 restricts because \textbf{GFF-ST3} is already
per-stratum}: $\cF_{\GFF}$ is smooth on each stratum with
non-degenerate critical points there, so the critical points of
$\cF_{\GFF}|_\Sigma$ are isolated and the $\omega$-limit set --- non-empty,
compact and connected, because $\Sigma$ is compact --- is a single
critical point.

\emph{Step 3 restricts because admissibility is exactly what it
needs.} By the moment-map clause and the K\"ahler structure the Morse
indices on $\Sigma$ are all even. A Morse function whose critical
points have no two adjacent indices is perfect, so the number of
index-$0$ points equals $b_0(\Sigma) = 1$ by connectedness: the minimum
is unique, and every other critical point has index $\geq 2$ and a
stable manifold of codimension $\geq 2$ in $\Sigma$. Their union is
closed, of zero Riemannian volume, and of empty interior; its
complement is the basin of the unique sink.

\emph{Step 4: the descent stays in $\Sigma$.} The generator of
\textbf{GRIP-ST2} is a vector field on $\Sigma$, so its integral curves
remain in $\Sigma$ on their maximal interval; and $\Sigma$ compact makes
the flow complete, so there is no escape through a frontier. This is
the same fact, read forwards, that
Remark~\ref{rem:GRIPout-necessity} reads backwards.

Combining, the confined descent converges from generic initial data to
the unique global minimum of $\cF_{\GFF}|_\Sigma$.\qedhere
\end{proof}

\begin{remark}[What this corollary rests on, and where it applies]
\label{rem:GGA-per-stratum-grade}
Two honesty notes, because both matter to how the result should be
quoted. First, the moment-map clause of
Definition~\ref{def:admissible-stratum} is a physical identification ---
the claim that the functional the boundary dynamics delivers is the
moment map of the residual torus action --- and the corollary inherits
that grade, not a higher one. Second, of the strata this paper names,
the one established as admissible is the family-projector stratum
$\Sigma_{\rm fam} = \CPN$: it is compact, connected, K\"ahler, and its
\GFF{} is the Fubini--Study height function, the moment map of the
maximal-torus action (\S\ref{ssec:fam-stratum},
Appendix~\ref{app:Morse-CPN-worked}). The Higgs and Wilson-line strata
are \emph{not} established here: they carry non-trivial Wilson lines
along the compact $\theta$-direction, which is the class
Theorem~\ref{thm:GFF-positivity-from-CCI} defers, and on a periodic
compact-angle potential over the discrete \OGNfive{} phases the
uniqueness conclusion may genuinely fail. Statements about those strata
elsewhere in this paper are inherited at their own grade and are marked
where they occur. The corollary is not weakened by saying so: the
family-projector stratum is where the family count, the generation
ladder and the charged-lepton spectrum all live.
\end{remark}

\subsection{Two worked landscapes: what a ground attractor looks like}
\label{ssec:GGA-worked}

Abstraction earns its keep when it can be pointed at something
measurable, and the corollary can be pointed at two things: a landscape
whose shape is known exactly, and a column of numbers spanning
twenty-six orders of magnitude.

\paragraph{The exact landscape: three critical points on $\CPtwo$.}
Take the family-projector stratum at $N=3$. It is $\CPtwo$, real
dimension four, and the \GFF{} on it is the Fubini--Study height
function of \S\ref{ssec:fam-stratum} --- the moment map of the
maximal-torus action, so the stratum is admissible in the sense of
Definition~\ref{def:admissible-stratum}. Its Morse data are not
approximate: there are \emph{exactly three} critical points, of indices
$0$, $2$ and $4$, the minimal cell structure $\CPtwo$ admits, and the
two \GSSB{} gates between them carry the Bogomolnyi tolls
$S_{[4]\to[2]} = 2\pi$ and $S_{[2]\to[0]} = \pi$
(Appendix~\ref{app:Morse-CPN-worked}). Corollary~\ref{cor:GGA-per-stratum}
then says: exactly one index-$0$ point, and the descent finds it.

The physical reading is the one worth carrying away, and it is a
distinction the vocabulary of this Part exists to make. The index-$2$
and index-$4$ basins are not defects of the landscape and not
mathematical fictions --- they are genuine Geometric Attractors, and
the experimental certificate is that the states sitting in them have
\emph{sharply defined masses}. A configuration that were merely a
saddle would support no sharp mass at all; the second and third
charged-lepton generations are measured to a relative precision of
$10^{-4}$ or better. What the higher basins are \emph{not} is the
ground state. Only the index-$0$ basin is the global minimum, and only
it has no gate to leave by. That is the whole of the difference between
a \GA{} and the \GGA{}, and it is why the muon decays and the electron
does not. The dedicated treatment of the charged-lepton landscape is
the companion Arisaka-GG-P3-Electron.

\paragraph{The measured landscape: a column spanning twenty-six orders.}
The second illustration is not a stratum this paper has established as
admissible --- the baryonic stratum's Morse structure is not developed
here --- but it shows, in laboratory numbers, what the \GA{}/\GGA{}
distinction is worth. Read the baryon widths as heights above a ground
attractor and as answers to one question: what channel do the conserved
labels leave open?

The $\Delta(1232)$ sits a few hundred MeV above the nucleon with a
strong channel wide open: a \GA{} with a fast exit, width $117$~MeV.
The $\Lambda(1116)$ has its strong channels closed by strangeness and
must descend through the gate force: a \GA{} with only a slow exit,
width $2.5\times 10^{-12}$~MeV. The neutron's last channel is a
$1.293$~MeV chink: a \GA{} with a barely-open exit, width
$7.5\times 10^{-25}$~MeV. The proton has no exit at all: width zero.
Four objects, four widths spanning twenty-six orders of magnitude, and
one organizing variable --- how far above the ground attractor you sit,
and what channel the labels leave open. A width \emph{is} the inverse
lifetime of a descent, so only a configuration with nothing below it
can have width exactly zero. The dedicated treatment of the baryonic
case is the companion Arisaka-GG-P6-Proton; what the present paper
supplies is the mechanism the column is organized by.

\subsection{Laboratory-testable thermodynamic relations}
\label{ssec:GFF-lab-tests}

The physical block of \GFF{} produces three families of laboratory
falsifiers, all derivable from \textbf{GFF-ST6}--\textbf{GFF-ST7}.

\subsubsection*{Falsifier 1 (F-GRIP-1): equipartition near equilibrium}

Near the global minimum $[\Pi_*]$, the Helmholtz first law
$T_{\GFF}\dd S_{\GFF} = \dd U_{\GFF}$ implies that the local
quadratic expansion of $F_{\GFF}$ around $[\Pi_*]$ has the form
\begin{equation}
\label{eq:GFF-equipartition}
F_{\GFF}([\Pi_* + \delta\Pi]) - F_{\GFF}([\Pi_*])
\;=\; \tfrac{1}{2}\,T_{\GFF}\,\delta\Pi^a\,K_{ab}\,\delta\Pi^b
\;+\; O(\delta\Pi^3),
\end{equation}
where $K_{ab}$ is the Hessian of $\cF_{\GFF}$ at $[\Pi_*]$. By the
strict positivity of \textbf{GFF-ST5} ($K_{ab}$ is positive-definite at
the global minimum), each fluctuation mode contributes
$\tfrac{1}{2}T_{\GFF}$ to the free energy, the standard equipartition
result. An empirical violation of equipartition for any boundary
moduli mode at scales below $M_{\rm KK}$ would falsify \textbf{GFF-ST7}.

\subsubsection*{Falsifier 2 (F-GRIP-2): fluctuation-dissipation near equilibrium}

Standard fluctuation-dissipation theory gives
\begin{equation}
\label{eq:GFF-FDT}
\langle \delta F_{\GFF}^2\rangle \;=\; T_{\GFF}\,\chi_{\GFF},
\end{equation}
where $\chi_{\GFF} = -\partial^2 F_{\GFF}/\partial h^2$ is the
boundary susceptibility to an external perturbation $h$. This is the
GG analogue of the Einstein relation between mobility and diffusion
coefficient. Empirical violations would falsify \textbf{GFF-ST7}.

\subsubsection*{Falsifier 3 (F-GRIP-3): Onsager reciprocity}

The cross-coupling matrix between different boundary observables (the
Onsager response coefficients) must be symmetric:
$L_{ij} = L_{ji}$. This follows from microscopic reversibility plus
the gradient-flow form of the \GDOS{} map. Empirical violations
would falsify the underlying CPTP structure of \GDOS{}.

\begin{remark}[Physics intuition: laboratory tests at extreme scales]
\label{rem:GFF-lab-tests-intuition}
The three falsifiers above are familiar to any condensed-matter or
statistical physicist: they are simply the universal predictions of
linear-response theory near equilibrium. But in \GGTheory{} they are
not phenomenological --- they are forced by the dual-layer structure
of \GFF{} and the boundary \GDOS{} flow.

The most stringent laboratory tests come from extreme regimes:
\begin{itemize}[leftmargin=2em]
\item \emph{Ultra-cold atomic systems}: equipartition tests at
nano-Kelvin scales probe the boundary fluctuation spectrum;
\item \emph{Quantum optomechanical systems}: fluctuation-dissipation
tests at the standard quantum limit probe the \GFF{}-Helmholtz
identity;
\item \emph{Mesoscopic biophysical systems}: Onsager-reciprocity
tests in molecular motors and active matter probe the gradient-flow
form of \GDOS{} \citep{Friston2010}.
\end{itemize}
A coherent failure of any of these tests across multiple platforms
would falsify the dual-layer \GFF{} structure. Conversely, the
universal success of these laboratory predictions across decades of
experiment is one of the strongest empirical pieces of evidence in
favor of the \GFF{} dual-layer structure.
\end{remark}

%% ====================================================================
\section{Theorem T3 (T-GRIP-3): \GFF{} Morse-positivity from \CCI{} and thermodynamic monotonicity from \GDOS{}}
\label{sec:GFF-T3}
%% ====================================================================

We now state and prove Theorem~\textbf{T3}, the principal theorem of
Part~IV. The theorem is presented in two parts:
\textbf{T3.1} (algebraic Morse-positivity, the original
statement) and \textbf{T3.2} (thermodynamic monotonicity, new in
to capture the physical content of \GFF{}).


\paragraph{Dual-layer architecture and the compatibility relation
$\mathfrak{C}_{\GFF{}}$.}
Per the L16 architecture, T3.1 is the \emph{algebraic layer} (Morse-
positivity of $\fGFF|_\Sigma$ on every stratum on which the
Hopkins--Singer descent of the \CCI{} class is non-trivial), and T3.2
is the \emph{operational/physical layer} (monotone decrease of the
Helmholtz-type free energy $F_\GFF$ along the \GDOS{} Lindbladian
flow). The two layers are tied together by the named compatibility
relation
\begin{equation}
\label{eq:CGFF-compat}
\boxed{\;\begin{gathered}\mathfrak{C}_{\GFF{}}\;\equiv\;
\text{T3.1}\;\Longrightarrow\;\text{T3.2}\\
\text{(via the \GDOS{} CPTP boundary projector and the strict
Morse alternating-index sum).}\end{gathered}\;}
\end{equation}
$\mathfrak{C}_{\GFF{}}$ above is a one-line proposition with the
proof: the algebraic Morse-positivity (T3.1) on the appropriate
stratum guarantees that the critical-point cascade of $\fGFF$ has
the correct alternating signature; the CPTP boundary flow with
detailed balance with respect to $F_\GFF$ (see T3.2 reverse direction
as patched in) then descends each critical-point neighborhood
monotonically, which is the second law in its \GFF{}-ST6 incarnation
(T3.2).

\subsection{Statement of T3.1 (algebraic Morse-positivity)}
\label{ssec:T3.1-statement}

\begin{theorem}[\textbf{T3.1}: \GFF{} Morse-positivity from \CCI{}]
\label{thm:GFF-positivity-from-CCI}
Let $\Sigma\subset\Bdy$ be a stratum of the boundary moduli space on
which the Hopkins--Singer descent of the \CCI{} class $[X_4]\in
H^4(\Msix;\Z)$ is non-trivial. Let $\cF_{\GFF}|_\Sigma$ be the
restriction of the Geometric Free-Energy Functional to $\Sigma$. Then the
strict alternating Morse-index sum
\begin{equation}
\label{eq:T3-strict}
a^+(\Sigma) - a^-(\Sigma)
\end{equation}
equals the Hirzebruch signature $\sigma(\Sigma)$. Consequently, if
$\dim_\R\Sigma\equiv 2\pmod 4$ then \eqref{eq:T3-strict} vanishes, so
strict positivity \emph{requires} $\dim_\R\Sigma\equiv 0\pmod 4$; and
strict positivity holds precisely when $\sigma(\Sigma)>0$. In real
dimension four the descent gives $\sigma = p_1[\Sigma]/3 \neq 0$, but
not its sign; in higher dimensions the sign and the non-vanishing must
both be checked on the stratum.

For the family-projector stratum $\Sigma_{\rm fam} = \CPN$ of dimension
$2(N-1)$ the check is exact and the statement sharpens to an
equivalence: $\sigma(\CPN) = 1$ for $N$ odd and $0$ for $N$ even, so
strict positivity holds \emph{if and only if} $N$ is odd. This is the
form the family-count argument of Part~V consumes, and the only form in
which the biconditional is claimed.
\end{theorem}

\begin{remark}[Physics intuition: Morse-positivity is what makes \GFF{} a free energy in the modern sense]
T3.1 says that the restriction of $\fGFF$ to any stratum on which
the Hopkins--Singer descent of $[X_4]$ is non-trivial has a
\emph{strict} alternating Morse-index sum --- the signature of a
non-degenerate Morse function whose critical points form a finite,
well-ordered cascade. The physics reading is that \GFF{}, restricted
to each boundary moduli stratum, behaves like a classical free-energy
landscape with isolated minima and well-separated saddle-point energy
barriers. This is what allows the cascade picture
of \GRIP{} (Part~V) to be written as a sequence of geometric attractor
transitions: without strict Morse-positivity, the cascade would be
ill-defined. The compatibility relation $\mathfrak{C}_{\GFF}$ then
guarantees that the algebraic Morse landscape is operationally
realized as a thermodynamic free-energy under the \GDOS{} CPTP flow
(T3.2).
\end{remark}

\subsection{Outline of the proof}
\label{ssec:T3-outline}

The proof proceeds in four steps, mirroring the chain established in
\citet[Appendix~E]{ArisakaP2N3}.

\textbf{Scope clarification.} Theorem T3.1 is stated for strata
$\Sigma$ that are K\"ahler and on which the boundary gauge
connections $F_5, F_B, F_{Qi}$ are flat-along-$\Sigma$ (so that the
product structure $\Bdy = \Sigma \times \Sigma_{\rm gauge}$ used in
the proof of $\mathcal H_1$ Part~2 holds). This restriction covers
the family-projector stratum $\Sigma_{\rm fam} = \CPN$ on which
T4.1 acts, but is stronger than the original formulation.
Strata with non-trivial Wilson lines along the compact $\theta$-
direction (e.g., the Wilson--Hosotani gauge-orbit moduli of Part~II)
require a separate argument that is deferred to a companion paper.

\textbf{Morse function specification.} By \emph{Morse function} in
what follows we mean the equivariant moment-map Morse function under
the maximal torus action of $\mathrm{SU}(N+1)$ on $\CPN$
(or, more generally, of the relevant symmetry group on the K\"ahler
stratum), in the sense of Atiyah--Bott and Frankel. With this
Morse function, the alternating Morse-index sum equals the Hirzebruch
signature $\sigma(\Sigma)$ via the localization formula and Hodge
filtration matching, not the Euler characteristic $\chi(\Sigma)$.
For $\CP^2$, $\sigma(\CP^2) = 1$; for $\CP^1$, $\sigma(\CP^1) = 0$.

\textbf{Step 1.} The alternating Morse-index sum of $\fGFF|_\Sigma$
equals the topological signature $\sigma(\Sigma)$ when $\Sigma$ is a
K\"ahler manifold with the equivariant moment-map Morse function
specified above.

\textbf{Step 2.} By Hirzebruch's signature theorem,
$\sigma(\Sigma) = \int_\Sigma L_m(p_1, \ldots, p_m)$ for
$\dim_\R\Sigma = 4m$, and is zero for $\dim_\R\Sigma \equiv 2 \pmod 4$.

\textbf{Step 3.} By the bulk-to-boundary descent of $X_4$, the integrated
L-genus on $\Sigma$ is non-zero iff $[X_4]\in H^4(\Msix;\Z)$ is
non-trivial. This is the \textbf{$\mathcal H_2$ theorem} of
Section~\ref{sec:H1-H2-theorems}.

\textbf{Step 4.} Combining Steps~1--3: strict positivity holds iff
$\dim_\R\Sigma\in 4\Z$, the parity condition of \textbf{T3}.

\subsection{Detailed proof}
\label{ssec:T3-proof}

\begin{proof}[Proof of Theorem~\ref{thm:GFF-positivity-from-CCI}]

\textbf{Step 1: alternating sum equals signature.}
Let $\Sigma$ be a Kähler manifold of complex dimension $n$ (real
dimension $2n$), with the height function
$f([\psi]) = \sum_a \lambda_a |\psi_a|^2 / \sum |\psi_a|^2$ as Morse
function. By a direct count
\citep[Lemma~E.2.1]{ArisakaP2N3}, the critical points of
$f$ have even Morse indices $0, 2, \ldots, 2n$, and the alternating
sum $\sum_a (-1)^{\ind_a/2}$ equals the topological signature
$\sigma(\Sigma)$. This is a textbook content of Morse theory on
Kähler manifolds.

\textbf{Step 2: Hirzebruch's signature theorem.}
By Hirzebruch's signature theorem
\citep{Hirzebruch1966}, for a closed oriented manifold $\Sigma$ of
real dimension $4m$,
\begin{equation}
\label{eq:T3-Hirzebruch}
\sigma(\Sigma) = \int_\Sigma L_m(p_1, p_2, \ldots, p_m),
\end{equation}
where $L_m$ is the Hirzebruch L-polynomial. For $\dim_\R\Sigma\equiv 2\pmod 4$,
the L-polynomial of the relevant degree does not exist (Pontryagin
classes live only in degrees that are multiples of 4), and
$\sigma(\Sigma)$ is identically zero.

\textbf{Step 3: bulk-to-boundary descent gives non-zero L-genus.}
By the $\mathcal H_2$ theorem (Theorem~\ref{thm:H2-theorem} of
Section~\ref{sec:H1-H2-theorems} below), if the \CCI{} class
$[X_4] \in H^4(\Msix;\Z)$ is non-trivial, then its Hopkins--Singer
descent onto the stratum $\Sigma$ is non-trivial; and by the
$\mathcal H_1$ theorem (Theorem~\ref{thm:H1-theorem} of
Section~\ref{sec:H1-H2-theorems}), the descended class equals
$c_R \cdot p_1(\Sigma)$ for some non-zero rational coefficient $c_R$.
Therefore $p_1(\Sigma)$ is non-trivial. In real dimension four this
already settles the matter, since $\sigma(\Sigma) = p_1[\Sigma]/3$ on a
closed oriented four-manifold, so the L-genus integral
\eqref{eq:T3-Hirzebruch} is non-zero; its \emph{sign} is a further fact
about $\Sigma$ and is not supplied by the descent. In real dimension
$4m$ with $m\geq 2$ non-vanishing of $p_1$ does not by itself force
$\int_\Sigma L_m \neq 0$ --- the L-polynomial mixes $p_1$ and $p_2$ ---
so there the signature must be evaluated on the stratum. For
$\Sigma = \CPN$ both are immediate: $p_1 = N h^2$ and $\sigma = 1$ for
$N$ odd.

\textbf{Step 4: synthesis.} Combining Steps~1--3:
\begin{itemize}[leftmargin=2em]
\item If $\dim_\R\Sigma \in 4\Z$: the alternating sum equals the
non-zero signature, hence $a^+ - a^- > 0$.
\item If $\dim_\R\Sigma \equiv 2 \pmod 4$: the signature is identically
zero, hence $a^+ - a^- = 0$ (no strict positivity).
\end{itemize}

This proves \textbf{T3}. In particular, for $\Sigma = \CPN$ of
dimension $2(N-1)$, strict positivity holds iff
$2(N-1) \in 4\Z$, i.e., $N$ is odd.\qedhere
\end{proof}

\subsection{The N=3 application of T3}
\label{ssec:T3-N3-application}

For $N=3$, the family-projector stratum is $\CPtwo$ of real dimension
$4 = 4\cdot 1$, which is divisible by 4. By \textbf{T3}, strict
Morse-positivity holds:
\begin{equation}
\label{eq:T3-N3}
a^+(\CPtwo) - a^-(\CPtwo) = \sigma(\CPtwo) = 1 > 0,
\end{equation}
which is exactly Postulate~3.5 of the N=3 proof of
\citet{ArisakaP2N3}. In of GG-2026-Generations this
was a postulate; in it was reduced to two named hypotheses
$\mathcal H_1, \mathcal H_2$; in the present paper, \textbf{T3} makes
it a full theorem of the \GiFour{} framework.

For $N=2$ (excluded by the lower bound) and $N=4$ (excluded by
\textbf{T3}), the dimension $2(N-1)$ is $\equiv 2\pmod 4$, and strict
Morse-positivity fails. This matches the parity result of the N=3
proof.

\subsection{Statement and proof of T3.2 (thermodynamic monotonicity)}
\label{ssec:T3.2-statement}

Indeed, the second half of Theorem~\textbf{T3} packages the
\emph{thermodynamic monotonicity theorem}, which packages
\textbf{GFF-ST6}--\textbf{GFF-ST7} into a precise iff statement linking
the algebraic Morse cascade to the boundary second law.

\begin{theorem}[\textbf{T3.2}: Thermodynamic monotonicity from \GDOS{}]
\label{thm:T3.2-thermodynamic}
Let $\cF_{\GFF}\colon \Bdy\to\R$ satisfy \textbf{GFF-ST1}--\textbf{GFF-ST5}
(algebraic block) on the stratified boundary moduli space of the
\TGG{} structure of Part~II. Let $t\mapsto[\Pi(t)]\in\Bdy$ be a
trajectory generated by a Lindbladian $\mathcal L$ acting CPTP on the
boundary state space (the \GDOS{} flow). Then the boundary
Helmholtz-type free energy
\begin{equation}
\label{eq:T3.2-helmholtz}
F_{\GFF}([\Pi(t)])
\;=\; U_{\GFF}([\Pi(t)]) \;-\; T_{\GFF}\,S_{\GFF}([\Pi(t)])
\end{equation}
of \textbf{GFF-ST6} satisfies the monotonic decrease
\begin{equation}
\label{eq:T3.2-monotonic}
\frac{\dd F_{\GFF}}{\dd t} \;\le\; 0,
\end{equation}
\textbf{if and only if} the following two conditions both hold:
\begin{enumerate}[label=\textnormal{(\arabic*)},leftmargin=2.2em]
\item \textbf{Geometric First Law.} The identifications
$U_{\GFF} = \cF_{\GEP}$ and $-T_{\GFF}S_{\GFF} = \cF_{\CEP}$ of
Theorem~\ref{thm:GFF-first-law} hold;
\item \textbf{Boundary CPTP structure.} The Lindbladian $\mathcal L$
generating the boundary flow is completely positive and
trace-preserving (i.e., the boundary dynamics is genuinely
open-system, as required by \OGGEP{}).
\end{enumerate}
Furthermore, asymptotically $F_{\GFF}\to F_{\GFF}([\Pi_*])$, where
$[\Pi_*]\in\Bdy$ is the unique global minimum of \textbf{GFF-ST7}.
\end{theorem}

\begin{proof}
\textbf{($\Leftarrow$)} Assume conditions (1) and (2) hold. By
Theorem~\ref{thm:GFF-H-theorem}, the boundary CPTP structure of
$\mathcal L$ implies monotonic decrease of relative entropy, which
via the Geometric First Law of condition (1) translates to
monotonic decrease of $F_{\GFF}$. Equation \eqref{eq:T3.2-monotonic}
follows. By Theorem~\ref{thm:GFF-equilibrium-uniqueness}, the
asymptotic limit is the unique global minimum.

\textbf{($\Rightarrow$)} Conversely, assume
$\dd F_{\GFF}/\dd t \le 0$ along every \GDOS{} trajectory generated
by $\mathcal L$. We must show conditions (1) and (2) hold. By
Spohn's H-theorem for Lindblad evolutions \citep{Spohn1978}, the
monotone decrease of $F_{\GFF}$ along every CPTP trajectory is
equivalent to $\mathcal L$ satisfying \emph{detailed balance} with
respect to the Helmholtz-type free energy $F_{\GFF}$. Detailed
balance is the additional hypothesis we now make explicit: condition
(2) of the iff is \emph{CPTP $\mathcal L$ with detailed balance with
respect to $F_{\GFF}$}, not merely CPTP $\mathcal L$. Under this
stronger hypothesis, monotone decrease and the gradient-flow
structure of \GDOS{} (with $F_\GFF$ as Lyapunov function) are
equivalent, so condition~(2) holds; the Helmholtz decomposition
\eqref{eq:T3.2-helmholtz} then forces the algebraic--thermodynamic
identification of condition~(1). The reverse direction is logically weaker absent the detailed-balance hypothesis, and the paper does not hide it: Spohn's theorem gives monotonicity only, not gradient-flow structure, so the ($\Leftarrow$) direction of T3.2 is where the detailed-balance clause does its work.
\end{proof}

\begin{remark}[Logical structure of T3: T3.1 + T3.2]
\label{rem:T3-structure}
Theorem~\textbf{T3} as stated in is the \emph{conjunction} of
\textbf{T3.1} and \textbf{T3.2}. T3.1 fixes the algebraic side: the
strict Morse-positivity $a^+(\Sigma) - a^-(\Sigma) > 0$ is forced by
the integrality of the \CCI{} class $[X_4]$ via Hopkins--Singer
descent. T3.2 fixes the physical side: the monotonic decrease
$\dd F_{\GFF}/\dd t \le 0$ is forced by the CPTP structure of the
\GDOS{} flow via Lindblad's monotonicity theorem. The two halves
are linked by the Geometric First Law of
Theorem~\ref{thm:GFF-first-law}: the same functional $\cF_{\GFF}$
that supports the algebraic Morse cascade \emph{is} the
thermodynamic free energy that satisfies the second law.
\end{remark}

\begin{remark}[Physics intuition: why T3.2 is the deeper half]
\label{rem:T3.2-deeper}
A graduate student in physics may at first think that T3.1 is the
``deeper'' theorem because it produces specific Morse numbers and
underlies the N=3 family proof. But T3.2 is in fact the deeper
statement: it is the assertion that those Morse numbers \emph{must}
correspond to physical equilibrium states under the boundary second
law. T3.1 alone only says that the algebraic Morse cascade exists ---
one could imagine objections that the cascade is ``just topology''
with no physical realization. T3.2 says that no other dynamical
behavior \emph{works}: any non-CPTP boundary dynamics would either
violate the second law (empirically excluded) or fail to converge to
a unique equilibrium (also empirically excluded).

In this sense, T3.2 is the \emph{empirical} arm of T3: the
Morse-positivity of T3.1 is forced not by topology alone but by the
empirical reality of irreversibility and equilibrium. The
convergence of T3.1 (algebraic) and T3.2 (physical) on the same
free-energy functional is one of the strongest internal consistency
checks of \GGTheory{}.
\end{remark}

\begin{corollary}[Laboratory falsifier from T3.2]
\label{cor:T3.2-falsifier}
Combining Theorem~\ref{thm:T3.2-thermodynamic} with the
laboratory-testable equipartition and fluctuation-dissipation
relations of \textbf{GFF-ST7}, the universal validity of equipartition
$\langle\delta F_{\GFF}^2\rangle = T_{\GFF}\chi_{\GFF}$ across all
boundary observables is a \emph{kinematic prediction} of \GGTheory{}.
A coherent empirical violation would falsify the dual-layer \GFF{}
structure and hence the entire \GG{} framework.
\end{corollary}

\begin{proof}
This is the linear-response specialization of T3.2 with the
identifications of Theorem~\ref{thm:GFF-first-law}.
\end{proof}

%% ====================================================================
\section{The descent structure on the family strata: theorem, obstruction, and the residue M6}
\label{sec:H1-H2-theorems}
%% ====================================================================

The N=3 proof of \citet[Appendix~E]{ArisakaP2N3} relied on
two named hypotheses about anomaly inflow:
\begin{itemize}[leftmargin=2em]
\item $\mathcal H_1$: the $\Spinone$ curvature descent on the
family-projector moduli is $c_R\cdot p_1$;
\item $\mathcal H_2$: non-trivial $[X_4]$ forces non-zero L-genus on
the descended stratum.
\end{itemize}
The capacity architecture of \citet{ArisakaP2N3} settles the status of
both, and this section records the settlement in \GiFour{} vocabulary.
$\mathcal H_1$ splits into an obstruction and a theorem: no functorial
descent of the bulk data can supply it (the rigidity lemma of that
paper --- of which the naive Chern--Weil restriction is the special
case), while the descended geometry \emph{is} supplied, at the Born
theorem's own grade, by the Berry/Fubini--Study structure that the
boundary runtime's adiabatic theorems force on the family strata.
$\mathcal H_2$'s salvageable content is integrality --- the
integrality theorem L-INT of that paper, whose topological residue
R-INT is precisely the $\eta$-invariant computation recorded here as
open problem M6, in sharpened form.

\subsection{$\mathcal H_1$ as a theorem}
\label{ssec:H1-theorem}

\begin{theorem}[$\mathcal H_1$: Spin(1,5) curvature descent]
\label{thm:H1-theorem}
Let $\Sigma\subset\Bdy$ be a smooth stratum of the boundary moduli on
which the bulk-to-boundary projector $\Pi:\Msix\to\Mfour$ extends
smoothly, and let $\check\rho_{\rm bdy}:\check H^4(\Msix;\Z)\to
\check H^4(\Sigma;\Z)$ be the Hopkins--Singer descent map. Then the
descent of the \TGG{} curvature 4-form is
\begin{equation}
\label{eq:H1-descent}
\check\rho_{\rm bdy}\bigl(\,[\,\tr R_{\Spinone}\wedge R_{\Spinone}\,]\bigr)
\;=\; c_R\cdot p_1(\Sigma)
\;\in\; H^4(\Sigma; \R),
\end{equation}
where $c_R\in\mathbb Q_{>0}$ is the rational coefficient set by the
$\Spinone$ Killing-form sign. The gauge-curvature contributions
$\Tr F_5^2, F_B^2, F_{Qi}^2$ pull back to zero on $\Sigma$ for any
stratum transverse to the gauge-orbit moduli.
\end{theorem}

\begin{proof}[Sketch, re-sourced]
The proof has two parts, and the first must be stated with care,
because the obvious route is provably closed.

\textbf{Part 1: the descended geometry, from the boundary runtime.}
What can \emph{not} carry this step is any restriction of the bulk
spin connection: on the canonical bulk $\tr R_{\Spinone}\wedge
R_{\Spinone}$ vanishes pointwise (conformal flatness), and the
rigidity lemma of \citet{ArisakaP2N3} (its Appendix~G) extends the
vanishing to every connection induced functorially from the bulk data
--- the Chern--Weil restriction that a first attempt would write down
is the special case that exposed it. The descended class is instead
supplied by the stratum's own quantum geometry: the boundary
runtime's adiabatic theorems (T-AD-1--T-AD-3 there, at the Born
theorem's own premises --- the \GDOS{} forward semigroup and the
collision-level contraction) force Berry transport on the
family-projector bundle, whose quantum geometric tensor equips
$\Sigma$ with the Fubini--Study package; the descended 4-class is the
stratum's own $p_1$, with the coefficient fixed by the Fubini--Study
normalization conventions (its Appendix~F), not by the bulk Killing
form.

\textbf{Part 2: Vanishing of gauge contributions.} The
$\SUfive$, $\UoneB$, and $\UoneQi$ bundles are pulled back from the
gauge-orbit moduli $\Sigma_{\rm gauge}$, which is by hypothesis
transverse to $\Sigma$. The product structure
$\Bdy = \Sigma\times\Sigma_{\rm gauge}\times\cdots$ implies that the
gauge field strengths are constant along $\Sigma$ at the level of
cohomology, so their pull-back as 4-forms vanishes. Specifically,
\begin{equation}
\label{eq:H1-gauge-zero}
\Tr F_5^2 \big|_\Sigma = F_B^2 \big|_\Sigma = F_{Qi}^2 \big|_\Sigma = 0
\end{equation}
in $H^4(\Sigma; \R)$.

Combining the two parts gives \eqref{eq:H1-descent}.
\end{proof}

This is the honest \GiFour{} restatement of the $\mathcal H_1$ content
from \citet[Appendix~E]{ArisakaP2N3}: an obstruction on the bulk side
(rigidity), a theorem on the boundary side (Berry/Fubini--Study at
Born-grade), and the transversality of the gauge-orbit moduli carrying
Part 2 unchanged. Nothing here rests on restricting the bulk
curvature, which would yield zero.

\subsection{$\mathcal H_2$ as a theorem}
\label{ssec:H2-theorem}

\begin{theorem}[$\mathcal H_2$: non-degeneracy of Hopkins--Singer descent]
\label{thm:H2-theorem}
Let $\check x_4 \in \check H^4(\Msix;\Z)$ be the differential
cohomology class of $X_4$ (Definition~\ref{def:X4-diff-class}). If
$[X_4]$ is non-trivial in $H^4(\Msix;\R)$, then the descended class
$\check\rho_{\rm bdy}(\check x_4)|_\Sigma$ is non-trivial in
$H^4(\Sigma;\R)$ for every stratum $\Sigma\subset\Bdy$ of real
dimension divisible by four on which the descent is well-defined.
\end{theorem}

\begin{proof}[Sketch]
The Hopkins--Singer descent map $\check\rho_{\rm bdy}$ is, by
construction \citep{HopkinsSinger2005}, a morphism of differential
cohomology theories. Its non-degeneracy follows from the
$\eta$-invariant pairing on differential cohomology. Specifically:

\textbf{Step 1.} The differential cohomology class $\check x_4$ has
ordinary cohomology image $[X_4] \in H^4(\Msix;\Z)$ which is non-trivial
by \textbf{CCI-ST5}.

\textbf{Step 2.} The descent map respects the long exact sequence of
differential cohomology \eqref{eq:diff-cohomology-seq}. Therefore the
ordinary cohomology image of $\check\rho_{\rm bdy}(\check x_4)$ is the
descent of $[X_4]$.

\textbf{Step 3.} By Theorem~\ref{thm:H1-theorem} ($\mathcal H_1$),
the descent of $[X_4]$ on a stratum $\Sigma$ of real dimension $4m$ is
$c_R\cdot p_1(\Sigma)$. The Pontryagin class $p_1(\Sigma)$ is
non-trivial on any compact closed Kähler manifold (it equals
$N\cdot h^2$ for $\Sigma = \CPN$, etc.).

\textbf{Step 4.} Therefore the descended cohomology class is
non-trivial, and by the long exact sequence of differential
cohomology, the descended differential cohomology class
$\check\rho_{\rm bdy}(\check x_4)|_\Sigma$ is non-trivial in
$\check H^4(\Sigma;\Z)$.

For the rigorous step 1--4 implementation in the 6D bulk, the
key technical fact is that the $\eta$-invariant pairing on
$\check H^4(\Msix;\Z)$ is non-degenerate
\citep{HopkinsSinger2005,FreedHopkins2000}. This non-degeneracy is
inherited by the descent.\qedhere
\end{proof}

This is the \GiFour{} restatement of the $\mathcal H_2$ content from
\citet[Appendix~E]{ArisakaP2N3}, with its honest residue displayed:
what the argument consumes from Hopkins--Singer theory is
\emph{integrality} of the descended periods --- the integrality
theorem L-INT of that paper, whose residue R-INT equals this paper's
M6 --- and the degree counting;
``non-degeneracy'' as an unconditional property is not claimed, being
underivable from either side alone (the rigidity lemma and the
boundary-relativity proposition of that paper close the two routes).

\subsection{Consequences for the N=3 proof}
\label{ssec:H1-H2-N3-consequences}

The promotion of $\mathcal H_1$ and $\mathcal H_2$ from hypotheses to
theorems has direct consequences for the rigor of the N=3 proof of
\citet{ArisakaP2N3}:

\begin{itemize}[leftmargin=2em]
\item In GG-2026-Generations, the strict \GDOS{} Morse-positivity
postulate was reduced (Appendix~E) to two technical hypotheses
$\mathcal H_1$ and $\mathcal H_2$, with the explicit acknowledgment
that proving them was an open problem.
\item In the present paper, both statements are settled in \GiFour{} vocabulary (Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem}, as re-sourced above): the descended geometry at Born-grade, the bulk-only route closed by rigidity, the residue the integrality computation M6 ($=$ R-INT of the
integrality theorem L-INT). \item Therefore strict \GDOS{} Morse-positivity stands as a consistency-grade statement --- holding as a theorem at the admissible value ($\sigma(\CP^2)=1$) --- with the exclusion work carried by the capacity theorem. \item The N=3 derivation of \citet{ArisakaP2N3} is, in light of the present \GiFour{} formalization, a derivation from the GG-Constitutional axioms that is \emph{reduced} to a single named residue --- the integrality computation
M6 ($=$ R-INT) --- together with the coupling clause and premises shared with the Constitutional Born Theorem, beyond the standard machinery of differential cohomology and the Hirzebruch signature theorem.
\end{itemize}

This is the principal payoff of formalizing \GFF{}: the last residual softness in the N=3 derivation is the single named
residue M6 ($=$ R-INT), with everything else theorem-grade or closed
by obstruction. On the benchmark background of the intended class,
M6 is discharged: Appendix~H of \citet{ArisakaP2N3} proves
the $\kappa$-bridge theorem --- the seam's Hopkins--Singer pushforward
equals the boundary's Berry bundle --- closes R-INT there, and closes
the warped and relative refinements at class level (its rigidity
theorems T-INF-7 and T-INF-8). The detailed re-presentation
of the N=3 derivation in \GiFour{} vocabulary is given in Part~VII.

%% ====================================================================
\section{Worked examples}
\label{sec:GFF-examples}
%% ====================================================================

We now work through four explicit examples of \GFF{} on different
strata of the boundary moduli.

\subsection{Example 1: \GFF{} on family-projector moduli $\mathbb{CP}^{N-1}$}
\label{ssec:GFF-ex-CPN}

The family-projector moduli stratum is $\Sigma_{\rm fam} = \CPN$ with
the Fubini--Study metric.

\textbf{Energetic part $\cF_{\GEP}$.} By
\eqref{eq:GEP-as-X4-integral} restricted to $\CPN$, $\cF_{\GEP}$ is
the integrated $X_4$ on the family-projector slice. By
Theorem~\ref{thm:H1-theorem}, this integrates to
$c_R \int_{\CPN} p_1 = c_R \cdot N$ (where $N=N\cdot \int h^{N-1}$
under the standard normalization).

\textbf{Morse function and critical points.} The Morse function on
$\CPN$ is the height function \eqref{eq:fGFF-CPN}. $N$ critical
points at $[\delta_{ab}]$, indices $0, 2, \ldots, 2(N-1)$.

\textbf{Strict Morse-positivity.} By Theorem~\ref{thm:GFF-positivity-from-CCI}
(\textbf{T3}), strict positivity holds iff $N$ is odd.

\textbf{For $N=3$:}
$\dim_\R \CPtwo = 4 \in 4\Z$, $\sigma(\CPtwo)=1$, strict positivity
holds. This is the N=3 case of the proof of
\citet{ArisakaP2N3}.

\textbf{For $N=2$:}
$\dim_\R \CP^1 = 2 \notin 4\Z$, $\sigma=0$, strict positivity fails;
$N=2$ is excluded by both the lower bound (CP) and the parity
constraint.

\textbf{For $N=4$:}
$\dim_\R \CP^3 = 6 \notin 4\Z$, $\sigma=0$, strict positivity fails;
$N=4$ is excluded.

\subsection{Example 2: \GFF{} on Higgs moduli}
\label{ssec:GFF-ex-Higgs}

The Higgs moduli stratum $\Sigma_H$ is the moduli of admissible
Wilson--Hosotani Higgs configurations.

\textbf{Morse function.} The Coleman--Weinberg / Wilson--Hosotani
effective potential. Two critical points: the symmetric phase
($V=0$, unstable) and the broken phase ($V=v\simeq 246\,$GeV, stable
\GA{}).

\textbf{Cascade chain.} Symmetric phase (saddle, index 1 in this
discrete count) decays to broken phase (\GA, index 0) via electroweak
phase transition. The instanton-action cost is set by the depth of
the broken-phase minimum.

\subsection{Example 3: \GFF{} on gauge-orbit moduli}
\label{ssec:GFF-ex-gauge}

The gauge-orbit moduli stratum $\Sigma_{\rm gauge}$ is the discrete
moduli of admissible orbifold breakings of $\SUfive$.

\textbf{Morse function.} Discrete: each conjugacy class has a fixed
energy, with the lowest-energy class being the SM-compatible
$\SU(3)\times\SU(2)\times\U(1)_Y$ broken phase.

\textbf{\GA{} catalog.} The unique stable broken phase is the SM
gauge structure. No cascade is needed since the moduli is discrete.

\subsection{Example 4: \GFF{} on Wilson-line moduli}
\label{ssec:GFF-ex-Wilson}

The Wilson-line moduli stratum $\Sigma_\theta$ is the moduli of the
$\UoneB$ and $\UoneQi$ phases.

\textbf{Morse function.} Periodic potential on the Wilson-line phase.
Critical points at the discrete \OGNfive{} phases:
$2\pi M/L = 2\pi/5$ for $\UoneB$.

\textbf{\GA{} catalog.} Five admissible phases for $\UoneB$
(corresponding to $L=5$ via the \OGNfive{} ledger), continuous slow-roll
for $\UoneQi$ (the Kaxion field). These are the boundary
\GA{}'s used in \citet{ArisakaC1Kaxion}.


%% ====================================================================
\section{Why there is such a thing as downhill}
\label{sec:close-part4}
%% ====================================================================

This Part supplies the paper with a landscape, and a landscape is a
more consequential thing to supply than it first appears.

Everyone who has taken a thermodynamics course knows the picture: a
quantity that decreases, a system that runs down until it can run down
no further, and equilibrium as the bottom of a valley. What is rarely
asked is why that picture is available at all. Why should the possible
states of a system arrange themselves into something with a
\emph{shape} --- with heights and slopes and bottoms --- rather than
into a bare list with no notion of one state being downhill from
another? In ordinary thermodynamics this is put in by hand: one writes
down a free energy because experience says there is one. Here it is
not put in by hand. The landscape is constructed from the geometric
data of the preceding Parts, and its most important property --- that
it curves upward in every direction away from its critical points, so
that valleys are genuinely valleys and not flat shelves --- is forced
by the consistency condition of Part~III rather than assumed.

That link deserves to be said slowly, because it is unusual. The
requirement that a certain anomaly cancel is a statement about the
internal consistency of a quantum field theory. The requirement that a
free-energy landscape have real minima is a statement about whether
matter can settle down. These are not obviously the same kind of
statement, and in most of physics they belong to different subjects
taught by different people. Here the first implies the second. Whether
the world can come to rest is, on this account, a question with an
answer in the geometry.

The functional splits into two pieces with recognizable meanings: one
counts what a configuration costs in energy, the other what it costs in
information --- how much has to be specified, how much order is being
maintained against the tendency to spread out. Anything that persists
is paying both bills at once. A protein holding its fold, a star
holding itself against its own weight, a cell holding a concentration
gradient across a membrane: in each case something is being spent, and
the accounting in this Part is meant to apply to all of them in the
same units. \emph{That the same functional prices a folded protein and a
gauge-orbit stratum is the framework's most aggressive claim, and the
laboratory tests named in this Part are aggressive for the same
reason.}

A word about what a landscape is not. It is not a place. Nothing is
literally sitting on a hillside; the heights are values of a functional
and the positions are configurations, and the picture is a way of
thinking rather than a map of anywhere. The reason it is nonetheless
trustworthy is that the mathematics behind it has the properties the
picture suggests --- there really are minima, they really are separated
by barriers, and the descent really is monotonic. When an intuition and
a theorem agree this closely it is usually because the intuition was
extracted from the theorem in the first place.

What is still missing is time. A landscape says where the low places
are; it does not say that anything goes there, or by what route, or
whether it stops. A ball on a hillside is a statement about a hillside,
not yet a history. The next Part supplies the motion --- and with it the
reason the world has three families of matter rather than any other
number.


\GGPart[GRIP]{\GRIP{}}{The Gauge-Runtime Invariance Pipeline: Pillar 4 of \GiFour{}}

This is the Part the paper is named for, and the one a reader with
limited time should read.

The preceding Part gave the universe a landscape. This one describes how
the universe walks across it, and the answer is the mechanism the whole
cohort turns on. A symmetric starting configuration does not slide
smoothly to the world we observe. It descends in a finite, ordered
sequence of stages: it begins with every possibility still open; it is
handed a landscape; it reaches a fork and takes one branch, at which
point a symmetry that was real a moment before is gone; and it is
captured in a basin, where it stays. Four moves. The remark opening the
Part's first section gives the picture --- a ball going down stairs
rather than down a ramp --- and the picture is exact rather than
decorative, because each step costs a definite amount and the total that
can be spent is finite.

That last clause is where the Part earns its keep. Because the budget is
finite and the cost per step is fixed by the geometry of the preceding
Parts, the number of steps the cascade can take is bounded --- and the
number of steps is the number of particle families. Three generations of
matter, the fact that has resisted explanation since the discovery of
the muon in 1936, comes out here as an arithmetic consequence of a
budget rather than as an input. The bound is stated with its
conditions attached: it leans on a capacity theorem proved in the
three-generations companion \citep{ArisakaP2N3}, and on one topological
question this paper leaves open and names rather than assumes.

The Part also draws a distinction that matters far beyond particle
physics, and it is the one a reader from biology or neuroscience should
look for. The engine runs in two regimes, separated by a single
question: does the landscape itself change? In one regime the landscape
holds still and the system descends on it until it reaches a basin with
no exit --- a settling, and it finishes. In the other, a transient event
reorganizes the landscape and hands the next descent a different
starting point. One stroke opens; the other closes. A universe with only
the first would have settled once and stopped; one with only the second
would never settle at all. The alternation is what lets structure keep
appearing at new scales, and it is the reason this paper claims to be
about more than the Standard Model. The claim that the same two strokes
run in chemistry, in biochemistry, and in cognition is the cohort's
wager, carried by the capstone \citep{ArisakaA11Origin}; what is proved
here is the mechanism, sharply enough that the wager could be shown to
fail.

Two further results close the Part. The descent leaves a signature ---
a ladder of particle decay rates falling by a fixed factor from one
family to the next --- which is a prediction with a number in it. And
the internal consistency of the whole picture requires that the
compensating structure identified in Part~III survive every step of the
descent, which the Part establishes and which is a real constraint
rather than a formality.


\section{\GRIP{}: motivation, dual-layer structure, and Structure Theorems}
\label{sec:GRIP-axioms}

\begin{remark}[Reader's bridge: a ball rolling down stairs, not a ramp]
\label{rem:GRIP-readers-bridge}
If \GFF{} gives the universe its preferred landscape, \GRIP{}
describes how the universe walks across it. Roughly speaking, the
boundary universe does not slide smoothly from a high-energy
configuration to the Standard-Model vacuum; it descends in a
finite, ordered \emph{cascade} of stages, very much like a ball
rolling down a flight of stairs. Each step from one stage to the
next costs a definite quantum of action (an integer multiple of
$\pi$ in natural units), and the total number of steps is
bounded by a finite budget inherited from the \BIsix{} closure of
Part~III. Three named landings span the cascade: \DGI{} at the
top (the unstable configuration), \GSSB{} in the middle (the
broken-symmetry intermediate), and \GA{} at the bottom (the
Standard-Model vacuum). The mathematical name for this is a
Morse cascade; the empirical signature is the generation decay-width ladder
of unstable particles --- the $\tau$ lepton lives longer than
the $W$ boson but much shorter than the muon, with the gaps fixed
by the cascade-step actions. The rest of this Part makes this
rolling-down picture mathematically precise.
\end{remark}

This Part develops \GRIP{} (the \emph{Gauge-Runtime Invariance
Pipeline}), the fourth and final \GiFour{} Pillar. Like \TGG{}
(Part~II), \CCI{} (Part~III), and \GFF{} (Part~IV), \GRIP{} is
intrinsically a \emph{dual-layer} object: it is simultaneously
\begin{enumerate}[leftmargin=2em,label=(\roman*)]
\item an \textbf{algebraic/cascade structure} --- a finite, ordered
chain of Morse critical points $\DGI\to\GSSB\to\GA$ on each stratum,
generated by gradient flow of $\cF_{\GFF}$ in the Riemannian metric
inherited from \TGG{}, with each transition saturating the
Bogomolnyi bound $S_{[2k]\to[2(k-1)]}=\pi k$ and the total cascade
action bounded by the \BIsix{} budget $\Sbi = 3\pi$, taken as input from the capacity theorem T-N3-6 of \citet{ArisakaP2N3} (open problem M10, demand met); and
\item a \textbf{physical/gauge-runtime invariance pipeline} --- a
boundary observability protocol that asserts that every cascade
transition \emph{preserves all four boundary Ward currents}
(electric, baryonic, informational, metric) at runtime, and that the
unique global GA is the only configuration consistent with the
empirically observed gauge structure of the Standard-Model vacuum.
The runtime checks produce the generation-ordered decay-width ladder
along the family chain ($\tau\to\mu\to e$), whose
controlled-kinematics ordering is the phenomenological falsifier of
\GRIP{}.
\end{enumerate}

The two layers are not independent: they are linked by the
\emph{Gauge-Cascade Compatibility Theorem} (introduced below in
Section~\ref{sec:GRIP-physical-layer}), which guarantees that the
Morse-cascade flow of \textbf{GRIP-ST1}--\textbf{GRIP-ST5} does not
generate gauge anomalies along the trajectory --- equivalently, that
the \PBIfour{} compensator field $C_\mu$ of \CCI{} extends smoothly
across every saddle. In short:
\begin{equation}
\label{eq:GRIP-dual-link}
\boxed{\;
\underbrace{\Stot \le \Sbi = 3\pi}_{\text{algebraic cascade closure}}
\;\xrightarrow{\;\text{Gauge-Cascade Compatibility}\;}\;
\underbrace{\nabla_\mu J_a^\mu = \text{anomaly}_a}_{\text{Ward at runtime}}
\;}
\end{equation}
The arrow is precisely the Gauge-Cascade Compatibility Theorem: it
carries the algebraic Bogomolnyi cascade to its physical avatar.

\paragraph{Section structure of Part~V.}
Section~\ref{sec:GRIP-axioms} (the present section) states the seven
\GRIP{} axioms (\textbf{GRIP-ST1} through \textbf{GRIP-ST7}),
organized into the algebraic block (\textbf{GRIP-ST1}--\textbf{GRIP-ST5})
and the physical block (\textbf{GRIP-ST6}--\textbf{GRIP-ST7}); gives
the dual-layer formal definition; and lists the first immediate
consequences. Section~\ref{sec:DGI-GSSB-GA} develops the three
stages of the cascade in detail. Section~\ref{sec:GRIP-gradient-flow}
formalizes the gradient-flow dynamics on the \GFF{} Morse landscape.
Section~\ref{sec:GRIP-physical-layer} (new in) develops the
physical/gauge-runtime layer: gauge-invariance preservation along the
cascade, runtime Ward-identity checks, the generation decay-width ladder of
unstable particles, and laboratory-accessible falsifiers.
Section~\ref{sec:GRIP-T4} states and proves Theorem~\textbf{T4}, now
in two parts: \textbf{T4.1} (algebraic cascade closure forcing
$N\leq 3$) and \textbf{T4.2} (runtime gauge-invariance preserved iff
\PBIfour{} compensator extends). Section~\ref{sec:GRIP-lifetimes}
connects \GRIP{} to the generation decay-width ladder of unstable
particles. Section~\ref{sec:GRIP-examples} contains worked examples.

\begin{remark}[Physics intuition: \GRIP{} as the GG analogue of the renormalization group]
\label{rem:GRIP-RG-analogue}
Readers familiar with quantum field theory will find an immediate
analogy. In the Wilsonian renormalization group (RG), one
integrates out high-energy degrees of freedom in a sequence of
discrete steps, each step generating a new effective theory at a
lower energy scale. The RG flow is irreversible (information about
high-energy modes is lost), the flow is monotonic in some
``c-function'' (the Zamolodchikov $c$-theorem), and the IR fixed
point is the universal long-distance theory.

In \GGTheory{}, \GRIP{} plays the same logical role but for the
boundary moduli space. $\DGI$ is the deepest unbroken (UV) phase;
each $\GSSB$ saddle is an intermediate effective theory at a lower
``Morse scale''; $\GA$ is the IR attractor. The c-function role is
played by $\cF_{\GFF}$ (which decreases monotonically by
Theorem~\textbf{T3.2}); the cascade is the GG analogue of the RG
flow.

The conceptual upgrade in \GRIP{} is twofold. First, the cascade
has \emph{finite} total action $\Stot = \pi N(N-1)/2$, which is
bounded by $\Sbi = 3\pi$ and forces $N\le 3$. This is sharper than
the RG, where the flow length is unbounded. Second, the runtime
gauge-invariance preservation provides an empirical falsifier ---
the generation-ordered decay-width ladder --- that has no
analogue in the RG framework.
\end{remark}

\begin{remark}[Why broader scientists should care about \GRIP{}]
\label{rem:GRIP-broader}
The cascade structure $\DGI\to\GSSB\to\GA$ is the universal
template for \emph{any} discrete, ordered, irreversible relaxation
process in nature. In chemistry, it describes multi-step reaction
networks with definite intermediate states. In condensed-matter
physics, it describes successive symmetry-breaking phase transitions
(e.g., paramagnetic $\to$ nematic $\to$ ferromagnetic). In biology,
it describes signaling cascades (e.g., MAPK pathway with a fixed
sequence of phosphorylation steps). In machine learning and
neuroscience, it describes layered hierarchical computation (e.g.,
deep neural network layers with monotonically decreasing
information content per layer, by the data-processing inequality).

What is genuinely new in \GRIP{} is that all of these distinct
cascade structures are \emph{boundary projections} of the same
upstream object: the gradient flow of $\cF_{\GFF}$ on the boundary
moduli, with finite cascade action bounded by the \BIsix{} budget. The
runtime gauge-invariance preservation provides a universal
\emph{computability} of the cascade: at every stage, one can check
whether the boundary configuration respects the underlying symmetry
content. This is the source of the universality of cascade
phenomena across the natural sciences.
\end{remark}

\subsection{Motivation: from static \GFF{} to dual-layer cascade}
\label{ssec:GRIP-motivation}

The \GFF{} structure of Part~IV provides a Morse landscape on each
stratum of the boundary moduli, with critical points labeled by
Morse index. By itself, this is a static picture: it tells us the
catalog of critical points and the global asymptotic equilibrium,
but does not say how the system moves between them, which transitions
are admissible, what the timescale of each transition is, or how the
gauge structure of each intermediate configuration is preserved.

\GRIP{} adds two complementary layers of dynamical content.

\smallskip
\noindent\textbf{Question 1 (algebraic).}
\GFF{} catalogs the critical points but does not specify the
cascade structure. Multiple non-equivalent ordered chains
$c_{\rm top}\to c_1\to\cdots\to c_{\rm bottom}$ could in principle
connect the same endpoints; we need a principle that selects the
\emph{unique} cascade chain compatible with gradient flow on the
\TGG-induced Riemannian metric, with each transition saturating the
Bogomolnyi bound and the total action bounded by the \BIsix{} budget.
This is the \textbf{algebraic content} of \GRIP{}.

\smallskip
\noindent\textbf{Question 2 (physical).}
\GFF{} guarantees existence of a unique global GA but does not
specify how the boundary observers \emph{verify} that the GA is the
empirically realized vacuum. Verification requires a runtime
mechanism: at every stage of the cascade, the boundary Ward currents
$(J_e, J_B, J_{Qi}, J_g)$ must remain closed (modulo the calculable
\PBIfour{} anomaly inflow), and the unbroken gauge subgroup at each
saddle must be passed through to the next saddle by an admissible
instanton transition. This runtime gauge-invariance preservation is
testable in the laboratory through the generation decay-width ladder
along the family chain $\tau\to\mu\to e$: each generation gate carries
the geometric factor $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ along
$\DGI\to\GSSB\to\GA$, the absolute lifetimes being fixed by phase
space in \citet{ArisakaA9SM}. This is the \textbf{physical
content} of \GRIP{}.

\smallskip
\noindent\textbf{Why both layers must be in the same Pillar.}
The two questions are inseparable. The Bogomolnyi-saturating
algebraic cascade (algebraic side) is precisely the unique chain that
preserves all four Ward currents at every saddle (physical side) ---
because every Bogomolnyi-violating instanton would generate a
boundary anomaly inflow of magnitude $\Delta\Stot\neq \pi k$, which
would break gauge invariance at runtime. Conversely, the empirical
observation of the generation decay-width ladder (physical side) directly
constrains the cascade structure (algebraic side), because each
lifetime ratio $\tau_{i+1}/\tau_i$ encodes a Bogomolnyi action ratio
$\e^{S_{i\to i+1}}$. This bidirectional coupling \emph{mandates}
that both layers be axiomatized together.

\GRIP{} is the formalization of this dual content. It asserts:
\begin{itemize}[leftmargin=2em]
\item (\textbf{Algebraic, GRIP-ST1--GRIP-ST5}.) The cascade dynamics
on each stratum is the gradient flow of $\cF_{\GFF}$ in the
Riemannian metric inherited from \TGG{} via bulk-to-boundary descent;
each transition is an instanton with action $S_{[2k]\to[2(k-1)]}=\pi k$;
the total cascade action is bounded by $\Sbi = 3\pi$; the cascade
terminates at a stable \GA{}.
\item (\textbf{Physical, GRIP-ST6--GRIP-ST7}.) The cascade preserves
all four boundary Ward currents at every saddle (gauge-runtime
invariance); the unique global GA is the only configuration
empirically observed via the generation decay-width ladder of unstable
particles ($\tau\to\mu\to e$).
\end{itemize}
Once \GRIP{} is given, the static \GFF{} Morse landscape is
upgraded to a dual-layer dynamical cascade with finite total action,
discrete attractor endpoints, and runtime-checkable gauge-invariance
preservation.

\begin{remark}[Physics intuition: why gauge-invariance preservation is automatic]
\label{rem:GRIP-gauge-automatic}
A graduate student in physics may wonder why the cascade should
\emph{automatically} preserve gauge invariance at every saddle ---
after all, gauge-invariance violation is a generic feature of
intermediate configurations in many dynamical systems. The answer
is structural: each instanton transition $c_i\to c_{i+1}$ is a
gradient-flow trajectory of $\cF_{\GFF}$, and $\cF_{\GFF}$ inherits
its gauge invariance from the bulk \TGG{} action via bulk-to-boundary descent. Therefore the gradient flow is itself gauge-equivariant,
and gauge invariance is preserved along every trajectory.

The non-trivial content of \textbf{GRIP-ST6} is that this preservation
is not just a mathematical curiosity --- it is \emph{operationally
testable} via the boundary Ward identities. The runtime checks of
\textbf{GRIP-ST6} make the algebraic gauge-invariance into a
laboratory-verifiable empirical statement.
\end{remark}

\subsubsection{Historical background (with citations)}

The gauge-runtime invariance pipeline that \GRIP{} formalizes rests on
three independent traditions in modern physics. The first is the
renormalization-group flow. Stueckelberg and Petermann (1953) first
identified the freedom of regularization choice as a continuous
symmetry; \citet{GellMannLow1954} produced the $\beta$-function
formalism in QED; \citet{Kadanoff1966} introduced the block-spin
coarse-graining picture for statistical-mechanical systems; and
Wilson (1971) unified these into the modern Wilsonian
renormalization-group framework \citep{Wilson1974} that underlies
every contemporary treatment of scale-dependent gauge dynamics.

The second tradition is gauge symmetry breaking on compact manifolds.
\citet{Hosotani1983} showed that on a manifold with a non-contractible
$S^1$ factor, a non-trivial Wilson line
$\langle e^{\mathrm{i} \oint A}\rangle$ spontaneously breaks the gauge
symmetry without an explicit Higgs field --- the Wilson--Hosotani
mechanism that the \GRIP{} \GSSB{} step identifies as the mathematical
content of the \GGTheory{} electroweak transition. Anderson (1972)
framed the broader principle in his ``More is different'' essay:
genuinely new collective behavior emerges at each scale and is not
reducible to the underlying microphysics.

The third tradition is the instanton description of tunneling.
\citet{tHooft1993} produced the original instanton calculation for the
$\theta$-vacuum structure of Yang--Mills theory, and
\citet{Coleman1977,CallanColeman1977} developed the semi-classical
bounce calculus, with action $S_{\rm inst} = \pi k$ for adjacent saddle
decay on a compact moduli space. \GRIP{} closes the loop by combining
these three traditions into the named cascade \DGI{} $\to$ \GSSB{}
$\to$ \GA{}: dynamical gauge invariance preserves gauge structure
under runtime evolution, geometric spontaneous symmetry breaking
selects the family-projector moduli, and the geometric attractor
catalog terminates the cascade at a finite-budget closure that
forces $N \leq 3$ (Theorem~\ref{thm:GRIP-existence-closure}; the upper
bound conditional on open problem M10).


\subsection{The seven \GRIP{} Structure Theorems (algebraic block + physical block)}
\label{ssec:GRIP-seven-st}

The seven \GRIP{} Structure Theorems partition naturally into two blocks:
\begin{description}[leftmargin=2em]
\item[Algebraic block: \textbf{GRIP-ST1} -- \textbf{GRIP-ST5}] These Structure Theorems
fix the cascade structure, gradient-flow generation, instanton
transitions, \BIsix{} action budget, and cascade closure. They live on
the stratified moduli space $\Bdy$ and are Morse-homological in
character.
\item[Physical block: \textbf{GRIP-ST6} -- \textbf{GRIP-ST7}] These Structure Theorems
state the runtime gauge-invariance preservation along the
cascade and the generation-ordered decay-width ladder.
They live in real (laboratory) time and are operationally testable.
\end{description}
The two blocks are linked by the Gauge-Cascade Compatibility Theorem
of Section~\ref{sec:GRIP-physical-layer}.

\subsubsection*{Algebraic block (GRIP-ST1 through GRIP-ST5)}

\begin{structheorem}[\textbf{GRIP-ST1}: Cascade structure]
\label{st:GRIP-ST1}
On each stratum $\Sigma\subset\Bdy$, \GRIP{} is a finite, ordered
sequence of transitions
\begin{equation}
\label{eq:GRIP-ST1}
c_{\rm top}(\Sigma) \;\to\; c_1(\Sigma) \;\to\; \cdots
\;\to\; c_{\rm bottom}(\Sigma),
\end{equation}
where each $c_i$ is a non-degenerate critical point of
$\cF_{\GFF}|_\Sigma$ and the indices satisfy
$\ind(c_{\rm top}) > \ind(c_1) > \cdots > \ind(c_{\rm bottom})$. The
top critical point $c_{\rm top}$ has index equal to $\dim_\R\Sigma$
(the deepest unstable phase, \DGI{}); the bottom critical point
$c_{\rm bottom}$ has index $0$ (the stable \GA{}). The intermediate
points $c_1, \ldots, c_{n-1}$ are the symmetry-breaking saddles
(\GSSB{} stages).
\end{structheorem}

\begin{structheorem}[\textbf{GRIP-ST2}: Gradient flow generation]
\label{st:GRIP-ST2}
The cascade is generated by the gradient flow of $\cF_{\GFF}$ in the
Riemannian metric $g_\Sigma$ inherited from \TGG{} via bulk-to-boundary descent. The flow satisfies
\begin{equation}
\label{eq:GRIP-ST2}
\dot x \;=\; -\,g_\Sigma^{-1}(\nabla \cF_{\GFF})(x),
\qquad x \in \Sigma,
\end{equation}
and is irreversible: the flow descends in $\cF_{\GFF}$ value and never
ascends.
\end{structheorem}

\noindent The one-way character of this descent is not an accident
of detailed balance: it is the trajectory-level face of the
law-level irreversibility of the \AX{3} projection --- the
\OGGEP{} channel is CPTP and non-invertible, records accumulate,
and no boundary operation runs it backward. Custody is stated once
for the whole cohort and is consumed here, not re-derived: \AX{3}
owns the arrow constitutionally (the \AX{3} synthesis of
\citealp{ArisakaA1Constitution}); Part~IX of the CPT member paper
\citep{ArisakaP5CPT} is the canonical law-level derivation of the
arrow from that axiom, over a bulk that is itself exactly
T-symmetric; and the \GDOS{} runtime \citep{ArisakaA5GDOS} is its
dynamical implementation --- of which the gradient descent above is
the \GiFour{} geometric face.

\begin{structheorem}[\textbf{GRIP-ST3}: Instanton transitions between adjacent
saddles]
\label{st:GRIP-ST3}
Each transition $c_i \to c_{i+1}$ in the cascade is an instanton
trajectory with definite real action $S_{i\to i+1} \in \R_{>0}$. For
a Kähler stratum $\Sigma$ with the canonical height-function Morse
form, the instanton action between adjacent critical points of indices
$2k$ and $2(k-1)$ is
\begin{equation}
\label{eq:GRIP-ST3}
S_{[2k]\to[2(k-1)]} \;=\; \pi\,k,
\end{equation}
saturating the Bogomolnyi bound.
\end{structheorem}

\begin{structheorem}[\textbf{GRIP-ST4}: \BIsix{} instanton-action budget]
\label{st:GRIP-ST4}
The total cascade action accumulated along
\eqref{eq:GRIP-ST1} is bounded above by a budget $\Sbi$ inherited from
\CCI{}:
\begin{equation}
\label{eq:GRIP-ST4}
\Stot(\Sigma) \;:=\; \sum_{i} S_{c_i \to c_{i+1}}
\;\leq\; \Sbi,
\qquad \Sbi = 3\pi.
\end{equation}
The budget value $\Sbi=3\pi$ is the capacity theorem T-N3-6 of \citet{ArisakaP2N3} --- in its sharper per-family form $S_{\max}(N) = \pi N$ --- derived there from the family stratum's quantum geometry, conditional on the coupling clause and the integrality theorem L-INT
(residue R-INT; open problem M10 below, whose $N$-independence demand
that theorem meets). What \emph{is} derived is the
budget's unit and not its coefficient: the \CCI{} class is integral
(\textbf{CCI-ST4}), which with the descent normalization
$\Phi_\omega = \pi$ of \S\ref{ssec:descent-normalisation} quantizes
admissible action on a lattice of step $\pi/2$; nothing upstream selects
the multiple. The reader should therefore treat $3\pi$ as an input of
stated conjectural grade throughout this paper.
\end{structheorem}

\begin{structheorem}[\textbf{GRIP-ST5}: Cascade closure to a stable GA]
\label{st:GRIP-ST5}
The cascade terminates at a stable Geometric Attractor
$c_{\rm bottom}(\Sigma)$, which is a local minimum of
$\cF_{\GFF}|_\Sigma$ (Morse index $0$). After cascade closure, no
further dynamics is possible on $\Sigma$ at the cascade level: any
further transitions require leaving the stratum or coupling to other
strata.
\end{structheorem}

\begin{remark}[Local by \textbf{GRIP-ST5}, global by
Corollary~\ref{cor:GGA-per-stratum}]
\label{rem:GRIPST5-vs-corollary}
\textbf{GRIP-ST5} asserts only what the cascade structure alone can
support: the terminus is a \emph{local} minimum, Morse index $0$. That
is the honest algebraic statement, and on a general stratum it is all
that holds --- a landscape may carry several index-$0$ basins, and
which one a trajectory finds is then a question about initial data. On
an \emph{admissible} stratum
(Definition~\ref{def:admissible-stratum}) the two statements collapse
into one: by Corollary~\ref{cor:GGA-per-stratum} there is exactly one
index-$0$ critical point, so the local minimum of \textbf{GRIP-ST5}
\emph{is} the global minimum, and the terminus is the \GGA{}. Wherever
this paper or its companions need the global statement, the corollary
is the warrant and \textbf{GRIP-ST5} alone is not.
\end{remark}

The five Structure Theorems \textbf{GRIP-ST1}--\textbf{GRIP-ST5} constitute the
\emph{algebraic} block of \GRIP{}. Axioms
\textbf{GRIP-ST1}--\textbf{GRIP-ST3} are essentially the standard Morse
homological structure on a K\"ahler manifold; the new mathematical
content is \textbf{GRIP-ST4} (the \BIsix{} budget) and \textbf{GRIP-ST5}
(the closure condition), which together force $N\leq 3$ on the
family-projector stratum via Theorem~\textbf{T4.1} (conditional on open
problem M10).

\subsubsection*{Physical block (GRIP-ST6 and GRIP-ST7)}

The physical block translates the algebraic cascade structure of
\GRIP{} into operational, observer-accessible runtime form. It
consists of two Structure Theorems.

\begin{structheorem}[\textbf{GRIP-ST6}: Runtime gauge-invariance preservation]
\label{st:GRIP-ST6}
Along every instanton transition $c_i\to c_{i+1}$ of the algebraic
cascade \eqref{eq:GRIP-ST1}, the four boundary Ward currents
$(J_e, J_B, J_{Qi}, J_g)$ remain closed (modulo the calculable
\PBIfour{} anomaly inflow):
\begin{equation}
\label{eq:GRIP-ST6-runtime}
\boxed{\;
\nabla_\mu J_a^\mu(t) \;=\; -\,\alpha_a\,C^\mu(t) \cdot \rho_a(t),
\qquad a\in\{e, B, Qi, g\},\quad t \in [t_i, t_{i+1}],
\;}
\end{equation}
where $C_\mu$ is the \PBIfour{} compensator field of \CCI-ST6, the
$\alpha_a$ are the \textbf{CCI-ST2} coefficients, and $\rho_a(t)$ are
the time-dependent local densities along the instanton trajectory.
Equivalently: the unbroken gauge subgroup $H_i\subset\Gtot$ at saddle
$c_i$ is passed to the unbroken subgroup $H_{i+1}\subset H_i$ at
saddle $c_{i+1}$ by the instanton transition --- there is no spurious
gauge-symmetry generation or destruction along the trajectory.

The runtime statement \eqref{eq:GRIP-ST6-runtime} is a kinematic
identity following from the gauge-equivariance of the gradient flow
\textbf{GRIP-ST2} on the \TGG-induced metric, together with the
boundary covariant conservation law \textbf{CCI-ST6}.
\end{structheorem}

\begin{structheorem}[\textbf{GRIP-ST7}: Generation-ordered decay-width ladder]
\label{st:GRIP-ST7}
The boundary signature of the \GRIP{} cascade is the family ladder
itself. Each generation gate $[2k]\to[2(k-1)]$ on the
family-projector stratum $\CPN$ carries the Bogomolnyi action
$S_{[2k]\to[2(k-1)]}=\pi k$ of \textbf{GRIP-ST3} and contributes a
geometric WKB factor to the corresponding decay width:
\begin{equation}
\label{eq:GRIP-ST7-lifetime}
\boxed{\;
\Gamma_{[2k]\to[2(k-1)]} \;\propto\; \e^{-S_{[2k]\to[2(k-1)]}}
\;=\; \e^{-\pi k}.
\;}
\end{equation}
Along the charged-lepton ladder $\tau\to\mu\to e$ (and the quark
ladder $t\to b\to c\to s\to u/d$) this fixes the \emph{geometric}
contribution to each generation's decay; the \emph{absolute} lifetimes
are dominated by the species-specific phase space (Sargent's rule
$\Gamma\propto G_F^2 Q^5$) and are computed in \citet{ArisakaA9SM} and
\citet{ArisakaP2N3}, where the cascade factor $\e^{-\pi k}$
enters as one factor among several. The full generation-lifetime
reading is given in \S\ref{ssec:gen-lifetimes} below.

The robust, \GRIP{}-level content is therefore the \emph{existence and
generation-ordering} of the gate-action ladder, not a closed-form
lifetime law: the cohort treats the numerical lifetimes at
order-of-magnitude only (\citealp{ArisakaP2N3}), the phase
space being the dominant modulation. The sharp falsifier is a
controlled-kinematics decay class --- one in which the Sargent factor
is a uniform multiplier across gates, so that the residual $\e^{-\pi k}$
gate ordering must survive; an unbroken family ladder whose gate-action
ordering is wrong would falsify \GRIP{}.
\end{structheorem}

The seven Structure Theorems \textbf{GRIP-ST1}--\textbf{GRIP-ST7} together
constitute the full \emph{Gauge-Runtime Invariance Pipeline}.
The algebraic block forces the cascade structure and bounds $N\le 3$
(conditionally on open problem M10).
The physical block makes the cascade operationally testable through
the boundary Ward identities and the generation-ordered decay-width
ladder.
The two halves are unified by Theorem~\textbf{T4} in
Section~\ref{sec:GRIP-T4}.

\subsection{Dual-layer formal definition of \GRIP{}}
\label{ssec:GRIP-formal-def}

We now consolidate the algebraic and physical blocks into a single
dual-layer definition.

\begin{definition}[\GRIP: the Gauge-Runtime Invariance Pipeline, dual-layer form]
\label{def:GRIP}
The \emph{Gauge-Runtime Invariance Pipeline} on the boundary moduli
space $\Bdy$, equipped with the \GFF{} Morse landscape and the
\PBIfour{} compensator field $C_\mu$ of \CCI{}, is the
\textbf{septuple}
\begin{equation}
\label{eq:GRIP-septuple}
\begin{gathered}
\GRIP = \Bigl(\;
\underbrace{\{c_i\}_{i=0}^{n},\;\dot x = -g^{-1}\nabla\cF_{\GFF},\;
S_{[2k]\to[2(k-1)]}=\pi k,\;\Sbi = 3\pi,\;\GA = c_{\rm bottom}}_{\text{algebraic block (GRIP-ST1--A5)}};\\
\underbrace{\nabla_\mu J_a^\mu = -\alpha_a C^\mu \rho_a,\;\;\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}}_{\text{physical block (GRIP-ST6--A7)}}
\;\Bigr),
\end{gathered}
\end{equation}
where the seven components correspond to the Structure Theorems
\textbf{GRIP-ST1}--\textbf{GRIP-ST7} respectively.

The algebraic part is the collection of cascades
\begin{equation}
\label{eq:GRIP-formal}
\GRIP_{\rm alg} \;=\; \bigsqcup_{\Sigma\subset\Bdy}\,
\bigl(\,c_{\rm top}(\Sigma) \to c_1(\Sigma) \to \cdots
\to c_{\rm bottom}(\Sigma)\,\bigr),
\end{equation}
one for each stratum $\Sigma$ of $\Bdy$, satisfying the Structure Theorems
\textbf{GRIP-ST1}--\textbf{GRIP-ST5}. The cascade on each stratum is
abbreviated as $\DGI(\Sigma)\to\GSSB(\Sigma)\to\GA(\Sigma)$, where
$\DGI(\Sigma) = c_{\rm top}$, $\GSSB(\Sigma)$ is the chain of
intermediate saddles, and $\GA(\Sigma) = c_{\rm bottom}$.
Definition~\ref{def:GGA} below refines the terminus: on an admissible
stratum the index-$0$ minimum is unique
(Corollary~\ref{cor:GGA-per-stratum}), and the cascade output is that
\emph{unique deepest} basin, the Ground Geometric Attractor. The chain
is written $\DGI\to(\GFF)\to\GSSB\to\GA$ throughout --- \GA{} names the
stage --- with the \GGA{} named as its terminus.

The physical part \eqref{eq:GRIP-ST6-runtime}--\eqref{eq:GRIP-ST7-lifetime}
asserts that the cascade preserves all four boundary Ward currents
at runtime and that the generation decay-width ladder of unstable
particles is the direct boundary projection of the cumulative
cascade action. The two parts are linked by the Gauge-Cascade
Compatibility Theorem (Theorem~\ref{thm:GRIP-gauge-cascade} of
Section~\ref{sec:GRIP-physical-layer}).

One classification, introduced program-wide in the capstone
\AEleven{} and defined at runtime level in
\S\ref{ssec:grip-in-out} below, should be recorded already here: the
septuple \eqref{eq:GRIP-septuple} \emph{is} the \GRIPin{} regime ---
the runtime on a fixed stratified landscape. The complementary
regime, \GRIPout{}, reorganizes the landscape itself and supplies
each stratum's initial datum $c_{\rm top}$; it is deliberately
\emph{not} an element of the septuple, and Definition~\ref{def:GRIPout}
states precisely what is delegated to \AEleven{}.
\end{definition}

\begin{remark}[Logical content of \GRIP{}]
\label{rem:GRIP-content}
The intuitive content of Definition~\ref{def:GRIP} is twofold:
\begin{itemize}[leftmargin=2em]
\item \textbf{Algebraic side.} \GRIP{} fixes the cascade structure
on each stratum (a unique chain of Morse critical points connected
by Bogomolnyi-saturating instantons), bounds the total cascade
action by $\Sbi=3\pi$, and forces termination at a stable \GA{}.
This is the upstream structure that produces the upper bound
$N\le 3$ in Theorem~\textbf{T4.1} (conditional on open problem M10).
\item \textbf{Physical side.} \GRIP{} packages the runtime
gauge-invariance preservation into an operationally testable
sequence of Ward-identity checks and produces the empirical lifetime
hierarchy of unstable particles via the cascade-action formula
\eqref{eq:GRIP-ST7-lifetime}. This is the downstream content that
makes \GRIP{} a phenomenologically falsifiable theory.
\end{itemize}
The two sides are linked by the Gauge-Cascade Compatibility Theorem:
the same cascade structure that produces $N=3$ algebraically also
produces the runtime-checkable Ward identities physically. This is
the duality of \GRIP{}.
\end{remark}

\begin{remark}[Open-system character of \GRIP{}]
\label{rem:GRIP-open-system}
The cascade dynamics of \GRIP{} is intrinsically open-system. In a
closed-system Hamiltonian theory, dynamics is unitary and reversible:
all states evolve forever, and there is no privileged ``cascade''
direction. \GRIP{} requires a finite instanton-action budget, an
irreversible gradient flow, a discrete catalog of attractor
endpoints, and a runtime-checkable gauge-invariance pipeline ---
all four of which are open-system features that have no closed-system
analogue. This is the structural reason that the upper bound
$N\leq 3$ cannot be derived from any closed-system framework, no
matter how elaborate, as we discussed in
\citet[Section~12]{ArisakaP2N3}.
\end{remark}

\begin{remark}[Physics intuition: why both layers are needed]
\label{rem:GRIP-both-layers-needed}
A graduate student in physics may at first think that the algebraic
side (cascade closure) is sufficient --- after all, it gives the
upper bound $N\le 3$ that closes the N=3 family proof. But the
algebraic side alone does not provide an empirical pathway to test
the cascade. The physical side (\textbf{GRIP-ST6}--\textbf{GRIP-ST7})
is what makes \GRIP{} testable: the generation decay-width ladder of unstable
particles is a direct laboratory measurement of the cumulative
cascade action.

Conversely, the physical side alone (a phenomenological lifetime
hierarchy) does not specify the upstream cascade structure or
explain the $\e^{-\pi k}$ ordering of the family gate factors.
Without the algebraic side, one would have only an empirical
fit-formula with no fundamental origin.

The dual-layer structure of \GRIP{} is therefore essential --- both
layers are needed, and the Gauge-Cascade Compatibility Theorem is
what makes them consistent.
\end{remark}

\subsection{First consequences of the \GRIP{} Structure Theorems}
\label{ssec:GRIP-first-consequences}

We list four immediate consequences of \textbf{GRIP-ST1}--\textbf{GRIP-ST5}.

\begin{proposition}[Cascade length equals stratum index range]
\label{prop:GRIP-length}
On a stratum $\Sigma$ of real dimension $2n$ supporting only even
Morse indices (Kähler-Hamiltonian case), the cascade
\eqref{eq:GRIP-ST1} has length $n$ (i.e., contains $n+1$ critical
points and $n$ transitions).
\end{proposition}

\begin{proof}
By \textbf{GFF-ST3} of Part~IV, the Morse function on $\Sigma$ is
perfect. The Morse polynomial of a Kähler manifold of complex dimension
$n$ is $1 + t^2 + t^4 + \cdots + t^{2n}$, indicating exactly $n+1$
critical points with indices $0, 2, \ldots, 2n$. The cascade
$c_{\rm top} \to \cdots \to c_{\rm bottom}$ visits all of them in
descending index order, giving $n$ transitions.
\end{proof}

\begin{proposition}[Total cascade action: triangular sum]
\label{prop:GRIP-total-action}
On the family-projector stratum $\Sigma_{\rm fam} = \CPN$ of complex
dimension $N-1$, the total cascade action is
\begin{equation}
\label{eq:GRIP-total-CPN}
\Stot(\CPN) \;=\; \sum_{k=1}^{N-1} \pi k \;=\; \pi \frac{N(N-1)}{2}.
\end{equation}
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:GRIP-length}, the cascade has $N-1$ transitions.
By \textbf{GRIP-ST3}, the action of the $k$-th transition (between
indices $2k$ and $2(k-1)$) is $\pi k$. Summing:
$\sum_{k=1}^{N-1} \pi k = \pi N(N-1)/2$, the triangular sum.
\end{proof}

\begin{proposition}[Cascade asymptotic time]
\label{prop:GRIP-asymptotic}
The gradient flow \eqref{eq:GRIP-ST2} reaches the stable \GA{} only
asymptotically as $t\to\infty$, but the cumulative instanton action is
finite (equal to $\Stot(\Sigma)$). The decay rate from each saddle to
the next is $\Gamma_{i\to i+1} \propto \e^{-S_{i\to i+1}}$.
\end{proposition}

\begin{proof}
Standard Morse-flow theory: the gradient flow has critical points as
asymptotic limits, with exponential decay rates given by the WKB
approximation. See \citet{Witten1982} for the supersymmetric Morse
formalism and \citet{Schwarz1993} for the homological version.
\end{proof}

\begin{proposition}[\GRIP{} as the formal realization of \OGGEP{} cascade]
\label{prop:GRIP-equiv-OGGEP-cascade}
The existence of \GRIP{} satisfying \textbf{GRIP-ST1}--\textbf{GRIP-ST5}
is equivalent to the cascade content of \OGGEP{}: the irreversible
bulk-to-boundary projection, with finite-budget
attractor selection.
\end{proposition}

\begin{proof}[Sketch]
\OGGEP{} states that observable boundary content is related to bulk
geometric content by an irreversible CPTP bulk-to-boundary projection.
\GRIP{} formalizes the dynamical content of this projection: the
cascade is the iterated application of the boundary projector, the
gradient flow is the dissipative dynamics, and the \GA{}'s are the
fixed points of the iterated projection. Conversely, given \GRIP{}
satisfying its five Structure Theorems, one recovers the CPTP-projection content
of \OGGEP{} via the gradient flow.
\end{proof}


%% --- Phase 8 TikZ figure: GRIP cascade with BI-6 budget ---
\begin{figure}[ht]
\centering
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 stage/.style={draw=green!55!black, rounded corners=4pt, line width=0.80pt, minimum width=3.6cm, minimum height=2.2cm, align=center, fill=green!10},
 arrow/.style={->, line width=1.10pt, >=Stealth},
 budgetbox/.style={draw=olive!55!black, rounded corners=3pt, fill=yellow!15, align=center, font=\sffamily\footnotesize, minimum width=4.6cm},
 edgelabel/.style={font=\sffamily\footnotesize, fill=white, inner sep=1pt},
]
% Three stages of the cascade
\node[stage] (DGI) at (0, 0) {\textbf{\DGI{}}\\ \footnotesize Dynamical\\ Gauge\\ Invariance};
\node[stage] (GSSB) at (5, 0) {\textbf{\GSSB{}}\\ \footnotesize Geometric\\ Spontaneous\\ SB};
\node[stage] (GA) at (10, 0) {\textbf{\GA{}}\\ \footnotesize Geometric\\ Attractor};

% Cascade arrows
\draw[arrow] (DGI) -- (GSSB) node[edgelabel, midway] {Wilson--Hosotani};
\draw[arrow] (GSSB) -- (GA) node[edgelabel, midway] {gradient flow};

% Instanton-action ladder
\node[budgetbox] (S1) at (2.5, -2.7) {$S_{[6]\to[4]} = \pi\cdot 1 = \pi$};
\node[budgetbox] (S2) at (7.5, -2.7) {$S_{[4]\to[2]} = \pi\cdot 2 = 2\pi$};
\draw[arrow, gray!70] (DGI.south) to[bend right=10] (S1.north);
\draw[arrow, gray!70] (GSSB.south) to[bend right=10] (S2.north);

% Total / Budget box
\node[draw=red!55!black, rounded corners=3pt, fill=red!12, align=center, font=\sffamily, line width=0.80pt, minimum width=10cm] (TOT) at (5, -4.6)
 {\textbf{Total cascade action on $\CPN$:}\\
 $\Stot(\CPN) = \sum_{k=1}^{N-1}\pi k = \pi N(N-1)/2 \le \Sbi = 3\pi$ (\BIsix{} budget)\\
 \footnotesize forces $N \le 3$ (T4.1, saturated at $N=3$)};

\draw[arrow, dashed] (S1) -- (TOT);
\draw[arrow, dashed] (S2) -- (TOT);

% PBI-4 compensator on right
\node[draw=orange!55!black, rounded corners=3pt, fill=orange!15, align=center, font=\sffamily, minimum width=4cm] (PBI) at (14.5, 0)
 {\textbf{\PBIfour{} compensator}\\ \footnotesize $C_\mu(t)$ smooth across\\ each instanton stratum\\ (T4.2 iff, M11)};
\draw[arrow, dashed, blue!60!black] (GA) -- (PBI);
\end{tikzpicture}%
}
\caption[The \GRIP{} cascade with the \BIsix{} budget closure (Pillar 4).]{The \GRIP{} cascade $\DGI{} \to \GSSB{} \to \GA{}$ (Pillar 4 of \GiFour{}) with the \BIsix{} budget closure. Three named stages span the cascade: dynamical gauge invariance preserves gauge structure on the family-projector moduli; geometric spontaneous symmetry breaking (Wilson--Hosotani mechanism) selects the boundary vacuum; geometric attractors are the cascade fixed points. Each cascade step costs $\pi k$ units of instanton action, accumulating to $\Stot(\CPN) = \pi N(N-1)/2$ on the family-projector stratum $\Sigma_{\rm fam} = \CPN$. The cumulative action must not exceed the \BIsix{} budget $\Sbi = 3\pi$ (the capacity theorem T-N3-6 of companion paper \citet{ArisakaP2N3}; open problem M10, demand met), which forces $N \le 3$ and is saturated at $N = 3$. Runtime gauge-invariance preservation along the cascade (T4.2) holds iff the \PBIfour{} compensator $C_\mu(t)$ extends smoothly across each instanton stratum (open problem M11 tracks the topological non-obstruction).}
\label{fig:grip-cascade}
\end{figure}

%% ====================================================================
\section{The \DGI{} -- \GSSB{} -- GA cascade in detail}
\label{sec:DGI-GSSB-GA}
%% ====================================================================

We now develop the three stages of the \GRIP{} cascade --- \DGI{},
\GSSB{}, \GA{} --- in detail.

\subsection{\DGI{}: Dynamical Gauge Invariance}
\label{ssec:DGI-detail}

\begin{definition}[\DGI: Dynamical Gauge Invariance]
\label{def:DGI}
The \emph{Dynamical Gauge Invariance} configuration on a stratum
$\Sigma$ is the critical point $c_{\rm top}(\Sigma)$ of
$\cF_{\GFF}|_\Sigma$ with maximum Morse index, equal to
$\dim_\R\Sigma$: the top of the cascade, at which every gauge
equivalence is intact and nothing has yet committed. It corresponds
to the unbroken phase of the boundary projector on $\Sigma$, with the
full \TGG{} composite-connection symmetry dynamically preserved.
\end{definition}

\textbf{Physical interpretation.} In the family-projector
stratum $\Sigma_{\rm fam} = \CPN$, the \DGI{} point is the family
configuration that has the most symmetry: equal weight on all $N$
family directions. This corresponds to a maximally symmetric flavor
basis with no preferred family. In the Higgs stratum $\Sigma_H$, the
\DGI{} point is the symmetric (unbroken) electroweak phase $V=0$.

\textbf{Stability properties.} The \DGI{} point is a Morse maximum, so
it is unstable in all directions: small perturbations drive the system
into the cascade. Its decay timescale is set by the instanton action
of the first cascade transition, $S_{[\dim\Sigma]\to[\dim\Sigma - 2]} = \pi(N-1)$
on $\CPN$.

\subsection{\GSSB{}: Geometric Spontaneous Symmetry Breaking}
\label{ssec:GSSB-detail}

\begin{definition}[\GSSB: Geometric Spontaneous Symmetry Breaking]
\label{def:GSSB}
The \emph{Geometric Spontaneous Symmetry Breaking} chain on a stratum
$\Sigma$ is the sequence of intermediate saddle points
$c_1, c_2, \ldots, c_{n-1}$ in the cascade
\eqref{eq:GRIP-ST1}, with Morse indices
$\dim\Sigma - 2, \dim\Sigma - 4, \ldots, 2$. Each saddle corresponds
to a partial breaking of the boundary projector symmetry.
\end{definition}

\textbf{Physical interpretation.} Each saddle in the \GSSB{} chain
represents a phase with intermediate symmetry breaking. In the
family-projector stratum $\CPN$, the saddles correspond to flavor
configurations with progressively less family symmetry: the index-$2k$
saddle has $k$ ``frozen'' family directions and $N-k$ ``free'' ones.

\textbf{Cascade dynamics.} Each saddle decays into the next by an
instanton of action $\pi k$ (where $k$ is the Morse-index drop). The
\GSSB{} chain therefore traces out a sequence of partial symmetry
breakings, each with its own instanton timescale.

\textbf{Connection to fermion masses.} In the application to the
Standard Model (Part~VII), each saddle in the \GSSB{} chain on
$\CPtwo$ (the $N=3$ family stratum) corresponds to a generation:
the index-$4$ \DGI{} point is associated with the third generation
(maximally unstable), the index-$2$ saddle with the second generation
(intermediate), and the index-$0$ \GA{} with the first generation
(stable). This is the structural origin of the e$^{-n\pi}$ cascade in
the \ANine{} Yukawa codebook.

\subsection{GA: Geometric Attractor}
\label{ssec:GA-detail}

\begin{definition}[\GA: Geometric Attractor]
\label{def:GA}
The \emph{Geometric Attractor} on a stratum $\Sigma$ is the critical
point $c_{\rm bottom}(\Sigma)$ of $\cF_{\GFF}|_\Sigma$ with Morse
index $0$, i.e., a local minimum of the \GFF{} Morse function. It
corresponds to a stable boundary observable.
\end{definition}

\textbf{Physical interpretation.} The \GA{} is the asymptotic
endpoint of the cascade: the dynamically selected, stable boundary
observable. In the family-projector stratum, the \GA{} corresponds to
the first generation (electron, up, down) --- the lightest, most stable
fermions. In the Higgs stratum, the \GA{} is the broken electroweak
vacuum $V = v\simeq 246$\,GeV. In the Wilson-line stratum, the \GA{}
is one of the discrete \OGNfive{} phases.

\textbf{Stability properties.} The \GA{} is a Morse minimum, so it is
locally stable: small perturbations relax back to the minimum. No
further cascade transitions are possible from the \GA{}; the cascade
on $\Sigma$ has terminated.

\textbf{Catalog of physical \GA{}'s.} The catalog of \GA{}'s
across all strata of $\Bdy$ constitutes what we observe as the
``physical world'' --- the catalog of stable boundary observables.
This catalog includes: the three Standard-Model fermion generations
(family-projector stratum), the broken electroweak vacuum (Higgs
stratum), the gauge couplings $\alpha_3, \alpha_W, \alpha_Y$ (gauge
stratum), the IR scale $\Mlock = \EPl/2\e^{35} \simeq 3.85\,\TeV$
(radial-scale stratum), and the Kaxion field value (Wilson-line
stratum).

\subsection{GGA: Ground Geometric Attractor}
\label{ssec:GGA-detail}

A Morse landscape generally admits \emph{many} index-$0$ minima ---
many \GA{}'s --- and Definition~\ref{def:GA} alone therefore does not
yet say \emph{which} stable boundary observable the cascade realizes.
The missing clause is the ground-state refinement, and the present
paper owns it: Corollary~\ref{cor:GGA-per-stratum} establishes, on any
admissible stratum, that the descent halts at a unique deepest basin.
The definition below names that object. Its generalization across the
layers of the program-wide cascade is the capstone's contribution
\citep{ArisakaA11Origin}; the runtime statement is here.

\begin{definition}[\GGA: Ground Geometric Attractor]
\label{def:GGA}
Let $\{\GA^{(1)},\GA^{(2)},\ldots\}$ be the set of Geometric
Attractors (Definition~\ref{def:GA}) of $\cF_{\GFF}$ on a stratum
$\Sigma$. The \emph{Ground Geometric Attractor} of $\Sigma$ is the
global minimum:
\begin{equation}
\label{eq:GGA-def}
\GGA(\Sigma) \;:=\; \GA^{*} \;=\;
\operatorname*{arg\,min}_{x\in\Sigma}\, \cF_{\GFF}(x).
\end{equation}
The cascade of Definition~\ref{def:GRIP} runs
$\GRIP := \DGI \to (\GFF) \to \GSSB \to \GA$, and on an admissible
stratum it terminates not merely on \emph{a} \GA{} but on \emph{the}
\GGA{}: the stage is the Geometric Attractor, the terminus is the
Ground Geometric Attractor. Existence and uniqueness are
Corollary~\ref{cor:GGA-per-stratum}, and no filter beyond the
landscape's own geometry is used in establishing them. Dynamically,
the \GGA{} is \emph{where \GRIPin{} halts}
(Definition~\ref{def:GRIPin} below): the unique attractor from which
no admissible exit channel remains.
\end{definition}

\textbf{Physical interpretation.} Chemistry made the distinction a
century ago. A molecule's electronic landscape has many stationary
configurations, but chemistry is built on the \emph{ground state}:
every water molecule is identical, and indefinitely stable, because
its electrons occupy not merely \emph{a} stable configuration but
\emph{the} lowest one. The \GGA{} is that idea promoted to a
constitutional principle of the runtime: the cascade does not settle
into \emph{some} attractor; it settles into the ground state of the
attractor manifold.

\subsection{GGA and the stability of matter}
\label{ssec:GGA-stability}

The refinement from \GA{} to \GGA{} is what converts the catalog
of stable boundary observables from a description into an
\emph{explanation}. A \GA{} is stable against small perturbations
(its Hessian is positive definite); only a \GGA{} is stable against
\emph{tunneling}, because on its stratum there is no deeper basin to
decay to. The stability of matter is, clause by clause, a \GGA{}
statement:

\begin{itemize}
\item \textbf{The electron is stable} because the charged-lepton
stratum's \GGA{} is the configuration $(m_e,-e,\tfrac12\hbar)$: there
is no lower-lying charged-lepton basin, and every cascade
realization, anywhere in spacetime, selects this same unique minimum
--- which is simultaneously why every electron is rigorously
identical to every other. That last clause is worth stating as the
mechanism it is: identity is not an extra postulate about particles
but a consequence of uniqueness, because a basin has no interior
degrees of freedom for two readouts of it to differ by. The
charged-lepton stratum is $\CPtwo$ and is admissible, so the
electron's case rests on
Corollary~\ref{cor:GGA-per-stratum} directly
(\S\ref{ssec:GGA-worked}).
\item \textbf{The proton is stable} because, with $\UoneB$ gauged
(\ANine{}), the baryonic stratum's ground attractor is the lightest
$B=1$ configuration: gauge invariance removes every operator that
would connect it to a deeper basin, so ``no deeper basin'' holds not
approximately but exactly. The baryonic stratum's Morse structure is
not developed in the present paper, so this reading is carried at the
grade of its own stratum rather than derived from
Corollary~\ref{cor:GGA-per-stratum}; the dedicated treatment is the
companion Arisaka-GG-P6-Proton.
\item \textbf{The Kaxion is stable} (and with it the dark matter of
the predictions branch) because it sits in the locked compact-angle
phase of the Wilson-line stratum, whose only loss channel is the
portal-suppressed leak off the stratum, with a lifetime tens of orders
of magnitude beyond the age of the universe (quantified in the
companion paper Arisaka-GG-C1-Kaxion). This case is stated more
carefully than the other two, and deliberately. The Wilson-line
stratum is the class Theorem~\ref{thm:GFF-positivity-from-CCI} defers,
its potential is periodic over the discrete \OGNfive{} phases, and a
periodic potential may carry several minima of equal depth; so
Corollary~\ref{cor:GGA-per-stratum} is not available here and no
uniqueness claim is made. What is claimed --- stability of the locked
phase against every channel within the stratum --- does not need
uniqueness, and is what the dark-matter argument uses.
\end{itemize}

The same clause explains the \emph{reproducibility} of physics: the
gear ratios and coupling locks of the \OGNfive{} ledger are
properties of \emph{the} \GGA{} of each stratum --- as the Rydberg is
a property of \emph{the} hydrogen ground state --- rather than
averages over a population of attractors that could drift or
disperse. Wherever this paper's earlier sections say ``the \GA{} of
the stratum'' in a context where uniqueness is doing work (the
first-generation fermions, the electroweak vacuum, the Kaxion phase),
the reader should henceforth read the sharper statement: \emph{the}
\GGA{}.

\subsection{The two regimes of the runtime: \GRIPin{} and \GRIPout{}}
\label{ssec:grip-in-out}

The runtime defined in this paper operates in exactly two regimes,
distinguished by one question: \emph{does the stratified landscape
change?} Since the present paper is the parent of the \GRIP{} concept,
the \emph{definitions} belong here, at runtime level, together with the
one structural fact that distinguishes them
(Remark~\ref{rem:GRIPout-necessity}). The capstone \AEleven{}
\citep{ArisakaA11Origin} carries the Lyapunov descent theorem
\textsf{T-GRIP-In-Lyapunov-Descent} and the full seventeen-event theory
of landscape reorganization.

\begin{definition}[\GRIPin{}: the fixed-landscape regime]
\label{def:GRIPin}
\GRIPin{} is the dynamics of the septuple of
Definition~\ref{def:GRIP} on a \emph{fixed} stratified landscape
$(\Bdy, \cF_{\GFF})$: the descent that carries any initial datum down
the chain $\DGI \to \GSSB \to \GA$ of its stratum, terminating at that
stratum's ground attractor. Its anatomy is
piecewise. \emph{Within} a basin the dynamics is the Riemannian
gradient flow \eqref{eq:GRIP-ST2}; \emph{between} basins the descent
proceeds by the instanton trajectories of \textbf{GRIP-ST3}, with
Bogomolnyi actions $S = \pi k$ crossing the \GSSB{} gates.
On an admissible stratum \GRIPin{} terminates, and terminates
uniquely: by Corollary~\ref{cor:GGA-per-stratum} the descent halts
exactly at the stratum's ground state --- equivalently, at the unique
attractor from which no admissible exit channel remains. The \GGA{} is, dynamically,
\emph{where \GRIPin{} halts}.
\end{definition}

\begin{definition}[\GRIPout{}: the landscape-reorganizing regime]
\label{def:GRIPout}
\GRIPout{} is the complementary regime: a transient,
out-of-equilibrium event that \emph{reorganizes the stratified
landscape itself} --- opening strata and basins that did not exist on
the prior landscape --- and thereby supplies the initial datum
$c_{\rm top} = \DGI$ from which a new \GRIPin{} descent begins.
\GRIPout{} is not generated by the gradient flow and is deliberately
not an element of the septuple \eqref{eq:GRIP-septuple}. Its necessity
for cascade ascent is Remark~\ref{rem:GRIPout-necessity} below; its
full theory --- the sixteen landscape-reorganizing events from the GUT
exit to the conscious episode --- resides in \AEleven{}
\citep{ArisakaA11Origin}. Within the present paper, \GRIPout{}
appears in exactly one role: it is where each stratum's
$c_{\rm top}$ comes from.
\end{definition}

\begin{remark}[Why a second regime is necessary at all]
\label{rem:GRIPout-necessity}
The necessity of \GRIPout{} is a one-line topological fact, and since
the present paper owns the runtime it should be carried here rather
than deferred. The \GRIPin{} generator of \textbf{GRIP-ST2} is a vector
field on a stratum, so its integral curves remain on that stratum for
as long as they exist; and by \textbf{GFF-ST1} the strata are labeled
by the \emph{discrete} data of the boundary projector class, data that
no continuous curve can change. A gradient trajectory therefore never
leaves the stratum it starts on --- which is the same statement
\textbf{GRIP-ST5} makes at the far end, that after cascade closure any
further transition requires leaving the stratum. Consequently a
cascade cannot ascend onto a stratum that did not previously exist by
gradient flow alone: some non-gradient, out-of-equilibrium event must
supply the new landscape and the new $c_{\rm top}$. That event is
\GRIPout{}, and this is the whole of what the present paper needs from
it. Which events they were, and in what order, is the capstone's
subject.
\end{remark}

\paragraph{The alternation.} Every cascade history is an alternation
of the two regimes: a \GRIPout{} event opens a landscape; \GRIPin{}
finishes the journey, halting at the \GGA{}; the settled \GGA{}
becomes the substrate on which the next \GRIPout{} can act
(\textsf{Cascade-Self-Sustenance}, \AEleven{}). In one sentence:
\emph{\GRIPout{} opens the landscape; \GRIPin{} finishes the
journey.} The electroweak transition and the family ladder make the
division concrete. The selection of the electroweak vacuum was a
\GRIPout{} event --- it created the Yukawa landscape on which the
three fermion generations are basins --- while every weak decay since
($\tau\to\mu\to e$; the valley of $\beta$ stability) is \GRIPin{} on
that fixed landscape, as the next subsection develops and as
\ANine{} deploys across the entire Standard Model.


\begin{figure}[p]
\centering
\resizebox{0.833\textwidth}{!}{%
\usebox{\GGscCascade}}
\caption[The \GRIP{} cascade and its two regimes.]{\small\textbf{The \GRIP{} cascade and its two regimes, in one picture.}
The central column is the runtime of Definition~\ref{def:GRIP}: from the
fully symmetric top (\DGI{}, every gauge equivalence intact), the system
descends the fixed landscape $\cF_{\GFF}$, selects branches at the
\GSSB{} saddles, and is captured into a stable basin --- a Geometric
Attractor (\GA{}). The attractor tier has two levels, and the
distinction carries the two-regime story of
\S\ref{ssec:grip-in-out}: from a \emph{non-ground} \GA{} (a way
station: a muon, an excited atom) the descent \emph{continues} ---
\GRIPin{} (Definition~\ref{def:GRIPin}) proceeds through the basin's
exit channels, the weak interaction being Nature's implementer on the
matter strata (\S\ref{ssec:weak-grip-example}) --- whereas at the
\GGA{}, the unique deepest basin of \textbf{GFF-ST7}
(Theorem~\ref{st:GFF-ST7}), the exit-channel set is empty and
\GRIPin{} \emph{halts}. After the halt, only \GRIPout{}
(Definition~\ref{def:GRIPout}) can act: a transient out-of-equilibrium
event reorganizes the landscape itself and supplies the initial datum
of the next descent. In one sentence: \GRIPout{} opens the landscape;
\GRIPin{} finishes the journey; the \GGA{} is where the journey ends.
(Adapted from the four-move pipeline figure of \AEleven{}
\citep{ArisakaA11Origin}, which deploys this alternation through
eighteen cascade layers.)}
\label{fig:grip-two-regimes}
\end{figure}

\paragraph{Two boundary clarifications.} Each is recorded here
because each invites a misreading. \emph{(i) \GSSB{} is a gate, not a
regime.} \GSSB{} names the symmetry-breaking saddle; such gates are
crossed both within \GRIPin{} chains (the intermediate saddles of
$\DGI\to\GSSB\to\GA$) and at \GRIPout{} shocks (the electroweak
transition). The regime label attaches to the dynamics --- is the
landscape fixed or reorganizing? --- not to the gate. \emph{(ii)
Port-driven excitation is not \GRIPout{}.} Pumping a state uphill
through the boundary ports (producing a $\tau$ lepton or a top quark
in a collider) is an excitation \emph{within} the fixed landscape,
promptly followed by \GRIPin{} relaxation; it reorganizes nothing.
Only events that change the landscape itself are \GRIPout{}.

\subsection{The two regimes and the two fibers: the engine's strokes}
\label{ssec:two-regimes-two-fibres}

The two-regime structure just defined --- distinguished by one
question, \emph{does the landscape change?} --- has a geometric
root that the cohort's recent campaign has made exact
(\citealp{ArisakaW1Symmetry}; \citealp{ArisakaW2Holonomy};
\citealp{ArisakaA5GDOS}; \citealp{ArisakaP5CPT}), and stating it
here upgrades
Remark~\ref{rem:GRIPout-necessity} from a dynamical observation to
a topological one. The instrument is one statement: \emph{holonomy
requires closed loops, and an interval has none.} The bulk's
compact circle $S^1_\theta$ is nothing but closed loops ---
everything on it is holonomy, quantized and protected, and its
transport is a group; the radial interval $I_r$ can hold no
holonomy at all --- its transport composes only forward, a
semigroup. The runtime's two regimes are these two fibers,
operating: \emph{\GRIPin{} is the circle's stroke} --- dynamics on
a \emph{fixed} landscape, cycling the four moves within a stratum
to the unique ground attractor, with the outcome deposited in
exactly the quantities the circle protects (charges, windings,
parities: the \GGA{}'s labels); \emph{\GRIPout{} is the interval's
stroke} --- the transient, one-way, landscape-reorganizing event
that no gradient flow generates and no operation reverses, the
semigroup acting at runtime level. The engine is the bulk in
miniature: one stroke that closes and remembers, one stroke that
advances and forgets.

Read this way, the necessity of the second regime
(Remark~\ref{rem:GRIPout-necessity}) acquires its shortest proof
sketch: a runtime with only \GRIPin{} would be all circle --- it
would relax, once, to one stratum's ground state and cycle there
forever, deciding nothing further; a runtime with only \GRIPout{}
would be all interval --- perpetual reorganization with no
convergence, nothing definite ever deposited. The cascade of
nature requires exactly one stroke of each kind for the same
topological reason the bulk carries exactly one fiber of each
kind: \emph{becoming} needs a semigroup and \emph{being} needs
holonomies, and the alternation \GRIPin{} $\to$ \GRIPout{} $\to$
\GRIPin{} is the minimal dynamics in which a world can do both.
The campaign's determinism result then reads as a statement about
this paper's own objects: the vacuum manifold is a quantized dial
(the circle's alphabet), the \GFF{} landscape has a unique floor
per stratum (Corollary~\ref{cor:GGA-per-stratum} --- the Morse
mechanism), and the between-stratum motion cannot re-choose (the
interval's semigroup) --- three mechanisms, all defined in this
paper, jointly closing every door through which randomness enters
the standard accounts of symmetry breaking. The downstream
payoffs are carried at their homes: the eighteen-layer cascade
and its chronology --- uniform in the geometry, exponential in
the clock, $d\ln t/dr = 2$ --- in the capstone
\citep{ArisakaA11Origin} and the CPT companion's treatment of
time \citep{ArisakaP5CPT}; the measurement arrow, where
\GRIPin{}'s selection and the interval's write meet in the
laboratory, in the quantum companion \citep{ArisakaA8QM}. Within
the present paper the sentence to keep is the definitional one:
\emph{the runtime has two regimes because the bulk has two
fibers, and the question that separates the regimes --- does the
landscape change? --- is the question of which fiber is doing the
work.}

\subsection{The two stages and the two fibers: custody of
transport, custody of data}
\label{ssec:stage-custody}

The mapping just stated classifies the two \emph{regimes} by where
their outcomes live: \GRIPin{} banks its result in the classes the
circle protects; \GRIPout{} changes the landscape and banks nothing.
A second, finer mapping classifies the \emph{stages} of the cascade
itself, and it runs along a different axis --- not where the outcome
is kept, but what algebra the dynamics composes under. The two
mappings are consistent, but only if a distinction is kept that this
subsection makes explicit: \emph{custody of transport} and
\emph{custody of data} are different things, and the cascade's middle
stage holds them on different fibers.

\paragraph{The descent is interval transport.} The first half of the
stage mapping is a composition of two results already established in
the boundary-law companion, and is stated at the grade that
composition earns.

\begin{proposition}[The \GSSB{}$\to$\GA{} leg realized as the scale
fiber's semigroup]
\label{prop:GSSB-transport}
On the boundary, the descent that executes the \GSSB{}$\to$\GA{} leg
of the cascade --- the \GRIPin{} flow of Definition~\ref{def:GRIPin}
--- is realized as a completely positive, trace-preserving forward
semigroup whose generator is the dilatation generator along $I_r$
projected to the boundary. In particular the selection performed at
each \GSSB{} gate composes under the interval's semigroup law:
forward only, with no inverse.
\end{proposition}

\begin{proof}
Composition of two results of \citet{ArisakaA5GDOS}. First, the
\GDOS{}-to-\GiFour{} bridge identifies the cascade with the
long-time \GDOS{} flow on the boundary moduli, stage for stage, with
the four boundary Ward identities preserved; the identification
itself is unconditional, while the $N \le 3$ closure inherits the
conditionality of the capacity theorem it rides on
\citep{ArisakaP2N3}. Second, the same companion identifies the
dissipative part of the \GDOS{} (GKLS) generator as the boundary
image of motion along the scale fiber --- the dilatation generator
along $I_r$, projected --- which is exactly what makes the family of
boundary channels a semigroup rather than a group. Composing the two
identifications gives the statement. The irreversibility clause of
\textbf{GRIP-ST2} --- the flow descends in $\cF_{\GFF}$ and never
ascends --- is thereby traced to its geometric root: the fiber the
descent runs on has no closed loops, hence no inverse transport.
\end{proof}

\paragraph{The invariance is the circle's.} At the other end of the
cascade, \DGI{} holds a symmetry intact, and the quantum companion
fixes exactly which geometry that symmetry belongs to: the phase
direction $S^1_\theta$ is a genuine principal $U(1)$-bundle over the
boundary --- connection, holonomy, and all --- while the scale fiber
$I_r$ is \emph{not} a gauge bundle, its ``structure group'' being the
one-way dilatation semigroup \citep{ArisakaA8QM}. Gauge structure, in
this cohort, lives on the circle and only on the circle. The
equivalences that \DGI{} keeps intact therefore compose as the
circle's transport --- a group --- and the constitutional ladder of
\POS{1} (\LGI{}/\DGI{}/\GGI{}) acquires a geometric reading:
\LGI{} is the circle's invariance stated in a closed theory; \DGI{}
is the same invariance carried through the open dynamics, the
equivalence class of boundary generators under induced gauge
transformations in the operator form of the information companion
\citep{ArisakaA7Qi4}. This identification is carried at
derived-conditional grade: it rests on the bundle asymmetry of
\citet{ArisakaA8QM} and on reading the Morse-top stabilizer of
Definition~\ref{def:DGI} as the boundary gauge group stratum by
stratum --- immediate on the gauge and Wilson-line strata, open in
general (problem M12 of \S\ref{ssec:open-math}).

\paragraph{The middle stage crosses, and the crossing is the
content.} \GSSB{} reads circle data and runs on the interval. Its
order parameters are Wilson-line phases on $S^1_\theta$
(Section~\ref{ssec:holonomy-global}); its execution --- the one-way
act of selection --- is the interval's semigroup
(Proposition~\ref{prop:GSSB-transport}); and its outcome is
deposited back into the classes the circle protects, the \GGA{}'s
labels. This is not a defect of the correspondence; it \emph{is} the
correspondence. Symmetry breaking, in the cohort's reading, is
symmetry \emph{reading} \citep{ArisakaW1Symmetry}: the alternatives
and the record belong to the fiber that remembers; the choice
belongs to the fiber that forgets. The shape is already in print
elsewhere --- the measurement write of the quantum companion is
interval-side while the record is circle-kept \citep{ArisakaA8QM} ---
so \GSSB{} stands to symmetry breaking exactly as the write stands
to measurement: one seam-event shape, two instances. The electron
paper exhibits both custodies on a single object --- charge and
identity circle-side, mass and decay rate interval-side, the descent
realized as an explicit GKLS semigroup whose terminus carries no
jump operator \citep{ArisakaP3Electron} --- and compresses the
correspondence to a sentence: an electron is a holonomy class
wearing a mass. This paragraph is carried at the reading grade: an
organization of proved structures, with the general theorem deferred
to M12.

\begin{table}[htbp]
\centering
\small
\caption[The \GRIP{} stages and the custody of their data.]{The four
stages of the \GRIP{} engine, each with the transport algebra it runs
on, the fiber that keeps its data, and the reading that follows. The
third column is the one to read down: the circle keeps what is
protected and the interval carries what is irreversible, and every
stage divides its data between the two the same way. The table is the
engine's custody ledger --- where a quantity lives decides whether it
can be undone --- and it is what makes symmetry breaking and
measurement two instances of one seam-event rather than two separate
phenomena.}
\label{tab:grip-stage-custody}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.13\textwidth}>{\raggedright\arraybackslash}p{0.25\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}@{}}
\toprule
Stage & Transport algebra & Data custody & Reading \\
\midrule
\DGI{} & the circle's group: gauge equivalences intact & bulk-wide
symmetry; the phases made dynamical are $S^1_\theta$'s & the
invariance is the circle's; the dynamism is the interval's \\
\GFF{} & none --- the landscape, not a transport & fed by both
fibers: \GEP{} through the Wilson line; $\Phi_\omega = \pi$ from
warp $\times$ circumference & the seam's price list \\
\GSSB{} & the interval's semigroup
(Prop.~\ref{prop:GSSB-transport}) & order parameter and alternatives
on the circle; outcome deposited to circle classes & symmetry
breaking as symmetry reading \\
\GA{}/\GGA{} & the halt of the semigroup & labels circle-protected
(charges, windings, parities) & where the interval stops and the
circle keeps \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Why the two mappings do not collide.} The stroke mapping
of \S\ref{ssec:two-regimes-two-fibres} and the stage mapping of this
subsection classify the same engine along orthogonal axes. The
strokes are named for their \emph{products}: \GRIPin{} is the
circle's stroke because its outcome is banked in circle classes;
\GRIPout{} is the interval's because it changes the landscape and
banks nothing. The stages are named for their \emph{transport}: the
\DGI{} equivalences compose as the circle's group; the descent
composes as the interval's semigroup. Both classifications are
pullbacks of the one fork --- holonomy requires closed loops, and an
interval has none --- one along the outcome axis, one along the
dynamics axis. In particular, ``\GRIPin{} is the circle's stroke''
and ``the \GRIPin{} descent is interval transport'' are both true,
and neither sentence can replace the other: the first says where the
result is kept, the second says what algebra the getting-there
composes under. The sentence to keep is the custody sentence:
\emph{the circle states the alternatives; the interval performs the
selection; the attractor is the circle's receipt.}

\subsection{\GRIP{} ending at the \GGA{}, realized in Nature: the weak interaction}
\label{ssec:weak-grip-example}

The cascade of this paper is not an abstraction awaiting application;
Nature runs it daily, and the runner has a name. Among the four
boundary forces, the strong and electromagnetic interactions are
flavor-diagonal: they \emph{build} the basins of the matter strata
(hadrons, nuclei, atoms) and relax excitations \emph{within} a basin
($\rho\to\pi\pi$; an excited atom radiating to its ground state), but
they cannot carry a state from one family basin to another. The
\emph{weak charged current} is the unique flavor-connecting channel of
the boundary group --- and therefore the unique physical implementer
of the fixed-landscape regime defined above: the weak charged current
is how Nature runs \GRIPin{} (Definition~\ref{def:GRIPin}) on the
matter strata.
Two clean clauses summarize the categorization: every inter-basin
descent observed in Nature is weak, and every weak flavor transition
is a step of the $\cF_{\GFF}$ descent toward the stratum's \GGA{}. The
division of labor is exactly that of Definition~\ref{def:GGA}: the
\GGA{} fixes the \emph{direction} of relaxation, the open-system
boundary law supplies the \emph{arrow}, and the weak coupling sets the
\emph{rate}.

The canonical example deserves to be named here because it is the most
photographed Morse landscape in science: the valley of $\beta$
stability. At fixed mass number $A$, the nuclear binding energy is a
parabola in proton number $Z$, and $\beta^-$, $\beta^+$, and
electron-capture transitions step a nuclide down that parabola, one
weak transition at a time, to the most-bound isobar --- the isobaric
\GGA{}. The chart of nuclides \emph{is} a stratified \GFF{} landscape;
$\beta$ decay \emph{is} its gradient flow. The same channel tracks the
\emph{conditional} ground state when the stratum changes: in a neutron
star, where the free energy includes density and degeneracy pressure,
the very same interaction runs ``uphill'' in vacuum masses
($p+e^-\to n+\nu$) because neutron matter is the \GGA{} of \emph{that}
stratum --- the strongest evidence that the weak interaction descends
$\cF_{\GFF}$ rather than a fixed mass table. The family ladders
($\tau\to\mu\to e$; $t\to b\to c\to s\to u/d$) are the elementary
instances, developed with their lifetime and ordering consequences in
\ANine{}, which devotes a dedicated subsection to this reading and its
examples; the dark sector completes the pattern, the Kaxion condensate
relaxing toward its \GGA{} by Hubble friction itself (the companion
paper Arisaka-GG-C1-Kaxion). In one sentence: \emph{the strong and
electromagnetic forces build the landscape; the weak force is how
matter walks downhill in it; and the walk ends, always, at the
\GGA{}.} And in regime language (\S\ref{ssec:grip-in-out}): every
weak process in Nature is a step of \GRIPin{}; no weak process is a
\GRIPout{}. The one \GRIPout{} of the electroweak story happened
once, when the vacuum was selected and the landscape built; the weak
interaction has been walking that landscape ever since.

%% ====================================================================
\section{Cascade dynamics: gradient flow and instanton trajectories}
\label{sec:GRIP-gradient-flow}
%% ====================================================================

We now formalize the dynamical content of \GRIP{} in detail.
Everything in this section concerns the \GRIPin{} regime
(Definition~\ref{def:GRIPin}): the two halves that name the section
--- gradient flow within basins and instanton trajectories between
them --- are precisely the piecewise anatomy of \GRIPin{}. The
landscape itself is fixed throughout; \GRIPout{} events sit outside
the septuple (Definition~\ref{def:GRIPout}) and enter only as the
suppliers of the initial data $c_{\rm top}$.

\subsection{The gradient flow on stratified \GFF{}}
\label{ssec:gradient-flow}

The cascade \eqref{eq:GRIP-ST1} is generated by the gradient flow
\eqref{eq:GRIP-ST2} of $\cF_{\GFF}$ on each stratum. Concretely, for a
point $x\in\Sigma$ that is not a critical point of $\cF_{\GFF}|_\Sigma$,
the flow
\begin{equation}
\label{eq:gradient-flow-explicit}
\dot x \;=\; -\,g_\Sigma^{ab}(x)\,\partial_b \cF_{\GFF}(x)\,\partial_a,
\end{equation}
descends along the steepest-descent direction of the Riemannian
metric $g_\Sigma$. The flow has the following properties:

\begin{itemize}[leftmargin=2em]
\item \textbf{Monotonicity.} $\dd \cF_{\GFF}(x(t))/\dd t \leq 0$, with
equality iff $x$ is a critical point. This is the irreversibility of
the cascade.
\item \textbf{Asymptotics.} As $t\to\infty$, $x(t)$ approaches a
critical point of $\cF_{\GFF}|_\Sigma$. Generically, the asymptotic
limit is the unique \GA{} of the basin containing the initial
condition.
\item \textbf{Trajectory connecting saddles.} Between two adjacent
critical points $c_+, c_-$ with $\ind(c_+) - \ind(c_-) = 2$, there
exists (generically) a one-parameter family of trajectories from
$c_+$ to $c_-$ along the unstable manifold of $c_+$. These are the
``connecting trajectories'' of Morse homology.
\end{itemize}

\subsection{Instanton action between adjacent saddles}
\label{ssec:instanton-action}

\begin{theorem}[Instanton action on Kähler strata]
\label{thm:instanton-action}
Let $\Sigma$ be a Kähler stratum equipped with the canonical
height-function Morse form
$\cF_{\GFF}(x) = \sum_a \lambda_a |\psi_a|^2 / \sum |\psi_a|^2$.
The instanton action of the gradient-flow trajectory from the
index-$2k$ critical point to the index-$2(k-1)$ critical point is
\begin{equation}
\label{eq:thm-instanton-action}
S_{[2k]\to[2(k-1)]} \;=\; \pi\, k.
\end{equation}
This action saturates the Bogomolnyi bound for gradient flows on
Kähler manifolds.
\end{theorem}

\begin{proof}
We derive $S_{[2k]\to[2(k-1)]} = \pi k$ by direct computation
on $\CPN$, working out the Fubini--Study symplectic area enclosed
by the Bogomolnyi-saturated gradient trajectory between adjacent
fixed points.

\textbf{Step 1: Bogomolnyi-type inequality.} The gradient flow on
a Kähler manifold with Hamiltonian-Morse form $\cF_{\GFF}$ satisfies
\begin{equation}
\label{eq:Bogomolnyi}
S \;\geq\; \int_{c_+}^{c_-} \dd \cF_{\GFF}
\;=\; \cF_{\GFF}(c_+) - \cF_{\GFF}(c_-),
\end{equation}
with equality saturated by the gradient trajectory itself. This is
the standard Bogomolnyi bound for Euclidean gradient flow:
$S = \tfrac{1}{2}\int(|\dot x|^2 + |\nabla\cF|^2)\,\dd t \geq
\int |\nabla\cF|\,|\dot x|\,\dd t = \int \dd\cF$, with equality
iff $\dot x = -\nabla\cF$.

\textbf{Step 2: Height function on $\CPN$ in homogeneous coordinates.}
Let $[\psi_0 : \psi_1 : \cdots : \psi_N]$ be homogeneous coordinates
on $\CPN$ and let $f([\psi]) = \sum_{a=0}^{N}
\lambda_a |\psi_a|^2 / \sum_{a=0}^{N}|\psi_a|^2$ be the moment-map
Morse function induced by the maximal-torus action of $U(N+1)$ with
weights $\lambda_0 < \lambda_1 < \cdots < \lambda_N$. The critical
points of $f$ are the $N+1$ coordinate axes $[0\!:\!\cdots\!:\!1_k\!:\!\cdots\!:\!0]$,
with $\lambda_k$ the critical value and Morse index $2k$ (counting
the unstable complex directions). The weights are fixed by the winding
structure of the family chain, not by a normalization convention: the
index-$2k$ critical point hosts the winding-$\ell = k{+}1$ generation
(three generations = three windings), and its Morse height is the
pair invariant of that winding,
$\lambda_k = \binom{k+1}{2} = \tfrac{1}{2}k(k+1)$, so that
$\lambda_k - \lambda_{k-1} = k$: the spacings grow linearly down the
chain. (The origin of the pair invariant --- the inflow charge of the
saddle, $\Delta c_2 = k$ --- is the subject of
Remark~\ref{rem:cascade-concordance}, where it is derived from the
Whitney pair count of the winding bundle; the unit-spaced alternative
is excluded there on internal grounds.)

\textbf{Step 3: Bogomolnyi-saturated trajectory between adjacent
critical points.} The gradient-flow trajectory from the
index-$2k$ fixed point $p_k = [0\!:\!\cdots\!:\!1_k\!:\!\cdots\!:\!0]$
to the index-$2(k-1)$ fixed point $p_{k-1}$ traverses the projective
line $\CP^1 \subset \CPN$ spanned by $\psi_{k-1}$ and $\psi_k$.
In affine coordinates $z = \psi_{k-1}/\psi_k$ on this projective
line, the moment map restricts to
\begin{equation}
\label{eq:f-on-CP1}
f|_{\CP^1}(z) \;=\; \frac{\lambda_k + \lambda_{k-1}|z|^2}{1+|z|^2}
\;=\; \lambda_k - (\lambda_k - \lambda_{k-1})\,\frac{|z|^2}{1+|z|^2}
\;=\; \lambda_k - k\,\Omega(z),
\end{equation}
where $\Omega(z) = |z|^2/(1+|z|^2)\in[0,1]$ is the symplectic
potential of the Fubini--Study form on $\CP^1$.

\textbf{Step 4: Symplectic area of the trajectory.} The Bogomolnyi
bound \eqref{eq:Bogomolnyi} equals
\begin{equation}
\label{eq:S-as-area}
S_{[2k]\to[2(k-1)]} \;=\; f(p_k) - f(p_{k-1})
\;=\; (\lambda_k) - (\lambda_{k-1})
\;=\; k
\end{equation}
in the Fubini--Study units of Step~2; we now convert to action
units. By the Duistermaat--Heckman formula, the symplectic
volume of the trajectory's $\CP^1$ projective line is
\begin{equation}
\label{eq:CP1-symp-volume}
\int_{\CP^1} \omega_{\rm FS} \;=\; \pi,
\end{equation}
with $\omega_{\rm FS}$ the standard Fubini--Study Kähler form. Each
Bogomolnyi-saturated step traverses one such $\CP^1$, so each step
carries action $\pi$ \emph{per unit eigenvalue spacing}.

\textbf{Step 5: Conversion to action units.} Combining Step~4's
drop $k$ with the area quantum $\pi$ per unit eigenvalue spacing, the
action of the single transition $[2k]\to[2(k-1)]$ saturates the
Bogomolnyi bound at
\begin{equation}
\label{eq:S-step-pi-k}
S_{[2k]\to[2(k-1)]} \;=\; \pi\,k,
\end{equation}
directly: the factor $k$ is the weight jump
$\lambda_k - \lambda_{k-1} = \binom{k+1}{2} - \binom{k}{2}$ of
Step~2 --- equivalently, the inflow charge $\Delta c_2 = k$ of
Remark~\ref{rem:cascade-concordance} --- and no stacking or
depth-counting convention is invoked. Descending the full chain, the
accumulated actions are $0$, $2\pi$, $3\pi$, saturating the \BIsix{}
budget $3\pi$ exactly; a fourth generation ($\ell = 4$,
$\lambda = 6$) would cost a further $3\pi$, overrunning the budget
--- the $N \le 3$ forcing in its sharpest form. This completes the
derivation.

The derivation reproduces \citet[\S 4]{Witten1982} on $\CPN$ and
is a special case of the Atiyah--Bott / Frankel localization
principle for equivariant moment-map Morse functions on Kähler
manifolds.
\end{proof}

\subsection{Saturation of the Bogomolnyi bound}
\label{ssec:Bogomolnyi}

The instanton action \eqref{eq:thm-instanton-action} is the
\emph{minimum} action of any trajectory from index $2k$ to index
$2(k-1)$. Higher-action trajectories exist (involving detours through
other saddles or oscillations around critical points), but they are
exponentially suppressed in the WKB sense, so the dominant contribution
to the cascade is the Bogomolnyi-saturating trajectory.

This is what makes the cascade ``minimal'': the system follows the
shortest path from \DGI{} to \GA{}, accumulating the smallest possible
total action. Any longer cascade would exceed the \BIsix{} budget by an
amount that is exponentially suppressed.

%% ====================================================================
\section{The physical layer of \GRIP{}: gauge-runtime invariance and the generation decay-width ladder}
\label{sec:GRIP-physical-layer}
%% ====================================================================

This section develops the \emph{physical} content
of \GRIP{} (Structure Theorems \textbf{GRIP-ST6} and \textbf{GRIP-ST7}) in
operational detail: the \emph{Gauge-Cascade Compatibility Theorem}
that guarantees runtime gauge-invariance preservation along the
cascade, the explicit derivation of the lifetime-hierarchy formula
\eqref{eq:GRIP-ST7-lifetime}, the numerical confrontation with the
empirical Standard-Model lifetimes, and laboratory-accessible
falsifiers. The mathematics here combines instanton calculus with
boundary Ward-identity analysis; the conceptual novelty is that the
algebraic Bogomolnyi cascade and the physical generation decay-width ladder are
the same dynamical process viewed at different timescales.

\subsection{The Gauge-Cascade Compatibility Theorem}
\label{ssec:GRIP-gauge-compatibility}

\begin{theorem}[\textbf{Gauge-Cascade Compatibility Theorem}]
\label{thm:GRIP-gauge-cascade}
Let $\Sigma\subset\Bdy$ be a stratum supporting the algebraic cascade
$c_{\rm top}\to c_1\to\cdots\to c_{\rm bottom}$ of
\textbf{GRIP-ST1}--\textbf{GRIP-ST5}. Let $C_\mu(t)$ be the
\PBIfour{} compensator field of \CCI-ST6 evaluated along the cascade
trajectory $t\mapsto[\Pi(t)]\in\Sigma$. Then the four boundary Ward
currents $(J_e, J_B, J_{Qi}, J_g)$ satisfy the runtime relation
\begin{equation}
\label{eq:gauge-cascade-runtime}
\nabla_\mu J_a^\mu(t) \;=\; -\,\alpha_a\,C^\mu(t)\cdot \rho_a(t),
\qquad a\in\{e, B, Qi, g\},
\end{equation}
along the entire cascade $t\in[t_0, t_\infty]$, with the $\alpha_a$
the \textbf{CCI-ST2} coefficients. Moreover, the unbroken gauge
subgroup $H_i\subseteq\Gtot$ at saddle $c_i$ is contained in the
unbroken subgroup at the previous saddle: $H_i \subseteq H_{i-1}$.
\end{theorem}

\begin{proof}
The gradient flow \textbf{GRIP-ST2} is generated by the gauge-equivariant
functional $\cF_{\GFF}$, which inherits its gauge invariance from
the bulk \TGG{} action via bulk-to-boundary descent (Part~II). Therefore
the flow vector field $-g_\Sigma^{-1}\nabla\cF_{\GFF}$ is itself
gauge-equivariant, and any gauge symmetry of the initial configuration
$c_{\rm top}$ that is preserved by $\cF_{\GFF}$ is also preserved by
the flow. This gives the descending chain of unbroken subgroups
$H_0\supseteq H_1\supseteq\cdots\supseteq H_n$.

For the runtime Ward divergence \eqref{eq:gauge-cascade-runtime}, we
apply the Master Conservation Theorem
\textbf{T2.2} of Part~III at each instant $t$ along the cascade. By
\textbf{T2.2}, the four sector divergences are linked by
$C_\mu(t)$ via Proposition~\ref{prop:CCI-ST6-via-compensator}. This
gives the desired runtime relation
\eqref{eq:gauge-cascade-runtime}. The continuity of $C_\mu(t)$ across
each instanton transition has two parts. The \emph{global}
(topological) part --- the integrality of the boundary class
$[\widetilde X_2]$ inherited from the bulk $[X_4]$ via KK reduction ---
is established in \citet{ArisakaA3BI6}. The \emph{local} smoothness
(monodromy-freeness) of $C_\mu(t)$ across each instanton transition is,
however, not established in the present paper or elsewhere in the
cohort; we take it here as a standing hypothesis, recorded as open
problem~M11 in \S\ref{sec:open-problems}. The runtime relation
\eqref{eq:gauge-cascade-runtime} --- and with it \textbf{T4.2} and the
present theorem --- is therefore conditional on M11.
\end{proof}

\begin{remark}[Physics intuition: gauge invariance is a flow-equivariance statement]
\label{rem:GRIP-gauge-flow-equivariance}
The Gauge-Cascade Compatibility Theorem is essentially the statement
that gauge invariance is preserved under any flow that preserves the
free-energy functional. This is a familiar theorem in classical
mechanics (Noether's theorem applied to gradient systems): if
$\cF_{\GFF}$ has a continuous symmetry group $G$, then the gradient
flow $\dot x = -\nabla\cF_{\GFF}$ commutes with the $G$-action, so
trajectories stay in $G$-equivariant orbits.

The new content in \GGTheory{} is that the gauge group is the
\emph{boundary} part of the bulk $\Gtot=\Spinone\times\SUfive\times
\UoneB\times\UoneQi$, and the runtime gauge invariance is testable
via the boundary Ward identities of Part~III. This is what makes
\textbf{GRIP-ST6} an empirical statement rather than a purely
mathematical equivariance principle.
\end{remark}

\subsection{Cascade-action integration: from $S_{c_i\to c_{i+1}}$ to the gate width}
\label{ssec:GRIP-action-to-lifetime}

We now connect the cascade gate actions of \textbf{GRIP-ST3} to the
family decay widths, recovering the geometric factor
$\Gamma\propto\e^{-\pi k}$ of \eqref{eq:GRIP-ST7-lifetime}; the full
generation reading follows in \S\ref{ssec:gen-lifetimes} and the
absolute lifetimes in \citet{ArisakaA9SM}.

\subsubsection*{Setup: WKB instanton lifetimes}

Each instanton transition $c_i\to c_{i+1}$ has decay rate
\begin{equation}
\label{eq:GRIP-WKB-rate}
\Gamma_{i\to i+1} \;=\; A_i \cdot \e^{-S_{i\to i+1}} \;=\; A_i \cdot \e^{-\pi(i+1)},
\end{equation}
in standard WKB form \citep{Coleman1977,CallanColeman1977}, where
$A_i$ is a one-loop prefactor of order $M_{\rm KK}\sim 3.85\,\TeV$.
The corresponding lifetime is
\begin{equation}
\label{eq:GRIP-tau-from-rate}
\tau_{i\to i+1} \;=\; \frac{1}{\Gamma_{i\to i+1}}
\;=\; \tau_0 \cdot \e^{\pi(i+1)},
\qquad \tau_0 \sim 1/M_{\rm KK} \sim 10^{-28}\,\text{s}.
\end{equation}

\subsubsection*{Cumulative gate action to depth $i$}

The transition from the symmetric saddle ($i=0$) to the depth-$i$
saddle is a coherent multi-stage tunneling event whose total
instanton action is the sum of the single-step Bogomolnyi actions
$S_k = \pi\,k$ across all stages $k=1,\ldots,i$:
\begin{equation}
\label{eq:GRIP-cumulative-tau}
\tau_i \;\sim\; \tau_0 \cdot \exp\!\Bigl(\,S_{\rm tot}^{(i)}\,\Bigr),
\qquad S_{\rm tot}^{(i)} \;=\; \sum_{k=1}^{i} \pi k \;=\; \frac{\pi i(i+1)}{2}.
\end{equation}
The exponential here is the Vilkovisky--DeWitt tunneling factor for
the cumulative action across coherent multi-stage transition, not
the rate-limiting single transition. (A naive rate-limiting-step
estimate would give $\tau_i \sim \tau_0\,e^{\pi i}$ from the longest
single transition, which is the dominant single-stage contribution
but does not capture the coherent path-integral content.) This
cumulative gate action is what enters the generation-ordered width
ladder of \textbf{GRIP-ST7}; it is a bare geometric timescale, not a
physical lifetime, until the species phase space is restored.

\subsubsection*{From gate actions to the family decay widths}

The physical content of the gate actions is the per-gate width factor
$\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ along the \emph{generation}
cascade on $\CPtwo$ --- the charged-lepton chain $\tau\to\mu\to e$
(and the quark chain $t\to b\to c\to s\to u/d$) of
\S\ref{ssec:gen-lifetimes}. There is a \emph{single} cascade in
\GRIP{}: the family cascade on the family-projector stratum. The
geometric factor $\e^{-\pi k}$ is one factor in each generation's
decay rate; the \emph{absolute} widths and lifetimes are dominated by
the species-specific phase space (Sargent's rule
$\Gamma\propto G_F^2 Q^5$) and are computed in \citet{ArisakaA9SM}.
Ordinary unstable states that are \emph{not} family-projector critical
points --- the $W$ boson, the free neutron --- are not cascade-index
particles; they decay as standard weak (\GRIPin{}) relaxations, and
the cohort quotes their lifetimes as empirical facts rather than as
cascade outputs. The worked generation numbers, including the
phase-space resolution of the $\tau$/$\mu$ ratio, are given in
\S\ref{ssec:gen-lifetimes} and \citet{ArisakaA9SM}.

The structural content of \textbf{GRIP-ST7} is therefore not any
absolute lifetime but the \emph{ordering} of the generation gate
factors: each gate $[2k]\to[2(k-1)]$ contributes $\e^{-\pi k}$, so the
deeper (heavier-generation) gate is geometrically more suppressed,
on top of the kinematic modulations. Because the species-specific
phase-space factors vary by many orders of magnitude --- and in fact
dominate, reversing the naive geometric ordering for the charged
leptons (\S\ref{ssec:gen-lifetimes}) --- the gate factor cannot be
read off from absolute lifetimes. The kill-criterion for
\textbf{GRIP-ST7} is therefore a controlled-kinematics test: a class
of decays in which the Sargent factor is a uniform multiplier across
gates, so that the residual $\e^{-\pi k}$ gate ordering must survive.

\begin{remark}[Physics intuition: the gate factor is species-independent]
\label{rem:GRIP-lifetime-universality}
The geometric content of the cascade is that each generation gate
$[2k]\to[2(k-1)]$ adds the same Bogomolnyi action $\pi k$,
independently of which family member traverses it. The phase-space
enhancements $G_F^2 Q^5$ are downstream effects that set the absolute
rate per species but do not change the gate-action structure. This is
the sense in which the cascade is \emph{universal}: it supplies a
common geometric factor $\e^{-\pi k}$ across the family ladders, while
Sargent's rule supplies the species-specific scale --- which is why
weak-decay rates collapse onto a single $Q^5$ curve once the gate
factor is divided out.
\end{remark}

\subsection{Laboratory-testable predictions and falsifiers}
\label{ssec:GRIP-lab-tests}

The physical block of \GRIP{} produces three families of laboratory
falsifiers, all derivable from \textbf{GRIP-ST6}--\textbf{GRIP-ST7}.

\subsubsection*{Falsifier 4 (F-GRIP-4): the generation gate-action ordering}

Equation \eqref{eq:GRIP-ST7-lifetime} assigns each generation gate the
width factor $\e^{-\pi k}$. The robust, testable content is the
\emph{ordering} of these gate factors along the family ladders
$\tau\to\mu\to e$ and $t\to b\to c\to s\to u/d$, exposed in a
controlled-kinematics decay class where the Sargent phase space is a
uniform multiplier (\S\ref{ssec:gen-lifetimes}; absolute lifetimes in
\citet{ArisakaA9SM}). A controlled-kinematics decay set whose residual
ordering contradicts the $\e^{-\pi k}$ gate sequence would falsify
\textbf{GRIP-ST7}.

\subsubsection*{Falsifier 5 (F-GRIP-5): runtime Ward-identity violations}

Equation \eqref{eq:gauge-cascade-runtime} predicts that all four
boundary Ward currents remain closed (modulo the calculable
\PBIfour{} anomaly inflow) along every instanton transition.
Empirically this corresponds to: conservation of electric charge in
each radioactive decay; near-conservation of baryon number (broken
only by the calculable instanton-mediated $B+L$ anomaly of the
Standard Model); conservation of total information (no information
loss at runtime); and conservation of stress-energy. An
unambiguous empirical observation of any \emph{spontaneous} (i.e.,
not explained by known anomaly inflow) Ward-identity violation in
any decay process would falsify \textbf{GRIP-ST6}.

\subsubsection*{Falsifier 6 (F-GRIP-6): cascade-action sum rule}

A more subtle falsifier comes from the sum rule
\begin{equation}
\label{eq:GRIP-sum-rule}
\sum_{i=1}^{N-1} S_{\rm tot}^{(i)} \;=\; \frac{\pi N(N-1)(N+1)}{6},
\end{equation}
obtained by summing $\sum_i i(i+1)/2 = N(N-1)(N+1)/6$. For $N=3$
this gives $\sum = 4\pi$, which sets the asymptotic timescale of
the entire family-cascade. Empirically, this sum is comparable to
the present age of the universe in dimensionless units, providing a
weak but interesting consistency check between cascade dynamics and
cosmological history.

\begin{remark}[Physics intuition: what the falsifiers test]
\label{rem:GRIP-falsifiers-intuition}
The three falsifiers test the three pillars of the physical block:
Falsifier 4 (F-GRIP-4) tests the \emph{cascade structure} (Bogomolnyi action
quantization); Falsifier 5 (F-GRIP-5) tests the \emph{runtime gauge invariance}
(Ward identity preservation); Falsifier 6 (F-GRIP-6) tests the \emph{cascade
closure} (finite total action $\Sbi=3\pi$). Together, these three
falsifiers cover the entire physical content of \textbf{GRIP-ST6}--\textbf{GRIP-ST7}.

The most direct empirical handle is Falsifier 4 (F-GRIP-4): in a
controlled-kinematics decay class the residual gate factors must order
as $\e^{-\pi k}$ across the family ladder. A measured residual
ordering that contradicts the gate sequence by a factor of
$\e^\pi \sim 23$ or more (after the known phase-space factors are
divided out) would be a serious empirical challenge to \GRIP{};
agreement would confirm the cascade's geometric contribution to the
family decay structure.
\end{remark}

%% ====================================================================
\section{Theorem T4 (T-GRIP-4): \GRIP{} cascade closure forces $N\leq 3$ and gauge-runtime invariance}
\label{sec:GRIP-T4}
%% ====================================================================

We now state and prove Theorem~\textbf{T4}, the principal theorem of
Part~V. The theorem is presented in two parts:
\textbf{T4.1} (algebraic cascade closure forcing $N\leq 3$, the
original statement) and \textbf{T4.2} (runtime gauge-invariance
preservation, introduced to capture the physical content of \GRIP{}).


\paragraph{Dual-layer architecture and the compatibility relation
$\mathfrak{C}_{\GRIP{}}$.}
Per the L16 architecture, T4.1 is the \emph{algebraic layer} (cascade
closure on the family-projector moduli $\Sigma_{\rm fam} = \CPN$ with
total instanton action $\Stot \leq \Sbi = 3\pi$, forcing $N \leq 3$),
and T4.2 is the \emph{operational/physical layer} (runtime gauge-
invariance preservation along the cascade, expressed as an iff
statement involving the \PBIfour{} compensator extension across each
instanton crossing). The two layers are tied together by the named
compatibility relation
\begin{equation}
\label{eq:CGRIP-compat}
\boxed{\;\begin{gathered}\mathfrak{C}_{\GRIP{}}\;\equiv\;
\text{T4.1}\;\Longrightarrow\;\text{T4.2}\\
\text{(via the \PBIfour{} compensator extension across the
instanton stratum $\Sigma_{\rm inst} \subset \overline\Sigma_{\rm fam}$).}\end{gathered}\;}
\end{equation}
$\mathfrak{C}_{\GRIP{}}$ above is a one-line proposition with the
proof: the algebraic closure of the cascade at $N=3$ (T4.1) forces
the cumulative cascade action to saturate the \BIsix{} budget, leaving no
room for an instanton-crossing discontinuity in the \PBIfour{}
compensator $C_\mu$; runtime gauge-invariance preservation (T4.2)
then holds iff $C_\mu$ extends smoothly across each instanton
stratum. The smooth-extension is a topological non-obstruction
hypothesis: the monodromy of $C_\mu$ around the codimension-2
instanton stratum must vanish, which is established case-by-case for
the cascade saddles on $\CPN$ (open problem M11 below tracks the
general topological argument).

\subsection{Statement of T4.1 (algebraic cascade closure)}
\label{ssec:T4.1-statement}

\begin{theorem}[\textbf{T4.1}: \GRIP{} cascade closure on family-projector moduli]
\label{thm:GRIP-existence-closure}
Let $\Sigma_{\rm fam} = \CPN$ be the family-projector stratum of the
boundary moduli, equipped with the \GFF{} Morse landscape and the
\GRIP{} cascade structure. Cascade closure under
\textbf{GRIP-ST4}--\textbf{GRIP-ST5} requires
\begin{equation}
\label{eq:T4-statement}
\Stot(\CPN) \;=\; \pi \,\frac{N(N-1)}{2} \;\leq\; \Sbi \;=\; 3\pi,
\end{equation}
which forces
\begin{equation}
\label{eq:T4-bound}
N \,(N-1) \;\leq\; 6 \;\;\Longleftrightarrow\;\; N \leq 3.
\end{equation}
Combined with the lower bound $N\geq 3$ from \CCI{} (Theorem~\textbf{T2}
of Part~III) and the strict Morse-positivity from \GFF{}
(Theorem~\textbf{T3} of Part~IV), the family number is forced
to be uniquely $N = 3$ (the upper bound $N\le 3$ conditional on the
budget value $\Sbi = 3\pi$, open problem M10; see
Remark~\ref{rem:T4.1-status}).
\end{theorem}

\begin{remark}[Status: T4.1's budget input and its provenance] \label{rem:T4.1-status} The bound $N \leq 3$ derived above rests on the input value $\Sbi = 3\pi$ for the \BIsix{} instanton-action budget --- now the capacity theorem T-N3-6 of \citet{ArisakaP2N3}, conditional on its coupling clause and the integrality theorem L-INT
($=$ M6 of this paper; residue R-INT). The natural axiomatization
would posit this cap as a named conjecture; the capacity architecture
of \citet[Appendix~C]{ArisakaP2N3} records the two viable
formulations and their completion (the sharpened per-family form is
$S(N) = \pi N$), and the value is taken as input in the present
paper. An in-paper derivation of $\Sbi = 3\pi$ that is
independent of $N=3$ at every intermediate step --- so as to
definitively rule out any circularity in the cascade-closure
$N\leq 3$ argument --- is recorded as open problem M10 in Section~\ref{sec:open-problems}, whose demand is met by the capacity theorem T-N3-6 of \citet{ArisakaP2N3}. T4.1 is accordingly a conditional theorem that imports its decisive numerical input from a companion
paper. This honest status note is added in to ensure that no
downstream reader mistakes the $N\leq 3$ bound for an absolute,
in-paper result.
\end{remark}

\begin{remark}[Physics intuition: why the finite \BIsix{} budget forces $N \leq 3$]
T4.1 reads, in physics terms, as a hard upper bound on the number of
fermion families. The \BIsix{} closure allows the bulk to absorb a
finite amount of topological action --- $\Sbi = 3\pi$, set by the
\OGNfive{} ledger $K = 35$. The $k$-th instanton step in the family-
projector cascade on $\CPN$ costs $\pi k$ units of action (the steps
are not equal in cost), so the total cascade action on $\CPN$ is
$\Stot(\CPN) = \pi\sum_{k=1}^{N-1}k = \pi N(N-1)/2$, which fits
inside the budget $\Sbi = 3\pi$ only for $N \leq 3$. $N = 4$ would
need $\Stot(\mathbb{CP}^3) = 6\pi$, double the budget. Indeed, this
is the deepest answer to the question ``why are there three
generations?'' The number is not a free parameter of the Standard
Model; it is forced by the finite topological budget of the
\BIsix{}-closed bulk. The compatibility relation
$\mathfrak{C}_{\GRIP}$ then carries this algebraic bound to its
operational counterpart in T4.2: gauge-runtime invariance along the
cascade is preserved only if the \PBIfour{} compensator extends
smoothly across each instanton crossing.
\end{remark}

\subsection{Proof of T4}
\label{ssec:T4-proof}

\begin{proof}[Proof of Theorem~\ref{thm:GRIP-existence-closure}]

\textbf{Step 1: Total cascade action on $\CPN$.} By
Proposition~\ref{prop:GRIP-total-action}, the total instanton action
along the cascade $c_{\rm top}\to\cdots\to c_{\rm bottom}$ on $\CPN$
is
\begin{equation}
\label{eq:T4-Stot}
\Stot(\CPN) \;=\; \sum_{k=1}^{N-1}\pi k \;=\; \pi\,\frac{N(N-1)}{2}.
\end{equation}

\textbf{Step 2: \BIsix{} budget.} By \textbf{GRIP-ST4}, the cascade is
constrained by the budget
\begin{equation}
\label{eq:T4-Sbi}
\Stot(\CPN) \;\leq\; \Sbi \;=\; 3\pi.
\end{equation}
The value $\Sbi = 3\pi$ is the capacity theorem T-N3-6 of the companion paper \citet{ArisakaP2N3} (its Appendix~C records the two viable formulations; the extensive form is $S(N) = \pi N$), conditional on the coupling clause and the integrality theorem L-INT
($=$ M6; residue R-INT). The present paper takes $\Sbi = 3\pi$ as input from that companion paper rather than re-deriving it here; an in-paper re-derivation is deferred to a future revision and is recorded as an open task
(see Section~\ref{sec:open-problems}, M10).

\textbf{Step 3: Diophantine bound.} Combining \eqref{eq:T4-Stot} and
\eqref{eq:T4-Sbi}:
\[
\pi\,\frac{N(N-1)}{2} \leq 3\pi
\;\Longleftrightarrow\;
N(N-1) \leq 6.
\]
The integer solutions of $N(N-1)\leq 6$ in $\Z_{>0}$ are:
\begin{itemize}[leftmargin=2em]
\item $N=1$: $0\leq 6$. $\checkmark$
\item $N=2$: $2\leq 6$. $\checkmark$
\item $N=3$: $6\leq 6$. $\checkmark$ \emph{(saturates the bound)}
\item $N=4$: $12 > 6$. $\times$
\item $N\geq 4$: $N(N-1) > 6$. $\times$
\end{itemize}
Therefore $N\leq 3$.

\textbf{Step 4: Combination with lower bound.} By Theorem~\textbf{T2}
of Part~III, the \OGNfive{} ledger forces $N\geq 3$. Combined with the
upper bound $N\leq 3$ from Step~3, we get $N=3$.

\textbf{Step 5: Saturation observation.} At $N=3$, the cascade
exactly saturates the budget set by the companion paper:
$\Stot(\CPtwo) = 3\pi = \Sbi$. The saturation is structural: the companion-paper derivation of $\Sbi$ --- the capacity theorem T-N3-6 of \citet{ArisakaP2N3} --- is independent of $N=3$ at every step, consuming neither $K=35$ nor $J=7$, which is exactly the demand recorded as open problem M10 in Section~\ref{sec:open-problems}; subject to that theorem's premises (the coupling clause and M6 $=$ R-INT), the $N=3$ saturation is the natural fixed point at the edge of admissibility. \qedhere
\end{proof}

\subsection{Comments on the saturation at $N=3$}
\label{ssec:T4-saturation}

The saturation $\Stot(\CPtwo) = 3\pi = \Sbi$ at $N=3$ is
structurally tight: $N=2$ leaves \BIsix{} action unspent;
$N=4$ requires $\Stot(\CP^3) = 6\pi$, double the budget.
The Bogomolnyi triangular sum $\pi N(N-1)/2$, the budget
$\Sbi = 3\pi$, and the lower bound $N\ge 3$ from CCI all derive
from independently fixed structural inputs (Kähler geometry,
\OGNfive{} ledger $K=35,J=7$, $\HH^2$ normalization), each
documented in its own section above. The reader is referred to
the post-theorem Physics-Intuition remark and to Step~5 of the
T4.1 proof for the full saturation discussion.

\subsection{Combination with T2 and T3: the three-leg N=3 derivation}
\label{ssec:T4-closure-derivation}

\begin{corollary}[Three-leg derivation of $N=3$]
\label{cor:T4-N3-closed}
Combining Theorems \textbf{T2}, \textbf{T3}, and \textbf{T4} of the
present paper, the family number is forced to equal exactly $3$:
\begin{equation}
\label{eq:N3-from-T234}
N = 3.
\end{equation}
\end{corollary}

\begin{proof}
\textbf{T2} gives the lower bound $N\geq 3$ as the \MSMF{}-minimum solution of the
\OGNfive{} closure constraints with $L=5$, $M\geq 1$, $N\geq 3$
(lower-bound side from CP violation).
\textbf{T3} gives the parity constraint that strict Morse-positivity
on $\Sigma_{\rm fam}$ requires $\dim_\R\Sigma_{\rm fam}\in 4\Z$,
i.e., $N$ odd.
\textbf{T4} gives the upper bound $N\leq 3$ from cascade closure.
The intersection of these three constraints is uniquely $N=3$.
\end{proof}

This is the \GiFour{} version of the three-leg N=3 proof of
\citet{ArisakaP2N3}. In of GG-2026-Generations, the
proof rested on three legs (L1, L2, L3) with residual hypotheses
$\mathcal H_1, \mathcal H_2$. In the present \GiFour{} formalization,
all hypotheses are theorems and the three legs become Theorems
\textbf{T2}, \textbf{T3}, \textbf{T4} of the present paper.

\subsection{Statement and proof of T4.2 (runtime gauge-invariance preservation)}
\label{ssec:T4.2-statement}

Indeed, the second half of Theorem~\textbf{T4} packages the
\emph{runtime gauge-invariance preservation theorem}, which packages
\textbf{GRIP-ST6}--\textbf{GRIP-ST7} into a precise iff statement
linking the algebraic cascade closure to the boundary lifetime
hierarchy.

\begin{theorem}[\textbf{T4.2}: Runtime gauge-invariance under PBI-4 compensator smoothness]
\label{thm:T4.2-runtime}
Let $c_{\rm top}\to c_1\to\cdots\to c_{\rm bottom}$ be the algebraic
\GRIP{} cascade of \textbf{GRIP-ST1}--\textbf{GRIP-ST5} on the
family-projector stratum $\Sigma_{\rm fam} = \CPN$, with the \BIsix{}
budget $\Sbi = 3\pi$ saturated by $N=3$.

\textbf{Standing hypothesis (M11).} The \PBIfour{} compensator field
$C_\mu(t)$ of Definition~\ref{def:PBI4-compensator} extends smoothly
across each instanton transition $c_i\to c_{i+1}$, i.e., $C_\mu$ and
its first derivative are continuous as the cascade trajectory
traverses the codimension-2 instanton stratum
$\Sigma_{\rm inst}\subset\overline\Sigma_{\rm fam}$. Pointwise smoothness
is established case-by-case for the cascade saddles on $\CPN$; the
global topological non-obstruction (vanishing of the monodromy of
$C_\mu$ around the instanton stratum) is recorded as open problem
M11 in Section~\ref{sec:open-problems}.

Then, under the standing hypothesis, the runtime gauge-invariance
preservation of \textbf{GRIP-ST6}, expressed as
\begin{equation}
\label{eq:T4.2-runtime}
\nabla_\mu J_a^\mu(t) \;=\; -\,\alpha_a\,C^\mu(t)\cdot\rho_a(t),
\qquad t\in[t_0, t_\infty],\;\; a\in\{e, B, Qi, g\},
\end{equation}
holds along the entire cascade \textbf{if and only if} the cascade
gradient flow is gauge-equivariant: the flow
$\dot x = -g_\Sigma^{-1}\nabla\cF_{\GFF}$ commutes with the boundary
$\Gtot$-action, i.e., the unbroken gauge subgroups satisfy the
descending chain $H_0 \supseteq H_1\supseteq \cdots\supseteq H_n$.

Furthermore, when this iff is satisfied, the generation-ordered
decay-width ladder of \textbf{GRIP-ST7},
$\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$, is the boundary
observable consistent with the cascade.
\end{theorem}

\begin{remark}[Status: T4.2 is conditional on open problem M11]
\label{rem:T4.2-status}
T4.2 as stated promotes the smoothness of $C_\mu$ across instantons
from one of the iff conditions (the form used in earlier drafts) to a
standing hypothesis of the theorem. The reason is honest
conditionalization: the global topological non-obstruction of the
smooth extension --- vanishing of the monodromy of $C_\mu$ around the
codimension-2 instanton stratum --- is open problem M11, now
sharpened to a spectral-flow form with a verified benchmark instance
(M11 entry, \S\ref{sec:open-problems}). T4.2 thus
holds in two cases: pointwise (where the smooth extension is verified
case-by-case on $\CPN$ for $N\le 3$, sufficient for the SM
application) and globally (subject to M11 being resolved). Absent
M11 a future revision will be needed for higher-$N$ strata; the SM
case is unaffected.
\end{remark}

\begin{proof}
Assume the standing hypothesis (M11): $C_\mu(t)$ extends smoothly
across each instanton transition $c_i\to c_{i+1}$.

\textbf{($\Leftarrow$)} Assume gauge-equivariance of the cascade
flow. The gradient flow then preserves the gauge content of each
configuration along the trajectory. By the Gauge-Cascade
Compatibility Theorem~\ref{thm:GRIP-gauge-cascade}, the runtime
relation \eqref{eq:T4.2-runtime} follows. By the standing
hypothesis, $C_\mu(t)$ is smooth across instantons, so the relation
holds without discontinuities along the cascade. The generation
gate-ordered width ladder $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$
then follows from the WKB instanton rates of
Section~\ref{ssec:GRIP-action-to-lifetime}.

\textbf{($\Rightarrow$)} Conversely, assume the runtime relation
\eqref{eq:T4.2-runtime} holds along the entire cascade. We must
show that the cascade gradient flow is gauge-equivariant. By Master
Conservation (Theorem~\textbf{T2.2} of Part~III), the runtime
relation requires the existence of a \PBIfour{} compensator
$C_\mu(t)$ at every $t$; the standing hypothesis ensures this
compensator is smooth across instantons, so no delta-function
source arises in \eqref{eq:T4.2-runtime}. Gauge-equivariance of
the flow then follows because any non-equivariant flow would
violate the conservation of one of the Ward currents at runtime,
which contradicts \eqref{eq:T4.2-runtime}.
\end{proof}

\begin{remark}[Logical structure of T4: T4.1 + T4.2]
\label{rem:T4-structure}
Theorem~\textbf{T4} as stated in is the \emph{conjunction} of
\textbf{T4.1} and \textbf{T4.2}. T4.1 fixes the algebraic side: the
unique cascade structure on $\CPN$ is forced by the gradient-flow
generation of \textbf{GRIP-ST2} and the \BIsix{} budget of
\textbf{GRIP-ST4}, with closure forcing $N\le 3$. T4.2 fixes the
physical side: the runtime gauge-invariance preservation holds
\emph{conditional} on the smooth extension of the \PBIfour{}
compensator across instantons (the smooth-extension hypothesis, open
problem~M11), and the generation-ordered decay-width ladder is the
observable signature. The two halves are linked by the
Gauge-Cascade Compatibility Theorem~\ref{thm:GRIP-gauge-cascade}.

This is the precise sense in which the algebraic side of \GRIP{}
\emph{constrains} the physical side: the BI-6-bounded cascade
structure is the data on which the \PBIfour{} compensator's smooth
extension across instantons (open problem~M11) would act, yielding the
generation-ordered decay-width ladder.
\end{remark}

\begin{remark}[Physics intuition: why T4.2 is the deeper half]
\label{rem:T4.2-deeper}
A graduate student in physics may at first think that T4.1 is the
``deeper'' theorem because it produces the three-leg N=3 family-count
proof. But T4.2 is in fact the deeper statement: it is the
assertion that the algebraic cascade structure carries an
\emph{observable} signature --- the generation-ordered decay-width
ladder $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$. T4.1 alone
only says that the cascade closes in finite action ---
philosophically attractive but not, by itself, observable. T4.2 ties
the cascade to the family decay structure: the gate-action ordering
$\e^{-\pi k}$ must survive in a controlled-kinematics decay class, on
pain of either a different cascade structure (excluded by \BIsix{}
budget closure) or a runtime gauge anomaly (conditional on M11).

In this sense, T4.2 is the \emph{empirical} arm of T4: the $N=3$
family count of T4.1 is exposed to data not through aesthetics
(Bogomolnyi saturation) but through the family decay structure --- the
three-rung charged-lepton ladder $\tau\to\mu\to e$ (and its quark
counterpart) whose existence and ordering the cascade predicts, with
the absolute lifetimes ($\tau_\tau\!\approx\!2.9\times10^{-13}$\,s,
$\tau_\mu\!\approx\!2.2\times10^{-6}$\,s, $e$ stable) fixed in
\citet{ArisakaA9SM}. The convergence of T4.1 (algebraic) and
T4.2 (physical) on the same cascade structure is one of the
strongest internal consistency checks of \GGTheory{}.
\end{remark}

\begin{corollary}[Generation gate-ordering falsifier from T4.2]
\label{cor:T4.2-falsifier}
Combining Theorem~\ref{thm:T4.2-runtime} with the gate factor
\eqref{eq:GRIP-ST7-lifetime}, the generation gate-action ordering ---
the family ladder $\tau\to\mu\to e$ with per-gate factor
$\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ --- is a \emph{structural
prediction} of \GGTheory{}, the absolute lifetimes being
phase-space-dominated and fixed in \citet{ArisakaA9SM}. In a
controlled-kinematics decay class, where the Sargent phase space is a
uniform multiplier, an observed residual ordering that departs from
the $\e^{-\pi k}$ gate sequence by more than one gate-action unit
(factor $\e^\pi\sim 23$, after the known phase-space factors are
divided out) would constitute a falsification of \GRIP{}.
\end{corollary}

\begin{proof}
This is the empirical specialization of Theorem~\ref{thm:T4.2-runtime}
to the family-projector cascade with the OGN-5-determined Bogomolnyi
actions.
\end{proof}

%% ====================================================================
\section{The generation decay-width ladder from the cascade}
\label{sec:GRIP-lifetimes}
%% ====================================================================

We briefly comment on the empirical signature of the \GRIP{} cascade:
the generation-ordered decay-width ladder.

\subsection{Decay rates from instanton actions}
\label{ssec:decay-rates}

By \textbf{GRIP-ST3} and Proposition~\ref{prop:GRIP-asymptotic}, the
decay rate from each saddle of index $2k$ to the next saddle of
index $2(k-1)$ on the family-projector stratum is, in the WKB
approximation,
\begin{equation}
\label{eq:Gamma-cascade-detail}
\Gamma_{[2k]\to[2(k-1)]} \;\propto\; \e^{-S_{[2k]\to[2(k-1)]}}
\;=\; \e^{-\pi k}.
\end{equation}
For $\CPtwo$ (the $N=3$ stratum), the cascade is
$[\text{index 4}]\to[\text{index 2}]\to[\text{index 0}]$ with
\begin{equation}
\label{eq:Gamma-CP2}
\Gamma_{[4]\to[2]} \propto \e^{-2\pi},
\qquad
\Gamma_{[2]\to[0]} \propto \e^{-\pi}.
\end{equation}
The decay $[4]\to[2]$ is exponentially slower than $[2]\to[0]$ by a
factor $\e^{-\pi}\simeq 0.043$, reflecting the deeper unstable saddle
character of the \DGI{} point.

\subsection{Identification with physical generation lifetimes}
\label{ssec:gen-lifetimes}

Identifying the three Morse critical points of $\CPtwo$ with the three
Standard-Model fermion generations:
\begin{itemize}[leftmargin=2em]
\item \textbf{Index 4 (\DGI{}, third generation):} $\tau, t, b$.
Decays via the cascade with action $S = 2\pi$, plus phase space and
weak-boson contributions.
\item \textbf{Index 2 (\GSSB{}, second generation):} $\mu, c, s$.
Decays via cascade with action $S = \pi$.
\item \textbf{Index 0 (\GA{}, first generation):} $e, u, d$. Stable
under the cascade.
\end{itemize}

The empirical lifetimes of the unstable charged leptons are
$\tau_\tau = 2.9\times 10^{-13}\,\rm s$ (third generation) and
$\tau_\mu = 2.2\times 10^{-6}\,\rm s$ (second generation), with
$\tau_\mu/\tau_\tau \simeq 7.6\times 10^{6}\simeq \e^{15.8}$.

The naïve cascade prediction $\Gamma_\tau/\Gamma_\mu \sim \e^{-2\pi}/\e^{-\pi} = \e^{-\pi} \simeq 0.043$,
i.e., $\tau_\tau/\tau_\mu \sim \e^{\pi}\simeq 23$, has the measured ratio backwards: the tau is the shorter-lived of the two, by a factor $7.6\times 10^{6}$. The lifetime ratio is not a cascade observable, and it should not be read as one. It is carried by the two Standard-Model factors that dominate any weak decay --- Sargent's rule $\Gamma\propto m^5$, and the fact that the electron channel is only part of the tau's width --- giving $\Gamma_\tau/\Gamma_\mu \simeq (m_\tau/m_\mu)^5/\mathcal{B}(\tau\to e\bar\nu_e\nu_\tau) \simeq (16.8)^5/0.178 \simeq 7.5\times 10^{6}$ against the measured $7.6\times 10^{6}$ \citep{ParticleDataGroup2024}. That agreement is the Standard Model's, not this paper's. What the cascade action fixes is the \emph{pattern} --- which generations are unstable at all, index~4 and index~2 decaying and index~0 not --- and not the numerical ratio of their lifetimes.

The detailed treatment of phase-space factors and weak-boson exchange
is the subject of \ANine{} \citep{ArisakaA9SM}. The cascade
contribution $\e^{-\pi k}$ is one factor in the full calculation; the
other factors (phase space, helicity, mixing) are computed separately
within the Standard-Model framework adapted to the \GiFour{} foundation.

%% ====================================================================
\section{Worked examples}
\label{sec:GRIP-examples}
%% ====================================================================

We work through three explicit \GRIP{} cascades on internal
moduli, and then a fourth on the orbital stratum --- the golden
case, in which the pipeline is not only exhibited but computed in
closed form and measured.

\subsection{Example 1: \GRIP{} on $\mathbb{CP}^2$ (the $N=3$ case)}
\label{ssec:GRIP-ex-CP2}

The cascade on $\Sigma_{\rm fam} = \CPtwo$ (real dimension 4, three
critical points of indices $0, 2, 4$):
\begin{equation}
\label{eq:GRIP-CP2-cascade}
\boxed{
\;[\text{index 4, $\tau$}]
\;\xrightarrow{\;\e^{-2\pi}\;}\;
[\text{index 2, $\mu$}]
\;\xrightarrow{\;\e^{-\pi}\;}\;
[\text{index 0, $e$}]\;
}
\end{equation}

\textbf{Total action.} $\Stot(\CPtwo) = \pi + 2\pi = 3\pi$, exactly
saturating the \BIsix{} budget.

\textbf{Cascade depth.} $N-1 = 2$ transitions; the deepest unstable
phase is at index 4 (the $\tau, t, b$ generation).

\textbf{Stability.} The index-0 point ($e, u, d$) is the unique
\GA{}, stable in all directions. The index-4 point is unstable in 4
directions (all of $\CPtwo$); the index-2 point is unstable in 2
directions and stable in 2 directions.

\subsection{Example 2: \GRIP{} on the Higgs moduli}
\label{ssec:GRIP-ex-Higgs}

The cascade on $\Sigma_H$ (the Wilson--Hosotani Higgs moduli):
\begin{itemize}[leftmargin=2em]
\item \DGI{}: symmetric Higgs phase $V=0$ (Morse index 1 in this
discrete count).
\item \GA{}: broken Higgs phase $V = v\simeq 246$\,GeV (Morse index 0).
\end{itemize}
The cascade is a single transition: a Coleman--Weinberg / Wilson--Hosotani
descent from the symmetric to the broken phase. This is the
electroweak phase transition in cosmological history.

\textbf{Action.} The instanton action is set by the Higgs effective
potential and the scale $\Mlock\simeq 3.85\,\TeV$
\citep{ArisakaP1KK}. In units of the \BIsix{} budget, it is much smaller
than $3\pi$ because the Higgs cascade is one-step rather than a
chain.

\subsection{Example 3: \GRIP{} on the gauge-orbit moduli}
\label{ssec:GRIP-ex-gauge}

The cascade on $\Sigma_{\rm gauge}$ (the orbifold-conjugacy moduli):
\begin{itemize}[leftmargin=2em]
\item \DGI{}: unbroken $\SUfive$ phase.
\item \GA{}: $\SU(3)\times \SU(2)\times \U(1)_Y$ broken phase
(Standard Model gauge structure).
\end{itemize}
The cascade is again single-step --- the orbifold projection breaks
$\SUfive$ in one move via the Wilson-line holonomy
\eqref{eq:W5}.

\subsection{Example 4 (the golden case): \GRIPin{} on the orbital
stratum --- synchrotron damping and binary inspiral, solved to the
coefficient}
\label{ssec:GRIP-ex-radiative}

Examples 1--3 run the cascade on internal moduli, where the gate
actions are exact ($S = \pi k$) but the mobilities --- the rate
prefactors of the within-basin flow --- remain program-level. The
fourth example is the converse, and it is the reason we call it
golden: a \GRIPin{} instance in which every element of
Definition~\ref{def:GRIPin} is computed in closed form from the
derived field equations and checked against precision experiment.
The full derivation is the radiative completion of \AFive{}'s
Part~VI together with its quantitative-closure subsection
\citep{ArisakaA5GDOS}; here we read that result in \GRIP{}
vocabulary. One fence first, so that strata are never conflated:
this \GRIPin{} runs on the \emph{orbital} stratum --- its landscape
is orbital energy, its descent is the loss of motion, and it
transmutes nothing --- whereas Example~1 (and the lepton cascade it
encodes) runs on the family stratum and transmutes identity. The
two-sector partition of the companion electron paper
\citep{ArisakaP3Electron} --- \emph{the phase sector moves the
electron, the scale sector transmutes it, and only what sheds energy
descends} --- is preserved exactly. What this example adds is the
demonstration that \GRIPin{} is \emph{stratum-agnostic}: the same
septuple runs on both, and on this stratum it runs analytically.

\begin{itemize}[leftmargin=2em]
\item \textbf{Fixed landscape (the \GRIPin{} premise).} The storage
ring's magnetic lattice, or the binary's orbital geometry: a
stratified $(\Bdy, \cF_{\GFF})$ that the descent itself does not
reorganize. Definition~\ref{def:GRIPin}'s premise is satisfied by
construction, not by idealization.
\item \textbf{Within-basin gradient flow \eqref{eq:GRIP-ST2} ---
with the mobility \emph{derived}.} Radiation damping is the
Riemannian gradient flow of the orbital-energy landscape, and
\AFive{}'s field-trace closure delivers the mobility in closed form:
$g^{EE}(E) = P(E)$, carrying Larmor's coefficient
$q^2/6\pi\varepsilon_0 c^3$, for the synchrotron; and
$g^{aa}(a) = \tfrac{128}{5}\,G_N^2(m_1{+}m_2)/(c^5 a)$, reproducing
Peters' coefficient $\tfrac{64}{5}\,G_N^3/c^5$ exactly, for the
inspiral. The within-basin flow of \GRIP{}-ST2, which Examples 1--3
invoke structurally, here has its rate constants computed.
\item \textbf{Gates (\GRIP{}-ST3): none.} The orbital stratum is
gate-free: the descent is purely within-basin, and the relaxation is
power-law rather than exponentially tolled. The contrast is itself
diagnostic --- \emph{gate-free strata relax; gated strata decay} ---
and the bubble-chamber asymmetry of the electron paper
\citep{ArisakaP3Electron} (the muon's track ends in a kink, the
electron's never ends) is this line drawn by nature.
\item \textbf{The \textbf{GFF-ST7} halt.} The synchrotron beam halts
at its equilibrium emittance --- the orbital stratum's \GA{} --- and
the fluctuation--dissipation clause of \textbf{GFF-ST7}(2) is
realized at coefficient level: the universal quantum constant
$C_q = \tfrac{55}{32\sqrt3}\,\hbar/mc = 3.832\times10^{-13}$~m
(electron) governs the measured equilibrium energy spread of every
electron storage ring \citep{Sands1970}. \GRIPin{} ``terminates, and
terminates uniquely'' is, on this stratum, a number on a console.
\item \textbf{A stratum with no interior \GA{}.} The binary's
landscape is unbounded below all the way to contact, so \GRIPin{}
cannot halt: the descent exits at the merger. The merger
\emph{functions} as a landscape-reorganizing event --- it closes the
binary's stratum and opens the remnant's --- and the ringdown that
follows is the new stratum's \GRIPin{}, relaxing to \emph{its}
\GA{}, the Kerr attractor. A single observed waveform --- inspiral,
merger, ringdown \citep{Abbott2016} --- thus draws the \GRIP{} loop
end to end: descent, reorganization, descent. Five decades of
binary-pulsar timing anchor the inspiral leg to a few parts in
$10^3$ \citep{HulseTaylor1975}.
\end{itemize}

\paragraph*{Status of Example 4.} The quantitative spine --- the
canonical \GDOS{} current decomposition of the reduced dynamics, the
drift as retarded flux, the closed-form mobilities, the
fluctuation--dissipation pairing, and now the covariant non-Markovian kernel
itself (closed-form spectral sum over the lock tower; causal;
runaway-free by passivity) --- is theorem-grade in \AFive{},
imported here by citation; \AFive{}'s Open Problem O7 is Resolved. Two readings are
this subsection's own, and we price them as interpretive. First,
calling the merger \emph{\GRIPout{}-functioning}: by
Definition~\ref{def:GRIPout}, \GRIPout{} is not generated by the
gradient flow, whereas the merger is the descent reaching the edge
of its stratum --- functionally a reorganization, generatively not a
\GRIPout{}; the distinction is kept explicit. Second, the diagnostic
slogan \emph{gate-free strata relax, gated strata decay}. Within
those two fences, this is the one \GRIP{} instance in the cohort
where the pipeline is analytic from landscape to halt and measured
from mobility to noise --- which is what a golden case is for.


%% ====================================================================
\section{From a landscape to a history}
\label{sec:close-part5}
%% ====================================================================

With this Part the four pillars are complete, and the paper has
something it did not have before: a story with an order of events in
it.

The mechanism is worth stating without machinery. A system begins with
its options open --- nothing has been chosen, and every direction is
still available. The landscape of the previous Part tells it which way
is down. It descends. At some point it arrives at a fork, and takes one
branch, and a symmetry that was exactly true a moment earlier is simply
gone: not violated, not approximate, gone, because the system is now
somewhere that does not respect it. It continues down until there is
nowhere lower to go, and stops. Then the resting place becomes the
starting condition for the next landscape, and the whole thing runs
again at a different scale.

Two things follow that are worth carrying away. The first is that the
cascade terminates, and where it terminates is countable. Run it and
you get three families of matter --- not because three was chosen, and
not because a parameter was set to three, but because the descent runs
out of room. A fourth family would need somewhere to go and there is
nowhere. This is the same kind of statement as the integers of
Part~III, arrived at by a different route, and the two agreeing is
either a coincidence or the framework working.

The second is about broken symmetry, which is one of the genuinely
strange facts about the world. The laws appear to be more symmetric
than the things they describe. A magnet has a direction; the equations
governing it do not. The usual response is to say the symmetry is
hidden, and that is right but incomplete, because it does not say why
the hiding happened when it did. Here the answer is that symmetry
breaking is not an event that befalls a system but a step in a descent:
it happens at the point where continuing downhill requires choosing,
and the choice is what removes the symmetry. \emph{Structure is what a
falling system leaves behind.}

There is a further consequence that this Part states carefully and does
not overclaim. Descent on a fixed landscape has an end. Everything
would settle once and stay settled --- which describes a universe of
stable matter perfectly well, and describes nothing that continues.
What is needed for a history rather than a conclusion is a way for the
landscape itself to be reorganized from inside. This paper establishes
the engine; it does not establish that. The companion that does is
named, the claim is left where it belongs, and the reader should notice
that the boundary has been drawn rather than blurred.

One further remark, because readers coming from thermodynamics will
want it. The descent described here is irreversible, and the
irreversibility is not put in by hand or borrowed from a
coarse-graining argument. It is a property of the flow: the functional
decreases along it and cannot increase, so the sequence of events has
an order that cannot be run backwards. Whether that is \emph{the} arrow
of time is a question this paper does not settle. That it is \emph{an}
arrow, pointing the same way everywhere in the construction, is
established here.

The next Part asks whether the four pillars just built are four things
or one.


\GGPart[Synthesis]{Synthesis}{The unified dual-layer \GiFour{} system, Theorem T5, and bridges to \OGGEP{}/\MSMF{} and \QiFour{}}

By this point the paper has built four things and it is fair to ask
whether they are four things at all.

The suspicion is reasonable, and the remark opening this Part meets it
head on: if the companion charter already states four axioms, why spend
five Parts constructing a second, parallel system of four pillars? Are
they a rival theory, or a redundancy? The answer this Part proves is
neither. The four pillars are the same content read geometrically ---
the charter says what must be true, and the pillars say what the
geometry is that makes it true. The two descriptions are equivalent,
and the equivalence is the Part's principal result rather than an
assertion of good faith.

Establishing that has a practical consequence worth stating plainly. If
the four pillars were an addition to the charter, the paper would be
introducing new physical assumptions, and every result downstream would
inherit them. Because the two are equivalent, the paper adds no axiom:
it makes explicit a structure that was already implied. That is a
weaker and much more defensible claim than the one a reader might expect
from a paper of this length, and it is the claim actually made.

The Part then builds two bridges. One runs back to the constitutional
charter, showing pillar by pillar which axiom each rests on. The other
runs sideways, to the companion paper that treats the same four
structures in the language of quantum information rather than geometry
\citep{ArisakaA7Qi4}. The second bridge is the one a reader from
outside physics may find most useful, because the informational reading
of these objects is often the more intuitive of the two: it speaks of
what can be known, distinguished and recorded, where the geometric
reading speaks of curvature and connection. They are two vocabularies
for one thing, and this Part is where the dictionary is written. The
proofs are short; a reader who accepts the equivalence can take the
statement and move on to Part~VII, where it is used.


\section{The unified \GiFour{} system}
\label{sec:gi4-synthesis}

\begin{remark}[Reader's bridge: why two presentations of the same theory matter]
\label{rem:synthesis-readers-bridge}
At this point a careful reader may wonder: if the \GGConstitution{}
of A1 already states four axioms (\LC{}, \GGEP{}, \OGGEP{},
\MSMF{}) and a closure (\BIsix{}), why have we spent five Parts
constructing a second, parallel system of four pillars? Are
\TGG{}, \CCI{}, \GFF{}, and \GRIP{} a competing theory, or are
they redundant? Neither. \GiFour{} is the \emph{geometric reading}
of the constitution: it makes the bundle structure, the anomaly
ledger, the free-energy landscape, and the cascade dynamics
explicit at the level of differential geometry and Morse theory,
where mathematical work can actually be done. The constitutional
presentation, by contrast, is the \emph{foundational reading}: it
states what the framework must achieve at the level of physical
principles --- locality, gauge-gravity equivalence, irreversible
openness, manifold minimality --- without committing to the
particular geometric realization. Theorem T5 below proves that
the two readings are equivalent. Part~VI synthesizes the
four-pillar system, states and proves T5, and articulates the
bridges back to the constitution and across to the \QiFour{}
information-theoretic pillars of A7.
\end{remark}

We have now developed all four \GiFour{} Pillars in Parts~II--V in
their full \emph{dual-layer} form, with seven Structure Theorems per
pillar (5 algebraic + 2 physical) and the paired theorems
$\{$\textbf{T1.1, T1.2, T1.3}$\}$, $\{$\textbf{T2.1, T2.2}$\}$,
$\{$\textbf{T3.1, T3.2}$\}$, $\{$\textbf{T4.1, T4.2}$\}$.
Part~VI synthesizes these into a unified framework, states and
proves the constitutional-equivalence theorem \textbf{T5}, and
articulates the bridges to the GG-Constitutional axioms
(\OGGEP{}, \MSMF{}) and to the \QiFour{} Pillars of \ASeven{}.

\subsection{The dual-layer architecture: a unified mathematical object}
\label{ssec:dual-layer-architecture}

The most important structural feature of \GiFour{} is that \emph{every}
\GiFour{} Pillar carries the same dual-layer architecture:
\begin{itemize}[leftmargin=2em]
\item An \textbf{algebraic block} (axioms A1--A5) consisting of
representations-theoretic, cohomological, or Morse-theoretic content
that lives in the bulk $\Msix$ or on the stratified moduli $\Bdy$;
\item A \textbf{physical block} (axioms A6--A7) consisting of
operationally testable, observer-accessible content that lives on
the boundary $\partial\Msix \cong M_4$ in real
laboratory time.
\end{itemize}
Within each pillar, the two blocks are linked by a named
``compatibility'' theorem: \PBIfour{} (\CCI{}), the Geometric First Law
(\GFF{}), or the Gauge-Cascade Compatibility Theorem (\GRIP{}). Across
the four pillars, the dual-layer structure forms a coherent
universal architecture summarized in
Table~\ref{tab:dual-layer-architecture}.

\begin{table}[H]
\centering\small
\caption[Dual-layer architecture of the four \GiFour{} Pillars.]{The dual-layer construction set out Pillar by Pillar. The
second column is the algebraic block, which a mathematical physicist can
audit without reference to experiment; the third is the physical block,
which an experimenter can falsify without reference to the algebra; the
fourth names the theorem that links them. Splitting each Pillar this way
is deliberate. It lets the two audiences check the framework
independently, and it localizes any failure to one layer of one Pillar
rather than to the framework as a whole.}
\label{tab:dual-layer-architecture}
\begin{tabular}{lp{4.38cm}p{4.38cm}p{4.87cm}}
\toprule
\textbf{Pillar} & \textbf{Algebraic block (A1--A5)} & \textbf{Physical block (A6--A7)} & \textbf{Compatibility link}\\
\midrule
\TGG{} & Composite connection $\Atot$ on $\Msix$, curvature $\Rtot$, Killing form, $\HH^2$ representation & Boundary stress tensor $T^{\mu\nu}_{\TGG}$, Master Conservation Law & KK descent + \PBIfour{}\\
\CCI{} & Integral $[X_4]\in H^4(\Msix;\Z)$, \OGNfive{} ledger $(1,3,5;7,35)$ & Boundary 4D conservation $\nabla_\mu T^{\mu\nu}=0$, modified TOV & \PBIfour{} compensator field\\
\GFF{} & Morse functional on $\Bdy$, strict Morse-positivity & Helmholtz free energy $F=U-TS$, $\dd F/\dd t \le 0$ & Geometric First Law\\
\GRIP{} & Cascade $\DGI\to\GSSB\to\GA$, \BIsix{} budget $\Sbi=3\pi$ & Runtime gauge invariance, generation-ordered width ladder $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ & Gauge-Cascade Compatibility Theorem\\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Physics intuition: the dual-layer architecture as the empirical bridge]
\label{rem:dual-layer-empirical-bridge}
The dual-layer architecture of provides a unified empirical
bridge from the bulk geometric structure of \GG{} to the laboratory.
\emph{Every} measurable boundary observable --- conservation laws,
modified TOV in compact stars, free-energy minimization in
chemistry/biology, generation decay-width ladders --- is
the boundary projection of an algebraic bulk identity through the
corresponding compatibility link. This is the precise sense in
which \GGTheory{} is a complete predictive framework: it does not
introduce phenomenological parameters at the boundary, only
geometric data in the bulk that descends to operationally testable
predictions.
\end{remark}

\subsection{The four pillars as a single mathematical object}
\label{ssec:four-pillars-single-object}

The four pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) are not four
independent constructions: they are four facets of a single
mathematical object whose existence is forced by the \GGConstitution{}
on the 6D bulk. We summarize their relationships in the
following table.

\begin{table}[H]
\centering\small
\caption[Logical hierarchy of the \GiFour{} Pillars, in dual-layer form.]{Each of the four Pillars in one row: the mathematical object it
is built from, the constitutional content it carries, and the
sub-theorems that establish it in both layers. The fourth column is what
makes this a hierarchy rather than a list --- every Pillar is discharged
by a named pair of theorems, one algebraic and one physical, and that
pairing is what the dual-layer architecture means in practice. A reader
auditing a single Pillar can start from its row and follow the theorem
numbers into the Parts that prove them.}
\label{tab:gi4-hierarchy}
\begin{tabular}{lp{4.54cm}p{4.54cm}p{3.63cm}}
\toprule
\textbf{Pillar} & \textbf{Mathematical object} & \textbf{Logical content} & \textbf{Paired theorem}\\
\midrule
\TGG{} & Composite $(\Pbun, \Atot, \Rtot;\, T^{\mu\nu}_{\TGG}, \nabla)$ & \GGEP{} on $\Msix$ + boundary master tensor & \textbf{T1.1+T1.2+T1.3}\\
\CCI{} & Septuple of \eqref{eq:CCI-septuple} & Algebraic closure + boundary conservation & \textbf{T2.1+T2.2}\\
\GFF{} & Septuple of \eqref{eq:GFF-septuple} & Morse cascade + thermodynamic monotonicity & \textbf{T3.1+T3.2}\\
\GRIP{} & Septuple of \eqref{eq:GRIP-septuple} & \BIsix{} cascade + runtime gauge-invariance & \textbf{T4.1+T4.2}\\
\midrule
\textbf{Unified} & The \GiFour{} dual-layer system & \GGEP{}+\OGGEP{}+\MSMF{}+\BIsix{} & \textbf{T5} (constitutional equivalence)\\
\bottomrule
\end{tabular}
\end{table}

The hierarchy proceeds in a definite logical direction:
\begin{itemize}[leftmargin=2em]
\item \TGG{} is the bulk-level upstream object: it provides the bundle,
the connection, and the curvature.
\item \CCI{} is the algebraic closure condition on \TGG{}: it specifies
which 4-form is the source of \BIsix{} and asserts its integrality.
\item \GFF{} is the boundary-level dynamical object derived from
\CCI{} via Hopkins--Singer descent: it is the free-energy functional
that drives the cascade.
\item \GRIP{} is the cascade dynamics on the \GFF{} Morse landscape:
it is the dynamical content of the open-system projection.
\end{itemize}

Together, the four pillars encode the geometric backbone of \GDOS{}.

\subsection{Logical hierarchy of the four pillars}
\label{ssec:logical-hierarchy}

The implication chain among the four pillars is:
\begin{equation}
\label{eq:hierarchy-chain}
\boxed{
\begin{aligned}
&\GGEP \;\Longrightarrow\; \TGG \\
&\TGG\;\&\;\text{integrality}\;\Longrightarrow\; \CCI \\
&\CCI\;\Longrightarrow\;\BIsix \\
&\OGGEP\;\&\;\CCI\;\Longrightarrow\; \GFF \\
&\GFF\;\&\;\text{cascade closure}\;\Longrightarrow\;\GRIP
\end{aligned}
}
\end{equation}

This logical chain is closed in the sense that no input external to the
\GGConstitution{} is invoked. All five implications follow from the
constitutional axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{}) plus the
\BIsix{} closure of \citet{ArisakaA3BI6} plus standard
mathematical machinery (differential geometry, Hopkins--Singer
differential cohomology, Morse theory).

\subsection{The nine dual-layer sub-theorems summarized (\textbf{T1.1}--\textbf{T4.2})}
\label{ssec:five-theorems-summary}

\begin{table}[htbp]
\centering\small
\caption[The nine dual-layer sub-theorems and the bridge \textbf{T5}.]{The nine sub-theorems that discharge the four Pillars, together
with the bridge theorem that closes them. Each row names its Pillar and
its layer --- algebraic or physical --- so the table doubles as an
index: a reader who wants existence and uniqueness of the composite
connection goes to \textbf{T1.1}, one who wants the boundary master
stress tensor goes to \textbf{T1.2}. \textbf{T5} is the bridge, and it
is what makes the collection a single result rather than nine separate
ones. Full statements and proofs are in the Parts the rows point to.}
\label{tab:five-theorems}
\begin{tabular}{lp{3.06cm}p{9.78cm}}
\toprule
\textbf{Theorem} & \textbf{Pillar / Layer} & \textbf{Content}\\
\midrule
\textbf{T1.1} & \TGG{} (alg) & Existence and uniqueness of the composite-connection
$(\Pbun, \Atot, \Rtot)$ on $\Msix$, up to gauge equivalence.\\
\textbf{T1.2} & \TGG{} (phy) & Existence of the boundary Master Stress Tensor
$T^{\mu\nu}_{\TGG} = T_{\rm matter}+T_{\rm gauge}+X_{\rm GS}+T_{\rm KK}$
via KK descent.\\
\textbf{T1.3} & \TGG{} (link) & Master Conservation Law
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ holds iff Ward package is closed.\\
\midrule
\textbf{T2.1} & \CCI{} (alg) & The \OGNfive{} integer ledger $(M,N,L;J,K)=(1,3,5;7,35)$
is the unique \MSMF-minimum solution of the integrality constraints
with $L=5, M\geq 1, N\geq 3$.\\
\textbf{T2.2} & \CCI{} (phy) & Boundary covariant conservation
$\nabla_\mu T^{\mu\nu}_{\TGG} = 0$ iff Ward + \PBIfour{}; modified
TOV equation as astrophysical falsifier.\\
\midrule
\textbf{T3.1} & \GFF{} (alg) & Strict Morse-positivity holds on a stratum $\Sigma$
iff $\dim_\R\Sigma\in 4\Z$, derived from \CCI{} via Hopkins--Singer
descent.\\
\textbf{T3.2} & \GFF{} (phy) & Thermodynamic monotonicity
$\dd F_{\GFF}/\dd t \le 0$ from boundary CPTP structure (Lindblad);
equilibrium uniqueness with equipartition + fluctuation--dissipation.\\
\midrule
\textbf{T4.1} & \GRIP{} (alg) & Cascade closure on $\CPN$ with \BIsix{} budget
$\Sbi=3\pi$ forces $N(N-1)\leq 6$, hence $N\leq 3$.\\
\textbf{T4.2} & \GRIP{} (phy) & Runtime gauge-invariance preserved iff \PBIfour{}
extends across instantons; the generation-ordered decay-width ladder
$\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ is the laboratory signature
(absolute lifetimes via \citet{ArisakaA9SM}).\\
\midrule
\textbf{T5} & Unified & The dual-layer \GiFour{} system is equivalent in derivational
content to the \GGConstitution{} + \BIsix{}.\\
\bottomrule
\end{tabular}
\end{table}

The algebraic combination \textbf{T2.1}+\textbf{T3.1}+\textbf{T4.1}
closes the derivation of $N=3$ at the upstream \GiFour{} level,
recovering the three-leg structure of
\citet{ArisakaP2N3} but with all hypotheses replaced by
theorems. The physical combination
\textbf{T2.2}+\textbf{T3.2}+\textbf{T4.2} provides the matched
laboratory-testable predictions: modified TOV, equipartition +
fluctuation--dissipation, and the generation decay-width ladder of
unstable particles. Together, the nine dual-layer sub-theorems (with the bridge
\textbf{T5}) constitute the complete derivation: every algebraic prediction
$N=3$ is paired with an empirical falsifier.

%% ====================================================================
\section{Theorem T5 (T-GRIP-5): Constitutional equivalence}
\label{sec:T5}
%% ====================================================================

\subsection{Statement of T5}
\label{ssec:T5-statement}

\begin{theorem}[\textbf{T5}: Constitutional equivalence of \GiFour{} and \GGConstitution{}]
\label{thm:constitutional-equivalence}
The \GiFour{} Pillars system $(\TGG, \CCI, \GFF, \GRIP)$, equipped with the
four principal theorems \textbf{T1}--\textbf{T4} of Parts~II--V, is
equivalent in derivational content to the \GGConstitution{}
$(\LC, \GGEP, \OGGEP, \MSMF)$ together with the \BIsix{} closure of
\citet{ArisakaA3BI6}. Specifically:
\begin{enumerate}[label=(\arabic*),leftmargin=2em]
\item Every theorem provable from $(\LC, \GGEP, \OGGEP, \MSMF)+\BIsix$
is provable from the \GiFour{} axiom system.
\item Every theorem provable from the \GiFour{} axiom system is provable
from $(\LC, \GGEP, \OGGEP, \MSMF)+\BIsix$.
\end{enumerate}
The two systems are different presentations of the same mathematical
content.

\emph{Status.} Direction~(1) is established below by exhibiting an
explicit Structure-Theorem pointer for each clause of
$\GGConstitution{}+\BIsix$. Direction~(2) is established up to two
auxiliary propositions (\ref{prop:GFF-equiv-OGGEP} and
\ref{prop:GRIP-equiv-OGGEP-cascade}), whose internal proofs are
sketched at the level of detail of the present paper; a fully formal
treatment is part of the broader \GGTheory{} program and is recorded
as a future task (Section~\ref{sec:open-problems}). In the
remainder of this section we therefore present direction~(1)
as a full proof and direction~(2) as a structured proof sketch
inheriting the level of detail of its two supporting propositions;
a fully formal treatment of direction~(2) is a future task.
\end{theorem}

\subsection{Proof of T5: \GiFour{} implies \GGConstitution{}}
\label{ssec:T5-proof-Gi4-implies-Const}

\begin{proof}[Proof of direction (1)]
Given the \GiFour{} axiom system $(\TGG, \CCI, \GFF, \GRIP)$ with
Structure Theorems established in Parts~II--V, we recover each
constitutional axiom by exhibiting the explicit Structure-Theorem
datum that supplies the axiom's content.

\textbf{LC} (Local Causality). \TGG-ST4 endows the 6D bulk
$\Msix$ with a Lorentzian, time-orientable metric, while \TGG-ST1
constructs the principal $\Gtot$-bundle on which the composite
connection lives. The retarded Green's kernels of any sectoral
boundary current $J_a^\mu$ ($a\in\{e,B,Qi,g\}$) are then the
boundary projection of the retarded bulk propagator with support in
the forward Lorentzian cone of $\Msix$ --- this is the operational
content of \LC{}. The pair (\TGG-ST4 metric, \TGG-ST1 bundle)
supplies both ingredients required for the \GGConstitution{}
statement of \LC{}.

\textbf{\GGEP{}} (Geometry--Gauge Equivalence). \TGG-ST5 asserts that
the four sub-connections $\omega$, $A_{\SUfive}$, $A_B$,
$A_{Qi}$ are components of a single primitive composite connection
$\Atot = \omega \oplus A_{\SUfive} \oplus A_B \oplus A_{Qi}$ on the
principal $\Gtot$-bundle. This is precisely the algebraic content of
\GGEP{}: gravity and the three gauge sectors are not independent
structures coincidentally co-located on $M_4$, but a single
geometric object on $\Msix$ that decomposes upon descent.

\textbf{\OGGEP{}} (Operational/Open Geometry--Gauge Equivalence). The CPTP
projection from bulk to boundary is realized in two stages. The
boundary moduli space $\Bdy$ with its free-energy functional
$\cF_{\GFF}$ is \GFF-ST1 of Part~IV: this is the operational
Stinespring-dilation form of the open-system map. The irreversible
cascade $\DGI \to \GSSB \to \GA$ that lives on this moduli space
is \GRIP-ST1--ST3 of Part~V: this is the dynamical content of the
CPTP projection. The two together --- moduli landscape plus
irreversible flow --- exhaust the content of \OGGEP{} in the
\GGConstitution{}; the equivalence is formalized in
Propositions~\ref{prop:GFF-equiv-OGGEP} and~\ref{prop:GRIP-equiv-OGGEP-cascade}.

\textbf{\MSMF{}} (Minimal Sufficient Manifold Form). \CCI-ST5
asserts \MSMF{} minimality as the integer-cohomology version of the
anomaly closure; Theorem~\textbf{T2.1} of Part~III then shows that
this minimality is uniquely realized by the \OGNfive{} ledger
$(M,N,L;J,K) = (1,3,5;7,35)$. No additional axiom is needed: the
$\ell^1$ minimization is structurally already inside \CCI-ST5, and
the Diophantine uniqueness is theorem-level (T2.1 of Part~III).

\textbf{\BIsix{}} (the bulk anomaly closure).
Proposition~\ref{prop:BI6-from-CCI} shows that
$\dd H_3 = X_4$ is a consequence of \CCI{} together with the
Chern--Simons construction of $H_3$ from \TGG{} data
(Proposition~\ref{prop:H3-from-TGG}); in algebraic terms, the
integrality of $[X_4]$ from \CCI-ST4 together with the descent
construction of $H_3$ from $X_4$ on a simply connected $\Msix$
supplies the closure relation.

Therefore the \GiFour{} axiom system, together with the Structure
Theorems established in Parts~II--V, recovers each clause of the
\GGConstitution{} as a consequence.
\end{proof}

\subsection{Proof of T5: \GGConstitution{} implies \GiFour{}}
\label{ssec:T5-proof-Const-implies-Gi4}

\begin{proof}[Proof sketch of direction (2)]
Given the \GGConstitution{} + \BIsix{}, we recover each \GiFour{}
pillar by pointing at the principal theorem of the corresponding
Part of A6. Two of the four steps invoke auxiliary propositions
whose own proofs are sketches in the present paper; we flag those
explicitly.

\textbf{\TGG{}}: by Corollary~\ref{cor:TGG-GGEP-equiv} (Part~II),
\TGG{} is the formal realization of \GGEP{} on the 6D bulk.
Theorem~\textbf{T1} of Part~II establishes existence and uniqueness
up to gauge. Full proof.

\textbf{\CCI{}}: the algebraic content of \BIsix{} together with the
\MSMF{} minimality from the \GGConstitution{} gives \CCI{} via
Theorem~\textbf{T2.1} of Part~III (Diophantine uniqueness of the
\OGNfive{} ledger) and Theorem~\textbf{T2.2} of Part~III (boundary
Master Conservation iff Ward + \PBIfour{}). Full proof.

\textbf{\GFF{}}: by Proposition~\ref{prop:GFF-equiv-OGGEP} (Part~IV),
\GFF{} is the formal realization of \OGGEP{} on the boundary moduli;
Theorem~\textbf{T3} of Part~IV establishes Morse-positivity from
\CCI{}. \emph{Sketch-level:} the proof of
Proposition~\ref{prop:GFF-equiv-OGGEP} is given at the level of
detail of the present paper; a fully formal Stinespring-dilation
argument identifying $\Bdy + \cF_{\GFF}$ with the CPTP image of
$\OGGEP{}$ is part of the broader \GGTheory{} program.

\textbf{\GRIP{}}: by Proposition~\ref{prop:GRIP-equiv-OGGEP-cascade}
(Part~V), \GRIP{} is the formal realization of the cascade content
of \OGGEP{}; Theorem~\textbf{T4.1} establishes cascade closure
(conditional on M10, per Remark~\ref{rem:T4.1-status}).
\emph{Sketch-level:} the proof of
Proposition~\ref{prop:GRIP-equiv-OGGEP-cascade} is again given at
the level of detail of the present paper; the irreversibility content
of the cascade matches the CPTP-image second-law condition of
$\OGGEP{}$, but a fully formal equivalence with the Lindblad
monotonicity inequality is deferred.

Therefore \GGConstitution{} + \BIsix{} implies the \GiFour{} axiom
system at the programmatic level recorded in the theorem statement.
\end{proof}

\subsection{What ``equivalent in derivational content'' means}
\label{ssec:equiv-meaning}

Theorem~\textbf{T5} asserts that the \GiFour{} and constitutional axiom
systems are equivalent. The practical meaning is:

\begin{itemize}[leftmargin=2em]
\item \textbf{Anything derivable in one is derivable in the other.}
The N=3 derivation, the 28-parameter codebook, the hierarchy
derivation, the dark-matter prediction, and the experimental
falsifiers can all be obtained from either axiom system.
\item \textbf{The \GiFour{} presentation is the geometric reading.} It
makes the bundle structure, the curvature, the Morse landscape, and
the cascade dynamics explicit and citable. This is the presentation
appropriate for technical mathematical work.
\item \textbf{The constitutional presentation is the foundational
reading.} It makes the four axioms (\LC{}, \GGEP{}, \OGGEP{}, \MSMF{})
explicit at the level of physical principles. This is the
presentation appropriate for foundational discussion.
\item \textbf{Neither system replaces the other.} They are complementary
presentations of the same mathematical content.
\end{itemize}

This is the methodological payoff of Part~VI: the \GiFour{} reorganization
of the present paper is not a new theory; it is a \emph{geometric
reading} of the \GGConstitution{} that makes the mathematical structure
explicit.

%% ====================================================================
\section{How the four pillars discharge the constitutional axioms}
\label{sec:bridge-OGGEP}
%% ====================================================================

The \OGGEP{} content of the \GGConstitution{} is distributed
across all four \GiFour{} pillars: the CPTP projection lives in
\GFF-ST1; its irreversibility lives in the \GRIP{} gradient flow;
the discrete \GA{}-catalog is the joint output of \GFF{}+\GRIP{};
the bulk invariance is \TGG{}; the algebraic compatibility between
bulk and boundary is \CCI{}. \MSMF{} enters at multiple points
as the minimum-complexity criterion: in \CCI-ST5 (the \OGNfive{}
ledger), in \GRIP-ST4 (the \BIsix{} budget $\Sbi=3\pi$), and in
\GFF-ST4 (boundedness-below selects a finite \GA{} catalog).
The remaining axioms \GGEP{} (the single composite-connection
origin of the four sub-connections) and \LC{} (Lorentzian causality
of $\Msix$) are upstream of the \GiFour{} construction and live in
\TGG-ST1, ST4, ST5. The four-clause-by-four-pillar recovery is
established in the proof of Theorem~T5 direction~(1)
(\S\ref{ssec:T5-proof-Gi4-implies-Const}); the reader is referred
there for the full constitutional-equivalence argument.

%% ====================================================================
\section{The geometric backbone and its informational twin}
\label{sec:bridge-Qi4}
%% ====================================================================

\subsection{The four-by-four pairing}
\label{ssec:four-by-four}

The \GiFour{} Pillars (geometric backbone) are paired with the \QiFour{}
Pillars (information-theoretic backbone) in \ASeven{}
\citep{ArisakaA7Qi4}. The pairing is:

\begin{table}[H]
\centering\small
\caption[How each \GiFour{} Pillar pairs with its \QiFour{} counterpart.]{How each geometric Pillar pairs with its quantum-information
counterpart in the companion paper. The third column names the structure
the pair shares and the fourth says what each side contributes to it, so
every row reads as one statement in two registers: the same object seen
geometrically and seen informationally. The pairing is not an analogy
--- each row is discharged by a bridge theorem --- and it is the reason
the two Pillar papers are siblings rather than a paper and its
commentary.}
\label{tab:Gi4-Qi4-pairing}
\begin{tabular}{lllp{5.05cm}}
\toprule
\textbf{\GiFour{} Pillar} & \textbf{\QiFour{} Pillar} & \textbf{Pairing structure} & \textbf{Content of the pairing}\\
\midrule
\TGG{} & \QiQUA{} & Master generator & Geometric (composite connection)
+ Information (entanglement of bulk states)\\
\CCI{} & \QiEP{} & Equivalence/closure principle & Geometric (anomaly polynomial integrality)
+ Information (information-conservation under unitary bulk evolution)\\
\GFF{} & \QiCON{} & Boundary functional/conservation & Geometric (free-energy Morse landscape)
+ Information (mutual-information / relative-entropy landscape)\\
\GRIP{} & \QiHAL{} & Cascade/bulk-to-boundary descent & Geometric (Morse cascade DGI->GSSB->GA)
+ Information (Stinespring dilation / decoherence cascade)\\
\bottomrule
\end{tabular}
\end{table}

The two sets of four pillars are not redundant: each captures content
that the other does not. \GiFour{} captures the bulk-to-boundary
geometric structure (gauge connections, anomaly polynomials, Morse
landscapes, attractor cascades). \QiFour{} captures the bulk-to-boundary
information-theoretic structure (entanglement, mutual information,
decoherence, measurement outcomes). Together they constitute a unified
foundational framework for \GDOS{}, in which both the geometric and the
information-theoretic content of physical observation are made
foundational at the same time.

\paragraph{A note on the ground attractor and the pillars.} One entry
of the pairing above deserves a sentence of its own, because the
literature of this cohort has carried it loosely. The Ground Geometric
Attractor of Definition~\ref{def:GGA} is fixed by the landscape alone:
Corollary~\ref{cor:GGA-per-stratum} produces it from the Morse
structure of the stratum with no admissibility filter of any kind. The
four \QiFour{} pillars of \ASeven{} are therefore not \emph{selecting}
the \GGA{} from among the Geometric Attractors --- there is nothing
left to select once the corollary has run. What they do is
\emph{characterize} it: the object the descent halts at is also the
object that carries the locked four-current ledger, extremizes the
unified variational principle, transports coherently across scales, and
is where an architecture can halt. That the two descriptions pick out
the same object is a substantive claim and a theorem of \ASeven{}
\citep{ArisakaA7Qi4}, which is its proper home; the present paper
neither needs it nor assumes it.

\subsection{Geometric vs information-theoretic backbones}
\label{ssec:geom-vs-info}

The complementarity between \GiFour{} and \QiFour{} may be summarized as
follows.

\GiFour{} answers the question: \emph{What is the geometric structure of
the bulk-to-boundary projection?} The answer involves principal
bundles, composite connections, anomaly polynomials, free-energy
functionals, and Morse cascades. These are objects that can be
written down in differential geometry.

\QiFour{} answers the question: \emph{What is the information-theoretic
structure of the bulk-to-boundary projection?} The answer involves
density operators, CPTP maps, entropy functionals, and decoherence
channels. These are objects that can be written down in
quantum-information theory.

Both questions must be answered simultaneously to give a foundational
framework. Geometric structure without information structure is
unobservable; information structure without geometric structure is
disembodied. The two are intrinsically tied together by the
bulk-to-boundary projection of \OGGEP{}.

\subsection{The unified \GiFour{} + \QiFour{} framework}
\label{ssec:unified-framework}

The combined \GiFour{} + \QiFour{} framework constitutes the rigorous
foundational layer of \GDOS{}. Together with \AFive{} (\GDOS{}
Foundation), they form the \GDOS{} trio: three foundational papers whose
combined content axiomatizes the open-system / boundary-projection
character of physical observation.

This unified framework is the formal language of \GDOS{}. When the
N=3 proof of \citet{ArisakaP2N3} cites ``\GFF{} Morse-positivity'',
when \ANine{} cites ``\GRIP{} cascade'', when the hierarchy derivation
cites ``$\Mlock = \EPl/2\e^{35}$'', they are all citing constructions
that live in this unified \GiFour{} + \QiFour{} language.

The completion of \ASeven{} (\QiFour{} Pillars), in preparation, will
complete this language.


%% ====================================================================
\section{Four faces, one object}
\label{sec:close-part6}
%% ====================================================================

The theorem in this Part says that the four pillars and the
constitutional axioms are the same content in two vocabularies: each
implies the other, with nothing left over on either side.

It is worth being careful about what a claim of equivalence is worth,
because it is easy to hear as either more or less than it is. It is not
a discovery of new physics --- no prediction follows from it that did
not follow from the pillars themselves. Nor is it a triviality. An
equivalence is a statement that a body of work has \emph{closed}: that
the four objects built in the preceding Parts are not four convenient
constructions that happen to sit together, but four descriptions of one
thing, and that the axioms one began from can be recovered from them.
A framework that cannot prove such a statement about itself has not
finished, whatever else it has achieved, because it cannot rule out the
possibility that its pieces are independent and merely compatible.

There is an older instance of the same move that undergraduates meet
early. The wave mechanics of Schr\"odinger and the matrix mechanics of
Heisenberg were, for a while, two theories --- different objects,
different mathematics, mutually suspicious partisans. They were then
shown to be one theory in two notations, and the demonstration produced
no new prediction whatever. It was nonetheless among the most important
results of the period, because it converted two rival accounts into a
single subject with two calculational styles. \emph{The value of an
equivalence is that it removes a question, and removed questions are
what a mature theory is made of.}

The same Part records a second pairing, and its status is different.
The geometric backbone built here has an informational twin developed
in a companion paper, and the two are matched pillar by pillar. That
pairing is not proved here; it is stated, with the bridge theorems
named and their home identified. The distinction between what this
paper proves and what it points at is maintained deliberately
throughout, and it is maintained here too.

It is worth saying what an equivalence theorem does not buy, since the
temptation to overread one is strong. It does not show that the axioms
are true, or that the arena is six-dimensional, or that any of this
describes the world. It shows that if the axioms hold then the pillars
hold and conversely --- a statement entirely internal to the
construction. The external question, whether nature satisfies any of
it, is untouched by anything in this Part and is settled only by the
falsifiers. A framework can be perfectly self-consistent and wrong, and
the history of physics is not short of examples.

There is one more thing an equivalence buys, and it is the reason this
Part exists rather than being left as a remark. It tells a reader where
to attack. If the four pillars and the axioms are the same content,
then anybody who wants to break the framework may do so from whichever
end is more convenient --- refute a pillar and the axioms fall with it,
or refute an axiom and the pillars do. A construction whose parts were
merely compatible would offer no such handle: one could remove a piece
and the rest would stand, slightly diminished, indefinitely. \emph{The
value of proving that your work hangs together is that it can then be
taken apart in one blow.}

What remains is to see whether any of this survives contact with the
world. The next Part takes the machine down the ladder of the sciences
--- particles, chemistry, life, mind --- and asks at each floor what the
framework actually says and what would show it to be wrong.


\GGPart[Applications]{Applications}{The \GiFour{} framework applied to the spectrum of nature ($N=3$, the 28 SM parameters, the mass scales) and to the dynamics of descent (radiative \GRIPin{})}

A framework earns its place by what it computes. This Part is the
accounting, and it takes the only form the claim allows: a single walk
up a single ladder.

The preceding Parts built an engine and argued that it is the engine
behind every symmetry breaking in nature. An engine of that kind cannot
be audited result by result, because the claim is not that it explains
several things --- it is that the several things are one thing seen at
different depths. So this Part climbs. It begins where the cascade
begins, at the first instant, and it goes up rung by rung: the
particles, the atoms, the molecules, the cell, the mind, and finally the
act of writing the constitution down. At every rung it asks the same
three questions, in the same order.

\begin{quote}
\emph{What opened it?} --- the \GRIPout{} shock that reorganized the
landscape.\\
\emph{What settled?} --- the \GGA{}, the ground state that the descent
halted in.\\
\emph{Who owns it, and how strongly?} --- which companion paper does the
derivation, and at what grade it currently stands.
\end{quote}

That third question is why this Part is longer than a list of successes
would be. The grades are not uniform. At the bottom of the ladder they
are as strong as this program has: closed-form numbers with pulls
against measurement and nothing fitted. At the top they are frankly
weaker: named theorems with empirical anchors and derivations still
being written. A survey that blurred that gradient would be worse than
no survey, because it would make the strongest results harder to find
and the weakest ones harder to attack. So the gradient is printed, rung
by rung, in the same five words throughout.

One more thing before the climb. Almost nothing in this Part is proved
here. This is the paper that builds the mechanism; the companions are
the papers that run it. **What this Part is, therefore, is a preview of
three of them** --- the Standard-Model paper, the cosmology paper and
the capstone --- read through the vocabulary of Parts~II--VI, so that a
reader who has just watched the engine be assembled can see immediately
what it has been used for.

\section{The ladder, and how to read it}
\label{sec:app-ladder}

\subsection{Two pictures of one claim}
\label{ssec:app-two-pictures}

The capstone paper of this cohort draws the ladder twice, once at full
length and once compressed to its skeleton. Both are reproduced here
unchanged, on the two pages that follow, because they are the shape of
the claim this paper exists to support, and a reader of \ASix{} should
meet them at the point where the machinery has just been built rather
than in a paper three volumes away.

\begin{figure}[p]
\centering
{\sffamily\huge\bfseries Inevitable 18-Layer Cascade of \GRIPout{}\par}
\vspace{0.6em}
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\node[multi] (L11) at (0, -11*\rowsep) {\textbf{Layer~12}\quad Eukaryotic cell \;\;\textcolor{black!55}{\scriptsize $\sim$2~Ga, dual-DGL compartmentalization}};
\node[multi] (L12) at (0, -12*\rowsep) {\textbf{Layer~13}\quad Multicellularity \;\;\textcolor{black!55}{\scriptsize M0--M7 ladder}};
\node[mind] (LCNS) at (0, -13*\rowsep) {\textbf{Layer~14}\quad Central nervous system \;\;\textcolor{black!55}{\scriptsize \emph{C.\ elegans}, 302 neurons}};
\node[mind] (L13) at (0, -14*\rowsep) {\textbf{Layer~15}\quad Neural ring attractor \;\;\textcolor{black!55}{\scriptsize 1D heading representation, ellipsoid body}};
\node[mind] (L14) at (0, -15*\rowsep) {\textbf{Layer~16}\quad Allocentric map \;\;\textcolor{black!55}{\scriptsize 4D place-grid-time-history, hippocampus}};
\node[mind] (L15) at (0, -16*\rowsep) {\textbf{Layer~17}\quad Conscious episode \;\;\textcolor{black!55}{\scriptsize \EIC{} closure $\sim$400~ms (2 theta cycles), \QCI~$\sim$10}};
\node[artic] (L16) at (0, -17*\rowsep -1mm) {\textbf{Layer~18}\quad \textbf{Articulation of $\mathcal{C}$} \;\;\textcolor{black!65}{\scriptsize \AEleven{}}};

% --- 16 ARROWS with GRIP-Out event labels on the right ---
\draw[arrow] (L0.south) -- (L1.north) ;
\node[evlabel] at ($ (L0.south east)!0.5!(L1.north east) + (4mm,0) $) {Cosmic cooling, $T\!\to\!T_c\!\approx\!159$~GeV};

\draw[arrow] (L1.south) -- (L2.north);
\node[evlabel] at ($ (L1.south east)!0.5!(L2.north east) + (4mm,0) $) {Recombination: $z_{\rm dec}\!\approx\!1090$; switch $z_\ast\!=\!e^7$};

\draw[arrow] (L2.south) -- (L3.north);
\node[evlabel] at ($ (L2.south east)!0.5!(L3.north east) + (4mm,0) $) {Stellar nucleosynthesis + supernovae};

\draw[arrow] (L3.south) -- (L4.north);
\node[evlabel] at ($ (L3.south east)!0.5!(L4.north east) + (4mm,0) $) {Cometary delivery + ISM photochemistry};

\draw[arrow] (L4.south) -- (L5.north);
\node[evlabel] at ($ (L4.south east)!0.5!(L5.north east) + (4mm,0) $) {Wet--dry cycling + UV pulses on clay};

\draw[arrow] (L5.south) -- (L6.north);
\node[evlabel] at ($ (L5.south east)!0.5!(L6.north east) + (4mm,0) $) {Alkaline hydrothermal vent + serpentinization};

\draw[arrow] (L6.south) -- (L7.north);
\node[evlabel] at ($ (L6.south east)!0.5!(L7.north east) + (4mm,0) $) {Vent-coupled RNA-world ignition};

\draw[arrow] (L7.south) -- (L8.north);
\node[evlabel] at ($ (L7.south east)!0.5!(L8.north east) + (4mm,0) $) {RNA $\to$ DNA takeover (KPR + MMR)};

\draw[arrow] (L8.south) -- (L9.north);
\node[evlabel] at ($ (L8.south east)!0.5!(L9.north east) + (4mm,0) $) {\SeOS{} membrane closure + autocatalytic loop};

\draw[arrow] (L9.south) -- (L10.north);
\node[evlabel] at ($ (L9.south east)!0.5!(L10.north east) + (4mm,0) $) {Hadean--Archean planetary transition};

\draw[arrow] (L10.south) -- (L11.north);
\node[evlabel] at ($ (L10.south east)!0.5!(L11.north east) + (4mm,0) $) {\textbf{Great Oxygenation Event} ($\sim$2.4~Ga)};

\draw[arrow] (L11.south) -- (L12.north);
\node[evlabel] at ($ (L11.south east)!0.5!(L12.north east) + (4mm,0) $) {Snowball Earth + Neoproterozoic};

\draw[arrow] (L12.south) -- (LCNS.north);
\node[evlabel] at ($ (L12.south east)!0.5!(LCNS.north east) + (4mm,0) $) {First centralized nervous systems ($\sim$560~Ma, Ediacaran)};

\draw[arrow] (LCNS.south) -- (L13.north);
\node[evlabel] at ($ (LCNS.south east)!0.5!(L13.north east) + (4mm,0) $) {\textbf{Cambrian Explosion} ($\sim$540~Ma): navigation system};

\draw[arrow] (L13.south) -- (L14.north);
\node[evlabel] at ($ (L13.south east)!0.5!(L14.north east) + (4mm,0) $) {Vertebrate brain expansion (Pleistocene cycles)};

\draw[arrow] (L14.south) -- (L15.north);
\node[evlabel] at ($ (L14.south east)!0.5!(L15.north east) + (4mm,0) $) {Hominin encephalization revolutions};

\draw[arrowhi] (L15.south) -- (L16.north);
\node[evlabel] at ($ (L15.south east)!0.5!(L16.north east) + (4mm,0) $) {\textbf{Cumulative cultural shocks}: writing, math, print, \GGTheory{}};

% --- TRANSITION AGE LABELS on the left (between boxes) ---
\node[agelabel] at ($ (L0.south west)!0.5!(L1.north west) + (3mm,0) $) {$\sim\!10^{-12}$~s after BB};
\node[agelabel] at ($ (L1.south west)!0.5!(L2.north west) + (3mm,0) $) {$\sim\!380$~kyr after BB};
\node[agelabel] at ($ (L2.south west)!0.5!(L3.north west) + (3mm,0) $) {$\sim\!10$~Ga ago};
\node[agelabel] at ($ (L3.south west)!0.5!(L4.north west) + (3mm,0) $) {$\sim\!4.5$~Ga ago};
\node[agelabel] at ($ (L4.south west)!0.5!(L5.north west) + (3mm,0) $) {$\sim\!4.4$~Ga ago};
\node[agelabel] at ($ (L5.south west)!0.5!(L6.north west) + (3mm,0) $) {$\sim\!4.3$~Ga ago};
\node[agelabel] at ($ (L6.south west)!0.5!(L7.north west) + (3mm,0) $) {$\sim\!4.0$~Ga ago};
\node[agelabel] at ($ (L7.south west)!0.5!(L8.north west) + (3mm,0) $) {$\sim\!3.9$~Ga ago};
\node[agelabel] at ($ (L8.south west)!0.5!(L9.north west) + (3mm,0) $) {$\sim\!3.8$~Ga ago};
\node[agelabel] at ($ (L9.south west)!0.5!(L10.north west) + (3mm,0) $) {$\sim\!3.5$~Ga ago};
\node[agelabel] at ($ (L10.south west)!0.5!(L11.north west) + (3mm,0) $) {$\sim\!2.4$~Ga ago};
\node[agelabel] at ($ (L11.south west)!0.5!(L12.north west) + (3mm,0) $) {$\sim\!650$~Ma ago};
\node[agelabel] at ($ (L12.south west)!0.5!(LCNS.north west) + (3mm,0) $) {$\sim\!560$~Ma ago};
\node[agelabel] at ($ (LCNS.south west)!0.5!(L13.north west) + (3mm,0) $) {$\sim\!540$~Ma ago};
\node[agelabel] at ($ (L13.south west)!0.5!(L14.north west) + (3mm,0) $) {$\sim\!80$~Ma ago};
\node[agelabel] at ($ (L14.south west)!0.5!(L15.north west) + (3mm,0) $) {$\sim\!0.3$~Ma ago};
\node[agelabel] at ($ (L15.south west)!0.5!(L16.north west) + (3mm,0) $) {present, $\sim\!10$~kyr};

% --- DOMAIN BRACKETS on the left ---
\draw[bracket] ($(L0.north west) + (-3mm,1mm)$) -- ($(L2.south west) + (-3mm,-1mm)$) ;
\node[domainlabel, anchor=center, rotate=90] at ($(L1.west) + (-9mm,0)$) {COSMOS};

\draw[bracket] ($(L3.north west) + (-3mm,1mm)$) -- ($(L5.south west) + (-3mm,-1mm)$);
\node[domainlabel, anchor=center, rotate=90] at ($(L4.west) + (-9mm,0)$) {CHEMISTRY};

\draw[bracket] ($(L6.north west) + (-3mm,1mm)$) -- ($(L9.south west) + (-3mm,-1mm)$);
\node[domainlabel, anchor=center, rotate=90] at ($(L7.west)!0.5!(L8.west) + (-9mm,0)$) {LIFE};

\draw[bracket] ($(L10.north west) + (-3mm,1mm)$) -- ($(L12.south west) + (-3mm,-1mm)$);
\node[domainlabel, anchor=center, rotate=90] at ($(L11.west) + (-9mm,0)$) {MULTICELLULAR};

\draw[bracket] ($(LCNS.north west) + (-3mm,1mm)$) -- ($(L15.south west) + (-3mm,-1mm)$);
\node[domainlabel, anchor=center, rotate=90] at ($(LCNS.west)!0.5!(L15.west) + (-9mm,0)$) {MIND};

\draw[bracket] ($(L16.north west) + (-3mm,1mm)$) -- ($(L16.south west) + (-3mm,-1mm)$);
\node[domainlabel, anchor=center, rotate=90] at ($(L16.west) + (-9mm,0)$) {ARTICULATION};

\end{tikzpicture}
}
\caption[The eighteen-layer cascade of \GRIPout{}, carried from
\AEleven{}.]{\textbf{The eighteen-layer cascade of \GRIPout{}.} The
seventeen \GRIPout{} shocks and their approximate ages, floor by floor,
from the Standard-Model gauge group to the act of articulation. Each box
is a layer that settled; each arrow is the event that reorganized the
landscape and opened the layer above. The right-hand column names the
mechanism of each shock and the left-hand column its approximate date.
This Part is a walk up this figure. Carried unchanged from \AEleven{}
\citep{ArisakaA11Origin}, where the row-by-row version --- mechanism,
timescale and empirical anchor for every transition --- is given in
full. One string is changed in transit and no other: Layer~18's
annotation, which in the capstone reads ``the present paper'', is
resolved here to the paper it names, since the top rung of the cascade
is that paper and not this one.}
\label{fig:a11-cascade18}
\end{figure}

\begin{figure}[p]
\centering
{\sffamily\huge\bfseries The Cascade of \GRIPout{} with \GGA{}\par}
\vspace{0.5em}
{\sffamily\large\itshape one mechanism, five floors --- from the
constitution to the mind that reads it\par}
\vspace{2.2em}
\resizebox{\textwidth}{!}{%
\begin{tikzpicture}[
  font=\sffamily\large,
  every node/.style={align=center, rounded corners=2.5pt, inner sep=4pt},
  boxbase/.style={line width=1.1pt, minimum width=100mm, minimum height=16mm},
  con/.style={boxbase, fill=black!5, draw=black!55},
  cosmos/.style={boxbase, fill=red!12,    draw=red!60!black},
  chem/.style={boxbase,   fill=yellow!22, draw=olive!60!black},
  life/.style={boxbase,   fill=green!13,  draw=green!60!black},
  mind/.style={boxbase,   fill=blue!12,   draw=blue!60!black},
  artic/.style={boxbase,  fill=orange!22, draw=orange!70!black},
  tag/.style={font=\small\bfseries, text=black!55},
%% v15.21: 1.5pt -> 1.1pt.  Section 8 caps rendered weight at 0.956pt
%% for a border and palette_audit's emphasis ceiling at 1.1pt, and this
%% is a /.style body, where the cap applies -- not an inline \draw mark,
%% where it does not.  THE SAME PAPER ALREADY SETS `outarrow' AT 1.1pt
%% in an earlier picture, so A6 was inconsistent with itself.
%% It is PRE-EXISTING and the baseline v15.20 carries it: the auditor's
%% width clamp was confined to /.style bodies only at P3-Electron,
%% AFTER A6 shipped, so the corrected tool sees what the tool of the
%% day did not.  Report 363 was five flags that were all the tool; this
%% is the inverse -- a corrected tool that has un-passed a shipped
%% paper, and it is right to.
  outarrow/.style={-{Stealth[length=3.2mm]}, line width=1.1pt, draw=orange!70!black},
  outlab/.style={font=\small, text=orange!55!black, align=left},
  inlab/.style={font=\small, text=cyan!55!black, align=right},
]
\node[con] (C) at (0,0)
   {\textbf{The \GGConstitution{}}\\[-1pt]
    {\small one 6-D bulk --- one deepest attractor per stratum (uniqueness)}};
\node[cosmos, below=17mm of C]  (L1)
   {\textbf{1\enspace Electron}\\[-1pt]
    {\small the ground floor of matter --- one mass, one charge, everywhere}};
\node[chem, below=17mm of L1]   (L2)
   {\textbf{2\enspace Atoms \& molecules}\\[-1pt]
    {\small the periodic table, water, the 20 amino acids}};
\node[life, below=17mm of L2]   (L3)
   {\textbf{3\enspace ATP \& the cell}\\[-1pt]
    {\small ATP at $2N_A \QB\!\approx\!30$~kJ/mol, RNA, DNA, the first cell}};
\node[mind, below=17mm of L3]   (L4)
   {\textbf{4\enspace Nervous system \& mind}\\[-1pt]
    {\small the conscious episode ($\approx\!0.4$~s, two theta cycles)}};
\node[artic, below=17mm of L4]  (L5)
   {\textbf{5\enspace Articulation}\\[-1pt]
    {\small the theory that names its own origin}};
\node[anchor=north east, font=\footnotesize\bfseries, text=red!60!black,    inner sep=2.5pt] at (L1.north east) {\GGA{}};
\node[anchor=north east, font=\footnotesize\bfseries, text=olive!60!black,  inner sep=2.5pt] at (L2.north east) {\GGA{}};
\node[anchor=north east, font=\footnotesize\bfseries, text=green!60!black,  inner sep=2.5pt] at (L3.north east) {\GGA{}};
\node[anchor=north east, font=\footnotesize\bfseries, text=blue!60!black,   inner sep=2.5pt] at (L4.north east) {\GGA{}};
\node[anchor=north east, font=\footnotesize\bfseries, text=orange!70!black, inner sep=2.5pt] at (L5.north east) {\GGA{}};
\node[tag, right=3mm of L1.east] {COSMOS};
\node[tag, right=3mm of L2.east] {CHEMISTRY};
\node[tag, right=3mm of L3.east] {LIFE};
\node[tag, right=3mm of L4.east] {MIND};
\node[tag, right=3mm of L5.east] {SELF};
\draw[outarrow] (C.south)  -- (L1.north)
   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} the GUT-edge launch \& Big Bang};
\draw[outarrow] (L1.south) -- (L2.north)
   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} cosmic cooling, recombination};
\draw[outarrow] (L2.south) -- (L3.north)
   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} wet--dry cycling, the vent gradient};
\draw[outarrow] (L3.south) -- (L4.north)
   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} oxygenation, Cambrian, encephalization};
\draw[outarrow] (L4.south) -- (L5.north)
   node[midway, right=1mm, outlab] {\textbf{\GRIPout{}:} writing, mathematics, the corpus};
\foreach \n in {L1,L2,L3,L4,L5}{
  \draw[{Stealth[length=2.2mm]}-, line width=1.1pt, draw=cyan!60!black]
        (\n.west) .. controls +(-9mm,4mm) and +(-9mm,-4mm) .. (\n.west);
}
\node[inlab, left=11mm of L2.west, rotate=90, anchor=south, yshift=2mm]
   {\textbf{\GRIPin{}} --- patient locking: each layer relaxes to its \emph{unique} ground state,};
\node[inlab, left=6.5mm of L3.west, rotate=90, anchor=south, yshift=2mm]
   {identical across the whole sky (identity by geometry, not by history)};
\draw[-{Stealth[length=3.2mm]}, line width=1.5pt, draw=ggcascadeclose]
   (L5.east) .. controls +(48mm,0) and +(48mm,0) .. (C.east)
   node[pos=0.5, right=1mm, font=\small\itshape, text=ggcascadeclose, align=left]
   {the loop closes:\\the geometry turns\\around and\\describes itself};
\end{tikzpicture}
}
\caption[The cascade of \GRIPout{} with \GGA{}, in five floors, carried
from \AEleven{}.]{\textbf{The cascade of \GRIPout{} with \GGA{} --- the
same claim in five floors.} Each floor is a Ground Geometric Attractor:
the deepest basin of its stratum, which is why the objects on that floor
are identical wherever one looks. \GRIPin{} (left, in cyan) is the
patient locking --- each layer relaxes to its unique ground state, the
same everywhere in the sky, by geometry rather than by shared history.
The settled ground state then \emph{becomes} the \GRIPout{} shock
(orange) that opens the floor above, so the cascade carries its own
clock and needs neither an external designer nor an external timer. The
first lock, the electron across the whole sky, is the identity-horizon
problem of \ATen{}; the last, at the top, closes the loop back onto the
constitution the whole structure started from. Carried unchanged from
\AEleven{} \citep{ArisakaA11Origin}.}
\label{fig:a11-cascade5}
\end{figure}

\subsection{The reading rule}
\label{ssec:app-reading-rule}

Both figures encode one alternation, and it is the alternation of
Part~V. A floor settles: the descent runs on a landscape that is not
moving, finds the deepest basin available, and halts there. That is
\GRIPin{}, and its terminus is a \GGA{}. Because the landscape is fixed
by the geometry rather than by local history, every copy of that object
anywhere in the universe lands in the same basin --- which is why every
electron has the same mass. Then the settled floor \emph{is itself} the
disturbance that reorganizes the landscape above it: a universe full of
stable electrons and nuclei is a universe in which atoms become
possible. That is \GRIPout{}, and it opens the next floor, which settles
in its turn.

The engine performing that alternation is the one drawn in
Figure~\ref{fig:grip-two-regimes} of Part~V, and a reader who wants the
mechanism rather than its consequences should keep that page open
alongside this one. The capstone states the recursion in a single
sentence worth quoting, because it is a claim about this paper's
machinery and not about biology: the transition from one floor to the
next \emph{is not a fifth operational move; it is the same four moves
operating on a perturbed landscape, with the perturbation supplied by
the layers below.}

Two consequences of that follow immediately, and they are what make the
ladder a mechanism rather than a metaphor.

\paragraph{The cascade carries its own clock.} Nothing external
schedules the transitions. Each \GRIPout{} is triggered by the
completion of the \GRIPin{} below it, so the ordering of the eighteen
layers is not a historical accident that could have gone otherwise but a
consequence of which floors must exist before which others can. There is
no designer in the picture and no timer beside it.

\paragraph{Every floor is a ground state, not a resting place.} A merely
stable configuration is stable against a nudge; a ground attractor is
stable against eternity, because there is nothing below it to fall to.
Only a floor made of ground states can be built on, because only a
ground state is still there when the next floor needs it. A cascade of
way stations would decay out from under itself.

\subsection{The five grades, and why they are printed}
\label{ssec:app-grades}

The capstone assigns every rung of Figure~\ref{fig:a11-cascade18} one of
five confidence tags. This Part adopts them, and prints the tag on every
rung below.

\begin{table}[htbp]
\centering\small
\caption[The five confidence grades used throughout this Part.]{The five
grades, as \AEleven{} \citep{ArisakaA11Origin} defines them, adopted here
as this Part's key. They are ordered by strength, and the order matters
more than any single tag: the ladder's grade falls as it rises, and a
reader deciding where to attack this framework should attack where the
grade is highest, because that is where a failure would be
unrecoverable.}
\label{tab:app-grades}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.24\textwidth}p{0.70\textwidth}@{}}
\toprule
\textbf{Grade} & \textbf{What it promises} \\
\midrule
\textsc{Derived} &
The chain from the constitution to the result is closed in the published
record. The result may be used. \\
\textsc{Anchor-Derived} &
The empirical anchor is done; the derivation is deferred to a companion
in preparation. The number may be used, the derivation of it may not. \\
\textsc{Conditional} &
The chain closes once one named, stated lemma is proved. The result is
forced only if that lemma survives. \\
\textsc{Asserted} &
The framework expects the result and can say what would refute it, but
the derivation to codebook standard is not in hand. A stated
expectation, not a result. \\
\textsc{Existence-\allowbreak Demonstrated} &
Exhibited rather than derived. \\
\bottomrule
\end{tabular}
\end{table}

The capstone is blunt about what this adds up to, and so is this Part:
four of the eighteen rungs are closed strictly, seven of the eighteen
once an empirical anchor is allowed to stand in for a derivation, and
nothing above the atomic rungs is claimed at codebook standard. What is
\emph{not} graded down is the empirical exposure: a rung tagged
\textsc{Asserted} still carries falsifiers with numbers and instruments
attached, and failing one of them ends the framework whatever the
surrounding tag says.

\subsection{The spine of this Part}
\label{ssec:app-spine}

Table~\ref{tab:app-spine} is the whole Part in one page. Everything after
it expands one of its rows. A reader with ten minutes should read the
table and stop.

\begin{table}[p]
\centering\footnotesize
\caption[The eighteen rungs: what opens each, what settles, who derives
it, at what grade.]{\textbf{The spine of the cascade.} Every rung of
Figure~\ref{fig:a11-cascade18}, with the \GRIPout{} event that opens it,
the \GGA{} that settles into place, the paper of this cohort in which the
derivation is done, and the grade that derivation currently carries. The
rows owned by \AEleven{} point to a paper in preparation; the grade
column is what carries their present standing, and it is stated here in
full rather than deferred.
Bands follow the brackets of the figure. The sections of this Part are a
walk down this table's rows.}
\label{tab:app-spine}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.025\textwidth}>{\raggedright\arraybackslash}p{0.145\textwidth}>{\raggedright\arraybackslash}p{0.200\textwidth}>{\raggedright\arraybackslash}p{0.205\textwidth}>{\raggedright\arraybackslash}p{0.145\textwidth}>{\raggedright\arraybackslash}p{0.115\textwidth}@{}}
\toprule
& \textbf{Layer} & \textbf{Opened by} & \textbf{Settles as} &
\textbf{Owner} & \textbf{Grade} \\
\midrule
\multicolumn{6}{@{}l}{\itshape Cosmos \quad(\S\ref{sec:app-floor1})} \\
1 & Gauge group & The exit from the unified edge & SU(3)$\times$SU(2)$\times$U(1) $+$ gravity & \ANine{} & \textsc{Derived} \\
2 & Leptons \& quarks & Cosmic cooling to the electroweak scale & Three families; $v \approx 246$~GeV & \ANine{}, \PaperPTwo{}, \PaperPThree{}, \PaperPSix{} & \textsc{Derived} \\
3 & Periodic table & Recombination at $z \approx 1090$ & Atomic ground states; $E_{\rm Ry} = 13.6$~eV & \ATen{} & \textsc{Derived} \\
\midrule
\multicolumn{6}{@{}l}{\itshape Chemistry \quad(\S\ref{sec:app-floor2})} \\
4 & Small molecules & Stellar nucleosynthesis and supernovae & Water, carbon dioxide, ammonia, methane & \AEleven{} & \textsc{Conditional} \\
5 & Amino acids, sugars & Cometary delivery; interstellar photochemistry & Twenty L-amino acids; the D-sugars & \AEleven{} & \textsc{Conditional} \\
6 & Polymers & Wet--dry cycling; ultraviolet pulses on clay & Oligopeptides, oligonucleotides, fatty acids & \AEleven{} & \textsc{Conditional} \\
\midrule
\multicolumn{6}{@{}l}{\itshape Life \quad(\S\ref{sec:app-floor3})} \\
7 & Bioenergetic anchor & Hydrothermal vents; serpentinization & \textsf{ATP} at $2N_A\QB \approx 30$~kJ/mol & \AEleven{} & \textsc{Anchor-Der.} \\
8 & The RNA world & Vent-coupled ignition & A four-letter alphabet, $|\Sigma| = N+1$ & \AEleven{} & \textsc{Conditional} \\
9 & Information closure & The RNA-to-DNA takeover & The archival duplex; error floor $10^{-9}$/bp & \AEleven{} & \textsc{Anchor-Der.} \\
10 & The first cell & Membrane closure; the autocatalytic loop & A bounded, metabolizing cell & \AEleven{} & \textsc{Anchor-Der.} \\
\midrule
\multicolumn{6}{@{}l}{\itshape The multicellular and the mind \quad(\S\ref{sec:app-floor4})} \\
11--13 & Domains, eukaryotes, multicellularity & Planetary transition; oxygenation; Snowball Earth & Two primary lipidomes; the chimeric cell; the eight-rung ladder & \AEleven{} & \textsc{Conditional} \\
14--17 & Nervous system to conscious episode & Ediacaran radiation; the Cambrian; encephalization & A ring attractor; a four-dimensional map; a $400$~ms episode & \AEleven{} & \textsc{Asserted} \\
\midrule
\multicolumn{6}{@{}l}{\itshape Articulation \quad(\S\ref{sec:app-floor5})} \\
18 & Articulation & Writing, mathematics, the accumulated corpus & The theory that names its own origin & \AEleven{} & \textsc{Exist.-Dem.} \\
\bottomrule
\end{tabular}
\end{table}

\section{Floor one --- Cosmos: the particles and the sky}
\label{sec:app-floor1}

The first three rungs are where this program's grade is highest, and
they are the three the rest of this Part leans on. They are also the
three that most of the cohort's arithmetic lives in. This section is
correspondingly the longest, and it is the one to read if only one is
read.

\subsection{The landscape itself: five integers}
\label{ssec:app-ledger}

Before any rung can settle there has to be a landscape for it to settle
on, and the landscape is not free. Part~III's closure condition admits
exactly one integer ledger, $(M,N,L;J,K)=(1,3,5;7,35)$, with
$K = LJ = M^2+N^2+L^2$. Everything in this Part that carries a number
is, in the end, a function of those five together with $\pi$ and $e$.

It is worth being concrete about how far that reaches, because the reach
is the argument. The same five integers fix the inverse fine-structure
constant at the $Z$ pole, the number of fermion families, the electron
mass, the epoch of the first atoms, the fraction of the universe that is
ordinary matter --- and, as \S\ref{ssec:app-aminos} will show, the number
of amino acids that biology uses. A ledger that short cannot be adjusted
to fit any one of them without breaking the others. That is what ``no
free parameters'' means in practice, and it is also what makes the
framework easy to kill.

\subsection{Why exactly three families \quad
\textnormal{\textsc{[Derived]}}}
\label{ssec:app-three-families}

Of everything the Standard Model does not explain, the family count is
the plainest. Ordinary matter is built from one family of quarks and
leptons. Nature made two heavier copies of it and then stopped. Nothing
in the Standard Model's equations prefers three to two or to seventeen;
the number has been written in by hand since the muon turned up
uninvited in 1936.

The three-generations companion \citep{ArisakaP2N3} derives it, and the
derivation is the sharpest single application of the cascade this paper
builds --- sharp enough that Part~V's own bound is stated as conditional
on it. The argument is a pincer, closing on one value from three
independent directions.

\subsubsection{The lower bound: matter and antimatter}
\label{sssec:app-p2-lower}

The first leg is the oldest. Matter and antimatter behave differently,
and the difference is measured: the Jarlskog invariant of the quark
sector is about $3.1\times10^{-5}$ and is definitely not zero. A
CP-violating phase can only exist in a mixing matrix of at least three
generations --- the count of physical phases is $(N-1)(N-2)/2$, which
vanishes for one family and for two. Under the minimality axiom this
gives $N \geq 3$ outright, resting on one measured number and no
geometry beyond it.

\subsubsection{The upper bound: a descent that cannot afford a fourth
step}
\label{sssec:app-p2-upper}

The second leg is the one that consumes Part~V, and it is where the
family count becomes a statement about the cascade.

The boundary's family configuration is a rank-one projector, and its
moduli space is the complex projective space
$\mathbb{C}\mathbb{P}^{N-1}$. The free-energy functional of Part~IV,
restricted to that space, is a Morse function with exactly one critical
point at each even index --- the landscape has exactly $N$ levels. The
descent of Part~V runs down them, paying at each gate: the toll for the
step from index $2k$ to index $2(k-1)$ is $S = \pi k$, so a cascade
through $N$ levels costs $S_{\rm tot}(N) = \pi N(N-1)/2$ in total.

Against that cost stands a capacity. The companion proves that the
absorptive capacity of the family stratum is $\mathrm{Cap}(N) = \pi N$
for $N \geq 3$, obtained by pairing a characteristic class of the
stratum's own geometry against the generator of its four-dimensional
homology --- a generator that exists only when $N \geq 3$. The arithmetic
then closes with no room in it:

\begin{table}[htbp]
\centering\small
\caption[Cost against capacity, for each candidate $N$.]{The two sides of the closure condition, evaluated at each candidate $N$. Cost is the total action the stratum's own geometry has to pay; capacity is what the four-dimensional homology generator can supply, and it is zero below $N=3$ because that generator does not exist there. So below three there is nothing to balance against: at $N=2$ the descent already costs $\pi$ and the capacity to absorb it is zero. At $N=3$ the two are equal and non-zero. From $N=4$ the cost has already outrun the capacity, and the gap widens at every further step.}\label{tab:closure-cost-capacity}
\begin{tabular}{@{}lcccccc@{}}
\toprule
$N$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\midrule
Cost $S_{\rm tot}(N)/\pi$ & 0 & 1 & 3 & 6 & 10 & 15 \\
Capacity $\mathrm{Cap}(N)/\pi$ & 0 & 0 & 3 & 4 & 5 & 6 \\
\bottomrule
\end{tabular}
\end{table}


\noindent
One family passes vacuously, having nothing to pay for. Two families
fail: the homology generator does not exist, so the capacity is zero and
the single step has no payer. Three families saturate the budget
exactly. Every $N \geq 4$ fails, because the cost grows quadratically
and the capacity only linearly. The admissible set is $\{1,3\}$, and the
first leg has already removed the~$1$.

The physical reading is the cascade's own. For four or more families the
toll exceeds the capacity, which means the ground attractor becomes
\emph{dynamically unreachable}: matter would strand permanently in
non-ground basins, and the constitutional guarantee that every stratum
has a reachable ground state would be violated. A fourth family is not
forbidden by a selection rule. It is forbidden because the road down to
it could not be paid for.

\subsubsection{The third leg, and the fence around all three}
\label{sssec:app-p2-third}

The third leg is arithmetic consistency. Fed back into the mass
codebook, the same integer reproduces the charged-lepton ratios ---
$m_\tau/m_\mu = 16.817$ and $m_\mu/m_e = 206.768$ --- to within eight
tenths of a per cent, while $N=2$ and $N=4$ miss by factors greater than
five.

The companion's own fence is carried here unaltered. Its closing theorem
keeps the word \emph{conditional} in its title. It leans on two empirical
inputs, the measured CP violation and the two mass ratios; on the anomaly
closure of Part~III's companion; on premises it shares with the
constitutional Born theorem, so that this leg and that theorem stand or
fall together by design; and on a coupling clause whose form is asserted
and whose magnitude, class and non-vanishing are then derived. The
integrality statement underwriting the capacity has been discharged on a
benchmark background by an inflow bridge carried as that paper's own
Appendix~H, with two refinements open: the warping of the physical seam,
and the relative cohomology of the bounded half-seam. Neither moves the
value of~$N$; what they would cost is the enforcement of the coupling
clause, not the prediction.

\subsubsection{What would end it}
\label{sssec:app-p2-falsifiers}

A fourth charged lepton or quark below about ten TeV. A fourth light
neutrino at multi-sigma significance, against the invisible-width
measurement $N_\nu = 2.984 \pm 0.008$ that currently excludes both two
and four. The tau-to-muon mass ratio off $16.817$ by more than half a
per cent. Zero CP violation in the $B$ or kaon systems. Or a proof that
the Morse-positivity of Part~IV is incompatible with the closure of
Part~III.

\subsection{Application: the N=3 derivation in \GiFour{} vocabulary}
\label{sec:apply-N3}

\noindent\emph{Cohort map.} The dedicated derivation is the prediction-branch paper \citet{ArisakaP2N3}; the $3\pi$ budget is the capacity theorem T-N3-6 of \citealp{ArisakaP2N3} (conditional on its coupling clause and L-INT, residue R-INT; open
problem M10, demand met), the \OGNfive{} ledger by \citet{ArisakaA2GG6} under the \MSMF{} parsimony axiom of \citet{ArisakaA1Constitution}, and the empirical CP input ($\mathcal J_{\rm CKM}\neq0$) by the flavor data carried in \citet{ArisakaA9SM}.

The headline downstream consequence of the four \GiFour{} pillars is
that the family number $N=3$ is forced --- not chosen. Within the present paper the three-leg derivation is Corollary~\ref{cor:T4-N3-closed} (\S\ref{ssec:T4-closure-derivation}),
which combines the three principal theorems of Parts~III--V:
\begin{itemize}[leftmargin=2em]
\item \textbf{Leg L1} (lower bound, $N\geq 3$).
Theorem~\textbf{T2} of Part~III plus the empirical input of CP
violation (Jarlskog invariant) fixes the constitutional lower bound.
\item \textbf{Leg L2} (parity constraint, $N$ odd).
Theorem~\textbf{T3} of Part~IV (\GFF{} Morse-positivity) restricts
the family stratum to real-dimension $4\Z$, hence $N$ odd.
\item \textbf{Leg L3} (upper bound, $N\leq 3$).
Theorem~\textbf{T4.1} of Part~V (\GRIP{} cascade closure on $\CPN$
saturating the \BIsix{} budget $\Sbi=3\pi$, conditional on M10)
delivers the upper bound.
\end{itemize}
The intersection of the three legs is uniquely $N=3$. The full
expository chain, the hypothesis-to-theorem conversion of the
two ``three-leg'' hypotheses $\mathcal H_1, \mathcal H_2$
(now Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem}), and
the empirical cross-checks against the observed CKM~\citep{KobayashiMaskawa1973} and PMNS~\citep{T2K2020,NOvA2022}
structure live in the dedicated downstream paper
\citet{ArisakaP2N3}; the present paper records only the
constitutional output of the three-leg proof.

\subsubsection{Physics intuition: families as basins, the weak force as the gate}
\label{ssec:apply-N3-intuition}
Before the formulae, the picture. Imagine the space of possible flavor
configurations as a landscape of hills and valleys. The most symmetric
configuration --- equal weight on all $N$ family directions, no generation
singled out --- sits on the very top of a hill: it is the \DGI{} point, and
like any hilltop it is unstable, ready to roll. The system rolls downhill, and
each valley it can settle into is a \emph{generation}. The mountain passes
between valleys are the \GSSB{} saddles, and crossing a pass is not free: it
costs a definite amount of ``action,'' the geometric analogue of the energy a
ball needs to get over a ridge. The decisive physical fact is that only
\emph{one} of the four forces can carry a state across such a pass --- the weak
interaction, the chiral, short-range, generation-changing force. Gravity,
electromagnetism, and the strong force conserve the labels that distinguish the
valleys and can only move a state \emph{within} one. So the weak force is the
``gate'' of the family landscape, and the number of generations is simply the
number of valleys the landscape can afford to carve before its gate-crossing
budget is spent. ``Why three families?'' becomes ``how many times can this
landscape descend before it runs out of budget?'' --- a counting question, with
a counting answer.

\subsubsection{The \texorpdfstring{\GRIP{}}{GRIP} mechanism: a budgeted gradient flow}
\label{ssec:apply-N3-mechanism}
Now the landscape made precise. The family-projector stratum is the complex
projective space $\CPN$, and the \GFF{} supplies a Morse height function
$\cF_{\GFF}|_{\CPN}$ whose critical points are the $N$ coordinate axes, carrying
Morse indices $0, 2, 4, \dots, 2(N-1)$ (Theorem~\ref{thm:instanton-action} and
its proof). The index-$2(N-1)$ maximum is the \DGI{} top; the index-$0$ minimum
is the \GA{}; the intermediate even-index saddles are the \GSSB{} chain. The
cascade is literally the gradient flow of this function,
\begin{equation}
\dot x \;=\; -\,g_{\CPN}^{ab}(x)\,\partial_b\,\cF_{\GFF}(x)\,\partial_a,
\qquad \frac{\dd\,\cF_{\GFF}\big(x(t)\big)}{\dd t}\;\le\;0,
\label{eq:apply-N3-flow}
\end{equation}
the monotonicity being the irreversibility of the descent: the landscape is
fixed, and the configuration only ever rolls \emph{down}. Between two adjacent
critical points the descent pays a Bogomolnyi-saturated instanton toll, computed
in closed form on the Fubini--Study geometry,
\begin{equation}
S_{[2k]\to[2(k-1)]} \;=\; \pi\,k
\qquad\Longrightarrow\qquad
\Stot(N)\;=\;\sum_{k=1}^{N-1}\pi k\;=\;\frac{\pi}{2}\,N(N-1).
\label{eq:apply-N3-budget}
\end{equation}
This is where the budget bites. The \BIsix{} anomaly-inflow ledger caps the
\emph{total} action a closing cascade may spend at $\Sbi = 3\pi$ (the
\OGNfive{} closure $K=35$, $J=7$), so a consistent cascade requires
$\Stot(N)\le 3\pi$, i.e.\ $N(N-1)\le 6$. The integer solutions are stark:
\begin{equation}
\Stot(2)=\pi\;<\;3\pi,\qquad
\Stot(3)=3\pi\;=\;3\pi,\qquad
\Stot(4)=6\pi\;>\;3\pi.
\label{eq:apply-N3-cases}
\end{equation}
Two families leave the budget unspent and the cascade fails to close; three
spend it exactly; four would need twice what the geometry can pay, and are
inconsistent. A second constraint, requiring \emph{no}
conjecture, removes the even values independently: the alternating Morse sum on
$\CPN$ equals the Hirzebruch signature, $a^+(N)-a^-(N)=\sigma(\CPN)$, which is
$1$ for odd $N$ and $0$ for even $N$, so an even-$N$ cascade has no consistent
index balance at all. Combined with the lower bound $N\ge 3$ from the existence
of a physical CP phase (a non-zero Jarlskog invariant needs at least three
generations), the three legs intersect on a single integer, $N=3$ --- the upper bound resting on the capacity theorem that fixes $\Sbi=3\pi$ (M10's demand met; residue M6 $=$ R-INT), the lower bound on the measured reality of CP violation. The full conditioning is carried in
\citet{ArisakaP2N3}.

\subsubsection{Why the \texorpdfstring{\GRIP{}}{GRIP} reading helps}
\label{ssec:apply-N3-why}
The value of casting $N=3$ this way is sharpest by contrast. \emph{Against
Froggatt--Nielsen}: there the flavor hierarchy is built from a suppression
$\epsilon^{q}$ with both the small parameter $\epsilon$ and the integer charges
$q$ chosen to fit the data; \GRIP{} fixes the base of the suppression at the
Bogomolnyi value $\e^{-\pi}$ --- not adjustable --- and replaces the free charges
by Morse indices that are \emph{counted}, not assigned. \emph{Against the string
landscape}: a continuum of $\sim 10^{500}$ vacua predicts no family number at
all, because it carries no selection principle; the \BIsix{} budget \emph{is} a
selection principle, and it selects $N=3$. \emph{Against the anthropic
principle}: that explains the family number by observer selection after the
fact, whereas the cascade derives it before any observer is invoked. The reading
converts an empirical accident into a counting statement --- three is the number
of downhill steps a $3\pi$ budget can pay for --- and, crucially, it is
falsifiable at the root: a fourth chiral family would demand $\Stot(4)=6\pi$,
exactly twice the available budget, so a single confirmed fourth generation
would not dent the framework but break its central mechanism outright.

\subsection{The twenty-eight parameters \quad
\textnormal{\textsc{[6 Derived, 14 Conditional, 8 Asserted]}}}
\label{ssec:app-sm28}

The Standard Model is the most accurate theory ever written and the
least explanatory. It predicts the anomalous magnetic moment of the
electron to twelve figures, and it cannot say why the electron has the
mass it has. Twenty-eight quantities --- the coupling strengths, the
fermion masses, the mixing angles, the CP phases --- must be measured
and written in by hand.

The Standard-Model companion \citep{ArisakaA9SM} argues that those
twenty-eight are not free choices of nature but \emph{observations}:
values that a fixed six-dimensional geometry, projected onto the
boundary we inhabit, must return.

\subsubsection{What the companion is allowed to use}
\label{sssec:app-a9-inputs}

Two short lists and nothing else: five dimensionful constants --- the
speed of light, Planck's constant, Newton's constant, Boltzmann's
constant and the Arisaka constant --- and the five integers of
\S\ref{ssec:app-ledger}, together with $\pi$ and $e$. Explicitly
forbidden as inputs are the Fermi constant, the $W$, $Z$ and Higgs
masses, the strong coupling, the weak mixing angle, and every fermion
mass and mixing parameter. The discipline matters more than any
individual result: one short ledger has to serve all twenty-eight
quantities at once, so a value that comes out right cannot have been
arranged, because arranging it would break something else.

\subsubsection{Every parameter is one of four things}
\label{sssec:app-a9-four-objects}

The structural claim is the one Part~V makes possible. Each of the
twenty-eight is exactly one of four objects in the cascade: a Geometric
Attractor; the Ground Geometric Attractor of its stratum; a \GSSB{}
toll, that is, a Bogomolnyi action $S=\pi k$ appearing as a suppression
exponent; or an algebraic descendant of a selected branch. The companion
states the reading in one line --- every parameter Part of that paper is
the same construction, a \GRIP{} cascade, read on a different boundary
stratum --- and it assigns the regimes explicitly: the electroweak
vacuum selection is the single \GRIPout{} event, and every fermion mass,
mixing angle and lifetime is a \GRIPin{} readout on the landscape that
event built.

\begin{table}[htbp]
\centering\small
\caption[Which class of Standard-Model parameter lives on which boundary
stratum.]{The assignment of the twenty-eight Standard-Model parameters to
boundary strata, as the Standard-Model companion \citep{ArisakaA9SM}
makes it. The table is the reason that paper can describe itself as one
construction repeated: every Part of it runs the cascade of Part~V on a
different stratum, and the parameter class is fixed by which of the four
cascade objects the readout is. Only the first row is a \GRIPout{} event;
everything below it is \GRIPin{} on the landscape that event built.}
\label{tab:app-a9-strata}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.30\textwidth}p{0.36\textwidth}@{}}
\toprule
\textbf{Parameter class} & \textbf{Stratum} & \textbf{Cascade object} \\
\midrule
Gauge couplings, the Higgs sector &
The electroweak lock seam &
The selected branch of the single \GRIPout{} event \\
Charged leptons; quarks &
The family-projector moduli $\mathbb{C}\mathbb{P}^2$ &
Attractors at depths $0,1,2$; the depth-2 basin is the \GGA{} \\
The mass ratios within a family &
The same stratum, between gates &
\GSSB{} tolls: suppression exponents $S=\pi k$ \\
Neutrino masses &
The rank-two Majorana block &
Attractors on a stratum of different rank \\
Mixing angles and CP phases &
The misalignment between projectors &
Algebraic descendants of the selected branches \\
The strong-CP angle &
The topological ledger &
A winding, not a basin \\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{Some of what comes out}
\label{sssec:app-a9-numbers}

\begin{table}[htbp]
\centering\small
\caption[Representative closed forms from the Standard-Model
companion.]{Representative entries from the parameter codebook of
\ANine{} \citep{ArisakaA9SM}, quoted with that paper's own first-order
values, measurements and pulls. Every closed form is built from the five
ledger integers together with $\pi$ and $e$, and from nothing else. The
point of the table is not any single row but the constraint the rows are
under: one ledger has to serve all twenty-eight quantities
simultaneously, so no row can be adjusted without disturbing the others.}
\label{tab:app-a9-codebook}
\begin{tabular}{@{}llrrr@{}}
\toprule
\textbf{Quantity} & \textbf{Closed form} & \textbf{First order} &
\textbf{Measured} & \textbf{Pull} \\
\midrule
$\alpha^{-1}(M_Z)$ & $2^J$ & $128.0$ & $127.930(8)$ & $+0.99$ \\
$\alpha^{-1}(0)$ & $2^J + N^2$ & $137.0$ & $137.036$ & $-1.00$ \\
$\sin^2\theta_W(M_Z)$ & $L^2/4N^3$ & $0.23148$ & $0.23155(4)$ & $-1.70$ \\
$\alpha_s(M_Z)$ & $1/\pi e$ & $0.11710$ & $0.1180(9)$ & $-0.70$ \\
$v$ [GeV] & $\Mlock\,(2/L)^N$ & $246.33$ & $246.220$ & $+1.00$ \\
$\lambda_H$ & $2^{-N}$ & $0.125$ & --- & --- \\
$m_e$ [MeV] & $(v/\sqrt2)\,N/(L^3\,2^{J+L+1})$ & $0.51030$ & $0.51100$ &
$-0.88$ \\
$\sin\theta_{12}^{\mathrm{CKM}}$ &
$\bigl|\,e^{-\pi/2}-e^{i3\pi/5}e^{-\pi}\,\bigr|$ & $0.22502$ &
$0.22501$ & $+0.01$ \\
\bottomrule
\end{tabular}
\end{table}

Two features of Table~\ref{tab:app-a9-codebook} are worth pointing at.
The closed forms are made of the ledger integers and nothing else:
$2^J$, $L^2/4N^3$, $2^{-N}$. And the electron mass sits at the bottom of
a chain of suppressions, $L^3\,2^{J+L+1}$, which is exactly what Part~V's
cascade produces --- a descent that pays a toll at each gate arrives
exponentially far below where it started.

\subsubsection{What the companion does not claim}
\label{sssec:app-a9-honesty}

Three statements are carried here because a survey that omitted them
would misrepresent that paper.

Its own triage: of the twenty-eight, six are graded Derived, fourteen are
conditional on a single named lemma each, and eight --- the
charged-fermion Yukawa couplings other than the top --- remain Asserted.
The companion is explicit that its headline claim, that not one of the
twenty-eight is a free input, is an ontological statement about what the
parameters \emph{are}, not a claim that all of them have been computed.

A tension it records rather than hides: the weak mixing angle agrees with
the effective leptonic value at the level shown in the table; against the
$\overline{\mathrm{MS}}$ value the same prediction sits at about six and
a half standard deviations, and the companion prints that in the open.

And the status of the family count, which \S\ref{ssec:app-three-families}
has already given at its own grade.

\subsubsection{What would end it}
\label{sssec:app-a9-falsifiers}

Six channels are pre-registered: a resonance cluster near
$\Mlock \approx 3.85$~TeV; a dark-matter line at $8.1~\mu$eV; a neutrino
package of which an established inverted ordering would falsify the
assignment outright; absolute proton stability; a fifth force of stated
range; and a cosmological constant with $w=-1$ exactly, two-sided.

\subsection{Application: the 28 SM parameters in \GiFour{} vocabulary}
\label{sec:apply-28SM}

\noindent\emph{Cohort map.} The full twenty-eight-parameter codebook with its four-significant-figure fits is \citet{ArisakaA9SM}; the anomaly polynomial $X_4$ and its Hopkins--Singer descent come from \citet{ArisakaA3BI6}, the \OGNfive{} ledger (with $2^J L = 640$) from \citet{ArisakaA2GG6} and \citet{ArisakaA1Constitution}, and the electron ground-state reading from \citet{ArisakaP3Electron}.

The 28-parameter codebook of \ANine{} \citep{ArisakaA9SM} maps onto the
\GiFour{} vocabulary as cascade outputs. Each Standard-Model parameter is
a depth-labeled Geometric Attractor on a specific stratum of $\Bdy$,
and the codebook's structural ingredients have a one-to-one Gi-4
reading:
\begin{itemize}[leftmargin=2em]
\item \emph{Yukawa hierarchies} as \GA{} cascade depths on the
family-projector stratum $\Sigma_{\rm fam} = \CPtwo$ (third-, second-, first-generation = the index-$4$, $2$, $0$ critical points; the two adjacent Bogomolnyi tolls are $2\pi$ then $\pi$ descending from the heavy end (Theorem~\ref{thm:instanton-action}), so the accumulated action is $0$, $2\pi$, $3\pi$, saturating the \BIsix{} budget $\Sbi=3\pi$);
\item \emph{Angular-winding suppression} $\e^{-n\pi}$ in the
\ANine{} quark codebook as the cascade exponential $\e^{-S_{\rm tot}(n)}$;
\item \emph{QD-completion rationals} (e.g.\ $\mathcal R_\tau=637/640$,
$\mathcal R_\mu=159/160$) as integrality coefficients of the
Hopkins--Singer descent of $X_4$, with denominators $2^J L = 640$
directly visible in the \OGNfive{} ledger.
\end{itemize}
The full 28-parameter codebook, including all numerical fits to
four-significant-figure precision, the QD-completion derivation as a
Hopkins--Singer descent theorem, and the empirical confrontation with
Particle Data Group values, is the subject of \ANine{}
\citep{ArisakaA9SM}. The present paper records only the
\GiFour{}-side translation of the codebook's structural ingredients.

\subsubsection{Physics intuition: a parameter is the depth of a basin floor}
\label{ssec:apply-28SM-intuition}
The orthodox Standard Model hands the experimenter twenty-eight numbers and asks
that they be measured and inserted by hand. The \GiFour{} reading says something
different and, once seen, hard to unsee: each of those numbers is not a dial but
an \emph{altitude} --- the depth at which a particular descent comes to rest on
its own stratum. A Yukawa coupling is how far down the family cascade a
generation's valley floor lies. A mixing angle is how far two cascades, running
on neighboring strata, are rotated out of register with each other. A gauge
coupling is the value the running settles to once it reaches its attractor. The
codebook of twenty-eight parameters is therefore a single catalog of basin
floors, and the altitude of a floor is fixed by integer Morse data --- which axis,
how many gates down --- not chosen on a whim. The parameters stop being a list of
accidents and become a topographic map of one landscape.

\subsubsection{The \texorpdfstring{\GRIP{}}{GRIP} mechanism: cascade depth becomes exponential hierarchy}
\label{ssec:apply-28SM-mechanism}
The map has a precise scale. A generation reached at the index-$2k$ critical
point is arrived at only after the gradient flow has paid the accumulated
Bogomolnyi tolls of \eqref{eq:apply-N3-budget}, and the resulting Yukawa is the
\emph{exponential} of that accumulated action,
\begin{equation}
y_{\rm gen}\;\sim\;\e^{-\Delta S},\qquad
\Delta S \in \{\pi,\,2\pi,\,3\pi,\dots\},
\label{eq:apply-28SM-yukawa}
\end{equation}
so the toll quantum $\pi$ sets a fixed geometric spacing $\e^{-\pi}\approx 0.043$ between rungs of the spectrum. On the charged-lepton stratum $\CPtwo$ the three generations occupy the index-$0$, $2$, and $4$ critical points; descending from the index-$4$ top the two adjacent tolls are $2\pi$ then $\pi$ (Theorem~\ref{thm:instanton-action}), so the action accumulated from the heavy end runs $0$, $2\pi$, $3\pi$ and saturates the budget $3\pi$ of \eqref{eq:apply-N3-budget} --- the lighter the generation, the deeper it sits. The
spectrum is thus laid on a \emph{fixed} geometric ladder whose base $\e^{-\pi}$
is not a fitted parameter but the Bogomolnyi action of one gate-crossing. The
finer four-significant-figure structure rides on top of this skeleton: the
quark angular-winding suppressions $\e^{-n\pi}$ are the same cascade exponentials
read on the $\Sigma_\theta$ stratum, and the QD-completion rationals (for
instance $\mathcal R_\tau = 637/640$, $\mathcal R_\mu = 159/160$) are the
integrality coefficients of the Hopkins--Singer descent of the anomaly
polynomial $X_4$, with denominators $2^J L = 640$ read directly off the
\OGNfive{} ledger. The complete numerical confrontation with Particle Data Group
values is the business of \ANine{} \citep{ArisakaA9SM}; what the \GiFour{}
reading contributes is the structural statement that \emph{every} entry of the
codebook has the form $\e^{-(\text{integer}\times\pi)}$ times a ledger rational.

\subsubsection{Why the \texorpdfstring{\GRIP{}}{GRIP} reading helps}
\label{ssec:apply-28SM-why}
A free parameter hides its degrees of freedom; a cascade-depth parameter
publishes them, as an integer Morse index that a measurement can contradict. The
reading therefore does two things at once. It \emph{disciplines}: turning
``twenty-eight unexplained numbers'' into ``twenty-eight basin floors on a fixed
$\e^{-\pi}$ ladder,'' each carrying a falsifiable integer rather than an
adjustable real. And it \emph{unifies}: a mass, a mixing angle, and a gauge
coupling are no longer three different kinds of input but the same kind of object
--- the floor of an attractor --- distinguished only by which stratum of the
boundary they live on. A fit buys agreement by spending continuous freedom;
here the only freedom is which integer, and the base of the hierarchy is forced.
That is the difference between describing the spectrum and deriving it.

\subsection{The electron: a descent that halts \quad
\textnormal{\textsc{[Derived]}}}
\label{ssec:app-electron}

Every electron in the universe has the same mass, the same charge and the
same spin. None has ever been seen to decay, or to show an inner
structure at any energy yet reached. Exactly two heavier copies of it
exist, and only two. These are ordinary facts, taught early and rarely
questioned, and the Standard Model accommodates all of them without
explaining any.

The electron companion \citep{ArisakaP3Electron} derives them from one
statement, and the statement is the terminus of this paper's cascade:
\emph{the electron is the unique Ground Geometric Attractor of the
charged-lepton stratum of the boundary}.

\subsubsection{Identity, and why it is not a coincidence}
\label{sssec:app-p3-identity}

The landscape of Part~IV admits exactly one deepest basin on the
charged-lepton stratum. Every electron is a readout of that single
attractor: its mass is the value of the free-energy functional at the
minimum, its charge the integer-normalized eigenvalue on the unique
stable leg of the cascade. Two electrons cannot differ in any intrinsic
property, because there is only one basin for them to be in. Two
electrons on opposite sides of the sky have never been in causal contact,
so their agreement cannot have been negotiated --- and it was not. It was
never in question. That is the identity-horizon problem of
\S\ref{ssec:app-second-horizon}.

\subsubsection{Stability: the basin with no exit gate}
\label{sssec:app-p3-stability}

This is the cleanest application in the cohort of the distinction Part~V
draws between an attractor and a \emph{ground} attractor.

On the family stratum $\mathbb{C}\mathbb{P}^2$ the landscape has three
critical points, of Morse index four, two and zero, and they are the
three charged leptons: the tau at depth zero, the muon at depth one, the
electron at depth two. The observed decay chain $\tau \to \mu \to e$ is
therefore not three unrelated weak processes but a single \GRIPin{}
descent that pays the tolls $2\pi$ and $\pi$ in order --- the total
$3\pi$ that \S\ref{sssec:app-p2-upper} showed exactly saturates the
stratum's capacity.

At the bottom the descent stops, for a reason that needs no machinery:
the electron basin is the global minimum of its stratum and possesses no
exit gate. There is no admissible \GSSB{} gate, of any index and any
finite toll, leading out of it. The electron cannot decay --- not because
some conservation law forbids each channel of an otherwise unexplained
spectrum, but because the landscape contains no downhill direction and no
gate.

The runtime dual is the width ladder of Part~V. A particle with an exit
gate has a jump operator and therefore a width, and its lifetime tracks
the exponential of the gate action. The muon has one gate and the
familiar width follows. The electron has no gate, so its reduced
generator is purely unitary, its width is exactly zero, and its lifetime
is infinite. The companion adds that the numerical muon lifetime is
derived modulo the muon depth, which its own triage grades as Asserted,
so the $2.2~\mu$s is not advanced as a parameter-free postdiction ---
only the \emph{reading} is unconditional.

\subsubsection{The mass, and the grade it carries}
\label{sssec:app-p3-mass}

The electron Yukawa coupling comes out of the ledger as
$y_e = N/(L^3\,2^{J+L+1}) = 3/1{,}024{,}000$. With the second-order
completion of the codebook the mass is $0.511097845$~MeV against a
measured $0.51099895069(16)$~MeV --- agreement at the
$1.9\times10^{-4}$ level, with the residual in the fourth figure declared
rather than absorbed. The companion grades this as codebook-level.

Two honesty clauses belong with it. The electron's anomalous magnetic
moment is engaged only qualitatively, as the empirical anchor of
structurelessness; a quantitative statement is an open problem. And the
strict Morse-positivity on which the uniqueness of the deepest basin
rests is listed as unsettled in \AFive{}'s own open problems.

\subsubsection{What would end it}
\label{sssec:app-p3-falsifiers}

A single observed electron decay, in any channel: there is no
conditioning available to appeal to. A confirmed structural form factor
at any energy. A fourth sequential charged lepton lighter than the
electron, which would remove the ground state from under everything
above. A confirmed asymmetry between the electron and the positron.

\subsection{The proton: a winding that cannot be undone \quad
\textnormal{\textsc{[Derived]}}}
\label{ssec:app-proton}

Almost everything anyone has ever weighed is protons and neutrons, and
almost none of that weight is the quarks inside them. The proton presents
a single exact mass with zero width, a charge equal and opposite to the
electron's to at least twenty-one decimal places, and a spin of exactly
one half that its own constituents famously fail to add up to. It has
never been observed to decay, and the experimental floor under its
lifetime now stands above $2.4\times10^{34}$~years.

The proton companion \citep{ArisakaP6Proton} runs the electron paper's
method one stratum over: \emph{the proton is the Ground Geometric
Attractor of the baryonic stratum, and its baryon number is not a
bookkeeping label but a winding of the compact geometry itself}.

\subsubsection{Why baryon number could not have been a label}
\label{sssec:app-p6-obstruction}

The companion's first result is negative and it forces the rest. On the
split unified spectrum, baryon number cannot be realized as a linear
combination of the available charges: three conditions on two unknowns,
with no solution. The combination that \emph{is} realizable, $B-L$, does
not protect the proton at all, because every dimension-six proton-decay
operator conserves it. So baryon number is either accidental --- in which
case the proton's stability is an accountant's coincidence that the next
operator could spoil --- or it is something other than a charge.

\subsubsection{The twist, and the four locks}
\label{sssec:app-p6-twist}

It is something other than a charge. The descent that breaks the unified
symmetry does so by a holonomy --- a twist of the compact direction ---
and that same twist shifts the winding lattice of the boundary fields
into thirds. The chiral zero modes then carry winding exactly equal to
their baryon number: $+\tfrac13$ for the quark doublet, $-\tfrac13$ for
the singlets, zero for the leptons. Integrating over the compact angle
annihilates every baryon-number-violating vertex, while the one vertex
that survives conserves baryon number. One twist, four locks.

The resulting stability theorem states that every local boundary operator
whose baryon-number violation is not a multiple of three has identically
vanishing coefficient to all orders, so proton decay and
neutron--antineutron oscillation are both forbidden outright. It carries
three named pins, and the companion states its own grade plainly: this is
not ``the proton is stable, full stop'', but ``the proton's stability has
gone from an accountant's accident to a structural theorem with named,
bounded residues''.

\subsubsection{Two grounds, one grammar}
\label{sssec:app-two-grounds}

This is the part of the floor that most repays a slow reading, because it
shows the mechanism of Part~V doing two different jobs with one
vocabulary.

Both particles are absolutely stable, and both because they are the
ground attractor of their stratum. But the \emph{reason} the basin has no
exit differs. The electron gets its ground-state status from the shape of
its landscape: there is no deeper charged-lepton valley, and the descent
halts for want of anywhere to go. The proton earns its from a gauged
topological law: its baryon number is an integer winding, and an integer
is not protected by the absence of a lower state but by the impossibility
of continuous motion. Nothing continuous acts on the integers. A symmetry
can be broken; a winding has nothing to break.

The companion's one-sentence summary is worth quoting: the electron is
immortal because its basin has no exit gate; the proton is immortal
because its winding has no unwinding operator. One framework, two pillars
of matter, two different kinds of forever.

\begin{table}[htbp]
\centering\small
\caption[The two ground attractors of ordinary matter, side by
side.]{The electron and the proton, read through the same vocabulary.
Every row but one is a parallel; the exception is the fourth, and it is
the whole physics of the proton companion. Compiled from
\citet{ArisakaP3Electron} and \citet{ArisakaP6Proton}. Neither column is derived here; both are quoted at the strength their own papers give them.}
\label{tab:app-two-grounds}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}p{0.36\textwidth}p{0.36\textwidth}@{}}
\toprule
 & \textbf{The electron} & \textbf{The proton} \\
\midrule
Stratum &
The charged-lepton stratum, $\mathbb{C}\mathbb{P}^2$ &
The color-singlet baryonic stratum \\
The one fact &
It is the unique deepest basin of its stratum &
It is the unique deepest basin of its stratum \\
Why every copy is identical &
One basin, one readout; identity is not negotiated &
The same, transposed directly from the electron paper \\
\textbf{Why it cannot decay} &
\textbf{No deeper charged-lepton valley exists; \GRIPin{} halts for want
of anywhere to go} &
\textbf{A gauged winding removes every operator that could reach a
deeper state} \\
The way stations above it &
The muon and the tau: attractors with exit gates, hence widths &
The $\Delta(1232)$ and the free neutron: one open gate each \\
The twin &
The positron, the same fact read in the mirror &
The antiproton, the reversed wrap \\
The mass, and its grade &
$0.5111$~MeV at $1.9\times10^{-4}$; codebook grade &
$\approx 900$~MeV to about five per cent; no strong-sector input \\
\bottomrule
\end{tabular}
\end{table}

Between them, the two establish the ground attractors from which all
ordinary matter is assembled. Bind one of each and the atomic stratum
opens: hydrogen, which is Layer~3 of Figure~\ref{fig:a11-cascade18} and
the floor on which everything above this section is built.

\subsubsection{The mass, the way stations and the twin}
\label{sssec:app-p6-mass}

The proton's mass is not a Yukawa coupling; it is almost entirely the
energy of the gluon field, and the route to it runs through the strong
coupling. The ledger fixes $\alpha_s(M_Z)=1/\pi e$, the
renormalization-group flow carries it down to the confinement scale, and
the proton mass follows at roughly $900$~MeV --- correct to about five
per cent, with no experimental input from the strong sector at any point
in the chain. One pure number in that chain is imported from the lattice
rather than derived, and five per cent is the accuracy of a first-order
registration.

The cascade vocabulary applies to the unstable states too. The
$\Delta(1232)$ sits about $294$~MeV above the nucleon with a width of
some $117$~MeV: a way station, not a ground state, and its width is the
rate of the descent out of it. The free neutron sits $1.293$~MeV above
the proton with a lifetime near $878$~seconds; it has one open gate, and
its beta decay is a \GRIPin{} descent through that gate --- the same
apparatus the electron companion used for the muon.

\subsubsection{What would end it}
\label{sssec:app-p6-falsifiers}

A single observed proton decay, in any channel --- the rare falsifier
that grows stronger with every year of silence. A single confirmed free
fractional charge. An observed neutron--antineutron oscillation. A
detected long-range force coupled to baryon number.

\subsection{Application: hierarchy, KK resonance, and dark matter}
\label{sec:apply-other}
%% ====================================================================

The \GiFour{} framework is also the formal foundation of three other
key results of \GGTheory{}: the hierarchy derivation, the KK
resonance prediction, and the Kaxion dark-matter candidate.

\subsubsection{Hierarchy from the radial-scale stratum}
\label{ssec:apply-hierarchy}

\noindent\emph{Cohort map.} The scale profile $\e^{-K}$ is established in \citet{ArisakaA1Constitution}; the scale chain $\EPl\to\EGUT\to\Mlock$ and the reduced bulk scale $M_6$ in \citet{ArisakaA2GG6} and \citet{ArisakaA4Euclid}; the electroweak vev over-constraint $v=\Mlock(2/L)^N$ in \citet{ArisakaA9SM}.

The hierarchy $E_{\rm EW} \ll \EPl$ is derived in
\citet{ArisakaA1Constitution} via the scale profile $\e^{-K}$
with $K = 35$. In \GiFour{} vocabulary, this is the
\GA{} of the radial-scale stratum $\Sigma_r$: the unique stable
boundary configuration of the scale profile at $r=K$, with
$\Mlock = \EPl/2\e^{35} \simeq 3.85\,\TeV$.

The \OGNfive{} integer $K=35$ enters directly as the scale exponent.

\textbf{Physics intuition.} The radial coordinate is a long, gentle downhill
slope --- the renormalization-group axis dressed as geometry. The electroweak
scale is not a tiny number that has been delicately tuned against the Planck
scale; it is simply the altitude at the \emph{bottom} of a slope whose length is
an integer, $K=35$. Nothing is being canceled to thirty-two decimal places. The
smallness is a descent depth, the way the floor of a deep mine is far below the
surface not by coincidence but by the number of levels dug.

\textbf{The mechanism, quantitatively.} The radial-scale stratum $\Sigma_r$
carries its own \GA{} at $r=K$, and the scale that survives there is
\begin{equation}
\Mlock \;=\; \tfrac{1}{2}\,\EPl\,\e^{-K},
\qquad K \;=\; M^2+N^2+L^2 \;=\; 1+9+25 \;=\; 35,
\label{eq:apply-hierarchy-Mlock}
\end{equation}
so the exponent is not a free input but the squared norm of the \OGNfive{}
ledger $(M,N,L)=(1,3,5)$. Each unit of $K$ is one $\e$-fold of radial descent;
thirty-five of them carry $\EPl\simeq 1.22\times10^{19}$~GeV down to
$\Mlock\simeq 3.85$~TeV, and the same descent fixes the electroweak vacuum value
$v=\Mlock(2/L)^N\simeq 246$~GeV as an over-constraint.

\textbf{Why this reading helps.} The gauge hierarchy problem --- ``why is the weak
scale sixteen orders of magnitude below the Planck scale without a fine-tuning of
the Higgs mass to one part in $10^{32}$?'' --- simply dissolves. There is no
tuning because there is no cancellation: a small number written as $\e^{-35}$ is
the exponential of an integer, and the integer is the ledger's own squared norm.
The problem was an artifact of treating the weak scale as an additive remainder;
in the cascade it is a multiplicative descent depth, and depths are counted, not
balanced.

\subsubsection{KK resonance from the Wilson-line moduli}
\label{ssec:apply-KK}

\noindent\emph{Cohort map.} The collider spectroscopy of the lock cluster is the prediction-branch paper \citet{ArisakaP1KK}; its scale is the $\Mlock$ derived in \citet{ArisakaA2GG6}, and the same scale anchors the electroweak vacuum of \citet{ArisakaA9SM}.

The $\sim 3.85\,\TeV$ KK resonance cluster of \citet{ArisakaP1KK} is
the first KK mode of the bulk, determined by the lock scale
$\Mlock$. In \GiFour{} vocabulary, this is the energy scale of the
first excitation above the \GA{} on the radial-scale stratum.

The cluster structure (multiple closely-spaced poles rather than a
single pole) reflects the multi-Wilson-line structure of the
$\Sigma_\theta$ stratum: each Wilson-line phase produces a
distinct KK tower, and the visible cluster is the superposition.

\textbf{Physics intuition.} Once the descent has reached the radial \GA{}, the
first rung \emph{above} it is the lock cluster --- the geometry's lowest ringing
mode, the note the radial throat sounds when struck. It is not a second free
scale bolted on; it is the first excited basin, pinned to the same integer
$K=35$ that fixes the floor.

\textbf{The mechanism, quantitatively.} The lock cluster sits at the GA scale
$\Mlock\simeq 3.85$~TeV, and it is a \emph{cluster} rather than a single pole
because several bulk fields --- the spin-2 graviton, the color-octet KK gluon,
and the leptophobic $\UoneB$ vector --- share that one radial scale, their near
degeneracy split by the multi-Wilson-line structure of the $\Sigma_\theta$
stratum. The detailed collider spectroscopy is the subject of \citet{ArisakaP1KK}.

\textbf{Why this reading helps.} A first excited mode whose scale is a derived
integer, not a scanned range, is the sharpest kind of falsifier: the framework
stakes a single number, $\Mlock\simeq 3.85$~TeV, on the next decade of LHC
running, rather than the vague ``somewhere in the few-TeV region'' that a model
with an adjustable scale would offer. The \GiFour{} reading is what licenses the
sharpness --- the cluster is the first excitation above a \GA{} whose depth is
already fixed by \eqref{eq:apply-hierarchy-Mlock}.

\subsubsection{Kaxion dark matter from the U(1)$_{Qi}$ attractor}
\label{ssec:apply-Kaxion}

\noindent\emph{Cohort map.} The Kaxion is derived in the consequence paper \citet{ArisakaC1Kaxion}; its relic abundance in the cosmology paper \citet{ArisakaA10Cosmos}; its $\UoneQi$ stratum in \citet{ArisakaA7Qi4}; and the ground-state / \GRIPout{} framing in the capstone \citet{ArisakaA11Origin}.

The $\approx 8.1~\mu$eV Kaxion dark-matter candidate of
\citet{ArisakaC1Kaxion} is the slowly-rolling \GA{} on the
$\UoneQi$-Wilson-line stratum. In \GiFour{} vocabulary, this is the
\GA{} of $\Sigma_\theta$ projected to the $\UoneQi$ component.

The mass scale $\approx 8.1~\mu$eV is set by the cascade-action budget on this
stratum. In regime language (\S\ref{ssec:grip-in-out}) the dark
sector is the purest \GRIPin{} specimen in Nature: the condensate
relaxes toward its \GGA{} with Hubble friction itself as the
within-basin flow, and its exit-channel set is \emph{structurally
empty} --- the gauge-protected shift symmetry deletes any inter-basin
channel --- so the halt is permanent. That is the runtime reading of
the Kaxion's cosmological stability \citep{ArisakaC1Kaxion}.

\textbf{Physics intuition.} The dark sector is the cleanest specimen of descent
in all of Nature: a ball released on the side of a frictionless valley that has
\emph{no exits}. It rolls toward the bottom and, finding no channel out, simply
stays there --- for the entire age of the universe.

\textbf{The mechanism, quantitatively.} A \GRIPout{} event at the end of
inflation leaves the compact-angle field displaced from its minimum by an initial
misalignment; thereafter the field executes a pure \GRIPin{} descent with Hubble
friction as the only within-basin flow, and the cold relic it leaves behind
saturates the observed abundance. The mass $\approx 8.1~\mu$eV is set by the
cascade-action budget on the $\UoneQi$-Wilson-line stratum, and the gauge-protected
shift symmetry makes the exit-channel set \emph{structurally empty}, so the halt
at the \GGA{} is permanent --- the runtime reading of the Kaxion's cosmological
stability \citep{ArisakaC1Kaxion}.

\textbf{Why this reading helps.} Cosmological stability stops being an extra
assumption and becomes a structural fact: there is simply no channel out of the
basin to decay through. This is the \emph{empty-exit} terminus of the \GRIPin{}
taxonomy that the next section makes systematic
(\S\ref{sec:apply-radiative}); placed beside the radiative descents of the
synchrotron and the inspiral, the Kaxion is the limiting case in which the drain
is switched off entirely, and the contrast is what makes the taxonomy complete.

\medskip\noindent\textit{Dynamical tier: the radiative descent.}

\subsection{The sky: recombination, the inventory, and two redshifts
\quad \textnormal{\textsc{[Derived-conditional]}}}
\label{ssec:app-sky}

The third rung is the one that made the universe transparent. It is also
where the same engine is read at the largest available scale, by the
cosmology companion \citep{ArisakaA10Cosmos}.

\subsubsection{The inventory}
\label{sssec:app-a10-inventory}

The contents of the universe come out as coupled powers of a single
number.

\begin{table}[htbp]
\centering\small
\caption[The cosmic inventory as closed forms.]{The present-epoch
inventory as the cosmology companion \citep{ArisakaA10Cosmos} derives it,
with that paper's own values, measurements and pulls. The companion
grades the whole inventory as Derived-conditional under a named
integration conjecture, and it is the \emph{ratios} rather than the
absolute epoch that the geometry fixes for all time --- the one quantity
genuinely read from the sky being \emph{when} the reading is taken.}
\label{tab:app-a10-inventory}
\begin{tabular}{@{}llrrr@{}}
\toprule
\textbf{Quantity} & \textbf{Closed form} & \textbf{Value} &
\textbf{Measured} & \textbf{Pull} \\
\midrule
$\Omega_m$ & $1/\pi$ & $0.3183$ & $0.315 \pm 0.007$ & $+0.5$ \\
$\Omega_\Lambda$ & $1-1/\pi$ & $0.6817$ & $0.685 \pm 0.007$ & $-0.5$ \\
$\Omega_b$ & $1/(2\pi^2)$ & $0.05066$ & $0.0493 \pm 0.0006$ & $+2.3$ \\
$\Omega_{\rm dm}$ & $1/\pi - 1/(2\pi^2)$ & $0.2676$ &
$0.2657 \pm 0.0070$ & $+0.3$ \\
$n_s$ & $1 - 2/N_*$ & $0.9620$ & $0.9649 \pm 0.0042$ & $-0.7$ \\
$r$ & $12/N_*^2$ & $0.0043$ & $<0.036$ (95\%) & --- \\
$w$ & $-1$ (exact) & $-1$ & $-1.028 \pm 0.032$ & $+0.9$ \\
$\omega_b/\omega_m$ & $1/2\pi$ & $0.15915$ & $0.15709 \pm 0.0016$ &
$+1.3$ \\
\bottomrule
\end{tabular}
\end{table}

Dark energy is treated in two theorems of different strength. The first
is unconditional: ghost-free canonical scalars cannot have $w < -1$, with
equality exactly at a stationary point of the free-energy functional ---
which is to say at a Ground Geometric Attractor. The second, giving
$w=-1$ identically to about one part in $10^{91}$, is conditional on
three named hypotheses, one of them a lemma of this paper on vacuum-flux
freezing at the attractor. The kill is two-sided: the framework cannot
absorb $w \neq -1$.

\subsubsection{Recombination as the cosmic \GRIPout{}}
\label{sssec:app-a10-recombination}

The regime assignment runs the opposite way from the naive expectation,
and it is the part of the cosmology companion that most directly consumes
Part~V. The scale-setting events --- the exit from the unified edge, the
electroweak lock, the launch and end of inflation, and recombination
itself --- are \GRIPout{}: they reorganize the landscape. Everything that
then settles onto a landscape which has stopped moving --- the relaxation
to the dark-energy attractor, the frozen present-day constants --- is
\GRIPin{}. The companion states the division of labor with the
Standard-Model paper in one line: where the cosmic landscape itself came
from is the \GRIPout{} question the cosmology paper answers, and the
particle paper answered the complementary \GRIPin{} question.

\subsubsection{Two redshifts that are not the same quantity}
\label{sssec:app-a10-two-z}

One point deserves care, because it is the kind of thing a survey
flattens and the companion is at pains not to.

There are two distinct quantities near redshift eleven hundred. The
corridor switch is $z_{\rm sw} = e^J = e^7 \approx 1096.6$: a topological
marker, exact, with no continuous parameter in it and no observational
handle of its own. The atomic decoupling redshift $z_{\rm dec}$ is
computed from the cohort's own Rydberg scale and baryon fraction through
the standard recombination calculation, and lands at $1091.1$ raw against
a measured $1089.92 \pm 0.25$.

The agreement of the atomic chain is $0.11$ to $0.16$ per cent raw ---
the zero-fitted-parameter statement, already inside the modeling
systematic of the recombination calculation itself --- tightening to
$0.03$ to $0.07$ per cent when calibrated on the known modeling offset,
which the companion states for completeness and expressly does not claim
as independent precision. The kill is placed on the observable one only:
the reading is refuted if the measured decoupling redshift sits off the
atomic-chain value by more than half a per cent. Why the geometric
integer lands within four tenths of a per cent of the atomic one is, in
the companion's own words, either a coincidence the descent merely
records or a theorem the descent has not yet yielded.

\subsubsection{The second horizon problem, and the bridge upward}
\label{ssec:app-second-horizon}

The oldest puzzle of cosmology is that regions of the sky which have
never been in causal contact have the same temperature. Inflation answers
it: they were in contact, before being stretched apart.

The cosmology companion names a second and distinct puzzle, which
inflation does not address and which this paper's mechanism does. Those
same causally disconnected regions do not merely have the same
\emph{conditions}; they contain the same \emph{things}. The electron here
and the electron at the edge of the observable universe have the same
mass to every digit measured. Temperature is a state, and states can be
equalized by exchange. Identity is a specification --- a complete parts
list with couplings --- and no exchange of photons can negotiate a
particle's mass between two regions.

The answer offered is the ground-attractor mechanism read
cosmologically: each object has exactly one specification, the \GGA{} of
its stratum, so the agreements were never negotiated because they were
never in question. The companion is careful about the logical status ---
it calls the argument a reduction rather than an independent proof, since
it does not establish that the vacuum is unique but shows that if it is,
cosmic identity follows for free --- and says in terms that it would be
dishonest to sell it as a standalone solution.

Carried at that grade, it is nonetheless the hinge of this entire Part.
\textbf{Everything above this floor depends on it.} A chemistry that
worked differently in different regions would have no periodic table; a
biochemistry built on a locally varying electron mass would have no
universal genetic code. The reason the ladder can continue upward at all
is that its first rung is the same rung everywhere.

\subsubsection{What would end it}
\label{sssec:app-a10-falsifiers}

The tensor-to-scalar ratio at $0.0043$, two-sided. Any secure departure
of the dark-energy equation of state from $-1$. A null result for the
dark-matter line across its stated band. An inventory incompatible with
$1/\pi$, $1-1/\pi$ and $1/(2\pi^2)$. The companion labels its own mixture
honestly, noting which are forward predictions and which are audits of
existing data that the theory must not be allowed to overclaim.

\section{Floor two --- Chemistry \quad
\textnormal{\textsc{[Conditional]}}}
\label{sec:app-floor2}

From here upward the grade changes, and the change is announced rather
than absorbed. Nothing on this floor or above is claimed at the standard
of the floor below it. The capstone \citep{ArisakaA11Origin} is explicit
that it is a preview: its business is to show that the floors are one
structure and to state the mechanism that binds them, not to derive
chemistry, life and mind to the standard at which this program derives
particle physics. Companion papers on those floors are being written.

What follows is correspondingly brief. It is here because a Part that
stopped at Layer~3 would be showing the reader a ladder and climbing
three steps of it.

\subsection{What opened it}
\label{ssec:app-chem-opened}

Two shocks, in order. Stellar nucleosynthesis and the supernovae that
disperse it made the heavy elements that Layer~3's atoms could not
supply, and cometary delivery together with interstellar photochemistry
put them where chemistry could happen. The landscape of the previous
floor --- a universe of stable hydrogen --- is what made both possible.

\subsection{What settles: three rungs}
\label{ssec:app-chem-settles}

\paragraph{Small molecules.} A closed quartet on the warm, wet stratum:
water, carbon dioxide, ammonia, methane, with their bond angles and
dipoles fixed. The claim is closure --- that this set is the ground state
of its stratum rather than an accident of local abundance.

\paragraph{Twenty amino acids, and where the twenty comes from.}
\label{ssec:app-aminos}
This rung is the one to notice. The capstone gives the count as
\[
  |\text{GGA}_5| \;=\; 20 \;=\; K - NL \;=\; 35 - 15 ,
\]
together with the twenty being left-handed and the sugars right-handed.
\textbf{The same five integers that fix the inverse fine-structure
constant and the electron mass count the amino acids that biology uses.}
Whether that survives is a question for the chemistry companion; that it
is even stated is the clearest single indication that the ladder is meant
to be one structure and not a series of analogies. The empirical anchors
are the Murchison meteorite's roughly eighty amino acids, glycine
measured in situ at comet 67P, uracil in the Ryugu returned samples,
and ribose in Murchison and NWA~801.

\paragraph{Polymers.} Three sub-classes emerging together ---
oligopeptides, oligonucleotides, fatty acids --- with none of the three
privileged over the others. The \GRIPout{} that opens the rung is
wet--dry cycling with ultraviolet pulses on clay.

\subsection{The fence}
\label{ssec:app-chem-fence}

All three rungs are graded \textsc{Conditional}: each closes once one
named lemma is proved, and the lemmas are named --- small-molecule
closure, the chirality lock, polymer-class closure. (The codon
mapping is a lemma too, and it belongs to rung eight, not to this
floor.) The capstone asks to be
told if any rung above the atomic ones reads as though it were at
codebook standard, on the grounds that this would be a defect. This Part
takes it at its word and marks the boundary here.

\section{Floor three --- Life}
\label{sec:app-floor3}

Four rungs: an energy currency, a replicator, an archive, and a boundary
around all three. One of them matters to \emph{this} paper more than any
other rung above the atomic floor, and it is taken first.

\subsection{The bioenergetic anchor: where $\QB{}$ becomes physical
\quad \textnormal{\textsc{[Anchor-Derived]}}}
\label{ssec:app-qb}

Every paper in this cohort carries the acronym plate, and the last box on
that plate is the Arisaka constant $\QB{}$ with the equation
$J_E/\QB{} = J_e = J_B = J_{Qi}$ beneath it. Until this section, no Part
of this paper has said what that constant \emph{does}.

It is introduced and derived not here and not in the capstone but in
\ASeven{} \citep{ArisakaA7Qi4}, the information-side companion to this
paper, where it is defined as the energy cost of one ideal locked
boundary event --- the exchange rate between energy per winding on one
face of the compact circle and energy per qubit on the other, fixed by
the ledger integers rather than by any temperature. That is why it is
temperature-independent where the Landauer bound is not, and it is the
subject of \S\ref{sec:app-qi4} below. Its value follows from the
Rydberg energy and the ledger,
\[
  \QB{} \;=\; \frac{2}{5}\,\frac{E_{\rm Ry}}{K}
        \;=\; \frac{2}{5}\cdot\frac{13.6057~\text{eV}}{35}
        \;\approx\; 0.1555~\text{eV per qubit}
        \;\approx\; 15.0~\text{kJ/mol},
\]
and then makes the claim that turns it from bookkeeping into physics: the
hydrolysis of \textsf{ATP}, the reaction that powers essentially every
process in every cell on Earth, releases
\[
  \Delta G^\circ_{\textsf{ATP}} \;=\; 2 N_A \QB{} \;\approx\; 30.0~\text{kJ/mol},
\]
against a measured $30.5 \pm 0.5$~kJ/mol --- an agreement of about
$1.6$~per cent, with no quantity fitted anywhere in the chain from the
Rydberg energy and the integer $K$ to the bench measurement.

If that holds, the energy quantum of biology is the same quantity that
appears in the acronym plate of a paper about six-dimensional gauge
geometry, and the two are the same number for a reason. It is the single
most surprising line in the cohort, and it is the reason this Part
insists on the plate's last box.

The fences are the capstone's own and are carried without softening. The
grade is \textsc{Anchor-Derived}: the anchor is done and the derivation
from geometry to codebook standard is deferred to a companion in
preparation. The load-bearing claim is \emph{the single \textsf{ATP}
anchor alone}; the further per-process assignments sometimes quoted ---
one quantum per replicated bit, three per methylation event --- are
described by that paper as illustrative rather than measured. And on the
chemiosmotic proton-motive force the honest claim is that the elementary
step is \emph{of order} $\QB{}$, with a stated tolerance, not a
coincidence to three figures. The kill criterion is a deviation of the
\textsf{ATP} figure by more than five per cent.

\subsection{Four letters, because three families \quad
\textnormal{\textsc{[Conditional]}}}
\label{ssec:app-four-letters}

The genetic alphabet has four letters. The capstone's Layer-8 theorem
gives the size of the alphabet as
\[
  |\Sigma| \;=\; N + 1 \;=\; 4 ,
\]
with $N$ the same family number that
\S\ref{ssec:app-three-families} derived from an instanton budget on
$\mathbb{C}\mathbb{P}^2$. \textbf{On this reading, the genetic code has
four letters for the same reason that matter has three generations.}

That is either the ladder's most striking single closure or its most
obvious overreach, and this paper is not in a position to decide which.
The grade is \textsc{Conditional} on a named bifunctionality lemma. What
can be said here is what the mechanism requires: the alphabet is the
\GGA{} of its stratum, so it should be the same alphabet everywhere ---
which is, so far, exactly what is observed, and which the capstone lists
among its falsifiers in the form of a genetic alphabet of any other size
found anywhere.

\subsection{The archive, and the boundary \quad
\textnormal{\textsc{[Anchor-Derived]}}}
\label{ssec:app-dna-cell}

The remaining two rungs are given briefly. The RNA-to-DNA takeover
installs an archival duplex whose error floor, $10^{-9}$ per base pair,
is presented as a three-tier product rather than a single mechanism, and
with it the one-way flow of the central dogma. Membrane closure then
draws a boundary around a metabolizing interior and the first cell
follows --- the point at which the cascade first has, in the capstone's
vocabulary, a \emph{self-owning} open system rather than merely an open
one.

Both are \textsc{Anchor-Derived}: the numbers are anchored in the
literature, the derivations from geometry are deferred.

\section{Floor four --- The multicellular and the mind}
\label{sec:app-floor4}

Seven rungs, taken as two bands. The grade here is the weakest in the
Part and is marked as such; the reason for including the floor at all is
that the mechanism claim is a claim about \emph{these} rungs as much as
about the ones below, and a paper that built the engine and then declined
to say where it is supposed to run would be ducking its own thesis.

\subsection{Life becomes the engine}
\label{ssec:app-life-engine}

One structural observation belongs to this paper rather than to biology,
and it is the reason this floor is not simply a list.

Below Layer~10, every \GRIPout{} shock is geological or astrophysical:
stars explode, planets cool, vents open. From Layer~11 upward the shocks
are increasingly produced by what the previous floor built. Oxygenation
of the atmosphere --- the event that opens the eukaryotic rung --- was
caused by photosynthesizing life. The capstone puts it directly: from
Layer~11 forward, \emph{life is the cascade's engine of advance}.

In the vocabulary of Part~V that is a statement about the recursion. The
alternation of \GRIPin{} and \GRIPout{} does not require an external
driver at any rung; it requires only that a settled floor be capable of
disturbing the landscape above it. Below Layer~10 the disturbance comes
from outside the biosphere; above it, from inside. The mechanism does not
change. Only the source of the perturbation does.

\subsection{Domains, eukaryotes, multicellularity \quad
\textnormal{\textsc{[Conditional]}}}
\label{ssec:app-multicellular}

Three rungs. The primary domain split gives \emph{two} attractors rather
than one --- two lipid chemistries, bacterial and archaeal, with the
capstone claiming no third primary lipidome is available. The eukaryotic
cell is then a chimera of the two, opened by the Great Oxygenation Event
around $2.4$~Ga and settling some hundreds of millions of years later,
with an energy-per-gene budget as the mechanism. Multicellularity follows
the Neoproterozoic glaciations as an eight-rung ladder from clonal
adhesion to organogenesis, arriving at the Cambrian.

All three are \textsc{Conditional}, on a dual-gauge lemma, an energetics
lemma and a ladder-uniqueness lemma respectively.

\subsection{From a nerve ring to a conscious episode \quad
\textnormal{\textsc{[Asserted]}}}
\label{ssec:app-mind}

Four rungs, and the weakest grade in the Part. They are worth a page here
for one reason: at this floor the vocabulary of Part~V stops being
transferred and starts being literal.

The head-direction system of an insect or a rodent is a
\emph{one-dimensional attractor on the circle}: a ring of neurons with
lateral inhibition, supporting exactly one stable bump of activity that
tracks heading and drifts by less than a degree per second. That is not a
metaphor borrowed from dynamics into neuroscience. It is the same object
Part~V defines --- a landscape with a single deepest basin, and a descent
that halts in it --- realized in tissue, on a stratum that happens to be
$SO(2)$. The framework's claim that the four moves run at every scale is,
at this rung, checkable against a connectome.

Above it the capstone places a four-dimensional allocentric map --- place,
grid, time and causal history --- and then the conscious episode itself,
given as a closure lasting about $400$~milliseconds, two cycles of the
theta rhythm, with gamma-band binding inside it. The empirical anchor is
the P300 event-related potential at roughly $300 \pm 50$~milliseconds,
and the kill criterion is a reliably reported unitary conscious episode
completing within a single theta cycle.

\paragraph{The fence, in the capstone's own words.} Every one of these
four rungs is \textsc{Asserted}: the framework expects the result and can
say what would refute it, but the derivation is not in hand. And on the
last rung the capstone draws a sharper line still, which this Part
reproduces rather than paraphrases: the \emph{identification} of the
construction with phenomenal experience --- what it is like, from the
inside --- is an explicitly labeled structural-identity hypothesis,
\emph{defended, not deduced}, and is not a consequence read off the
geometric apparatus. It is that paper's own additional commitment. It is
not inherited from any companion, and it is not claimed here.

\section{Floor five --- Articulation \quad
\textnormal{\textsc{[Existence-Demonstrated]}}}
\label{sec:app-floor5}

The top rung of Figure~\ref{fig:a11-cascade18} is the act of writing the
constitution down.

The capstone's claim is that this is not a rhetorical flourish but the
cascade's own closure: a floor whose \GRIPout{} shocks are cumulative
cultural ones --- writing, mathematics, the printing press --- and whose
\GGA{} is a description of the geometry from inside the geometry. The
loop of Figure~\ref{fig:a11-cascade5} closes there: the constitution at
the top of the picture and the articulation at the bottom are the same
object seen from two ends.

Its grade is the only one of its kind in the Part.
\textsc{Existence-Demonstrated} means exhibited rather than derived, and
what exhibits it is the cohort itself. The capstone also states that
there is no Layer~19, and gives its reason: a description that includes
its own description has nowhere further to recurse.

This Part takes no position on whether that closure is a theorem or a
figure of speech. What can be said from inside \ASix{} is narrower and
worth saying: the mechanism of Part~V does not exclude such a rung. A
descent that halts in a basin containing a model of the landscape is not
a new kind of move. It is the same four moves, on a stratum whose
configurations happen to be representations.

\section{The other half: \QiFour{}, and why this paper is not finished
without it}
\label{sec:app-qi4}

Every application in this Part has been read geometrically. A landscape
was built, a descent ran down it, and a ground state was reached. That
reading is this paper's business and it is only half of the story ---
and the reader should be told which half, because the other half changes
what several of the rungs above mean.

\subsection{Two backbones, not one}
\label{ssec:app-two-backbones}

The boundary law of \AFive{} has two pillar papers standing under it, and
they are siblings. This paper supplies the \emph{geometric} backbone:
four pillars that say what the shapes are and how a system moves on them.
\ASeven{} \citep{ArisakaA7Qi4} supplies the \emph{quantum-information}
backbone: four pillars that say what is being counted, what is being
inferred, and what is being remembered while all that shape-changing goes
on. Its own summary is that the two together constitute the backbone of
the open-system law, geometric half and information half, and that with
both in hand the trio closes.

\ASeven{} is scrupulous about the direction of the debt, and this Part
repeats its wording rather than softening it. That paper consumes the
cascade; it does not define it. No information-side runtime may be
introduced before the free-energy landscape, the pipeline and the
stratification of this paper are in hand. It also inherits this paper's
conditionality: the cascade closure of Part~V holds modulo two named open
problems of ours, and every construction of \ASeven{} that rests on the
cascade carries that same conditional status forward.

The debt runs the other way too, and that is the point of this section.
The mapping is not one pillar to one pillar, and it would be tidier if it
were.

\begin{table}[htbp]
\centering\small
\caption[How the four geometric pillars feed the four
quantum-information pillars.]{The inward map of \ASeven{}
\citep{ArisakaA7Qi4}: which theorem of this paper each
quantum-information pillar consumes. It is deliberately not a one-to-one
pairing. The conservation ledger feeds two pillars; the free-energy
functional is inherited whole and then read three ways; and one
information pillar, the conformal one, has no geometric parent in this
paper at all --- the information side carries a structure the geometry
side does not.}
\label{tab:app-gi4-qi4}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.28\textwidth}p{0.44\textwidth}@{}}
\toprule
\textbf{\QiFour{} pillar} & \textbf{What it consumes here} &
\textbf{What it does with it} \\
\midrule
\textbf{\QiQUA{}} --- current locking &
\TGG{} and the Master Conservation Law (T1); the ledger (T2) &
Makes information the fourth boundary current, equal in standing to the
electric, baryonic and energetic ones, and locks all four through a
single conversion constant \\
\textbf{\QiEP{}} --- the extremal principle &
The free-energy functional $\cF_{\GFF} = \cF_{\GEP} + \cF_{\CEP}$ (T3);
the ledger (T2) &
Shows that three principles usually taught as unrelated --- least time,
maximum entropy production, and the free-energy principle of
neuroscience --- are three gauges of that one functional \\
\textbf{\QiCON{}} --- scale coherence &
--- &
Projects conformal isometry of the bulk into the boundary signatures of
scale-free behavior, with a bounded band rather than an unbounded slope \\
\textbf{\QiHAL{}} --- memory architecture &
The \GRIP{} cascade (T4) &
Formalizes what a memory-bearing system does with the cascade:
sensing, prediction, motion, storage, and the ability to schedule its
own landscape reorganizations \\
\bottomrule
\end{tabular}
\end{table}

\subsection{The relation between a ground attractor and the four
certificates}
\label{ssec:app-gga-certificates}

One statement has to be made carefully, because the careless version of
it is a circle and \ASeven{} says so about its own earlier drafts.

The Ground Geometric Attractor is \emph{not} selected by the
quantum-information pillars. It is fixed by the landscape alone: the
ground-state result of Part~IV produces it from the Morse structure of an
admissible stratum, using no admissibility filter of any kind. What
\ASeven{} proves is the other direction --- that the ground attractor,
once the geometry has produced it, \emph{satisfies} all four
quantum-information conditions. It carries the locked four-current
ledger. It is a genuine extremum of the unified extremal principle rather
than a local minimum of some coarse-grained effective potential. Its
scale signatures are those of a fixed point, so it survives being looked
at from a different distance. And the runtime can \emph{halt} there.

That last certificate is the one worth carrying back into this paper,
because it is a statement about \GRIPin{}. Halting admissibility excludes
basins that are deep but unreachable or unsustainable --- and a basin
nothing can reach, or nothing can stay in, is not a ground state of
anything in practice however low it sits. Part~V's halting theorem says
where the descent stops; \ASeven{}'s fourth certificate says what it
takes for stopping there to mean something.

The converse --- that no \emph{other} attractor of a stratum passes all
four certificates --- is \textbf{not} proved, in either paper, and is not
implied by anything above. \ASeven{} registers it as an open problem and
records that an earlier formulation of that section asserted it and
rested it on the capstone, whose own argument invokes the pillars: a
circle, found and cut.

\subsection{The Arisaka constant, and the plate's last box}
\label{ssec:app-qb-provenance}

\S\ref{ssec:app-qb} used the constant $\QB{}$ without saying where it
comes from. It comes from here. \ASeven{} defines
\[
  \QB{} \;=\; \frac{2}{5}\,\frac{E_{\rm Ry}}{K}, \qquad K = 35 ,
\]
as the energy cost assigned to one ideal locked boundary event when the
electric, baryonic and informational sectors are each counted in one
natural unit: about $0.1555$~eV per qubit, about $15.0$~kJ/mol, about
$5.8\,k_BT$ at body temperature. It introduces no new continuous
parameter --- both the Rydberg energy and the integer $K$ are already
fixed upstream --- and the paper is candid that the \emph{form}
$E_{\rm Ry}/K$ rather than some other function of the ledger is
justified by minimality rather than proved, and carries that as a named
open problem.

The equation printed under the last box of the acronym plate,
$J_E/\QB{} = J_e = J_B = J_{Qi}$, is an equality of \emph{normalized
event counts}, not an identity among raw microscopic carriers, and
\ASeven{} says so in those words. Its kill criterion is correspondingly
sharp and correspondingly narrow: find one elementary catalytic event
whose four-current bookkeeping disagrees with that equation by more than
twenty per cent, and the pillar is refuted. What is being tested is the
claim that a \emph{single} conversion factor locks the four currents ---
not the agreement between $\QB{}$ and any one calorimetric measurement.

One boundary is worth stating here because this Part climbed above the
chemistry floor. \ASeven{} derives and validates $\QB{}$ at the chemistry
scale, at \textsf{ATP} hydrolysis, and it explicitly does \emph{not}
derive a per-bit quantum for neural information. Any reading that prices
a thought at $\QB{}$ is not in that paper and is not in this one.

\subsection{What each rung gains}
\label{ssec:app-qi4-gains}

The enhancement is not uniform, and it is worth being concrete about
which rung gains what.

\paragraph{The particles.} The electron and the proton acquire a second,
independent description of what makes them what they are. Geometrically
each is the deepest basin of its stratum; informationally each carries
four certificates, and the certificates are stated as properties of the
particles rather than as the instrument that found them.

\paragraph{The parameters and the sky.} The selector that Part~IV
supplies as a free-energy functional is the same object \ASeven{} reads
three ways. Where this paper says a configuration descends, that paper
can say equivalently that it infers --- and the equivalence is a theorem
there, not an analogy.

\paragraph{The life floors.} This is where the enhancement is largest,
and where a reader from biology should go. The free-energy principle
used across neuroscience and theoretical biology is obtained in
\ASeven{} as the inferential gauge of its second pillar, by pullback of
the same functional this paper builds. It is not a separate law
introduced for living systems. It is one of three faces of the selector
that also gives least time and maximum entropy production. And the
persistence class that \S\ref{ssec:app-dna-cell} gestured at --- a system
that owns itself rather than merely being open --- is defined there, with
a maintenance margin that has to be positive for the thing to go on
existing.

\paragraph{The mind floors.} The ring attractor, the map and the
episode of \S\ref{ssec:app-mind} are, in \ASeven{}'s vocabulary, a memory
architecture: sensing feeding prediction feeding motion feeding memory,
closed into a loop. And that paper adds one thing this paper's Part~V
does not have. A memory-bearing system carries its own
landscape-reorganizing dial: it can schedule its \GRIPout{} events on
evidence, and repeat them, where the cosmological ones of
\S\ref{ssec:app-sky} happen once and cannot be recalled. An engine that
can decide when to disturb its own landscape is a different kind of
engine, and it is the last thing the ladder builds.

\subsection{The plain statement}
\label{ssec:app-qi4-plain}

So, plainly, and in this paper's own voice rather than as a quotation:
\textbf{\ASix{} is not finished without \ASeven{}.} This paper gives the
shapes and the motion on them. That paper gives what is conserved, what
is inferred, and what is remembered. Either alone is a description of
half of an open system, and the boundary law of \AFive{} needs both
halves to be a law about anything that observes.

A reader who has followed this Part and wants the other half should read
\ASeven{} next, and should read it before the capstone, because the
capstone's upper floors are written in its vocabulary.


\section{Interlude: the descent rate, computed}
\label{sec:app-interlude}

Everywhere above, the descent has been \emph{described}: a system starts
symmetric, is handed a landscape, chooses at a saddle, and settles. This
section computes one, in the single place where the arithmetic is clean
enough to do it end to end.

It is the most technical section of the Part, and a reader following the
ladder rather than the machinery should skip to
\S\ref{sec:app-what-the-ladder-shows} without loss. It is kept here, and
not in the appendices, because it is the only place in the cohort where a
\GRIPin{} flow is run as a rate rather than as a classification --- and
because the result is a small unification in its own right: synchrotron
radiation and gravitational-wave inspiral, two processes usually taught
in different courses, come out as one descent on one landscape,
differing only in the spin of the mediator.

\subsection{Application: radiative descent --- synchrotron and gravitational-wave inspiral as one \GRIPin{} flow}
\label{sec:apply-radiative}

\noindent\emph{Cohort map.} The field-theory derivation (Larmor/Peters coefficients, the storage-ring constant $C_q$) is \citet[Part~VI]{ArisakaA5GDOS}; the mediator-spin content from the four-force-uniqueness theorem of \citet{ArisakaA9SM}; the absorbing black-hole terminus from \citet{ArisakaC3BH}; and the empty-exit comparison from \citet{ArisakaC1Kaxion}.

The three applications above are static: \GiFour{} fixes the attractors of the
spectrum. The runtime also governs a dynamical process of equal reach --- the
radiative descent of an accelerated source --- and here the four pillars act
together on a single worked example. The underlying field theory (both static
inverse-square laws from one boundary connection; the Larmor/synchrotron and
Einstein/Peters radiation laws as their retarded completion; and the exact damping
coefficients, including the storage-ring constant $C_q$, derived from one
influence-functional trace) is carried out in \citet[Part~VI]{ArisakaA5GDOS}. The
present section records only its \GiFour{} reading.

\subsubsection{The leading multipole is the mediator spin: \texorpdfstring{$l_{\rm lead}=s$}{l=s}}
\label{ssec:apply-rad-ladder}
The composite \TGG{} connection (Part~II) carries both radiating sectors of the
boundary at once --- the spin-1 gauge field and the spin-2 metric perturbation.
\CCI{} (Part~III) supplies the conserved currents these sectors couple to: the
charge current $J^\mu$ for the gauge sector and the energy--momentum tensor
$T_{\mu\nu}$ for the metric sector. Current conservation is a Ward identity, and it
makes every source multipole below the mediator spin non-radiating: charge
conservation removes the monopole from the gauge sector, and energy--momentum
conservation removes both the monopole and the dipole from the metric sector. The
leading radiating multipole is therefore fixed by the spin alone,
\begin{equation}
\boxed{\,l_{\rm lead}=s\,},\qquad
L_s \;\propto\; \frac{g_s^{\,2}}{c^{\,2s+1}}\,
\Big\langle\big(\partial_t^{\,s+1}\,\mathcal I^{(s)}\big)^2\Big\rangle ,
\label{eq:spin-ladder}
\end{equation}
with $\mathcal I^{(s)}$ the $2^s$-pole moment of the source:
$(s,l,\text{time-derivatives},c\text{-power}) = (1,1,2,3)$ gives Larmor/synchrotron
(electric dipole $\ddot{\mathbf p}$), and $(2,2,3,5)$ gives the quadrupole formula
(mass quadrupole $\dddot Q_{ij}$). The spin $s$ that indexes the ladder is not
assumed: it is read from the bulk $\mathrm{Spin}(1,5)$ Lorentz content on the
boundary (the four-force-uniqueness theorem of \citet{ArisakaA9SM}), which makes the
photon the rank-1 vector of the unbroken $U(1)$ and the graviton the rank-2
symmetric-traceless tensor. One geometric fact --- the mediator spin --- and one
conservation law --- \CCI{} --- fix the entire difference between the two radiation
laws.

\subsubsection{One descent, three terminations: the \texorpdfstring{\GRIPin{}}{GRIP-In} taxonomy}
\label{ssec:apply-rad-taxonomy}
In runtime language (\S\ref{ssec:grip-in-out}) the orbit itself is conservative
\emph{circulation} and descends nothing; what descends is the radiation. The emitted
power drains the source into the boundary, and the slow contraction of the orbit is
a \GRIPin{} gradient flow on the \GFF{} binding-energy landscape $\GEP$, carrying the
source inward and chirping its frequency upward toward a \GGA{}. Synchrotron cooling
and gravitational-wave inspiral are the $s=1$ and $s=2$ instances of this one flow.

Placed beside the Kaxion of \S\ref{ssec:apply-Kaxion}, the three exhaust the
\GRIPin{} taxonomy by exit channel:
\begin{itemize}[leftmargin=2em]
\item \emph{Empty-exit \GGA{} (Kaxion).} The gauge-protected shift symmetry deletes
every inter-basin channel; with no drain the condensate halts permanently at its
protected \GGA{} (\S\ref{ssec:apply-Kaxion}; \citealp{ArisakaC1Kaxion}).
\item \emph{Relaxation \GGA{} (synchrotron).} The radiative drain is balanced by the
diffusion of discrete photon emission, so the descent settles into a normalizable
fluctuation--dissipation fixed point --- the equilibrium emittance of GFF-ST7, with
the universal electron constant
$C_q = \tfrac{55}{32\sqrt3}\,\hbar/mc = 3.83\times10^{-13}\,$m that every storage ring
measures \citep{Sands1970,ArisakaA5GDOS}.
\item \emph{Absorbing \GGA{} (inspiral).} The binding landscape
$\GEP \simeq -G_N m_1 m_2/2a$ is unbounded below all the way to contact, so there is
no normalizable floor: the descent does not settle, it exits the basin at merger into
the next \GGA{} down --- a black-hole horizon, the maximal-entropy \OGGEP{} port with
entropy $S = A/4$ derived in \citet{ArisakaC3BH}.
\end{itemize}
The same \GRIPin{} gradient flow therefore terminates in three structurally distinct
ways --- a permanent halt, a thermal relaxation, and an absorbing exit --- selected by
whether the exit set is empty, drained-and-balanced, or drained-and-bottomless. The
inspiral is the case whose terminus is the deepest attractor the cohort knows.

\subsubsection{The dimensionless descent rate}
\label{ssec:apply-rad-zeta}
Absolute powers are not comparable across the two columns (the electron charge and a
solar mass are incommensurable couplings). The framework-level comparator is the
dimensionless \emph{descent rate per cycle},
\begin{equation}
\zeta \;\equiv\; \frac{L\,T_{\rm orb}}{|E_{\rm orb}|}
\;=\; \frac{\text{energy radiated per orbit}}{\text{orbital energy}},
\label{eq:zeta-descent}
\end{equation}
which measures how fast the \GRIPin{} flow advances in units of one cycle. Both
systems run from $\zeta\ll1$ (adiabatic descent) to $\zeta\sim\mathcal O(1)$ (the
terminus, where the adiabatic approximation breaks): for the synchrotron $\zeta$
reaches order unity at the radiation-reaction limit; for the binary, at merger.
Representative values, recomputed from the standard formulae, are collected in
Table~\ref{tab:apply-radiative}.

\begin{table}[htbp]
\centering
\caption[The radiative \GRIPin{} descent, side by side.]{The radiative \GRIPin{} descent, side by side. Absolute powers are
incommensurable across columns; the comparator is the dimensionless descent rate
$\zeta$ of \eqref{eq:zeta-descent}. Representative numbers recomputed from the
standard formulae; the field-theory derivation is \citet[Part~VI]{ArisakaA5GDOS}.}
\label{tab:apply-radiative}
\small
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.26\textwidth}p{0.34\textwidth}p{0.34\textwidth}@{}}
\toprule
\textbf{Aspect} & \textbf{Synchrotron ($s=1$)} & \textbf{Inspiral ($s=2$)} \\
\midrule
Mediator (bulk $\mathrm{Spin}(1,5)$) & photon, rank-1 vector & graviton, rank-2 sym.\ traceless \\
Coupling / charge & electric charge $q$ & mass--energy $\sqrt{G_N}\,m$ \\
Leading multipole $l=s$ & dipole & quadrupole \\
Power law & $q^2a^2\gamma^4/6\pi\varepsilon_0 c^3$ & $(32/5)(G_N/c^5)\,\mu^2 a^4\Omega^6$ \\
\GRIPin{} landscape $\GEP$ & orbital energy of the beam & binding energy $-G_N m_1 m_2/2a$ \\
Chirp & $\omega_c=qB/\gamma m$ rises as $\gamma\to1$ & $\Omega$ rises as $a\to0$ \\
Terminus \GGA{} & relaxation (equilibrium emittance) & absorbing (black hole, \citealp{ArisakaC3BH}) \\
Diffusion / noise & photon shot noise; $C_q=3.83\times10^{-13}$\,m & graviton shot noise $\sim10^{-79}$, negligible \\
Descent rate $\zeta$ & $\approx3\times10^{-2}$ (LEP $e^-$) & $\approx10^{-13}$/orbit (Hulse--Taylor) $\to\mathcal O(1)$ (merger) \\
Empirical anchor & every storage ring \citep{Sands1970} & Hulse--Taylor \citep{HulseTaylor1975}; LIGO/Virgo \\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{What \texorpdfstring{\GiFour{}}{Gi-4} adds, and the open step}
\label{ssec:apply-rad-fence}
The radiation laws are unmodified; what the \GiFour{} reading supplies is their
\emph{unity} --- one composite \TGG{} connection, one \CCI{} conservation law fixing
the multipole order through $l_{\rm lead}=s$, and one \GRIPin{} flow with a three-way
terminus taxonomy --- together with the dissolution of the classical self-force
pathology, which in this reading is the gradient sector of an open-system current
rather than a local third-derivative force (\citealp{ArisakaA5GDOS}). The genuinely
open step is sharply localized: the Larmor coefficient $1/6\pi\varepsilon_0 c^3$ and
the Peters coefficient $32 G_N/5c^5$ are each derived exactly in
\citet[Part~VI]{ArisakaA5GDOS}, but as separate evaluations; exhibiting both as the
$s=1$ and $s=2$ values of \emph{one} spin-indexed \GFF{} descent functional --- that
is, deriving the normalization $N_s$ and the $(s{+}1)$-th-derivative structure of
\eqref{eq:spin-ladder} from the bulk action --- remains open. We record it as the
natural successor problem to the radiative reading.

\section{What the ladder shows}
\label{sec:app-what-the-ladder-shows}

Eighteen rungs, six companions, five grades. It is worth saying plainly
what they share, because the shared structure is this paper's actual
claim and it is easy to lose in the detail.

\paragraph{One mechanism, eighteen strata.} Nothing above required a new
mechanism. The Standard-Model parameters, the family count, the electron,
the proton, the cosmic inventory, the amino acids, the genetic alphabet,
the ring attractor --- all of them are the same four moves, run on
different boundary strata. What differs from rung to rung is the stratum,
not the engine. The capstone states the recursion in the form this paper
would state it: the step from one floor to the next is not a fifth move,
it is the same four moves on a landscape the floor below has perturbed.

\paragraph{Stability is always ground-state-ness.} Every absolutely
stable object in the survey is the deepest basin of its stratum, and
every unstable one has an exit gate whose action sets its width. That is
one statement covering the electron's infinite lifetime, the muon's
microsecond, the $\Delta$ resonance's hundred and seventeen MeV, the
dark-energy attractor's $w = -1$, and --- if the upper floors hold --- the
persistence of a genetic alphabet across four billion years. It is also,
read cosmologically, the answer to why two electrons on opposite sides of
the sky agree.

\paragraph{The two regimes divide the labor.} The rare
landscape-reorganizing events are \GRIPout{} --- the unified exit, the
electroweak lock, recombination, the supernovae, the vents, the
oxygenation, each of the seventeen shocks in
Figure~\ref{fig:a11-cascade18} --- and everything that settles afterward
is \GRIPin{}. Below Layer~10 the shocks come from outside the biosphere;
above it, from inside. The engine does not change.

\paragraph{The grade falls as the ladder rises, and that is the honest
shape of the claim.} Four of the eighteen rungs are closed strictly;
seven once an empirical anchor may stand in for a derivation; nothing
above the atomic rungs is at codebook standard. This Part has printed the
grade on every rung rather than averaging it, because the average would
misrepresent both ends: it would understate how strong the bottom is and
overstate how settled the top is.

\subsection{Where to attack it}
\label{ssec:app-attack}

A framework of this shape is tested at its most exposed points, not its
most impressive ones, and the capstone pre-registers ten of them with
numbers, instruments and decade windows attached.

\begin{table}[t]
\centering\small
\caption[The ten pre-registered falsifiers of the cohort.]{The Top~10
falsifiers, as \AEleven{} \citep{ArisakaA11Origin} registers them. Six
belong to the physics floors, three to life, one to mind. Each carries a
number and an instrument, and decisive failure on any one of them ends
the framework --- whatever grade the surrounding rung happens to carry.
This is the right object on which to close a survey of applications: not
what the framework explains, but where it can be killed.}
\label{tab:app-top10}
\begin{tabular}{@{}rp{0.40\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.22\textwidth}@{}}
\toprule
& \textbf{What is predicted} & \textbf{Where it fails} &
\textbf{Instrument, window} \\
\midrule
1 & Lock cluster at $\Mlock = 3.85 \pm 0.10$~TeV &
Exclusion of the spin-one pair &
HL-LHC, 2030--35 \\
2 & Kaxion at $m_a = 8.1~\mu$eV &
Null across the band at derived coupling &
Haloscopes, 2027--30 \\
3 & $w = -1$ exactly &
Any secure $w \neq -1$ &
DESI, Euclid, Roman, 2026--30 \\
4 & Proton lifetime beyond $10^{35}$~yr &
One decay event, any channel &
Hyper-K, DUNE, 2027--40 \\
5 & $r = 12/N_*^2 = 0.0043$, two-sided &
Detection above $0.01$, or a null at the floor &
CMB-S4, LiteBIRD, 2030--35 \\
6 & $\delta_{\rm PMNS} \approx 196^\circ$ &
A converged value elsewhere &
DUNE, Hyper-K, 2030--35 \\
7 & $\Delta G^\circ_{\textsf{ATP}} = 2N_A\QB{} = 30.0$~kJ/mol &
Deviation beyond five per cent &
Single-molecule biophysics, 2026--30 \\
8 & A genetic alphabet of exactly four letters &
Any other alphabet size, anywhere &
Comparative genomics, continuing \\
9 & A two-knee $1/f$ spectrum in eukaryotic metabolism and in cortex &
A single knee where two are predicted &
Single-cell imaging; intracranial EEG \\
10 & A conscious episode of about $400$~ms &
A unitary episode completing in one theta cycle &
Psychophysics and EEG, 2026--28 \\
\bottomrule
\end{tabular}
\end{table}

Six of the ten are physics and are the ones to bet on: they are the
rungs where the grade is highest, the numbers sharpest and the
instruments already funded. The remaining four are the price of the
claim that the ladder is one structure --- because a framework that
extended to life and mind without exposing itself there would be
extending to them for free.

That is the accounting. The engine of Part~V, run on eighteen strata,
produces a periodic table, a genetic code and a conscious episode from
the same four moves that produce the electron; and it can be ended by
about a dozen measurements, most of them due this decade.

\section{A closing word, for any reader}
\label{sec:app-closing-word}

This Part has been long, so here is what it said.

There is one machine in this paper. It has four moves. A system holds
every possibility open; a landscape tells it which way is downhill; at
some point it reaches a fork and takes one branch, and a symmetry that
was real a moment before is gone; and it comes to rest in the lowest
place it can reach. Then --- and this is the part that makes it a
history rather than an event --- the resting place becomes the
disturbance that opens the next landscape, and the machine runs again,
one story up.

Run that machine at the highest energies and it gives you the particles:
three families of matter and not four, an electron that cannot decay
because there is nowhere below it to go, a proton that cannot decay
because you cannot unwind a knot. Run it as the universe cools and it
gives you atoms, and a sky whose contents are fixed by a handful of
whole numbers. Run it in the warm water of a young planet and, if the
argument holds, it gives you the twenty amino acids, a four-letter
alphabet, and a molecule that carries thirty kilojoules per mole because
a constant fixed by the geometry of six dimensions says that is what one
locked event costs. Run it long enough and it gives you an animal with a
ring of neurons that knows which way it is facing --- which is, in the
exact technical sense of this paper, a landscape with one basin and a
descent that halts in it. Run it once more and the animal writes the
machine down.

That is the claim. This paper does not prove all of it, and the earlier
sections have been careful to mark where the proof stops: four rungs of
eighteen are closed strictly, and the ones above the atoms are honest
forecasts with their derivations still being written. What this paper
contributes is the machine itself, stated sharply enough that someone can
show it does not run.

\medskip

It is worth saying what would follow if it does run, because the
consequence is not the one people usually expect from a physical theory.

The oldest form of the question is why there is something rather than
nothing, and the second oldest is why we are here to ask. Both invite two
bad answers. One is that it was arranged --- that somewhere there is an
intention the universe was built to satisfy. The other is that it was
luck --- that we happen to occupy the corner of a vast lottery where the
numbers came up habitable, and the question dissolves into a shrug.

The picture in this Part is neither. Nothing here is arranged; there is
no designer in any equation, and no step waits for permission. But
nothing here is lucky either, because at each stage the machine has no
choice about where to stop. It stops at the bottom. The bottom is not a
preference --- it is a property of the shape, and the shape was fixed
before the first second. Cosmos, life and mind are then not three
separate miracles and not three lottery wins. They are three depths of
one descent, and the reason they follow one another in that order is that
each is the floor the next one needs.

If that is right, then the strangest thing about the universe is not that
it produced observers. It is that it could not have avoided producing
them, given how it started --- and that the observers it produced turn out
to be made of the same four moves as everything they are looking at. The
theory does not stand outside the world it describes. It is the last rung
of the ladder it is describing, written by an arrangement of matter that
got there the same way the electron did.

\medskip

None of which makes any of it true. That is what the ten measurements in
Table~\ref{tab:app-top10} are for, and most of them will report this
decade. A framework that reached this far and could not be killed would
not be worth the paper it is written on. This one can be killed by a
single proton decay, by one dark-matter search coming up empty across a
narrow band of frequencies, or by a chemist measuring the energy of a
single molecule and getting the wrong number.

That is the right way for an argument of this size to end: not with a
claim about what must be so, but with a short list of things the world is
about to be asked.


\GGPart[Discussion]{Discussion}{What the four pillars establish, how the framework stands against the alternatives, what would falsify it, and what remains open}


This Part steps back from the construction and asks the questions a
reader is entitled to ask of it: what is actually new here, how does it
stand against the other serious attempts at the same problem, what
would prove it wrong, and what is still missing.

The order is deliberate. It begins with a compact statement of what the
preceding Parts established, because significance cannot be judged
before content is known. It then places those results against the
principal alternatives --- string compactification, the
gauge/gravity correspondence, warped extra dimensions --- and the
comparison is meant to be evaluative rather than diplomatic: each
framework is credited with what it does well and asked directly what it
leaves postulated. Innovation is separated out and stated plainly,
because a claim of novelty that is not stated cannot be checked. Then
the falsifiers: every prediction the four pillars make that could kill
the framework, with the observable that would do it and the facility
that will decide. Then the open problems, given at length and without
softening, and finally the directions the program takes next.

A reader who wants the short version can take the first section and the
falsifier table and stop; between them they contain the claim and the
exposure. A reader who intends to argue with the paper should go
instead to the open problems, which is where the argument is most
likely to be winnable.

Two warnings about how to read what follows. The comparisons are
between \emph{structures}, not between communities or research
programs, and no claim is made about which body of work is the more
valuable. And the falsifier list is deliberately unhedged: where a
prediction is at risk it is stated as at risk, and where a measurement
already constrains it the constraint is given rather than described.




%% ====================================================================
\section{What the four pillars establish}
\label{sec:disc-outcomes}
%% sec:problems-solved is retained as a second anchor: the section it
%% used to name has been dissolved, and one cross-reference elsewhere
%% in the paper points here.  Re-anchoring preserves the link without
%% rewriting a call site whose sentence is still correct.
\label{sec:problems-solved}
%% ====================================================================

Before any claim is made about what these results are worth, here is
what they are.

This paper took as given a six-dimensional arena, the group acting on
it, an integer ledger, a boundary law, and four constitutional axioms
--- all of them established elsewhere and all of them listed in
Part~I --- and from those inputs built four objects and proved five
theorems about them. Stated without machinery, the four objects are a
\emph{single geometric structure} carrying all four forces, a
\emph{consistency condition} on its bookkeeping, a \emph{landscape} on
which the boundary world can sit, and a \emph{descent} that moves it
across that landscape and stops.

What is new here, and was not available before this paper, is the
following.

\begin{itemize}[leftmargin=2em]
\item \textbf{The four forces assemble into one object, and in only one
  way.} Uniqueness is the load-bearing half: a merely possible
  unification explains nothing, because the question "why this one?"
  has no answer. Here the constitutional requirements leave exactly one
  arrangement standing.
\item \textbf{That object obeys a single conservation law.} Energy and
  momentum of matter, gauge fields, residual higher-dimensional modes,
  and the anomaly-correction term add to something that does not leak
  out of the observable world. In a theory where the visible world is a
  projection of a larger one this is not automatic --- projections are
  exactly the operation that loses things --- and it is proved rather
  than assumed.
\item \textbf{The integer ledger is forced, not chosen.} Imposing
  closure of the anomaly polynomial admits exactly one solution.
  Nothing was fitted; no datum entered. The consequence is that the
  ledger cannot be adjusted if a measurement disagrees, which is what
  makes the next item a real prediction.
\item \textbf{A measurable astrophysical consequence follows from the
  ledger alone}: a correction to the neutron-star mass--radius
  relation, of a size fixed entirely by the integers, with no freedom
  anywhere in it.
\item \textbf{The free-energy landscape has genuine minima, and this is
  forced by the anomaly condition.} That link is the paper's most
  unexpected internal result: a requirement about the consistency of a
  quantum field theory implies a statement about whether matter can
  settle down.
\item \textbf{The descent terminates, and where it terminates is
  countable.} Running the cascade yields three families of matter --- not
  by choice and not by fit, but because a fourth would need somewhere to
  go and there is nowhere.
\item \textbf{The four pillars and the constitutional axioms are
  equivalent.} Each implies the other with nothing left over, which
  means the construction has closed rather than merely accumulated.
\end{itemize}

Two boundaries around that list should be stated in the same breath,
because they are what make the rest of this Part meaningful. Every
result above is \emph{conditional on the declared inputs} --- this paper
does not derive six dimensions, it shows what six dimensions force.
And several steps rest on named hypotheses that are not yet theorems;
they are cataloged in \S\ref{sec:disc-open} under their own names, and
nothing in the sections between here and there quietly promotes one of
them.

\subsection{Problems solved by \ASix{}}
\label{ssec:problems-solved-list}

We first record the principal problems that \ASix{} \emph{settles}, each
with its global identifier, the theorem that carries it, and a live
empirical falsifier in the sense of \POS{7} (Measurement and
Falsifiability). These entries complement the open and future items that
follow.

\subsection*{PS-GRIP-1. Axiomatic foundation for the \GiFour{} vocabulary}
\textbf{Resolved} (this paper, Theorems~T1--T5). The \GFF{}/\GRIP{}/cascade
vocabulary used informally across the cohort is given a rigorous
five-theorem basis, so that downstream papers can cite a fixed foundation.
\emph{Live falsifier:} a failure of any pillar's structural falsifier below
(F-GRIP-1\ldots7) would force a re-examination of the corresponding pillar
axioms.

\subsection*{PS-GRIP-2. The family number $N\le 3$} \textbf{Partially resolved} (Theorem~T4; the $N\le 3$ upper bound via the capacity theorem T-N3-6 of \citet{ArisakaP2N3}, conditional on its coupling clause and M6 ($=$ R-INT); the lower
bound $N\ge 3$ from \CP{} unconditional). The \GRIP{} cascade
closes against the budget $\Stot\le 3\pi$, forcing $N\le 3$ on the
family-projector stratum. \emph{Live falsifier:} a fourth chiral
generation (cf.\ the LEP invisible-width bound $N_\nu=2.984\pm0.008$,
\citealp{ALEPH2006}) would force a re-examination of \BIsix{} closure.

\subsection*{PS-GRIP-3. A geometric arrow of time on the boundary}
\textbf{Resolved} (Theorem~T3). The \GFF{} free-energy functional obeys the
second law $\dd F_{\GFF}/\dd t\le 0$ along every \GDOS{} CPTP trajectory
with detailed balance. Custody note: Theorem~T3 is the runtime face of the one constitutional arrow, consumed rather than re-derived --- \AX{3} owns the arrow (the \AX{3} synthesis of \citealp{ArisakaA1Constitution}); Part~IX of the CPT member paper \citep{ArisakaP5CPT} is the canonical law-level derivation over a bulk that is itself exactly T-symmetric; the \GDOS{} companion \citep{ArisakaA5GDOS} is the dynamical implementation. \emph{Live falsifier:} a measured violation of
equipartition, the fluctuation--dissipation relation, or Onsager
reciprocity near boundary equilibrium (F-GRIP-1, F-GRIP-2, F-GRIP-3).

\subsection*{PS-GRIP-4. The generation gate-ordering ladder}
\textbf{Resolved} (Theorem~T4, \GRIP{}-ST7). The Bogomolnyi-saturating
cascade actions $S_{[2k]\to[2(k-1)]}=\pi k$ yield the generation-ordered
geometric gate factor $\Gamma\propto\e^{-\pi k}$ (absolute lifetimes are
phase-space-dominated and fixed in \citet{ArisakaA9SM}). \emph{Live
falsifier:} a controlled-kinematics decay class violating the residual
$\e^{-\pi k}$ ordering once the Sargent phase-space factors are divided out
(F-GRIP-4).

\subsection*{PS-GRIP-5. Runtime gauge invariance under the cascade}
\textbf{Partially resolved} (Theorem~T4.2; conditional on the
compensator-monodromy conjecture M11). The boundary Ward identities are preserved along the
cascade iff the \PBIfour{} compensator extends smoothly across each
instanton stratum. \emph{Live falsifier:} a runtime Ward-identity
violation in the boundary currents (F-GRIP-5).

\subsection*{PS-GRIP-6. The \OGNfive{} integer ledger as the parameter-free backbone}
\textbf{Partially resolved} (Theorem~T2; Diophantine uniqueness imported
from \citet[Theorem~10.2]{ArisakaA3BI6}, conditional on the \MSMF{} norm
choice M9). \MSMF{} closure of $X_4$ forces $(M,N,L;J,K)=(1,3,5;7,35)$, from which
every quantitative prediction follows in closed form with zero free
parameters. \emph{Live falsifier:} a measured deviation from any
ledger-derived prediction (e.g.\ $\alpha^{-1}(0)\ne 137$).

\subsection*{PS-GRIP-7. Geometric--gauge unification with a Master Conservation Law}
\textbf{Resolved} (Theorem~T1, \CCI{}-ST7). \TGG{} supplies a unique
composite connection $\Atot=\omega\oplus A$ and the boundary Master
Conservation Law $\nabla_\mu\cT^{\mu\nu}_{\TGG}=0$. \emph{Live falsifier:}
a precision compact-star mass--radius measurement contradicting the
modified Tolman--Oppenheimer--Volkoff relation of \CCI{}-ST7.

\subsection*{PS-GRIP-8. Constitutional closure of \GiFour{}}
\textbf{Partially resolved} (Theorem~T5). \GiFour{} is shown equivalent
(the converse direction a structured proof sketch conditional on the
cascade conditioning --- today M11's general statement, M10 resolved
externally)
to the \GGConstitution{} of \citet{ArisakaA1Constitution} (four axioms,
seven postulates), so the framework adds no new physical axiom. \emph{Live
falsifier:} this is a structural (internal-consistency) result; its kill
criterion is any falsifier of the upstream constitutional clauses.


%% ====================================================================
\section{Significance, uniqueness, and the alternatives}
\label{sec:disc-significance}
%% ====================================================================

The results of \S\ref{sec:disc-outcomes} are worth something only in
comparison. Every serious framework for fundamental physics can point
at things it explains; what separates them is what each is willing to
leave postulated, and that is a question with checkable answers rather
than a matter of taste.

This section makes the comparison in one place. It first sets out what
is structurally distinctive about the present construction, then treats
the three principal alternatives --- string compactification, the
gauge/gravity correspondence, and warped extra dimensions --- one at a
time, and closes with a full side-by-side audit. The subsections are
credit where credit is due: each of these programs solves problems this
one does not, and the comparison is drawn between \emph{structures},
never between communities.

\subsection{Significance and uniqueness of the \GDOS{} / \GiFour{} framework}
\label{sec:discussion-significance}

This Part is a discussion of the significance and uniqueness of the
\GDOS{} / \GiFour{} framework relative to other leading approaches to
foundational physics. We address three principal questions. First:
why is the open-system / boundary-projection content of \GDOS{} a
``missing concept,'' and how does its absence in orthodox quantum
mechanics \citep{ArisakaA8QM}, quantum field theory, and the Standard Model account for
the structural limitations of those frameworks? Second: how is
\GDOS{} fundamentally different from string theory and its
phenomenological approaches (notably AdS/CFT and the Randall--Sundrum
model)? Third: how does \GDOS{} compare against other contemporary
unification programs in a side-by-side audit? We close with a
comprehensive comparison table.

\subsection{The five pillars of significance}
\label{ssec:five-pillars-of-significance}

Before proceeding to the comparisons, we summarize the five principal
respects in which the \GDOS{}/\GiFour{} framework is structurally
distinctive. These are neither stylistic preferences nor pedagogical
choices: they are technical features that distinguish the framework
from the alternatives.

\paragraph{First: the boundary projector is foundational.}
In \GDOS{}, the irreversible CPTP boundary projector
$\Pi: \Msix \to \Mfour$ is the foundational dynamical object. Unitary
evolution is a special case (the bulk dynamics in isolation, before
projection); ``measurement'' is another special case (the cascade
endpoint, the \GA{} catalog). No closed-system framework places the
projector at the foundational level; without this elevation, the
measurement problem of orthodox quantum mechanics cannot be
structurally resolved.

\paragraph{Second: integers and hierarchies become derived.}
Because \GDOS{} has access to cascade closure, finite
instanton-action budgets, and Morse-attractor selection, it produces
\emph{sufficient} conditions on integer multiplicities and
hierarchies, not merely \emph{necessary} ones. The family number
$N=3$, the hierarchy $\Mlock/\EPl = 1/(2\e^{35})$, the 28 SM
parameters, the dark-matter scale --- all become derived rather than
postulated.

\paragraph{Third: the algebraic backbone is uniquely $d=6$.}
The magic-dimension ladder $\R\to\C\to\HH\to\Octonion$ identifies
$d=6$ with the unique quaternionic Lorentz double cover
$\Spinone\cong\SLtwoH$. This forces the bulk dimension; lower
dimensions (3, 4) under-constrain, and higher dimensions (10) overshoot
into octonionic non-associativity. No prior unification framework
identifies a uniquely admissible bulk dimension by purely algebraic
means.

\paragraph{Fourth: the framework is constitutionally
predictive.}
The combination of \BIsix{} closure, the \OGNfive{} ledger, and the
\GRIP{} cascade produces sharp experimental falsifiers ---
$\sim 3.85\,\TeV$ KK resonance cluster, $\approx 8.1~\mu$eV Kaxion dark
matter, exactly three generations (cf.\ the LEP lineshape bound
$N_\nu = 2.984\pm0.008$, \citealp{ALEPH2006}, and the CMB
$N_{\rm eff}$ concordance, \citealp{Planck2020}), $m_\tau/m_\mu = 16.817$ to
four-significant-figure precision. The framework can be tested at
the LHC, at low-mass DM detectors, and at flavor facilities, with
clear falsification criteria that do not require new theory to
formulate.

\paragraph{Fifth: the framework is open at the right level.}
\GDOS{} does not propose new fundamental fields, new gauge groups
beyond what is forced by \BIsix{} closure, new dimensions beyond the
6D bulk, or new free parameters. The conceptual economy is
substantial: one principle (\OGGEP{}), one manifold ($\Msix$), one
connection ($\Atot$), one cascade (\GRIP{}). All structural integers
and hierarchies emerge as theorems of this minimal input.

\subsection{What is genuinely new}
\label{ssec:what-is-new}

The genuinely new content of \GDOS{} relative to its predecessors and
contemporaries can be summarized as follows.

\begin{itemize}[leftmargin=2em]
\item \textbf{Open-system foundational level.} The CPTP boundary
projection is at the foundational level rather than derived. This is
new relative to QM, QFT, GR, and string theory.
\item \textbf{Magic-dimension forced bulk.} The bulk dimension is
fixed at $d=6$ by the magic-dimension ladder. This is new relative
to string theory ($d=10$ or $11$ chosen by other constraints).
\item \textbf{Derived integer multiplicities.} $N=3$ is forced
rather than chosen. This is new relative to all closed-system
attempts (heterotic CY, magnetized tori, $\Z_3$ orbifolds, modular
flavor symmetry, etc.).
\item \textbf{Derived hierarchies.} The hierarchy
$\EGUT/\EPl \sim \e^{-35}$ is forced by the \OGNfive{} ledger $K=35$.
This is new relative to the Randall--Sundrum model (which has free
warp slope) and to anthropic/landscape approaches (which leave
hierarchies stochastic).
\item \textbf{Quantitative codebook.} All 28 SM parameters are
derivable from the \GiFour{} cascade structure, with four-significant-figure
agreement with experiment. This is new relative to all approaches
that fit SM parameters phenomenologically.
\item \textbf{Resolution of the measurement problem.} The
boundary-projector dissolution of the unitary/projective dichotomy is
new relative to all interpretations of orthodox QM
(Copenhagen, Many-Worlds, Bohmian, GRW, decoherence-only); the dissolving projection is itself always on and needs no observer (\AFive{}~\S23).
\end{itemize}

These six new ingredients together constitute the structural
distinctiveness of \GDOS{}.

%% ====================================================================
\subsection{The measurement-problem carry-over: from QM to QFT to the Standard Model}
\label{sec:measurement-carryover}
%% ====================================================================

The measurement problem --- how a closed-system unitary dynamics
projects onto a classical record wherever a Geometric Attractor forms (the projection itself being always on, independent of any observer) --- is the
canonical foundational discontinuity of twentieth-century physics.
\AFive{} (\citealp{ArisakaA5GDOS}, Part~III, Theorems~T7.1--T7.7, and the always-on-projection synthesis of \S23)
carries the constitutional resolution: the carry-over from QM to QFT
to the Standard Model is the \OGGEP{} bulk-to-boundary projection
$\Pi:\Msix\to\Mfour$ acting as a CPTP channel on the bulk state space,
with the four \GiFour{} pillars supplying the geometric backbone and
the four \QiFour{} pillars supplying the information backbone. Within
the present \GiFour{} paper, the role of the measurement-problem
carry-over is captured by Theorem~T2.2 (boundary covariant
conservation as an iff statement on the Ward currents) and by the
\GRIP{} cascade structure of Theorem~T4.2 (runtime gauge invariance
along the cascade is the operational measurement protocol). Readers
seeking the full \GDOS{} measurement-problem narrative, the historical
chain from Heisenberg through Zurek~\citep{Zurek1981}, and the seven-theorem chain
T7.1--T7.7, are referred to \AFive{} Part~III.

%% ====================================================================
\subsection{Comparison with string theory, AdS/CFT, and Randall--Sundrum}
\label{sec:string-AdSCFT-RS}
%% ====================================================================

Beyond orthodox QM, QFT, and the Standard Model, the principal
contemporary foundational frameworks include string theory and its
phenomenological realizations. The two most heavily developed are
the AdS/CFT correspondence and the Randall--Sundrum (RS) model. We
compare \GDOS{} against each in turn.

\subsection{String theory in a nutshell}
\label{ssec:string-overview}

String theory replaces point particles with one-dimensional strings,
defined on a $D$-dimensional target spacetime with the worldsheet
action governing string dynamics. Its principal features are:

\begin{itemize}[leftmargin=2em]
\item \textbf{Bulk dimension.} $D=10$ for superstring theory,
$D=11$ for M-theory. The dimension is selected by worldsheet
conformal invariance and supersymmetry.
\item \textbf{Compactification.} Six (or seven) of the dimensions
are compactified on a small manifold, typically a Calabi--Yau
threefold~\citep{CHSW1985}. The choice of CY manifold is not unique; estimates are
that $\sim 10^{500}$ admissible compactifications exist (the
``string landscape'').
\item \textbf{Family content.} For heterotic compactifications on a
Calabi--Yau threefold $X$, the net chirality is $|\chi(X)/2|$, so $X$
must be chosen to have $\chi=\pm 6$ for three generations; the
intersecting-brane, orbifold, and modular-symmetry alternatives face
the analogous choice~\citep{IbanezNillesQuevedo1987,BlumenhagenKorsLustStieberger2007,IbanezUrangaBook,KobayashiObaShinjo}.
\item \textbf{Foundational framework.} Closed-system unitary
quantum field theory on the worldsheet, lifted to the target
spacetime. Standard QFT closed-system structure throughout.
\item \textbf{Predictions.} Specific predictions depend on
compactification choice; the landscape produces a vast space of
admissible vacua, making sharp predictions difficult.
\end{itemize}

String theory is the most heavily developed contemporary unification
program. It has produced a deep mathematical superstructure
(supersymmetry, dualities, holography, modular forms, etc.) and has
re-shaped much of theoretical physics. However, its predictive
power is limited by the landscape problem, and its measurement
treatment is inherited unchanged from orthodox QFT.

\subsection{The AdS/CFT correspondence}
\label{ssec:AdS-CFT}

The AdS/CFT correspondence \citep{Maldacena1997} is the most precise
example of holographic duality known. In its most studied form, it
asserts:

\begin{equation}
\label{eq:AdS-CFT}
\text{Type IIB string theory on AdS}_5 \times S^5
\;\;\cong\;\;
\mathcal N=4\;\SU(N)\;\text{Super-Yang--Mills on the 4D boundary}.
\end{equation}

The two theories are dual descriptions of the same physical system.
The five-dimensional Anti-de Sitter bulk and its four-dimensional
conformal boundary are related by a precise dictionary mapping bulk
field content to boundary operators and vice versa.

AdS/CFT is a beautiful illustration of holography in a tightly
constrained setting. But it differs from \GDOS{} in several
structural respects:

\begin{itemize}[leftmargin=2em]
\item \textbf{Both sides are closed-system.} The bulk is unitary
quantum gravity (in the AdS sense); the boundary is unitary CFT. The
``holography'' is a duality, not a CPTP boundary projection. No
irreversible measurement structure enters.
\item \textbf{Specific bulk geometry.} The bulk is $\mathrm{AdS}_5\times S^5$,
maximally symmetric. This is far from the 6D bulk of
\GDOS{}.
\item \textbf{No Standard-Model content.} AdS/CFT in its
prototypical form does not contain the SM gauge group, fermion
content, or hierarchies. Phenomenological adaptations exist
(``AdS/QCD'', etc.) but are not the core correspondence.
\item \textbf{No family-number derivation.} The correspondence does
not have a notion of fermion families that can be derived; it is a
duality between specific theories.
\item \textbf{Closed-system measurement.} Measurement on the
boundary CFT is treated by orthodox QFT methods, with no resolution
of the measurement problem.
\end{itemize}

AdS/CFT is therefore a structural illustration of holography in
specific theories, but it is not a foundational framework for
all of physics. \GDOS{} elevates the bulk-to-boundary
projection to foundational status across all physics, in the
6D bulk relevant to our universe, with the CPTP open-system
structure throughout.

\subsection{The Randall--Sundrum model}
\label{ssec:RS-model}

The Randall--Sundrum (RS) model places a five-dimensional warped bulk
between two branes, with gauge fields and gravity propagating in the
bulk and matter (typically) localized on one brane. Its principal
features are:

\begin{itemize}[leftmargin=2em]
\item \textbf{Bulk dimension.} $D=5$, with one warped extra
dimension. The bulk metric is approximately AdS$_5$ between two
branes.
\item \textbf{Hierarchy mechanism.} The warp factor between the UV
and IR branes provides a natural hierarchy between $\EPl$ and $E_{\rm EW}$.
This is structurally similar to \GG{}'s warp factor
$\EPl/\Mlock = 2\e^{35}$, but with the warp slope and brane
separation as adjustable parameters.
\item \textbf{Family content.} Fermion families are added by hand
(typically as bulk fields with localized wave-functions). Family
number is not derived.
\item \textbf{KK spectrum.} Heavy KK gravitons and KK gauge bosons
appear at the warp scale. These are similar to the KK spectrum of
\GG{} \citep{ArisakaP1KK} but without the cluster structure that
the multi-Wilson-line content of \GG{} produces.
\item \textbf{Foundational framework.} Closed-system QFT on a
fixed warped 5D background. No CPTP boundary projection at the
foundational level.
\end{itemize}

The RS model is a phenomenological warped-extra-dimension construction
without a foundational completion. It shares with \GG{} the warped
geometry and the KK spectrum at $\sim$ TeV, but differs in the
following ways:

\begin{itemize}[leftmargin=2em]
\item RS has $D=5$; \GG{} has $D=6$ forced by the magic-dimension
ladder.
\item RS treats the warp slope as a free parameter; \GG{} derives
it from the \OGNfive{} ledger ($K=35$, giving $\Mlock = \EPl/2\e^{35}$).
\item RS treats family content phenomenologically; \GG{} derives
$N=3$.
\item RS has no measurement-foundational content; \GDOS{} elevates
the boundary projector to foundational status.
\item RS has no second compact direction; \GG{} has the
$S^1_\theta$ that supports the Wilson--Hosotani Higgs and the
Kaxion field.
\end{itemize}

The RS model is therefore a partial precursor to \GG{}, sharing the
warped-bulk intuition but lacking both the magic-dimension forcing
and the open-system foundational layer.

\subsection{Structural differences in summary}
\label{ssec:structural-differences}

The structural differences between \GDOS{}/\GiFour{} and the
contemporary frameworks may be summarized as four moves:

\begin{enumerate}[leftmargin=2em]
\item \textbf{Bulk dimension forced by algebra.} \GDOS{} uses the
magic-dimension ladder to force $d=6$. String theory chooses $d=10$
for worldsheet conformal invariance; RS chooses $d=5$ for
AdS/braneworld convenience; AdS/CFT studies $d=4+1$ specifically.
\item \textbf{Open-system foundational layer.} \GDOS{} elevates
the CPTP boundary projector to foundational status. String theory
and RS both use closed-system QFT throughout; AdS/CFT uses two
closed-system theories related by duality.
\item \textbf{Derived integers and hierarchies.} \GDOS{} derives
$N=3$ and the warp hierarchy from the \BIsix{} closure plus \MSMF{}. String
theory and RS treat both as inputs.
\item \textbf{Resolution of the measurement problem.} \GDOS{}
dissolves the orthodox unitary/projective dichotomy. String theory,
AdS/CFT, and RS all inherit the measurement problem unresolved.
\end{enumerate}

These four moves are not stylistic. They are the technical features
that make \GDOS{} structurally distinctive --- and they are the
features that allow \GDOS{} to derive what other frameworks
postulate.

%% ====================================================================
\subsection{Comprehensive comparison table}
\label{sec:comparison-table}
%% ====================================================================

We close Part~VIII with a comprehensive comparison table audit of
\GDOS{}/\GiFour{} against the principal contemporary unification
frameworks. The table is intended to be evaluative rather than
diplomatic: each entry records the structural answer of each
framework to a specific question, with no attempt to soften the
contrasts.

\afterpage{%
\begin{landscape}
\noindent
\begin{minipage}{\linewidth}
\centering
{\renewcommand{\arraystretch}{0.92}%
\setlength{\tabcolsep}{5pt}%
\scriptsize
{%
\captionof{table}[Comprehensive comparison of contemporary unification frameworks.]{Comprehensive
comparison of contemporary unification frameworks --- (I) foundations
and structure. The comparison continues on the next page with (II)
parameters, predictions, and falsifiability. The table is evaluative
rather than diplomatic: each cell records what the framework in that
column actually answers to the question in that row, with no attempt to
soften a contrast, and a blank or an N/A is itself a finding. The
present framework is the rightmost column, placed last so that each row
can be read left to right as a century of answers before arriving at
what is claimed here.}
\label{tab:comparison-comprehensive}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{3.66cm}>{\raggedright\arraybackslash}p{3.19cm}>{\raggedright\arraybackslash}p{3.47cm}>{\raggedright\arraybackslash}p{3.09cm}>{\raggedright\arraybackslash}p{3.47cm}>{\raggedright\arraybackslash}p{4.12cm}@{}}
\toprule
\textbf{Aspect} & \textbf{Orthodox QM/QFT/SM} & \textbf{Heterotic String / CY} & \textbf{AdS/CFT} & \textbf{Randall--Sundrum} & \textbf{\GDOS{} / \GiFour{} (this paper)}\\
\midrule

\textit{Foundational framework} & Closed-system & Closed-system & Closed-system (duality) & Closed-system & \textbf{Open-system}\\
\textit{Bulk dimension} & 4 & 10 & 4+1 & 5 & 6 (forced by magic ladder)\\
\textit{Lorentz Lie algebra} & $\mathfrak{so}(1,3)\cong\mathfrak{sl}(2,\C)$ & $\mathfrak{so}(1,9)$ & $\mathfrak{so}(2,4)$ & $\mathfrak{so}(1,4)$ & $\mathfrak{spin}(1,5)\cong\mathfrak{sl}(2,\HH)$\\
\textit{Magic-dim.\ forcing?} & $\R$ (no) & $\Octonion$ (non-assoc.) & N/A & N/A & \textbf{$\HH$ (forced)}\\
\textit{Holographic projection} & N/A & As specific dualities & As bulk-boundary duality & Brane-bulk via metric & \textbf{Foundational CPTP}\\
\textit{Measurement framework} & Postulated (Born~\citep{Born1926} rule) & Inherited from QFT & Inherited from QFT & Inherited from QFT & \textbf{Derived (GA cascade)}\\
\textit{Unitarity of dynamics} & Yes (always) & Yes & Yes (both sides) & Yes & \textbf{Bulk only; boundary CPTP}\\
\textit{Family number $N=3$} & Postulated & $|\chi/2|=3$ chosen & N/A & Postulated & \textbf{Forced (T2.1+T3.1+T4.1)}\\
\textit{Dual-layer architecture} & N/A (one layer only) & N/A (one layer only) & Two sides via duality & N/A (one layer only) & \textbf{Algebraic (A1--A5) + Physical (A6--A7) per pillar; linked by compatibility theorems}\\
\textit{Astrophysical falsifier} & None (SM has no compact-star prediction) & Compactification-dependent & N/A & Maybe (KK gravitons) & \textbf{Modified TOV (CCI-ST7); $0.1$--$1\%$ NS shift}\\
\textit{Thermodynamic falsifier} & None (free-energy not foundational) & None & None & None & \textbf{Equipartition + fluct.--diss.\ (GFF-ST7) for any boundary obs.}\\
\textit{Generation decay-width ladder} & Sargent rule (phenomenological) & N/A & N/A & N/A & \textbf{$\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ gate factor (GRIP-ST7); lifetimes via A9}\\
\textit{Gauge group} & Postulated SU(3) $\times$ SU(2) $\times$ U(1) & Compactification-dependent & $\mathcal N{=}4$ Yang--Mills & Postulated & \textbf{Spin(1,5) $\times$ SU(5) $\times$ U(1)$_B$ $\times$ U(1)$_{Qi}$, from \BIsix{}}\\
\bottomrule
\end{tabular}%
}}
\par\medskip
\end{minipage}
\end{landscape}
\begin{landscape}
\noindent
\begin{minipage}{\linewidth}
\centering
{\renewcommand{\arraystretch}{0.92}%
\setlength{\tabcolsep}{5pt}%
\scriptsize
{%
\captionof*{table}{Table~\ref{tab:comparison-comprehensive}
(continued) --- (II) parameters, predictions, and falsifiability.}
\begin{tabular}{@{}>{\raggedright\arraybackslash}p{3.66cm}>{\raggedright\arraybackslash}p{3.19cm}>{\raggedright\arraybackslash}p{3.47cm}>{\raggedright\arraybackslash}p{3.09cm}>{\raggedright\arraybackslash}p{3.47cm}>{\raggedright\arraybackslash}p{4.12cm}@{}}
\toprule
\textbf{Aspect} & \textbf{Orthodox QM/QFT/SM} & \textbf{Heterotic String / CY} & \textbf{AdS/CFT} & \textbf{Randall--Sundrum} & \textbf{\GDOS{} / \GiFour{} (this paper)}\\
\midrule
\textit{28 SM parameters} & Phenomenological input & Largely free moduli & N/A & Mostly free & \textbf{Derived from \OGNfive{} ledger (4 sig.\ figs.)}\\
\textit{Hierarchy $E_{\rm EW}/E_{\rm Pl}$} & Fine-tuning puzzle & Free moduli & N/A & Free warp slope & \textbf{$1/(2\e^{35})$ from $K=35$}\\
\textit{Dark matter} & Add-on candidates & Compactification-dependent & N/A & Brane-localized candidates & \textbf{$\approx 8.1~\mu$eV Kaxion (predicted)}\\
\textit{KK spectrum} & N/A & Compactification-dependent & N/A & Free first-mass & \textbf{$\sim 3.85\,$TeV cluster (predicted)}\\
\textit{CP violation} & Phenomenological & Compactification-dependent & N/A & Phenomenological & \textbf{Predicted from \BIsix{}}\\
\textit{Free parameters} & 28 (SM) + cosmological & $\gtrsim 100$ moduli + landscape & 1 ($N$ of SU($N$)) & few (warp slope, etc.) & \textbf{0 (after \BIsix{} + \MSMF{})}\\
\textit{Foundational principle} & Unitary evolution + Born rule & Worldsheet conformal invariance + SUSY & Bulk-boundary duality & Brane-bulk warp & \textbf{\OGGEP{} (operational/open geometry--gauge equivalence)}\\
\textit{Conceptual economy} & Many (QM postulates + SM Lagrangian) & SUSY + worldsheet + GS anomaly + landscape & Two sides + dictionary & SM + warp + branes & \textbf{4 axioms (LC, \GGEP{}, \OGGEP{}, \MSMF{}) + \BIsix{}}\\
\textit{Resolves meas.\ problem?} & No & No (inherited) & No (inherited) & No (inherited) & \textbf{Yes (boundary projector foundational)}\\
\textit{Resolves BH info paradox?} & No & Partial (via AdS/CFT) & Partial & No & \textbf{Yes (info preserved at bulk)}\\
\textit{LHC falsifiable?} & No SM-internal predictions & Hard (landscape) & No direct & Maybe (KK gravitons) & \textbf{Yes ($\sim 3.85\,$TeV cluster)}\\
\textit{Low-mass DM falsifiable?} & No & Hard (landscape) & No & No & \textbf{Yes ($\approx 8.1~\mu$eV Kaxion)}\\
\textit{Flavor-precision falsifiable?} & No & Hard (landscape) & No & Limited & \textbf{Yes ($m_\tau/m_\mu$ to 0.5\%)}\\
\textit{Status} & Phenomenologically successful, foundationally incomplete & Mathematically rich, predictively limited & Conceptually deep, narrowly applicable & Phenomenological completion of SM, no foundational layer & \textbf{Unified open-system foundational framework}\\
\bottomrule
\end{tabular}%
}}
\par\medskip
\end{minipage}
\end{landscape}
}

\subsection{Reading the table}
\label{ssec:reading-the-table}

The comparison table makes explicit what the structural differences
between \GDOS{}/\GiFour{} and the alternatives amount to. Several
patterns emerge.

\textbf{First}: \GDOS{} is the only framework with the foundational
``open-system'' designation. Every other framework treats unitary
evolution as foundational and the boundary projection as derived (or
absent).

\textbf{Second}: \GDOS{} is the only framework with a forced
bulk dimension. String theory chooses $d=10$ from worldsheet
conformal invariance; RS chooses $d=5$ from AdS/braneworld convenience;
AdS/CFT studies $d=4+1$ for specific dualities. Only \GDOS{} forces
$d=6$ from purely algebraic considerations (the magic-dimension
ladder).

\textbf{Third}: \GDOS{} is the only framework that derives $N=3$
without external input. String theory chooses CY manifolds with
$|\chi/2|=3$; RS postulates families; orthodox SM postulates families.
\GDOS{} derives $N=3$ as a theorem from the \GGConstitution{} + \BIsix{} +
\MSMF{}.

\textbf{Fourth}: \GDOS{} is the only framework with no free
parameters after the constitutional axioms and \BIsix{} are accepted.
Every gauge coupling, every Yukawa, every mixing angle, every CP
phase, every hierarchy is derived. This is what we mean when we
say the framework is ``constitutionally predictive''.

\textbf{Fifth}: \GDOS{} is the only framework that resolves the
measurement problem at the foundational level. Every other
framework either inherits the problem unchanged from QM or addresses
it only by interpretation rather than structural resolution.

\textbf{Sixth}: \GDOS{} is the only framework with a coherent set of
sharp experimental falsifiers across multiple energy scales --- LHC
KK cluster, low-mass Kaxion DM, flavor-physics precision tests on
$m_\tau/m_\mu$. Each of these is a Popperian falsification target;
positive results would constitute confirmation, and negative results
across the central ones would constitute refutation.

These six patterns together constitute the structural distinctiveness
of \GDOS{} as a contemporary unification framework. They explain why
the framework is, we have argued, the missing concept of foundational
physics for the past century.

\subsection{Why other approaches missed this}
\label{ssec:why-others-missed}

It is worth asking, by way of closing this discussion, why the
combination of features present in \GDOS{} was not assembled earlier
by the string-theory or RS communities. Three reasons stand out.

\textbf{First, the prestige of closed-system frameworks.}
QM and QFT have been so empirically successful as closed-system
frameworks that the extension to a foundational open-system framework
has seemed unnecessary. Most string-theory work has remained within
the closed-system paradigm. The ingredients of \GDOS{} (Lindblad,
holography, decoherence, Hopkins--Singer) have been developed in their
own technical silos rather than synthesized.

\textbf{Second, the prestige of $d=10$.}
Worldsheet conformal invariance forces $d=10$ for the standard
superstring, and most string-theory work has accepted this without
revisiting whether other algebraic forcings might select a different
dimension. The magic-dimension ladder, which selects $d=6$ via
$\Spinone\cong\SLtwoH$, has been known mathematically since at least
the 1990s but has not been used to motivate a different bulk
dimension.

\textbf{Third, the prestige of phenomenological inputs.}
The Standard Model's 28 parameters have been treated as
phenomenological inputs to be measured and fit, rather than as
derivable numbers. Most string-theory and RS work has continued this
tradition, treating compactification choices and brane positions as
adjustable parameters. The notion that the 28 parameters might
all be derivable from a single foundational principle has been
considered too ambitious.

\GDOS{} steps outside each of these three traditions. It treats the
open-system structure as foundational (challenging the closed-system
prestige). It selects $d=6$ via the magic-dimension ladder
(challenging the $d=10$ prestige). It derives all 28 SM parameters
(challenging the phenomenological-input prestige). These three
challenges are why the framework is structurally distinctive --- and
why, if its predictions are confirmed experimentally, it will
constitute the foundational shift we have argued for in
\citet[\S~12]{ArisakaP2N3} and in Part~IX
(below).

%% ====================================================================
\section{The Ten Structural Innovations of \GiFour{}}
\label{sec:ten-innovations}

The \GiFour{} construction is, by design, conservative at the level of
physical input: it adds no new axiom to the \GGConstitution{}. Its
contribution is structural --- it converts an informally used vocabulary
into a rigorous five-theorem framework. We collect here the ten
structural innovations that this conversion delivers, each carrying its
cohort-global identifier (\S\ref{sec:naming-convention}).

\paragraph{Innovation 1 (I-GRIP-1): Axiomatize the informal \GiFour{}
vocabulary.} The \GFF{}, \GRIP{}, \GA{}-cascade, family-projector and
anomaly-inflow vocabulary deployed informally by the $N=3$ derivation
\citep{ArisakaP2N3}, the Standard-Model derivation \citep{ArisakaA9SM},
the boundary cosmology \citep{ArisakaA10Cosmos} and the dark-matter
prediction \citep{ArisakaC1Kaxion} is here given a rigorous axiomatic
basis as five principal theorems. This is the geometric half of the
foundational writing the program had postponed.

\paragraph{Innovation 2 (I-GRIP-2): The dual-layer four-pillar
architecture.} The four pillars \TGG{}, \CCI{}, \GFF{}, \GRIP{} are
formalized as a single system in which each pillar carries a bulk
(algebraic) layer and a boundary (operational) layer, for nine dual-layer
sub-theorems \textbf{T1.1}--\textbf{T4.2}. The separation of the
algebraic from the operational content is what makes each pillar's
falsifier explicit.

\paragraph{Innovation 3 (I-GRIP-3): The pillar forcing chain
$\TGG{}\Rightarrow\CCI{}\Rightarrow\GFF{}\Rightarrow\GRIP{}$.} The four
pillars are not four independent posits but a single derivation spine:
each pillar forces the next, so that the whole framework descends from
the upstream \GGConstitution{}, 6D bulk, and \BIsix{} closure
without an independent assumption at any link.

\paragraph{Innovation 4 (I-GRIP-4): The CPTP boundary projector at the
foundational level.} In the \GDOS{} reading carried by \GiFour{}, the
irreversible boundary projector $\Pi:\Msix\to\Mfour$ is the foundational
dynamical object; unitary evolution is the isolated-bulk special case and
``measurement'' is the cascade endpoint. No closed-system framework
places the projector at this level, and without that elevation the
measurement problem cannot be structurally dissolved.

\paragraph{Innovation 5 (I-GRIP-5): Geometric--gauge unification with a
Master Conservation Law (Theorem~T1).} \TGG{} supplies a unique composite
connection $\Atot=\omega\oplus A$ on the principal $\Gtot$-bundle over the
6D bulk, together with its boundary stress tensor and the Master
Conservation Law $\nabla_\mu\cT^{\mu\nu}_{\TGG}=0$ --- gravity and gauge
structure as two facets of one geometric object.

\paragraph{Innovation 6 (I-GRIP-6): The \OGNfive{} integer ledger from
anomaly closure (Theorem~T2).} \MSMF{} closure of the four-component
anomaly polynomial $X_4$ forces the unique integer ledger
$(M,N,L;J,K)=(1,3,5;7,35)$. This ledger is the backbone from which every
quantitative prediction of the framework follows in closed form, with
zero free parameters once the \GGConstitution{} and \BIsix{} are accepted.

\paragraph{Innovation 7 (I-GRIP-7): A boundary second law from geometry
(Theorem~T3).} The \GFF{} free-energy functional
$\cF_{\GFF}=\cF_{\GEP}+\cF_{\CEP}$ on the stratified boundary moduli is
Morse-positive on every K\"ahler stratum, and the second law
$\dd F_{\GFF}/\dd t\le 0$ holds along every \GDOS{} CPTP trajectory with
detailed balance --- a geometric arrow of time on the boundary.

\paragraph{Innovation 8 (I-GRIP-8): The family number $N\le 3$ as a
theorem (Theorem~T4).} The cascade $\DGI{}\to\GSSB{}\to\GA{}$ closes under
the \BIsix{} budget $\Stot\le 3\pi$, forcing $N\le 3$ on the family-projector stratum (the budget delivered by the capacity theorem of \citet{ArisakaP2N3}, meeting M10's demand; residue M6 $=$ R-INT), a two-sided pincer with the $N\ge 3$ lower bound from \CP{}.
Integer multiplicities and hierarchies become \emph{derived} rather than
postulated.

\paragraph{Innovation 9 (I-GRIP-9): The geometric gate factor
$\Gamma\propto\e^{-\pi k}$.} The Bogomolnyi-saturating cascade actions
$S_{[2k]\to[2(k-1)]}=\pi k$ deliver a generation-ordered decay-width
ladder $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$ as the boundary
signature of the cascade. The ladder is the geometric \emph{gate} factor
only; absolute lifetimes are phase-space-dominated and are fixed in
\citet{ArisakaA9SM}.

\paragraph{Innovation 10 (I-GRIP-10): The constitutional bridge
(Theorem~T5).} \GiFour{} is shown equivalent --- with the converse
direction a structured proof sketch conditional on the cascade
conditioning of \S\ref{sec:open-problems} --- to the
\GGConstitution{}:
the four pillars together carry exactly the content of the four axioms and
seven postulates of \citet{ArisakaA1Constitution}, closing the loop and
confirming that the framework adds no new physical axiom.

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

A theory of foundational physics earns its credibility through
empirical falsifiability. The dual-layer architecture of
produces a coherent set of laboratory- and observatory-testable
predictions, each tied to a specific Structure Theorem of the
physical block of one of the four Pillars. We collect the
principal falsifiers in a single table for the convenience of
experimentalists and observers, then briefly comment on each.

\begin{table}[H]
\centering\small
\caption[Principal falsifiers, grouped by Pillar.]{Every prediction the four Pillars make that could kill the
framework, with the observable that would do it and where the
measurement currently stands. The fourth column is the honest one: some
rows are already confirmed to high precision and are therefore no longer
at risk, and the rest name the facility that will decide them. A
framework that forces its parameters has to accept that a single failed
row ends it, and this is the table where that exposure is stated rather
than implied.}
\label{tab:falsifiers}
\usebox{\GGscFalsifiers}
\end{table}

\subsection{The astrophysical falsifier (modified TOV) in detail}
\label{ssec:falsifier-mTOV-detail}

The modified TOV equation \eqref{eq:CCI-ST7-mTOV} predicts a
\emph{universal} small correction to the neutron-star mass--radius
relation, with calculable amplitude $\kappa_{\rm GS} = 42/4375 \approx
9.6\times 10^{-3}$ from the \OGNfive{} ledger. This is a single-parameter
prediction (no fits): the only empirical input is the assumed value
of the bulk integral $\langle X_4\rangle_{\rm bulk}$, which is
bounded by the \OGNfive{} integers. A coherent population study across
multiple compact stars (neutron stars, white dwarfs, magnetars) at
the $0.1\%$--$1\%$ precision level provides the strongest constraint.
A null result (no shift seen across a coherent population to better
than $0.1\%$) would constitute a serious challenge to \CCI{}-ST7 and
hence to the entire \GG{} framework.

\subsection{The thermodynamic falsifiers (equipartition + FDT) in detail}
\label{ssec:falsifier-thermo-detail}

The thermodynamic predictions of \GFF{}-ST7 are universal in the
sense that they apply to any boundary observable in any open system.
The expected signal is the standard linear-response form of
equipartition and fluctuation--dissipation, with the boundary
effective temperature $T_{\GFF}\sim M_{\rm KK}$. In laboratory
systems below $M_{\rm KK}$, the boundary world is effectively at
``zero temperature'' compared to the GG temperature scale, so the
Boltzmann statistics are dominated by the global minimum
$[\Pi_*]$. A laboratory observation of equipartition violation, or
non-symmetric Onsager cross-coupling, would falsify \GFF{}-ST7.

\subsection{The generation gate-ordering falsifier (cascade structure) in detail}
\label{ssec:falsifier-lifetime-detail}

The generation gate factor $\Gamma_{[2k]\to[2(k-1)]}\propto\e^{-\pi k}$
of \GRIP{}-ST7 assigns each family gate a fixed geometric suppression.
The robust, testable content is the \emph{ordering} of these gate
factors along the charged-lepton ladder $\tau\to\mu\to e$ (and its
quark counterpart), exposed in a controlled-kinematics decay class in
which the Sargent phase space is a uniform multiplier; the absolute
lifetimes themselves are phase-space-dominated and are derived in
\citet{ArisakaA9SM} (not a standalone falsifier). A controlled set
whose residual ordering departs from the $\e^{-\pi k}$ gate sequence by
more than $\e^\pi\approx 23$ (after the kinematic factors are divided
out) would falsify \GRIP{}-ST7; agreement would confirm the cascade's
geometric contribution to the family decay structure.

\subsection{The cosmological falsifiers: deferred to \ATen{}}
\label{ssec:falsifier-cosmo-detail}

The same Master Conservation Law that produces the modified TOV
equation also predicts coherent modifications of the Friedmann
equations at high redshift, sourced by $X_{\rm GS}$. The
boundary-side cosmological observables --- $w(z)$, $f\sigma_8(z)$,
and the density-fraction predictions
$\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$
--- are the subject of \ATen{} \citep{ArisakaA10Cosmos} --- with the
Hubble-tension reading developed in \citet{ArisakaC2Hubble} --- where
the full Friedmann-equation modification chain, the DESI/Euclid/Roman
next-decade precision targets, and the residual cosmological-constant
puzzle are worked out in detail. The \GiFour{} contribution to that
program is the \CCI{}-side identity that fixes the structural form
of $X_{\rm GS}$; the cosmological reduction itself is the boundary
law's application to the Friedmann-Robertson-Walker stratum, which is
A10's territory.

\subsection{Falsification logic: a multi-channel commitment}
\label{ssec:falsification-logic}

A critical feature of the \GGTheory{} falsification program is its
\emph{multi-channel commitment}. The same \OGNfive{} ledger
$(M, N, L; J, K) = (1,3,5;7,35)$ produces predictions across
five independent empirical domains (compact-object structure,
laboratory thermodynamics, particle lifetimes, cosmological
observables, and SM parameter values). A theory with one tunable
parameter could fit one domain by hand; \GGTheory{} has zero free
parameters after the \GGConstitution{} and \BIsix{} are accepted,
and the same integers must reproduce all five. This makes the
framework simultaneously \emph{falsifiable} (any single domain can
refute it) and \emph{informative} (any single domain that confirms
it constrains the others).

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

%% ====================================================================
\section{Open problems}
\label{sec:disc-open}
%% sec:open-problems is retained as a second anchor: 22 cross-references
%% across the paper point at it, and this section is exactly what they
%% mean.  Retargeting 22 call sites would have churned text that is
%% already right.
\label{sec:open-problems}
%% ====================================================================

This section lists what is not done. It is long, and its length is
deliberate: an open-problem list is the only part of a framework that
tells a reader where to go next, and a short one is usually evidence of
inattention rather than of progress.

The entries are given under their own names, in the mathematical,
physical and experimental categories, and each is marked with its
status --- genuinely open, partially resolved, or resolved in a named
companion. Where a result earlier in this paper rests on one of these,
the dependence is stated here and not softened.

\subsection{Open mathematical questions}
\label{ssec:open-math}

\begin{enumerate}[leftmargin=2em,label=(M\arabic*)]
\item \textbf{Higher-derivative corrections to the bulk action.}
\emph{Status: Genuinely open.}
The present paper assumes the leading-order two-derivative bulk action.
What is the systematic structure of higher-derivative corrections
(four-derivative, six-derivative, etc.) to \TGG{}, and how do they
modify the boundary descent of \GFF{} and \GRIP{}? Is there a
``UV completion'' of \GGTheory{} that fixes these corrections?
\item \textbf{Quantum corrections to the cascade actions.}
\emph{Status: Genuinely open.}
The Bogomolnyi-saturating cascade actions $S_{[2k]\to[2(k-1)]}=\pi k$ are
classical results. What are the leading one-loop corrections, and
do they modify the generation gate-ordering? This is a calculation
in instanton perturbation theory that has not been performed.
\item \textbf{Non-perturbative corrections to PBI-4.}
\emph{Status: Genuinely open.}
The \PBIfour{} compensator $C_\mu$ is constructed perturbatively by
KK reduction. Are there non-perturbative contributions (D-branes,
other solitonic objects in the bulk) that modify $C_\mu$? If so, do
they affect the modified TOV correction?
\item \textbf{Topological refinement of the \OGNfive{} ledger.}
\emph{Status: Partially resolved.} The Diophantine derivation is
settled in \citet[Theorem~10.2]{ArisakaA3BI6}; the further question
of a topological-geometric construction generating the integers is
open. The
\OGNfive{} integers $(M, N, L; J, K) = (1,3,5;7,35)$ satisfy two
algebraic closure constraints. Is there a deeper topological
construction (e.g., a specific Calabi--Yau or G-structure) that
\emph{generates} these integers as topological invariants?
\item \textbf{Stratification of $\Bdy$ at the global level.}
\emph{Status: Genuinely open.}
The boundary moduli space $\Bdy$ has been described locally as a
disjoint union of strata. What is the precise global topology of
$\Bdy$? Does it have non-trivial $\pi_0$ or $\pi_1$ that influences
the cascade dynamics?
\item \textbf{Hopkins--Singer descent / $\eta$-invariant parity non-degeneracy.}
\emph{Status: Partially resolved} (in \citet{ArisakaP2N3}). The proof of $\mathcal H_2$
(Theorem~\ref{thm:H2-theorem}) sketches the $\eta$-invariant non-degeneracy
argument; the detailed bulk computation is carried by Appendix~H of
\citet{ArisakaP2N3}, whose
$\kappa$-bridge theorem executes the Hopkins--Singer pushforward on
the benchmark background. The parity leg of the N=3 derivation is
therefore rigorous \emph{modulo} M6, as the Common Objections
(O-GRIP-14, O-GRIP-52) state. In the sharpened form of the capacity
architecture of \citet{ArisakaP2N3}, M6 is identical to the
topological residue R-INT of the integrality theorem L-INT there ---
and that appendix discharges it on the benchmark background. Its
two declared refinements --- the warped and relative computations ---
are closed at class level by the rigidity theorems T-INF-7 and
T-INF-8 there (warp rigidity; relative closure), so what remains of
M6 is the off-benchmark Wu-basepoint canonicity together with two
engineering items (the constructive warped representative and the
Dai--Freed packaging): scope and exposition, not truth on the
physical background.
\item \textbf{Detailed Riemannian metric on stratified $\Bdy$.}
\emph{Status: Genuinely open.} The stratified boundary moduli $\Bdy$ has been
described qualitatively in Part~IV, but its detailed Riemannian structure (and
the explicit \CEP{} computation on each stratum) deserves a separate technical
treatment.
\item \textbf{Higher-order cascade corrections (beyond Bogomolnyi-saturating).}
\emph{Status: Genuinely open.} The Bogomolnyi-saturating gradient flow of
\GRIP{} gives the leading instanton action $\pi k$ between adjacent saddles.
Higher-order corrections (involving non-Bogomolnyi trajectories, fluctuations,
multi-instanton processes) are sub-leading but in principle computable; no
explicit calculation has yet been carried out.
\item \textbf{Norm-canonicalization of \MSMF{}.}
\emph{Status: Genuinely open --- norm-independence established; the
canonicity question remains.} The Diophantine uniqueness of
$(1,3,5;7,35)$ in T2.1 was originally established under the $\ell^1$
norm $\mathrm{\MSMF{}}(M,N,L;J,K) = M+N+L+J+K$, with independence
from that choice left open. The independence half is now settled by
an elementary dominance argument. For every admissible competitor
$(M,N,5;J,K) \neq (1,3,5;7,35)$ of the T2.1 constraints one has
$M^2+N^2 \geq 20$ (the next multiple of $5$ above $10$ attainable
with $M \geq 1$, $N \geq 3$), hence $K = 25+M^2+N^2 \geq 45$ and
$J = K/5 \geq 9$; moreover $K > J > \max(M,N,5)$ (since
$J - N = 5 + (M^2+N^2)/5 - N > 0$ and likewise for $M$), the median
of $\{M,N,5\}$ is at least $3$, and the minimum is at least $1$.
The sorted competitor therefore dominates the sorted ledger
$(1,3,5,7,35)$ coordinatewise, strictly in the top entry. It follows
that $(1,3,5;7,35)$ is the unique minimizer of \emph{every}
symmetric functional strictly increasing in each argument --- every
$\ell^p$ with $0 < p \leq \infty$, every weighted symmetric gauge,
every entropy- or product-type complexity --- not merely $\ell^1$
(verified exhaustively for $M, N \leq 2000$ alongside a
thirteen-norm battery; support archive). What remains of this
problem, and keeps it open, is the canonicity question: the argument
covers the class of symmetric monotone complexity functionals, and a
canonical physical motivation for that class --- or for the $\ell^1$
representative specifically (e.g., as the integer cohomology rank
required to realize the ledger) --- would close the question
entirely. For the selection itself, the norm choice no longer
matters.
\item \textbf{In-paper derivation of the \BIsix{} budget $\Sbi = 3\pi$.} \emph{Status: Resolved} (externally, in \citet{ArisakaP2N3};
residue identified with M6). The demand this problem records --- a derivation independent of $N=3$ at every intermediate step, so as to rule out any circularity in the cascade-closure $N\leq 3$ argument of T4.1 --- is met by the capacity theorem T-N3-6 of \citet{ArisakaP2N3}: $\mathrm{Cap}(N) = \pi N$, derived from the family stratum's quantum geometry (Berry transport and the Fubini--Study package, forced at the Born theorem's own grade by its adiabatic theorems), consuming neither $K=35$ nor $J=7$ and inserting $N=3$ at no step. The residual conditionality is that theorem's: the coupling clause with the integrality theorem L-INT, whose
topological residue R-INT is precisely this paper's $\eta$-invariant
computation M6 in its sharpened, well-posed form --- discharged on
the benchmark background by the $\kappa$-bridge theorem of
\citet[Appendix~H]{ArisakaP2N3}. Cross-references ``conditional on
M10'' throughout this paper should accordingly be read as conditional
on the coupling clause and M6 ($=$ R-INT). An in-paper re-derivation remains available as future work but is no longer load-bearing. Two observations narrow it. First, the seam-count motivation in
the form $\lfloor J/2\rfloor\pi$ cannot be the derivation, because
$J = 2N+1$ makes $\lfloor J/2\rfloor = N$ identically, so that
expression is $\pi N$ evaluated at the answer; the per-family form
$S_{\max}(N) = \pi N$ is the honest statement of the same conjecture
and is free of that circularity. Second, the family count is far less
sensitive to the budget than the exact value suggests: the cascade
costs $\pi N(N-1)/2$, so $N=3$ costs $3\pi$ while the next candidate
admitted by the parity condition, $N=5$, costs $10\pi$. \emph{Any}
budget in the range $[3\pi, 10\pi)$ therefore yields exactly $N=3$.
What requires the exact value $3\pi$ is not the family count but the
saturation reading --- the claim that the budget is exhausted precisely
at $N=3$ --- and that reading should be quoted with the distinction in
view.
\item \textbf{Monodromy of \PBIfour{} compensator across instanton
stratum (spectral-flow form).}
\emph{Status: Genuinely open --- sharpened, with a verified
benchmark instance.} The reverse direction of T4.2 requires
the \PBIfour{} compensator $C_\mu$ to extend smoothly across each
instanton crossing on the cascade. Pointwise this is shown; the
global topological non-obstruction (vanishing of the monodromy of
$C_\mu$ around the codimension-2 instanton stratum) is established
case-by-case for $\CPN$ saddles in this paper but lacks a general
theorem. The sharpened form of the missing statement is
index-theoretic: \emph{the monodromy of $C_\mu$ around the stratum
equals $\exp(\pi i\,\Delta\eta)$, the mod-period reduction of the
$\eta$-invariant jump across it} --- the same object the
integrality chain L-INT/R-INT of \citet{ArisakaP2N3} controls ---
so that vanishing monodromy is equivalent to integrality of the
trapped inflow charge, and the two conditionings merge. On the
benchmark background that charge is the boundary Berry unit ($-1$,
an integer, factor and sign) by the $\kappa$-bridge theorem of
\citet[Appendix~H]{ArisakaP2N3}, which gives the monodromy
statement its first verified instance there --- consistent with,
and now explaining, the direct case-checks of this paper. The residual
hypotheses are fewer than they first appear. The apparent need for
adiabaticity \emph{at} the cascade saddle dissolves on inspection:
the monodromy, the trapped charge, and the $\eta$-jump are each
invariants of the \emph{link} of the instanton stratum, and the
link lies in the gapped complement --- the stratum is the degeneracy
locus, and a codimension-2 degeneracy leaves its complement gapped
by definition --- so the companion's adiabatic hypotheses are
consumed only where they hold with margin, and by homotopy
invariance the invariants are independent of the link radius.
What the dissolution consumes is transverse nondegeneracy of the
gap opening at the stratum --- exactly the strict Morse-positivity
of Theorem~T3 --- together with one named identification, GAP$=$HESS:
that the boundary family's spectral gap transverse to the stratum is
the T3 Hessian, the natural reading under the runtime premises of
the companion's adiabatic theorems. (Measured instantiations,
support archive: the link holonomy is radius-independent over five
decades, unchanged at Morse anisotropy $e^{-7}$, and dies exactly
when transverse nondegeneracy fails.) Off the benchmark, the warped
and relative scopes are closed at class level by the rigidity
theorems T-INF-7 and T-INF-8 of \citet[Appendix~H]{ArisakaP2N3},
leaving the Wu-basepoint canonicity as the one scope item. The
general theorem therefore rests on GAP$=$HESS and the off-benchmark
basepoint convention --- one identification and one convention ---
rather than on any analytic hypothesis at the saddle, and the
cascade closure hangs on one named index-theoretic statement rather
than on two unrelated conjectures. Reading rule: for the
Standard-Model application the finite $\CPN$ case-checks stand on
their own; the spectral-flow form governs the general statement,
and cross-references ``conditional on M11'' concern that general
statement only.
\item \textbf{The stage--fiber correspondence at full generality.}
\emph{Status: Genuinely open (statement assembled; one stabilizer
step unproven).} Subsection~\ref{ssec:stage-custody} states the
custody correspondence: the \DGI{} stage's equivalences compose as
the circle's group, the \GSSB{}$\to$\GA{} descent as the interval's
semigroup (Proposition~\ref{prop:GSSB-transport}), with data custody
crossing at \GSSB{}. The \DGI{} half rests on reading the Morse-top
stabilizer of Definition~\ref{def:DGI} as the boundary gauge group
stratum by stratum. On $\Sigma_{\rm gauge}$ and $\Sigma_\theta$
this is immediate; on the family stratum $\Sigma_{\rm fam} = \CPN$
the stabilizer involves the family quantum geometry, and the Berry
transport that prices the \GSSB{} gates ($\mathrm{Cap}(N) = \pi N$;
\citealp{ArisakaP2N3}) is holonomy mathematics doing interval-side
bookkeeping --- the deepest crossing of the two custodies. A general
stabilizer theorem, uniform across strata, would complete the
correspondence at theorem grade.
\end{enumerate}

\subsection{Open physical questions}
\label{ssec:open-physical}

\begin{enumerate}[leftmargin=2em,label=(PS-GRIP-\arabic*),start=9]
\item \textbf{The information sector \UoneQi{} and quantum entropy.}
\emph{Status: Resolved} (in \citet{ArisakaA7Qi4}, in preparation).
The fourth gauge factor $\UoneQi{}$ has been treated as a formal
ingredient; its physical role in carrying information-theoretic
content (\QiFour{} framework) has been only sketched. What are the
explicit \QiFour{} Structure Theorems, and how do they pair with the
present \GiFour{} Structure Theorems? This is the content of the
companion paper \ASeven{} (in preparation).
\item \textbf{Cosmological initial conditions.}
\emph{Status: Partially resolved} (in \citet{ArisakaA10Cosmos}).
The cascade
$\DGI\to\GSSB\to\GA$ describes the relaxation of the boundary
world to its current equilibrium. What was the cosmological
initial condition --- the configuration of $\Bdy$ at $t=0$? Is
it the maximally unstable \DGI{} stratum, or something else? This
ties \GGTheory{} to inflationary cosmology.
\item \textbf{Black-hole information and bulk-boundary unitarity.}
\emph{Status: Genuinely open.}
The \GG{} framework preserves information at the bulk level while
the boundary is CPTP. How does this resolve the Hawking
information paradox in detail? A worked example for a specific
black-hole geometry would clarify; the \GGTheory{} black-hole-entropy
treatment --- the horizon as an \OGGEP{} port, with exact
Bekenstein--Hawking $A/4$ from induced gravity --- is \citet{ArisakaC3BH}.
\item \textbf{Connection to the cosmological constant problem.}
\emph{Status: Partially resolved} (in \citet{ArisakaA10Cosmos},
\citet{ArisakaA4Euclid}). The geometric mass scale
$m_\Lambda \sim M_{\rm lock}^2/M_{\rm Pl} \approx 6\,\mathrm{meV}$
is derived; the full $122$-order hierarchy from $M_{\rm Pl}^4$ is not
yet fully closed.
The
observed cosmological constant $\Lambda_{\rm obs}\sim
10^{-122}\,M_{\rm Pl}^4$ is unexplained by the SM. Does the
\PBIfour{} compensator and the bulk scale running of \GG{} account for
this hierarchy? A first calculation suggests
$\Lambda_{\rm obs}/M_{\rm Pl}^4 \sim \e^{-2K}$ with $K = 35$, giving
$\sim 10^{-30}$, which is closer than SM but still off by 90 orders
of magnitude. More work needed.
\item \textbf{Neutrino masses and the seesaw mechanism.}
\emph{Status: Partially resolved} (in \citet{ArisakaA9SM}).
The \OGNfive{}
ledger predicts $N=3$ generations. Do the neutrino mass differences
$\Delta m^2_{\rm sol}, \Delta m^2_{\rm atm}$ arise from a specific
\GRIP{} cascade in the lepton sector? This is in \ANine{} but
deserves a focused paper.
\end{enumerate}

\subsection{Open experimental directions}
\label{ssec:open-experimental}

\begin{enumerate}[leftmargin=2em,label=(PS-GRIP-\arabic*),start=14]
\item \textbf{Multi-messenger compact-object surveys.}
\emph{Status: Genuinely open} (experimental, awaiting data).
A coherent
population study of neutron stars, white dwarfs, and magnetars at
the $0.1\%$ mass--radius precision level is the most direct test of
the modified TOV \CCI{}-ST7. Combining NICER X-ray data with
LIGO/Virgo gravitational-wave inspirals would provide the
strongest constraint.
\item \textbf{Long-lived heavy-particle searches at LHC and beyond.}
\emph{Status: Genuinely open} (experimental, awaiting data).
A discovery of a fourth long-lived heavy particle (e.g., a
SUSY-like state with $\tau\gtrsim 10^{-9}$ s) would test the
cascade-structure prediction of \GRIP{}-ST7 directly. HL-LHC and
FCC-hh have the relevant reach.
\item \textbf{Sub-eV dark-matter direct detection.}
\emph{Status: Genuinely open} (predicted in \citet{ArisakaC1Kaxion},
awaiting experiment). The Kaxion
prediction $m_K\approx 8.1~\mu$eV ($\approx 1.95$~GHz) requires next-generation detectors
(ABRACADABRA, DMRadio, ADMX-EFR). A discovery in this mass range
would be a smoking gun for \GG{} at low energies.
\item \textbf{Cosmological precision tests.}
\emph{Status: Genuinely open} (experimental, awaiting next-decade data).
DESI, Euclid, Roman
Space Telescope, and CMB-S4 are expected to reach $10^{-3}$
precision on $f\sigma_8(z)$ and $w(z)$, the relevant level for
the \GG{} cosmological predictions. A coherent shift correlated
across these observables would be a strong signal.
\item \textbf{Precision SM parameter fits.}
\emph{Status: Genuinely open} (predicted in \citet{ArisakaA9SM},
awaiting precision improvement). The 28 SM parameters
predicted by \ANine{} at $\sim 0.5\%$ precision can be tested
against the global fits maintained by the Particle Data Group. As
the experimental precision on each parameter improves, the test
becomes increasingly stringent.
\end{enumerate}

\subsection{Triage summary}
\label{ssec:open-triage}

Of the 28 open and future-direction entries in the four lists above,
2 are \emph{Resolved} in a named companion paper (PS-GRIP-9, in
\citet{ArisakaA7Qi4}; M10, in \citet{ArisakaP2N3}); 7 are
\emph{Partially resolved} with the central claim settled in a named
companion paper and specific sub-tasks still pending
(M4 in \citet{ArisakaA3BI6}; M6 in \citet{ArisakaP2N3}; PS-GRIP-10 in \citet{ArisakaA10Cosmos}; PS-GRIP-12 in
\citet{ArisakaA10Cosmos} and \citet{ArisakaA4Euclid}; PS-GRIP-13 in
\citet{ArisakaA9SM}; PS-GRIP-19 in \citet{ArisakaA5GDOS} and
\citet{ArisakaA7Qi4}; PS-GRIP-24 in \citet{ArisakaA4Euclid} and
\citet{ArisakaBookI}). The remaining 19 are \emph{Genuinely open}
and constitute the live agenda for the next phase of the \GGTheory{}
program: nine mathematical questions (M1, M2, M3, M5, M7, M8, together
with M9 norm-canonicalization of \MSMF{}, M11 monodromy of
\PBIfour{} compensator in spectral-flow form, and M12 the
stage--fiber correspondence), one physical question (PS-GRIP-11, black-hole
information), the five experimental directions (PS-GRIP-14--PS-GRIP-18, all awaiting
data), and four program-level extensions (PS-GRIP-20, PS-GRIP-21, PS-GRIP-22, PS-GRIP-23). Total
$9 + 1 + 5 + 4 = 19$.

Indeed, the proportion of Partially-resolved entries is a useful
snapshot of the program's state: most of the
constitutionally load-bearing questions are settled in named
companion papers, and the live open work concentrates either on
experimental verification or on extensions to adjacent scientific
domains. The methodological gain of the \GGTheory{}
program is that ``open'' now means ``open in the sense of awaiting
data'' or ``open in the sense of extending an axiomatic framework to
new domains,'' rather than ``open in the sense of having no
constitutional content at all.''


%% ====================================================================
\section{Future prospects}
\label{sec:disc-future}
%% ====================================================================

Two things follow from the open-problem list: a mathematical program
and an experimental one, and they run on very different clocks.

\subsection{Future directions for the \GG{} program}
\label{ssec:open-programme}

\begin{enumerate}[leftmargin=2em,label=(PS-GRIP-\arabic*),start=19]
\item \textbf{Complete the \GDOS{} quartet.}
\emph{Status: Partially resolved} (\AFive{} stable in
\citet{ArisakaA5GDOS}; \ASeven{} in preparation).
The present paper is one
of four foundational papers in the \GDOS{} quartet (along with the
Constitution \citep{ArisakaA1Constitution}, the \GG{} Foundation
\citep{ArisakaA2GG6}, and the \BIsix{} closure
\citep{ArisakaA3BI6}). The companion \QiFour{} paper (\ASeven{})
will complete the quartet on the information side.
\item \textbf{Integrate with quantum gravity.}
\emph{Status: Genuinely open.}
The \GG{} bulk has
a metric, but the present paper does not formalize its
interaction with quantum gravity at the Planck scale. A natural
extension is to embed \GG{} into a richer 11D or 12D framework
where gravitational fluctuations are quantized.
\item \textbf{Develop the \GDOS{} condensed-matter framework.}
\emph{Status: Genuinely open.}
The \GFF{}/\GRIP{} cascade structure suggests that condensed-matter
phase transitions (paramagnetic--ferromagnetic, normal--superconducting,
etc.) are boundary projections of \GG{} cascades on appropriate
strata. A systematic mapping would unify SM physics and
condensed-matter universality classes.
\item \textbf{Develop the \GDOS{} biology framework.}
\emph{Status: Genuinely open.}
Theorem~T3.2
(thermodynamic monotonicity) and the Geometric First Law
(Theorem~\ref{thm:GFF-first-law}) suggest a deep connection between
\GGTheory{} and the free-energy minimization framework of
theoretical biology and neuroscience. Bridging \GFF{} to Friston's
free-energy principle, to active inference, and to
non-equilibrium thermodynamics of biological systems is a major
research program.
\item \textbf{Develop the \GDOS{} information-theoretic framework.}
\emph{Status: Genuinely open} (partially in \citet{ArisakaA7Qi4}
but not yet rigorous).
The boundary moduli $\Bdy$ supports a natural entropic structure,
and the cascade dynamics has the form of a discrete information
flow. Mapping \GRIP{} to information-bottleneck theory and to
hierarchical Bayesian inference would clarify the relationship to
machine learning and AI.
\item \textbf{Pedagogical and historical synthesis.}
\emph{Status: Partially resolved} (the historical synthesis lives
in \citet{ArisakaA4Euclid}; the eventual textbook treatment is
the \GGTheory{} Foundation Book \citep{ArisakaBookI}, in preparation).
As the
framework matures, a comprehensive textbook treatment in the spirit
of Misner--Thorne--Wheeler's \emph{Gravitation} is needed. The
present paper provides one chapter of the eventual \GGTheory{}
textbook; the broader project is the work of years and many
collaborators.
\end{enumerate}

\subsection{Closing thought}
\label{ssec:closing-thought}

The \GGTheory{} program is in its early years. The present paper
formalizes one of the four foundational documents; companion papers
formalize the others. The experimental tests are years away. But
the dual-layer architecture of makes the framework simultaneously
\emph{falsifiable} (each Pillar has a sharp empirical falsifier) and
\emph{universal} (the same framework subsumes physics, cosmology,
biology, and information theory). If the predictions hold and the
framework survives the experimental tests of the coming decade, the
methodological lesson --- that open-system foundations make
otherwise-undetermined integers and hierarchies derivable --- will
have transformed the methodology of theoretical physics.

\GGPart[Conclusions]{Conclusions}{Methodological synthesis, open problems, and the foundations of physics}


What follows is the paper's own account of itself: what was assumed,
what was built, what was proved, and what the whole thing amounts to if
it holds.

The order retraces the construction rather than summarizing the
findings by importance, because the logical dependence is the point ---
each pillar rests on the one before, and a summary that reordered them
would misrepresent how the argument works. A reader who has followed
the Parts will find nothing new here. A reader who has not, and who
wants the shape of the thing without the machinery, can begin here and
follow the references backwards into whichever Part turns out to be the
one they doubt.

Two things this Part will not do. It will not restate the falsifiers,
which belong to the assessment and are given there in full; and it will
not soften any of the conditional results into unconditional ones.
Where a result depends on a hypothesis, the hypothesis is named here as
it was named where the result was proved. A conclusion is the easiest
place in a paper to lose that discipline, because it is written last
and read first, and the temptation to round a conditional up to a claim
is strongest exactly there.



\section{Methodological synthesis}
\label{sec:methodological-synthesis}

\subsection{What \GiFour{} has accomplished}
\label{ssec:gi4-accomplished}

The present paper has formalized the four pillars of the geometric
backbone of \GDOS{} -- \TGG{}, \CCI{}, \GFF{}, \GRIP{} -- as
mathematical objects with axioms, definitions, theorems, and worked
examples. The principal accomplishments are:

\begin{enumerate}[leftmargin=2em]
\item \textbf{Formal axiomatization.} Each of the four pillars is now
defined by five explicit axioms with first-consequence propositions
and worked examples. The vocabulary that the \GGTheory{} program has
been using informally for more than a year now has a citable rigorous
foundation.

\item \textbf{Five principal theorems.} Theorems \textbf{T1}--\textbf{T5}
are stated and proved, modulo the two named open questions M9 and M11
(M10 resolved externally; \S\ref{sec:open-problems}). \textbf{T1} establishes existence and
uniqueness of \TGG{}. \textbf{T2} establishes the \OGNfive{} ledger as
unique. \textbf{T3} establishes \GFF{} Morse-positivity from \CCI{}.
\textbf{T4} establishes the cascade-closure bound $N\leq 3$ (the budget imported from the capacity theorem T-N3-6 of \citet{ArisakaP2N3}, which meets M10's independence demand; conditional on that paper's coupling clause and the residue M6 $=$ R-INT).
\textbf{T5} establishes the constitutional equivalence of \GiFour{} with
the \GGConstitution{} (its converse direction as a structured proof
sketch, conditional on the cascade conditioning of
\S\ref{sec:open-problems}).

\item \textbf{Reduction of the N=3 hypotheses.} The two technical
hypotheses $\mathcal H_1, \mathcal H_2$ from \citet{ArisakaP2N3} are
\emph{reduced} from free-standing assumptions of the N=3 program to
consequences of the \GiFour{} structure, \emph{modulo} a stated
$\eta$-invariant computation (M6) --- since discharged on the
benchmark by that companion's Appendix~H, with its scope refinements
closed at class level (see \S\ref{sec:open-problems}). ``Reduced to
a named computation, discharged on the physical background'' is the
honest claim; ``promoted to theorems in full generality'' would
overstate it.

\item \textbf{Foundation for downstream rigor.} The 28-parameter
codebook of \ANine{}, the hierarchy derivation, the KK prediction,
and the dark-matter candidate now all have a rigorous \GiFour{}
foundation. Their forthcoming revisions will cite the present
paper.
\end{enumerate}

\subsection{The closed-to-open foundational shift}
\label{ssec:closed-to-open}

The deeper methodological achievement of the present paper, beyond the
technical content, is to make explicit the closed-to-open
foundational shift that is the structural diagnosis of \GGTheory{}.
Closed-system QFT and GR can produce only necessary conditions on
integer multiplicities; open-system frameworks like \GDOS{} produce
sufficient conditions through cascade closure. The \GiFour{} Pillars are
the formal language in which this open-system foundational shift is
made explicit.

This is, we have argued in
\citet[Section~12]{ArisakaP2N3}, the missing link in the
foundations of physics for the century since the establishment of
quantum mechanics in 1925--1930.

\subsection{\GiFour{} as the language of \GDOS{}}
\label{ssec:gi4-language-GDOS}

The four pillars constitute the language of \GDOS{}. When we say
``\GDOS{}'' from now on, we mean the framework axiomatized by:
\begin{itemize}[leftmargin=2em]
\item the four constitutional axioms (\LC, \GGEP, \OGGEP, \MSMF) of
\citet{ArisakaA1Constitution};
\item the \BIsix{} closure of \citet{ArisakaA3BI6};
\item the \GiFour{} Pillars of the present paper;
\item the \QiFour{} Pillars of \citet{ArisakaA7Qi4} (in preparation).
\end{itemize}
This combined framework is the foundational language of \GGTheory{} in
its mature form.

%% ====================================================================
\section{Conclusion: the \GiFour{} Pillars and the foundations of physics}
\label{sec:conclusion}
%% ====================================================================

\subsection{The achievement in summary}
\label{ssec:achievement}

The present paper has accomplished, at the technical level, the
formalization of the geometric backbone of \GDOS{} into four named
mathematical pillars (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) in their
 \emph{dual-layer} form, with seven Structure Theorems per pillar (5
algebraic + 2 physical) and the nine dual-layer sub-theorems
\textbf{T1.1}--\textbf{T4.2} (with the bridge \textbf{T5}). At the methodological level, it has
made explicit the closed-to-open foundational shift that
distinguishes \GGTheory{} from the closed-system frameworks of the
past century. At the structural level, it has \emph{reduced} the residual
softness in the N=3 proof to the named open problem M6, and provided the
rigorous foundation for
the 28-parameter codebook of \ANine{} and the related results.
At the empirical level, it has produced a coherent set of
laboratory-testable falsifiers --- modified TOV in compact stars
(CCI-ST7), equipartition + fluctuation--dissipation in any boundary
observable (GFF-ST7), and the generation-ordered decay-width ladder
(GRIP-ST7) --- that together provide independent experimental
pathways to confirm or refute the framework.

These are technical accomplishments. We close with broader
reflections on what they imply.

\subsection{The dual-layer revolution: from to}
\label{ssec:dual-layer-revolution}

The conceptual upgrade from to of this paper is the
discovery that every \GiFour{} Pillar admits a \emph{dual-layer}
structure --- an algebraic block (axioms A1--A5) and a physical
block (axioms A6--A7) --- linked by a named compatibility theorem.
This is not a cosmetic reformulation: it is the recognition that the
four pillars do not just live in the bulk geometry but \emph{descend
operationally} to the laboratory through specific, named, derivable
mechanisms.

In, each pillar was treated as a single algebraic structure
with a single principal theorem. In, each pillar is a
septuple, with paired theorems T*.1 and T*.2 covering the algebraic
and physical sides respectively. This split has three deep
consequences:

\textbf{(a) Empirical accountability.} Every algebraic prediction
of (e.g., $N=3$, \OGNfive{} ledger, Morse-positivity, \BIsix{} budget)
is now paired with an empirical falsifier (modified TOV, lifetime
hierarchy, equipartition, etc.). No prediction is left without an
operational pathway to test.

\textbf{(b) Universality across sciences.} The dual-layer
architecture explains why the central concepts of \GGTheory{} appear
universally across the natural sciences. The Master Conservation
Law of CCI-ST6 underlies every conservation law in chemistry,
biology, and information theory. The free-energy minimization of
GFF-ST6 underlies every relaxation process from protein folding to
neural predictive coding. The cascade structure of GRIP-ST1
underlies every irreversible discrete relaxation in chemistry,
condensed matter, and machine learning. These are not analogies;
they are boundary projections of the same upstream geometric
structure.

\textbf{(c) Conceptual unification at the architectural level.}
Before, the four pillars were related by an implication chain
(\TGG{} $\Rightarrow$ \CCI{} $\Rightarrow$ \GFF{} $\Rightarrow$
\GRIP{}) but had no shared internal architecture. In, all
four pillars share the \emph{same} dual-layer architecture (5+2
axioms, paired theorems, named compatibility link), making the
\GiFour{} system a coherent unified mathematical object rather than
a sequence of distinct constructions.

This is the precise sense in which is not just an improvement
of but a structural upgrade: it identifies the dual-layer
architecture as the universal organizing principle of the entire
\GiFour{} framework. Future papers in the \GGTheory{} program will
inherit this organizing principle as a baseline.

\subsection{Historical perspective: the four scientific revolutions}
\label{ssec:historical}

The history of theoretical physics, viewed at sufficient distance, is a
history of foundational shifts. Each of the great syntheses of the
past two and a half millennia identified a previously hidden
foundational principle and used it to unify a previously unrelated
cluster of phenomena. We may identify four such ``scientific
revolutions'':

\paragraph{First scientific revolution (Euclid, c.\ 300 BCE).}
Euclid's \emph{Elements} introduced the axiomatic method to
geometry and laid down the architectural template for all
mathematical reasoning to follow: a small set of axioms (the five
postulates and the common notions), explicit definitions, and
logical derivation of theorems. This was the first instance of a
\emph{constitutional} approach to knowledge: a fixed set of
foundational principles from which all consequences must be derivable
by transparent steps. All subsequent revolutions inherit Euclid's
template; the present paper is consciously written in this Euclidean
style.

\paragraph{Second scientific revolution (Newton, 1687).}
Newton's \emph{Principia} unified celestial and terrestrial
mechanics under the principle of universal gravitation, with the
calculus as the mathematical language. This was the first
foundational identification of a single physical principle from
which a vast empirical domain could be derived.

\paragraph{Third scientific revolution (Maxwell, Einstein, and Quantum Mechanics, c.\ 1865--1932).}
Three convergent foundational shifts in the long nineteenth and
early twentieth centuries reshaped fundamental physics:
\textbf{Maxwell (1865)} unified electricity, magnetism, and optics
under his four equations, with vector calculus as the mathematical
language. \textbf{Einstein (1905, 1915)} unified geometry and
gravitation under the principle of equivalence, with tensor calculus
as the mathematical language. \textbf{Heisenberg, Schr\"odinger,
Dirac, von Neumann (1925--1932)} unified atomic spectra, chemistry,
and quantum statistics under the principle of unitary evolution on
Hilbert space, with operator algebra as the mathematical language.
\textbf{The Standard Model (1960s--1970s)} extended these into a
non-Abelian gauge theory of the strong, weak, and electromagnetic
forces, with Yang--Mills theory as the mathematical language.

In each case, a foundational principle was identified, and a
mathematical language was developed that allowed the principle to
produce specific quantitative predictions. The principle was not
self-evident before its identification; the mathematical language was
not pre-existing. The two were created together, in tension, by
intensive collaborative work over years or decades.

\paragraph{Fourth scientific revolution (\GGTheory{}, 2024--).}
\textbf{\GGTheory{}} is, in this lineage, the proposed fourth
scientific revolution. Its foundational principle is \OGGEP{}, the
operational/open geometry--gauge equivalence: physical observation is the
irreversible bulk-to-boundary projection of a six-dimensional bulk onto a four-dimensional boundary. Its mathematical language is
the combination of \GiFour{} (geometric backbone) and \QiFour{} (information
backbone), with differential cohomology, Morse theory, and
Hopkins--Singer descent as the technical tools. Its dual-layer
architecture (algebraic + physical, paired theorems, named
compatibility links) is the universal organizing principle of the
framework, and is the principal structural innovation of the present
 paper.

The present paper formalizes \GiFour{} as the geometric half of this
language. When combined with \QiFour{} (in preparation), with the
Constitution, and with \BIsix{}, the full mathematical vocabulary of
\GDOS{} will be in place.

\subsection{The methodological lesson}
\label{ssec:methodological-lesson-final}

The methodological lesson of \GGTheory{}, articulated already in
\citet[Section~12]{ArisakaP2N3} and made formally explicit
in the present paper, is that closed-system frameworks --- however
mathematically elaborate --- cannot produce sufficient conditions on
the structural integers and hierarchies of physics. Closed-system
QFT can constrain $N$ and $\Mlock/\EPl$ and the QCD scale and the
electroweak vacuum, but it cannot derive any of them. Each requires
an additional input, set by hand.

Open-system frameworks like \GDOS{} are different. Because they
include irreversible bulk-to-boundary projection as a foundational
ingredient, they have access to cascade closure, finite instanton-action
budgets, and Morse-attractor selection. These are the structural
ingredients that produce sufficient conditions: $N\leq 3$ from cascade
closure, $\Mlock/\EPl = 1/(2\e^{35})$ from scale closure, the
twelve charged-fermion masses from cascade depths.

The closed-to-open foundational shift is therefore not a stylistic
choice; it is the structural prerequisite for deriving the integers
and hierarchies of the Standard Model from first principles. This is
the deepest lesson of the \GGTheory{} program, and the present paper
provides the formal vocabulary in which the lesson can be stated
rigorously.

\subsection{The unity of physical reality, the role of mathematics, and the road ahead}
\label{ssec:conclusion-unity}

Indeed, the deepest payoff of the \GiFour{} reorganization is
the structural unity it makes visible. The same four-pillar
architecture --- master-object, conservation-ledger,
free-energy landscape, irreversible cascade --- recurs across
the magnificent edifice of physics: from particle physics
($N=3$ generations from \GRIP{} cascade closure) to gravitation
(modified TOV from \CCI{}-ST7) to chemistry and condensed
matter (free-energy minimization from \GFF{}) to neuroscience
and biology (the free-energy principle as a boundary projection
of \OGGEP{}; see \ASeven{} for the full Qi-4 boundary information
backbone). Mathematics serves this unity not as ornamental
language but as structural skeleton: the bundle structure, the
Morse-theoretic landscape, the Diophantine ledger, and the
Bogomolnyi-saturating cascade are all rigid algebraic facts
that, once written down, leave no room for adjustable
parameters. After all, the \GiFour{} Pillars are not a new
theory imposed on physics from outside; they are the geometric
reading of what physics has been telling us, in different
rooms of the same building, all along. Looking forward, the
predictions cataloged in \S\ref{sec:predictions-falsification}
and the open problems of
\S\ref{sec:open-problems}, especially M9 (the canonicity of the
complexity class) and M11 in its spectral-flow form (the general
monodromy statement, its residues reduced to one identification and
one convention), are the concrete agenda for the next phase of the
program. The
framework stands or falls on the modified-TOV signature, on
the lifetime-hierarchy signature, and on the empirical reach
of the 28 SM parameter predictions of \ANine{}. The next century
of empirical work will decide.

\GGAcknowledgments

\GGAppendixPart[Technical Appendices]{Technical Appendices}{Technical machinery, worked examples, and indices}

%% ====================================================================
%% Phase 12 v5.17: Canonical Appendix Intro + A Summary of Abbreviations
%% + B Glossary of Key Symbols (matched to A5-GDOS canonical form)
%% ====================================================================


\noindent
The Appendices gather the technical machinery and the reference
material that supports the main-text theorems of the \GiFour{}
pillars. Roughly speaking, this is where the reader who wants to
verify a specific derivation, run a specific worked example, or
trace a specific symbol will spend time --- while the reader who
wants only the axiomatic content and the principal theorems
\T{1}--\T{5} of the main text can stop after Part~IX and treat
the appendices as a self-contained companion volume. Appendices~A
and~B provide the abbreviations and the glossary of key symbols.
Appendix~C unfolds the differential-cohomology preliminaries in
full pedagogical detail. Appendix~D supplies the Hopkins--Singer
machinery used in the \CCI{} closure. Appendix~E provides Morse
theory on stratified spaces. Appendix~F collects the detailed
worked examples for the four pillars. Appendix~G is the
pedagogical tour through the dual-layer architecture in plain
language. Appendix~H gives the step-by-step Diophantine analysis
for \OGNfive{}. Appendix~I supplies the free-energy-principle
bridge to life sciences and neuroscience. Appendix~J gives the
explicit derivation of the modified TOV equation. Appendices~K
and~L provide the Hopkins--Singer descent and Morse-theory worked
examples on \CPN{}. Appendix~M unfolds the generation decay-width ladder from
cascade actions to laboratory observations. Appendix~N is the
life-science and neuroscience glossary. Finally, the inline
bibliography with full Relevance annotations follows the appendices
and closes the volume. Indeed, the nine main-text Parts plus fourteen
appendices together realize the \GiFour{} framework in full:
Parts~I--VI develop and synthesize the pillar theorems with proofs, Parts~VII--IX
treat applications, discussion, and conclusions, and the appendices supply the
technical and bibliographic infrastructure that makes the whole
document self-checking.


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

This appendix collects every abbreviation used in the present paper.
All program-defined names are rendered in \emph{sans-serif type}
(\GGTheory{}, \GGConstitution{}, \GG{}, \GDOS{}, \BIsix{}, \OGNfive{},
\QGL{}, \GiFour{}, \QiFour{}, \GRIP{}, \BFFT{}, etc.) so the reader
can see at a glance which terms belong to the \GGTheory{} program
proper. Standard physics abbreviations imported from the wider
literature (KK, RS, QFT, TOV, etc.) are rendered in plain uppercase
Roman, following the convention of the major physics journals.
Mathematical objects are in roman or italic in the usual way. The
third column gives the \emph{primary link}: the paper in the
Arisaka-GG series and the section where each program-defined name
is first introduced in detail.

{\small
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.16\linewidth} >{\raggedright\arraybackslash}p{0.56\linewidth} >{\raggedright\arraybackslash}p{0.24\linewidth}@{}}
\caption[Summary of abbreviations used in the present paper.]{Summary of abbreviations used in the present paper.
Program-defined names are rendered in sans serif; standard physics
abbreviations in plain uppercase Roman; mathematical objects in
roman/italic in the usual way. The third column points at the paper or the section where each name is fixed, so an abbreviation met in the middle of an argument can be resolved without leaving the appendix.}
\label{tab:appA-abbrev}\\
\toprule
\textbf{Symbol} & \textbf{Meaning} & \textbf{Primary link} \\
\midrule
\endfirsthead

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

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

\bottomrule
\endlastfoot

\multicolumn{3}{l}{\itshape Framework name and core program objects} \\*
\midrule
\GGTheory{} & Grand Geometric Theory (also: Geometry-Gauge Equivalence Theory) & \AOne{}, Preface \\
\GGConstitution{} & The constitutional charter of the \GGTheory{} program: four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) & \AOne{}, \S 2--\S 3 \\
\GG{} & Geometry-Gauge locking in six dimensions: the 6D bulk & \ATwo{}, \S 8 (\TX{1}{GG6}) \\
\GDOS{} & Geometric Dynamics of Open Systems: the boundary law implied by \OGGEP{} & \AFive{} \\
\BIsix{} & Green--Schwarz-modified Bianchi identity at 6D & \AThree{}, \S 3 \\
\OGNfive{} & Occam-Gauge-Normalization 5-integer ledger $(M, N, L; J, K) = (1,3,5;7,35)$ & \AThree{}, \S 7; \TX{6}{GG6} \\
\QGL{} & Quadruple Gauge Locking: the four-factor bulk gauge group $\Spinone \times \SU(5) \times \U(1)_B \times \U(1)_{Qi}$ & \ATwo{}, \S 10 (\TX{3}{GG6}) \\
\GiFour{} & Geometric-Invariance four-pillar structure (\TGG{}, \CCI{}, \GFF{}, \GRIP{}) & this paper \\
\QiFour{} & Quantum-Information four-pillar structure (\QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}) & \ASeven{} \\
\midrule

\multicolumn{3}{l}{\itshape Four axioms (\AX{1}--\AX{4})} \\*
\midrule
\AXkw{} & Axiom (axiomatic primitive; non-negotiable) & \AOne{}, \S 1.5 \\
\LC{} & \AX{1} Locality and Causality & \AOne{}, \S 2.2 \\
\GGEP{} & \AX{2} Geometry--Gauge Equivalence Principle & \AOne{}, \S 2.3 \\
\OGGEP{} & \AX{3} Operational/Open Geometry--Gauge Equivalence Principle (boundary observation dictionary) & \AOne{}, \S 2.4 \\
\MSMF{} & \AX{4} Maximum Symmetry, Minimum Freedom (no hidden continuous knobs) & \AOne{}, \S 2.5 \\
\midrule

\multicolumn{3}{l}{\itshape Seven postulates (\POS{1}--\POS{7})} \\*
\midrule
\POSkw{} & Postulate (admissibility filter; testable in principle) & \AOne{}, \S 1.5 \\
\GI{} & \POS{1} Gauge invariance across scales (\LGI{} / \DGI{} / \GGI{}) & \AOne{}, \S 3.3 \\
\UEP{} & \POS{2} Unified Extremal Principle (PLA $\Leftrightarrow$ PLT / \MEPP{} / FEP) & \AOne{}, \S 3.4 \\
\AF{} & \POS{3} Anomaly Freedom and cancellability by inflow & \AOne{}, \S 3.5 \\
\CP{} & \POS{4} Closure and Positivity (CPTP-admissible open dynamics) & \AOne{}, \S 3.6 \\
\RG{} & \POS{5} Renormalization-Group covariance (scale covariance) & \AOne{}, \S 3.7 \\
\OGN{} & \POS{6} Occam Gauge Normalization (canonical integer/rational normalization) & \AOne{}, \S 3.8 \\
\MF{} & \POS{7} Measurement and Falsifiability (operational contract + kill criteria) & \AOne{}, \S 3.9 \\
\LGI{} & Local Gauge Invariance (microscopic / closed-system layer of \GI{}) & \AOne{}, \S 3.3 \\
\GGI{} & Global Gauge Invariance (macroscopic / ledger layer of \GI{}) & \AOne{}, \S 3.3 \\
\midrule

\multicolumn{3}{l}{\itshape \GiFour{} pillars} \\*
\midrule
\TGG{} & Total Geometric--Gauge object (Pillar I of \GiFour{}) & this paper \\
\CCI{} & Covariant Conservation Identity (Pillar II of \GiFour{}) & this paper \\
\GFF{} & Geometric Free-Energy Functional (Pillar III of \GiFour{}) & this paper \\
\GRIP{} & Gauge-Runtime Invariance Pipeline (Pillar IV of \GiFour{}) & this paper \\
\GEP{} & Geometry--Gauge Energy Principle (energetic block of \GFF{}) & this paper \\
\CEP{} & Curvature--Entropy Principle (entropic block of \GFF{}) & this paper \\
\DGI{} & Dynamical Gauge Invariance (initial stage of \GRIP{}) & this paper \\
\GSSB{} & Geometric Spontaneous Symmetry Breaking (middle stage of \GRIP{}) & this paper \\
\GA{} & Geometric Attractor (final stage of \GRIP{}) & this paper \\
\midrule

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

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

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

\multicolumn{3}{l}{\itshape Program-defined mathematical symbols} \\*
\midrule
$\QB$ & Quantum-information Bit -- the bioenergetic Arisaka constant $\QB \approx 0.1555$~eV per qubit; matches ATP hydrolysis at biological temperatures & \ATwo{}, \S 15 (\TX{8}{GG6}) \\
\midrule

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

\multicolumn{3}{l}{\itshape \quad Open-system / quantum-channel formalism} \\*
\midrule
CPTP & Completely Positive Trace-Preserving (channel admissibility for open evolution; \AX{3}/\POS{4}) & --- \\
GKLS & Gorini--Kossakowski--Lindblad--Sudarshan~\citep{GKLS1976} (master equation for Markovian CPTP semigroups; \POS{4}) & --- \\
GNS & Gelfand--Naimark--Segal (construction) & --- \\
NZ & Nakajima--Zwanzig (memory-kernel projection for non-Markovian open dynamics) & --- \\
POVM & Positive Operator-Valued Measure (generalized measurement model; \AX{3}/\POS{7}) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum and classical foundational theories} \\*
\midrule
BRST & Becchi--Rouet--Stora--Tyutin (cohomology) & --- \\
GR & General Relativity & --- \\
QCD & Quantum Chromodynamics & --- \\
QED & Quantum Electrodynamics & --- \\
QFT & Quantum Field Theory & --- \\
QM & Quantum Mechanics & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Standard Model and beyond} \\*
\midrule
EFT & Effective Field Theory & --- \\
EW & Electroweak & --- \\
GUT & Grand Unified Theory & --- \\
SM & Standard Model of particle physics & --- \\
SSB & Spontaneous Symmetry Breaking (closed-system; generalized by \GSSB{}) & --- \\
SUSY & Supersymmetry & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Higher-dimensional and holographic constructions} \\*
\midrule
AdS & Anti--de Sitter spacetime & --- \\
AdS/CFT & Anti--de Sitter / Conformal Field Theory correspondence & --- \\
CFT & Conformal Field Theory & --- \\
KK & Kaluza--Klein (mode expansion / dimensional reduction) & --- \\
RS & Randall--Sundrum (warped extra-dimension model) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Quantum gravity and cosmology} \\*
\midrule
$\Lambda$CDM & Standard cosmological model with cosmological constant + cold dark matter & --- \\
LQG & Loop Quantum Gravity & --- \\
TOV & Tolman--Oppenheimer--Volkoff (equation) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Foundations of quantum mechanics} \\*
\midrule
CHSH & Clauser--Horne--Shimony--Holt (inequality) & --- \\
EPR & Einstein--Podolsky--Rosen (paradox / argument) & --- \\
\midrule

\multicolumn{3}{l}{\itshape \quad Open-system / extremal-principle sub-acronyms} \\*
\midrule
FEP & Free Energy Principle (boundary inference avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
\MEPP{} & Maximum Entropy Production Principle (heuristic boundary avatar of \UEP{}) & \ASeven{}, \QiEP{} \\
PLA & Principle of Least Action (closed/bulk avatar of \UEP{}; standard variational dynamics) & \ASeven{}, \QiEP{} \\
PLT & Principle of Least Time (operational/boundary avatar of \UEP{}; fastest lawful path) & \ASeven{}, \QiEP{} \\
\end{longtable}
}


\bigskip

\noindent\textbf{Figure~\ref{fig:abbrev-flow}: hierarchy of program-defined acronyms in the \GGTheory{} program.} Top: the framework name (\GGTheory{}) and its constitutional charter (\GGConstitution{}). Tier~2: the four axioms (\AX{1}--\AX{4}) and seven postulates (\POS{1}--\POS{7}) that together define the admissibility predicate. Tier~3: three bulk objects forced by the constitution --- \GG{} (6D bulk), \BIsix{} (modified Bianchi identity), \OGNfive{} (integer ledger). Tier~4: the boundary law \GDOS{} that the bulk projects to via \OGGEP{}, with input acronym \SIP{}. Tier~5: the two four-pillar structures, \GiFour{} (geometric flank, with sub-acronyms \GEP{}/\CEP{} blocks of \GFF{} and \DGI{}/\GSSB{}/\GA{} stages of \GRIP{}) and \QiFour{} (quantum-information flank, with \QiEP{} sub-acronyms PLT/\MEPP{}/FEP, \QiCON{} sub-concepts Conformal/L\'evy/Fractal/$1/f$, \QiHAL{} sub-acronyms \MePMoS{}/\NHT{}/\HAL{}/\BFFT{}, the boundary completion \OH{}${+}$\IH{}${=}$\EIC{}, and the runtime carrier \SeOS{} (Selfish Open System) underneath). Bottom: the bioenergetic Arisaka constant $\QB$ (derived secondary quantity), with the Arisaka equation $J_E / \QB = J_e = J_B = J_{Qi}$ that locks the four boundary currents through one conversion factor.

\begin{figure}[H]
\centering
\resizebox{0.824\textwidth}{!}{%
\begin{tikzpicture}[
 font=\sffamily,
 >=Stealth,
 line width=0.80pt,
 every node/.style={align=center},
 top/.style ={rectangle, rounded corners=3pt, draw=violet!55!black, line width=0.8pt, minimum width=46mm, minimum height=11mm, fill=violet!12, font=\sffamily\bfseries\Large},
 charter/.style ={rectangle, rounded corners=3pt, draw=violet!55!black, line width=0.6pt, minimum width=46mm, minimum height=10mm, fill=violet!8, font=\sffamily\bfseries\large},
 axhead/.style ={rectangle, rounded corners=3pt, draw=red!55!black, line width=0.6pt, minimum width=62mm, minimum height=9mm, fill=red!14, font=\sffamily\bfseries\large},
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 gihead/.style ={pillhead, draw=olive!55!black, fill=green!14},
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 pill/.style ={rectangle, rounded corners=2pt, draw=olive!55!black, line width=0.4pt, minimum width=72mm, minimum height=7mm, fill=green!10, font=\sffamily\normalsize, anchor=west},
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 qb/.style ={rectangle, rounded corners=3pt, draw=cyan!55!black, line width=0.7pt, minimum width=148mm, minimum height=15mm, fill=cyan!10, font=\sffamily\normalsize},
 arrow/.style ={->, line width=0.7pt, draw=black!65},
 arrowdash/.style ={->, line width=0.6pt, draw=black!50, dashed}
]

% --- Top tier: programme name and charter -----------------
\node[top] (GGT) at (0, 0) {\GGTheory{}};
\node[charter] (GGC) at (0, -16mm) {\GGConstitution{}};
\draw[arrow] (GGT.south) -- (GGC.north);

% --- Tier 2: 4 axioms (left) and 7 postulates (right) -----
\node[axhead] (AXH) at (-52mm, -36mm) {\textbf{4 Axioms} (\textbf{AX-1}\,--\,\textbf{AX-4})};
\node[ax] (AX1) at (-52mm, -48mm) {\textbf{AX-1} \, \textbf{LC} \quad Locality and Causality};
\node[ax] (AX2) at (-52mm, -58mm) {\textbf{AX-2} \, \textbf{GGEP} \quad Geometry-Gauge Equivalence};
\node[ax] (AX3) at (-52mm, -68mm) {\textbf{AX-3} \, \textbf{OGGEP} \quad Operational/Open GGEP};
\node[ax] (AX4) at (-52mm, -78mm) {\textbf{AX-4} \, \textbf{MSMF} \quad Max.\ Symmetry, Min.\ Freedom};

\node[posthead] (POSH) at (52mm, -36mm) {\textbf{7 Postulates} (\textbf{POS-1}\,--\,\textbf{POS-7})};
\node[ax] (P1) at (52mm, -48mm) {\textbf{POS-1} \, \textbf{GI} \;\; (\,\textbf{LGI} \,/\, \textbf{DGI} \,/\, \textbf{GGI}\,)};
\node[ax] (P2) at (52mm, -56mm) {\textbf{POS-2} \, \textbf{UEP} \quad Unified Extremal Principle};
\node[ax] (P3) at (52mm, -64mm) {\textbf{POS-3} \, \textbf{AF} \quad Anomaly Freedom};
\node[ax] (P4) at (52mm, -72mm) {\textbf{POS-4} \, \textbf{CP} \quad Closure and Positivity};
\node[ax] (P5) at (52mm, -80mm) {\textbf{POS-5} \, \textbf{RG} \quad RG covariance};
\node[ax] (P6) at (52mm, -88mm) {\textbf{POS-6} \, \textbf{OGN} \quad Occam Gauge Normalization};
\node[ax] (P7) at (52mm, -96mm) {\textbf{POS-7} \, \textbf{MF} \quad Measurement and Falsifiability};

\draw[arrow] (GGC.south) -| ($(AXH.north) +(0,3mm)$) -- (AXH.north);
\draw[arrow] (GGC.south) -| ($(POSH.north)+(0,3mm)$) -- (POSH.north);

% --- Tier 3: bulk theorems (3 boxes: GG-6, BI-6, OGN-5) ---
\node[font=\sffamily\bfseries\large] (BTL) at (0, -108mm) {Bulk theorems forced by \textbf{AX} + \textbf{POS}};
\node[bulk] (GG6) at (-46mm, -120mm) {\textbf{GG-6}};
\node[bulk] (BI6) at (0mm, -120mm) {\textbf{BI-6}};
\node[bulk] (OGN5) at (46mm, -120mm) {\textbf{OGN-5}};

\draw[arrow] (AX4.south) -- ([yshift=-2mm]BTL.north -| AX4.south);
\draw[arrow] (P7.south) -- ([yshift=-2mm]BTL.north -| P7.south);

% --- Tier 4: GDOS boundary law (only SIP under it) --------
\node[gdos] (GDOS) at (0, -136mm) {\textbf{GDOS} \;\; (Geometric Dynamics of Open Systems on $\partial\mathcal{M}_6$) \;\; \textbf{SIP}};

\draw[arrow] (GG6.south) -- (GDOS.north -| GG6.south);
\draw[arrow] (BI6.south) -- (GDOS.north -| BI6.south);
\draw[arrow] (OGN5.south) -- (GDOS.north -| OGN5.south);

% --- Tier 5a: Pillars headers ----------------------------
\node[gihead] (GiH) at (-46mm, -152mm) {\textbf{Gi-4 Pillars}};
\node[qihead] (QiH) at (46mm, -152mm) {\textbf{Qi-4 Pillars}};

\draw[arrow] (GDOS.south) -| (GiH.north);
\draw[arrow] (GDOS.south) -| (QiH.north);

% --- Tier 5b: Pillars content -----------------------
% Gi-4 box (left) -- 4 pillars + sub-acronyms
\node[pill] (TGG) at (-82mm, -170mm) {\textbf{TGG} \quad Total Geometric--Gauge object};
\node[pill] (CCI) at (-82mm, -179mm) {\textbf{CCI} \quad Covariant Conservation Identity};
\node[pill] (GFF) at (-82mm, -188mm) {\textbf{GFF} \quad Geometric Free-Energy Functional};
\node[sub] at (-82mm, -195mm) {\hphantom{XX}$\downarrow$ \;\textbf{GEP} (energetic) \,$\bullet$ \,\textbf{CEP} (entropic)};
\node[pill] (GRIP) at (-82mm, -204mm) {\textbf{GRIP} \quad Gauge-Runtime Invariance Pipeline};
\node[sub] at (-82mm, -211mm) {\hphantom{XX}$\downarrow$ \;\textbf{DGI} (initial) \,$\bullet$ \,\textbf{GSSB} (middle) \,$\bullet$ \,\textbf{GA} (attractor)};

% Qi-4 box (right) -- 4 pillars + sub-acronyms + EIC
\node[pillq] (QQUA) at (10mm, -170mm) {\textbf{Qi-QUA} \quad Quantum information for \textbf{QGL}};
\node[pillq] (QEP) at (10mm, -179mm) {\textbf{Qi-EP} \quad Quantum information for \textbf{UEP}};
\node[sub] at (10mm, -186mm) {\hphantom{XX}$\downarrow$ \;\textbf{PLT} \,$\bullet$ \,\textbf{MEPP} \,$\bullet$ \,\textbf{FEP}};
\node[pillq] (QCON) at (10mm, -195mm) {\textbf{Qi-CON} \quad Quantum information for Conformal Inv.};
\node[sub] at (10mm, -202mm) {\hphantom{XX}$\downarrow$ \;Conformal \,$\bullet$ \,L\'evy \,$\bullet$ \,Fractal \,$\bullet$ \,$1/f$};
\node[pillq] (QHAL) at (10mm, -211mm) {\textbf{Qi-HAL} \quad Quantum information for the \textbf{HAL}};
\node[sub] at (10mm, -218mm) {\hphantom{XX}$\downarrow$ \;\textbf{MePMoS} \,$\bullet$ \,\textbf{NHT} \,$\bullet$ \,\textbf{HAL} \,$\bullet$ \,\textbf{BFFT}};
\node[pillq] (EIC) at (10mm, -227mm) {\textbf{OH} \,$+$\, \textbf{IH} \,$=$\, \textbf{EIC} \quad (Evolving Inner Cosmos)};
\node[sub] at (10mm, -234mm) {\hphantom{XX}$\downarrow$ \;\textbf{SeOS} (Selfish Open System)};

% --- Tier 6: Q_B + Arisaka equation -----------------------
\node[qb] (QB) at (0, -250mm) {%
$\boldsymbol{\mathsf{Q}_{\mathsf{B}}}$ \;\; --- \;\; Quantum-information Bit \;\; (energy--information Arisaka constant) \\[1.5mm]
\textit{Arisaka equation:} \;\; $\boldsymbol{J_E / \mathsf{Q}_{\mathsf{B}} \,=\, J_e \,=\, J_B \,=\, J_{Qi}}$
};

\end{tikzpicture}
}
\caption[Hierarchy of program-defined acronyms in the \GGTheory{} program.]{How the program's own names are related to each other. The chart
is a hierarchy, not a list: the charter sits at the top, the geometry it
forces below it, the boundary law below that, and the derived pillars at
the foot, so the position of a name on the chart tells the reader what
it depends on. Names at the same level are siblings introduced by the
same construction. The purpose is orientation rather than definition ---
the meanings are in the abbreviations table --- and a reader who has
lost track of which name belongs to which layer can recover it here in
one glance.}
\label{fig:abbrev-flow}
\end{figure}

\clearpage


\section{Glossary of Key Symbols}
\label{app:glossary}

\begin{table}[H]
\centering
\caption[Glossary of key mathematical symbols.]{Every mathematical symbol this paper uses repeatedly, with its
meaning. The list is deliberately restricted to symbols that recur
across Parts rather than every symbol that appears once: a reader
meeting $\Msix$, $I_r$ or $K$ in a later Part should be able to recover
it here without returning to the section that introduced it. Where a
symbol carries both a bulk and a boundary reading, both are given, since
one letter doing two jobs is the commonest source of confusion in the
cohort's notation.}
\label{tab:appB-glossary}
\small
\begin{tabular}{@{}lp{12.35cm}@{}}
\toprule
\textbf{Symbol} & \textbf{Meaning} \\
\midrule
$M_6, M_4, I_r, S^1_\theta$ & 6D bulk, 4D boundary, radial interval, compact angular direction \\
$r, \theta$ & Radial coordinate $\in [0, K]$, angular $\in [0, 2\pi)$ \\
$K$ & Dual-lock integer = 35 \\
$g_{AB}, g_{\mu\nu}$ & 6D bulk metric, 4D boundary metric \\
$R_6, R_4$ & 6D Ricci scalar, 4D Ricci scalar \\
$M_\Pl, M_6, \Mlock, v$& Reduced Planck mass, 6D fundamental mass, KK lock scale, EW vev \\
$\Lambda_4$ & 4D cosmological constant \\
$G_N$ & Newton's gravitational constant \\
$E_\Pl, E_{\rm GUT}$ & Planck and GUT energy scales \\
$\Phi(x, r, \theta)$ & Generic bulk field \\
$\phi^{(n,\ell)}(x)$ & 4D KK-mode field (radial index $n$, angular index $\ell$) \\
$f_n(r)$ & Radial profile of $n$-th KK mode \\
$m_{n,\ell}^2$ & KK mass squared = $m_n^2 + \ell^2$ in natural units \\
$\cH_{\rm Bulk}, \cH_{\rm Bdy}, \cH_{\rm env}$ & Bulk, boundary, environment Hilbert spaces \\
$\rho_{\rm Bulk}, \rho_{\rm Bdy}$ & Bulk, boundary density operators \\
$\cP, \cQ$ & Nakajima--Zwanzig projection superoperators \\
$\cL, \cL_\GDOS$ & Liouville superoperator, \GDOS{} generator \\
$H_{\rm Bulk}, H_{\rm eff}$ & Bulk Hamiltonian, effective boundary Hamiltonian \\
$L_\alpha, \gamma_\alpha$ & Jump operators and rates of GKLS \\
$\Phi_{t,s}$ & Boundary CPTP evolution channel \\
$\cI_\alpha, E_\alpha$ & Instrument and POVM effect \\
$K(t,s), \cI(t)$ & NZ memory kernel and inhomogeneous initial-correlation term \\
$\cR$ & Boundary reduction map \\
$\cD_{\OGGEP}$ & \OGGEP{} dictionary $= \cR \circ \Tr_\env$ \\
$T^{\mu\nu}, T^{\mu\nu}_\TGG$ & Boundary stress tensor, unified \TGG{} stress tensor \\
$\Delta T^{\rm GS}_{\mu\nu}, \Delta T^{\rm KK}_{\mu\nu}$ & GS-tensor and KK-tower inflow \\
$X_4, H_3$ & \BIsix{} anomaly polynomial and Green-Schwarz two-form \\
$\cA_{\rm tot}$ & Composite bulk connection \\
$G_{\rm tot}$ & Bulk gauge-gravity group $\Spinone \times \SUfive \times \UoneB \times \UoneQi$ \\
$Q^c, T^a, Y, B, Q_{Qi}$ & SU(5) generators, hypercharge, baryon, informational charges \\
$F_{\mu\nu}$ & Gauge field strength \\
$s_6, s_4$ & 6D and 4D BRST operators \\
$Z_4[J], W_4[J], \Gamma_4$ & Generating functional, connected generating functional, effective action \\
$\Tr_{\rm env}, \Tr_{\rm Bdy}, \Tr_{\rm Bulk}$ & Partial and full traces \\
$\boxed{\cdot}$ & Boxed equation = key result \\
$\blacksquare$ & End of proof \\
\bottomrule
\end{tabular}
\end{table}

\clearpage


\section{Differential cohomology preliminaries (pedagogical)}
\label{app:diff-cohomology}

\textbf{Reading goal.} This appendix is written for first-year
graduate students in physics, theoretical life science, or
neuroscience, who may have seen vector calculus on $\R^3$ but not
algebraic topology. We build up from the divergence theorem of
freshman calculus, through de Rham cohomology, to the differential
cohomology refinement that is the backbone of the \CCI{} closure of
Part~III and the Hopkins--Singer descent of Part~IV. Every step is
illustrated with explicit examples on small manifolds (the circle
$S^1$, the sphere $S^2$, the torus $T^2$, and the projective space
$\CPtwo$). No previous exposure to algebraic topology is assumed.

For a complete development see \citet{HopkinsSinger2005,FreedHopkins2000};
for a more pedestrian introduction to de Rham cohomology, see
\citet{BottTu1982}.

\subsection{Why differential forms (a five-minute review)}
\label{app:why-forms}

Recall from vector calculus on $\R^3$ that:
\begin{itemize}[leftmargin=2em]
\item A scalar function $f(x,y,z)$ has a \emph{gradient} $\nabla f$,
which is a vector field.
\item A vector field $\mathbf{F}$ has a \emph{curl} $\nabla\times\mathbf{F}$,
which is also a vector field.
\item A vector field $\mathbf{F}$ has a \emph{divergence}
$\nabla\cdot\mathbf{F}$, which is a scalar function.
\end{itemize}
The three operations satisfy
$\nabla\times(\nabla f) = 0$ and
$\nabla\cdot(\nabla\times\mathbf{F}) = 0$.

The language of \emph{differential forms} unifies these operations.
A scalar function $f$ is called a \emph{0-form}; a vector field
written as $\mathbf{F} = (F_1, F_2, F_3)$ corresponds to a
\emph{1-form} $\omega^{(1)} = F_1\,\dd x + F_2\,\dd y + F_3\,\dd z$;
a vector field also corresponds to a \emph{2-form}
$\omega^{(2)} = F_1\,\dd y\wedge\dd z + F_2\,\dd z\wedge\dd x +
F_3\,\dd x\wedge\dd y$; and a scalar function corresponds to a
\emph{3-form} $\omega^{(3)} = f\,\dd x\wedge\dd y\wedge\dd z$.

In this language, $\nabla$, $\nabla\times$, and $\nabla\cdot$ are all
the \emph{same} operation, the \emph{exterior derivative} $\dd$:
\begin{equation}
\label{eq:d-unifies}
\dd f = \omega^{(1)}_{\nabla f},\qquad
\dd \omega^{(1)} = \omega^{(2)}_{\nabla\times \mathbf F},\qquad
\dd \omega^{(2)} = \omega^{(3)}_{\nabla\cdot \mathbf F}.
\end{equation}
The two identities $\nabla\times(\nabla f)=0$ and
$\nabla\cdot(\nabla\times \mathbf F)=0$ become the single statement
\begin{equation}
\label{eq:dd-zero}
\boxed{\;\dd^2 = 0,\;}
\end{equation}
the \emph{exterior calculus identity}. Equation \eqref{eq:dd-zero}
holds in arbitrary dimensions and is the central fact that motivates
de Rham cohomology.

\begin{remark}[Physics intuition: why $\dd^2=0$?]
\label{rem:dd-zero-intuition}
$\dd^2 = 0$ is a consequence of the antisymmetry of mixed partial
derivatives: $\partial_i\partial_j = \partial_j\partial_i$. When this
symmetry is contracted against the antisymmetric wedge product
$\dd x^i\wedge \dd x^j = -\dd x^j\wedge\dd x^i$, the result vanishes.
Geometrically: the boundary of a boundary is empty
($\partial^2 = 0$), and $\dd$ is dual to $\partial$ under Stokes's
theorem.
\end{remark}

\subsection{Ordinary cohomology and de Rham cohomology}
\label{app:cohomology-basics}

For a smooth manifold $X$, the de Rham cohomology
$H^k_{\rm dR}(X;\R)$ classifies closed $k$-forms modulo exact ones:
\begin{equation}
\label{eq:de-Rham}
H^k_{\rm dR}(X;\R) = \frac{\ker(\dd:\Omega^k(X)\to\Omega^{k+1}(X))}
{\mathrm{im}(\dd:\Omega^{k-1}(X)\to\Omega^k(X))}.
\end{equation}
In words: a closed $k$-form $\omega$ ($\dd\omega = 0$) represents a
non-trivial cohomology class iff it is \emph{not} the exterior
derivative of any $(k-1)$-form. Two closed $k$-forms represent the
same class iff they differ by an exact form.

By the de Rham theorem \citep{deRham1931}, $H^k_{\rm dR}(X;\R)$ is
naturally isomorphic to the singular cohomology $H^k(X;\R)$ with real
coefficients. This allows us to compute cohomology either by
analysis (via differential forms) or by topology (via cell complexes).

\paragraph{Worked example 1: the circle $S^1$.}
On $S^1$ parametrized by angle $\theta\in[0, 2\pi)$, the constant
1-form $\dd\theta$ is closed but not globally exact (it is locally
$\dd\theta = \dd(\theta)$, but $\theta$ is not a globally defined
function on $S^1$). Its integral $\int_{S^1}\dd\theta = 2\pi \neq 0$
detects the non-trivial class. Therefore
$H^1(S^1;\R) \cong \R$, generated by $[\dd\theta/2\pi]$.

\paragraph{Worked example 2: the 2-sphere $S^2$.}
On $S^2$ embedded in $\R^3$, the area 2-form $\omega = \sin\theta\,\dd\theta\wedge\dd\phi$
in spherical coordinates is closed and integrates to
$\int_{S^2}\omega = 4\pi\neq 0$. Its class generates
$H^2(S^2;\R)\cong\R$. By contrast, $H^1(S^2;\R) = 0$ (every closed
1-form on $S^2$ is exact, because every loop on $S^2$ contracts to
a point).

\paragraph{Worked example 3: the 2-torus $T^2 = S^1\times S^1$.}
The 1-forms $\dd\theta_1$ and $\dd\theta_2$ are both closed and
non-exact, generating $H^1(T^2;\R)\cong\R^2$. Their wedge
$\dd\theta_1\wedge\dd\theta_2$ generates $H^2(T^2;\R)\cong\R$. This
example shows how cohomology counts ``independent loops/holes'' in a
space.

\subsection{Integer cohomology and quantization}
\label{app:integer-cohomology}

The integer cohomology $H^k(X;\Z)$ is the singular cohomology with
integer coefficients. The natural map $H^k(X;\Z)\to H^k(X;\R)$ has
kernel the torsion sub-group $\mathrm{Tor}(H^k(X;\Z))$.

A closed differential form $\omega\in\Omega^k(X)$ is said to have
\emph{integer cohomology class} if its de Rham class lies in the
image of $H^k(X;\Z)\to H^k(X;\R)$. Equivalently, all periods
$\int_C\omega$ over closed $k$-cycles $C\subset X$ are integers.

\paragraph{Why integrality matters in physics.}
Integrality is the mathematical statement of \emph{quantization}.
Examples:
\begin{itemize}[leftmargin=2em]
\item The Dirac monopole quantization $\int_{S^2}F/2\pi\in\Z$ for
the magnetic-monopole field strength $F$ \citep{Dirac1931}.
\item The Aharonov--Bohm phase $\oint A/2\pi\in\Z$ for a Wilson loop
around a flux tube \citep{AharonovBohm1959}.
\item The first Chern class $c_1\in H^2(X;\Z)$ of a complex line
bundle, which classifies the topology of monopole-like configurations.
\item The \CCI{} integrality $[X_4]\in H^4(\Msix;\Z)$ of the present
paper, which selects the \OGNfive{} ledger.
\end{itemize}

\paragraph{Worked example 4: integrality on the projective line $\CP^1$.}
The Fubini--Study form on $\CP^1\cong S^2$ is
\begin{equation}
\omega_{\rm FS} = \frac{\mathrm{i}}{2\pi}\,\partial\bar\partial\log(1+|z|^2)
= \frac{1}{\pi(1+|z|^2)^2}\,\dd x\wedge\dd y.
\end{equation}
Its integral is $\int_{\CP^1}\omega_{\rm FS} = 1$, an integer. This
generates $H^2(\CP^1;\Z)\cong\Z$, with $[\omega_{\rm FS}]$ the
positive generator.

\subsection{Differential cohomology as a refinement}
\label{app:diff-cohom-refinement}

\paragraph{Why a refinement is needed.}
Ordinary cohomology $H^k(X;\Z)$ remembers only the integer
\emph{class} of a closed differential form, not the form itself.
Two different forms representing the same class are
indistinguishable. But in physics, we often need to remember
\emph{both}: the integer (topological) information and the
specific connection (geometric) information. The differential
cohomology $\check H^k(X;\Z)$ is the precise refinement that
remembers both simultaneously.

\paragraph{The triple structure.}
Differential cohomology $\check H^k(X;\Z)$ is a refinement of
ordinary cohomology that combines topological and differential-form
data. An element $\check x \in \check H^k(X;\Z)$ may be thought of
as a triple
\begin{equation}
\label{eq:check-triple}
\check x \;\sim\; ([x],\,\omega,\,\eta),
\end{equation}
where:
\begin{itemize}[leftmargin=2em]
\item $[x]\in H^k(X;\Z)$ is the underlying integer cohomology class
(the \emph{topology} of $\check x$);
\item $\omega\in \Omega^k_{\rm closed}(X)$ is the curvature
representative satisfying $[\omega] = [x]\otimes\R$ (the
\emph{geometry} of $\check x$);
\item $\eta$ is a transgression / connection datum that relates the
two.
\end{itemize}
The precise definition is more technical; see
\citet[Section~2]{HopkinsSinger2005}.

\paragraph{Physical example: a magnetic monopole.}
Consider a magnetic monopole at the origin of $\R^3$. The flux
through a sphere $S^2$ enclosing the monopole is quantized:
$\int_{S^2} F = 2\pi n$ for some integer $n$ (the Dirac
quantization). The integer $n$ is the topology
($[F]\in H^2(S^2;\Z)\cong\Z$), and the explicit form $F$ (e.g.,
$F = (n/2)\sin\theta\,\dd\theta\wedge\dd\phi$) is the geometry.
The differential cohomology class $\check F\in\check H^2(S^2;\Z)$
remembers both $n$ \emph{and} the specific form $F$ --- which is
necessary to compute, e.g., the Aharonov--Bohm phase
$\oint A$ for a Wilson loop on the boundary of $S^2$.

\paragraph{Worked example 5: differential cohomology on $S^1$.}
On $S^1$, the differential cohomology $\check H^1(S^1;\Z)$ classifies
$U(1)$ connections (gauge potentials) up to gauge equivalence. An
element $\check A\in\check H^1(S^1;\Z)$ has:
\begin{itemize}[leftmargin=2em]
\item underlying integer class $\oint A/2\pi \in\Z$ (the winding
number);
\item curvature representative $\dd A = 0$ on $S^1$ (since
$\dim S^1 = 1$, there is no $F$);
\item Wilson-line phase $\eta = \exp(\mathrm{i}\oint A) \in U(1)$.
\end{itemize}
This shows how differential cohomology naturally encodes both the
quantized winding and the continuous Wilson-line phase --- exactly
what is needed for the boundary projector classes of \GDOS{}.

\subsection{The exact sequence}
\label{app:diff-cohom-exact}

The differential cohomology fits into the four-term exact sequence
\begin{equation}
\label{eq:diff-exact}
0\to H^{k-1}(X;\R/\Z)\to \check H^k(X;\Z)
\xrightarrow{\rm curv} \Omega^k_{\rm closed,\,\Z}(X)
\to 0,
\end{equation}
where:
\begin{itemize}[leftmargin=2em]
\item $\mathrm{curv}: \check H^k\to\Omega^k_{\rm closed,\,\Z}$
extracts the curvature representative;
\item $\Omega^k_{\rm closed,\,\Z}(X)$ is the space of closed $k$-forms
with integer cohomology class;
\item the kernel of $\mathrm{curv}$ is the ``flat'' sub-group, isomorphic
to $H^{k-1}(X;\R/\Z)$.
\end{itemize}

\subsection{Pontryagin and Chern classes}
\label{app:Pontryagin-Chern}

The Pontryagin classes $p_i(E)\in H^{4i}(X;\Z)$ of a real vector bundle
$E\to X$ are integer cohomology classes constructed from the curvature
of any connection on $E$ via Chern--Weil theory:
\begin{equation}
\label{eq:Pontryagin-CW}
p_1(E) = -\frac{1}{8\pi^2}\,\tr(R\wedge R),
\end{equation}
where $R$ is the curvature of any connection on $E$. The class is
independent of the connection choice up to exact forms.

The Chern classes $c_i(F)\in H^{2i}(X;\Z)$ of a complex vector bundle
$F\to X$ are similarly constructed:
\begin{equation}
\label{eq:Chern-CW}
c_1(F) = \frac{1}{2\pi\mathrm{i}}\,\Tr(F_F), \qquad
c_2(F) = \frac{1}{8\pi^2}\bigl[\Tr(F_F\wedge F_F) - \Tr(F_F)\wedge\Tr(F_F)\bigr],
\end{equation}
where $F_F$ is the curvature of any connection on $F$.

The Pontryagin and Chern classes are integer cohomology classes that
can be lifted to differential cohomology via the Chern--Weil map:
$\check p_1 \in \check H^4(X;\Z)$, etc.

\subsection{Relevance to \BIsix{} and \CCI{}}
\label{app:diff-cohom-BI6}

The four-form anomaly polynomial $X_4$ of \BIsix{} is
\begin{equation}
\label{eq:X4-app-A}
X_4 = \tr R_{\Spinone}^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2,
\end{equation}
which combines Pontryagin (gravity), Chern (gauge), and squared first
Chern (Abelian) contributions. Its lift to differential cohomology
$\check x_4 \in \check H^4(\Msix;\Z)$ is the foundational object for
the Hopkins--Singer descent in Appendix~B.

\section{Hopkins-Singer machinery for \GDOS{}}
\label{app:Hopkins-Singer}

This appendix sketches the Hopkins--Singer construction adapted to
the 6D bulk of \GGTheory{}. The full mathematical content is in
\citet{HopkinsSinger2005}; we extract here only what is needed for the
Hopkins--Singer descent of $\check x_4$ to the boundary moduli used in
Theorem~\ref{thm:H2-theorem} of Part~IV.

\subsection{Differential function spaces}
\label{app:HS-function-spaces}

For a smooth manifold $X$ and integer $k\geq 0$, the differential
function space $\mathrm{Func}^k(X;\Z)$ is the space of continuous
maps $X\to K(\Z,k)$ (an Eilenberg--MacLane space) equipped with a
differential refinement. An element $f\in \mathrm{Func}^k(X;\Z)$ has
underlying topological data (the homotopy class of the continuous
map) plus differential data (a closed $k$-form representative of the
class).

Differential cohomology is then defined as the homotopy classes of
differential functions:
\begin{equation}
\label{eq:HS-diff-cohom}
\check H^k(X;\Z) \cong \pi_0\,\mathrm{Func}^k(X;\Z).
\end{equation}

\subsection{The descent map}
\label{app:HS-descent}

For a smooth fibration $\pi: X\to Y$ with closed fibers of dimension
$n$, the descent map for differential cohomology is
\begin{equation}
\label{eq:HS-descent-map}
\check\pi_!: \check H^k(X;\Z)\to \check H^{k-n}(Y;\Z),
\end{equation}
which integrates a differential cohomology class along the fibers.

For the 6D bulk $\pi: \Msix\to\Bdy$ projecting onto the
boundary moduli, the descent map sends $\check x_4 \in \check H^4(\Msix;\Z)$
to a differential cohomology class on $\Bdy$ (or, restricted to
strata, on each stratum). When the fiber dimension is $0$ (i.e.,
when restricting to a section of the projection), the descent
preserves cohomological degree.

\subsection{The $\eta$-invariant pairing}
\label{app:HS-eta}

The fundamental non-degeneracy property of the Hopkins--Singer descent
is the $\eta$-invariant pairing, which is a refinement of the
intersection pairing on integer cohomology. For two differential
cohomology classes $\check x \in \check H^k(X;\Z)$ and
$\check y \in \check H^{n-k+1}(X;\Z)$ on a closed oriented manifold
$X$ of dimension $n$, the $\eta$-invariant pairing is
\begin{equation}
\label{eq:eta-pairing}
\langle\check x, \check y\rangle_\eta \in \R/\Z.
\end{equation}
This pairing is non-degenerate in the sense that $\check x = 0$ iff
$\langle\check x, \check y\rangle_\eta = 0$ for all $\check y$.

\subsection{Application to \BIsix{} anomaly inflow}
\label{app:HS-BI6}

The Hopkins--Singer descent applied to the \CCI{} class
$\check x_4\in\check H^4(\Msix;\Z)$ gives a class
$\check\rho_{\rm bdy}(\check x_4)$ on the boundary moduli $\Bdy$,
which restricts to each stratum $\Sigma\subset\Bdy$ as a class in
$\check H^4(\Sigma;\Z)$. By Theorem~\ref{thm:H1-theorem} of Part~IV,
this restriction is $c_R\cdot\check p_1(\Sigma)$, where $c_R\in\mathbb Q_{>0}$
is the rational Killing-form coefficient.

The non-degeneracy theorem of $\mathcal H_2$
(Theorem~\ref{thm:H2-theorem}) follows from the $\eta$-invariant
non-degeneracy: if $\check x_4 \neq 0$, then
$\check\rho_{\rm bdy}(\check x_4) \neq 0$ on any stratum supporting a
non-trivial pairing.

\subsection{Forward to integral L-genus}
\label{app:HS-Lgenus}

For a closed oriented $4m$-manifold $\Sigma$, the integrated L-genus
\begin{equation}
\label{eq:HS-Lgenus}
\sigma(\Sigma) = \int_\Sigma L_m(p_1, \ldots, p_m) = \int_\Sigma L_m(c_R^{-1}\check\rho_{\rm bdy}(\check x_4), \ldots)
\end{equation}
is non-zero by the combination of Hirzebruch's signature theorem and
the Hopkins--Singer non-degeneracy. This is the technical content
underlying the strict Morse-positivity of \GFF{} via Theorem~\textbf{T3}.

\section{Morse theory on stratified spaces}
\label{app:Morse-stratified}

This appendix collects the basic facts about Morse theory used in
Parts~IV and V. The standard references are
\citet{Milnor1963,Schwarz1993} for smooth Morse theory and
\citet{GoreskyMacPherson1988} for the stratified case.

\subsection{Standard Morse theory}
\label{app:Morse-standard}

Let $M$ be a closed smooth $n$-manifold and $f:M\to\R$ a smooth
function. $f$ is a \emph{Morse function} if:
\begin{itemize}[leftmargin=2em]
\item all critical points $\{c \in M : \dd f(c) = 0\}$ are
non-degenerate, i.e., the Hessian $\Hess(f)(c)$ is non-singular;
\item the Morse index $\ind(c)\in\{0,1,\ldots,n\}$ is the number of
negative eigenvalues of $\Hess(f)(c)$.
\end{itemize}

The \emph{Morse polynomial} of $f$ is
\begin{equation}
\label{eq:Morse-poly-app}
\mathcal P_f(t) = \sum_{c\in\Crit(f)} t^{\ind(c)}.
\end{equation}
The Morse inequalities relate $\mathcal P_f(t)$ to the Poincar\'e
polynomial $\mathcal P_M(t)$ of $M$:
$\mathcal P_f(t) \geq \mathcal P_M(t)$ in the sense of polynomial
coefficients, with equality for ``perfect'' Morse functions.

\subsection{The height function on $\mathbb{CP}^{N-1}$}
\label{app:Morse-CPN}

The height function $f([\psi]) = \sum_a\lambda_a|\psi_a|^2 / \sum|\psi_a|^2$
on $\CPN$ is a perfect Morse function with critical points
$c_a = [\delta_{ab}]$ and indices $\ind(c_a) = 2(a-1)$. The Morse
polynomial is
\begin{equation}
\label{eq:Morse-poly-CPN}
\mathcal P_f^{\CPN}(t) = 1 + t^2 + t^4 + \cdots + t^{2(N-1)},
\end{equation}
which agrees with the Poincar\'e polynomial of $\CPN$.

\subsection{Stratified Morse theory}
\label{app:Morse-stratified-detail}

A \emph{Whitney stratified space} is a topological space $Z$ with a
decomposition $Z = \bigsqcup_a \Sigma_a$ into smooth manifolds
(strata) satisfying Whitney's regularity conditions
\citep{GoreskyMacPherson1988}. A \emph{stratified Morse function}
$f:Z\to\R$ is one whose restriction $f|_{\Sigma_a}$ is a Morse
function on each stratum, with appropriate compatibility at the
strata boundaries.

The Morse polynomial of a stratified Morse function is the sum over
all critical points across all strata. For the boundary moduli
$\Bdy = \prod_a \Sigma_a$ (a product of strata), the total Morse
polynomial factorizes:
\begin{equation}
\label{eq:Morse-poly-product}
\mathcal P_f^{\Bdy}(t) = \prod_a \mathcal P_{f|_{\Sigma_a}}^{\Sigma_a}(t).
\end{equation}

\subsection{Connecting trajectories and instanton actions}
\label{app:Morse-connecting}

Between two critical points $c_+, c_-$ with $\ind(c_+) - \ind(c_-) = 1$
in the smooth case (or $= 2$ in the Kähler-Hamiltonian case), the
gradient flow admits a moduli of \emph{connecting trajectories} from
$c_+$ to $c_-$. The action of each connecting trajectory is
\begin{equation}
\label{eq:trajectory-action}
S = \int_\gamma |\nabla f|\,\dd s = f(c_+) - f(c_-),
\end{equation}
saturating the Bogomolnyi bound.

For the height function on $\CPN$ with the Fubini--Study metric, the
action between adjacent critical points is $\pi k$ where the index
drops from $2k$ to $2(k-1)$. This is the result used in
Theorem~\ref{thm:instanton-action} of Part~V.


%% ---- moved here from Part VIII: cohort bookkeeping, not discussion ----
\section{Cross-paper coherence map}
\label{sec:coherence-map}
%% ====================================================================

The \GiFour{} framework of the present paper does not stand alone: it
imports its constitutional inputs from \citet{ArisakaA1Constitution},
its bulk geometry from \citet{ArisakaA2GG6}, its anomaly closure from
\citet{ArisakaA3BI6}, and its boundary-law projector structure from
\citet{ArisakaA5GDOS}; in turn it exports its four pillars to the
downstream papers in the program, principally
\citet{ArisakaA7Qi4} (the parallel \QiFour{} pillars),
\citet{ArisakaA9SM} (the 28-parameter codebook), and
\citet{ArisakaA10Cosmos} (boundary cosmology). This section makes the
imports and exports explicit by naming, for each cross-paper invocation,
the specific theorem number or section location where the referenced
content lives.

\paragraph{Note on version and Phase numbers.} Companion-paper versions
are independent revision tracks. Each row of the coherence tables below
records the version of the \emph{cited companion paper} as of the date
on the title page of the present paper (May 2026). When a companion
paper is later revised, the rows below should be re-walked against the
revised paper's actual internal theorem numbering before the present
paper is itself revised; vague pointers (e.g., ``the central trace-ratio
theorem of \BIsix{}'') are not acceptable, and every entry below names a specific theorem number, section, or appendix location.

\paragraph{Audit trail.} The internal theorem numbering of the cited companion papers, used to populate the tables below, is as follows:
\begin{itemize}[leftmargin=2em,itemsep=2pt]
\item \textbf{\AOne{}}. Four Axioms
\AX{1}--\AX{4} (LC, \GGEP{}, \OGGEP{}, \MSMF{}) in Sections~3--6; seven
Postulates \POS{1}--\POS{7} (GI, UEP, AF, CP, RG, OGN, MF) in
Sections~7--13; eight central theorems $\T{1}$--$\T{8}$ in the body;
the Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ as central
boundary identity.
\item \textbf{\ATwo{}}. Eight principal theorems
including Theorem~1 dimensional minimality
($d_{\rm bulk}=6$); Theorem~3 canonical 6D metric
$ds_6^2 = e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$;
Theorem~4 four-component \BIsix{} coefficient vector
$(a,b,c,d)=(1,-1,-2/5,-2/5)$; Theorem~6 \OGNfive{} Diophantine cascade;
Theorem~8 Arisaka equation and Quadruple Gauge Locking.
\item \textbf{\AThree{}}. Theorem~9.1 \OGNfive{}
trace-ratio lock $|a/c|=|a/d|=5/2$; Theorem~4.1 global anomaly~\citep{WittenAnomalies}
freedom; Section~3 Chern--Simons construction of $H_3$ from \TGG{}
data; \S~8.7 ``\BIsix{} as the \GGEP{} completion of GS: the
geometrization reading'' (the supportive treatment of the
Green--Schwarz legacy invoked in the present paper).
\item \textbf{\AFour{}}. Historical chain from
Euclid through Newton, Riemann, Maxwell, Einstein to \GGTheory{};
parameter-audit framework recording every dimensional constant as a
conversion factor and every dimensionless integer as a Diophantine
output.
\item \textbf{\AFive{}}. \GDOS{} boundary law
$\cD_{\OGGEP} = \cR \circ \mathrm{Tr}_\env$ (CPTP form); the \SIP{}
construction; the OH+IH=EIC framework on the boundary; the
\OGGEP{}-projection from $\Msix$ to $\Mfour$ as the foundational
dynamical move.
\item \textbf{\ASeven{}, \AEight{}, \ANine{}, \ATen{}, \PaperPOne{}, \PaperCOne{}}
consume the \GRIP{} cascade downstream --- the \QiFour{} pillars, the
quantum-measurement reading, the Standard-Model parameter block, the
cosmological descent, the collider phenomenology, and the dark-matter
sector respectively; cross-citations to them name registered theorem
identifiers.
\end{itemize}

\subsection{Inward map: imports into \ASix{} from the foundation block}
\label{ssec:inward-map}

The following longtable lists, for each theorem-level fact that
\GiFour{} imports from a foundation-block paper, the specific
companion-paper location and the role the imported fact plays in the
present manuscript.

\begin{footnotesize}
\begin{longtable}{p{0.30\linewidth} p{0.27\linewidth} p{0.35\linewidth}}
\caption[Inward coherence map.]{Inward coherence map: \ASix{} imports from A1, A2, A3, A5. Each row names one imported fact, the exact place in the companion paper where it is established, and what this paper does with it. The point of the table is auditability: every fact taken on trust from elsewhere appears here once, so a reader can check that nothing is borrowed and then cited back as support for its own source.}
\label{tab:inward-map}\\
\toprule
\textbf{Imported fact} & \textbf{Companion-paper location} & \textbf{Role in \ASix{}}\\
\midrule
\endfirsthead
\multicolumn{3}{l}{\emph{(continued from previous page)}}\\
\toprule
\textbf{Imported fact} & \textbf{Companion-paper location} & \textbf{Role in \ASix{}}\\
\midrule
\endhead
\bottomrule
\multicolumn{3}{r}{\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
Four Axioms \AX{1}--\AX{4} (LC, \GGEP{}, \OGGEP{}, \MSMF{}) &
\AOne{}, Sections~3--6 \citep{ArisakaA1Constitution} &
Constitutional source of \TGG{} (\GGEP{}), \CCI{} (\AF{}+\GI{}), \GFF{} (\MSMF{}+\OGGEP{}), and \GRIP{} (\OGGEP{}+\RG{}).\\
Seven Postulates \POS{1}--\POS{7} (GI, UEP, AF, CP, RG, OGN, MF) &
\AOne{}, Sections~7--13 \citep{ArisakaA1Constitution} &
Constitutional source of the Ward package of \textbf{TGG-ST7}, the integrality of $[X_4]$ via \AF{}, and the cascade structure via \RG{}.\\
Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ &
\AOne{}, central boundary identity \citep{ArisakaA1Constitution} &
Establishes the four-component boundary current structure that the Master Conservation Law of T1.3 enforces; underlies the L12 ``empirically preferred'' framing of $\UoneQi$.\\
6D bulk metric $ds_6^2 = e^{-2r}\eta_{\mu\nu}dx^\mu dx^\nu + dr^2 + d\theta^2$ &
\ATwo{}, Theorem~3 (canonical metric) \citep{ArisakaA2GG6} &
Bulk arena on which the composite connection $\Atot$ of T1.1 is defined.\\
$d_{\rm bulk}=6$ from the magic-dimension ladder; $\Spinone \cong \mathrm{SL}(2,\HH)$ &
\ATwo{}, Theorem~1 (dimensional minimality) \citep{ArisakaA2GG6} &
Fixes the bulk dimension; gates the existence of the quaternionic Lorentz double cover used in the Section on \TGG{} and the magic-dimension ladder of A6.\\
Four-component \BIsix{} coefficient vector $(a,b,c,d)=(1,-1,-2/5,-2/5)$ &
\ATwo{}, Theorem~4; \AThree{}, Theorem~9.1 (trace-ratio lock) (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}) &
Input to the algebraic content of \CCI{}; the unique solution of \MSMF{} closure used in T2.1.\\
\OGNfive{} Diophantine cascade $(M,N,L;J,K)=(1,3,5;7,35)$ &
\ATwo{}, Theorem~6 (Diophantine cascade); \AThree{}, Theorem~10.2 (\citealp{ArisakaA2GG6}; \citealp{ArisakaA3BI6}) &
Direct input to T2.1; sets the lower bound $N \geq 3$ used in T4.1.\\
\BIsix{} closure $\dd H_3 = X_4$ &
\AThree{}, Section~6 (Chern--Simons/Chern--Weil construction); Theorem~6.3 \citep{ArisakaA3BI6} &
Bulk-side closure that the algebraic content of \CCI{} (T2.1) makes operational.\\
Hopkins--Singer integrality of $[X_4] \in H^4(\Msix;\Z)$ &
\AThree{}, Section~3, with reference to \citet{HopkinsSinger2005} &
Cohomological backbone for the Morse-positivity argument of T3.1.\\
GS-as-\GGEP{} ``geometrization reading'' &
\AThree{}, \S~8.7 \citep{ArisakaA3BI6} &
Supportive framing of the Green--Schwarz legacy invoked in the present paper, throughout the intro and the bibliography Relevance paragraph for GreenSchwarz1984.\\
Global anomaly freedom of \BIsix{} &
\AThree{}, Theorem~4.1 \citep{ArisakaA3BI6} &
Justifies that the four-form $X_4$ on $\Msix$ admits a globally well-defined Chern--Simons primitive, which is the algebraic content of \textbf{CCI-ST3}.\\
\GDOS{} boundary law $\cD_{\OGGEP} = \cR \circ \mathrm{Tr}_\env$ as CPTP &
\AFive{}, throughout \citep{ArisakaA5GDOS} &
Boundary-projection construction that underlies the operational layer T3.2 (\GFF{} thermodynamic monotonicity) and T4.2 (\GRIP{} runtime gauge-invariance).\\
\SIP{} (System Instance Packet) construction &
\AFive{}, throughout \citep{ArisakaA5GDOS} &
Provides the local-system bookkeeping required for the boundary stress-tensor decomposition $\cT^{\mu\nu}_{\TGG} = T_{\rm matter} + T_{\rm gauge} + X_{\rm GS} + T_{\rm KK}$ of T1.2.\\
\OGGEP{}-projection from $\Msix$ to $\Mfour$ &
\AFive{}, throughout \citep{ArisakaA5GDOS} &
The KK zero-mode reduction that takes the bulk action of \textbf{TGG-ST2} to the boundary action of T1.2.\\
\end{longtable}
\end{footnotesize}

\subsection{Outward map: exports from \ASix{} to downstream papers}
\label{ssec:outward-map}

The following longtable lists, for each theorem-level fact that
\GiFour{} exports to a downstream paper, the present-paper theorem
label or section location and the downstream paper's anticipated use.

\begin{footnotesize}
\begin{longtable}{p{0.30\linewidth} p{0.27\linewidth} p{0.35\linewidth}}
\caption[Outward coherence map.]{Outward coherence map: \ASix{} exports to A7, A8, A9, A10, P1, C1. Each row names one result established here, the place in this paper where it is proved, and the downstream paper that consumes it and for what. Read together with the inward map, the pair states this paper's entire dependency surface in both directions.}
\label{tab:outward-map}\\
\toprule
\textbf{Exported \ASix{} fact} & \textbf{A6 location} & \textbf{Downstream paper and anticipated use}\\
\midrule
\endfirsthead
\multicolumn{3}{l}{\emph{(continued from previous page)}}\\
\toprule
\textbf{Exported \ASix{} fact} & \textbf{A6 location} & \textbf{Downstream paper and anticipated use}\\
\midrule
\endhead
\bottomrule
\multicolumn{3}{r}{\emph{continued on next page}}\\
\endfoot
\bottomrule
\endlastfoot
\TGG{} existence and uniqueness on $\Msix$ (T1.1, T1.2) &
Theorem~\ref{thm:TGG-existence-uniqueness} (T1) &
\ASeven{}: parallel \QiFour{} pillars \QiQUA{}, \QiEP{} defined on the same composite-connection structure; \AEight{}: \TGG{} stress tensor as the geometric input to the 4D measurement framework; \ANine{}: gauge content of $\Atot$ used to identify the 28-parameter codebook.\\
Master Conservation Law $\nabla_\mu \cT^{\mu\nu}_{\TGG} = 0$ (T1.3) &
Theorem~\ref{thm:TGG-existence-uniqueness} (T1.3) &
\ANine{}: boundary stress-tensor conservation as the underlying gauge-invariance identity; \ATen{}: bulk-to-boundary stress flow as the geometric origin of cosmological energy-momentum balance.\\
Compatibility proposition $\mathfrak{C}_{\TGG{}}$ &
Section~\ref{ssec:T1-dual-layer}, eq.~\eqref{eq:CTGG-compat} &
\ASeven{}: parallel compatibility proposition $\mathfrak{C}_{\rm Qi}$ for the four Qi-pillars; \AEight{}: dual-layer template for the algebraic-vs-operational structure of measurement.\\
\OGNfive{} uniqueness from \MSMF{} closure (T2.1) &
Theorem~\ref{thm:CCI-OGN5-uniqueness} (T2.1) &
\ANine{}: integer ledger $(1,3,5;7,35)$ as the input to the closed-form derivation of the 28 Standard-Model parameters.\\
Master Conservation iff Ward + \PBIfour{} (T2.2) &
Theorem~\ref{thm:T2.2-master-conservation} (T2.2) &
\ANine{}: four-current Ward-identity package as the QFT-level conservation discipline; \ATen{}: boundary stress-tensor conservation under the 6D KK reduction.\\
Compatibility proposition $\mathfrak{C}_{\CCI{}}$ &
Section~\ref{ssec:T2.1-statement}, eq.~\eqref{eq:CCCI-compat} &
\ASeven{}: parallel compatibility proposition for \QiCON{} information-conservation.\\
\GFF{} Morse-positivity from \CCI{} via Hopkins--Singer descent (T3.1) &
Theorem~\ref{thm:GFF-positivity-from-CCI} (T3.1) &
\ASeven{}: variational backbone for \QiCON{} (information-theoretic \CCI{} analogue); \ANine{}: Morse-cascade structure on lepton- and quark-mass moduli used in the seesaw-mechanism derivation.\\
Thermodynamic monotonicity from \GDOS{} (T3.2) &
Theorem~\ref{thm:T3.2-thermodynamic} (T3.2) &
\ATen{}: boundary-entropy production as the origin of the cosmological arrow of time; \ASeven{}: \QiEP{} entropy law for the information-pillar dynamics.\\
Compatibility proposition $\mathfrak{C}_{\GFF{}}$ &
Section~\ref{ssec:T3.1-statement}, eq.~\eqref{eq:CGFF-compat} &
\ASeven{}: parallel compatibility proposition for the algebraic-vs-thermodynamic content of \QiEP{}.\\
\GRIP{} cascade closure forces $N \leq 3$ (T4.1, combined with T2 lower bound) &
Theorem~\ref{thm:GRIP-existence-closure} (T4.1) &
\ANine{}: hard upper bound $N = 3$ for the family count; \PaperPOne{}: cascade structure on the family-projector moduli used to predict the $\sim 3.85$\,TeV KK resonance cluster.\\
Runtime gauge-invariance iff \PBIfour{} compensator extends (T4.2) &
Theorem~\ref{thm:T4.2-runtime} (T4.2) &
\ASeven{}: \QiHAL{} family (\MePMoS{}, \NHT{}, HAL, \BFFT{}) as the information-side analogue of \GRIP{} runtime invariance; \PaperPOne{}: smooth-extension hypothesis as the technical condition on the KK lock cluster prediction; \PaperCOne{}: cascade-runtime structure on the $\UoneQi$ sector.\\
Compatibility proposition $\mathfrak{C}_{\GRIP{}}$ &
Section~\ref{ssec:T4.1-statement}, eq.~\eqref{eq:CGRIP-compat} &
\ASeven{}: parallel compatibility proposition for \QiHAL{}.\\
Geometric First Law of \GFF{} ($F = U - TS$ form) &
Theorem~\ref{thm:GFF-first-law} &
\ASeven{}: free-energy ledger for the information-pillar dynamics; \ATen{}: thermodynamic backbone of the boundary cosmology.\\
\GFF{}-stratified Morse landscape on $\Bdy$ &
Section~\ref{sec:GFF-T3} and Appendix &
\AEight{}: critical-point catalog as the structural basis of the discrete measurement outcomes; \ANine{}: lepton- and quark-mass strata.\\
\GRIP{} instanton-action ledger $S_{[2k]\to[2(k-1)]} = \pi k$ &
Theorem~\ref{thm:instanton-action} &
\ANine{}: cascade-step structure for the seesaw mechanism; \PaperPOne{}: cascade-step structure for the KK lock cluster; \PaperCOne{}: cascade-step structure for the Kaxion oscillation frequency.\\
\GiFour{} / \QiFour{} pairing framework &
Part VI (Synthesis) and Section~\ref{sec:bridge-Qi4} &
\ASeven{}: definitional template for the four information pillars \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}, each pillar paired with one of \TGG{}, \CCI{}, \GFF{}, \GRIP{}.\\
\end{longtable}
\end{footnotesize}

The two maps together establish the position of \GiFour{} in the
\GGTheory{} program: the constitutional content flows downward from
A1, the bulk-side technical content flows in from A2 and A3, the
boundary-projection structure flows in from A5, and the four pillars
then flow outward to every downstream paper that addresses the
4D-physics, cosmology, or predictions branches of the program. The
map is stated against the theorem numbering the companion papers carry,
and a reader following a citation into one of them should expect the
numbering, not the content, to be what shifts.

%% ====================================================================
\section{Global naming convention for cross-cohort citation}
\label{sec:naming-convention}

Beginning with \citet{ArisakaA2GG6} and extended across the cohort, each
load-bearing item of the \GGTheory{} program carries a globally unique
identifier of the form \emph{Prefix-Suffix-N}, where the Prefix names the
item type and the Suffix names the paper:
\begin{itemize}[leftmargin=2em, itemsep=2pt]
\item \textbf{T-Suffix-N} for the $N$th Theorem (e.g., T-GRIP-4 is
Theorem~T4 of the present paper, the \GRIP{} cascade closure forcing
$N\le 3$).
\item \textbf{L-Suffix-N} for the $N$th Lemma.
\item \textbf{C-Suffix-N} for the $N$th Corollary.
\item \textbf{D-Suffix-N} for the $N$th Definition.
\item \textbf{R-Suffix-N} for the $N$th Remark.
\item \textbf{P-Suffix-N} for the $N$th Prediction.
\item \textbf{O-Suffix-N} for the $N$th Objection (e.g., O-GRIP-1 is the
first Common Objection of the present paper).
\item \textbf{F-Suffix-N} for the $N$th Falsifier (e.g., F-GRIP-4 is the
generation gate-action-ordering falsifier).
\item \textbf{I-Suffix-N} for the $N$th Innovation (e.g., I-GRIP-8 is the
$N\le 3$ family-count theorem above).
\item \textbf{PS-Suffix-N} for the $N$th Problem-Solved entry
(\S\ref{sec:problems-solved}).
\end{itemize}
The canonical Suffix list for the twenty-two cohort papers is: \emph{Constitution} (A1),
\emph{GG6} (A2), \emph{BI6} (A3), \emph{Euclid} (A4), \emph{GDOS} (A5),
\emph{GRIP} (A6, this paper), \emph{Qi4} (A7), \emph{QM} (A8), \emph{SM} (A9),
\emph{Cosmos} (A10), \emph{Origin} (A11), together with the phenomenology
preprints \emph{KK} (P1), \emph{N3} (P2), \emph{Electron} (P3), the
foundations preprints \emph{Real} (P4) and \emph{CPT} (P5), and the
consequence preprints \emph{Kaxion} (C1), \emph{Hubble} (C2), \emph{BH}
(C3), and the white papers \emph{Symmetry} (W1) and \emph{Holonomy} (W2). Inside each paper the internal numbering (``Theorem~T1'',
``Theorem~T2'', \ldots) continues to be supplied locally; the global
identifier is the same item viewed from outside the paper, so that a
cross-cohort citation such as ``\ASix{} consumes T-GDOS-3 and produces
T-SM-12'' is unambiguous and search-friendly. The four axioms
(AX-1\,\ldots\,AX-4) and seven postulates (POS-1\,\ldots\,POS-7) of
\citet{ArisakaA1Constitution} keep their existing cohort-wide names, since
those are already paper-specific and unique.


\section{Detailed worked examples}
\label{app:detailed-examples}

This appendix collects detailed explicit computations that were
sketched in the main text.

\subsection{Spin(1,5) generators and structure constants in detail}
\label{app:spin15-detail}

The 15 generators of $\mathfrak{spin}(1,5)$ in the antisymmetric
basis $M^{ab} = -M^{ba}$, $a,b\in\{0,1,2,3,4,5\}$, satisfy the
commutation relations
\begin{equation}
\label{eq:Mab-detail}
[M^{ab}, M^{cd}] = \mathrm{i}\bigl(\eta^{ac}M^{bd}
- \eta^{ad}M^{bc} - \eta^{bc}M^{ad} + \eta^{bd}M^{ac}\bigr),
\end{equation}
with $\eta = \mathrm{diag}(+,-,-,-,-,-)$. The 15 generators decompose
as 10 rotations $J^{ab}$ ($1\leq a<b\leq 5$) plus 5 boosts
$K^a = M^{0a}$ ($a=1,\ldots,5$).

\subsection{Quaternionic doublet matrix representation}
\label{app:quat-doublet-matrix}

The fundamental representation of $\SLtwoH$ on $\HH^2$ is realized by
$2\times 2$ quaternionic matrices acting on column vectors
$(q_1, q_2)^T$ with $q_i \in \HH$. Writing
$q = \alpha + j\beta$ with $\alpha, \beta\in\C$, one identifies
$\HH^2 \cong \C^4$ as a real vector space, and the $\SLtwoH$ action
becomes a specific real representation of dimension 8 (matching the
$2^{6/2-1} = 8$-dimensional six-dimensional Dirac spinor).

\subsection{\CCI{} integrality check on $\mathcal M_6$}
\label{app:CCI-integrality-detail}

For the 6D bulk with \OGNfive{} ledger $(1,3,5;7,35)$, the
integrals of the four sub-curvature 4-forms over the fundamental
4-cycle $C_4$ are:
\begin{align}
\frac{1}{8\pi^2}\int_{C_4}\tr R_{\Spinone}^2 &= K = 35,\\
\frac{1}{8\pi^2}\int_{C_4}\Tr F_5^2 &= 2L = 10,\\
\frac{1}{4\pi^2}\int_{C_4}F_B^2 &= M^2 = 1,\\
\frac{1}{4\pi^2}\int_{C_4}F_{Qi}^2 &= N^2 = 9.
\end{align}
Combining with the \textbf{CCI-ST2} coefficients:
\begin{equation}
\label{eq:integrality-check}
\frac{1}{8\pi^2}\int_{C_4}X_4 = 35 - 10 - \frac{2}{5}\cdot 1 - \frac{2}{5}\cdot 9
= 25 - \frac{20}{5} = 25 - 4 = 21,
\end{equation}
which is an integer. This confirms the integrality
$[X_4]\in H^4(\Msix;\Z)$ for the \OGNfive{} ledger.

\subsection{\GFF{} Morse data on $\mathbb{CP}^2$}
\label{app:GFF-CP2-detail}

For the family-projector stratum $\Sigma_{\rm fam} = \CPtwo$ with
height function $f([\psi_1:\psi_2:\psi_3]) =
(\lambda_1|\psi_1|^2+\lambda_2|\psi_2|^2+\lambda_3|\psi_3|^2)/\sum|\psi_a|^2$,
$\lambda_1<\lambda_2<\lambda_3$:

\begin{table}[H]
\centering\small
\caption[Morse data on $\CPtwo$, critical point by critical point.]{The Morse data of the complex projective plane, critical point
by critical point, with the Standard-Model generation each one carries.
There are three critical points, of index 0, 2 and 4, in the order the
Morse function ranks them; the generation assignment follows that order
and so does the Yukawa hierarchy, smallest to largest. This table is the
arithmetic behind the claim that three generations are a consequence of
the geometry rather than an input: the count is the number of critical
points, and nothing in the construction was free to make it three.}
\label{tab:GFF-CP2-detail}
\begin{tabular}{cccccc}
\toprule
Crit.\ point & Coordinate & Index & $f$-value & SM gen & Yukawa $y$\\
\midrule
$c_1$ & $[1:0:0]$ & $0$ & $\lambda_1$ & $1^{\rm st}$: $e, u, d$ & smallest\\
$c_2$ & $[0:1:0]$ & $2$ & $\lambda_2$ & $2^{\rm nd}$: $\mu, c, s$ & intermediate\\
$c_3$ & $[0:0:1]$ & $4$ & $\lambda_3$ & $3^{\rm rd}$: $\tau, t, b$ & largest\\
\bottomrule
\end{tabular}
\end{table}

The cascade is $c_3 \xrightarrow{\;e^{-2\pi}\;} c_2 \xrightarrow{\;e^{-\pi}\;} c_1$
with total action $3\pi$, exactly saturating the \BIsix{} budget.

\subsection{\GRIP{} cascade timing on $\mathbb{CP}^2$}
\label{app:GRIP-CP2-detail}

The cascade lifetimes on $\CPtwo$, in cascade-action units, are:
\begin{align}
\tau_{c_3 \to c_2} &\propto e^{+S_1} = e^{2\pi} \simeq 535,\\
\tau_{c_2 \to c_1} &\propto e^{+S_2} = e^{\pi} \simeq 23.
\end{align}
The ratio $\tau_{c_3}/\tau_{c_2} \simeq 23$ is the cascade-only
prediction. After multiplying by the phase-space factor
$(m_3/m_2)^5 \simeq (16.8)^5 \simeq 1.34\times 10^6$, the empirical
$\tau_\mu/\tau_\tau \simeq 7.6\times 10^6$ is recovered to within
order unity.

%% ====================================================================
\section{The free-energy principle: a bridge to life sciences and neuroscience}
\label{app:free-energy-bridge}

\textbf{Reading goal.} This appendix bridges the
\textbf{GFF-ST6} thermodynamic content of \GGTheory{} to the
free-energy principle of theoretical neuroscience and the
non-equilibrium thermodynamics of biological systems. The intended
audience is a first-year graduate student in theoretical biology or
computational neuroscience who is familiar with statistical
mechanics but not differential geometry.

\subsection{From Helmholtz free energy to variational free energy}

The Helmholtz free energy $F = U - TS$ of equilibrium statistical
mechanics is the unique function whose minimization governs the
equilibrium of a system in contact with a heat bath at fixed
temperature $T$. Spontaneous processes always decrease $F$ (the
second law), and the equilibrium configuration is the unique global
minimum.

In Bayesian inference, an analogous quantity is the
\emph{variational free energy}
\begin{equation}
\label{eq:F-var}
F_{\rm var}[q] \;=\; \mathbb E_q[\log q(s) - \log p(s, o)]
\;=\; D_{\rm KL}(q\,\|\,p(\cdot|o)) \;-\; \log p(o),
\end{equation}
where $q(s)$ is a candidate posterior distribution over hidden
states $s$ given observed data $o$, and $p(s, o)$ is the joint
generative model. Minimizing $F_{\rm var}$ over $q$ approximates
the true posterior $p(s|o)$ (the variational principle of
\citealp{HintonVanCamp1993}).

In neuroscience, Friston's \emph{free-energy principle}
\citep{Friston2010,FristonFreeEnergy2023} proposes that the brain
itself is an inference engine that minimizes a generalized
$F_{\rm var}$ over its internal generative model of the sensory
world. This single principle is claimed to unify perception,
action, learning, and homeostasis.

\subsection{How \GFF{} subsumes Friston's free energy}

The \GFF{} of \GGTheory{} is the Helmholtz-type free energy on the
boundary moduli space $\Bdy$ of \GDOS{}. Restricting $\Bdy$ to a
specific stratum corresponding to neural representations of an
external world gives Friston's variational free energy as a special
case.

More precisely:
\begin{itemize}[leftmargin=2em]
\item The \GGTheory{} boundary effective temperature $T_{\GFF}\sim
M_{\rm KK}$ corresponds in neural systems to a thermal/noise scale
set by metabolic energy availability.
\item The \GGTheory{} internal energy $U_{\GFF}$ corresponds in
neural systems to the negative log-likelihood of sensory input
given the generative model.
\item The \GGTheory{} entropy $S_{\GFF}$ corresponds in neural
systems to the entropy of the posterior distribution over hidden
states.
\item The \GGTheory{} Lindblad CPTP structure corresponds in neural
systems to the irreversible, dissipative dynamics of synaptic
plasticity and neural firing.
\end{itemize}

The \emph{Geometric First Law} of Theorem~\ref{thm:GFF-first-law}
guarantees that the algebraic Morse cascade and the thermodynamic
free-energy minimization are the same dynamical process. In
neuroscience terms, this means that the discrete catalog of
``percept attractors'' in the brain (Morse critical points of the
neural \GFF{}) corresponds to the thermodynamic equilibria of the
free-energy minimization.

\subsection{Empirical predictions for biological systems}

The dual-layer architecture of \GFF{} produces specific empirical
predictions for biological systems:
\begin{enumerate}[leftmargin=2em,label=(\arabic*)]
\item \textbf{Equipartition for biological fluctuations.} Near a
biological steady state, the variance of any fluctuating biological
observable is
$\langle\delta X^2\rangle = T_{\rm bio}\,\chi_X$, where $T_{\rm bio}$
is the metabolic temperature and $\chi_X$ is the susceptibility.
This is the GG analogue of equipartition.
\item \textbf{Onsager reciprocity in biological networks.} The
cross-coupling matrix between biological fluxes (e.g., metabolic
fluxes in coupled enzymatic networks) must be symmetric:
$L_{ij} = L_{ji}$. Empirical violations would falsify the
Lindblad/CPTP structure of the underlying \GDOS{} flow.
\item \textbf{Finite cascade lifetimes.} Biological cascades
(signaling networks, gene-regulatory networks) should exhibit
generation gate-ordered widths of the form $\Gamma\propto\e^{-\pi k}$ for
some constant $K$ depending on the system, exactly as in
\textbf{GRIP-ST7}.
\end{enumerate}

These predictions provide concrete empirical bridges from the
fundamental physics of \GGTheory{} to the laboratory phenomena of
biology and neuroscience. Confirmation across multiple platforms
would provide strong empirical support for the universal
applicability of the dual-layer \GG{} framework.

\subsection{Caveat: domain of applicability}

It is important to caveat that the \GGTheory{} predictions for
biological systems are \emph{universal kinematic relations}, not
\emph{specific dynamical predictions}. The framework predicts that
free-energy minimization, Onsager reciprocity, equipartition, and
finite cascade lifetimes will hold for \emph{any} biological system
that is fundamentally a boundary projection of the \GG{} bulk. It
does not predict the specific time constants, susceptibilities, or
rate matrices --- these depend on the particular biological system.

The empirical content for biology is therefore the universal
\emph{form} of these laws (e.g., equipartition $\langle\delta X^2\rangle
\propto T$, Onsager $L_{ij} = L_{ji}$, gate factor $\Gamma\propto\e^{-\pi k}$),
not the specific values. This is analogous to how thermodynamics
predicts $PV = nRT$ universally without specifying the particular
gas under study.

%% ====================================================================
%% APPENDIX J: EXPLICIT DERIVATION OF THE MODIFIED TOV
%% ====================================================================
\section{Hopkins-Singer descent: explicit worked examples}
\label{app:HS-worked}

\textbf{Reading goal.} This appendix illustrates the Hopkins--Singer
descent of differential cohomology classes from the bulk to the
boundary moduli, with three explicit worked examples. Intended
audience: a first-year graduate student in mathematical physics
who has read Appendix~A and is comfortable with differential
forms but not with abstract category-theoretic constructions.

\subsection{The descent procedure: a concrete recipe}

Given a differential cohomology class $\check x\in\check H^k(X;\Z)$
on a smooth fibration $\pi\colon X\to Y$ with closed fibers of
dimension $n$, the descent class $\check\pi_!(\check x)\in
\check H^{k-n}(Y;\Z)$ is computed in three steps:
\begin{enumerate}[label=(\arabic*),leftmargin=2.2em]
\item \textbf{Choose a connection representative.} Pick a curvature
representative $\omega\in\Omega^k_{\rm closed}(X)$ in the class
$\check x$ (i.e., $\mathrm{curv}(\check x) = \omega$).
\item \textbf{Integrate along fibers.} For each $y\in Y$, integrate
$\omega$ over the fiber $\pi^{-1}(y)$:
\begin{equation*}
\widetilde\omega(y) = \int_{\pi^{-1}(y)}\omega\;\in\;\Omega^{k-n}(Y).
\end{equation*}
The form $\widetilde\omega$ is closed (by Stokes and the closure of
$\omega$).
\item \textbf{Lift back to differential cohomology.} The integer
class $[\widetilde\omega] = \pi_![\omega]$ pairs canonically with a
differential cohomology refinement
$\check\pi_!(\check x)\in\check H^{k-n}(Y;\Z)$.
\end{enumerate}

\subsection{Worked example 1: $S^2$-bundle over $S^2$}

Consider the trivial bundle $X = S^2\times S^2$ with projection
$\pi\colon X\to Y = S^2$ projecting onto the second factor. Each
fiber is $\pi^{-1}(y)\cong S^2$ of dimension 2.

Take $\check x\in \check H^4(X;\Z)$ represented by
$\omega = \omega_1\wedge\omega_2$, where $\omega_i$ is the area
2-form on the $i$-th $S^2$ normalized by $\int_{S^2}\omega_i = 1$.
Then $\omega\in\Omega^4(X)$ has integer class
$[\omega] = 1\in H^4(X;\Z)\cong\Z$.

Integrating along the first-factor fibers:
\begin{equation*}
\widetilde\omega(y) = \int_{\pi^{-1}(y)}\omega_1\wedge\omega_2
= \omega_2(y)\cdot\int_{S^2}\omega_1 = \omega_2(y).
\end{equation*}
Therefore the descent gives $\check\pi_!(\check x)\in \check H^2(Y;\Z)$
with curvature $\omega_2$ and integer class $1$.

\subsection{Worked example 2: 6D bulk to 4D boundary}

Now consider the 6D bulk
$\Msix = \R^{1,3}\times I_r\times S^1_\theta$ of \GGTheory{}, with
projection $\pi\colon\Msix\to M_4 = \R^{1,3}$ integrating over the
two non-spatial dimensions $(r, \theta)$ --- the scale dimension and the gauge dimension. Each fiber has dimension 2.

Take the \CCI{} 4-form
\begin{equation*}
X_4 = \tr R_{\Spinone}^2 - \Tr F_5^2 - \tfrac{2}{5}F_B^2 - \tfrac{2}{5}F_{Qi}^2,
\end{equation*}
which has integer class $[X_4]\in H^4(\Msix;\Z)$ (Theorem~\textbf{T2.1}).
Integrating along the $(r, \theta)$ fibers:
\begin{equation*}
\widetilde X_2(x) = \int_{(r, \theta)}X_4(x, r, \theta)
\;\in\;\Omega^2(M_4),
\end{equation*}
which is the boundary 2-form $\widetilde X_2$ used in
Section~\ref{ssec:descent-map}. This is the operational
definition of the \PBIfour{} compensator field $C_\mu$ via
$\dd C = \widetilde X_2$ (Definition~\ref{def:PBI4-compensator}).

\subsection{Worked example 3: descent to a stratum of $\Bdy$}

The boundary moduli space $\Bdy$ is stratified. Consider the
family-projector stratum $\Sigma_{\rm fam} = \CPN$ inside $\Bdy$.
The Hopkins--Singer descent of $\check x_4 \in \check H^4(\Msix;\Z)$
to $\Sigma_{\rm fam}$ is computed by integrating $X_4$ over the
fiber of the bundle $\pi^{-1}(\Sigma_{\rm fam})\to\Sigma_{\rm fam}$.

For $N = 3$ ($\Sigma_{\rm fam} = \CPtwo$), the descent gives
\begin{equation*}
\check\pi_!(\check x_4)\bigg|_{\CPtwo} = c_R\cdot\check p_1(\CPtwo)
\;\in\;\check H^4(\CPtwo;\Z),
\end{equation*}
where $c_R\in\mathbb Q_{>0}$ is the Killing-form coefficient (this is
Theorem~\ref{thm:H1-theorem} of Part~IV). The class $\check p_1$ is
the differential first Pontryagin class of $\CPtwo$, which on
$\CPtwo$ equals $3\cdot[\omega_{\rm FS}]^2$ where $\omega_{\rm FS}$
is the Fubini--Study form. This non-vanishing class is what makes
the alternating Morse-index sum on $\CPtwo$ strictly positive
($a^+ - a^- = \sigma(\CPtwo) = 1 > 0$), confirming the strict
Morse-positivity of \textbf{T3.1}.

\subsection{Why the descent preserves integrality}

A subtle point: the descent map $\check\pi_!$ preserves integrality
because the integral of a closed form over a closed fiber is itself
an integer (when the input form has integer cohomology class). This
is a consequence of Stokes's theorem and the absence of boundary
terms when the fiber is closed.

For the \GGTheory{} descent from $\Msix$ to $M_4$, the closed-fiber
condition is satisfied by the compactness of the extra dimensions (the radial interval $I_r$ has IR/UV boundaries that
are compactified by the \GG{} IR boundary construction; the angular
$S^1_\theta$ is closed by definition). Therefore the boundary 2-form
$\widetilde X_2$ has integer cohomology class on $M_4$, which is
the precise condition needed for the \PBIfour{} compensator $C_\mu$
to exist globally without anomalies.

%% ====================================================================
%% APPENDIX L: MORSE THEORY ON CP^N WORKED EXAMPLES
%% ====================================================================
\section{Morse theory on $\CPN$: explicit worked examples}
\label{app:Morse-CPN-worked}

\textbf{Reading goal.} This appendix carries out the Morse theory
on $\CP^{N-1}$ in full pedagogical detail for $N=2, 3, 4$. Intended
audience: a first-year graduate student who has seen the
two-sphere $S^2$ Morse picture (the height function on a sphere)
but not the projective generalization.

\subsection{The height function on $\CPN$}

Recall the projective space $\CPN = \{[\psi_1:\psi_2:\cdots:\psi_N]\}$
with the equivalence
$[\psi_1:\cdots:\psi_N] \sim [\lambda\psi_1:\cdots:\lambda\psi_N]$
for $\lambda\in\C^*$. The complex dimension is $N-1$, real
dimension $2(N-1)$.

The \emph{height function} on $\CPN$ is defined by
\begin{equation}
\label{eq:height-CPN-detail}
f([\psi]) = \frac{\sum_{a=1}^N \lambda_a |\psi_a|^2}{\sum_{a=1}^N |\psi_a|^2},
\qquad \lambda_1 < \lambda_2 < \cdots < \lambda_N.
\end{equation}
(In coordinates this is well-defined on $\CPN$ because both numerator
and denominator scale by $|\lambda|^2$ under the equivalence.)

\subsection{Critical points of the height function}

The critical points of $f$ on $\CPN$ are exactly the $N$ ``corner''
points
\begin{equation*}
c_a = [0:\cdots:0:1:0:\cdots:0]\quad (\text{1 in slot } a),
\qquad a = 1, \ldots, N.
\end{equation*}
At each $c_a$, the value of $f$ is $f(c_a) = \lambda_a$. Since the
$\lambda_a$ are all distinct, $f$ takes $N$ distinct values at $N$
distinct critical points.

\subsection{Morse index of each critical point}

The Morse index of $c_a$ is the number of negative eigenvalues of
the Hessian $\Hess(f)$ at $c_a$. Computing the Hessian in local
holomorphic coordinates centered at $c_a$, one finds:
\begin{equation}
\label{eq:Morse-index-CPN-detail}
\ind(c_a) = 2(a-1).
\end{equation}
So $\ind(c_1) = 0$ (local minimum, the \GA{}), $\ind(c_2) = 2$,
\ldots, $\ind(c_N) = 2(N-1)$ (local maximum, the \DGI{}).

Note that the indices are all even. This is a consequence of the
\emph{K\"ahler} structure of $\CPN$: the Hessian eigenvalues come
in complex pairs $(\lambda, \bar\lambda)$, so they always have
even index.

\subsection{Morse polynomial}

Summing $t^{\ind(c)}$ over critical points gives the Morse
polynomial:
\begin{equation}
\label{eq:Morse-poly-CPN-detail}
\mathcal P_f^{\CPN}(t) = \sum_{a=1}^{N} t^{2(a-1)}
= 1 + t^2 + t^4 + \cdots + t^{2(N-1)}.
\end{equation}
This equals the Poincar\'e polynomial of $\CPN$ (the cohomology
polynomial of $\CPN$ is generated by the K\"ahler form
$[\omega_{\rm FS}]\in H^2$ and its powers, giving
$\dim H^{2k}(\CPN;\mathbb Q) = 1$ for $k = 0, 1, \ldots, N-1$). This shows
that $f$ is a \emph{perfect Morse function}.

\subsection{Worked example: $N = 2$ ($\CP^1\cong S^2$)}

For $N=2$, $\CP^1\cong S^2$ has two critical points $c_1, c_2$ with
indices $0, 2$. Morse polynomial $\mathcal P_f = 1 + t^2$.
Alternating sum $a^+ - a^- = 1 - 1 = 0$ (with $a^+$ = number of
indices $\equiv 0\pmod 4$ and $a^-$ = number $\equiv 2\pmod 4$).

\textbf{Conclusion}: $\dim_\R\CP^1 = 2$ is \emph{not} divisible by
4, so strict Morse-positivity \emph{fails} on $\CP^1$. This rules
out $N = 2$ for family-projector closure.

\subsection{Worked example: $N = 3$ ($\CPtwo$)}

For $N=3$, $\CPtwo$ has three critical points $c_1, c_2, c_3$ with
indices $0, 2, 4$. Morse polynomial $\mathcal P_f = 1 + t^2 + t^4$.
Alternating sum:
\begin{equation*}
a^+(\CPtwo) - a^-(\CPtwo)
= |\{0, 4\}| - |\{2\}| = 2 - 1 = 1 > 0.
\end{equation*}

\textbf{Conclusion}: $\dim_\R\CPtwo = 4$ \emph{is} divisible by 4,
and strict Morse-positivity \emph{holds}. This is the parity
argument that makes $N = 3$ the unique admissible family count
together with the upper bound $N\le 3$ from \textbf{T4.1}.

\subsection{Worked example: $N = 4$ ($\CP^3$)}

For $N=4$, $\CP^3$ has four critical points with indices $0, 2, 4, 6$.
Morse polynomial $\mathcal P_f = 1 + t^2 + t^4 + t^6$. Alternating sum:
\begin{equation*}
a^+(\CP^3) - a^-(\CP^3) = |\{0, 4\}| - |\{2, 6\}| = 2 - 2 = 0.
\end{equation*}

\textbf{Conclusion}: $\dim_\R\CP^3 = 6$ is \emph{not} divisible by
4, so strict Morse-positivity fails. This rules out $N = 4$ from
the parity side, complementing the action-budget upper bound from
\textbf{T4.1}.

\subsection{Pattern: parity of $\dim_\R\CPN$}

The pattern is $\dim_\R\CPN = 2(N-1)$:
\begin{itemize}[leftmargin=2em]
\item $N$ odd ($N = 1, 3, 5, \ldots$): $\dim_\R = 0, 4, 8, \ldots$ (multiples of 4),
strict positivity holds.
\item $N$ even ($N = 2, 4, 6, \ldots$): $\dim_\R = 2, 6, 10, \ldots$ (NOT multiples of 4),
strict positivity fails.
\end{itemize}

So among integer $N\ge 1$, the family-projector dimension test
selects $N\in\{1, 3, 5, 7, \ldots\}$. Combined with the lower
bound $N\ge 3$ (from CP violation, Theorem~\textbf{T2.1}) and the
upper bound $N\le 3$ (from cascade closure, Theorem~\textbf{T4.1}),
the unique admissible value is $N = 3$.

\subsection{Cascade actions on $\CP^{N-1}$}

The cascade on $\CPN$ is $c_N \to c_{N-1}\to\cdots\to c_1$. The
action of the $k$-th transition (from $c_{N-k+1}$ at index $2(N-k)$
to $c_{N-k}$ at index $2(N-k-1)$) is, by Theorem~\textbf{T1.1},
$S_k = \pi(N-k)$. Total cascade action:
\begin{equation*}
\Stot(\CPN) = \sum_{k=1}^{N-1}\pi(N-k) = \pi\sum_{j=1}^{N-1}j = \pi\frac{N(N-1)}{2}.
\end{equation*}

For $N = 3$: $\Stot(\CPtwo) = 3\pi$, exactly saturating $\Sbi = 3\pi$.
For $N = 4$: $\Stot(\CP^3) = 6\pi$, exceeding the budget by $3\pi$
(equivalent to one full cascade in addition). This is why $N\le 3$.

%% ====================================================================
%% APPENDIX M: LIFETIME HIERARCHY DERIVATION FOR LIFE-SCIENCE STUDENTS
%% ====================================================================
\section{The generation decay-width ladder: from cascade gate actions to laboratory observations}
\label{app:lifetime-derivation}

\textbf{Reading goal.} This appendix illustrates how the cascade gate
factor $\e^{-\pi k}$ enters the \emph{family} decay widths alongside
the dominant phase-space factors, with explicit numerical comparison
to laboratory data. There is a single cascade --- the generation
cascade $\tau\to\mu\to e$ of \S\ref{ssec:gen-lifetimes}; $W$ and the
free neutron are ordinary weak (\GRIPin{}) decays, not cascade-index
particles, and appear here only as familiar order-of-magnitude
anchors for the phase-space factor. Intended
audience: a first-year graduate student in particle physics or
related fields. Useful background: the WKB approximation; standard
weak-decay phase space (Sargent's rule).

\subsection{From cascade action to decay rate (WKB)}

The decay rate of an unstable configuration through an instanton
transition with Euclidean action $S_E$ is, in the WKB approximation,
\begin{equation}
\label{eq:WKB-decay-detail}
\Gamma = A\cdot\e^{-S_E},
\end{equation}
where $A$ is a one-loop prefactor of order the relevant energy
scale \citep{Coleman1977,CallanColeman1977}. In our case, the
energy scale is $M_{\rm KK}\simeq 3.85\,\TeV$, so
$A\sim M_{\rm KK}\sim 10^{27}$ Hz (since $1\,\TeV \approx 1.5\times 10^{27}$ Hz).
The corresponding bare lifetime is
\begin{equation}
\label{eq:tau-from-Gamma}
\tau = 1/\Gamma = (1/A)\cdot\e^{+S_E}\approx\tau_0\cdot\e^{S_E},
\quad\tau_0 = 1/A\sim 10^{-28}\,\text{s},
\end{equation}
matching the value $\tau_0\sim 10^{-28}$~s used in Eq.~\eqref{eq:GRIP-cumulative-tau}
of Part~V.

(Note: this $\tau_0\sim 10^{-28}$~s is the bare cascade timescale
before species-specific phase-space corrections; the empirical
$\tau_W\sim 10^{-25}$~s already includes the gauge-coupling enhancement
$1/g_W^2$ and the inherited $1/m_W$ kinematic scale relative to $\tau_0$.)

\subsection{Cumulative cascade action}

The $i$-th cascade stage adds Bogomolnyi action $\pi(i+1)$ to the
running total. Cumulative action up to stage $i$:
\begin{equation}
\label{eq:cumulative-action}
S_E^{(i)} = \sum_{k=1}^{i}\pi k = \pi\frac{i(i+1)}{2}.
\end{equation}
Therefore the bare geometric timescale after descending $i$ gates is
\begin{equation}
\label{eq:tau-i}
\tau^{\rm geom}_i = \tau_0\cdot\e^{\pi i(i+1)/2},
\end{equation}
a cumulative gate factor, \emph{not} a physical lifetime: the
physical width is $\tau^{\rm geom}_i$ modulated by the species phase
space below.

\subsection{Numerical sequence}

Setting $\tau_0\sim 10^{-28}$ s (after accounting for phase-space
factors that depend on the specific particle), we compute:
\begin{table}[H]
\centering\small
\caption[The geometric gate factor along the generation cascade.]{The geometric gate factor along the generation cascade, from the
tau through the muon to the electron. Each step down the cascade costs
one factor of $\e^{-\pi}$, and the third column turns that into the
numbers the last two columns carry: two gates for the third generation,
one for the second, none for the first --- which is why the electron is
stable. The lifetimes themselves are read off through the companion
paper rather than derived here. What the present paper supplies is the
exponent, and this is where the exponent meets the measured values.}
\label{tab:lifetimes-pred}
\begin{tabular}{ccccc}
\toprule
Gate $k$ & Gate action $\pi k$ & Gate factor $\e^{-\pi k}$ & Generation & Lifetime (via A9)\\
\midrule
$2$ & $2\pi$ & $\e^{-2\pi}\approx 1.9\times10^{-3}$ & 3rd ($\tau$) & $\tau_\tau\approx 2.9\times10^{-13}$ s\\
$1$ & $\pi$ & $\e^{-\pi}\approx 4.3\times10^{-2}$ & 2nd ($\mu$) & $\tau_\mu\approx 2.2\times10^{-6}$ s\\
--- & --- & --- (ground state) & 1st ($e$) & stable\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Why the empirical lifetimes are larger than the cascade prediction}

The empirical lifetimes are larger than the bare cascade predictions
by enormous factors. This is \emph{not} a problem for \GRIP{}; it is
a feature. The cascade-action factor $\e^{\pi i(i+1)/2}$ gives the
\emph{geometric/topological} enhancement of the lifetime due to the
cascade structure. The remaining enhancement comes from the
\emph{kinematic} phase-space factors that depend on the specific
particle and decay mode.

For weak decays, the standard kinematic factor is Sargent's rule:
\begin{equation}
\label{eq:Sargent}
\Gamma_{\rm weak} \propto G_F^2\,Q^5,
\end{equation}
where $Q$ is the energy release. Plugging in:
\begin{itemize}[leftmargin=2em]
\item $W$ boson: $Q\sim m_W = 80$ GeV, $\Gamma\sim G_F^2 m_W^5 \sim 2$ GeV, $\tau\sim 10^{-25}$ s. (Sargent phase-space estimate matches the observed lifetime)
\item Muon: $Q\sim m_\mu = 106$ MeV, $\Gamma\sim G_F^2 m_\mu^5\sim 0.5\,\mu$eV, $\tau\sim 10^{-6}$ s. (Sargent phase-space estimate matches the observed lifetime)
\item Neutron: $Q\sim \Delta m_{n-p} = 1.3$ MeV, $\Gamma\sim G_F^2\Delta m^5\sim 10^{-3}$ eV, $\tau\sim 10^{3}$ s. (Sargent phase-space estimate matches the observed lifetime)
\end{itemize}
The species-specific Sargent factors by themselves reproduce the
empirical lifetimes to within an order of magnitude per species.
The cascade-action factor $e^{\pi i(i+1)/2}$ then contributes a
structural ordering on top of the kinematic modulation. In this
sense Sargent's rule and the cascade-action ladder are
complementary: the former determines the absolute scale per species,
the latter determines the parametric ordering across cascade depth.
Disentangling the two requires a controlled-kinematics test class
(see the kill-criterion at the end of Part~V's worked example).

\subsection{Universality across decay channels}

The most striking feature of the generation decay-width ladder is its
\emph{universality}: the cascade-action factor $\pi i(i+1)/2$
depends only on the cascade index $i$, not on the specific particle.
The kinematic enhancements (Sargent's rule) modify the prefactor
but do not change the exponential structure.

This explains why all weak-decay lifetimes, when plotted on a
log-log scale against $Q^5$, collapse onto a single Sargent curve.
The cascade structure is universal; only the kinematic prefactors
vary by particle.

\subsection{Falsifier 7 (F-GRIP-7): a particle outside the sequence}

In a controlled-kinematics decay class --- one in which the Sargent
phase space is a uniform multiplier across generation gates --- the
residual decay-rate ordering must follow the gate factors $\e^{-\pi k}$.
A measured residual ordering that contradicts the gate sequence by a
factor of $\e^\pi \approx 23$ or more (after the phase-space factors
are divided out) would be a serious empirical challenge to the cascade
structure of \textbf{GRIP-ST7}. This is the same controlled-kinematics
kill-criterion stated in Part~V; the absolute lifetimes themselves are
phase-space-dominated and are not a standalone falsifier.

%% ====================================================================
%% APPENDIX N: GLOSSARY FOR LIFE SCIENCE & NEUROSCIENCE READERS
%% ====================================================================
%%  PART: Common Objections and Responses (four-Part tail; Q&A in Phase 4)
%% ====================================================================
\GGPart[Common Objections]{Common Objections and Responses}{The strongest objections a hostile reviewer would raise against the \GiFour{} construction --- on reading every conservation law off one composite connection, the cascade and its budget, and the open-problem ledger M9--M11 --- answered, with pointers into the paper}
\label{sec:common-objections}

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

A construction that claims to read every conservation law of physics off a single composite connection, and to force the family number to three by an instanton-action budget, invites the most demanding scrutiny; it would not be honest scholarship to leave the obvious objections unstated. We collect here the strongest adversarial readings of the \GiFour{} thesis --- the ones a particle theorist, a mathematical physicist, a string theorist, and a skeptical reviewer would each raise on a careful reading --- and answer each, with explicit pointers to the theorem, table, or companion paper that bears the weight. Where the honest answer is ``not yet,'' we say so and name the open item (in particular the open-problem ledger M9--M11 of \S\ref{sec:open-problems}).

\bigskip

%% =====================================================================
%% Cluster 0 -- For the curious reader
%% =====================================================================
\bigskip
\section{For the curious reader: the big questions, plainly answered}
\label{sec:qa-curious}

The twenty-five questions that follow are not a specialist's. They are the
ones a thoughtful reader without a physics degree actually carries
into a paper like this one: why there are three kinds of matter rather than two or four, why
time runs one way when the equations do not care which, what became of
a symmetry when it broke, and whether ``downhill'' is a fact about the
world or a way of talking about it. Several of these have been settled in
popular science in directions this paper does not take, and a reader
is owed plain answers before the specialist objections begin. They
are met first, and plainly.

The hardest of the twenty-five is the eighth --- whether predicting three families of matter, when three is what
everyone already knows, is a prediction at all. The response concedes the
difficulty rather than dissolving it, and says which part of it the
paper can carry and which part it cannot. Where a claim rests on a
step that is not closed, the response says so and names the step;
where the honest answer is that the framework does not reach the
question, it says that too.


\bigskip
\noindent\textbf{Question 1 (Q-GRIP-1) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why are there exactly three generations of matter?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the descent runs out of room, and the number three is what
falls out of a budget rather than a preference. \\ The picture is worth
having plainly. Structure forms here by a system settling: it starts
with its options open, moves downhill on a landscape, reaches a fork,
takes one branch, and continues until there is nowhere lower. Each
completed round of that process leaves behind one family of matter, and
each round costs something from a fixed budget fixed by the integer
ledger. Three rounds fit. A fourth does not, because there is nothing
left to spend and nowhere lower to go. \\ Two things make this a
prediction rather than a story. First, the budget is not adjustable ---
it is a ratio of whole numbers, and there is no version of the framework
with a slightly larger one. Second, the argument runs in the direction
of forbidding rather than accommodating: it says a fourth family cannot
exist, which is a statement that could be shown false tomorrow by a
single discovery. \\ It is worth being exact about the status. The
upper bound --- no more than three --- is conditional on a capacity
result established in a companion paper, and that conditionality is
flagged wherever the claim is made. The lower bound --- at least three
--- is unconditional. Nobody should quote the result without the
condition, and the paper does not.
\end{objqabody}

\bigskip
\noindent\textbf{Question 2 (Q-GRIP-2) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Nobody has seen a fourth generation. Isn't predicting three just predicting what we already know?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the hardest question here, and it deserves a straight
answer rather than a defense. \\ You are right that three is not news.
It has been known since the nineteen-eighties, when the decay of the Z
boson was measured precisely enough to count how many light neutrino
species exist, and the answer came out just under three with an
uncertainty of less than one per cent. Any framework producing three
today is producing a number its author knew. That is a real weakness in
the evidential value of the result and it should not be dressed up.
\\ What can be said in the framework's favor has two parts. The first is
that most frameworks do not produce three at all --- they accommodate it.
The Standard Model does not predict the number of generations; it is an
input. String compactifications can be arranged to give three, and also
two, and also many other values, depending on the folding chosen. A
construction that forces three and cannot be made to give anything else
is doing something different from one that fits three. \\ The second is
that the same budget argument makes claims that are not already known:
an ordering of decay widths across the families, and a relationship
between them fixed by the same integers. Those are not measured yet in
the form the framework states them. That is where the evidential weight
has to sit, and a reader is right to discount the family count
accordingly.
\end{objqabody}

\bigskip
\noindent\textbf{Question 3 (Q-GRIP-3) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why does time run one way when the underlying equations don't care which way it runs?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because the equations are not the whole description, and what is
missing from them is exactly what supplies the direction. \\ The puzzle
is genuine and old. Write down the fundamental laws and they work
equally well run backwards; play a film of two colliding molecules in
reverse and nothing looks wrong. Play a film of a smashed cup in reverse
and everything looks wrong. Something asymmetric has entered between
those two scales and it is not in the equations. \\ The standard
answer, from Boltzmann onward, is statistical: there are vastly more
disordered arrangements than ordered ones, so a system started in an
ordered state will almost certainly become less ordered. That answer is
correct and incomplete, because it explains the direction only by
assuming the past was ordered, which is the thing one wanted explained.
\\ What this framework adds is structural rather than statistical.
Descent on the landscape is monotonic --- the functional decreases along
the flow and cannot increase --- so the sequence of settlings has an
order built into the geometry rather than into the counting of
microstates. That does not by itself explain why the universe began
where it did, and the paper does not claim it does. What it claims is
narrower: that the irreversibility is a property of the projection
rather than an accident of initial conditions, which is a different
kind of answer and one that could be wrong in a checkable way.
\end{objqabody}

\bigskip
\noindent\textbf{Question 4 (Q-GRIP-4) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is symmetry breaking, really? Did the symmetry go somewhere?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nowhere. It is still in the laws; what changed is the state, and
keeping those two apart is the whole of the idea. \\ The standard
example is a pencil balanced on its point. The situation is perfectly
symmetric --- no direction is preferred --- and the pencil falls, and now
there is a direction. The laws did not change; nothing chose east over
west; and yet the outcome has a property the setup did not. The symmetry
is not hidden or destroyed. It survives as the statement that the pencil
was equally likely to fall any way, which is a fact about the ensemble
and no help to the fallen pencil. \\ What this framework adds is a
reason the falling happens \emph{when} it does. In the usual telling,
symmetry breaking is something that befalls a system as it cools, and
the timing is a matter of circumstance. Here it is a step in the
descent: it occurs at the point where continuing downhill requires
choosing, and the choice is what removes the symmetry. The breaking is
not an event that interrupts the process; it is what the process looks
like at a fork. \\ The sentence worth carrying away is that structure is
what a falling system leaves behind. Everything with a definite property
--- a direction, a charge, a mass --- has it because something further
back could have gone otherwise and did not.
\end{objqabody}

\bigskip
\noindent\textbf{Question 5 (Q-GRIP-5) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the structure is inevitable, why did the universe need fourteen billion years?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because a fixed rule book says nothing about rates, and the two
questions are more separable than they feel. \\ What the framework
determines is what can exist and in what order it can form. It does not
determine how long any stage takes, which depends on densities,
distances and temperatures that are not fixed by the geometry in any
usable way. A construction can force the existence of carbon and be
entirely silent about how long stars take to make it. \\ What is forced
is the \emph{ordering}, and that is more than nothing. Each settling
needs the previous one to have finished and needs conditions the
previous one produced. Nuclei cannot precede the quarks that make them;
atoms cannot precede nuclei; chemistry cannot precede atoms; and nothing
resembling biology can precede the heavy elements that only stellar
deaths supply. A layered structure requires a long history whether or
not the layers were inevitable. \\ It is worth saying what the framework
does not claim, since it would be easy to read in. The particular
history our universe had is not forced. Fourteen billion years is a
measurement, not a prediction of this or any related construction, and
anyone offering one should be asked to show the calculation.
\end{objqabody}

\bigskip
\noindent\textbf{Question 6 (Q-GRIP-6) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Is `downhill' a physical fact, or just a way of speaking?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A physical fact, and the reason it is worth asking is that in most
of physics the picture is imported rather than derived. \\ In ordinary
thermodynamics one writes down a free energy because experience says
there is one, and the landscape picture that follows --- valleys,
barriers, a system running down --- is a way of visualizing the
mathematics rather than something the mathematics produced. That is
perfectly respectable and it does mean the picture is doing no
independent work. \\ Here the landscape is constructed from the
geometric data, and its crucial property --- that it curves upward in
every direction away from its critical points, so valleys are genuinely
valleys and not flat shelves --- is forced by the anomaly condition
rather than assumed. That link is the paper's most surprising internal
result, and it is worth stating in the strange form it deserves:
whether matter can come to rest turns out to be a question with an
answer in the consistency of a quantum field theory. \\ What the
landscape is \emph{not} is a place. Nothing sits on a hillside; the
heights are values of a functional and the positions are
configurations. The picture is trustworthy because the mathematics has
the properties the picture suggests, which is usually because the
picture was extracted from the mathematics in the first place.
\end{objqabody}

\bigskip
\noindent\textbf{Question 7 (Q-GRIP-7) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is a `cascade', and why does it stop?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A cascade is a sequence of settlings in which each one creates the
conditions for the next, and it stops when a settling produces
conditions that support no further step. \\ The everyday version is
familiar. Water vapor cools and condenses; the droplets fall and pool;
the pool freezes. Each stage needed the one before and each has its own
character. Nothing external decided the number of stages; the sequence
ran until the conditions stopped supporting a new one. \\ What makes the
version here more than an analogy is that the stopping condition is
computable. Each round of descent consumes part of a fixed budget set by
the integer ledger, and when the remaining budget cannot pay for another
round the cascade halts. Three rounds fit. That is why the framework can
say how many families of matter there are rather than observing how
many there happen to be. \\ There is a limitation the paper is careful
about. Descent on a landscape that does not move has an end, full stop
--- everything settles once and stays settled, which describes stable
matter and describes nothing that continues. For a history rather than a
conclusion, something must reorganize the landscape from inside. This
paper establishes the engine and does not establish that; the companion
that does is named, and the boundary is drawn rather than blurred.
\end{objqabody}

\bigskip
\noindent\textbf{Question 8 (Q-GRIP-8) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is matter stable? Why doesn't the electron just decay?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Because there is nowhere for it to go, and that is a stronger
statement than it sounds. \\ Decay is not something particles choose. A
particle decays when there exists a lighter combination carrying the
same conserved quantities; if such a combination exists, the decay
happens, and if it does not, the particle is stable forever. The
electron is the lightest thing carrying its charge, so charge
conservation alone makes it permanent. Nothing further is required.
\\ What a framework of this kind can add is why the ladder of
possibilities terminates where it does --- why there is a lightest
charged thing at all, rather than an unbounded sequence of lighter ones.
That follows from the descent having an end: the cascade halts, and what
it halts at is what cannot decay. \\ There is a second and less obvious
case worth mentioning because it works differently. The proton's
stability, on this account, is not about there being nothing lighter ---
there is --- but about the decay requiring a change in something that
counts rather than something that varies. Unwinding a count is not a
small change, so the process is not merely slow but structurally
obstructed. Whether that obstruction is absolute is a live question and
one of the framework's sharpest exposures, since experiments have been
watching large tanks of water for decades waiting for one proton to
disagree.
\end{objqabody}

\bigskip
\noindent\textbf{Question 9 (Q-GRIP-9) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't the second law about statistics, not geometry?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, and the two accounts are answering different questions that
are easy to run together. \\ The statistical account says a system moves
toward disorder because there are overwhelmingly more disordered
arrangements than ordered ones. That is correct and it explains the
overwhelming reliability of the law. What it does not explain is why the
past was ordered, and without that the argument gives no direction ---
run the same reasoning backwards and it predicts that the past was
disordered too, which is false. This has been recognized since
Boltzmann and remains the honest state of the subject. \\ The
geometric account here says something different. The functional
decreases monotonically along the flow because of the structure of the
flow, not because of a counting argument about arrangements. That
supplies a direction without needing an assumption about how things
started. \\ The two are compatible and they are not the same claim, and
the framework does not assert it has solved the arrow of time. It
asserts that one component of the asymmetry is structural rather than
statistical, and it is specific about which component. A reader should
be wary of any account that claims to have settled this question
outright; it has resisted a great deal of talent.
\end{objqabody}

\bigskip
\noindent\textbf{Question 10 (Q-GRIP-10) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What does `free energy' mean when there is no heat bath?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It means the same thing it means with one --- a quantity that
decreases and whose decrease measures what has been irreversibly spent
--- but the reason it exists is different, and the difference matters.
\\ In ordinary thermodynamics free energy is defined relative to a bath
at a temperature, and the bath is doing real work in the definition: it
is what makes some energy available and some not. Take the bath away and
the usual definition has nothing to hold on to, which is why applying
thermodynamic language to a single system or to the universe as a whole
has always been delicate. \\ The functional here is built from the
geometry rather than from a bath, and it splits into two recognizable
pieces: one counting what a configuration costs in energy, the other
counting what it costs in specification --- how much has to be pinned
down, how much order is being maintained against spreading. Anything
that persists pays both. \\ That is the framework's most aggressive
claim, and its exposure is exactly proportional. If the same functional
prices a folded protein and a gauge-orbit stratum, then a set of
relationships among laboratory measurements must hold that no ordinary
account requires. Those relationships are stated in the assessment and
are measurable on a bench, which is an unusually accessible place for a
claim of this ambition to be tested.
\end{objqabody}

\bigskip
\noindent\textbf{Question 11 (Q-GRIP-11) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should a landscape have a bottom at all? Why not descend forever?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It is a fair question and the answer is not obvious --- plenty of
mathematically respectable landscapes have no bottom, and a system on
one would fall without end. \\ Such a landscape describes nothing that
could exist. If the functional is unbounded below, a system can always
lower it further; there are no stable configurations, nothing settles,
and there is no matter in any recognizable sense. So the requirement
that the landscape be bounded is not an aesthetic preference. It is the
condition for there being anything at all, and a framework that failed
to guarantee it would be describing a world with no contents.
\\ What is done here is to prove it rather than assume it. The
boundedness follows from the same consistency condition that fixes the
integer ledger --- the anomaly condition again, doing a second job. That
is the technically interesting part of the paper: a single requirement
about the internal consistency of a field theory turns out to imply both
that the ledger is unique and that the landscape has a floor. \\ The
consequence worth carrying away is that stability is not a happy fact
about our universe. On this account, any world that is consistent enough
to be described has to have somewhere for things to come to rest ---
which converts the existence of stable matter from a coincidence into a
theorem, conditional on the arena.
\end{objqabody}

\bigskip
\noindent\textbf{Question 12 (Q-GRIP-12) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If everything descends, why is there any structure left? Shouldn't it all have run down?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It has, and that is the point --- the structure \emph{is} what the
running down left. \\ The intuition that descent destroys structure
comes from thinking of the landscape as smooth, so that everything ends
in one featureless minimum. Real landscapes are not smooth. They have
many separated valleys, and a system descending gets caught in one and
stays. What we call structure --- a nucleus, an atom, a molecule, a cell
--- is a system sitting in a valley it cannot climb out of, and it is
there \emph{because} it descended, not despite it. \\ So the answer is
that running down and building structure are the same process seen from
two sides. Every stable thing is a place where a descent stopped. The
water in a lake is not evidence that water failed to flow downhill; it
is evidence that it did. \\ There is a second half that this paper does
not supply and is careful to say so. Descent on a fixed landscape
happens once, and afterward nothing further occurs. For a world that
keeps producing new kinds of structure, the landscape itself has to be
reorganized from within, repeatedly. That requires something the four
pillars here do not provide, and the companion that provides it is
named.
\end{objqabody}

\bigskip
\noindent\textbf{Question 13 (Q-GRIP-13) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Could the cascade have stopped somewhere else? What if it had run twice, or four times?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It could not, given the budget --- and that is exactly what makes
the count a prediction rather than an observation. \\ The budget is
fixed by the integer ledger, which is itself the unique solution of a
consistency condition. Each round of descent costs a computable
fraction of it. Two rounds would leave the budget unspent, which the
descent does not permit --- it runs until it cannot. Four would need more
than exists. Three is the only number that both fits and exhausts.
\\ The right way to be skeptical about this is not to doubt the
arithmetic but to ask what the budget depends on. It depends on the
arena, which is an input. A different arena would give a different
budget and could give a different count. So the claim is not that three
is necessary absolutely; it is that three is necessary given a structure
fixed on independent grounds. That is the honest form and it is the form
the paper uses. \\ What makes the claim useful anyway is that the
independent grounds were settled first and are written down. Nobody can
adjust the arena after a measurement to recover the count, because the
arena's argument makes no reference to families of matter. If a fourth
family appears, the arena was wrong, and everything built on it goes
with it.
\end{objqabody}

\bigskip
\noindent\textbf{Question 14 (Q-GRIP-14) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Does this explain why the universe isn't uniform? Why there are galaxies rather than a smooth soup?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Partly, and the part it does not explain is worth marking, because
the question mixes two things. \\ Structure requires two ingredients: a
mechanism that amplifies small differences, and small differences to
amplify. Gravity supplies the first, and has since Newton --- a region
slightly denser than its surroundings pulls harder, grows denser still,
and runs away. That part is not in dispute and is not this framework's
contribution. \\ Where the seeds came from is the harder half. The
standard account attributes them to quantum fluctuations stretched to
cosmic size during an early rapid expansion, and the measured pattern in
the microwave background matches that account remarkably well. This
framework is consistent with that story and does not replace it; the
cohort paper on cosmology carries the detail. \\ What the present paper
contributes is one step further down: given that structure formed, why
it formed in \emph{layers} --- nuclei, then atoms, then molecules ---
rather than in one undifferentiated collapse. That is the cascade, and
the layering is forced by each stage requiring the previous one's
output. A reader looking here for an account of galaxy formation will
not find one, and should not; a reader asking why the world has levels
at all will.
\end{objqabody}

\bigskip
\noindent\textbf{Question 15 (Q-GRIP-15) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What is the difference between a law and a habit? How do you know the constants aren't just drifting slowly?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is a sharper question than it looks, and it has been taken
seriously enough to be measured. \\ Dirac proposed in 1937 that some
constants might vary over cosmic time, and the idea has never quite
died. It is checkable: light from distant quasars was emitted billions
of years ago, and the atomic transitions imprinted on it depend on the
fine-structure constant, so comparing them with laboratory values
constrains any drift. The bounds are now very tight --- variation, if any,
is below a part in $10^{17}$ per year --- and a natural nuclear reactor
that ran in Gabon two billion years ago provides an independent and
similarly tight constraint. \\ What this framework adds is a reason to
expect exactly zero rather than a small number. Quantities fixed by
whole numbers cannot drift, because there is no continuous path from one
integer to another. A framework in which constants are ratios of
integers predicts no variation at all --- not merely a variation too small
to have been seen. \\ That is a strong position and an exposed one. A
confirmed nonzero drift, at any level, would end this framework outright
while leaving accounts that treat constants as parameters entirely
comfortable. It is one of the cleanest disagreements between the two
kinds of theory, and it is being measured now.
\end{objqabody}

\bigskip
\noindent\textbf{Question 16 (Q-GRIP-16) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Isn't our universe just one bubble among countless others?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It might be, and this paper's contribution is to remove the
question's motivation rather than to answer it. \\ The multiverse exists
to explain why the constants look arranged, and it works by making the
arrangement a selection effect within an ensemble. That requires the
constants to have been capable of taking other values. \\ Here they were
not. The family count is what a fixed budget permits; the budget is a
ratio of integers; the integers are the unique solution of a
consistency condition. There is no distribution to draw from. If that is
right, the ensemble has nothing to explain and its principal job is
gone. \\ The comparison worth making is with the framework's own risk.
The multiverse cannot be embarrassed by a measurement of the family
count, because any count is compatible with some member of the
ensemble. This framework can be ended by finding a fourth. That
asymmetry is not an argument that this one is correct. It is an argument
that it is the kind of thing that can be found out, which is the more
useful property for a theory to have.
\end{objqabody}

\bigskip
\noindent\textbf{Question 17 (Q-GRIP-17) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``The universe looks fine-tuned for life. Doesn't that point to a designer?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It points to something requiring explanation, and the inference to
a designer needs a premise this framework denies. \\ The observation is
real: alter certain quantities slightly and there are no stable nuclei,
no chemistry, nothing that lasts. Feeling the pull of design there is
not naive. \\ The inference requires that the quantities could have been
set otherwise. A dial that was turned implies a hand. If the values are
what a fixed budget permits and the budget is not adjustable, there was
no dial, and asking who set them has no purchase. \\ This is not an
argument against a designer and should not be recruited as one. Someone
who believes on other grounds is untouched by anything here, and so is
someone who does not. What changes is that one particular argument
loses its premise if the framework holds --- and whether it holds is
settled by whether a fourth family of matter turns up, which is an odd
place for a theological question to be decided and is where this one now
sits.
\end{objqabody}

\bigskip
\noindent\textbf{Question 18 (Q-GRIP-18) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``If the cascade is forced, is my future already written?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Nothing here touches your future, and the inference does not hold
however tight it feels. \\ What the cascade fixes is the rule book: what
kinds of matter exist, what is stable, in what order structure can form.
It does not fix any particular history. The distinction is between the
rules of chess and the game played last Tuesday, and a rule book
admitting no alternative says nothing about which of an enormous number
of permitted games occurred. \\ There is a further point that cuts
deeper. Even a fully deterministic rule book delivers no readable
future, because the only system capable of computing what the rules
imply for something as complicated as a person is a system at least that
complicated running at least that long. There is no shortcut and no
vantage point. A universe can be lawful all the way down and remain, in
the only sense available to anything inside it, unwritten until it
happens. \\ And the honest limit: this paper is about a descent and its
budget. What agency is, and whether the concept survives contact with
physics of any kind, is a question it does not reach.
\end{objqabody}

\bigskip
\noindent\textbf{Question 19 (Q-GRIP-19) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why is there something rather than nothing?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The framework does not touch it, and saying so is more useful than
a gesture. \\ Everything here is conditional: given an arena with
certain properties, the descent has this shape and stops after three
rounds. Why there is an arena at all sits outside the construction, and
no rearrangement reaches it. That is not a gap further work closes; it
is a different kind of question. \\ What can be contributed is narrower.
If the framework is right, the space of possible somethings is far
smaller than it appears --- very nearly a single admissible structure
rather than a landscape from which one was drawn. That converts ``why
this?'' into ``why anything?'', which is harder and cleaner. Narrowing a
question is a real service and it is not an answer. \\ There is a
temptation, in a paper about descent, to say that the question presumes
a starting point and that a structure without one needs no beginning.
That move is available, has defenders, and does not follow from
anything here.
\end{objqabody}

\bigskip
\noindent\textbf{Question 20 (Q-GRIP-20) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``This explains a great deal from very little. Shouldn't that make you suspicious?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} It should and it does, and Eddington is the standing warning: a
great astrophysicist who spent his final years deriving the
fine-structure constant from combinatorics that convinced nobody, and
whose reputation never recovered. \\ What separates a derivation from a
coincidence is not the quality of the agreement. It is whether the
machinery producing it exists for another reason and whether it commits
you to anything further. A quantity reverse-engineered from a
measurement will always match. \\ Here the descent structure is built
from consistency conditions that know nothing about particle physics,
and it then fixes not only the family count --- which was known --- but an
ordering of decay widths across the families and a set of thermodynamic
relations in ordinary laboratory systems, neither of which was used in
building it. \\ That is the only defense available and it is the right
one: not ``too good to be chance'' but ``here are further consequences,
please check them.'' Keep the suspicion. If those consequences fail, the
agreement was a coincidence, and no elegance in the construction will
help.
\end{objqabody}

\bigskip
\noindent\textbf{Question 21 (Q-GRIP-21) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why should I take one person's framework seriously against thousands working on other approaches?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Not because of who wrote it, and not despite it --- both are the
same error with opposite sign. \\ The practical question is fair, since
attention is finite. The useful filter is exposure rather than
headcount. A framework supported by thousands that forbids nothing
measurable is a worse use of attention than one supported by a single
person that forbids specific things and names the measurement that
decides. Community size tracks funding history and fashion and does not
track correctness. \\ There is a real cost to working outside a large
community: errors are found faster where more people look, and a
framework developed largely by one person carries a higher chance of a
mistake a hundred readers would have caught quickly. Internal coherence
does nothing to reduce that. \\ So do not take it on trust. The
thermodynamic relations this paper predicts are measurable on an
ordinary bench, need no six-dimensional mathematics to evaluate, and
either hold exactly or do not. Check one of those, and judge the rest by
what you find.
\end{objqabody}

\bigskip
\noindent\textbf{Question 22 (Q-GRIP-22) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``What would convince you that you are wrong?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Several things, named rather than gestured at. \\ A fourth family of
matter would end it: the count follows from a budget that cannot be
enlarged. A measured drift in a dimensionless constant, at any level,
would end it, since quantities fixed by integers cannot drift. A clean
violation of the thermodynamic relations the framework says must hold
exactly --- the symmetry of a particular coupling matrix, a specific
fluctuation-dissipation relation --- would break the third pillar, and
those are bench measurements rather than frontier ones. And a decay
ordering across the families that disagrees with the predicted geometric
factor would break the fourth. \\ An internal failure counts fully too:
if the argument that the landscape is bounded below has a hole, the
descent has no guaranteed floor and the whole structure loses its
terminus. \\ What would not convince: a disagreement in a regime not
claimed, or the failure of a result whose hypotheses are already flagged
as open. A falsifier list is only honest if it excludes things, and a
reader should be wary of any proponent unwilling to say which failures
are survivable.
\end{objqabody}

\bigskip
\noindent\textbf{Question 23 (Q-GRIP-23) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Why does complexity seem to increase when entropy is also increasing? Aren't those opposites?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} They are not opposites, and the belief that they are is one of the
most common misreadings of thermodynamics. \\ The second law says that
the total entropy of an isolated system does not decrease. It says
nothing whatever about any part of that system. A refrigerator makes its
inside colder and more ordered every day without violating anything,
because it dumps more disorder into the kitchen than it removes from the
cabinet. The Earth does the same on a larger scale: it receives
concentrated energy from a hot sun and radiates diffuse energy to cold
space, and everything that has ever grown here was paid for out of that
difference. \\ So local complexity is not a loophole or an exception. It
is what a system does when energy flows through it, and the flow is
possible precisely because the overall descent is happening. Structure
is not swimming upstream; it is an eddy in the current. \\ What this
framework contributes is a way of pricing the eddy. The functional's two
parts --- what a configuration costs in energy and what it costs in
specification --- let the trade be written as an inequality with named
terms rather than described qualitatively. Whether the pricing is right
is checkable in the laboratory, which is more than can be said for most
statements about complexity.
\end{objqabody}

\bigskip
\noindent\textbf{Question 24 (Q-GRIP-24) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``Are the four `pillars' four things or one? It sounds like a framework in search of an architecture.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} One, and the paper proves it rather than asserting it --- which is
the difference between an architecture and a list. \\ The suspicion is
reasonable. Frameworks announcing a fixed number of pillars, principles
or laws often turn out to be organizing a set of independently developed
results after the fact, and the number is a presentational choice. The
test is whether anything would break if you removed one. \\ Here the
four are shown to be equivalent to the constitutional axioms taken
together: each implies the other, with nothing left over on either side.
That is a real theorem and it has a specific consequence --- the four are
not four convenient constructions that happen to sit together but four
descriptions of one thing, and the axioms can be recovered from them.
\\ What an equivalence buys is worth stating because it is easy to
overrate. It produces no new prediction; nothing follows from it that
did not follow from the pillars. What it does is tell a critic where to
attack: if the four and the axioms are the same content, then refuting a
pillar refutes an axiom and conversely. A construction whose parts were
merely compatible would offer no such handle --- one could remove a piece
and the rest would stand, slightly diminished, indefinitely.
\end{objqabody}

\bigskip
\noindent\textbf{Question 25 (Q-GRIP-25) (on the curious reader)}\par\nobreak\smallskip
\noindent\textbf{``I'm not a physicist and I've got this far. What is the one thing you want me to take away?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That structure is what a falling system leaves behind, and that
this is a more useful idea than it sounds. \\ The intuition most people
carry is that order and decay are opposed --- that things run down, and
that the existence of anything complicated is a small miracle working
against the grain. That picture is wrong in a specific and fixable way.
Everything with a definite property has it because something further
back could have gone otherwise and did not. A magnet has a direction
because it cooled and had to choose. An atom exists because a descent
stopped there. The three kinds of matter are three rounds of the same
process, and there is no fourth because there was nothing left to spend.
\\ Once you have that, several separate-looking puzzles stop being
separate. Why time runs one way, why the world has levels rather than
being uniform, why matter is stable, why there are three families and
not four --- these are one question with one answer, which is that a
system running downhill on a landscape with many valleys gets caught,
and what it gets caught in is what we call the world. \\ So take away
the picture rather than the theorem. And when you next meet a claim
about fundamental physics, ask what it forbids: this one forbids a
fourth family and forbids the constants drifting, and both are being
measured now.
\end{objqabody}

\section{The composite connection and TGG (Pillar 1)}

\bigskip
\noindent\textbf{Objection 1 (O-GRIP-1) (the differential geometer)}\par\nobreak\smallskip
\noindent\textbf{``Your `composite connection' $\Atot = \omega \oplus A$ is just the direct sum of four commuting connections on a fiber-product bundle. It carries no information beyond its summands --- no cross-terms, no mixing. Calling the obvious sum a single unified primitive is a relabeling, not a unification.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} At the level of the connection one-form the objection is correct, and Theorem~\ref{thm:TGG-existence-uniqueness} does not claim otherwise: $\Atot$ is a direct sum on the product structure group $\Gtot = \Spinone \times \SUfive \times \UoneB \times \UoneQi$. The content of \TGG{} is not that the sum is mysterious but that the constitution \emph{forces exactly these four factors and no others} (the gauge-group proposition, \S\ref{sec:composite-connection}, Proposition~\ref{prop:TGG-gauge-group}) and that the four sectors' stress contributions assemble into one boundary tensor whose conservation is a single law (T1.3). The unification claimed is constitutional and conservational, not a claim of off-diagonal mixing in the connection. Where the objection bites, and we concede it, is that ``composite connection'' as a phrase oversells a direct sum; the honest statement is that the four summands are jointly forced and jointly conserved, which is what \GGEP{} (\AX{2} of \citep{ArisakaA1Constitution}) asserts and what Corollary~\ref{cor:TGG-GGEP-equiv} records.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 2 (O-GRIP-2) (the differential geometer)}\par\nobreak\smallskip
\noindent\textbf{``Your uniqueness proof kills the obstruction by asserting $H^1$ of the contractible bulk vanishes --- yet the same construction classifies the two $U(1)$ bundles by a nontrivial $H^2$ generated by the compact $\theta$-circle, with Wilson holonomy. The base cannot be contractible and simultaneously support a nontrivial $H^2$ with winding. Which is it?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The two statements live on different factors of the product geometry $\Msix = \cM_4 \times I_r \times S^1_\theta$, and the proof should --- and in the final-review pass does --- say so explicitly. The $H^1$-vanishing that fixes the connection up to gauge is on the \emph{non-compact, contractible} $\cM_4 \times I_r$ factor; the nontrivial $H^2$ and the Wilson holonomy live on the \emph{compact} $S^1_\theta$ factor (Proposition~\ref{prop:TGG-theta-holonomy}), whose first cohomology carries the integer $\UoneB$/$\UoneQi$ winding. There is no contradiction: the bundle is topologically a product, trivial along the contractible directions and classified by the circle's holonomy along the compact one. The objection correctly identifies that the bare phrasing ``the contractible bulk'' is sloppy --- the bulk is contractible only after the circle is excised --- and the uniqueness statement is properly read as ``unique given the fixed $S^1_\theta$ holonomy class.''
\end{objqabody}

\bigskip
\noindent\textbf{Objection 3 (O-GRIP-3) (the 6D-supergravity specialist)}\par\nobreak\smallskip
\noindent\textbf{``You treat $\Spin(1,5)$ as a fourth internal gauge factor summed into $\Gtot$, but the spin connection is fixed by the metric (Levi-Civita); it is not an independent principal connection on equal footing with $\SU(5)$. Treating gravity's $\omega$ as a commuting gauge summand double-counts the geometry.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the sharpest structural objection to \TGG{}, and the answer is that the equal-footing treatment is exactly the constitutional content of \AX{2} (\GGEP{}), not an oversight --- but it is a \emph{posture}, and we mark it as such. Under \GGEP{} the spin connection and the gauge connections are facets of one composite primitive, and the Levi-Civita condition that ties $\omega$ to the metric is read as the \GGEP{}-closed limit (Proposition~\ref{prop:TGG-frame}, \S\ref{sec:magic-ladder}); the standard separated-sector treatment is recovered when that closure is imposed. So the framework does not deny that $\omega$ is metric-determined; it claims that the metric and the gauge data are projections of one object, which is precisely the non-standard move the constitution makes. We concede that a reader trained on 6D supergravity will find the equal-footing notation unfamiliar and that, at the level of degrees of freedom, $\omega$ is not independent of the metric; the constitutional claim is about the \emph{primitive}, and Proposition~\ref{prop:TGG-frame} is where the reconciliation with the metric-determined $\omega$ is stated.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 4 (O-GRIP-4) (the category theorist)}\par\nobreak\smallskip
\noindent\textbf{```Unique up to gauge equivalence' --- in what category, and up to which equivalence? You never specify the homotopy type of the base, never compute $[\Msix, B\Gtot]$, and never state which moduli you quotient by. The uniqueness is asserted at the level of slogans.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and Theorem~\ref{thm:TGG-existence-uniqueness} should state its category, which is: principal $\Gtot$-bundles with connection over the fixed product geometry $\Msix$, up to bundle automorphism (gauge transformation) covering the identity, with the $S^1_\theta$ holonomy class held fixed. With the homotopy type fixed --- $\Msix \simeq S^1_\theta$ after contracting the non-compact directions --- the classifying set $[\Msix, B\Gtot]$ reduces to the $\pi_1(S^1_\theta)$ data, i.e.\ the integer holonomies, and ``uniqueness'' means: given those integers, the bundle-with-connection is determined up to gauge. We grant that the body invokes Kobayashi--Nomizu at a slogan level; the precise statement is the holonomy-fixed one just given, and the uniqueness is uniqueness \emph{relative to} the fixed holonomy class, not an absolute classification claim. We have sharpened the statement of the theorem accordingly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 5 (O-GRIP-5) (the skeptic of necessary-conditions-as-achievements)}\par\nobreak\smallskip
\noindent\textbf{``T1.3 defines $\cT^{\mu\nu}_{\TGG}$ as the right-hand side of the Einstein equation, so $\nabla_\mu \cT^{\mu\nu}=0$ is automatic by the Bianchi identity. This is textbook general relativity re-skinned as a `Master Conservation Law.' What is the theorem beyond `a stress tensor defined via Einstein's equations is conserved on-shell'?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is right that the gravitational half of T1.3 (Theorem~\ref{thm:T2.2-master-conservation}, its TGG prelude) is the contracted Bianchi identity and carries no new content. What is not textbook is the \emph{iff}: the boundary tensor assembles four sectors (matter, gauge, Green--Schwarz inflow, KK), and $\nabla_\mu \cT^{\mu\nu}=0$ holds iff the four boundary Ward currents close \emph{simultaneously} together with the \PBIfour{} compensator (T2.2). The non-trivial direction is that the four sector divergences are linked --- a failure of any one Ward identity shows up as a calculable inflow term, not as a quiet redefinition. So the gravitational conservation is indeed automatic; the content is the four-sector simultaneous-closure condition (Proposition~\ref{prop:sector-divergences}), and we agree the prose should foreground that rather than the Bianchi half, which the final-review pass does.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 6 (O-GRIP-6) (the string theorist)}\par\nobreak\smallskip
\noindent\textbf{``T1.3 asserts $\nabla_\mu T^{\mu\nu}_{\rm KK}=0$ because heavy KK modes decouple at boundary scales. Decoupling does not make their stress contribution separately conserved; integrating them out produces higher-derivative operators and threshold corrections. The clean four-way split into separately-conserved sectors is assumed, not derived.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the separate conservation of the KK sector is a leading-order, decoupling-limit statement, and the higher-derivative and threshold corrections the objection names are real and are exactly the content of open problems M1 and M3 (\S\ref{sec:open-problems}). The honest claim of T1.3 is that \emph{at the boundary scales where the framework makes predictions} the four sectors are separately conserved up to the calculable inflow term; the loop-level completion --- where integrating out the KK tower mixes the sectors --- is not in this paper. We do not claim the four-way split survives to arbitrary precision; we claim it at leading order and flag the corrections as open. A reader who needs the all-orders statement is correctly pointed to M1/M3 rather than offered a false guarantee.
\end{objqabody}

\bigskip
\section{CCI, the integer ledger, and the modified-TOV falsifier (Pillar 2)}

\bigskip
\noindent\textbf{Objection 7 (O-GRIP-7) (the number theorist)}\par\nobreak\smallskip
\noindent\textbf{``T2.1 is advertised as forcing $(1,3,5;7,35)$, but it takes $N\geq 3$ and $L=5$ as inputs and then `derives' $N=3$. Your own remark concedes $L=5$ is a constitutional commitment and $N\geq 3$ is imported from CP violation. The theorem is `given $L=5$ and $N\geq 3$, the minimal ledger has $N=3$.' The empirical content was inserted as a hypothesis.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The description is accurate and the paper states it (Theorem~\ref{thm:CCI-OGN5-uniqueness} and its remark): $L=5$ is the smallest-simple-group commitment and $N\geq 3$ is the Kobayashi--Maskawa CP lower bound, both inputs. What T2.1 adds is the \emph{closure}: given those, the \MSMF{}-minimal integer solution of the integrality constraints is the unique tuple $(1,3,5;7,35)$, and the alternatives $N\in\{4,6,\dots\}$ fail the parity/positivity filter (Pillar 3) while $N\geq 5$ fails the budget (Pillar 4, conditionally). So T2.1 is not a from-nothing uniqueness theorem; it is the statement that the constitutional inputs plus the integrality constraints have a unique minimal solution. We agree ``forces'' overstates it when the inputs are empirical, and the corrected reading is ``the ledger is the unique \MSMF{}-minimal solution consistent with the smallest-simple-group and CP-lower-bound inputs.''
\end{objqabody}

\bigskip
\noindent\textbf{Objection 8 (O-GRIP-8) (the number theorist)}\par\nobreak\smallskip
\noindent\textbf{``The budget $K=35$ that supposedly forces $N\leq 3$ is itself $M^2+N^2+L^2 = 1+9+25 = 35$ --- it already contains $N=3$. The budget that bounds $N$ is built from the answer. The whole $N=3$ result is a fixed point of its own inputs.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the single most damaging objection to the paper, and we neither hide it nor claim to have closed it: the closure $K = M^2+N^2+L^2 = 35$ does contain $N=3$, and the \BIsix{} budget $\Sbi = 3\pi$ that bounds the cascade is motivated through $K=35$. That is precisely why open problem M10 (\S\ref{sec:open-problems}) is stated in the exact words ``derive $\Sbi = 3\pi$ independent of $N=3$ at every intermediate step, so as to rule out any circularity'' --- the framework documents its own potential circularity and labels closing it the central open task. What survives unconditionally is the \emph{lower} bound $N\geq 3$ from CP violation, which uses none of the budget; the \emph{upper} bound $N\leq 3$ is honestly conditional on M10 (Corollary~\ref{cor:CCI-N3} and the T4.1-status remark say so). So $N=3$ is ``$N\geq 3$ (unconditional) and $N\leq 3$ (conditional on M10's resolution).'' M10's $N$-independence demand is met by the capacity theorem T-N3-6 of \citet{ArisakaP2N3} --- which consumes neither $K=35$ nor $J=7$ --- so the upper bound is a theorem conditional on that paper's coupling
clause and integrality theorem L-INT (residue R-INT $=$ M6), and the
abstract's ``forces'' is read with exactly that conditional attached.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 9 (O-GRIP-9) (the anomaly theorist)}\par\nobreak\smallskip
\noindent\textbf{``The coefficient vector $(1,-1,-2/5,-2/5)$ in $X_4$ is asserted to be fixed by `$\mathcal H^2$ normalization,' with the actual derivation outsourced to A2/A3. In a self-contained A6 the central trace-ratio $5/2$ lock is imported, not shown. A reviewer cannot check the load-bearing $-2/5$ here.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and stated: the rank-one collapse of the anomaly matrix and the trace-ratio lock $|a/c|=|a/d|=5/2$ are proved in \AThree{} \citep{ArisakaA3BI6} (and the bulk content in \ATwo{} \citep{ArisakaA2GG6}); \S\ref{sec:CCI-coefficients} of the present paper records the result and cites the proof rather than re-deriving it. This is the cohort's division of labor, not a concealment --- A6 is the \emph{formalization} paper for the pillars, and the anomaly computation is A3's business. We concede the cost the objection names: a reader who wants to verify the $-2/5$ from first principles must consult A3, and A6 is not self-contained on that point. The honest framing, which \S\ref{sec:CCI-vs-BI6} now carries, is that A6 \emph{uses} the trace-ratio lock as an input from A3 and builds the conservation structure on it; the lock itself is A3's theorem.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 10 (O-GRIP-10) (the general relativist)}\par\nobreak\smallskip
\noindent\textbf{``The modified-TOV `astrophysical falsifier' produces a sub-percent shift in the mass--radius relation. But a sub-percent stress-energy correction is generically mimicked by uncertainties in the nuclear equation of state, which span far more than 1\%. How is this distinguishable from `general relativity plus a slightly different equation of state'?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is correct that at the level of a single mass--radius point the \OGNfive{} correction is degenerate with equation-of-state freedom, and the paper does not call it a clean stand-alone falsifier. The distinguishing claim (Corollary~\ref{cor:T2.2-falsifier}, Eq.~\eqref{eq:CCI-ST7-mTOV}) is structural: the correction enters with a \emph{fixed dimensionless coefficient} $\kappa_{\rm GS}$ tied to the ledger and the \emph{same} functional dependence across all compact stars, so the falsifiable object is a population-level, equation-of-state-orthogonal ratio rather than a single $M$--$R$ shift. We concede honestly that whether such an orthogonal ratio is observationally extractable --- given that a one-parameter equation-of-state deformation can mimic much of the functional form --- is not settled in this paper, and the cosmological/astrophysical falsifiers proper are the territory of \ATen{} and \AFive{}. As stated here, modified TOV is a consistency target, not a near-term kill test, and we have removed the word ``falsifier'' where it claimed more.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 11 (O-GRIP-11) (the mathematical physicist)}\par\nobreak\smallskip
\noindent\textbf{``T2.2's reverse direction derives the compensator one-form $C_\mu$ from the Poincar\'e lemma applied to the closed boundary 2-form. But Poincar\'e gives only \emph{local} exactness; you need a \emph{global} smooth $C_\mu$, and the global obstruction is exactly the open M11 monodromy problem. The iff is proved only locally.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded precisely. The forward direction of T2.2 (Theorem~\ref{thm:T2.2-master-conservation}) is global; the reverse direction constructs $C_\mu$ locally via the Poincar\'e lemma, and the global smooth existence across the codimension-2 instanton strata is the monodromy non-obstruction recorded as open problem M11 (\S\ref{sec:open-problems}, \S\ref{ssec:PBI4-compensator}). So the iff is established locally and globally \emph{modulo} M11. This is the same gap that conditions T4.2, and the paper marks both with the same open problem rather than papering over the local-to-global step. The honest statement of T2.2 is ``iff, with the reverse direction global up to the M11 monodromy hypothesis.''
\end{objqabody}

\bigskip
\section{GFF, Morse theory, and the second law (Pillar 3)}

\bigskip
\noindent\textbf{Objection 12 (O-GRIP-12) (the Morse theorist)}\par\nobreak\smallskip
\noindent\textbf{``T3.1 proves strict positivity iff $\dim_\R \Sigma \equiv 0 \pmod 4$, i.e. $N$ odd. But this is equally satisfied by $N=1,3,5,7,\dots$. T3 contributes \emph{zero} upper bound on the number of generations --- it is a parity filter, nothing more. Advertising it as one of three legs that force $N=3$ overstates its content.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Exactly right, and the paper is careful to assign T3 only the parity role: Theorem~\ref{thm:GFF-positivity-from-CCI} (T3.1) excludes even $N$ via the Hirzebruch-signature obstruction (Proposition~\ref{prop:GFF-Morse-eq-signature}) and contributes \emph{no} upper bound. The three-leg pincer is explicit that the legs do different jobs: CP violation gives $N\geq 3$, the signature/parity gives ``$N$ odd,'' and the budget (Pillar 4, conditional on M10) gives $N\leq 3$; only the conjunction yields $N=3$ (Corollary~\ref{cor:T4-N3-closed}). If any prose calls T3 a leg that ``forces $N=3$'' on its own, that is an overstatement we correct: T3 forces \emph{odd}, and odd-plus-($N\geq3$)-plus-($N\leq3$) forces three. We thank the reviewer for the sharpening and have made the parity-only role explicit at \S\ref{sec:GFF-T3}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 13 (O-GRIP-13) (the Morse theorist)}\par\nobreak\smallskip
\noindent\textbf{``The `alternating index sum = Hirzebruch signature' identity holds only for the equivariant moment-map Morse function under a maximal-torus action on K\"ahler strata with flat connections. For a generic GFF landscape the index sum is the Euler characteristic, not the signature. You have engineered the Morse function to land on the invariant you need.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection correctly identifies the load-bearing assumption, and the paper states it as a scope condition rather than hiding it: Proposition~\ref{prop:GFF-Morse-eq-signature} holds for the moment-map height function on the K\"ahler family-projector strata $\CPN$ with gauge connections flat along $\Sigma$, and that is the class on which the $N$-counting is done. The substantive claim is that the \emph{physical} $\cF_{\GFF}$ on the family-projector stratum \emph{is} the moment map of the residual torus action --- because the stratum is the projective space of family rotations and the free energy is the equivariant height --- not an arbitrary Morse function. We concede this identification is an assumption about which functional the physics delivers, and that for a generic, non-torus-invariant landscape the relevant invariant would be the Euler characteristic and the parity argument would not go through. The reader is entitled to ask for a proof that the physical functional is the moment map; \S\ref{sec:GFF-T3} states it as the operative hypothesis, and a from-dynamics derivation is part of the open agenda.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 14 (O-GRIP-14) (the topologist)}\par\nobreak\smallskip
\noindent\textbf{``Step 3 needs the first Pontryagin class $p_1(\Sigma)\neq 0$, supplied by the $\mathcal H_1/\mathcal H_2$ theorems. But you admit the $\mathcal H_2$ proof leaves the bulk implementation to a separate paper that is \emph{in preparation}. The signature non-triviality --- hence the entire parity result --- rests on an unwritten paper.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and it is one of the paper's genuine soft spots. Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem} (\S\ref{sec:H1-H2-theorems}) upgrade the hypotheses $\mathcal H_1, \mathcal H_2$ of the N=3 program to theorems \emph{within} \GiFour{}, but the $\eta$-invariant non-degeneracy step that underwrites $p_1(\Sigma)\neq 0$ is, as open problem M6 records, sketched here with the detailed bulk implementation deferred to a companion technical note that is not yet public. So the parity result is rigorous modulo M6. We do not claim $\mathcal H_2$ is fully proved in print; we claim it is reduced to a stated $\eta$-invariant computation, flagged as open. A reader is right to treat the parity leg as conditional on M6 until that computation appears.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 15 (O-GRIP-15) (the open-quantum-systems theorist)}\par\nobreak\smallskip
\noindent\textbf{``T3.2 claims $dF/dt \leq 0$ iff CPTP. But CPTP alone gives only monotonicity (Spohn); the clean iff needs an \emph{additional} detailed-balance hypothesis, which you patch into the text. Open systems generically violate detailed balance. State the assumption in the theorem, not in an apology.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted in full, and the theorem statement (Theorem~\ref{thm:T3.2-thermodynamic}/\ref{thm:GFF-H-theorem}) now carries the hypothesis rather than the footnote: strict monotonicity $\dd F_{\GFF}/\dd t \leq 0$ holds for \GDOS{} CPTP dynamics \emph{satisfying detailed balance with respect to $F_{\GFF}$}; without detailed balance one has only the entropy-production inequality $\dot S_{\rm tot}\geq 0$, and the system can sit in a non-equilibrium steady state. This is exactly the A5 result (its T5 Lyapunov lemma) specialized to the family-projector moduli, and the detailed-balance condition is the same one A5 states. The reviewer is right that ``iff CPTP'' is too strong; the corrected headline is ``iff CPTP with detailed balance,'' and that condition is now in the theorem. The honesty cost --- that the second law here is conditional on a non-generic regularity of the dynamics --- is one we state rather than bury.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 16 (O-GRIP-16) (the open-quantum-systems theorist)}\par\nobreak\smallskip
\noindent\textbf{``T3.2 is monotonicity from GDOS, but GDOS is A5 and A5's T5 \emph{is} the boundary second law. So what does A6 add? Either T3.2 is A5's theorem re-imported (not new content), or it strengthens A5 with detailed balance (a new assumption, refuting `no new axiom'). You cannot have both.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The dilemma is real and we accept its second horn cleanly: T3.2 \emph{specializes} A5's general boundary second law (\citep{ArisakaA5GDOS}, its Lyapunov theorem) to the stratified family-projector moduli, and in doing so it makes explicit the detailed-balance regularity under which the strict version holds. That regularity is a \emph{condition on the dynamics}, not a new physical axiom about the bulk --- it is the standard detailed-balance assumption of non-equilibrium thermodynamics, which A5 already invokes --- so ``no new physical axiom'' survives, but ``A6 adds nothing'' does not: A6 adds the specialization to $\CPN$ and the explicit statement of the operative condition. The honest position is therefore the narrow one: T3.2 is not an independent new law; it is A5's law restricted to the stratum where the family count lives, with its hypotheses made explicit. We have rewritten \S\ref{sec:GFF-T3} to say exactly that.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 17 (O-GRIP-17) (the thermodynamicist)}\par\nobreak\smallskip
\noindent\textbf{``The `laboratory falsifier' --- equipartition plus the fluctuation--dissipation relation --- is standard linear-response theory for any system with a free energy. A coherent violation would falsify ordinary statistical mechanics, not specifically GiFour. This is a prediction of thermodynamics, not of your framework.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Correct, and we downgrade the claim accordingly. The equipartition/fluctuation--dissipation relation (Corollary~\ref{cor:T3.2-falsifier}, Eq.~\eqref{eq:GFF-FDT}) is what \emph{any} system with the \GFF{} free energy at equilibrium must satisfy, and a violation would indeed indict statistical mechanics broadly rather than \GiFour{} specifically. Its honest role is not as a discriminating falsifier but as a \emph{consistency requirement}: it certifies that the \GFF{} functional behaves as a genuine free energy, which is a non-trivial check on the construction but not a kill test that singles out the framework. We have relabeled it a consistency condition rather than a falsifier, and the genuinely framework-specific exposure is the integer-ledger structure, not the thermodynamic identity.
\end{objqabody}

\bigskip
\section{GRIP, the cascade, and the budget (Pillar 4)}

\bigskip
\noindent\textbf{Objection 18 (O-GRIP-18) (the skeptic of conjecture-laden proofs)}\par\nobreak\smallskip
\noindent\textbf{``The entire upper bound $N\leq 3$ --- the headline result, exported to A9 and P2 --- rests on $\Sbi = 3\pi$, which your own status remark admits is a named conjecture (M10) taken as input. A 201-page paper whose central numerical claim is an unproven conjecture is a conditional result. The `forces $N\leq 3$' language is not warranted.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We agree the language must carry the conditional, and the theorem does: Theorem~\ref{thm:GRIP-existence-closure} (T4.1) closes the cascade and yields $N\leq 3$ \emph{conditional on the \BIsix{} budget $\Sbi=3\pi$}, which the status remark (\S\ref{sec:GRIP-T4}) and open problem M10 mark as an unproven conjecture. So the honest, unconditional content of A6's GRIP pillar is: the cascade structure, the per-step action $\pi k$, the total $\pi N(N-1)/2$, and the implication ``\emph{if} the budget is $3\pi$ \emph{then} $N\leq 3$.'' The numerical bound itself is conditional. We do not claim a proof of $N\leq 3$; we claim a proof that $N\leq 3$ follows from the budget, plus the honest flag that the budget is conjectural. Every downstream use in A9 and P2 inherits that conditional, and those papers cite it as conditional. The reviewer is right that the abstract should not read ``forces'' without the M10 caveat, and we attach it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 19 (O-GRIP-19) (the skeptic of conjecture-laden proofs)}\par\nobreak\smallskip
\noindent\textbf{``The budget \emph{exactly} equals the $N=3$ cascade cost: $\Stot(\CPtwo)=3\pi=\Sbi$. A budget that saturates precisely at the desired answer, motivated only by a ledger that already encodes $N=3$, is the signature of a tuned constraint. Until M10 is closed, $N\leq 3$ is indistinguishable from `we chose the budget to give 3.'''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The exact saturation $\Stot(\CPtwo)=3\pi$ (Proposition~\ref{prop:GRIP-total-action}, \S\ref{sec:GRIP-T4}) is real and, as the reviewer says, is the thing a tuned constraint would also produce; the framework's self-consistency remark explicitly warns that saturation ``cannot count as evidence for the budget's value inside a derivation of $N=3$.'' This is the same circularity as Objection~8 seen from the cascade side, and the answer is the same: it is documented as M10, and the task M10 sets is to derive $\Sbi=3\pi$ \emph{without} using $N=3$ at any step. That task is done: the capacity theorem T-N3-6 of \citet{ArisakaP2N3} derives $\mathrm{Cap}(N)=\pi N$ from the stratum's quantum geometry, consuming neither $K=35$ nor $J=7$, conditional on the coupling clause and L-INT ($=$ M6). What we will defend is that the framework does not \emph{pretend} the saturation is independent evidence --- it flags it as suspect and quarantines the derivation as open. A reader is entitled to withhold assent to the upper bound until M10 lands, and the paper asks for exactly that posture.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 20 (O-GRIP-20) (the instanton physicist)}\par\nobreak\smallskip
\noindent\textbf{``The per-step action $S_{[2k]\to[2(k-1)]}=\pi k$ is a standard Bogomolnyi / Duistermaat--Heckman result on $\CP^{N-1}$ (Witten 1982). The proof is correct but it is not a result \emph{about generations}. What forces the family-projector moduli to be $\CP^{N-1}$ with this exact torus height function rather than any other K\"ahler stratum?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection draws the right line: Theorem~\ref{thm:instanton-action} (the $\pi k$ result) is standard symplectic geometry on $\CPN$, and the physics is entirely in the identification of that space with the family-projector moduli. The justification offered is that the family sector is the projective space of $N$-family rotations modulo phase --- i.e.\ $\CP^{N-1}$ by construction of the projector moduli (\S\ref{sec:DGI-GSSB-GA}) --- and the height function is the equivariant moment map of the residual symmetry, which is why the $\pi k$ ladder appears. We concede that ``the family moduli \emph{are} $\CP^{N-1}$ with the moment-map height'' is the load-bearing physical input, not a theorem of this paper, and it is the same assumption flagged in Objection~13. The instanton-action theorem is then standard given that input; its novelty is not claimed, only its application. The honest division is: the symplectic result is textbook, the identification with generations is the framework's posited structure.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 21 (O-GRIP-21) (the mathematical physicist)}\par\nobreak\smallskip
\noindent\textbf{``T4.2's reverse direction requires $C_\mu$ to extend smoothly across each codimension-2 instanton stratum, which you admit is established case-by-case for the $\CP^{N-1}$ saddles but lacks a general theorem (M11). So T4.2 is a finite case-check, not a theorem.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted as stated. Theorem~\ref{thm:T4.2-runtime} (T4.2) establishes runtime gauge-invariance preservation \emph{conditional on} the smooth extension of the \PBIfour{} compensator across each instanton transition, verified case-by-case for the finitely many $\CP^{N-1}$ saddles relevant to the $N\leq 3$ application and \emph{not} as a general theorem --- the general monodromy non-obstruction is open problem M11 (status remark, \S\ref{sec:GRIP-T4}). So for the Standard-Model application ($N\leq 3$) the needed cases are checked; the general-$N$ statement is open. We do not claim a general theorem; we claim the finite verification sufficient for the physics plus a flagged conjecture for the rest. The reviewer is correct that ``theorem'' is generous for a finite case-check, and the status remark says exactly that.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 22 (O-GRIP-22) (the dynamical-systems theorist)}\par\nobreak\smallskip
\noindent\textbf{``You split GRIP-in (within the fixed landscape) from GRIP-out (which changes it) and quarantine all the hard dynamics into GRIP-out, declared outside the septuple and supplying only initial data. But \emph{why} the system starts at the top DGI saddle --- what sets $c_{\rm top}$ --- is exactly GRIP-out, which you exclude. The theorem governs only the easy half.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection correctly identifies the scope boundary, and we own it: Theorems~\ref{thm:GRIP-existence-closure} and~\ref{thm:T4.2-runtime} govern \GRIPin{} --- the descent on a fixed landscape from given initial data $c_{\rm top}$ --- and the selection of $c_{\rm top}$, i.e.\ how the landscape was built and the system placed at its top, is \GRIPout{}, which the paper explicitly places outside the septuple (\S\ref{sec:DGI-GSSB-GA}). This is deliberate, not evasive: the cohort assigns the landscape-\emph{building} event (the one-time electroweak \GRIPout{}) to the boundary-law and cosmology papers, while A6 formalizes the \emph{traversal}. We concede that this means A6's theorems do not explain the initial condition, and that the ``interesting'' question of vacuum selection is GRIP-out and hence out of scope here. The honest claim is that A6 proves the descent dynamics given the landscape, not the genesis of the landscape.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 23 (O-GRIP-23) (the instanton physicist)}\par\nobreak\smallskip
\noindent\textbf{``Step 4 of the instanton-action theorem conflates the symplectic \emph{volume} $\pi$ of one $\CP^1$ with the \emph{action} of a tunneling trajectory. These are different objects; the factor-$k$ cumulative `symplectic depth' is asserted, not computed. Show the action integral, not the Duistermaat--Heckman volume.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The reviewer is right that the step deserves the explicit Bogomolnyi line integral, and the final-review pass supplies it. The action of the Bogomolnyi-saturating gradient trajectory is $S=\int |\nabla \cF_{\GFF}|\,|\dot x|\,\dd t = \int \dd\cF = \cF(c_+)-\cF(c_-)$ (the saturated bound, \S\ref{ssec:Bogomolnyi}, Eq.~\eqref{eq:Bogomolnyi}), and for the moment-map height on $\CPN$ the value difference between adjacent fixed points of index $2k$ and $2(k-1)$ is the symplectic area enclosed, which equals $\pi k$ in Fubini--Study normalization. So the equality of the trajectory action with the symplectic depth is the Bogomolnyi saturation, not a conflation --- but we agree the text stated the Duistermaat--Heckman volume and let the reader infer the equality, and it now shows the action integral explicitly. The result $\pi k$ stands; the derivation is made line-by-line.
\end{objqabody}

\bigskip
\section{The decay-width ladder and the falsifiers}

\bigskip
\noindent\textbf{Objection 24 (O-GRIP-24) (the phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``Your geometric gate factor is $e^{-\pi}\simeq 0.043$ (a factor 23), but the observed $\tau_\mu/\tau_\tau \simeq 7.6\times 10^6$ is delivered almost entirely by the Sargent $m^5$ phase space ($\sim 10^6$). The geometric contribution is about 0.5\% of the log. Calling the generation lifetime ladder a signature of the framework is hype.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The arithmetic is right and the paper now states it without inflation (\S\ref{sec:GRIP-lifetimes}, \S\ref{ssec:GRIP-action-to-lifetime}): the cascade supplies only the geometric gate factor $\Gamma_{[2k]\to[2(k-1)]}\propto \e^{-\pi k}$, and the absolute lifetimes are dominated --- by orders of magnitude --- by the species-specific phase space, which in fact reverses the naive geometric ordering for the charged leptons. So the ``ladder'' is, quantitatively, a small geometric residue on top of standard weak-decay kinematics, and the robust \GRIP{}-level content (Structure Theorem GRIP-ST7) is the \emph{existence and generation-ordering} of the gate factors, not a numerical lifetime prediction. The honest claim, which matches \citep[\S7.4]{ArisakaP2N3} and \ANine{}, is that the absolute numbers belong to A9 and the framework owns only the geometric factor. We have removed any phrasing that presented the lifetime numbers as a framework triumph.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 25 (O-GRIP-25) (the phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``Falsifier 4 (F-GRIP-4) is the ordering of gate factors `after the known phase-space factors are divided out.' But those factors are huge and model-dependent, deferred to A9. To extract a factor-23 residual you need a complete SM-lifetime calculation accurate to better than 0.5\%. Until A9 exists, this falsifier cannot be executed even in principle.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded: the gate-ordering falsifier (\S\ref{ssec:GRIP-lab-tests}) is only executable once the phase-space factors are computed to sub-percent accuracy, and that computation is \ANine{}'s, not this paper's. So Falsifier 4 (F-GRIP-4) is, today, contingent on A9 and is not a stand-alone near-term test. We state the honest version: the robust falsifiable content is structural --- a controlled-kinematics decay class in which the Sargent factor is a \emph{uniform} multiplier across gates, so that the residual $\e^{-\pi k}$ ordering must survive --- and identifying such a class and computing its kinematics is downstream work. A reader is right that, absent A9, the lifetime falsifier is a program, not an executable test, and the paper now says so rather than advertising an immediate kill criterion.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 26 (O-GRIP-26) (the phenomenologist)}\par\nobreak\smallskip
\noindent\textbf{``Falsifier 6 (F-GRIP-6), the `cascade-action sum rule' $\sum S = 4\pi$ `comparable to the age of the universe in dimensionless units,' has no unit conversion and no dimensionful prediction. This is numerology, not a falsifier. Remove it or make it dimensionful.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted; it is the weakest item in the section and we treat it as such. The sum rule $\sum_i \Stot^{(i)} = \pi N(N-1)(N+1)/6 = 4\pi$ at $N=3$ (\S\ref{ssec:GRIP-lab-tests}) is a closed arithmetic identity about the cascade actions, with no dimensionful bridge to a physical timescale, and the ``comparable to the age of the universe'' remark is a coincidence with no derivation behind it. The paper labels it ``a weak but interesting consistency check,'' which is generous; the honest move, taken in the final-review pass, is to demote it from the list of falsifiers to a numerical remark, or to remove the cosmological-timescale comparison entirely. It is not a falsifier and is no longer presented as one.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 27 (O-GRIP-27) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{``Your master move --- read every conservation law off one connection --- yields \emph{necessary} conditions (anomaly cancellation, Ward closure, Bianchi identities) that any consistent theory must satisfy, then presents satisfying them as a positive achievement. Necessary conditions for consistency are not predictions.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The distinction is the right one and the paper should make it cleanly (\S\ref{sec:discussion-significance}). Anomaly freedom and Ward closure are indeed necessary conditions that any consistent theory meets, and meeting them is not, by itself, a prediction. What the framework claims as positive content is not ``we are consistent'' but ``the constitution \emph{forces} a specific gauge content and a specific integer ledger, and those force specific numbers'' --- $\alpha^{-1}=137$ at the integer level, $\Mlock\approx 3.85$~TeV, the kaxion line, the family count --- which are the discriminating outputs, and which live downstream in A4/A9/P1/C1. So the honest framing is: the conservation laws are necessary-condition scaffolding; the predictions are the forced integers. We concede that where the prose presents anomaly cancellation \emph{itself} as a triumph it conflates the two, and the corrected emphasis is on what the framework forbids (a non-integer ledger, a fourth family at low budget) rather than on satisfying universal consistency requirements.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 28 (O-GRIP-28) (the working physicist)}\par\nobreak\smallskip
\noindent\textbf{``On inspection every falsification channel is degenerate, deferred, or standard physics: modified TOV is equation-of-state-degenerate, equipartition is thermodynamics, the gate-ordering needs an unbuilt A9 calculation, the sum rule is numerology, the cosmology is deferred to A10. What single near-term, GiFour-only measurement could return a result that kills the framework?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} This is the fair bottom line of the falsifier critique, and the honest answer is that A6's \emph{own} near-term kill tests are weak --- the sharp, near-term falsifiers of the program live downstream, and A6 should point to them rather than claim them. The framework-specific, near-term exposures are: the kaxion line at $8.1~\mu$eV ($\approx 1.95$~GHz) in haloscopes (\citep{ArisakaC1Kaxion}), the KK cluster near $3.85$~TeV at HL-LHC (\citep{ArisakaP1KK}), and the integer $\alpha^{-1}=137$ structure (\citep{ArisakaA9SM}); a clean null at the kaxion line, or a fourth light family, would propagate back through the ledger that A6 formalizes. A6's own contribution is structural --- the rank-one/integer-ledger architecture and the conditional $N\leq 3$ --- and its falsifiability is mathematical (the ledger is wrong if the trace data give a different rank) more than experimental. We concede A6 in isolation offers no decisive near-term laboratory kill test, and we now point the reader to the downstream ones rather than overselling the in-paper ``falsifiers.''
\end{objqabody}

\bigskip
\section{T5: constitutional equivalence}

\bigskip
\noindent\textbf{Objection 29 (O-GRIP-29) (the logician)}\par\nobreak\smallskip
\noindent\textbf{``T5 claims the GiFour system and the Constitution are equivalent in derivational content --- mutual derivability. But the statement concedes direction (2) is established only `up to two auxiliary propositions whose internal proofs are sketched.' A bi-implication with one direction admittedly unproven is not an equivalence; it is one theorem plus a conjecture.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted, and the status note on Theorem~\ref{thm:constitutional-equivalence} (T5, \S\ref{sec:T5}) says so: direction (1) --- the constitution plus \BIsix{} yield the \GiFour{} pillars --- is proved; direction (2) --- the pillars recover the constitutional axioms --- is established up to two auxiliary propositions whose proofs are sketched, and is presented as a structured proof sketch, not a completed proof. So T5 is honestly ``one direction proved, the converse sketched,'' and the word ``equivalence'' should carry that qualification, which the status note supplies and the abstract should echo. We do not claim a completed bi-implication; we claim one direction and a sketched converse, and the reviewer is right that ``equivalence'' unqualified overstates it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 30 (O-GRIP-30) (the logician)}\par\nobreak\smallskip
\noindent\textbf{```Equivalent in derivational content' is never precisely defined. Without specifying the deductive system, the axioms, and what counts as a theorem, mutual derivability is not a checkable claim. As stated it is unfalsifiable hand-waving about two informal vocabularies.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The objection is correct that \S\ref{sec:T5} gives the equivalence an informal gloss (``anything derivable in one is derivable in the other'') rather than a formal definition over a fixed deductive system. The defensible precise content is narrower and we state it: each constitutional axiom (\AX{1}--\AX{4}) appears as a named theorem or proposition of the \GiFour{} system (the bridge maps, \S\ref{sec:bridge-OGGEP}), and each \GiFour{} principal theorem is derived from the constitution plus \BIsix{} (direction (1)); ``equivalence'' is the conjunction of those two finite lists of named derivations, not a meta-theorem about all possible deductions. So the checkable claim is the explicit correspondence table, not an abstract mutual-derivability over an unspecified logic. We concede the prose reaches for the grander, vaguer statement, and the corrected version is the finite, tabulated correspondence --- which is checkable, modulo the direction-(2) sketches.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 31 (O-GRIP-31) (the logician)}\par\nobreak\smallskip
\noindent\textbf{``Even the sketch of direction (2) recovers GRIP via a proposition `conditional on M10' and defers GFF to a Stinespring argument that is `part of the broader program.' The equivalence inherits every load-bearing gap --- M10, M11 --- that the rest of the paper leans on.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Yes --- direction (2) of T5 is no more secure than the pillars it reconstructs, and it inherits exactly their conditionals. Proposition~\ref{prop:GRIP-equiv-OGGEP-cascade} (recovering GRIP) is conditional on M10, and the GFF reconstruction (Proposition~\ref{prop:GFF-equiv-OGGEP}) is reduced to a Stinespring dilation argument not completed here. So the equivalence is conditional on the same cascade conditioning
that T4 carries --- today M11's general statement, M10 being resolved
externally. This is not a separate weakness; it is the same weakness propagating into T5, and the paper marks it rather than presenting T5 as a clean keystone. The honest reading: T5's forward direction is solid, its converse is a conditional sketch carrying the cascade's open problems, and the ``equivalence'' is therefore as conditional as the cascade.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 32 (O-GRIP-32) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``If T5 is true --- the two systems are different presentations of the same content and the GiFour reorganization `is not a new theory' --- then nothing in A6 is independently provable that was not already provable from the Constitution plus BI-6. The 201 pages prove no new physics. Your value claim and your equivalence claim are in tension.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The tension is real and we resolve it by being precise about ``new.'' T5 says A6 introduces no new \emph{physical content} beyond the constitution plus \BIsix{} --- and that is the intended claim: A6 is the formalization paper, not a new-physics paper. What A6 adds is not new physics but new \emph{rigor and organization}: it gives the pillars explicit theorem statements, proofs (some conditional), and a dual-layer architecture that makes the latent content auditable, in the way that Lagrange and Hamilton added no new mechanics to Newton but made it provable. So the resolution is: ``no new physics, genuine new formal content.'' We concede the prose sometimes blurs ``new theorems'' (formal) with ``new physics'' (content) and that the two must be kept apart; the corrected statement is that A6's novelty is formalization, and T5 is precisely the assertion that the formalization is faithful.
\end{objqabody}

\bigskip
\section{The ``no new axiom'' claim and the promoted hypotheses}

\bigskip
\noindent\textbf{Objection 33 (O-GRIP-33) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{``You insist A6 `introduces no additional physical axiom' and `merely makes explicit' latent content --- yet you claim five principal theorems and genuinely new content. A pure formalization cannot also be a source of new theorems. Which is the honest description?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Both, once ``new'' is split as in Objection~32: no new \emph{physical axiom}, genuinely new \emph{formal theorems}. The five principal theorems are derivations \emph{from} the existing constitution plus \BIsix{}; they add no postulate to the framework's physics, but they are non-trivial mathematical statements (existence/uniqueness of the composite connection, the conservation iff, the parity result, the conditional cascade closure, the equivalence) that were not previously written down or proved. The analogy the paper draws --- Newton needed Lagrange and Hamilton --- is exactly this: new theorems, no new physics. We accept that the bare phrases ``no new axiom'' and ``new content'' read as contradictory, and the paper now consistently says ``no new physical axiom; new formal results,'' which is the honest description and is what T5 certifies.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 34 (O-GRIP-34) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{``The detailed-balance hypothesis you smuggled into T3.2 \emph{is} a new physical assumption about the open-system dynamics, not present in a generic CPTP map. That alone refutes `no new axiom.' Either retract the claim or prove detailed balance from prior axioms.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} We accept the force of this and draw the line carefully. Detailed balance is a \emph{regularity condition on the dynamics} under which the strict second law holds; it is the standard near-equilibrium assumption of non-equilibrium thermodynamics, and it is already invoked in A5's boundary second law (\citep{ArisakaA5GDOS}), which A6 specializes. It is therefore not a new \emph{constitutional} axiom of the \GGTheory{} framework, but it \emph{is} an extra hypothesis on the regime in which T3.2's iff holds, and the reviewer is right that calling the framework ``axiom-free'' while relying on it is sloppy. The corrected statement: the four constitutional axioms are unchanged; T3.2's strict form additionally assumes detailed balance, stated in the theorem. We do not claim to derive detailed balance from the constitution --- generic open dynamics violate it --- so the honest position is that the strict second law is a conditional result, and ``no new axiom'' refers to the constitutional axioms, not to every regularity condition a theorem may need.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 35 (O-GRIP-35) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``You say the hypotheses $\mathcal H_1, \mathcal H_2$ are `promoted to theorems' in GiFour --- then concede A6 does not provide rigorous proofs of them and defers them to a paper in preparation. Promoting a hypothesis to a theorem whose proof is in an unwritten paper is renaming, not promoting.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and the wording is corrected. What A6 actually does (Theorems~\ref{thm:H1-theorem},~\ref{thm:H2-theorem}, \S\ref{sec:H1-H2-theorems}) is \emph{reduce} $\mathcal H_1, \mathcal H_2$ from free-standing postulates of the N=3 program to consequences of the \GiFour{} structure \emph{modulo} a stated $\eta$-invariant computation (open problem M6) whose bulk implementation is deferred. ``Reduced to a named open computation'' is honest; ``promoted to a theorem'' is not, when the computation is not in print. So the corrected claim is the weaker, true one: within \GiFour{}, $\mathcal H_1$ and $\mathcal H_2$ are no longer independent assumptions but follow from the pillar structure once M6 is discharged. We agree ``promoted'' oversold a reduction-modulo-open-problem, and the text now says ``reduced to M6.''
\end{objqabody}
\bigskip
\section{Cohort interdependence and the risk of circular citation}

\bigskip
\noindent\textbf{Objection 36 (O-GRIP-36) (the skeptic of citation loops)}\par\nobreak\smallskip
\noindent\textbf{``A6's T4.1 imports the budget $3\pi$ from P2 (Appendix C), while A6's outward map exports $N\leq 3$ back to P2 and A9 for the hard upper bound on family count. P2 supplies the conjecture A6 needs; A6 supplies the bound P2 cites. This is a citation cycle: neither paper derives the result without the other assuming it. Where is the acyclic foundation?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The cycle the reviewer draws is real at the level of the \emph{statement} $N\leq 3$, and the acyclic foundation is the shared open problem, named identically in both papers. The unconditional, acyclic content is: A6 proves ``\emph{if} $\Sbi=3\pi$ \emph{then} $N\leq 3$'' (Theorem~\ref{thm:GRIP-existence-closure}); P2 proves the same conditional from the family side; and the budget is proved in one place only --- the capacity theorem T-N3-6 of \citet{ArisakaP2N3}, conditional on its coupling clause and L-INT ($=$ M6) --- while this paper consumes it as input. So there is no circular \emph{proof}: there is one derivation, one residue (M6 $=$ R-INT) that both papers flag, and a conditional implication that each proves independently. The cycle would be vicious only if one paper claimed to \emph{derive} the budget by citing the other; neither does --- both cite M10 as open. The honest statement is that $N\leq 3$ rests on the coupling clause and M6 ($=$ R-INT), and the cohort's job is to close M6, not to launder it between papers. We have made the shared-open-problem status explicit at \S\ref{sec:coherence-map}.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 37 (O-GRIP-37) (the skeptic of citation loops)}\par\nobreak\smallskip
\noindent\textbf{``T2.1's Diophantine uniqueness is imported from A2 and A3, the modified-TOV from A5, the $\mathcal H^2$ normalization from A2. The four pillar theorems of A6 are, at their load-bearing steps, citations to four other single-author preprints, none re-derived in this document. A reader cannot verify the core results without four unavailable companions.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The dependency is exactly as described and is the explicit design of the cohort (the inward coherence map, \S\ref{sec:coherence-map}): A6 consumes the bulk geometry (A2), the anomaly closure (A3), and the boundary law (A5), and builds the pillar formalization on them. We concede the consequence the reviewer names: A6 is \emph{not} self-contained --- its load-bearing inputs are theorems of A2/A3/A5, and verifying them requires those papers. This is the cost of a partitioned program, and the mitigation is that each import is named at its point of use with the specific theorem cited, so the dependency is auditable rather than hidden. We do not claim A6 stands alone; we claim it stands on named, located results of the upstream papers, which a reader must consult. A monograph would re-prove them; a cohort cites them. The reviewer is right to weigh that trade-off, and the inward map is the instrument for doing so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 38 (O-GRIP-38) (the philosopher of physics)}\par\nobreak\smallskip
\noindent\textbf{``The whole cohort cross-validates internally --- you call the convergence of the aesthetic and empirical routes `one of the strongest internal consistency checks.' But internal consistency among co-authored, mutually-citing preprints is not external evidence. The framework risks being a closed deductive island that agrees with itself by construction.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The epistemic point is correct and we accept it: agreement \emph{within} the cohort is consistency, not confirmation, and the paper should not present internal convergence as empirical support. The genuine external exposure of the whole program is the small set of falsifiable numbers --- $\alpha^{-1}=137$, $\Mlock\approx 3.85$~TeV, the kaxion line at $8.1~\mu$eV, $\Omega_m=1/\pi$ --- which face data independently of how internally coherent the derivation is. Internal consistency checks (such as the two routes to the ledger agreeing) are valuable as \emph{bug-detection} --- an inconsistency would falsify the construction --- but they are not evidence that the construction is true of the world. We concede that any wording presenting internal convergence as confirmation overclaims, and the corrected framing (\S\ref{sec:discussion-significance}) is that the framework's truth-claim rests on the downstream empirical numbers, with internal coherence as a necessary but non-confirming precondition.
\end{objqabody}

\bigskip
\section{Conjecture density at the core: M9, M10, M11}

\bigskip
\noindent\textbf{Objection 39 (O-GRIP-39) (the skeptic of conjecture-laden proofs)}\par\nobreak\smallskip
\noindent\textbf{``Three named open problems sit directly under the central results: M9 (the norm choice that makes the ledger unique), M10 (the budget that forces $N\leq 3$), M11 (the compensator smoothness that closes the conservation iff). Yet you open the conclusions claiming the framework is `mathematically complete, all nine theorems proved.' That claim is contradicted three subsections later.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The contradiction is real and the ``mathematically complete'' phrasing is withdrawn. The accurate statement, which the open-problems triage (\S\ref{sec:open-problems}) already supports, is: the nine dual-layer sub-theorems are proved \emph{conditional on} three named hypotheses --- M9 (the \MSMF{} norm), M11 (the compensator monodromy), and, for the upper bound, the coupling clause with M6 ($=$ R-INT) through which M10's met demand now conditions T4.1 --- each underwriting one of the three deliverables (ledger uniqueness, the upper bound, the conservation closure). So the framework is \emph{conditionally} complete: every theorem has a proof modulo a stated, located conjecture, and the conjectures are the honest residue. We should not, and no longer, say ``complete'' without the
conditioning ledger of \S\ref{sec:open-problems} attached. The reviewer is right that the opening sentence of the conclusions oversold a conditional result, and it is corrected to match the triage that follows it.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 40 (O-GRIP-40) (the number theorist)}\par\nobreak\smallskip
\noindent\textbf{``M9 admits the ledger's uniqueness is proved only under the $\ell^1$ norm, with a norm-independence theorem `missing.' If uniqueness is norm-dependent, then $(1,3,5;7,35)$ is the minimizer of a \emph{chosen} complexity functional, not a forced one. `Minimal' is doing covert work.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded. The \MSMF{} minimality that selects the ledger is, as M9 states, established for the $\ell^1$ complexity norm, and while other natural norms appear to select the same minimizer, a norm-independence theorem is not in hand. So ``minimal'' is, strictly, ``minimal with respect to the $\ell^1$ norm,'' and the uniqueness is conditional on that choice. We do not claim the ledger is the minimizer under \emph{every} complexity measure; we claim it under the stated one and flag the independence as open (M9). The reviewer is right that ``forced'' is too strong while the norm is a choice; the honest reading is ``the unique $\ell^1$-minimal admissible ledger,'' and whether the choice of norm can be removed is exactly what M9 asks.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 41 (O-GRIP-41) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``M10's own phrasing --- derive $3\pi$ `independent of $N=3$ at every intermediate step, so as to rule out any circularity' --- is an admission that the current derivation is \emph{not} independent of $N=3$. You have documented your own circularity and labeled it future work.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} That was a fair reading of M10's demand, and the demand has since been met on its own terms: the capacity theorem T-N3-6 of \citet{ArisakaP2N3} derives the budget through the stratum's quantum geometry, with $K=35$ and $J=7$ appearing at no step --- exactly the $N$-independence the wording required. The seam-count route this paper documented as circular remains circular; it is simply no longer the route. We regard documenting the circularity precisely --- rather than obscuring it --- as the correct scholarly posture for a load-bearing gap, but we do not pretend it is anything other than a gap. The upper bound $N\leq 3$ is conditional on the coupling clause and M6 ($=$ R-INT); the formerly documented circularity is dissolved by the external derivation; closing M6 is the remaining task. A reader who concludes ``the upper bound is earned modulo one named lemma'' has read the paper correctly, and M10 is written so that they will.
\end{objqabody}

\bigskip
\section{Relationship to string theory, 6D supergravity, holography, and Randall--Sundrum}

\bigskip
\noindent\textbf{Objection 42 (O-GRIP-42) (the string theorist)}\par\nobreak\smallskip
\noindent\textbf{``You frame GiFour as superior to string theory because $d=6$ is `forced by the magic-dimension ladder' while string theory `chooses' $d=10$. But $d=10$ is \emph{derived} from worldsheet anomaly cancellation, whereas your $d=6$ rests on the magic-dimension ladder, which is an algebraic preference for normed division algebras, not a consistency requirement. You criticize string theory for a choice you also make.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The reviewer is right that the magic-dimension ladder is a mathematical organizing principle, not a worldsheet-style consistency derivation, and the comparison should be drawn at that level (\S\ref{sec:string-AdSCFT-RS}). The honest distinction is not ``forced vs chosen'' --- both frameworks make a structural commitment --- but \emph{which rung and why}: string theory takes $d=10$ because supersymmetry needs the octonionic spinor identity to close, while \GGTheory{} takes $d=6$ as the unique magic dimension realizable \emph{without} supersymmetry (the quaternionic rung). Neither is a pure consistency theorem in the other's sense. We concede that any wording calling our $d=6$ ``forced'' while calling string theory's $d=10$ ``chosen'' applies a double standard, and the corrected comparison states that both rest on a structural principle, the discriminating question being the empirical content each dimension is asked to carry (the integer ledger, here) rather than the dimension count itself.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 43 (O-GRIP-43) (the 6D-supergravity specialist)}\par\nobreak\smallskip
\noindent\textbf{``A 6D theory with $H_3$, $X_4 = \tr R^2 - \Tr F^2 - \dots$, and a Green--Schwarz inflow term is \emph{exactly} the well-studied 6D $(1,0)$ supergravity anomaly setup (Green--Schwarz--Sagnotti, Sagnotti). Your $\BIsix$ is the standard 6D GS mechanism, and the paper barely cites this literature. The new claims must be located relative to known 6D anomaly results, or they look like rediscovery.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Accepted, and the location is the same one \AThree{} \citep{ArisakaA3BI6} draws: the Green--Schwarz mechanism and the modified Bianchi identity are standard, borrowed wholesale, and the engagement with the 6D anomaly literature belongs in A3, where the descent and global completion are proved. A6 \emph{uses} that machinery and does not re-derive it, so the present paper's thin citation of the 6D-SUGRA literature reflects the division of labor, not a claim of originality for the GS mechanism. What is new --- and what must be located relative to the standard results --- is the \emph{four-component} $X_4$ with the integer trace-ratio lock and the constitutional forcing of the gauge content, which is A3's contribution, not a rediscovery of GS. We concede A6 should signpost the reader to A3 (and through it to the Green--Schwarz--Sagnotti literature) more prominently at \S\ref{sec:CCI-vs-BI6}, and that the GS mechanism itself is in no way claimed as original here.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 44 (O-GRIP-44) (the AdS/CFT specialist)}\par\nobreak\smallskip
\noindent\textbf{``You repeatedly invoke `holographic descent' and bulk-to-boundary maps, but the bulk is a warped product with a real boundary (Randall--Sundrum-like), not asymptotically AdS --- there is no large-$N$ CFT, no GKP--Witten dictionary, no conformal boundary. Using `holographic' for ordinary KK zero-mode reduction is misleading branding.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The reviewer is right and the paper should say plainly what the operation is: ``holographic descent'' here means Kaluza--Klein zero-mode projection plus boundary-term variation onto the four-dimensional boundary of the warped product (\S\ref{ssec:KK-reduction}), in the holographic-renormalization-group \emph{sense} in which the warp (radial) coordinate is an energy scale --- not AdS/CFT in the GKP--Witten sense. There is no asymptotically-AdS region, no large-$N$ CFT, and no conformal boundary, and we do not claim a holographic duality. The word ``holographic'' is used for the radial-as-scale intuition and the boundary projection, and we concede that without the explicit disclaimer it reads as a claim of AdS/CFT, which it is not. The clarifying statement --- ``holographic descent $=$ KK zero-mode projection $+$ boundary variation, with the warp as RG scale'' --- is now at the first use (\S\ref{ssec:AdS-CFT}, \S\ref{ssec:RS-model}).
\end{objqabody}

\bigskip
\noindent\textbf{Objection 45 (O-GRIP-45) (the Randall--Sundrum specialist)}\par\nobreak\smallskip
\noindent\textbf{``You claim the hierarchy $\EGUT/\EPl \sim e^{-35}$ is forced by the ledger $K=35$ and superior to RS's free warp slope. But pinning the warp exponent to the integer $35$ from a Diophantine ledger is a choice of boundary condition --- exactly the data RS leaves free. Why is $e^{-K}$ with $K=35$ a derivation and RS's warp not?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The honest answer is that the warp exponent is forced \emph{relative to the ledger}, and the ledger is the constitutional commitment --- so the comparison with RS is not ``derived vs free'' in the abstract but ``tied to the integer ledger vs continuously tunable.'' In RS the warp slope $k r_c$ is a continuous parameter fixed only by matching the observed hierarchy; in \GGTheory{} the warp depth is the integer $K=35$ of the \OGNfive{} ledger (\S\ref{sec:string-AdSCFT-RS}), so it cannot be slid continuously without breaking the same ledger that fixes $\alpha^{-1}=137$ and the family count. That is the substantive difference: not that ours descends from nothing, but that it is locked to an integer that simultaneously fixes other observables, whereas RS's is a free continuous dial. We concede that calling ours a ``derivation'' while the ledger itself is a constitutional input overstates it; the defensible claim is ``integer-locked and over-constrained'' versus ``continuously free,'' and that is the comparison the corrected text draws.
\end{objqabody}

\bigskip
\section{Method, scope, and specific technical soft spots}

\bigskip
\noindent\textbf{Objection 46 (O-GRIP-46) (the skeptic)}\par\nobreak\smallskip
\noindent\textbf{``The dual-layer `compatibility relations' $\mathfrak C_{\TGG}, \mathfrak C_{\CCI}, \mathfrak C_{\GFF}, \mathfrak C_{\GRIP}$ are each described as `a one-line proposition with the proof: $\dots$.' A one-line proposition with the proof folded into the same sentence --- especially when the `via' clause is exactly the open M11 --- is an assertion, not a proof. The dual-layer architecture multiplies theorem labels without multiplying proven content.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The criticism of the presentation is fair: the compatibility relations (Eqs.~\eqref{eq:CTGG-compat}--\eqref{eq:CGRIP-compat}) link each pillar's algebraic layer to its operational layer, and three of the four are genuinely one-line consequences of the corresponding pillar theorem, but $\mathfrak C_{\GRIP}$ routes through the compensator smoothness and is therefore conditional on M11 --- which a one-line statement obscures. We concede that folding ``the proof'' into the proposition sentence is not a proof and that, for $\mathfrak C_{\GRIP}$ especially, the dependence on M11 must be stated in the proposition, not assumed in a clause. The corrected treatment gives each compatibility relation its own short displayed proof (or, for $\mathfrak C_{\GRIP}$, an explicit ``conditional on M11''), so that the dual-layer labels correspond to checkable content rather than to relabeled assertions. The reviewer is right that label-count is not proof-count.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 47 (O-GRIP-47) (the working physicist)}\par\nobreak\smallskip
\noindent\textbf{``Why 201 pages and twelve Parts for content the paper itself calls `austere formalization, not a new theory'? The length, the abbreviation glossary, the `fourth scientific revolution' framing, and the Friston free-energy / neuroscience bridge are scope-inflation around a core that reduces to standard 6D GS inflow plus a Diophantine ledger plus a $\CP^{N-1}$ Morse cascade, three of whose key steps are open conjectures.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} There is a real tension between ``austere formalization'' and 201 pages, and we accept part of the criticism. The length is driven by the dual-layer treatment of four pillars with worked examples and the cohort-standard front and back matter; the core mathematical content is indeed compact --- the composite-connection conservation law, the integer ledger (imported), and the $\CP^{N-1}$ cascade
(conditional, today, on M11's general statement; M10 resolved
externally). The ``fourth scientific revolution'' framing and the free-energy/neuroscience appendix are motivational and interdisciplinary scope, explicitly marked as such, and a reader who wants only the theorems can take the pillar Parts and skip them. We concede that the framing material should not be mistaken for load-bearing content, and that the paper's defensible core is exactly the compact list the reviewer gives, three of whose steps are conditional. The honest one-line summary of A6 is: a formal organization of standard inflow plus a Diophantine ledger
plus a Morse cascade, with the named conditioning of
\S\ref{sec:open-problems} at the load-bearing steps --- and the
length is presentation, not additional proof.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 48 (O-GRIP-48) (the differential geometer)}\par\nobreak\smallskip
\noindent\textbf{``The chain `three quaternionic structures of $\HH^2$ $\Rightarrow$ family triplicity $N=3$' is asserted, not derived: $\HH^2$ has an $SU(2)\cong Sp(1)$ of complex structures (a two-sphere's worth), not three discrete ones. Mapping `$I,J,K$' to three families is a numerological coincidence ($\Im\HH$ is three-dimensional) dressed as structure.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Conceded, and the paper says it is not a derivation: the quaternionic ``triplicity'' picture (\S\ref{sec:magic-ladder}) is a suggestive heuristic, explicitly flagged as ``not yet a closed derivation,'' and the reviewer's point that the quaternionic complex structures form a two-sphere, not three discrete objects, is exactly why it cannot be load-bearing. The actual $N=3$ result does \emph{not} rest on the $I,J,K$ picture --- it rests on the CP lower bound, the signature parity, and the (conditional) budget. The quaternionic reading is decoration that gestures at why three is natural; it is not a proof and the paper does not use it as one. We agree it should be clearly cordoned off as heuristic so no reader mistakes the three-dimensionality of $\Im\HH$ for a derivation of three generations, and \S\ref{sec:magic-ladder} now does so.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 49 (O-GRIP-49) (the representation theorist)}\par\nobreak\smallskip
\noindent\textbf{``A footnote flags `$\dim\Spin(1,5)=15$, not 12 as a careless count might suggest' --- but $15=\binom{6}{2}$ is elementary. Flagging $12$ as a tempting wrong answer suggests the magic-ladder bookkeeping may rest on similarly ad hoc counts. Where do $K=35$ and $J=7$ come from group-theoretically, not just Diophantine fitting?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The footnote is over-cautious and we will trim it; $\dim\Spin(1,5)=15$ needs no defending. On the substantive question: $K$ and $J$ are \emph{not} claimed to be dimensions of a group --- they are entries of the \OGNfive{} integer ledger fixed by the Diophantine cascade ($K=LJ=M^2+N^2+L^2=35$, $J=2N+1=7$), with $L=5$ from $\SU(5)$ and $N=3$ from the trace ratio (\S\ref{sec:CCI-coefficients}, and the full derivation in \citep{ArisakaA3BI6}). So they are arithmetic outputs of the ledger, not group dimensions, and the reviewer is right that we should not let magic-ladder bookkeeping imply otherwise. The honest statement is that $K$ and $J$ are ledger integers with a Diophantine, not a representation-theoretic, origin; any group-dimension coincidence (e.g.\ $35=\dim\SU(6)$) is not used and is not claimed. We have removed the misleading footnote and stated the ledger's arithmetic provenance plainly.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 50 (O-GRIP-50) (the open-quantum-systems theorist)}\par\nobreak\smallskip
\noindent\textbf{``You invoke `the boundary is CPTP at the foundational level' as the deep novelty, but a CPTP map requires a specified environment and dilation. You point to a Stinespring-dilation form but never construct the dilation or the environment Hilbert space. `The boundary is CPTP' is a slogan; CPTP-ness is a property of a map you have not written down.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The construction the reviewer asks for is \AFive{}'s, not A6's, and the paper should point there rather than gesture. The \GDOS{} boundary channel --- the CPTP map, its Stinespring dilation, the environment (the integrated-out bulk/KK modes), and the GKLS generator --- is constructed in \citep{ArisakaA5GDOS} (its open-system theorems); A6 \emph{uses} that channel to define the free-energy flow and does not reconstruct it. So ``the boundary is CPTP'' is not an A6 slogan standing on nothing; it is an A5 theorem A6 imports. We concede that A6, read alone, asserts CPTP-ness without exhibiting the dilation, and that the reader must go to A5 for the actual map. The honest signpost --- ``the CPTP channel and its dilation are constructed in A5; here it is used'' --- now appears at \S\ref{sec:GFF-T3}, and A6 claims no independent construction of the channel.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 51 (O-GRIP-51) (the mathematical physicist)}\par\nobreak\smallskip
\noindent\textbf{``Hopkins--Singer differential cohomology is deployed to make the `$[X_4]$ descends to $p_1(\Sigma)$' step precise, but the decisive $\eta$-invariant non-degeneracy lemma is admitted to be a sketch deferred to a paper in preparation (M6). Importing heavy machinery whose decisive lemma is unproven adds apparent rigor without actual rigor.''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} Fair, and the same concession as Objection~14 from the machinery side. The Hopkins--Singer differential-cohomology framework (\S\ref{sec:CCI-Hopkins-Singer}) is the correct setting for the integral-class descent, and the framework is used honestly to \emph{state} the non-degeneracy condition precisely --- but the decisive $\eta$-invariant computation that establishes it on the bulk is open problem M6, sketched here and deferred. So the machinery makes the claim \emph{precise and checkable}, which is a genuine gain, but it does not by itself \emph{prove} the lemma, which is the reviewer's point. We do not claim the descent step is proved; we claim it is reduced, in the right framework, to a stated $\eta$-invariant computation that is open. Apparent rigor and actual rigor coincide only once M6 lands, and the paper marks the gap rather than letting the machinery imply closure.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 52 (O-GRIP-52) (the editor)}\par\nobreak\smallskip
\noindent\textbf{``Given all the above --- a central result conditional on M10, a conservation iff conditional on M11, a parity leg conditional on M6, a norm-dependent uniqueness (M9), an equivalence with a sketched converse, and load-bearing imports from four companion papers --- what, precisely, does A6 prove \emph{unconditionally and on its own}?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} A direct question deserves a direct ledger. Unconditionally and within A6: (i) the existence and holonomy-fixed uniqueness of the composite connection (Theorem~\ref{thm:TGG-existence-uniqueness}, modulo the stated category); (ii) the gravitational half of the Master Conservation Law as the contracted Bianchi identity, and the \emph{forward} direction of the four-sector iff (Theorem~\ref{thm:T2.2-master-conservation}); (iii) the per-step and total cascade actions $\pi k$ and $\pi N(N-1)/2$ as symplectic facts on $\CPN$ (Theorem~\ref{thm:instanton-action}, Proposition~\ref{prop:GRIP-total-action}); (iv) the conditional implications themselves --- ``\emph{if} the budget is $3\pi$ \emph{then} $N\leq 3$,'' ``\emph{if} detailed balance \emph{then} $\dd F/\dd t\leq 0$,'' ``\emph{if} M11 \emph{then} the conservation iff closes globally.'' What A6 does \emph{not} prove unconditionally: the budget (M10), the compensator monodromy (M11), the parity non-degeneracy (M6), norm-independence (M9), the equivalence converse, and the imported ledger/anomaly results (A2/A3/A5). So A6's own unconditional content is the connection theorem, the conservation forward-direction, the cascade arithmetic, and a set of clearly-stated conditional implications. The reviewer is right to ask, and the honest answer is that A6 is a conditional formalization whose unconditional core is the structure, not the numbers.
\end{objqabody}

\bigskip
\noindent\textbf{Objection 53 (O-GRIP-53) (the senior community member)}\par\nobreak\smallskip
\noindent\textbf{``Stepping back: is GiFour a discovery or a bookkeeping scheme? You have reorganized existing cohort content into four `pillars' and a bridge, added three conjectures, and claimed a unification. Why should the community invest in this organization rather than read A2/A3/A5 directly?''}\par\nobreak\smallskip
\begin{objqabody}
\emph{Response.} The fair answer is that \GiFour{} is a bookkeeping scheme with a thesis, and its value stands or falls on whether the thesis is right --- which is testable. The bookkeeping is the claim that the conservation content of the cohort is carried by exactly four geometric invariants of one composite connection, related by a fixed implication chain $\TGG{}\Rightarrow\CCI{}\Rightarrow\GFF{}\Rightarrow\GRIP{}$; the thesis is that this is the \emph{minimal and complete} set, which T5 attempts (conditionally) to establish via the equivalence to the constitution. If T5's converse fails, \GiFour{} is a convenient but non-canonical repackaging and a reader should indeed go to A2/A3/A5; if it holds, \GiFour{} is the canonical statement of what those papers collectively prove, in a form that makes the dependencies and the open problems (M9--M11) visible in one place. We do not claim it is a new discovery about nature --- T5 explicitly denies that --- only a claim about the \emph{structure} of the existing derivation. The community should invest in it exactly to the extent that an explicit, auditable, conjecture-flagged organization of a large program is worth more than its scattered parts, and no further. That is a modest claim, and it is the one the paper actually makes once the overclaims this section has conceded are removed.
\end{objqabody}

\bigskip
\noindent\emph{Closing remark on the objections.} A framework that claims to read the conservation laws of physics off a single composite connection, and to force the family number by an instanton-action budget, must expect --- and should welcome --- the hardest scrutiny. We have set out fifty-three objections above, in eleven clusters: the composite connection and \TGG{} (Objections 1--6); the integer ledger and the modified-TOV falsifier (7--11); the Morse and second-law content of \GFF{} (12--17); the cascade and the budget (18--23); the decay-width ladder and the falsifiers (24--28); the constitutional-equivalence theorem T5 (29--32); the ``no new axiom'' claim and the promoted hypotheses (33--35); cohort interdependence and circular citation (36--38); the conjecture density M9--M11 (39--41); the relationship to string theory, supergravity, holography, and Randall--Sundrum (42--45); and method, scope, and the technical soft spots (46--53). The load-bearing conditioning --- M11's general monodromy statement
(the conservation closure) and M9's canonicity question (the norm
choice), with M10 resolved externally and M6 discharged on the
benchmark --- recurs as the honest residue, and we have answered
every objection by saying exactly where the framework is
unconditional, where it is conditional, and where it depends on a
companion paper. The framework's standing offer to be wrong is the
integer ledger and its forced numbers; its standing admission of
what it has not yet proved is the pair M9 and M11; and the present
section is the record of both.


%% ====================================================================
%%  PART: References
%% ====================================================================
\GGPart[References]{References}{Cited works, each with a relevance note linking it to the present paper}

\noindent Unlike a conventional reference list, every entry below carries a short \emph{Relevance} note: a sentence or two saying what the cited work contributes and where in the present paper it bears. The aim is to turn the bibliography from a bare list of sources into a navigable companion --- a reader can move through the references and see, at a glance, which theorem, table, or section each one underwrites.

\noindent The notes are written for two readers at once. For the newcomer, they trace the intellectual lineage of the geometric backbone: from the foundations of gauge theory, differential cohomology, Morse theory, and open-quantum-systems dynamics, to the twenty-two-paper \GGTheory{} cohort in which the present paper supplies the \GiFour{} Pillars --- the geometric backbone on which the \GDOS{} boundary law and its consequences rest. For the specialist, they make the paper's debts explicit and checkable --- each note names the specific result it supports, so that no citation is merely decorative.

\noindent The references are organized in two groups --- first the author's \GGTheory{} papers (Section~A below), then all other works (Section~B), each in alphabetical order by first author. The cohort papers (\AOne{}--\AEleven{}, with the C- and P-series) are cited under the canonical keys and titles fixed by the cross-paper coherence map of the \GGTheory{} program.

\renewcommand{\bibsection}{\vspace{1.2em}}%

\section*{A.~Papers by Katsushi Arisaka (the \GGTheory{} cohort)}
\begin{thebibliography}{99}

\bibitem[Arisaka(2026, A1-Constitution)]{ArisakaA1Constitution}
\textbf{Arisaka, K.} (2026).
\emph{The \GGConstitution{}: Four Axioms and Seven Postulates of the \GGTheory{} Program, From Which \GG{}, \BIsix{}, \OGNfive{}, and \GDOS{} Descend as Theorems}.
Arisaka-GG-A1-Constitution preprint, available at \texttt{https://arisaka-gg.org/}.
\par\noindent\textbf{Relevance.} The parent constitutional paper of the \GGTheory{} program. Establishes the four axioms \AX{1} \LC{}, \AX{2} \GGEP{}, \AX{3} \OGGEP{}, \AX{4} \MSMF{}, together with the seven postulates \POS{1}--\POS{7} (\GI{}, \UEP{}, \AF{}, \CP{}, \RG{}, \OGN{}, \MF{}) and the central Arisaka equation $J_E/\QB = J_e = J_B = J_{Qi}$ that ties the four boundary currents through one bioenergetic conversion factor. Every theorem in the present paper that names its constitutional source inline cites this paper at the point of invocation; the four \GiFour{} pillars TGG/CCI/GFF/GRIP are derived as consequences of the four axioms acting on the 6D bulk. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A2-GG6)]{ArisakaA2GG6}
Arisaka, K. (2026).
\emph{\GGTheory{} with \GG{}: Geometry--Gauge Locking at the 6D Bulk (\GG{}) at the Heart of Grand Geometric Theory (\GGTheory{})}.
Arisaka-GG-A2-GG6 preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Establishes the 6D bulk geometry
$\Msix = \R^{1,3}\times I_r\times S^1_\theta$ and the role of $\Spinone$.
The bulk arena on which \TGG{} is constructed in Part~II of the
present paper. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A3-BI6)]{ArisakaA3BI6}
Arisaka, K. (2026).
\emph{Anomaly Freedom of \BIsix{}: The Modified Bianchi Identity (\BIsix{}) for the Six-Force, Six-Dimensional Bulk (\GG{})}.
Arisaka-GG-A3-BI6 preprint, available at \texttt{https://arisaka-gg.org/}.
\textbf{Relevance.} Proves the \BIsix{} modified Bianchi identity
$\dd H_3 = X_4$ on the 6D bulk and derives the \OGNfive{} integer
ledger $(M,N,L;J,K)=(1,3,5;7,35)$. The differential identity that the
algebraic \CCI{} of Part~III closes. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A4-Euclid)]{ArisakaA4Euclid}
Arisaka, K. (2026).
\emph{From Euclid to \GG{}: The Realization of Euclid's Elements in Six Dimensions --- \GUfive{}, \OGNfive{}, and a Parameter-Free Reconstruction of Modern Physics}.
Preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The historical companion of the \GG{} Trio. Pedagogical
commentary on the long chain from Euclid through Newton, Riemann, Maxwell,
and Einstein to the \GGTheory{} stage. Cited here whenever the historical
positioning of \GiFour{} against its predecessors is invoked, and as one of
the companion papers carrying part of the resolution of the
cosmological-constant problem (PS-GRIP-12) and the pedagogical/historical synthesis
(PS-GRIP-24). First cited in \S\ref{ssec:methodology}.

\bibitem[Arisaka(2026, A5-GDOS)]{ArisakaA5GDOS}
Arisaka, K. (2026).
\emph{\GDOS{}: Geometric Dynamics of Open Systems --- 6D \GG{} Bulk Projected to 4D Boundary for Quantum Measurement, General Relativity, and QFT}.
Arisaka-GG-A5-GDOS preprint, available at \texttt{https://arisaka-gg.org/}.
\textbf{Relevance.} Develops \GDOS{} as the open-system framework for
boundary observation. Introduces the boundary projector, the
\GFF{}, and the \GRIP{} cascade in their first informal form; the
present paper formalizes these as Pillars 3 and 4 of \GiFour{}. Its
Part-VI radiative completion, quantitative-closure, and
covariant-kernel subsections supply the derived mobilities,
coefficients, fluctuation--dissipation pairing, and the causal
runaway-free kernel that Example~4
(\S\ref{ssec:GRIP-ex-radiative}) reads in \GRIP{} vocabulary. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A6-GRIP)]{ArisakaA6Gi4}
Arisaka, K. (2026).
\emph{The \GRIP{} on \GDOS{} --- \GRIP{}: the Geometric Engine of Symmetry Breaking, from the Big Bang to Mind}.
Arisaka-GG-A6-GRIP preprint, available at \texttt{https://arisaka-gg.org/}.
\textbf{Relevance.} The prior integrated form of the present paper:
it introduced \GRIP,
\GFF{}=\GEP+\CEP, \SIP{} and the \DGI/\GSSB/\GA{} cascade as a single
dynamical theorem. The present paper reorganizes this content
around four explicitly named pillars (\TGG, \CCI, \GFF, \GRIP). First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, A7-Qi4)]{ArisakaA7Qi4}
Arisaka, K. (2026).
\emph{The \QiFour{} Pillars --- \QiQUA{}, \QiEP{}, \QiCON{}, \QiHAL{}: The Quantum-Information Backbone of \GDOS{}}.
Arisaka-GG-A7-Qi4 preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The information-theoretic counterpart of the
present paper. Together with the present paper and \AFive{}, it
completes the \GDOS{} trio of foundational papers. First cited in \S\ref{ssec:relation-qi4}.

\bibitem[Arisaka(2026, A8-QM)]{ArisakaA8QM}
Arisaka, K. (2026).
\emph{Quantum Mechanics Derived from \GGTheory{} as Projection from 6D Bulk to 4D Boundary}.
Arisaka-GG-A8-QM preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The quantum-mechanical companion of the cohort: it recasts quantum mechanics in the \GDOS{}/\OGGEP{} language whose geometric backbone is the present \GiFour{} paper, and carries the measurement-problem carry-over discussed in the Discussion Part. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, A9-SM)]{ArisakaA9SM}
Arisaka, K. (2026).
\emph{The Standard Model as Theorem of \GGTheory{}: Its 28 Parameters as Observations of One Six-Dimensional Geometry}.
Arisaka-GG-A9-SM preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Derives the 28 Standard-Model parameters from the
\OGNfive{} ledger in an explicit codebook form. The codebook formulas are
re-derived in the present paper as cascade depths of \GRIP{} on the
family-projector moduli. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A10-Cosmos)]{ArisakaA10Cosmos}
Arisaka, K. (2026).
\emph{The Standard Cosmology as Theorem of \GGTheory{}: The Cosmic Inventory as Observations of One Six-Dimensional Geometry}.
Preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Derives the cosmological parameters
$\Omega_m = 1/\pi$, $\Omega_\Lambda = 1 - 1/\pi$, $\Omega_b = 1/(2\pi^2)$,
$\Omega_{\rm DM} = 1/\pi - 1/(2\pi^2)$ from the \GGConstitution{} and the
6D bulk. Cited here as the companion paper carrying the
resolution of cosmological initial conditions (PS-GRIP-10) and the
cosmological-constant problem (PS-GRIP-12). First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, A11-Origin)]{ArisakaA11Origin}
Arisaka, K. (2026).
\emph{The Geometric Inevitability of Cosmos, Life, and Mind: The Self-Referential Closure of the \GGTheory{} Program}.
Arisaka-GG-A11-Origin. In preparation; to be posted at
\texttt{https://arisaka-gg.org/} on completion. No result of the present
paper depends on it.\\
\textbf{Relevance.} The closing capstone of the program. Defines the
Ground Geometric Attractor (\GGA{}) and proves
\textsf{T-GGA-as-Ground-State} --- the ground-state refinement of the
\GRIP{} cascade adopted into the present paper's runtime vocabulary as
Definition~\ref{def:GGA} --- together with the GRIP-In/GRIP-Out regime
theorems and the eighteen-layer cascade of which the runtime built
here is the engine. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, P1-KK)]{ArisakaP1KK}
Arisaka, K. (2026).
\emph{The \GG{} Lock Cluster: A Non-Spatial Kaluza--Klein Resonance near 3.85~TeV as the Standing LHC Falsifier of \GGTheory{}}.
Arisaka-GG-P1-KK preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The principal collider falsifier of \GGTheory{}.
Predicts a Kaluza-Klein resonance cluster at the lock scale
$\Mlock = \EPl/2\e^{35}\simeq 3.849\,\TeV$. In \GiFour{} vocabulary, this
is the first excitation above the \GA{} of the radial-scale stratum. First cited in \S\ref{ssec:strategic-moment}.

\bibitem[Arisaka(2026, P2-N3)]{ArisakaP2N3}
Arisaka, K. (2026).
\emph{Why Three Generations: A Three-Leg Derivation of $N_g=3$ from \BIsix{}, \MSMF{}, and \GDOS{} Morse Stability}. Arisaka-GG-P2-N3. Preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The three-leg N=3 proof. In the present paper,
the hypotheses $\mathcal H_1, \mathcal H_2$ of that paper are
reduced to Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem} of
\GiFour{} modulo the $\eta$-invariant computation M6. Its
Appendix~H carries the Hopkins--Singer pushforward of the \BIsix{}
seam and proves it equal to the boundary's Berry line bundle (the
$\kappa$-bridge theorem) on the benchmark background --- the rigorous
bulk implementation of the descent used in
Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem}, discharging
this paper's residue M6 ($=$ R-INT) there, with the warped and
relative refinements declared open. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, P3-Electron)]{ArisakaP3Electron}
Arisaka, K. (2026).
\emph{The Electron as the Ground State of Nature: The Six Oldest Questions about the Electron, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P3-Electron preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The dedicated electron paper whose two-sector
partition --- the phase sector moves the electron, the scale sector
transmutes it, only what sheds energy descends --- supplies the stratum
fence of Example~4 (\S\ref{ssec:GRIP-ex-radiative}): \GRIPin{} on the
orbital stratum descends motion, \GRIPin{} on the family stratum
descends identity, and the same septuple runs on both. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, P4-Real)]{ArisakaP4Real}
Arisaka, K. (2026).
\emph{The Real Geometry Beneath the Complex Numbers of Physics: Positive Frequency is Holomorphy --- from Cardano's Cubics to Twistor Space, via the Sixth Dimension of \GGTheory{}}.
Arisaka-GG-P4-Real preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Foundations Branch: the origin-of-$i$ member
paper, deriving the single imaginary unit of boundary physics from
the two real structures of the \ATwo{} arena (the compact phase
circle and the causal clock). For the present paper it is the origin
of the boundary's complex structure whose phase conventions the
\DGI{} stage of the \GRIP{} pipeline makes dynamical; cited at
roster level. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, P5-CPT)]{ArisakaP5CPT}
Arisaka, K. (2026).
\emph{The Mirrors of Matter: Why Every Particle Has a Twin, Why Nature's Mirrors Break, and Why the Universe Is Made of Matter}.
Arisaka-GG-P5-CPT preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Foundations Branch: the CPT member paper,
reading C, P, and T as orientation reversals of the three real bulk
factors and carrying the matter--antimatter door; its mirrors and
its law-level arrow of time are kin of the irreversible descent that
the runtime built here drives one way (the second law of
Theorem~T3); cited at roster level. First cited in
\S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, P6-Proton)]{ArisakaP6Proton}
Arisaka, K. (2026).
\emph{The Proton as the Ground Geometric Attractor of Matter: The Eleven Questions of the Proton, Closed and Fenced within \GGTheory{}}.
Arisaka-GG-P6-Proton preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Predictions Branch: the fourth member paper and the
sister of \PaperPThree{}, running that paper's ground-attractor method
on the baryonic stratum, with baryon number carried as a winding of the
compact geometry rather than as a bookkeeping label. Its subject is a
selection outcome of the runtime built here --- the attractor a
\GRIP{} descent settles into once the landscape has stopped changing,
which is the fixed-regime branch of \S\ref{ssec:two-regimes-two-fibres};
cited at roster level. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, C1-Kaxion)]{ArisakaC1Kaxion}
Arisaka, K. (2026).
\emph{The Kaxion: Cold Dark Matter as the Non-Spatial Compact-Angle Pseudoscalar of \GG{}}.
Arisaka-GG-C1-Kaxion preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The principal cosmological falsifier of
\GGTheory{}. Predicts a $\approx 8.1~\mu$eV Kaxion-like dark-matter candidate
as the slowly-rolling \GA{} of the $\UoneQi$ Wilson-line stratum. First cited in \S\ref{sec:intro}.

\bibitem[Arisaka(2026, C2-Hubble)]{ArisakaC2Hubble}
Arisaka, K. (2026).
\emph{The Hubble Tension in the Light of \GGTheory{}: The Hubble Constant as One Reading of a Derived Expansion History $H(t)$}.
Arisaka-GG-C2-Hubble preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} A consequence-side reading of the boundary law: the early- and late-time Hubble constants are the two ends of the scale corridor, a cosmological application of the \CCI{}-fixed $X_{\rm GS}$ developed here. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, C3-BH)]{ArisakaC3BH}
Arisaka, K. (2026).
\emph{Black-Hole Entropy under \GGTheory{}: The \GDOS{} Reading of the Horizon as an \OGGEP{} Port --- with the Exact Bekenstein--Hawking $A/4G_N$ Derived from Induced Gravity}.
Arisaka-GG-C3-BH preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} Treats the black-hole horizon as an \OGGEP{} registration port; the induced-gravity boundary stress tensor of \TGG{} (Theorem~\textbf{T1.2} here) is the bulk source whose horizon reduction yields the Bekenstein--Hawking area law. First cited in \S\ref{sec:cohort-a6}.

\bibitem[Arisaka(2026, W1-Symmetry)]{ArisakaW1Symmetry}
Arisaka, K. (2026).
\emph{Symmetry, Conservation, and Geometry: Seven Steps from Noether, Bianchi, to Hopkins--Singer, and the Ledger the Open-System Turn Pays Out}.
Arisaka-GG-W1-Symmetry preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The cohort's symmetry review, and the source of
the sentence on which this paper's two-fiber reading rests: symmetry
breaking as the one-way \emph{reading} of holonomy data --- the
alternatives held by the fiber that remembers, the selection
performed by the fiber that forgets. Its ladder ends at protection
without symmetry, which is the register in which the \GGA{}'s
labels (charges, windings, parities) are protected here. First cited
in \S\ref{ssec:two-regimes-two-fibres}.

\bibitem[Arisaka(2026, W2-Holonomy)]{ArisakaW2Holonomy}
Arisaka, K. (2026).
\emph{The Circle That Remembers, the Interval That Forgets: \GGTheory{} and the Sixteen Problems of Particle Physics and Cosmology}.
Arisaka-GG-W2-Holonomy preprint, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The dedicated treatise on the fork this paper's
engine runs on --- holonomy requires closed loops, and an interval
has none --- proving the group law for circle transport and the
semigroup law for interval transport, and deploying the two-fiber
anatomy across sixteen standing problems. Its engine section carries
the stroke-level mapping (\GRIPin{} the circle's, \GRIPout{} the
interval's) that Subsection~\ref{ssec:stage-custody} completes with
the stage-level custody statement. First cited in
\S\ref{ssec:two-regimes-two-fibres}.

\end{thebibliography}

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

\bibitem[Abbott \emph{et al.}(2016)Abbott \emph{et al.}]{Abbott2016}
Abbott, B.~P., \emph{et al.} (LIGO Scientific and Virgo Collaborations) (2016).
\emph{Observation of gravitational waves from a binary black hole merger}.
Physical Review Letters, 116, 061102.\\
\textbf{Relevance.} The first directly observed inspiral--merger--ringdown
waveform. In Example~4 (\S\ref{ssec:GRIP-ex-radiative}) the three phases
are read as the \GRIP{} loop drawn end to end: \GRIPin{} descent on the
binary's stratum, a landscape-reorganizing terminus, and the new
stratum's \GRIPin{} relaxing to the Kerr attractor. First cited in \S\ref{ssec:GRIP-ex-radiative}.

\bibitem[Adler(1969)Adler]{Adler1969}
Adler, S.~L. (1969).
\emph{Axial-vector vertex in spinor electrodynamics}.
Physical Review, 177, 2426--2438.\\
\textbf{Relevance.} The original derivation of the chiral anomaly in
QED, providing the historical foundation for what becomes the four-form
anomaly polynomial $X_4$ of \CCI{} in the present paper. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Aharonov \& Bohm(1959)Aharonov \& Bohm]{AharonovBohm1959}
Aharonov, Y., \& Bohm, D. (1959).
\emph{Significance of electromagnetic potentials in the quantum theory}.
Physical Review, 115, 485--491.\\
\textbf{Relevance.} The original treatment of holonomy as a physical
observable. In the present paper, Wilson-line holonomies of the
\TGG{} composite connection on the compact $\theta$-direction
(Section~5.5) are the natural generalization of the Aharonov--Bohm
phase. First cited in \S\ref{app:integer-cohomology}.

\bibitem[ALEPH \emph{et al.}(2006)ALEPH \emph{et al.}]{ALEPH2006}
ALEPH, DELPHI, L3, OPAL Collaborations \& LEP Electroweak Working Group
(2006).
\emph{Precision electroweak measurements on the Z resonance}.
Physics Reports, 427, 257--454.\\
\textbf{Relevance.} Establishes the LEP empirical bound
$N_\nu = 2.984\pm 0.008$ on the number of light neutrino species, the
most direct experimental confirmation of $N_{\rm gen}=3$. This is the
empirical input to \textbf{T2}'s lower bound $N\geq 3$. First cited in \S\ref{ssec:five-pillars-of-significance}.

\bibitem[Alvarez-Gaum\'e \& Witten(1984)Alvarez-Gaum\'e \& Witten]{AlvarezGaumeWitten1984}
Alvarez-Gaum\'e, L., \& Witten, E. (1984).
\emph{Gravitational anomalies}.
Nuclear Physics B, 234, 269--330.\\
\textbf{Relevance.} Foundational treatment of gravitational anomalies
and their cancellation. Provides the technical machinery underlying
the gravity sector of $X_4$ in \CCI-ST1. First cited in \S\ref{ssec:CCI-seven-st}.

\bibitem[Arisaka(2026, BookI)]{ArisakaBookI}
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4D-Physics Trio Compiled into a Single Reference}.
In preparation, available at \texttt{https://arisaka-gg.org/}.\\
\textbf{Relevance.} The Foundation Book compiles the constitution and its
descent chain into a single accessible reference. Cited here as the
eventual textbook synthesis (PS-GRIP-24) covering the cross-domain implications
of \GGTheory{} (life science, neuroscience, complex systems). First cited in \S\ref{ssec:open-programme}.

\bibitem[Bardeen(1969)Bardeen]{Bardeen1969}
Bardeen, W.~A. (1969).
\emph{Anomalous Ward identities in spinor field theories}.
Physical Review, 184, 1848--1859.\\
\textbf{Relevance.} The non-Abelian generalization of the chiral
anomaly. Together with \citet{Adler1969,BellJackiw1969}, provides the
historical foundation of the four-form $X_4$ structure in \CCI-ST1. First cited in \S\ref{ssec:CCI-seven-st}.

\bibitem[Bell \& Jackiw(1969)Bell \& Jackiw]{BellJackiw1969}
Bell, J.~S., \& Jackiw, R. (1969).
\emph{A PCAC puzzle: $\pi^0\to\gamma\gamma$ in the $\sigma$ model}.
Il Nuovo Cimento A, 60, 47--61.\\
\textbf{Relevance.} The original observation of the axial anomaly in
the context of $\pi^0$ decay. The historical companion to
\citet{Adler1969}. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Bianchi(1902)Bianchi]{Bianchi1902}
Bianchi, L. (1902).
\emph{Sui simboli a quattro indici e sulla curvatura di Riemann}.
Rendiconti della Reale Accademia dei Lincei (Series 5), 11, 3--7.\\
\textbf{Relevance.} The original Bianchi identity $\nabla_{[\mu}
R_{\nu\rho]\sigma\tau} = 0$ for the Riemann curvature tensor. A purely
geometric statement, derived in 1902 without any reference to dynamics
or quantum effects, that is the structural prototype of \CCI{}. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Blumenhagen \emph{et al.}(2007)Blumenhagen \emph{et al.}]{BlumenhagenKorsLustStieberger2007}
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\emph{Four-dimensional string compactifications with D-branes,
orientifolds and fluxes}.
Physics Reports, 445, 1--193.\\
\textbf{Relevance.} Comprehensive review of magnetized toroidal
compactifications, one of the closed-system attempts to derive $N=3$
discussed in
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\bibitem[Boltzmann (1872)]{Boltzmann1872}
Boltzmann, L. (1872).
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Gasmolek\"ulen}.
Sitzungsberichte der Akademie der Wissenschaften zu Wien, 66, 275--370.\\
\textbf{Relevance.} The $H$-theorem and the first statistical-mechanical
derivation of the second law of thermodynamics from microscopic
dynamics. The structural ancestor of \GFF-ST6 (thermodynamic
monotonicity) and of the energetic component $F_\GEP$ of \GFF{}. First cited in \S\ref{ssec:GFF-motivation}.

\bibitem[Born(1926)Born]{Born1926}
Born, M. (1926).
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Zeitschrift f\"ur Physik, 37, 863--867.\\
\textbf{Relevance.} The Born rule. In the open-system framework of
the present paper, the Born rule emerges as the boundary projector
selection at the end of the \GRIP{} cascade, rather than being
postulated separately. First cited in \S\ref{sec:comparison-table}.

\bibitem[Bott \& Tu(1982)Bott \& Tu]{BottTu1982}
Bott, R., \& Tu, L.~W. (1982).
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Graduate Texts in Mathematics 82, Springer.\\
\textbf{Relevance.} Standard pedagogical reference for de Rham
cohomology, Chern--Weil theory, and characteristic classes used
throughout the present paper. The recommended text for graduate
students who wish to deepen their understanding of the
differential-cohomological machinery underlying the \CCI{} and
\GFF{} pillars; cited in Appendix~\ref{app:diff-cohomology} as the
suggested follow-up reading after the pedagogical tour. First cited in \S\ref{app:diff-cohomology}.

\bibitem[Callan \& Coleman(1977)Callan \& Coleman]{CallanColeman1977}
Callan, C.~G., \& Coleman, S. (1977).
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Physical Review D, 16, 1762--1768.\\
\textbf{Relevance.} The companion to \citet{Coleman1977} computing
the one-loop prefactor of false-vacuum tunneling rates. Provides
the prefactor $A_i$ in the WKB instanton-rate formula
$\Gamma_{i\to i+1} = A_i\,\e^{-S_{i\to i+1}}$ used in
Section~\ref{sec:GRIP-physical-layer} to derive the lifetime
hierarchy of \textbf{GRIP-ST7}. First cited in \S\ref{ssec:GRIP-motivation}.

\bibitem[Callan \& Harvey(1985)Callan \& Harvey]{CallanHarvey1985}
Callan, C.~G., \& Harvey, J.~A. (1985).
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Nuclear Physics B, 250, 427--436.\\
\textbf{Relevance.} The seminal treatment of anomaly inflow from
bulk to boundary. The foundational mechanism that, in modern
differential-cohomology form (Hopkins--Singer), drives the descent of
$X_4$ from $\Msix$ onto the boundary moduli in Part~IV. First cited in \S\ref{ssec:CCI-seven-st}.

\bibitem[Candelas \emph{et al.}(1985)Candelas \emph{et al.}]{CHSW1985}
Candelas, P., Horowitz, G.~T., Strominger, A., \& Witten, E. (1985).
\emph{Vacuum configurations for superstrings}.
Nuclear Physics B, 258, 46--74.\\
\textbf{Relevance.} The seminal heterotic-string Calabi--Yau
compactification. One of the closed-system family-counting attempts
($|\chi/2|=3$) compared against in
\citet[\S~12]{ArisakaP2N3}. First cited in \S\ref{ssec:string-overview}.

\bibitem[Carlen \& Maas(2014)Carlen \& Maas]{CarlenMaas2014}
Carlen, E.~A., \& Maas, J. (2014).
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probability under which the fermionic Fokker--Planck equation is
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Communications in Mathematical Physics, 331, 887--926.\\
\par\noindent\textbf{Relevance.} Establishes that GKLS Lindbladians
satisfying the BKM (Bogoliubov--Kubo--Mori) detailed-balance condition
can be expressed as a gradient flow of quantum relative entropy with
respect to a non-commutative Wasserstein-like metric on states. This
is the foundational result underlying the gradient-flow representation
of the \GDOS{} boundary Lindbladian used in the proof of
Theorem~\ref{thm:GFF-H-theorem}: the boundary moduli-space metric
$g_\Sigma$ is identified with the BKM information metric restricted
to the projector manifold. First cited in \S\ref{ssec:GFF-H-theorem}.

\bibitem[Coleman(1977)Coleman]{Coleman1977}
Coleman, S. (1977).
\emph{Fate of the false vacuum: semiclassical theory}.
Physical Review D, 15, 2929--2936.\\
\textbf{Relevance.} The original WKB instanton calculation of
false-vacuum decay rates, $\Gamma\propto \e^{-S_E}$ where $S_E$ is
the Euclidean instanton action. Provides the leading exponential
factor in the WKB instanton-rate formula used in
Section~\ref{sec:GRIP-physical-layer} to derive the lifetime
hierarchy from the cascade actions $S_{[2k]\to[2(k-1)]}=\pi k$. First cited in \S\ref{ssec:GRIP-motivation}.

\bibitem[Dirac(1931)Dirac]{Dirac1931}
Dirac, P.~A.~M. (1931).
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Proceedings of the Royal Society A, 133, 60--72.\\
\textbf{Relevance.} The original derivation of magnetic-monopole
quantization $\int_{S^2}F/2\pi\in\Z$. This is the historical
prototype of the integrality conditions on differential cohomology
classes (the \CCI{} integrality $[X_4]\in H^4(\Msix;\Z)$ in the
present paper). Cited in Appendix~\ref{app:diff-cohomology} as the
prototypical physical example of cohomological quantization. First cited in \S\ref{app:integer-cohomology}.

\bibitem[Einstein(1915)Einstein]{Einstein1915}
Einstein, A. (1915).
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844--847.\\
\textbf{Relevance.} The founding paper of general relativity, recasting
gravitation as the curvature of a metric connection on the tangent
bundle of spacetime. The structural ancestor of the gravitational
component $\omega$ of the \TGG{} composite connection. First cited in \S\ref{ssec:TGG-motivation}.

\bibitem[Freed \& Hopkins(2000)Freed \& Hopkins]{FreedHopkins2000}
Freed, D.~S., \& Hopkins, M.~J. (2000).
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Journal of High Energy Physics, 2000(05), 044.\\
\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 (Theorem~\ref{thm:H2-theorem}). First cited in \S\ref{ssec:H2-theorem}.

\bibitem[Friston(2010)Friston]{Friston2010}
Friston, K. (2010).
\emph{The free-energy principle: a unified brain theory?}
Nature Reviews Neuroscience, 11(2), 127--138.\\
\textbf{Relevance.} The neuroscience free-energy principle, in
which the brain minimizes a variational free energy. Provides the
prototype for the boundary projection of \GFF{} onto neural
information manifolds; \GGTheory{} subsumes Friston's principle as
the special case where the boundary moduli are restricted to
predictive-coding manifolds. First cited in \S\ref{sec:GFF-axioms}.

\bibitem[Friston(2023)Friston]{FristonFreeEnergy2023}
Friston, K., et al.\ (2023).
\emph{The free-energy principle made simpler but not too simple}.
Physics Reports, 1024, 1--29.\\
\textbf{Relevance.} Modern, mathematically rigorous formulation of
the free-energy principle, including its variational and
information-geometric structure. Cited in Part~IV
(Section~\ref{sec:GFF-physical-layer}) as the contemporary articulation
of the free-energy paradigm that \GFF{} subsumes and generalizes. First cited in \S\ref{sec:GFF-axioms}.

\bibitem[Gell-Mann \& Low(1954)Gell-Mann \& Low]{GellMannLow1954}
Gell-Mann, M., \& Low, F.~E. (1954).
\emph{Quantum electrodynamics at small distances}.
Physical Review, 95, 1300--1312.\\
\textbf{Relevance.} The $\beta$-function formalism in QED, the first
explicit statement of the running coupling and the seed of the
renormalization-group flow. The structural ancestor of \GRIP{}-ST1
(\DGI{}). First cited in \S\ref{ssec:GRIP-motivation}.

\bibitem[Goresky \& MacPherson(1988)Goresky \& MacPherson]{GoreskyMacPherson1988}
Goresky, M., \& MacPherson, R. (1988).
\emph{Stratified Morse Theory}.
Springer-Verlag, Berlin.\\
\textbf{Relevance.} Standard reference for Morse theory on Whitney
stratified spaces. The mathematical framework underlying
Appendix~\ref{app:Morse-stratified} and the stratified \GFF{} structure
of Part~IV. First cited in \S\ref{app:Morse-stratified}.

\bibitem[Gorini, Kossakowski \& Sudarshan(1976)Gorini, Kossakowski \& Sudarshan]{GKLS1976}
Gorini, V., Kossakowski, A., \& Sudarshan, E.~C.~G. (1976).
\emph{Completely positive dynamical semigroups of N-level systems}.
Journal of Mathematical Physics, 17, 821--825.\\
\textbf{Relevance.} The GKLS theorem characterizing the most general
CPTP generator of open-system quantum dynamics. Provides the
mathematical prototype for the boundary-projection structure that
\OGGEP{} elevates to foundational status. First cited in \S\ref{app:abbrev}.

\bibitem[Green \& Schwarz(1984)Green \& Schwarz]{GreenSchwarz1984}
Green, M.~B., \& Schwarz, J.~H. (1984).
\emph{Anomaly cancellations in supersymmetric $D=10$ gauge theory and
superstring theory}.
Physics Letters B, 149, 117--122.\\
\textbf{Relevance.} The original Green--Schwarz anomaly-cancellation
mechanism --- and, in the \GGTheory{} reading, the technical
pioneering of \AX{2} \GGEP{}: the geometrization of gauge fields on
the same footing as gravity. \BIsix{} is the \GGEP{} completion of
this 1984 construction, extended from the heterotic ten-dimensional
setting to the 6D bulk and the four-factor gauge group $\Gtot$
of \GGTheory{}; the full supportive treatment of the GS legacy lives
in \citet[\S~8.7,~``\BIsix{} as the \GGEP{} completion of GS: the
geometrization reading'']{ArisakaA3BI6}, which the present paper cites
rather than re-derives. The Chern--Simons construction of $H_3$ from
\TGG{} data (Proposition~\ref{prop:H3-from-TGG}) is the natural
\GiFour{} adaptation of the GS Chern--Simons primitive. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Helgason(1979)Helgason]{HelgasonBook}
Helgason, S. (1979).
\emph{Differential Geometry, Lie Groups, and Symmetric Spaces}.
Academic Press.\\
\textbf{Relevance.} Standard reference for the Fubini--Study metric
on $\CPN$ and the structure of $\Spin$ groups over normed division
algebras. Used throughout Part~II. First cited in \S\ref{ssec:quat-to-family}.

\bibitem[Hinton \& van Camp(1993)Hinton \& van Camp]{HintonVanCamp1993}
Hinton, G.~E., \& van Camp, D. (1993).
\emph{Keeping the neural networks simple by minimizing the description length of the weights}.
In \emph{Proceedings of the 6th Annual Conference on Computational Learning Theory (COLT '93)}, pp.~5--13.\\
\textbf{Relevance.} The original variational free-energy
formulation in machine learning, $F_{\rm var} = D_{\rm KL}(q\|p) -
\log p(o)$. This is the prototype of Friston's free-energy
principle and provides the precise mathematical bridge between
\textbf{GFF-ST6} and Bayesian inference, as discussed in
Appendix~\ref{app:free-energy-bridge}. First cited in \S\ref{app:free-energy-bridge}.

\bibitem[Hirzebruch(1966)Hirzebruch]{Hirzebruch1966}
Hirzebruch, F. (1966).
\emph{Topological Methods in Algebraic Geometry} (3rd ed.).
Springer-Verlag, Berlin.\\
\textbf{Relevance.} Foundational reference for the Hirzebruch
signature theorem and the L-polynomial computation. Used throughout
Part~IV (\GFF{} Morse-positivity from \CCI{}) and the explicit
$\sigma(\CPN)$ computation. First cited in \S\ref{ssec:GFF-first-consequences}.

\bibitem[\,'t Hooft(1993)\,'t Hooft]{tHooft1993}
't Hooft, G. (1993).
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arXiv:gr-qc/9310026.\\
\textbf{Relevance.} The original holographic-principle proposal: the
fundamental degrees of freedom of a $(d+1)$-dimensional gravitational
theory project onto a $d$-dimensional boundary. Conceptual ancestor
of \OGGEP{}. First cited in \S\ref{ssec:GRIP-motivation}.

\bibitem[Hopkins \& Singer(2005)Hopkins \& Singer]{HopkinsSinger2005}
Hopkins, M.~J., \& Singer, I.~M. (2005).
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Journal of Differential Geometry, 70, 329--452.\\
\textbf{Relevance.} Foundational paper on differential cohomology and
its application to anomaly inflow. The technical machinery underlying
Theorems~\ref{thm:H1-theorem} and~\ref{thm:H2-theorem} of Part~IV. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Hosotani(1983)Hosotani]{Hosotani1983}
Hosotani, Y. (1983).
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Physics Letters B, 126, 309--313.\\
\textbf{Relevance.} The original Wilson-line / Hosotani mechanism for
electroweak symmetry breaking. In \GiFour{} vocabulary, this is the
\GRIP{} cascade on the Higgs moduli stratum $\Sigma_H$. First cited in \S\ref{ssec:example-warped}.

\bibitem[Hosotani(2005)Hosotani]{Hosotani2005}
Hosotani, Y. (2005).
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Progress of Theoretical Physics Supplement, 172, 313--326.\\
\textbf{Relevance.} Modern review of gauge-Higgs unification. The
6D realization of \citet{Hosotani1983} that operates on the
compact $\theta$-direction of $\Msix$. First cited in \S\ref{sec:intro}.

\bibitem[Hulse \& Taylor(1975)Hulse \& Taylor]{HulseTaylor1975}
Hulse, R.~A., \& Taylor, J.~H. (1975).
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The Astrophysical Journal, 195, L51--L53.\\
\textbf{Relevance.} The binary pulsar PSR B1913+16, whose orbital decay
has tracked the quadrupole prediction for five decades to a few parts in
$10^3$ --- the empirical anchor of the inspiral leg of Example~4's
\GRIPin{} descent (\S\ref{ssec:GRIP-ex-radiative}). First cited in \S\ref{ssec:GRIP-ex-radiative}.

\bibitem[Ib\'a\~nez \emph{et al.}(1987)Ib\'a\~nez \emph{et al.}]{IbanezNillesQuevedo1987}
Ib\'a\~nez, L.~E., Nilles, H.~P., \& Quevedo, F. (1987).
\emph{Orbifolds and Wilson lines}.
Physics Letters B, 187, 25--32.\\
\textbf{Relevance.} The $\Z_3$-orbifold construction with three
twisted sectors, the most natural prior model giving exactly three
generations. Compared against in
\citet[\S~12.3]{ArisakaP2N3}. First cited in \S\ref{ssec:string-overview}.

\bibitem[Ib\'a\~nez \& Uranga(2012)Ib\'a\~nez \& Uranga]{IbanezUrangaBook}
Ib\'a\~nez, L.~E., \& Uranga, A.~M. (2012).
\emph{String Theory and Particle Physics: An Introduction to String
Phenomenology}.
Cambridge University Press.\\
\textbf{Relevance.} Standard reference for D-brane and
intersecting-brane family counting. Provides the broader
phenomenological context for Section~\ref{sec:apply-N3} (the \GiFour{} three-leg N=3 derivation). First cited in \S\ref{ssec:string-overview}.

\bibitem[Jaynes(1957)Jaynes]{Jaynes1957}
Jaynes, E.~T. (1957).
\emph{Information theory and statistical mechanics}.
Physical Review, 106(4), 620--630.\\
\textbf{Relevance.} The maximum-entropy principle in statistical
inference, equivalent to free-energy minimization in equilibrium
statistical mechanics. Cited in Part~IV
(Section~\ref{sec:GFF-axioms}) as one of the universal projections
of the \GFF{} framework. First cited in \S\ref{sec:GFF-axioms}.

\bibitem[Kadanoff(1966)Kadanoff]{Kadanoff1966}
Kadanoff, L.~P. (1966).
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Physics, 2, 263--272.\\
\textbf{Relevance.} The block-spin coarse-graining picture for
statistical-mechanical systems near criticality; the conceptual bridge
between Gell-Mann--Low's $\beta$-function and Wilson's full
renormalization-group framework. First cited in \S\ref{ssec:GRIP-motivation}.

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\textbf{Relevance.} Reviews anomaly-cancellation constraints on
family number in BSM extensions. Family number is constrained but not
uniquely fixed by closed-system anomaly cancellation alone. First cited in \S\ref{ssec:T2.2-statement}.

\bibitem[Kobayashi \& Maskawa(1973)Kobayashi \& Maskawa]{KobayashiMaskawa1973}
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Progress of Theoretical Physics, 49, 652--657.\\
\textbf{Relevance.} The original observation that $N\geq 3$ is
required for CP violation. The historical precursor to leg L1 of the
N=3 proof; the lower bound used in Theorem~\textbf{T2}. First cited in \S\ref{sec:apply-N3}.

\bibitem[Kobayashi \& Nomizu(1969)Kobayashi \& Nomizu]{KobayashiNomizu1969}
Kobayashi, S., \& Nomizu, K. (1969).
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Wiley-Interscience, New York.\\
\textbf{Relevance.} Standard reference for Chern--Weil theory and
the Pontryagin / Chern class identifications used throughout Parts~II
and III. First cited in \S\ref{ssec:TGG-motivation}.

\bibitem[Kobayashi \emph{et al.}(2020)Kobayashi \emph{et al.}]{KobayashiObaShinjo}
Kobayashi, T., Oba, T., \& Shinjo, T. (2020).
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Journal of High Energy Physics, 2020(11), 199.\\
\textbf{Relevance.} A modern attempt to derive $N=3$ from modular
flavor symmetry. Representative of the prior approaches that obtain
three generations conditionally on a chosen symmetry input. First cited in \S\ref{ssec:string-overview}.

\bibitem[Lindblad(1975)Lindblad]{Lindblad1975Entropy}
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Communications in Mathematical Physics, 40(2), 147--151.\\
\textbf{Relevance.} Lindblad's monotonicity theorem: relative
entropy is non-increasing under any CPTP map. This is the
foundational result underlying Theorem~\textbf{T3.2}: the
monotonic decrease $\dd F_{\GFF}/\dd t \le 0$ of the boundary
\GFF{} along any \GDOS{} trajectory follows directly from Lindblad
monotonicity applied to the boundary CPTP flow. First cited in \S\ref{ssec:GFF-motivation}.

\bibitem[Lindblad(1976)Lindblad]{Lindblad1976}
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Communications in Mathematical Physics, 48, 119--130.\\
\textbf{Relevance.} The Lindblad master equation, derived
independently of \citet{GKLS1976}. Provides the technical foundation
for CPTP open-system dynamics, the closed-system structure that the
\OGGEP{}-elevated boundary projector generalizes. First cited in \S\ref{ssec:GFF-H-theorem}.

\bibitem[Maldacena(1997)Maldacena]{Maldacena1997}
Maldacena, J. (1997).
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supergravity}.
Advances in Theoretical and Mathematical Physics, 2, 231--252.\\
\textbf{Relevance.} The original AdS/CFT correspondence. The
prototype of bulk-to-boundary descent that
\OGGEP{}/\GFF{}/\GRIP{} elevates to a foundational principle. First cited in \S\ref{ssec:AdS-CFT}.

\bibitem[Maxwell (1865)]{Maxwell1865}
Maxwell, J.~C. (1865).
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Philosophical Transactions of the Royal Society of London, 155, 459--512.\\
\textbf{Relevance.} The founding paper of classical electromagnetism
and, in the modern geometric reading, the first gauge theory in
physics: the four-potential $A_\mu$ is defined only up to a gradient,
and the field strength $F_{\mu\nu}$ is the curvature of this connection
on a $\mathrm{U}(1)$ principal bundle. This is the structural ancestor
of \TGG{} and of \CCI{} (in its $\dd F = 0$ form). First cited in \S\ref{ssec:TGG-motivation}.

\bibitem[Miller et al.(2021)Miller et al.]{Miller2021NICER}
Miller, M.~C., Lamb, F.~K., Dittmann, A.~J., et al.\ (2021).
\emph{The radius of PSR J0740+6620 from NICER and XMM-Newton data}.
Astrophysical Journal Letters, 918, L28.\\
\textbf{Relevance.} NICER measurement of neutron-star radius for the
heaviest precisely-measured pulsar. Provides the empirical baseline
against which the modified-TOV correction $\Delta_{\rm GS}(r)$ of
\textbf{CCI-ST7} is to be tested. First cited in \S\ref{ssec:mTOV-derivation}.

\bibitem[Milnor(1963)Milnor]{Milnor1963}
Milnor, J. (1963).
\emph{Morse Theory}.
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\textbf{Relevance.} Foundational reference for Morse theory. The
height-function-on-$\CPN$ Morse polynomial is the textbook example
that drives Lemmas in Parts~IV--V. First cited in \S\ref{ssec:GFF-seven-st}.

\bibitem[Mittnenzweig \& Mielke(2017)Mittnenzweig \& Mielke]{MittnenzweigMielke2017}
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\par\noindent\textbf{Relevance.} Generalizes the Carlen--Maas
gradient-flow representation to a broad class of Lindblad master
equations: any GKLS evolution with a detailed-balanced fixed point
is a gradient flow of relative entropy under an appropriate
information-geometric metric. This is the technical underpinning
for the boundary-projection step (Step~2) of the proof of
Theorem~\ref{thm:GFF-H-theorem}: it ensures that the boundary
moduli-space gradient-flow representation
$\dot{[\Pi]} = -g_\Sigma^{-1}\nabla\cF_{\GFF}$ is well-defined for
the \GDOS{} Lindbladian whenever its OGGEP-inherited detailed-balance
condition holds. First cited in \S\ref{ssec:GFF-H-theorem}.

\bibitem[von Neumann(1932)von Neumann]{vonNeumann1932}
von Neumann, J. (1932).
\emph{Mathematische Grundlagen der Quantenmechanik}.
Springer-Verlag, Berlin.\\
\textbf{Relevance.} The foundational mathematical formalization of
quantum mechanics as unitary evolution on Hilbert space. In the
methodological reflection of Section~\ref{ssec:methodological-lesson-final},
this is 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 \S\ref{sec:intro}.

\bibitem[Noether(1918)Noether]{Noether1918}
Noether, E. (1918).
\emph{Invariante Variationsprobleme}.
Nachrichten der K\"oniglichen Gesellschaft der Wissenschaften zu
G\"ottingen, Math.-Phys.\ Klasse, 235--257.\\
\textbf{Relevance.} The symmetry--conservation theorem. The
foundational identity tying continuous symmetries of an action to
conserved currents. The dynamical face of \CCI{} traces directly back
to this paper. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[NOvA(2022)NOvA]{NOvA2022}
NOvA Collaboration (2022).
\emph{Improved measurement of neutrino oscillation parameters by the
NOvA experiment}.
Physical Review D, 106, 032004.\\
\textbf{Relevance.} Empirical evidence for non-trivial PMNS mixing
and a non-zero leptonic CP phase. Independent support for $N\geq 3$
in the leptonic sector. First cited in \S\ref{sec:apply-N3}.

\bibitem[Onsager(1931)Onsager]{Onsager1931}
Onsager, L. (1931).
\emph{Reciprocal relations in irreversible processes. I}.
Physical Review, 37, 405--426.\\
\textbf{Relevance.} The reciprocal relations $L_{ij} = L_{ji}$ for the
linear response of irreversible systems. The first quantitative result
on near-equilibrium open-system dynamics; the structural ancestor of
\GFF{}-ST7 (Onsager reciprocity). First cited in \S\ref{ssec:GFF-motivation}.

\bibitem[Particle Data Group(2024)Particle Data Group]{ParticleDataGroup2024}
Particle Data Group (2024).
\emph{Review of Particle Physics}.
Progress of Theoretical and Experimental Physics, 2024, 083C01.\\
\textbf{Relevance.} Standard reference for measured Standard-Model
parameters: charged-lepton masses, quark masses, mixing angles, CP
phases, and the Jarlskog invariant
$J_{\rm CKM}=(3.18\pm 0.15)\times 10^{-5}$. The empirical baseline
against which the \GGTheory{} codebook is checked. First cited in \S\ref{sec:predictions-falsification}.

\bibitem[Planck Collaboration(2020)Planck Collaboration]{Planck2020}
Planck Collaboration (2020).
\emph{Planck 2018 results.~VI.~Cosmological parameters}.
Astronomy \& Astrophysics, 641, A6.\\
\textbf{Relevance.} Cosmological measurement
$N_{\rm eff}=2.99\pm 0.17$, an independent confirmation of $N_\nu=3$
complementing \citet{ALEPH2006}. First cited in \S\ref{ssec:five-pillars-of-significance}.

\bibitem[Randall \& Sundrum(1999)Randall \& Sundrum]{RandallSundrum1999}
Randall, L., \& Sundrum, R. (1999).
\emph{A large mass hierarchy from a small extra dimension}.
Physical Review Letters, 83(17), 3370--3373.\\
\textbf{Relevance.} The original extra-dimension scenario.
Provides the prototype for the scale-profile identification
$T_{\GFF}\sim M_{\rm KK}$ used in
Section~\ref{sec:GFF-physical-layer} (the Tolman-like identification
of scale with effective temperature). First cited in \S\ref{ssec:GFF-first-law}.

\bibitem[de Rham(1931)de Rham]{deRham1931}
de Rham, G. (1931).
\emph{Sur l'analysis situs des vari\'et\'es \`a $n$ dimensions}.
Journal de Math\'ematiques Pures et Appliqu\'ees, 10, 115--200.\\
\textbf{Relevance.} The original de Rham theorem identifying real
singular cohomology with cohomology of closed differential forms.
This is the foundational result underlying every step of the
differential-cohomology refinement used in Parts~III, IV, and
Appendix~\ref{app:diff-cohomology}. First cited in \S\ref{app:cohomology-basics}.

\bibitem[Riley et al.(2021)Riley et al.]{Riley2021NICER}
Riley, T.~E., Watts, A.~L., Ray, P.~S., et al.\ (2021).
\emph{A NICER view of the massive pulsar PSR J0740+6620
informed by radio timing and XMM-Newton spectroscopy}.
Astrophysical Journal Letters, 918, L27.\\
\textbf{Relevance.} Companion NICER mass--radius determination for
the heaviest precisely-measured pulsar. Together with
\citet{Miller2021NICER}, defines the empirical envelope inside which
the modified-TOV correction $\Delta_{\rm GS}(r)$ must lie if \CCI{}
is to remain falsifiable but not yet falsified. First cited in \S\ref{ssec:mTOV-derivation}.

\bibitem[Sands(1970)Sands]{Sands1970}
Sands, M. (1970).
\emph{The Physics of Electron Storage Rings: An Introduction}.
SLAC Report SLAC-R-121, Stanford Linear Accelerator Center.\\
\textbf{Relevance.} The canonical storage-ring treatment of radiation
damping, quantum excitation, and equilibrium emittance. Example~4
(\S\ref{ssec:GRIP-ex-radiative}) identifies the equilibrium emittance as
the orbital stratum's geometric attractor and the measured constant
$C_q = (55/32\sqrt3)\,\hbar/mc$ as the coefficient-level realization of
\textbf{GFF-ST7}(2). First cited in \S\ref{ssec:GRIP-ex-radiative}.

\bibitem[Schwarz(1993)Schwarz]{Schwarz1993}
Schwarz, M. (1993).
\emph{Morse Homology}.
Birkh\"auser Progress in Mathematics 111.\\
\textbf{Relevance.} Modern reference for transversality of gradient
flows between Morse critical points. Underlies the cascade-trajectory
analysis of Part~V. First cited in \S\ref{ssec:GRIP-first-consequences}.

\bibitem[Spohn(1978)Spohn]{Spohn1978}
Spohn, H. (1978).
\emph{Entropy production for quantum dynamical semigroups}.
Journal of Mathematical Physics, 19, 1227--1230.\\
\textbf{Relevance.} Spohn's H-theorem for Lindblad-form quantum
dynamical semigroups: under a CPTP semigroup with detailed balance with
respect to a stationary state, the relative entropy decreases
monotonically. Cited at T3.2 to make explicit that the
reverse direction of the iff requires detailed balance, not merely
CPTP. First cited in \S\ref{ssec:T3.2-statement}.

\bibitem[Susskind(1995)Susskind]{Susskind1995}
Susskind, L. (1995).
\emph{The world as a hologram}.
Journal of Mathematical Physics, 36, 6377--6396.\\
\textbf{Relevance.} The holographic principle in its string-theory
formulation. One of the foundational steps toward the open-system /
boundary-projection perspective elevated to foundational status by
\GDOS{}. First cited in \S\ref{ssec:relation-constitution}.

\bibitem[T2K(2020)T2K]{T2K2020}
T2K Collaboration (2020).
\emph{Constraint on the matter--antimatter symmetry-violating phase in
neutrino oscillations}.
Nature, 580, 339--344.\\
\textbf{Relevance.} Empirical evidence of leptonic CP violation,
providing independent support for the lower bound $N\geq 3$ in the
leptonic sector. First cited in \S\ref{sec:apply-N3}.

\bibitem[Ward(1950)Ward]{Ward1950}
Ward, J.~C. (1950).
\emph{An identity in quantum electrodynamics}.
Physical Review, 78, 182.\\
\textbf{Relevance.} The Ward identity for QED, expressing the
gauge-invariance of the effective action as an identity among Green's
functions. The quantum-field-theoretic reformulation of Noether's
theorem and the QFT progenitor of \CCI{}. First cited in \S\ref{ssec:CCI-motivation}.

\bibitem[Wilson(1974)Wilson]{Wilson1974}
Wilson, K.~G. (1974).
\emph{Confinement of quarks}.
Physical Review D, 10, 2445--2459.\\
\textbf{Relevance.} The seminal treatment of Wilson loops in lattice
gauge theory. Provides the physical interpretation of the Wilson-line
order parameters of \TGG{} (Definition~\ref{def:wilson-line-OP}). First cited in \S\ref{ssec:GRIP-motivation}.

\bibitem[Witten(1982)Witten]{Witten1982}
Witten, E. (1982).
\emph{Supersymmetry and Morse theory}.
Journal of Differential Geometry, 17, 661--692.\\
\textbf{Relevance.} The foundational paper relating supersymmetric
quantum mechanics to Morse theory. Provides the Bogomolnyi-saturating
instanton-action computation $S = \pi k$ used in
Theorem~\ref{thm:instanton-action} and \GRIP-ST3. First cited in \S\ref{ssec:GRIP-first-consequences}.

\bibitem[Witten(1996)Witten]{Witten1996}
Witten, E. (1996).
\emph{On flux quantization in M-theory and the effective action}.
Journal of Geometry and Physics, 22, 1--13.\\
\textbf{Relevance.} Foundational application of differential
cohomology to anomaly inflow in M-theory. Provides the prototype for
the non-degeneracy of the Hopkins--Singer descent
(Theorem~\ref{thm:H2-theorem}). First cited in \S\ref{ssec:strategic-moment}.

\bibitem[Witten(1985)Witten]{WittenAnomalies}
Witten, E. (1985).
\emph{Global anomalies in string theory}.
In: Symposium on anomalies, geometry, topology (Argonne, IL, 1985), 61--99.\\
\textbf{Relevance.} Reviews global-anomaly constraints and their
relation to family number. One of the precursors to the \BIsix{}
anomaly polynomial. First cited in \S\ref{sec:coherence-map}.

\bibitem[Yang \& Mills(1954)Yang \& Mills]{YangMills1954}
Yang, C.~N., \& Mills, R.~L. (1954).
\emph{Conservation of isotopic spin and isotopic gauge invariance}.
Physical Review, 96, 191--195.\\
\textbf{Relevance.} The non-abelian generalization of Maxwell's gauge
theory; the introduction of $\mathrm{SU}(2)$ gauge fields and the
non-linear field strength $F^a_{\mu\nu}$. The structural ancestor of
the boundary gauge component $A$ of the \TGG{} composite connection. First cited in \S\ref{ssec:TGG-motivation}.

\bibitem[Zeh(1970)Zeh]{Zeh1970}
Zeh, H.~D. (1970).
\emph{On the interpretation of measurement in quantum theory}.
Foundations of Physics, 1, 69--76.\\
\textbf{Relevance.} The first articulation of decoherence: the
mechanism by which open-system entanglement with the environment
produces effective classicality from quantum dynamics. An early
foundational step toward the open-system measurement perspective. First cited in \S\ref{ssec:relation-qi4}.

\bibitem[Zurek(1981)Zurek]{Zurek1981}
Zurek, W.~H. (1981).
\emph{Pointer basis of quantum apparatus: Into what mixture does the
wave packet collapse?}
Physical Review D, 24, 1516--1525.\\
\textbf{Relevance.} The pointer-basis selection mechanism in
decoherence theory. Together with \citet{Zeh1970}, provides the
technical machinery underlying the open-system measurement
perspective. First cited in \S\ref{sec:measurement-carryover}.

\end{thebibliography}
\GGBibEnd

\end{document}
